**Meet the editor**

Dr. Farzad Ebrahimi was born in Qazvin, Iran, in 1979. He graduated in mechanical engineering, from the University of Tehran, Iran, in 2002. He received his Msc and PhD in mechanical engineering, with a specialization in applied design from the University of Tehran, Iran, in 2009. Since 2002, he has been working at the "Smart Materials and Structures Lab" Research Center of the fac-

ulty of mechanical engineering at the University of Tehran. Now he is an assistant professor in the department of mechanical engineering at Imam Khomeini International University, Qazvin. He has authored more than 30 papers and also authored/edited 6 books in the fields of "Vibration Analysis", "functionally graded structures" and "Piezoelectric Transducers".

Contents

**Preface IX** 

**Section 1 Applications of FEA in** 

**Section 2 Applications of FEA in** 

B.H. Wu

Chapter 6 **Flexural Behavior of** 

Chapter 7 **Finite Element Analysis of** 

Samia Dridi

**"Fluid Mechanics and Heat Transfer" 1** 

**for Flow in Porous Media: Resin Transfer Molding 3**  Jamal Samir, Jamal Echaabi and Mohamed Hattabi

**"Structural Mechanics and Composite Materials" 63** 

A. Ghorbanpour Arani, R. Kolahchi, A. A. Mosalaei Barzoki,

Chapter 4 **Nonlinear Large Deflection Analysis of Stiffened Plates 87**  Khosrow Ghavami and Mohammad Reza Khedmati

**of Cylindrical Vessels Under In-Plane Moment 115** 

**Functionally Graded Sandwich Composite 131**  Mrityunjay R. Doddamani and Satyabodh M Kulkarni

**Bias Extension Test of Dry Woven 155** 

Chapter 1 **Control Volume Finite Element Methods** 

Chapter 2 **Electromagnetic and Fluid Analysis of Collisional Plasmas 31**  Antonis P. Papadakis

Chapter 3 **Finite Element Analysis of Functionally Graded Piezoelectric Spheres 65** 

A. Loghman and F. Ebrahimi

Chapter 5 **3D Nonlinear Finite Element Plastic Analysis** 

## Contents

## **Preface XI**


X Contents


## Preface

The advent of high-speed electronic digital computers has given tremendous impetus to all numerical methods for solving engineering problems. Finite element analysis (FEA) form one of the most versatile classes of such methods, and were originally developed in the field of structural analysis but now it has been extended as a general method of solution to many complex engineering and physical science problems. In the past few decades, the FEA has been developed into a key indispensable technology in the modeling and simulation of various engineering systems. In the development of an advanced engineering system, engineers have to go through a very rigorous process of modeling, simulation, visualization, analysis, designing, prototyping, testing, and finally, fabrication/construction. As such, techniques related to modeling and simulation in a rapid and effective way play an increasingly important role in building advanced engineering systems, and therefore the application of the FEA has multiplied rapidly.

This book collects original and innovative research studies on recent applications of Finite Element Analysis in mechanical engineering through original and innovative research studies exhibiting various investigation directions. Through its 16 chapters the reader will have access to works related to five major topics of mechanical engineering namely, fluid mechanics and heat transfer, machine elements analysis and design, machining and product design, wave propagation and failure-analysis and structural mechanics and composite materials. It is meant to provide a small but valuable sample of contemporary research activities around the world in this field and it is expected to be useful to a large number of researchers.

The present book is a result of contributions of experts from international scientific community working in different aspects of **Finite Element Analysis**. The introductions, data, and references in this book will help the readers know more about this topic and help them explore this exciting and fast-evolving field. The text is addressed not only to researchers, but also to professional engineers, students and other experts in a variety of disciplines, both academic and industrial seeking to gain a better understanding of what has been done in the field recently, and what kind of open problems are in this area.

#### X Preface

I am honored to be editing such a valuable book, which contains contributions of a selected group of researchers describing the best of their work. I would like to express my sincere gratitude to all of them for their outstanding chapters. I also wish to acknowledge the InTech editorial staff, in particular, Oliver kurelic for indispensable technical assistance in book preparation and publishing.

#### **Dr. Farzad Ebrahimi**

Faculty of Engineering Mechanical Engineering Department International University of Imam Khomeini Qazvin, I.R.Iran

X Preface

I am honored to be editing such a valuable book, which contains contributions of a selected group of researchers describing the best of their work. I would like to express my sincere gratitude to all of them for their outstanding chapters. I also wish to acknowledge the InTech editorial staff, in particular, Oliver kurelic for indispensable

> **Dr. Farzad Ebrahimi** Faculty of Engineering

> > Qazvin, I.R.Iran

Mechanical Engineering Department International University of Imam Khomeini

technical assistance in book preparation and publishing.

**Section 1** 

**Applications of FEA in** 

**"Fluid Mechanics and Heat Transfer"** 

## **Applications of FEA in "Fluid Mechanics and Heat Transfer"**

**Chapter 1** 

© 2012 Samir et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Samir et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Control Volume Finite Element Methods for** 

Jamal Samir, Jamal Echaabi and Mohamed Hattabi

implementation of the Newton-Raphson linearization technique.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/46167

**1. Introduction** 

and its neighbors.

providing the details of the derivation.

**Flow in Porous Media: Resin Transfer Molding** 

The method of solution for the own problem is based on the control volume finite element approach which has been shown to be particularly well suited for a fast and efficient

The basic idea of the control volume finite element approach is to obtain a discretized equation that mimics the governing mass conservation equation locally. A volume of influence, referred to as a control volume, is assigned to each node. The discretized equation for a given node then consists of a term describing the change in fluid mass storage for that volume which is balanced by the term representing the divergence of the fluid mass flux in the volume. The fluid mass flux will depend on the physical properties associated with the volume and the difference in the value of the primary variable between the node in question

Discretization of the subsurface and the surface flow equations is identical except for the difference in dimensionality. For the sake of clarity, we present in this chapter, a detailed description of the control volume finite element method applied to discretize a simplified prototype continuity equation in Liquid composite molding (LCM) . The final discretized equations for all subsurface domains and for surface flow are then presented without

Liquid composite molding (LCM) processes are routinely considered as a viable option to manufacture composite parts. In this process, a fibrous preform is placed in a mold. The mold is sealed and a liquid thermoset resin is injected to impregnate the fibrous preform. All LCM processes involve impregnation of the resin into a bed of fibrous network. The goal is to saturate all the empty spaces between the fibers with the resin before the resin gels and then solidifies. In RTM, resin is injected slowly and little or no heat transfer and chemical

## **Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding**

Jamal Samir, Jamal Echaabi and Mohamed Hattabi

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/46167

## **1. Introduction**

The method of solution for the own problem is based on the control volume finite element approach which has been shown to be particularly well suited for a fast and efficient implementation of the Newton-Raphson linearization technique.

The basic idea of the control volume finite element approach is to obtain a discretized equation that mimics the governing mass conservation equation locally. A volume of influence, referred to as a control volume, is assigned to each node. The discretized equation for a given node then consists of a term describing the change in fluid mass storage for that volume which is balanced by the term representing the divergence of the fluid mass flux in the volume. The fluid mass flux will depend on the physical properties associated with the volume and the difference in the value of the primary variable between the node in question and its neighbors.

Discretization of the subsurface and the surface flow equations is identical except for the difference in dimensionality. For the sake of clarity, we present in this chapter, a detailed description of the control volume finite element method applied to discretize a simplified prototype continuity equation in Liquid composite molding (LCM) . The final discretized equations for all subsurface domains and for surface flow are then presented without providing the details of the derivation.

Liquid composite molding (LCM) processes are routinely considered as a viable option to manufacture composite parts. In this process, a fibrous preform is placed in a mold. The mold is sealed and a liquid thermoset resin is injected to impregnate the fibrous preform. All LCM processes involve impregnation of the resin into a bed of fibrous network. The goal is to saturate all the empty spaces between the fibers with the resin before the resin gels and then solidifies. In RTM, resin is injected slowly and little or no heat transfer and chemical

© 2012 Samir et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Samir et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

reaction takes place until the mold is filled. Thus, the entire cycle can be viewed as two separate events, fill and subsequent cure.

Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding 5

On the whole, our study is concerned with the simulation of isothermal filling of moulds in RTM process while adopting the CVFEM and VOF method, taking into account the presence of obstacles and the thickness variation of the reinforcement. The elaborated code allows calculating the position of the injection points and vents, and injection pressure in order to optimize the process parameters. We present a mesh generator for 1D, 2D and 2.5 D geometries for one or several points of injection and vents, and a numerical method for the simulation of the RTM process. Numerical examples are used to validate and assess the applicability of the developed code in the case of anisotropic reinforcements, multilayered,

In RTM process and when filling the mould, the resin flow passes through a bed of fibers, The process of injection in the mould is treated as part of the flows of fluids inside a porous

On the basis of partial saturation concept, the mass balance at a point within the domain of an isothermal incompressible fluid flow inside a fiber preform can be expressed as [24] :

> . *<sup>s</sup> <sup>q</sup> <sup>t</sup>*

(1)

(2)

(3)

(4)

is the porosity of

is the porosity and the saturation level *s* 

is 1 for a fully saturated node and its value ranges between 0 and 1 for a partially saturated point. If the transient term on the left hand side of the above equation is removed (saturate

> 0 *i q x*

*i i <sup>q</sup> <sup>v</sup>* 

As the fluid flows through the pores of the preform, the interstitial velocity of the resin is

Using the assumptions that the preform is a porous medium and that the flow is quasi-

*ij*

*x*

*K P*

*j*

*i*

*q*

case), the following equation for quasi-steady state situation is obtained:

Where *<sup>i</sup> v* the intrinsic phase average resin velocity within the pores and

steady state, the momentum equation can be replaced by Darcy's law:

several injection points and the existence of inserts.

**2.1. Continuity equations and Darcy law** 

Where q is the volumetric flow rate per unit area,

**2. Mathematical formalism** 

medium.

given by :

the solid.

The mould filling is considered as one of the most critical and complicated stages throughout the entire RTM process. It has a great influence on the performance and quality of the final parts. However, it is hard to understand effects of the filling parameters on the flow front pattern during mold filling. Therefore, it is necessary to understand interrelationship among filling parameters, flow behavior during RTM, and physical properties of the final parts.

The present study concerns the numerical modeling of resin transfer molding techniques (RTM). From a mechanical point of view, these processes can be treated in the same way as the problems of fluids in porous media. Some of authors use methods based on the systems of curvilinear coordinates adapted to a border. However, this approach becomes limited during divisions or fusion of the flow fronts [1-4]. The modeling methods currently elaborated are based on a control volume finite element method (CVFEM). This type of approach was first presented by Wang and others and was adopted in the case of thin shell injection molding [5]. Fracchia was the first to apply the CVFEM to simulate the RTM process [6] and other researchers also followed this approach [7-16]. The application of these methods generate several commercial software: RTM-FLOT ( no longer supported ), PAM-RTM, MyRTM and LIMS.

In the last years, despite the introduction of alternative methods for simulation of flow in porous media BEM (Trochu [17]) and SPH (Krawczak [18]), the CVFEM method has been, usually, used to simulate resin flow in the RTM process. However, this numerical approach has some inherent drawbacks. First, the flow front is difficult to define with the exact location because of using fixed mesh system. This problem in the resin flow front location limits the accuracy of CVFEM method [19]. Mass conservation problems have also been reported with the use of this numerical approach [20,21]. Researchers have addressed these numerical problems and put forward methods to improve the conventional CVFEM method [19, 22].

In this study, the simulation of the resin flow in the RTM process is developed by the control volume finite element method (CVFEM) coupled with the equation of the free surface location. The equation is solved at each time step using nonconforming linear finite elements on triangles, which allow the conservation of the resin flow rate along interelement boundaries [23]. At each time step, the velocity and pressure in the saturated domain is calculated. The effective velocity is used to update the front position. The filling algorithm determines the time increment needed to fill up completely at least one new element, then the boundary condition is updated and the flow front is advanced for the next iteration. The flow front is refined in an adaptive manner at each time step by using our mesh generator to add new nodes, to get a smoother flow front and reduce the error in the pressure at the flow front of a CVFEM simulation of resin flow in a porous medium. Thus, the position of the flow front, the time-lapse and the rate of the unsaturated zone are calculated at every step. Our results will be compared with the experimental and analytical models in the literature.

On the whole, our study is concerned with the simulation of isothermal filling of moulds in RTM process while adopting the CVFEM and VOF method, taking into account the presence of obstacles and the thickness variation of the reinforcement. The elaborated code allows calculating the position of the injection points and vents, and injection pressure in order to optimize the process parameters. We present a mesh generator for 1D, 2D and 2.5 D geometries for one or several points of injection and vents, and a numerical method for the simulation of the RTM process. Numerical examples are used to validate and assess the applicability of the developed code in the case of anisotropic reinforcements, multilayered, several injection points and the existence of inserts.

## **2. Mathematical formalism**

4 Finite Element Analysis – Applications in Mechanical Engineering

separate events, fill and subsequent cure.

properties of the final parts.

RTM, MyRTM and LIMS.

models in the literature.

reaction takes place until the mold is filled. Thus, the entire cycle can be viewed as two

The mould filling is considered as one of the most critical and complicated stages throughout the entire RTM process. It has a great influence on the performance and quality of the final parts. However, it is hard to understand effects of the filling parameters on the flow front pattern during mold filling. Therefore, it is necessary to understand interrelationship among filling parameters, flow behavior during RTM, and physical

The present study concerns the numerical modeling of resin transfer molding techniques (RTM). From a mechanical point of view, these processes can be treated in the same way as the problems of fluids in porous media. Some of authors use methods based on the systems of curvilinear coordinates adapted to a border. However, this approach becomes limited during divisions or fusion of the flow fronts [1-4]. The modeling methods currently elaborated are based on a control volume finite element method (CVFEM). This type of approach was first presented by Wang and others and was adopted in the case of thin shell injection molding [5]. Fracchia was the first to apply the CVFEM to simulate the RTM process [6] and other researchers also followed this approach [7-16]. The application of these methods generate several commercial software: RTM-FLOT ( no longer supported ), PAM-

In the last years, despite the introduction of alternative methods for simulation of flow in porous media BEM (Trochu [17]) and SPH (Krawczak [18]), the CVFEM method has been, usually, used to simulate resin flow in the RTM process. However, this numerical approach has some inherent drawbacks. First, the flow front is difficult to define with the exact location because of using fixed mesh system. This problem in the resin flow front location limits the accuracy of CVFEM method [19]. Mass conservation problems have also been reported with the use of this numerical approach [20,21]. Researchers have addressed these numerical problems and put forward methods to improve the conventional CVFEM method [19, 22].

In this study, the simulation of the resin flow in the RTM process is developed by the control volume finite element method (CVFEM) coupled with the equation of the free surface location. The equation is solved at each time step using nonconforming linear finite elements on triangles, which allow the conservation of the resin flow rate along interelement boundaries [23]. At each time step, the velocity and pressure in the saturated domain is calculated. The effective velocity is used to update the front position. The filling algorithm determines the time increment needed to fill up completely at least one new element, then the boundary condition is updated and the flow front is advanced for the next iteration. The flow front is refined in an adaptive manner at each time step by using our mesh generator to add new nodes, to get a smoother flow front and reduce the error in the pressure at the flow front of a CVFEM simulation of resin flow in a porous medium. Thus, the position of the flow front, the time-lapse and the rate of the unsaturated zone are calculated at every step. Our results will be compared with the experimental and analytical

#### **2.1. Continuity equations and Darcy law**

In RTM process and when filling the mould, the resin flow passes through a bed of fibers, The process of injection in the mould is treated as part of the flows of fluids inside a porous medium.

On the basis of partial saturation concept, the mass balance at a point within the domain of an isothermal incompressible fluid flow inside a fiber preform can be expressed as [24] :

$$
\rho \frac{\partial \mathbf{\hat{s}}}{\partial t} = -\nabla.q \tag{1}
$$

Where q is the volumetric flow rate per unit area, is the porosity and the saturation level *s*  is 1 for a fully saturated node and its value ranges between 0 and 1 for a partially saturated point. If the transient term on the left hand side of the above equation is removed (saturate case), the following equation for quasi-steady state situation is obtained:

$$\frac{\partial \overline{q}}{\partial \mathbf{x}\_i} = 0 \tag{2}$$

As the fluid flows through the pores of the preform, the interstitial velocity of the resin is given by :

$$
\sigma\_i = \frac{q\_i}{\phi} \tag{3}
$$

Where *<sup>i</sup> v* the intrinsic phase average resin velocity within the pores and is the porosity of the solid.

Using the assumptions that the preform is a porous medium and that the flow is quasisteady state, the momentum equation can be replaced by Darcy's law:

$$q\_i = -\frac{K\_{ij}}{\mu} \frac{\partial P}{\partial \boldsymbol{\alpha}\_j} \tag{4}$$

where is the fluid viscosity, *Kij* is the permeability tensor of the preform, and *P* is the fluid pressure.

Assuming that the resin is incompressible and substituting (4) into (2) gives the governing differential equation of the flow :

$$\frac{\partial}{\partial \mathbf{x}\_i} \left( \frac{K\_{ij}}{\mu} \frac{\partial P}{\partial \mathbf{x}\_j} \right) = 0 \tag{5}$$

Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding 7

contains no point from *P* in its interior. The union of all Delaunay simplices forms the Delaunay diagram, DT (*P*). If the set *P* is not degenerate then the DT (*P*) is a simplex

The geometric dual of Delaunay Diagram is the *Voronoi Diagram*, which consists of a set of polyhedra *V*1, . . . , *V*n, one for each point in *P* , called the Voronoi Polyhedra. Geometrically, Vi is the set of points *<sup>d</sup> p* whose Euclidean distance to *pi* is less than or equal to that of any other point in *P*. We call *pi* the center of *polyhedra Vi*. For more

The DT has some very desired properties for mesh generation. For example, among all triangulations of a point set in 2D, the DT maximizes the smallest angle, it contains the nearest-neighbors graph, and the minimal spanning tree. Thus Delaunay triangulation is very useful for computer graphics and mesh generation in two dimensions. Moreover, discrete maximum principles will only exist for Delaunay triangulations. Chew [28] and Ruppert [29] have developed Delaunay refinement algorithms that generate provably good

In processes such as Resin Transfer Moulding (RTM), numerical simulations are usually performed on a fixed mesh, on which the numerical algorithm predict the displacement of the flow front. Error estimations can be used in the numerical algorithm to optimize the mesh for the finite element analysis. The mesh can also be adapted during mould filling to follow the shape of the moving boundary. In CVFEM, the calculation domain is first discretized using finite elements, and then each element is further divided into subvolumes. For the discretization of the calculation domain in FEM, we developed a mesh generator (figure 2) allowing to generate 2 and 2.5 dimensional, unstructured Delaunay and

decomposition of the convex hull of *P*.

discussion, see [26, 27].

meshes for 2D domains.

**3.2. Discretization domain** 

**Figure 1.** A Delaunay triangulation in the plane with circumcircles.

constrained Delaunay triangulations in general domains.

This second order partial differential equation can be solved when the boundary conditions are prescribed. Two common boundary conditions for the inlet to the mould are either a prescribed pressure condition:

$$P\_{inlet} = P\_{inlet}(t) \tag{6}$$

Or a prescribed flow rate condition:

$$\mathbf{Q}\_n = \mathbf{n}\_i \frac{\mathbf{K}\_{ij}}{\mu} \frac{\partial \mathbf{P}}{\partial \mathbf{x}\_j} \tag{7}$$

Where *Qn* is the volumetric flow rate and *<sup>i</sup> n* is the normal vector to the inlet.

The boundary conditions along the flow front are as follows:

$$P\_{front} = 0\tag{8}$$

Since the resin cannot pass through the mould wall, the final boundary condition necessary to solve equation (5) is that the velocity normal to the wall at the boundary of the mould must be zero:

$$
\overline{\boldsymbol{\upsilon}} \cdot \overline{\boldsymbol{n}} = 0 \tag{9}
$$

Where n is the vector normal to the mould wall.

### **3. Discretization of the domain by CV /FEM -VOF method**

#### **3.1. Delaunay triangulations**

In mathematics and computational geometry, a Delaunay triangulation for a set P of points in the plane is a triangulation DT (*P*) such that no point in *P* is inside the circumcircle (figure 1)of any triangle in DT (P). Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid triangles with high aspect ratio [25].

Suppose *P* = {*p1,….., pn*} is a point set in *d* dimensions. The convex hull of *d+1* affinely independent points from P forms a *Delaunay simplex* if the circumscribed ball of the simplex contains no point from *P* in its interior. The union of all Delaunay simplices forms the Delaunay diagram, DT (*P*). If the set *P* is not degenerate then the DT (*P*) is a simplex decomposition of the convex hull of *P*.

The geometric dual of Delaunay Diagram is the *Voronoi Diagram*, which consists of a set of polyhedra *V*1, . . . , *V*n, one for each point in *P* , called the Voronoi Polyhedra. Geometrically, Vi is the set of points *<sup>d</sup> p* whose Euclidean distance to *pi* is less than or equal to that of any other point in *P*. We call *pi* the center of *polyhedra Vi*. For more discussion, see [26, 27].

The DT has some very desired properties for mesh generation. For example, among all triangulations of a point set in 2D, the DT maximizes the smallest angle, it contains the nearest-neighbors graph, and the minimal spanning tree. Thus Delaunay triangulation is very useful for computer graphics and mesh generation in two dimensions. Moreover, discrete maximum principles will only exist for Delaunay triangulations. Chew [28] and Ruppert [29] have developed Delaunay refinement algorithms that generate provably good meshes for 2D domains.

**Figure 1.** A Delaunay triangulation in the plane with circumcircles.

## **3.2. Discretization domain**

6 Finite Element Analysis – Applications in Mechanical Engineering

is the fluid viscosity, *Kij* is the permeability tensor of the preform, and *P* is the

Assuming that the resin is incompressible and substituting (4) into (2) gives the governing

*i j K P x x* 

This second order partial differential equation can be solved when the boundary conditions are prescribed. Two common boundary conditions for the inlet to the mould are either a

() *inlet inlet P Pt* (6)

*ij*

*<sup>K</sup> <sup>P</sup> Q n x*

Since the resin cannot pass through the mould wall, the final boundary condition necessary to solve equation (5) is that the velocity normal to the wall at the boundary of the mould

In mathematics and computational geometry, a Delaunay triangulation for a set P of points in the plane is a triangulation DT (*P*) such that no point in *P* is inside the circumcircle (figure 1)of any triangle in DT (P). Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid triangles with high aspect

Suppose *P* = {*p1,….., pn*} is a point set in *d* dimensions. The convex hull of *d+1* affinely independent points from P forms a *Delaunay simplex* if the circumscribed ball of the simplex

*j*

(7)

<sup>0</sup> *front <sup>P</sup>* (8)

*v n*. 0 (9)

*n i*

Where *Qn* is the volumetric flow rate and *<sup>i</sup> n* is the normal vector to the inlet.

**3. Discretization of the domain by CV /FEM -VOF method** 

The boundary conditions along the flow front are as follows:

Where n is the vector normal to the mould wall.

**3.1. Delaunay triangulations** 

 

0 *ij*

(5)

where

differential equation of the flow :

prescribed pressure condition:

Or a prescribed flow rate condition:

fluid pressure.

must be zero:

ratio [25].

In processes such as Resin Transfer Moulding (RTM), numerical simulations are usually performed on a fixed mesh, on which the numerical algorithm predict the displacement of the flow front. Error estimations can be used in the numerical algorithm to optimize the mesh for the finite element analysis. The mesh can also be adapted during mould filling to follow the shape of the moving boundary. In CVFEM, the calculation domain is first discretized using finite elements, and then each element is further divided into subvolumes. For the discretization of the calculation domain in FEM, we developed a mesh generator (figure 2) allowing to generate 2 and 2.5 dimensional, unstructured Delaunay and constrained Delaunay triangulations in general domains.

Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding 9

(10)

known. Thus, the mesh density can affect seriously the accuracy of numerical solution of the

For any control volume and after integration equation 5, we obtained the following

*cv cs ij i*

**Figure 4.** Treatment of the flow front during the fixed meshing method.

 Experiment front Numerical

Node

*xx x*

0 *ij ij*

 

are the control surface, the control volume and the normal vector of the

 *f* = 0 0 < *f* < 1 *f* = 1

*K K P P <sup>d</sup> n ds*

control surface, respectively. CV and CS represent the control volume and the control

Moreover, in every iteration, the calculation matrix contains only elements that have at least one node with a filling ratio unity *f=1*. This approach requires a rigor during the

During the resolution of the pressure field, we adopted Galerkin's approximation to

Using the procedure outlined by Reddy [30], the finite element formulation of Equation (5)

*e*

*ij*

*e j i*

*x x*

 

*K K d*

*ee e KP F ij j i* (11)

(12)

development of code. However, the time of treatment of the problem is ameliorated.

flow front.

relationship:

Where *s*,Ω, *n*

surface domains respectively.

**4. Numerical simulation** 

represent the distribution of the field of pressure.

**4.1. Pressure** 

was found to be:

Where

**Figure 2.** Discretization of calculation domain

#### **3.3. Domain discretization CV/FEM**

To use the method CV/FEM coupled with VOF, the mould is first divided into finite elements. Around each nodal location, a control volume is constructed by subdividing the elements into smaller volumes. These control volumes are used to track the location of the flow front.

The calculation domain is in a finite number of triangular elements. After connecting the centroides of the elements with the middles of the elements borders, the calculation domain another time being divided in a number of polygonal control volume, as indicated in figure 3. The borders of any element of the control volume constitute the control surface.

**Figure 3.** Discretization of the calculation domain during CV/FEM

#### **3.4. Resin front tracking**

The control volumes can be empty, partially filled, or completely filled. The amount of fluid in each control volume is monitored by a quantity called the fill factor. It is the ratio of the volume of fluid to the total volume of the control volume. The fill factor takes values from 0 to 1 where 0 represents totally empty and 1 represents totally full. The control volume method tracks the flow front by determining which control volumes are partially filled and connecting them to form the flow front. The numerical flow front is composed from the nodes with partially filled control volumes as shown in figure 4. The location of the fluid in the control volume cannot be identified, therefore the exact shape of the flow front is not known. Thus, the mesh density can affect seriously the accuracy of numerical solution of the flow front.

For any control volume and after integration equation 5, we obtained the following relationship:

$$\int\_{cv} \frac{\partial}{\partial \mathbf{x}\_i} \left( \frac{K\_{ij}}{\mu} \frac{\partial P}{\partial \mathbf{x}\_j} \right) d\Omega = \int\_{cs} \frac{K\_{ij}}{\mu} \frac{\partial P}{\partial \mathbf{x}\_i} \stackrel{\cdot}{n} ds = 0 \tag{10}$$

Where *s*,Ω, *n* are the control surface, the control volume and the normal vector of the control surface, respectively. CV and CS represent the control volume and the control surface domains respectively.

**Figure 4.** Treatment of the flow front during the fixed meshing method.

Moreover, in every iteration, the calculation matrix contains only elements that have at least one node with a filling ratio unity *f=1*. This approach requires a rigor during the development of code. However, the time of treatment of the problem is ameliorated.

#### **4. Numerical simulation**

#### **4.1. Pressure**

8 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 2.** Discretization of calculation domain

**3.3. Domain discretization CV/FEM** 

flow front.

To use the method CV/FEM coupled with VOF, the mould is first divided into finite elements. Around each nodal location, a control volume is constructed by subdividing the elements into smaller volumes. These control volumes are used to track the location of the

The calculation domain is in a finite number of triangular elements. After connecting the centroides of the elements with the middles of the elements borders, the calculation domain another time being divided in a number of polygonal control volume, as indicated in figure

The control volumes can be empty, partially filled, or completely filled. The amount of fluid in each control volume is monitored by a quantity called the fill factor. It is the ratio of the volume of fluid to the total volume of the control volume. The fill factor takes values from 0 to 1 where 0 represents totally empty and 1 represents totally full. The control volume method tracks the flow front by determining which control volumes are partially filled and connecting them to form the flow front. The numerical flow front is composed from the nodes with partially filled control volumes as shown in figure 4. The location of the fluid in the control volume cannot be identified, therefore the exact shape of the flow front is not

 Control Surface Control Volume

3. The borders of any element of the control volume constitute the control surface.

**Figure 3.** Discretization of the calculation domain during CV/FEM

 Node Elements rims

**3.4. Resin front tracking** 

During the resolution of the pressure field, we adopted Galerkin's approximation to represent the distribution of the field of pressure.

Using the procedure outlined by Reddy [30], the finite element formulation of Equation (5) was found to be:

$$\mathbb{E}\left[\left.K\_{ij}^{e}\right]\right]\left\{P\_{j}^{e}\right\}=\left\{F\_{i}^{e}\right\}\tag{11}$$

Where

$$K\_{ij}^{\epsilon} = \int\_{\Omega\_{\epsilon}} \frac{K\_{\alpha\beta}}{\mu} \frac{\partial \,\nu\_{\ j}}{\partial \mathbf{x}\_{\alpha}} \frac{\partial \,\nu\_{j}}{\partial \mathbf{x}\_{\beta}} d\Omega \tag{12}$$

And

$$F\_i^\epsilon = \int\_{\Omega\_\epsilon} f \, \Psi\_i \, d\Omega + \int\_{\Gamma\_\epsilon} Q\_n \, \Psi\_i \, d\Gamma \tag{13}$$

Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding 11

(16)

(17)

After the flow rates in each control volume have been calculated, the fill factors can be updated. Given the current time step, the fill factors from the previous step, the calculated

*i i* 1 *e en*

The time step for the next iteration must be calculated before the solution can proceed. The optimal time step would be where the fluid just fills one control volume. If a larger step were chosen, the flow front would over-run the control volume and a loss of mass from the system would result. The time to fill the partially filled control volume "n" is calculated

(1 ) *n n*

*f V <sup>t</sup> Q*

Once *<sup>n</sup> t* has been calculated for all the partially filled control volumes, the smallest *t* is

The numerical schemes used in mould filling simulations are usually based on a time dependent resolution of an unsteady (free surface) boundary value problem. The boundary of the filled area in the mould cavity is constantly evolving, and it is difficult to generate a mesh suitable for all the successive calculation steps of a filling simulation. The fluid front cannot be approximated with a fine precision with an isotropic mesh. Such a mesh would have to be very fine everywhere in the geometrical domain in order to provide an accurate approximation. This would lead to time consuming calculations, although a fine mesh is required only in the vicinity of the flow front and near the inlet gates. For this reason, several researchers have proposed to construct a new mesh of the fluid saturated domain at each time step (Bechet et al. [31] for Eulerian scheme, Muttin et al. [32] for Lagrangian schemes). This approach is long in terms of computer time and rather complex, especially in the case of obstacles, merging flow fronts and for 3D problems Kang and Lee [19] proposed an algorithm, referred to as the Floating Imaginary Nodes and Elements (FINE) method, to get a smoother flow front and reduce the error in the pressure at the flow front of a CVFEM simulation of resin flow in a porous medium. With the FINE method, imaginary new nodes were added at the estimated flow front and the flow front elements were divided into two

*e en*

*n*

*t Q f f <sup>V</sup>* 

*n*

*f* is the fill factor, *t* is the time step, *Vn* is the volume of the control volume, and

flow rate, and the volume of each CV, the new fill factors can be calculated with:

*n n*

*4.3.2. Fill factor calculations* 

the superscripts indicate time level.

*4.3.3. Time step calculation* 

with the following relation:

**5. Adaptive mesh** 

chosen as the time step for the next iteration.

where *<sup>n</sup>*

Here *e* is the domain of an element. *<sup>e</sup>* is the surface of an element, *<sup>e</sup> <sup>j</sup> P* is the pressure at each node, f is a volumetric source term, *Qn* is a specified fluid flux through the face of the element, and *i* is a linear interpolation function.

### **4.2. Velocity**

After the pressures are calculated, the velocities are calculated at the centroid of each element using the volumetric flux equation 4 :

$$w\_i = -\frac{K\_{ij}}{\mu \phi} \frac{\partial P}{\partial \boldsymbol{\chi}\_j} \tag{14}$$

#### **4.3. Calculation of the parameters of filling**

The control volume method tracks the flow front by identifying the controls volume partially filled, and connecting them to form the flow front. The numerical flow front is made from the nodes with the partially filled control volumes.

#### *4.3.1. Flow rate calculation*

It is assumed that the velocity of the fluid is constant throughout each element (figure.5).

$$Q\_{en} = \overline{\upsilon}\_e \,\overline{a}\_{en} = -\int\_0^A \frac{K\_{ij}}{\mu} \frac{\partial P}{\partial \mathbf{x}\_j} \cdot \overline{n} \, ds \tag{15}$$

Where *Qen* is the volumetric flow rate in the control volume (n) from element (e), *en v* is the fluid velocity in the element, and *en a* is the area vector for the sub-volume.

**Figure 5.** Calculation of the filling velocity (CV/FEM)

#### *4.3.2. Fill factor calculations*

10 Finite Element Analysis – Applications in Mechanical Engineering

element, and *i* is a linear interpolation function.

element using the volumetric flux equation 4 :

**4.3. Calculation of the parameters of filling** 

**Figure 5.** Calculation of the filling velocity (CV/FEM)

*4.3.1. Flow rate calculation* 

made from the nodes with the partially filled control volumes.

*e*

Here *e* is the domain of an element. *<sup>e</sup>* is the surface of an element, *<sup>e</sup>*

*e e*

each node, f is a volumetric source term, *Qn* is a specified fluid flux through the face of the

After the pressures are calculated, the velocities are calculated at the centroid of each

*ij*

*x*

The control volume method tracks the flow front by identifying the controls volume partially filled, and connecting them to form the flow front. The numerical flow front is

It is assumed that the velocity of the fluid is constant throughout each element (figure.5).

*en e en <sup>o</sup> <sup>j</sup> <sup>K</sup> <sup>P</sup> Q va n ds*

fluid velocity in the element, and *en a* is the area vector for the sub-volume.

Where *Qen* is the volumetric flow rate in the control volume (n) from element (e), *en v* is the

*A ij*

*x*

*K P*

*j*

*i*

*v*

*i i ni F fd Q d* (13)

(14)

(15)

*<sup>j</sup> P* is the pressure at

And

**4.2. Velocity** 

After the flow rates in each control volume have been calculated, the fill factors can be updated. Given the current time step, the fill factors from the previous step, the calculated flow rate, and the volume of each CV, the new fill factors can be calculated with:

$$f\_n^{i+1} = f\_n^i + \frac{\Delta t \sum\_{\epsilon} Q\_{\epsilon n}}{V\_n} \tag{16}$$

where *<sup>n</sup> f* is the fill factor, *t* is the time step, *Vn* is the volume of the control volume, and the superscripts indicate time level.

#### *4.3.3. Time step calculation*

The time step for the next iteration must be calculated before the solution can proceed. The optimal time step would be where the fluid just fills one control volume. If a larger step were chosen, the flow front would over-run the control volume and a loss of mass from the system would result. The time to fill the partially filled control volume "n" is calculated with the following relation:

$$
\Delta t\_n = \frac{(1 - f\_n)V\_n}{\sum\_{e} Q\_{en}} \tag{17}
$$

Once *<sup>n</sup> t* has been calculated for all the partially filled control volumes, the smallest *t* is chosen as the time step for the next iteration.

### **5. Adaptive mesh**

The numerical schemes used in mould filling simulations are usually based on a time dependent resolution of an unsteady (free surface) boundary value problem. The boundary of the filled area in the mould cavity is constantly evolving, and it is difficult to generate a mesh suitable for all the successive calculation steps of a filling simulation. The fluid front cannot be approximated with a fine precision with an isotropic mesh. Such a mesh would have to be very fine everywhere in the geometrical domain in order to provide an accurate approximation. This would lead to time consuming calculations, although a fine mesh is required only in the vicinity of the flow front and near the inlet gates. For this reason, several researchers have proposed to construct a new mesh of the fluid saturated domain at each time step (Bechet et al. [31] for Eulerian scheme, Muttin et al. [32] for Lagrangian schemes). This approach is long in terms of computer time and rather complex, especially in the case of obstacles, merging flow fronts and for 3D problems Kang and Lee [19] proposed an algorithm, referred to as the Floating Imaginary Nodes and Elements (FINE) method, to get a smoother flow front and reduce the error in the pressure at the flow front of a CVFEM simulation of resin flow in a porous medium. With the FINE method, imaginary new nodes were added at the estimated flow front and the flow front elements were divided into two

separate regions: the area of resin and the area of air. Thus, the flow front element was refined in an adaptive manner at each time step. In this study, the generation of the mesh is realized by a code developed by our team, included like a module of the code of numerical simulation. The mesh generator allows to discretize the field in unstructured triangular elements with possibility of local refinement (static and dynamic) and inclusion of inserts. The development of the mesh generator code and its use in our solution allows the refinement of the critical zone (0<f<1) in each iteration.

Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding 13

**Figure 6.** Refinement of the triangular element.

**Figure 7.** Refinement by inserting new nodes in the triangulation .

condition of refinement as :

Delaunay refinement applied to elements of the flow front (figure.8).

In the present approach, a local mesh generator module has been developed and integrated to the principal code of numerical simulation of the RTM process. This approach is useful to carry out the instruction of the meshing at each step time during the execution of the program, which improves calculation time. The meshing technique adopted is based on two concepts. The first one is the restricted triangulations of Delaunay and the second is the

Starting from the reference level of meshing G0, we define for each element E1,ps, the

*Condition node E f*

1, 1,

*node E f*

 

*i ps node j ps node*

: / 1

/ 1

*i j*

In the CVFEM process, the numerical flow front is composed from the nodes with partially filled control volumes. So, since the location of the fluid in the control volume is not known, the exact shape of the flow front cannot be identified. In our numerical code, for the zone positioned in the flow front, we have developed a technique of local refinement. This technique uses an iterative method to refine repeatedly an initial triangulation of Delaunay, by inserting new nodes in the triangulation until satisfying size criteria and the shape of the elements (figure.6). The number of calculation points necessary to characterize accurately the deformations of the flow front decreases, and thus, the numerical computing time is reduced.

The refinement of position and shape of the front flow consists in adding the new nodes to the initials meshes (triangulation of Delaunay) in the zone of the flow front. The integration of these refined nodes, in the computer code, is conditioned by the value of the filling rate. The algorithm adopted for the mesh generator makes possible to generate first the standard Delaunay elements and initial nodes for the calculation domain. Then, in a second time, these elements are re-meshed by a technique of addition of nodes *(*figure.7*)*. The criteria to be respected during refinement are:

**The meshing coincides at the interfaces:** the meshing technique used implies the coincidence of the grid at the interfaces between the neighbouring elements (of the first standard grid). The triangulation of Delaunay is based directly on the contour nodes discretization which are discretized only once.

**An automatic identification of the nodes and the elements of refinement:** During the refinement of the meshes created, the initial nodes and elements are identified by a traditional technique of programming called "the coloring". This technique consists in affecting a particular code to define the elements of the sub-domains. During the resolution of the equations of the linear system, the integration of the nodes and elements resulting from refinement, in the numerical code, is conditioned by the value of the filling rate. Only, the elements with partially filled nodes are taken into account.

**Association of the sub-domains to the principal element:** During the refinement of the initial elements, the numerical algorithm affects a code to each sub-domain realized by the mesh generator. This code corresponds to the principal element generating the subdomains. Thus, the mesh code generates a structure of data that permits to associate the subdomains, resulting from the refinement process, to their principal element.

**Figure 6.** Refinement of the triangular element.

refinement of the critical zone (0<f<1) in each iteration.

reduced.

be respected during refinement are:

discretization which are discretized only once.

the elements with partially filled nodes are taken into account.

domains, resulting from the refinement process, to their principal element.

separate regions: the area of resin and the area of air. Thus, the flow front element was refined in an adaptive manner at each time step. In this study, the generation of the mesh is realized by a code developed by our team, included like a module of the code of numerical simulation. The mesh generator allows to discretize the field in unstructured triangular elements with possibility of local refinement (static and dynamic) and inclusion of inserts. The development of the mesh generator code and its use in our solution allows the

In the CVFEM process, the numerical flow front is composed from the nodes with partially filled control volumes. So, since the location of the fluid in the control volume is not known, the exact shape of the flow front cannot be identified. In our numerical code, for the zone positioned in the flow front, we have developed a technique of local refinement. This technique uses an iterative method to refine repeatedly an initial triangulation of Delaunay, by inserting new nodes in the triangulation until satisfying size criteria and the shape of the elements (figure.6). The number of calculation points necessary to characterize accurately the deformations of the flow front decreases, and thus, the numerical computing time is

The refinement of position and shape of the front flow consists in adding the new nodes to the initials meshes (triangulation of Delaunay) in the zone of the flow front. The integration of these refined nodes, in the computer code, is conditioned by the value of the filling rate. The algorithm adopted for the mesh generator makes possible to generate first the standard Delaunay elements and initial nodes for the calculation domain. Then, in a second time, these elements are re-meshed by a technique of addition of nodes *(*figure.7*)*. The criteria to

**The meshing coincides at the interfaces:** the meshing technique used implies the coincidence of the grid at the interfaces between the neighbouring elements (of the first standard grid). The triangulation of Delaunay is based directly on the contour nodes

**An automatic identification of the nodes and the elements of refinement:** During the refinement of the meshes created, the initial nodes and elements are identified by a traditional technique of programming called "the coloring". This technique consists in affecting a particular code to define the elements of the sub-domains. During the resolution of the equations of the linear system, the integration of the nodes and elements resulting from refinement, in the numerical code, is conditioned by the value of the filling rate. Only,

**Association of the sub-domains to the principal element:** During the refinement of the initial elements, the numerical algorithm affects a code to each sub-domain realized by the mesh generator. This code corresponds to the principal element generating the subdomains. Thus, the mesh code generates a structure of data that permits to associate the sub-

**Figure 7.** Refinement by inserting new nodes in the triangulation .

In the present approach, a local mesh generator module has been developed and integrated to the principal code of numerical simulation of the RTM process. This approach is useful to carry out the instruction of the meshing at each step time during the execution of the program, which improves calculation time. The meshing technique adopted is based on two concepts. The first one is the restricted triangulations of Delaunay and the second is the Delaunay refinement applied to elements of the flow front (figure.8).

Starting from the reference level of meshing G0, we define for each element E1,ps, the condition of refinement as :

$$\text{Condition } \mathfrak{R}: \begin{cases} \exists \, node\_i \in E\_{1,ps} \;/\; f\_{node\_i} = 1\\ \exists \, node\_j \in E\_{1,ps} \;/\; f\_{node\_j} < 1 \end{cases}$$

Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding 15

The operation for detecting which elements of G0 must be refined, is repeated at each iteration of the resolution process. If the condition has value "true" for an element E1,ps of the level G0, the numerical algorithm applies the refinement process and create subdomains. Thus, the element E1,ps is divided into an number of sub-domains (7 in figure.9), in each direction to preserve good properties of connection between the domains, which creates a new meshing level Gl. The solutions at the refined Gl are initialized by a prolongation of the same numerical algorithm of the G0 level. The system of equation is solved successively at each level of grid. Each refined element has a pressure boundary conditions of Dirichlet type. These operations are repeated at each iteration of the numerical

The advantage of this technique is to increase the local precision of calculations while preserving the properties of a meshing. On the other hand, the resolution process at each meshing level can be carried out starting from the same system of equations, the same

Large complex structures need computational models that accurately capture both the geometric and physical phenomena. This may involve the use of flow simulations as warranted by geometry, thickness, and fiber preforms employed. There also exists a need for accuracy improvements by refining the discretization of the computational domain. All these have serious impact on the computational time and power requirements. Physical modeling and computational algorithms and methodologies play an important role in computational times. For large-scale computations, it becomes critical to have algorithms that are physically accurate and permit faster solution of the computational domain. It is not only essential to design efficient parallel algorithms, data structures, and communication strategies for highly scalable parallel computing, it is also very important to have improved computational algorithms and methodologies to further improve the computational

In this study, the algorithm adopted uses techniques for the optimization of the execution time. For example, the management of the memory by the dynamic allocation of the tables and matrices allow the optimization of the resources machines. Also, the utilization of the pointers in the definition of the variables of the problems ensures the code the adaptation to the size of the data to be treated. In the same way, the adoption of algorithm of the sparse matrix for the inversion impacts the processing time seriously. Also in every iteration, the calculation matrix is dynamic and contains only elements that have a node with a filling ratio unity f=1. This approach imposes a rigor during the development of code, however, the

In this study, the numerical adopted is based on the various computational steps involved in adaptive meshing that accurately capture both the geometric and physical phenomena., CVFEM methodology for the update of the filled regions and refinement technique for flow

resolution until the mould is completely filled.

approximations and the same solver.

performance of large-scale simulations.

time of treatment of the problem is ameliorated.

front advancement are summarized below.

**6. Numerical algorithm** 

**Figure 8.** Identification of the elements for refinement

**Figure 9.** Example of numerical simulation in RTM process using refinement technique.

The operation for detecting which elements of G0 must be refined, is repeated at each iteration of the resolution process. If the condition has value "true" for an element E1,ps of the level G0, the numerical algorithm applies the refinement process and create subdomains. Thus, the element E1,ps is divided into an number of sub-domains (7 in figure.9), in each direction to preserve good properties of connection between the domains, which creates a new meshing level Gl. The solutions at the refined Gl are initialized by a prolongation of the same numerical algorithm of the G0 level. The system of equation is solved successively at each level of grid. Each refined element has a pressure boundary conditions of Dirichlet type. These operations are repeated at each iteration of the numerical resolution until the mould is completely filled.

The advantage of this technique is to increase the local precision of calculations while preserving the properties of a meshing. On the other hand, the resolution process at each meshing level can be carried out starting from the same system of equations, the same approximations and the same solver.

## **6. Numerical algorithm**

14 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 8.** Identification of the elements for refinement

*f=1*

*f<1*

*f<1*

**Figure 9.** Example of numerical simulation in RTM process using refinement technique.

Large complex structures need computational models that accurately capture both the geometric and physical phenomena. This may involve the use of flow simulations as warranted by geometry, thickness, and fiber preforms employed. There also exists a need for accuracy improvements by refining the discretization of the computational domain. All these have serious impact on the computational time and power requirements. Physical modeling and computational algorithms and methodologies play an important role in computational times. For large-scale computations, it becomes critical to have algorithms that are physically accurate and permit faster solution of the computational domain. It is not only essential to design efficient parallel algorithms, data structures, and communication strategies for highly scalable parallel computing, it is also very important to have improved computational algorithms and methodologies to further improve the computational performance of large-scale simulations.

In this study, the algorithm adopted uses techniques for the optimization of the execution time. For example, the management of the memory by the dynamic allocation of the tables and matrices allow the optimization of the resources machines. Also, the utilization of the pointers in the definition of the variables of the problems ensures the code the adaptation to the size of the data to be treated. In the same way, the adoption of algorithm of the sparse matrix for the inversion impacts the processing time seriously. Also in every iteration, the calculation matrix is dynamic and contains only elements that have a node with a filling ratio unity f=1. This approach imposes a rigor during the development of code, however, the time of treatment of the problem is ameliorated.

In this study, the numerical adopted is based on the various computational steps involved in adaptive meshing that accurately capture both the geometric and physical phenomena., CVFEM methodology for the update of the filled regions and refinement technique for flow front advancement are summarized below.

Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding 17

**Number of principal Nodes** 

**Number of secondary Nodes** 

**Time Consuming** 

In the first case (figure.11), we present examples prove the accuracy provided by our adaptive meshing technique. These examples treat an injection mold for radial rectangular 400mmx400mm with a uniform thickness of 4mm. The injection is located at the center of the mold. The meshing generates the following data (table.1). The running time is related to

> **Number of secondary elements**

**<sup>A</sup>**No 160 0 97 0 12s

**<sup>B</sup>**Yes 160 800 97 506 27s

**<sup>C</sup>**Yes 160 3520 97 1972 97s

In the second case (figure.12), we present two examples for comparison. These examples are related to the gain of the execution time for our adaptive meshing technique. These two examples treat an injection mold for radial rectangular 400mmx400mm with a uniform thickness of 4 mm. The first example is related to a discretization without adaptive meshing. This model has the same level of precision as the second example related to the adaptive mesh (same total number of nodes and elements). The meshing generates the following data

(table.2). The running time is related to a machine CPU 2Ghz Core Duo 2GB RAM

**7. Results and discussion** 

a machine CPU 2Ghz Core Duo 2GB RAM

**Number of principal elements** 

**Figure 11.** Numerical simulation with different type of meshing

**Using Technique Adaptive Mesh** 

**7.1. Adaptive meshing** 

**Example** 

**Example** 

**Example** 

**Table 1.** Data of meshing

**Figure 10.** Numerical Algorithm of simulation of filling of moulds in RTM process

## **7. Results and discussion**

## **7.1. Adaptive meshing**

16 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 10.** Numerical Algorithm of simulation of filling of moulds in RTM process

No

**Compute velocity**

**CV Mesh**

**Compute Pressure**

**Identification Nodes of calculus**

 **FEM** 

**CV**

**MEF Mesh** 

**Compute domain's properties**

**Definition of Calculation domain**

**Start**

**Compute Flow Rate** 

**Compute time step**

**Update fill factor**

**Identification flow front position**

**Mold completely filled**

**Refinement position**

Yes

**End**

In the first case (figure.11), we present examples prove the accuracy provided by our adaptive meshing technique. These examples treat an injection mold for radial rectangular 400mmx400mm with a uniform thickness of 4mm. The injection is located at the center of the mold. The meshing generates the following data (table.1). The running time is related to a machine CPU 2Ghz Core Duo 2GB RAM


**Table 1.** Data of meshing

**Figure 11.** Numerical simulation with different type of meshing

In the second case (figure.12), we present two examples for comparison. These examples are related to the gain of the execution time for our adaptive meshing technique. These two examples treat an injection mold for radial rectangular 400mmx400mm with a uniform thickness of 4 mm. The first example is related to a discretization without adaptive meshing. This model has the same level of precision as the second example related to the adaptive mesh (same total number of nodes and elements). The meshing generates the following data (table.2). The running time is related to a machine CPU 2Ghz Core Duo 2GB RAM

18 Finite Element Analysis – Applications in Mechanical Engineering


Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding 19

**Pressure Front's position Filling time** 

*KP t*

*<sup>µ</sup>*

The mould used has rectangular cavity of dimensions 400x400 mm2, and the thickness is uniform 4mm. The resin is injected from the central point of the mould, the reinforcement is isotropic and the permeability is the same one in the various directions. Under these conditions, the form and the position of the flow front can be obtained analytically by the equation 18. The comparison of the two front's kinetics (analytical and numerical) is

presented in figure.14. It shows a good concordance with the solution obtained.

2

0 10 20 30 Time (s)

With *rf* is the radius of the front flow in a time *t.* 

*f f*

0

*r*

**7.3. Validation of the results relating to RTM flow with variable thickness** 

2 2 0

<sup>1</sup> ln ( ) 2 4

RTM process can be used to produce pieces with complex geometry. In the industry of the composite, the plates employed often consist of reinforcements with a variable number of plies and stacking sequences. A correct simulation of this process requires taking into account all these parameters. Lonné makes a modeling according to a formalism derived

*r r <sup>K</sup> rr PP t*

*f f inlet*

(18)

2

*µL*

*KP* 

> Analytical Numerical

*inlet*

2 *<sup>f</sup>*

0 500 1000 Position (mm)

*t*

0,00

0,50

1,00

Pressure (Bar)

1,50

2,00

2,50

**Table 3.** Analytical solutions of unidirectional simulations in RTM

*<sup>x</sup> <sup>t</sup>* <sup>2</sup> *inlet f*

*f*

*x*

Analytical Num erical Experimental

**Figure 13.** Unidirectional Validation

0

200

400

600

Front's Position (mm)

800

1000

1200

( ) (1 ) ( ) *inlet*

*<sup>x</sup> Px P*

*f*

*7.2.2. Bidirectional validation* 

*7.3.1. Analytical validation* 

**Table 2.** Data of meshing

**Figure 12.** Numerical simulation with different type of meshing

## **7.2. Validation of the results relating to RTM flow with uniform thickness**

In the validation examples presented, the moulds used, have a rectangular cavity, with dimensions: (1000 × 200) mm2 and (400 × 400) mm2, respectively. The first example is relative to the unidirectional validation (1D) of our numerical results, whereas the second is used in radial injection (2D). The fluid viscosity μ = 0,109 Pa.s, the pressure injection Pinj = 2 × 105 Pa, the permeability K = 2,65 × 10−10 m2 and the porosity is *φ*= 0,696.

## *7.2.1. Unidirectional validation*

The mould used has a rectangular cavity of dimensions (1000 × 200) mm2, the thickness is uniform and equal to 4 mm. The resin is injected from left side of the mould and the vents are placed on the right-side of the cavity. Under these conditions, the kinetics of the flow can be obtained analytically by the equations of table.3. The comparison of the three kinetics of the front flow obtained is presented in figure 13. It shows a good concordance with the solution obtained.


**Table 3.** Analytical solutions of unidirectional simulations in RTM

**Figure 13.** Unidirectional Validation

**Number of principal elements** 

**Figure 12.** Numerical simulation with different type of meshing

the permeability K = 2,65 × 10−10 m2 and the porosity is *φ*= 0,696.

*7.2.1. Unidirectional validation* 

solution obtained.

**7.2. Validation of the results relating to RTM flow with uniform thickness** 

In the validation examples presented, the moulds used, have a rectangular cavity, with dimensions: (1000 × 200) mm2 and (400 × 400) mm2, respectively. The first example is relative to the unidirectional validation (1D) of our numerical results, whereas the second is used in radial injection (2D). The fluid viscosity μ = 0,109 Pa.s, the pressure injection Pinj = 2 × 105 Pa,

The mould used has a rectangular cavity of dimensions (1000 × 200) mm2, the thickness is uniform and equal to 4 mm. The resin is injected from left side of the mould and the vents are placed on the right-side of the cavity. Under these conditions, the kinetics of the flow can be obtained analytically by the equations of table.3. The comparison of the three kinetics of the front flow obtained is presented in figure 13. It shows a good concordance with the

**Number of secondary elements** 

**<sup>A</sup>**No 3680 0 2069 0 452s

**<sup>B</sup>**Yes 160 3520 97 1972 97s

**Number of principal Nodes** 

**Number of secondary Nodes** 

**Time Consuming** 

**Using Technique Adaptive Mesh** 

**Example** 

**Example** 

**Table 2.** Data of meshing

#### *7.2.2. Bidirectional validation*

The mould used has rectangular cavity of dimensions 400x400 mm2, and the thickness is uniform 4mm. The resin is injected from the central point of the mould, the reinforcement is isotropic and the permeability is the same one in the various directions. Under these conditions, the form and the position of the flow front can be obtained analytically by the equation 18. The comparison of the two front's kinetics (analytical and numerical) is presented in figure.14. It shows a good concordance with the solution obtained.

$$\frac{r\_f^2}{2}\ln\left(\frac{r\_f}{r\_0}\right) + \frac{1}{4}(r\_0^2 - r\_f^2) = \frac{K}{\sigma\mu}(P\_f - P\_{inlet})t\tag{18}$$

With *rf* is the radius of the front flow in a time *t.* 

#### **7.3. Validation of the results relating to RTM flow with variable thickness**

#### *7.3.1. Analytical validation*

RTM process can be used to produce pieces with complex geometry. In the industry of the composite, the plates employed often consist of reinforcements with a variable number of plies and stacking sequences. A correct simulation of this process requires taking into account all these parameters. Lonné makes a modeling according to a formalism derived

from the Thomson-Haskell method for the prediction of these geometrical variations on the ultrasound transmission [33].

Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding 21

idealized media structures [44–47]. Most models may not give accurate prediction of permeabilities since fibrous mats used in RTM are often more complex than the idealized unit cell patterns used in theoretical derivations. Thus due to the lack of adequate predictive

A number of permeability measurement methods have been developed; however, there is no standardized measurement technique for RTM applications. Trevino et al. [48] and Young et al. [49] determined unidirectional permeability using two pressure transducers at each of the inlet and the exit in conjunction with an equation based on Darcy's law applied to their flow geometry. Calhoun et al. [50] presented a technique based on placing several pressure transducers at various locations inside the mold. Adams and Rebenfeld [51–53] developed a technique that quantifies the planar permeability using the position and shape of the advancing resin front as a function of time. A transparent mold was used to enable the monitoring of the advancing front. Other techniques based also on the observation of the moving resin front are common in the literature [43,51-59]. However, transparent mold walls may not have enough rigidity to avoid deflection, which has been shown to perturbate the measured permeability values [45,48,49,54]. In this context, the University of Plymouth radial flow permeability appartus was enhanced by the use of a laminate of two 25 mm toughened float glass sheets [60] as the upper mould tool to achieve a similar stiffness to the

In this study, the variation of the plies number (figure.15.a) and the stacking sequence are modeled by the variation of permeability and porosity. During the standard approach, these parameters are defined as an intrinsic property of the global discretized domain. In our approach, the permeability and porosity are defined at the level of the element. The comparison of the two kinetic fronts (analytical and numerical) is presented in figure.15.b ; it

The analytical model for this type of reinforcement is indeed the prolongation of the linear model already adopted in the case of a medium with isotropic permeability. To ensure the accuracy of our numerical results, the elements of the initial meshing belong only to the one of the two zones. Also, in the static refinement of meshing, the creation of the new refined elements respects the condition of the uniformity of the permeability within the element (the

In addition, to illustrate the thickness effect, we present within the framework of the analytical validation, the case of a radial flow through a multi-thickness reinforcement.

In the setting of a bi-dimensional flow, we used the reinforcement with variation of the

models, permeability of RTM preforms are usually determined experimentally.

aluminium mould base [61].

shows a good concordance with the solution obtained.

element of refinement must belong only to the one of the two zones).

1. Reinforcement with a constant thickness (10 plies) (Figure 16.a) 2. Reinforcement with half 10 plies and half 20 plies (Figure 16.b) 3. Reinforcement with ¼ 10 plies and ¾ 20 plies (Figure 16.c)

number of plies. The mould cavity had a uniform thickness.

**Figure 14.** Bidirectional Validation with K11=K22=K

The reinforcement variation generates different pressures when closing the mold. Under the impact of the compressibility or the relaxation of the mold plates, a variation occurs in the volume and the pores distribution through the fabric and influences permeability and porosity. The mechanical performance of resin transfer molding depend on the fiber volume fraction [34], microstructure of the preform [35–36], void content [37], and impregnation parameters [38]. In most cases, mechanical properties of composite parts can be improved by increasing fiber volume fraction. Higher fiber volume fractions, however, require increased injection pressure and longer time to fill up the mould, which may significantly affect the properties of the final part. Patel et al. [39] molded composite parts containing glass fibers at constant injection pressure.

The study of Chen and al [40] showed that the initial compressibility of reinforcements is essentially related to that of the pores. This compressibility or « relaxation » effect directly influences the global volume and the distribution of the pores. During the mold closing, the variation of the reinforcement thickness generates, under the compressibility or the relaxation effects, a variation of the pores volume and their distribution through the fabric. The works of Buntain and Bickerton [41] were oriented to the way that compressibility affects permeability. Their results clearly showed that permeability (a property required to be perfectly controlled for a correct simulation of the flow front and the distribution of the pressure) was closely related to the pore volumetric fraction. Several models have been proposed to estimate the value of the permeability for various porous media. Capillary models such as those proposed by Carman [42] and Gutowski et al. [43] use the fiber radius and porosity to predict the permeability, but several discrepancies with experimental data have been reported [43–47]. Theoretical models have also been developed for different idealized media structures [44–47]. Most models may not give accurate prediction of permeabilities since fibrous mats used in RTM are often more complex than the idealized unit cell patterns used in theoretical derivations. Thus due to the lack of adequate predictive models, permeability of RTM preforms are usually determined experimentally.

20 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 14.** Bidirectional Validation with K11=K22=K

glass fibers at constant injection pressure.

ultrasound transmission [33].

from the Thomson-Haskell method for the prediction of these geometrical variations on the

The reinforcement variation generates different pressures when closing the mold. Under the impact of the compressibility or the relaxation of the mold plates, a variation occurs in the volume and the pores distribution through the fabric and influences permeability and porosity. The mechanical performance of resin transfer molding depend on the fiber volume fraction [34], microstructure of the preform [35–36], void content [37], and impregnation parameters [38]. In most cases, mechanical properties of composite parts can be improved by increasing fiber volume fraction. Higher fiber volume fractions, however, require increased injection pressure and longer time to fill up the mould, which may significantly affect the properties of the final part. Patel et al. [39] molded composite parts containing

The study of Chen and al [40] showed that the initial compressibility of reinforcements is essentially related to that of the pores. This compressibility or « relaxation » effect directly influences the global volume and the distribution of the pores. During the mold closing, the variation of the reinforcement thickness generates, under the compressibility or the relaxation effects, a variation of the pores volume and their distribution through the fabric. The works of Buntain and Bickerton [41] were oriented to the way that compressibility affects permeability. Their results clearly showed that permeability (a property required to be perfectly controlled for a correct simulation of the flow front and the distribution of the pressure) was closely related to the pore volumetric fraction. Several models have been proposed to estimate the value of the permeability for various porous media. Capillary models such as those proposed by Carman [42] and Gutowski et al. [43] use the fiber radius and porosity to predict the permeability, but several discrepancies with experimental data have been reported [43–47]. Theoretical models have also been developed for different A number of permeability measurement methods have been developed; however, there is no standardized measurement technique for RTM applications. Trevino et al. [48] and Young et al. [49] determined unidirectional permeability using two pressure transducers at each of the inlet and the exit in conjunction with an equation based on Darcy's law applied to their flow geometry. Calhoun et al. [50] presented a technique based on placing several pressure transducers at various locations inside the mold. Adams and Rebenfeld [51–53] developed a technique that quantifies the planar permeability using the position and shape of the advancing resin front as a function of time. A transparent mold was used to enable the monitoring of the advancing front. Other techniques based also on the observation of the moving resin front are common in the literature [43,51-59]. However, transparent mold walls may not have enough rigidity to avoid deflection, which has been shown to perturbate the measured permeability values [45,48,49,54]. In this context, the University of Plymouth radial flow permeability appartus was enhanced by the use of a laminate of two 25 mm toughened float glass sheets [60] as the upper mould tool to achieve a similar stiffness to the aluminium mould base [61].

In this study, the variation of the plies number (figure.15.a) and the stacking sequence are modeled by the variation of permeability and porosity. During the standard approach, these parameters are defined as an intrinsic property of the global discretized domain. In our approach, the permeability and porosity are defined at the level of the element. The comparison of the two kinetic fronts (analytical and numerical) is presented in figure.15.b ; it shows a good concordance with the solution obtained.

The analytical model for this type of reinforcement is indeed the prolongation of the linear model already adopted in the case of a medium with isotropic permeability. To ensure the accuracy of our numerical results, the elements of the initial meshing belong only to the one of the two zones. Also, in the static refinement of meshing, the creation of the new refined elements respects the condition of the uniformity of the permeability within the element (the element of refinement must belong only to the one of the two zones).

In addition, to illustrate the thickness effect, we present within the framework of the analytical validation, the case of a radial flow through a multi-thickness reinforcement.

In the setting of a bi-dimensional flow, we used the reinforcement with variation of the number of plies. The mould cavity had a uniform thickness.


Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding 23

**Figure 16.** a) Flow Front for a bi-dimensional injection with Reinforcement with a constant thickness, b) Flow Front for a bi-dimensional injection with Reinforcement half 10 plies and half 20 plies, c) Flow

(c)

(a)

(b)

The pressure distribution directly influences the injection pressure, the time needed to fill up the mould, the position of the injection points and vents and especially the mechanical properties of the final piece. The approach adopted by our team, use the permeability at the element's level. The thickness variation of the element directly influences its permeability

Front for a bi-dimensional injection with Reinforcement ¼ 10 plies and ¾ 20 plies

**reinforcement** 

**7.4. Simulation of the pressure distribution for a multiple – Thickness** 

**Figure 15.** a**)** Reinforcement with multiple thickness, b) Evolution of the pressure-position for multiple thickness

For a reinforcement with uniform thickness (figure16.a), the permeability in the principal directions is constant. The shape of the flow front for a radial injection is a circle whose center is the point of injection. For the case of two different reinforcements thicknesses (figure 16.b, and 16.c), the position and the shape of the flow are variable according to the resistance presented by the fabric (permeability).

Thus, for the figure 16.b where each half of the reinforcement is characterized by a fixed value of the permeability, it is quite clear that the solution of the Darcy law concerning the shape of flow front, in each zone, has a form of half-circles spaced according to the value of the permeability. The connection between the two shapes respects the continuity of the flow front. Finally, and in order to ensure the required precision, we proceeded to each step of time, with a refinement in the zone of connection of the two parts. The same approach is adopted for the figure16 .c.

resistance presented by the fabric (permeability).

adopted for the figure16 .c.

thickness

**Figure 15.** a**)** Reinforcement with multiple thickness, b) Evolution of the pressure-position for multiple

(b)

(a)

For a reinforcement with uniform thickness (figure16.a), the permeability in the principal directions is constant. The shape of the flow front for a radial injection is a circle whose center is the point of injection. For the case of two different reinforcements thicknesses (figure 16.b, and 16.c), the position and the shape of the flow are variable according to the

Thus, for the figure 16.b where each half of the reinforcement is characterized by a fixed value of the permeability, it is quite clear that the solution of the Darcy law concerning the shape of flow front, in each zone, has a form of half-circles spaced according to the value of the permeability. The connection between the two shapes respects the continuity of the flow front. Finally, and in order to ensure the required precision, we proceeded to each step of time, with a refinement in the zone of connection of the two parts. The same approach is

**Figure 16.** a) Flow Front for a bi-dimensional injection with Reinforcement with a constant thickness, b) Flow Front for a bi-dimensional injection with Reinforcement half 10 plies and half 20 plies, c) Flow Front for a bi-dimensional injection with Reinforcement ¼ 10 plies and ¾ 20 plies

## **7.4. Simulation of the pressure distribution for a multiple – Thickness reinforcement**

The pressure distribution directly influences the injection pressure, the time needed to fill up the mould, the position of the injection points and vents and especially the mechanical properties of the final piece. The approach adopted by our team, use the permeability at the element's level. The thickness variation of the element directly influences its permeability

and its porosity. The integration of these parameter's variation inside the code of resolution, gives more precision.

Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding 25

**Figure 18.** a) 2D Simulation without taking into account the effect of thickness variation, b) 2D

(b)

(a)

Simulation when taking into account the effect of thickness variation

**Figure 19.** 2.5D simulation of pressure distribution with thickness variation

The model we treated is a wiring cover made of glass fibers and an Isophthalic Polyester resin. The model has the specificity of including a square insert inside the piece, with the possibility to vary the thickness around the insert (square zone, Figure 17).

**Figure 17.** Piece with insert and reinforcement multiple thickness

During the moulding in RTM process, the resin injected infiltrates in empty spaces between fibers. However, a minor modification of the characteristics of the preform in specific places (around the insert for example), can cause significant deviations in the flow and the results on the final properties of the part can be disastrous generating the rejection of the whole process.

In order to simulate the filling process for 3D Dimension, logically a 3D model would be required. Since the thickness of composite parts is often much smaller than its length and width, thin part assumptions can be used for these simulation models [62]. For example, the resin flow in the thickness direction (here denoted as z) is neglected. Therefore, these models, although they describe 3D geometries, they are often called 2.5 D flow models [63].

The figures presented in 18.a, 18.b and 19 clearly show the impact of taking into account the variation thickness on the accuracy of the simulation of the flow front and the distribution of the pressure for 2.5 D models. These elements are paramount to control the parameters of the moulding and to obtain the required properties of the final part. the increase in thickness on a particular area by inserting new plies of reinforcement, leading to decreased permeability and increased porosity.

During numerical resolution, after the pressures are calculated, the velocities are calculated at the centroid of each element. Thus, given the reduced permeability and increased porosity, the velocity decreases while increased the thickness. In Figure 18.b we note the delaying of flow front in this area.

gives more precision.

process.

permeability and increased porosity.

delaying of flow front in this area.

and its porosity. The integration of these parameter's variation inside the code of resolution,

The model we treated is a wiring cover made of glass fibers and an Isophthalic Polyester resin. The model has the specificity of including a square insert inside the piece, with the

During the moulding in RTM process, the resin injected infiltrates in empty spaces between fibers. However, a minor modification of the characteristics of the preform in specific places (around the insert for example), can cause significant deviations in the flow and the results on the final properties of the part can be disastrous generating the rejection of the whole

In order to simulate the filling process for 3D Dimension, logically a 3D model would be required. Since the thickness of composite parts is often much smaller than its length and width, thin part assumptions can be used for these simulation models [62]. For example, the resin flow in the thickness direction (here denoted as z) is neglected. Therefore, these models, although they describe 3D geometries, they are often called 2.5 D flow models [63].

The figures presented in 18.a, 18.b and 19 clearly show the impact of taking into account the variation thickness on the accuracy of the simulation of the flow front and the distribution of the pressure for 2.5 D models. These elements are paramount to control the parameters of the moulding and to obtain the required properties of the final part. the increase in thickness on a particular area by inserting new plies of reinforcement, leading to decreased

During numerical resolution, after the pressures are calculated, the velocities are calculated at the centroid of each element. Thus, given the reduced permeability and increased porosity, the velocity decreases while increased the thickness. In Figure 18.b we note the

possibility to vary the thickness around the insert (square zone, Figure 17).

**Figure 17.** Piece with insert and reinforcement multiple thickness

**Figure 18.** a) 2D Simulation without taking into account the effect of thickness variation, b) 2D Simulation when taking into account the effect of thickness variation

**Figure 19.** 2.5D simulation of pressure distribution with thickness variation

The approach adopted during this study ensures flexibility during the simulation of the heterogeneities of the problem. The properties of the medium are calculated at the level of the element of the mesh. Having developed an adaptive mesh generator specific to the team, we are able to ensure a better adaptation of the elements of grid to describe physical specificities generated in each problem. The results obtained on the figures of this paragraph show the relevance of the present approach.

Control Volume Finite Element Methods for Flow in Porous Media: Resin Transfer Molding 27

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## **8. Conclusion**

During this study, we developed a mesh generator and a numerical code to simulate the filling of an isothermal mould in RTM process, by adopting the CV/FEM and VOF method. An adaptive static meshing to model the variation of the thickness, and dynamic to refine the flow front are used. This approach is useful to carry out the meshing at each step of time during the resolution process, which improves precision and calculation time. The algorithm adopted treats permeability and porosity at the level of the element of the mesh. The effects of the variations of the plies number and the stacking sequence, around the inserts for example, are modeled by the variation of permeability and porosity. The flows around obstacles and through the reinforced area near inserts are simulated by the present formalism with a local refinement of the meshing. The results obtained during the numerical simulation show a good concordance with the results: analytical, experimental and numerical. We can note the effectiveness of the numerical model developed in the predicted flow front and the distribution of pressure. An excellent reproduction of the form of the front and a good precision of its position are obtained. In our next studies, we will be interested in applying a similar approach to simulate saturation effects.

## **Author details**

Jamal Samir, Jamal Echaabi and Mohamed Hattabi *Applied Research Team on Polymers, Department of Mechanical Engineering, ENSEM, Hassan II University, Casa Blanca, Oasis, Casablanca, Morocco* 

## **9. References**


[4] Nielsen D, Pitchumani R. "Intelligent model-based control of preform permeation in liquid composite molding processes, with online optimization". Compos Part A: Appl Sci Manuf, 2001; 32(12):1789-803.

26 Finite Element Analysis – Applications in Mechanical Engineering

show the relevance of the present approach.

**8. Conclusion** 

**Author details** 

**9. References** 

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The approach adopted during this study ensures flexibility during the simulation of the heterogeneities of the problem. The properties of the medium are calculated at the level of the element of the mesh. Having developed an adaptive mesh generator specific to the team, we are able to ensure a better adaptation of the elements of grid to describe physical specificities generated in each problem. The results obtained on the figures of this paragraph

During this study, we developed a mesh generator and a numerical code to simulate the filling of an isothermal mould in RTM process, by adopting the CV/FEM and VOF method. An adaptive static meshing to model the variation of the thickness, and dynamic to refine the flow front are used. This approach is useful to carry out the meshing at each step of time during the resolution process, which improves precision and calculation time. The algorithm adopted treats permeability and porosity at the level of the element of the mesh. The effects of the variations of the plies number and the stacking sequence, around the inserts for example, are modeled by the variation of permeability and porosity. The flows around obstacles and through the reinforced area near inserts are simulated by the present formalism with a local refinement of the meshing. The results obtained during the numerical simulation show a good concordance with the results: analytical, experimental and numerical. We can note the effectiveness of the numerical model developed in the predicted flow front and the distribution of pressure. An excellent reproduction of the form of the front and a good precision of its position are obtained. In our next studies, we will be

interested in applying a similar approach to simulate saturation effects.

*Applied Research Team on Polymers, Department of Mechanical Engineering, ENSEM,* 

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[34] Naik, R.A. "Failure Analysis of Woven and Braided Fabric Reinforced Composites",

[35] Wang, Y. "Effect of Consolidation Method on the Mechanical Properties of Nonwoven Fabric Reinforced Composites", Applied Composite Materials, 6(1): 19–34. 1999. [36] Wang, Y. and Li, J. "Properties of Composites Reinforced with E-Glass Nonwoven

ISSN 0029-5981 CODEN IJNMBH vol. 36, no12, pp. 2001-2015 (26 ref.)1993

[28] L. P. Chew. Guaranteed-quality triangular meshes. TR-89-983, Cornell, 1989.

	- [54] Bickerton, S., Sozer, E.M., Graham, P.J. and Advani, S.G "Fabric Structure and Mold Curvature Effects on Preform Permeability and Mold Filling in the RTM Process". Part I. Experiments, Composites Part A, 31(5): 423–438. . 2000.

**Chapter 2** 

© 2012 Papadakis, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Papadakis, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Electromagnetic and Fluid Analysis of** 

For the analysis of collisional plasmas one needs to analyze the different constituent particles which characterize these discharges, mainly the charged particles such as electrons, positive and negative ions as well as the neutral gas particles. Due to the presence of the charged particles, one needs to calculate also the electromagnetic fields within the plasma which are characterized by the Maxwell's equations. The general Maxwell equations can be characterized by substituting a scalar and vector potential equations using Lorentz Gauge transformation to calculate the electric and magnetic fields. In the case of electrostatic fields, Maxwell equations reduce to Poisson equation that characterizes the electric field in the absence of a magnetic field, whereas in the magnetostatic case, a steady current exists invariant in time with the magnetic field related quantities considered constant. At high frequencies, the electromagnetic wavelength is small and if one is outside this single wavelength, the electric and magnetic field are directly coupled to each other such that if one of the two parameters is calculated, then the other is known. In the case of low frequencies, where the wavelength is high, if you are within the near field region, the electric and magnetic fields are completely independent, therefore the solution of both electric and magnetic field equations is necessary to calculate the electric and magnetic field

The charged particles behaviour is characterized using the continuity conservation equations of mass, momentum and energy for electrons, positive and negative ions including convective, diffusive and source term phenomena. Regarding the behaviour of the neutral gas particles within collisional plasmas, the Navier-Stokes equations, which are the conservation equations for mass, momentum and energy for the neutral gas particles need to be solved including convective, diffusive as well as source terms phenomena such as shear stresses, momentum transfer by elastic collisions, Lorentz forces, Joule heating and

**Collisional Plasmas** 

Additional information is available at the end of the chapter

distribution within the collisional plasma.

Antonis P. Papadakis

http://dx.doi.org/10.5772/48328

**1. Introduction** 


## **Electromagnetic and Fluid Analysis of Collisional Plasmas**

Antonis P. Papadakis

30 Finite Element Analysis – Applications in Mechanical Engineering

Composites Part A, 31A(12); 1433-1441. 2000.

reinforcement, Composites Part A, A27(4); 255-261. 1996.

(2000).

pp. 445-457

38 (1), 2007, pp. 51-60

I. Experiments, Composites Part A, 31(5): 423–438. . 2000.

[54] Bickerton, S., Sozer, E.M., Graham, P.J. and Advani, S.G "Fabric Structure and Mold Curvature Effects on Preform Permeability and Mold Filling in the RTM Process". Part

[55] Bickerton, S., Sozer, E.M., Simacek, P. and Advani, S.G. "Fabric Structure and Mold Curvature Effects on Preform Permeability and Mold Filling in the RTM Process". Part II. Predictions and Comparisons with Experiments, Composites Part A, 31(5): 439–458.

[56] Gauvin, R., Trochu, F., Lemenn, Y. and Diallo, L. "Permeability Measurement and Flow Simulation Through Fiber Reinforcement", Polymer Composites, 17(1): 34–42. 1996. [57] Sheard, J., Senft, V., Mantell, S.C. and Vogel, J.H. "Determination of Corner and Edge Permeability in Resin Transfer Molding", Polymer Composites, 19(1): 96–105. 1998. [58] Ferland, P., Guittard, D. and Torchu, F. "Concurrent Methods for Permeability Measurement in Resin Transfer Molding", Polymer Composites, 17(1): 149–158. 1996. [59] Dungan, F.D., Senoguz, M.T., Sastry, A.M. and Faillaci, D.A. "Simulation and Experiments on Low Pressure" Permeation of Fabrics: Part I-3D Modeling of

Unbalanced Fabric, Journal of Composite Materials, 35(14): 1250–1284. 2001.

[60] Pearce N.R.L., Summerscales J. and Guild F.J., Improving the resin transfer moulding process for fabric-reinforced composites by modification of the fibre architecture,

[61] Carter E.J., Fell A.W., Griffin P.R. and Summerscales J., Data validation procedures for the automated determination of the two-dimensional permeability tensor of a fabric

[62] M.J. Buntain, S. Bickerton "Compression flow permeability measurement: a continuous technique" Composites Part A: Applied Science and Manufacturing, Vol. 34 (5), 2003,

[63] Josef F.A. Kessels, Attie S. Jonker, Remko Akkerman, "Fully 2.1/2D flow modeling of resin infusion under flexible tooling using unstructured meshes and wet and dry compaction properties" Composites Part A: Applied Science and Manufacturing, Vol. Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48328

## **1. Introduction**

For the analysis of collisional plasmas one needs to analyze the different constituent particles which characterize these discharges, mainly the charged particles such as electrons, positive and negative ions as well as the neutral gas particles. Due to the presence of the charged particles, one needs to calculate also the electromagnetic fields within the plasma which are characterized by the Maxwell's equations. The general Maxwell equations can be characterized by substituting a scalar and vector potential equations using Lorentz Gauge transformation to calculate the electric and magnetic fields. In the case of electrostatic fields, Maxwell equations reduce to Poisson equation that characterizes the electric field in the absence of a magnetic field, whereas in the magnetostatic case, a steady current exists invariant in time with the magnetic field related quantities considered constant. At high frequencies, the electromagnetic wavelength is small and if one is outside this single wavelength, the electric and magnetic field are directly coupled to each other such that if one of the two parameters is calculated, then the other is known. In the case of low frequencies, where the wavelength is high, if you are within the near field region, the electric and magnetic fields are completely independent, therefore the solution of both electric and magnetic field equations is necessary to calculate the electric and magnetic field distribution within the collisional plasma.

The charged particles behaviour is characterized using the continuity conservation equations of mass, momentum and energy for electrons, positive and negative ions including convective, diffusive and source term phenomena. Regarding the behaviour of the neutral gas particles within collisional plasmas, the Navier-Stokes equations, which are the conservation equations for mass, momentum and energy for the neutral gas particles need to be solved including convective, diffusive as well as source terms phenomena such as shear stresses, momentum transfer by elastic collisions, Lorentz forces, Joule heating and

© 2012 Papadakis, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Papadakis, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

thermal conduction. Having established the necessary equations, the implementation of the solution procedure for solving Poisson, charged particle continuity and Navier-Stokes equations is presented. Thereafter, the Finite Element-Flux Corrected Transport method (FE-FCT) formulation follows in two-dimensional Cartesian, two-dimensional cylindrical axisymmetric and three-dimensional Cartesian coordinates, comprising of the predictorcorrector step based on the Taylor-Galerkin finite element method to calculate the high and low order schemes. The validation of the fluid flow equations using the FE-FCT is performed using the shock tube type problem, the shock wave incident on a wedge test case and the energy source term that result in sound and shock wave generation. Having validated the fluid model thoroughly, an adaptive mesh generation technique is discussed which reduces computational needs significantly and at the same time guaranteeing the stability and accuracy of the results. Finally, different collisional plasma configurations are analyzed including avalanche, primary and secondary streamer propagations and finally heating effects in constant voltage Dielectric Barrier Discharges and normal and abnormal glow discharges in ambient atmospheric air.

## **2. Model description**

The complete plasma model in its multidimensional form consists of the Maxwell equations to account for the electromagnetic field, the continuity equations for charges to account for the charged particles (electrons, positive and negative ions) and the Navier-Stokes equations to account for the neutral gas charges.

#### **2.1. Maxwell's equations**

The Maxwell equations consist of the following four differential equations in macroscopic form:

$$\nabla \cdot \mathbf{D} = \rho\_c \tag{1}$$

Electromagnetic and Fluid Analysis of Collisional Plasmas 33

( ) *E uBx* (5)

*<sup>c</sup> E* (6)

*<sup>c</sup>* (7)

*B A x* (8)

*.B .( A) x* 0 (9)

*<sup>t</sup>* (10)

*<sup>t</sup>* (11)

*<sup>t</sup>* (12)

*<sup>t</sup>* (13)

*<sup>t</sup>* (14)

*J* 

*B H* 

where εc and μc are the dielectric and magnetic permeability constants. One can define a

which satisfies equation (2) above of Maxwell equations which basically states that no

since the divergence of a curl is zero satisfying the magnetic monopole constraint.

 

 *A x*( )0 *E*

The term in the parenthesis of equation (11) above has no curl present implying that a

 *<sup>A</sup> <sup>E</sup> <sup>V</sup>*

*E J <sup>B</sup>* <sup>2</sup>

*c x*

() 0 *c*

Substituting equation (13) into Amperes law of electromagnetism equation (4) gives:

 *<sup>A</sup> <sup>E</sup> <sup>V</sup>*

*A) <sup>E</sup>* ( *<sup>x</sup>*

*x*

Now by substituting equation (8) into Faraday's equation (3), one gets:

*D*

vector potential **A** that **B** is the curl of:

magnetic monopoles exist such that:

By rewriting equation (10) above, one gets:

potential V exists such that:

where :

which gives the electric field to be:

where:

$$
\nabla \cdot \mathbf{B} = 0 \tag{2}
$$

$$
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \tag{3}
$$

$$
\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} \tag{4}
$$

where **D** is the electric field strength, **B** is the magnetic flux density, **E** is the electric field strength, **H** is the magnetic field strength, ρc is the net charge density, **J** is the current density and t is the time.

Equation (1) is known as the Gauss's law equation for electricity, equation (2) the Gauss's law for magnetism, equation (3) as the Michael-Faraday's or Faraday's law of induction and finally equation (4) shows the Ampere's circuital law with Maxwell's correction.

where:

32 Finite Element Analysis – Applications in Mechanical Engineering

glow discharges in ambient atmospheric air.

to account for the neutral gas charges.

**2. Model description** 

**2.1. Maxwell's equations** 

form:

and t is the time.

thermal conduction. Having established the necessary equations, the implementation of the solution procedure for solving Poisson, charged particle continuity and Navier-Stokes equations is presented. Thereafter, the Finite Element-Flux Corrected Transport method (FE-FCT) formulation follows in two-dimensional Cartesian, two-dimensional cylindrical axisymmetric and three-dimensional Cartesian coordinates, comprising of the predictorcorrector step based on the Taylor-Galerkin finite element method to calculate the high and low order schemes. The validation of the fluid flow equations using the FE-FCT is performed using the shock tube type problem, the shock wave incident on a wedge test case and the energy source term that result in sound and shock wave generation. Having validated the fluid model thoroughly, an adaptive mesh generation technique is discussed which reduces computational needs significantly and at the same time guaranteeing the stability and accuracy of the results. Finally, different collisional plasma configurations are analyzed including avalanche, primary and secondary streamer propagations and finally heating effects in constant voltage Dielectric Barrier Discharges and normal and abnormal

The complete plasma model in its multidimensional form consists of the Maxwell equations to account for the electromagnetic field, the continuity equations for charges to account for the charged particles (electrons, positive and negative ions) and the Navier-Stokes equations

The Maxwell equations consist of the following four differential equations in macroscopic

 *D* 

where **D** is the electric field strength, **B** is the magnetic flux density, **E** is the electric field strength, **H** is the magnetic field strength, ρc is the net charge density, **J** is the current density

Equation (1) is known as the Gauss's law equation for electricity, equation (2) the Gauss's law for magnetism, equation (3) as the Michael-Faraday's or Faraday's law of induction and

finally equation (4) shows the Ampere's circuital law with Maxwell's correction.

 *<sup>B</sup> <sup>E</sup>*

 *<sup>D</sup> <sup>H</sup> <sup>J</sup> <sup>t</sup>*

*<sup>c</sup>* (1)

*t* (3)

(4)

*B* 0 (2)

$$J = \sigma(E + \mathfrak{u}\mathfrak{x}\mathfrak{B})\tag{5}$$

$$\mathbf{D} = \boldsymbol{\varepsilon}\_c \cdot \mathbf{E} \tag{6}$$

$$\mathbf{B} = \mu\_c \cdot \mathbf{H} \tag{7}$$

where εc and μc are the dielectric and magnetic permeability constants. One can define a vector potential **A** that **B** is the curl of:

$$\mathcal{B} = \nabla \chi \mathcal{A} \tag{8}$$

which satisfies equation (2) above of Maxwell equations which basically states that no magnetic monopoles exist such that:

$$
\nabla \cdot \mathbf{B} = \nabla . (\nabla \times \mathbf{A}) = 0 \tag{9}
$$

since the divergence of a curl is zero satisfying the magnetic monopole constraint.

Now by substituting equation (8) into Faraday's equation (3), one gets:

$$\frac{\partial(\nabla \chi \mathcal{A})}{\partial t} = -\nabla \chi \mathcal{E} \tag{10}$$

By rewriting equation (10) above, one gets:

$$\nabla \mathbf{x} (\mathbf{E} + \frac{\partial \mathcal{A}}{\partial t}) = \mathbf{0} \tag{11}$$

The term in the parenthesis of equation (11) above has no curl present implying that a potential V exists such that:

$$E + \frac{\partial \mathcal{A}}{\partial t} = -\nabla V \tag{12}$$

which gives the electric field to be:

$$E = -\frac{\partial \mathcal{A}}{\partial t} - \nabla V \tag{13}$$

Substituting equation (13) into Amperes law of electromagnetism equation (4) gives:

$$\frac{\partial E}{\partial t} - c^2(\nabla x \mathbf{B}) - \frac{\mathbf{J}}{\varepsilon\_c} = 0 \tag{14}$$

where :

$$c^2 = \frac{1}{\mu\_c \varepsilon\_c} \tag{15}$$

Electromagnetic and Fluid Analysis of Collisional Plasmas 35

*t* (24)

*B A x* (26)

*<sup>c</sup> c V* (28)

*V* (29)

*V* 0 (30)

*t* (27)

(25)

Now using the Lorentz Gauge tranformation from equation (20) into equation (23) above

 *)* <sup>2</sup> <sup>2</sup> 1 (

*<sup>c</sup> <sup>t</sup> <sup>V</sup>*

Finally, equations (21) and (25) are respectively the vector and scalar potential equations of

In the electrostatic case, one assumes that ∂/∂t , **J**, **H** and **B** are all zero, and equation (25)

*c*

 2 2 2 <sup>0</sup> *<sup>c</sup>*

<sup>2</sup> *<sup>c</sup>*

which is the Poisson equation. In the presence of no free charges, ρ = 0 and the above

<sup>2</sup>

In the magnetostatic case, there are steady currents in the system under consideration, which generate magnetic fields (ferromagnetic media are ignored), therefore ones sets ∂/∂t=0 and assumes that **J**, **H** and **B** are constant vectors. Substituting these relations into the scalar

*<sup>A</sup> <sup>E</sup> <sup>V</sup>*

one can calculate the electric and magnetic field of time-dependent Maxwell's equations.

