**Meet the editor**

Dr. Charles Attah Osheku is the current director of the Centre for Space Transport and Propulsion, Epe, Lagos, an activity centre of the National Space Research and Development Agency, Nigeria. He obtained his BSc degree in Mechanical Engineering from the Obafemi Awolowo University and his MSc and PhD degrees in Mechanical Engineering (Engineering Mechanics) from

the University of Lagos. Dr. Osheku is a research expert in the fields of rocket and missile solid fuel physics; solid rocket engine design; structural-, flow- and combustion-induced vibrations; aircraft noise modelling, and dissipation and vibration of laminated structures. Dr. Osheku has 66 scientific papers to his credit, published in reputable international journals and ASME International Offshore Mechanics Conferences. He is a registered engineer and a professional member of ASME and AIAA.

Contents

**Preface VII**

**Applications 3** Charles Attah Osheku

**Section 2 Theory of Lamination 17**

Chapter 1 **Introductory Chapter: Laminations - Theory and**

**and Toughening Mechanisms 19**

Chapter 3 **Bending of Laminated Composite Plates in**

**Layerwise Theory 51** Marina Rakočević

**Section 3 Application of Lamination 71**

Dionysios E. Mouzakis

**Plastic Film 73**

Chapter 2 **Multiscale Hierarchical Structure and Laminated Strengthening**

Baoxi Liu, Lujun Huang, Lin Geng and Fuxing Yin

Chapter 4 **Dynamic Modeling of a Serial Link Robot Laminated with**

Norimitsu Sakagami and Mizuho Shibata

Chapter 5 **Advanced Technologies in Manufacturing 3D-Layered Structures for Defense and Aerospace 89**

**Section 1 Introduction 1**

## Contents

**Preface XI**


Preface

The field of lamination has developed significantly over the past 5000 years. Nowadays, we have a humongous array of structures and technological systems where composite lami‐ nates are applied. These range from aviation, aerospace/space to civil and marine structures. From the viewpoint of structural mechanics, an interface slip motion between two laminated structures, such as beam plate and plate in the presence of dry friction, can be utilized for slip damping systems. By scientific definition, slip damping is a mechanism exploited for dissipating noise and vibration energy in machine structures and systems. Researchers have developed several mathematical models for noise dissipation, minimization and complete vibration isolation laminated mechanisms. The purpose of this book is to describe new con‐ cepts of producing laminated structures and possible modern engineering applications.

The introductory chapter of this book offers new engine design concepts on how laminated enclosures can be useful in aerospace, machine and ocean structures. Chapter 2 presents re‐ search results on how higher strength can be achieved in titanium matrix composites by ad‐ justing the multi-scale hierarchical structure. In Chapter 3, the proposed higher-order zigzag theory is expected to ease the determination of deflections and stresses in composite sand‐ wich laminate analysis. In Chapter 4, the dynamic modelling of a serial link robot laminated with plastic film to improve waterproofing and dust-proofing of serial link robots is the main focus. Chapter 5 emphasizes the importance of additive 3D manufacturing techniques

A number of technological applications are possible in the areas of machine structures, pow‐ er plants, ship turbines, military jets and commercial aircrafts with suitable laminated mate‐ rials in conjunction with intelligent manufacturing systems to produce high-tech devices/ systems that are more comfortable for use by humans especially in the modern-day world

> **Charles A. Osheku, PhD** Director/Chief Executive Officer

Centre for Space Transport and Propulsion

Federal Ministry of Science and Technology LASU, Epe Campus, Epe, Lagos, Nigeria

National Space Research and Development Agency

and their robust applications in aeronautical and defence structural systems.

with advanced 3D manufacturing techniques.

## Preface

The field of lamination has developed significantly over the past 5000 years. Nowadays, we have a humongous array of structures and technological systems where composite lami‐ nates are applied. These range from aviation, aerospace/space to civil and marine structures.

From the viewpoint of structural mechanics, an interface slip motion between two laminated structures, such as beam plate and plate in the presence of dry friction, can be utilized for slip damping systems. By scientific definition, slip damping is a mechanism exploited for dissipating noise and vibration energy in machine structures and systems. Researchers have developed several mathematical models for noise dissipation, minimization and complete vibration isolation laminated mechanisms. The purpose of this book is to describe new con‐ cepts of producing laminated structures and possible modern engineering applications.

The introductory chapter of this book offers new engine design concepts on how laminated enclosures can be useful in aerospace, machine and ocean structures. Chapter 2 presents re‐ search results on how higher strength can be achieved in titanium matrix composites by ad‐ justing the multi-scale hierarchical structure. In Chapter 3, the proposed higher-order zigzag theory is expected to ease the determination of deflections and stresses in composite sand‐ wich laminate analysis. In Chapter 4, the dynamic modelling of a serial link robot laminated with plastic film to improve waterproofing and dust-proofing of serial link robots is the main focus. Chapter 5 emphasizes the importance of additive 3D manufacturing techniques and their robust applications in aeronautical and defence structural systems.