2 2 2 2 <sup>2</sup> 0 *<sup>c</sup> V c c V*

the Maxwell's equations which when solved by using the equations (26) and (27):

*t*

*V*

*c c*

gives:

and:

becomes:

*2.1.1. Electrostatic case* 

*2.1.2. Magnetostatic case* 

equation (29) becomes the Laplace equation:

and vector potential equations (21) and (25) of Maxwell's gives:

with c being the speed of propagation within the medium. From vector calculus, one has that:

$$
\nabla \times \mathbf{B} = \nabla \times \nabla \times \mathbf{A} = \nabla \left(\nabla \cdot \mathbf{A}\right) - \nabla^2 \mathbf{A} \tag{16}
$$

Substituting equation (16) above and equation (13) into equation (14) gives:

$$\frac{\partial(-\frac{\partial \mathcal{A}}{\partial t} - \nabla V)}{\partial t} - c^2(\nabla(\nabla \cdot \mathcal{A}) - \nabla^2 \mathcal{A}) - \frac{\mathcal{J}}{\mathcal{E}\_c} = 0 \tag{17}$$

which gives:

$$
\varepsilon \frac{\partial^2 A}{\partial t^2} - \nabla(\frac{\partial V}{\partial t}) + c^2 \nabla^2 A - c^2 (\nabla \left( \nabla \cdot A \right) - \frac{J}{\varepsilon\_c} = 0 \tag{18}
$$

Rearranging equation (18) above gives:

$$\frac{\partial^2 \mathcal{A}}{\partial t^2} - c^2 \nabla^2 \mathcal{A} + c^2 \nabla(\nabla \cdot \mathcal{A} + \frac{1}{c^2} \frac{\partial V}{\partial t}) - \frac{J}{\varepsilon\_c} = 0 \tag{19}$$

In order to make the choice of vector potential **A** not arbitrary, one imposes the Lorentz gauge which sets the parenthesis of equation (19) equal to zero:

$$(\nabla \cdot \mathcal{A} + \frac{1}{c^2} \frac{\partial V}{\partial t}) = 0 \tag{20}$$

with equation (20) above becoming:

$$\frac{\partial^2 \mathcal{A}}{\partial t^2} - c^2 \nabla^2 \mathcal{A} - \frac{\mathbf{J}}{\varepsilon\_c} = 0 \tag{21}$$

Equation (21) above is Maxwell's equation for the vector potential. On another note, by combining equations (1) and (13), one gets:

$$\nabla \cdot \left( -\nabla V - \frac{\partial \mathcal{A}}{\partial t} \right) = \frac{\rho\_c}{\varepsilon\_c} \tag{22}$$

which gives:

$$-\nabla^2 V - \frac{\partial(\nabla \mathcal{A})}{\partial t} = \frac{\rho\_c}{\varepsilon\_c} \tag{23}$$

Now using the Lorentz Gauge tranformation from equation (20) into equation (23) above gives:

$$-\nabla^2 V + \frac{\partial(\frac{1}{c^2}\frac{\partial V}{\partial t})}{\partial t} = \frac{\rho\_c}{\varepsilon\_c} \tag{24}$$

$$c\frac{\partial^2 V}{\partial t^2} - c^2 \nabla^2 V - \frac{c^2}{\varepsilon} \rho\_c = 0 \tag{25}$$

Finally, equations (21) and (25) are respectively the vector and scalar potential equations of the Maxwell's equations which when solved by using the equations (26) and (27):

$$\mathcal{B} = \nabla \chi A \tag{26}$$

and:

34 Finite Element Analysis – Applications in Mechanical Engineering

that:

which gives:

which gives:

 <sup>2</sup> <sup>1</sup> *c c*

with c being the speed of propagation within the medium. From vector calculus, one has

*<sup>J</sup> A A)* 2 2

( 0

<sup>1</sup> ( )0

*V*

In order to make the choice of vector potential **A** not arbitrary, one imposes the Lorentz

 

Equation (21) above is Maxwell's equation for the vector potential. On another note, by

 

 *A)* <sup>2</sup> ( . *<sup>c</sup>*

*V*

*V*

*A J <sup>A</sup>*

2 2 <sup>2</sup> 0

*c*

*c*

*A <sup>c</sup>*

*c*

*c*

*A* <sup>2</sup> <sup>1</sup> ( )0 *<sup>V</sup>*

 *A J A A*

22 2 2 2

*c c*

*t*

2

22 2 <sup>2</sup> () ( 0

*A J A A*

*c c*

Substituting equation (16) above and equation (13) into equation (14) gives:

*V t c*

*V*

*A*

( )

gauge which sets the parenthesis of equation (19) equal to zero:

2

2

Rearranging equation (18) above gives:

with equation (20) above becoming:

combining equations (1) and (13), one gets:

*c* (15)

*B A AA* 2 (16)

*c*

*t c <sup>t</sup>* (19)

*c*

*<sup>c</sup> <sup>t</sup>* (20)

(21)

*<sup>t</sup>* (22)

*<sup>t</sup>* (23)

*<sup>t</sup> <sup>t</sup>* (18)

*c*

*t* (17)

$$E = -\nabla V - \frac{\partial \mathcal{A}}{\partial t} \tag{27}$$

one can calculate the electric and magnetic field of time-dependent Maxwell's equations.

#### *2.1.1. Electrostatic case*

In the electrostatic case, one assumes that ∂/∂t , **J**, **H** and **B** are all zero, and equation (25) becomes:

$$-c^2\nabla^2V - \frac{c^2}{c}\rho\_c = 0\tag{28}$$

$$
\nabla^2 V = -\frac{\rho\_c}{\varepsilon\_c} \tag{29}
$$

which is the Poisson equation. In the presence of no free charges, ρ = 0 and the above equation (29) becomes the Laplace equation:

$$
\nabla^2 V = 0\tag{30}
$$

#### *2.1.2. Magnetostatic case*

In the magnetostatic case, there are steady currents in the system under consideration, which generate magnetic fields (ferromagnetic media are ignored), therefore ones sets ∂/∂t=0 and assumes that **J**, **H** and **B** are constant vectors. Substituting these relations into the scalar and vector potential equations (21) and (25) of Maxwell's gives:

$$-c^2\nabla^2V - \frac{c^2}{\varepsilon}\rho\_c = 0\tag{31}$$

Electromagnetic and Fluid Analysis of Collisional Plasmas 37

where α the ionisation coefficient, η the attachment coefficient, βep the recombination coefficient between electrons and positive ions, βpn the recombination coefficient between positive and negative ions and De the electron diffusion coefficient. The positive and negative ion diffusion coefficients are omitted since their effect is negligible compared to

The conservation of momentum and energy density for electrons, positive and negative ions

 **<sup>s</sup>**

**W ) W W** <sup>s</sup> **MT** ( .( ) . *<sup>s</sup> s s*

Subscripts s, p and n represent the electron, positive and negative ions, and the various terms are similar to the ones used for the neutral gas particle conservation equations that are

The Navier-Stokes equations which account for the neutral gas particles are described by

*s P x*

*s s <sup>P</sup> f x t* (46)

*t* (45)

 .( ) **<sup>v</sup>** *<sup>S</sup>*

 **<sup>s</sup> v) vv MT J B** ( .( ) .

 **<sup>s</sup>** .( ) .( ) .( ( )) . . ( **v Q v I MT v J E J B).v** *ts*

where ρ the neutral gas density, **v** the neutral gas velocity vector, S the neutral gas density source term, P the pressure, **τ** the shear stress tensor which comprises of the τii and τjj components, **MTs** the momentum transfer of charged to the neutral particles due to elastic collisions, ε the neutral gas thermal energy density, **Q** is the thermal conductivity term, fts the percentage of energy density of charged particle with subscript s that is transferred as thermal energy to the neutral particles due to inelastic collisions and **J** the current density, **JxB** is the Lorentz force and **(JxB).v** is the magnetic Lorentz force acting on the energy of

 .( ) .( ) .( ( )) . . *<sup>s</sup> s s s ss s s ts s*

**s s s**

*W Q W P I MT v f J E t* (43)

*P*

  *s*

*s s*

*t* (44)

*t* (42)

equations are described below in equations (42) and (43) and have the general form:

electron diffusion for the time scales considered here.

The neutral gas density source term is calculated by:

**2.3. Navier-Stokes equations** 

*2.3.1. Navier-Stokes general form* 

equations (44) to (46) below:

the flow.

explained below.

which is again the Poisson equation:

$$
\nabla^2 V = -\frac{\rho\_c}{\varepsilon\_c} \tag{32}
$$

and:

$$-c^2 \nabla^2 A - \frac{\mathbf{J}}{\mathcal{E}\_c} = 0\tag{33}$$

$$
\nabla^2 \mathcal{A} = \frac{\mathbf{J}}{c^2 \varepsilon\_c} \tag{34}
$$

$$
\nabla^2 \mathcal{A} = -\mu \mathcal{J} \tag{35}
$$

which is the magnetic equivalent of the electrostatic Poisson equation.

#### **2.2. Continuity equations for charges**

The continuity equations which comprise of the conservation of mass for electrons, positive and negative ions are described in equations (36) to (38) below:

$$\frac{\partial \mathbf{N}\_e}{\partial t} + \nabla. (\mathbf{N}\_e \mathbf{W}\_e) = \mathbf{S}\_e \tag{36}$$

$$\frac{\partial \mathbf{N}\_p}{\partial t} + \nabla.(\mathbf{N}\_p \mathbf{W}\_\mathbf{p}) = \mathbf{S}\_p \tag{37}$$

$$\frac{\partial \mathbf{N}\_n}{\partial t} + \nabla. (\mathbf{N}\_n \mathbf{W}\_n) = \mathbf{S}\_n \tag{38}$$

where Ne, Np, Nn are respectively the electron, positive and negative ion densities, **We**, **Wp** and **Wn** the corresponding velocity vectors and Se, Sp, Sn are the source terms for the electrons, positive and negative ions respectively, which are calculated according to equations (39) to (41) below:

$$\mathbf{S}\_{\varepsilon} = a \mathbf{N}\_{\varepsilon} \parallel \mathbf{W}\_{\mathbf{e}} \parallel - \eta \mathbf{N}\_{\varepsilon} \parallel \mathbf{W}\_{\mathbf{e}} \parallel - \beta\_{cp} \mathbf{N}\_{\varepsilon} \mathbf{N}\_{p} + \nabla. (\mathbf{D}\_{\varepsilon} \nabla \mathbf{N}\_{\varepsilon}) \tag{39}$$

$$S\_p = \alpha N\_e \parallel \mathbf{W\_e} \parallel -\beta\_{ep} N\_e N\_p - \beta\_{pn} N\_p N\_n \tag{40}$$

$$S\_n = \eta N\_e \mid \mathbf{W\_e} \mid -\beta\_{pn} N\_p N\_n \tag{41}$$

where α the ionisation coefficient, η the attachment coefficient, βep the recombination coefficient between electrons and positive ions, βpn the recombination coefficient between positive and negative ions and De the electron diffusion coefficient. The positive and negative ion diffusion coefficients are omitted since their effect is negligible compared to electron diffusion for the time scales considered here.

The conservation of momentum and energy density for electrons, positive and negative ions equations are described below in equations (42) and (43) and have the general form:

$$\frac{\partial(\rho\_s \mathbf{W\_s})}{\partial t} + \nabla. (\rho\_s \mathbf{W\_s} \mathbf{W\_s}) = -\nabla P\_s - \nabla. \mathbf{r\_s} - \sum\_s \mathbf{M} \mathbf{T\_s} \tag{42}$$

$$\frac{\partial \varepsilon\_s}{\partial t} + \nabla. (\varepsilon\_s W\_s) = -\nabla. (Q\_s) - \nabla. (W\_s (P\_s I + \tau\_s)) - \sum\_s M T\_s,\\ \upsilon - \sum\_s f\_{ts} I\_s E \tag{43}$$

Subscripts s, p and n represent the electron, positive and negative ions, and the various terms are similar to the ones used for the neutral gas particle conservation equations that are explained below.

#### **2.3. Navier-Stokes equations**

36 Finite Element Analysis – Applications in Mechanical Engineering

which is again the Poisson equation:

**2.2. Continuity equations for charges** 

equations (39) to (41) below:

and:

<sup>2</sup> *<sup>c</sup>*

 *<sup>J</sup> <sup>A</sup>* 2 2 <sup>0</sup> *c*

> *<sup>J</sup> <sup>A</sup>*<sup>2</sup> 2 *<sup>c</sup> c*

 *A J* 

The continuity equations which comprise of the conservation of mass for electrons, positive

**Wn** .( ) *<sup>n</sup>*

*<sup>N</sup> N S*

where Ne, Np, Nn are respectively the electron, positive and negative ion densities, **We**, **Wp** and **Wn** the corresponding velocity vectors and Se, Sp, Sn are the source terms for the electrons, positive and negative ions respectively, which are calculated according to

 

 

**Wp** .( ) *<sup>p</sup>*

*<sup>N</sup> N S*

*e e*

*p p*

*n n*

**W W e e** | | | | .( ) *e e <sup>e</sup> ep e p e e S N N NN D N* (39)

**We** | | *<sup>p</sup> <sup>e</sup> ep e p pn p n S N NN NN* (40)

**We** | | *n e pn p n S N NN* (41)

*N S*

**We** .( ) *<sup>e</sup>*

which is the magnetic equivalent of the electrostatic Poisson equation.

*N*

and negative ions are described in equations (36) to (38) below:

*c*

*<sup>c</sup> c V* (31)

*V* (32)

*c* (33)

2 (35)

*t* (36)

*t* (37)

*t* (38)

(34)

 2 2 2 <sup>0</sup> *<sup>c</sup>*

#### *2.3.1. Navier-Stokes general form*

The Navier-Stokes equations which account for the neutral gas particles are described by equations (44) to (46) below:

$$\frac{\partial \rho}{\partial t} + \nabla.(\rho \mathbf{v}) = \mathbf{S} \tag{44}$$

$$\frac{\partial(\rho \mathbf{v})}{\partial t} + \nabla.(\rho \mathbf{v} \mathbf{v}) = -\nabla P - \nabla.\mathbf{r} + \sum\_{s} \mathbf{M} \mathbf{T}\_{s} + \mathbf{J}x \mathbf{B} \tag{45}$$

$$\frac{\partial \mathcal{E}}{\partial t} + \nabla . (\mathcal{E} \mathbf{v}) = -\nabla . (\mathbf{Q}) - \nabla . (\mathbf{v} (P \mathbf{I} + \mathbf{r})) + \sum\_{s} \mathbf{M} \mathbf{T}\_{s} \mathbf{v} + \sum\_{s} f\_{ls} \mathbf{J} \cdot \mathbf{E} + (\mathbf{J} \times \mathbf{B}) \mathbf{v} \tag{46}$$

where ρ the neutral gas density, **v** the neutral gas velocity vector, S the neutral gas density source term, P the pressure, **τ** the shear stress tensor which comprises of the τii and τjj components, **MTs** the momentum transfer of charged to the neutral particles due to elastic collisions, ε the neutral gas thermal energy density, **Q** is the thermal conductivity term, fts the percentage of energy density of charged particle with subscript s that is transferred as thermal energy to the neutral particles due to inelastic collisions and **J** the current density, **JxB** is the Lorentz force and **(JxB).v** is the magnetic Lorentz force acting on the energy of the flow.

The neutral gas density source term is calculated by:

$$\mathcal{S} = m(-\alpha \mathbf{N}\_{\varepsilon} \mid \mathbf{W}\_{\mathbf{e}} \mid -\eta \mathbf{N}\_{\varepsilon} \mid \mathbf{W}\_{\mathbf{e}} \mid +\mathbf{\mathcal{Q}} \boldsymbol{\rho}\_{pn} \mathbf{N}\_{p} \mathbf{N}\_{n} + \boldsymbol{\rho}\_{ep} \mathbf{N}\_{\varepsilon} \mathbf{N}\_{p}) \tag{47}$$

where m is the neutral gas particle mass which is constant.

The thermal conductivity, current density and pressure are calculated respectively by equations (48) to (50):

$$\mathbf{Q} = -k\nabla T \tag{48}$$

Electromagnetic and Fluid Analysis of Collisional Plasmas 39

*<sup>v</sup>* (55)

(56)

(57)

(58)

 *x y v*

*Q*

*x x x <sup>x</sup> y x*

*v vv P*

> *v v P v*

> >

**W W e e**

*2.3.3. Two-dimensional cylindrical axisymmetric coordinates of Navier-Stokes* 

*s s*

*xx x xy y*

*x*

( )

0 *xx*

*x yx*

*x*

*L*

are formulated as follows:

*<sup>T</sup> K vv*

*MT x x x <sup>M</sup>*

*K*

*y*

*K*

*L*

*m N N NN NN*

( 2 ) { } ( ){ } { } ( ){ } . {} { } ( ){ } ( ){ .}

*ss s ts s x s y x y*

**J E JB JB**

The Navier-Stokes equations in two-dimensional cylindrical (r, z) axisymmetric coordinates

 *r v*

*z*

*K*

*z r z*

*v v v*

*z*

( )

*z z*

*vv P P v*

 1 1 ( ) () *<sup>Q</sup> Kr K Lr L r z rz <sup>M</sup>*

*Q*

*r r r*

*v vv P*

> *z r r*

*v v P v*

( )

*r*

*K*

*MT y x y f MT x v MT y v x x v x y v*

 

*<sup>e</sup>*|| || *<sup>e</sup> ep e p pn p n*

**J B J B** *y x y*

*v v v*

( )

*vv P P v*

*y*

0 *xy*

*<sup>T</sup> K vv*

 

*tr r zr r z* (59)

*vz* (60)

(61)

 

*yy y yx x*

*y y*

*y yy*

*y*

$$\mathbf{J} = q\_s \mathbf{N}\_s \mathbf{W}\_s \tag{49}$$

$$P = \text{NKT} \tag{50}$$

where k is the thermal conductivity, T is the neutral gas temperature, qs, Ns, **Ws** the charge, density per unit volume and velocity vector of charged particle s, N the number of neutral gas particles per unit volume and K the Boltzmann constant.

The shear stress tensors are calculated by:

$$
\sigma\_{ii} = -\mu (2\frac{\partial v\_i}{\partial \mathbf{x}\_i} - \frac{2}{3}\nabla.\mathbf{v}) \tag{51}
$$

$$
\sigma\_{ij} = -\mu(\frac{\partial \upsilon\_i}{\partial \mathbf{x}\_j} + \frac{\partial \upsilon\_j}{\partial \mathbf{x}\_i}) \tag{52}
$$

where the viscosity coefficient, and subscript i and j represent the different degrees of freedom in the different space directions.

The momentum transfer of charged to the neutral particles due to elastic collisions **MTs** is calculated by:

$$\mathbf{MT}\_{\mathbf{s}} = \mathbf{N}\_{s} \frac{m}{m + m\_{s}} \frac{q\_{s}}{\mu\_{s}} (\mathbf{W}\_{\mathbf{s}} - \mathbf{v}) \tag{53}$$

where μs is the mobility of charged particle s.

The numerical algorithm and its implementation are presented below in two-dimensional Cartesian, two-dimensional axisymmetric cylindrical and three-dimensional Cartesian coordinates in the following sections.

#### *2.3.2. Two-dimensional Cartesian coordinates of Navier-Stokes*

The Navier-Stokes equations in two-dimensional Cartesian (x, y) coordinates are formulated as follows:

$$\frac{\partial \mathbf{Q}}{\partial t} + \frac{\partial \mathbf{K}\_x}{\partial \mathbf{x}} + \frac{\partial \mathbf{K}\_y}{\partial y} + \frac{\partial \mathbf{L}\_x}{\partial \mathbf{x}} + \frac{\partial \mathbf{L}\_y}{\partial y} = \mathbf{M} \tag{54}$$

#### Electromagnetic and Fluid Analysis of Collisional Plasmas 39

$$\mathcal{Q} = \begin{bmatrix} \rho \\ \rho v\_x \\ \rho v\_y \\ \varepsilon \end{bmatrix} \tag{55}$$

$$\begin{aligned} \boldsymbol{K}\_{\boldsymbol{x}} &= \begin{bmatrix} \rho \boldsymbol{\upsilon}\_{\boldsymbol{x}} \\ \rho \boldsymbol{\upsilon}\_{\boldsymbol{x}} \boldsymbol{\upsilon}\_{\boldsymbol{x}} + P \\ \rho \boldsymbol{\upsilon}\_{\boldsymbol{y}} \boldsymbol{\upsilon}\_{\boldsymbol{x}} \\ (\boldsymbol{\varepsilon} + P) \boldsymbol{\upsilon}\_{\boldsymbol{x}} \end{bmatrix} \boldsymbol{K}\_{\boldsymbol{y}} = \begin{bmatrix} \rho \boldsymbol{\upsilon}\_{\boldsymbol{y}} \\ \rho \boldsymbol{\upsilon}\_{\boldsymbol{x}} \boldsymbol{\upsilon}\_{\boldsymbol{y}} \\ \rho \boldsymbol{\upsilon}\_{\boldsymbol{y}} \boldsymbol{\upsilon}\_{\boldsymbol{y}} + P \\ (\boldsymbol{\varepsilon} + P) \boldsymbol{\upsilon}\_{\boldsymbol{y}} \end{bmatrix} \tag{56}$$

$$L\_x = \begin{bmatrix} 0 \\ \tau\_{xx} \\ \tau\_{yx} \\ -K\frac{\partial T}{\partial x} + \tau\_{xx}\upsilon\_x + \tau\_{xy}\upsilon\_y \\ -K\frac{\partial T}{\partial x} + \tau\_{xx}\upsilon\_x + \tau\_{xy}\upsilon\_y \end{bmatrix} L\_y = \begin{bmatrix} 0 \\ \tau\_{xy} \\ \tau\_{yy} \\ -K\frac{\partial T}{\partial y} + \tau\_{yy}\upsilon\_y + \tau\_{yx}\upsilon\_x \\ \end{bmatrix} \tag{57}$$

$$M = \begin{bmatrix} m(-\alpha N\_e \mid \mathbf{W\_e} \mid -\eta N\_e \mid \mathbf{W\_e} \mid +\beta\_{cp} N\_e N\_p + 2\beta\_{pn} N\_p N\_n) \\ & MT\_s \{ \mathbf{x} \} + (\mathbf{JxB} \rangle \langle \mathbf{x} \rangle \\ & MT\_s \{ y \} + (\mathbf{JxB} \} \langle y \rangle \\ \sum\_s f\_b \mathbf{J}, \mathbf{E} + \sum\_s MT\_s \{ \mathbf{x} \} \upsilon\_x + \sum\_s MT\_s \{ y \} \upsilon\_y + (\mathbf{JxB} \} \langle \mathbf{x} \rangle \upsilon\_x + (\mathbf{JxB} \} \langle y, \upsilon\_y \rangle \end{bmatrix} \tag{58}$$

#### *2.3.3. Two-dimensional cylindrical axisymmetric coordinates of Navier-Stokes*

38 Finite Element Analysis – Applications in Mechanical Engineering

equations (48) to (50):

The thermal conductivity, current density and pressure are calculated respectively by

where k is the thermal conductivity, T is the neutral gas temperature, qs, Ns, **Ws** the charge, density per unit volume and velocity vector of charged particle s, N the number of neutral

> 

where the viscosity coefficient, and subscript i and j represent the different degrees of

The momentum transfer of charged to the neutral particles due to elastic collisions **MTs** is

*<sup>m</sup> <sup>q</sup> <sup>N</sup> m m*

The numerical algorithm and its implementation are presented below in two-dimensional Cartesian, two-dimensional axisymmetric cylindrical and three-dimensional Cartesian

The Navier-Stokes equations in two-dimensional Cartesian (x, y) coordinates are formulated

*M tx yxy* (54)

 *y y x x <sup>Q</sup> K L K L*

<sup>2</sup> (2 . ) <sup>3</sup> *i*

*i v x*

 ( )*<sup>j</sup> <sup>i</sup>*

*j i v v x x*

 **MTs s** (**W v)** *<sup>s</sup>*

*s s*

**v**

 

**W W e e** ( 2 || || ) *S m N N <sup>e</sup> <sup>e</sup> pn p n ep e p NN NN* (47)

 

**Q** *k T* (48)

**<sup>s</sup> J W** *s s q N* (49)

*P NKT* (50)

(51)

(52)

(53)

where m is the neutral gas particle mass which is constant.

gas particles per unit volume and K the Boltzmann constant.

*ii*

*ij*

 

> 

*s*

The shear stress tensors are calculated by:

freedom in the different space directions.

where μs is the mobility of charged particle s.

*2.3.2. Two-dimensional Cartesian coordinates of Navier-Stokes* 

coordinates in the following sections.

calculated by:

as follows:

The Navier-Stokes equations in two-dimensional cylindrical (r, z) axisymmetric coordinates are formulated as follows:

$$\frac{\partial \mathcal{Q}}{\partial t} + \frac{1}{r} \frac{\partial (K\_r r)}{\partial r} + \frac{\partial K\_z}{\partial z} + \frac{1}{r} \frac{\partial (L\_r r)}{\partial r} + \frac{\partial L\_z}{\partial z} = M \tag{59}$$

$$\mathbf{Q} = \begin{bmatrix} \rho \\ \rho v\_r \\ \rho vz \\ \varepsilon \end{bmatrix} \tag{60}$$

$$\begin{aligned} \boldsymbol{K}\_r &= \begin{bmatrix} \rho \boldsymbol{v}\_r \\ \rho \boldsymbol{v}\_r \boldsymbol{v}\_r + P \\ \rho \boldsymbol{v}\_z \boldsymbol{v}\_r \\ (\boldsymbol{\varepsilon} + P) \boldsymbol{v}\_r \end{bmatrix} \boldsymbol{K}\_z = \begin{bmatrix} \rho \boldsymbol{v}\_z \\ \rho \boldsymbol{v}\_r \boldsymbol{v}\_z \\ \rho \boldsymbol{v}\_z \boldsymbol{v}\_z + P \\ (\boldsymbol{\varepsilon} + P) \boldsymbol{v}\_z \end{bmatrix} \tag{61} \end{aligned} \tag{61}$$

$$L\_r = \begin{bmatrix} 0 \\ \tau\_{rr} \\ \tau\_{zr} \\ -K\frac{\partial T}{\partial r} + \tau\_{rr}\upsilon\_r + \tau\_{rz}\upsilon\_z \end{bmatrix} L\_z = \begin{bmatrix} 0 \\ \tau\_{rz} \\ \tau\_{zz} \\ -K\frac{\partial T}{\partial z} + \tau\_{zz}\upsilon\_z + \tau\_{zz}\upsilon\_r \end{bmatrix} \tag{62}$$

$$M = \begin{bmatrix} \min\{-\alpha \mathbf{N}\_e \mid \mathbf{W}\_e \mid -\eta \mathbf{N}\_e \mid \mathbf{W}\_e \mid +\rho\_{vp}N\_eN\_p + 2\rho\_{pn}N\_pN\_n\} \\\\ MT\_s\{r\} + \frac{P}{r} + (\mathbf{JxB})\{r\}\upsilon\_r \\\\ MT\_s\langle z\rangle + (\mathbf{JxB})\{z\}\upsilon\_z \\ \sum\_s f\_{js}\mathbf{J}\_s + \sum\_s MT\_s\{r\}\upsilon\_r + \sum\_s MT\_s\{z\}\upsilon\_z + (\mathbf{JxB})\{r\}\upsilon\_r + (\mathbf{JxB})\{z\}\upsilon\_z\} \end{bmatrix} \tag{63}$$

#### *2.3.4. Three-dimensional Cartesian coordinates of Navier-Stokes*

The Navier-Stokes equations in three-dimensional Cartesian (x, y, z) coordinates are formulated as follows:

$$\frac{\partial \mathbf{Q}}{\partial t} + \frac{\partial \mathbf{K}\_x}{\partial \mathbf{x}} + \frac{\partial \mathbf{K}\_y}{\partial y} + \frac{\partial \mathbf{K}z}{\partial z} + \frac{\partial \mathbf{L}\_x}{\partial \mathbf{x}} + \frac{\partial \mathbf{L}\_y}{\partial y} + \frac{\partial \mathbf{L}\_z}{\partial z} = \mathbf{M} \tag{64}$$

$$\mathbf{Q} = \begin{bmatrix} \rho \\ \rho v\_x \\ \rho v\_y \\ \rho v\_z \\ \varepsilon \end{bmatrix} \tag{65}$$

Electromagnetic and Fluid Analysis of Collisional Plasmas 41

(67)

(68)

0

*xx*

*x yx*

*x*

*L*

*L*

*L*

 

*M MT y x y v*

**3. Implementation of the solution procedure** 

*y yy*

*y*

*z yz*

*z*

**W W e e**

*zx*

*<sup>T</sup> K vvv*

*xy*

0

*zy*

*<sup>T</sup> K vvv*

0

*xz*

*zz*

*<sup>T</sup> K vvv*

 

*<sup>e</sup>*|| || *<sup>e</sup> ep e p pn p n s x s y s z ss s s ts s x s y s z x y z*

( 2 ) { } ( ){ } { } ( ){ .} { } ( ){ } . {} { } { } ( ){ } ( ){ .} ( ){ }

**J B J B J B J E JB JB JB**

*MT z x z v f MT x v MT y v MT z v x x v x y v x z v*

The solution procedure follows the pattern shown in Figure 1 for a single time step. It assumes that an initial distribution of electrons and positive ions at a given neutral gas temperature, velocity, density and applied voltage exists. The charged particle densities at time n (ρn) are fed to Poisson's equation (PO) to obtain the electric field distribution at time n (En). The electric field and neutral gas parameters (Nn) are used to calculate the transport parameters (TP) at time n (TPn). Then, the transport parameters are passed to the charge particle continuity equation (CON) to calculate the charge densities at time n+1/2 (ρn+1/2). Next, the charge densities (ρn), electric field (En) and transport properties (TPn) at time n are passed to the Navier-Stokes solver (NS) to calculate the neutral gas properties at time n+1/2

*m N N NN NN MT x x x v*

*zx x zy y zz z*

 

*yx x yy y yz z*

*xx x xy y xz z*

$$\begin{aligned} \mathbf{K}\_{x} = \begin{bmatrix} \rho \upsilon\_{x} \\ \rho \upsilon\_{x} \upsilon\_{x} + P \\ \rho \upsilon\_{x} \upsilon\_{y} \\ \rho \upsilon\_{x} \upsilon\_{z} \\ (\varepsilon + P) \upsilon\_{x} \end{bmatrix} \mathbf{K}\_{y} = \begin{bmatrix} \rho \upsilon\_{y} \\ \rho \upsilon\_{y} \upsilon\_{x} + P \\ \rho \upsilon\_{y} \upsilon\_{z} + P \\ \rho \upsilon\_{y} \upsilon\_{z} \\ (\varepsilon + P) \upsilon\_{y} \end{bmatrix} \mathbf{K} \mathbf{z} = \begin{bmatrix} \rho \upsilon\_{z} \\ \rho \upsilon\_{z} \upsilon\_{x} \\ \rho \upsilon\_{z} \upsilon\_{y} \\ \rho \upsilon\_{z} \upsilon\_{z} + P \\ (\varepsilon + P) \upsilon\_{z} \end{bmatrix} \tag{66}$$

The two-step Lax-Wendroff technique comprising of the predictor-corrector steps is used for time stepping. The charged particle continuity and Navier-Stokes are discretised using Taylor-Galerkin Finite Elements [1], whereas Poisson's equation using Galerkin Finite Elements. Accuracy and efficiency for these long calculations are a crucial factor. The Flux Corrected Transport method (FCT) [2-4] ensures that accurate and efficient results are obtained free from inaccuracies building up and from non-physical oscillations.

 0 0 0 *xx x yx zx xx x xy y xz z xy y yy zy yx x yy y yz z xz z yz zz zx x zy y zz z L <sup>T</sup> K vvv x L <sup>T</sup> K vvv y L <sup>T</sup> K vvv z* (67) **W W e e J B J B J B J E JB JB JB** ( 2 ) { } ( ){ } { } ( ){ .} { } ( ){ } . {} { } { } ( ){ } ( ){ .} ( ){ } *<sup>e</sup>*|| || *<sup>e</sup> ep e p pn p n s x s y s z ss s s ts s x s y s z x y z m N N NN NN MT x x x v M MT y x y v MT z x z v f MT x v MT y v MT z v x x v x y v x z v* (68)

#### **3. Implementation of the solution procedure**

40 Finite Element Analysis – Applications in Mechanical Engineering

*L*

formulated as follows:

0 *rr*

*r zr*

*r*

*<sup>P</sup> MT r x r v <sup>M</sup> <sup>r</sup>*

*2.3.4. Three-dimensional Cartesian coordinates of Navier-Stokes* 

*x x x x xy*

*v vv P*

> *x z x*

*v v P v*

( )

*K vv*

*<sup>T</sup> K vv*

 

**W W e e**

0 *rz*

*z zz*

*z*

*L*

*m N N NN NN*

( 2 )

{ } ( ){ } { } ( ){ .} . {} { } ( ){ } ( ){ .}

*s r s z ss s ts s r s z r z*

 

The Navier-Stokes equations in three-dimensional Cartesian (x, y, z) coordinates are

*v*

*Q v*

 *x y z*

*v v v*

*y y x*

( )

*y z y*

*v v P v*

The two-step Lax-Wendroff technique comprising of the predictor-corrector steps is used for time stepping. The charged particle continuity and Navier-Stokes are discretised using Taylor-Galerkin Finite Elements [1], whereas Poisson's equation using Galerkin Finite Elements. Accuracy and efficiency for these long calculations are a crucial factor. The Flux Corrected Transport method (FCT) [2-4] ensures that accurate and efficient results are

obtained free from inaccuracies building up and from non-physical oscillations.

*K vv P*

*y yy*

*v*

 *y y x xz <sup>Q</sup> K LL K L Kz <sup>M</sup>*

*MT z x z v f MT r v MT z v x r v x z v*

 

*<sup>e</sup>*|| || *<sup>e</sup> ep e p pn p n*

**J B J E JB JB**

**J B**

*tx y zxyz* (64)

(65)

*z z x z y z z*

*v v v*

*z*

(66)

( )

*vv P P v*

*Kz v v*

 

*<sup>T</sup> K vv*

  (62)

(63)

*zz z zr r*

*rr r rz z*

The solution procedure follows the pattern shown in Figure 1 for a single time step. It assumes that an initial distribution of electrons and positive ions at a given neutral gas temperature, velocity, density and applied voltage exists. The charged particle densities at time n (ρn) are fed to Poisson's equation (PO) to obtain the electric field distribution at time n (En). The electric field and neutral gas parameters (Nn) are used to calculate the transport parameters (TP) at time n (TPn). Then, the transport parameters are passed to the charge particle continuity equation (CON) to calculate the charge densities at time n+1/2 (ρn+1/2). Next, the charge densities (ρn), electric field (En) and transport properties (TPn) at time n are passed to the Navier-Stokes solver (NS) to calculate the neutral gas properties at time n+1/2

(Nn+1/2). This completes the half time step solution. Subsequently, the charge particle densities at time n+1/2 are passed to the Poisson solver to calculate the electric field at time n+1/2 (En+1/2). The electric field and neutral gas parameters at time n+1/2 are used to calculate the transport properties at time n+1/2 (TPn+1/2). Consequently the transport properties are fed to the charged particle continuity solver to calculate the charge densities at time n+1 (ρn+1). Finally, the Navier-Stokes solver uses the transport parameters, the electric field and the charge densities at time n+1/2 to calculate the neutral gas parameters at time n+1 (Nn+1) and this process is repeated continuously to proceed forward in time.

Electromagnetic and Fluid Analysis of Collisional Plasmas 43

(69)

*L* are respectively the convective

continuity equations via production and loss processes, and via momentum and kinetic

The fluid transport equations take the general form in the two-dimensional Cartesian

*j j*

 *K* and

*M*

*Q* at a first order approximation, thereby resulting in the half time

*<sup>A</sup> ttt Q Q Kb Kc A M* (70)

*tx x*

*Q K L*

1 1

*j j j j*

The Taylor Galerkin scheme is used to develop the FE-FCT scheme, with the time discretization preceding the space discretization. The time stepping is performed in two

The advective predictor step is formulated using the Taylor series expansion for the

*n nn n n*

where Ae is the area of a triangular element, Δt is the time step, bi and ci are the linearly interpolated piecewise shape functions. The velocities at time (n+1/2) are assumed to be the

The corrector step utilizes the Taylor series expansion for the full time step to the second

 1/2 1/2 1/2 1/2 2 2 22

*i i ii n nnn*

*n e e ee*

*c x y xy tb tc tb tc DQ K K L L*

 1/2 33 3 3

*<sup>e</sup> i i <sup>i</sup> <sup>i</sup> i ie ii i i*

11 1 1 34 4 6

 *<sup>n</sup>* 1/2 *Q* :

 2 2

**4. Finite element formulation of continuity and Navier-Stokes fluid** 

energy exchange between charged and neutral particles.

**4.1. Two-dimensional Cartesian coordinates** 

*Q* is the unknown or independent variable,

steps, the advective predictor and the corrector step.

*M* is the source term.

average value of the nodal velocities at time n.

**equations** 

coordinates as:

and diffusive fluxes,

unknown variable

*4.1.2. Corrector step* 

order approximation to get:

*4.1.1. Advective predictor step* 

values of the independent variable

where

In fluid analysis, the collision coefficients and drift velocities of the charged particles can be calculated either using experimental or numerical techniques. Experimentally they are calculated as a function of the ratio of the electric field to the number of neutral gas particles per unit volume (E/N), depending also on the density and pressure of the neutral gas that has a direct effect on the frequency of collisions between the charged and neutral gas particles.

Numerically, commercial software packages calculate the transport properties by solving the electron distribution Boltzmann equation by utilizing the collisional cross sectional data and such software examples are ELENDIF [5], BOLSIG [6], and BOLSIG+ [7] which are used to calculate the above transport parameters [8].

**Figure 1.** Solution procedure of the complete model

The charged particle continuity equations are coupled to Poisson's equation via the electric field strength and to the Navier-Stokes equations via the number of neutral gas particles per unit volume and their pressure. Poisson's equation is coupled to the continuity equations via the net charge density. As far as the Navier-Stokes equations are concerned, they are coupled to Poisson's equation via the electric field strength and to the charged particle continuity equations via production and loss processes, and via momentum and kinetic energy exchange between charged and neutral particles.

## **4. Finite element formulation of continuity and Navier-Stokes fluid equations**

#### **4.1. Two-dimensional Cartesian coordinates**

42 Finite Element Analysis – Applications in Mechanical Engineering

to calculate the above transport parameters [8].

PO

 n

**Start of Time Step**

NS

**Figure 1.** Solution procedure of the complete model

**End of Time Step**

particles.

n

 n+1

 n+1

 n

(Nn+1/2). This completes the half time step solution. Subsequently, the charge particle densities at time n+1/2 are passed to the Poisson solver to calculate the electric field at time n+1/2 (En+1/2). The electric field and neutral gas parameters at time n+1/2 are used to calculate the transport properties at time n+1/2 (TPn+1/2). Consequently the transport properties are fed to the charged particle continuity solver to calculate the charge densities at time n+1 (ρn+1). Finally, the Navier-Stokes solver uses the transport parameters, the electric field and the charge densities at time n+1/2 to calculate the neutral gas parameters at time

In fluid analysis, the collision coefficients and drift velocities of the charged particles can be calculated either using experimental or numerical techniques. Experimentally they are calculated as a function of the ratio of the electric field to the number of neutral gas particles per unit volume (E/N), depending also on the density and pressure of the neutral gas that has a direct effect on the frequency of collisions between the charged and neutral gas

Numerically, commercial software packages calculate the transport properties by solving the electron distribution Boltzmann equation by utilizing the collisional cross sectional data and such software examples are ELENDIF [5], BOLSIG [6], and BOLSIG+ [7] which are used

CON TP

The charged particle continuity equations are coupled to Poisson's equation via the electric field strength and to the Navier-Stokes equations via the number of neutral gas particles per unit volume and their pressure. Poisson's equation is coupled to the continuity equations via the net charge density. As far as the Navier-Stokes equations are concerned, they are coupled to Poisson's equation via the electric field strength and to the charged particle

TP CON

n+1/2

<sup>n</sup>

PO

n+1/2

n+1/2

NS

n+1/2

n+1 (Nn+1) and this process is repeated continuously to proceed forward in time.

The fluid transport equations take the general form in the two-dimensional Cartesian coordinates as:

$$\frac{\partial \bar{\mathcal{Q}}}{\partial t} + \sum\_{j=1}^{2} \frac{\partial \bar{\mathcal{K}}\_j}{\partial \mathbf{x}\_j} + \sum\_{j=1}^{2} \frac{\partial \bar{\mathcal{L}}\_j}{\partial \mathbf{x}\_j} = \bar{\mathcal{M}} \tag{69}$$

where *Q* is the unknown or independent variable, *K* and *L* are respectively the convective and diffusive fluxes, *M* is the source term.

The Taylor Galerkin scheme is used to develop the FE-FCT scheme, with the time discretization preceding the space discretization. The time stepping is performed in two steps, the advective predictor and the corrector step.

#### *4.1.1. Advective predictor step*

The advective predictor step is formulated using the Taylor series expansion for the unknown variable *Q* at a first order approximation, thereby resulting in the half time values of the independent variable *<sup>n</sup>* 1/2 *Q* :

$$\overset{-}{\mathbf{Q}}^{n+1/2} = \frac{A\_e}{\mathbf{3}} \sum\_{i=1}^3 \overset{-}{Q\_i}\_i + \frac{\Delta t}{\mathbf{4}} \sum\_{i=1}^3 \overset{-}{K\_i}\_i b\_i + \frac{\Delta t}{\mathbf{4}} \sum\_{i=1}^3 \overset{-}{K\_i}\_i c\_i - \frac{\Delta t}{6} A\_e \sum\_{i=1}^3 \overset{-}{M\_i}\_i \tag{70}$$

where Ae is the area of a triangular element, Δt is the time step, bi and ci are the linearly interpolated piecewise shape functions. The velocities at time (n+1/2) are assumed to be the average value of the nodal velocities at time n.

#### *4.1.2. Corrector step*

The corrector step utilizes the Taylor series expansion for the full time step to the second order approximation to get:

$$D\_{\boldsymbol{c}} \boldsymbol{\Delta} \overset{-}{Q}^{n} = -\frac{\boldsymbol{\Delta t} b\_{i}^{\epsilon}}{2} \boldsymbol{K}\_{\boldsymbol{x}}^{n+1/2} - \frac{\boldsymbol{\Delta t c}\_{i}^{\epsilon}}{2} \boldsymbol{K}\_{\boldsymbol{y}}^{n+1/2} - \frac{\boldsymbol{\Delta t b}\_{i}^{\epsilon}}{2} \boldsymbol{L}\_{\boldsymbol{x}}^{n+1/2} - \frac{\boldsymbol{\Delta t c}\_{i}^{\epsilon}}{2} \boldsymbol{L}\_{\boldsymbol{y}}^{n+1/2}$$

$$+\Delta t M^{n+1/2} \frac{A\_\epsilon}{\mathfrak{D}} + \Delta t \int\_{\Gamma\_\epsilon} (K\_n^{n+1/2} + L\_n^n) \mathcal{N}\_i^\epsilon d\Gamma \tag{71}$$

where Dc is the consistent mass matrix, *<sup>n</sup>*1/2 *Kn* and *n <sup>n</sup> L* are the convective term at time n+1/2 and diffusive term at time n fluxes outward in the normal direction of the boundary, <sup>Γ</sup>e is the boundary surrounding element e and *<sup>e</sup> Ni* is the interpolation function. Writing the above equation in the general form results in:

$$D\_c \stackrel{-}{\mathbf{A}} \stackrel{\text{\textquotedblleft}}{\mathbf{Q}} = \mathbf{B}^{\text{\textquotedblleft}} \tag{72}$$

Electromagnetic and Fluid Analysis of Collisional Plasmas 45

*r*

(78)

*r* (79)

(77)

The Taylor Galerkin scheme is used to develop the FE-FCT scheme, with the time discretization preceding the space discretization. The time stepping is performed in two

The advective predictor step is formulated using the Taylor series expansion for the

*i ii <sup>e</sup> <sup>e</sup> t t <sup>t</sup> <sup>Q</sup> rQ M K c K b K <sup>A</sup>*

1 11 <sup>1</sup> (3 )( ) (( ) ( ) ) ( ) <sup>12</sup> 2 4 <sup>6</sup>

 1/2 3 33 '

where Ae is the area of a triangular element, Δt is the time step, bi and ci are the linearly

 ' *<sup>i</sup> i*

*e*

*r*

The velocities at time (n+1/2) are assumed to be the average value of the nodal velocities at

The corrector step utilizes the Taylor series expansion for the full time step to the second

 

 

The above discretization results in the following equations:

( ) 2 3

*e n n nn i e <sup>e</sup> ee e <sup>z</sup> n n <sup>i</sup>*

1/2 1/2

*nn n n eee iii e eee r zr <sup>c</sup> bcb r D Q tr K tr K tr L*

222

1/2 1/2

*e*

*c A t r L t r M t r K L N dA* (80)

*A*

*r*

*r r*

*n nn n n n*

 *<sup>n</sup>* 1/2 *Q* :

*r* is the r-coordinate of the centroid of the element e given by:

*e i*

*Q* at a first order approximation, thereby resulting in the half time

*i i zr r ii ii i*

steps, the advective predictor and the corrector step.

*4.2.1 Advective predictor step* 

values of the independent variable

interpolated piecewise shape functions.

*r* is given by:

unknown variable

where ' *i*

where

time n.

 *e*

*4.2.2. Corrector step* 

order approximation to get:

where *Q* is the vector of nodal increments, *<sup>n</sup> B* is the vector of added element contributions to the nodes. Instead of using a consistent mass matrix, a lumped mass matrix (Dl) is used for computational purposes, resulting in:

$$D\_l \stackrel{-^{a,n}}{\!\!\!D} - (D\_c - D\_l) \stackrel{-^{a-1,n}}{\!\!\!D} = \mathbb{B}^n \tag{73}$$

where α is an integer representing the iteration number.

To calculate the low order scheme, mass diffusion is added of the form:

$$\mathcal{L}\_d (D\_c - D\_l) \stackrel{\cdots}{Q}^n \tag{74}$$

to get the low order scheme:

*n*

$$\stackrel{-a,n}{D\_l \Lambda} \stackrel{-a,n}{\mathbf{Q}} - (\stackrel{-a-1,n}{D\_c} - \stackrel{a-1,n}{\mathbf{Q}}\_d - \stackrel{-n}{c\_d}(\stackrel{-}{D\_c} - \stackrel{-}{D\_l})\stackrel{\mathbf{Q}}{\mathbf{Q}}\_d = \mathbf{B}^n \tag{75}$$

where cd is the variable diffusion coefficient and is dependent on the mesh size, time step and speed of an element. A detailed analysis of the above formulation procedure is found in [9].

#### **4.2. Two-dimensional cylindrical axisymmetric coordinates**

The fluid transport equations take the general form in the two-dimensional cylindrical axisymmetric coordinates as:

$$\frac{\partial \bar{\mathcal{Q}}}{\partial t} + \frac{1}{r} \frac{\partial (\bar{\mathcal{K}}\_r r)}{\partial r} + \frac{\partial (\bar{\mathcal{K}}\_z)}{\partial z} + \frac{1}{r} \frac{\partial (\bar{\mathcal{L}}\_r r)}{\partial r} + \frac{\partial (\bar{\mathcal{L}}\_z)}{\partial z} = \bar{\mathcal{M}} \tag{76}$$

where *Kr Kz r L <sup>r</sup> L* are respectively the radial and axial convective and diffusive fluxes, and r is the radial coordinate in cylindrical axisymmetric coordinates.

The Taylor Galerkin scheme is used to develop the FE-FCT scheme, with the time discretization preceding the space discretization. The time stepping is performed in two steps, the advective predictor and the corrector step.

#### *4.2.1 Advective predictor step*

44 Finite Element Analysis – Applications in Mechanical Engineering

where Dc is the consistent mass matrix,

above equation in the general form results in:

(Dl) is used for computational purposes, resulting in:

where α is an integer representing the iteration number.

To calculate the low order scheme, mass diffusion is added of the form:

where

 *n*

to get the low order scheme:

axisymmetric coordinates as:

where

 *Kr Kz r L*   1/2 1/2 ( ) <sup>3</sup> *e n e n ne*

> *<sup>n</sup>*1/2 *Kn* and

n+1/2 and diffusive term at time n fluxes outward in the normal direction of the boundary, <sup>Γ</sup>e is the boundary surrounding element e and *<sup>e</sup> Ni* is the interpolation function. Writing the

> *n*

contributions to the nodes. Instead of using a consistent mass matrix, a lumped mass matrix

 , 1, ( ) *a n a n*

( )

 

() () *a n a n n*

where cd is the variable diffusion coefficient and is dependent on the mesh size, time step and speed of an element. A detailed analysis of the above formulation procedure is found in [9].

The fluid transport equations take the general form in the two-dimensional cylindrical

1 1 ( ) ( ) ( ) () *<sup>Q</sup> Kr K Lr L r z rz <sup>M</sup>*

 

, 1,

**4.2. Two-dimensional cylindrical axisymmetric coordinates** 

r is the radial coordinate in cylindrical axisymmetric coordinates.

*Q* is the vector of nodal increments, *<sup>n</sup> B* is the vector of added element

*n*

*<sup>n</sup> D Q D D Q c D DQ B l c l dc l* (75)

*tr r z r r z* (76)

*<sup>r</sup> L* are respectively the radial and axial convective and diffusive fluxes, and

*n ni*

 *n*

*<sup>A</sup> tM t K L N d* (71)

*<sup>n</sup> DQ B <sup>c</sup>* (72)

*<sup>n</sup> DQ D D Q B l cl* (73)

*dc l c D DQ* (74)

*<sup>n</sup> L* are the convective term at time

The advective predictor step is formulated using the Taylor series expansion for the unknown variable *Q* at a first order approximation, thereby resulting in the half time values of the independent variable *<sup>n</sup>* 1/2 *Q* :

$$\overset{\text{-}\,^{n+1/2}\mathbf{Q}}{\mathbf{Q}} = \frac{1}{12} \sum\_{i=1}^{3} (\mathbf{\hat{3}} + \mathbf{r}\_{i}^{\dagger}) (\overset{-}{\mathbf{Q}}\_{i} + \frac{\Delta t}{2} \overset{-}{\mathbf{M}}^{n}) - \frac{\Delta t}{4A\_{\varepsilon}} \sum\_{i=1}^{3} (\mathbf{(K}\_{z})\_{i} c\_{i} + (\mathbf{K}\_{r})\_{i} b\_{i}) - \frac{\Delta t}{6r\_{\varepsilon}} \sum\_{i=1}^{3} (\mathbf{K}\_{r})\_{i} \tag{77}$$

where Ae is the area of a triangular element, Δt is the time step, bi and ci are the linearly interpolated piecewise shape functions.

where ' *i r* is given by:

$$
\sigma\_i^{\cdot} = \frac{r\_i}{r\_e} \tag{78}
$$

where *e r* is the r-coordinate of the centroid of the element e given by:

$$\bar{r\_e} = \sum\_{i=1}^{3} \frac{r\_i}{3} \tag{79}$$

The velocities at time (n+1/2) are assumed to be the average value of the nodal velocities at time n.

#### *4.2.2. Corrector step*

The corrector step utilizes the Taylor series expansion for the full time step to the second order approximation to get:

$$
\stackrel{-}{r}\_{\epsilon} \stackrel{-}{D}\_{\epsilon} \Delta \stackrel{-}{Q} = -\Delta t \, r\_{\epsilon} \frac{b\_{i}^{\epsilon}}{2} \stackrel{-}{K}\_{\epsilon} \, -\Delta t \, r\_{\epsilon} \frac{c\_{i}^{\epsilon}}{2} \stackrel{-}{K}\_{\epsilon} \, -\Delta t \, r\_{\epsilon} \frac{b\_{i}^{\epsilon}}{2} \stackrel{-}{L}\_{\epsilon}
$$

$$
$$

The above discretization results in the following equations:

$$\stackrel{\text{\tiny}}{D\_{\text{f}}} \stackrel{\text{\tiny}}{\Lambda} \stackrel{\text{\tiny}}{\text{Q}}^n = \stackrel{\text{\tiny}}{\text{B}} \tag{81}$$

Electromagnetic and Fluid Analysis of Collisional Plasmas 47

*n nn n n n*

where Ve is the volume of a tetrahedral element, bi, ci and di are the linearly interpolated piecewise shape functions. The velocities of the charged species at time (n+1/2) are assumed

The corrector step utilizes the Taylor series expansion for the full time step to the second

 1/2 1/2 1/2 1/2 1/2 1/2 3 3 3 333

*i i i iii n n nnnn*

*n ni*

*<sup>V</sup> tM t K L N dA* (86)

*<sup>n</sup> DQ D D Q B l cl* (88)

*<sup>n</sup> DQ B <sup>c</sup>* (87)

*dc l c D DQ* (89)

*n e e e eee*

*c x y z xyz tb tc td tb tc td DQ K K K L L L*

> 1/2 1/2 ( ) <sup>4</sup> *e n e n ne*

*A*

where Ae is the boundary surrounding element e. Writing the above equation in the general

 *n*

Instead of using a consistent mass matrix, a lumped mass matrix is used for computational

 , 1, ( ) *a n a n*

( )

 

() () *a n a n n*

In order to validate the fluid flow equations and the methodology developed, the Euler equations which are the Navier-Stokes equations without diffusion and source terms were tested using the shock tube type problem, where different parameters, such as density,

, 1,

**5. Fluid flow validation of the above FE-FCT formulation** 

*n*

*<sup>n</sup> D Q D D Q c D DQ B l c l dc l* (90)

To calculate the low order scheme, mass diffusion is added of the form:

to be the average value of the nodal velocities at time n.

*4.3.2. Corrector step* 

form results in:

purposes, resulting in:

to get the low order scheme:

order approximation to get:

 1/2 44 4 4 4

*<sup>e</sup> iii <sup>i</sup> <sup>i</sup> i i ie ii i i i*

*<sup>V</sup> tttt Q Q Kb Kc Kd V M* (85)

11 1 1 1 4 12 12 12 8

where:

$$\stackrel{\cdot}{D}\_{c} = \stackrel{\cdot}{r} D\_{c} \tag{82}$$

and

$$\stackrel{\text{'''}}{B} = \stackrel{\text{'''}}{r} \stackrel{\text{'''}}{B}^{\text{!!}} \tag{83}$$

where *r* denoting a matrix of *re* entries everywhere, *Dc* is the consistent mass matrix and '*n*

'

*B* is the vector of added element contributions to the nodes in the cylindrical axisymmetric case. In order to implement the Cartesian and axisymmetric cases together, the half time *<sup>n</sup>* 1/2

step values *Q* have to be calculated slightly differently, and the consistent mass matrix

Dc and lumped matrix Dl need to be multiplied by *re* in the cylindrical axisymmetric case, Finally, during the limiting procedure, the antidiffusive element contribution, the consistent

and lumped matrices need to be multiplied by *re* for each element.

#### **4.3. Three-dimensional Cartesian coordinates**

The fluid transport equations take the general form in the three-dimensional Cartesian coordinates as:

$$
\frac{\partial \bar{\mathcal{Q}}}{\partial t} + \sum\_{j=1}^{3} \frac{\partial \bar{\mathcal{K}}\_j}{\partial \mathbf{x}\_j} + \sum\_{j=1}^{3} \frac{\partial \bar{\mathcal{L}}\_j}{\partial \mathbf{x}\_j} = \bar{\mathcal{M}} \tag{84}
$$

The Taylor Galerkin scheme is used to develop the FE-FCT scheme, with the time discretization preceding the space discretization. The time stepping is performed in two steps, the advective predictor and the corrector step.

#### *4.3.1. Advective predictor step*

The advective predictor step is formulated using the Taylor series expansion for the unknown variable *Q* at a first order approximation, thereby resulting in the half time values of the independent variable *<sup>n</sup>* 1/2 *Q* :

Electromagnetic and Fluid Analysis of Collisional Plasmas 47

$$\overset{\text{\\_}}{\text{Q}}^{n+1/2} = \frac{V\_{\varepsilon}}{4} \sum\_{i=1}^{4} \overset{\text{\\_}}{Q\_{i}} + \frac{\Delta t}{12} \sum\_{i=1}^{4} \overset{\text{\\_}}{K\_{i}} \overset{\text{\\_}}{b\_{i}} + \frac{\Delta t}{12} \sum\_{i=1}^{4} \overset{\text{\\_}}{K\_{i}} \overset{\text{\\_}}{c\_{i}} + \frac{\Delta t}{12} \sum\_{i=1}^{4} \overset{\text{\\_}}{K\_{i}} \overset{\text{\\_}}{d\_{i}} - \frac{\Delta t}{8} V\_{\varepsilon} \sum\_{i=1}^{4} \overset{\text{\\_}}{M\_{i}} \tag{85}$$

where Ve is the volume of a tetrahedral element, bi, ci and di are the linearly interpolated piecewise shape functions. The velocities of the charged species at time (n+1/2) are assumed to be the average value of the nodal velocities at time n.

#### *4.3.2. Corrector step*

46 Finite Element Analysis – Applications in Mechanical Engineering

where:

and

where

'*n* step values

coordinates as:

*r* denoting a matrix of

*<sup>n</sup>* 1/2

Dc and lumped matrix Dl need to be multiplied by

**4.3. Three-dimensional Cartesian coordinates** 

steps, the advective predictor and the corrector step.

*4.3.1. Advective predictor step* 

values of the independent variable

unknown variable

and lumped matrices need to be multiplied by

 ' ' *n n*

> '

> '*n*

*B* is the vector of added element contributions to the nodes in the cylindrical axisymmetric case. In order to implement the Cartesian and axisymmetric cases together, the half time

Finally, during the limiting procedure, the antidiffusive element contribution, the consistent

The fluid transport equations take the general form in the three-dimensional Cartesian

*j j*

*Q K L*

1 1

*j j j j*

The Taylor Galerkin scheme is used to develop the FE-FCT scheme, with the time discretization preceding the space discretization. The time stepping is performed in two

The advective predictor step is formulated using the Taylor series expansion for the

 *<sup>n</sup>* 1/2 *Q* :

 3 3

'

*Q* have to be calculated slightly differently, and the consistent mass matrix

*re* for each element.

*M*

*Q* at a first order approximation, thereby resulting in the half time

*tx x* (84)

*re* entries everywhere,

*DQ B <sup>c</sup>* (81)

*<sup>c</sup> D rDc* (82)

*<sup>n</sup> B rB* (83)

*Dc* is the consistent mass matrix and

*re* in the cylindrical axisymmetric case,

The corrector step utilizes the Taylor series expansion for the full time step to the second order approximation to get:

$$D\_c \overset{-}{\Delta}\_c^{-n} = -\frac{\Delta t b\_i^\epsilon}{3} K\_x^{n+1/2} - \frac{\Delta t c\_i^\epsilon}{3} K\_y^{n+1/2} - \frac{\Delta t d\_i^\epsilon}{3} K\_z^{n+1/2} - \frac{\Delta t b\_i^\epsilon}{3} L\_x^{n+1/2} - \frac{\Delta t c\_i^\epsilon}{3} L\_y^{n+1/2} - \frac{\Delta t d\_i^\epsilon}{3} L\_z^{n+1/2}$$

$$+ \Delta t M^{n+1/2} \frac{V\_\epsilon}{4} + \Delta t \int\_{A\_\epsilon} (K\_n^{n+1/2} + L\_n^n) N\_l^\epsilon dA \tag{86}$$

where Ae is the boundary surrounding element e. Writing the above equation in the general form results in:

$$D\_c \stackrel{-}{\mathbf{Q}} \stackrel{\mu}{\mathbf{Q}}^{\pi} = \mathbf{B}^{\pi} \tag{87}$$

Instead of using a consistent mass matrix, a lumped mass matrix is used for computational purposes, resulting in:

$$D\_l \stackrel{-a,n}{\sim} - (D\_c - D\_l) \stackrel{-a-1,n}{\sim} = \mathcal{B}^n \tag{88}$$

To calculate the low order scheme, mass diffusion is added of the form:

$$c\_d (D\_c - D\_l) \stackrel{- \text{ } n \text{ } \tag{89}}{\text{ }}$$

to get the low order scheme:

$$\stackrel{-a,n}{D\_l \Lambda} \stackrel{-a,n}{\quad} - (D\_c - D\_l) \stackrel{-a-1,n}{\Lambda} \stackrel{-a,n}{\quad} - c\_d (D\_c - D\_l) \stackrel{-a}{\mathcal{Q}}^n = \mathcal{B}^n \tag{90}$$

#### **5. Fluid flow validation of the above FE-FCT formulation**

In order to validate the fluid flow equations and the methodology developed, the Euler equations which are the Navier-Stokes equations without diffusion and source terms were tested using the shock tube type problem, where different parameters, such as density, diffusion coefficient and mesh size were considered. Furthermore, a shock wave incident on a wedge was also tested in the Cartesian case. Finally, energy source terms are introduced into the two-dimensional Cartesian and axisymmetric cylindrical geometries, which result in sound and shock wave generation, resembling the heating expected during the development of a spark and an arc plasma gas discharge.

Electromagnetic and Fluid Analysis of Collisional Plasmas 49

**Figure 3.** Comparison of analytical and experimental results using the finer mesh at three different

The two-dimensional Cartesian Euler equations in air with γ equal to 1.4 are tested by introducing a heating source term of magnitude 1x1012 J/m2 for a duration of 3x10-8 s within a square block of size 0.002 m x 0.002 m, with the simulation being run for 5000 steps at a constant time step increments of 3x10-8 s. The contour plot of the momentum in the x direction is shown in Figure 4, where at the exterior of the wave front, the wave is moving outwards in the x direction with maximum momentum. At the interior of the wave front, there is a sudden decrease and a change in the direction of the momentum. This is due to the fact that the inertial effects of the sudden explosion cause an overexpansion of the wave, therefore a rarefraction wave is created moving inwardly. Figure 5 shows a contour plot of the energy at a time of 9x10-5 s. High energies occur at the shock front of the wave with low energies forming behind the shock front, again due to the overexpansion of the wave, which

A shock wave of Mach number 2 in air with γ equal to 1.4 is incident on a wedge angle of 46o degrees. The neutral gas is at a temperature of 300 oK and a pressure of 30 kPa. The simulation is run for 13000 time steps of 1x10-7 s duration. The simulation is terminated 0.1 m before the shock wave hits the boundary on the right. Figure 6 shows the Cartesian contour plots of the density at a time of 1.3x10-3 s with the results being compared with the benchmark test case of a shock wave impinging on a wedge Takayama (1997) [11]. The results are found to be in good agreement capturing the shock wave created at (0.9 m, 0.7 m)

instants in time t = 0.1, 0.3 and 0.5 s. □ line: Analytical solution, + line: Mesh 3, 5000 nodes

**5.2. Heating source term test case** 

are in agreement with blast wave theory Baker (1973) [10].

**5.3. Shock wave incident on a wedge test case** 

coordinates.

## **5.1. Riemann test case**

The Euler equations are validated in one-dimensional Cartesian coordinates using the shock tube or Riemann test case in air with the ratio of the specific heats, γ, equal to 5/3, where analytical results are available. Figure 2 shows a one-dimensional comparison between the analytical and numerical solution of density for three different meshes at times t = 0.1 and 0.5 s. It is shown that in the case of the finer mesh, the overshoot observed is reduced. Figure 3 shows the analytical and numerical solution for density for the finest mesh (Mesh 3) at three different instants in time of t = 0.1, 0.3 and 0.5. A direct comparison shows clearly that the Euler solver developed is capable of simulating the propagation of shock waves and that the results are of adequate accuracy. This is also verified by calculating the maximum percentage error for the three different meshes as shown in Table 1 below.


**Table 1.** Maximum density error for three different meshes

**Figure 2.** Comparison of analytical and numerical results using three different meshes at two different instants in time t = 0.1 and 0.5 s. □ line: Analytical solution, ◊ line: Mesh 1, 500 nodes, x line: Mesh 2, 1000 nodes and + line: Mesh 3, 5000 nodes

**Figure 3.** Comparison of analytical and experimental results using the finer mesh at three different instants in time t = 0.1, 0.3 and 0.5 s. □ line: Analytical solution, + line: Mesh 3, 5000 nodes

#### **5.2. Heating source term test case**

48 Finite Element Analysis – Applications in Mechanical Engineering

development of a spark and an arc plasma gas discharge.

**Table 1.** Maximum density error for three different meshes

1000 nodes and + line: Mesh 3, 5000 nodes

**5.1. Riemann test case** 

diffusion coefficient and mesh size were considered. Furthermore, a shock wave incident on a wedge was also tested in the Cartesian case. Finally, energy source terms are introduced into the two-dimensional Cartesian and axisymmetric cylindrical geometries, which result in sound and shock wave generation, resembling the heating expected during the

The Euler equations are validated in one-dimensional Cartesian coordinates using the shock tube or Riemann test case in air with the ratio of the specific heats, γ, equal to 5/3, where analytical results are available. Figure 2 shows a one-dimensional comparison between the analytical and numerical solution of density for three different meshes at times t = 0.1 and 0.5 s. It is shown that in the case of the finer mesh, the overshoot observed is reduced. Figure 3 shows the analytical and numerical solution for density for the finest mesh (Mesh 3) at three different instants in time of t = 0.1, 0.3 and 0.5. A direct comparison shows clearly that the Euler solver developed is capable of simulating the propagation of shock waves and that the results are of adequate accuracy. This is also verified by calculating the maximum

> Mesh number Maximum density error 1 1.6177 % 2 1.0895 % 3 0.2345 %

**Figure 2.** Comparison of analytical and numerical results using three different meshes at two different instants in time t = 0.1 and 0.5 s. □ line: Analytical solution, ◊ line: Mesh 1, 500 nodes, x line: Mesh 2,

percentage error for the three different meshes as shown in Table 1 below.

The two-dimensional Cartesian Euler equations in air with γ equal to 1.4 are tested by introducing a heating source term of magnitude 1x1012 J/m2 for a duration of 3x10-8 s within a square block of size 0.002 m x 0.002 m, with the simulation being run for 5000 steps at a constant time step increments of 3x10-8 s. The contour plot of the momentum in the x direction is shown in Figure 4, where at the exterior of the wave front, the wave is moving outwards in the x direction with maximum momentum. At the interior of the wave front, there is a sudden decrease and a change in the direction of the momentum. This is due to the fact that the inertial effects of the sudden explosion cause an overexpansion of the wave, therefore a rarefraction wave is created moving inwardly. Figure 5 shows a contour plot of the energy at a time of 9x10-5 s. High energies occur at the shock front of the wave with low energies forming behind the shock front, again due to the overexpansion of the wave, which are in agreement with blast wave theory Baker (1973) [10].

#### **5.3. Shock wave incident on a wedge test case**

A shock wave of Mach number 2 in air with γ equal to 1.4 is incident on a wedge angle of 46o degrees. The neutral gas is at a temperature of 300 oK and a pressure of 30 kPa. The simulation is run for 13000 time steps of 1x10-7 s duration. The simulation is terminated 0.1 m before the shock wave hits the boundary on the right. Figure 6 shows the Cartesian contour plots of the density at a time of 1.3x10-3 s with the results being compared with the benchmark test case of a shock wave impinging on a wedge Takayama (1997) [11]. The results are found to be in good agreement capturing the shock wave created at (0.9 m, 0.7 m) coordinates.

Electromagnetic and Fluid Analysis of Collisional Plasmas 51

**Figure 6.** Contour plot of the density at a time of 1.3x10-3 s

The two-dimensional Euler solver is validated in the cylindrical axisymmetric case using numerical blast wave tests that involve the release of sudden burst of energy by focusing a laser beam within small volumes in air with γ equal to 1.4, leading to the development of shock waves. Jiang et al (1998) [12] has analyzed both experimentally and numerically the propagation of micro-blast waves in ambient air. The duration of the pulse was of the order of 18 ns and the total amount of energy released was measured to be 1.38 J. This sudden release of energy raises instantaneously the pressure, density and temperature to values many times higher than those of the ambient conditions, thereby creating highly

Figures 7 and 8 compare respectively one-dimensional plots of the density against time along the axis of symmetry obtained by Jiang and the authors for the same mesh and different diffusion coefficients and for different mesh and the same diffusion coefficient. Given the diversities of the two numerical schemes, the results show to be in good

compressive acoustic waves that will in turn generate strong shock waves.

**5.4. Micro-blast waves test case** 

agreement.

**Figure 4.** Contour plot of the x-momentum at time t = 9x10-5 s

**Figure 5.** Contour plot of the energy at time t = 9x10-5s

**Figure 6.** Contour plot of the density at a time of 1.3x10-3 s

#### **5.4. Micro-blast waves test case**

50 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 4.** Contour plot of the x-momentum at time t = 9x10-5 s

**Figure 5.** Contour plot of the energy at time t = 9x10-5s

The two-dimensional Euler solver is validated in the cylindrical axisymmetric case using numerical blast wave tests that involve the release of sudden burst of energy by focusing a laser beam within small volumes in air with γ equal to 1.4, leading to the development of shock waves. Jiang et al (1998) [12] has analyzed both experimentally and numerically the propagation of micro-blast waves in ambient air. The duration of the pulse was of the order of 18 ns and the total amount of energy released was measured to be 1.38 J. This sudden release of energy raises instantaneously the pressure, density and temperature to values many times higher than those of the ambient conditions, thereby creating highly compressive acoustic waves that will in turn generate strong shock waves.

Figures 7 and 8 compare respectively one-dimensional plots of the density against time along the axis of symmetry obtained by Jiang and the authors for the same mesh and different diffusion coefficients and for different mesh and the same diffusion coefficient. Given the diversities of the two numerical schemes, the results show to be in good agreement.

Electromagnetic and Fluid Analysis of Collisional Plasmas 53

2

l is the shape function of node j in

is

*N NU N U d* (91)

For an adaptive mesh generator, an error indicator is necessary to decide on the amount of refinement. The error indicator used by the author is the one used by Lohner [13], which in

element l, Uj is the value of the variable chosen to be used as error indicator at node j, Ei

the error indicator value at node i, and is a factor varying from 0 to 1. The value of is added as noise filter, so that any loss of monotonicity such as wiggles or ripples are not refined. The above error indicator is dimensionless, fast to calculate, varies from 0 to 1 such that prefixed tolerances and many variables as error indicators can be used at the same time.

In this case, an adaptive mesh algorithm in two-dimensional Cartesian, and twodimensional cylindrical axisymmetric coordinates has been developed by the author in order to analyze plasmas much faster and more accurately. This is achieved by constructing a computer resource efficient algorithm, which is automated in providing the necessary results. Generally, the numerical solution of time dependent differential equations is classified into the categories of static and dynamic. For static methods, any addition of nodes, and edge swap and topological movement of nodes is performed at a fixed time. In the dynamic case, or moving mesh methods, a mesh equation is introduced that involves node velocities such that a fixed number of nodes are moved in such a way that the nodes are always concentrated near regions of rapid variation of the solution. Thereby the simultaneous solution of the differential and mesh equations is necessary, having the advantage that no interpolation between existing and future meshes is necessary [14]. The author of this paper has adopted the static method, which is more widely used and tested in

The algorithm incorporates an innovative element quality improvement procedure that has the ability to guarantee the generation of new meshes, as well as the treatment of existing bad element quality meshes, to nearly ideal standards, guaranteeing a minimum element quality of 0.85, and above 0.90 of average element quality in uniform and non-uniform geometric domains. Furthermore, another novelty of the paper is also the utilisation of an interpolation method between meshes which is very fast, making the adaptive meshing more attractive. The re-meshing times are decided optimally according to the maximum speeds of the fastest particle within the simulation to guarantee that the desired mesh resolution is in place at all times, whereas the numerical diffusion error due to the interpolation between meshes is minimized by interpolating between meshes of similar size.

Figure 9 shows a flow chart of the implementation procedure of the adaptive mesh developed by the author. The first step is to create a reference coarse mesh that will be used

,

*i j j k lj l j*

( | |[| | (| || |)] )

( .)

*NNd U*

*i j kl j*

2

multidimensional form is calculated as 999999follows:

*E*

where Ni

the literature.

k is the shape function of node i in element k, Nj

*i k l*

,

*k l*

**Figure 7.** Non-dimensionalized density (ρ\* ) in radial direction at t1 = 5.36 μs, t2 = 6.65 μs, t3 = 10.05 μs, t4 = 12.58 μs. - line: Jiang, ◊ line : Mesh 2 : 27570 nodes, Diff. Coeff., 0.0012, x line: Mesh 2:, 27570 nodes, Diff. Coeff., 0.002

**Figure 8.** Non-dimensionalized density (ρ\* ) in the radial direction at t1 = 5.36 μs, t2 = 6.65 μs, t3 = 10.05 μs, t4 = 12.58 μs. - line: Jiang, ◊ line: Mesh 2, Diff. Coeff., 0.002, x line: Mesh 1, Diff. Coeff., 0.002

### **6. Adaptive mesh algorithm**

In order to reduce computational needs considerably, an adaptive mesh generator has been developed, rendering possible the analysis of heating effects both in short and long gaps. For an adaptive mesh generator, an error indicator is necessary to decide on the amount of refinement. The error indicator used by the author is the one used by Lohner [13], which in multidimensional form is calculated as 999999follows:

52 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 7.** Non-dimensionalized density (ρ\*

**Figure 8.** Non-dimensionalized density (ρ\*

**6. Adaptive mesh algorithm** 

Diff. Coeff., 0.002

) in radial direction at t1 = 5.36 μs, t2 = 6.65 μs, t3 = 10.05 μs, t4

) in the radial direction at t1 = 5.36 μs, t2 = 6.65 μs, t3 = 10.05

= 12.58 μs. - line: Jiang, ◊ line : Mesh 2 : 27570 nodes, Diff. Coeff., 0.0012, x line: Mesh 2:, 27570 nodes,

μs, t4 = 12.58 μs. - line: Jiang, ◊ line: Mesh 2, Diff. Coeff., 0.002, x line: Mesh 1, Diff. Coeff., 0.002

In order to reduce computational needs considerably, an adaptive mesh generator has been developed, rendering possible the analysis of heating effects both in short and long gaps.

$$E^i = \sqrt{\frac{\sum\_{k,l} \left( \int\_{\Omega} N\_k^i N\_l^j d\Omega \,\mathrm{II}\_j \right)^2}{\sum\_{k,l} \left( \int\_{\Omega} N\_k^i \, | \, \| \, N\_l^j \| \, \mathrm{I}\_j \| + \varepsilon (\| \, N\_l^j \| \, \| \, \mathrm{II}\_j \|) \right) \mathrm{d}\Omega\right)^2}}\tag{91}$$

where Ni k is the shape function of node i in element k, Nj l is the shape function of node j in element l, Uj is the value of the variable chosen to be used as error indicator at node j, Ei is the error indicator value at node i, and is a factor varying from 0 to 1. The value of is added as noise filter, so that any loss of monotonicity such as wiggles or ripples are not refined. The above error indicator is dimensionless, fast to calculate, varies from 0 to 1 such that prefixed tolerances and many variables as error indicators can be used at the same time.

In this case, an adaptive mesh algorithm in two-dimensional Cartesian, and twodimensional cylindrical axisymmetric coordinates has been developed by the author in order to analyze plasmas much faster and more accurately. This is achieved by constructing a computer resource efficient algorithm, which is automated in providing the necessary results. Generally, the numerical solution of time dependent differential equations is classified into the categories of static and dynamic. For static methods, any addition of nodes, and edge swap and topological movement of nodes is performed at a fixed time. In the dynamic case, or moving mesh methods, a mesh equation is introduced that involves node velocities such that a fixed number of nodes are moved in such a way that the nodes are always concentrated near regions of rapid variation of the solution. Thereby the simultaneous solution of the differential and mesh equations is necessary, having the advantage that no interpolation between existing and future meshes is necessary [14]. The author of this paper has adopted the static method, which is more widely used and tested in the literature.

The algorithm incorporates an innovative element quality improvement procedure that has the ability to guarantee the generation of new meshes, as well as the treatment of existing bad element quality meshes, to nearly ideal standards, guaranteeing a minimum element quality of 0.85, and above 0.90 of average element quality in uniform and non-uniform geometric domains. Furthermore, another novelty of the paper is also the utilisation of an interpolation method between meshes which is very fast, making the adaptive meshing more attractive. The re-meshing times are decided optimally according to the maximum speeds of the fastest particle within the simulation to guarantee that the desired mesh resolution is in place at all times, whereas the numerical diffusion error due to the interpolation between meshes is minimized by interpolating between meshes of similar size.

Figure 9 shows a flow chart of the implementation procedure of the adaptive mesh developed by the author. The first step is to create a reference coarse mesh that will be used

as a reference mesh for future refinement and coarsening of the meshes. The second step is to apply the initial conditions on the reference coarse mesh, and calculate the amount of refinement by multiplying the error by a constant factor in each element. The refinement factor is decided on a basis of experience, and trial and error. Then a newly adapted initial mesh is developed using a freely available two-dimensional software package which uses the divide and conquer method to refine elements [15]. Since the elements that are created from this package are of bad element equality, they are first treated using adaptive mesh techniques. Specifically, the h-refinement/coarsening technique is firstly used, which includes the edge swap and node additions/removals to improve the interconnectivity between adjacent nodes. Secondly, the r-refinement technique, which involves jiggling of the mesh, follows, i.e. the movement of nodes around the geometry in a controlled way, such that the overall element quality is improved.

Electromagnetic and Fluid Analysis of Collisional Plasmas 55

that the correct resolution is provided at all times. Then in order to decide on the appropriate mesh to run the simulations, the results are interpolated from the adapted initial mesh back to the reference coarse mesh, and then the error is calculated to create a final mesh. Then this mesh is treated as above, using edge swap and node additions/removals operations to improve the interconnectivity between adjacent nodes, and finally mesh jiggling operations to improve the overall element quality, creating the adapted final mesh. Then one needs not to interpolate from the reference coarse mesh to the created adapted final mesh, but instead from the created adapted initial mesh to the created adapted final mesh. This operation is performed due to the fact that interpolation is a source of numerical diffusion on the results, and by interpolating just once, instead of twice, makes the results during the interpolation processes more accurate. Then having decided on the adapted final mesh and having interpolated the results, then one runs the simulation forward in time and the procedure is repeated many times. This completes the implementation of the adaptive mesh algorithm. For the implementation of the above adaptive mesh algorithm, three tools are necessary, which are (a) the error calculation, (b) the element quality improvement algorithm, and (c) the algorithm for interpolation between

**7. Application of electromagnetic and fluid code on microplasmas** 

In this section, results are presented showing the avalanche and streamer propagation in a gap of 1 cm in an RF applicator at 40 MHz. Figures 10 and 11 refer to times t1 = 50, t2 = 55, t3 = 62.5, t4 = 63.5 ns. Figure 10 shows the radial field along the symmetry axis from times t1 to t4. It is shown that during the time interval t1 to t2, its magnitude increases to a value of approximately 0.5 x 105 V/m due to the net charge that exists and that it stays fairly constant at the bottom electrode end, but gradually increases at the upper electrode end, with both radial fields always extending closer towards the electrodes as time progresses. It is interesting to note that the radial field is positive during this time at the upper end and negative towards the bottom end. This initiates the propagation of streamers towards the electrodes. At time t3, the radial field reverses at both ends, when compared with the time interval of t1 to t2, with the radial field magnitude being larger than before and this is due to the streamer impinging on the two electrodes. At time t4, the radial field increases even further at the bottom electrode side, but changes sign at the upper electrode side. This is because the streamer that hits the upper electrode forms earlier than the one that hits the lower electrode, as the discharge is overall exposed to a larger negative voltage cycle. The change of sign of the radial field at the upper electrode is due to the absorption of the electrons into the electrode and the start of the formation of a cathode fall region of net positive charge near it. Figure 11 shows the axial electric field along the symmetry axis from times t1 to t4. At time t1, at the middle of the gap, the axial electric field gets distorted from the Laplacian field which initiates the streamer propagation. During the time interval t1 to t2, the distortion increases in magnitude and spreads out further towards the two electrodes. At

meshes tools, and are thoroughly explained below.

**7.1. RF applicator at 40 MHz** 

**Figure 9.** Flow chart of the implementation procedure of the adaptive mesh generator

Once the adapted initial mesh is created that can be used for the solution of the differential equations, an interpolation of the results from the reference coarse mesh to the adapted initial mesh is performed. Having defined the mesh to be used and calculated the corresponding values that all the variables have at the nodes of this mesh, the simulation proceeds forward in time, until an indicator signifies that the solution has reached the outer boundary of the refined region, and a re-meshing operation needs to be performed. The indicator calculates the maximum distance that the fastest particle travels, and ensures that it does not exceed the geometric tolerance of the initial coarse mesh, thereby it is guaranteed that the correct resolution is provided at all times. Then in order to decide on the appropriate mesh to run the simulations, the results are interpolated from the adapted initial mesh back to the reference coarse mesh, and then the error is calculated to create a final mesh. Then this mesh is treated as above, using edge swap and node additions/removals operations to improve the interconnectivity between adjacent nodes, and finally mesh jiggling operations to improve the overall element quality, creating the adapted final mesh. Then one needs not to interpolate from the reference coarse mesh to the created adapted final mesh, but instead from the created adapted initial mesh to the created adapted final mesh. This operation is performed due to the fact that interpolation is a source of numerical diffusion on the results, and by interpolating just once, instead of twice, makes the results during the interpolation processes more accurate. Then having decided on the adapted final mesh and having interpolated the results, then one runs the simulation forward in time and the procedure is repeated many times. This completes the implementation of the adaptive mesh algorithm. For the implementation of the above adaptive mesh algorithm, three tools are necessary, which are (a) the error calculation, (b) the element quality improvement algorithm, and (c) the algorithm for interpolation between meshes tools, and are thoroughly explained below.

## **7. Application of electromagnetic and fluid code on microplasmas**

#### **7.1. RF applicator at 40 MHz**

54 Finite Element Analysis – Applications in Mechanical Engineering

such that the overall element quality is improved.

Initial Mesh

Edge swap and Node Addition/Removals (h-ref.)

> Mesh Jiggling (r-ref.)

Apply Initial Conditions to the Adapted Initial Mesh

Create Adapted Initial Mesh

Reference Coarse Mesh

Error Calculation

**Figure 9.** Flow chart of the implementation procedure of the adaptive mesh generator

Run the Simulation

Interpolation from Adapted Initial Mesh to the Adapted Final Mesh

Once the adapted initial mesh is created that can be used for the solution of the differential equations, an interpolation of the results from the reference coarse mesh to the adapted initial mesh is performed. Having defined the mesh to be used and calculated the corresponding values that all the variables have at the nodes of this mesh, the simulation proceeds forward in time, until an indicator signifies that the solution has reached the outer boundary of the refined region, and a re-meshing operation needs to be performed. The indicator calculates the maximum distance that the fastest particle travels, and ensures that it does not exceed the geometric tolerance of the initial coarse mesh, thereby it is guaranteed

Decision Time to Re-mesh

Interpolation from the Adapted Initial Mesh back to the Reference Coarse Mesh

Error Calculation

Create Final Mesh

Edge Swapping and Node Addition/Removals (h-ref.)

Mesh Jiggling (r-ref.)

Create Adapted Final Mesh

as a reference mesh for future refinement and coarsening of the meshes. The second step is to apply the initial conditions on the reference coarse mesh, and calculate the amount of refinement by multiplying the error by a constant factor in each element. The refinement factor is decided on a basis of experience, and trial and error. Then a newly adapted initial mesh is developed using a freely available two-dimensional software package which uses the divide and conquer method to refine elements [15]. Since the elements that are created from this package are of bad element equality, they are first treated using adaptive mesh techniques. Specifically, the h-refinement/coarsening technique is firstly used, which includes the edge swap and node additions/removals to improve the interconnectivity between adjacent nodes. Secondly, the r-refinement technique, which involves jiggling of the mesh, follows, i.e. the movement of nodes around the geometry in a controlled way,

> In this section, results are presented showing the avalanche and streamer propagation in a gap of 1 cm in an RF applicator at 40 MHz. Figures 10 and 11 refer to times t1 = 50, t2 = 55, t3 = 62.5, t4 = 63.5 ns. Figure 10 shows the radial field along the symmetry axis from times t1 to t4. It is shown that during the time interval t1 to t2, its magnitude increases to a value of approximately 0.5 x 105 V/m due to the net charge that exists and that it stays fairly constant at the bottom electrode end, but gradually increases at the upper electrode end, with both radial fields always extending closer towards the electrodes as time progresses. It is interesting to note that the radial field is positive during this time at the upper end and negative towards the bottom end. This initiates the propagation of streamers towards the electrodes. At time t3, the radial field reverses at both ends, when compared with the time interval of t1 to t2, with the radial field magnitude being larger than before and this is due to the streamer impinging on the two electrodes. At time t4, the radial field increases even further at the bottom electrode side, but changes sign at the upper electrode side. This is because the streamer that hits the upper electrode forms earlier than the one that hits the lower electrode, as the discharge is overall exposed to a larger negative voltage cycle. The change of sign of the radial field at the upper electrode is due to the absorption of the electrons into the electrode and the start of the formation of a cathode fall region of net positive charge near it. Figure 11 shows the axial electric field along the symmetry axis from times t1 to t4. At time t1, at the middle of the gap, the axial electric field gets distorted from the Laplacian field which initiates the streamer propagation. During the time interval t1 to t2, the distortion increases in magnitude and spreads out further towards the two electrodes. At

time t3, there is a sudden decrease of the axial electric field at the two electrodes, leaving a nearly constant plateau in the middle of the gap due to the streamer impinging on the two electrodes. At a later stage of t4,, the ripple profile reverses shape and starts to increase again at an ever higher rate, which corresponds to the times when the streamers charge is accumulated on the two electrodes.

Electromagnetic and Fluid Analysis of Collisional Plasmas 57

**7.2. Heating effects of normal and abnormal glow discharge** 

almost ambient temperature just outside it, where it remains almost constant.

development.

6.75 ns at V= 5600 V

The development of a glow discharge in atmospheric pressure air and its associated Joule heating are analyzed by solving the Poisson, charged particle continuity and Navier-Stokes equations using the FE-FCT method in two-dimensional cylindrical axisymmetric coordinates. An applied direct current voltage of 20% above the breakdown voltage is applied at the anode in a 1 mm air gap between two parallel plate electrodes and a normal glow discharge is shown to consist of the cathode fall, negative glow, positive column and anode regions. Neutral gas heating occurs with the initiation of the glow discharge, with the temperature at the anode shown to increase by only a few degrees oK as shown in Figure 12 due to the low electron densities and axial fields at the adjacency of the anode, whereas at the proximity of the cathode the temperature increases by approximately 180 oK as shown in Figure 13, with the temperature maximum at the cathode fall region, reducing abruptly at

The numerical results for the development of a glow discharge and its associated Joule heating by modeling its transition from a single electron, in a uniform applied electric field in atmospheric air, to a fully fledged state which consists of the cathode fall, negative glow, positive column and anode regions are presented. It is shown that the positive column travels towards the anode in the form of a return wave and maximum heating by approximately 180 oK occurs at the cathode fall region during the glow discharge

**Figure 12.** One-dimensional plots of the temperature along the symmetry axis close to the cathode at t =

**Figure 10.** One-dimensional plots of the radial field at times t1 to t4 along the symmetry axis

**Figure 11.** One-dimensional plots of the axial field at times t1 to t4 along the symmetry axis

## **7.2. Heating effects of normal and abnormal glow discharge**

56 Finite Element Analysis – Applications in Mechanical Engineering

accumulated on the two electrodes.

time t3, there is a sudden decrease of the axial electric field at the two electrodes, leaving a nearly constant plateau in the middle of the gap due to the streamer impinging on the two electrodes. At a later stage of t4,, the ripple profile reverses shape and starts to increase again at an ever higher rate, which corresponds to the times when the streamers charge is

**Figure 10.** One-dimensional plots of the radial field at times t1 to t4 along the symmetry axis

**Figure 11.** One-dimensional plots of the axial field at times t1 to t4 along the symmetry axis

The development of a glow discharge in atmospheric pressure air and its associated Joule heating are analyzed by solving the Poisson, charged particle continuity and Navier-Stokes equations using the FE-FCT method in two-dimensional cylindrical axisymmetric coordinates. An applied direct current voltage of 20% above the breakdown voltage is applied at the anode in a 1 mm air gap between two parallel plate electrodes and a normal glow discharge is shown to consist of the cathode fall, negative glow, positive column and anode regions. Neutral gas heating occurs with the initiation of the glow discharge, with the temperature at the anode shown to increase by only a few degrees oK as shown in Figure 12 due to the low electron densities and axial fields at the adjacency of the anode, whereas at the proximity of the cathode the temperature increases by approximately 180 oK as shown in Figure 13, with the temperature maximum at the cathode fall region, reducing abruptly at almost ambient temperature just outside it, where it remains almost constant.

The numerical results for the development of a glow discharge and its associated Joule heating by modeling its transition from a single electron, in a uniform applied electric field in atmospheric air, to a fully fledged state which consists of the cathode fall, negative glow, positive column and anode regions are presented. It is shown that the positive column travels towards the anode in the form of a return wave and maximum heating by approximately 180 oK occurs at the cathode fall region during the glow discharge development.

**Figure 12.** One-dimensional plots of the temperature along the symmetry axis close to the cathode at t = 6.75 ns at V= 5600 V

Electromagnetic and Fluid Analysis of Collisional Plasmas 59

and the primary streamer are analyzed as shown in Figures 16a-f, which show the mesh plots and the corresponding electron densities along the symmetry axis at three different instances in time. Figures 16a and 16b show respectively the initial mesh refinement and electron density distribution along the symmetry axis, whereas Figures 16c and 16d show the mesh and electron density during cathode directed primary streamer propagation, and Figures 16e and 16f show the mesh and electron density distributions just before the streamer hits the cathode. In the simulations, the photoemission effect is included in the calculation using the model developed by the authors [17], whereas photoionization and impact ionization phenomena are excluded. It has been shown that the avalanche and streamer propagation are captured by the adaptive mesh generator in an optimum way.

**Figure 14.** Plot of the two-dimensional cylindrical axisymmetric neutral gas temperature at time

**Figure 15.** Plot of the two-dimensional axial electric field distribution at a time t = 11.8 ns

t = 0.3263 μs

**Figure 13.** One-dimensional plots of the temperature along the anode at four instances in time at V = 5600 V

#### **7.3. Secondary streamers**

A constant voltage of 130 kV is applied on a dielectric barrier discharge configuration with the two metallic electrodes placed 4 mm apart, and with each one covered by alumina dielectric of 1 mm thickness, leaving an air gap of 2 mm for the discharge to develop. The two-dimensional neutral gas temperature distribution at time t = 0.3263 μs is shown in Figure 14, with most of the heating within the microplasma occurring along the symmetry axis where most of the activity takes place [8]. The column of heated air extends to a radial distance of 0.2 mm, and the temperature is raised approximately 120 oK. It is also observed that striations occur between the anode and the cathode. Along the dielectric electrodes, the temperature is larger than in the inter-electrode gap, but smaller than that at the symmetry axis. On average, higher neutral gas temperatures are observed at the cathode due to the arrival of the primary and secondary streamer charges on the dielectric cathode [16]. The two-dimensional axial electric field distribution at time t = 11.8 ns is shown below in Figure 15. Along the symmetry axis, there exists a cathode fall, negative glow, positive column and anode regions, similar to those of a normal glow discharge. A secondary streamer is shown to form at a radial distance of 1.5 mm that travels from the anode towards the cathode, extending radially up to the outer boundaries of the discharge and at later stages, when it hits the cathode, the sheath region spreads throughout the cathode dielectric surface.

#### **7.4. Adaptive mesh results on DC avalanche and streamers in ambient air**

The adaptive mesh generator has been tested in a geometric configuration that comprises of two metallic parallel-plates that are placed 1 mm apart in ambient atmospheric air. With a single electron released at the cathode as initial condition, the development of the avalanche and the primary streamer are analyzed as shown in Figures 16a-f, which show the mesh plots and the corresponding electron densities along the symmetry axis at three different instances in time. Figures 16a and 16b show respectively the initial mesh refinement and electron density distribution along the symmetry axis, whereas Figures 16c and 16d show the mesh and electron density during cathode directed primary streamer propagation, and Figures 16e and 16f show the mesh and electron density distributions just before the streamer hits the cathode. In the simulations, the photoemission effect is included in the calculation using the model developed by the authors [17], whereas photoionization and impact ionization phenomena are excluded. It has been shown that the avalanche and streamer propagation are captured by the adaptive mesh generator in an optimum way.

58 Finite Element Analysis – Applications in Mechanical Engineering

5600 V

**7.3. Secondary streamers** 

**Figure 13.** One-dimensional plots of the temperature along the anode at four instances in time at V =

A constant voltage of 130 kV is applied on a dielectric barrier discharge configuration with the two metallic electrodes placed 4 mm apart, and with each one covered by alumina dielectric of 1 mm thickness, leaving an air gap of 2 mm for the discharge to develop. The two-dimensional neutral gas temperature distribution at time t = 0.3263 μs is shown in Figure 14, with most of the heating within the microplasma occurring along the symmetry axis where most of the activity takes place [8]. The column of heated air extends to a radial distance of 0.2 mm, and the temperature is raised approximately 120 oK. It is also observed that striations occur between the anode and the cathode. Along the dielectric electrodes, the temperature is larger than in the inter-electrode gap, but smaller than that at the symmetry axis. On average, higher neutral gas temperatures are observed at the cathode due to the arrival of the primary and secondary streamer charges on the dielectric cathode [16]. The two-dimensional axial electric field distribution at time t = 11.8 ns is shown below in Figure 15. Along the symmetry axis, there exists a cathode fall, negative glow, positive column and anode regions, similar to those of a normal glow discharge. A secondary streamer is shown to form at a radial distance of 1.5 mm that travels from the anode towards the cathode, extending radially up to the outer boundaries of the discharge and at later stages, when it

hits the cathode, the sheath region spreads throughout the cathode dielectric surface.

**7.4. Adaptive mesh results on DC avalanche and streamers in ambient air** 

The adaptive mesh generator has been tested in a geometric configuration that comprises of two metallic parallel-plates that are placed 1 mm apart in ambient atmospheric air. With a single electron released at the cathode as initial condition, the development of the avalanche

**Figure 14.** Plot of the two-dimensional cylindrical axisymmetric neutral gas temperature at time t = 0.3263 μs

**Figure 15.** Plot of the two-dimensional axial electric field distribution at a time t = 11.8 ns

Electromagnetic and Fluid Analysis of Collisional Plasmas 61

The electromagnetic and fluid analysis of collisional plasmas has been thoroughly discussed including the necessary differential conservation equations to characterize such plasmas. The finite-element formulations were presented for the solution of these equations and the implementation procedure to couple the above set of equations was discussed. Thereafter, the FE-FCT algorithm was validated against theoretical and experimental fluid flow results and it was then used in a variety of collisional plasma configurations to study different

[1] Georghiou GE, Morrow R, Metaxas AC (1999) An Improved Finite Element Flux-

[2] Zalesak S (1979) Fully Multidimensional Flux-Corrected Transport Algorithms For

[3] Lohner R (1987) Finite Element flux-corrected transport (fem-fct) for the euler navier

[4] Georghiou GE, Morrow R, Metaxas AC (2000) Two-dimensional simulation of

[5] Morgan WL, Penetrante BMA (1990) ELENDIF: A time-dependent Boltzmann solver for

[6] BOLSIG 2011 CPAT (2011) Available: http://www.siglo-kinema.com/bolsig.htm.

[7] BOLSIG+ 2011 CPAT (2011) Available: http://www.bolsig.laplace.univ-tlse.fr/. Accessed

[8] Papadakis AP, Rossides S, Metaxas AC (2011) Microplasmas: A Review. Open Applied

[9] Papadakis AP (2004) Modelling of Gas Discharges using Finite Elements: Incorporation of Navier-Stokes equations. Ph.D. dissertation, Department of Electrical Engineering,

[12] Jiang Z, Moosad KPB, Takayama K, Onodera O (1998) Numerical and experimental study of a micro-blast wave generated by pulsed laser beam. Shock waves 8: 337-349. [13] Lohner R (1987) An adaptive finite element scheme for transient problems in CFD.

[10] Baker WE (1973) Explosions in Air. University of Texas Press, Austin and London. [11] Takayama K, Jiang Z (1997) Shock wave reflection over wedges: a benchmark test for

streamers using the FE-FCT algorithm. J. Phys. D: Appl. Phys. 33: 27-32.

partially ionized plasmas. Comput. Phys. Commun. 58: 127-152.

*Department of Electrical Engineering, Frederick University, Nicosia, Cyprus* 

Corrected Transport Algorithm. J. Comput. Phys. 148: 605-620.

stokes equations. Int. J. Numer. Meth. Fl. 7: 1093-1109.

Fluids. J. Comput. Phys. 31: 335-362.

Accessed 2011 Mar. 10.

Physics Journal 4: 45-63.

University of Cambridge, Trinity College.

cfd and experiments. Shock Waves, 7: 191-203.

Comput. Method. Appl. M. 61: 323-338, North Holland.

2011 Mar. 10.

**8. Conclusions** 

plasma phenomena.

**Author details** 

**9. References** 

Antonis P. Papadakis

**Figure 16.** (a) Two-dimensional cylindrical mesh at time t = 0 s at a voltage of 5600 V, (b) Onedimensional electron distribution along the symmetry axis at time t = 0 s at a voltage of 5600 V, (c) Twodimensional cylindrical mesh at time t = 3.59 ns at a voltage of 5600 V, (d) One-dimensional electron distribution along the symmetry axis at time t = 3.59 ns at a voltage of 5600 V, (e) Two-dimensional cylindrical mesh at time t = 4.54 ns at a voltage of 5600 V, (f) One-dimensional electron distribution along the symmetry axis at time t = 4.54 ns at a voltage of 5600 V

## **8. Conclusions**

60 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 16.** (a) Two-dimensional cylindrical mesh at time t = 0 s at a voltage of 5600 V, (b) One-

along the symmetry axis at time t = 4.54 ns at a voltage of 5600 V

dimensional electron distribution along the symmetry axis at time t = 0 s at a voltage of 5600 V, (c) Twodimensional cylindrical mesh at time t = 3.59 ns at a voltage of 5600 V, (d) One-dimensional electron distribution along the symmetry axis at time t = 3.59 ns at a voltage of 5600 V, (e) Two-dimensional cylindrical mesh at time t = 4.54 ns at a voltage of 5600 V, (f) One-dimensional electron distribution

(e) (f)

(a) (b)

(c) (d)

The electromagnetic and fluid analysis of collisional plasmas has been thoroughly discussed including the necessary differential conservation equations to characterize such plasmas. The finite-element formulations were presented for the solution of these equations and the implementation procedure to couple the above set of equations was discussed. Thereafter, the FE-FCT algorithm was validated against theoretical and experimental fluid flow results and it was then used in a variety of collisional plasma configurations to study different plasma phenomena.

## **Author details**

Antonis P. Papadakis *Department of Electrical Engineering, Frederick University, Nicosia, Cyprus* 

## **9. References**

	- [14] Huang W, Ren Y, Russell RD (1994) Moving mesh methods based on moving mesh partial differential equations. J. Comput. Phys. 113: 279-290.

**Section 2** 

**Applications of FEA in** 

**"Structural Mechanics and Composite Materials"** 


## **Applications of FEA in**

62 Finite Element Analysis – Applications in Mechanical Engineering

triangulations. Algorithmica 2: 137-151.

34: 200-208.

partial differential equations. J. Comput. Phys. 113: 279-290.

dielectric barrier discharge. IEEE Trans. Plasma Sci. 40: 811 - 820.

[14] Huang W, Ren Y, Russell RD (1994) Moving mesh methods based on moving mesh

[15] Dwyer R (1989) A faster divide and conquer algorithm for constructing Delaunay

[16] Papadakis AP (2012) Numerical analysis of the heating effects of an atmospheric air

[17] Georghiou GE, Morrow R, Metaxas AC (2001) The effect of photoemission on the streamer development and propagation in short uniform gaps. J. Phys. D: Appl. Phys. **"Structural Mechanics and Composite Materials"** 

**Chapter 3** 

© 2012 Ebrahimi et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Ebrahimi et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Finite Element Analysis of Functionally** 

A. Ghorbanpour Arani, R. Kolahchi, A. A. Mosalaei Barzoki,

A smart structure typically comprises of one or more active (or functional) materials. These active materials act in a unique way in which couple at least two of the following elds to provide the required functionality: mechanical, electrical, magnetic, thermal, chemical and optical. Through this coupling, these materials have the ability to change their shape, respond to external stimuli and vary their physical, geometrical and rheological properties. In modern technologies there has been an intense interest in FGPMs which are used in smart structures. It is well known that piezoelectric materials produce an electric field when deformed, and undergo deformation when subjected to an electric field. The coupling nature of piezoelectric materials has conducted wide applications in electro-mechanical and electric devices, such as electro-mechanical actuators, sensors and transducers. For example, piezoelectric actuators can be used to modify the shape of an airfoil, thereby reducing transverse vortices [1], or to maintain proper tension with overhead electrical wires on a

For homogeneous piezoelectric media, problems of radially-polarized piezoelectric bodies were considered and solved analytically by Chen [3]. Sinha [4] obtained the solution of the problem of static radial deformation of a piezoelectric spherical shell and under a given voltage difference between these surfaces, coupled with a radial distribution of temperature from the inner to the outer surface. Ghorbanpour et al. [5] investigated the stress and electric potential fields in piezoelectric hollow spheres. Stress in piezoelectric hollow sphere under thermal environment was developed by Saadatfar and Rastgoo [6]. Dai and Wang [7] presented the thermo-electro-elastic transient responses in piezoelectric hollow structures. Dai and Fu [8] studied the electromagneto transient stress and perturbation of magnetic

field vector in transversely isotropic piezoelectric solid spheres.

**Graded Piezoelectric Spheres** 

Additional information is available at the end of the chapter

A. Loghman and F. Ebrahimi

http://dx.doi.org/10.5772/52932

locomotive pantograph [2].

**1. Introduction** 

**Chapter 3** 

## **Finite Element Analysis of Functionally Graded Piezoelectric Spheres**

A. Ghorbanpour Arani, R. Kolahchi, A. A. Mosalaei Barzoki, A. Loghman and F. Ebrahimi

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52932

## **1. Introduction**

A smart structure typically comprises of one or more active (or functional) materials. These active materials act in a unique way in which couple at least two of the following elds to provide the required functionality: mechanical, electrical, magnetic, thermal, chemical and optical. Through this coupling, these materials have the ability to change their shape, respond to external stimuli and vary their physical, geometrical and rheological properties. In modern technologies there has been an intense interest in FGPMs which are used in smart structures. It is well known that piezoelectric materials produce an electric field when deformed, and undergo deformation when subjected to an electric field. The coupling nature of piezoelectric materials has conducted wide applications in electro-mechanical and electric devices, such as electro-mechanical actuators, sensors and transducers. For example, piezoelectric actuators can be used to modify the shape of an airfoil, thereby reducing transverse vortices [1], or to maintain proper tension with overhead electrical wires on a locomotive pantograph [2].

For homogeneous piezoelectric media, problems of radially-polarized piezoelectric bodies were considered and solved analytically by Chen [3]. Sinha [4] obtained the solution of the problem of static radial deformation of a piezoelectric spherical shell and under a given voltage difference between these surfaces, coupled with a radial distribution of temperature from the inner to the outer surface. Ghorbanpour et al. [5] investigated the stress and electric potential fields in piezoelectric hollow spheres. Stress in piezoelectric hollow sphere under thermal environment was developed by Saadatfar and Rastgoo [6]. Dai and Wang [7] presented the thermo-electro-elastic transient responses in piezoelectric hollow structures. Dai and Fu [8] studied the electromagneto transient stress and perturbation of magnetic field vector in transversely isotropic piezoelectric solid spheres.

© 2012 Ebrahimi et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Ebrahimi et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In-homogenity was considered in a number of studies. Elastic analysis of internally pressurized thick-walled spherical pressure vessels of functionally graded materials (FGMs) investigated by You et al. [9]. Analytical solution for a non-homogeneous isotropic piezoelectric hollow sphere was presented by Ding et al. [10]. Effect of material inhomogeneity on electro-thermo-mechanical behaviors of functionally graded piezoelectric rotating cylinder was considered by Ghorbanpour et al. [11]. Wang and Xu [12] studied the effect of material inhomogeneity on electromechanical behaviors of functionally graded piezoelectric spherical structures. Magnetothermoelastic problems of FGM spheres are studied by Ghorbanpour et al. [13].

Finite Element Analysis of Functionally Graded Piezoelectric Spheres 67

εε ε , *<sup>M</sup> <sup>T</sup>* (2)

0

*r*

 

(5)

*r* is considered. The sphere

(6)

and a

(4)

(3)

, e and Te are the fourth-order elasticity tensor, the dielectric permittivity

tensor, third order tensor of piezoelectric coefficient and transpose of it, respectively.

Assuming total strain tensor to be the sum of mechanical ( *M* ) and thermal (*T* ) strains [19, 20]

ε ,

<sup>ε</sup> , <sup>ε</sup> , <sup>0</sup>

It is also noted that the electric field tensor E can be written in terms of electric potential

E . *grad* 

distributed temperature field *T r* (Fig. 1). It is assumed that, only the radial displacement

 , 0, . *U ur U U <sup>r</sup> r* 

 

is subjected to an internal and external pressures *<sup>i</sup> P* and *<sup>o</sup> P* , an electric potential

*Ur* is nonzero and electric potential is the functions of radial coordinate *r* , Thus

*w*

*M T*

cot <sup>0</sup>

*r* and outer radius *<sup>o</sup>*

 

, ,

*u*

*r*

,

sin

,

,

sin

sin

**3. Formulation for electrothermoelastic FGPM spheres** 

A hollow FGPM sphere with an inner radius *<sup>i</sup>*

, , ,

*u v v r r v w*

*r*

*r rr u w w r r*

,

 

*r*

 

cot

*w u v*

*r*

*r*

*r rr T <sup>v</sup> <sup>u</sup> <sup>T</sup> r r T*

*rr*

where <sup>E</sup> C ,

where

as [21]

Sladek et al. [14] derived Local integral equations for numerical solution of 3-D problems in linear elasticity of FGMs viewed as 2-D axisymmetric problems while the meshless local Petrov-Galerkin method was applied to transient dynamic problems in 3D axisymmetric piezoelectric solids with continuously non-homogeneous material properties subjected to mechanical and thermal loads by Sladek et al.[15]. They concluded that this method is promising for numerical analysis of multi-eld problems like piezoelectric or thermoelastic problems, which cannot be solved efficiently by the conventional boundary element method.

Motivated by these ideas, new applications of piezoelectric sensors and actuators are being introduced and expanded for a number of geometric configurations. In this chapter, a hollow sphere composed of a radially polarized transversely isotropic piezoelectric material, e.g., PZT-4, which is subjected to mechanical and thermal loads, together with a potential difference induced by electrodes attached to the inner and outer surfaces of the annular sphere is considered. All mechanical, thermal and piezoelectric properties of the FGPM hollow sphere, except for the Poisson's ratio, are assumed to depend on the radius *r* and expressed in terms of its power functions. Hence, the equation of equilibrium in the radially polarized form is reduced to a system of second–order ordinary differential equation and is solved analytically for four different sets of boundary conditions. Finally, the thermal stresses, electric potential and displacement distributions are shown for different material in-homogeneity Also, a three-dimensional finite element analysis of asymmetric closed and open spheres with different boundary conditions subjected to an internal pressure and a uniform temperature field has also been carried out using ANSYS software.

## **2. Electromechanical coupling**

The subsequent characterization of electromechanical coupling covers the various classes of piezoelectric materials. Details with respect to definition and determination of the constants describing these materials have been standardized by the Institute of Electrical and Electronics Engineers [16]. Stresses σ and strains ε on the mechanical side, as well as flux density D and field strength E on the electrostatic side, may be arbitrarily combined as follows [17,18]

$$
\begin{Bmatrix} \mathbf{c} \\ \mathbf{D} \end{Bmatrix} = \begin{bmatrix} \mathbf{C}^{E} & -\mathbf{e} \\ \mathbf{e}^{T} & \mathbf{e}^{e} \end{bmatrix} \begin{Bmatrix} \mathbf{e} \\ \mathbf{E} \end{Bmatrix}' \tag{1}
$$

where <sup>E</sup> C , , e and Te are the fourth-order elasticity tensor, the dielectric permittivity tensor, third order tensor of piezoelectric coefficient and transpose of it, respectively.

Assuming total strain tensor to be the sum of mechanical ( *M* ) and thermal (*T* ) strains [19, 20]

$$\boldsymbol{\mathfrak{e}} = \boldsymbol{\mathfrak{e}}^{\boldsymbol{M}} + \boldsymbol{\mathfrak{e}}^{\boldsymbol{T}} \, \, \, \, \, \, \tag{2}$$

where

66 Finite Element Analysis – Applications in Mechanical Engineering

studied by Ghorbanpour et al. [13].

**2. Electromechanical coupling** 

follows [17,18]

In-homogenity was considered in a number of studies. Elastic analysis of internally pressurized thick-walled spherical pressure vessels of functionally graded materials (FGMs) investigated by You et al. [9]. Analytical solution for a non-homogeneous isotropic piezoelectric hollow sphere was presented by Ding et al. [10]. Effect of material inhomogeneity on electro-thermo-mechanical behaviors of functionally graded piezoelectric rotating cylinder was considered by Ghorbanpour et al. [11]. Wang and Xu [12] studied the effect of material inhomogeneity on electromechanical behaviors of functionally graded piezoelectric spherical structures. Magnetothermoelastic problems of FGM spheres are

Sladek et al. [14] derived Local integral equations for numerical solution of 3-D problems in linear elasticity of FGMs viewed as 2-D axisymmetric problems while the meshless local Petrov-Galerkin method was applied to transient dynamic problems in 3D axisymmetric piezoelectric solids with continuously non-homogeneous material properties subjected to mechanical and thermal loads by Sladek et al.[15]. They concluded that this method is promising for numerical analysis of multi-eld problems like piezoelectric or thermoelastic problems, which cannot be solved efficiently by the conventional boundary element method. Motivated by these ideas, new applications of piezoelectric sensors and actuators are being introduced and expanded for a number of geometric configurations. In this chapter, a hollow sphere composed of a radially polarized transversely isotropic piezoelectric material, e.g., PZT-4, which is subjected to mechanical and thermal loads, together with a potential difference induced by electrodes attached to the inner and outer surfaces of the annular sphere is considered. All mechanical, thermal and piezoelectric properties of the FGPM hollow sphere, except for the Poisson's ratio, are assumed to depend on the radius *r* and expressed in terms of its power functions. Hence, the equation of equilibrium in the radially polarized form is reduced to a system of second–order ordinary differential equation and is solved analytically for four different sets of boundary conditions. Finally, the thermal stresses, electric potential and displacement distributions are shown for different material in-homogeneity Also, a three-dimensional finite element analysis of asymmetric closed and open spheres with different boundary conditions subjected to an internal pressure and a

uniform temperature field has also been carried out using ANSYS software.

The subsequent characterization of electromechanical coupling covers the various classes of piezoelectric materials. Details with respect to definition and determination of the constants describing these materials have been standardized by the Institute of Electrical and Electronics Engineers [16]. Stresses σ and strains ε on the mechanical side, as well as flux density D and field strength E on the electrostatic side, may be arbitrarily combined as

> <sup>σ</sup> C e <sup>ε</sup> , D E <sup>e</sup> *E T*

(1)

$$\varepsilon = \begin{bmatrix} \mathcal{E}\_{rr} \\ \mathcal{E}\_{\theta\theta} \\ \mathcal{E}\_{\zeta\zeta} \\ \mathcal{E}\_{r\zeta} \\ \mathcal{E}\_{r\zeta} \\ \mathcal{E}\_{\zeta\theta} \\ \mathcal{E}\_{r\theta} \end{bmatrix}, \tag{3}$$

$$\mathbf{c}^{M} = \begin{pmatrix} u\_{,r} \\ \frac{\pi r\_{,\theta}}{r \sin \zeta} + \frac{\cot \zeta}{r} v + \frac{u}{r} \\ \frac{\upsilon\_{,\zeta}}{r} + \frac{u}{r} \\ \frac{u\_{,\zeta}}{r} + \upsilon\_{,r} - \frac{\upsilon}{r} \\ \frac{\upsilon\_{,\theta}}{r} + \frac{w\_{,\zeta}}{r} - \frac{\cot \zeta}{r} w \\ \frac{\upsilon\_{,\theta}}{r \sin \zeta} + \frac{w\_{,\zeta}}{r} - \frac{\cot \zeta}{r} w \\ \frac{u\_{,\theta}}{r \sin \zeta} + w\_{,r} - \frac{w}{r} \end{pmatrix}, \qquad \varepsilon^{T} = \begin{pmatrix} -a\_{,r}T \\ -a\_{,\theta}T \\ -a\_{,\zeta}T \\ 0 \\ 0 \\ 0 \end{pmatrix}, \tag{4}$$

It is also noted that the electric field tensor E can be written in terms of electric potential as [21]

$$
\mathbf{E} = -\mathbf{g} \, \text{rad } \boldsymbol{\phi}.\tag{5}
$$

#### **3. Formulation for electrothermoelastic FGPM spheres**

A hollow FGPM sphere with an inner radius *<sup>i</sup> r* and outer radius *<sup>o</sup> r* is considered. The sphere is subjected to an internal and external pressures *<sup>i</sup> P* and *<sup>o</sup> P* , an electric potential and a distributed temperature field *T r* (Fig. 1). It is assumed that, only the radial displacement *Ur* is nonzero and electric potential is the functions of radial coordinate *r* , Thus

$$\mathcal{U}\mathcal{U}\_r = \mathfrak{u}\begin{pmatrix} r \\ \end{pmatrix}, \ \mathcal{U}\_\zeta = \mathcal{U}\_\theta = 0, \ \phi = \phi\begin{pmatrix} r \\ \end{pmatrix}. \tag{6}$$

**Figure 1.** Hollow FGPM sphere subject to uniform temperature field, uniform internal pressure, uniform external pressure and applied voltage V.

The equilibrium equation of the FGPM sphere in the absence of body force and the Maxwell's equation for free electric charge density are [18, 22]

$$
\sigma\_{rr,r} + \frac{2(\sigma\_{rr} - \sigma\_{\theta\theta})}{r} = 0,\tag{7}
$$

Finite Element Analysis of Functionally Graded Piezoelectric Spheres 69

 

(12)

0 22 11

(14)

 

 

 

 

12 13 21 22 33 32 23 12 13 *CCC CC CC ee* , ,, . (13)

0 0 11 0

*i r*

(15)

(16)

( )

 

 

 

*r*

(17)

(18)

1 22 0

0, *<sup>r</sup>*

0. *<sup>r</sup>*

,

( ) <sup>1</sup>

*CCC E <sup>U</sup> <sup>T</sup> C CE*

*CC E <sup>U</sup> <sup>T</sup>*

,

( ) <sup>1</sup>

*U T*

 

,, , , , , .

*D*

2

Before substituting the component of the electric field in Maxwell's equation, appropriate

 <sup>0</sup> , *<sup>r</sup>* 

in which *<sup>r</sup>* represents the general properties of the sphere such as the elastic, piezoelectric, dielectric coefficients and thermal conductivity, and 0 corresponds to the value of the

 

*C e ir c i E i E C*

*r o rr*

*u r <sup>r</sup> <sup>C</sup> <sup>D</sup> U r <sup>D</sup> rr r E E*

2

,

*r*

coefficients at the outer surface. Substituting Eqs. (14) and (17) into Eq. (12), yields

1221 2 32 23 2 122

1

*D*   

, , 1,2,3 , 1,2,3 , ,

 

11 12 13 11 21 22 23 12 31 32 33 13 11 12 13 11

*rr rr r*

*CCC e T r CCC e T r CCC e T r*

*rr rr*

For transversely isotropic properties, when the concerned axis of rotation is oriented in the radial direction, the elasticity and piezoelectric coefficient tensors are summarized to [25]

*D eee E*

It is appropriate to introduce the following dimensionless quantities as

22 22 0

*CC E*

*ii i*

power functions for all properties are assumed as [26]

*r*

The solution of Eq. (16) is

*r*

*D EEE*

**4. Electrothermoelastic analysis of FGPM spheres** 

*ii i*

 

 

*ii i i*

1 1

Using the above dimensionless variables, Eqs. (7) and (8) can be expressed as

,

*r*

$$D\_{rr,r} + \frac{2}{r} D\_{rr} = 0,\tag{8}$$

where , *ii i r* is the stress tensor and *Drr* is the radial electric displacement.

Also, the radial and circumferential strain and the relation between electric field and electric potential are reduced to

$$
\boldsymbol{\varepsilon}\_{rr} = \boldsymbol{\mu}\_{,r} \tag{9}
$$

$$
\mathcal{E}\_{\theta\theta} = \mathcal{E}\_{\xi\xi} = \frac{\mu}{r},
\tag{10}
$$

$$E\_{rr} = -\phi\_{,r}.\tag{11}$$

The constitutive relations of spherically radially polarized piezoelectric media and the component of radial electric displacement vector also can be written as [23, 24]

$$
\begin{bmatrix}
\sigma\_{rr} \\
\sigma\_{\theta\theta} \\
\sigma\_{\theta\theta} \\
\sigma\_{\zeta\zeta} \\
D\_{rr}
\end{bmatrix} = \begin{bmatrix}
\mathbb{C}\_{11} & \mathbb{C}\_{12} & \mathbb{C}\_{13} & e\_{11} \\
\mathbb{C}\_{21} & \mathbb{C}\_{22} & \mathbb{C}\_{23} & e\_{12} \\
\mathbb{C}\_{31} & \mathbb{C}\_{32} & \mathbb{C}\_{33} & e\_{13} \\
e\_{11} & e\_{12} & e\_{13} & -e\_{11} \\
\end{bmatrix} \begin{bmatrix}
\varepsilon\_{rr} - a\_r T(r) \\
\varepsilon\_{\theta\theta} - a\_\theta T(r) \\
\varepsilon\_{\zeta\zeta} - a\_\zeta T(r) \\
E\_{rr}
\end{bmatrix} \tag{12}
$$

For transversely isotropic properties, when the concerned axis of rotation is oriented in the radial direction, the elasticity and piezoelectric coefficient tensors are summarized to [25]

$$\mathbf{C}\_{12} = \mathbf{C}\_{13} = \mathbf{C}\_{21'} \quad \mathbf{C}\_{22} = \mathbf{C}\_{33'} \quad \mathbf{C}\_{32} = \mathbf{C}\_{23'} \quad \qquad \mathbf{e}\_{12} = \mathbf{e}\_{13}. \tag{13}$$

It is appropriate to introduce the following dimensionless quantities as

$$\begin{aligned} \sigma\_i &= \frac{\sigma\_{ii}}{C\_{22}} \left( i = r\_r \theta \right), & c\_i &= \frac{C\_{1i}}{C\_{22}} \left( i = 1, 2, 3 \right), & E\_i &= \frac{e\_{1i}}{E\_0} \left( i = 1, 2, 3 \right), & E\_0 &= \sqrt{C\_{22} \cdot \Xi\_{11}}, \\\ \mathrm{LI} &= \frac{u\_r}{r\_i}, & \xi &= \frac{r}{r\_i}, \quad \eta = \frac{r\_o}{r\_i}, \quad \beta &= \frac{\beta\_1}{E\_0}, & \Phi &= \frac{\phi}{\phi\_0}, \quad \phi\_0 = r\_i \sqrt{\frac{C\_{22}}{\varepsilon\_{11}}}, & D\_r &= \frac{D\_{rr}}{E\_0}. \end{aligned} \tag{14}$$

Using the above dimensionless variables, Eqs. (7) and (8) can be expressed as

$$
\sigma\_{r,\xi} + \frac{2\left(\sigma\_r - \sigma\_\theta\right)}{\xi} = 0,\tag{15}
$$

$$D\_{r\_r, \xi} + \frac{2D\_r}{\xi} = 0.\tag{16}$$

Before substituting the component of the electric field in Maxwell's equation, appropriate power functions for all properties are assumed as [26]

$$
\Gamma\_r = \Gamma\_0 \left( \boldsymbol{\xi} \right)^\gamma,\tag{17}
$$

in which *<sup>r</sup>* represents the general properties of the sphere such as the elastic, piezoelectric, dielectric coefficients and thermal conductivity, and 0 corresponds to the value of the coefficients at the outer surface. Substituting Eqs. (14) and (17) into Eq. (12), yields

$$
\begin{Bmatrix}
\sigma\_r \\ \sigma\_\theta \\ \sigma\_\zeta \\ D\_r \end{Bmatrix} = \xi^\gamma \begin{Bmatrix} \mathbf{C}\_1 & \mathbf{C}\_2 & \mathbf{C}\_2 & E\_1 \\ \mathbf{C}\_2 & \mathbf{1} & \mathbf{C}\_3 & E\_2 \\ \mathbf{C}\_2 & \mathbf{C}\_3 & \mathbf{1} & E\_2 \\ E\_1 & E\_2 & E\_2 & -\mathbf{1} \end{Bmatrix} \begin{Bmatrix} \mathbf{U}\_{,\bar{\xi}} - \xi^\gamma a\_r T(\xi) \\ \mathbf{U} \\ \bar{\xi} \\ \mathbf{U} - \xi^{\gamma \gamma} a\_\theta T(\xi) \\ \bar{\xi} \\ \Phi\_{,\bar{\xi}} \end{Bmatrix} \tag{18}
$$

#### **4. Electrothermoelastic analysis of FGPM spheres**

The solution of Eq. (16) is

68 Finite Element Analysis – Applications in Mechanical Engineering

uniform external pressure and applied voltage V.

where , *ii i r*

potential are reduced to

Maxwell's equation for free electric charge density are [18, 22]

, *<sup>u</sup>*

, . *rr r E*

**Figure 1.** Hollow FGPM sphere subject to uniform temperature field, uniform internal pressure,

,

*rr r r*

,

 

<sup>2</sup> 0, *D D rr r rr <sup>r</sup>*

Also, the radial and circumferential strain and the relation between electric field and electric

, , *rr r* 

> 

The constitutive relations of spherically radially polarized piezoelectric media and the

 

component of radial electric displacement vector also can be written as [23, 24]

The equilibrium equation of the FGPM sphere in the absence of body force and the

2( ) 0, *rr*

is the stress tensor and *Drr* is the radial electric displacement.

*r*

(7)

(8)

*u* (9)

(10)

(11)

$$D\_r = \frac{A\_1}{\xi^2},$$
 
$$\xi^2$$

Finite Element Analysis of Functionally Graded Piezoelectric Spheres 71

(27)

*r*

2 2 12 1 1 2

 

 

*u u* (29)

, *Uu u <sup>g</sup> <sup>p</sup>* (31)

)( )

*<sup>r</sup> EE E C B*

is defined as

(30)

1

(32)

*B*

 

The particular solution of the differential Eq. (24) may be obtained as

where

in which *R*

*r*

  1 2 1 2 , *q q <sup>p</sup> uuu* 

2 1

*Wq q Wq q*

1 2 1 2 ( ) ( ) , , (,) (,) *q q R R*

1 2

*g g g g*

*u u*

1 3 3 3 5 4

 

 

2 1 2 1 21 . ( 1)( 1) (3 )(3 ) ( 3)( 3) *<sup>p</sup> D D <sup>D</sup> u A BB q q q q qq*

The complete solution for *<sup>m</sup> U* in terms of the non-dimensional radial coordinate is written as

where *K*<sup>1</sup> , *K*<sup>2</sup> and *A*1 are unknown constants. Substituting the displacement from Eq. (31)

2

2 11 1 2 1 1 1 2 12 1 2 1 2 2

*C Cq EE Eq K C Cq EE Eq K*

2 12 1 1 4 2 12 1 1

 

 

2 2 (3 )( ) 2( ) ( ) ( 3)( 3) 1

*C EE E C D C EE E C*

2 2 2

1 21

<sup>1</sup> <sup>2</sup> 1 1 2 2

*q q*

2 2 2 1

2

*C*

1 1

*E A*

1 2

(,) . ( )( )

 (28)

1 2

1 2

 

*Wq q*

Combining Eqs. (25)-(29) one can obtain the particular solution as

into Eq. (20) the radial and circumferential stresses are obtained as

() 2 2 2 2

2( (3 )(3 )

 

2 1

*q q*

2 2 (1 )( )

( 1)( 1)

*C EE E C D*

2 2 (3 )( )

 

2 1

*q q*

2 12 1 15

*q q*

*C EE E C D*

2 1

 

 

2 2 12 1 13 2

is the expression on the right hand side of Eq. (24) and *W*

*u u*

  

where *A*<sup>1</sup> is a constant. Substituting Eq. (19) into Eq. (18), we obtain

$$
\begin{Bmatrix} \sigma\_r \\ \sigma\_\theta \end{Bmatrix} = \xi^{\prime \prime} \left[ \left( \begin{bmatrix} \mathbf{C}\_1 & \mathbf{C}\_2 & \mathbf{C}\_2 \\ \mathbf{C}\_2 & \mathbf{1} & \mathbf{C}\_3 \end{bmatrix} + \begin{bmatrix} E\_1^2 & E\_1 E\_2 & E\_1 E\_2 \\ E\_1 E\_2 & E\_1^2 & E\_2^2 \end{bmatrix} \right) \begin{Bmatrix} \begin{Bmatrix} \mathbf{U}\_{,\xi} - \xi^{\prime \prime} a\_r T(\xi) \\ \mathbf{U}\_{,\xi} - \xi^{\prime \prime} a\_\theta T(\xi) \\ \tilde{\xi} \\ \tilde{\xi} \\ \frac{\mathbf{U}}{\xi} - \xi^{\prime \prime} a\_\theta T(\xi) \end{Bmatrix} \right) - \begin{Bmatrix} E\_1 \\ E\_2 \end{Bmatrix} \begin{Bmatrix} A\_1 \xi^{-\prime - 2} \\ A\_1 \xi^{-\prime - 2} \end{Bmatrix} (20) \right] \right] \tag{20}
$$

In this study a distributed temperature field due to steady-state heat conduction has been considered. Using Eq. (17) for the thermal conductivity property, the heat conduction equation without any heat source is written in spherical coordinate as [22, 27]

$$\frac{1}{\xi^2} \Big( \mathcal{K}\_0 \, \xi^{\gamma+2} \, T(\xi)\_{,\xi} \Big)\_{,\xi} = 0,\tag{21}$$

$$\begin{aligned} \text{at } \xi &= 1 & T(\xi) &= T\_{a'}\\ \text{at } \xi &= \eta & T(\xi)\_{,\xi} + hT(\xi) &= 0 \end{aligned} \tag{22}$$

where *h* is the ratio of the convective heat-transfer coefficient and *K*0 is the nominal heat conductivity coefficient. Integrating Eq. (21) twice yields

$$T(\xi) = -\frac{B\_1}{\gamma + 1} \xi^{-\gamma - 1} + B\_{2\prime} \tag{23}$$

Constants 1 *B* and <sup>2</sup> *B* are obtained using thermal boundary conditions which shown in Eq. (22).

Finally, substituting Eq. (20) and (24) into Eq. (15) yields the following non-homogeneous Cauchy differential equation

$$\xi^2 \frac{\partial^2 \mathcal{U}}{\partial \ \xi^2} + D\_1 \xi \frac{\partial \mathcal{U}}{\partial \xi} + D\_2 \mathcal{U} = D\_4 B\_1 + D\_5 B\_2 \xi^{\left(1+\gamma\right)} + D\_3 A\_1 \xi^{-\left(1+\gamma\right)} \tag{24}$$

where 1,...8 *D i <sup>i</sup>* are defined in Appendix A.

The exact solution for Eq. (24) is written as follows

$$\mu\_{\mathcal{g}} = \underbrace{K\_1 \exp(q\_1 \xi)}\_{u\_{\mathcal{g}1}} + \underbrace{K\_2 \exp(q\_2 \xi)}\_{u\_{\mathcal{g}2}} \prime \tag{25}$$

$$q\_{1\prime}q\_2 = \frac{(1 - D\_1) \pm \sqrt{\left(D\_1 - 1\right)^2 - 4D\_2}}{2}.\tag{26}$$

The particular solution of the differential Eq. (24) may be obtained as

$$
u\_p = \xi^{q\_1} u\_1 + \xi^{q\_2} u\_2. \tag{27}$$

where

70 Finite Element Analysis – Applications in Mechanical Engineering

*r*

where *A*<sup>1</sup> is a constant. Substituting Eq. (19) into Eq. (18), we obtain

2 2

equation without any heat source is written in spherical coordinate as [22, 27]

conductivity coefficient. Integrating Eq. (21) twice yields

 

*g*

1 2

*q q*

Cauchy differential equation

where 1,...8 *D i <sup>i</sup>* are defined in Appendix A.

The exact solution for Eq. (24) is written as follows

( ) <sup>1</sup>

, <sup>2</sup>

2 3 2 1 2

*CCC E EE EE <sup>U</sup> <sup>E</sup> T A C C EE E E E*

In this study a distributed temperature field due to steady-state heat conduction has been considered. Using Eq. (17) for the thermal conductivity property, the heat conduction

> <sup>2</sup> <sup>2</sup> 0 , , <sup>1</sup> *K T*( ) 0,

1 () ,

 

> 

where *h* is the ratio of the convective heat-transfer coefficient and *K*0 is the nominal heat

 

,

1 1 <sup>2</sup> ( ) , <sup>1</sup> *<sup>B</sup> T B*

 

 

Constants 1 *B* and <sup>2</sup> *B* are obtained using thermal boundary conditions which shown in Eq. (22).

Finally, substituting Eq. (20) and (24) into Eq. (15) yields the following non-homogeneous

 <sup>2</sup> 2 1 1 2 1 2 41 52 3 1 *U U D DU DB DB DA*

> 1 2 <sup>1122</sup> exp( ) exp( ), *g g*

> > <sup>2</sup> 11 2

*u u uK q K q* 

(1 ) 1 4 , . <sup>2</sup> *DD D*

(25)

(26)

(24)

*<sup>a</sup> at T T at T hT*

1 <sup>2</sup> , *<sup>r</sup> <sup>A</sup> <sup>D</sup>* 

12 12 122 1 2

 

( ) ( ) 0,

 

(19)

 

(20)

( )

 

( )

(21)

(23)

 

(22)

 

 

<sup>1</sup>

*U T*

*<sup>U</sup> <sup>T</sup>*

1 2 2

*r*

$$\mu\_1 = -\int \frac{\xi^{q\_2} R(\xi)}{W(q\_{1'}, q\_2)}, \qquad \mu\_2 = \int \frac{\xi^{q\_1} R(\xi)}{W(q\_{1'}, q\_2)},\tag{28}$$

in which *R* is the expression on the right hand side of Eq. (24) and *W* is defined as

$$\mathcal{W}(q\_1, q\_2) = \begin{vmatrix} u\_{\mathcal{g}1} & u\_{\mathcal{g}2} \\ \binom{u\_{\mathcal{g}1}}{\mathcal{g}1} & \binom{u\_{\mathcal{g}2}}{\mathcal{g}2} \end{vmatrix} . \tag{29}$$

Combining Eqs. (25)-(29) one can obtain the particular solution as

$$u\_p = \frac{D\_3 \xi^{1-\gamma}}{(q\_2 + \gamma - 1)(q\_1 + \gamma - 1)} A\_1 + \frac{D\_5 \xi^{3+\gamma}}{(3 - q\_2 + \gamma)(3 - q\_1 + \gamma)} B\_2 + \frac{D\_4 \xi^3}{(q\_2 - 3)(q\_1 - 3)} B\_1. \tag{30}$$

The complete solution for *<sup>m</sup> U* in terms of the non-dimensional radial coordinate is written as

$$
\mathcal{U} = \mathfrak{u}\_{\mathcal{X}} + \mathfrak{u}\_{p'} \tag{31}
$$

where *K*<sup>1</sup> , *K*<sup>2</sup> and *A*1 are unknown constants. Substituting the displacement from Eq. (31) into Eq. (20) the radial and circumferential stresses are obtained as

 <sup>1</sup> <sup>2</sup> 1 1 2 2 2 11 1 2 1 1 1 2 12 1 2 1 2 2 2 2 2 1 2 12 1 1 4 2 12 1 1 1 2 1 2 2 2 2 12 1 15 2 2 1 () 2 2 2 2 2 2 (3 )( ) 2( ) ( ) ( 3)( 3) 1 2 2 (3 )( ) 2( (3 )(3 ) *q q r r C Cq EE Eq K C Cq EE Eq K C EE E C D C EE E C B q q C EE E C D C q q* 2 2 12 1 1 2 2 2 12 1 13 2 1 1 2 1 )( ) 2 2 (1 )( ) 2 ( 1)( 1) *<sup>r</sup> EE E C B C EE E C D E A q q* (32)

 1 2 2 1 2 2 1 1 3 21 1 21 2 1 3 2 1 22 22 2 2 22 1 3 2 2 12 4 3 2 2 12 1 2 1 2 2 2 3 2 12 2 5 ( ) 2 1 2 1 323 1 ( 2 1) ( ) ( 3)( 3) 1 2 1 (3 )( ) ( (3 )(3 ) *q q r m m C Cq EEq E K C E EEq Cq K C C E EE D C E C EE B q q C E EE C D q q* 2 2 3 2 2 12 2 2 3 12 2 12 2 3 2 2 1 2 1 1 2) ( ) 2 2 1 (1 )( ) ( 1)( 1) *<sup>r</sup> C E C EE B C EE E EE C D E A q q* (33)

Finite Element Analysis of Functionally Graded Piezoelectric Spheres 73

,

(37)

*r*0

2 e -6 1 / *K*

. In this case

1.5 their maximum values

2 e - 5 1 / *K*

are shown in Fig. 3. Electric potentials satisfy the

*7500* <sup>3</sup> *kg m*/

 11 12 13 14 1 21 22 23 24 2 31 32 33 34 3 41 42 43 44 4

*K mmmm b K mmmm b*

*A mmmm b A mmmm b*

The numerical results are showing the variation of stresses, electric potential and displacement across the thickness of the FGPM sphere for different material inhomogenity

piezoelectric material PZT-4 has been selected because of its technical applications. Mechanical and electrical properties of piezoelectric material, PZT-4 are tabulated in Table

> 15.1 <sup>2</sup> *C m*/

Results of the first case are illustrated in Figs. 2 to 5. Radial stresses for different material in-

boundary conditions at the inner and outer surfaces of the FGPM sphere. The maximum absolute values of radial stresses belong to a material identified by in-homogeneity

the minimum absolute values of which belong to 1.5

there is no imposed electric potential however, the induced electric potentials for different

electrical boundary conditions at the inner and outer surfaces of the FGPM sphere. It is also obvious that higher induced electric potentials belong to higher absolute values of compressive radial stresses. In this case circumferential stresses shown in Fig. 4 are highly

located at the inner and their minimum values located at the outer surfaces of the sphere. Displacements are illustrated in Fig. 5 for all material properties. Displacements are positive throughout the thickness and they smoothly decrease from their maximum value at the

. Presented results are for the two cases of different boundary conditions with

1.3 . The plots in these figures correspond to 323 *<sup>i</sup> T K* and 298 *<sup>o</sup> T K* . The


are shown in Fig. 2. Radial stresses satisfy the mechanical

*inv*

where the *mij* and *<sup>i</sup> b* ( , 1,...4) *i j* are defined in Appendix B

*Property C*<sup>11</sup> *C*<sup>12</sup> *C*<sup>22</sup> *C*<sup>23</sup> <sup>11</sup> *e* <sup>12</sup> *e* <sup>11</sup> *e*

77.8 *Gpa*

139 *Gpa*

**Table 1.** Mechanical, electrical and thermal properties for PZT-4

tensile throughout thickness and except for the material

**5. Numerical results and discussion** 

**5.1. Analytical solution** 

*Gpa*

homogeneity parameters

material in-homogeneity parameters

74.3 *Gpa*

parameter

1[28].

aspect ratio

*PZT-4* 115

*5.1.1. Case 1* 

parameter 1.5 

Substituting *U* from above into the last term of Eq. (18), ( , ) and combining with Eq. (19) and performing the integrating, electric potential is obtained as

$$\begin{aligned} \Phi(\xi) &= \left( \left( \frac{2E\_2}{q\_1} + E\_1 \right) \xi^{q\_1} \right) K\_1 + \left( \left( \frac{2E\_2}{q\_2} + E\_1 \right) \xi^{q\_2} \right) K\_2 \\ &+ \left( \frac{\left( \varphi E\_1 - E\_1 - 2E\_2 \right) D\_3 \xi^{1-\gamma}}{(q\_2 + \gamma - 1)(q\_1 + \gamma - 1)(1 - \gamma)} + \frac{\xi^{-\gamma - 1}}{(1 + \gamma)} \right) A\_1 \\ &+ \left( \frac{(E\_1 + \sum\_{\mathcal{G}}' E\_2) D\_4 \xi^{\mathcal{G}}}{(q\_2 - 3)(q\_1 - 3)} + \frac{(2E\_2 \alpha\_\theta + E\_1 \alpha\_r) \ln(\xi)}{\gamma + 1} \right) B\_1 \\ &+ \left( \frac{\left( 3E\_1 + \gamma E\_1 + 2E\_2 \right) D\_5 \xi^{\mathcal{G} + 3}}{(q\_2 + \gamma - 1)(q\_1 + \gamma - 1)(3 + \gamma)} - \frac{(2E\_2 \alpha\_\theta + E\_1 \alpha\_r) \xi^{\mathcal{G} + 1}}{\gamma + 1} \right) B\_2 + A\_{2'} \end{aligned} \tag{34}$$

where *<sup>r</sup>* , and are radial stress, hoop stress and electric potential, respectively. Two sets of mechanical and electrical loading boundary conditions are considered in this investigation which in normalized form are written as

$$\text{case 1}: \sigma\_r(1) = -1, \ \sigma\_r(\eta) = 0, \quad \Phi(1) = 0, \ \Phi(\eta) = 0 \tag{35}$$

$$\text{case } \mathcal{D}: \sigma\_r(1) = 0, \quad \sigma\_r(\eta) = 0, \quad \Phi(1) = 1, \quad \Phi(\eta) = 0 \tag{36}$$

In case 1, the FGPM hollow sphere is subjected to an internal uniform pressure without any imposed electric potential and external pressure. However in this case the induced electric potential existed across the thickness. In this case, the sphere acts as an sensor. In the second case, an electrical potential difference is applied between the inner and outer surfaces of the sphere without any internal and external pressures. In this case, the sphere acts as an actuator.

For the above mentioned cases 1, and 2 the system of linear algebraic equations for the constants *K*<sup>1</sup> , *K*<sup>2</sup> , *A*1 and *A*2 of the Eqs. (32), (33) and (34) can be written in the following from

$$
\begin{bmatrix} K\_1 \\ K\_2 \\ A\_1 \\ A\_2 \end{bmatrix} = inv \left[ \begin{bmatrix} m\_{11} & m\_{12} & m\_{13} & m\_{14} \\ m\_{21} & m\_{22} & m\_{23} & m\_{24} \\ m\_{31} & m\_{32} & m\_{33} & m\_{34} \\ m\_{41} & m\_{42} & m\_{43} & m\_{44} \end{bmatrix} \right] \times \begin{bmatrix} b\_1 \\ b\_2 \\ b\_3 \\ b\_4 \end{bmatrix}' \tag{37}
$$

where the *mij* and *<sup>i</sup> b* ( , 1,...4) *i j* are defined in Appendix B

## **5. Numerical results and discussion**

#### **5.1. Analytical solution**

72 Finite Element Analysis – Applications in Mechanical Engineering

 

2 1

*q q*

 

2 1

2 1

*q q*

Substituting *U* from above into the last term of Eq. (18), ( ,

2 1

*q q*

where *<sup>r</sup>* , 

actuator.

from

2 1

*q q*

1 1 23

*E E ED*

2

2 2 ( )

3 1 24 2 1

 

( 1)( 1)(1 ) (1 )

 

3 2

2 1

investigation which in normalized form are written as

*q q*

1 1 25

*E E ED*

and performing the integrating, electric potential is obtained as

 

2 2 1 (1 )( ) ( 1)( 1)

*C EE E EE C D*

(3 )(3 )

*m m*

3 2 12 2 5

2

2 1 (3 )( )

*C E EE C D q q*

2 2 2

3 21 1 21 2 1 3 2 1 22 22 2 2 22 1

*C Cq EEq E K C E EEq Cq K*

(

 

1 2

1 2) ( )

 

*q q*

1

(34)

(35)

(36)

(33)

 

*B*

*r*

 

2 2 3 2 2 12 2

*<sup>r</sup> C E C EE B*

 

> 

) and combining with Eq. (19)

2 2 1 1

2 1

*E A*

1

*A*

2 1

 

and are radial stress, hoop stress and electric potential, respectively. Two

 

1

*<sup>r</sup> E E B A*

 

1

2 2

 

2 2 1 2 11 12

 

> >

*E E EK EK*

3

 

 

 

 

(2 ) , ( 1)( 1)(3 ) <sup>1</sup>

sets of mechanical and electrical loading boundary conditions are considered in this

case 1 : (1) 1, ( ) 0, (1) 0, ( ) 0 *r r*

case 2 : (1) 0, ( ) 0, (1) 1, ( ) 0 *r r*

In case 1, the FGPM hollow sphere is subjected to an internal uniform pressure without any imposed electric potential and external pressure. However in this case the induced electric potential existed across the thickness. In this case, the sphere acts as an sensor. In the second case, an electrical potential difference is applied between the inner and outer surfaces of the sphere without any internal and external pressures. In this case, the sphere acts as an

For the above mentioned cases 1, and 2 the system of linear algebraic equations for the constants *K*<sup>1</sup> , *K*<sup>2</sup> , *A*1 and *A*2 of the Eqs. (32), (33) and (34) can be written in the following

1 1

*r*

 

*q q*

1 2

 

*q q*

<sup>2</sup> ( ) (2 )ln( ) <sup>3</sup> ( 3)( 3) 1

 

*E ED E E <sup>B</sup>*

323 1 ( 2 1) ( ) ( 3)( 3) 1

 

( ) 2 1 2 1

3 2 2 12 4 3 2 2 12

*C C E EE D C E C EE*

 

3 12 2 12 2 3 2

 

 

> The numerical results are showing the variation of stresses, electric potential and displacement across the thickness of the FGPM sphere for different material inhomogenity parameter . Presented results are for the two cases of different boundary conditions with aspect ratio 1.3 . The plots in these figures correspond to 323 *<sup>i</sup> T K* and 298 *<sup>o</sup> T K* . The piezoelectric material PZT-4 has been selected because of its technical applications. Mechanical and electrical properties of piezoelectric material, PZT-4 are tabulated in Table 1[28].


**Table 1.** Mechanical, electrical and thermal properties for PZT-4

#### *5.1.1. Case 1*

Results of the first case are illustrated in Figs. 2 to 5. Radial stresses for different material inhomogeneity parameters are shown in Fig. 2. Radial stresses satisfy the mechanical boundary conditions at the inner and outer surfaces of the FGPM sphere. The maximum absolute values of radial stresses belong to a material identified by in-homogeneity parameter 1.5 the minimum absolute values of which belong to 1.5 . In this case there is no imposed electric potential however, the induced electric potentials for different material in-homogeneity parameters are shown in Fig. 3. Electric potentials satisfy the electrical boundary conditions at the inner and outer surfaces of the FGPM sphere. It is also obvious that higher induced electric potentials belong to higher absolute values of compressive radial stresses. In this case circumferential stresses shown in Fig. 4 are highly tensile throughout thickness and except for the material 1.5 their maximum values located at the inner and their minimum values located at the outer surfaces of the sphere. Displacements are illustrated in Fig. 5 for all material properties. Displacements are positive throughout the thickness and they smoothly decrease from their maximum value at the inner surface to their minimum value at the outer surface of the FGPM sphere. Maximum values of displacements belong to 1.5 and minimum values belong to 1.5 

Finite Element Analysis of Functionally Graded Piezoelectric Spheres 75

.

= -1.5 = -0.5 = 0.0001 = 0.5 = 1.5

= -1.5 = -0.5 = 0.0001 = 0.5 = 1.5

> .

**Figure 4.** Case 1: Distributions of the electric potential for different values of


1.6

1.8

2

2.2

2.4

Dimensionless displacement

2.6

2.8

3

3.2

Dimensionless electric potential

1 1.05 1.1 1.15 1.2 1.25 1.3

1 1.05 1.1 1.15 1.2 1.25 1.3

Dimensionless radius

Dimensionless radius

**Figure 5.** Case 1: Distributions of the radial displacement for different values of

**Figure 2.** Case 1: Distributions of the radial stress for different values of .

**Figure 3.** Case 1: Distributions of the circumferential stress for different values of .

**Figure 4.** Case 1: Distributions of the electric potential for different values of .

= -1.5 = -0.5 = 0.0001 = 0.5 = 1.5

**Figure 2.** Case 1: Distributions of the radial stress for different values of

**Figure 3.** Case 1: Distributions of the circumferential stress for different values of

values of displacements belong to 1.5


1

1.5

2

2.5

Dimensionless circumferential stress

3

3.5






Dimensionless radial stress





0

inner surface to their minimum value at the outer surface of the FGPM sphere. Maximum

1 1.05 1.1 1.15 1.2 1.25 1.3

1 1.05 1.1 1.15 1.2 1.25 1.3

Dimensionless radius

Dimensionless radius

and minimum values belong to 1.5

.

> = -1.5 = -0.5 = 0.0001 = 0.5 = 1.5

> > .

**Figure 5.** Case 1: Distributions of the radial displacement for different values of .

### *5.1.2. Case 2*

Results of this case are illustrated in Figs. 6 to 9. In this case the imposed electric potential satisfies the electrical boundary conditions at the inner and outer surfaces of the sphere. The maximum electric potentials belong to 1.5 the minimum values of which belong to 1.5 . In this case there is no applied pressure at the inner and outer surfaces of the sphere however the induced compressive radial stresses satisfy the free mechanical boundary conditions. The maximum absolute values of the induced compressive radial stresses belong to the same maximum value of electric potential. Circumferential induced stresses are compressive throughout thickness for different material in-homogeneity parameters . However, for negative parameters the minimum values of circumferential stresses located at the inner surface while for positive parameters their minimum values located at the outer surface of the FGPM sphere. The induced displacement is negative across the thickness for all material parameters. Their minimum values located at the inner and their maximum values located at the outer surfaces of the FGPM sphere. It is interesting to compare the induced radial and circumferential stresses in this case with the residual stresses locked in the sphere during the autofrettage process of spheres made of uniform material. One might conclude that by easily imposing an electric potential there is no need to autofrettage these vessels.

Finite Element Analysis of Functionally Graded Piezoelectric Spheres 77

= -1.5 = -0.5 = 0.0001 = 0.5 = 1.5

> .

.

= -1.5 = -0.5 = 0.0001 = 0.5 = 1.5

results and the only small differences are due to thermal stresses which are not considered

1 1.05 1.1 1.15 1.2 1.25 1.3

1 1.05 1.1 1.15 1.2 1.25 1.3

Dimensionless radius

Dimensionless radius

**Figure 7.** Case 2: Distributions of the circumferential stress for different values of

**Figure 8.** Case 2: Distributions of the electric potential for different values of

by Wang and Xu .

0





Dimensionless displacement






0.1

0.2

0.3

0.4

0.5

Dimensionless electric potential

0.6

0.7

0.8

0.9

1

**Figure 6.** Case 2: Distributions of the radial stress for different values of .

#### **5.2. Validation**

The results of this investigation are validated with the recently published paper by Wang and Xu [12] which is shown in Figs. 10 and 11. There are a very good agreement among the results and the only small differences are due to thermal stresses which are not considered by Wang and Xu .

76 Finite Element Analysis – Applications in Mechanical Engineering

maximum electric potentials belong to 1.5

at the inner surface while for positive parameters

However, for negative parameters

Results of this case are illustrated in Figs. 6 to 9. In this case the imposed electric potential satisfies the electrical boundary conditions at the inner and outer surfaces of the sphere. The

 1.5 . In this case there is no applied pressure at the inner and outer surfaces of the sphere however the induced compressive radial stresses satisfy the free mechanical boundary conditions. The maximum absolute values of the induced compressive radial stresses belong to the same maximum value of electric potential. Circumferential induced stresses are compressive throughout thickness for different material in-homogeneity parameters

surface of the FGPM sphere. The induced displacement is negative across the thickness for all material parameters. Their minimum values located at the inner and their maximum values located at the outer surfaces of the FGPM sphere. It is interesting to compare the induced radial and circumferential stresses in this case with the residual stresses locked in the sphere during the autofrettage process of spheres made of uniform material. One might conclude that

by easily imposing an electric potential there is no need to autofrettage these vessels.

the minimum values of which belong to

the minimum values of circumferential stresses located

.

their minimum values located at the outer

.

**Figure 6.** Case 2: Distributions of the radial stress for different values of

The results of this investigation are validated with the recently published paper by Wang and Xu [12] which is shown in Figs. 10 and 11. There are a very good agreement among the

1 1.05 1.1 1.15 1.2 1.25 1.3

Dimensionless radius

= -1.5 = -0.5 = 0.0001 = 0.5 = 1.5

**5.2. Validation** 







Dimensionless circumferential stress





*5.1.2. Case 2* 

**Figure 7.** Case 2: Distributions of the circumferential stress for different values of .

**Figure 8.** Case 2: Distributions of the electric potential for different values of .

Finite Element Analysis of Functionally Graded Piezoelectric Spheres 79

1.5 . A controlled mesh in which very fine elements

**Figure 11.** Case 2: Comparison of the electric potential distributions with Ref. [12] for homogeneous

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Dimensionless radius

In order to develop the one-dimensional solution to a three-dimensional approach finite element analysis of a sphere subjected to an internal pressure and a uniform temperature field has been carried out using ANSYS finite element software. A three-dimensional element identified by solid 191 is selected because it is an appropriate element for the FGPM structures. Sphere has been divide into eight layers by a controlled mesh system along radius and the mechanical, electrical and thermal properties are functionally defined

are located at the supports where stress concentration existed is employed in this method. However farther from the supports a coarser mesh is dominated. In this work, two cases for

In this case, consider a sphere with two asymmetric simply supported boundary conditions on the outer surface of the sphere as shown in Fig. 12. For this boundary condition dimensionless effective stresses versus normalized radius at two cross sections (i.e. A-A and B-B) are depicted in Fig. 13. Section A-A is selected to pass through supported point on the outer surface of the sphere and section B-B is an arbitrary section as shown in Fig. 12. It can be seen from this figure that the maximum effective stress for the above mentioned sections occur at the inner surface of the sphere and the effective stresses are decreasing with

1.3 . Total dimensionless displacement versus dimensionless radius

piezoelectric hollow sphere.

**5.3. Finite element solution** 

0

0.1

0.2

0.3

0.4

0.5

Dimensionless electric potential

0.6

0.7

0.8

0.9

1

Present Work Wang and Xu [12]

according to power law Eq. (17) for

sphere are considered as follows:

*5.3.1. Three- dimensional sphere* 

increasing radius for

**Figure 9.** Case 2: Distributions of the radial displacement for different values of .

**Figure 10.** Case 2: Comparison of the radial stress distributions with Ref. [12] for homogeneous piezoelectric hollow sphere.

**Figure 11.** Case 2: Comparison of the electric potential distributions with Ref. [12] for homogeneous piezoelectric hollow sphere.

#### **5.3. Finite element solution**

78 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 9.** Case 2: Distributions of the radial displacement for different values of

= -1.5 = -0.5 = 0.0001 = 0.5 = 1.5

1 1.05 1.1 1.15 1.2 1.25 1.3

Dimensionless radius

**Figure 10.** Case 2: Comparison of the radial stress distributions with Ref. [12] for homogeneous

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Dimensionless radius

piezoelectric hollow sphere.


0

0.2

0.4

Dimensionless radial stress

0.6

0.8

1


Dimensionless displacement

.

Present Work Wang and Xu [12] In order to develop the one-dimensional solution to a three-dimensional approach finite element analysis of a sphere subjected to an internal pressure and a uniform temperature field has been carried out using ANSYS finite element software. A three-dimensional element identified by solid 191 is selected because it is an appropriate element for the FGPM structures. Sphere has been divide into eight layers by a controlled mesh system along radius and the mechanical, electrical and thermal properties are functionally defined according to power law Eq. (17) for 1.5 . A controlled mesh in which very fine elements are located at the supports where stress concentration existed is employed in this method. However farther from the supports a coarser mesh is dominated. In this work, two cases for sphere are considered as follows:

#### *5.3.1. Three- dimensional sphere*

In this case, consider a sphere with two asymmetric simply supported boundary conditions on the outer surface of the sphere as shown in Fig. 12. For this boundary condition dimensionless effective stresses versus normalized radius at two cross sections (i.e. A-A and B-B) are depicted in Fig. 13. Section A-A is selected to pass through supported point on the outer surface of the sphere and section B-B is an arbitrary section as shown in Fig. 12. It can be seen from this figure that the maximum effective stress for the above mentioned sections occur at the inner surface of the sphere and the effective stresses are decreasing with increasing radius for 1.3 . Total dimensionless displacement versus dimensionless radius

for 1.3 at two cross sections of A-A and B-B are demonstrated in Fig. 14. As can be seen from this figure the maximum displacement occur at the inner surface of the sphere and displacement value is decreasing with increasing dimensionless radius so that for section A-A, the zero value of displacement at the outer surface satisfies the boundary condition at this point.

Finite Element Analysis of Functionally Graded Piezoelectric Spheres 81

Section A-A Section B-B

**Figure 14.** Total displacement distribution along the radius of asymmetric closed sphere.

are clamped-clamped, clamped-simply and simply-simply supported respectively.

The geometry and loading condition as well as its boundary conditions are shown in Fig. 15. Three different boundary conditions are considered in this case. These boundary conditions

1 1.05 1.1 1.15 1.2 1.25 1.3

Dimensionless radius

The solution obtained by the software clearly indicates the most critical region of the sphere. In the most critical region normalized effective stress distribution and the total displacements are plotted in Figs. 16 and 17 along normalized radius at all node points for the above mentioned three boundary conditions. Fig 16 shows that in general the effective stresses are decreasing along radius to an absolute minimum and then increasing to their maximum values located at the outer surface of the vessel. For simply-simply supported boundary condition this absolute minimum is located near the outer surface of the vessel, however for the clamped-clamped condition it is nearly at the middle surface of the vessel. For the clamped-simply supported condition this minimum is somewhere between the

It has been found that the magnitude of effective stresses at all node points are higher for the clamped-clamped condition and are lower for the simply-simply supported condition. It can be observed from Fig. 17 that the maximum displacements for the three boundary conditions are located at the inner surface and they are decreasing to zero value at the outer surface of the sphere. It is also found that the displacement curve for simply-simply support

*5.3.2. Three- dimensional open sphere* 

0

0.5

1

1.5

2

2.5

Dimensionless displacement

3

3.5

4

condition is higher than other boundary conditions.

previous two cases.

**Figure 12.** A schematic of asymmetric thick-walled sphere with simply-simply supported.

**Figure 13.** Effective stress distribution along the radius of asymmetric closed sphere.

**Figure 14.** Total displacement distribution along the radius of asymmetric closed sphere.

## *5.3.2. Three- dimensional open sphere*

80 Finite Element Analysis – Applications in Mechanical Engineering

 at two cross sections of A-A and B-B are demonstrated in Fig. 14. As can be seen from this figure the maximum displacement occur at the inner surface of the sphere and displacement value is decreasing with increasing dimensionless radius so that for section A-A, the zero value of displacement at the outer surface satisfies the boundary condition at

**Figure 12.** A schematic of asymmetric thick-walled sphere with simply-simply supported.

**Figure 13.** Effective stress distribution along the radius of asymmetric closed sphere.

0.5

1

1.5

2

2.5

Dimensionless effective stress

3

3.5

4

1 1.05 1.1 1.15 1.2 1.25 1.3

Section A-A Section B-B

Dimensionless radius

for 1.3 

this point.

The geometry and loading condition as well as its boundary conditions are shown in Fig. 15. Three different boundary conditions are considered in this case. These boundary conditions are clamped-clamped, clamped-simply and simply-simply supported respectively.

The solution obtained by the software clearly indicates the most critical region of the sphere. In the most critical region normalized effective stress distribution and the total displacements are plotted in Figs. 16 and 17 along normalized radius at all node points for the above mentioned three boundary conditions. Fig 16 shows that in general the effective stresses are decreasing along radius to an absolute minimum and then increasing to their maximum values located at the outer surface of the vessel. For simply-simply supported boundary condition this absolute minimum is located near the outer surface of the vessel, however for the clamped-clamped condition it is nearly at the middle surface of the vessel. For the clamped-simply supported condition this minimum is somewhere between the previous two cases.

It has been found that the magnitude of effective stresses at all node points are higher for the clamped-clamped condition and are lower for the simply-simply supported condition. It can be observed from Fig. 17 that the maximum displacements for the three boundary conditions are located at the inner surface and they are decreasing to zero value at the outer surface of the sphere. It is also found that the displacement curve for simply-simply support condition is higher than other boundary conditions.

Finite Element Analysis of Functionally Graded Piezoelectric Spheres 83

Clamped-Clamped Clamped-Simply Simply-Simply

**Figure 17.** Total displacement distribution along the radius of the open sphere with different boundary

1 1.05 1.1 1.15 1.2 1.25 1.3

Dimensionless radius

In this research, the electro-thermo-mechanical behavior of radially polarized FGPM hollow sphere was investigated. An analytic solution technique was developed for the electrothermo-mechanical problem, where stresses were produced under combined thermomechanical and electrical loading conditions. Variation of normalized stresses, electric potential and displacement of four sets of boundary conditions for different material

stresses and electric potentials satisfy the mechanical and electrical boundary conditions at the inner and outer surfaces of the FGPM sphere. It was concluded that higher absolute values of compressive radial stresses are associated with the higher induced electric potentials throughout the thickness in all cases. It was found that the induced radial and circumferential stresses of an imposed electric potential is similar to the residual stresses locked in the sphere during the autofrettage process of these vessels. Therefore, one might concluded that by easily imposing an electric potential there is no need to autofrettage these vessels. It was interesting to see that the compressive circumferential stresses due to an external pressure were very similar to the induced circumferential stresses resulted from imposing an electric potential. Moreover a three-dimensional finite element analysis of an asymmetric sphere subjected to an internal pressure and a uniform temperature field has been carried out using ANSYS software. In this study closed and opened spheres with different boundary conditions were considered. The finite element analysis indicated that the values of effective stress and total displacement at all node points along the thickness of

were plotted against dimensionless radius. In general, radial

conditions.

**6. Conclusions** 

in-homogeneity parameters

0

0.5

1

1.5

Dimensionless total displacement

2

2.5

3

**Figure 15.** The three-dimensional nite element model for open sphere subjected to internal pressure with clamped-clamped boundary conditions.

**Figure 16.** Effective stresses distribution along the radius of the open sphere with different boundary conditions.

**Figure 17.** Total displacement distribution along the radius of the open sphere with different boundary conditions.

## **6. Conclusions**

82 Finite Element Analysis – Applications in Mechanical Engineering

with clamped-clamped boundary conditions.

conditions.


0

1

2

Dimensionless effective stress

3

4

5

**Figure 15.** The three-dimensional nite element model for open sphere subjected to internal pressure

Clamped-Clamped Clamped-Simply Simply-Simply

**Figure 16.** Effective stresses distribution along the radius of the open sphere with different boundary

1 1.05 1.1 1.15 1.2 1.25 1.3

Dimensionless radius

In this research, the electro-thermo-mechanical behavior of radially polarized FGPM hollow sphere was investigated. An analytic solution technique was developed for the electrothermo-mechanical problem, where stresses were produced under combined thermomechanical and electrical loading conditions. Variation of normalized stresses, electric potential and displacement of four sets of boundary conditions for different material in-homogeneity parameters were plotted against dimensionless radius. In general, radial stresses and electric potentials satisfy the mechanical and electrical boundary conditions at the inner and outer surfaces of the FGPM sphere. It was concluded that higher absolute values of compressive radial stresses are associated with the higher induced electric potentials throughout the thickness in all cases. It was found that the induced radial and circumferential stresses of an imposed electric potential is similar to the residual stresses locked in the sphere during the autofrettage process of these vessels. Therefore, one might concluded that by easily imposing an electric potential there is no need to autofrettage these vessels. It was interesting to see that the compressive circumferential stresses due to an external pressure were very similar to the induced circumferential stresses resulted from imposing an electric potential. Moreover a three-dimensional finite element analysis of an asymmetric sphere subjected to an internal pressure and a uniform temperature field has been carried out using ANSYS software. In this study closed and opened spheres with different boundary conditions were considered. The finite element analysis indicated that the values of effective stress and total displacement at all node points along the thickness of

the open sphere were the highest and lowest for the clamped-clamped condition, respectively.

Finite Element Analysis of Functionally Graded Piezoelectric Spheres 85

[1] P. Destuynder, A few remarks on the controllability of an aeroacoustic model using

[2] H.W. Jiang, F. Schmid, W. Brand, G.R. Tomlinson, Controlling pantograph dynamics

[3] W.Q. Chen, Problems of radially polarized piezoelastic bodies, Int. J. solids struct. 36

[4] D.K. Sinha, Note on the radial deformation of a piezoelectric, polarized spherical shell

[5] A. Ghorbanpour, S. Golabi, M. Saadatfar, Stress and electric potential fields in

[6] M. Saadatfar, A. Rastgoo, Stress in piezoelectric hollow sphere under thermal

[7] H.L. Dai, X. Wang, Thermo-electro-elastic transient response, Int. J. solids struct. 42

[8] H.L. Dai, Y.M. Fu, Electromagnetotransient stress and perturbation of magnetic field vector in transversely isotropic piezoelectric solid sphere, Mater. Sci. Eng. B 129 (2006)

[9] L.H. You, J.J. Zhang, X.Y. You, Elastic analysis of internally pressurized thick-walled spherical pressure vessels of functionally graded materials, Int. J. Press. Vess. Pip. 82

[10] H.J. Ding, H.M. Wang, W.Q. Chen, Analytical solution for a non-homogeneous

[11] A. Ghorbanpour Arani, R. Kolahchi, A.A. Mosallaie Barzoki, Effect of material inhomogeneity on electro-thermo-mechanical behaviors of functionally graded

[12] H.M. Wang, Z.X. Xu, Effect of material inhomogeneity on electromechanical behaviors of functionally graded piezoelectric spherical structures, Comput. Mater. Sci. 48 (2010)

[13] A. Ghorbanpour, M. Salari, H. Khademizadeh, A. Arefmanesh, Magnetothermoelastic

[14] V. Sladek, J. Sladek, Ch. Zhang, Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method, Comput.

[15] J. Sladek, V. Sladek, P. Solek, A. Saez, Dynamic 3D axisymmetric problems in continuously non-homogeneous piezoelectric solids, Int. J. solids struct. 45 (2008) 4523–

[16] Institute of Electrical and Electronics Engineers. Standard on Piezoelectricity, Std (1978)

[17] Y.C. Fungn, Foundations of Solid Mechanics, Prentice-Hall, New York, 1965. [18] H.F. Tiersten, Linear Piezoelectric Plate Vibrations, Plenum Press, New York, 1969

isotropic piezoelectric hollow sphere, Arch. Appl. Mech. 73 (2003) 49–62.

piezoelectric rotating cylinder, J. Appl. Math. Model. 35 (2011) 2771–2789.

problems of FGM spheres, Arch. Appl. Mech. (2010) 189-200.

piezo-devices, Int. J. Holnicki-Szulc. (1999) 53–62.

environment, J. Mech. Sci. Tech, 22 (2008) 1460–1467.

using smart technology, Int. J. Holnicki-Szulc. (1999) 125–132.

with a symmetrical distribution, J. Acoust. Soc. 34 (1962) 1073–1075.

piezoelectric smart spheres, J. Mech. Sci. Tech. 20 (2006) 1920–1933.

**7. References** 

(1998) 4317-4332.

(2005) 1151–1171.

(2005) 347–354.

86–92.

440–445.

4542.

Mater. Sci. 28 (2003) 494-504.

176-1978 IEEE, New York.

## **Appendix A**

$$D\_1 = \frac{\mathcal{Y}\left(\mathcal{C}\_1 + E\_1^2\right) + 2\mathcal{C}\_1 + 2E\_1^2}{\mathcal{C}\_1 + E\_1^2}, \\ D\_2 = \frac{\mathcal{Y}\left(2\mathcal{C}\_2 + 2E\_1E\_2\right) + 2\mathcal{C}\_2 + 2E\_1E\_2 - 2 - 2\mathcal{C}\_3 - 4E\_2^2}{\mathcal{C}\_1 + E\_1^2}, \\ D\_3 = \frac{2E\_2}{\mathcal{C}\_1 + E\_1^2}$$

$$D\_{s} = \frac{\left(\gamma + 1\right)\left(2\mathcal{C}\_{z}a\_{\boldsymbol{\theta}} + \mathcal{C}\_{i}a\_{r} + \mathcal{E}\_{i}\left(2\mathcal{E}\_{z}a\_{\boldsymbol{\theta}} + \mathcal{E}\_{i}a\_{r}\right)\right) + 2\left(2\mathcal{C}\_{z}a\_{\boldsymbol{\theta}} + \mathcal{C}\_{i}a\_{r}\right) + 2\left(\mathcal{E}\_{i} - \mathcal{E}\_{z}\right)\left(2\mathcal{E}\_{z}a\_{\boldsymbol{\theta}} + \mathcal{E}\_{i}a\_{r}\right)\left(\left(1 + \mathcal{C}\_{z}\right)a\_{\boldsymbol{\theta}} + \mathcal{C}\_{z}a\_{r}\right) + \mathcal{E}\_{i}\left(2\mathcal{C}\_{z}a\_{\boldsymbol{\theta}} + \mathcal{E}\_{i}a\_{r}\right)\right)}{\left(\mathcal{C}\_{z} + \mathcal{E}\_{i}^{2}\right)\left(\mathcal{Y} + 1\right)}$$

$$D\_{s} = \frac{\mathcal{Y}\left(4\mathcal{C}\_{z}a\_{\boldsymbol{\theta}} - 2\mathcal{C}\_{i}a\_{r} - 2\mathcal{E}\_{i}\left(2\mathcal{E}\_{z}a\_{\boldsymbol{\theta}} + \mathcal{C}\_{i}a\_{r}\right)\right) - 2\left(2\mathcal{C}\_{z}a\_{\boldsymbol{\theta}} + \mathcal{C}\_{i}a\_{r}\right) - \left(2\mathcal{E}\_{z} - \mathcal{E}\_{z}\right)\left(2\mathcal{E}\_{z}a\_{\boldsymbol{\theta}} + \mathcal{E}\_{i}a\_{r}\right) + \left(2\left(1 + \mathcal{C}\_{z}\right)a\_{\boldsymbol{\theta}} + 2\mathcal{C}\_{z}a\_{r}\right)\right)}{\left(\mathcal{C}\_{z} + \mathcal{E}\_{i}^{2}\right)}$$

## **Appendix B**

*mij ji* )4,...1,( are the coefficient of *K*<sup>1</sup> , *K*<sup>2</sup> , *A*1 and *A*2 which defined as *qEEEqCCm* 2 <sup>12111211</sup> 2 2 , <sup>2</sup> 2 <sup>12121212</sup> 2 2 *qEEEqCCm* , <sup>1</sup> 2 1 31 2 212 1 <sup>13</sup> 2 )1)(1( ))(1(22 *<sup>E</sup> qq EEC DCE <sup>m</sup>* 1 1 2 <sup>12111221</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> *<sup>q</sup> qEEEqCCm* , 1 2 2 <sup>12121222</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> *<sup>q</sup> qEEEqCCm* , <sup>2</sup> 1 2 1 31 2 212 1 <sup>23</sup> 2 )1)(1( ))(1(22 *E qq EEC DCE m* , <sup>1</sup> 1 2 31 <sup>2</sup> *<sup>E</sup> q E m* , )1( 1 )1)(1)(1( 2 2 1 3211 33 *qq DEEE m* , <sup>1</sup> <sup>1</sup> 1 2 41 <sup>2</sup> *<sup>q</sup> <sup>E</sup> q E m* , <sup>2</sup> <sup>1</sup> 2 2 42 <sup>2</sup> *<sup>q</sup> <sup>E</sup> q E m* )1()1)(1)(1( 2 <sup>1</sup> 2 1 1 3211 43 *qq DEEE m* , *m*<sup>41</sup> 0 , *m*<sup>24</sup> 0 , *m*<sup>34</sup> 1 , *m*<sup>44</sup> 1 *ib* )2,1( *<sup>i</sup>* are correspond to cases 1 and 24 which denoted as follows: )1( *b*<sup>1</sup> *<sup>r</sup>* , )( *b*<sup>1</sup> *<sup>r</sup>* 

## **Author details**

A. Ghorbanpour Arani, R. Kolahchi, A. A. Mosalaei Barzoki and A. Loghman *Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran* 

F. Ebrahimi

*Department of Mechanical Engineering, Faculty of Engineering, International University of Imam Khomeini, Qazvin, Iran* 

### **7. References**

84 Finite Element Analysis – Applications in Mechanical Engineering

4 2

5 2

 

2 <sup>12111211</sup> 2 2 , <sup>2</sup>

2

<sup>1</sup>

 

 

*qEEEqCCm*

2 1

*qq*

2 1

3211

*qq*

)1)(1)(1( 2

*DEEE*

212 1 <sup>23</sup> 2 )1)(1(

3211

*DEEE*

2 1

2 1

*qq*

**Author details** 

*Khomeini, Qazvin, Iran* 

*qq*

212 1 <sup>13</sup> 2 )1)(1( ))(1(22 *<sup>E</sup>*

*EEC DCE <sup>m</sup>* 

)1()1)(1)(1(

*EEC DCE*

respectively.

33

43

)( *b*<sup>1</sup> *<sup>r</sup>* 

F. Ebrahimi

**Appendix B** 

*D*

*D*

**Appendix A** 

the open sphere were the highest and lowest for the clamped-clamped condition,

11 1 1 2 12 2 12 3 2 <sup>2</sup> 1 2 2 22 3

 2 1 1 2 1 2 1 12 2 1 3 2

 2 1 1 2 1 2 1 12 2 1 3 2

1 1 4 2 22 2 2 2 2 21 2 *rrr r r C C EE E C C EE E E C C*

<sup>12121212</sup> 2 2 *qEEEqCCm* ,

1

41

*Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran* 

*Department of Mechanical Engineering, Faculty of Engineering, International University of Imam* 

*m*

*C E*

1 1 1 1 1 1

1 *rr r r r C C EE E C C EE E E C C*

2

<sup>12121222</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> *<sup>q</sup> qEEEqCCm* ,

<sup>2</sup> *<sup>E</sup> q E m* ,

 <sup>1</sup> 1 2

> 

31

<sup>2</sup> *<sup>q</sup> <sup>E</sup>*

1 2

*q E*

   

   

 

1

<sup>2</sup> *<sup>q</sup> <sup>E</sup>*

 

2 2

*<sup>r</sup>* ,

*q E*

 

2 2

 

, <sup>2</sup> <sup>1</sup>

*m*

42

 

*C E C E C E*

2 2 <sup>2</sup>

 

*mij ji* )4,...1,( are the coefficient of *K*<sup>1</sup> , *K*<sup>2</sup> , *A*1 and *A*2 which defined as

31

1 2

*m* , <sup>1</sup> <sup>1</sup>

2 <sup>1</sup>

1

 

2

<sup>2</sup>

))(1(22

1

<sup>12111221</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> *<sup>q</sup> qEEEqCCm* ,

31

)1( 1

 

A. Ghorbanpour Arani, R. Kolahchi, A. A. Mosalaei Barzoki and A. Loghman

*ib* )2,1( *<sup>i</sup>* are correspond to cases 1 and 24 which denoted as follows: )1( *b*<sup>1</sup>

*m* , *m*<sup>41</sup> 0 , *m*<sup>24</sup> 0 , *m*<sup>34</sup> 1 , *m*<sup>44</sup> 1

 

*E*

*m* ,

2 2 2 2 2 2 22 4 <sup>2</sup> , , *C E C E C EE C EE C E <sup>E</sup> D D <sup>D</sup>*

1 2 2 2 2 2 2 1

 

1 1

 

*C E*

 

	- [19] A. Manonukul, F.P.E. Dunne, D. Knowles, S. Williams, Multiaxial creep and cyclic plasticity in nickel-base superalloy C263, Int. J. Plasticity, 21 (2005) 1–20.

**Chapter 4** 

© 2012 Ghavami and Khedmati, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Ghavami and Khedmati, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Nonlinear Large Deflection Analysis of** 

Stiffened plates are basic structural members in marine structures as shown in Figure 1, and include also aeronautic and space shuttles among other structures. Due to the simplicity in their fabrication and high strength-to-weight ratio, stiffened plates are also widely used for construction of land based structures such as box girder and plate girder bridges. The stiffened plate has a number of one-sided stiffeners in either one or both directions, the latter configuration being also called a grillage (Figure 2). Ultimate limit state design of Stiffened plates' structures requires accurate knowledge about their behaviour when subjected to

One of the most important loads applied on stiffened plates is the longitudinal in plane axial compression arising for instance from longitudinal bending of the ship hull girder as presented in Figure 3. The need to improve our knowledge of the buckling modes of such plates was emphasised after the collapse of several offshore structures and some ships in Brazil as well as the failure of several box girder bridges in the seventies of the twentieth century, Merrison Committee [1], Crisfield [2], Murray [3], Frieze, et.al. [4]. Stiffened plates are efficient structures, as a large increment of the strength is created by a small addition of weight in the form of stiffeners. However the collapse mechanisms of stiffened plates under predominantly compressive load present a complex engineering problem due to the large number of possible combinations of plate and stiffener geometry, materials, boundary conditions and loading. The design of such structure has to meet several requirements such as minimization of the weight and maximization of the buckling load. Thus, the designer of this structure is confronted with the problem of satisfying two conflicting objectives; such problems are called multi-objective or vector optimisation problems. In general, the objective-functions do not attain their optimum in a common point of the feasible points,

Khosrow Ghavami and Mohammad Reza Khedmati

Additional information is available at the end of the chapter

**Stiffened Plates** 

http://dx.doi.org/10.5772/48368

extreme loading conditions.

Brosowski & Ghavami [5, 6].

**1. Introduction** 


## **Chapter 4**

## **Nonlinear Large Deflection Analysis of Stiffened Plates**

Khosrow Ghavami and Mohammad Reza Khedmati

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48368

## **1. Introduction**

86 Finite Element Analysis – Applications in Mechanical Engineering

loadings. Compos: Part B. 39 (2008) 986–998.

2005.

981.

489–500.

New York, 2001.

Systems, Boston, 2010.

Model. 34 (2010) 343–357.

Sci. Tech. 23 (2009) 45–53.

[19] A. Manonukul, F.P.E. Dunne, D. Knowles, S. Williams, Multiaxial creep and cyclic

[20] H. Martin, ELASTICITY Theory, Applications and Numerics, Elsevier Inc, London,

[21] H.J. Ding, W.Q. Chen, Three Dimensional Problems of Piezoelasticity, Nova Science,

[22] M. Sadeghian, H. Ekhteraei Toussi, Axisymmetric yielding of functionally graded spherical vessel under thermo-mechanical loading, Comput. Mater. Sci. 50 (2011) 975–

[23] A. Salehi-Khojin, N. Jalili, A comprehensive model for load transfer in nanotube reinforced piezoelectric polymeric composites subjected to electro-thermo-mechanical

[24] Zh. Li, Ch. Wang, Ch. Chen, Effective electromechanical properties of transversely isotropic piezoelectric ceramics with microvoids, Comput. Mater. Sci. 27 (2003) 381–392. [25] N. Jalili, Piezoelectric-Based Vibration Control from Macro to Micro/Nano Scale

[26] M.H. Babaei, Z.T. Chen, Analytical solution for the electromechanical behavior of a rotating functionally graded piezoelectric hollow shaft, Arch. Appl. Mech. 78 (2008)

[27] H.L., Dai, L. Hong, Y. M., Fu, X. Xiao, Analytical solution for electro-magneto-thermoelastic behaviors of a functionally graded piezoelectric hollow cylinder, J. Appl. Math.

[28] M. Saadatfar, A.S. Razavi, Piezoelectric hollow cylinder with thermal gradient, J. Mech.

plasticity in nickel-base superalloy C263, Int. J. Plasticity, 21 (2005) 1–20.