A number of technological applications are possible in the areas of machine structures, pow‐ er plants, ship turbines, military jets and commercial aircrafts with suitable laminated mate‐ rials in conjunction with intelligent manufacturing systems to produce high-tech devices/ systems that are more comfortable for use by humans especially in the modern-day world with advanced 3D manufacturing techniques.

#### **Charles A. Osheku, PhD**

Director/Chief Executive Officer Centre for Space Transport and Propulsion National Space Research and Development Agency Federal Ministry of Science and Technology LASU, Epe Campus, Epe, Lagos, Nigeria

**Section 1**

**Introduction**

**Section 1**

## **Introduction**

**Chapter 1**

Provisional chapter

**Introductory Chapter: Laminations - Theory and**

DOI: 10.5772/intechopen.74457

Historically, the earliest use of laminated materials was by the Mesopotamians around 3400 BC. Similarly, plywood laminates from wood strips glued together at different angles were constructed for structural applications. Nevertheless, for several centuries, primitive lamination methods were used for the construction of bows and canoes. Specifically, the invention of Bakelite in 1907 heralded a new era for composite and laminated material applications. For instance, at the inception of the aircraft by the Orville and Wilbur Wright brothers, it was made with wood and fabrics. The need for better aerodynamic performance and efficiency supported by advances in aerospace engineering led to the use of lighter materials with strength that matches that of steel. The development of this field has progressed significantly to the point of the ubiquitous availability of laminates in materials for aviation, civil structures and automotive uses. In particular, carbon fiber reinforced plastics are in use in modern aircraft aerospace systems. In modern engineering practices, the use of layered sandwich structural flat plate members is increasing steadily in aerospace, civil, mechanical and offshore structures due to their high specific strength and stiffness. Such structural applications have significantly led to reduction in vulnerability of warships to blasts, ballistics, bombs and fire attacks and to enhance superior resistance to fatigue crack propagation, impact

From the viewpoint of structural mechanics, an interface slip motion between two laminated structures, such as beam, beam plate and plate in the presence of dry friction can be utilized for slip damping systems. By scientific definition, slip damping is a mechanism exploited for dissipating noise and vibration energy in machine structures and systems. There are several engineering procedures to effect or simulate such a damping phenomenon. For example, the introduction of constrained, unconstrained and even viscoelastic layers has been very helpful in this respect. One good technique is layered construction

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

> © 2018 The Author(s). Licensee InTech. This chapter is 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.

Introductory Chapter: Laminations - Theory and

**Applications**

Applications

1. Introduction

damage and local buckling.

Charles Attah Osheku

Charles Attah Osheku

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74457

#### **Introductory Chapter: Laminations - Theory and Applications** Introductory Chapter: Laminations - Theory and Applications

DOI: 10.5772/intechopen.74457

Charles Attah Osheku Charles Attah Osheku

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74457

## 1. Introduction

Historically, the earliest use of laminated materials was by the Mesopotamians around 3400 BC. Similarly, plywood laminates from wood strips glued together at different angles were constructed for structural applications. Nevertheless, for several centuries, primitive lamination methods were used for the construction of bows and canoes. Specifically, the invention of Bakelite in 1907 heralded a new era for composite and laminated material applications. For instance, at the inception of the aircraft by the Orville and Wilbur Wright brothers, it was made with wood and fabrics. The need for better aerodynamic performance and efficiency supported by advances in aerospace engineering led to the use of lighter materials with strength that matches that of steel. The development of this field has progressed significantly to the point of the ubiquitous availability of laminates in materials for aviation, civil structures and automotive uses. In particular, carbon fiber reinforced plastics are in use in modern aircraft aerospace systems. In modern engineering practices, the use of layered sandwich structural flat plate members is increasing steadily in aerospace, civil, mechanical and offshore structures due to their high specific strength and stiffness. Such structural applications have significantly led to reduction in vulnerability of warships to blasts, ballistics, bombs and fire attacks and to enhance superior resistance to fatigue crack propagation, impact damage and local buckling.

From the viewpoint of structural mechanics, an interface slip motion between two laminated structures, such as beam, beam plate and plate in the presence of dry friction can be utilized for slip damping systems. By scientific definition, slip damping is a mechanism exploited for dissipating noise and vibration energy in machine structures and systems. There are several engineering procedures to effect or simulate such a damping phenomenon. For example, the introduction of constrained, unconstrained and even viscoelastic layers has been very helpful in this respect. One good technique is layered construction

made possible by externally applied pressure at the interface of two structural members. An arrangement of this nature could either be jointed or fastened by appropriate bolting methods. Within that concept, interface pressure profiles must assume a significant role subject to interfacial slip motion to waste or dissipate energy from the induced vibration arising from any form of excitation.