Stiffened plates are basic structural members in marine structures as shown in Figure 1, and include also aeronautic and space shuttles among other structures. Due to the simplicity in their fabrication and high strength-to-weight ratio, stiffened plates are also widely used for construction of land based structures such as box girder and plate girder bridges. The stiffened plate has a number of one-sided stiffeners in either one or both directions, the latter configuration being also called a grillage (Figure 2). Ultimate limit state design of Stiffened plates' structures requires accurate knowledge about their behaviour when subjected to extreme loading conditions.

One of the most important loads applied on stiffened plates is the longitudinal in plane axial compression arising for instance from longitudinal bending of the ship hull girder as presented in Figure 3. The need to improve our knowledge of the buckling modes of such plates was emphasised after the collapse of several offshore structures and some ships in Brazil as well as the failure of several box girder bridges in the seventies of the twentieth century, Merrison Committee [1], Crisfield [2], Murray [3], Frieze, et.al. [4]. Stiffened plates are efficient structures, as a large increment of the strength is created by a small addition of weight in the form of stiffeners. However the collapse mechanisms of stiffened plates under predominantly compressive load present a complex engineering problem due to the large number of possible combinations of plate and stiffener geometry, materials, boundary conditions and loading. The design of such structure has to meet several requirements such as minimization of the weight and maximization of the buckling load. Thus, the designer of this structure is confronted with the problem of satisfying two conflicting objectives; such problems are called multi-objective or vector optimisation problems. In general, the objective-functions do not attain their optimum in a common point of the feasible points, Brosowski & Ghavami [5, 6].

© 2012 Ghavami and Khedmati, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Ghavami and Khedmati, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Nonlinear Large Deflection Analysis of Stiffened Plates 89

For the analysis of such structural elements, the theory of orthotropic plate can be used to predict the global buckling stresses but not the local buckling and the interaction between the plate and the stiffeners, for the predominantly in-plane loading. In stiffened plates the initial imperfections due to the fabrication are inevitable. The buckling mechanism of stiffened plates depends, strongly, on the direction of initial bows, i.e. whether they are towards the plate or the stiffener. In the former case, the collapse is sudden due to buckling of the stiffener in contrast to the latter case, where a gradual failure occurs. Despite a substantial amount of theoretical research into the ultimate load behaviour of stiffened plates subjected to predominately in-plane loading, the accuracy and reliability of the predicted collapse load considering all the variables is not yet well confirmed. Specifically, in the available literature, no systematic theoretical and experimental investigation of the geometrical shape of the stiffeners cross-section on the ultimate buckling load behaviour of the stiffened plates, the interaction between the stiffeners and the plate, which was is the

The buckling behaviour of stiffened plates under different loading conditions which has been the topic of the authors investigation, both experimentally and numerically, during last three decades has been reviewed concisely in this chapter. Chen et al. [7] carried out experimental investigations on 12 stiffened plates under in-plane longitudinal compression, purely or in combination with lateral load. The specimens were in different damage conditions: seven "as-built", two "dented" and three "corroded". Hu and Jiang [8] simulated some of the tests made by Chen et. al. [7], using the commercial program ADINA [9] and in-house program VAST [10], both based on the FEM. The former was used to analyse the "as-built" and "dented" stiffened plates, whereas the "corroded" specimens were analysed using VAST [4]. It was found, that in most cases the FEM produced similar responses to those of experimental results up to the loss of structural continuity. Grondin et al. [11] made a parametric study on the buckling behaviour of stiffened plates using the FEM-based commercial program ABAQUS [12]. Sheikh et. al. [13] extended the studies in [11] to investigate the combined effect of in-plane compression and bending using the same program. In these studies, only tee-shape stiffeners, plate aspect ratios, plate-to-stiffener cross-sectional area ratio with different

All the cited studies, either experimentally or numerically, investigated the strength behaviour of longitudinally stiffened plates with specific boundary conditions. The continuity of both plates and stiffeners in thin-walled structures, composed of stiffened plates, leads to an interaction among the adjacent panels. Among the several available experimental investigations, two series of well executed experimental data on longitudinally multi-stiffened steel plates, with and without transversal stiffeners subjected to uniform axial in-plane load carried out to study the buckling and post-buckling up to final failure have been chosen. The first series are those of Ghavami [14] where the influences of stiffener cross-section of the type rectangular (R), L and T, as shown in Figure 4, have been investigated. The spacing of the stiffeners and the presence of rigid transversal stiffeners on the buckling behaviour up to collapse have also been studied. The second series of Tanaka &

objectives of this chapter is being presented.

initial imperfections of the plates were investigated.

**Figure 1.** Some examples of thin-walled structures

**Figure 2.** Structure of stiffened plates of the grillage type

**Figure 3.** In-plane loading of stiffened plates when longitudinal bending of ship hull girder

For the analysis of such structural elements, the theory of orthotropic plate can be used to predict the global buckling stresses but not the local buckling and the interaction between the plate and the stiffeners, for the predominantly in-plane loading. In stiffened plates the initial imperfections due to the fabrication are inevitable. The buckling mechanism of stiffened plates depends, strongly, on the direction of initial bows, i.e. whether they are towards the plate or the stiffener. In the former case, the collapse is sudden due to buckling of the stiffener in contrast to the latter case, where a gradual failure occurs. Despite a substantial amount of theoretical research into the ultimate load behaviour of stiffened plates subjected to predominately in-plane loading, the accuracy and reliability of the predicted collapse load considering all the variables is not yet well confirmed. Specifically, in the available literature, no systematic theoretical and experimental investigation of the geometrical shape of the stiffeners cross-section on the ultimate buckling load behaviour of the stiffened plates, the interaction between the stiffeners and the plate, which was is the objectives of this chapter is being presented.

88 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 1.** Some examples of thin-walled structures

**Figure 2.** Structure of stiffened plates of the grillage type

**Figure 3.** In-plane loading of stiffened plates when longitudinal bending of ship hull girder

The buckling behaviour of stiffened plates under different loading conditions which has been the topic of the authors investigation, both experimentally and numerically, during last three decades has been reviewed concisely in this chapter. Chen et al. [7] carried out experimental investigations on 12 stiffened plates under in-plane longitudinal compression, purely or in combination with lateral load. The specimens were in different damage conditions: seven "as-built", two "dented" and three "corroded". Hu and Jiang [8] simulated some of the tests made by Chen et. al. [7], using the commercial program ADINA [9] and in-house program VAST [10], both based on the FEM. The former was used to analyse the "as-built" and "dented" stiffened plates, whereas the "corroded" specimens were analysed using VAST [4]. It was found, that in most cases the FEM produced similar responses to those of experimental results up to the loss of structural continuity. Grondin et al. [11] made a parametric study on the buckling behaviour of stiffened plates using the FEM-based commercial program ABAQUS [12]. Sheikh et. al. [13] extended the studies in [11] to investigate the combined effect of in-plane compression and bending using the same program. In these studies, only tee-shape stiffeners, plate aspect ratios, plate-to-stiffener cross-sectional area ratio with different initial imperfections of the plates were investigated.

All the cited studies, either experimentally or numerically, investigated the strength behaviour of longitudinally stiffened plates with specific boundary conditions. The continuity of both plates and stiffeners in thin-walled structures, composed of stiffened plates, leads to an interaction among the adjacent panels. Among the several available experimental investigations, two series of well executed experimental data on longitudinally multi-stiffened steel plates, with and without transversal stiffeners subjected to uniform axial in-plane load carried out to study the buckling and post-buckling up to final failure have been chosen. The first series are those of Ghavami [14] where the influences of stiffener cross-section of the type rectangular (R), L and T, as shown in Figure 4, have been investigated. The spacing of the stiffeners and the presence of rigid transversal stiffeners on the buckling behaviour up to collapse have also been studied. The second series of Tanaka &

Endo [15], where the behaviour of stiffened plates have three and two flat bars for longitudinal and transversal stiffeners respectively, were analysed. Besides, owing to the recent progress in the field of finite element method and available powerful FEM programs, it has been possible to assess the structural behaviour of the considered plates and stiffeners subjected to any combination of loads.

Nonlinear Large Deflection Analysis of Stiffened Plates 91

thickness of the plate was t=4.4mm for the longitudinally stiffened plates with one and two rectangular (R), L and T stiffeners, designated as P1R, P1L, P1T and P2R, P2L, P2T respectively. The thickness of the plates, stiffened longitudinally, as for the series II and III but with one or two transversal stiffeners of T sections, P1R1T, P1L1T, P1T1T, P2R1T, P2L1T, P2T1T and P2R2T, P2L2T, P2T2T respectively was equal to 4.8mm. The span between the simple supports for all models was 650mm in both directions. In each group one isotropic plate, P1, P2 was also tested as a reference model. However the supports for the longitudinally stiffened plates, series II and III were not continuously simply supported but were very closely discretized simply supported and those with transversal stiffeners had continuously simply supported boundary conditions. A summary of material properties

**No. Definition Test models I** Unstiffened plate P1, P2

section P1R, P1L, P1T

section P2R, P2L, P2T

P1R1T, P1L1T, P1T1T

P2R1T, P2L1T, P2T1T

P2R2T, P2L2T, P2T2T

**II** Plate with one longitudinal stiffener of R, L and T cross-

**III** Plate with two longitudinal stiffeners of R, L and T cross-

**IV** Plate as series II but with addition of one transversal

**<sup>V</sup>**Plate as series III but with addition of one transversal

**VI** Plate as series II but with addition of two transversal

stiffener at the mid-span

stiffener at the mid-span

stiffeners at 1/3 of span

and test results is given in Table 3.

**Figure 5.** Ghavami's testing rig

**Table 1.** Definition of Ghavami test models

**Series** 

**Figure 4.** Ghavami's test models

Therefore one of the principal aims of this chapter is to present the applicability of the finite element method to simulate test results. The Finite Element Method (FEM) technique is employed to trace a full-range of elastic-plastic behaviour of the stiffened plates. It is seen that the FEM-based software is capable and accurate enough to simulate the test results. With the availability of high memory and high speed PCs', FEM programs become fast and cheap means to predict the buckling and post-buckling behaviour of stiffened plates with different configurations up to collapse. Successful simulations using FEM-based software means, that plate with different dimensions under various types of loading combinations and damages can be studied numerically. Besides, validated simulations using such programs enhance estimation of the ultimate strength analysis of box-like thin-walled structures composed of plates and stiffened plates.

## **2. Ghavami's experiments**

Ghavami [14] tested a total number of 17 plate models of overall dimensions B=L=750 mm in a specially designed testing rig as shown in Figure 5. The models were divided into six series, with their definition and dimensions summarised in Tables 1 and 2. The average thickness of the plate was t=4.4mm for the longitudinally stiffened plates with one and two rectangular (R), L and T stiffeners, designated as P1R, P1L, P1T and P2R, P2L, P2T respectively. The thickness of the plates, stiffened longitudinally, as for the series II and III but with one or two transversal stiffeners of T sections, P1R1T, P1L1T, P1T1T, P2R1T, P2L1T, P2T1T and P2R2T, P2L2T, P2T2T respectively was equal to 4.8mm. The span between the simple supports for all models was 650mm in both directions. In each group one isotropic plate, P1, P2 was also tested as a reference model. However the supports for the longitudinally stiffened plates, series II and III were not continuously simply supported but were very closely discretized simply supported and those with transversal stiffeners had continuously simply supported boundary conditions. A summary of material properties and test results is given in Table 3.

**Figure 5.** Ghavami's testing rig

90 Finite Element Analysis – Applications in Mechanical Engineering

subjected to any combination of loads.

**Figure 4.** Ghavami's test models

structures composed of plates and stiffened plates.

**2. Ghavami's experiments** 

Endo [15], where the behaviour of stiffened plates have three and two flat bars for longitudinal and transversal stiffeners respectively, were analysed. Besides, owing to the recent progress in the field of finite element method and available powerful FEM programs, it has been possible to assess the structural behaviour of the considered plates and stiffeners

Therefore one of the principal aims of this chapter is to present the applicability of the finite element method to simulate test results. The Finite Element Method (FEM) technique is employed to trace a full-range of elastic-plastic behaviour of the stiffened plates. It is seen that the FEM-based software is capable and accurate enough to simulate the test results. With the availability of high memory and high speed PCs', FEM programs become fast and cheap means to predict the buckling and post-buckling behaviour of stiffened plates with different configurations up to collapse. Successful simulations using FEM-based software means, that plate with different dimensions under various types of loading combinations and damages can be studied numerically. Besides, validated simulations using such programs enhance estimation of the ultimate strength analysis of box-like thin-walled

Ghavami [14] tested a total number of 17 plate models of overall dimensions B=L=750 mm in a specially designed testing rig as shown in Figure 5. The models were divided into six series, with their definition and dimensions summarised in Tables 1 and 2. The average


**Table 1.** Definition of Ghavami test models


92 Finite Element Analysis – Applications in Mechanical Engineering

**Table 2.** Dimensions of plate and stiffeners in Ghavami test models

The testing rig was constructed within the Structural and Material Laboratory of PUC-Rio and is shown in Figs. 5 and 6. Out of plane deflections of plates and stiffeners were measured principally by mechanical dial gauges fixed at specific points mounted on the testing rig, as shown in Figure 6. In all models electrical linear strain gauges or rosettes measured the strains. More details on the test rig, test models and the process of the tests can be found in reference [14]. In each test the maximum ultimate collapse stress ult was calculated by dividing the ultimate load Pu to the overall cross-section of the plate Ap and stiffeners As as given by eqn (1):

$$
\sigma\_{\rm ult} = P\_{\rm u} / \left(\mathbf{A}\_{\rm p} + \mathbf{A}\_{\rm s}\right) \tag{1}
$$

Nonlinear Large Deflection Analysis of Stiffened Plates 93

**Maximum shortening**  **Collapse stress** 

 / *ult Yp*

**Maximum deflection**

*Ys* / *W t <sup>o</sup>* max *W t* / max *U t* /

**Test model** 

*E*

**Material properties Measured** 

**Table 3.** Summary of material properties and results for Ghavami test models

**Figure 6.** Stiffened plate model positioned in the Ghavami's testing rig

Tanaka & Endo [15] carried out a series of experimental and numerical investigations on the ultimate compressive strength of plates having three and two flat-bars stiffeners welded longitudinally and transversally respectively. A total of 12 tests were performed. The test

**3. Tanaka & Endos' experiments** 

*Yp* **deflection**

**P1** 1.81 218 ---- 61 278 0.38 42.2 **P1R** 1.81 218 390 69 188 0.31 70.2 **P1L** 1.99 227 270 36 123 0.34 66.5 **P1T** 1.99 227 170 9 33 0.27 60.0 **P2R** 1.95 224 390 25 117 0.41 66.0 **P2L** 2.21 223 270 19 142 0.30 74.0 **P2T** 2.21 223 270 3 128 0.37 74.0 **P2** 1.78 220 ---- 20 121 0.40 48.2 **P1R1T** 1.85 219 326 21 123 0.33 74.0 **P1L1T** 1.91 225 326 27 27 0.48 71.1 **P1T1T** 1.75 219 273 33 121 0.33 72.1 **P2R1T** 1.75 219 326 70 52 0.64 88.6 **P2L1T** 1.89 227 326 40 33 0.60 84.6 **P2T1T** 1.78 220 273 23 30 0.51 89.1 **P2R2T** 1.91 225 326 32 18 0.63 86.2 **P2L2T** 1.89 227 326 28 53 0.56 97.4 **P2T2T** 2.09 218 273 32 16 0.51 103.2

<sup>5</sup> *MPa* 10 *MPa MPa* % % % %

The squash load P was calculated by multiplying the yield stress of the plate sq Yp and the stiffener Ys with their appropriate cross-section areas as eqn (2):

$$\mathbf{P\_{sq}} = \sigma\_{\mathbf{Y}p} \mathbf{A\_p} + \sigma\_{\mathbf{Y}s} \mathbf{A\_s} \tag{2}$$

The test results together with those of maximum initial W0 , final Wmax deflections and inplane shortening Umax are given in Table 3.


Nonlinear Large Deflection Analysis of Stiffened Plates 93

**Table 3.** Summary of material properties and results for Ghavami test models

**Figure 6.** Stiffened plate model positioned in the Ghavami's testing rig

## **3. Tanaka & Endos' experiments**

92 Finite Element Analysis – Applications in Mechanical Engineering

**Table 2.** Dimensions of plate and stiffeners in Ghavami test models

stiffener Ys with their appropriate cross-section areas as eqn (2):

stiffeners As as given by eqn (1):

plane shortening Umax are given in Table 3.

**Plate Longitudinal stiffener Transverse stiffener** *L b t wt wh <sup>f</sup> t <sup>f</sup> b wt t wt h ft t ft b mm mm mm mm mm mm mm mm mm mm mm*

**P1** 650 650 4.4 ---- ---- ---- ---- ---- ---- ---- ---- **P1R** 650 325 4.4 7.0 30.0 ---- ---- ---- ---- ---- ---- **P1L** 650 325 4.4 6.4 30.0 3.9 16.4 ---- ---- ---- ---- **P1T** 650 325 4.4 6.4 30.0 4.8 26.4 ---- ---- ---- ---- **P2R** 650 217 4.4 7.0 30.0 ---- ---- ---- ---- ---- ---- **P2L** 650 216 4.4 6.4 30.0 19.5 16.4 ---- ---- ---- ---- **P2T** 650 217 4.4 6.4 30.0 20.0 26.4 ---- ---- ---- ---- **P2** 650 650 4.8 ---- ---- ---- ---- ---- ---- ---- ---- **P1R1T** 325 325 4.8 5.1 30.0 ---- ---- 4.7 41.1 4.1 35.3 **P1L1T** 325 325 4.8 5.2 30.2 3.4 14.8 4.8 40.4 4.1 34.2 **P1T1T** 325 325 4.8 4.6 30.0 3.8 25.3 4.9 40.4 4.2 35.2 **P2R1T** 325 216 4.8 5.1 30.0 ---- ---- 4.7 40.7 3.8 35.7 **P2L1T** 325 217 4.8 5.1 30.2 17.2 14.6 4.6 40.6 4.1 35.9 **P2T1T** 325 217 4.8 4.7 28.8 13.5 25.0 4.7 39.6 4.1 34.8 **P2R2T** 216 216 4.8 5.0 30.1 ---- ---- 4.7 40.4 4.1 35.7 **P2L2T** 217 217 4.8 5.1 30.0 17.0 14.9 4.7 40.6 4.1 35.5 **P2T2T** 216 216 4.8 4.6 29.8 13.0 24.8 4.8 40.6 4.1 35.5

The testing rig was constructed within the Structural and Material Laboratory of PUC-Rio and is shown in Figs. 5 and 6. Out of plane deflections of plates and stiffeners were measured principally by mechanical dial gauges fixed at specific points mounted on the testing rig, as shown in Figure 6. In all models electrical linear strain gauges or rosettes measured the strains. More details on the test rig, test models and the process of the tests can be found in reference [14]. In each test the maximum ultimate collapse stress ult was calculated by dividing the ultimate load Pu to the overall cross-section of the plate Ap and

The squash load P was calculated by multiplying the yield stress of the plate sq Yp and the

 

The test results together with those of maximum initial W0 , final Wmax deflections and in-

 

ult u p s P/A A (1)

sq Yp p Ys s P AA (2)

**Test model** 

> Tanaka & Endo [15] carried out a series of experimental and numerical investigations on the ultimate compressive strength of plates having three and two flat-bars stiffeners welded longitudinally and transversally respectively. A total of 12 tests were performed. The test

specimen was designed so that the longitudinally stiffened plates located in the middle of whole test specimens could fail. The test specimens were intended to fail by local plate buckling or tripping of longitudinal stiffeners. A typical test rig from the Tanaka & Endo study is shown in Figure 7. A stiffened plate model positioned in their testing rig is presented schematically in Figure 8. To account for the effect of adjacent panels on the collapse behaviour of central panel, three-span models with two adjacent (dummy) stiffened panels and supported by two transverse frames were employed. The thickness of plate and stiffeners in two adjacent panels was 1.2-1.3 times that of plate and stiffeners in the central panel. Table 4, where a=1080mm is the span length of the plate with average plate thickness between t=4.38mm to t=6.15mm, represents geometric and material properties for the Tanaka & En'os' test structures. The boundaries of stiffened plates were continuously simply supported and the in-plane axial compression load was applied longitudinally. The maximum measured initial deflections in the plate were ranging between 0.1-0.4 mm. The ultimate collapse strength and squash load were calculated in the same manner using equations 1 and 2 as those considered by Ghavami [14].

Nonlinear Large Deflection Analysis of Stiffened Plates 95

**Figure 8.** Stiffened plate model positioned in the Tanaka & Endo's testing rig

Since the test specimens in all above-reported experiments, had large deflections and plastic deformations, finite element analyses had to be performed using the software offering combined geometrical and material non-linear capabilities. In this study, the commercially available finite element code, ANSYS [16] was adopted. In the control menu of ANSYS solver, the options of "large deflection" and "arc-length method" are activated. The arclength method is used to trace the non-linear large deflection response of the models.

Both plate and stiffeners are modelled using SHELL43 elements selected from ANSYS library of elements. The SHELL43 element in Figure 9 is a so-called plastic large strain element and categorised in the family of four-node quadrilateral elements. Each node has three translational degrees of freedom in the nodal x, y and z directions as well as three rotational degrees of freedom about the nodal x, y and z-axes. The chosen element allows for elastic, perfectly plastic, with strain hardening or strain softening, large strain and large

A convergence study indicated that in the finite element mesh of isotropic and stiffened plates respectively, assuming 10 ( / ) *a b* mesh divisions along local plate panels and 10

**4. Finite element simulations** 

**4.1. Shell element formulation** 

**Figure 9.** Shell43 element of the ANSYS FEM program

**4.2. Finite element mesh and boundary conditions** 

deflection response [16].


**Table 4.** Geometric and material properties of Tanaka & Endo tests

**Figure 7.** Tanaka & Endo's test model

**Figure 8.** Stiffened plate model positioned in the Tanaka & Endo's testing rig

## **4. Finite element simulations**

94 Finite Element Analysis – Applications in Mechanical Engineering

equations 1 and 2 as those considered by Ghavami [14].

**t (mm)**

**Table 4.** Geometric and material properties of Tanaka & Endo tests

*wh* **(mm)**

*wt* **(mm)**

**D0** 1440 6.15 110.0 9.77 11.26 0.101 234.2 287.1 205.8 **D0A** 1440 5.65 110.0 10.15 10.84 0.250 249.9 196.0 205.8 **D1** 1200 5.95 110.0 10.19 10.79 0.143 253.8 250.9 205.8 **D2** 1560 5.95 110.0 10.19 10.79 0.288 253.8 250.9 205.8 **D3** 1080 1440 5.95 103.5 11.84 8.74 0.312 253.8 326.3 205.8 **D4** 1440 5.95 118.5 7.98 14.85 0.119 253.8 284.2 205.8 **D4A** 1440 5.65 118.5 8.08 14.67 0.379 249.9 274.4 205.8 **D10** 1200 4.38 65.0 4.38 14.84 0.515 442.0 442.0 205.8 **D11** 1200 4.38 90.0 4.38 20.55 0.503 442.0 442.0 205.8 **D12** 1440 4.38 65.0 4.38 14.84 0.523 442.0 442.0 205.8

*w w h t*

*A*<sup>03</sup> **(mm)**

 *Yp* **(MPa)** 

 *Ys* **(MPa)** 

*E* **(GPa)** 

**b (mm)**

**Structure No.** 

**a (mm)**

**Figure 7.** Tanaka & Endo's test model

specimen was designed so that the longitudinally stiffened plates located in the middle of whole test specimens could fail. The test specimens were intended to fail by local plate buckling or tripping of longitudinal stiffeners. A typical test rig from the Tanaka & Endo study is shown in Figure 7. A stiffened plate model positioned in their testing rig is presented schematically in Figure 8. To account for the effect of adjacent panels on the collapse behaviour of central panel, three-span models with two adjacent (dummy) stiffened panels and supported by two transverse frames were employed. The thickness of plate and stiffeners in two adjacent panels was 1.2-1.3 times that of plate and stiffeners in the central panel. Table 4, where a=1080mm is the span length of the plate with average plate thickness between t=4.38mm to t=6.15mm, represents geometric and material properties for the Tanaka & En'os' test structures. The boundaries of stiffened plates were continuously simply supported and the in-plane axial compression load was applied longitudinally. The maximum measured initial deflections in the plate were ranging between 0.1-0.4 mm. The ultimate collapse strength and squash load were calculated in the same manner using

Since the test specimens in all above-reported experiments, had large deflections and plastic deformations, finite element analyses had to be performed using the software offering combined geometrical and material non-linear capabilities. In this study, the commercially available finite element code, ANSYS [16] was adopted. In the control menu of ANSYS solver, the options of "large deflection" and "arc-length method" are activated. The arclength method is used to trace the non-linear large deflection response of the models.

## **4.1. Shell element formulation**

Both plate and stiffeners are modelled using SHELL43 elements selected from ANSYS library of elements. The SHELL43 element in Figure 9 is a so-called plastic large strain element and categorised in the family of four-node quadrilateral elements. Each node has three translational degrees of freedom in the nodal x, y and z directions as well as three rotational degrees of freedom about the nodal x, y and z-axes. The chosen element allows for elastic, perfectly plastic, with strain hardening or strain softening, large strain and large deflection response [16].

**Figure 9.** Shell43 element of the ANSYS FEM program

## **4.2. Finite element mesh and boundary conditions**

A convergence study indicated that in the finite element mesh of isotropic and stiffened plates respectively, assuming 10 ( / ) *a b* mesh divisions along local plate panels and 10

mesh divisions across them is sufficient to capture accurately the buckling and plastic collapse behaviour. Respectively *a* and *b* represent the length and breadth of local plate panels. In order to model the stiffener's web and flange, respectively 6 to 7 and 5 to 6 elements are sufficient. However, the purpose of this study was to simulate the testing results and finer meshes were therefore used. In the case of Tanaka & Endo tests, to reduce the number of mesh divisions and speed up the time of analysis, a rational assumption was made. The transverse stiffeners or frames for the case of Tanaka & Endo tests were not modelled for simplicity; instead the nodes on the line of attachment of the transverse stiffeners were constrained from translational movement out of plate plane. Furthermore, the translational movements of these nodes along the axis perpendicular to the line of attachment of transverse stiffeners were coupled with each other. Transverse frames were modelled in the case of Ghavami's tests. In both Ghavami and Tanaka & Endo's tests, the stiffened plates were loaded in axial compression along the stiffeners. Also in their tests the simply supported boundary conditions were assumed in the models. Figs. 10 and 11 show typical finite element models with the simulated boundary conditions, used for the analysis of Tanaka & Endo test specimens and Ghavami P2L2T test specimen (as an example).

Nonlinear Large Deflection Analysis of Stiffened Plates 97

references [14] and [15], a special procedure was employed. Uniform lateral pressure was applied first on the stiffened plate model and a linear elastic finite element analysis was carried out. This analysis was repeated in a trial and error sequence of calculations so that the magnitude of maximum deflection of plate reached that, measured by Ghavami. It is assumed that this procedure would simulate both the residual welding stresses and initial geometrical imperfection. After satisfying this condition, the information concerning the coordinates of nodal points, element coordinates and boundary conditions was transferred to a new finite element mesh for the geometrical and material non-linear response analysis

It should be emphasised that the pattern of initial deflections induced in the Ghavami's specimens [14] were nearly matching the pattern produced by this procedure. For the case of Tanaka & Endo's tests, first an eigenvalue buckling analysis was made using ANSYS, in order to capture the three-wave buckling mode deflection of the specimens [15]. Then the deflection pattern in this mode was scaled to the same pattern with the maximum magnitude of initial deflection, A03, (Table 4) before testing, which has been reported by Tanaka & Endo [15]. Nonlinear response analysis under the action of longitudinal in-plane

It is evident that strain-hardening effect has an important influence on the non-linear behaviour of isotropic and stiffened plates respectively. The degree of such an influence is a function of several factors including plate and stiffener slenderness. In this chapter, experimental material behaviour for both plate and stiffener are modelled as a bi- linear elastic-plastic with strain-hardening rate of *E* / 65 , as seen in Figure 12. *E* is modulus of elasticity of material. This value was obtained through an extensive study of elastic-plastic large deflection analyses made by Khedmati [17] and presents an average value for the strain-hardening rate. The application of *E* / 65 predicts the collapse load with sufficient

under the action of longitudinal in-plane compression.

compression was performed on this model.

**Figure 12.** Assumed bi-linear behaviour for the material

**4.4. Material properties** 

**Figure 10.** Finite element model of Tanaka & Endo's test specimens

**Figure 11.** Finite element model of Ghavami's P2L2T test specimen

## **4.3 Imperfections**

Welding residual stresses were not modelled specifically in this study. However, in order to simulate the complex pattern of residual welding stresses and initial deflections stated in references [14] and [15], a special procedure was employed. Uniform lateral pressure was applied first on the stiffened plate model and a linear elastic finite element analysis was carried out. This analysis was repeated in a trial and error sequence of calculations so that the magnitude of maximum deflection of plate reached that, measured by Ghavami. It is assumed that this procedure would simulate both the residual welding stresses and initial geometrical imperfection. After satisfying this condition, the information concerning the coordinates of nodal points, element coordinates and boundary conditions was transferred to a new finite element mesh for the geometrical and material non-linear response analysis under the action of longitudinal in-plane compression.

It should be emphasised that the pattern of initial deflections induced in the Ghavami's specimens [14] were nearly matching the pattern produced by this procedure. For the case of Tanaka & Endo's tests, first an eigenvalue buckling analysis was made using ANSYS, in order to capture the three-wave buckling mode deflection of the specimens [15]. Then the deflection pattern in this mode was scaled to the same pattern with the maximum magnitude of initial deflection, A03, (Table 4) before testing, which has been reported by Tanaka & Endo [15]. Nonlinear response analysis under the action of longitudinal in-plane compression was performed on this model.

**Figure 12.** Assumed bi-linear behaviour for the material

#### **4.4. Material properties**

96 Finite Element Analysis – Applications in Mechanical Engineering

mesh divisions across them is sufficient to capture accurately the buckling and plastic collapse behaviour. Respectively *a* and *b* represent the length and breadth of local plate panels. In order to model the stiffener's web and flange, respectively 6 to 7 and 5 to 6 elements are sufficient. However, the purpose of this study was to simulate the testing results and finer meshes were therefore used. In the case of Tanaka & Endo tests, to reduce the number of mesh divisions and speed up the time of analysis, a rational assumption was made. The transverse stiffeners or frames for the case of Tanaka & Endo tests were not modelled for simplicity; instead the nodes on the line of attachment of the transverse stiffeners were constrained from translational movement out of plate plane. Furthermore, the translational movements of these nodes along the axis perpendicular to the line of attachment of transverse stiffeners were coupled with each other. Transverse frames were modelled in the case of Ghavami's tests. In both Ghavami and Tanaka & Endo's tests, the stiffened plates were loaded in axial compression along the stiffeners. Also in their tests the simply supported boundary conditions were assumed in the models. Figs. 10 and 11 show typical finite element models with the simulated boundary conditions, used for the analysis

of Tanaka & Endo test specimens and Ghavami P2L2T test specimen (as an example).

**Figure 10.** Finite element model of Tanaka & Endo's test specimens

**Figure 11.** Finite element model of Ghavami's P2L2T test specimen

Welding residual stresses were not modelled specifically in this study. However, in order to simulate the complex pattern of residual welding stresses and initial deflections stated in

**4.3 Imperfections** 

It is evident that strain-hardening effect has an important influence on the non-linear behaviour of isotropic and stiffened plates respectively. The degree of such an influence is a function of several factors including plate and stiffener slenderness. In this chapter, experimental material behaviour for both plate and stiffener are modelled as a bi- linear elastic-plastic with strain-hardening rate of *E* / 65 , as seen in Figure 12. *E* is modulus of elasticity of material. This value was obtained through an extensive study of elastic-plastic large deflection analyses made by Khedmati [17] and presents an average value for the strain-hardening rate. The application of *E* / 65 predicts the collapse load with sufficient accuracy. Poisson's ratio, , in all experimental investigation and FEM analysis was considered to be equal to 0.3.

Nonlinear Large Deflection Analysis of Stiffened Plates 99

**Collapse mode** 

( )

*ult FEM ult EXPERIMENT*

**P1R** 1.11

**P1L** 0.84

**P1T** 0.87

**P2R** 1.05

**P2L** 1.22

transverse frame

**Table 5.** Summary of finite element simulation results for some of Ghavami test models without

**Test model** 

( )

## **5. Large deflection behaviour of the tested plates**

A summary of the results obtained through the finite element simulation of the experimental research carried out by Ghavami is given in Tables 5 and 6 and that of Tanaka & Endo is presented in Table 7. In these tables, the collapse modes from FEM analyses are, also, presented. A comparison of the experimental and those obtained results from FEM results present a very good agreement. The maximum differences varied between 16 percents and 22 percents for the series II and III (Table 5) of Ghavami's experimental result. These two extreme differences are related to the plates with L shape stiffener, which does not have a symmetrical geometrical shape. The simple assumption considered in the FEM simulation of complex pattern of initial imperfections (including both initial deflections and welding residual stresses) inherent in the experimental investigation, in addition to not having perfect simply supported boundary conditions in these two series must have led to those higher discrepancies. It should be emphasized that it was possible to trace the curve of average stress-average strain relationship for any combination of plate and stiffener. Finite element simulation results for Ghavami's test models without transverse frame show that the collapse has occurred following the buckling instability of local plate panels (Table 5). This was well predicted by FEM for test specimen P2R with only 5 percent difference. Detailed information concerning the behaviour of each of the Ghavami's test specimens are well documented in the References [18-22].

In the analysis of Tanaka & Endo's tests, the longitudinally stiffened plate located in the middle of the test specimens were simulated assuming all edges straight and having simply supported conditions. The same boundary conditions were considered in FEM analysis. In such cases, finite element simulation results, described also well the interactive buckling of plates and stiffeners in most of the cases, (Table 7). The smallest value of stiffener web height-to-web thickness ratio belongs to model D3, while the biggest value of this ratio corresponds to models D4 and D4A. Model D3 has failed due to local deformation in the plate, while in the case of models D4 and D4A the collapse has been produced by large plastic deformations both in the plate and stiffeners. Interactive buckling in both plate and stiffeners can be observed in other models, where the level of plastic deformations, in the plate varies among them. The ultimate strength predicted by FEM are well consistent as compared with those obtained by Tanaka & Endo [15]. This could be related to the initial deflection of the test specimens which was presented in FEM with a good accuracy.

A summary of results for three tests from each series of VI, V and VI that had perfect simply supported boundary is presented in Table 6. It can be noted that the difference between FEM and those of experimental results had only a difference of up to 5 percent. In the following, the results of FEM for P1R1T, P2R1T and P2L2T of the Ghavami's models with transverse frame are discussed in details.


well documented in the References [18-22].

transverse frame are discussed in details.

**5. Large deflection behaviour of the tested plates** 

, in all experimental investigation and FEM analysis was

A summary of the results obtained through the finite element simulation of the experimental research carried out by Ghavami is given in Tables 5 and 6 and that of Tanaka & Endo is presented in Table 7. In these tables, the collapse modes from FEM analyses are, also, presented. A comparison of the experimental and those obtained results from FEM results present a very good agreement. The maximum differences varied between 16 percents and 22 percents for the series II and III (Table 5) of Ghavami's experimental result. These two extreme differences are related to the plates with L shape stiffener, which does not have a symmetrical geometrical shape. The simple assumption considered in the FEM simulation of complex pattern of initial imperfections (including both initial deflections and welding residual stresses) inherent in the experimental investigation, in addition to not having perfect simply supported boundary conditions in these two series must have led to those higher discrepancies. It should be emphasized that it was possible to trace the curve of average stress-average strain relationship for any combination of plate and stiffener. Finite element simulation results for Ghavami's test models without transverse frame show that the collapse has occurred following the buckling instability of local plate panels (Table 5). This was well predicted by FEM for test specimen P2R with only 5 percent difference. Detailed information concerning the behaviour of each of the Ghavami's test specimens are

In the analysis of Tanaka & Endo's tests, the longitudinally stiffened plate located in the middle of the test specimens were simulated assuming all edges straight and having simply supported conditions. The same boundary conditions were considered in FEM analysis. In such cases, finite element simulation results, described also well the interactive buckling of plates and stiffeners in most of the cases, (Table 7). The smallest value of stiffener web height-to-web thickness ratio belongs to model D3, while the biggest value of this ratio corresponds to models D4 and D4A. Model D3 has failed due to local deformation in the plate, while in the case of models D4 and D4A the collapse has been produced by large plastic deformations both in the plate and stiffeners. Interactive buckling in both plate and stiffeners can be observed in other models, where the level of plastic deformations, in the plate varies among them. The ultimate strength predicted by FEM are well consistent as compared with those obtained by Tanaka & Endo [15]. This could be related to the initial

deflection of the test specimens which was presented in FEM with a good accuracy.

A summary of results for three tests from each series of VI, V and VI that had perfect simply supported boundary is presented in Table 6. It can be noted that the difference between FEM and those of experimental results had only a difference of up to 5 percent. In the following, the results of FEM for P1R1T, P2R1T and P2L2T of the Ghavami's models with

accuracy. Poisson's ratio,

considered to be equal to 0.3.

**Table 5.** Summary of finite element simulation results for some of Ghavami test models without transverse frame

Nonlinear Large Deflection Analysis of Stiffened Plates 101

Collapse mode

**Tanaka & Endo Present**

( )

( )

*ult FEM ult EXPERIMENT*

**Structure No.** 

( )

*ult FEM ult EXPERIMENT*

**D0** 0.977 1.014

**D0A** 1.028 1.065

**D1** 0.869 0.911

**D2** 0.936 0.944

**D3** 0.860 0.853

**D4** 0.792 0.866

**D4A** 0.866 0.960

**Table 7.** Summary of results for some of Tanaka & Endo tests

( )

**Table 6.** Summary of finite element simulation results for some of Ghavami test models with transverse frame


**Table 7.** Summary of results for some of Tanaka & Endo tests

( )

*ult FEM ult EXPERIMENT*

**P1R1T** 1.05

**P2R1T** 1.02

**P2L2T** 1.02

frame

**Table 6.** Summary of finite element simulation results for some of Ghavami test models with transverse

**Test model** 

( )

**Collapse mode** 

### **5.1. P1R1T Ghavami model**

The relative undimensional average stress-average strain relationship obtained by FEM analysis for P1R1T model is shown in Figure 13. The P1R1T model failed because of torsional buckling and plastic failure mechanism of the longitudinal stiffener (R). The torsional failure of the stiffener is induced in the FEM model shortly before the collapse of the model due to work softening as can be seen in Figure 13. A comparison between the collapse modes of the experimental model, Figure 14 (left) and that of FEM analysis, Figure 14 (right) is presented. It can be observed that the simulation of plate deformations by FEM analysis is almost identical to the failure mode occurred in the test specimen. The work hardening of the model started at about y =0.8 and reached the ultimate buckling stress at y =1.0 (, y is the average strain and the yield strain respectively). The ultimate buckling strength of this model is about 80 percent of the plate yield strength, as can be seen in Figure 13 in turn it is close to the experimental results presented in Table 3. The FEM result overestimated the experimental one by only 5 percent. This mainly could be related to the discrepancy in the consideration of initial welding and initial deflection in REM analysis.

Nonlinear Large Deflection Analysis of Stiffened Plates 103

As it can be seen from the relative average stress-average strain relationship of P2R1T model (Figure 15), the work hardening of the test model started at about y =0.8 and reached the ultimate buckling stress at y =0,93 percent in relation to the plate material yield strength. Then the work softening or unloading started at y =1.0 together with the local plastic deformations in the post-ultimate buckling region. The P2R1T model failed under axial compression load due to the buckling in both plate and stiffeners. Such a failure was predominant in upper part of the transverse T frame, as can be observed in Figure 16 (left). In the lower part of the transverse T stiffener, the plastic deformation in the plate and stiffeners was not very large. The comparison of FEM results with that of the experimental one, presented in Figure 16 present a relatively perfect prediction of the ultimate buckling modes. The FEM result overestimated the experimental one by only 2 percent. This can be also related

mainly to consideration of the initial welding and initial deflection in the FEM analysis.

**Figure 15.** Average stress-average strain relationship and spread of yielding at collapse and final step

**Figure 16.** Deflected mode at collapse for Ghavami P2R1T model obtained by experiment (left) and

**5.2. P2R1T Ghavami model** 

of calculation for Ghavami P2R1T model

FEA (right)

**Figure 13.** Average stress-average strain relationship and spread of yielding at collapse and final step of calculation for Ghavami P1R1T model

**Figure 14.** Deflected mode at collapse for Ghavami P1R1T model obtained by experiment (left) and FEA (right)

### **5.2. P2R1T Ghavami model**

102 Finite Element Analysis – Applications in Mechanical Engineering

The relative undimensional average stress-average strain relationship obtained by FEM analysis for P1R1T model is shown in Figure 13. The P1R1T model failed because of torsional buckling and plastic failure mechanism of the longitudinal stiffener (R). The torsional failure of the stiffener is induced in the FEM model shortly before the collapse of the model due to work softening as can be seen in Figure 13. A comparison between the collapse modes of the experimental model, Figure 14 (left) and that of FEM analysis, Figure 14 (right) is presented. It can be observed that the simulation of plate deformations by FEM analysis is almost identical to the failure mode occurred in the test specimen. The work hardening of the model started at about y =0.8 and reached the ultimate buckling stress at y =1.0 (, y is the average strain and the yield strain respectively). The ultimate buckling strength of this model is about 80 percent of the plate yield strength, as can be seen in Figure 13 in turn it is close to the experimental results presented in Table 3. The FEM result overestimated the experimental one by only 5 percent. This mainly could be related to the discrepancy in the consideration of initial welding and initial deflection in REM analysis.

**Figure 13.** Average stress-average strain relationship and spread of yielding at collapse and final step

**Figure 14.** Deflected mode at collapse for Ghavami P1R1T model obtained by experiment (left) and

**5.1. P1R1T Ghavami model** 

of calculation for Ghavami P1R1T model

FEA (right)

As it can be seen from the relative average stress-average strain relationship of P2R1T model (Figure 15), the work hardening of the test model started at about y =0.8 and reached the ultimate buckling stress at y =0,93 percent in relation to the plate material yield strength. Then the work softening or unloading started at y =1.0 together with the local plastic deformations in the post-ultimate buckling region. The P2R1T model failed under axial compression load due to the buckling in both plate and stiffeners. Such a failure was predominant in upper part of the transverse T frame, as can be observed in Figure 16 (left). In the lower part of the transverse T stiffener, the plastic deformation in the plate and stiffeners was not very large. The comparison of FEM results with that of the experimental one, presented in Figure 16 present a relatively perfect prediction of the ultimate buckling modes. The FEM result overestimated the experimental one by only 2 percent. This can be also related mainly to consideration of the initial welding and initial deflection in the FEM analysis.

**Figure 15.** Average stress-average strain relationship and spread of yielding at collapse and final step of calculation for Ghavami P2R1T model

**Figure 16.** Deflected mode at collapse for Ghavami P2R1T model obtained by experiment (left) and FEA (right)

### **5.3. P2L2T Ghavami model**

As it can be seen in Figure 17 which presents the relative average stress-average strain, relationship of P2R2T model, a small work hardening started at about y =0.88 of the plate yield stress and reached the ultimate buckling stress of 100 percent. Then a plastic deformation started at the y =1.0 up to y =1.7 generating several local plate. After this stage the work softening or unloading started with the expansion of local plastic deformations in the post-ultimate buckling region. The P2R2T model finally failed due to the buckling induced in both plate and longitudinal L stiffeners in the centre of the stiffened plate as can be noted well in Figure 18 (left). The P2L2T model showed a high strength under in-plane compression load. The FEM deflected form in Figure 18 (right) simulated well the experimental results. The FEM result overestimated the experimental one by only 2 percent as can be seen in Table 6. This could be related principally to the initial welding and initial deflection.

Nonlinear Large Deflection Analysis of Stiffened Plates 105

**6. Large deflection behaviour of Stiffened plates subjected to combined** 

For the stiffened plates in the bottom structure of ships, the basic load case for buckling

transverse compression arising from the bending of double bottom under lateral

The continuous plate was assumed to be simply-supported along the stiffener lines with no out-of-plane deflection. In reality, however, the stiffener is also subjected to lateral pressure, and it may collapse prior to the failure of the panels. The focus of the present chapter is concentrated on the buckling and plastic collapse behaviour of continuously stiffened plates subjected to combined biaxial compression and lateral pressure with the main objective of identification of the collapse modes of the plates subjected to mentioned combination of

A series of elasto-plastic large deflection FEM analyses is performed on continuous stiffened plates with flat-bar, tee-bar, and angle-bar stiffeners of the same flexural rigidity. The buckling/plastic collapse behaviour and ultimate strength of stiffened plates are hereby assessed so that both the material and geometrical nonlinearities are taken into account.

Local plate panels with length, *a* , of 2400 mm and breadth, *b* , of 800 mm are considered, and their thickness, *t* , changes from 13mm, 15mm, and 20 mm. Yield stress of the material,

 *<sup>Y</sup>* , is taken as 313.6 MPa, and bilinear stress-strain relationship is assumed with the kinematical strain-hardening rate of *E* /65, where *E* is Young's modulus of the material. *E* is considered as 205.8GPa. The cross-sectional geometries of stiffeners are given in Table 8. In each group, the stiffeners have the same moment of inertia. A triple span-double bay model is applied for the analysis of buckling/plastic collapse behaviour of continuous stiffened plate with symmetrical stiffeners (ABDC in Figure 20). When a stiffener has an unsymmetrical geometry, a triple span-triple bay model is used (ABFE in Figure 20) [23].

design consists of the following loads applied simultaneously (Figure 19): longitudinal compression arising from the overall hull girder bending,

local bending arising from the direct action of lateral pressure.

**in-plane compression and lateral pressure** 

**Figure 19.** Basic loads applied on ship stiffened plates

pressure, and

loading condition.

**Figure 17.** Average stress-average strain relationship and spread of yielding at collapse and final step of calculation for Ghavami P2L2T model

**Figure 18.** Deflected mode at collapse for Ghavami P2L2T model obtained by experiment (left) and FEA (right)

## **6. Large deflection behaviour of Stiffened plates subjected to combined in-plane compression and lateral pressure**

For the stiffened plates in the bottom structure of ships, the basic load case for buckling design consists of the following loads applied simultaneously (Figure 19):


**Figure 19.** Basic loads applied on ship stiffened plates

104 Finite Element Analysis – Applications in Mechanical Engineering

As it can be seen in Figure 17 which presents the relative average stress-average strain, relationship of P2R2T model, a small work hardening started at about y =0.88 of the plate yield stress and reached the ultimate buckling stress of 100 percent. Then a plastic deformation started at the y =1.0 up to y =1.7 generating several local plate. After this stage the work softening or unloading started with the expansion of local plastic deformations in the post-ultimate buckling region. The P2R2T model finally failed due to the buckling induced in both plate and longitudinal L stiffeners in the centre of the stiffened plate as can be noted well in Figure 18 (left). The P2L2T model showed a high strength under in-plane compression load. The FEM deflected form in Figure 18 (right) simulated well the experimental results. The FEM result overestimated the experimental one by only 2 percent as can be seen in Table 6. This could be related principally to the initial welding and

**Figure 17.** Average stress-average strain relationship and spread of yielding at collapse and final step

**Figure 18.** Deflected mode at collapse for Ghavami P2L2T model obtained by experiment (left) and

**5.3. P2L2T Ghavami model** 

initial deflection.

FEA (right)

of calculation for Ghavami P2L2T model

The continuous plate was assumed to be simply-supported along the stiffener lines with no out-of-plane deflection. In reality, however, the stiffener is also subjected to lateral pressure, and it may collapse prior to the failure of the panels. The focus of the present chapter is concentrated on the buckling and plastic collapse behaviour of continuously stiffened plates subjected to combined biaxial compression and lateral pressure with the main objective of identification of the collapse modes of the plates subjected to mentioned combination of loading condition.

A series of elasto-plastic large deflection FEM analyses is performed on continuous stiffened plates with flat-bar, tee-bar, and angle-bar stiffeners of the same flexural rigidity. The buckling/plastic collapse behaviour and ultimate strength of stiffened plates are hereby assessed so that both the material and geometrical nonlinearities are taken into account.

Local plate panels with length, *a* , of 2400 mm and breadth, *b* , of 800 mm are considered, and their thickness, *t* , changes from 13mm, 15mm, and 20 mm. Yield stress of the material, *<sup>Y</sup>* , is taken as 313.6 MPa, and bilinear stress-strain relationship is assumed with the kinematical strain-hardening rate of *E* /65, where *E* is Young's modulus of the material. *E* is considered as 205.8GPa. The cross-sectional geometries of stiffeners are given in Table 8. In each group, the stiffeners have the same moment of inertia. A triple span-double bay model is applied for the analysis of buckling/plastic collapse behaviour of continuous stiffened plate with symmetrical stiffeners (ABDC in Figure 20). When a stiffener has an unsymmetrical geometry, a triple span-triple bay model is used (ABFE in Figure 20) [23].

106 Finite Element Analysis – Applications in Mechanical Engineering


Nonlinear Large Deflection Analysis of Stiffened Plates 107

(4)

(3)

(5)

The lateral pressure ranging from 0 to 60 metres water head initially is applied up to a specified value always perpendicularly to the plate surface. Then biaxial compression is

Three types of initial imperfections as described in the following are accounted for:

100 *<sup>p</sup>*


**6.1. Plates with flat-bar stiffener subjected to combined longitudinal** 

and spread of yielding at ultimate strength are presented in Figure 23.

buckling of plate with three buckling half waves.

The characteristics of the collapse behaviour can be summarised as follows:

Average stress-average strain relationships for continuous stiffened plates with flat-bar stiffeners subjected to combined longitudinal compression and variable levels of lateral pressure, are shown in Figure 22 for the plate thickness of *t* =13 mm. The deflection mode



<sup>0</sup> sin sin


 <sup>0</sup> sin 1000 *<sup>s</sup> a x <sup>W</sup>*

 <sup>0</sup> sin 1000 *<sup>w</sup> a x <sup>h</sup>*

*t mx <sup>y</sup> <sup>W</sup>*

 *a b*

*a*

*a*

exerted proportionally by uniform forced displacements.

where *m* is the number of buckling half-waves in the plate,

The welding residual stresses are not considered.

**Figure 21.** Initial imperfections in the stiffened plate models

**compression and lateral pressure** 

21(b)):

**Table 8.** Cross-sectional geometries of stiffeners

**Figure 20.** Stiffened plate model for FEM analysis

The considered boundary conditions are as follows:


The lateral pressure ranging from 0 to 60 metres water head initially is applied up to a specified value always perpendicularly to the plate surface. Then biaxial compression is exerted proportionally by uniform forced displacements.

Three types of initial imperfections as described in the following are accounted for:


$$\mathcal{W}\_{p0} = \frac{t}{100} \sin \frac{m \pi \chi}{a} \sin \frac{\pi \chi}{b} \tag{3}$$

where *m* is the number of buckling half-waves in the plate,

106 Finite Element Analysis – Applications in Mechanical Engineering

**Table 8.** Cross-sectional geometries of stiffeners

**Figure 20.** Stiffened plate model for FEM analysis

triple bay model).

The considered boundary conditions are as follows:

perpendicular directions is assumed to be uniform.

 Periodically continuous conditions are imposed at the same y-coordinate along the transverse edges (i.e. along AC and BD in double bay model and along AE and BF in

 Symmetry conditions are imposed along the longitudinal edges of double bay model (i.e. along AB and CD). But periodically continuous conditions are defined at the same x-coordinate along the longitudinal edges of triple bay model (i.e. along AB and EF). Although transverse frames are not modelled, the out-of-plane deformations of plate and stiffener are restrained along the junction lines of them and the transverse frame. To consider the plate continuity, in-plane movement of the plate edges in their - initial deflection in the stiffener with the maximum magnitude of *a* /1000 (Figure 21(b)):

$$\mathcal{W}\_{s0} = \frac{a}{1000} \sin \frac{\pi \chi}{a} \tag{4}$$


$$
\phi\_0 h\_w = \frac{a}{1000} \sin \frac{\pi \chi}{a} \tag{5}
$$

The welding residual stresses are not considered.

**Figure 21.** Initial imperfections in the stiffened plate models

## **6.1. Plates with flat-bar stiffener subjected to combined longitudinal compression and lateral pressure**

Average stress-average strain relationships for continuous stiffened plates with flat-bar stiffeners subjected to combined longitudinal compression and variable levels of lateral pressure, are shown in Figure 22 for the plate thickness of *t* =13 mm. The deflection mode and spread of yielding at ultimate strength are presented in Figure 23.

The characteristics of the collapse behaviour can be summarised as follows:



Nonlinear Large Deflection Analysis of Stiffened Plates 109

compression. With a further increase in the applied lateral pressure, however, the deteriorating effect of lateral pressure, i.e. enhancing yielding at stiffener becomes more

Average stress-average strain relationships for continuous stiffened plates with tee-bar stiffeners of type 2 subjected to combined longitudinal compression and variable levels of lateral pressure, are shown in Figure 24(a) for the plate thickness of *t* =13 mm. Fundamental collapse behaviours and ultimate strength of stiffened plates with tee-bar stiffeners are almost the same as those for the flat-bar stiffener, but strength reduction in the post-ultimate range is smaller comparing with Figure 22(b). This is because the horizontal bending rigidity

**Figure 24.** Comparison of average stress-average strain relationships for a continuous stiffened plate

Average stress-average strain relationships and collapse modes obtained for the continuous stiffened plates with angle-bar stiffeners are shown in Figure 24(b) and Figure 25,

Unlike the flat-bar or tee-bar stiffeners having symmetrical cross-sectional shape, the anglebar stiffener deflects to the same horizontal and vertical directions in all adjacent spans (Figure 25). This flexural-torsional deflection of stiffener clamped at both ends constrains the panel deformation, resulting in larger ultimate strength and smaller strength reduction in

**6.3. Plates with angle-bar stiffeners subjected to combined longitudinal** 

under combined longitudinal thrust and lateral pressure (plate: 2400x800x13 mm)

the post-ultimate range than those for flat-bar or tee-bar stiffeners.

predominant and the ultimate strength starts to decrease considerably.

**compression and lateral pressure** 

of tee-bar is much greater than that of flat-bar.

**compression and lateral pressure** 

respectively, for the plate thickness of *t* =13 mm.

**6.2. Plates with tee-bar stiffener subjected to combined longitudinal** 


**Figure 22.** Comparison of average stress-average strain relationships for a continuous stiffened plate under combined longitudinal thrust and lateral pressure (plate: 2400x800x13 mm)

**Figure 23.** Change in the deflection mode at ultimate strength for a continuous stiffened plate under combined longitudinal thrust and lateral pressure (plate: 2400x800x13 mm, stiffener: flat-bar of type 2)

For plates with flat-bar stiffeners of type 1 having smaller flexural rigidity, as the plate thickness is increased, the ultimate strength is increased with the increase of lateral pressure up to a certain value. This is because the collapse mode changes from Eulerian buckling mode to a clamped mode in which the plate itself exhibits a higher resistance to longitudinal compression. With a further increase in the applied lateral pressure, however, the deteriorating effect of lateral pressure, i.e. enhancing yielding at stiffener becomes more predominant and the ultimate strength starts to decrease considerably.

## **6.2. Plates with tee-bar stiffener subjected to combined longitudinal compression and lateral pressure**

108 Finite Element Analysis – Applications in Mechanical Engineering

deformation of stiffener gets decreased.

deformation is produced in the stiffener web.




**Figure 22.** Comparison of average stress-average strain relationships for a continuous stiffened plate

**Figure 23.** Change in the deflection mode at ultimate strength for a continuous stiffened plate under combined longitudinal thrust and lateral pressure (plate: 2400x800x13 mm, stiffener: flat-bar of type 2)

For plates with flat-bar stiffeners of type 1 having smaller flexural rigidity, as the plate thickness is increased, the ultimate strength is increased with the increase of lateral pressure up to a certain value. This is because the collapse mode changes from Eulerian buckling mode to a clamped mode in which the plate itself exhibits a higher resistance to longitudinal

under combined longitudinal thrust and lateral pressure (plate: 2400x800x13 mm)

stiffened plate is increased with a decrease in the post-ultimate strength.

Average stress-average strain relationships for continuous stiffened plates with tee-bar stiffeners of type 2 subjected to combined longitudinal compression and variable levels of lateral pressure, are shown in Figure 24(a) for the plate thickness of *t* =13 mm. Fundamental collapse behaviours and ultimate strength of stiffened plates with tee-bar stiffeners are almost the same as those for the flat-bar stiffener, but strength reduction in the post-ultimate range is smaller comparing with Figure 22(b). This is because the horizontal bending rigidity of tee-bar is much greater than that of flat-bar.

**Figure 24.** Comparison of average stress-average strain relationships for a continuous stiffened plate under combined longitudinal thrust and lateral pressure (plate: 2400x800x13 mm)

## **6.3. Plates with angle-bar stiffeners subjected to combined longitudinal compression and lateral pressure**

Average stress-average strain relationships and collapse modes obtained for the continuous stiffened plates with angle-bar stiffeners are shown in Figure 24(b) and Figure 25, respectively, for the plate thickness of *t* =13 mm.

Unlike the flat-bar or tee-bar stiffeners having symmetrical cross-sectional shape, the anglebar stiffener deflects to the same horizontal and vertical directions in all adjacent spans (Figure 25). This flexural-torsional deflection of stiffener clamped at both ends constrains the panel deformation, resulting in larger ultimate strength and smaller strength reduction in the post-ultimate range than those for flat-bar or tee-bar stiffeners.

Nonlinear Large Deflection Analysis of Stiffened Plates 111

**Figure 27.** Change in the deflection mode at ultimate strength for a continuous stiffened plate under combined transverse thrust and lateral pressure (plate: 2400x800x13 mm, stiffener: flat-bar of type 2)

**6.5. Stiffened plates subjected to combined biaxial compression and lateral** 

A series of FEM analyses is performed on a continuous stiffened plate with flat-bar stiffeners subjected to combined biaxial compression and lateral pressure. The results are shown in Figure 28. The dotted lines are loading paths for different ratios of applied biaxial displacements. The solid line is the obtained envelope of all loading paths representing the

**Figure 28.** Interaction curves for a continuous stiffened plate subjected to combined biaxial thrust and

lateral pressure (plate: 2400x800 mm, stiffener: flat-bar of type 2)

**pressure** 

ultimate strength interaction curve.

**Figure 25.** Change in the deflection mode at ultimate strength for a continuous stiffened plate under combined longitudinal thrust and lateral pressure (plate: 2400x800x13 mm, stiffener: angle-bar of type 2)

It is to be noted here that although an angle-bar stiffener is quite effective from the viewpoint of buckling/plastic collapse strength, it should be carefully used from the view point of fatigue strength [24].

## **6.4. Stiffened plates subjected to combined transverse compression and lateral pressure**

The results for the continuous stiffened plates with flat-bar stiffeners of type 2 subjected to combined transverse compression are shown in Figs. 26 and 27.

**Figure 26.** Comparison of average stress-average strain relationships for a continuous stiffened plate under combined transverse thrust and lateral pressure

When lateral pressure is small, the local rectangular panels collapse as if they were simplysupported along the edges, accompanied by some rotation of stiffeners. With an increase in lateral pressure, the collapse mode changes from the simply-supported mode to the alledges clamped mode. These behaviours are basically the same as those observed for continuous plate simply-supported along stiffener lines. Since the stiffener is not subjected to compression, its deflection is small compared to the panel deflection.

#### Nonlinear Large Deflection Analysis of Stiffened Plates 111

110 Finite Element Analysis – Applications in Mechanical Engineering

point of fatigue strength [24].

**pressure** 

**Figure 25.** Change in the deflection mode at ultimate strength for a continuous stiffened plate under combined longitudinal thrust and lateral pressure (plate: 2400x800x13 mm, stiffener: angle-bar of type 2)

It is to be noted here that although an angle-bar stiffener is quite effective from the viewpoint of buckling/plastic collapse strength, it should be carefully used from the view

**6.4. Stiffened plates subjected to combined transverse compression and lateral** 

The results for the continuous stiffened plates with flat-bar stiffeners of type 2 subjected to

**Figure 26.** Comparison of average stress-average strain relationships for a continuous stiffened plate

When lateral pressure is small, the local rectangular panels collapse as if they were simplysupported along the edges, accompanied by some rotation of stiffeners. With an increase in lateral pressure, the collapse mode changes from the simply-supported mode to the alledges clamped mode. These behaviours are basically the same as those observed for continuous plate simply-supported along stiffener lines. Since the stiffener is not subjected

to compression, its deflection is small compared to the panel deflection.

combined transverse compression are shown in Figs. 26 and 27.

under combined transverse thrust and lateral pressure

**Figure 27.** Change in the deflection mode at ultimate strength for a continuous stiffened plate under combined transverse thrust and lateral pressure (plate: 2400x800x13 mm, stiffener: flat-bar of type 2)

## **6.5. Stiffened plates subjected to combined biaxial compression and lateral pressure**

A series of FEM analyses is performed on a continuous stiffened plate with flat-bar stiffeners subjected to combined biaxial compression and lateral pressure. The results are shown in Figure 28. The dotted lines are loading paths for different ratios of applied biaxial displacements. The solid line is the obtained envelope of all loading paths representing the ultimate strength interaction curve.

**Figure 28.** Interaction curves for a continuous stiffened plate subjected to combined biaxial thrust and lateral pressure (plate: 2400x800 mm, stiffener: flat-bar of type 2)

It is seen that each interaction curve basically consists of two parts; a semi-horizontal region in which the stiffened plate behaves as if it were under combined transverse compression and lateral pressure, and a semi-vertical region where the behaviour as in the case of combined longitudinal compression and lateral pressure is dominant.

Nonlinear Large Deflection Analysis of Stiffened Plates 113

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College, London.

24(9): 353-369.

Inc.: Pawtucket, RI.

164 (in Japanese).

Hiroshima University.

## **7. Conclusion**

Basing the results of this chapter on the analysis of 29 experimental investigation, on stiffened steel plates subjected to uniform axial compression load up to final failure, by the Finite Element program ANSYS, the following conclusions may be drawn. The selected element SHELL43, could trace full-range, elastic-plastic behaviour of the stiffened plates. The capability of the non-linear FEM to perform the analysis of stiffened plates has been demonstrated through the accurate simulation of the Ghavami and Tanaka & Endo tests. Although some simplifying assumptions for the simulation of initial imperfections and residual welding stresses were made for reducing the calculation volume and speeding up the analysis, the accuracy of the collapse load obtained through FEM simulations is relatively in good consistency with the test results. The differences were higher in cases of not having perfect simply supported boundary conditions as in series II and III of Ghavami's test. It was shown also, that obtaining deflection mode is possible at any step of loading. This allows predicting the local buckling of stiffened plates with relatively good precision.

For small value of lateral pressure, the local panel and stiffener tend to collapse in a simplysupported mode. With an increase in the applied pressure, they are likely to fail in a clamped mode. Angle-bar stiffener has larger stiffening effects than those of flat-bar and teebar stiffeners having the same flexural rigidity, from the view point of ultimate strength.

## **Author details**

Khosrow Ghavami *Department of Civil Engineering, Pontificia Universidade Católica (PUC-Rio), Rio de Janeiro, Brazil* 

Mohammad Reza Khedmati *Faculty of Marine Technology, Amirkabir University of Technology, Tehran, Iran* 

## **Acknowledgement**

The authors would like to thank the sponsoring organizations for their financial supports and special thanks are also due to the students who executed the projects through the years.

## **8. References**

[1] Merrison Committee (1973) Inquiry into Basis of Design and Methods of Erection of Steel Box Girder Bridges. Report of the Committee – Appendix 1: Interim Design and Workmanship Rules. Her Majesty's Stationary Office, London.

Corresponding Author

[2] Crisfield MA (1975) Full-range Analysis of Steel Plates Stiffened Plating under Uniaxial Compression. Proc. Civ. Engrs. 59(2): 595-624.

112 Finite Element Analysis – Applications in Mechanical Engineering

**7. Conclusion** 

**Author details** 

Khosrow Ghavami

Mohammad Reza Khedmati

**Acknowledgement** 

**8. References** 

Corresponding Author

 

combined longitudinal compression and lateral pressure is dominant.

predicting the local buckling of stiffened plates with relatively good precision.

*Faculty of Marine Technology, Amirkabir University of Technology, Tehran, Iran* 

Workmanship Rules. Her Majesty's Stationary Office, London.

It is seen that each interaction curve basically consists of two parts; a semi-horizontal region in which the stiffened plate behaves as if it were under combined transverse compression and lateral pressure, and a semi-vertical region where the behaviour as in the case of

Basing the results of this chapter on the analysis of 29 experimental investigation, on stiffened steel plates subjected to uniform axial compression load up to final failure, by the Finite Element program ANSYS, the following conclusions may be drawn. The selected element SHELL43, could trace full-range, elastic-plastic behaviour of the stiffened plates. The capability of the non-linear FEM to perform the analysis of stiffened plates has been demonstrated through the accurate simulation of the Ghavami and Tanaka & Endo tests. Although some simplifying assumptions for the simulation of initial imperfections and residual welding stresses were made for reducing the calculation volume and speeding up the analysis, the accuracy of the collapse load obtained through FEM simulations is relatively in good consistency with the test results. The differences were higher in cases of not having perfect simply supported boundary conditions as in series II and III of Ghavami's test. It was shown also, that obtaining deflection mode is possible at any step of loading. This allows

For small value of lateral pressure, the local panel and stiffener tend to collapse in a simplysupported mode. With an increase in the applied pressure, they are likely to fail in a clamped mode. Angle-bar stiffener has larger stiffening effects than those of flat-bar and teebar stiffeners having the same flexural rigidity, from the view point of ultimate strength.

*Department of Civil Engineering, Pontificia Universidade Católica (PUC-Rio), Rio de Janeiro, Brazil* 

The authors would like to thank the sponsoring organizations for their financial supports and special thanks are also due to the students who executed the projects through the years.

[1] Merrison Committee (1973) Inquiry into Basis of Design and Methods of Erection of Steel Box Girder Bridges. Report of the Committee – Appendix 1: Interim Design and

	- [20] Conci A. (1983) Instabilidade Ate O Colapso De Placas De Aco Enrijecidas Em Dual Direcoes. MSc thesis, Pontificia Universidade Catolica, Rio de Janeiro.

**Chapter 5** 

© 2012 Wu, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Wu, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**3D Nonlinear Finite Element Plastic Analysis of** 

Cylindrical vessels with nozzles are common structural components in many industries, such as power engineering, petrochemical, etc. Under the applied pressure and piping loads, a high stress concentration is caused by the geometric discontinuity. The connection region of the vessel and nozzle will become the weakest location of the entire structure. Therefore, it is necessary to have an accurate design method for the structure. The plastic limit design method is one possibility. In design, a gross plastic deformation is prevented by restricting the allowable load relative to the plastic limit load of the vessels. Therefore, the key step in limit design is to determine the plastic limit load of the vessels under different

The plastic limit load estimate for practical engineering materials (with strain hardening and geometrical strengthening) can be determined by the twice-elastic-slope (TES) criterion [1]. Many approaches to determine the plastic limit load have been contributed by a number of authors employing analytical, experimental and finite element methods for components under internal pressure, and nozzle or branch pipe loading. Ellyin[2] [3] reported experimental results for the elastic-plastic behavior and plastic limit loads of five teeshaped cylinder-cylinder intersections under internal pressure, and in-plane or out-of-theplane moment. The results indicated that the out-of-plane loading case was the critical one. Schoreder[4] provided experimental limit branch moment loads on 4-in. ANSI B16.9 Tees using different limit load criteria. The results of the study showed that the branch moment capacity for the tee models was greater than the theoretical limit load of the equivalent nominal size straight pipe. Junker[5] performed inelastic finite element analyses to estimate the limit moment for a cylindrical vessel with a nozzle subjected to in-plane and out-of-plane moment loadings. The results indicated that the predicted limit moment levels agree to within 10% with the experimental results and that the finite element

**Cylindrical Vessels Under In-Plane Moment** 

Additional information is available at the end of the chapter

B.H. Wu

http://dx.doi.org/10.5772/46166

**1. Introduction** 

loads.


## **3D Nonlinear Finite Element Plastic Analysis of Cylindrical Vessels Under In-Plane Moment**

B.H. Wu

114 Finite Element Analysis – Applications in Mechanical Engineering

Universidade Catolica, Rio de Janeiro.

Naval Arch. 85:121-128 (in Japanese).

[20] Conci A. (1983) Instabilidade Ate O Colapso De Placas De Aco Enrijecidas Em Dual

[21] Rocha SAS. (1982) Comportamento Ultimo De Placas Enrijecidas. MSc thesis, Pontificia

[22] Ghavami K. (1986) The Collapse of Continuously Welded Stiffened Plates Subjected to Uniaxil Compression Load. In Proc. Inelastic Behaviour of Plates and Shells, Simp. Rio de Janeiro, 1985, eds. L. Bevilacqua, R. Feijoo & R. Valid, Springer Berlin: 404-415. [23] Yao T, Fujikubo M, Yanagihara D, Irisawa M. (1998) Consideration on FEM Modelling for Buckling/Plastic Collapse Analysis of Stiffened Plates. Trans. of the West-Japan Soc.

[24] Kawano H, Kuramoto Y, Sakai D, Hashimoto K, Inoue S, Fushimi A, Hagiwara K. (1992) Some Considerations on Basic Behaviour of Asymmetric Sectional Frame Under Uniform Pressure. Trans. of the West-Japan Soc. Naval Arch. 83:161-166 (in Japanese).

Direcoes. MSc thesis, Pontificia Universidade Catolica, Rio de Janeiro.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/46166

## **1. Introduction**

Cylindrical vessels with nozzles are common structural components in many industries, such as power engineering, petrochemical, etc. Under the applied pressure and piping loads, a high stress concentration is caused by the geometric discontinuity. The connection region of the vessel and nozzle will become the weakest location of the entire structure. Therefore, it is necessary to have an accurate design method for the structure. The plastic limit design method is one possibility. In design, a gross plastic deformation is prevented by restricting the allowable load relative to the plastic limit load of the vessels. Therefore, the key step in limit design is to determine the plastic limit load of the vessels under different loads.

The plastic limit load estimate for practical engineering materials (with strain hardening and geometrical strengthening) can be determined by the twice-elastic-slope (TES) criterion [1]. Many approaches to determine the plastic limit load have been contributed by a number of authors employing analytical, experimental and finite element methods for components under internal pressure, and nozzle or branch pipe loading. Ellyin[2] [3] reported experimental results for the elastic-plastic behavior and plastic limit loads of five teeshaped cylinder-cylinder intersections under internal pressure, and in-plane or out-of-theplane moment. The results indicated that the out-of-plane loading case was the critical one. Schoreder[4] provided experimental limit branch moment loads on 4-in. ANSI B16.9 Tees using different limit load criteria. The results of the study showed that the branch moment capacity for the tee models was greater than the theoretical limit load of the equivalent nominal size straight pipe. Junker[5] performed inelastic finite element analyses to estimate the limit moment for a cylindrical vessel with a nozzle subjected to in-plane and out-of-plane moment loadings. The results indicated that the predicted limit moment levels agree to within 10% with the experimental results and that the finite element

© 2012 Wu, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Wu, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

method gives a reasonably accurate determination of limit moments for cylindrical vessels with nozzles under in-plane and out-of-plane moments. Moffat[6] performed an experimental study of branch connections subjected to external moment loadings. Further, in 1991, Moffat et al.[7] provided extensive numerical results for the effective stress factor of branch junctions under internal pressure and external moment loads. An empirical formula was presented using polynomial equations. Rodabaugh[8] [9] contributed a valuable review of limit loads for pipe connections in pressure vessels and piping. In the review, a comprehensive overview of pipe connections was provided. Other studies [10]~[13] carried out important works on plastic limit analysis of cylindrical vessels under external nozzle loadings.

3D Nonlinear Finite Element Plastic Analysis of Cylindrical Vessels Under In-Plane Moment 117

 D(mm) L(mm) L' (mm) T(mm) d(mm) l(mm) t(mm) d/ D t /T D / T L1 500 1000 500 8 86 1000 3 0.172 0.375 62.5 L2 500 1000 500 8 123 1000 4 0.246 0.50 62.5 L3 500 1000 500 8 214 1000 5 0.428 0.625 62.5

The materials of the cylinder and the nozzle are Q235-A (low carbon steel, similar to A36-77) and 20# (low carbon steel, similar to A106-80 GrA), respectively. Detailed chemical composition and mechanical properties of materials are given in Reference [14]. Figure 2

The local strains and nozzle displacements under in-plane moment on nozzle were measured to obtain the elastic stresses and deformation characteristics of the model vessels.The in-plane moment is applied as a force at the end of the nozzle, and the moment

Strain values at the typical measurement points of the cylinders and nozzles were measured using two element strain rosettes. Strain gages were mounted in the axial direction on the outside and inside surfaces for the cylinders, and on the outside surface only for the nozzles.

Displacement values under in-plane moment (longitudinal loading direction) for the selected measurement points on the nozzle were measured using mechanical displacement sensors to obtain the load-displacement relationship in the elastic and plastic stages of the nozzles. A total of three displacement sensors were installed at locations along the nozzle

Figure 4 is a photograph of test vessel No. L1 during the test. The figure shows an obvious

Figure 3 illustrates the detailed locations of the strain gages for test vessel No. L2.

length direction. The locations of the displacement sensors are indicated in Figure1.

longitudinal deformation of the nozzle under the in-plane moment on the nozzle.

shows the engineering stress-strain curve of materials for Q235-A and 20# Steel.

**Table 1.** Dimensions of model vessels

**Figure 2.** The curve of engineering stress-strain

arm is the distance to the surface of the vessel.

**2.2. Experimental setup** 

The objective of this paper is to determine the plastic limit moment by both experiment and finite element analysis for cylindrical vessels under in-plane moment loading on the nozzle. Based on these results, a parametric analysis is carried out and an empirical formula is proposed.

## **2. Experimental study**

## **2.1. Model vessels**

Three model vessels with different d/D ratios were designed and fabricated for the experimental study. Every model vessel consisted of a cylinder, nozzle, and flanges for fixing and loading bars. Figure 1 shows a typical configuration, and dimensions for the model vessels are listed in Table 1.

**Figure 1.** Arrangement of model vessels (mm)


**Table 1.** Dimensions of model vessels

116 Finite Element Analysis – Applications in Mechanical Engineering

nozzle loadings.

**2. Experimental study** 

model vessels are listed in Table 1.

**Figure 1.** Arrangement of model vessels (mm)

**2.1. Model vessels** 

proposed.

method gives a reasonably accurate determination of limit moments for cylindrical vessels with nozzles under in-plane and out-of-plane moments. Moffat[6] performed an experimental study of branch connections subjected to external moment loadings. Further, in 1991, Moffat et al.[7] provided extensive numerical results for the effective stress factor of branch junctions under internal pressure and external moment loads. An empirical formula was presented using polynomial equations. Rodabaugh[8] [9] contributed a valuable review of limit loads for pipe connections in pressure vessels and piping. In the review, a comprehensive overview of pipe connections was provided. Other studies [10]~[13] carried out important works on plastic limit analysis of cylindrical vessels under external

The objective of this paper is to determine the plastic limit moment by both experiment and finite element analysis for cylindrical vessels under in-plane moment loading on the nozzle. Based on these results, a parametric analysis is carried out and an empirical formula is

Three model vessels with different d/D ratios were designed and fabricated for the experimental study. Every model vessel consisted of a cylinder, nozzle, and flanges for fixing and loading bars. Figure 1 shows a typical configuration, and dimensions for the The materials of the cylinder and the nozzle are Q235-A (low carbon steel, similar to A36-77) and 20# (low carbon steel, similar to A106-80 GrA), respectively. Detailed chemical composition and mechanical properties of materials are given in Reference [14]. Figure 2 shows the engineering stress-strain curve of materials for Q235-A and 20# Steel.

**Figure 2.** The curve of engineering stress-strain

#### **2.2. Experimental setup**

The local strains and nozzle displacements under in-plane moment on nozzle were measured to obtain the elastic stresses and deformation characteristics of the model vessels.The in-plane moment is applied as a force at the end of the nozzle, and the moment arm is the distance to the surface of the vessel.

Strain values at the typical measurement points of the cylinders and nozzles were measured using two element strain rosettes. Strain gages were mounted in the axial direction on the outside and inside surfaces for the cylinders, and on the outside surface only for the nozzles. Figure 3 illustrates the detailed locations of the strain gages for test vessel No. L2.

Displacement values under in-plane moment (longitudinal loading direction) for the selected measurement points on the nozzle were measured using mechanical displacement sensors to obtain the load-displacement relationship in the elastic and plastic stages of the nozzles. A total of three displacement sensors were installed at locations along the nozzle length direction. The locations of the displacement sensors are indicated in Figure1.

Figure 4 is a photograph of test vessel No. L1 during the test. The figure shows an obvious longitudinal deformation of the nozzle under the in-plane moment on the nozzle.

3D Nonlinear Finite Element Plastic Analysis of Cylindrical Vessels Under In-Plane Moment 119

Figure 5 indicates the elastic-plastic load deformation response at the three measurement

The intersection area of the nozzle and cylinder also produced an evident plastic deformation: the root of the nozzle moved into the vessel wall in the radial direction; as the cylinder yielded on the compression side of the cylinder and nozzle. Figure 6 shows an actual deformation state for test vessels No.L1and No.L2 under in-plane moment of 4.2kN.m

Experimental plastic limit moments were obtained by the use of load versus displacement plots of the measurement points on the nozzle and load against strain curves of the key gauges located near the junction of the cylinder and nozzle. The plastic limit load is defined by applying the twice elastic slope criterion provided by the ASME Boiler and Pressure

Figure 7 illustrates some typical load-displacement curves and corresponding limit moment

Figure 8 shows some typical load-strain curves and relevant limit moment of the strain gage

**2.3. Experimental results** 

and 7.5kN.m.

Vessel Code[1].

for the sensor No.3.

No.6 as an example for the experiment vessels.

*2.3.2. Plastic limit moment* 

*2.3.1. Plastic deformation behavior* 

points on the nozzle for model vessel No. L2.

**Figure 5.** Load and elastic-plastic deformation response of nozzle

**Figure 3.** Locations of steain gages for model No. L2

**Figure 4.** Photo of model No L1 during the test

## **2.3. Experimental results**

118 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 3.** Locations of steain gages for model No. L2

**Figure 4.** Photo of model No L1 during the test

## *2.3.1. Plastic deformation behavior*

Figure 5 indicates the elastic-plastic load deformation response at the three measurement points on the nozzle for model vessel No. L2.

**Figure 5.** Load and elastic-plastic deformation response of nozzle

The intersection area of the nozzle and cylinder also produced an evident plastic deformation: the root of the nozzle moved into the vessel wall in the radial direction; as the cylinder yielded on the compression side of the cylinder and nozzle. Figure 6 shows an actual deformation state for test vessels No.L1and No.L2 under in-plane moment of 4.2kN.m and 7.5kN.m.

#### *2.3.2. Plastic limit moment*

Experimental plastic limit moments were obtained by the use of load versus displacement plots of the measurement points on the nozzle and load against strain curves of the key gauges located near the junction of the cylinder and nozzle. The plastic limit load is defined by applying the twice elastic slope criterion provided by the ASME Boiler and Pressure Vessel Code[1].

Figure 7 illustrates some typical load-displacement curves and corresponding limit moment for the sensor No.3.

Figure 8 shows some typical load-strain curves and relevant limit moment of the strain gage No.6 as an example for the experiment vessels.

3D Nonlinear Finite Element Plastic Analysis of Cylindrical Vessels Under In-Plane Moment 121

**Figure 8.** Load-strains curves for test models (Strains gage No.6)

displacement measuring or strain measuring.

Table 2 is a summary and comparison of the experimental plastic limit moment for the three model vessels. From the table, it is seen that the limit moments obtained by two measurement methods (displacement measuring and strain measuring) are in agreement. The maximum difference between the limit moments obtained from the two methods is 9.76%. For the same model vessel, the limit moments obtained using different measuring points also are in good agreement regardless of whether the measuring method is

**Figure 7.** Load-displacement curves for test models (Sensor No.3)

**Figure 8.** Load-strains curves for test models (Strains gage No.6)

**Figure 6.** Load-deformation of the test vessels

**Figure 7.** Load-displacement curves for test models (Sensor No.3)

Table 2 is a summary and comparison of the experimental plastic limit moment for the three model vessels. From the table, it is seen that the limit moments obtained by two measurement methods (displacement measuring and strain measuring) are in agreement. The maximum difference between the limit moments obtained from the two methods is 9.76%. For the same model vessel, the limit moments obtained using different measuring points also are in good agreement regardless of whether the measuring method is displacement measuring or strain measuring.



3D Nonlinear Finite Element Plastic Analysis of Cylindrical Vessels Under In-Plane Moment 123

The boundary conditions for the numerical simulation are the same as those used in the experimental portion of this study. One end of the cylinder is fixed, while the other end is free. All nodes on the symmetry section (longitudinal plane of the vessel) are constrained against displacement in the direction perpendicular to the symmetry plane. The FEA models were loaded a force at the end of the nozzle along the longitudinal direction of the vessels like in the experiment. For the purpose of comparing the results of the element analysis with those of the experiments, the same materials and loading increments as those of the test vessels were used. As a nonliner geometry analysis, the multilinear isotropic type of hardening behavior is used.A multi-linear elastic-plastic material model defined by nine

Material Point 1 2 3 4 5 6 7 8 9

For the same loading condition as the experiment, the nozzle produced an obvious bending deformation and a depression of the cylinder on the compression side (in the longitudinal section) occurs. Figure 10 shows the local deformation characteristics which are consistent

Figure 11 indicates the load-displacement plots of simulation vessels from the sensor No.3

At the same location as that of the experiment, load strain curves were plated in Figure 12, and corresponding limit moments for simulation vessels of strain gage No.6 were also

with those of the experiment (see Figure 4) for model vessel No. L2.

and corresponding limit moments on the nozzles.

355 357 378 403 432 463 485 510 519

1830 12232 20620 32698 49778 74576 96572 129445 158602

319 334 376 415 464 512 553 584 612

1477 16600 29567 48896 77066 113430 157103 198948 262455

**3.2. Boundary and loading conditions** 

points, as given in Table 3, was employed.

Q235-A Stress

20# Stress

**Table 3.** Material model

**3.3. Analysis results** 

*3.3.1. Plastic deformation* 

*3.3.2. Plastic limit moment* 

shown in Figure 12.

(MPa)

Strain (με)

(MPa)

Strain (με)

**Table 2.** Experimental results of plastic limit moment (kN.m)

## **3. Finite element analysis**

## **3.1. Models and mesh**

A static nonlinear finite element analysis of the experimental model vessels was carried out using the ANSYS code [15].Three-dimensional isoperimetric solid elements defined by eight nodes were used to generate the FEA mesh of the cylinder, nozzle and weld seam between the cylinder and nozzle. Two elements were used across the thickness of the shell.Due to the symmetry of the structure and loading, only one-half of the test vessel was modeled. Figure 9 illustrates the finite element mesh of the model vessel No.L2.

**Figure 9.** Finite element mesh for model vessel No.L2

## **3.2. Boundary and loading conditions**

122 Finite Element Analysis – Applications in Mechanical Engineering

By load-strain method Strain gage

**Table 2.** Experimental results of plastic limit moment (kN.m)

9 illustrates the finite element mesh of the model vessel No.L2.

**Figure 9.** Finite element mesh for model vessel No.L2

No.1

Strain gage No.6

Strain gage No.11

Strain gage No.16

Difference of the average value (%) 9.76 6.32 2.03

A static nonlinear finite element analysis of the experimental model vessels was carried out using the ANSYS code [15].Three-dimensional isoperimetric solid elements defined by eight nodes were used to generate the FEA mesh of the cylinder, nozzle and weld seam between the cylinder and nozzle. Two elements were used across the thickness of the shell.Due to the symmetry of the structure and loading, only one-half of the test vessel was modeled. Figure

By load-displacement method

**3. Finite element analysis** 

**3.1. Models and mesh** 

Models L1 L2 L3

Sensor No.1 5.96 10.81 33.42 Sensor No.2 5.77 10.53 33.22 Sensor No.3 5.50 10.46 32.41 Average value 5.74 10.60 33.02

Average value 5.18 9.93 32.35

5.05 9.73 33.22

5.37 9.89 32.21

5.15 10.33 31.09

5.16 9.77 32.86

The boundary conditions for the numerical simulation are the same as those used in the experimental portion of this study. One end of the cylinder is fixed, while the other end is free. All nodes on the symmetry section (longitudinal plane of the vessel) are constrained against displacement in the direction perpendicular to the symmetry plane. The FEA models were loaded a force at the end of the nozzle along the longitudinal direction of the vessels like in the experiment. For the purpose of comparing the results of the element analysis with those of the experiments, the same materials and loading increments as those of the test vessels were used. As a nonliner geometry analysis, the multilinear isotropic type of hardening behavior is used.A multi-linear elastic-plastic material model defined by nine points, as given in Table 3, was employed.


**Table 3.** Material model

### **3.3. Analysis results**

## *3.3.1. Plastic deformation*

For the same loading condition as the experiment, the nozzle produced an obvious bending deformation and a depression of the cylinder on the compression side (in the longitudinal section) occurs. Figure 10 shows the local deformation characteristics which are consistent with those of the experiment (see Figure 4) for model vessel No. L2.

### *3.3.2. Plastic limit moment*

Figure 11 indicates the load-displacement plots of simulation vessels from the sensor No.3 and corresponding limit moments on the nozzles.

At the same location as that of the experiment, load strain curves were plated in Figure 12, and corresponding limit moments for simulation vessels of strain gage No.6 were also shown in Figure 12.

3D Nonlinear Finite Element Plastic Analysis of Cylindrical Vessels Under In-Plane Moment 125

Models L1 L2 L3

Difference of the average value (%) 8.70 9.83 6.85

Sensor No.1 5.27 11.67 30.38 Sensor No.2 5.39 11.49 30.52 Sensor No.3 5.53 11.32 30.64 Average value 5.40 11.49 30.51

Strain gage No.1 4.95 10.37 28.62 Strain gage No.6 4.93 10.25 27.99 Strain gage No.11 4.81 10.46 28.11 Strain gage No.16 5.02 10.36 28.94 Average value 4.93 10.36 28.42

**Figure 12.** Load-strain plots for simulation models (Strain gage No. 6)

By load-displacement method

By load-strain method

**Table 4.** FEA Results for the Plastic Limit Moment (kN.m)

**Figure 10.** Local deformation for model NO. L2 (Mi=13.05 kN.m)

**Figure 11.** Load-displacement plots for simulation models (Sensor No. 3)

The plastic limit moments for the three model vessels by FEA are listed in Table 4.

**Figure 10.** Local deformation for model NO. L2 (Mi=13.05 kN.m)

**Figure 11.** Load-displacement plots for simulation models (Sensor No. 3)

The plastic limit moments for the three model vessels by FEA are listed in Table 4.

**Figure 12.** Load-strain plots for simulation models (Strain gage No. 6)


**Table 4.** FEA Results for the Plastic Limit Moment (kN.m)

## **4. Summary and comparison**

From Tables 2 and 4, it is seen that the plastic limit moments from the displacement measurements are consistent with those determined from the strain measurements, whether by experimental method or finite element analysis.

3D Nonlinear Finite Element Plastic Analysis of Cylindrical Vessels Under In-Plane Moment 127

[8] (kN.m)

D=500mm D=1000mm D=1500mm D=2000mm

1 2.25 16.43 64.37 156.52 2 3.11 22.56 88.98 216.47 3 3.52 25.58 97.96 231.89 4 3.12 24.27 79.05 185.34 5 5.63 43.73 161.53 398.26 6 18.31 142.05 539.24 1307.58 7 6.51 50.14 158.86 367.96 8 11.52 85.09 282.45 632.46 9 9.47 73.76 275.15 608.14 10 11.67 83.05 276.31 650.87 11 58.02 434.86 1536.42 3500.24 12 36.21 256.45 853.13 2068.57 13 13.37 109.21 363.01 548.12 14 27.52 204.18 680.76 1358.54 15 52.54 360.04 1221.85 2808.47 16 118.34 836.42 2804.57 6618.21

0.1≤d/D≤0.4,0.25≤t/T≤1,50≤D/T≤125,500mm≤D≤2000mm

Model **d/ D t/T D/T** 1 0.1 0.25 50 2 0.1 0.5 75 3 0.1 0.75 100 4 0.1 1.0 125 5 0.2 0.25 75 6 0.2 0.5 50 7 0.2 0.75 125 8 0.2 1.0 100 9 0.3 0.25 100 10 0.3 0.5 125 11 0.3 0.75 50 12 0.3 1.0 75 13 0.4 0.25 125 14 0.4 0.5 100 15 0.4 0.75 75 16 0.4 1.0 50

*Mb*=d2tσ<sup>s</sup>

This equation holds for the following range of parameters:

**Table 6.** Primary parameters of the FE models

**Table 7.** Limit loads of the FE models (kN.m)

Models Limit Moment

Using an average of the values from different measuring points and displacement sensors, a summary and comparison of the twice-elastic-slope plastic limit load by experiment and finite element analysis for the three model vessels are shown in Table 5.


**Table 5.** Summary and comparison of plastic limit moment

From Table 5, it can be seen that the results for the plastic limit moment from experiment and finite element analysis including displacement and strain measuring techniques are in good agreement. The difference between experiment and finite element analysis is within 10%.

## **5. Parametric analysis and correlation equation**

## **5.1. Parametric finite element modeling**

On the basis of the previous studies, parametric modeling was performed. The sets of parameters of the analysis models used in this part of the study are given in Table 6. A total of 64 configurations were analyzed to investigate the relationship between the various geometric parameters and plastic limit moments of the vessel-nozzle structures. The data in Table 7 were obtained from the finite element analysis of 64 models.

## **5.2. Correlation equation**

A correlation equation for plastic limit load of cylindrical vessels under in-plane moment on the nozzle can be obtained by using the data from Table 7 and the software package Statistica (non-linear regression). The resulting correlation equation for the plastic limit moment on the nozzle is as follows:

$$M\_{iL} = \left[ -0.0148 + 0.704 \left( \frac{d}{D} \right) + 1.845 \left( \frac{d}{D} \right)^2 \right] \left( \frac{d}{D} \right)^{-1.761} \left( \frac{t}{T} \right)^{-0.555} \left( \frac{D}{T} \right)^{-0.476} M\_b$$

Where, *Mb* is the limit load of a straight nozzle,

$$M \coloneqq \mathbf{d}^2 \mathbf{t} \,\,\sigma \,\, ^{[8]} \,\, (\mathbf{k} \,\text{N.m})$$

This equation holds for the following range of parameters:

126 Finite Element Analysis – Applications in Mechanical Engineering

by experimental method or finite element analysis.

**Table 5.** Summary and comparison of plastic limit moment

**5. Parametric analysis and correlation equation** 

Table 7 were obtained from the finite element analysis of 64 models.

**5.1. Parametric finite element modeling** 

**5.2. Correlation equation** 

moment on the nozzle is as follows:

Where, *Mb* is the limit load of a straight nozzle,

From Tables 2 and 4, it is seen that the plastic limit moments from the displacement measurements are consistent with those determined from the strain measurements, whether

Using an average of the values from different measuring points and displacement sensors, a summary and comparison of the twice-elastic-slope plastic limit load by experiment and

> Limit-moment by FEA (kN.m)

> > loadstrain method

average

loaddisplace ment method

2 1.761 0.555 0.476

Difference of the average value) (%)

finite element analysis for the three model vessels are shown in Table 5.

loaddisplace ment method

Limit-moment by Experiment (kN.m)

> loadstrain method

L1 62.5 0.172 0.375 5.74 5.18 5.46 5.40 4.93 5.17 5.31 L2 62.5 0.246 0.50 10.60 9.93 10.27 11.49 10.36 10.93 6.43 L3 62.5 0.428 0.625 33.02 32.35 32.69 30.51 28.42 29.47 9.85

From Table 5, it can be seen that the results for the plastic limit moment from experiment and finite element analysis including displacement and strain measuring techniques are in good agreement. The difference between experiment and finite element analysis is within 10%.

On the basis of the previous studies, parametric modeling was performed. The sets of parameters of the analysis models used in this part of the study are given in Table 6. A total of 64 configurations were analyzed to investigate the relationship between the various geometric parameters and plastic limit moments of the vessel-nozzle structures. The data in

A correlation equation for plastic limit load of cylindrical vessels under in-plane moment on the nozzle can be obtained by using the data from Table 7 and the software package Statistica (non-linear regression). The resulting correlation equation for the plastic limit

> [ 0.0148 0.704 1.845 ] *iL <sup>b</sup> d dd t D <sup>M</sup> <sup>M</sup>*

*D DD T T* 

average

**4. Summary and comparison** 

Model

No. D /T d/ D t/T


0.1≤d/D≤0.4,0.25≤t/T≤1,50≤D/T≤125,500mm≤D≤2000mm

**Table 6.** Primary parameters of the FE models


**Table 7.** Limit loads of the FE models (kN.m)

In order to verify the accuracy of the correlation equation, the plastic limit moments of three models were calculated and compared with those from the experimental and FEA studies as seen in Table 8.

3D Nonlinear Finite Element Plastic Analysis of Cylindrical Vessels Under In-Plane Moment 129

*Department of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing,* 

[1] ASME Boiler and Pressure Vessel Code, 2004 edition, section Ⅷ, Division 2, American

[2] Ellyin, F., 1976, "Experimental Investigation of Limit Loads of Nozzles in Cylindrical

[3] Ellyin, F., 1977, "An Experimental Study of Elastic-plastic Response of Branch Pipe Tee Connection Subjected to Internal Pressure, External Couples and Combined Loadings,"

[4] Schroeder, J., 1985, "Experimental Limit Couples for Branch Moment Loads on 4-in. ANSI B16.9 Tees,"Welding Research Council Bulletin, No.304, WRC, New York. [5] Junker, A.T., 1985, "Finite Element Determination of the Limit Load of a Nozzle in a Cylindrical Vessel Due to In-plane and Out-of-plane Moments," Swanson Engineering Associates Corporation (SEAC) Report No.SEAC-TR-312,Rev.2, to the Pressure Vessel

[6] Moffat, D.G., 1985, "Experimental Stress Analysis of Four Fabricated Equal Diameter Branch Pipe Intersections Subjected to Moment Loadings and Implications on Branch

[7] Moffat, D.G., 1991, Mwenifumbo, J.A.M., Xu, S. H. and Mistry, T., "Effective Stress Factors for Piping Branch Junctions Due to Internal Pressure and External Moment

[8] Rodabaugh, E.C., 1988, "A Review of Area Replacement Rules for Pipe Connections in Pressure Vessels and Piping,"Welding Research Council Bulletin, No.335, WRC, New

[9] Rodabaugh, E.C., Interpretive report on limit analysis and plastic behavior of piping

[10] Kalnins, A., 2001, "Guidelines for Sizing of Vessels by Limit Analysis,"Welding

[11] Chapuliot, S., Moulin, D., and Plancg, D., 2002, "Mechanical Behavior of a Branch Pipe Subjected to Out-of-plane Bending Load," ASME Journal of Pressure Vessel

[12] Mourad Hashem, M., Maher, Y.A., 2002, "Limit-Load analysis of Pipe Bend under Outof-plane Moment and Internal Pressure," ASME Journal of Pressure Vessel Technology,

[13] Widera, G.E.O., Wei, Z., 1998, "Parametric Finite Element of Large Diameter Shell Intersection," Part 2, External Loading, PVRC Project No.96-20AS, WRC, New York.

Vessels,"Welding Research Council Bulletin, No.219, WRC, New York.

Junction Design," Proc Instn. Mech. Engrs., Vol.199, No.4, PP. 261~284.

Loads," Journal of Strain Analysis, Vol.26, No.2, PP.85~101.

products, Welding Research Council Bullition No.254,1976

Research Council Bulletin, No.464, WRC, New York.

Technology, Vol.124, PP.1~14.

Vol.124, PP.7~32.

Welding Research Council Bulletin, No.230, WRC, New York.

**Author details** 

**7. References** 

York.

*People's Republic of China* 

Society of Mechanical Engineering, New York.

Research Committee, WRC, New York.

B.H. Wu


**Table 8.** Comparison of plastic limit moment (kN.m)

## **6. Conclusions**

Experiments and comparative 3D-nonlinear finite element analyses of cylindrical vesselnozzle connections subjected to an in-plane moment on the nozzle were carried out. In addition, an extensive geometric parameter study of such joints was performed with FEA.The following general conclusions can be drawn from these studies:


## **Author details**

#### B.H. Wu

128 Finite Element Analysis – Applications in Mechanical Engineering

5.55 11.53 33.63

5.46 10.27 32.69

) 1.62% 10.93% 2.80%

**Table 8.** Comparison of plastic limit moment (kN.m)

external loads on the nozzle.

FE calculated limit moments is about 10%.

agreement for the analyzed geometry.

an in-plane load on the nozzle.

seen in Table 8.

Limit-moment by Regression Equation (M *EQU* )

Limit- moment by Experiment (M *EXP* )

Difference

( *EQU EXP EQU*

**6. Conclusions** 

*M M M*

In order to verify the accuracy of the correlation equation, the plastic limit moments of three models were calculated and compared with those from the experimental and FEA studies as

Models L1 L2 L3 Models L1 L2 L3

Limit- moment by Regression Equation (M *EQU* )

Limit- moment by FEA (M *FEA* )

Difference

( *EQU FEA EQU*

*M M M*

Experiments and comparative 3D-nonlinear finite element analyses of cylindrical vesselnozzle connections subjected to an in-plane moment on the nozzle were carried out. In addition, an extensive geometric parameter study of such joints was performed with

1. The experimental and 3D-nonlinear finite element analyses including load-strain and load-displacement of the nozzle responses provide effective and reliable determination methods for the pressure vessel-nozzle (or main-branch pipe) connections subjected to

2. The plastic limit moments determined from experimental and finite element analysis studies are in good agreement. The maximum error of the average experimental versus

3. The limit moments from load-displacement and load-strain curves are in reasonable

4. The plastic limit moments from experimental and finite element analysis studies are

5. The correlation equation provided in this paper can be used to determine, within reasonable limits, the plastic limit moment of cylinder-nozzle connections subjected to

greatly increased with an increase of d/D, t/T ratios of the model vessels.

FEA.The following general conclusions can be drawn from these studies:

5.55 11.53 33.63

5.17 10.93 29.47

) 6.85% 5.20% 12.57%

*Department of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing, People's Republic of China* 

## **7. References**

	- [14] Sang, Z.F., Wang, Z.L., Xue, L.P., Widera, G..E.O., 2005, "Plastic Limit Loads of Nozzles in Cylindrical Vessels under Out-of-plane Moment Loading," International Journal of Pressure Vessels and Piping, 82, PP. 638-648.

**Chapter 6** 

© 2012 Doddamani and Kulkarni, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Doddamani and Kulkarni, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

properly cited.

**Flexural Behavior of** 

http://dx.doi.org/10.5772/51134

**1. Introduction** 

**Functionally Graded Sandwich Composite** 

During World War II, the british made a bomber De Havilland *Mosquito* which served in Europe, Middle and Far East and on the Russian front. Designed as a bomber, it excelled not only in this field but also as a fighter aircraft, mine layer, path finder in military transport and photo reconnaissence. It was constructed during the *Battle of Britain* and the first prototype made its maiden flight in november 1940, less than a year after the design project is started. From an engineering viewpoint, it has one spectacular feature - the fuselage is made of a molded plywood-balsa sandwich material, which is strong and yet lightweight and equally important in times of war, its components are readily available unlike aluminium ones. The importance of the Mosquito in the war effort proved the value of the new sandwich materials [1]. Sandwich composites are popular due to high specific strength and stiffness. The concept of sandwiches came in as early as the year 1849 AD but their potential realized mainly during Second World War as mentioned earlier. Sandwiches are composed of two stiff, strong and thin faces (skins) bonded to a light, thick weaker core. Faces sustain in-plane and bending loads, while the core resist transverse shear forces and keep the facings in place. These provide increased flexural rigidity and strength by virtue of their geometry. The high specific strength and stiffness make them ideal in structural design [2-3]. Developments in aviation posed requirement of lightweight, high strength and highly damage tolerant materials. Sandwich composites, fulfilling these requirements became the

first choice for many applications including ground transport and marine vessels [4].

Sandwich panels are used in a variety of engineering applications including aircraft, construction and transportation where strong, stiff and light structures are required [5]. The applicability of sandwiches could be improved if it contains a FG core which might help to distribute the stresses due to bending or in progressive absorption of energy under impact loading [6]. It is required to study the behavior of sandwich panels under these types of

Mrityunjay R. Doddamani and Satyabodh M Kulkarni

Additional information is available at the end of the chapter

[15] Swanson Analysis System Inc., "ANSYS Engineering Analysis Systems User's Manual".

## **Flexural Behavior of Functionally Graded Sandwich Composite**

Mrityunjay R. Doddamani and Satyabodh M Kulkarni

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51134

## **1. Introduction**

130 Finite Element Analysis – Applications in Mechanical Engineering

Pressure Vessels and Piping, 82, PP. 638-648.

[14] Sang, Z.F., Wang, Z.L., Xue, L.P., Widera, G..E.O., 2005, "Plastic Limit Loads of Nozzles in Cylindrical Vessels under Out-of-plane Moment Loading," International Journal of

[15] Swanson Analysis System Inc., "ANSYS Engineering Analysis Systems User's Manual".

During World War II, the british made a bomber De Havilland *Mosquito* which served in Europe, Middle and Far East and on the Russian front. Designed as a bomber, it excelled not only in this field but also as a fighter aircraft, mine layer, path finder in military transport and photo reconnaissence. It was constructed during the *Battle of Britain* and the first prototype made its maiden flight in november 1940, less than a year after the design project is started. From an engineering viewpoint, it has one spectacular feature - the fuselage is made of a molded plywood-balsa sandwich material, which is strong and yet lightweight and equally important in times of war, its components are readily available unlike aluminium ones. The importance of the Mosquito in the war effort proved the value of the new sandwich materials [1]. Sandwich composites are popular due to high specific strength and stiffness. The concept of sandwiches came in as early as the year 1849 AD but their potential realized mainly during Second World War as mentioned earlier. Sandwiches are composed of two stiff, strong and thin faces (skins) bonded to a light, thick weaker core. Faces sustain in-plane and bending loads, while the core resist transverse shear forces and keep the facings in place. These provide increased flexural rigidity and strength by virtue of their geometry. The high specific strength and stiffness make them ideal in structural design [2-3]. Developments in aviation posed requirement of lightweight, high strength and highly damage tolerant materials. Sandwich composites, fulfilling these requirements became the first choice for many applications including ground transport and marine vessels [4].

Sandwich panels are used in a variety of engineering applications including aircraft, construction and transportation where strong, stiff and light structures are required [5]. The applicability of sandwiches could be improved if it contains a FG core which might help to distribute the stresses due to bending or in progressive absorption of energy under impact loading [6]. It is required to study the behavior of sandwich panels under these types of

© 2012 Doddamani and Kulkarni, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Doddamani and Kulkarni, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

failures with a functionally graded material (FGM) as core to explore their new application in bullet proofing and crash worthiness. FGM's are new class of materials where property is function of geometry such as thickness, length etc [7]. These are the materials whose composition and microstructure are not uniform in space, but gradually vary following a predetermined law [8-11]. FGM's differ from composites in the sense that property is uniform in a particular direction throughout the composite. The concept of FGM's is proposed as early as 1984 by material scientists as a means of preparing thermal barrier materials [12]. Closest to FGM's is laminated composites with variation in laminate properties but they possess distinct interfaces across which properties change abruptly [13]. For example, a rocket motor casing can be made with a material system such that the inside is made of a refractory material, the outside is made of a strong metal, and the transition from the refractory material to the metal is gradual through the thickness [14]. FGM's possess a number of advantages that make them attractive in many applications, including a potential reduction of in-plane and transverse through-the-thickness stresses, an improved residual stress distribution, enhanced thermal properties, higher fracture toughness, and reduced stress intensity factors. It is worth mentioning that the distribution of the material in functionally graded structures may be designed to various spatial specifications (1). Currently, advanced processing methods to introduce compositional gradients into various material systems are being developed by materials scientists [15-17]. A typical particulate composite with prescribed variation in distribution of constituent phases could be a representative FGM. The FGM concept could be borrowed in making sandwiches with FG core which exhibit resistance (stiffness) proportional to the applied load can serve some applications better than regular sandwiches, like a spring with varying stiffness. Such a sandwich could be realized by using a particulate composite with varying volume fraction of constituents.

Flexural Behavior of Functionally Graded Sandwich Composite 133

1. To prepare functionally graded rubber cores with varying fly ash reinforcement.

orientation) as per L9 orthogonal array at three levels.

point loading condition.

the response is averaged out for these five.

sandwiches.

consideration.

**3. Processing details** 

**3.1. Plan of experiment** 

2. To plan the experiments using DOE for processing FG sandwiches with different factors (weight fraction of fly ash, core to total sandwich thickness - C/H ratio and jute skin

3. To study the effect of above parameters on mechanical properties of sandwich three

4. To identify the most influential factor governing the mechanical behavior of FG

5. To validate the gradation observed through finite element (FE) modeling using spring

6. Comparison of Experimental and FE results for properties of sandwich under

Developed FG cores are utilized in sandwiches to characterize FG sandwiches for their suitability in real world applications. Sandwiches are prepared as per design of experiments approach so that multiple factors (fly ash weight fraction, C/H ratio and jute skin orientation) at three different levels can be simultaneously analyzed. Further, these sandwiches are subjected to bending test. Another set of samples called confirmatory set is made with 25% and 35% filler by weight. Five samples are subjected to mechanical test and

Furthermore, experimental values are compared with results of FE analysis. ANSYS 5.4 package is used to achieve this objective. Analysis are carried out with three gradation variations namely uniform, linear and piecewise linear. Young's modulus is computed for FG cores using FE approach and is compared with experimental result. Specific bending strength is the properties focused in simulating sandwich behavior. Finally, elaborate

This section presents properties of starting material used, procedures followed for preparing FG composites and their sandwiches. Details of reagents / chemicals used at different stages like for sample curing are also described. Characteristics of the reinforcements used are also enlisted. As outlined in the objectives and scope of the work in the preceding section, the objective of the present investigation is to study the properties of functionally graded sandwiches. This section lists materials and their properties and methods adopted for

In this work experiments are designed based on Taguchi's DOE approach for FG sandwiches [32]. Factors and levels chosen for planning the experiments for FG sandwiches are presented in Table 1. Table 2 shows orthogonal array for sandwich. Table 3 presents

coding of samples bearing varying content of filler, C/H ratio and jute orientation.

discussion on fractured samples is presented as the last segment of this work.

processing composites with varying content of the filler.

analogy for variations in property like uniform, linear and piecewise linear.

7. Visual inspection of fractured FG sandwiches under different tests.

The flexural behavior of sandwich beams has been studied extensively by many investigators [18-23]. Studies on three point bend tests have been conducted in flexural [24- 25] and short beam shear test configurations [26]. An experimental investigation of failure of piecewise FG of sandwiches subjected to three point bending is carried out by Avila [27]. In addition, fiber reinforced syntactic foams [28-30] and syntactic foam core sandwich composites have also been studied for bending properties [31]. Specific properties of sandwich with complaint FG core needs attention as it is yet to be reported.

## **2. Objectives and scope**

From the foregoing literature survey, clear is the fact that the research reports on development of low cost materials for bullet proofing and energy absorption is hardly available. A low cost ash filled functionally graded polymer system is proposed for applications like ballistic energy absorption. The perusal of sandwich literature review prompted a thorough and systematic study on these sandwiches by performing experimental characterization of flexural properties. Therefore the work undertaken pursues the following objectives:


Developed FG cores are utilized in sandwiches to characterize FG sandwiches for their suitability in real world applications. Sandwiches are prepared as per design of experiments approach so that multiple factors (fly ash weight fraction, C/H ratio and jute skin orientation) at three different levels can be simultaneously analyzed. Further, these sandwiches are subjected to bending test. Another set of samples called confirmatory set is made with 25% and 35% filler by weight. Five samples are subjected to mechanical test and the response is averaged out for these five.

Furthermore, experimental values are compared with results of FE analysis. ANSYS 5.4 package is used to achieve this objective. Analysis are carried out with three gradation variations namely uniform, linear and piecewise linear. Young's modulus is computed for FG cores using FE approach and is compared with experimental result. Specific bending strength is the properties focused in simulating sandwich behavior. Finally, elaborate discussion on fractured samples is presented as the last segment of this work.

## **3. Processing details**

132 Finite Element Analysis – Applications in Mechanical Engineering

of constituents.

**2. Objectives and scope** 

the following objectives:

failures with a functionally graded material (FGM) as core to explore their new application in bullet proofing and crash worthiness. FGM's are new class of materials where property is function of geometry such as thickness, length etc [7]. These are the materials whose composition and microstructure are not uniform in space, but gradually vary following a predetermined law [8-11]. FGM's differ from composites in the sense that property is uniform in a particular direction throughout the composite. The concept of FGM's is proposed as early as 1984 by material scientists as a means of preparing thermal barrier materials [12]. Closest to FGM's is laminated composites with variation in laminate properties but they possess distinct interfaces across which properties change abruptly [13]. For example, a rocket motor casing can be made with a material system such that the inside is made of a refractory material, the outside is made of a strong metal, and the transition from the refractory material to the metal is gradual through the thickness [14]. FGM's possess a number of advantages that make them attractive in many applications, including a potential reduction of in-plane and transverse through-the-thickness stresses, an improved residual stress distribution, enhanced thermal properties, higher fracture toughness, and reduced stress intensity factors. It is worth mentioning that the distribution of the material in functionally graded structures may be designed to various spatial specifications (1). Currently, advanced processing methods to introduce compositional gradients into various material systems are being developed by materials scientists [15-17]. A typical particulate composite with prescribed variation in distribution of constituent phases could be a representative FGM. The FGM concept could be borrowed in making sandwiches with FG core which exhibit resistance (stiffness) proportional to the applied load can serve some applications better than regular sandwiches, like a spring with varying stiffness. Such a sandwich could be realized by using a particulate composite with varying volume fraction

The flexural behavior of sandwich beams has been studied extensively by many investigators [18-23]. Studies on three point bend tests have been conducted in flexural [24- 25] and short beam shear test configurations [26]. An experimental investigation of failure of piecewise FG of sandwiches subjected to three point bending is carried out by Avila [27]. In addition, fiber reinforced syntactic foams [28-30] and syntactic foam core sandwich composites have also been studied for bending properties [31]. Specific properties of

From the foregoing literature survey, clear is the fact that the research reports on development of low cost materials for bullet proofing and energy absorption is hardly available. A low cost ash filled functionally graded polymer system is proposed for applications like ballistic energy absorption. The perusal of sandwich literature review prompted a thorough and systematic study on these sandwiches by performing experimental characterization of flexural properties. Therefore the work undertaken pursues

sandwich with complaint FG core needs attention as it is yet to be reported.

This section presents properties of starting material used, procedures followed for preparing FG composites and their sandwiches. Details of reagents / chemicals used at different stages like for sample curing are also described. Characteristics of the reinforcements used are also enlisted. As outlined in the objectives and scope of the work in the preceding section, the objective of the present investigation is to study the properties of functionally graded sandwiches. This section lists materials and their properties and methods adopted for processing composites with varying content of the filler.

## **3.1. Plan of experiment**

In this work experiments are designed based on Taguchi's DOE approach for FG sandwiches [32]. Factors and levels chosen for planning the experiments for FG sandwiches are presented in Table 1. Table 2 shows orthogonal array for sandwich. Table 3 presents coding of samples bearing varying content of filler, C/H ratio and jute orientation.


Flexural Behavior of Functionally Graded Sandwich Composite 135

From the standpoint of cost, availability, and the scarce literature prompted for going in for an elastomeric material which is naturally occurring and known by the name 'natural rubber' as the matrix material. Further it is reinforced with fly ash and is used as core in

As many of the polymeric systems for developing FGM's are generally with the tag of expensiveness associated, it is decided to examine the gradation in composition and its subsequent mechanical behavior when an abundantly available lower density possessing fly ashes are the filler materials for the core. Fly ashes are fine particulate waste products derived during generation of power in a thermal power plant. These have aspect ratios closer to unity and hence are expected to display near isotropic characteristics. These inexpensive and possessing good mechanical properties, when used with well established matrix systems help to reduce the cost of the system and at the same time either retain or improve specific and desirable mechanical properties. Fly ash has attracted interest [33-34] lately, because of the abundance in terms of the volume of the material generated and the environmental-linked problems in the subsequent disposal. Fly ash mainly consists of alumina and silica, which are expected to improve the composite properties. Fly ash also consists to some extent hollow spherical particles termed as cenosphere which aid in maintenance of lower density values for the composite, a feature of considerable significance in weight-specific applications [35-36]. Again, as the fillers do not come under irregular shape, the resin spread, is better and as the ashes are essentially a mixture of solid, hollow and composite particles displaying near isotropic properties, developing newer and utilitarian systems using them should be an interesting and challenging task [37].

Compositional details of a fly ash particle are tabulated in Table 4.

**Table 4.** Compositional details of fly ash particle

*3.2.2. Skin used in sandwich* 

Constituent Wt. % SiO2 63 Al2O3 26.55 CaO 0.42 Fe2O3 6.7 TiO2 2.47

Further on, in this effort, for the skins too, it is decided to employ instead of the well explored man-made fibers like glass, carbon or aramid a fairly strong but naturally occurring one going by the name 'jute fiber' and known for its inexpensiveness. Jute is an attractive natural fiber for use as reinforcement in composite because of its low cost, renewable nature and much lower energy requirement for processing. In comparison to glass fibers jute has higher specific modulus and lower specific gravity as against that of

*3.2.1. Core for FG sandwich* 

sandwich.

**Table 1.** Factors and Levels selected for sandwich with FG core


**Table 2.** L9 Orthogonal array for FG Sandwich


**Table 3.** Description of sample codes used for sandwiches

Experimentation is done with due considerations to all the above parameters with both configurations of gradation namely rubber up and ash up. In each trial minimum of five replicates are tested. Average of the measured parameters for each set of replicates is subjected to statistical ANOVA to find the most influential factor governing the behavior using Minitab release 14 statistical analysis tool.

## **3.2. Materials**

Details of materials used for main constituents of sandwiches (core and skin) are presented hereafter.

## *3.2.1. Core for FG sandwich*

134 Finite Element Analysis – Applications in Mechanical Engineering

(Factor 1)

**Table 1.** Factors and Levels selected for sandwich with FG core

Core to thickness ratio (Factor 2)

Parameters Weight Fraction (%) C/H Ratio Orientation

/900

, 300/600

, 450/450)

Level 1 20 0.4 00/900 Level 2 30 0.6 300

1 20 0.4 00

4 30 0.4 300

5 30 0.6 450

6 30 0.8 00

9 40 0.8 300

W Indicates factor 1 (Wt. fraction of fly ash) a Levels of factor 1 in % (20, 30, 40) R Indicates factor 2 (C/H ratio) b Levels of factor 2 (0.4, 0.6, 0.8) O Indicates factor 3 (Fiber Orientation in skin)

Experimentation is done with due considerations to all the above parameters with both configurations of gradation namely rubber up and ash up. In each trial minimum of five replicates are tested. Average of the measured parameters for each set of replicates is subjected to statistical ANOVA to find the most influential factor governing the behavior

Details of materials used for main constituents of sandwiches (core and skin) are presented

Sample code Description

c Levels of factor 3 (00

**Table 3.** Description of sample codes used for sandwiches

using Minitab release 14 statistical analysis tool.

**3.2. Materials** 

hereafter.

WaRbOc Sandwich specification

7 40 0.4 450/450 8 40 0.6 00

2 20 0.6 300/600 3 20 0.8 450

Level 3 40 0.8 450

Orientation of Jute Fabric (Factor 3)

/600

/450

/900

/450

/600

/450

/900

/900

/600

Details Wt Fraction of Fly ash %

**Table 2.** L9 Orthogonal array for FG Sandwich

Experiment No.

From the standpoint of cost, availability, and the scarce literature prompted for going in for an elastomeric material which is naturally occurring and known by the name 'natural rubber' as the matrix material. Further it is reinforced with fly ash and is used as core in sandwich.

As many of the polymeric systems for developing FGM's are generally with the tag of expensiveness associated, it is decided to examine the gradation in composition and its subsequent mechanical behavior when an abundantly available lower density possessing fly ashes are the filler materials for the core. Fly ashes are fine particulate waste products derived during generation of power in a thermal power plant. These have aspect ratios closer to unity and hence are expected to display near isotropic characteristics. These inexpensive and possessing good mechanical properties, when used with well established matrix systems help to reduce the cost of the system and at the same time either retain or improve specific and desirable mechanical properties. Fly ash has attracted interest [33-34] lately, because of the abundance in terms of the volume of the material generated and the environmental-linked problems in the subsequent disposal. Fly ash mainly consists of alumina and silica, which are expected to improve the composite properties. Fly ash also consists to some extent hollow spherical particles termed as cenosphere which aid in maintenance of lower density values for the composite, a feature of considerable significance in weight-specific applications [35-36]. Again, as the fillers do not come under irregular shape, the resin spread, is better and as the ashes are essentially a mixture of solid, hollow and composite particles displaying near isotropic properties, developing newer and utilitarian systems using them should be an interesting and challenging task [37]. Compositional details of a fly ash particle are tabulated in Table 4.


**Table 4.** Compositional details of fly ash particle

## *3.2.2. Skin used in sandwich*

Further on, in this effort, for the skins too, it is decided to employ instead of the well explored man-made fibers like glass, carbon or aramid a fairly strong but naturally occurring one going by the name 'jute fiber' and known for its inexpensiveness. Jute is an attractive natural fiber for use as reinforcement in composite because of its low cost, renewable nature and much lower energy requirement for processing. In comparison to glass fibers jute has higher specific modulus and lower specific gravity as against that of

glass fiber. Jute reinforced plastics offer attractive propositions for cost-effective applications [38]. These in the form of laminates have much better properties than their neat resin counterparts [39]. Better properties of woven jute fabric reinforced composites demonstrated their potential for use in a number of consumable goods in an earlier literature [40]. Substantial increases in flexural modulus and strength with small amounts of reinforcement of unidirectional jute have also been reported [41]. Keeping these things in mind a bidirectionally woven jute fabric is used in different orientations. Table 5 gives the brief overview of comparison between glass fibers and jute fibers.

Flexural Behavior of Functionally Graded Sandwich Composite 137

With these materials in hand, FG sandwiches are prepared for mechanical testing.

FG cores used in the present work are produced using the following procedure. The gradation in the core is expected due to differential settling of the particles with different densities at different depths in the rubber matrix. A measured quantity of natural latex is mixed with pre-weighed amounts of fly ash, sulphur (vulcanizer) and zinc oxide (catalyst) [42] by adopting gentle stirring for about 1 hour. The mold employed for preparation of core specimen is completely covered on all sides with teflon sheet. Subsequently, silicone releasing agent is applied to facilitate ease of removal of the cast sample at a later stage. The mixture is then slowly decanted into the mold cavity followed by curing at 90°C in an oven for about 5-6 hours. The cured rigid plate sample is withdrawn from the mold and the edges trimmed. Figure 1 presents one such FG sample which in turn will be used as core in

As regards the sandwich skins, a bi-directional woven jute fabric procured from M/S Barde Agencies, Belgaum, Karnataka, India is used. This fabric is cut into layers of dimensions depending on the sandwich sample size in required orientation. Thickness of each fabric piece is 0.5 mm. All the layers of jute fabric are heated in an oven at 700C for 5-10 minutes to remove moisture present. The jute stack thickness to form the thin skin, on either side of FG core, is computed. This enables one to arrive at the required number of fabric layers to be used, as thickness of each layer is known. Based on required C/H ratio number of fabric

> Number of jute layers above core

Sandwich thickness - H (mm)

Number of jute layers below core

**Table 7.** Jute layer arrangement for achieving C/H ratios in sandwich

0.4 4 6 6 10 0.6 6 4 4 10 0.8 8 2 2 10

With this background data on hand to begin with, the required fabric pieces are dipped in mixture of epoxy and K-6 hardener and placed on base plate forming the bottom stack of the

**4. Processing of FG sandwich** 

**Figure 1.** Functionally graded core sample drawn out of mold

layers to be used are determined (Table 7).

Core thickness - C (mm)

sandwiches.

C/H Ratio


**Table 5.** Mechanical Properties of Glass and Jute Fibers

The major drawback of natural fiber reinforced composites is due to its affinity towards moisture. Many experimental studies have shown that compatible coupling agents are capable of either slowing down or preventing the de-bonding process and hence moisture absorption even under severe environmental conditions such as exposure to boiling water. Jute fibers/fabrics can be modified chemically through graft co-polymerization and through incorporation of different resin systems by different approaches.

## *3.2.3. Matrix for skin*

For fabricating both the skins and core a matrix system is required. A thermosetting epoxy is chosen for this purpose as far as the skins are concerned. The adhesive used in present work consists of a medium viscosity epoxy resin (LAPOX L-12) and a room temperature curing polyamine hardener (K-6) supplied by ATUL India Ltd. Epoxy resin is selected as the material for the matrix system because of its wide application, good mechanical properties, excellent corrosion resistance and ease of processing. Some details including density of the constituents of the matrix system chosen are listed in Table 6.


\* As suggested in the manufacturer's catalogue

**Table 6.** Details of the constituents of matrix used for skin in sandwich

With these materials in hand, FG sandwiches are prepared for mechanical testing.

## **4. Processing of FG sandwich**

136 Finite Element Analysis – Applications in Mechanical Engineering

overview of comparison between glass fibers and jute fibers.

incorporation of different resin systems by different approaches.

constituents of the matrix system chosen are listed in Table 6.

name

Diglycidyl Ether of bisphenol A (DGEBA)

> Tri ethylene Tetra amine (TETA)

**Table 6.** Details of the constituents of matrix used for skin in sandwich

Constituent Trade name Chemical

L-12

As suggested in the manufacturer's catalogue

Resin LAPOX

Hardener K-6

\*

**Table 5.** Mechanical Properties of Glass and Jute Fibers

*3.2.3. Matrix for skin* 

glass fiber. Jute reinforced plastics offer attractive propositions for cost-effective applications [38]. These in the form of laminates have much better properties than their neat resin counterparts [39]. Better properties of woven jute fabric reinforced composites demonstrated their potential for use in a number of consumable goods in an earlier literature [40]. Substantial increases in flexural modulus and strength with small amounts of reinforcement of unidirectional jute have also been reported [41]. Keeping these things in mind a bidirectionally woven jute fabric is used in different orientations. Table 5 gives the brief

The major drawback of natural fiber reinforced composites is due to its affinity towards moisture. Many experimental studies have shown that compatible coupling agents are capable of either slowing down or preventing the de-bonding process and hence moisture absorption even under severe environmental conditions such as exposure to boiling water. Jute fibers/fabrics can be modified chemically through graft co-polymerization and through

For fabricating both the skins and core a matrix system is required. A thermosetting epoxy is chosen for this purpose as far as the skins are concerned. The adhesive used in present work consists of a medium viscosity epoxy resin (LAPOX L-12) and a room temperature curing polyamine hardener (K-6) supplied by ATUL India Ltd. Epoxy resin is selected as the material for the matrix system because of its wide application, good mechanical properties, excellent corrosion resistance and ease of processing. Some details including density of the

> Epoxide equivalent

Density

182 - 192 1162 ATUL India


(kg/m3) Supplier Parts by

weight

Ltd. <sup>100</sup>

Property E-glass Jute Specific Gravity 2.5 1.3 Tensile Strength (MN/m2) 3400 442 Young's Modulus (MN/m2) 72 55.5 Specific Strength (MN/m2) 1360 340 Specific Modulus (GN/m2) 28.8 42.7 FG cores used in the present work are produced using the following procedure. The gradation in the core is expected due to differential settling of the particles with different densities at different depths in the rubber matrix. A measured quantity of natural latex is mixed with pre-weighed amounts of fly ash, sulphur (vulcanizer) and zinc oxide (catalyst) [42] by adopting gentle stirring for about 1 hour. The mold employed for preparation of core specimen is completely covered on all sides with teflon sheet. Subsequently, silicone releasing agent is applied to facilitate ease of removal of the cast sample at a later stage. The mixture is then slowly decanted into the mold cavity followed by curing at 90°C in an oven for about 5-6 hours. The cured rigid plate sample is withdrawn from the mold and the edges trimmed. Figure 1 presents one such FG sample which in turn will be used as core in sandwiches.

**Figure 1.** Functionally graded core sample drawn out of mold

As regards the sandwich skins, a bi-directional woven jute fabric procured from M/S Barde Agencies, Belgaum, Karnataka, India is used. This fabric is cut into layers of dimensions depending on the sandwich sample size in required orientation. Thickness of each fabric piece is 0.5 mm. All the layers of jute fabric are heated in an oven at 700C for 5-10 minutes to remove moisture present. The jute stack thickness to form the thin skin, on either side of FG core, is computed. This enables one to arrive at the required number of fabric layers to be used, as thickness of each layer is known. Based on required C/H ratio number of fabric layers to be used are determined (Table 7).


**Table 7.** Jute layer arrangement for achieving C/H ratios in sandwich

With this background data on hand to begin with, the required fabric pieces are dipped in mixture of epoxy and K-6 hardener and placed on base plate forming the bottom stack of the

sandwich. Now, the earlier mentioned procedure-wise made FG core dipped in resin mixture is placed on the bottom stack of skins. Finally, over such an arrangement, the remaining layers of jute fabrics having undergone the same procedure for fabrication are stacked to constitute the top skin. A procedure of this nature should help in ensuring a greater degree of spread of the resin on the fibrillar jute. Following this, the excess resin is made to come out by a squeezing operation that is aided by tightening of the mold top plate. The mold assembly is then cured at room temperature for about 24-26 hours. The sandwich sample is withdrawn from the mold and trimmed to the required size. Similarly numbers of samples are made with various core thickness and orientation in skin as schematically illustrated in Figure 2. Figure 2 (a) shows top view with different orientations and while the front view with varying core thickness to total sandwich thickness (C/H ratio) is presented in Figure 2 (b).

Flexural Behavior of Functionally Graded Sandwich Composite 139

*bending E <sup>g</sup>* (1)

(2)

*bending u x g*

to span ratio of the tested sandwich samples is 1:16. The crosshead displacement rate is maintained at 2 mm/min. The load deflection data is recorded at equal intervals up to a

point at which the specimen shows the first sign of failure.

**Figure 3.** (a). Rubber Up condition in FG core, (b). Ash Up condition in FG core

From load deflection data, bending modulus and strength are estimated using relations 1 and 2 respectively and the mean of five samples in each sandwich configuration is used for

Flexural modulus Specific bending modulus Weight density x

Ultimate strength Specific bending strength Weight density

**Figure 4.** Sandwich sample mounted on flexural test set-up

inference.

**Figure 2.** (a). Orientation of jute strands in the sandwich skins, (b). Variation of C/H ratio considered for analysis

## **5. Experimental details**

The mechanical testing of sandwich composites to obtain parameters such as strength, stiffness etc. is a time consuming and often difficult process. It is, however, an essential process, and can be somewhat simplified by the testing of simple structures such as flat coupons. The data obtained from these tests can then be directly related with varying degrees of simplicity and accuracy to any structural shape. The test methods outlined in this section merely represent a small selection available to the composites scientist. Various FG sandwiches fabricated are characterized for three point bending condition. Influence of rubber up (rubber rich region towards the top) and ash up (ash rich region below the loading point) configurations are critically analyzed. Expected gradation in FG cores is presented in Figure 3 (rubber up and ash up).

The three point bending test is carried out in accordance with ASTM C 393 [42] using Instron universal testing machine of model 4206 with loading capacity ranging from 0.1 N to 150 kN. Figure 4 shows the sandwich sample mounted on flexural test set-up. The thickness to span ratio of the tested sandwich samples is 1:16. The crosshead displacement rate is maintained at 2 mm/min. The load deflection data is recorded at equal intervals up to a point at which the specimen shows the first sign of failure.

**Figure 3.** (a). Rubber Up condition in FG core, (b). Ash Up condition in FG core

**Figure 4.** Sandwich sample mounted on flexural test set-up

138 Finite Element Analysis – Applications in Mechanical Engineering

in Figure 2 (b).

analysis

**5. Experimental details** 

presented in Figure 3 (rubber up and ash up).

sandwich. Now, the earlier mentioned procedure-wise made FG core dipped in resin mixture is placed on the bottom stack of skins. Finally, over such an arrangement, the remaining layers of jute fabrics having undergone the same procedure for fabrication are stacked to constitute the top skin. A procedure of this nature should help in ensuring a greater degree of spread of the resin on the fibrillar jute. Following this, the excess resin is made to come out by a squeezing operation that is aided by tightening of the mold top plate. The mold assembly is then cured at room temperature for about 24-26 hours. The sandwich sample is withdrawn from the mold and trimmed to the required size. Similarly numbers of samples are made with various core thickness and orientation in skin as schematically illustrated in Figure 2. Figure 2 (a) shows top view with different orientations and while the front view with varying core thickness to total sandwich thickness (C/H ratio) is presented

**Figure 2.** (a). Orientation of jute strands in the sandwich skins, (b). Variation of C/H ratio considered for

The mechanical testing of sandwich composites to obtain parameters such as strength, stiffness etc. is a time consuming and often difficult process. It is, however, an essential process, and can be somewhat simplified by the testing of simple structures such as flat coupons. The data obtained from these tests can then be directly related with varying degrees of simplicity and accuracy to any structural shape. The test methods outlined in this section merely represent a small selection available to the composites scientist. Various FG sandwiches fabricated are characterized for three point bending condition. Influence of rubber up (rubber rich region towards the top) and ash up (ash rich region below the loading point) configurations are critically analyzed. Expected gradation in FG cores is

The three point bending test is carried out in accordance with ASTM C 393 [42] using Instron universal testing machine of model 4206 with loading capacity ranging from 0.1 N to 150 kN. Figure 4 shows the sandwich sample mounted on flexural test set-up. The thickness From load deflection data, bending modulus and strength are estimated using relations 1 and 2 respectively and the mean of five samples in each sandwich configuration is used for inference.

$$\text{Specific bending modulus} = \frac{\text{Flexural modulus}}{\text{Weight density}} = \frac{E\_{bending}}{\rho \times g} \tag{1}$$

$$\text{Specific bending strength} = \frac{\text{Ultimate strength}}{\text{Weight density}} = \frac{\sigma\_{u\_{\text{ bending}}}}{\left(\rho \propto g\right)}\tag{2}$$

where 3 6 *bending u M B H* and 4 *FL <sup>M</sup>*

## **5.1. Details of finite element modelling**

As outlined earlier, FE model helps to model the constituents of the FG composites and their sandwiches to study the interactions of these in load transfer and mechanisms influencing their failure. To understand and predict the effect of material as well as geometrical parameters on the mechanical behavior of FG fly ash filled rubber composites and their sandwiches finite element analysis can be a very effective technique. Towards this, a simple disctretized model is built in the software ANSYS® representing FG composites with properties varying from top layer to bottom representing gradation.

Flexural Behavior of Functionally Graded Sandwich Composite 141

**Figure 6.** Spring analogy for gradation in modulus of core material

**Figure 7.** FG rubber core configurations used in FEA

0.65 (upper) 0.75 (middle) 0.88 (bottom)

fly ash

20% 0.7575

Fly ash distributions taken into account for uniform configuration are 20%, 30% and 40% through the thickness. For these weight fractions Young's modulus is estimated using inverse rule of mixtures For skins, young's modulus is estimated by preparing five tensile samples of jute/epoxy with orientations of 00/900, 300/600 and 450/450 which are subsequently tested as per ASTM D3039 [44] guidelines. Density of skins is determined experimentally using procedure outlined in ASTM D792 [45]. Table 8 presents properties of core and skin used in the FE analysis. Results of FE analysis are compared with experimental values.

Wt. % of Element

1168.4

1163.9 (L1) 1167.5 (L2) 1172.5 (L3)

0.71 (L2) 1165.2 (L2) 0.79 (L3) 1168.2 (L3)

0.88 (L4) 1173.5 (L4)

1162.8 (L1)

2D Plane 42

FG Core

0.65 (L1)

Young's modulus (GPa) Density (Kg/m3) U L PL\* U L PL\*

Static analysis is performed using FEM software ANSYS 5.4. In this analysis a two dimensional model of a FG system is constructed and meshed with 4-node PLANE42 element. Three different mesh sizes are tested with 4-node elements to check the convergence of the model, based on which medium mesh size (element edge length is taken as 0.5) is selected. Number of nodes and elements used in the analysis are 800 and 5000 respectively.

Finite element values are compared with experimental ones for bending behavior of FG sandwich. At the contact surfaces of the layers and between layers and faces of sandwich glue conditions are applied to eliminate relative movement of layers with respect of each other. Furthermore, nodes are merged at the interface allowing proper coupling between layers and interfaces. Figure 5 shows finite element mesh with boundary conditions as a typical case considered for three point bending analysis. Skins are being represented by top and bottom portions of the structure whereas in between are the four layers having graded properties.

**Figure 5.** Finite element mesh with boundary condition for FG sandwich

While modeling gradation in ANSYS 5.4, the analogy of springs is used having differing stiffness (K1 < K2 < K3 < K4) from the top layer to bottom (Figure 6).

Sandwiches with FG core are modeled in FEA package ANSYS 5.4 [43] as emphasized before. Three different gradations of filler U (uniform), L (linear) and PL (piecewise linear) are considered during modeling of FG cores (Figure 7). Young's modulus and density of FG cores are determined for different weight fractions of fly ash from constituent properties are provided as input to FEA (Table 8).

**Figure 6.** Spring analogy for gradation in modulus of core material

**5.1. Details of finite element modelling** 

4 *FL <sup>M</sup>*

properties varying from top layer to bottom representing gradation.

**Figure 5.** Finite element mesh with boundary condition for FG sandwich

stiffness (K1 < K2 < K3 < K4) from the top layer to bottom (Figure 6).

provided as input to FEA (Table 8).

As outlined earlier, FE model helps to model the constituents of the FG composites and their sandwiches to study the interactions of these in load transfer and mechanisms influencing their failure. To understand and predict the effect of material as well as geometrical parameters on the mechanical behavior of FG fly ash filled rubber composites and their sandwiches finite element analysis can be a very effective technique. Towards this, a simple disctretized model is built in the software ANSYS® representing FG composites with

Static analysis is performed using FEM software ANSYS 5.4. In this analysis a two dimensional model of a FG system is constructed and meshed with 4-node PLANE42 element. Three different mesh sizes are tested with 4-node elements to check the convergence of the model, based on which medium mesh size (element edge length is taken as 0.5) is selected. Number of nodes and elements used in the analysis are 800 and 5000

Finite element values are compared with experimental ones for bending behavior of FG sandwich. At the contact surfaces of the layers and between layers and faces of sandwich glue conditions are applied to eliminate relative movement of layers with respect of each other. Furthermore, nodes are merged at the interface allowing proper coupling between layers and interfaces. Figure 5 shows finite element mesh with boundary conditions as a typical case considered for three point bending analysis. Skins are being represented by top and bottom portions of the structure whereas in between are the four layers having graded

While modeling gradation in ANSYS 5.4, the analogy of springs is used having differing

Sandwiches with FG core are modeled in FEA package ANSYS 5.4 [43] as emphasized before. Three different gradations of filler U (uniform), L (linear) and PL (piecewise linear) are considered during modeling of FG cores (Figure 7). Young's modulus and density of FG cores are determined for different weight fractions of fly ash from constituent properties are

where 3

respectively.

properties.

*bending u*

6

and

*M B H*


**Figure 7.** FG rubber core configurations used in FEA

Fly ash distributions taken into account for uniform configuration are 20%, 30% and 40% through the thickness. For these weight fractions Young's modulus is estimated using inverse rule of mixtures For skins, young's modulus is estimated by preparing five tensile samples of jute/epoxy with orientations of 00/900, 300/600 and 450/450 which are subsequently tested as per ASTM D3039 [44] guidelines. Density of skins is determined experimentally using procedure outlined in ASTM D792 [45]. Table 8 presents properties of core and skin used in the FE analysis. Results of FE analysis are compared with experimental values.



Flexural Behavior of Functionally Graded Sandwich Composite 143

(3)

Experimental density values are subjected to statistical analysis (MINITAB 14) to propose

<sup>3</sup> Density Kg / m 1099 11.6 Fly ash weight %

for this might be lower specific weight with increasing skin orientation.

29.7 C / H Ratio [0.459 Jute Orientation]

Equation 3 comes handy, which predicts density for large number of samples with varying combination of factors within the range of chosen levels without experimentation. Density increases with filler content as well as with C/H ratio (core to thickness ratio) being positive coefficients while shows a decreasing trend with increase in jute orientation. Obvious reason

Three point bending behavior of a FG sandwich composite is investigated under flexural loading condition. Results are analyzed for specific modulus and specific bending strength. Load deflection data is traced all along the path. The load and corresponding deflection data is noted at equal intervals up to a maximum load at which the specimen shows the first sign of failure (point 'A'). The load and deflections obtained during testing are plotted. A typical

Load-displacement consists of an initial linear part followed by a nonlinear portion (Figure 8). A nonlinear mechanics of materials analysis that accounts for the combined effect of the nonlinear behavior of the facings and core materials (material nonlinearity) and the large deflections of the beam (geometric nonlinearity) are observed. The nonlinear load-deflection behavior of the beams is attributed to the combined effect of material and geometric nonlinearity. The material nonlinearity of the sandwich beam is due to the nonlinear normal stress-strain behavior of the facing material and the FG core. For long beam spans, even though there is a geometric nonlinearity effect, the overall load-deflection curve of the beam

For long beam spans the nonlinearity of the load-deflection curve is mainly due to the combined effect of the facings nonlinearity and the large deflections of the beam. Both effects, however, have a small contribution to the load-deflection behavior, which shows a small deviation from linearity. Some of the general observations made are listed below.

1. The load decreases sharply after the end of the elastic region due to failure initiation in

2. All samples have shown small linear region (B to C) before skin failure in compressive

3. Variation in displacement value at which peak load is observed for various types of FG

From load deflection data the average specific modulus and strength for five samples (Table

regression equation which is presented in equation 3.

load deflection curve is shown in Figure 8.

does not deviate much from linearity.

sandwich composites (A to B).

sandwiches is considerable.

**6.1. Specific bending modulus** 

4. The failure originates on the tensile side.

10) are estimated using equations 1 and 2.

side.

L-layer, \*L1-top layer (rubber rich), L4-bottom layer (ash rich)

**Table 8.** Core and skin properties used in FEA

Bending tested samples are subjected to visual observation using regular photography technique for FG sandwich. These methods came in handy during the characterization of failures especially in impact failed samples.

## **6. Results and discussion**

FG sandwiches are tested for Density, the results of which are presented in Table 9.


**Table 9.** Density results of FG sandwiches

Experimental density values are subjected to statistical analysis (MINITAB 14) to propose regression equation which is presented in equation 3.

$$\begin{aligned} \text{Density } \left( \text{Kg} / \text{m}^3 \right) &= 1099 + \left[ 11.6 \times \text{Fly ash weight } \% \right] + \\ + \left[ 29.7 \times \text{C/H Ratio} \right] - \left[ 0.459 \times \text{Jute Orionation} \right] \end{aligned} \tag{3}$$

Equation 3 comes handy, which predicts density for large number of samples with varying combination of factors within the range of chosen levels without experimentation. Density increases with filler content as well as with C/H ratio (core to thickness ratio) being positive coefficients while shows a decreasing trend with increase in jute orientation. Obvious reason for this might be lower specific weight with increasing skin orientation.

Three point bending behavior of a FG sandwich composite is investigated under flexural loading condition. Results are analyzed for specific modulus and specific bending strength. Load deflection data is traced all along the path. The load and corresponding deflection data is noted at equal intervals up to a maximum load at which the specimen shows the first sign of failure (point 'A'). The load and deflections obtained during testing are plotted. A typical load deflection curve is shown in Figure 8.

Load-displacement consists of an initial linear part followed by a nonlinear portion (Figure 8). A nonlinear mechanics of materials analysis that accounts for the combined effect of the nonlinear behavior of the facings and core materials (material nonlinearity) and the large deflections of the beam (geometric nonlinearity) are observed. The nonlinear load-deflection behavior of the beams is attributed to the combined effect of material and geometric nonlinearity. The material nonlinearity of the sandwich beam is due to the nonlinear normal stress-strain behavior of the facing material and the FG core. For long beam spans, even though there is a geometric nonlinearity effect, the overall load-deflection curve of the beam does not deviate much from linearity.

For long beam spans the nonlinearity of the load-deflection curve is mainly due to the combined effect of the facings nonlinearity and the large deflections of the beam. Both effects, however, have a small contribution to the load-deflection behavior, which shows a small deviation from linearity. Some of the general observations made are listed below.


### **6.1. Specific bending modulus**

142 Finite Element Analysis – Applications in Mechanical Engineering

0.68 (upper) 0.865 (middle) 1.15 (bottom)

0.71 (upper) 1.015 (middle) 1.66 (bottom)

L-layer, \*L1-top layer (rubber rich), L4-bottom layer (ash rich)

**Table 8.** Core and skin properties used in FEA

failures especially in impact failed samples.

**6. Results and discussion** 

**Table 9.** Density results of FG sandwiches

fly ash

30% 0.89

40% 1.1

FG Core

0.68 (L1)

0.71 (L1)

Jute / Epoxy skin Orientation Ex (GPa) Ey (GPa) Density (Kg/m3) 00/900 3.25 2.5 1468 300/600 1.63 1.25 1451.2 450/450 2.29 1.77 1444.3

Young's modulus (GPa) Density (Kg/m3) U L PL\* U L PL\*

Wt. % of Element

1330.2

1444.7

Bending tested samples are subjected to visual observation using regular photography technique for FG sandwich. These methods came in handy during the characterization of

Sandwich code Trial-1 Trial-2 Trial-3 Trial-4 Trial-5 Density (Kg/m3) W20R0.4O0 1325.6 1328.9 1329.4 1332.8 1330.8 1329.5 W20R0.6O30 1333.5 1334.8 1336.2 1336.4 1331.6 1334.5 W20R0.8O45 1342.8 1350.7 1348.6 1345.5 1348.9 1347.3 W30R0.4O30 1465.8 1464.6 1460.3 1462.1 1463.2 1463.2 W30R0.6O45 1435.2 1435.9 1431.9 1432.8 1433.7 1433.9 W30R0.8O0 1467.1 1466.9 1469.3 1470.5 1467.2 1468.2 W40R0.4O45 1547.6 1549.8 1551.7 1550.6 1548.8 1549.7 W40R0.6O0 1599.5 1598.8 1595.6 1594.4 1596.2 1596.9 W40R0.8O30 1564.1 1561.8 1562.4 1560.9 1563.8 1562.6

FG sandwiches are tested for Density, the results of which are presented in Table 9.

1324.5 (L1) 1331.1 (L2) 1336.9 (L3)

1435.2 (L1) 1445.7 (L2) 1452.6 (L3)

0.79 (L2) 1328.4 (L2) 0.94 (L3) 1334.6 (L3)

1.15 (L4) 1337.2 (L4)

0.88 (L2) 1442.8 (L2) 1.15 (L3) 1450.6 (L3)

1.66 (L4) 1455.2 (L4)

1323.9 (L1)

1434.9 (L1)

From load deflection data the average specific modulus and strength for five samples (Table 10) are estimated using equations 1 and 2.

Flexural Behavior of Functionally Graded Sandwich Composite 145

153.1

151.4

192.21

159.53

120.1

110.34

159.21

121.44

119.3

117.17

154.45

125.45

Sp. flexural modulus (MPa/Nm-1) Sp. flexural strength (MPa/Nm-1)

Up Avg. Ash Up Avg. Rubber Up Avg. Ash Up Avg.

148.7

149.3

188.98

155.23

6018.2

1692.71

4065.98

6062.65

6520.7 6070.4 151.8 123.3 6521.4 6060.9 161.32 127.56 6518.2 6058.5 164.2 128.9 6515.2 6073.15 165.1 126.05

It can be clearly seen from the table that, rubber up configuration registered higher results compared to ash up condition for both the properties in the range of 7 to 30%. Constrained straining and resisting forces set up in the FG core might be the reasons for such an

4370.28 4068.63 193.5 155.29 4365.39 4065.98 199.7 152.8 4360.87 4059.3 191.49 155.7 4360.86 4075.86 187.38 149.25

2138.69 1692.67 148.7 121.56 2141.92 1688.2 152.4 117.17 2139.26 1690.4 159.3 120.23 2140.42 1690.3 147.3 116.55

6562.8 6018.2 149.2 121.3 6570.4 6018.2 157.3 118.3 6560.55 6020.9 151.2 121.54 6560.21 6015.5 159.1 115.26

Sample coding

W30R0.8O0

W40R0.4O45

W40R0.6O0

W40R0.8O30

Rubber

6559.3

2134.3

4372.5

6515.5

6562.65

2138.92

4365.98

6518.2

observation in bending test as depicted in Figure 9.

**Figure 9.** Loads acting on FG sandwich in bending test

**Table 10.** Specific bending modulus and strength for FG sandwich

6018.2

1702

4060.12

6050.3

**Figure 8.** Load-deflection behavior under three point bend test for sandwich



**Table 10.** Specific bending modulus and strength for FG sandwich

Sample coding

W20R0.4O0

W20R0.6O30

W20R0.8O45

W30R0.4O30

W30R0.6O45

Rubber

3945.23

5322.6

7391.4

2996.2

4043.3

**Figure 8.** Load-deflection behavior under three point bend test for sandwich

3410.9

4540.15

6150.4

2390.31

3533.59

3953.07

5319.4

7387.91

3001.3

4045.36

Sp. flexural modulus (MPa/Nm-1) Sp. flexural strength (MPa/Nm-1)

Up Avg. Ash Up Avg. Rubber Up Avg. Ash Up Avg.

132.7

85.3

54.59

141.4

94.76

128.1

88.1

54.59

141.4

101.23

103.55

72.3

45.23

115.43

80.17

98.81

70.7

48.75

113.26

78.75

3410.29

4545.36

6155.14

2398.92

3533.57

4047.6 3528.61 99.14 81.34 4042.4 3531.75 95.35 75.46 4045.36 3523.16 106.4 79.1 4048.12 3550.73 110.5 77.68

3003.1 2398.92 141.4 116.23 3004.5 2398.92 145.5 113.26 3001.3 2398.92 149.1 111.59 3001.4 2407.53 129.6 109.79

7387.91 6160.73 54.5 50.4 7377.4 6155.14 57.7 47.9 7393.4 6155.14 58.1 46.8 7389.42 6154.28 48.06 53.42

5306.8 4545.39 88.1 70.7 5321.3 4539.4 83.5 71.9 5322.9 4544.95 92.4 70.7 5323.4 4556.9 91.2 67.9

3933.7 3404.2 128.1 105.32 3961.5 3419.16 127.7 95.47 3963.7 3416.4 128.1 99.1 3961.2 3400.8 123.9 90.61

> It can be clearly seen from the table that, rubber up configuration registered higher results compared to ash up condition for both the properties in the range of 7 to 30%. Constrained straining and resisting forces set up in the FG core might be the reasons for such an observation in bending test as depicted in Figure 9.

**Figure 9.** Loads acting on FG sandwich in bending test

Rubber up condition of FG core in sandwich represents ash rich region on tensile side. Crack initiation is observed to be from tensile region to compressive region in pre sent loading case. In rubber up condition, as stiffer zone is near tensile region, sandwich can take up higher loads resulting in better performance compared to homogenous cores and ash up condition in FG core. Thereby, such sandwiches are excellent examples of optimized designs.

Flexural Behavior of Functionally Graded Sandwich Composite 147

Results of specific bending strength from Table 10 are statistically analyzed and are used to

Level 1 38.6 42.92 43.84 Level 2 42.27 41.56 41.99 Level 3 44.44 40.83 39.48 Effect 5.85 2.09 4.36 Rank 1 3 2

From SN response Table, it can be seen that specific bending strength behavior is prominently governed by fly ash weight % followed by orientation and C/H ratio. Figure 11

From SN response plot shown in Figure 11, the best combination for specific strength is a sample with fly ash content of 40%, C/H of 0.4 and orientation of 00/900. Reasons for this could be stiffening effect due to high modulus filler and larger skin-epoxy component for lower C/H ratios. Similar results are observed for ash up configuration. Even though W20R0.8O45 and W40R0.6O0 are showing higher values (Table 10) for modulus and strength respectively, inference on basis of these will not lead to an appropriate conclusion. The reason being these values are merely based on average of means. Inference on the grounds of SN analysis leads to a meaningful conclusion as it takes means and data spread into account. By the SN ratio analysis the best sandwich configurations are W20R0.8O0 and W40R0.4O0 for specific modulus and strength respectively. Similar observation is noted for ash

presents SN plot for specific bending strength incase of rubber up condition.

Fly ash weight % C/H ratio Orientation

**6.2. Specific bending strength** 

rank the variables as presented in Table 12.

**Table 12.** SN ratio table for specific bending strength (Rubber Up)

**Figure 11.** Plot of SN ratio in specific bending strength (Rubber Up)

Developed FG sandwiches can be used in practical cases wherein structures are continuously subjected to bending loads. Depending upon whether load is acting downwards or upwards sandwiches can be suitable placed with either rubber up or ash up configuration as regards to FG cores.

Figure 10 shows the signal to noise (SN) response plot for specific bending modulus with respect to the parameters under study. Response of SN ratio in Specific bending modulus for Rubber Up condition is presented in Table 11.

**Figure 10.** Variation of SN ratio in specific bending modulus (Rubber Up)


**Table 11.** SN ratio table for specific bending modulus (Rubber Up)

From the data analysis, vide response Table 11, it is seen that C/H ratio and fly ash % exhibit greater influence compared to the orientation. It is further observed from the Table and Figure 10 that samples with fly ash content of 20%, C/H of 0.8 and an orientation of 00/900 possess highest specific bending modulus. This could be due to higher C/H ratio implying larger rubber rich region imparting higher modulus to sandwich system.

## **6.2. Specific bending strength**

146 Finite Element Analysis – Applications in Mechanical Engineering

configuration as regards to FG cores.

for Rubber Up condition is presented in Table 11.

**Figure 10.** Variation of SN ratio in specific bending modulus (Rubber Up)

**Table 11.** SN ratio table for specific bending modulus (Rubber Up)

larger rubber rich region imparting higher modulus to sandwich system.

designs.

Rubber up condition of FG core in sandwich represents ash rich region on tensile side. Crack initiation is observed to be from tensile region to compressive region in pre sent loading case. In rubber up condition, as stiffer zone is near tensile region, sandwich can take up higher loads resulting in better performance compared to homogenous cores and ash up condition in FG core. Thereby, such sandwiches are excellent examples of optimized

Developed FG sandwiches can be used in practical cases wherein structures are continuously subjected to bending loads. Depending upon whether load is acting downwards or upwards sandwiches can be suitable placed with either rubber up or ash up

Figure 10 shows the signal to noise (SN) response plot for specific bending modulus with respect to the parameters under study. Response of SN ratio in Specific bending modulus

Fly ash weight % C/H ratio Orientation

Level 1 74.61 69.36 73.69 Level 2 72.68 73.15 73.45 Level 3 71.90 76.66 72.04 Effect 2.71 7.30 1.66 Rank 2 1 3

From the data analysis, vide response Table 11, it is seen that C/H ratio and fly ash % exhibit greater influence compared to the orientation. It is further observed from the Table and Figure 10 that samples with fly ash content of 20%, C/H of 0.8 and an orientation of 00/900 possess highest specific bending modulus. This could be due to higher C/H ratio implying Results of specific bending strength from Table 10 are statistically analyzed and are used to rank the variables as presented in Table 12.


**Table 12.** SN ratio table for specific bending strength (Rubber Up)

From SN response Table, it can be seen that specific bending strength behavior is prominently governed by fly ash weight % followed by orientation and C/H ratio. Figure 11 presents SN plot for specific bending strength incase of rubber up condition.

**Figure 11.** Plot of SN ratio in specific bending strength (Rubber Up)

From SN response plot shown in Figure 11, the best combination for specific strength is a sample with fly ash content of 40%, C/H of 0.4 and orientation of 00/900. Reasons for this could be stiffening effect due to high modulus filler and larger skin-epoxy component for lower C/H ratios. Similar results are observed for ash up configuration. Even though W20R0.8O45 and W40R0.6O0 are showing higher values (Table 10) for modulus and strength respectively, inference on basis of these will not lead to an appropriate conclusion. The reason being these values are merely based on average of means. Inference on the grounds of SN analysis leads to a meaningful conclusion as it takes means and data spread into account. By the SN ratio analysis the best sandwich configurations are W20R0.8O0 and W40R0.4O0 for specific modulus and strength respectively. Similar observation is noted for ash up configuration. Regression equation is proposed based on the experimental data for specific bending properties are presented in equations 4-7.

Flexural Behavior of Functionally Graded Sandwich Composite 149

The breaking load taken from experiment is applied on FE model. For this applied load, maximum stress (von misses criteria) is recorded and finally specific strength is determined by taking the ratio of maximum stress to the weight of sample. The specific strength values obtained from FEA for three variations in gradation (Uniform-U, Linear-L and Piecewise

W20R0.4O0 115.4 119.6 132.75 128.1 3.50 W20R0.6O30 78.2 81.5 92.58 88.1 4.84 W20R0.8O45 46.9 48.9 58.58 54.59 6.81 W30R0.4O30 125.1 130.2 147.34 141.4 4.03 W30R0.6O45 84.4 88.8 110.38 101.23 8.29 W30R0.8O0 129.7 137.6 160.88 153.1 4.84 W40R0.4O45 126.6 131.3 169.11 151.4 10.47 W40R0.6O0 175.2 179.5 201.42 192.21 4.57 W40R0.8O30 140.2 145.6 165.7 159.53 3.72

It is significant to note that the experimental results for specific bending strength match well with FEA values especially for the ones with PL gradation. It is observed that bending strength obtained from FEA is slightly higher than experimental values. This could be due to inability of modeling inhomogenities creeping in during the processing of samples which

Within the elastic region of the load-displacement curve (Figure 8), where no damage is induced, the responses of all specimens to the applied loads are quite similar. This is visible in the form of nearly constant slope in the elastic region of the load-displacement curves. It is observed that the failure starts in the form of crack origination on the tensile side of the specimen as displacement increases. On further loading, the skin of the sandwich composite that is on the tensile side tends to fracture, causing the final failure of the specimen. However, it is not significant enough to lead to the final failure of the specimen. It is observed that the entire specimen fractures at a much later instant of skin fracture. Appearance of small linear region (B to C in Figure 8) at the end in the load-displacement curves is due to stiffening of FG core before final failure. During the loading process, deformation also takes place in the compression side of the specimen. Cracks initiate from the tensile side and propagate to the compressive side within the core in all sandwiches.

It is worth discussing the mode of failure. Sandwich samples tested under bending did not display the distinct separation into pieces at failure. The FG core being compliant is

Experimental % Error with PL U L PL

linear-PL) and with experimental approach is presented in Table 13.

FEA

**Table 13.** Specific bending strength (MPa/Nm-1) results for sandwich

may result in lowering specific strength.

**6.4. Discussion on fractured samples** 

Sandwich configuration

$$\begin{aligned} \text{(29.8 \times C/H Ratio)} & \begin{bmatrix} \text{(29.8 \times C/H Ratio)} \end{bmatrix} \\ & - \begin{bmatrix} \text{29.8 \times C/H Ratio} \end{bmatrix} - \begin{bmatrix} \text{0.912 \times Jute Orionation} \end{bmatrix} \end{aligned} \tag{7}$$

## **6.3. Finite element analysis**

Specific bending strength is estimated by simulating the sample and loading (Gupta et al. 2008) in FEA. Figure 12 represents the plot for bending stress in the sample for one typical loading case.

**Figure 12.** Bending stress in x-direction for typical case in FG sandwich

The breaking load taken from experiment is applied on FE model. For this applied load, maximum stress (von misses criteria) is recorded and finally specific strength is determined by taking the ratio of maximum stress to the weight of sample. The specific strength values obtained from FEA for three variations in gradation (Uniform-U, Linear-L and Piecewise linear-PL) and with experimental approach is presented in Table 13.


**Table 13.** Specific bending strength (MPa/Nm-1) results for sandwich

It is significant to note that the experimental results for specific bending strength match well with FEA values especially for the ones with PL gradation. It is observed that bending strength obtained from FEA is slightly higher than experimental values. This could be due to inability of modeling inhomogenities creeping in during the processing of samples which may result in lowering specific strength.

## **6.4. Discussion on fractured samples**

148 Finite Element Analysis – Applications in Mechanical Engineering

**6.3. Finite element analysis** 

loading case.

specific bending properties are presented in equations 4-7.

**Figure 12.** Bending stress in x-direction for typical case in FG sandwich

up configuration. Regression equation is proposed based on the experimental data for

 

 

(4)

(5)

(6)

(7)

Specific Bending Modulus Rubber Up 1151 – 60.6 Weight % of Fly ash 480 C / H Ratio – 8.38 Jute Orientation

Specific Bending Strength Rubber Up 70.4 3.87 Weight % of Fly ash – 44.7 C / H Ratio – 1.19 Jute Orientation

Specific Bending Modulus Ash Up 342 – 38.2 Weight % of Fly ash 8945 C / H Ratio – 14.2 Jute Orientation

Specific Bending Strength Ash Up 54.2 2.98 Weight % of Fly ash – 29.8 C / H Ratio – 0.912 Jute Orientation

Specific bending strength is estimated by simulating the sample and loading (Gupta et al. 2008) in FEA. Figure 12 represents the plot for bending stress in the sample for one typical

Within the elastic region of the load-displacement curve (Figure 8), where no damage is induced, the responses of all specimens to the applied loads are quite similar. This is visible in the form of nearly constant slope in the elastic region of the load-displacement curves. It is observed that the failure starts in the form of crack origination on the tensile side of the specimen as displacement increases. On further loading, the skin of the sandwich composite that is on the tensile side tends to fracture, causing the final failure of the specimen. However, it is not significant enough to lead to the final failure of the specimen. It is observed that the entire specimen fractures at a much later instant of skin fracture. Appearance of small linear region (B to C in Figure 8) at the end in the load-displacement curves is due to stiffening of FG core before final failure. During the loading process, deformation also takes place in the compression side of the specimen. Cracks initiate from the tensile side and propagate to the compressive side within the core in all sandwiches.

It is worth discussing the mode of failure. Sandwich samples tested under bending did not display the distinct separation into pieces at failure. The FG core being compliant is

observed to be successfully absorbing media. Basically two types of failure mechanisms observed are skin cracking and delamination between skins and core. Figure 13 shows the failed sandwich specimens with their failure modes.

Flexural Behavior of Functionally Graded Sandwich Composite 151

higher value of specific strength. Rubber up configuration registered higher results compared to ash up condition for modulus and strength. The ash up condition recorded about 30% increase in strength. Increasing fly ash weight fraction rendered an increase in bending strength of about 29% for rubber up condition. Specific strength values estimated from FEA for bending loads match well with experimental results especially for piecewise

 and Satyabodh M Kulkarni *Mechanical Engineering, National Institute of Technology Karnataka, Surathkal, India* 

Graduate school of the university of Florida; 2005.

Theoretical and Applied Mechanics 2008;35(1-3),105-118.

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gradation.

**Author details** 

**8. References** 

1997.

1998.

1999.

Corresponding Author

 \*

Mrityunjay R. Doddamani\*

**Figure 13.** Sandwich failure modes under three point bending loads

The sandwich beams failed at the center of the two supporting rollers. In this portion of the beam, the shear force is zero and only the pure bending exists. Thus, the sandwich samples are capable of resisting higher bending moment. As the load on the specimen is increased, failures first start under the loads in tensile region and then they propagate towards the compressive zone through compliant FG core. All the samples failed under skin tension or compression and skin - core debonding. The sandwiches with higher C/H ratio have shown skin - core debonding. FG core takes up most of the load applied for higher C/H ratios (lesser skin thickness). Since core is made up of rubber composite being compliant in nature, relative movements are set up with respect to skin resulting in inter laminar shear stresses. As magnitude of these stresses crosses the adhesive strength delamination creeps in. Some sandwich samples are seen to be intact even after the first sign of failure. These samples exhibited a spring back effect. Samples bearing lower C/H ratio have failed mainly because of skin cracking along the jute orientation. Few samples failed due to shearing at skin-core interface displayed step formation.

## **7. Conclusions**

This section highlights the significant conclusions drawn from the results presented earlier. Major inferences from both experimental and finite element investigations are discussed below.

Density of FG sandwiches increases with filler content and C/H ratio while decreases with jute orientation. An experimental investigation of sandwiches under bending loads for specific modulus and specific strength shows that C/H ratio and fly ash weight fraction are the influential factors respectively. Specific bending modulus in both cases (i.e. rubber up and ash up) the sample W20R0.8O0 registered the higher value while W40R0.4O0 shows higher value of specific strength. Rubber up configuration registered higher results compared to ash up condition for modulus and strength. The ash up condition recorded about 30% increase in strength. Increasing fly ash weight fraction rendered an increase in bending strength of about 29% for rubber up condition. Specific strength values estimated from FEA for bending loads match well with experimental results especially for piecewise gradation.

## **Author details**

150 Finite Element Analysis – Applications in Mechanical Engineering

failed sandwich specimens with their failure modes.

**Figure 13.** Sandwich failure modes under three point bending loads

interface displayed step formation.

**7. Conclusions** 

below.

observed to be successfully absorbing media. Basically two types of failure mechanisms observed are skin cracking and delamination between skins and core. Figure 13 shows the

The sandwich beams failed at the center of the two supporting rollers. In this portion of the beam, the shear force is zero and only the pure bending exists. Thus, the sandwich samples are capable of resisting higher bending moment. As the load on the specimen is increased, failures first start under the loads in tensile region and then they propagate towards the compressive zone through compliant FG core. All the samples failed under skin tension or compression and skin - core debonding. The sandwiches with higher C/H ratio have shown skin - core debonding. FG core takes up most of the load applied for higher C/H ratios (lesser skin thickness). Since core is made up of rubber composite being compliant in nature, relative movements are set up with respect to skin resulting in inter laminar shear stresses. As magnitude of these stresses crosses the adhesive strength delamination creeps in. Some sandwich samples are seen to be intact even after the first sign of failure. These samples exhibited a spring back effect. Samples bearing lower C/H ratio have failed mainly because of skin cracking along the jute orientation. Few samples failed due to shearing at skin-core

This section highlights the significant conclusions drawn from the results presented earlier. Major inferences from both experimental and finite element investigations are discussed

Density of FG sandwiches increases with filler content and C/H ratio while decreases with jute orientation. An experimental investigation of sandwiches under bending loads for specific modulus and specific strength shows that C/H ratio and fly ash weight fraction are the influential factors respectively. Specific bending modulus in both cases (i.e. rubber up and ash up) the sample W20R0.8O0 registered the higher value while W40R0.4O0 shows Mrityunjay R. Doddamani\* and Satyabodh M Kulkarni *Mechanical Engineering, National Institute of Technology Karnataka, Surathkal, India* 

## **8. References**


<sup>\*</sup> Corresponding Author

sandwich beams with functionally graded viscoelastic core. Sharjah, United Arab Emirates, 125 - 128.

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[38] Mohan R, Kishore, Shridhar MK, Rao RMVGK. Compressive strength of jute-glass hybrid fibre composites. Journal of Materials Science Letters 1982;2(2),99-102. [39] Shah AN, Lakkad, SC. Mechanical properties of jute-reinforced plastics. Fibre Science

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[41] Mohan, Rengarajan, Kishore. Jute-Glass sandwich composites. Journal of Reinforced

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[14] Venkataraman S, Sankar BV. 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference 2001: conference proceeding. Analysis of sandwich beams with functionally graded core. AIAA-2001-1281, Seattle,

[15] Kirigulige MS, Kitey R, Tippur HV. Dynamic fracture behaviour of model sandwich structures with functionally graded core: a feasibility study. Composites Science and

[16] Pollien A, Conde Y, Pambaguian L, Mortensen A. Graded open-cell aluminium foam core sandwich beams. Materials Science and Engineering: A 2005;404(1-2),9-18. [17] Gupta N. A functionally graded syntactic foam material for high energy absorption

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**Chapter 7** 

© 2012 Dridi, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Dridi, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the composite industry, the shearing behaviour of dry woven plays a crucial role in fabric formability when doubly curved surfaces must be covered [1-9]. The ability of fabric to

It has been proved that shear rigidity can be calculated from the tensile properties along a 45° bias direction. Bias Extension tests are simple to perform and provide reasonably repeatable results [13-14]. Extensive investigations have been carried out on the textile fabric

The tests were conducted simply using two pairs of plates, clamping a rectangular piece of woven material such that the two groups of yarns are orientated ±45° to the direction of external tensile force. The ratio between the initial length and width of the specimen is

λ = l0/w0 (see Figure 1a). In the case of λ =2, the deformed conguration of the material can be represented by Figure1b, which includes seven regions. Triangular regions C adjacent to the xture remain undeformed, while the central square region A and other four triangular regions B undergo

The present chapter focuses on numerical analysis of Bias Extension test using an

In the first, analytical responses of the Bias Extension test and the traction test on 45° are developed using the proposed model. Strain and stress states in specimen during these tests

orthotropic hyperelastic continuum model of woven fabric.

shear within a plain enables it to t three-dimensional surfaces without folds [10-12].

**Finite Element Analysis of** 

Additional information is available at the end of the chapter

Samia Dridi

http://dx.doi.org/10.5772/46161

**1. Introduction** 

in Bias Extension test [15]

dened as aspect ratio:

shear deformation [16-17].

are detailed.

**Bias Extension Test of Dry Woven** 

[45] ASTM D792. Standard test methods for density and specific gravity (Relative density) of plastics by displacement. ASTM International, PA, USA, 2008.

**Chapter 7** 

## **Finite Element Analysis of Bias Extension Test of Dry Woven**

Samia Dridi

154 Finite Element Analysis – Applications in Mechanical Engineering

composite materials. ASTM International, PA, USA, 2008.

of plastics by displacement. ASTM International, PA, USA, 2008.

[44] ASTM D3039. Standard Test Method for Tensile properties of polymer matrix

[45] ASTM D792. Standard test methods for density and specific gravity (Relative density)

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/46161

## **1. Introduction**

In the composite industry, the shearing behaviour of dry woven plays a crucial role in fabric formability when doubly curved surfaces must be covered [1-9]. The ability of fabric to shear within a plain enables it to t three-dimensional surfaces without folds [10-12].

It has been proved that shear rigidity can be calculated from the tensile properties along a 45° bias direction. Bias Extension tests are simple to perform and provide reasonably repeatable results [13-14]. Extensive investigations have been carried out on the textile fabric in Bias Extension test [15]

The tests were conducted simply using two pairs of plates, clamping a rectangular piece of woven material such that the two groups of yarns are orientated ±45° to the direction of external tensile force. The ratio between the initial length and width of the specimen is dened as aspect ratio:

λ = l0/w0 (see Figure 1a).

In the case of λ =2, the deformed conguration of the material can be represented by Figure1b, which includes seven regions. Triangular regions C adjacent to the xture remain undeformed, while the central square region A and other four triangular regions B undergo shear deformation [16-17].

The present chapter focuses on numerical analysis of Bias Extension test using an orthotropic hyperelastic continuum model of woven fabric.

In the first, analytical responses of the Bias Extension test and the traction test on 45° are developed using the proposed model. Strain and stress states in specimen during these tests are detailed.

© 2012 Dridi, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Dridi, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Figure 1.** Kinematic of Bias Extension test, a: Initial state, b: Deformed state

In the second, the proposed model is implanted into Abaqus/Explicit to simulate the Bias Extension test of three aspect ratios.

Exploiting numerical results, we studied the effect of the ratio between shearing and traction rigidities on homogeneities of stress and strain in the central zone of three Finite Element Models (FEM).

## **2. The proposed hyperelastic model**

One of significant characteristics of the woven structure is the existence of two privileged material directions: warp and weft. We considered that the fabric is a continuous structure having two privileged material directions defined by the two unit tensors *M***1** and *M***2** as follows:

$$
\mathbf{M}\_1 = \vec{M}\_1 \otimes \vec{M}\_1 \; ; \; \mathbf{M}\_2 = \vec{M}\_2 \otimes \vec{M}\_2 \tag{1}
$$

Finite Element Analysis of Bias Extension Test of Dry Woven 157

*I 1 2 g E:Eg* (6)

   , <sup>2</sup> *<sup>g</sup>* )

(7)

1 2 12 *W WI I I* () ( , , ) *E* (5)

. *I12* measured the sliding in the plane ( <sup>1</sup> *<sup>g</sup>*

<sup>1</sup> ( ) <sup>2</sup>

*S= g g g Eg g Eg* (8)

*W kI kI k II kI* (9)

can be presented by one of flowing

(11)

1 1 12 2 2 2 12 1 3 ( )( ) ( ) *kI k I kI k I k <sup>1</sup> 2 12 21 S g g g Eg g Eg* (10)

<sup>1</sup> ( )

2

We choose following invariants to present the strain energy function:

), are defined as follows:

*WWW I I II* 

> 1 1 2 2

between two groups of yarns. *k3* presented the shearing rigidity of woven.

2

is the angle between *M*<sup>1</sup>

The second Piola Kirchhoff stress tensor *S* can be written as:

*1*



Green Lagrange strain tensor *E* in the base *<sup>i</sup> g*

This leads to the constitutive equation:

*Ii* measured elongations along directions *<sup>i</sup> g*

, <sup>2</sup> *<sup>g</sup>*

*<sup>i</sup> <sup>I</sup> <sup>i</sup> g :E <sup>i</sup>* 1..2 ; 1/2 12

witch is the angle variation between warp and weft direction. Components *ij <sup>g</sup> <sup>E</sup>* of *E* in the

 *,i IE*

and *M*<sup>2</sup>

2 12 12

A simplified hyperplastic model is proposed. It is based on following assumptions:

 *1 2 12 21*

> 22 2 1 1 2 2 12 1 2 3 12

So *k1* and *k2* presented tensile rigidities in yarns directions. *k12* described the interaction

The relation between components *Sgij* of second Piola Kirchhoff stress tensor *S* and *Egij* of

11 11 1 12 22 12 2 22 3 12 12

*g g g g g g*

*S E k k S kk E S E k* 

0 0

0 0

1 12 12 1 2 1 1 ( 1) 1..2, cos( ) 2 2 *i gij <sup>g</sup> I E*

.

are yarns extensions (ratio between deformed and initial lengths) along directions

Where

1 and <sup>2</sup> 

of <sup>1</sup> *g*

 and 2 *<sup>g</sup>* . 

expressions

reference system ( <sup>1</sup> *g*

Where *M*<sup>1</sup> and *<sup>M</sup>*<sup>2</sup> are two unit vectors carried by two yarns directions. The sign indicate the tensor product. In the reference configuration, these privileged material directions are supposed to be orthogonal and they are defined by *g***1** and *g***2** presented by Equation 2.

$$\mathbf{g}\_1 = \vec{\mathbf{g}}\_1 \otimes \vec{\mathbf{g}}\_1 \quad \text{ } \mathbf{g}\_2 = \vec{\mathbf{g}}\_2 \otimes \vec{\mathbf{g}}\_2 \tag{2}$$

In Lagrangian formulation, the hyperelastic behavior is defined by the strain energy function *W*(*E*) depending of Green-Lagrange tensor components [18-21].

The second Piola Kirchhoff stress tensor *S* derives is presented in Equation 3:

$$\mathbf{S} = \frac{\partial W}{\partial E} \tag{3}$$

The physical behaviour is completely defined by the choice of *W*(*E*). The woven structures is very thin, we are interested more particularly in plane solicitations (plane stress or strain) in the plan ( <sup>1</sup> *g* , <sup>2</sup> *<sup>g</sup>* ). We supposed that *W*(*E*) is an isotropic function of variables (*E,* **g1**, **g2**). Using the representation theorems of isotropic functions, strain energy function *W*(*E*) depends of invariants:

$$\text{g}\_i \text{:} \text{E} \quad \text{g}\_i \text{:} \text{E}^2 \quad \text{tr}(\text{E}^3) \text{ (} i = 1..2 \text{)} \tag{4}$$

We choose following invariants to present the strain energy function:

$$\mathcal{W}(\mathbf{E}) = \mathcal{W}(I\_{1'}I\_{2'}I\_{12}) \tag{5}$$

Where

156 Finite Element Analysis – Applications in Mechanical Engineering

Extension test of three aspect ratios.

a

**2. The proposed hyperelastic model** 

Models (FEM).

follows:

Where *M*<sup>1</sup>

the plan ( <sup>1</sup> *g*

, <sup>2</sup> *<sup>g</sup>*

depends of invariants:

and *<sup>M</sup>*<sup>2</sup>

**Figure 1.** Kinematic of Bias Extension test, a: Initial state, b: Deformed state

In the second, the proposed model is implanted into Abaqus/Explicit to simulate the Bias

**B**

**B**

Exploiting numerical results, we studied the effect of the ratio between shearing and traction rigidities on homogeneities of stress and strain in the central zone of three Finite Element

One of significant characteristics of the woven structure is the existence of two privileged material directions: warp and weft. We considered that the fabric is a continuous structure having two privileged material directions defined by the two unit tensors *M***1** and *M***2** as

the tensor product. In the reference configuration, these privileged material directions are

In Lagrangian formulation, the hyperelastic behavior is defined by the strain energy

*W S*

The physical behaviour is completely defined by the choice of *W*(*E*). The woven structures is very thin, we are interested more particularly in plane solicitations (plane stress or strain) in

Using the representation theorems of isotropic functions, strain energy function *W*(*E*)

). We supposed that *W*(*E*) is an isotropic function of variables (*E,* **g1**, **g2**).

supposed to be orthogonal and they are defined by *g***1** and *g***2** presented by Equation 2.

function *W*(*E*) depending of Green-Lagrange tensor components [18-21].

The second Piola Kirchhoff stress tensor *S* derives is presented in Equation 3:

*<sup>i</sup> g :E* , *<sup>2</sup>*

*M M* 1 1 *M1* ; *M M* 2 2 *M2* (1)

**A** 

**C C** 

b

**B**

**B**

1 1 *<sup>g</sup> <sup>g</sup> <sup>1</sup> <sup>g</sup>* , 2 2 *<sup>g</sup> <sup>g</sup> <sup>2</sup> <sup>g</sup>* (2)

*E* (3)

*<sup>i</sup> g :E* , *tr*( ) *<sup>3</sup> <sup>E</sup>* ( 1..2) *<sup>i</sup>* (4)

are two unit vectors carried by two yarns directions. The sign indicate

$$I\_i = \mathbf{g}\_i \mathbf{:} \mathbf{E} \ \left( i = 1..2 \right) \; ; \; I\_{12} = \frac{1}{2} (\mathbf{g}\_1 \mathbf{E} \mathbf{:} \mathbf{E} \mathbf{g}\_2)^{1/2} \tag{6}$$

*Ii* measured elongations along directions *<sup>i</sup> g* . *I12* measured the sliding in the plane ( <sup>1</sup> *<sup>g</sup>* , <sup>2</sup> *<sup>g</sup>* ) witch is the angle variation between warp and weft direction. Components *ij <sup>g</sup> <sup>E</sup>* of *E* in the reference system ( <sup>1</sup> *g* , <sup>2</sup> *<sup>g</sup>* ), are defined as follows:

$$I\_i = E\_{g\bar{i}\bar{j}} = \frac{1}{2}(\delta\_1^2 - 1) \text{ , } i = 1..2, I\_{12} = \left| E\_{g12} \right| = \frac{1}{2}\delta\_1 \delta\_2 \left| \cos(\theta) \right|\tag{7}$$

1 and <sup>2</sup> are yarns extensions (ratio between deformed and initial lengths) along directions of <sup>1</sup> *g* and 2 *<sup>g</sup>* . is the angle between *M*<sup>1</sup> and *M*<sup>2</sup> .