Within this context, researchers have developed a number of interesting mathematical models to derive the advantages discussed above for noise dissipation, minimization or complete vibration isolation. For instance, in the paper published by Osheku et al. [1], they employed a mathematical model to discuss the effect of structural vibration on the propagation of acoustic pressure waves through a cantilevered 3D laminated beam-plate enclosure. On the other hand, slip damping with heterogeneous sandwich composite viscoelastic beam-plate smart systems as a model for dissipation of vibration and active noise control mechanism in ship and floating structures was studied effectively by Olunloyo and Osheku in [2]. A scientific discussion of interfacial slip through layered laminates is described as shown in Figure 1.

As a demonstration of the application of laminations to noise reduction, Figure 2 shows the problem geometry subject to governing equations as derived in [1]:

$$\nabla^2 P - \frac{\partial^2 P}{c\_1^2 \partial t^2} - \frac{2\mu\_x \partial^2 P}{c\_1^2 \partial t \partial x} - \frac{\mu\_x^2}{c\_1^2} \frac{\partial^2 P}{\partial x^2} = -\frac{\partial^2 W\_1}{\rho \partial t^2} \tag{1}$$

<sup>D</sup> <sup>∂</sup><sup>4</sup> W<sup>1</sup> <sup>∂</sup>x<sup>4</sup> <sup>þ</sup> <sup>2</sup> <sup>∂</sup><sup>4</sup>

For the case Ω2,

For the case Ω3,

For the case Ω4,

<sup>D</sup> <sup>∂</sup><sup>4</sup> W<sup>3</sup> <sup>∂</sup>x<sup>4</sup> <sup>þ</sup> <sup>2</sup> <sup>∂</sup><sup>4</sup>

W<sup>1</sup> ∂x<sup>2</sup>∂y<sup>2</sup> þ

W<sup>3</sup> ∂x<sup>2</sup>∂y<sup>2</sup> þ

Figure 2. Problem geometry of 3D composite structure.

∂4 W<sup>1</sup> ∂y<sup>4</sup>

> ∂2 W<sup>2</sup> <sup>∂</sup>x<sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup>

∂4 W<sup>3</sup> ∂y<sup>4</sup>

> ∂2 W<sup>4</sup> <sup>∂</sup>x<sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup>

<sup>∀</sup>α<sup>1</sup> <sup>¼</sup> <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>α</sup><sup>3</sup> <sup>¼</sup> <sup>α</sup><sup>4</sup> <sup>¼</sup> <sup>6</sup><sup>μ</sup>

equations in Fourier finite double sine transforms plane, viz.:

þ r1h ∂2 W<sup>1</sup> ∂t

þ r1h ∂2 W<sup>3</sup> ∂t

> ∂2 W<sup>4</sup> ∂t <sup>2</sup> ¼ α<sup>4</sup>

The boundary stresses through the domains Ω1, Ω<sup>2</sup> and Ω<sup>3</sup> are evaluated from the following

∂2 W<sup>2</sup> ∂t <sup>2</sup> ¼ α<sup>2</sup>

<sup>2</sup> � <sup>μ</sup>H<sup>2</sup> 2

<sup>2</sup> � <sup>μ</sup>H<sup>2</sup> 2

∂p xð Þ ; 0 ∂x

∂p xð Þ ; 0 ∂x

∂P xð Þ ; 0

∂P xð Þ ; 0

Eh<sup>2</sup> ; <sup>β</sup><sup>1</sup> <sup>¼</sup> <sup>β</sup><sup>2</sup> <sup>¼</sup> <sup>β</sup><sup>3</sup> <sup>¼</sup> <sup>β</sup><sup>4</sup> <sup>¼</sup> <sup>r</sup>bh

<sup>þ</sup> <sup>μ</sup>H<sup>2</sup> 2

Introductory Chapter: Laminations - Theory and Applications

http://dx.doi.org/10.5772/intechopen.74457

5

<sup>þ</sup> <sup>μ</sup>H<sup>2</sup> 2

∂pð Þ 0; y ∂y

<sup>∂</sup><sup>x</sup> (3)

∂pð Þ 0; y ∂y

<sup>∂</sup><sup>x</sup> (5)

EI (6)

¼ 0 (2)

¼ 0 (4)

while the governing equations for each vibrating boundary are namely:

For the case Ω1,

Figure 1. Mechanism of interfacial slip geometry.

Figure 2. Problem geometry of 3D composite structure.

$$D\left(\frac{\partial^4 W\_1}{\partial \mathbf{x}^4} + 2\frac{\partial^4 W\_1}{\partial \mathbf{x}^2 \partial \mathbf{y}^2} + \frac{\partial^4 W\_1}{\partial \mathbf{y}^4}\right) + \rho\_1 h \frac{\partial^2 W\_1}{\partial t^2} - \left(\frac{\mu H\_2}{2} \frac{\partial p(\mathbf{x}, \mathbf{0})}{\partial \mathbf{x}} + \frac{\mu H\_2}{2} \frac{\partial p(\mathbf{0}, \mathbf{y})}{\partial \mathbf{y}}\right) = \mathbf{0} \tag{2}$$

For the case Ω2,

made possible by externally applied pressure at the interface of two structural members. An arrangement of this nature could either be jointed or fastened by appropriate bolting methods. Within that concept, interface pressure profiles must assume a significant role subject to interfacial slip motion to waste or dissipate energy from the induced vibration