The second Piola Kirchhoff stress tensor *S* can be written as:

$$\mathbf{S} = \frac{\partial W}{\partial I\_1}\mathbf{g}\_1 + \frac{\partial W}{\partial I\_2}\mathbf{g}\_2 + \frac{\partial W}{\partial I\_{12}}\frac{1}{2I\_{12}}(\mathbf{g}\_1\mathbf{E}\mathbf{g}\_2 + \mathbf{g}\_2\mathbf{E}\mathbf{g}\_1) \tag{8}$$

A simplified hyperplastic model is proposed. It is based on following assumptions:


$$\mathcal{W} = \frac{1}{2}k\_1 I\_1^2 + \frac{1}{2}k\_2 I\_2^2 + k\_{12} I\_1 I\_2 + k\_3 I\_{12}^2 \tag{9}$$

This leads to the constitutive equation:

$$\mathbf{S} = (k\_1 I\_1 + k\_{12} I\_2)\mathbf{g}\_1 + (k\_2 I\_2 + k\_{12} I\_1)\mathbf{g}\_2 + k\_3(\mathbf{g}\_1 \mathbf{E} \mathbf{g}\_2 + \mathbf{g}\_2 \mathbf{E} \mathbf{g}\_1) \tag{10}$$

So *k1* and *k2* presented tensile rigidities in yarns directions. *k12* described the interaction between two groups of yarns. *k3* presented the shearing rigidity of woven.

The relation between components *Sgij* of second Piola Kirchhoff stress tensor *S* and *Egij* of Green Lagrange strain tensor *E* in the base *<sup>i</sup> g* can be presented by one of flowing expressions

$$
\begin{bmatrix} \mathbf{S}\_{\mathcal{S}^{11}} \\ \mathbf{S}\_{\mathcal{S}^{22}} \\ \mathbf{S}\_{\mathcal{S}^{12}} \end{bmatrix} = \begin{bmatrix} k\_1 & k\_{12} & 0 \\ k\_{12} & k\_2 & 0 \\ 0 & 0 & k\_3 \end{bmatrix} \begin{bmatrix} E\_{\mathcal{S}^{11}} \\ E\_{\mathcal{S}^{22}} \\ E\_{\mathcal{S}^{12}} \end{bmatrix} \tag{11}
$$

$$
\begin{bmatrix} E\_{\mathcal{g}11} \\ E\_{\mathcal{g}22} \\ E\_{\mathcal{g}12} \end{bmatrix} = \begin{bmatrix} c\_1 & c\_{12} & 0 \\ c\_{12} & c\_2 & 0 \\ 0 & 0 & c\_3 \end{bmatrix} \begin{bmatrix} S\_{\mathcal{g}11} \\ S\_{\mathcal{g}22} \\ S\_{\mathcal{g}12} \end{bmatrix} \tag{12}
$$

Finite Element Analysis of Bias Extension Test of Dry Woven 159

(15)

(18)

(20)

1 1

2

2

*S , F* (14)

*<sup>F</sup> P fS <sup>S</sup>* (16)

*E* (17)

, are as follows:

*f*

/ /

1 2

The components of the Green–Lagrange strain tensor *E,* in the base *<sup>i</sup> e*

1 11

( 4( )

0

 

0

*ψ<sup>0</sup>* by 45°, Equation 20 became like the following:

45 2 3 1 2 12 1 2 2 4( ) *kk k <sup>C</sup> k kk k* 

1 2 12

0

/

Where:

Let

Where

as follows:

Where:

0 *<sup>F</sup> <sup>P</sup>*

related to *S* by:

0 *i i* 0 0 0 *e e*

*S f f*

0 01 ; ; () *L B <sup>f</sup> f f tg L Bf* 

0

12 22 *ie E E E E* 

The response of the model presented by Equation 8 for this solicitation can be summarised

; () ; () ( ) *f E P E EE g E <sup>C</sup>* 

4 4 2 0 1 0 2 0 3 12 0 2

3 12 1 2 0 12

 , 3 12 1 2 12 45

*c c cc c C*

(2 2 )sin (2 )

*C c c cc*

( ) 2( )

*<sup>g</sup> <sup>C</sup>*

<sup>1</sup> ( ) cos ( ) sin ( ) ( )sin (2 ) <sup>2</sup>

0 2 2

*<sup>S</sup>* where *F* is the tensile force and *So* is the initial cross section of the specimen. *P* is

1

11 12

2 2 22 2 11 1 22 2 1 12 1 2 1; 2 1 ; 2 *Ef Ef f Ef* 

22 0 11 12 0 11

0 1 0 2 0 3 12 0

 

sin(2 ) cos ( ) sin ( ) ( )cos(2 )

*c c cc*

The tensile test on 45° is a particular case of out-axes tensile tests where *ψ0=45).* To replacing

0

(2 2 ) 4 *c c cc c C*

45

, <sup>2</sup>

 

> 

> > 45

<sup>45</sup> 2 *c <sup>g</sup> <sup>C</sup>* (21)

 (19)

Where:

$$\mathcal{L}\_1 = \frac{k\_2}{k\_1 k\_2 - k\_{12}^2}, \mathcal{L}\_2 = \frac{k\_1}{k\_1 k\_2 - k\_{12}^2}, \mathcal{L}\_3 = \frac{1}{k\_3}, \mathcal{L}\_{12} = \frac{-k\_{12}}{k\_1 k\_2 - k\_{12}^2} \tag{13}$$

#### **2.1. Out-axes tensile test: Tensile test on 45°**

In tis parts the proposed hyperelastic model is used to study the mechanical behaviour during the out-axes tensile test of the dry woven.

Out-axes tensile test is a tensile test exerted on a fabric but according to a direction which is not necessarily warp or weft directions [22]. In the case of anisotropic behavior stress and strains tensors have not, in general, the same principal directions. During this test, the simple is subjected to a shearing. Particular precautions must be taken to ensure a relative homogeneity of the test [23].

We considered a tensile test along a direction <sup>1</sup> *E* forming an angle *ψ0* with orthotropic direction *<sup>i</sup> g* (Figure.2).

**Figure 2.** Kinematics of Out-axes tensile test, a: Reference configuration, b: Deformed configuration.

In the base *<sup>i</sup> e* , components of the second Piola Kirchhoff tensor *S* and the Gradient of transformation tensor *F* are as follows [23]

Finite Element Analysis of Bias Extension Test of Dry Woven 159

$$\mathbf{S}\_{/\vec{e}\_i} = \begin{bmatrix} \mathbf{S} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix}, \mathbf{F}\_{/\vec{e}\_i} = \begin{bmatrix} f\_1 & f\_1 \mathcal{V} \\ \mathbf{0} & f\_2 \end{bmatrix} \tag{14}$$

Where:

(12)

158 Finite Element Analysis – Applications in Mechanical Engineering

**2.1. Out-axes tensile test: Tensile test on 45°** 

during the out-axes tensile test of the dry woven.

We considered a tensile test along a direction <sup>1</sup> *E*

homogeneity of the test [23].

(Figure.2).

direction *<sup>i</sup> g*

In the base *<sup>i</sup> e*

*B0* 

transformation tensor *F* are as follows [23]

*L0* 

Where:

11 11 1 12 22 12 2 22 3 12 12

2 1 12 1 2 2 2 3 12 2 1 2 12 1 2 12 3 1 2 12 *kk k* 1

*, , ,* (13)

forming an angle *ψ0* with orthotropic

*<sup>F</sup> <sup>B</sup>*

*kk k kk k k kk k*

In tis parts the proposed hyperelastic model is used to study the mechanical behaviour

Out-axes tensile test is a tensile test exerted on a fabric but according to a direction which is not necessarily warp or weft directions [22]. In the case of anisotropic behavior stress and strains tensors have not, in general, the same principal directions. During this test, the simple is subjected to a shearing. Particular precautions must be taken to ensure a relative

**Figure 2.** Kinematics of Out-axes tensile test, a: Reference configuration, b: Deformed configuration.

(a) (b)

*F*

*ψ<sup>0</sup>*

, components of the second Piola Kirchhoff tensor *S* and the Gradient of

*g g g g g g*

*E S c c E cc S E S c* 

0 0

*c c cc*

0 0

$$f\_1 = \frac{L}{L\_0}; f\_2 = \frac{B}{B\_0}; \gamma = \frac{f\_2}{f\_1} t \mathbf{g}(\xi) \tag{15}$$

Let 0 *<sup>F</sup> <sup>P</sup> <sup>S</sup>* where *F* is the tensile force and *So* is the initial cross section of the specimen. *P* is related to *S* by:

$$P = \frac{F}{S\_0} = f\_1 S \tag{16}$$

The components of the Green–Lagrange strain tensor *E,* in the base *<sup>i</sup> e* , are as follows:

$$\mathbf{E}\_{\sqrt{\vec{e}}\_{i}} = \begin{bmatrix} E\_{11} & E\_{12} \\ E\_{12} & E\_{22} \end{bmatrix} \tag{17}$$

Where

$$\text{:} \, \mathbf{2} \, \mathbf{E}\_{11} = f\_1^2 - \mathbf{1}; \, \mathbf{2} \, \mathbf{E}\_{22} = f\_2^2 - \mathbf{1} + f\_1^2 \boldsymbol{\gamma}^2; \, \mathbf{2} \, \mathbf{E}\_{12} = f\_1^2 \boldsymbol{\gamma} \tag{18}$$

The response of the model presented by Equation 8 for this solicitation can be summarised as follows:

$$P = \frac{f\_1 E\_{11}}{\mathcal{C}(\boldsymbol{\nu}\_0)} ; \; E\_{22} = -\nu(\boldsymbol{\nu}\_0) E\_{11} ; \; E\_{12} = \mathcal{g}(\boldsymbol{\nu}\_0) E\_{11} \tag{19}$$

Where:

$$\begin{aligned} \mathsf{C}(\boldsymbol{\nu}\_{0}) &= c\_{1}\cos^{4}(\boldsymbol{\nu}\_{0}) + c\_{2}\sin^{4}(\boldsymbol{\nu}\_{0}) + \frac{1}{2}(c\_{3} + c\_{12})\sin^{2}(2\boldsymbol{\nu}\_{0}) \\ \nu(\boldsymbol{\nu}\_{0}) &= \frac{\left[ (2c\_{3} + 2c\_{12} - c\_{1} - c\_{2})\sin^{2}(2\boldsymbol{\nu}\_{0}) - c\_{12} \right]}{4\mathsf{C}(\boldsymbol{\nu}\_{0})} \\ \boldsymbol{g}(\boldsymbol{\nu}\_{0}) &= \frac{\sin(2\boldsymbol{\nu}\_{0}) \left[ c\_{1}\cos^{2}(\boldsymbol{\nu}\_{0}) - c\_{2}\sin^{2}(\boldsymbol{\nu}\_{0}) - (c\_{3} + c\_{12})\cos(2\boldsymbol{\nu}\_{0}) \right]}{2\mathsf{C}(\boldsymbol{\nu}\_{0})} \end{aligned} \tag{20}$$

The tensile test on 45° is a particular case of out-axes tensile tests where *ψ0=45).* To replacing *ψ<sup>0</sup>* by 45°, Equation 20 became like the following:

$$\mathcal{C}\_{45} = \frac{1}{2k\_3} + \frac{k\_1 + k\_2 + 2k\_{12}}{4(k\_1 k\_2 - k\_{12}^2)} \; , \; \nu\_{45} = \frac{\left\lfloor (2c\_3 + 2c\_{12} - c\_1 - c\_2) - c\_{12} \right\rfloor}{4C\_{45}} \; , \; \mathcal{g}\_{45} = -\frac{c\_2}{2C\_{45}} \tag{21}$$

*S1* and *S2* are respectively the maximum and the minimum Eigen values of Piola Kirchhoff tensor *S* .In Tensile test on 45°, Equation 14 shows that:

$$\frac{S\_2}{S\_1} = 0\tag{22}$$

Finite Element Analysis of Bias Extension Test of Dry Woven 161

During the Bias Extension test, the pure shearing occurred in the central zone A and the

 

 

1

*f*

The Gradient of Transformation tensor *F* is presented by Equation 27:

2

1 0 ( ) *<sup>i</sup>* 0 1 *<sup>e</sup> <sup>S</sup>* 

1 0 ( ) *<sup>i</sup>* 0 1 *<sup>e</sup> <sup>E</sup>* 

The internal power per unit of volume in zone A is defined by Equation 32:

To calculate to internal power per unit of volume in zone B we replace

*a SE k S E* 

*b SE k* 

*f*

2 cos( ) 2 2 <sup>2</sup>

cos( ) sin( ) 0 <sup>0</sup> 2 2

Using the proposed model, components *Sij ,Eij* of the second Piola Kirchhoff stress and Green

*<sup>E</sup>*where <sup>1</sup> ( ) sin( ) <sup>2</sup>

2 1 <sup>1</sup> *<sup>S</sup>*

2 1 <sup>1</sup> *<sup>E</sup>*

Where *S1* and *S2* are respectively the maximum and the minimum Eigen values of the second Piola Kirchhoff tensor *S* and *E1* and *E2* are respectively the maximum and the

<sup>1</sup> : 2 ( ) ( ) sin(2 ) <sup>4</sup> *A A*

<sup>1</sup> : 2 ( ) ( ) sin( ) <sup>228</sup>

*S* where <sup>3</sup>

<sup>0</sup> 0 cos( ) sin( ) 2 2

 

, as follows:

*E =*

3

3

 

 *B B S E* (33)

*F* (27)

*a*

*D d*

*D*

<sup>1</sup> ( ) sin( ) <sup>2</sup> *S k* 

*<sup>S</sup>* (30)

*<sup>E</sup>* (31)

 by 2 

(32)

in Equation 32:

(28)

(29)

(26)

shear angle is defined by Equation 26:

/ ,

*Ei ei*

Lagrange strain tensors are given, in the base *<sup>i</sup> e*

Thus

And

/

/

minimum Eigen values of Green Lagrange tensor *E.* 

The expression of the applied force *F* is deducted from Equation 16:

$$F = PS\_0 = \frac{2k\_3 S\_0 f\_1 (f\_1^2 - 1)(k\_1 k\_2 - k\_{12}^2)}{2k\_1 k\_2 - 2k\_{12}^2 + k\_3 (k\_1 + k\_2 - 2k\_{12})} \tag{23}$$

For a balanced woven (*k1*=*k2*=*k*) where the interaction between yarns is neglected (*k12*=0), the expression of *F* became:

$$F = \frac{k\_3 k S\_0 f\_1 (f\_1^2 - 1)}{k + k\_3} \tag{24}$$

The ratio between the minimum and the maximum Eigen values of Green Lagrange tensor *E.*, in the tensile test with 45°, is given by Equation 25:

$$\frac{E\_2}{E\_1} = -\frac{k - k\_3}{k\_3 + k} \tag{25}$$

#### **2.2. Bias extension test**

To explainer the pure shearing test of woven fabric, it has been noted that woven cloths in general deform as a pin-jointed-net (PJN) [24-28]. Yarns are considered to be inextensible and fixed at each cross-over point, rotating about these points like it is shown in Figure 3.

**Figure 3.** Kinematics Pure shear a: Reference configuration, b: Deformed configuration

During the Bias Extension test, the pure shearing occurred in the central zone A and the shear angle is defined by Equation 26:

$$
\varphi = \frac{\pi}{2} - \theta = \frac{\pi}{2} - 2a\cos(\frac{D+d}{\sqrt{2D}}) \tag{26}
$$

The Gradient of Transformation tensor *F* is presented by Equation 27:

$$\mathbf{F}\_{f\acute{e}t,\acute{e}t} = \begin{bmatrix} f\_1 & 0\\ 0 & f\_2 \end{bmatrix} = \begin{bmatrix} \cos(\frac{\wp}{2}) + \sin(\frac{\wp}{2}) & 0\\ 0 & \cos(\frac{\wp}{2}) - \sin(\frac{\wp}{2}) \end{bmatrix} \tag{27}$$

Using the proposed model, components *Sij ,Eij* of the second Piola Kirchhoff stress and Green Lagrange strain tensors are given, in the base *<sup>i</sup> e* , as follows:

$$\mathbf{S}\_{/\overline{\epsilon}\_i} = \mathbf{S}(\boldsymbol{\wp}) \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \text{ where } \begin{aligned} \mathbf{S}(\boldsymbol{\wp}) = \frac{1}{2} k\_3 \sin(\boldsymbol{\wp}) \end{aligned} \tag{28}$$

$$E\_{/c\_i} = E(\rho) \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \quad \text{where} \quad E(\rho) = \frac{1}{2} \sin(\rho) \tag{29}$$

Thus

160 Finite Element Analysis – Applications in Mechanical Engineering

expression of *F* became:

**2.2. Bias extension test** 

l

tensor *S* .In Tensile test on 45°, Equation 14 shows that:

*E.*, in the tensile test with 45°, is given by Equation 25:

D

The expression of the applied force *F* is deducted from Equation 16:

0 2

*S1* and *S2* are respectively the maximum and the minimum Eigen values of Piola Kirchhoff

*kS f f kk k F PS*

For a balanced woven (*k1*=*k2*=*k*) where the interaction between yarns is neglected (*k12*=0), the

*k kS f f* ( 1) *<sup>F</sup> k k* 

The ratio between the minimum and the maximum Eigen values of Green Lagrange tensor

2 3 1 3 *E k k E kk* 

To explainer the pure shearing test of woven fabric, it has been noted that woven cloths in general deform as a pin-jointed-net (PJN) [24-28]. Yarns are considered to be inextensible and fixed at each cross-over point, rotating about these points like it is shown in Figure 3.

l

*F F* 

D+d

**Figure 3.** Kinematics Pure shear a: Reference configuration, b: Deformed configuration

(a) (b)

2 2 3 0 1 1 1 2 12

1 2 12 3 1 2 12 2 ( 1)( ) 2 2 ( 2)

*kk k k k k k*

2 3 01 1

3

*<sup>S</sup>* (22)

(23)

(24)

(25)

$$\frac{S\_2}{S\_1} = -1\tag{30}$$

And

$$\frac{E\_2}{E\_1} = -1\tag{31}$$

Where *S1* and *S2* are respectively the maximum and the minimum Eigen values of the second Piola Kirchhoff tensor *S* and *E1* and *E2* are respectively the maximum and the minimum Eigen values of Green Lagrange tensor *E.* 

The internal power per unit of volume in zone A is defined by Equation 32:

$$
tau = \mathbf{S}\_A : \dot{\mathbf{E}}\_A = \mathfrak{L}\mathbf{S}(\varphi)\dot{\mathbf{E}}(\varphi) = \frac{1}{4}k\_3 \sin(2\varphi)\dot{\varphi} \tag{32}
$$

To calculate to internal power per unit of volume in zone B we replace by 2 in Equation 32:

$$
\rho \alpha \phi = \mathbf{S}\_{\mathsf{B}} : \dot{E}\_{\mathsf{B}} = 2S(\frac{\rho}{2})\dot{E}(\frac{\rho}{2}) = \frac{1}{8}k\_3 \sin(\rho)\dot{\rho} \tag{33}
$$

**Figure 4.** Kinematic of Bias Extension Test, a: initial configuration, b: deformed configuration

The total internal power in the specimen is given by Equation 34:

$$P\text{int} = Va.\alpha\alpha + Vb.\alpha b \tag{34}$$

Finite Element Analysis of Bias Extension Test of Dry Woven 163

U1≠0 U2=0 U3=0

component of the second Piola Kirchhoff tensor *S*, and the Green Lagrange tensor *E*

The fabric is modelling by rectangular part meshed by continuum element (M3D4R).The

[29-30] compared the numerical results for the biased mesh and the aligned mesh and they proved that by using the biased mesh (Figure 5b), where the fibres are run diagonally across the rectangular element, neither the deformation profile nor the reaction forces are predicted

**Figure 5.** FEM mesh for the Bias Extension simulation, a: boundary condition of FEM, b: biased mesh, c:

(b) (c)

(a)

In order to simplify the problem, we used a balanced woven (*k1*=*k2*=*k*=700 N/mm2) and we ignored the interaction between extension in yarns direction (*k12*=0). The analysis is done for three different FEM with the same thickness of 0.2mm. Dimensions of FEM are presented in

MEF Length(mm) Width(mm) Aspect ratio: λ

1 100 50 2 2 150 50 3 3 200 50 4

projected in 1 2 (,) *<sup>g</sup> <sup>g</sup>* . We can also drew curves of Fore versus displacement.

boundary condition of model is presented in Figure 5a.

correctly, for this we used the aligned mesh (Figure 5c).

aligned mesh.

U1=0 U2=0 U3=0

table 1.

**Table 1.** Dimensions of samples

Where *Va* and *Vb* are respectively the initial volume in zones A and B defined as follows

$$\begin{aligned} Vb &= e\_0 w\_0^2\\ Va &= e\_0 (Dw\_0 - \frac{w\_0^2}{2}) = e\_0 Dw\_0 - \frac{1}{2} Vb \end{aligned} \tag{35}$$

The External power is defined as:

$$P\text{ext} = F.\dot{d} = \frac{1}{2} F D f\_2 \dot{\phi} \tag{36}$$

The equality between internal and external powers allows to determinate the expression of applied force F given by Equation 37:

$$F = e\_0 w\_0 f\_1 k\_3 \sin(\phi) [1 + \frac{1}{4(\lambda - 1)\cos(\phi)} (1 - 2\cos(\phi))] \tag{37}$$

Where <sup>0</sup> 0 *L w* is the aspect ratio.

#### **3. Numerical simulation of Bias Extension test**

In this section, we simulated the Bias Extension test (BE) using the hyperelastic proposed model implanted into Abaqus/Explicit thought user material subroutine (VUMAT). Out put of the VUMAT are stress components of Cauchy tensor projected in the Green-Nagdi basis, component of the second Piola Kirchhoff tensor *S*, and the Green Lagrange tensor *E* projected in 1 2 (,) *<sup>g</sup> <sup>g</sup>* . We can also drew curves of Fore versus displacement.

The fabric is modelling by rectangular part meshed by continuum element (M3D4R).The boundary condition of model is presented in Figure 5a.

[29-30] compared the numerical results for the biased mesh and the aligned mesh and they proved that by using the biased mesh (Figure 5b), where the fibres are run diagonally across the rectangular element, neither the deformation profile nor the reaction forces are predicted correctly, for this we used the aligned mesh (Figure 5c).

**Figure 5.** FEM mesh for the Bias Extension simulation, a: boundary condition of FEM, b: biased mesh, c: aligned mesh.

In order to simplify the problem, we used a balanced woven (*k1*=*k2*=*k*=700 N/mm2) and we ignored the interaction between extension in yarns direction (*k12*=0). The analysis is done for three different FEM with the same thickness of 0.2mm. Dimensions of FEM are presented in table 1.


**Table 1.** Dimensions of samples

162 Finite Element Analysis – Applications in Mechanical Engineering

B

A C

*D/2*

 *L0/2*

*w0*

B

**Figure 4.** Kinematic of Bias Extension Test, a: initial configuration, b: deformed configuration

(a) (b)

2 0 0 0 00 <sup>1</sup> ( ) 2 2

*<sup>w</sup> Va e Dw e Dw Vb*

<sup>1</sup> . <sup>2</sup> *Pext F d FDf*

The equality between internal and external powers allows to determinate the expression of

<sup>1</sup> sin( )[1 (1 2cos( ))] 4( 1)cos( ) *F ew fk*

In this section, we simulated the Bias Extension test (BE) using the hyperelastic proposed model implanted into Abaqus/Explicit thought user material subroutine (VUMAT). Out put of the VUMAT are stress components of Cauchy tensor projected in the Green-Nagdi basis,

Where *Va* and *Vb* are respectively the initial volume in zones A and B defined as follows

2 0 0

 

2

 

*<sup>=</sup>*(37)

(35)

(*D+d*)/2

<sup>A</sup> <sup>B</sup>

B

C

(36)

  (34)

The total internal power in the specimen is given by Equation 34:

*Vb e w*

int . . *P Va a Vb b*

0 013

**3. Numerical simulation of Bias Extension test** 

The External power is defined as:

applied force F given by Equation 37:

is the aspect ratio.

Where <sup>0</sup>

0 *L w* 

This analysis is realised on four values of the ratio between shearing and tensile rigidities <sup>3</sup> ( 0.007,0.02,0.1,0.3,1) *k <sup>k</sup>* along three paths in FEM (see Figure 6).

Finite Element Analysis of Bias Extension Test of Dry Woven 165

*k*

*<sup>k</sup>* =1), *E1* is homogenous and it conformed to the predicted value

=0.007 and *U1*=40mm.

*<sup>k</sup>* ), the central

**Figure 7.** Deformed mesh with contour of Green Lagrange shear strain *E12* for <sup>3</sup> *<sup>k</sup>*

FEM2

FEM3

in the case of isotropic elastic material. To decreasing the ratio of rigidities ( <sup>3</sup> *<sup>k</sup>*

Figure 8 shows the variation of the maximum principal *E1* of Green Lagrange along the first path. We noticed that *E1* is symmetric with regard to the centre of the FEM. For the higher

B

FEM1 A

C

**3.1. Strain state** 

value of ratio of rigidities ( <sup>3</sup> *<sup>k</sup>*

The first path is longitudinal line in the middle of FEM. It joined zones A and C, the second path is along the yarn direction and the third path is transversal middle line Flowing results are illustrated for a displacement of 10% of initial length.

The deformed mesh with the contour of the Green Lagrange shear strain is shown in Figure7. We noticed that appearance of three discernible deformation zones of the Bias Extension test in three FEM. No significant deformation occurred in zone C. The main mode of deformation in zone A is the shearing. The most deformation of the fabric occurs in this zone.

In to order to study homogeneities of stress and strain states, we compared the analytical and the numerical results of strain and stress along three paths of Figure (6).

**Figure 7.** Deformed mesh with contour of Green Lagrange shear strain *E12* for <sup>3</sup> *<sup>k</sup> k* =0.007 and *U1*=40mm.

#### **3.1. Strain state**

164 Finite Element Analysis – Applications in Mechanical Engineering

*<sup>k</sup>* along three paths in FEM (see Figure 6).

are illustrated for a displacement of 10% of initial length.

<sup>3</sup> ( 0.007,0.02,0.1,0.3,1)

**Figure 6.** Different paths used in analysis

in this zone.

*k*

This analysis is realised on four values of the ratio between shearing and tensile rigidities

The first path is longitudinal line in the middle of FEM. It joined zones A and C, the second path is along the yarn direction and the third path is transversal middle line Flowing results

Path 2 Path 1

Path3

The deformed mesh with the contour of the Green Lagrange shear strain is shown in Figure7. We noticed that appearance of three discernible deformation zones of the Bias Extension test in three FEM. No significant deformation occurred in zone C. The main mode of deformation in zone A is the shearing. The most deformation of the fabric occurs

In to order to study homogeneities of stress and strain states, we compared the analytical

and the numerical results of strain and stress along three paths of Figure (6).

Figure 8 shows the variation of the maximum principal *E1* of Green Lagrange along the first path. We noticed that *E1* is symmetric with regard to the centre of the FEM. For the higher value of ratio of rigidities ( <sup>3</sup> *<sup>k</sup> <sup>k</sup>* =1), *E1* is homogenous and it conformed to the predicted value in the case of isotropic elastic material. To decreasing the ratio of rigidities ( <sup>3</sup> *<sup>k</sup> <sup>k</sup>* ), the central

zone characterised by the higher value of *E1*. En addition, we observed the appearance of two zones where the strain is not more important. In the first hand, to comparing with the analytical value of *E1* in the central zone, the numerical values of *E1* is closely to that predicted in the Bias Extension test for the few shearing rigidity. Zones C coincided with ends of the path where the deformation was not more significant. In another hand, we remarked that in the central zone of the path, the deformation is not homogenous especially in FEM1and Finite Element Analysis of Bias Extension Test of Dry Woven 167

**Figure 8.** Variation of Maximum principal of Green Lagrange strain *E1* along the path1

FEM2. For more analyse the strains state in FEM, Figure 9 presented the evolution of <sup>2</sup> 1 *E <sup>E</sup>* ,

along the first path. It is clear that to decreasing <sup>3</sup> *<sup>k</sup> <sup>k</sup>* , the value of <sup>2</sup> 1 *E <sup>E</sup>* tends to (-1) in three FEM.

This proved that, in spite of the low displacement, the deformation in Bias Extension test is influenced by the ratio between shearing and tensile rigidities of the woven.

#### **3.2. Stress state**

Comparing the numerical and the analytical values of <sup>2</sup> 1 *S S* , we determinate the stress state in different FEM for an displacement of 10% along the first path .

Figure10 show that to decreasing <sup>3</sup> *<sup>k</sup> <sup>k</sup>* , the value of <sup>2</sup> 1 *S <sup>S</sup>* decrease but never achieved (-1).

Indeed, if this simulation is interpreted like a Bias Extension test, <sup>2</sup> 1 *S <sup>S</sup>* should be verifying Equation 30 in the central zone. However the ratio of principal strain is approximately equal to 0. So it is conforming to Equation 22, and the stress state is the traction state.

In addition to varying the value of <sup>3</sup> *<sup>k</sup> <sup>k</sup>* , we evaluated the ratio of strain versus the ratio of stress in the central element of FEM. In Figure12, it can be noticed that in FEM1, to reducing the value of <sup>3</sup> *<sup>k</sup> <sup>k</sup>* , <sup>2</sup> 1 *E <sup>E</sup>* tend to (-1) and it conformed to the predicted value by Equation 31 for a few values of <sup>3</sup> *<sup>k</sup> k* . But <sup>2</sup> 1 *S S* have a negative value and it remain different to (-1). In FEM2, it was visibly that <sup>2</sup> 1 *S <sup>S</sup>* stayed proximity null for different value of <sup>3</sup> *<sup>k</sup> k* thus it verified Equation 22 but <sup>2</sup> 1 *E <sup>E</sup>* tend to (-1) for few values of <sup>3</sup> *<sup>k</sup> <sup>k</sup>* . In FEM3, it was clear that for few value of <sup>3</sup> *<sup>k</sup> k* , 2 1 *E <sup>E</sup>* tend to (-1), but the <sup>2</sup> 1 *S <sup>S</sup>* had positive values. Consequently, the shearing deformation in Bias Extension test depends of the ratio of rigidities between shearing and tensile, but the stress state is always the tensile stress.

along the first path. It is clear that to decreasing <sup>3</sup> *<sup>k</sup>*

Comparing the numerical and the analytical values of <sup>2</sup>

in different FEM for an displacement of 10% along the first path .

Indeed, if this simulation is interpreted like a Bias Extension test, <sup>2</sup>

to 0. So it is conforming to Equation 22, and the stress state is the traction state.

*<sup>S</sup>* stayed proximity null for different value of <sup>3</sup> *<sup>k</sup>*

**3.2. Stress state** 

the value of <sup>3</sup> *<sup>k</sup>*

few values of <sup>3</sup> *<sup>k</sup>*

was visibly that <sup>2</sup>

1 *E*

*<sup>E</sup>* tend to (-1), but the <sup>2</sup>

22 but <sup>2</sup>

2 1 *E*

Figure10 show that to decreasing <sup>3</sup> *<sup>k</sup>*

In addition to varying the value of <sup>3</sup> *<sup>k</sup>*

. But <sup>2</sup> 1 *S*

*<sup>E</sup>* tend to (-1) for few values of <sup>3</sup> *<sup>k</sup>*

1 *S*

1 *S*

stress state is always the tensile stress.

*<sup>k</sup>* , <sup>2</sup> 1 *E*

*k*

zone characterised by the higher value of *E1*. En addition, we observed the appearance of two zones where the strain is not more important. In the first hand, to comparing with the analytical value of *E1* in the central zone, the numerical values of *E1* is closely to that predicted in the Bias Extension test for the few shearing rigidity. Zones C coincided with ends of the path where the deformation was not more significant. In another hand, we remarked that in the central zone of the path, the deformation is not homogenous especially in FEM1and

FEM2. For more analyse the strains state in FEM, Figure 9 presented the evolution of <sup>2</sup>

This proved that, in spite of the low displacement, the deformation in Bias Extension test is

*<sup>k</sup>* , the value of <sup>2</sup>

Equation 30 in the central zone. However the ratio of principal strain is approximately equal

stress in the central element of FEM. In Figure12, it can be noticed that in FEM1, to reducing

Bias Extension test depends of the ratio of rigidities between shearing and tensile, but the

influenced by the ratio between shearing and tensile rigidities of the woven.

*<sup>k</sup>* , the value of <sup>2</sup>

1 *S*

1 *S*

*<sup>E</sup>* tend to (-1) and it conformed to the predicted value by Equation 31 for a

*S* have a negative value and it remain different to (-1). In FEM2, it

*<sup>S</sup>* had positive values. Consequently, the shearing deformation in

1 *E*

1 *E <sup>E</sup>* ,

*<sup>E</sup>* tends to (-1) in three FEM.

*S* , we determinate the stress state

*<sup>S</sup>* decrease but never achieved (-1).

*<sup>k</sup>* , we evaluated the ratio of strain versus the ratio of

*<sup>k</sup>* . In FEM3, it was clear that for few value of <sup>3</sup> *<sup>k</sup>*

1 *S*

*<sup>S</sup>* should be verifying

*k* thus it verified Equation

*k* ,

**Figure 8.** Variation of Maximum principal of Green Lagrange strain *E1* along the path1

Finite Element Analysis of Bias Extension Test of Dry Woven 169

**Figure 10.** Variation of <sup>2</sup>

1 *S*

*<sup>S</sup>* along path 1.

**Figure 9.** Variation of <sup>2</sup> 1 *E <sup>E</sup>* along the path 1

**Figure 10.** Variation of <sup>2</sup> 1 *S <sup>S</sup>* along path 1.

**Figure 9.** Variation of <sup>2</sup>

1 *E*

*<sup>E</sup>* along the path 1

Finite Element Analysis of Bias Extension Test of Dry Woven 171

In this section, we compared between the numerical and the predicted values of the angle

Using the proposed model, the numerical angle between yarns is given by the following

In the case of the Bias Extension test, the predict angle between yarns in the central zone A is

2arcos( ) 2 *<sup>B</sup>*

2 12 22 11 (2 ) arcos( ) (2 1)(2 1) *g*

(38)

(39)

(40)

*g g*

*D d D*

11 45

11 22

(1 ) arcos( ) ( 1)

*E E*

*E*

2

*E E E*

**3.3. Angle between yarns** 

expression:

FEM1.

50

55

60

65

70

75

80

85

90

given by Equation 39:

between yarns, along the first path.

*N*

*T*

The predict angle between yarns in the Tensile test in 45° is given by Equation 41:

**Figure 12.** Comparison between Numerical and Predicted angles between yarns along the path 1 in

Anlytical\_T45\_k3/k=0.007

0 20 40 60 80 100 Distance between nodes

BE Numerical\_k3/k=1 Analytical\_T45\_k3/k=1 Numerical\_k3/k=0.3 Analytical\_T45\_k3/k=0.3 Numerical \_k3/k=0.1 Analytical\_T45 k3/k=0.1 Numerical\_k3/k=0.02 Analytical\_T45\_k3/k=0.02 Numerical\_k3/k=0.007

**Figure 11.** Variation of <sup>2</sup> 1 *S S* versus <sup>2</sup> 1 *E E* along path 1.

#### **3.3. Angle between yarns**

170 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 11.** Variation of <sup>2</sup>

1 *S S*

versus <sup>2</sup>

1 *E*

*E* along path 1.

In this section, we compared between the numerical and the predicted values of the angle between yarns, along the first path.

Using the proposed model, the numerical angle between yarns is given by the following expression:

$$\theta\_N = \arccos(\sqrt{\frac{(2E\_{\lg 12})^2}{(2E\_{\lg 22} + 1)(2E\_{\lg 11} + 1)}}) \tag{38}$$

In the case of the Bias Extension test, the predict angle between yarns in the central zone A is given by Equation 39:

$$\theta\_B = 2 \arccos(\frac{D+d}{\sqrt{2D}}) \tag{39}$$

The predict angle between yarns in the Tensile test in 45° is given by Equation 41:

$$\theta\_T = \arccos(\frac{E\_{11}(1-\nu\_{45})}{\sqrt{\left(E\_{11}+E\_{22}+1\right)^2}})\tag{40}$$

**Figure 12.** Comparison between Numerical and Predicted angles between yarns along the path 1 in FEM1.

Figure 12 demonstrated that the value of the angle between yarns was not uniform in the central zone of the FEM and it was not null in ends of the path1. For three FEM, the numerical angle between yarns tend to verify the predict angle (solid line) in the Bias Extension test for the lower value of <sup>3</sup> *<sup>k</sup> <sup>k</sup>* This is another reason to justify the influence of the ration of rigidities on the shearing deformation of woven.

#### **3.4. Elongation of yarns**

Under the pin-joint assumption for trellising deformation mode, the edge length of the membrane element should remain unchanged during the deformation; thus the Green Lagrange stretch *Eg11* and *Eg22* should be null in Bias Extension test:

$$E\_{\mathfrak{g}11} = E\_{\mathfrak{g}22} = 0\tag{41}$$

Finite Element Analysis of Bias Extension Test of Dry Woven 173

*E* along the third path. Like the first path,

subjected a few elongation. These stretches depend of the value of the ratio between shearing and tensile rigidities of woven. Same previous analyses are taken also along the

versus <sup>2</sup>

In this work, an orthotropic hyperelastic model test of woven fabric is developed and implanted into Abaqus/explicit to simulate Bias-Extension at low displacement. The analysis of numerical answers along longitudinal and transversal middle paths, proved, in the first hand, that to decreasing the ratio between shearing and tensile rigidities, the state deformation became to be conform to that predicted by the proposed model in the Bias Extension test for all FEM. In another hand, the angle between yarns tends to verify the predicted angle during the Bias Extension test. Although the stress state, is conform to the expected analysis of Traction test on 45°. The analysis of Green Lagrange stretching strain in the yarns direction, demonstrated that there was an elongation of yarns during test for different shearing rigidity. This elongation was exactly conforming to the predicted analytical elongation in the Traction test in 45°. Curves of Force versus displacement of the

0 5 10 15 20 25 30 35 40 True distance betwen nodes

1 *E*

Analytical BE Numerical\_k3/k=1 Analytical\_T45\_k3/k=1 Numerical\_k3/k=0.3 Analytical\_T45\_k3/k=0.3 Numerical\_k3/k=0.1 Annalytical\_T45\_k3/k=0.1 Numerical\_k3/k=0.02 Analytical\_T45\_k3/k=0.02 Numerical\_k3/k=0.007

*<sup>k</sup>* , the shearing is the utmost deformation. But in all cases, the Bias

1 *S S*

Analytical\_T45\_k3/k=0.007

third vertical path (Path3) and same results are verified.

Extension test is characterized by the tensile state.

**Figure 13.** Variation of *Eg11* along the path 2in FEM1.

**4. Conclusion** 

0

0.02

Eg11

0.04

0.06

Figure13 represented the evolution of <sup>2</sup>

for few values of <sup>3</sup> *<sup>k</sup>*

In Tensile test on 45°, warp and weft yarns are submitted respectively to Green Lagrange deformations 11 *<sup>g</sup> <sup>E</sup>* and 22 *<sup>g</sup> <sup>E</sup>* as follows:

$$E\_{\mathcal{G}\_{11}} = \frac{1}{2} \{ \left| \mathbf{F}, \bar{\mathbf{g}}\_{1} \right|^{2} - 1 \} = \frac{k\_{1} - k\_{12}}{2(k\_{1}k\_{2} - k\_{12}^{2})} \frac{E\_{11}}{\mathbf{C}\_{45}} \tag{42}$$

$$E\_{\mathcal{g}\_{22}} = \frac{1}{2} \{ \mathbf{F}, \vec{\mathcal{g}}\_{\cdot \cdot} \Big|^{2} - 1 \} = \frac{k\_2 - k\_{12}}{2(k\_1 k\_2 - k\_{12}^2)} \frac{E\_{11}}{C\_{45}} \tag{43}$$

In the case of balanced fabric without coupling between elongations in yarns directions, the warp and weft yarns are submitted to the same elongation:

So

$$E\_{\chi11} = E\_{\chi22} = E \tag{44}$$

Where

$$E = \frac{k\_3}{k + k\_3} E\_{11} \tag{45}$$

In Figure 13, we compared numerical stretch deformation along the second path and the predicted elongation in yarn direction.

In the first hand, we noticed that the numerical elongation was not null. It became more important by increasing the value of <sup>3</sup> *<sup>k</sup> k* in all FEM. In another hand, numerical value of elongation is closely conforming to the expected value in the tensile test in 45° for different values of <sup>3</sup> *<sup>k</sup> k* in all FEM. This analysis provided that during Bias Extension test, yarns are subjected a few elongation. These stretches depend of the value of the ratio between shearing and tensile rigidities of woven. Same previous analyses are taken also along the third vertical path (Path3) and same results are verified.

Figure13 represented the evolution of <sup>2</sup> 1 *S S* versus <sup>2</sup> 1 *E E* along the third path. Like the first path,

for few values of <sup>3</sup> *<sup>k</sup> <sup>k</sup>* , the shearing is the utmost deformation. But in all cases, the Bias Extension test is characterized by the tensile state.

**Figure 13.** Variation of *Eg11* along the path 2in FEM1.

## **4. Conclusion**

172 Finite Element Analysis – Applications in Mechanical Engineering

ration of rigidities on the shearing deformation of woven.

Lagrange stretch *Eg11* and *Eg22* should be null in Bias Extension test:

11 1

22 2

warp and weft yarns are submitted to the same elongation:

<sup>1</sup> ( . 1)

<sup>1</sup> ( . 1)

Extension test for the lower value of <sup>3</sup> *<sup>k</sup>*

deformations 11 *<sup>g</sup> <sup>E</sup>* and 22 *<sup>g</sup> <sup>E</sup>* as follows:

predicted elongation in yarn direction.

important by increasing the value of <sup>3</sup> *<sup>k</sup>*

**3.4. Elongation of yarns** 

So

Where

values of <sup>3</sup> *<sup>k</sup>*

Figure 12 demonstrated that the value of the angle between yarns was not uniform in the central zone of the FEM and it was not null in ends of the path1. For three FEM, the numerical angle between yarns tend to verify the predict angle (solid line) in the Bias

Under the pin-joint assumption for trellising deformation mode, the edge length of the membrane element should remain unchanged during the deformation; thus the Green

In Tensile test on 45°, warp and weft yarns are submitted respectively to Green Lagrange

2 2( ) *<sup>g</sup> kk E E g kk k <sup>C</sup>* 

2 2( ) *<sup>g</sup> kk E E g kk k <sup>C</sup>* 

In the case of balanced fabric without coupling between elongations in yarns directions, the

3

In Figure 13, we compared numerical stretch deformation along the second path and the

In the first hand, we noticed that the numerical elongation was not null. It became more

elongation is closely conforming to the expected value in the tensile test in 45° for different

11 3 *<sup>k</sup> E E k k*

*k* in all FEM. This analysis provided that during Bias Extension test, yarns are

<sup>2</sup> 1 12 11

<sup>2</sup> 2 12 11

2 1 2 12 45

2 1 2 12 45

*<sup>k</sup>* This is another reason to justify the influence of the

11 22 0 *g g E E* (41)

*<sup>F</sup>* (42)

*<sup>F</sup>* (43)

*g g* 11 22 *EEE* (44)

*k* in all FEM. In another hand, numerical value of

(45)

In this work, an orthotropic hyperelastic model test of woven fabric is developed and implanted into Abaqus/explicit to simulate Bias-Extension at low displacement. The analysis of numerical answers along longitudinal and transversal middle paths, proved, in the first hand, that to decreasing the ratio between shearing and tensile rigidities, the state deformation became to be conform to that predicted by the proposed model in the Bias Extension test for all FEM. In another hand, the angle between yarns tends to verify the predicted angle during the Bias Extension test. Although the stress state, is conform to the expected analysis of Traction test on 45°. The analysis of Green Lagrange stretching strain in the yarns direction, demonstrated that there was an elongation of yarns during test for different shearing rigidity. This elongation was exactly conforming to the predicted analytical elongation in the Traction test in 45°. Curves of Force versus displacement of the Traction test in 45° applied to of the central zone A is closely to the numerical answers. We are able to adjust both curves by coefficients of adjustment.

Finite Element Analysis of Bias Extension Test of Dry Woven 175

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Research Journal 66:622-633.

Composite Part A 33:1095–1105.

This study allowed to verify analytical hypothesis adopted to interpret the Bias Extension test. The comparison between in Bias Extension test, the shearing deformation depends of the ratio between shearing and tensile rigidities of fabric. In Spite of the low displacement, this test presented always a stress state.

## **Author details**

Samia Dridi *Department of Mechanics of Structures and Materials of the ISAE, Institute Superieur of the Aeronautique and the Space, Edouard Belin, Toulouse, France* 

## **5. References**


[12] Launay J, Hivet G, Duong A.V, Boisse P.(2008) Experimental Analysis of the Influence of Tensions on in Plane Shear Behaviour of Woven Composite Reinforcements. Composites Science and Technology 68:506–515.

174 Finite Element Analysis – Applications in Mechanical Engineering

this test presented always a stress state.

**Author details** 

**5. References** 

Textiles 13:26-30.

and technologies 1:6-13.

and Technology 66: 919–933.

Samia Dridi

are able to adjust both curves by coefficients of adjustment.

*Department of Mechanics of Structures and Materials of the ISAE,* 

Bias Extension at Low Loads. Fibres & Textiles 16:59-63

Woven Fabrics. Fibres& Textiles 29:333-338.

fabrics. Composite Structures 61:341–352.

Rigidity of Woven Fabrics. Textile Research Journal 76:145-151.

Plain Fabrics. The Journal of the Textile Institute 86:495-497.

experiments. Composites science and technology 64:1453–65.

*Institute Superieur of the Aeronautique and the Space, Edouard Belin, Toulouse, France* 

Traction test in 45° applied to of the central zone A is closely to the numerical answers. We

This study allowed to verify analytical hypothesis adopted to interpret the Bias Extension test. The comparison between in Bias Extension test, the shearing deformation depends of the ratio between shearing and tensile rigidities of fabric. In Spite of the low displacement,

[1] Naujokaitytė L, Strazdienė E, Domskienė J (2008) Investigation of Fabric Behaviour in

[2] Zheng J, Komatsu T, Yazaki Y, Takater M, Inui S, Shimizu Y. (2006) Evaluating Shear

[3] Alamdar-Yazdi A (2004) A New Method to Evaluate Low-stress Shearing Behavior of

[4] Dobb B, Russell S (1995). A System for the Quantitative Comparison of Wrinkling in

[5] Domskienė J, Strazdienė E (2005) Investigation of Fabric Shear Behavior. Fibres &

[6] Mahar T.J, Ajiki I, Dhingra R.C, Postle R(1989) Fabric Mechanical and Physical Properties Relevant to Clothing Manufacture. International journal of clothing science

[7] Lebrun G, Bureau M.N, Denault J (2003) Evaluation of bias-extension and picture-frame test methods for the measurement of intraply shear properties of PP/glass commingled

[8] Sharma S.B, Sutcliffe M.P.F, Chang S.H (2003) Characterisation of Material Properties for Draping of Dry Woven Composite Material. Composites Part A 34:1167–1175. [9] Harrison P, Clifford M.J, Long A.C (2004).Shear Characterisation of Viscous Woven Textile Composites: a comparison between picture frame and bias extension

[10] Potluri P, Perez Ciurezu D.A, Ramgulam R.B (2006) Measurement of Meso-scale Shear Deformations for Modelling Textile Composites. Composite Part A 37:303–314. [11] Lomov S.V, Verpoest I (2006) Model of Shear of Woven Fabric and Parametric Description of Shear Resistance of Glass Woven Reinforcements. Composites Science

	- [30] Tenthijie R.H.W, Akkeman R (2008) Solution to Inter Ply Shear Locking in Finite Element Analyses of Fibre Reinforced Material. Composite Part A 39:1167-1176.

**Section 3** 

**Applications of FEA in** 

**"Wave Propagation and Failure-Analysis"** 

## **Applications of FEA in "Wave Propagation and Failure-Analysis"**

176 Finite Element Analysis – Applications in Mechanical Engineering

[30] Tenthijie R.H.W, Akkeman R (2008) Solution to Inter Ply Shear Locking in Finite Element Analyses of Fibre Reinforced Material. Composite Part A 39:1167-1176.

**Chapter 0**

**Chapter 8**

**Perfectly Matched Layer for Finite Element**

Numerical analysis of scattering and propagation of elastic waves in solids gives insight into physical phenomena under operation of ultrasonic devices such as electromechanical filters and resonators, nondestructive testing with ultrasonic waves and seismic prospecting. To take anisotropy of solids and complex structures of composite solids into account, commercial simulator based on the finite element method are available. Numerical models for finite element analysis (FEA) must be bounded and infinite half spaces of models should be replaced

Perfectly matched layer (PMLs) is one of popular absorbing boundary conditions for truncating the computational domain of open regions without reflection of oblique incident waves. In 1994, Berenger invented a PML for electromagnetic waves in the finite difference time domain (FD-TD) method by a splitting field method.[1] Because fields in Berenger's PML do not satisfy the Maxwell's equations, two concepts have been introduced for implementation in the finite element method (FEM) of electromagnetic wave problems: the analytic continuation or the complex coordinate stretching[2, 3](CCS) and anisotropic PMLs.[4] Nowadays PMLs for electromagnetic waves are widely used in the FD-TD method

Extension of PMLs to elastic waves in isotropic solids in the Cartesian coordinate first appeared in 1996.[5, 6] In the cylindrical and spherical coordinates, PMLs were presented by using splitting field method in isotropic solids in 1999[7] and by using analytic continuation in anisotropic solids in 2002.[8] Recently validity and usefulness of PMLs derived from the analytic continuation in piezoelectric solids was demonstrated. [9–11] Hastings et. al.[5] reported better performance of PMLs by the FD-TD method than the second-order absorbing boundary condition (ABC) of Peng and Toksöz: in the range of the incident angle from 0◦ to 80◦, reflection powers of S- or P-wave, which is excited by a pure S- or P-wave line source

> ©2012 Hasegawa and Shimada, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original

©2012 Hasegawa and Shimada, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Analysis of Elastic Waves in Solids**

Additional information is available at the end of the chapter

with finite domains and absorbing boundary conditions.

work is properly cited.

Koji Hasegawa and Takao Shimada

http://dx.doi.org/10.5772/46162

**1. Introduction**

and the FEM.

## **Perfectly Matched Layer for Finite Element Analysis of Elastic Waves in Solids**

Koji Hasegawa and Takao Shimada

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/46162

## **1. Introduction**

Numerical analysis of scattering and propagation of elastic waves in solids gives insight into physical phenomena under operation of ultrasonic devices such as electromechanical filters and resonators, nondestructive testing with ultrasonic waves and seismic prospecting. To take anisotropy of solids and complex structures of composite solids into account, commercial simulator based on the finite element method are available. Numerical models for finite element analysis (FEA) must be bounded and infinite half spaces of models should be replaced with finite domains and absorbing boundary conditions.

Perfectly matched layer (PMLs) is one of popular absorbing boundary conditions for truncating the computational domain of open regions without reflection of oblique incident waves. In 1994, Berenger invented a PML for electromagnetic waves in the finite difference time domain (FD-TD) method by a splitting field method.[1] Because fields in Berenger's PML do not satisfy the Maxwell's equations, two concepts have been introduced for implementation in the finite element method (FEM) of electromagnetic wave problems: the analytic continuation or the complex coordinate stretching[2, 3](CCS) and anisotropic PMLs.[4] Nowadays PMLs for electromagnetic waves are widely used in the FD-TD method and the FEM.

Extension of PMLs to elastic waves in isotropic solids in the Cartesian coordinate first appeared in 1996.[5, 6] In the cylindrical and spherical coordinates, PMLs were presented by using splitting field method in isotropic solids in 1999[7] and by using analytic continuation in anisotropic solids in 2002.[8] Recently validity and usefulness of PMLs derived from the analytic continuation in piezoelectric solids was demonstrated. [9–11] Hastings et. al.[5] reported better performance of PMLs by the FD-TD method than the second-order absorbing boundary condition (ABC) of Peng and Toksöz: in the range of the incident angle from 0◦ to 80◦, reflection powers of S- or P-wave, which is excited by a pure S- or P-wave line source

©2012 Hasegawa and Shimada, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Hasegawa and Shimada, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 180 Finite Element Analysis – Applications in Mechanical Engineering Perfectly Matched Layer for Finite Element Analysis of Elastic Waves in Solids <sup>3</sup>

and propagating in a two-dimensional infinite isotropic solid modeled by a rectangular solid with its opposite sides attached sponge mediums and other sides imposed ABC or loaded PMLs, from the PML side are suppressed below -45 and -80 dB with 8 and 16 grid spaces of the PML region, respectively. On the other hand, reflection power level at the computational domain edge imposed ABC is in the range of -90 dB to -10 dB. This implies that PMLs yield more superior approximation of perfect matching than the ABC with thickening PML and increasing the number of grid spaces.

In this chapter, we also examine PML performance of FE-models in the frequency range with scattering problems of elastic waves in an isotropic solid as field analysis in the thick layer in the one dimension. To the best of our knowledge, quantification of reflection power generated by FE-discretization has not attracted attention. Recently, for electromagnetic waves, we reported that the reflection power caused by discretization can be computed by the equivalent transmission line with its impedance and propagation constant determined by discretized wave numbers.[15] Because, for elastic waves, the transfer matrix is popular, we explain the reflection from PMLs by the transfer matrix of elastic waves, and confirm that numerical results of FE-models may be predicted by replacement of propagation constants of elastic

Perfectly Matched Layer for Finite Element Analysis of Elastic Waves in Solids 181

**2. Derivation of perfectly matched layers for elastic waves by using**

A particle displacement vector **u**, particle velocity vector **v**, density of momenta **P**, stress

**<sup>u</sup>** <sup>=</sup> *<sup>u</sup><sup>i</sup> <sup>∂</sup> ∂x<sup>i</sup>*

**<sup>v</sup>** <sup>=</sup> *<sup>v</sup><sup>i</sup> <sup>∂</sup> ∂x<sup>i</sup>*

represent the tensor product and the cross product, respectively. *d* is the exterior differential

Changing the coordinate gives relations of tensor components: for a tensor with a tensor type

*<sup>∂</sup>xk* <sup>⊗</sup> *dxβ*<sup>1</sup> ∧···∧ *dxβ<sup>q</sup>* , the relation of tensor components is

*∂xβ*<sup>1</sup> *<sup>∂</sup>Xα*<sup>1</sup> ··· *<sup>∂</sup>xβ<sup>q</sup>*

*dT* ¯¯ <sup>=</sup> *<sup>∂</sup>***<sup>P</sup>** *∂t*

¯¯ are given as follows:

, (1)

, (2)

. (6)

*Xα*1···*α<sup>q</sup>*

*∂*

. (7)

*<sup>∂</sup>Xi* <sup>⊗</sup> *dXα*<sup>1</sup> ∧···∧

*<sup>∂</sup>x<sup>i</sup>* <sup>⊗</sup> *dx<sup>α</sup>* <sup>∧</sup> *dx<sup>β</sup>* <sup>∧</sup> *dxγ*, (3)

*<sup>∂</sup>x<sup>i</sup>* <sup>⊗</sup> *dx<sup>α</sup>* <sup>∧</sup> *dxβ*, (4)

*<sup>∂</sup>x<sup>i</sup>* <sup>⊗</sup> *dx<sup>α</sup>* <sup>=</sup> *<sup>d</sup>***u**, (5)

(*i* = 0, 1, 2) are the contravariant and covariant basis vectors, ⊗ and ∧

*<sup>∂</sup>Xα<sup>q</sup> <sup>V</sup><sup>k</sup>*

*xβ*<sup>1</sup> ···*β<sup>q</sup>*

**complex coordinate stretching and differential form**

**<sup>P</sup>** <sup>=</sup> <sup>1</sup> 3! *Pi αβγ ∂*

*T* ¯¯ <sup>=</sup> <sup>1</sup> 2 *Ti αβ ∂*

*F* ¯¯ = *F<sup>i</sup> α ∂*

of the contravariant of rank 1 and the covariant of rank q, *V* = *V<sup>i</sup>*

*<sup>X</sup>α*1···*α<sup>q</sup>* <sup>=</sup> *<sup>∂</sup>X<sup>i</sup>*

*∂x<sup>k</sup>*

*Vi*

waves in PML with discretized wave numbers.

¯¯ and displacement gradient tensor *F*

**2.1. Differential form**

tensor *T*

where *<sup>∂</sup>*

*dXα<sup>q</sup>* = *V<sup>k</sup>*

*<sup>∂</sup>xi* and *dx<sup>i</sup>*

*xβ*<sup>1</sup> ···*β<sup>q</sup>*

*∂*

operator. Newton's equation of motion is

We recommend that readers who are unfamiliar with PMLs consult Basu and Chopra[12] about explanations and finite element (FE) implementation of PMLs for time-harmonic elastodynamics, Michler et al.[13] about derivation of material constants of PML in FE method by analytic continuation, and Taflove and Hagness[14] about PMLs for electromagnetic waves in FD-TD method.

Although PML is one of attractive artificial materials, two questions of PMLs derived from the analytic continuation are left: why are the particle displacements in the complex coordinate identical to those in the real coordinate and why must we multiply stress tensors by the Jacobian of the coordinate transformation?

For replying to the questions, we will examine a derivation of PMLs for elastic waves in the Cartesian, the cylindrical and the spherical coordinates from the differential form on manifolds. Our results reveal that the components of stress tensors and the particle displacement vectors in the analytic continuation are not transformed to the real space.[15] In addition, the rule for determining PML parameters in the Cartesian coordinate holds in the cylindrical and spherical coordinates.[16]

Mathematical models of PMLs, which are given by differential equations and boundary conditions, are exactly perfect matching medium. In numerical models, however, discretizing PMLs changes phase velocities of propagating waves and generates reflection waves from the PML region.[17] Furthermore, approximation of infinite regions with finite thick layers also generates reflection waves from the PML terminal.[1, 17, 18]

Estimating matching performance and optimizing parameters of PMLs in a numerical domain are required before solving problems. Chew and Jin investigated dependence of PML's performance on attenuation parameters of FE analysis of electromagnetic wave problems.[18] For FD-TD method Collino and Monk also carried out such an investigation.[19] Recently, Bermúdez et al. investigated absorbing functions for time harmonic Helmholtz equations in the Cartesian and cylindrical coordinates under the condition of ignoring reflection caused by FE-discretization and showed the advantages of non-integrable absorbing functions over conventional functions of power series.[20, 21]

Most of these investigations of optimizing attenuation parameters of PML employed numerical analysis of scattering problems in the two dimensions such as plane or cylindrical wave scattering problems. For tackling optimization problem of PML parameters, plane wave scattering problem is appropriate because required resource of computation is small. For FEA, Chew and Jin[18] modeled scattering of plane waves as electromagnetic field analysis in the thick layer in the one dimension. But this model has not been applied to elastic wave scattering.

In this chapter, we also examine PML performance of FE-models in the frequency range with scattering problems of elastic waves in an isotropic solid as field analysis in the thick layer in the one dimension. To the best of our knowledge, quantification of reflection power generated by FE-discretization has not attracted attention. Recently, for electromagnetic waves, we reported that the reflection power caused by discretization can be computed by the equivalent transmission line with its impedance and propagation constant determined by discretized wave numbers.[15] Because, for elastic waves, the transfer matrix is popular, we explain the reflection from PMLs by the transfer matrix of elastic waves, and confirm that numerical results of FE-models may be predicted by replacement of propagation constants of elastic waves in PML with discretized wave numbers.

## **2. Derivation of perfectly matched layers for elastic waves by using complex coordinate stretching and differential form**

#### **2.1. Differential form**

2 Will-be-set-by-IN-TECH

and propagating in a two-dimensional infinite isotropic solid modeled by a rectangular solid with its opposite sides attached sponge mediums and other sides imposed ABC or loaded PMLs, from the PML side are suppressed below -45 and -80 dB with 8 and 16 grid spaces of the PML region, respectively. On the other hand, reflection power level at the computational domain edge imposed ABC is in the range of -90 dB to -10 dB. This implies that PMLs yield more superior approximation of perfect matching than the ABC with thickening PML and

We recommend that readers who are unfamiliar with PMLs consult Basu and Chopra[12] about explanations and finite element (FE) implementation of PMLs for time-harmonic elastodynamics, Michler et al.[13] about derivation of material constants of PML in FE method by analytic continuation, and Taflove and Hagness[14] about PMLs for electromagnetic waves

Although PML is one of attractive artificial materials, two questions of PMLs derived from the analytic continuation are left: why are the particle displacements in the complex coordinate identical to those in the real coordinate and why must we multiply stress tensors by the

For replying to the questions, we will examine a derivation of PMLs for elastic waves in the Cartesian, the cylindrical and the spherical coordinates from the differential form on manifolds. Our results reveal that the components of stress tensors and the particle displacement vectors in the analytic continuation are not transformed to the real space.[15] In addition, the rule for determining PML parameters in the Cartesian coordinate holds in the

Mathematical models of PMLs, which are given by differential equations and boundary conditions, are exactly perfect matching medium. In numerical models, however, discretizing PMLs changes phase velocities of propagating waves and generates reflection waves from the PML region.[17] Furthermore, approximation of infinite regions with finite thick layers also

Estimating matching performance and optimizing parameters of PMLs in a numerical domain are required before solving problems. Chew and Jin investigated dependence of PML's performance on attenuation parameters of FE analysis of electromagnetic wave problems.[18] For FD-TD method Collino and Monk also carried out such an investigation.[19] Recently, Bermúdez et al. investigated absorbing functions for time harmonic Helmholtz equations in the Cartesian and cylindrical coordinates under the condition of ignoring reflection caused by FE-discretization and showed the advantages of non-integrable absorbing functions over

Most of these investigations of optimizing attenuation parameters of PML employed numerical analysis of scattering problems in the two dimensions such as plane or cylindrical wave scattering problems. For tackling optimization problem of PML parameters, plane wave scattering problem is appropriate because required resource of computation is small. For FEA, Chew and Jin[18] modeled scattering of plane waves as electromagnetic field analysis in the thick layer in the one dimension. But this model has not been applied to elastic wave

increasing the number of grid spaces.

Jacobian of the coordinate transformation?

cylindrical and spherical coordinates.[16]

conventional functions of power series.[20, 21]

scattering.

generates reflection waves from the PML terminal.[1, 17, 18]

in FD-TD method.

A particle displacement vector **u**, particle velocity vector **v**, density of momenta **P**, stress tensor *T* ¯¯ and displacement gradient tensor *F* ¯¯ are given as follows:

$$\mathbf{u} = u^{i} \frac{\partial}{\partial \mathbf{x}^{i}}\tag{1}$$

$$\mathbf{v} = v^i \frac{\partial}{\partial \mathbf{x}^i} \,\tag{2}$$

$$\mathbf{P} = \frac{1}{\Im l} P^{i}\_{\alpha\beta\gamma} \frac{\partial}{\partial \mathbf{x}^{i}} \otimes d\mathbf{x}^{\alpha} \wedge d\mathbf{x}^{\beta} \wedge d\mathbf{x}^{\gamma} \tag{3}$$

$$\tilde{T} = \frac{1}{2} T^{i}\_{\alpha\beta} \frac{\partial}{\partial \mathbf{x}^{i}} \otimes d\mathbf{x}^{\alpha} \wedge d\mathbf{x}^{\beta} \,. \tag{4}$$

$$\bar{\vec{F}} = F\_a^i \frac{\partial}{\partial \mathbf{x}^i} \otimes d\mathbf{x}^a = d\mathbf{u}\_\prime \tag{5}$$

where *<sup>∂</sup> <sup>∂</sup>xi* and *dx<sup>i</sup>* (*i* = 0, 1, 2) are the contravariant and covariant basis vectors, ⊗ and ∧ represent the tensor product and the cross product, respectively. *d* is the exterior differential operator. Newton's equation of motion is

$$d\bar{\bar{T}} = \frac{\partial \mathbf{P}}{\partial t}. \tag{6}$$

Changing the coordinate gives relations of tensor components: for a tensor with a tensor type of the contravariant of rank 1 and the covariant of rank q, *V* = *V<sup>i</sup> Xα*1···*α<sup>q</sup> ∂ <sup>∂</sup>Xi* <sup>⊗</sup> *dXα*<sup>1</sup> ∧···∧ *dXα<sup>q</sup>* = *V<sup>k</sup> xβ*<sup>1</sup> ···*β<sup>q</sup> ∂ <sup>∂</sup>xk* <sup>⊗</sup> *dxβ*<sup>1</sup> ∧···∧ *dxβ<sup>q</sup>* , the relation of tensor components is

$$V^{i}\_{Xa\_{1}\cdots a\_{q}} = \frac{\partial X^{i}}{\partial x^{k}} \frac{\partial x^{\beta\_{1}}}{\partial X^{a\_{1}}} \cdots \frac{\partial x^{\beta\_{q}}}{\partial X^{a\_{q}}} V^{k}\_{\mathbf{x}\beta\_{1}\cdots\beta\_{q}}.\tag{7}$$

Using CCS[2, 3, 8] given by *<sup>X</sup><sup>i</sup>* <sup>=</sup> *<sup>x</sup><sup>i</sup> <sup>s</sup>*˜*i*(*τ*)*d<sup>τ</sup>* <sup>=</sup> *<sup>x</sup><sup>i</sup> s*˜*i*R(*τ*) + j*s*˜*i*I(*τ*)*dτ* with the two real functions *s*˜*i*R(*τ*) and *s*˜*i*I(*τ*) , we have the relation

$$V\_{X\mathfrak{a}\_1\cdots\mathfrak{a}\_q}^{i} = \mathfrak{s}\_i(\mathfrak{x}^i) \left[ \mathfrak{s}\_{\mathfrak{a}\_1}(\mathfrak{x}^{\mathfrak{a}\_1}) \cdots \mathfrak{s}\_{\mathfrak{a}\_q}(\mathfrak{x}^{\mathfrak{a}\_q}) \right]^{-1} V\_{\mathfrak{x}\mathfrak{a}\_1\cdots\mathfrak{a}\_q}^{i}.\tag{8}$$

and the stiffness is

Here, *s*<sup>0</sup> = *s*˜0, *s*<sup>1</sup> = *s*˜1

consult Zheng and Huang.[8]

governing equations are

system.

*C*PML

*R*

<sup>−</sup>*ρω*2*ur* <sup>=</sup> <sup>1</sup>

<sup>−</sup>*ρω*2*u<sup>θ</sup>* <sup>=</sup> <sup>1</sup>

<sup>−</sup>*ρω*2*uz* <sup>=</sup> <sup>1</sup>

equations by *s*0*s*1*s*<sup>2</sup> and using the assumption of *u*<sup>c</sup>

*r ∂ ∂r*

*r ∂ ∂r*

*r ∂ ∂r*

of stress tensors in the PMLs as *T*PMLA

governing equations in the PML:

<sup>−</sup>*ρ*PMLA*ω*2*ur* <sup>=</sup> <sup>1</sup>

<sup>−</sup>*ρ*PMLA*ω*2*u<sup>θ</sup>* <sup>=</sup> <sup>1</sup>

<sup>−</sup>*ρ*PMLA*ω*2*uz* <sup>=</sup> <sup>1</sup>

eqs.(17)∼(19).

*r ∂ ∂r*

*r ∂ ∂r*

*r ∂ ∂r*

(*rs*1*s*2*T*<sup>c</sup>

(*rs*1*s*2*T*<sup>c</sup>

(*rs*1*s*2*T*<sup>c</sup>

*rr*) + <sup>1</sup> *r* (

*<sup>θ</sup>r*) + <sup>1</sup> *r* (ˆ *<sup>θ</sup>* • *<sup>∂</sup>r*<sup>ˆ</sup>

*zr*) + <sup>1</sup> *r*

coordinate (*R*, Θ, *Z*), and *s*<sup>0</sup> = *s*˜0, *s*<sup>1</sup> = *s*˜1

*ijkl* <sup>=</sup> *<sup>s</sup>*0*s*1*s*2*sk sisjsl*

**2.3. Derivation of PML constants by the analytic continuation**

*R <sup>r</sup>* , *s*<sup>2</sup> = *s*˜2

(*r*, *θ*, *φ*) with its complex coordinate (*R*, Θ, Φ). In addition, *si* = *s*˜*<sup>i</sup>* in the Cartesian coordinate

Eqs. (15) and (16) show that PML parameters for elastic waves in solids in the cylindrical and spherical coordinates may be calculated by the same procedure in the Cartesian coordinate.

For simplicity, we present a procedure of deriving material constants in only cylindrical coordinates by the analytic continuation[8] below. Note that in spherical coordinates the same procedure may be applied. We recommend that the reader who is interesting in the procedure

First we consider Newton's equation of motion. In a cylindrical coordinate (*r*, *θ*, *z*) , the

*r* ( *∂Tr<sup>θ</sup>*

*r* (ˆ *<sup>θ</sup>* • *<sup>∂</sup>r*<sup>ˆ</sup>

*r ∂Tz<sup>θ</sup> ∂θ* <sup>+</sup>

Here, we use phasor notation. The time dependences of the fields are exp(j*ωt*) where *ω* is angular frequency. Applying CCS with a complex coordinate (*R*, Θ, *Z*), multiplying the CCS

> *∂*(*s*0*s*2*T*<sup>c</sup> *rθ*) *∂θ* <sup>+</sup> *<sup>r</sup>*<sup>ˆ</sup> • *<sup>∂</sup>* <sup>ˆ</sup>

*∂*(*s*0*s*2*T*<sup>c</sup> *zθ* ) *∂θ* <sup>+</sup>

Here, the mass density of PML is defined as *ρ*PMLA = *s*0*s*1*s*2*ρ*. When we rewrite components

*ij* <sup>=</sup> *<sup>s</sup>*0*s*1*s*<sup>2</sup>

*∂θ* (*s*0*s*2*T*<sup>c</sup>

*sj T*c

*∂θ* <sup>+</sup> *<sup>r</sup>*<sup>ˆ</sup> • *<sup>∂</sup>* <sup>ˆ</sup>

*∂θ Tr<sup>θ</sup>* <sup>+</sup>

*∂Tzz*

*θ*

*∂Tθθ*

*<sup>i</sup>* <sup>=</sup> *ui*, *<sup>R</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>r</sup>*ˆ, <sup>Θ</sup><sup>ˆ</sup> <sup>=</sup> <sup>ˆ</sup>

*θ ∂θ* (*s*0*s*2*T*<sup>c</sup>

*<sup>r</sup>θ*) + *<sup>∂</sup>*(*s*0*s*2*T*<sup>c</sup>

*∂*(*s*0*s*1*T*<sup>c</sup> *zz*) *θθ*)

*∂θ <sup>T</sup>θθ*) + *<sup>∂</sup>Trz*

*∂θ* ) + *<sup>∂</sup>Tθ<sup>z</sup>*

*<sup>∂</sup><sup>z</sup>* , (17)

*<sup>∂</sup><sup>z</sup>* , (18)

*θ* and *Z*ˆ = *z*ˆ, we get the

*rz*) *∂z*

*θz*) *<sup>∂</sup><sup>z</sup>* , (21)

, (20)

*<sup>∂</sup><sup>z</sup>* . (19)

*θθ*)) + *<sup>∂</sup>*(*s*0*s*1*T*<sup>c</sup>

*∂θ* ) + *<sup>∂</sup>*(*s*0*s*1*T*<sup>c</sup>

*ij*, we may identify eqs.(20)∼(22) to

*<sup>∂</sup><sup>z</sup>* . (22)

(*rTrr*) + <sup>1</sup>

(*rTθr*) + <sup>1</sup>

(*rTzr*) + <sup>1</sup>

*Cijkl* (no summation). (16)

Perfectly Matched Layer for Finite Element Analysis of Elastic Waves in Solids 183

*<sup>R</sup>* sin <sup>Θ</sup> *<sup>r</sup>* sin *<sup>θ</sup>* in the spherical coordinate system

*<sup>r</sup>* , *s*<sup>2</sup> = *s*˜2 in the cylindrical coordinate system (*r*, *θ*, *z*) with its complex

Here j is the imaginary unit.

#### **2.2. PMLs in the Cartesian, the cylindrical and the spherical coordinates**

In the complex coordinate stretching (CCS), we consider that the real coordinate (*x*0, *x*1, *x*2) is (*x*, *y*, *z*), (*r*, *θ*, *z*) or (*r*, *θ*, *φ*) for the Cartesian, the cylindrical or the spherical coordinate, respectively. Assuming that the same constitutive equations in the real Cartesian, cylindrical and spherical coordinate exist in the complex coordinate, (*X*0, *X*1, *X*2), we have

$$\mathbf{P}^{\mathbf{c}} = \rho \mathbf{v}^{\mathbf{c}}\,\,\,\,\,\tag{9}$$

$$\begin{split} T\_{ij}^{\mathbf{c}} &= \mathbb{C}\_{ijkl} \mathbf{S}\_{kl}^{\mathbf{c}} \\ &= \mathbb{C}\_{ijkl} (F\_{kl}^{\mathbf{c}} + F\_{lk}^{\mathbf{c}}) / 2 \\ &= \mathbb{C}\_{ijkl} F\_{kl}^{\mathbf{c}} .\end{split}\tag{10}$$

Here, the superscript c denotes the value in the complex coordinate and the mass density *ρ* and the stiffness *Cijkl* (*i*, *j*, *k*, *l* = *X*0, *X*1, *X*2) are the values corresponding to original material parameters, mass density and stiffness constants, of its PML in the Cartesian, the cylindrical and the spherical coordinates. Using eq. (8) to eqs.(2)-(5), and recalling that the base differentials *dξ*, *dη* and *dζ* of the general orthogonal coordinate system (*ξ*, *η*, *ζ*) are dual to the unit vectors <sup>ˆ</sup> *ξ hξ* , *η*ˆ *<sup>h</sup><sup>η</sup>* and <sup>ˆ</sup> *ζ hζ* , we have

$$
\upsilon\_i^\mathbb{C} = s\_i \upsilon\_i \text{ (no summation)}.\tag{11}
$$

$$P\_{\dot{l}}^{\mathbb{C}} = \frac{s\_{\dot{l}}}{s\_{0}s\_{1}s\_{2}}P\_{\text{i}} \text{ (no summation)},\tag{12}$$

$$T\_{i\circ}^{\mathbb{C}} = \frac{s\_i s\_j}{s\_0 s\_1 s\_2} T\_{i\circ} \text{ (no summation)},\tag{13}$$

$$F\_{i\bar{j}}^{\xi} = \frac{s\_{\bar{i}}}{s\_{\bar{j}}} F\_{i\bar{j}} \text{ (no summation)}.\tag{14}$$

Here *si* <sup>=</sup> *<sup>h</sup>*<sup>c</sup> *i h*r *i s*˜*<sup>i</sup>* with *h*<sup>r</sup> *<sup>i</sup>* and *<sup>h</sup>*<sup>c</sup> *<sup>i</sup>* being scale factors of general orthogonal coordinate systems (*x*0, *x*1, *x*2) and (*X*0, *X*1, *X*2), respectively. Note that the scale factors *hi* are given by follows: in the cylindrical coordinate (*r*, *θ*, *z*) *h*<sup>0</sup> = 1, *h*<sup>1</sup> = *r*, *h*<sup>2</sup> = 1 , and in the spherical coordinate (*r*, *θ*, *φ*) *h*<sup>0</sup> = 1, *h*<sup>1</sup> = *r*, *h*<sup>2</sup> = *r* sin *θ*. In addition, in the Cartesian coordinate, *h*<sup>0</sup> = *h*<sup>1</sup> = *h*<sup>2</sup> = 1.

The quotient rule and eqs. (9)- (14) yield PML material constants: the mass density *ρ*PML is

$$
\rho^{\rm PML} = s\_0 s\_1 s\_2 \rho \tag{15}
$$

and the stiffness is

4 Will-be-set-by-IN-TECH

*<sup>s</sup>*˜*i*(*τ*)*d<sup>τ</sup>* <sup>=</sup> *<sup>x</sup><sup>i</sup>*

*<sup>s</sup>*˜*α*<sup>1</sup> (*xα*<sup>1</sup> )···*s*˜*α<sup>q</sup>* (*xα<sup>q</sup>* )

In the complex coordinate stretching (CCS), we consider that the real coordinate (*x*0, *x*1, *x*2) is (*x*, *y*, *z*), (*r*, *θ*, *z*) or (*r*, *θ*, *φ*) for the Cartesian, the cylindrical or the spherical coordinate, respectively. Assuming that the same constitutive equations in the real Cartesian, cylindrical

*kl*

Here, the superscript c denotes the value in the complex coordinate and the mass density *ρ* and the stiffness *Cijkl* (*i*, *j*, *k*, *l* = *X*0, *X*1, *X*2) are the values corresponding to original material parameters, mass density and stiffness constants, of its PML in the Cartesian, the cylindrical and the spherical coordinates. Using eq. (8) to eqs.(2)-(5), and recalling that the base differentials *dξ*, *dη* and *dζ* of the general orthogonal coordinate system (*ξ*, *η*, *ζ*) are dual

(*x*0, *x*1, *x*2) and (*X*0, *X*1, *X*2), respectively. Note that the scale factors *hi* are given by follows: in the cylindrical coordinate (*r*, *θ*, *z*) *h*<sup>0</sup> = 1, *h*<sup>1</sup> = *r*, *h*<sup>2</sup> = 1 , and in the spherical coordinate (*r*, *θ*, *φ*) *h*<sup>0</sup> = 1, *h*<sup>1</sup> = *r*, *h*<sup>2</sup> = *r* sin *θ*. In addition, in the Cartesian coordinate, *h*<sup>0</sup> = *h*<sup>1</sup> = *h*<sup>2</sup> = 1. The quotient rule and eqs. (9)- (14) yield PML material constants: the mass density *ρ*PML is

*kl* <sup>+</sup> *<sup>F</sup>*<sup>c</sup> *lk*)/2 −<sup>1</sup> *Vi xα*1···*α<sup>q</sup>*

*s*˜*i*R(*τ*) + j*s*˜*i*I(*τ*)*dτ* with the two real

, (9)

*kl*. (10)

*<sup>i</sup>* = *sivi* (no summation), (11)

*<sup>i</sup>* being scale factors of general orthogonal coordinate systems

*Pi* (no summation), (12)

*Tij* (no summation), (13)

*Fij* (no summation). (14)

*ρ*PML = *s*0*s*1*s*2*ρ* (15)

. (8)

Using CCS[2, 3, 8] given by *<sup>X</sup><sup>i</sup>* <sup>=</sup> *<sup>x</sup><sup>i</sup>*

Here j is the imaginary unit.

to the unit vectors <sup>ˆ</sup>

Here *si* <sup>=</sup> *<sup>h</sup>*<sup>c</sup>

*i h*r *i*

*s*˜*<sup>i</sup>* with *h*<sup>r</sup>

*ξ hξ* , *η*ˆ *<sup>h</sup><sup>η</sup>* and <sup>ˆ</sup> *ζ hζ*

functions *s*˜*i*R(*τ*) and *s*˜*i*I(*τ*) , we have the relation

*Vi*

*<sup>X</sup>α*1···*α<sup>q</sup>* <sup>=</sup>*s*˜*i*(*x<sup>i</sup>*

) 

**2.2. PMLs in the Cartesian, the cylindrical and the spherical coordinates**

and spherical coordinate exist in the complex coordinate, (*X*0, *X*1, *X*2), we have

**P**<sup>c</sup> = *ρ***v** <sup>c</sup>

*ij* <sup>=</sup> *CijklS*<sup>c</sup>

= *Cijkl*(*F*<sup>c</sup>

= *CijklF*<sup>c</sup>

*T*c

, we have

*ij* <sup>=</sup> *sisj s*0*s*1*s*<sup>2</sup>

*v*c

*P*c *<sup>i</sup>* <sup>=</sup> *si s*0*s*1*s*<sup>2</sup>

*T*c

*F*c *ij* <sup>=</sup> *si sj*

*<sup>i</sup>* and *<sup>h</sup>*<sup>c</sup>

$$\mathcal{C}^{\text{PML}}\_{ijkl} = \frac{\text{s}\_{0}\text{s}\_{1}\text{s}\_{2}\text{s}\_{k}}{\text{s}\_{i}\text{s}\_{j}\text{s}\_{l}}\mathcal{C}\_{ijkl} \text{ (no summation)}.\tag{16}$$

Here, *s*<sup>0</sup> = *s*˜0, *s*<sup>1</sup> = *s*˜1 *R <sup>r</sup>* , *s*<sup>2</sup> = *s*˜2 in the cylindrical coordinate system (*r*, *θ*, *z*) with its complex coordinate (*R*, Θ, *Z*), and *s*<sup>0</sup> = *s*˜0, *s*<sup>1</sup> = *s*˜1 *R <sup>r</sup>* , *s*<sup>2</sup> = *s*˜2 *<sup>R</sup>* sin <sup>Θ</sup> *<sup>r</sup>* sin *<sup>θ</sup>* in the spherical coordinate system (*r*, *θ*, *φ*) with its complex coordinate (*R*, Θ, Φ). In addition, *si* = *s*˜*<sup>i</sup>* in the Cartesian coordinate system.

Eqs. (15) and (16) show that PML parameters for elastic waves in solids in the cylindrical and spherical coordinates may be calculated by the same procedure in the Cartesian coordinate.

#### **2.3. Derivation of PML constants by the analytic continuation**

For simplicity, we present a procedure of deriving material constants in only cylindrical coordinates by the analytic continuation[8] below. Note that in spherical coordinates the same procedure may be applied. We recommend that the reader who is interesting in the procedure consult Zheng and Huang.[8]

First we consider Newton's equation of motion. In a cylindrical coordinate (*r*, *θ*, *z*) , the governing equations are

$$-\rho\omega^2 u\_r = \frac{1}{r}\frac{\partial}{\partial r}(rT\_{rr}) + \frac{1}{r}(\frac{\partial T\_{r\theta}}{\partial \theta} + \hat{r}\bullet\frac{\partial \hat{\theta}}{\partial \theta}T\_{\theta\theta}) + \frac{\partial T\_{rz}}{\partial z},\tag{17}$$

$$-\rho\omega^2\mathbf{u}\_{\theta} = \frac{1}{r}\frac{\partial}{\partial r}(rT\_{\theta r}) + \frac{1}{r}(\dot{\theta}\bullet\frac{\partial\hat{r}}{\partial\theta}T\_{r\theta} + \frac{\partial T\_{\theta\theta}}{\partial\theta}) + \frac{\partial T\_{\theta z}}{\partial z},\tag{18}$$

$$-\rho\omega^2 u\_z = \frac{1}{r}\frac{\partial}{\partial r}(rT\_{zr}) + \frac{1}{r}\frac{\partial T\_{z\theta}}{\partial \theta} + \frac{\partial T\_{zz}}{\partial z}.\tag{19}$$

Here, we use phasor notation. The time dependences of the fields are exp(j*ωt*) where *ω* is angular frequency. Applying CCS with a complex coordinate (*R*, Θ, *Z*), multiplying the CCS equations by *s*0*s*1*s*<sup>2</sup> and using the assumption of *u*<sup>c</sup> *<sup>i</sup>* <sup>=</sup> *ui*, *<sup>R</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>r</sup>*ˆ, <sup>Θ</sup><sup>ˆ</sup> <sup>=</sup> <sup>ˆ</sup> *θ* and *Z*ˆ = *z*ˆ, we get the governing equations in the PML:

$$-\rho^{\rm PMLA}\omega^2 u\_r = \frac{1}{r}\frac{\partial}{\partial r}(rs\_1s\_2T^\varepsilon\_{rr}) + \frac{1}{r}(\frac{\partial(s\_0s\_2T^\varepsilon\_{r\theta})}{\partial \theta} + \mathfrak{f}\bullet\frac{\partial\bar{\theta}}{\partial \theta}(s\_0s\_2T^\varepsilon\_{\theta\theta})) + \frac{\partial(s\_0s\_1T^\varepsilon\_{rz})}{\partial z},\tag{20}$$

$$-\rho^{\rm PMLA}\omega^2\mu\_{\theta} = \frac{1}{r}\frac{\partial}{\partial r}(rs\_1s\_2T^{\xi}\_{\theta r}) + \frac{1}{r}(\dot{\theta}\bullet\frac{\partial\theta}{\partial\theta}(s\_0s\_2T^{\xi}\_{r\theta}) + \frac{\partial(s\_0s\_2T^{\xi}\_{\theta\theta})}{\partial\theta}) + \frac{\partial(s\_0s\_1T^{\xi}\_{\theta z})}{\partial z},\tag{21}$$

$$-\rho^{\rm PMLA}\omega^2 u\_z = \frac{1}{r}\frac{\partial}{\partial r}(rs\_1s\_2T^\xi\_{zr}) + \frac{1}{r}\frac{\partial(s\_0s\_2T^\xi\_{z\theta})}{\partial \theta} + \frac{\partial(s\_0s\_1T^\xi\_{zz})}{\partial z}.\tag{22}$$

Here, the mass density of PML is defined as *ρ*PMLA = *s*0*s*1*s*2*ρ*. When we rewrite components of stress tensors in the PMLs as *T*PMLA *ij* <sup>=</sup> *<sup>s</sup>*0*s*1*s*<sup>2</sup> *sj T*c *ij*, we may identify eqs.(20)∼(22) to eqs.(17)∼(19).

#### 6 Will-be-set-by-IN-TECH 184 Finite Element Analysis – Applications in Mechanical Engineering Perfectly Matched Layer for Finite Element Analysis of Elastic Waves in Solids <sup>7</sup>

Next we consider the displacement gradient <sup>∇</sup>**u**<sup>c</sup> = [Γ<sup>c</sup> *kl*] in the complex coordinate (*R*, Θ, *Z*). Using definition *<sup>d</sup>***<sup>u</sup>** <sup>=</sup> <sup>∇</sup>**<sup>u</sup>** • (*x*ˆ*<sup>i</sup> hidx<sup>i</sup>* ) and the assumption of *u*<sup>c</sup> *<sup>i</sup>* <sup>=</sup> *ui*, *<sup>R</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>r</sup>*ˆ, <sup>Θ</sup><sup>ˆ</sup> <sup>=</sup> <sup>ˆ</sup> *θ*, *Z*ˆ = *z*ˆ, and applying CCS with a complex coordinate (*R*, Θ, *Z*), we have the relation:

<sup>∇</sup>**u**<sup>c</sup> <sup>=</sup> ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *∂u*<sup>c</sup> *R ∂R* 1 *R* � *∂u*<sup>c</sup> *R <sup>∂</sup>*<sup>Θ</sup> <sup>+</sup> *<sup>R</sup>*<sup>ˆ</sup> · *<sup>∂</sup>*Θ<sup>ˆ</sup> *<sup>∂</sup>*<sup>Θ</sup> *<sup>u</sup>*<sup>c</sup> Θ � *∂u*<sup>c</sup> *R ∂Z ∂u*<sup>c</sup> Θ *∂R* 1 *R* � <sup>Θ</sup><sup>ˆ</sup> · *<sup>∂</sup>R*<sup>ˆ</sup> *<sup>∂</sup>*<sup>Θ</sup> *<sup>u</sup>*<sup>c</sup> *<sup>R</sup>* <sup>+</sup> *<sup>∂</sup>u*<sup>c</sup> Θ *∂*Θ � *∂u*<sup>c</sup> Θ *∂Z ∂u*<sup>c</sup> *Z ∂R* 1 *R ∂u*<sup>c</sup> *Z ∂*Θ *∂u*<sup>c</sup> *Z ∂Z* ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ 1 *s*0 *∂ur <sup>∂</sup><sup>r</sup>* <sup>1</sup> *s*1 1 *r* � *<sup>∂</sup>ur ∂θ* <sup>+</sup> *<sup>r</sup>*<sup>ˆ</sup> · *<sup>∂</sup>* <sup>ˆ</sup> *θ ∂θ u<sup>θ</sup>* � <sup>1</sup> *s*2 *∂ur ∂z* 1 *s*0 *∂u<sup>θ</sup> <sup>∂</sup><sup>r</sup>* <sup>1</sup> *s*1 1 *r* � ˆ *<sup>θ</sup>* · *<sup>∂</sup>r*<sup>ˆ</sup> *∂θ ur* <sup>+</sup> *<sup>∂</sup>u<sup>θ</sup> ∂θ* � <sup>1</sup> *s*2 *∂u<sup>θ</sup> ∂z* 1 *s*0 *∂uz <sup>∂</sup><sup>r</sup>* <sup>1</sup> *s*1 1 *r ∂uz ∂θ* 1 *s*2 *∂uz ∂z* ⎤ ⎥ ⎥ ⎥ <sup>⎦</sup> . (23)

*i j*

0 0 <sup>1</sup> *s*2 0

1 1 <sup>1</sup> *s*2 1

2 2 <sup>1</sup> *s*2 2

1 2 *μ*

2 1 *μ*

2 0 *μ*

0 2 *μ*

0 1 *μ*

1 0 *μ*

 *λ sisk*

*C*PMLA *ijkl* =

coordinate.

gives *T*˜

*ij* <sup>=</sup> *<sup>T</sup>*˜

**frequency domain**

respectively.

*C*PML

 *u*1,2 *s*2 2

 *u*1,2 *s*2 2

 *u*2,0 *s*2 0

 *u*2,0 *s*2 0

 *u*0,1 *s*2 1

 *u*0,1 *s*2 1

*<sup>δ</sup>ijδkl*<sup>+</sup> *<sup>μ</sup> s*2 *j*

however, we have a symmetric stress tensor, *Tij* = *Tji* (*i* = *j*).

where j is the imaginary unit and *ω* is the angular frequency.

*C*PML ijkl = ( *<sup>λ</sup> s*2 i

the stiffness component of its PML *C*PML

*Tij s*0*s*1*s*<sup>2</sup>

*μ*

*μ*

*μ*

*μ*

*μ*

*μ*

*<sup>δ</sup>ilδjk*

*ji* (*<sup>i</sup>* <sup>=</sup> *<sup>j</sup>*) and we predict that rotational forces may be observed. With *<sup>C</sup>*PML

*ijkl* :analytic continuation

Perfectly Matched Layer for Finite Element Analysis of Elastic Waves in Solids 185

*<sup>s</sup>*<sup>1</sup> <sup>+</sup> *<sup>u</sup>*2,2 *<sup>s</sup>*<sup>2</sup> )

*<sup>s</sup>*<sup>2</sup> <sup>+</sup> *<sup>u</sup>*0,0 *<sup>s</sup>*<sup>0</sup> )

*<sup>s</sup>*<sup>0</sup> <sup>+</sup> *<sup>u</sup>*1,1 *<sup>s</sup>*<sup>1</sup> )

*s*0*s*1*s*<sup>2</sup> (no summation). (25)

*ijkl* and *<sup>C</sup>*PMLA

*δ*il*δ*jk)*sxsysz*. (26)

*ijkl* . *<sup>C</sup>*PMLA *ijkl*

*ijkl* ,

*s*2 0 + *<sup>λ</sup> <sup>s</sup>*<sup>0</sup> ( *<sup>u</sup>*1,1

*s*2 1 + *<sup>λ</sup> <sup>s</sup>*<sup>1</sup> ( *<sup>u</sup>*2,2

*s*2 2 + *<sup>λ</sup> <sup>s</sup>*<sup>2</sup> ( *<sup>u</sup>*0,0

*s*2 *u*1,2 *<sup>s</sup>*<sup>2</sup> <sup>+</sup> *<sup>u</sup>*2,1 *s*1 

*s*1 *u*1,2 *<sup>s</sup>*<sup>2</sup> <sup>+</sup> *<sup>u</sup>*2,1 *s*1 

*s*0 *u*2,0 *<sup>s</sup>*<sup>0</sup> <sup>+</sup> *<sup>u</sup>*0,2 *s*2 

*s*2 *u*2,0 *<sup>s</sup>*<sup>0</sup> <sup>+</sup> *<sup>u</sup>*0,2 *s*2 

*s*1 *u*0,1 *<sup>s</sup>*<sup>1</sup> <sup>+</sup> *<sup>u</sup>*1,0 *s*0 

*s*0 *u*0,1 *<sup>s</sup>*<sup>1</sup> <sup>+</sup> *<sup>u</sup>*1,0 *s*0 

*ijkl* :differential form *<sup>C</sup>*PMLA

[(*<sup>λ</sup>* <sup>+</sup> <sup>2</sup>*μ*)*u*0,0 <sup>+</sup> *<sup>λ</sup>*(*u*1,1 <sup>+</sup> *<sup>u</sup>*2,2)] (*<sup>λ</sup>* <sup>+</sup> <sup>2</sup>*μ*) *<sup>u</sup>*0,0

[(*<sup>λ</sup>* <sup>+</sup> <sup>2</sup>*μ*)*u*1,1 <sup>+</sup> *<sup>λ</sup>*(*u*2,2 <sup>+</sup> *<sup>u</sup>*0,0)] (*<sup>λ</sup>* <sup>+</sup> <sup>2</sup>*μ*) *<sup>u</sup>*1,1

[(*<sup>λ</sup>* <sup>+</sup> <sup>2</sup>*μ*)*u*2,2 <sup>+</sup> *<sup>λ</sup>*(*u*0,0 <sup>+</sup> *<sup>u</sup>*1,1)] (*<sup>λ</sup>* <sup>+</sup> <sup>2</sup>*μ*) *<sup>u</sup>*2,2

+ *<sup>u</sup>*2,1 *s*2 1

+ *<sup>u</sup>*2,1 *s*2 1

+ *<sup>u</sup>*0,2 *s*2 2

+ *<sup>u</sup>*0,2 *s*2 2

+ *<sup>u</sup>*1,0 *s*2 0

+ *<sup>u</sup>*1,0 *s*2 0

Table 1 shows all components of the stress tensor computed with *C*PML

**Table 1.** Components of a stress tensor in a PML material of an isotropic solid in the Cartesian

*<sup>δ</sup>ikδjl*<sup>+</sup> *<sup>μ</sup> sisj*

**3. Reflection from PMLs discretized for finite element models in the**

We consider a plane elastic wave propagating in a half infinite isotropic solid attached with its PML backed with a vacuum region as shown in Fig.1. Here *θ* is the incident angle, *θp* and *θ<sup>s</sup>* are propagation angles of P-waves and SV- or SH-waves, *L* is thickness of the PML, **k**<sup>i</sup> and **k***r*,*<sup>m</sup>* (*m* = 0, 1, 2) are wave vectors of the incident wave and reflected P-, SV- and SH-waves,

We use the phasor notation and assume that the time dependences of all fields are exp(j*ωt*),

When the stiffness component of the isotropic solid *C*ijkl (i, j, k, l = *x*, *y*, *z*) is given by *C*ijkl = *λδ*ij*δ*kl + *μ*(*δ*ik*δ*jl + *δ*jl*δ*ik) where *λ* and *μ* are the Lamé constants and *δ*ij is the Kronecker delta,

> *<sup>δ</sup>*ik*δ*jl <sup>+</sup> *<sup>μ</sup> s*2 i

ijkl is

*<sup>δ</sup>*ij*δ*kl <sup>+</sup> *<sup>μ</sup> s*2 j

Hence we have Γ<sup>c</sup> *kl* <sup>=</sup> <sup>1</sup> *sl* Γ*kl*.

Using the quotient rule and the constitutive equation, *T*<sup>c</sup> *ij* <sup>=</sup> *Cijkl*Γ<sup>c</sup> *kl*, we get the constitutive equation of the PML in the real coordinate (*r*, *θ*, *z*): *T*PMLA *ij* <sup>=</sup> *<sup>s</sup>*0*s*1*s*<sup>2</sup> *sjsl Cijkl*Γ*kl*. Therefore, we may define the stiffness of the PML derived by the analytic continuation: *C*PMLA *ijkl* <sup>=</sup> *<sup>s</sup>*0*s*1*s*<sup>2</sup> *sjsl Cijkl*.

## **2.4. Comparison with PML material constants derived from differential forms and the analytic continuation**

By the analytic continuation, Zheng and Huang[8] derived the mass density and stiffness of PML in the cylindrical and spherical coordinates: *ρ*PMLA = *s*0*s*1*s*2*ρ* and *C*PMLA *ijkl* <sup>=</sup> *<sup>s</sup>*0*s*1*s*<sup>2</sup> *sjsl Cijkl*. The mass density agree with our result, eq. (15), because multiplying the stress tensors by the Jacobian of the coordinate transformation, *s*0*s*1*s*<sup>2</sup> , adjusts the mass density. We note that the form of eq. (15) is also derived from eq. (6) with the tensor type of mass density being covariant of rank 3, i.e. 3-form. The stiffness is different from eq. (16) because in the analytic continuation, the manipulation of the coordinate transformation corresponding to the part of stress tensor and the particle displacement vector, contravariant of rank 1, is excluded. This fact can be confirmed by the derivation procedure presented in the previous section for the cylindrical coordinate:we put *T*PMLA *ij* <sup>=</sup> *<sup>s</sup>*0*s*1*s*<sup>2</sup> *sj T*c *ij* and use the assumption *ui* <sup>=</sup> *<sup>u</sup>*<sup>c</sup> *i* .

To show a difference between PML material constants, we consider an isotropic solid with following stiffness constants in the Cartesian coordinate (*x*0, *x*1, *x*2): *Cijkl* = *λδijδkl* + *μ*(*δikδjl* + *δilδjk*). Here, *λ* and *μ* are the Lamé constants of an isotropic solid, the subscripts *i*, *j*, *k* and *l* denote the *xi*-, *xj*- , *xk*- and *xl*-axis, respectively, and *δij* is the Kronecker delta. Components of stiffness tensors derived from the differential form and analytic continuation, *C*PML *ijkl* and *C*PMLA *ijkl* , respectively, are given by

$$\mathbf{C}\_{ijkl}^{\rm PML} = \left[ (\frac{\lambda}{s\_i^2} \delta\_{ij} \delta\_{kl} + \frac{\mu}{s\_j^2} \delta\_{ik} \delta\_{jl} + \frac{\mu}{s\_i^2} \delta\_{il} \delta\_{jk} \right] s\_0 s\_1 s\_2 \quad \text{(no summation)},\tag{24}$$


6 Will-be-set-by-IN-TECH

1 *R ∂u*<sup>c</sup> *Z ∂*Θ

> *s*1 1 *r ∂uz ∂θ*

) and the assumption of *u*<sup>c</sup>

*<sup>∂</sup>*<sup>Θ</sup> *<sup>u</sup>*<sup>c</sup> Θ � *∂u*<sup>c</sup> *R ∂Z*

> � *∂u*<sup>c</sup> Θ *∂Z*

> > *∂u*<sup>c</sup> *Z ∂Z*

> > > 1 *s*2 *∂uz ∂z*

*ij* <sup>=</sup> *Cijkl*Γ<sup>c</sup>

*ij* and use the assumption *ui* <sup>=</sup> *<sup>u</sup>*<sup>c</sup>

*ij* <sup>=</sup> *<sup>s</sup>*0*s*1*s*<sup>2</sup>

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

> ⎤ ⎥ ⎥ ⎥

*<sup>R</sup>* <sup>+</sup> *<sup>∂</sup>u*<sup>c</sup> Θ *∂*Θ

*∂θ* <sup>+</sup> *<sup>r</sup>*<sup>ˆ</sup> · *<sup>∂</sup>* <sup>ˆ</sup> *θ ∂θ u<sup>θ</sup>* � <sup>1</sup> *s*2 *∂ur ∂z*

> *∂θ ur* <sup>+</sup> *<sup>∂</sup>u<sup>θ</sup> ∂θ* � <sup>1</sup> *s*2 *∂u<sup>θ</sup> ∂z*

*kl*] in the complex coordinate (*R*, Θ, *Z*).

*<sup>i</sup>* <sup>=</sup> *ui*, *<sup>R</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>r</sup>*ˆ, <sup>Θ</sup><sup>ˆ</sup> <sup>=</sup> <sup>ˆ</sup>

<sup>⎦</sup> . (23)

*kl*, we get the constitutive

*ijkl* <sup>=</sup> *<sup>s</sup>*0*s*1*s*<sup>2</sup>

*i* .

*s*0*s*1*s*<sup>2</sup> (no summation), (24)

*sjsl Cijkl*.

*ijkl* and

*sjsl Cijkl*.

*sjsl Cijkl*Γ*kl*. Therefore, we may

*ijkl* <sup>=</sup> *<sup>s</sup>*0*s*1*s*<sup>2</sup>

*θ*, *Z*ˆ = *z*ˆ,

Next we consider the displacement gradient <sup>∇</sup>**u**<sup>c</sup> = [Γ<sup>c</sup>

<sup>∇</sup>**u**<sup>c</sup> <sup>=</sup>

=

Using the quotient rule and the constitutive equation, *T*<sup>c</sup>

equation of the PML in the real coordinate (*r*, *θ*, *z*): *T*PMLA

*hidx<sup>i</sup>*

*∂u*<sup>c</sup> *R ∂R* 1 *R* � *∂u*<sup>c</sup> *R <sup>∂</sup>*<sup>Θ</sup> <sup>+</sup> *<sup>R</sup>*<sup>ˆ</sup> · *<sup>∂</sup>*Θ<sup>ˆ</sup>

*∂u*<sup>c</sup> Θ *∂R* 1 *R* � <sup>Θ</sup><sup>ˆ</sup> · *<sup>∂</sup>R*<sup>ˆ</sup> *<sup>∂</sup>*<sup>Θ</sup> *<sup>u</sup>*<sup>c</sup>

*∂u*<sup>c</sup> *Z ∂R*

1 *s*0 *∂ur <sup>∂</sup><sup>r</sup>* <sup>1</sup> *s*1 1 *r* � *<sup>∂</sup>ur*

1 *s*0 *∂u<sup>θ</sup> <sup>∂</sup><sup>r</sup>* <sup>1</sup> *s*1 1 *r* � ˆ *<sup>θ</sup>* · *<sup>∂</sup>r*<sup>ˆ</sup>

1 *s*0 *∂uz*

define the stiffness of the PML derived by the analytic continuation: *C*PMLA

PML in the cylindrical and spherical coordinates: *ρ*PMLA = *s*0*s*1*s*2*ρ* and *C*PMLA

*ij* <sup>=</sup> *<sup>s</sup>*0*s*1*s*<sup>2</sup> *sj T*c

*<sup>∂</sup><sup>r</sup>* <sup>1</sup>

**2.4. Comparison with PML material constants derived from differential forms**

By the analytic continuation, Zheng and Huang[8] derived the mass density and stiffness of

The mass density agree with our result, eq. (15), because multiplying the stress tensors by the Jacobian of the coordinate transformation, *s*0*s*1*s*<sup>2</sup> , adjusts the mass density. We note that the form of eq. (15) is also derived from eq. (6) with the tensor type of mass density being covariant of rank 3, i.e. 3-form. The stiffness is different from eq. (16) because in the analytic continuation, the manipulation of the coordinate transformation corresponding to the part of stress tensor and the particle displacement vector, contravariant of rank 1, is excluded. This fact can be confirmed by the derivation procedure presented in the previous section for the

To show a difference between PML material constants, we consider an isotropic solid with following stiffness constants in the Cartesian coordinate (*x*0, *x*1, *x*2): *Cijkl* = *λδijδkl* + *μ*(*δikδjl* + *δilδjk*). Here, *λ* and *μ* are the Lamé constants of an isotropic solid, the subscripts *i*, *j*, *k* and *l* denote the *xi*-, *xj*- , *xk*- and *xl*-axis, respectively, and *δij* is the Kronecker delta. Components of stiffness tensors derived from the differential form and analytic continuation, *C*PML

> *<sup>δ</sup>ikδjl*<sup>+</sup> *<sup>μ</sup> s*2 *i δilδjk* �

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎢ ⎣

and applying CCS with a complex coordinate (*R*, Θ, *Z*), we have the relation:

Using definition *<sup>d</sup>***<sup>u</sup>** <sup>=</sup> <sup>∇</sup>**<sup>u</sup>** • (*x*ˆ*<sup>i</sup>*

Hence we have Γ<sup>c</sup>

*kl* <sup>=</sup> <sup>1</sup> *sl* Γ*kl*.

**and the analytic continuation**

cylindrical coordinate:we put *T*PMLA

*ijkl* , respectively, are given by

*C*PML *ijkl* = � ( *λ s*2 *i*

*<sup>δ</sup>ijδkl*<sup>+</sup> *<sup>μ</sup> s*2 *j*

*C*PMLA

**Table 1.** Components of a stress tensor in a PML material of an isotropic solid in the Cartesian coordinate.

$$\mathbf{C}\_{ijkl}^{\rm PMLA} = \left[ \frac{\lambda}{s\_l s\_k} \delta\_{lj} \delta\_{kl} + \frac{\mu}{s\_j^2} \delta\_{ik} \delta\_{jl} + \frac{\mu}{s\_l s\_j} \delta\_{il} \delta\_{jk} \right] s\_0 s\_1 s\_2 \quad \text{(no summation)}.\tag{25}$$

Table 1 shows all components of the stress tensor computed with *C*PML *ijkl* and *<sup>C</sup>*PMLA *ijkl* . *<sup>C</sup>*PMLA *ijkl* gives *T*˜ *ij* <sup>=</sup> *<sup>T</sup>*˜ *ji* (*<sup>i</sup>* <sup>=</sup> *<sup>j</sup>*) and we predict that rotational forces may be observed. With *<sup>C</sup>*PML *ijkl* , however, we have a symmetric stress tensor, *Tij* = *Tji* (*i* = *j*).

## **3. Reflection from PMLs discretized for finite element models in the frequency domain**

We consider a plane elastic wave propagating in a half infinite isotropic solid attached with its PML backed with a vacuum region as shown in Fig.1. Here *θ* is the incident angle, *θp* and *θ<sup>s</sup>* are propagation angles of P-waves and SV- or SH-waves, *L* is thickness of the PML, **k**<sup>i</sup> and **k***r*,*<sup>m</sup>* (*m* = 0, 1, 2) are wave vectors of the incident wave and reflected P-, SV- and SH-waves, respectively.

We use the phasor notation and assume that the time dependences of all fields are exp(j*ωt*), where j is the imaginary unit and *ω* is the angular frequency.

When the stiffness component of the isotropic solid *C*ijkl (i, j, k, l = *x*, *y*, *z*) is given by *C*ijkl = *λδ*ij*δ*kl + *μ*(*δ*ik*δ*jl + *δ*jl*δ*ik) where *λ* and *μ* are the Lamé constants and *δ*ij is the Kronecker delta, the stiffness component of its PML *C*PML ijkl is

$$\mathcal{C}^{\rm FML}\_{\rm ijkl} = (\frac{\lambda}{s\_{\rm i}^2} \delta\_{\rm ij} \delta\_{\rm kl} + \frac{\mu}{s\_{\rm j}^2} \delta\_{\rm ik} \delta\_{\rm jl} + \frac{\mu}{s\_{\rm i}^2} \delta\_{\rm il} \delta\_{\rm jk}) s\_{\rm x} s\_{\rm y} s\_{\rm z}. \tag{26}$$

**Figure 1.** Reflection by the plane boundary between an isotropic solid and its PML.

Here *si* (*i* = *x*, *y*, *z*) is a coordinate stretching factor of *i*-direction.[2] The mass density of the PML *ρ*PML is given by

$$
\rho^{\rm PML} = s\_{\rm x} s\_{\rm y} s\_{\rm z} \rho. \tag{27}
$$

**3.1. Numerical procedure**

**Figure 2.** Line element with (n+1)-nodes.

incident and reflected plane waves:

Derivative of (33) with respect to *x* is

⎡ ⎣ *R*0 *R*1 *R*2 ⎤

and for **r** = **0** we have

*∂ ∂x*

**u** = ∑ *l*

⎦ = [*K*]

−1

⎛

⎡ ⎣

⎜⎜⎜⎝ j**k***<sup>i</sup>* · *x*ˆ

**r** = **0**, we have

Because finite element formulation of a thick plate with line elements as shown in Fig. 2 is well known and we use COMSOL MultiPhysics for FEA, we explain the Robin condition at *x* = 0 and a formula for reflection coefficients. In the half isotropic solid, the field distribution,

*1 2 n n+1*

components of the particle displacement and stress, can be expressed by superposition of

Where *Rl*, **k***r*,*l*, **u***l*(*l* = 0, 1, 2) and **u***<sup>i</sup>* are reflection constants, the wave vector, the particle displacement vectors of reflection P-, SV- and SH-waves and the incident wave respectively, which are given by the solutions of the Christoffel equation for the isotropic solid. When

> ⎡ ⎣ *R*0 *R*1 *R*2

where the superscript *T* denotes transpose and ˆ*i*(*i* = *x*, *y*, *z*) is the unit vector of the *i*-direction.

(−j**k***r*,*<sup>l</sup>* · *<sup>x</sup>*ˆ)*Rl***u***l*e−j**k***r*,*<sup>l</sup>*·**<sup>r</sup>** + (−j**k***<sup>i</sup>* · *<sup>x</sup>*ˆ)**u***i*e−j**k***i*·**<sup>r</sup>**

⎤ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎣ *∂x*ˆ·**u** *∂x* |*x*=−0 *∂y*ˆ·**u** *∂x* |*x*=−0 *∂z*ˆ·**u** *∂x* |*x*=−0

⎤ ⎥ ⎥ ⎥ ⎦ ⎞

*x*ˆ · **u***i*(−**0**) *y*ˆ · **u***i*(−**0**) *z*ˆ · **u***i*(−**0**)

⎤

*x*ˆ · **u**<sup>0</sup> *x*ˆ · **u**<sup>1</sup> *x*ˆ · **u**<sup>2</sup> *y*ˆ · **u**<sup>0</sup> *y*ˆ · **u**<sup>1</sup> *y*ˆ · **u**<sup>2</sup> *z*ˆ · **u**<sup>0</sup> *z*ˆ · **u**<sup>1</sup> *z*ˆ · **u**<sup>2</sup>

⎤ ⎥ ⎥

Perfectly Matched Layer for Finite Element Analysis of Elastic Waves in Solids 187

[**u**(*x*)] = [*x*<sup>ˆ</sup> · **<sup>u</sup>**(*x*) *<sup>y</sup>*<sup>ˆ</sup> · **<sup>u</sup>**(*x*) *<sup>z</sup>*<sup>ˆ</sup> · **<sup>u</sup>**(*x*) ]*<sup>T</sup>* (36)

*Rl***u***l*(**r**)e−j**k***r*,*<sup>l</sup>*·**<sup>r</sup>** + **u***i*e−j**k***i*·**<sup>r</sup>**

. (33)

⎦ + [**u***i*(−**0**)], (34)

<sup>⎦</sup> , (35)

, (37)

⎟⎟⎟⎠ . (38)

*h*

**u**(**r**) = ∑ *l*

[**u**(−**0**)] = [*L*]

[*L*] =

⎡ ⎢ ⎢ ⎣

*3.1.1. Finite element analysis*

Here *ρ* is the mass density of the isotropic solid. For examining absorbing performance of PMLs in the *x* direction, taking an assumption of considering fields being consisted by plane waves propagating on the *x*-*y* plane, we have a differential equation in one variable *x*: from Newton's equation of motion and constitutive equation we get the differential equation in the PML

$$\mathcal{L}\_{\text{ijkl}}^{\text{PML}} \frac{\partial}{\partial \mathbf{x}\_j} \left( \frac{\partial u\_k}{\partial \mathbf{x}\_l} \right) = -\omega^2 \rho^{\text{PML}} u\_l \tag{28}$$

where *ui* is the component of the particle displacement in the *i*-direction (*i* = *x*, *y*, *z*).

In this case, we may choose the coordinate stretching factor as follows:

$$\mathbf{s}\_{\mathbf{x}} = \mathbf{1} - \mathbf{j}\mathbf{s}\_{xI}(\mathbf{x}),$$
 
$$\mathbf{s}\_{\mathbf{y}} = \mathbf{s}\_{\mathbf{z}} = \mathbf{1}.\tag{29}$$

Here *sx I*(*x*) is the imaginary part of *sx* and therefore a real function, which controls absorbing performance of propagating waves in PMLs.

Boundary conditions at the interface of isotropic solid and PML, *x* = 0, are the nonslip condition and the continuous condition of the normal component of the stress:[15]

$$
u\_i(-0) = s\_i 
u\_i(+0),\tag{30}$$

$$T\_{\rm ix}(-0) = \frac{s\_{\rm i}s\_{\rm x}}{s\_{\rm x}s\_{\rm y}s\_{\rm z}}T\_{\rm ix}(+0), \quad i = \propto \,\!\!y \,\!\!z. \tag{31}$$

At the terminal of PML, *x* = *L*, the boundary condition is

$$\frac{\mathbf{s}\_{i}\mathbf{s}\_{\mathbf{x}}}{\mathbf{s}\_{\mathbf{x}}\mathbf{s}\_{\mathbf{y}}\mathbf{s}\_{\mathbf{z}}}T\_{\text{ix}}(L) = \mathbf{0}, \quad \mathbf{i} = \mathbf{x}\_{\prime}\mathbf{y}\_{\prime}\mathbf{z}.\tag{32}$$

#### **3.1. Numerical procedure**

8 Will-be-set-by-IN-TECH

*x*

θp

*y*

*x*=0 *x*=*L*

Here *si* (*i* = *x*, *y*, *z*) is a coordinate stretching factor of *i*-direction.[2] The mass density of the

Here *ρ* is the mass density of the isotropic solid. For examining absorbing performance of PMLs in the *x* direction, taking an assumption of considering fields being consisted by plane waves propagating on the *x*-*y* plane, we have a differential equation in one variable *x*: from Newton's equation of motion and constitutive equation we get the differential equation in the

*sx* = 1 − j*sx I*(*x*),

Here *sx I*(*x*) is the imaginary part of *sx* and therefore a real function, which controls absorbing

Boundary conditions at the interface of isotropic solid and PML, *x* = 0, are the nonslip

condition and the continuous condition of the normal component of the stress:[15]

*sxsysz*

*L*

Isotropic solid PML vacuum

θ

<sup>s</sup> or **k**r,2

**Figure 1.** Reflection by the plane boundary between an isotropic solid and its PML.

*z*

**k***i*

*C*PML ijkl

In this case, we may choose the coordinate stretching factor as follows:

*Tix*(−0) = *sisx*

*sisx sxsysz*

At the terminal of PML, *x* = *L*, the boundary condition is

performance of propagating waves in PMLs.

*∂ ∂xj*  *∂uk ∂xl*

where *ui* is the component of the particle displacement in the *i*-direction (*i* = *x*, *y*, *z*).

**k**r,0

**k**r,1

PML *ρ*PML is given by

PML

P-wave

SV- or SH-wave

*ρ*PML = *sxsyszρ*. (27)

*sy* = *sz* = 1. (29)

*Tix*(+0), *i* = *x*, *y*, *z*. (31)

*Tix*(*L*) = 0, *i* = *x*, *y*, *z*. (32)

*ui*(−0) = *siui*(+0), (30)

<sup>=</sup> <sup>−</sup>*ω*2*ρ*PML*ui* (28)

θ

#### *3.1.1. Finite element analysis*

Because finite element formulation of a thick plate with line elements as shown in Fig. 2 is well known and we use COMSOL MultiPhysics for FEA, we explain the Robin condition at *x* = 0 and a formula for reflection coefficients. In the half isotropic solid, the field distribution,

**Figure 2.** Line element with (n+1)-nodes.

components of the particle displacement and stress, can be expressed by superposition of incident and reflected plane waves:

$$\mathbf{u}(\mathbf{r}) = \sum\_{l} R\_{l} \mathbf{u}\_{l}(\mathbf{r}) \mathbf{e}^{-j\mathbf{k}\_{\prime l} \cdot \mathbf{r}} + \mathbf{u}\_{l} \mathbf{e}^{-j\mathbf{k}\_{l} \cdot \mathbf{r}}.\tag{33}$$

Where *Rl*, **k***r*,*l*, **u***l*(*l* = 0, 1, 2) and **u***<sup>i</sup>* are reflection constants, the wave vector, the particle displacement vectors of reflection P-, SV- and SH-waves and the incident wave respectively, which are given by the solutions of the Christoffel equation for the isotropic solid. When **r** = **0**, we have

$$[\mathbf{u}(-\mathbf{0})] = [L] \begin{bmatrix} R\_0 \\ R\_1 \\ R\_2 \end{bmatrix} + [\mathbf{u}\_i(-\mathbf{0})],\tag{34}$$

$$[L] = \begin{bmatrix} \hat{\mathbf{x}} \cdot \mathbf{u}\_0 \ \hat{\mathbf{x}} \cdot \mathbf{u}\_1 \ \hat{\mathbf{x}} \cdot \mathbf{u}\_2 \\\\ \hat{\mathbf{y}} \cdot \mathbf{u}\_0 \ \hat{\mathbf{y}} \cdot \mathbf{u}\_1 \ \hat{\mathbf{y}} \cdot \mathbf{u}\_2 \\\\ \hat{\mathbf{z}} \cdot \mathbf{u}\_0 \ \hat{\mathbf{z}} \cdot \mathbf{u}\_1 \ \hat{\mathbf{z}} \cdot \mathbf{u}\_2 \end{bmatrix},\tag{35}$$

$$\mathbf{u}\left[\mathbf{u}(\mathbf{x})\right] = \left[\mathbf{\hat{x}} \cdot \mathbf{u}(\mathbf{x})\,\,\,\theta \cdot \mathbf{u}(\mathbf{x})\,\,\,\mathcal{E} \cdot \mathbf{u}(\mathbf{x})\right]^T \tag{36}$$

where the superscript *T* denotes transpose and ˆ*i*(*i* = *x*, *y*, *z*) is the unit vector of the *i*-direction. Derivative of (33) with respect to *x* is

$$\frac{\partial}{\partial \mathbf{x}} \mathbf{u} = \sum\_{l} (-j\mathbf{k}\_{r/l} \cdot \mathbf{\hat{x}}) \mathbf{\hat{k}}\_{l} \mathbf{u}\_{l} \mathbf{e}^{-j\mathbf{k}\_{r/l} \cdot \mathbf{r}} + (-j\mathbf{k}\_{i} \cdot \mathbf{\hat{x}}) \mathbf{u}\_{i} \mathbf{e}^{-j\mathbf{k}\_{i} \cdot \mathbf{r}},\tag{37}$$

and for **r** = **0** we have

$$
\begin{bmatrix} R\_0 \\ R\_1 \\ R\_2 \end{bmatrix} = \left[ \mathbf{K} \right]^{-1} \left( \mathbf{j} \mathbf{k}\_l \cdot \mathbf{\hat{x}} \begin{bmatrix} \mathbf{\hat{x}} \cdot \mathbf{u}\_l(-\mathbf{0}) \\ \mathbf{\hat{y}} \cdot \mathbf{u}\_l(-\mathbf{0}) \\ \mathbf{\hat{z}} \cdot \mathbf{u}\_l(-\mathbf{0}) \end{bmatrix} + \begin{bmatrix} \frac{\partial \mathbf{\hat{x}} \cdot \mathbf{u}}{\partial x}|\_{x=-0} \\ \frac{\partial \mathbf{\hat{y}} \cdot \mathbf{u}}{\partial x}|\_{x=-0} \\ \frac{\partial \mathbf{\hat{z}} \cdot \mathbf{u}}{\partial x}|\_{x=-0} \end{bmatrix} \right). \tag{38}
$$

Here

$$\mathbf{k}[K] = -\mathbf{j} \begin{bmatrix} \mathbf{k}\_{r,0} \cdot \hat{\mathbf{x}}\hat{\mathbf{x}} \cdot \mathbf{u}\_0 \ \mathbf{k}\_{r,1} \cdot \hat{\mathbf{x}}\hat{\mathbf{x}} \cdot \mathbf{u}\_1 \ \mathbf{k}\_{r,2} \cdot \hat{\mathbf{x}}\hat{\mathbf{x}} \cdot \mathbf{u}\_2 \\\\ \mathbf{k}\_{r,0} \cdot \hat{\mathbf{x}}\hat{\mathbf{y}} \cdot \mathbf{u}\_0 \ \mathbf{k}\_{r,1} \cdot \hat{\mathbf{x}}\hat{\mathbf{y}} \cdot \mathbf{u}\_1 \ \mathbf{k}\_{r,2} \cdot \hat{\mathbf{x}}\hat{\mathbf{y}} \cdot \mathbf{u}\_2 \\\\ \mathbf{k}\_{r,0} \cdot \hat{\mathbf{x}}\hat{\mathbf{z}} \cdot \mathbf{u}\_0 \ \mathbf{k}\_{r,1} \cdot \hat{\mathbf{x}}\hat{\mathbf{z}} \cdot \mathbf{u}\_1 \ \mathbf{k}\_{r,2} \cdot \hat{\mathbf{x}}\hat{\mathbf{z}} \cdot \mathbf{u}\_2 \end{bmatrix}. \tag{39}$$

Order of finite element Discretized wave number *β*˜

<sup>3</sup>*<sup>h</sup>* cos−<sup>1</sup>

*<sup>h</sup>* cos−<sup>1</sup>

<sup>2</sup>*<sup>h</sup>* cos−<sup>1</sup>

⎤

Here **u**(+0) and **u***i*(−0) are particle displacements at PML's incident side given by FEA

The finite element approximation of the propagating elastic fields changes the propagation constant given by the Christoffel equation, which is called the intrinsic wave number *β*PML,

elements with the polynomial interpolate function as shown in Fig.2 after Scott[22]. Here *β* and *h* are the *x*-component of the intrinsic wave number of P-, SV- or SH-wave propagating in the PML and equal interval between nodes, respectively. Figure 3 shows the difference of the discretized wave number and the intrinsic wave number as the function of the *x*-axis

Because the structure shown in Fig.1 is a layered structure where the propagation constants in the isotropic solid and its PML are given as the intrinsic wave numbers and discretized wave numbers respectively, we can compute the reflection coefficient by the transfer matrix.

In this section, we consider the fields composed of P- and SV-waves propagating on the *x*-*y* plane with the same *y*-component *ky* of the P- and SV-wave numbers only since SH-waves are not coupled with P- or SV-waves and SH-wave scattering problem is straightforward. Assuming that field distributions do not vary in the *z*-direction, we have the particle

> � 4 ∑ *i*=1 *Ai*,*<sup>m</sup> fx*,*<sup>i</sup> sx*

*ux*,*<sup>m</sup>* = e−j*kyy*

� <sup>6</sup>−2(*β*PML*h*)<sup>2</sup> 6+(*β*PML*h*)<sup>2</sup>

Perfectly Matched Layer for Finite Element Analysis of Elastic Waves in Solids 189

� <sup>15</sup>−26(*β*PML*h*)<sup>2</sup>+3(*β*PML*h*)<sup>4</sup> 15+4(*β*PML*h*)<sup>2</sup>+(*β*PML*h*)<sup>4</sup>

� <sup>2800</sup>−11520(*β*PML*h*)<sup>2</sup>+4860(*β*PML*h*)<sup>4</sup>−324(*β*PML*h*)<sup>6</sup> 2800+1080(*β*PML*h*)<sup>2</sup>+270(*β*PML*h*)<sup>4</sup>+81(*β*PML*h*)<sup>6</sup>

� <sup>19845</sup>−148680(*β*PML*h*)<sup>2</sup>+134064(*β*PML*h*)<sup>4</sup>−28800(*β*PML*h*)<sup>6</sup>+1280(*k*PML*h*)<sup>8</sup> 19845+10080(*β*PML*h*)<sup>2</sup>+3024(*β*PML*h*)<sup>4</sup>+768(*β*PML*h*)<sup>6</sup>+256(*β*PML*h*)<sup>8</sup>

⎦ [**u**(+**0**)] − [**u***i*(−**0**)]

PML. Table 2 shows discretized wave numbers for nodal finite

e−j*kx*,*isx <sup>x</sup>*

�

, (47)

1 <sup>1</sup>

2 <sup>1</sup>

<sup>4</sup>*<sup>h</sup>* cos−<sup>1</sup>

⎤

⎦ = [*L*]

solution and known incident field vector of displacements.

−1 ⎛ ⎝ ⎡ ⎣

3 <sup>1</sup>

4 <sup>1</sup>

**Table 2.** Discretized wave number in PML.

⎡ ⎣ *R*0 *R*1 *R*2

relation derived from eq. (34):

*3.1.2. Discretized wave number*

to discretized wave number *β*˜

propagation constant.

*3.1.3. Transfer matrix analysis*

displacements in the solid:[15]

PML

�

�

⎠ . (46)

�

�

⎞

Using eqs.(34) and (38), we have

$$[\mathbf{u}(-0)] = [L][K]^{-1} \begin{bmatrix} \frac{\partial \hat{\mathbf{x}} \cdot \mathbf{u}}{\partial x} \vert\_{x=-0} \\ \frac{\partial \boldsymbol{\theta} \cdot \mathbf{u}}{\partial x} \vert\_{x=-0} \\ \frac{\partial \hat{\mathbf{z}} \cdot \mathbf{u}}{\partial x} \vert\_{x=-0} \end{bmatrix}$$

$$+ \left( [I] + \mathbf{j} \mathbf{k}\_i \cdot \hat{\mathbf{x}}[L][K]^{-1} \right) \begin{bmatrix} \hat{\mathbf{x}} \cdot \mathbf{u}\_i(-\mathbf{0}) \\ \hat{\boldsymbol{\theta}} \cdot \mathbf{u}\_i(-\mathbf{0}) \\ \hat{\boldsymbol{\omega}} \cdot \mathbf{u}\_i(-\mathbf{0}) \end{bmatrix}.\tag{40}$$

Equation (31) yields

$$\left. \frac{\partial [\mathbf{u}]}{\partial \mathbf{x}} \right|\_{\mathbf{x} = -0} = [\mathbb{C}\_{i1}]^{-1} [\mathbb{C}\_{t1}] \left. \frac{\partial [\mathbf{u}]}{\partial \mathbf{x}} \right|\_{\mathbf{x} = +0} \tag{41}$$

$$\begin{aligned} \left[ \mathbf{C}\_{i1} \right] = \begin{bmatrix} \lambda + 2\mu & 0 \ 0 \\ 0 & \mu \ 0 \\ 0 & 0 \ \mu \end{bmatrix} \end{aligned} \tag{42}$$

$$\begin{bmatrix} \mathbb{C}\_{t1} \end{bmatrix} = \begin{bmatrix} \lambda + 2\mu & 0 \ 0 & \\ 0 & \mu \ 0 \\ 0 & 0 \ \mu \end{bmatrix} \tag{43}$$

$$\begin{bmatrix} \mathbb{C}\_{t1} \end{bmatrix} = \begin{bmatrix} \lambda + 2\mu & 0 & 0\\ 0 & \frac{s\_y}{s\_x}\mu & 0\\ 0 & 0 & \frac{s\_z}{s\_x}\mu \end{bmatrix} . \tag{44}$$

Substituting eqs. (30) and (41) into eq. (40), we get the Robin condition:

$$\left[L\right][K]^{-1}[\mathbb{C}\_{i1}]^{-1}[\mathbb{C}\_{t1}]\left.\frac{\partial[\mathbf{u}]}{\partial \mathbf{x}}\right|\_{\mathbf{x}=\mathbf{+0}} - [s][\mathbf{u}(+\mathbf{0})]$$

$$=-\left([I] + \mathrm{jk}\_{\mathrm{ix}}[L][\mathbb{K}]^{-1}\right)[\mathbf{u}\_{i}(-\mathbf{0})].\tag{45}$$

After we solve the distributions of the particle displacements in the PML by COMSOL MultiPhysics, the reflection coefficients *Rl*(*l* = 0, 1, 2) are computed with the following


**Table 2.** Discretized wave number in PML.

relation derived from eq. (34):

10 Will-be-set-by-IN-TECH

[**u**(−0)] = [*L*][*K*]

[*I*] + j**k***<sup>i</sup>* · *x*ˆ[*L*][*K*]

= [*Ci*1]

[*Ct*1] =

⎡ ⎢ ⎣

<sup>−</sup>1[*Ct*1] *<sup>∂</sup>*[**u**] *∂x* � � � � *x*=+0

[*I*] + j*kix*[*L*][*K*]

After we solve the distributions of the particle displacements in the PML by COMSOL MultiPhysics, the reflection coefficients *Rl*(*l* = 0, 1, 2) are computed with the following

[*Ct*1] =

Substituting eqs. (30) and (41) into eq. (40), we get the Robin condition:

= − �

[*L*][*K*] −1 [*Ci*1] [*Ci*1] =

⎡ ⎢ ⎣

**k***r*,0 · *x*ˆ*x*ˆ · **u**<sup>0</sup> **k***r*,1 · *x*ˆ*x*ˆ · **u**<sup>1</sup> **k***r*,2 · *x*ˆ*x*ˆ · **u**<sup>2</sup> **k***r*,0 · *x*ˆ*y*ˆ · **u**<sup>0</sup> **k***r*,1 · *x*ˆ*y*ˆ · **u**<sup>1</sup> **k***r*,2 · *x*ˆ*y*ˆ · **u**<sup>2</sup> **k***r*,0 · *x*ˆ*z*ˆ · **u**<sup>0</sup> **k***r*,1 · *x*ˆ*z*ˆ · **u**<sup>1</sup> **k***r*,2 · *x*ˆ*z*ˆ · **u**<sup>2</sup>

−1

−1 � ⎡ ⎢ ⎢ ⎣

<sup>−</sup>1[*Ct*1] *<sup>∂</sup>*[**u**] *∂x* � � � � *x*=+0

*λ* + 2*μ* 0 0 0 *μ* 0 0 0 *μ*

*λ* + 2*μ* 0 0 0 *sy sx μ* 0

0 0 *sz*

*sx μ*

−1 �

*λ* + 2*μ* 0 0 0 *μ* 0 0 0 *μ*

> ⎤ ⎥

> ⎤ ⎥

− [*s*][**u**(+**0**)]

⎡ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎣

*∂x*ˆ·**u** *∂x* |*x*=−0 *∂y*ˆ·**u** *∂x* |*x*=−0 *∂z*ˆ·**u** *∂x* |*x*=−0

*x*ˆ · **u***i*(−**0**) *y*ˆ · **u***i*(−**0**) *z*ˆ · **u***i*(−**0**) ⎤ ⎥ ⎥

⎤ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥

⎤ ⎥ <sup>⎦</sup> . (39)

<sup>⎦</sup> . (40)

, (41)

<sup>⎦</sup> , (42)

<sup>⎦</sup> , (43)

<sup>⎦</sup> . (44)

[**u***i*(−**0**)]. (45)

[*K*] = −j

+ �

> *∂*[**u**] *∂x* � � � � *x*=−0

Using eqs.(34) and (38), we have

Equation (31) yields

⎡ ⎢ ⎢ ⎣

Here

$$
\begin{bmatrix} R\_0 \\ R\_1 \\ R\_2 \end{bmatrix} = [L]^{-1} \left( \begin{bmatrix} s\_x & 0 & 0 \\ 0 & s\_y & 0 \\ 0 & 0 & s\_z \end{bmatrix} \begin{bmatrix} \mathbf{u}(+\mathbf{0}) \end{bmatrix} - \begin{bmatrix} \mathbf{u}\_i(-\mathbf{0}) \end{bmatrix} \right). \tag{46}
$$

Here **u**(+0) and **u***i*(−0) are particle displacements at PML's incident side given by FEA solution and known incident field vector of displacements.

#### *3.1.2. Discretized wave number*

The finite element approximation of the propagating elastic fields changes the propagation constant given by the Christoffel equation, which is called the intrinsic wave number *β*PML, to discretized wave number *β*˜ PML. Table 2 shows discretized wave numbers for nodal finite elements with the polynomial interpolate function as shown in Fig.2 after Scott[22]. Here *β* and *h* are the *x*-component of the intrinsic wave number of P-, SV- or SH-wave propagating in the PML and equal interval between nodes, respectively. Figure 3 shows the difference of the discretized wave number and the intrinsic wave number as the function of the *x*-axis propagation constant.

#### *3.1.3. Transfer matrix analysis*

Because the structure shown in Fig.1 is a layered structure where the propagation constants in the isotropic solid and its PML are given as the intrinsic wave numbers and discretized wave numbers respectively, we can compute the reflection coefficient by the transfer matrix.

In this section, we consider the fields composed of P- and SV-waves propagating on the *x*-*y* plane with the same *y*-component *ky* of the P- and SV-wave numbers only since SH-waves are not coupled with P- or SV-waves and SH-wave scattering problem is straightforward. Assuming that field distributions do not vary in the *z*-direction, we have the particle displacements in the solid:[15]

$$\mu\_{\mathbf{x},m} = \mathbf{e}^{-jk\_y y} \left( \sum\_{i=1}^{4} A\_{i,m} \frac{f\_{\mathbf{x},i}}{s\_{\mathbf{x}}} \mathbf{e}^{-jk\_{\mathbf{x},\mathbf{S}\_{\mathbf{x}}} \mathbf{x}} \right) \tag{47}$$

**Figure 3.** Phase error and attenuation error as a function of 2*π*/(*βh*) for 1st-, 2nd-, 3rd-, and 4th order elements.

$$\mu\_{y,m} = \mathbf{e}^{-jk\_yy} \left( \sum\_{i=1}^{4} A\_{i,m} f\_{y,i} \mathbf{e}^{-jk\_{x,i}s\_{X}x} \right). \tag{48}$$

Here,

[*s*] =

[*Y*0] =

*Y*3*i*,0 = −j

*Y*4*i*,0 = −j

[*Y*1] =

*Y*3*i*,1 = −j

*Y*4*i*,1 = −j

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

[*T*(*x*)] =

� *kx*,*isx* 1 *sx*

�

*kysx<sup>μ</sup> fx*,*<sup>i</sup> sx*

�

�

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

00 0 <sup>1</sup>

*sz*

*kx*,*i*(*λ* + 2*μ*)*fx*,*<sup>i</sup>* + *kyλ fy*,*<sup>i</sup>*

�

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

*kyμ fx*,*<sup>i</sup>* + *kx*,*iμ fy*,*<sup>i</sup>*

*Y*11,1 *Y*12,1 *Y*13,1 *Y*14,1

*Y*21,1 *Y*22,1 *Y*23,1 *Y*24,1

*Y*31,1 *Y*32,1 *Y*33,1 *Y*34,1

*Y*41,1 *Y*42,1 *Y*43,1 *Y*44,1

(*λ* + 2*μ*)

+ *kx*,*isx*

e−j*kx*,1*sx <sup>x</sup>* 000

0 e−j*kx*,2 *sx <sup>x</sup>* 0 0

0 0e−j*kx*,3*sx <sup>x</sup>* 0

0 0 0e−j*kx*,4 *sx <sup>x</sup>*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

*Y*1*i*,0 = *fx*,*i*, (53)

*Y*2*i*,0 = *fy*,*i*, (54)

*Y*1*i*,1 = *fx*,*i*/*sx*, (58)

*Y*2*i*,1 = *fy*,*i*, (59)

+ *ky* 1 *sx λ fy*,*<sup>i</sup>* �

*fx*,*<sup>i</sup> sx*

> 1 *sx μ fy*,*<sup>i</sup>* �

�

*Y*11,0 *Y*12,0 *Y*13,0 *Y*14,0 *Y*21,0 *Y*22,0 *Y*23,0 *Y*24,0 *Y*31,0 *Y*32,0 *Y*33,0 *Y*34,0 *Y*41,0 *Y*42,0 *Y*43,0 *Y*44,0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

, (51)

Perfectly Matched Layer for Finite Element Analysis of Elastic Waves in Solids 191

, (52)

, (55)

, (56)

, (57)

, (60)

. (62)

, (61)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Here, *kx*,*<sup>i</sup>* is the *x*-component of the intrinsic wave number for the isotropic solid and the discretized wave number for the PML, the subscripts *i* = 1 and *i* = 3 denote P-waves propagating to +*x*- and −*x*-direction respectively, *i* = 2 and *i* = 4 denote SV-waves propagating to +*x*- and −*x*-direction respectively, and *Ai*,*m*(*i* = 1, 2, 3, 4, *m* = 0, 1) is the amplitude at *x* = 0 in the isotropic solid (*m* = 0) or PML (*m* = 1). *fx*,*<sup>i</sup>* and *fy*,*<sup>i</sup>* are shown in Table 3. Here *θp* and *θs* are angles between the *x*-direction and the wave vectors of P-waves or SV-waves as shown in Fig. 1. In the isotropic region, we set *sx* = 1.

$$\begin{array}{l c c c} \hline \hline i & 1 & 2 & 3 & 4 \\ \hline f\_{\chi,i} \cos \theta\_p & -\sin \theta\_s & -\cos \theta\_p \sin \theta\_s \\ f\_{y,i} \sin \theta\_p & \cos \theta\_s & \sin \theta\_p & \cos \theta\_s \\ \hline \end{array}$$

**Table 3.** Displacement directions of P- and SV-waves, *fx*,*<sup>i</sup>* and *fy*,*i*.

Using the boundary conditions at *x* = 0 and *x* = *L*, and eliminating *Ai*,1, we get the relation

$$
\begin{bmatrix} A\_{3,0} \\ A\_{4,0} \end{bmatrix} = \begin{bmatrix} X\_{31} \ X\_{32} \\ X\_{41} \ X\_{42} \end{bmatrix} \begin{bmatrix} X\_{11} \ X\_{12} \\ X\_{21} \ X\_{22} \end{bmatrix}^{-1} \begin{bmatrix} A\_{1,0} \\ A\_{2,0} \end{bmatrix} \tag{49}
$$

where [*X*] is the square matrix with four columns and rows given by

$$f[X] = [Y\_0]^{-1}[\mathbf{s}][Y\_1][T(L)][Y\_1]^{-1}[\mathbf{s}]^{-1}.\tag{50}$$

Here,

12 Will-be-set-by-IN-TECH

1st order 2nd order 3rd order 4th order

Phase constant Attenuation constant

<sup>10</sup> <sup>50</sup> <sup>100</sup> <sup>500</sup> <sup>1000</sup> <sup>3000</sup> 10-10

**Figure 3.** Phase error and attenuation error as a function of 2*π*/(*βh*) for 1st-, 2nd-, 3rd-, and 4th order

Here, *kx*,*<sup>i</sup>* is the *x*-component of the intrinsic wave number for the isotropic solid and the discretized wave number for the PML, the subscripts *i* = 1 and *i* = 3 denote P-waves propagating to +*x*- and −*x*-direction respectively, *i* = 2 and *i* = 4 denote SV-waves propagating to +*x*- and −*x*-direction respectively, and *Ai*,*m*(*i* = 1, 2, 3, 4, *m* = 0, 1) is the amplitude at *x* = 0 in the isotropic solid (*m* = 0) or PML (*m* = 1). *fx*,*<sup>i</sup>* and *fy*,*<sup>i</sup>* are shown in Table 3. Here *θp* and *θs* are angles between the *x*-direction and the wave vectors of P-waves or

> *i* 12 34 *fx*,*<sup>i</sup>* cos *θ<sup>p</sup>* − sin *θ<sup>s</sup>* − cos *θ<sup>p</sup>* sin *θ<sup>s</sup> fy*,*<sup>i</sup>* sin *θ<sup>p</sup>* cos *θ<sup>s</sup>* sin *θ<sup>p</sup>* cos *θ<sup>s</sup>*

Using the boundary conditions at *x* = 0 and *x* = *L*, and eliminating *Ai*,1, we get the relation

<sup>−</sup>1[*s*][*Y*1][*T*(*L*)][*Y*1]

 *X*<sup>11</sup> *X*<sup>12</sup> *X*<sup>21</sup> *X*<sup>22</sup> −<sup>1</sup>

<sup>−</sup>1[*s*]

*A*1,0 *A*2,0 

*X*<sup>31</sup> *X*<sup>32</sup> *X*<sup>41</sup> *X*<sup>42</sup>

 4 ∑ *i*=1

*uy*,*<sup>m</sup>* = e−j*kyy*

SV-waves as shown in Fig. 1. In the isotropic region, we set *sx* = 1.

**Table 3.** Displacement directions of P- and SV-waves, *fx*,*<sup>i</sup>* and *fy*,*i*.

*A*3,0

 = 

where [*X*] is the square matrix with four columns and rows given by

[*X*]=[*Y*0]

*A*4,0

2π/(β*h*)

*Ai*,*<sup>m</sup> fy*,*i*e−j*kx*,*isx <sup>x</sup>*

. (48)

<sup>−</sup>1. (50)

(49)

10-9

elements.

10-8

10-7

10-6

10-5

Relative error[%]

10-4

10-3

10-2

10-1

$$\begin{aligned} [s] = \begin{bmatrix} s\_{\chi} & 0 & 0 & 0\\ 0 & s\_{\chi} & 0 & 0\\ 0 & 0 & \frac{s\_{\chi}}{s\_{\chi}s\_{z}} & 0\\ 0 & 0 & 0 & \frac{1}{s\_{z}} \end{bmatrix}' \end{aligned} \tag{51}$$

$$\begin{aligned} [Y\_0] = \begin{bmatrix} Y\_{11,0} \ Y\_{12,0} \ Y\_{13,0} \ Y\_{14,0} \\ Y\_{21,0} \ Y\_{22,0} \ Y\_{23,0} \ Y\_{24,0} \\ Y\_{31,0} \ Y\_{32,0} \ Y\_{33,0} \ Y\_{34,0} \\ Y\_{41,0} \ Y\_{42,0} \ Y\_{43,0} \ Y\_{44,0} \end{bmatrix}, \end{aligned} \tag{52}$$

$$Y\_{1i,0} = f\_{\mathbf{x},i\nu} \tag{53}$$

$$Y\_{2\text{i},0} = f\_{y,\text{i}\prime} \tag{54}$$

$$Y\_{3i,0} = -\mathbf{j}\left(k\_{x,i}(\lambda + 2\mu)f\_{x,i} + k\_y\lambda f\_{y,i}\right),\tag{55}$$

$$Y\_{4i,0} = -\mathbf{j}\left(k\_y \mu f\_{x,i} + k\_{x,i} \mu f\_{y,i}\right) \tag{56}$$

$$\begin{aligned} \begin{bmatrix} Y\_{11,1} \ Y\_{12,1} \ Y\_{13,1} \ Y\_{14,1} \end{bmatrix} \end{aligned} $$
 
$$\begin{bmatrix} Y\_{1} \end{bmatrix} = \begin{bmatrix} Y\_{21,1} \ Y\_{22,1} \ Y\_{23,1} \ Y\_{24,1} \\\\ Y\_{31,1} \ Y\_{32,1} \ Y\_{33,1} \ Y\_{34,1} \\\\ Y\_{41,1} \ Y\_{42,1} \ Y\_{43,1} \ Y\_{44,1} \end{bmatrix}' \tag{57}$$

$$Y\_{1i,1} = f\_{\mathbf{x},i} / s\_{\mathbf{x},i} \tag{58}$$

$$Y\_{2i,1} = f\_{y,i\nu} \tag{59}$$

$$Y\_{3i,1} = -\mathbf{j}\left(k\_{\mathbf{x},i}\mathbf{s}\_{\mathbf{x}}\frac{1}{s\_{\mathbf{x}}}(\lambda + 2\mu)\frac{f\_{\mathbf{x},i}}{s\_{\mathbf{x}}} + k\_{\mathbf{y}}\frac{1}{s\_{\mathbf{x}}}\lambda f\_{\mathbf{y},i}\right),\tag{60}$$

$$Y\_{4i,1} = -\mathbf{j}\left(k\_{\mathcal{Y}}s\_{\mathcal{X}}\mu \frac{f\_{\mathbf{x},i}}{s\_{\mathcal{X}}} + k\_{\mathbf{x},i}s\_{\mathbf{x}}\frac{1}{s\_{\mathcal{X}}}\mu f\_{\mathbf{y},i}\right),\tag{61}$$

$$\begin{bmatrix} T(\mathbf{x}) \end{bmatrix} = \begin{bmatrix} \mathbf{e}^{-\mathbf{j}\mathbf{k}\_{\mathbb{X}}\mathbf{s}\_{\mathbb{X}}\mathbf{x}} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{e}^{-\mathbf{j}\mathbf{k}\_{\mathbb{X}}\mathbf{s}\_{\mathbb{X}}\mathbf{x}} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{e}^{-\mathbf{j}\mathbf{k}\_{\mathbb{X}}\mathbf{s}\_{\mathbb{X}}\mathbf{x}} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{e}^{-\mathbf{j}\mathbf{k}\_{\mathbb{X}}\mathbf{s}\_{\mathbb{X}}\mathbf{x}} \end{bmatrix}. \tag{62}$$

The reflection coefficients at the boundary *x* = 0, we obtain (*A*3,0/*A*1,0 and *A*4,0/*A*1,0) with *A*2,0 = 0 when the incident wave is the P-wave and in the case of SV-wave incidence we obtain (*A*3,0/*A*2,0 and *A*4,0/*A*2,0) with *A*1,0 = 0.

#### **3.2. Computed results**

Figure 4 shows the computed results of the reflection coefficient dependence on 2*π*/(*βh*) in the case of the SH-wave incidence with incident angle *θ* = 0, the attenuation coefficient *sx I*(*x*) = 0.1 and normalized thickness *ksL* = 24*π*. Here *ks* is the intrinsic wave number of the SH-wave in the isotropic solid. Decreasing the interval between adjacent nodal points *h*, the reflection coefficient approaches the value estimated by the truncation effect which caused by the reflection at the PML end terminal and can be estimated by attenuated waves in the PML, 20 log10(exp(2*ksLs*2*I*)) = 20 × 4.8*π* log10 e = 131dB. A higher order element causes lower reflection because of a better approximation of the intrinsic wave number. Figure 5 shows dependence on the incident angle. Smaller intervals of finite element nodes, *ksh* = 0.1*π*, gives a better approximation than *ksh* = 0.2*π*. Increasing the incident angle, *β* decreases and the approximation of the intrinsic wave number with discretized wave number becomes better. Hence, the reflection by FE-discretization decreases. However, incident angle becomes larger than the angle such as about 63.5 degrees for 1st order element, reflection increases because decreasing *β* yields decreasing of wave attenuation in PMLs and the reflection by the truncation effect increases. Figures 4 and 5 show that the results of the transfer matrix agree well those of FEA and we confirm that the reflection of the FE-model of the PML may be explained with the discretized wave number and the truncation effect.

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> -140

 θ[deg]

Perfectly Matched Layer for Finite Element Analysis of Elastic Waves in Solids 193

 θ[deg]

FEM

FEM

1st order(*N*=240) 2nd order(*N*=120) 3rd order(*N*=80) 4th order(*N*=60)

1st order(*N*=120) 2nd order(*N*=60) 3rd order(*N*=40) 4th order(*N*=30)

SH-wave incident angle

discretized wavenumber

Calculated by

(a) *ksh* = 0.2*π*.

Truncation effect only(*N* )

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> -140

SH-wave incident angle

**Figure 5.** Dependence of reflection coefficients of the SH-wave perpendicular incidence with the attenuation coefficient *sxI*(*x*) = 0.1 and normalized thickness *ksL* = 24*π* on 2*π*/(*βh*). Here *N* is the

(b) *ksh* = 0.1*π*.






Reflection coefficients[dB]

number of FEs.



0




Reflection coefficients[dB]



0

Calculated by

discretized wavenumber

Truncation effect only(*N* )

We consider an isotropic solid and its PML with the Poisson ratio *σ*=0.3 in this section.

**Figure 4.** Dependence of SH-wave perpendicular incidence on 2*π*/(*βh*) for *ksL* = 24*π* and *sxI*(*x*) = 0.1.

14 Will-be-set-by-IN-TECH

The reflection coefficients at the boundary *x* = 0, we obtain (*A*3,0/*A*1,0 and *A*4,0/*A*1,0) with *A*2,0 = 0 when the incident wave is the P-wave and in the case of SV-wave incidence we obtain

Figure 4 shows the computed results of the reflection coefficient dependence on 2*π*/(*βh*) in the case of the SH-wave incidence with incident angle *θ* = 0, the attenuation coefficient *sx I*(*x*) = 0.1 and normalized thickness *ksL* = 24*π*. Here *ks* is the intrinsic wave number of the SH-wave in the isotropic solid. Decreasing the interval between adjacent nodal points *h*, the reflection coefficient approaches the value estimated by the truncation effect which caused by the reflection at the PML end terminal and can be estimated by attenuated waves in the PML, 20 log10(exp(2*ksLs*2*I*)) = 20 × 4.8*π* log10 e = 131dB. A higher order element causes lower reflection because of a better approximation of the intrinsic wave number. Figure 5 shows dependence on the incident angle. Smaller intervals of finite element nodes, *ksh* = 0.1*π*, gives a better approximation than *ksh* = 0.2*π*. Increasing the incident angle, *β* decreases and the approximation of the intrinsic wave number with discretized wave number becomes better. Hence, the reflection by FE-discretization decreases. However, incident angle becomes larger than the angle such as about 63.5 degrees for 1st order element, reflection increases because decreasing *β* yields decreasing of wave attenuation in PMLs and the reflection by the truncation effect increases. Figures 4 and 5 show that the results of the transfer matrix agree well those of FEA and we confirm that the reflection of the FE-model of the PML may be

explained with the discretized wave number and the truncation effect.

We consider an isotropic solid and its PML with the Poisson ratio *σ*=0.3 in this section.

Calculated by

FEM

discretized wavenumber

1st order 2nd order 3rd order 4th order

<sup>10</sup> <sup>50</sup> <sup>100</sup> <sup>500</sup> <sup>1000</sup> <sup>3000</sup> -140

2π/(β*h*)

**Figure 4.** Dependence of SH-wave perpendicular incidence on 2*π*/(*βh*) for *ksL* = 24*π* and *sxI*(*x*) = 0.1.

(*A*3,0/*A*2,0 and *A*4,0/*A*2,0) with *A*1,0 = 0.



Reflection coefficients[dB]




**3.2. Computed results**

**Figure 5.** Dependence of reflection coefficients of the SH-wave perpendicular incidence with the attenuation coefficient *sxI*(*x*) = 0.1 and normalized thickness *ksL* = 24*π* on 2*π*/(*βh*). Here *N* is the number of FEs.

1st order(*N*=120) 2nd order(*N*=60)

FEM

Calculated by

discretized wavenumber


0




Reflection coefficients[dB]



0

Reflection coefficients[dB]

Reflection coefficients[dB]

0

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> -140

θ [deg]

θ [deg]

θ [deg]

P-wave SV-wave

Perfectly Matched Layer for Finite Element Analysis of Elastic Waves in Solids 195

P-wave incident angle

(a) *sx I*(*x*) = 0.1.

1st order(*N*=120) 2nd order(*N*=60)

P-wave SV-wave

FEM Calculated by

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> -140

P-wave incident angle

(b) *sx I*(*x*) = 0.2.

1st order(*N*=120) 2nd order(*N*=60)

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> -200

P-wave incident angle

(c) *sx I*(*x*) = 0.3.

discretized wavenumber

P-wave SV-wave

Calculated by

FEM

**Figure 7.** Dependence on P-wave incident angle *θ<sup>i</sup>* for *ksh* = 0.2*π* and *ksL* = 24*π*.

discretized wavenumber

**Figure 6.** Dependence of reflection coefficients on 2*π*/(*βh*) in the case of perpendicular P-wave incident, *ksL* = 24*π* and *sxI*(*x*) = 0.1.

Next, we consider P-wave or SV-wave scattering problems. Figure 6 shows the computed result of the reflection coefficient dependence on 2*π*/(*βh*) in case of the P-wave perpendicular incidence with the attenuation coefficient *sx I*(*x*) = 0.1 and normalized thickness *ksL* = 24*π*. Because the result of the SV-wave perpendicular incidence is the same result of the SH-vave perpendicular incidence owing to a symmetry of the problem, Fig. 4 also shows the result of SV-wave incidence with the attenuation coefficient *sx I*(*x*) = 0.1 and normalized thickness *ksL* = 24*π*. Both cases also approaches the value estimated by the truncation effect, -70.0dB and -131dB. Note that the wave number of the P-wave is *<sup>μ</sup>*/(*<sup>λ</sup>* <sup>+</sup> <sup>2</sup>*μ*) = <sup>√</sup>2/7 times of the SV-wave wave number. Dependencies of P- and SV-wave reflections on P- and SV-wave incident angle are shown in Figs. 7 and 8, respectively. Reflection coefficients of incident waves computed by the transfer matrix except the range that is larger than the critical angle, about 32.3 degrees, of SV-wave incidence are good agreement with the results of FEA. However, reflection coefficients of converted waves from incident waves are smaller for the SV-wave excited by the incident P-wave and larger for the P-wave than the results of FEA. We still can not explain this discrepancy.

Increasing *sx I*, the reflection coefficient of P-wave in case of the P-wave incidence decreases in the incident angle range that is larger than 59 degrees. In the lower range, the reflection does not decrease because FE-discretization effect dominates the reflection. In the case of SV-wave incidence, we confirm that the P-wave converted from the SV-wave is amplified in the incident angle range that is lager than the critical angle when P-wave's wave number is zero in the isotropic region because of PML's intrinsic characteristics for non-propagating waves.

16 Will-be-set-by-IN-TECH

FEM

Calculated by

discretized wavenumber

1st order 2nd order 3rd order 4th order

<sup>10</sup> <sup>50</sup> <sup>100</sup> <sup>500</sup> <sup>1000</sup> <sup>3000</sup> -75

2π/(β*h*)

**Figure 6.** Dependence of reflection coefficients on 2*π*/(*βh*) in the case of perpendicular P-wave incident,

Next, we consider P-wave or SV-wave scattering problems. Figure 6 shows the computed result of the reflection coefficient dependence on 2*π*/(*βh*) in case of the P-wave perpendicular incidence with the attenuation coefficient *sx I*(*x*) = 0.1 and normalized thickness *ksL* = 24*π*. Because the result of the SV-wave perpendicular incidence is the same result of the SH-vave perpendicular incidence owing to a symmetry of the problem, Fig. 4 also shows the result of SV-wave incidence with the attenuation coefficient *sx I*(*x*) = 0.1 and normalized thickness *ksL* = 24*π*. Both cases also approaches the value estimated by the truncation effect, -70.0dB and -131dB. Note that the wave number of the P-wave is *<sup>μ</sup>*/(*<sup>λ</sup>* <sup>+</sup> <sup>2</sup>*μ*) = <sup>√</sup>2/7 times of the SV-wave wave number. Dependencies of P- and SV-wave reflections on P- and SV-wave incident angle are shown in Figs. 7 and 8, respectively. Reflection coefficients of incident waves computed by the transfer matrix except the range that is larger than the critical angle, about 32.3 degrees, of SV-wave incidence are good agreement with the results of FEA. However, reflection coefficients of converted waves from incident waves are smaller for the SV-wave excited by the incident P-wave and larger for the P-wave than the results of FEA. We

Increasing *sx I*, the reflection coefficient of P-wave in case of the P-wave incidence decreases in the incident angle range that is larger than 59 degrees. In the lower range, the reflection does not decrease because FE-discretization effect dominates the reflection. In the case of SV-wave incidence, we confirm that the P-wave converted from the SV-wave is amplified in the incident angle range that is lager than the critical angle when P-wave's wave number is zero in the isotropic region because of PML's intrinsic characteristics for non-propagating


still can not explain this discrepancy.

waves.



Reflection coefficients[dB]

*ksL* = 24*π* and *sxI*(*x*) = 0.1.



**Figure 7.** Dependence on P-wave incident angle *θ<sup>i</sup>* for *ksh* = 0.2*π* and *ksL* = 24*π*.

**Author details**

*Institute of Technology, Muroran, Japan*

waves. J. Comput. Phys. 114 : 185-200.

*Division of Information and Electronic Engineering, Graduate School of Engineering, Muroran*

Perfectly Matched Layer for Finite Element Analysis of Elastic Waves in Solids 197

*Department of Electrical and Electronic Engineering, Tsuyama National College of Technology,*

[1] J.P. Berenger (1994) A perfectly matched layer for the absorption of electromagnetic

[2] W.C.Chew and W.H. Weedon (1994) A 3D perfectly matched medium for modified Maxwell's equations with stretched coordinates. Microwave Opt. Tech. Lett. 7 : 599-604. [3] F.L.Teixeira and W.C.Chew (2000) Complex space approach to perfectly matched layers:

[4] Z.S.Sacks, D.M.Kingsland, R. Lee and J.F.Lee (1995) A perfectly matched anisotropic absorber for use as an absorbing boundary condition. IEEE Trans. Antennas Propag. 43:

[5] F.D.Hastings, J.B.Schneider and S.L.Broschat (1996) Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation. J Acoust. Soc.

[6] W.C.Chew and Q.H.Liu (1996) Perfectly matched layers for elastodynamics: a new

[7] Q.H.Liu (1999) Perfectly matched layers for elastic waves in cylindrical and spherical

[8] Y.Zheng and X.Huang (2002) Anisotropic perfectly matched layers for elastic waves in cartesian and curvilinear coordinates. MIT Earth Resources Laboratory Industry

[9] S.Ballandras,D.Gachon, J.Masson and W.Daniau (2007) Development of absorbing conditions for the analysis of finite dimension elastic wave-guides. Proc. Int. Freq. Contr.

[10] M.Mayer, S.Zaglmayr, K.Wagner and J. Schöberl (2007) Perfectly matched layer finite element simulation of parasitic acoustic wave radiation in microacoustic devices. Proc.

[11] Y.Li, O.B.Matar, V.Preobrazhensky and P. Pernod (2008) Convolution-Perfectly Matched Layer(C-PML) absorbing boundary condition for wave propagation in piezoelectric

[12] U.Basu and A.K.Chopra (2003) Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation.

a review and some new developments. Int. J. Numer. Model. 13 : 441-455.

absorbing bounary condition. J. Comput. Acoust. 4 : 341-359.

coordinates. J. Acoust. Soc. Am. 105: 2075-2084.

solid. Proc. IEEE Ultrason. Symp.: 1568-1571.

Comput. Methods Appl. Mech. Eng. 192 : 1337-1375.

Koji Hasegawa

Takao Shimada

*Tsuyama, Japan*

**5. References**

1460-1463.

Am. 100: 3061-3069.

Consortium Meeting.

IEEE Ultrason. Symp.: 702-706.

Symp.: 729-732.

**Figure 8.** Dependence on SV-wave incident angle *θ<sup>i</sup>* for *ksh* = 0.2*π* and *ksL* = 24*π*.

#### **4. Conclusions**

In this chapter, first, PMLs in the Cartesian, the cylindrical and the spherical coordinates for elastic waves in solids were derived from differential forms on manifolds. Our results show that PML parameters in any orthogonal coordinate system for elastic waves in solids may be determined by the same procedure in the Cartesian coordinates. Next, scattering of elastic waves in an isotropic solid was analyzed by field analysis in the thick layer in the one dimension. Numerical results show that the reflection from PMLs by the transfer matrix of elastic waves approximates the numerical results of FE-models successfully. We concluded that the reflection by FE discritization may be explained by FE-approximation of the intrinsic wave number.

## **Author details**

#### Koji Hasegawa

18 Will-be-set-by-IN-TECH

P-wave SV-wave FEM

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> -140

SV-wave incident angle

P-wave SV-wave

FEM

**Figure 8.** Dependence on SV-wave incident angle *θ<sup>i</sup>* for *ksh* = 0.2*π* and *ksL* = 24*π*.

(a) *sx I*(*x*) = 0.1.

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> -140

SV-wave incident angle

(b) *sx I*(*x*) = 0.3.

In this chapter, first, PMLs in the Cartesian, the cylindrical and the spherical coordinates for elastic waves in solids were derived from differential forms on manifolds. Our results show that PML parameters in any orthogonal coordinate system for elastic waves in solids may be determined by the same procedure in the Cartesian coordinates. Next, scattering of elastic waves in an isotropic solid was analyzed by field analysis in the thick layer in the one dimension. Numerical results show that the reflection from PMLs by the transfer matrix of elastic waves approximates the numerical results of FE-models successfully. We concluded that the reflection by FE discritization may be explained by FE-approximation of the intrinsic





Reflection coefficients[dB]

**4. Conclusions**

wave number.




0




Reflection coefficients[dB]



0

1st order(*N*=120) 2nd order(*N*=60)

discretized wavenumber

Calculated by

θ [deg]

1st order(*N*=120) 2nd order(*N*=60)

> θ [deg]

Calculated by discretized wavenumber

*Division of Information and Electronic Engineering, Graduate School of Engineering, Muroran Institute of Technology, Muroran, Japan*

#### Takao Shimada

*Department of Electrical and Electronic Engineering, Tsuyama National College of Technology, Tsuyama, Japan*

## **5. References**

	- [13] C.Michler, L.Demkowicz, J.Kurtz and D.Pardo (2007) Improving the performance of perfectly matched layers by means of hp-adaptivity. Num. Methods Partial Diffe. Equa. 23 : 832-858.
	- [14] A.Taflove and S.C.Hagness (2005) *Computational electrodynamics*(Artech House, Boston) 3rd ed. Ch7, p.273.
	- [15] T.Shimada and K.Hasegawa: (2010) Perfectly matched layers for elastic waves propagating in anisotropic solids. IEICE Trans. J93-C : 215-223 [in Japanese].
	- [16] T. Shimada and K. Hasegawa: (2010) Perfectly matched layers in the cylindrical and spherical coordinates. Jpn. J. Appl. Phys. 49 : 07HB08.
	- [17] T.Shimada, K.Hasegawa, and S.Sato (2009) Absorbing characteristics of electromagnetic plane waves in perfectly matched layers discretized by finite element method. Keisan Suri Kogaku Ronbunshu 9 : 04-091211 [in Japanese].
	- [18] W.C.Chew and J.M.Jin (1996) Perfectly matched layer in the discretized space: an analysis and optimaization. Electromagnetics 16 : 325-340.
	- [19] F.Collino and B.P.Monk: (1998) Optimizing the perfectly matched layer. Comput. Methods. Appl. Mech. Eng. 164 : 157-171.
	- [20] A.Berumúdez, L.Hervella-Nieto, A.prieto, and R.Rodrïguez: (2007) An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems. J. Compu. Phys. 223 : 469-488.
	- [21] A.Berumúdez, L.Hervella-Nieto, A.prieto, and R.Rodrïguez (2010) Perfectly matched layers for time-haromonic second order elliptic problems. Arch. Comput. Methods Eng. 17 : 77-107.
	- [22] W.R. Scott, Jr. (1994) Errors due to spatial discretization and numerical precision in the finite-element method. IEEE Trans. Antennas Propagat. 42: 1565-1570.

© 2012 Leonardi and Buonsanti, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Leonardi and Buonsanti, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

the Lamè elastic constants,

the media density

**Modeling Dynamics Behaviour of Materials:** 

The study of the dynamic behaviour of materials falls in a multidisciplinary area where many different disciplines converge. The definition of the state of the solid body subject to various actions is very different from the conditions of almost static load, or single dynamic

Complex dynamic actions (i. e. explosion, travelling waves, etc.) request an approach where both inertia and kinetics of the material are fundamental elements to describe the variable

A first section, where particular attention about the shock waves-induced phase transformations and chemical changes will be given. A modelling coupled multifield processes will be introduced in the multiphase solids case through constitutive assumption,

A second part in which some applications of finite element analysis to multi-physics

When an elastic media is subject, over one or more points, to fast actions then media acceleration results. The strain field resulting is carried out within the media by elastic

then, in the isotropic, homogeneous and elastic media we have the follow motion equation:

waves, and so the new and variable stress field should be equilibrated [1, 2].

 and 

**Theoretical Framework and Applications** 

Giovanni Leonardi and Michele Buonsanti

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50642

answer in terms of stress and strain.

The topics covered in the chapter are the following:

dynamic problems is presented and discussed.

**2. Waves equations** 

Let us call *u* the displacement field,

energy balance and mass transfer and a reaction-diffusion model.

**1. Introduction** 

load.

## **Modeling Dynamics Behaviour of Materials: Theoretical Framework and Applications**

Giovanni Leonardi and Michele Buonsanti

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50642

## **1. Introduction**

20 Will-be-set-by-IN-TECH

[13] C.Michler, L.Demkowicz, J.Kurtz and D.Pardo (2007) Improving the performance of perfectly matched layers by means of hp-adaptivity. Num. Methods Partial Diffe. Equa.

[14] A.Taflove and S.C.Hagness (2005) *Computational electrodynamics*(Artech House, Boston)

[15] T.Shimada and K.Hasegawa: (2010) Perfectly matched layers for elastic waves propagating in anisotropic solids. IEICE Trans. J93-C : 215-223 [in Japanese]. [16] T. Shimada and K. Hasegawa: (2010) Perfectly matched layers in the cylindrical and

[17] T.Shimada, K.Hasegawa, and S.Sato (2009) Absorbing characteristics of electromagnetic plane waves in perfectly matched layers discretized by finite element method. Keisan

[18] W.C.Chew and J.M.Jin (1996) Perfectly matched layer in the discretized space: an

[19] F.Collino and B.P.Monk: (1998) Optimizing the perfectly matched layer. Comput.

[20] A.Berumúdez, L.Hervella-Nieto, A.prieto, and R.Rodrïguez: (2007) An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic

[21] A.Berumúdez, L.Hervella-Nieto, A.prieto, and R.Rodrïguez (2010) Perfectly matched layers for time-haromonic second order elliptic problems. Arch. Comput. Methods Eng.

[22] W.R. Scott, Jr. (1994) Errors due to spatial discretization and numerical precision in the

finite-element method. IEEE Trans. Antennas Propagat. 42: 1565-1570.

spherical coordinates. Jpn. J. Appl. Phys. 49 : 07HB08.

Suri Kogaku Ronbunshu 9 : 04-091211 [in Japanese].

scattering problems. J. Compu. Phys. 223 : 469-488.

Methods. Appl. Mech. Eng. 164 : 157-171.

analysis and optimaization. Electromagnetics 16 : 325-340.

23 : 832-858.

17 : 77-107.

3rd ed. Ch7, p.273.

The study of the dynamic behaviour of materials falls in a multidisciplinary area where many different disciplines converge. The definition of the state of the solid body subject to various actions is very different from the conditions of almost static load, or single dynamic load.

Complex dynamic actions (i. e. explosion, travelling waves, etc.) request an approach where both inertia and kinetics of the material are fundamental elements to describe the variable answer in terms of stress and strain.

The topics covered in the chapter are the following:

A first section, where particular attention about the shock waves-induced phase transformations and chemical changes will be given. A modelling coupled multifield processes will be introduced in the multiphase solids case through constitutive assumption, energy balance and mass transfer and a reaction-diffusion model.

A second part in which some applications of finite element analysis to multi-physics dynamic problems is presented and discussed.

## **2. Waves equations**

When an elastic media is subject, over one or more points, to fast actions then media acceleration results. The strain field resulting is carried out within the media by elastic waves, and so the new and variable stress field should be equilibrated [1, 2].

Let us call *u* the displacement field, and the Lamè elastic constants, the media density then, in the isotropic, homogeneous and elastic media we have the follow motion equation:

© 2012 Leonardi and Buonsanti, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Leonardi and Buonsanti, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

$$
\rho \ddot{\boldsymbol{\mu}} = \mu \nabla^2 \boldsymbol{\mu} + (\boldsymbol{\mathcal{X}} + \boldsymbol{\mu}) \nabla (\nabla \boldsymbol{\mu}) \tag{1}
$$

Modeling Dynamics Behaviour of Materials: Theoretical Framework and Applications 201

(6)

*x y y x H yx x y y y H yx*

*y* (8)

(7)

effects, belongs to the wave normal direction and therefore lies in the vertical plane. Otherwise, the particle motion, due to shear effects, present components in either vertical or horizontal plane. Introducing the functions H and , called the Helmholtz potential

> ( / ) ( / ); ( / ) ( / ); ( ) ( / ); ( / ) ( / ) 0.

2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 22

( 2 )[( / ) ( / )] 2 [( / ) ( / ) / )] ( 2 )[( / ) ( / )] 2 [( / ) ( / ) / )]

 

0; 0 *yy yx yz*

In according with [5], we see that the problem above defined can be uncoupled and therefore resolve the motion problem into two parts, namely the first one is plane strain, such that *uz* = 0, *ux*, *uy* . The second one is the secondary wave motion where only

From now on, we wish to study the interaction of elastic waves with discontinuities or

Particularly, we focus the attention over the scattering of compression waves against absorbed obstacles [3, 5], as well as inclusion, in elastic half-space. The propagation and reflection of waves, generated by dynamical forcing over the external surface, against inner surfaces or discontinuity [4-6] has, also, great interest in seismology, structural foundations since the vibratory phenomenon represents a very important further load condition for

Moreover this building framework picks up more general problems, for example voids, flaws or stress raise in half-space constituent materials. The approach adopted follows the

The impact between two solid elements represents the simplified condition for the generation of shock waves. In the specific case of parallel impact the two surfaces enter in contact simultaneously and all the points of the two surfaces enter in contact at the same

stated assumptions and hypothesis , that is expansion of the wave fields in series.

*x zy z yz y x y*

*xx x yy x*

*u x H yu y H x u Hy H x H x H x*

functions, the governing equations related on this approach follow:

2 2 22 2

*xy H y H x*

boundaries of more complex shape than that of the half-space framework.

[2( / ) ( / ) ( / )]

Moreover at Eq. (7) the boundary conditions should be summed:

The stress-displacement relations are given by:

22 2

*xy z z*

)]

*H y H yx*

[2( / ) ( /

*yz x y*

global stability and strength [7-9].

**3. Shock waves** 

time.

*uz* .

 

 

 

 

In the Eq. (1) the symbol represents the nabla differential operator. Under the volume forces equal to zero one possible solution to the Eq. (1) has the form:

$$\mathbf{u}(\mathbf{x}, \mathbf{t}) = \mathbf{u}(\mathbf{t} \text{ - } \mathbf{u}\infty \text{ / } \mathbf{c}) \tag{2}$$

where *n* is a constant unit vector and *c* represents the velocity. Representing the Eq. (2) the solution to the Eq. (1) we observe that the Eq. (2) is a plane wave equation scattering in *n* direction, with *c* velocity. When a direction is fixed, for example *x*1, then we have:

$$
\mu(\mathbf{x}, t) = \mu[t - (\mathbf{x}\_1 / \mathbf{c})] \tag{3}
$$

Generally, an elastic wave can be of two kinds, longitudinal (volume wave) or transversal (shear wave), and respective velocities go with the function:

$$\alpha = \text{sqr}[(\mathcal{A} + \mathcal{D}\mu) \mid \rho\text{)}; \qquad \beta = \text{sqr}(\mu \mid \rho \text{)}\tag{4}$$

As a principle, we have an elastic wave's emission when a fast, and localized variation on the body force exists.

In this case, the Eq. (1) appears as a Green tensor, that is a second order tensor time dependent, *Gij* (*x*, *t*; , ). Neglecting isotropic source, the dynamics source gives out longitudinal and transversal waves with various amplitude according to the direction.

The wave shape represents the signal shape reproduced by the source, in other words the temporal course of the source namely, the *F*(*t*) function.

Since the Green tensor calculation appears with difficulty, through known references, it becomes possible to proceed by the Helmholtz potential method, and therefore to derive, for convolution, the Green tensor final form [3]:

$$\begin{split} G\_{ij}(\mathbf{x}, t; 0, 0) &= (1 / 4 \pi \rho) (\Im \gamma\_i \gamma\_j - \delta\_{ij}) r^{-3} t [H(t - r^{\alpha - 1}) - H(t - r \beta^{-1})] + \\ &+ [(1 / 4 \pi \rho \alpha^2) \gamma\_j \gamma\_j \delta(t - r \alpha^{-1})] + [(1 / 4 \pi \rho \beta^2) (\delta\_{ij} - \gamma\_i \gamma\_j) \delta r^{-1} (t - r \beta^{-1})] \end{split} \tag{5}$$

Where i and j are the director cosine of *x*, the *x* varied position and *ij* the Kronecker delta.

The Eq. (5) is composed by 3 terms, all depending on the distance. We have the first one, called close field, while the other two called away fields. We observe not separable waves in the close field while, in the away field longitudinal and transversal waves appear distinct. All of this allows, in the next modelling to consider only the close field and then longitudinal and transversal motions together.

Here we consider plane waves travelling in an elastic half-space and, without loss of generality, we affirm that the wave normal lies in the vertical plane of the half-space. Referring to the infinite space case, we assume that the particle motion, due to dilatational effects, belongs to the wave normal direction and therefore lies in the vertical plane. Otherwise, the particle motion, due to shear effects, present components in either vertical or horizontal plane. Introducing the functions H and , called the Helmholtz potential functions, the governing equations related on this approach follow:

$$\begin{aligned} \mu\_x &= \{\partial\Phi / \partial\mathbf{x}\} + \{\partial H\_z / \partial y\}; & \mu\_y &= \{\partial\Phi / \partial y\} + \{\partial H\_z / \partial \mathbf{x}\}; \\ \mu\_y &= \{\partial H\_z \partial y\} + \{\partial H\_y / \partial \mathbf{x}\}; & \{\partial H\_x / \partial \mathbf{x}\} + \{\partial H\_y / \partial \mathbf{x}\} &= 0. \end{aligned} \tag{6}$$

The stress-displacement relations are given by:

200 Finite Element Analysis – Applications in Mechanical Engineering

2

direction, with *c* velocity. When a direction is fixed, for example *x*1, then we have:

*sqr*[( 2 ) / ];

   

In the Eq. (1) the symbol represents the nabla differential operator. Under the volume

where *n* is a constant unit vector and *c* represents the velocity. Representing the Eq. (2) the solution to the Eq. (1) we observe that the Eq. (2) is a plane wave equation scattering in *n*

Generally, an elastic wave can be of two kinds, longitudinal (volume wave) or transversal

As a principle, we have an elastic wave's emission when a fast, and localized variation on

In this case, the Eq. (1) appears as a Green tensor, that is a second order tensor time

The wave shape represents the signal shape reproduced by the source, in other words the

Since the Green tensor calculation appears with difficulty, through known references, it becomes possible to proceed by the Helmholtz potential method, and therefore to derive, for

> ( , ;0,0) (1 / 4 )(3 ) [ ( ) ( )] [(1 / 4 ) ( )] [(1 / 4 )( ) ( )]

*G xt r tHt r Ht r*

*i j ij i j*

The Eq. (5) is composed by 3 terms, all depending on the distance. We have the first one, called close field, while the other two called away fields. We observe not separable waves in the close field while, in the away field longitudinal and transversal waves appear distinct. All of this allows, in the next modelling to consider only the close field and then

Here we consider plane waves travelling in an elastic half-space and, without loss of generality, we affirm that the wave normal lies in the vertical plane of the half-space. Referring to the infinite space case, we assume that the particle motion, due to dilatational

2 1 2 11

 

*t r r tr*

longitudinal and transversal waves with various amplitude according to the direction.

 

 

). Neglecting isotropic source, the dynamics source gives out

31 1

the *x* varied position and

 

(5)

*ij* the Kronecker

*uu u* ( )( ) (1)

*u(x, t) = u(t - nx / c)* (2)

<sup>1</sup> *uxt ut x c* ( , ) [ ( / )] (3)

*sqr*(/) (4)

 

forces equal to zero one possible solution to the Eq. (1) has the form:

(shear wave), and respective velocities go with the function:

temporal course of the source namely, the *F*(*t*) function.

*ij i j ij*

 

> 

, 

convolution, the Green tensor final form [3]:

 

Where i and j are the director cosine of *x*,

longitudinal and transversal motions together.

the body force exists.

dependent, *Gij* (*x*, *t*;

delta.

$$\begin{aligned} \sigma\_{xx} &= (\lambda + 2\mu)[(\hat{\circ}^2 \Phi / \hat{\alpha}^2) + (\hat{\circ}^2 \Phi / \hat{\alpha}^2)] - 2\mu[(\hat{\circ}^2 \Phi / \hat{\alpha}^2) - (\hat{\circ}^2 \Phi / \hat{\alpha}^2) - \hat{\circ}^2 H\_{\text{x}} / \hat{\alpha} \hat{\alpha} \infty]] \\ \sigma\_{yy} &= (\lambda + 2\mu)[(\hat{\circ}^2 \Phi / \hat{\alpha}^2) + (\hat{\circ}^2 \Phi / \hat{\alpha}^2)] - 2\mu[(\hat{\circ}^2 \Phi / \hat{\alpha}^2) - (\hat{\circ}^2 \Phi / \hat{\alpha}^2) - \hat{\circ}^2 H\_{\text{x}} / \hat{\alpha} \hat{\rho} \infty]] \\ \sigma\_{xy} &= \mu[2(\hat{\circ}^2 \Phi / \hat{\alpha} \hat{\alpha}) + (\hat{\circ}^2 H\_{\text{x}} / \hat{\alpha} \hat{y}^2) - (\hat{\circ}^2 H\_{\text{x}} / \hat{\alpha} \hat{x}^2)] \\ \sigma\_{yz} &= \mu[2(\hat{\circ}^2 H\_{\text{x}} / \hat{\alpha}^2 \hat{y}) + (\hat{\circ}^2 H\_{\text{y}} / \hat{\alpha} \hat{y} \hat{\alpha})] \end{aligned} \tag{7}$$

Moreover at Eq. (7) the boundary conditions should be summed:

$$
\sigma\_{yy} = \tau\_{yx} = \tau\_{yz} = 0; \qquad y = 0 \tag{8}
$$

In according with [5], we see that the problem above defined can be uncoupled and therefore resolve the motion problem into two parts, namely the first one is plane strain, such that *uz* = 0, *ux*, *uy* . The second one is the secondary wave motion where only *uz* .