Within this context, researchers have developed a number of interesting mathematical models to derive the advantages discussed above for noise dissipation, minimization or complete vibration isolation. For instance, in the paper published by Osheku et al. [1], they employed a mathematical model to discuss the effect of structural vibration on the propagation of acoustic pressure waves through a cantilevered 3D laminated beam-plate enclosure. On the other hand, slip damping with heterogeneous sandwich composite viscoelastic beam-plate smart systems as a model for dissipation of vibration and active noise control mechanism in ship and floating structures was studied effectively by Olunloyo and Osheku in [2]. A scientific discussion of interfacial slip through layered laminates is described as

As a demonstration of the application of laminations to noise reduction, Figure 2 shows the

P

<sup>1</sup>∂t∂<sup>x</sup> � <sup>u</sup><sup>2</sup>

x c2 1

∂2 P <sup>∂</sup>x<sup>2</sup> ¼ � <sup>∂</sup><sup>2</sup>

W<sup>1</sup> r∂t

<sup>2</sup> (1)

<sup>2</sup> � <sup>2</sup>ux∂<sup>2</sup>

c2

problem geometry subject to governing equations as derived in [1]:

while the governing equations for each vibrating boundary are namely:

∇2 <sup>P</sup> � <sup>∂</sup><sup>2</sup> P c2 1∂t

arising from any form of excitation.

4 Lamination - Theory and Application

shown in Figure 1.

For the case Ω1,

Figure 1. Mechanism of interfacial slip geometry.

$$\frac{\partial^2 W\_2}{\partial \mathbf{x}^2} + \beta\_2 \frac{\partial^2 W\_2}{\partial t^2} = \alpha\_2 \frac{\partial P(\mathbf{x}, 0)}{\partial \mathbf{x}} \tag{3}$$

For the case Ω3,

$$D\left(\frac{\partial^4 W\_3}{\partial \mathbf{x}^4} + 2\frac{\partial^4 W\_3}{\partial \mathbf{x}^2 \partial y^2} + \frac{\partial^4 W\_3}{\partial y^4}\right) + \rho\_1 h \frac{\partial^2 W\_3}{\partial t^2} - \left(\frac{\mu H\_2}{2} \frac{\partial p(\mathbf{x}, \mathbf{0})}{\partial \mathbf{x}} + \frac{\mu H\_2}{2} \frac{\partial p(\mathbf{0}, y)}{\partial y}\right) = \mathbf{0} \tag{4}$$

For the case Ω4,

$$
\frac{
\partial^2 W\_4}{
\partial \mathbf{x}^2} + \beta\_2 \frac{
\partial^2 W\_4}{
\partial t^2} = \alpha\_4 \frac{
\partial P(\mathbf{x}, 0)}{
\partial \mathbf{x}}
\tag{5}
$$

$$\forall \alpha\_1 = \alpha\_2 = \alpha\_3 = \alpha\_4 = \frac{6\mu}{Eh^2}; \; \beta\_1 = \beta\_2 = \beta\_3 = \beta\_4 = \frac{\rho bh}{EI} \tag{6}$$

The boundary stresses through the domains Ω1, Ω<sup>2</sup> and Ω<sup>3</sup> are evaluated from the following equations in Fourier finite double sine transforms plane, viz.:

$$\begin{aligned} \tilde{P}^{F\_{p}F\_{z}}(0,\lambda\_{m},\lambda\_{k},\mathbf{s}) &= \iiint\_{0} [\mu P\_{av}] \sin\left(\frac{m\pi y}{b}\right) \sin\left(\frac{k\pi z}{c}\right) dydz, \\ \tilde{P}^{F\_{y}F\_{z}}(a,\lambda\_{m},\lambda\_{k},\mathbf{s}) &= \iiint\_{0} [\mu P\_{av}] \sin\left(\frac{m\pi y}{b}\right) \sin\left(\frac{k\pi z}{c}\right) dydz \\ \tilde{p}^{F\_{v}F\_{r}}(1,\lambda\_{1},\mathbf{1},\mathbf{1},\mathbf{0}) &= \iiint [P\_{av}] \, \cdot \, \cdot \, \left(n\pi \mathbf{x}\right) \, \cdot \, \cdot \, \left(m\pi \mathbf{y}\right) \, \cdot \, \cdot \end{aligned} \tag{7}$$

In the meantime, a first-order approximation follows as:

Goodman and Klumpp [4] becomes:

upper and lower laminates.

E

E

<sup>2</sup> ð Þ <sup>2</sup><sup>z</sup> � <sup>h</sup> <sup>∂</sup><sup>2</sup>

<sup>2</sup> ð Þ <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>h</sup> <sup>∂</sup><sup>2</sup>

ð Þ σ<sup>x</sup> <sup>1</sup>ð Þ¼� x; z; t

ð Þ σ<sup>x</sup> <sup>2</sup>ð Þ¼� x; z; t

This gives Eq. (11) as:

ð x

∂2

Wð Þ x; τ ∂ξ

<sup>2</sup> þ B

2

0

and on introducing the nondimensionalized parameters, viz.:

0

ΔUð Þ¼ x

and

ΔU1ð Þ¼ x; z; t ΔU1ð Þþ 0; 0þ; t z � α1ð Þx

ΔU2ð Þ¼ x; z; t ΔU2ð Þ� 0; 0�; t z þ α2ð Þx

The associated interfacial slip motion is then given by the equation below which on following

where ξ is a dummy axial spatial variable of integration across the interface and 0+, 0� denote the origin of the transverse spatial variable for each layer; while subscripts 1 and 2 refer to the

For the contrived geometry, the derived corresponding spatial bending stresses are namely:

<sup>1</sup> � <sup>μ</sup><sup>0</sup> μm � �

<sup>1</sup> � <sup>μ</sup><sup>0</sup> μm � �

<sup>F</sup> ; <sup>P</sup><sup>0</sup> <sup>¼</sup> <sup>P</sup><sup>0</sup>

� � <sup>W</sup> <sup>ξ</sup>; <sup>τ</sup> � � � <sup>W</sup>ð Þ <sup>1</sup>; <sup>τ</sup> � � <sup>þ</sup> <sup>2</sup>μP<sup>0</sup> <sup>1</sup> <sup>þ</sup> εξ � � <sup>ξ</sup> � <sup>1</sup> � � " #d<sup>ξ</sup> (19)

<sup>F</sup>=bh

; t <sup>¼</sup> <sup>2</sup>πτ ω0

B2 0 μ0

B2 0 μ0

Here, ΔU1ð Þ 0; 0þ; t and ΔU2ð Þ 0; 0�; t must be zero at the fixed end.

ð x

0

W xð Þ ; t ∂x<sup>2</sup> þ

W xð Þ ; t ∂x<sup>2</sup> þ

> <sup>1</sup> � <sup>μ</sup><sup>0</sup> μm

<sup>Δ</sup><sup>U</sup> <sup>¼</sup> <sup>Δ</sup>Uð Þ<sup>x</sup> Ebh<sup>2</sup> L2

For the static case, the governing differential equation is described in [2] as

<sup>Δ</sup>U xð Þ¼ ; <sup>0</sup> <sup>E</sup>�<sup>1</sup>

h 2 � � <sup>∂</sup>W xð Þ ; <sup>t</sup>

Introductory Chapter: Laminations - Theory and Applications

http://dx.doi.org/10.5772/intechopen.74457

h 2 � � <sup>∂</sup>W xð Þ ; <sup>t</sup>

ΔU xð Þ¼ ; z; t ΔU1ð Þ� x; z; t ΔU2ð Þ x; z; t (15)

ð Þ <sup>σ</sup><sup>x</sup> <sup>1</sup>ð Þ� <sup>ξ</sup>; <sup>0</sup>þ; <sup>t</sup> ð Þ <sup>σ</sup><sup>x</sup> <sup>2</sup>ð Þ <sup>ξ</sup>; <sup>0</sup>þ; <sup>t</sup> � �d<sup>ξ</sup> (16)

<sup>½</sup>W xð Þ� ; <sup>t</sup> W Lð Þ ; <sup>t</sup> � þ <sup>μ</sup>P0ð Þ <sup>1</sup> <sup>þ</sup> <sup>ε</sup><sup>x</sup> ð Þ <sup>x</sup> � <sup>L</sup>

<sup>½</sup>W xð Þ� ; <sup>t</sup> W Lð Þ ; <sup>t</sup> � � <sup>μ</sup>P0ð Þ <sup>1</sup> � <sup>ε</sup><sup>x</sup> ð Þ <sup>x</sup> � <sup>L</sup>

h (17)

h (18)

(20)

<sup>∂</sup><sup>x</sup> (13)

7

<sup>∂</sup><sup>x</sup> (14)

$$\tilde{P}^{F\_{x}F\_{y}}(\lambda\_{n},\lambda\_{m},0,s) = \iint\limits\_{0\neq 0} \left[\frac{P\_{av}}{s}\right] \sin\left(\frac{n\pi\mathbf{x}}{a}\right) \sin\left(\frac{m\pi y}{b}\right) dx dy$$

$$\_{h.c.}$$

$$\hat{P}^{F\_x F\_y}(\lambda\_\text{tr}, \lambda\_\text{tr}, \mathbf{c}, \mathbf{s}) = -\prod\_{0 \neq 0}^{b} \left[ \frac{P\_{\text{av}}}{\mathbf{s}} \right] \sin \left( \frac{m \pi \mathbf{x}}{a} \right) \sin \left( \frac{k \pi y}{b} \right) d\mathbf{x} dy,\tag{8}$$

$$\tilde{P}^{F\_z F\_z}(\lambda\_n, b, \lambda\_k, s) = \int\_0^l \left[ \frac{P\_{av}}{s} \right] \sin \left( \frac{n \pi x}{a} \right) \sin \left( \frac{k \pi z}{c} \right) dx dz,\tag{9}$$

$$\tilde{P}^{F\_t F\_t}(\lambda\_n, 0, \lambda\_k, \mathbf{s}) = -\int\_0^a \int\_0^c \left[\frac{P\_{av}}{s}\right] \sin\left(\frac{n\pi \mathbf{x}}{a}\right) \sin\left(\frac{k\pi \mathbf{z}}{c}\right) d\mathbf{x} d\mathbf{z},\tag{10}$$

subject to interfacial slip mechanism as modulated by the pressure profile as discussed in the paper. In the meantime, when such a slip mechanism is to be examined under the influence of an electromagnetic field, the details of such a problem were discussed in Osheku [3].