From now on, we wish to study the interaction of elastic waves with discontinuities or boundaries of more complex shape than that of the half-space framework.

Particularly, we focus the attention over the scattering of compression waves against absorbed obstacles [3, 5], as well as inclusion, in elastic half-space. The propagation and reflection of waves, generated by dynamical forcing over the external surface, against inner surfaces or discontinuity [4-6] has, also, great interest in seismology, structural foundations since the vibratory phenomenon represents a very important further load condition for global stability and strength [7-9].

Moreover this building framework picks up more general problems, for example voids, flaws or stress raise in half-space constituent materials. The approach adopted follows the stated assumptions and hypothesis , that is expansion of the wave fields in series.

#### **3. Shock waves**

The impact between two solid elements represents the simplified condition for the generation of shock waves. In the specific case of parallel impact the two surfaces enter in contact simultaneously and all the points of the two surfaces enter in contact at the same time.

The true profile of a shock wave is complex. In the following Figure 1, it is possible to observe the difference between the ideal and the true profile where, for the latter, it is clear the dependence form the characteristics of the material and the pressure applied at contact.

Modeling Dynamics Behaviour of Materials: Theoretical Framework and Applications 203

(9)

(10)

(11)

(13)

(14)

, and are

*ii ii E T* (12)

On a theoretical point of view, we classify the problem as the propagation of a shock wave, where a uniform contact pressure is applied on a plane solid surface in an elastic semi space. Given the geometrical origin *x* = 0 and the beginning of the phenomenon at time *t* = 0, after a laps of time *t* the shock front divides the space in two regions, one undisturbed, the other compressed and accelerated. Therefore the flow equation is reduced to the jump condition:

> ( )( ) *s op s s*

*v vv v*

*V*

with *vs* the wave propagation speed, *vo* and *vp* are the speeds of the particles respectively

The clear result is the introduction of a pressure step which travels across the medium, with

In the case of impact the contact time tends to zero, therefore *ti* = *tp* where *ti* is the impact time and *tp* is the plasticization time. In theory, the problem can be represented as two

First phase: transversal speed at the centre of the body remains constant. This phase is

Second phase: a concentrated plasticization begins which starts to expand from the core to

<sup>2</sup> / 6 *o o t vr M* 

Where *μ* is the Lame's material constant and the associated maximum permanent

2 2 <sup>0</sup> 3 /4 *<sup>c</sup>*

In regards to the mechanical proprieties of the medium subject to the impact actions in the case, the response of solids made of asphalt mixtures can be divided, accordingly to [12, 13] in three groups: elastic visco-elastic and visco-plastic. In the one-dimensional case we have:

*e* 1

( ) *ve ii ii E AT t*

( )( ) *vp ii ii E BT f Nt*

*pr p*

behind and in front of the shock front, and, *Vs* is the specific volume of the medium.

changes of shape which depend on the mechanical proprieties of the element.

necessary to absorb the remaining kinetic energy in the body.

the external part of the body. The time it takes is given by the expression:

transversal displacement in the contact zone can be approximated as:

where *f*(*N*), *A*(*T*), *B*(*T*) are functions of the stress in the viscose phase. *A*, *B*,

constants determined at constant temperature.

In the multidimensional case the equations above become:

successive phases.

*P*

An impulsive stress on contact has an initial, middle and final pressure value. Initially it's a shock wave (discontinuity in compression); the mean reaction is characterized from a slow variation of pressure and the final from a dissolution which tends to the undisturbed state.

**Figure 1.** (*a*) idealized and *(b)* generic realistic shock wave profile (from Meyers [10]).

In a previous paper the authors has investigated the waves generation after the impact on a granular plate [9]. The study has been developed, initially from a microscopic point of view and subsequently on macroscale.

The effects are strictly linked with material degradation associated with damage evolution. In accordance to [11] the shock waves can induce phase transitions in the solid, (Figure 2), then transitions form elastic to plastic response (in our case plasticization of the mixture binding component).

**Figure 2.** Pressure distribution in a pulse propagating through a material undergoing a phase transformation and a transition from elastic to inelastic behaviour (from Meyers [10]).

On a theoretical point of view, we classify the problem as the propagation of a shock wave, where a uniform contact pressure is applied on a plane solid surface in an elastic semi space. Given the geometrical origin *x* = 0 and the beginning of the phenomenon at time *t* = 0, after a laps of time *t* the shock front divides the space in two regions, one undisturbed, the other compressed and accelerated. Therefore the flow equation is reduced to the jump condition:

202 Finite Element Analysis – Applications in Mechanical Engineering

The true profile of a shock wave is complex. In the following Figure 1, it is possible to observe the difference between the ideal and the true profile where, for the latter, it is clear the dependence form the characteristics of the material and the pressure applied at contact.

An impulsive stress on contact has an initial, middle and final pressure value. Initially it's a shock wave (discontinuity in compression); the mean reaction is characterized from a slow variation of pressure and the final from a dissolution which tends to the undisturbed state.

**Figure 1.** (*a*) idealized and *(b)* generic realistic shock wave profile (from Meyers [10]).

and subsequently on macroscale.

binding component).

In a previous paper the authors has investigated the waves generation after the impact on a granular plate [9]. The study has been developed, initially from a microscopic point of view

The effects are strictly linked with material degradation associated with damage evolution. In accordance to [11] the shock waves can induce phase transitions in the solid, (Figure 2), then transitions form elastic to plastic response (in our case plasticization of the mixture

**Figure 2.** Pressure distribution in a pulse propagating through a material undergoing a phase transformation and a transition from elastic to inelastic behaviour (from Meyers [10]).

$$
\Delta P = \frac{(\upsilon\_s - \upsilon\_o)(\upsilon\_p - \upsilon\_s)}{V\_s} \tag{9}
$$

with *vs* the wave propagation speed, *vo* and *vp* are the speeds of the particles respectively behind and in front of the shock front, and, *Vs* is the specific volume of the medium.

The clear result is the introduction of a pressure step which travels across the medium, with changes of shape which depend on the mechanical proprieties of the element.

In the case of impact the contact time tends to zero, therefore *ti* = *tp* where *ti* is the impact time and *tp* is the plasticization time. In theory, the problem can be represented as two successive phases.

First phase: transversal speed at the centre of the body remains constant. This phase is necessary to absorb the remaining kinetic energy in the body.

Second phase: a concentrated plasticization begins which starts to expand from the core to the external part of the body. The time it takes is given by the expression:

$$t = \mu \upsilon^{o} r^{2} / \,\epsilon M^{o} \tag{10}$$

Where *μ* is the Lame's material constant and the associated maximum permanent transversal displacement in the contact zone can be approximated as:

$$
\rho \alpha \equiv 3p\_0^2 r^2 \;/\, 4\mu p\_c \tag{11}
$$

In regards to the mechanical proprieties of the medium subject to the impact actions in the case, the response of solids made of asphalt mixtures can be divided, accordingly to [12, 13] in three groups: elastic visco-elastic and visco-plastic. In the one-dimensional case we have:

$$E\_{il}^{\epsilon} = \epsilon^{-1} T\_{il} \tag{12}$$

$$E\_{ii}^{ve} = A(T\_{ii})t^{\alpha} \tag{13}$$

$$E\_{i\bar{i}}^{vp} = B(T\_{i\bar{i}})f(\mathcal{N})t^{\beta} \tag{14}$$

where *f*(*N*), *A*(*T*), *B*(*T*) are functions of the stress in the viscose phase. *A*, *B*, , and are constants determined at constant temperature.

In the multidimensional case the equations above become:

$$\mathbb{E}\left[E\right]^{\epsilon} = \mathbb{E}K\mathbb{I}\cdot\left[T\right] \tag{15}$$

Modeling Dynamics Behaviour of Materials: Theoretical Framework and Applications 205

*<sup>z</sup>* pressure distribution under the concentrated

, *z*.

(21)

(22)

, respectively, Poisson's and

*rz* and the

*<sup>z</sup>* and

We follow the volume element theory RVE, it's possible to represent a non-homogeneous solid with periodic microstructure. Particularly in the transition toward the micro-scale our RVE can be represented by more granular elements joint by means of an asphalt mixture, so considerations are applied on the contact area among two granular elements. In this manner

Following Sneddon's solution [14] type we model the physics of impact by means of a rigid frictionless asymmetric concentrated impact, with generic concave profile described by the

To reproduce a possible genuine model becomes fundamental to describe the single load conditions since blast action, fundamentally, can be decomposed in thermal and shock wave's loads. Here we develop the theoretical assumptions in both cases just starting with

From a structural point of view the tunnel can be considered as well as a half thick-walled cylinder subject to internal and external pressures. So we consider a half cylinder of inner radius a and outer radius b and subject to an internal pressure *pa* and an external pressure *pb*. We choose, as the closest to real behaviour, the plane stresses condition so that the ends of

Assuming the *z*-axis as the revolution axis, the deformation becomes symmetrical respect to

0 *r r r r* 

*uu u u E EE rr r z* 

Young's modulus) after some simple calculations we get the basic equations governing the

*pa pb a b p p u r*

2 2 2 2

2 2 22 2

*a b ab*

*pa pb p p b a b a rb a*

2 2 2 2

1 1 *a b a b*

2 2 2 2

 

*b a r b a*

; ; *rrz r r*

> and

, *z*) represents the displacements field over the shell.

the homogenization problems can be satisfied.

function *f*(*r*). We find respectively, the

the cylinder be free to expand.

where the function *u*(*r*,

thick-walled half-cylinder:

impact and the displacement on the surface.

**4. Shell structures and blast loading** 

some structural considerations about the thick shell behaviour.

the *z*-axis. Consequently it's convenient to use cylindrical coordinates *r*,

 

According to [16-18] the plane stresses conditions involve

while the deformation field *E* as the components in the form:

Introducing the Lame's constitutive equations (with

*r*

*r*

1

equilibrium conditions, without body forces become.

$$\left[ \left[ E \right]^{\text{pt}} = A t^{\alpha} \left[ H \right] \cdot \left[ T^{\text{lt}} \right] \tag{16}$$

where *K* is the deformability matrix, *H K* and [ ] *<sup>h</sup> ijk T TT*

In regard to the visco-elastic part, it has to be specified that the critical points which arise in this phase with the load time can be resolved using the Perzyna theory [13]. For an associated visco-plastic flow we have:

$$\{E\}^{\text{vp}} = \gamma \cdot \phi(t) \cdot \hat{\circ} F / \hat{\circ} t \tag{17}$$

where:


Passing to the numerical implementation, in the case of reduced load intervals, an iterative procedure, as a Newton-Raphson, can be applied.

$$A[T]^\mathfrak{t} = [K][A[E]]^\mathfrak{t} - A\alpha t^{\alpha - 1}[K][T^\mathfrak{t}]^\mathfrak{t} - A[E^{vp}]^\mathfrak{t} \tag{18}$$

and, after the rightful developing, stress and strain in approximated as:

$$\left[\left[T\right]\right]^{\iota-1} = \left[T\right]^{\iota} + A\left[T\right]^{\iota} \tag{19}$$

$$\left[\left[E^{vp}\right]^{\imath+1} = \left[E^{vp}\right] + A\{E^{vp}\}^{\imath}\right.\tag{20}$$

Therefore it follows the link between micro-scale effects and material behaviours at macroscale.

So we focus the micromechanics of the damage processes because the nonlinear response of typical engineering materials is almost entirely dependent on the primary change in the concentration, distribution, orientation and defects in its structural composition.

The relation between the continuum damage mechanics and the fracture mechanics is very complicated, in essence, a question of scale. The important role of scale can be clarified by an energetic point of view.

In view of an approximated continuum theory with the physical foundation of micromechanical models, a promising strategy would consist of combining the best features of both models. In this approach we consider only the first layer of the pavement package because, at micro-scale, damage distribution at the edge of the body, where surface degradation is of importance, is expected to be significantly different from the damage distribution far from the edge in the body.

We follow the volume element theory RVE, it's possible to represent a non-homogeneous solid with periodic microstructure. Particularly in the transition toward the micro-scale our RVE can be represented by more granular elements joint by means of an asphalt mixture, so considerations are applied on the contact area among two granular elements. In this manner the homogenization problems can be satisfied.

Following Sneddon's solution [14] type we model the physics of impact by means of a rigid frictionless asymmetric concentrated impact, with generic concave profile described by the function *f*(*r*). We find respectively, the *<sup>z</sup>* pressure distribution under the concentrated impact and the displacement on the surface.

## **4. Shell structures and blast loading**

204 Finite Element Analysis – Applications in Mechanical Engineering

associated visco-plastic flow we have:


procedure, as a Newton-Raphson, can be applied.

where:

scale.

energetic point of view.

distribution far from the edge in the body.

cycles;


where *K* is the deformability matrix, *H K* and [ ] *<sup>h</sup>*


and, after the rightful developing, stress and strain in approximated as:

[ ] [ ]·[ ] *<sup>e</sup> E KT* (15)

(17)

<sup>1</sup> [] [] [] *nnn T T AT* (19)

<sup>1</sup> [] [] [] *vp n n vp vp E E AE* (20)

(16)

(18)

[ ] [ ]·[ ] *ve <sup>h</sup> E At H T* 

In regard to the visco-elastic part, it has to be specified that the critical points which arise in this phase with the load time can be resolved using the Perzyna theory [13]. For an

> [ ] · ( )· / *vp E tF t*

Passing to the numerical implementation, in the case of reduced load intervals, an iterative

<sup>1</sup> [ ] [ ][ [ ]] [ ][ ] [ ] *<sup>n</sup> <sup>n</sup> hn n vp AT K AE A t K T AE* 

Therefore it follows the link between micro-scale effects and material behaviours at macro-

So we focus the micromechanics of the damage processes because the nonlinear response of typical engineering materials is almost entirely dependent on the primary change in the

The relation between the continuum damage mechanics and the fracture mechanics is very complicated, in essence, a question of scale. The important role of scale can be clarified by an

In view of an approximated continuum theory with the physical foundation of micromechanical models, a promising strategy would consist of combining the best features of both models. In this approach we consider only the first layer of the pavement package because, at micro-scale, damage distribution at the edge of the body, where surface degradation is of importance, is expected to be significantly different from the damage

concentration, distribution, orientation and defects in its structural composition.

is a fluidity parameter associated to the loading times and the number of loading

*ijk T TT*

To reproduce a possible genuine model becomes fundamental to describe the single load conditions since blast action, fundamentally, can be decomposed in thermal and shock wave's loads. Here we develop the theoretical assumptions in both cases just starting with some structural considerations about the thick shell behaviour.

From a structural point of view the tunnel can be considered as well as a half thick-walled cylinder subject to internal and external pressures. So we consider a half cylinder of inner radius a and outer radius b and subject to an internal pressure *pa* and an external pressure *pb*. We choose, as the closest to real behaviour, the plane stresses condition so that the ends of the cylinder be free to expand.

Assuming the *z*-axis as the revolution axis, the deformation becomes symmetrical respect to the *z*-axis. Consequently it's convenient to use cylindrical coordinates *r*, , *z*.

According to [16-18] the plane stresses conditions involve *<sup>z</sup>* and *rz* and the equilibrium conditions, without body forces become.

$$\frac{\partial \sigma\_r}{\partial r} + \frac{\sigma\_r - \sigma\_\theta}{r} = 0 \tag{21}$$

while the deformation field *E* as the components in the form:

$$E\_{\partial} = \frac{1}{r} \frac{\partial u\_{\partial}}{\partial \theta} + \frac{u\_{r}}{r}; \quad E\_{r} = \frac{\partial u\_{r}}{\partial r}; \quad E\_{r} = \frac{\partial u\_{z}}{\partial z} \tag{22}$$

where the function *u*(*r*, , *z*) represents the displacements field over the shell.

Introducing the Lame's constitutive equations (with and , respectively, Poisson's and Young's modulus) after some simple calculations we get the basic equations governing the thick-walled half-cylinder:

$$\begin{aligned} \mu\_r &= \frac{1-\nu}{\epsilon} \frac{p\_a a^2 - p\_b b^2}{b^2 - a^2} r + \frac{1+\nu}{\epsilon} \frac{a^2 b^2}{r} \frac{p\_a - p\_b}{b^2 - a^2} \\\ \sigma\_r &= \frac{p\_a a^2 - p\_b b^2}{b^2 - a^2} - \frac{b^2 a^2}{r^2} \frac{p\_a - p\_b}{b^2 - a^2} \end{aligned}$$

$$
\sigma\_0 = \frac{p\_a a^2 - p\_b b^2}{b^2 - a^2} + \frac{b^2 a^2}{r^2} \frac{p\_a - p\_b}{b^2 - a^2} \tag{23}
$$

Modeling Dynamics Behaviour of Materials: Theoretical Framework and Applications 207

If the temperature T is positive and if the external temperature is equal to zero then the

After numerous disasters in the building and structures, the fire-structure question was developed for many researchers, which has reproduced a large and specific literature. For instance it is our opinion, referring at some as important in [16-19]. From now we will deepen the other coupled action namely the structural effects after the burst. According to [10] the interaction of a detonating explosive with a material in contact with it or in close proximity is extremely complex, since it evolves detonation waves, shock waves, expanding

The question was developed, principally, by military requirement which study has

It's our interest some basic assumption linked to the effective problem that requires us namely, only actions from shock waves. So, we affirm the following basic assumptions:

b. The shear modulus is assumed to be zero and so it responds to the wave as a fluid, and

Now we will consider the dynamic behaviour of thick-wall cylindrical shell under internal

Let *pc* be the collapse pressure, then the shell is subject to a symmetrical internal pressure

Supposing the pressure load symmetric, then the yielding is controlled by force in the shell

let *v*° be the spherically symmetric outwards impulsive velocity, we then find the radial

<sup>2</sup> / *<sup>o</sup> Nt r vt <sup>c</sup>*

/ 4 *<sup>o</sup> f c*

 

After some calculations we have the associated permanent radial displacement field over

 

= *Nc* (with *Nc* the fully plastic membrane forces). Neglecting the elastic effects, the

be the generalized membrane forces, at the yielding point we have

be the transverse displacement of the shell middle plane and

(27)

*rv N* (28)

 = *N*=*Nc*.

. Again we assume a perfectly rigid

*t t\** where *t\** is the

, while *p* = 0 when *t*

developed the computational apparatus, for instance the Gurney equation [25-26].

a. A shock is a discontinuous surface and has no apparent thickness.

c. Body forces and heat conduction at the shock front are negligible.

For major clarify we consider, as the second phase, the time as

the theory can be restricted to higher pressures.

e. Material do not undergo phase transformations.

dynamics response consist of two phases motion with *N*

d. There is no elastic-plastic behaviour.

pressure produced by shock wave.

pulse, in the interval time 0 *t*

plastic material behaviour.

response duration time. Let

displacement

the shell.

middle plane. So, let *N*

*N*

radial stress is always compressive, like other stresses in the inner surface.

gases, and their interrelationships.

Under these conditions, we recall either of the specific conditions of the internal and external pressure loads. In the first case (internal pressure) the above equations becomes:

$$\begin{aligned} \sigma\_r &= \frac{p\_a a^2}{b^2 - a^2} \left( 1 - \frac{b^2}{r^2} \right) \\ \sigma\_\vartheta &= \frac{p\_a a^2}{b^2 - a^2} \left( 1 + \frac{b^2}{r^2} \right) \end{aligned} \tag{24}$$

From the equations above, a consideration can be drawn about the circumferential stress ( tensile stress), which is at its greatest on the inner surface and is always greater than *pa*. In the second case (external pressure) the general equations assume the form:

$$\begin{aligned} \sigma\_r &= -\frac{p\_b}{b^2 - a^2} \left( 1 - \frac{a^2}{r^2} \right) \\ \sigma\_\vartheta &= \frac{p\_b b^2}{b^2 - a^2} \left( 1 + \frac{a^2}{r^2} \right) \end{aligned} \tag{25}$$

The stress paths, when no inner holes were present, are uniformly distributed in the cylinder. From now we will be able to describe the coupling actions over the thick-walled half-cylinder shell and for this we run recalling some basic thermo-elasticity assumption. There is a large literature over the question but we prefer to follow [18-20].

We focus the consistence of thermal stresses induced in thick-walled half-cylinder when the temperature field is symmetrical about the z-axis. In this case we suppose the temperature T as radius function only and independent from z then plane strain *Ez* = 0. With analogous considerations as above, the basic equations, for the coupled problem, can be written as.

$$\begin{aligned} \mu\_r &= \frac{(1+\nu)\alpha}{(1-\nu)r} \left\{ \int\_a^r Tr dr + \frac{(1-2\nu)r^2 + a^2}{b^2 - a^2} \right\} \left[ Tr dr \right] \\ \sigma\_r &= \frac{\alpha \epsilon}{(1-\nu)} \frac{1}{r^2} \left\{ \frac{r^2 - a^2}{b^2 - a^2} \Big|\_a^b Tr dr - \int\_a^r Tr dr \right\} \\ \sigma\_\theta &= \frac{\alpha \epsilon}{(1-\nu)} \frac{1}{r^2} \left\{ \frac{r^2 - a^2}{b^2 - a^2} \Big|\_a^b Tr dr + \int\_a^r Tr dr - Tr r^2 \right\} \\ \sigma\_z &= \frac{\alpha \epsilon}{(1-\nu)} \left\{ \frac{2}{b^2 - a^2} \Big| \frac{r dr d\tau - Tr}{b^2 - a^2} \right\} \end{aligned} \tag{26}$$

If the temperature T is positive and if the external temperature is equal to zero then the radial stress is always compressive, like other stresses in the inner surface.

After numerous disasters in the building and structures, the fire-structure question was developed for many researchers, which has reproduced a large and specific literature. For instance it is our opinion, referring at some as important in [16-19]. From now we will deepen the other coupled action namely the structural effects after the burst. According to [10] the interaction of a detonating explosive with a material in contact with it or in close proximity is extremely complex, since it evolves detonation waves, shock waves, expanding gases, and their interrelationships.

The question was developed, principally, by military requirement which study has developed the computational apparatus, for instance the Gurney equation [25-26].

It's our interest some basic assumption linked to the effective problem that requires us namely, only actions from shock waves. So, we affirm the following basic assumptions:


206 Finite Element Analysis – Applications in Mechanical Engineering

 

2 2 2 2

Under these conditions, we recall either of the specific conditions of the internal and external pressure loads. In the first case (internal pressure) the above equations becomes:

*a*

*a*

From the equations above, a consideration can be drawn about the circumferential stress (

tensile stress), which is at its greatest on the inner surface and is always greater than *pa*. In

*b*

*b*

22 2 2 2 22 2

*p a ba r p b a ba r*

 

1

 

The stress paths, when no inner holes were present, are uniformly distributed in the cylinder. From now we will be able to describe the coupling actions over the thick-walled half-cylinder shell and for this we run recalling some basic thermo-elasticity assumption.

We focus the consistence of thermal stresses induced in thick-walled half-cylinder when the temperature field is symmetrical about the z-axis. In this case we suppose the temperature T as radius function only and independent from z then plane strain *Ez* = 0. With analogous considerations as above, the basic equations, for the coupled problem, can

> 2 2 22 2

*u Trdr Trdr r b a*

(1 ) (1 2 )

(1 )

 

*r*

*r*

*z*

(1 )

(1 )

(1 )

1

1

2 2

*b*

*a*

22 2

*rb a*

2 2

*b a*

2

*rb a*

1

*r*

the second case (external pressure) the general equations assume the form:

*r*

There is a large literature over the question but we prefer to follow [18-20].

be written as.

2 2 22 2 *a b ab pa pb p p b a b a rb a* 

> 2 2 22 2

1

*p a b ba r p a b ba r*

 

 

> 2 2 22 2

2

2 2

*r a*

2

(26)

2 2

*r b*

*a a b r*

*r a Trdr Trdr*

*a a b r*

*a a*

*Trdr Tr*

*r a Trdr Trdr Tr*

1

(23)

(24)

(25)

e. Material do not undergo phase transformations.

Now we will consider the dynamic behaviour of thick-wall cylindrical shell under internal pressure produced by shock wave.

Let *pc* be the collapse pressure, then the shell is subject to a symmetrical internal pressure pulse, in the interval time 0 *t* , while *p* = 0 when *t* . Again we assume a perfectly rigid plastic material behaviour.

Supposing the pressure load symmetric, then the yielding is controlled by force in the shell middle plane. So, let *N* be the generalized membrane forces, at the yielding point we have *N* = *Nc* (with *Nc* the fully plastic membrane forces). Neglecting the elastic effects, the dynamics response consist of two phases motion with *N* = *N*=*Nc*.

For major clarify we consider, as the second phase, the time as *t t\** where *t\** is the response duration time. Let be the transverse displacement of the shell middle plane and let *v*° be the spherically symmetric outwards impulsive velocity, we then find the radial displacement

$$
\rho = N\_c t^2 \;/\; \mu r - v^o t \tag{27}
$$

After some calculations we have the associated permanent radial displacement field over the shell.

$$
\rho\_f = -\mu r v^o \,/\, 4N\_c \tag{28}
$$

## **5. Applications examples**

In the following, we will give two examples of finite element applications to dynamic analysis in particular, interaction problems of structures systems subject to explosion waves and impact loads are presented [9] [15].

Modeling Dynamics Behaviour of Materials: Theoretical Framework and Applications 209

RP 1 RP 2

**Figure 4.** Three-dimensional view of the finite element model.

**<sup>X</sup> <sup>Y</sup> Z**

(with no gaps).

transversal direction.

= 0) on the sides parallel to x-axis.

profiles along the transverse center line.

without resorting to potentially costly field experiments.

created to be the same size as the wheel imprint of an Airbus A321.

The degree of mesh refinement is the most important factor in estimating an accurate stress field in the pavement: the finest mesh is required near the loads to capture the stress and strain gradients. The mesh presented has 126245 nodes and 29900 quadratic hexahedral elements of type C3D20R (continuum 3-dimensional 20 node elements with reduced integration). Quadratic elements yield better solution than linear interpolation elements [29]. The loads (vertical and horizontal) were uniformly applied to the element, which was

In this example the surface was considered to be free from any discontinuities (with no cracks) or unevenness, and the interface between layers was considered to be fully bonded

The model was constrained at the bottom (encastre: U1 = U2 = U3 = UR1 = UR2 = UR3 = 0); X-Symm (U1 = UR2 = UR3 = 0) on the sides parallel to y-axis; and Y-Symm (U2 = UR1 = UR3

The results of the non-linear FE analysis are illustrated in the following figures. Figure 5 shows the Mises stress distribution for the considered FE model at the landing aircraft impact instant and Figure 6 presents the results of pavement surface deflection along

Finally, in the graph of Figure 7 are plotted the predicted transversal surface deflection

This example shows how finite element analysis of pavement structures, if validated, can be extremely useful, because it can be used directly to estimate pavement response parameters

If accurate correlations between the theoretically-calculated and the field-measured response parameters can be obtained, then the finite element model can be used to simulate

## **5.1. Impact loads on flexible pavement**

In this example we assessed the effects of a heavy impact caused by aircraft landing gear wheels on a flexible airport pavement.

Flexible pavements are usually idealized as closed systems consisting of several layers; so the surface, base, sub-base and sub-grade material were modelled using 3-D finite elements. While an elastic constitutive model was assumed for the granular layers and the base course, a time hardening creep model was incorporated to simulate the viscoelastic behaviour of the HMA surface layer

The aircraft considered in the model was the Airbus 321 [26]. The most common way of applying wheel loads in a finite element analysis is to apply pressure load to a circular or rectangular equivalent contact area with uniform tyre pressure [27]. To investigate the impact simulation in exceptional condition, the dynamic parameters of an "hard" landing, that caused the broken of some gear components, were considered [28]. Starting from this, considering the damping effect of the gear system, it is possible to calculate the acceleration graph during the landing (Figure 3).

**Figure 3.** Acceleration graph.

As shown in Figure 3 the peak acceleration value, during the hard landing, is 1.99 m/s2. This value of acceleration was used in the finite element model to calculate the maximum wheel load.

The finite element model has the following dimensions: 10 m in *x* and *y* directions and 2.5 m in the *z*- direction. The three-dimensional view of finite element model is shown in Figure 4.

**Figure 4.** Three-dimensional view of the finite element model.

In the following, we will give two examples of finite element applications to dynamic analysis in particular, interaction problems of structures systems subject to explosion waves

In this example we assessed the effects of a heavy impact caused by aircraft landing gear

Flexible pavements are usually idealized as closed systems consisting of several layers; so the surface, base, sub-base and sub-grade material were modelled using 3-D finite elements. While an elastic constitutive model was assumed for the granular layers and the base course, a time hardening creep model was incorporated to simulate the viscoelastic

The aircraft considered in the model was the Airbus 321 [26]. The most common way of applying wheel loads in a finite element analysis is to apply pressure load to a circular or rectangular equivalent contact area with uniform tyre pressure [27]. To investigate the impact simulation in exceptional condition, the dynamic parameters of an "hard" landing, that caused the broken of some gear components, were considered [28]. Starting from this, considering the damping effect of the gear system, it is possible to calculate the acceleration

As shown in Figure 3 the peak acceleration value, during the hard landing, is 1.99 m/s2. This value of acceleration was used in the finite element model to calculate the maximum wheel

The finite element model has the following dimensions: 10 m in *x* and *y* directions and 2.5 m in the *z*- direction. The three-dimensional view of finite element model is shown in Figure 4.

**5. Applications examples** 

and impact loads are presented [9] [15].

wheels on a flexible airport pavement.

behaviour of the HMA surface layer

graph during the landing (Figure 3).

**Figure 3.** Acceleration graph.

load.

**5.1. Impact loads on flexible pavement** 

The degree of mesh refinement is the most important factor in estimating an accurate stress field in the pavement: the finest mesh is required near the loads to capture the stress and strain gradients. The mesh presented has 126245 nodes and 29900 quadratic hexahedral elements of type C3D20R (continuum 3-dimensional 20 node elements with reduced integration). Quadratic elements yield better solution than linear interpolation elements [29].

The loads (vertical and horizontal) were uniformly applied to the element, which was created to be the same size as the wheel imprint of an Airbus A321.

In this example the surface was considered to be free from any discontinuities (with no cracks) or unevenness, and the interface between layers was considered to be fully bonded (with no gaps).

The model was constrained at the bottom (encastre: U1 = U2 = U3 = UR1 = UR2 = UR3 = 0); X-Symm (U1 = UR2 = UR3 = 0) on the sides parallel to y-axis; and Y-Symm (U2 = UR1 = UR3 = 0) on the sides parallel to x-axis.

The results of the non-linear FE analysis are illustrated in the following figures. Figure 5 shows the Mises stress distribution for the considered FE model at the landing aircraft impact instant and Figure 6 presents the results of pavement surface deflection along transversal direction.

Finally, in the graph of Figure 7 are plotted the predicted transversal surface deflection profiles along the transverse center line.

This example shows how finite element analysis of pavement structures, if validated, can be extremely useful, because it can be used directly to estimate pavement response parameters without resorting to potentially costly field experiments.

If accurate correlations between the theoretically-calculated and the field-measured response parameters can be obtained, then the finite element model can be used to simulate

pavement response utilizing measurements from strain gages. In particular, the proposed model has clearly confirmed the need and importance of 3-Dimensional finite element analyses on flexible pavements to consider the behaviour of the structure under high stress.

Modeling Dynamics Behaviour of Materials: Theoretical Framework and Applications 211

In the following application a 3-D simulation of tunnel structures under Blast loading is

The Finite Element model was based on a single track railway tunnel system consisting of concrete tunnel tube with the section dimensions reported in Figure 8. The tunnel was about 10 m below the ground surface. The model extended 150 m in the longitudinal direction of the tunnel, while the length and height of the model were of 26.8 m. The finite Element model was fixed at the base and roller boundaries were imposed to the four side. The modelled tunnel structure is surrounded by soil and this load represents the starting state of stress. Drucker-Prager elasto-plastic model was used to model the soil. For the characterization of the reinforced concrete of the tunnel structure it was considered a C50/60 class concrete having thermal characteristics according with the indications of the Eurocode

A fundamental aspect in the study of fire resistance in underground structures is the definition beforehand of the fire scenario taken in the analysis, therefore choosing the best fit standard curve. A standard curve is the cellulose curve defined in several standards, e.g. ISO 834 [23]. Specific temperature curves have been developed in some countries to simulate hydrocarbon fires in tunnels. Examples of such curves are the RABT/ZTV Tunnel Curve in Germany [31] and the Rijkswaterstaat Tunnel Curve (RWS curve) in The Netherlands (based

*5.00*

**5.2. Confined explosions** 

**Figure 8.** Rail tunnel section.

proposed.

2 [30].

**Figure 5.** Mises stress at the instant of impact.


**Figure 6.** Displacement contours at the instant of impact.

**Figure 7.** Predicted deflection profiles (y-direction).

## **5.2. Confined explosions**

210 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 5.** Mises stress at the instant of impact.

**Figure 6.** Displacement contours at the instant of impact.

**Figure 7.** Predicted deflection profiles (y-direction).

pavement response utilizing measurements from strain gages. In particular, the proposed model has clearly confirmed the need and importance of 3-Dimensional finite element analyses on flexible pavements to consider the behaviour of the structure under high stress.

> (Avg: 75%) S, Mises

> > +1.093e+03 +2.440e+05 +4.870e+05 +7.299e+05 +9.728e+05 +1.216e+06 +1.459e+06 +1.702e+06 +1.945e+06 +2.188e+06 +2.430e+06 +2.673e+06 +2.916e+06

In the following application a 3-D simulation of tunnel structures under Blast loading is proposed.

The Finite Element model was based on a single track railway tunnel system consisting of concrete tunnel tube with the section dimensions reported in Figure 8. The tunnel was about 10 m below the ground surface. The model extended 150 m in the longitudinal direction of the tunnel, while the length and height of the model were of 26.8 m. The finite Element model was fixed at the base and roller boundaries were imposed to the four side. The modelled tunnel structure is surrounded by soil and this load represents the starting state of stress. Drucker-Prager elasto-plastic model was used to model the soil. For the characterization of the reinforced concrete of the tunnel structure it was considered a C50/60 class concrete having thermal characteristics according with the indications of the Eurocode 2 [30].

**Figure 8.** Rail tunnel section.

A fundamental aspect in the study of fire resistance in underground structures is the definition beforehand of the fire scenario taken in the analysis, therefore choosing the best fit standard curve. A standard curve is the cellulose curve defined in several standards, e.g. ISO 834 [23]. Specific temperature curves have been developed in some countries to simulate hydrocarbon fires in tunnels. Examples of such curves are the RABT/ZTV Tunnel Curve in Germany [31] and the Rijkswaterstaat Tunnel Curve (RWS curve) in The Netherlands (based

on laboratory scale tunnel tests performed by TNO in 1979 [32]). In the considered model the HC curve was used to simulate the fire action.

Modeling Dynamics Behaviour of Materials: Theoretical Framework and Applications 213

**Figure 10.** Temperature distribution (°C) *t* = 1800 sec.

**Figure 11.** Mises stress (Pa) of the tunnel at the explosion instant.

the hypothesis of the mechanical process being dynamic.

The topics developed in this chapter belong to multi-physics problems and consequently represent a great computational weight on the results. Again, further complexities arise in

In the almost static case the strain, in any instant of time, is in a situation of almost equilibrium with the loads; instead in the dynamic case the stress state is variable in space, therefore there are portions of the solid under stress against others in almost absence of stress. In other words the stress travels inside the solid as a stress-wave and it becomes a fundamental parameter for the description of the behaviour of the material. The dynamic processes in materials involve different scientific disciplines and areas, as materials science, shock physics/chemistry, mechanics combustion, applied mathematics and large scale

**6. Conclusions** 

computation.

The blast overpressure was generated from an instantaneous release of 50 m3 LPG rail tanker at 326K. The pressure-time curve was assumed to be of triangular shape, the duration of which was obtained from CONWEB reflected pressure diagram [33]. To calculate the decay of blast overpressure during the longitudinal direction of the tunnel the Energy Concentration Factor (ECF) method was used [34].

During the propagation of the blast wave over the first 75 m from the BLAVE to the tunnel opening, the blast overpressure falls from 1700 kPa (vapor pressure at 326 K) down to approximately 86 kPa. This decay is solely through the intense energy dissipation in the strong leading shock of the blast wave.

The 3-Dimensional model is representative of a tunnel section 300 meters long. This model was implemented by quadratic tetrahedral type elements [31] obtaining 95003 elements and 147528 nodes as shown in Figure 9.

**Figure 9.** Meshed model.

The analysis was carried out in two steps [35]. The first step obtained the initial stress state caused by soil load and fire and the second step analysed the dynamic response under blast loading. Consequently the following load conditions were considered in the FE analysis:


Therefore, on the base of this analysis, the distribution of the temperature inside of the structure is known. In Figure 10 the temperature distribution is showed.

Subsequently, the mechanical behaviour of the models was analysed introducing also to thermal stress, the explosion load. Figure 11 shows the deformation and the Mises stress of tunnel section, from meddle, where explosion is localized, to the tunnel opening.

Modeling Dynamics Behaviour of Materials: Theoretical Framework and Applications 213

**Figure 10.** Temperature distribution (°C) *t* = 1800 sec.

**Figure 11.** Mises stress (Pa) of the tunnel at the explosion instant.

## **6. Conclusions**

212 Finite Element Analysis – Applications in Mechanical Engineering

the HC curve was used to simulate the fire action.

Concentration Factor (ECF) method was used [34].

strong leading shock of the blast wave.

147528 nodes as shown in Figure 9.

**Figure 9.** Meshed model.

and to the fire thermal stress;

FE analysis:

on laboratory scale tunnel tests performed by TNO in 1979 [32]). In the considered model

The blast overpressure was generated from an instantaneous release of 50 m3 LPG rail tanker at 326K. The pressure-time curve was assumed to be of triangular shape, the duration of which was obtained from CONWEB reflected pressure diagram [33]. To calculate the decay of blast overpressure during the longitudinal direction of the tunnel the Energy

During the propagation of the blast wave over the first 75 m from the BLAVE to the tunnel opening, the blast overpressure falls from 1700 kPa (vapor pressure at 326 K) down to approximately 86 kPa. This decay is solely through the intense energy dissipation in the

The 3-Dimensional model is representative of a tunnel section 300 meters long. This model was implemented by quadratic tetrahedral type elements [31] obtaining 95003 elements and

The analysis was carried out in two steps [35]. The first step obtained the initial stress state caused by soil load and fire and the second step analysed the dynamic response under blast loading. Consequently the following load conditions were considered in the

1. from time *t* = 0 to time *t* = 120 min the tunnel was subjected to the surrounding soil load

Therefore, on the base of this analysis, the distribution of the temperature inside of the

Subsequently, the mechanical behaviour of the models was analysed introducing also to thermal stress, the explosion load. Figure 11 shows the deformation and the Mises stress of

2. at the instant *t* = 2 sec the structure was subjected to the blast over pressure.

tunnel section, from meddle, where explosion is localized, to the tunnel opening.

structure is known. In Figure 10 the temperature distribution is showed.

The topics developed in this chapter belong to multi-physics problems and consequently represent a great computational weight on the results. Again, further complexities arise in the hypothesis of the mechanical process being dynamic.

In the almost static case the strain, in any instant of time, is in a situation of almost equilibrium with the loads; instead in the dynamic case the stress state is variable in space, therefore there are portions of the solid under stress against others in almost absence of stress. In other words the stress travels inside the solid as a stress-wave and it becomes a fundamental parameter for the description of the behaviour of the material. The dynamic processes in materials involve different scientific disciplines and areas, as materials science, shock physics/chemistry, mechanics combustion, applied mathematics and large scale computation.

Certainly, in developing this approach we have had the opportunity to deepen analysis about strength of materials and structures, and damage and fracture at micro and macroscale.

Modeling Dynamics Behaviour of Materials: Theoretical Framework and Applications 215

[12] Lu Y., Wright P. Numerical approach of visco-elastoplastic analysis for asphalt

[13] Krishnan J., Rajagopal K. On the mechanical behavior of asphalt, Mechanics of

[15] Buonsanti M, Leonardi G, Scopelliti F. 3-D Simulation of shock waves generated by

[16] Timoshenko S, Woinowsky-Krieger S, Woinowsky S. Theory of plates and shells: McGraw-hill New York; 1959. C. Truesdell, Mechanics of solids: Springer Verlag, 1973. [17] Sadd M. Elasticity Theory, Applications, and Numerics. 2005. Burlington, MA: Elsevier

[18] Srinath LS. Advanced mechanics of solids: Tata McGraw-Hill Publishing Company

[19] Doghri I. Mechanics of deformable solids: linear, nonlinear, analytical, and

[20] Crozier DA, Sanjayan JG. Tests of load-bearing slender reinforced concrete walls in fire.

[21] O'meagher A, Bennetts I. Modelling of concrete walls in fire. Fire safety journal. 1991;

[22] Pesavento F, Gawin D, Majorana CE, Witek A, Schrefler B. Modelling of thermal damaging of concrete structures during fire. In: VII International Conference

[23] Schrefler B, Brunello P, Gawin D, Majorana C, Pesavento F. Concrete at high temperature with application to tunnel fire. Computational Mechanics. 2002; 29(1): 43-

[24] Gurney RW. The Initial Velocities of Fragments from Bombs, Shell and Grenades. DTIC

[25] Kennedy J, editor. Behavior and Utilization of Explosives in Engineering Design. 1972.

[29] Kuo CM, Hall KT, Darter MI. Three-dimensional finite element model for analysis of concrete pavement support. Transportation Research Record. 1995; 1505:119-27. [30] Committee E. Eurocode2: Design of concrete structures-Part 1-2: General rules-

[31] Forschungsgesellschaft für Straßen - und Verkehrswesen. Richtlinien für Ausstattung

[32] Instituut TNO voor Bouwmaterialen en Bouwconstructies. Rapport betreffende de beproeving van het gedrag van twee isolatiematerialenter bescherming van tunnels

[33] Hyde D. CONWEP, Conventional Weapons Effects Program. US Army Engineer

[14] Little R. W., Keer L. Elasticity. Journal of Applied Mechanics. 1974; 41: 7.

dense explosive in shell structures. Procedia Engineering. 2011; 10: 1550-5.

mixtures, Computers & Structures. 1998; 69: 139-147.

computational aspects: Springer Verlag; 2000.

ACI Structural Journal. 2000; 97(2).

Computational Plasticity, 2003.

[26] AIRBUS. Airplane Characteristics A321. 1995.

Structural fire design. ENV 1992-1-2, 1995.

und Betrieb von Tunneln (RABT). Ausgabe1985.

tegen brand. Rapport B-80-33. Delft, The Netherlands 1980.

Waterways Experiment Station, Vicksburg, MS. 1992.

[28] AAIB. AAIB Bulletin: 6/2009 EW/C2008/07/02. 2009.

[27] Huang Y. Pavement analysis and design: Prentice Hall; 1993.

materials. 2005; 37: 1085-1100.

Butterworth-Heinemann..

Limited; 2009.

17(4): 315-35.

Document, 1943.

51.

In the second section of the chapter, two numerical simulations relatively to one impact against air field pavements and one explosion in tunnel structures have been presented. Both simulations assume the problems as multi-field, and the results are quantitatively adequate.

## **Author details**

Giovanni Leonardi\* and Michele Buonsanti *Faculty of Engineering, University of Reggio Calabria, Department of Mechanics and Materials – MECMAT, Reggio Calabria, Italy* 

## **7. References**


<sup>\*</sup> Corresponding Author


and Michele Buonsanti

*Department of Mechanics and Materials – MECMAT, Reggio Calabria, Italy* 

[5] Graff K. F. Wave Motion in Elastic Solids, Dover Pbs. New York; 1991.

International Journal of Engineering Science. 1970; 8: 857-874.

[10] Meyers M. Dynamic behavior of materials. Wiley-Interscience; 1994.

*Faculty of Engineering, University of Reggio Calabria,* 

Vibrations, 5-9 July 2009, Kracòw, Poland; 2006.

Foundations Division. ASCE. 1968; 94: 951-979.

Springer-Verlag Berlin; 1984.

Press, Cambridge; 1988.

Physics. 1977; 49:523-579.

New York; 1972.

2011.

Corresponding Author

 \*

macroscale.

adequate.

**Author details** 

Giovanni Leonardi\*

**7. References** 

Certainly, in developing this approach we have had the opportunity to deepen analysis about strength of materials and structures, and damage and fracture at micro and

In the second section of the chapter, two numerical simulations relatively to one impact against air field pavements and one explosion in tunnel structures have been presented. Both simulations assume the problems as multi-field, and the results are quantitatively

[1] Buonsanti M., Cirianni F., Leonardi G. Study of the barriers for the mitigation of railway vibrations. In: ICSV16 Proceedings 16th International Conference of Sound and

[2] Truesdell C. Mechanics of Solids Waves in Elastic and Viscoelastic Solids, vol. IV,

[3] Hudson J. A. The Excitation and Propagation of Elastic Waves, Cambridge University

[4] Love A. E. H. A Treatise on the Mathematical Theory of Elasticity, IV Ed., Dover Pbs.

[6] Wood R. D. Diffraction of Transient Horizontal Shear Waves by a Finite Rigid Ribbon.

[7] Wood R. D., Pao Y.H. Diffractions and Horizontal Shear Waves by a Parabolic Cylinder and Dynamic Stress Concentration. Journal Applied Mechanics. 1966; 33: 785-792. [8] Woods R. D. Screening of Surface Waves in Soils, Journal of the Soil Mechanics and

[9] Buonsanti M., Cirianni F., Leonardi G., Scopelliti F. Impact on Granular Plate. In: De Roeck G., Degrande G., Lombaert G., Muller G. (eds.) Eurodyn2011: proceedings of the 8th International Conference on Structural Dynamics, EURODYN2011, Leuven, Belgium,

[11] Duvall G., Graham R. Phase transitions under shock-wave loading, Reviews of Modern

	- [34] Silvestrini M, Genova B, Leon Trujillo F. Energy concentration factor. A simple concept for the prediction of blast propagation in partially confined geometries. Journal of Loss Prevention in the Process Industries. 2009; 22(4): 449-54.

**Chapter 10** 

© 2012 Karayan et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Karayan et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Finite Element Analysis Applications** 

Dwi Marta Nurjaya, Ahmad Ashari and Homero Castaneda

The use of the finite element in engineering applications has grown rapidly in recent years. Finite element analysis (FEA) is based on numerical computation that calculates all parameters and boundaries given. Supported with powerful computer processors and continuous software development, the finite element method is rapidly advancing. The use of finite element analysis is not limited to the engineering field, as there are also medical and geospatial applications. The early development of the finite element can be traced back to the work of Courant [1], followed by the work of Martin [2], which applied the solutions for structural analyses at Boeing Company in the 1950s. Further work by Argyris, Clough, Turner, and Zienkiewicz developed the governing mathematical equation for the finite

The numerical simulation introduced in the 70s for the stress model on concrete as published by Hillerborg [4] is a clear example of the FEA concept. Huiskes *et al.* [5] also stated that the finite element has been used in a structural stress analysis of human bones for biomechanic applications. Zienkiewicz *et al.* [6] applied the finite element method to the linear and non-linear problems encountered during the analysis of a reactor vessel. Gallagher [7] studied brittle material design through use of the finite element method, which incorporated thermal and elastic analysis aspects of the overall design. Miller *et al.* [8] used the finite element method during the study of crack stability of a turbine blade and proposed a hypothesis based on material strength characteristics, plastic zone size/history,

In the mid 80s, Oritz *et al.* [9] proposed a method that aimed to enhance the performance of general classes of elements that undergo strain localization. An overview of the application of the finite element in machining from the 70-90s was well documented by Mackerle [10].

**in Failure Analysis: Case Studies** 

Ahmad Ivan Karayan, Deni Ferdian, Sri Harjanto,

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51024

**1. Introduction** 

element method [3].

and residual plastic strains.

[35] Buonsanti M, Leonardi G, 3-D Simulation of tunnel structures under Blast loading. Archive of Mechanics and Engineering, in press, 2012.

**Chapter 10** 

## **Finite Element Analysis Applications in Failure Analysis: Case Studies**

Ahmad Ivan Karayan, Deni Ferdian, Sri Harjanto, Dwi Marta Nurjaya, Ahmad Ashari and Homero Castaneda

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51024

## **1. Introduction**

216 Finite Element Analysis – Applications in Mechanical Engineering

Prevention in the Process Industries. 2009; 22(4): 449-54.

Archive of Mechanics and Engineering, in press, 2012.

[34] Silvestrini M, Genova B, Leon Trujillo F. Energy concentration factor. A simple concept for the prediction of blast propagation in partially confined geometries. Journal of Loss

[35] Buonsanti M, Leonardi G, 3-D Simulation of tunnel structures under Blast loading.

The use of the finite element in engineering applications has grown rapidly in recent years. Finite element analysis (FEA) is based on numerical computation that calculates all parameters and boundaries given. Supported with powerful computer processors and continuous software development, the finite element method is rapidly advancing. The use of finite element analysis is not limited to the engineering field, as there are also medical and geospatial applications. The early development of the finite element can be traced back to the work of Courant [1], followed by the work of Martin [2], which applied the solutions for structural analyses at Boeing Company in the 1950s. Further work by Argyris, Clough, Turner, and Zienkiewicz developed the governing mathematical equation for the finite element method [3].

The numerical simulation introduced in the 70s for the stress model on concrete as published by Hillerborg [4] is a clear example of the FEA concept. Huiskes *et al.* [5] also stated that the finite element has been used in a structural stress analysis of human bones for biomechanic applications. Zienkiewicz *et al.* [6] applied the finite element method to the linear and non-linear problems encountered during the analysis of a reactor vessel. Gallagher [7] studied brittle material design through use of the finite element method, which incorporated thermal and elastic analysis aspects of the overall design. Miller *et al.* [8] used the finite element method during the study of crack stability of a turbine blade and proposed a hypothesis based on material strength characteristics, plastic zone size/history, and residual plastic strains.

In the mid 80s, Oritz *et al.* [9] proposed a method that aimed to enhance the performance of general classes of elements that undergo strain localization. An overview of the application of the finite element in machining from the 70-90s was well documented by Mackerle [10].

© 2012 Karayan et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Karayan et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The application of the method to localize fractures was studied by Broberg [11], and further work was continued by Borst *et al.* [12]. Nowadays, finite element analysis is widely used by the engineering field in fluid dynamic and electrical applications.

Finite Element Analysis Applications in Failure Analysis: Case Studies 219

1970, was run on a massive mainframe computer that was less powerful than the personal computers of today [16]. The early software was only capable of simulating the 2D beam model, but it eventually progressed towards the modern 3D solid model of today as

The applications of finite element analysis in engineering failure analysis are in continuous evolution as more factors related to the failure event are taken into account. For example, in the case of turbine blades involved in jet engine failure, engineers should incorporate the thermal effect and load effect received by the blade in the simulation, not to mention other possibilities from foreign particles that could initiate the failure. Therefore, FEA calculations now incorporate a combination of multiple physical environments. Current research related to the application of the finite element in failure analysis has been published in peerreviewed journals, and can be seen in publications such as *Engineering Failure Analysis* 

In general, a finite element method consists of three phases: (1) pre-processing, where the analyst creates the finite element mesh and applies certain parameters or boundaries to the model, (2) solver/solution, where the program runs the governing mathematical equation that was created by the model, and (3) post-processing, where the result is evaluated and

Some limitations must be recognized in the finite element method. The ability to define and analyze the system and the model will determine the quality of the simulation results. The evaluations of any failure event need a comprehensive approach to reveal the root cause of the failure. FEA is a tool where multidisciplinary fields are combined to help find or support the solution with better accuracy for a final conclusion. Below are three case studies that

The use of FEA for mechanical failure has become a necessary tool that is easily obtained, as the commercial software is readily available for users. FEA is widely used in failure analysis assessment when analyzing and characterizing the quantitative and qualitative approaches used to determine the root cause of an event leading to a failure. The software is given historical background information related to the failure along with other boundary conditions. Below is an example of a mechanical failure analysis assessment that used finite

In the first case, the finite element method was used to analyze the stress distribution of a failed 28MW horizontal hydroturbine shaft [17]. The data corresponded to the fractography and metallographic observations. The finite element analysis was performed for normal conditions as well as the type of high load conditions that would be experienced during the

The shaft was constructed by the joining of a forged and cast part by slag welding. A crack was observed at the radius area of the casted part. The part was given an estimated 200,000

illustrate how finite element analysis can be utilized in failure analysis.

element analysis software to verify the findings regarding the failure event.

computer hardware developed.

validated for further interpretation.

**3. FEA in mechanical failure** 

start-up period.

*Journal*® and *Failure Analysis and Prevention Journal*®.

In general, the finite element analysis is widely used in the pre-production manufacturing process to determine the most cost-effective decision based on the analysis. Finite element simulations allow comparison between different "designs." Finite element analysis can simulate operational and environmental conditions and formulate modifications without creating a physical prototype.

In order to adequately determine the root cause of material failure, two outcomes are required: the answer and an explanation. The failure of a mechanical component is usually associated with materials, the environment, a third party, or human error. An investigation through metallurgical failure analysis is usually conducted to reveal the root cause and the failure mechanism. Work by Griffith [13] on fracture mechanics created a breakthrough in the understanding of the material fracture mechanism. In certain cases, a conventional failure analysis approach is not enough to reveal the failure, and therefore, a more comprehensive analysis through the finite element is needed. Prawoto *et al.* [14] explained that the use of the finite element is an effective approach when the causes of failure are determined using qualitative metallographic and fractographic testing. The finite element requires an understanding of how a component works and will support the correct information and data. The quality of the data provided is a key element for the successful outcome of the simulation.

## **2. The finite element in failure analysis**

The use of finite element analysis in mechanical applications has heavily increased in recent years. The continuous efforts to improve calculations and analyses so that models accurately incorporate actual conditions have been rewarded as a consequence of computer tool evolution. The finite element analysis has become an important tool for improving the design quality in numerous applications. The finite element analysis is a computer-based technique to solve problems using numerical solutions. The analysis includes a method based on creating a geometrical model of the structure that is divided into individual nodes or elements.

Finite element modeling provides different insights into the engineering analysis that cannot be obtained with the classical failure analysis method. Classical commercial software such as ABAQUS® and ANSYS® has been widely used to analyze failures or defects and reconstruct possible root causes. The images and animation produced by the software help to give a better understanding by visualizing the root cause behind the failure event. Previously, the development of the finite element evolved slowly due to a lack of tools to solve mathematical equations, and therefore, the method remained dormant until the computer era.

The early finite element software was commissioned by NASA in mid 1960s, which introduced NASTRAN® as an application that helped to design more efficient structures for vehicles that were developed by them [15]. The ANSYS® software, which was released in 1970, was run on a massive mainframe computer that was less powerful than the personal computers of today [16]. The early software was only capable of simulating the 2D beam model, but it eventually progressed towards the modern 3D solid model of today as computer hardware developed.

The applications of finite element analysis in engineering failure analysis are in continuous evolution as more factors related to the failure event are taken into account. For example, in the case of turbine blades involved in jet engine failure, engineers should incorporate the thermal effect and load effect received by the blade in the simulation, not to mention other possibilities from foreign particles that could initiate the failure. Therefore, FEA calculations now incorporate a combination of multiple physical environments. Current research related to the application of the finite element in failure analysis has been published in peerreviewed journals, and can be seen in publications such as *Engineering Failure Analysis Journal*® and *Failure Analysis and Prevention Journal*®.

In general, a finite element method consists of three phases: (1) pre-processing, where the analyst creates the finite element mesh and applies certain parameters or boundaries to the model, (2) solver/solution, where the program runs the governing mathematical equation that was created by the model, and (3) post-processing, where the result is evaluated and validated for further interpretation.

Some limitations must be recognized in the finite element method. The ability to define and analyze the system and the model will determine the quality of the simulation results. The evaluations of any failure event need a comprehensive approach to reveal the root cause of the failure. FEA is a tool where multidisciplinary fields are combined to help find or support the solution with better accuracy for a final conclusion. Below are three case studies that illustrate how finite element analysis can be utilized in failure analysis.

## **3. FEA in mechanical failure**

218 Finite Element Analysis – Applications in Mechanical Engineering

**2. The finite element in failure analysis** 

creating a physical prototype.

computer era.

the engineering field in fluid dynamic and electrical applications.

The application of the method to localize fractures was studied by Broberg [11], and further work was continued by Borst *et al.* [12]. Nowadays, finite element analysis is widely used by

In general, the finite element analysis is widely used in the pre-production manufacturing process to determine the most cost-effective decision based on the analysis. Finite element simulations allow comparison between different "designs." Finite element analysis can simulate operational and environmental conditions and formulate modifications without

In order to adequately determine the root cause of material failure, two outcomes are required: the answer and an explanation. The failure of a mechanical component is usually associated with materials, the environment, a third party, or human error. An investigation through metallurgical failure analysis is usually conducted to reveal the root cause and the failure mechanism. Work by Griffith [13] on fracture mechanics created a breakthrough in the understanding of the material fracture mechanism. In certain cases, a conventional failure analysis approach is not enough to reveal the failure, and therefore, a more comprehensive analysis through the finite element is needed. Prawoto *et al.* [14] explained that the use of the finite element is an effective approach when the causes of failure are determined using qualitative metallographic and fractographic testing. The finite element requires an understanding of how a component works and will support the correct information and data. The quality of the data provided is a key element for the successful outcome of the simulation.

The use of finite element analysis in mechanical applications has heavily increased in recent years. The continuous efforts to improve calculations and analyses so that models accurately incorporate actual conditions have been rewarded as a consequence of computer tool evolution. The finite element analysis has become an important tool for improving the design quality in numerous applications. The finite element analysis is a computer-based technique to solve problems using numerical solutions. The analysis includes a method based on creating a

Finite element modeling provides different insights into the engineering analysis that cannot be obtained with the classical failure analysis method. Classical commercial software such as ABAQUS® and ANSYS® has been widely used to analyze failures or defects and reconstruct possible root causes. The images and animation produced by the software help to give a better understanding by visualizing the root cause behind the failure event. Previously, the development of the finite element evolved slowly due to a lack of tools to solve mathematical equations, and therefore, the method remained dormant until the

The early finite element software was commissioned by NASA in mid 1960s, which introduced NASTRAN® as an application that helped to design more efficient structures for vehicles that were developed by them [15]. The ANSYS® software, which was released in

geometrical model of the structure that is divided into individual nodes or elements.

The use of FEA for mechanical failure has become a necessary tool that is easily obtained, as the commercial software is readily available for users. FEA is widely used in failure analysis assessment when analyzing and characterizing the quantitative and qualitative approaches used to determine the root cause of an event leading to a failure. The software is given historical background information related to the failure along with other boundary conditions. Below is an example of a mechanical failure analysis assessment that used finite element analysis software to verify the findings regarding the failure event.

In the first case, the finite element method was used to analyze the stress distribution of a failed 28MW horizontal hydroturbine shaft [17]. The data corresponded to the fractography and metallographic observations. The finite element analysis was performed for normal conditions as well as the type of high load conditions that would be experienced during the start-up period.

The shaft was constructed by the joining of a forged and cast part by slag welding. A crack was observed at the radius area of the casted part. The part was given an estimated 200,000

hours of operating life, and failure was detected after 163,411 hours in service. A visual examination of the fracture surface showed a fatigue pattern with obvious ratchet marks. Further observation of the fracture surface showed visible distorted fatigue lines around numerous gas holes and areas of increased porosity.

Finite Element Analysis Applications in Failure Analysis: Case Studies 221

**Leaks**

**Figure 1.** Big and small leak near the backing bar at about 6 o'clock position viewed from the inner side

**Backing Bar** 

**Backing Bar** 

**Big Leak Beneath the Strong Back** 

**Figure 2.** Big and small leak near the backing bar at about 6 o'clock position viewed from the inner side

**Small Leak Beneath the Strong Back** 

of seawater inlet pipe.

of seawater inlet pipe.

An original document indicated that the shaft was heat-treated to complete an austenization process. However, metallographic examination showed cast ferrite-pearlite with an undissolved dendrite structure, which might indicate that an improper heat treatment process occurred. In addition, a large non-metallic inclusion was also observed in the cast part.

A linear FEA was used to determine the stress state of the turbine shaft and shaft flange. The commercial ANSYS® software was used for the finite element modeling. The model represents a discrete continuum by an 8-node finite element with three degree of freedom that comprises 49,430 nodes and 47,547 elements. All of the boundary conditions were incorporated into the model.

A numerical calculation to determine the shaft flange stress states was performed for two characteristic load cases, where one load was taken from the manufacturer's documentation and the second load was the static load that occurs during start-up. Calculation of the finite element for both load conditions showed that the maximum stress was at the crack initiation site at the shaft flange.

The data collected from the chemical composition test and mechanical test showed that the material did not comply with the minimum standard required. Therefore, the crack location where the failure occurred was more susceptible to stress. Finite element analysis showed a high distribution of stress at the failed area. The finite element analysis revealed that the obtained tensile stress value on the shaft flange transition radius due to the load in case 2 was higher than recommended, and was characterized by the stress intensity factor at the crack tip being higher than the material threshold.

It was concluded that corrosion fatigue was the cause of the shaft failure [16]. The root cause for this case was improper corrosion protection at the failed area and a lack of periodical inspection, both of which were necessary due to the high stress on the region.

## **4. FEA in corrosion failure**

Karayan *et al.* [18] studied the failure of a seawater inlet pipe. The failure was first characterized by a small leak at approximately the 4-8 o'clock position. A schematic drawing of the inlet pipe showing the backing bar near the leak location is shown in Figure 1. This backing bar was installed on the welded surface. The visual examination of this failed pipe is shown in Figure 2. In order to find out the root cause of failure, a number of laboratory tests were performed. The results showed that the failure was caused by cavitation, as evident by the presence of a crater-like surface near the backing bar (Fig. 3). These localized craters seemed unusual since they were only noticed near the backing bar. An additional tool such as finite element analysis was used to determine why this was the case. A finite

numerous gas holes and areas of increased porosity.

crack tip being higher than the material threshold.

**4. FEA in corrosion failure** 

part.

incorporated into the model.

site at the shaft flange.

hours of operating life, and failure was detected after 163,411 hours in service. A visual examination of the fracture surface showed a fatigue pattern with obvious ratchet marks. Further observation of the fracture surface showed visible distorted fatigue lines around

An original document indicated that the shaft was heat-treated to complete an austenization process. However, metallographic examination showed cast ferrite-pearlite with an undissolved dendrite structure, which might indicate that an improper heat treatment process occurred. In addition, a large non-metallic inclusion was also observed in the cast

A linear FEA was used to determine the stress state of the turbine shaft and shaft flange. The commercial ANSYS® software was used for the finite element modeling. The model represents a discrete continuum by an 8-node finite element with three degree of freedom that comprises 49,430 nodes and 47,547 elements. All of the boundary conditions were

A numerical calculation to determine the shaft flange stress states was performed for two characteristic load cases, where one load was taken from the manufacturer's documentation and the second load was the static load that occurs during start-up. Calculation of the finite element for both load conditions showed that the maximum stress was at the crack initiation

The data collected from the chemical composition test and mechanical test showed that the material did not comply with the minimum standard required. Therefore, the crack location where the failure occurred was more susceptible to stress. Finite element analysis showed a high distribution of stress at the failed area. The finite element analysis revealed that the obtained tensile stress value on the shaft flange transition radius due to the load in case 2 was higher than recommended, and was characterized by the stress intensity factor at the

It was concluded that corrosion fatigue was the cause of the shaft failure [16]. The root cause for this case was improper corrosion protection at the failed area and a lack of periodical

Karayan *et al.* [18] studied the failure of a seawater inlet pipe. The failure was first characterized by a small leak at approximately the 4-8 o'clock position. A schematic drawing of the inlet pipe showing the backing bar near the leak location is shown in Figure 1. This backing bar was installed on the welded surface. The visual examination of this failed pipe is shown in Figure 2. In order to find out the root cause of failure, a number of laboratory tests were performed. The results showed that the failure was caused by cavitation, as evident by the presence of a crater-like surface near the backing bar (Fig. 3). These localized craters seemed unusual since they were only noticed near the backing bar. An additional tool such as finite element analysis was used to determine why this was the case. A finite

inspection, both of which were necessary due to the high stress on the region.

**Figure 1.** Big and small leak near the backing bar at about 6 o'clock position viewed from the inner side of seawater inlet pipe.

**Figure 2.** Big and small leak near the backing bar at about 6 o'clock position viewed from the inner side of seawater inlet pipe.

Finite Element Analysis Applications in Failure Analysis: Case Studies 223

**Figure 4.** Finite element analysis of the inner pipe showing the orifice effect and the eddy zone near the

The last case is a trunkline that burst during service [19]. This incident produced a significant impact on the gas production of the company and also on the environment. The failure was characterized by a mesa-like attack and wall thinning at the 5-7 o'clock position on the inner surface. The location of failure is shown in Figure 5. A reddish brown corrosion product was noticed on the surface, but there with no indication of the occurrence of an H2S attack found in the material. This trunkline, carrying the gas with a total pressure of 905 psi, spanned across the jungle in the descending position. The analysis of the gas composition is listed in Table 1. The topography and characteristics of the soil in which the trunkline was

backing bar correlated with the actual leaks on the inlet pipe.

located and also the material specification data is listed in Table 2.

**Figure 3.** Surface morphologies of brown crater – like surface taken from 6 – 3 o'clock position.

element analysis was executed based on the pipe dimension and actual fluid conditions such as velocity, pressure, temperature, and implicit parameters. Because there were no data for the initial height of the unwanted backing bar, the authors assumed that the initial height was the highest backing bar found on the specimen. Interestingly, the failure location predicted by the finite element analysis matched up with the actual evidence (Fig. 4). It precisely showed that the failure could be located around the backing bar where the eddy zone was formed in this area. This suggests that sometimes the results obtained from laboratory tests cannot explain why a failure occurs, although evidence indicates the existence of a certain problem. In this case, a finite element analysis is the only tool that can help a failure analyst find the root cause of failure. As can be seen from Figure 4, the leaks and crater-like surfaces found in the area near the backing bar were attributed to the formation of eddy zones in this area. The length of the eddy zone predicted that the area that might suffer from a flow-induced attack.

**Figure 3.** Surface morphologies of brown crater – like surface taken from 6 – 3 o'clock position.

that might suffer from a flow-induced attack.

element analysis was executed based on the pipe dimension and actual fluid conditions such as velocity, pressure, temperature, and implicit parameters. Because there were no data for the initial height of the unwanted backing bar, the authors assumed that the initial height was the highest backing bar found on the specimen. Interestingly, the failure location predicted by the finite element analysis matched up with the actual evidence (Fig. 4). It precisely showed that the failure could be located around the backing bar where the eddy zone was formed in this area. This suggests that sometimes the results obtained from laboratory tests cannot explain why a failure occurs, although evidence indicates the existence of a certain problem. In this case, a finite element analysis is the only tool that can help a failure analyst find the root cause of failure. As can be seen from Figure 4, the leaks and crater-like surfaces found in the area near the backing bar were attributed to the formation of eddy zones in this area. The length of the eddy zone predicted that the area

**Figure 4.** Finite element analysis of the inner pipe showing the orifice effect and the eddy zone near the backing bar correlated with the actual leaks on the inlet pipe.

The last case is a trunkline that burst during service [19]. This incident produced a significant impact on the gas production of the company and also on the environment. The failure was characterized by a mesa-like attack and wall thinning at the 5-7 o'clock position on the inner surface. The location of failure is shown in Figure 5. A reddish brown corrosion product was noticed on the surface, but there with no indication of the occurrence of an H2S attack found in the material. This trunkline, carrying the gas with a total pressure of 905 psi, spanned across the jungle in the descending position. The analysis of the gas composition is listed in Table 1. The topography and characteristics of the soil in which the trunkline was located and also the material specification data is listed in Table 2.

Finite Element Analysis Applications in Failure Analysis: Case Studies 225

Item Description Result

Q ± 3 MM WC -

GOR -

P (Psi) 750

T (F) 140

H2S 0.00 CO2 (%) 1.9406

SRB content -

Chloride (%) 0.7635

Water (%) 0.0898

trestle Laydown

Uniform/localized/pitting/etc pitting/localized

external/internal/both Internal

On 6 o'clock/others 6 o'clock

Elevation Descending Any river/road crossing No

Seam position -

On seam/not -

Soil pH -

Any trees/bushes Yes

WT inspection not yet

Document Work over on wells using the line No job in last one year

A visual examination showed that the burst area was located at the 6 o'clock position of the (a) downstream part and the (b) upstream part (Fig. 6). Some points on the inner surface of the downstream part at the 5-7 o'clock position showed surface degradation with wall thinning and pits (Fig. 7). Some points on the inner surface of the upstream part at the 5-7 o'clock position also showed surface degradation with wall thinning and pits (Fig. 8). A

uniform attack at the burst area from the 5-7 o'clock position is shown in Figure 9.

Laydown/buried/support w/ith

Property

Fluid Composition

Pipe Arrangement

> Corrosion Form

Environment

**Table 2.** Fluid properties and failed trunkline condition.

**Figure 5.** The specimen sent to the laboratory for analysis is indicated by arrow


**Table 1.** Result of gas analysis in trunkline



**Figure 5.** The specimen sent to the laboratory for analysis is indicated by arrow

Hydrogen Oxygen Nitrogen Carbon Dioxide Hydrogen Sulfide

Methane Ethane Propane Iso-Butane n-Butane Iso-Pentane n-Pentane Hexanes Heptanes Octanes Nonanes Decanes

Undecanes plus

**Table 1.** Result of gas analysis in trunkline

H2 O2 N2 CO2 H2S C1 C2 C3 iC4 nC4 iC5 C5 C6 C7 C8 C9 C10 C11+

Component Mole %

0.0000 0.0084 3.6036 1.9406 0.0000 82.7838 6.7222 3.0077 0.6144 0.6610 0.2384 0.1418 0.1352 0.0982 0.0306 0.0072 0.0069 0.0000

A visual examination showed that the burst area was located at the 6 o'clock position of the (a) downstream part and the (b) upstream part (Fig. 6). Some points on the inner surface of the downstream part at the 5-7 o'clock position showed surface degradation with wall thinning and pits (Fig. 7). Some points on the inner surface of the upstream part at the 5-7 o'clock position also showed surface degradation with wall thinning and pits (Fig. 8). A uniform attack at the burst area from the 5-7 o'clock position is shown in Figure 9.

Finite Element Analysis Applications in Failure Analysis: Case Studies 227

**Figure 7.** Wall thinning and pits at 5 – 7 o'clock position of downstream part.

**Figure 8.** Wall thinning and pits at 5 – 7 o'clock position of upstream part.

**Figure 9.** Uniform attack at the burst area from 5-7 o'clock position.

**Figure 6.** (a) Downstream part and (b) upstream part showing the burst located at 6 o'clock position.

Finite Element Analysis Applications in Failure Analysis: Case Studies 227

**Figure 7.** Wall thinning and pits at 5 – 7 o'clock position of downstream part.

226 Finite Element Analysis – Applications in Mechanical Engineering

(a)

(b)

**Figure 6.** (a) Downstream part and (b) upstream part showing the burst located at 6 o'clock position.

**Figure 8.** Wall thinning and pits at 5 – 7 o'clock position of upstream part.

**Figure 9.** Uniform attack at the burst area from 5-7 o'clock position.

The metallographic preparation and macroetching were performed on the perimeter of the trunkline, and the results showed that the trunkline was made of seamless pipe (Figs. 10 and 11). The thinning area shown in Figure 9 was an ERW (electric resistance welding)-free area (Fig. 10). This information indicates that the failure could not be attributed to the ERW pipe issue.

Finite Element Analysis Applications in Failure Analysis: Case Studies 229

**Figure 12.** Microstructure of trunkline showing a typical seamless microstructure with equiaxed grains.

**P (%)** 

0.030 (max) < 0.003

**S (%)** 

0.030 (max) < 0.003

524 517 (min)

**Ti (%)** 

0.040 < 0.002

**Mn (%)** 

1.40 0.594

**Material** *Average HRB Approxmate UTS Based on Conversion (MPa)*

The finite element analysis was executed around the overfill, and the results (Fig. 13) showed that the area at which the pipe burst had a dead or eddy zone due to the excessive overfill (Figs. 14 and 15). Because there was no information about the initial overfill height, we assumed an overfill height of 1 cm for the finite element simulation. This selection was based on the fact that there was one point at the remaining overfill that had a height of 1 cm. The other areas were mostly degraded, with heights of less than 0.5 cm. Severe corrosion

The dark phase is pearlite and the light one is ferrite.

**Material <sup>C</sup>**

**Table 3.** Chemical composition of trunkline

was noticed on every peak of overfill.

API 5L X60 PSL 1 [20] Failed Trunkline

Failed trunkline API 5L X60 [20]

**(%)** 

0.22 0.165

> 82 -

**Table 4.** Converted tensile strength in comparison with API 5LX60 specification.

**Figure 10.** Results of macroetching showing the absence of ERW.

**Figure 11.** Macrograph of 5-7 o'clock position showing the absence of ERW.

The microstructure of the trunkline was taken from the cross-section, and the results are displayed in Figure 12. As can be seen, the trunkline is composed of ferrite (light phase) and pearlite (dark phase) with equiaxed grains, which is typical of seamless pipe. The chemical composition of the trunkline was tested using an optical emission spectrometer and the results (Table 3) show that this trunkline is composed of an API 5L X60 steel [20]. Due to the insufficient geometry of the trunkline, the mechanical property of the trunkline was only examined by hardness testing. In order to verify that the specification of this material was API 5L, the resultant hardness values were converted to tensile strength (Table 4).

**Figure 12.** Microstructure of trunkline showing a typical seamless microstructure with equiaxed grains. The dark phase is pearlite and the light one is ferrite.


**Table 3.** Chemical composition of trunkline

228 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 10.** Results of macroetching showing the absence of ERW.

**Figure 11.** Macrograph of 5-7 o'clock position showing the absence of ERW.

The microstructure of the trunkline was taken from the cross-section, and the results are displayed in Figure 12. As can be seen, the trunkline is composed of ferrite (light phase) and pearlite (dark phase) with equiaxed grains, which is typical of seamless pipe. The chemical composition of the trunkline was tested using an optical emission spectrometer and the results (Table 3) show that this trunkline is composed of an API 5L X60 steel [20]. Due to the insufficient geometry of the trunkline, the mechanical property of the trunkline was only examined by hardness testing. In order to verify that the specification of this material was API 5L, the resultant hardness values were converted to tensile strength

issue.

(Table 4).

The metallographic preparation and macroetching were performed on the perimeter of the trunkline, and the results showed that the trunkline was made of seamless pipe (Figs. 10 and 11). The thinning area shown in Figure 9 was an ERW (electric resistance welding)-free area (Fig. 10). This information indicates that the failure could not be attributed to the ERW pipe


**Table 4.** Converted tensile strength in comparison with API 5LX60 specification.

The finite element analysis was executed around the overfill, and the results (Fig. 13) showed that the area at which the pipe burst had a dead or eddy zone due to the excessive overfill (Figs. 14 and 15). Because there was no information about the initial overfill height, we assumed an overfill height of 1 cm for the finite element simulation. This selection was based on the fact that there was one point at the remaining overfill that had a height of 1 cm. The other areas were mostly degraded, with heights of less than 0.5 cm. Severe corrosion was noticed on every peak of overfill.

Finite Element Analysis Applications in Failure Analysis: Case Studies 231

**Figure 14.** The inner side of circle area in Fig. 11 showing an excessive overfill (arrow) in the failed pipe

**Figure 15.** An excessive overfill and severe surface degradation beside overfill at 6 o'clock position.

The sweet environment of this system with a CO2 partial pressure of 17.56 psig may influence the presence of corrosion. The relationship between corrosion tendency and CO2 partial pressure in the sweet environment with pH 7 or less has been reported elsewhere [22]. When the CO2 partial pressure is less than 7 psig, the system is non-corrosive. When it

at 12 o'clock position.

**Figure 13.** Finite element analysis showing the eddy zone due to overfill in comparison with the actual failed area of pipe.

As can be seen in Table 1, the only corrosive gas in the fluids flowing in this failed trunkline was CO2. Carbon dioxide systems are one of the most common environments in the oil and gas field industry where corrosion occurs. In a relatively slow reaction, carbon dioxide forms a weak acid known as carbonic acid (H2CO3) in water, but the corrosion rate of CO2 is greater than that of carbonic acid. Cathodic depolarization may occur, and other attack mechanisms may also occur. The presence of salts is relatively unimportant in sweet (CO2) service [21], and thus, the presence of chloride in this system (0.7635%) did not significantly contribute to the failure.

**Figure 13.** Finite element analysis showing the eddy zone due to overfill in comparison with the actual

As can be seen in Table 1, the only corrosive gas in the fluids flowing in this failed trunkline was CO2. Carbon dioxide systems are one of the most common environments in the oil and gas field industry where corrosion occurs. In a relatively slow reaction, carbon dioxide forms a weak acid known as carbonic acid (H2CO3) in water, but the corrosion rate of CO2 is greater than that of carbonic acid. Cathodic depolarization may occur, and other attack mechanisms may also occur. The presence of salts is relatively unimportant in sweet (CO2) service [21], and thus, the presence of chloride in this system (0.7635%) did not significantly

failed area of pipe.

contribute to the failure.

**Figure 14.** The inner side of circle area in Fig. 11 showing an excessive overfill (arrow) in the failed pipe at 12 o'clock position.

**Figure 15.** An excessive overfill and severe surface degradation beside overfill at 6 o'clock position.

The sweet environment of this system with a CO2 partial pressure of 17.56 psig may influence the presence of corrosion. The relationship between corrosion tendency and CO2 partial pressure in the sweet environment with pH 7 or less has been reported elsewhere [22]. When the CO2 partial pressure is less than 7 psig, the system is non-corrosive. When it

is somewhere between 7-30 psig, corrosion in the system may be present. Lastly, when it is higher than 30 psig, the system is corrosive [22]. Temperature and flow regime are closely linked because CO2 corrosion is dynamic and very sensitive to electrochemical and physical imbalances (especially fluctuating pressure, temperature, and volume). Steady-state (P,T,V,) conditions tend to promote protective film compaction, and therefore, passivation and low corrosion rates. Lower temperatures <120°F (approximately 50°C) tend to promote patchy corrosion with softer multi-layered iron carbonate (siderite) scales that provide some barrier protection up to 140-160°F (60-70°C). Above these temperatures, damaging localized corrosion is observed as films lose stability and spall off, giving rise to galvanic mesa attack [23]. The failed pipe we studied with an operating temperature of 140°F might have formed a protective film. However, the phenomenon of film removal and its effect on the failure of this trunkline was not evident in our laboratory test. Our finite element analysis in the failed area in Figure 13 shows the unstable and chaotic flow in the failed area (called the dead or eddy zone). As illustrated, the eddy zone was triggered by the excessive overfill. This suggests that the pipe surface in this eddy area was severely attacked by flow. The presence of a passive carbonate layer could not protect the surface from this type of flow-induced attack that led to a mesa attack.

Finite Element Analysis Applications in Failure Analysis: Case Studies 233

[2] H. C. Martin, Large Deflection and Stability Analysis by Direct Stiffness Method, JPL Technical Report No. 32-931, California Institute of Technology, August 1966. [3] R. W. Clough, Early History Of The Finite Element Method From The View Point of a Pioneer, International Journal For Numerical Methods In Engineering, 60, 2004, pp.

[4] A. Hillerborg, M. Modéer, P.-E. Petersson, *Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements*, Cement and Concrete

[5] R. Huiskes , E.Y.S. Chao, *A survey of finite element analysis in orthopedic biomechanics: The* 

[6] O.C. Zienkiewicz, D.R.J. Owen, D.V. Phillips, G.C. Nayak , Finite element methods in the analysis of reactor vessels, Nuclear Engineering and Design, 20(2) 1972, pp. 507–

[7] R.H. Gallagher, Finite element analysis in brittle material design, Journal of the Franklin

[8] R.E. Miller Jr., B.F. Backman, H.B. Hansteen, C.M. Lewis, R.A. Samuel, S.R. Varanasi, Recent advances in computerized aerospace structural analysis and design Computers

[9] M.Ortiz, Y.Leroy, A.Needleman, *A finite element method for localized failure analysis, Computer Methods in Applied Mechanics and Engineering*, Vol 61(2) 1987, pp. 189-214. [10] J. Mackerle, *Finite-element analysis and simulation of machining: a bibliography (1976–1996)*, Journal of Materials Processing Technology, Vol 86, Issues 1-3, 1998, pp. 17-44. [11] K.B. Broberg, *The foundations of fracture mechanics*, Engineering Fracture

[12] Borst, R. De; Sluys, L.J. Muhlhaus, H.-B. Pamin, J. *Fundamental Issues In Finite Element Analyses Of Localization Of Deformation Engineering Computations*, Int J for Computer-

[14] Y.Prawoto, *Quantitative failure analysis using a simple finite element approach*. Journal of

[17] D. Momcilovic, Z. Odanovic, R.Mitrovic, I.Atanasovska, T.Vuherer, Failure analysis of

[18] A.I. Karayan, A. Hersuni, D. Adisty, A. Yatim, Failure analysis of seawater inlet pipe,

[19] A.I. Karayan, Failure Analysis of Trunkline: An Internal Report, Center for Materials

[22] Garverick L, Corrosion in the petrochemical industry, pp. 92, ASM International, 1994.

hydraulic turbine shaft, Engineering Failure Analysis, 20 (2012) pp. 54–66.

Journal of Failure analysis and Prevention, 11(2011) pp. 481-486.

*first decade*, Journal of Biomechanics, Vol 16(6) 1983, pp. 385-409.

283–287.

541.

Research,Vol 6(6), 1976, pp. 773-781.

Institute, 290(6) 1970, pp. 523–537.

& Structures, 7(2) 1977, pp. 315–326.

Mechanics,Vol16(4) 1982, pp. 497-515.

[15] http://en.wikipedia.org/wiki/Nastran

Aided Engineering, Vol 10(2), 1993 , pp. 99-121(23). [13] Griffith: Phil Trans Roy Soc 1921, v221, pp. 163-198.

Failure Analysis and Prevention, 10 (1) 2010. pp. 8-10.

[16] http://www.odonnellconsulting.com/forensicfea.html

Processing and Failure Analysis (CMPFA), 2011. [20] American Petroleum Institute (API) 5L Standard.

[21] http://octane.nmt.edu/WaterQuality/corrosion/CO2.aspx

## **5. Conclusions**

The three case studies discussed in this chapter have clearly shown us that finite element analysis (FEA) is an excellent and powerful tool that can be employed in failure analysis. Finite element analysis provides a failure analyst with more quantitative and qualitative information about the causes of failure. Although visual examinations and laboratory tests may not be able to determine a failure mechanism, the results of finite element analysis will support all data obtained from these tests. As long as a competent analyst running the finite element analysis is given sufficient data and has good knowledge of the system under study, the results of FEA will be reliable, although they should always be validated with experimental or real condition information.

## **Author details**

Ahmad Ivan Karayan and Homero Castaneda *Department of Chemical and Biomolecular Engineering, The University of Akron, USA* 

Deni Ferdian, Sri Harjanto, Dwi Marta Nurjaya and Ahmad Ashari *Department of Metallurgy and Materials Engineering, University of Indonesia, Indonesia* 

Deni Ferdian, Sri Harjanto, Dwi Marta Nurjaya and Ahmad Ashari *Center for Materials Processing and Failure Analysis (CMPFA), University of Indonesia, Indonesia* 

## **6. References**

[1] F. Williamson, *Richard Courant and the finite element method: A further look,* Historia Mathematica, Vol 7(4), 1980, pp. 369-378.


attack that led to a mesa attack.

experimental or real condition information.

Ahmad Ivan Karayan and Homero Castaneda

Mathematica, Vol 7(4), 1980, pp. 369-378.

**5. Conclusions** 

**Author details** 

**6. References** 

is somewhere between 7-30 psig, corrosion in the system may be present. Lastly, when it is higher than 30 psig, the system is corrosive [22]. Temperature and flow regime are closely linked because CO2 corrosion is dynamic and very sensitive to electrochemical and physical imbalances (especially fluctuating pressure, temperature, and volume). Steady-state (P,T,V,) conditions tend to promote protective film compaction, and therefore, passivation and low corrosion rates. Lower temperatures <120°F (approximately 50°C) tend to promote patchy corrosion with softer multi-layered iron carbonate (siderite) scales that provide some barrier protection up to 140-160°F (60-70°C). Above these temperatures, damaging localized corrosion is observed as films lose stability and spall off, giving rise to galvanic mesa attack [23]. The failed pipe we studied with an operating temperature of 140°F might have formed a protective film. However, the phenomenon of film removal and its effect on the failure of this trunkline was not evident in our laboratory test. Our finite element analysis in the failed area in Figure 13 shows the unstable and chaotic flow in the failed area (called the dead or eddy zone). As illustrated, the eddy zone was triggered by the excessive overfill. This suggests that the pipe surface in this eddy area was severely attacked by flow. The presence of a passive carbonate layer could not protect the surface from this type of flow-induced

The three case studies discussed in this chapter have clearly shown us that finite element analysis (FEA) is an excellent and powerful tool that can be employed in failure analysis. Finite element analysis provides a failure analyst with more quantitative and qualitative information about the causes of failure. Although visual examinations and laboratory tests may not be able to determine a failure mechanism, the results of finite element analysis will support all data obtained from these tests. As long as a competent analyst running the finite element analysis is given sufficient data and has good knowledge of the system under study, the results of FEA will be reliable, although they should always be validated with

*Department of Chemical and Biomolecular Engineering, The University of Akron, USA* 

*Department of Metallurgy and Materials Engineering, University of Indonesia, Indonesia* 

*Center for Materials Processing and Failure Analysis (CMPFA), University of Indonesia, Indonesia* 

[1] F. Williamson, *Richard Courant and the finite element method: A further look,* Historia

Deni Ferdian, Sri Harjanto, Dwi Marta Nurjaya and Ahmad Ashari

Deni Ferdian, Sri Harjanto, Dwi Marta Nurjaya and Ahmad Ashari

	- [23] Sing B, Krishnathasan K. Pragmatic effects of flow on corrosion prediction, NACE corrosion confeence and expo 2009, Paper No. 09275.

**Section 4** 

**Applications of FEA in** 

**"Machine Elements Analysis and Design"** 

## **Applications of FEA in "Machine Elements Analysis and Design"**

234 Finite Element Analysis – Applications in Mechanical Engineering

corrosion confeence and expo 2009, Paper No. 09275.

[23] Sing B, Krishnathasan K. Pragmatic effects of flow on corrosion prediction, NACE

**Chapter 11** 

© 2012 Darji and Vakharia, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Darji and Vakharia, licensee InTech. This is a paper distributed under the terms of the Creative Commons

**Development of Graphical Solution to** 

**Hollow Cylindrical Roller Bearing Using** 

Technological progress creates increasingly arduous conditions for rolling mechanisms. Advances in many fields including gas turbine design, aeronautics, space and atomic power, involve extreme operating speeds, load, temperatures, environments which increases power and load on machinery and demand high strength to weight ratio of the rolling element bearings. Also bearing stiffness is an important parameter in the designing. Bearing design calculations require a good understanding of the Hertzian contact stress due to which high stress concentration is produced which greatly influence the fatigue life and dominate the upper speed limits as in the case of solid rolling elements. Since being originally introduced, cylindrical rolling element bearings have been significantly improved, in terms of their performance and working life. A major objective has been to decrease the Hertz contact stresses at the roller–raceway interfaces, because these are the most heavily stressed areas in a bearing. It has been shown that bearing life is inversely proportional to the stress raised to the ninth power (even higher). Whereas making the rollers hollow which are flexible

enough reduces stress concentration and finally increase the fatigue life of bearing.

Investigators have proposed that under large normal loads a hollow element with a sufficiently thin wall thickness will deflect appreciably more than a solid element of the same size. An improvement in load distribution and thus load capacity may be realized, as well as contact stress is also reduced considerably by using a bearing with hollow rollers. Since for hollow roller bearing no method is available for the calculation of hollowness, contact stresses and deformation. The contact stresses in hollow members are often calculated by using the same equations and procedures as for solid specimens. This

**Determine Optimum Hollowness of** 

**Elastic Finite Element Analysis** 

Additional information is available at the end of the chapter

P.H. Darji and D.P. Vakharia

http://dx.doi.org/10.5772/46160

approach seems to be incorrect.

**1. Introduction** 