The underlying principle is such that during bending, each half of the sandwich elastic in transverse magnetic field has its neutral plane that does not necessarily coincide with the geometric mid-plane through the interface because of the frictional stresses. These are located at <sup>z</sup> <sup>¼</sup> <sup>α</sup>1ð Þ<sup>x</sup> <sup>h</sup> <sup>2</sup> and <sup>z</sup> <sup>¼</sup> <sup>α</sup>2ð Þ<sup>x</sup> <sup>h</sup> <sup>2</sup> on layers (1) and (2) respectively, where α1ð Þx and α2ð Þx are generalized functions of x. The use of Taylor series approximation leads to the following expressions for the microscopic slip or displacements of the two adjacent opposite points A1 and A2 to the corresponding A1 <sup>1</sup> and A2 <sup>1</sup> namely:

$$\begin{split} \Delta \mathcal{U}\_{1}(\mathbf{x}, z, t) &= \Delta \mathcal{U}\_{1}(0, 0+, t) + \left(z - a\_{1}(\mathbf{x})\frac{h}{2}\right) \frac{\partial \mathcal{W}(\mathbf{x}, t)}{\partial \mathbf{x}} + \frac{1}{2} \left(z - a\_{1}(\mathbf{x})\frac{h}{2}\right)^{2} \frac{\partial^{2} \mathcal{W}(\mathbf{x}, t)}{\partial \mathbf{x}^{2}} \\ &+ \frac{1}{6} \left(z - a\_{1}(\mathbf{x})\frac{h}{2}\right)^{3} \frac{\partial^{3} \mathcal{W}(\mathbf{x}, t)}{\partial \mathbf{x}^{3}} \end{split} \tag{11}$$

and

$$\begin{split} \Delta \mathcal{U}\_2(\mathbf{x}, z, t) &= \Delta \mathcal{U}\_2(0, 0-, t) - \left( z + a\_2(\mathbf{x}) \frac{h}{2} \right) \frac{\partial \mathcal{W}(\mathbf{x}, t)}{\partial \mathbf{x}} - \frac{1}{2} \left( z + a\_1(\mathbf{x}) \frac{h}{2} \right)^2 \frac{\partial^2 \mathcal{W}(\mathbf{x}, t)}{\partial \mathbf{x}^2} \\ &- \frac{1}{6} \left( z + a\_1(\mathbf{x}) \frac{h}{2} \right)^3 \frac{\partial^3 \mathcal{W}(\mathbf{x}, t)}{\partial \mathbf{x}^3} \end{split} \tag{12}$$

In the meantime, a first-order approximation follows as:

$$
\Delta \mathcal{U}\_1(\mathbf{x}, z, t) = \Delta \mathcal{U}\_1(\mathbf{0}, \mathbf{0} +, t) + \left(z - a\_1(\mathbf{x})\frac{h}{2}\right) \frac{\partial \mathcal{W}(\mathbf{x}, t)}{\partial \mathbf{x}} \tag{13}
$$

$$
\Delta \mathcal{U}\_2(\mathbf{x}, z, t) = \Delta \mathcal{U}\_2(0, 0-, t) - \left(z + a\_2(\mathbf{x})\frac{h}{2}\right) \frac{\partial \mathcal{W}(\mathbf{x}, t)}{\partial \mathbf{x}} \tag{14}
$$

Here, ΔU1ð Þ 0; 0þ; t and ΔU2ð Þ 0; 0�; t must be zero at the fixed end.

$$
\Delta \mathcal{U}(\mathbf{x}, z, t) = \Delta \mathcal{U}\_1(\mathbf{x}, z, t) - \Delta \mathcal{U}\_2(\mathbf{x}, z, t) \tag{15}
$$

The associated interfacial slip motion is then given by the equation below which on following Goodman and Klumpp [4] becomes:

$$
\Delta \mathcal{U}(\mathbf{x}, \mathbf{0}) = E^{-1} \int\_0^\mathbf{x} [(\sigma\_x)\_1(\xi, \mathbf{0} +, t) - (\sigma\_x)\_2(\xi, \mathbf{0} +, t)] d\xi \tag{16}
$$

where ξ is a dummy axial spatial variable of integration across the interface and 0+, 0� denote the origin of the transverse spatial variable for each layer; while subscripts 1 and 2 refer to the upper and lower laminates.

For the contrived geometry, the derived corresponding spatial bending stresses are namely:

$$\rho(\sigma\_{\mathbf{x}})\_1(\mathbf{x}, z, t) = -\frac{E}{2}(2z - h)\frac{\partial^2 \mathcal{W}(\mathbf{x}, t)}{\partial \mathbf{x}^2} + \frac{B\_0^2}{\mu\_0} \left(1 - \frac{\mu\_0}{\mu\_\mathrm{m}}\right) [\mathcal{W}(\mathbf{x}, t) - \mathcal{W}(L, t)] + \mu P\_0 (1 + \varepsilon \overline{\mathbf{x}}) \frac{(\mathbf{x} - L)}{h} \tag{17}$$

and

<sup>P</sup>~FyFz <sup>0</sup>; <sup>λ</sup>m; <sup>λ</sup><sup>k</sup> ð Þ¼ ;<sup>s</sup>

6 Lamination - Theory and Application

<sup>P</sup>~FyFz <sup>a</sup>; <sup>λ</sup>m; <sup>λ</sup><sup>k</sup> ð Þ¼ ;<sup>s</sup>

<sup>P</sup>~FxFy ð Þ¼ <sup>λ</sup>n; <sup>λ</sup>m; <sup>0</sup>;<sup>s</sup>

<sup>P</sup>~FxFy ð Þ¼� <sup>λ</sup>n; <sup>λ</sup>m; <sup>c</sup>;<sup>s</sup>

<sup>P</sup>~FxFz <sup>λ</sup>n; <sup>b</sup>; <sup>λ</sup><sup>k</sup> ð Þ¼ ;<sup>s</sup>

<sup>P</sup>~FxFz <sup>λ</sup>n; <sup>0</sup>; <sup>λ</sup><sup>k</sup> ð Þ¼� ;<sup>s</sup>

<sup>2</sup> and <sup>z</sup> <sup>¼</sup> <sup>α</sup>2ð Þ<sup>x</sup> <sup>h</sup>

þ 1

� 1

ΔU1ð Þ¼ x; z; t ΔU1ð Þþ 0; 0þ; t z � α1ð Þx

ΔU2ð Þ¼ x; z; t ΔU2ð Þ� 0; 0�; t z þ α2ð Þx

<sup>6</sup> <sup>z</sup> <sup>þ</sup> <sup>α</sup>1ð Þ<sup>x</sup>

<sup>6</sup> <sup>z</sup> � <sup>α</sup>1ð Þ<sup>x</sup>

and A2 to the corresponding A1

at <sup>z</sup> <sup>¼</sup> <sup>α</sup>1ð Þ<sup>x</sup> <sup>h</sup>

and

ð b ðc

μPav

μPav

Pav s � �

> Pav s � �

Pav s � �

> Pav s � �

subject to interfacial slip mechanism as modulated by the pressure profile as discussed in the paper. In the meantime, when such a slip mechanism is to be examined under the influence of

The underlying principle is such that during bending, each half of the sandwich elastic in transverse magnetic field has its neutral plane that does not necessarily coincide with the geometric mid-plane through the interface because of the frictional stresses. These are located

generalized functions of x. The use of Taylor series approximation leads to the following expressions for the microscopic slip or displacements of the two adjacent opposite points A1

<sup>1</sup> namely:

W xð Þ ; t ∂x<sup>3</sup>

W xð Þ ; t ∂x<sup>3</sup>

h 2 � � <sup>∂</sup>W xð Þ ; <sup>t</sup>

h 2 � � <sup>∂</sup>W xð Þ ; <sup>t</sup>

∂x

þ 1

<sup>∂</sup><sup>x</sup> � <sup>1</sup>

<sup>2</sup> <sup>z</sup> � <sup>α</sup>1ð Þ<sup>x</sup>

<sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>α</sup>1ð Þ<sup>x</sup>

h 2 � �<sup>2</sup> ∂<sup>2</sup>

h 2 � �<sup>2</sup> ∂<sup>2</sup>

� � sin <sup>m</sup>π<sup>y</sup>

� � sin <sup>m</sup>π<sup>y</sup>

b � �

b � �

sin <sup>n</sup>π<sup>x</sup> a � �

sin <sup>m</sup>π<sup>x</sup> a � �

sin <sup>n</sup>π<sup>x</sup> a � �

> sin <sup>n</sup>π<sup>x</sup> a � �

sin <sup>k</sup>π<sup>z</sup> c � �

sin <sup>k</sup>π<sup>z</sup> c � �

sin <sup>m</sup>π<sup>y</sup> b � �

> sin <sup>k</sup>π<sup>y</sup> b � �

sin <sup>k</sup>π<sup>z</sup> c � �

> sin <sup>k</sup>π<sup>z</sup> c � �

<sup>2</sup> on layers (1) and (2) respectively, where α1ð Þx and α2ð Þx are

dydz,

dydz

(7)

dxdy

dxdy, (8)

dxdz, (9)

dxdz, (10)

W xð Þ ; t ∂x<sup>2</sup>

W xð Þ ; t ∂x<sup>2</sup>

(11)

(12)

0

ð b

0

ð b

0

ð b

0

ða

ðc

0

ða

ðc

0

an electromagnetic field, the details of such a problem were discussed in Osheku [3].

<sup>1</sup> and A2

h 2 � �<sup>3</sup> ∂<sup>3</sup>

h 2 � �<sup>3</sup> ∂<sup>3</sup> 0

0

0

ða

0

0

ðc

0

ðc

$$\rho\_1(\sigma\_\mathbf{x})\_2(\mathbf{x}, z, t) = -\frac{E}{2}(2\varepsilon + h)\frac{\partial^2 \mathcal{W}(\mathbf{x}, t)}{\partial \mathbf{x}^2} + \frac{B\_0^2}{\mu\_0} \left(1 - \frac{\mu\_0}{\mu\_\mathbf{m}}\right) [\mathcal{W}(\mathbf{x}, t) - \mathcal{W}(L, t)] - \mu P\_0 (1 - \varepsilon \overline{\mathbf{x}}) \frac{(\mathbf{x} - L)}{h} \tag{18}$$

This gives Eq. (11) as:

$$\Delta LI(\overline{\mathbf{x}}) = \int\_0^{\overline{\mathbf{x}}} \left[ \frac{\partial^2 \overline{W}(\overline{\mathbf{x}}, \tau)}{\partial \overline{\xi}^2} + \overline{B} \left( 1 - \frac{\mu\_0}{\mu\_m} \right) [\overline{W}(\overline{\xi}, \tau) - \overline{W}(1, \tau)] + 2\mu P\_0 (1 + \varepsilon \overline{\xi}) (\overline{\xi} - 1) \right] d\overline{\xi} \tag{19}$$

and on introducing the nondimensionalized parameters, viz.:

$$
\Delta U = \frac{\Delta \mathcal{U}(\overline{\mathbf{x}}) E b h^2}{L^2 F}; \overline{P}\_0 = \frac{P\_0}{\mathbb{P}\_{\text{fb}}}; t = \frac{2\pi \tau}{\omega\_0} \tag{20}
$$

For the static case, the governing differential equation is described in [2] as

$$EI\frac{d^4W}{d\mathbf{x}^4} + \frac{bh}{\mu\_0} \left(1 - \frac{\mu\_0}{\mu\_m}\right) B\_o^2 \frac{d^2W}{d\mathbf{x}^2} = \frac{1}{2}bh\mu \frac{dP}{d\mathbf{x}}\tag{21}$$

subject to the form computed as in [3] as:

$$
\overline{P}\_{av} = \overline{P}\_o \int\_0^1 (1 + \varepsilon \overline{\pi}) d\overline{\pi} = \overline{P}\_o \left( 1 + \frac{\varepsilon}{2} \right) \tag{22}
$$

In the meantime, several derivatives of this analysis can be conjured for comparative analysis with literature.

For the case of a layer of sandwich homogenous magnetoelastic beam-plate of thickness h with non-uniform pressure at the interface, the formulated equation governing the vibration takes the form:

$$D\frac{\partial^4 W}{\partial \mathbf{x}^4} + \frac{h}{\mu\_0} \left(1 - \frac{\mu\_0}{\mu\_m}\right) B\_0^{-2} \frac{\partial^2 W}{\partial \mathbf{x}^2} - \frac{\sigma\_c h^3 B\_0^{-2}}{12} \frac{\partial^3 W}{\partial \mathbf{t} \partial \mathbf{x}^2} + \rho h \frac{\partial^2 W}{\partial t^2} = \frac{1}{2} h \mu \frac{\partial P}{\partial \mathbf{x}} \tag{23}$$

$$\forall D = \frac{Eh^3}{12(1-v^2)}\tag{24}$$

so that for the case corresponding to uniform interfacial pressure, Eq. (23) reduces to the following:

$$D\frac{\partial^4 W}{\partial \mathbf{x}^4} + \frac{h}{\mu\_0} \left(1 - \frac{\mu\_0}{\mu\_m}\right) B\_0^{-2} \frac{\partial^2 W}{\partial \mathbf{x}^2} - \frac{\sigma\_c h^3 B\_0^{-2}}{12} \frac{\partial^3 W}{\partial t \partial \mathbf{x}^2} + \rho h \frac{\partial^2 W}{\partial t^2} = 0 \tag{25}$$

Nonetheless, a theoretical investigation by [5] on the proposed uses of lamination theory via slip damping was hinged on the following pictorial representation in Figure 3.

Figure 3 illustrates a slip damping theory and its application in ship hull subject to Eq. (26). Detailed solution methods of this boundary value mathematical physics problem are outlined in [5]. The intent here is to demonstrate possible means to enhance moving ocean craft stability subject to turbulent flow conditions in Figure 4. Meanwhile, the governing equation is depicted as follows,

$$\left(\frac{c\_1 + c\_2}{4}\right)\frac{\partial^5 W}{\partial t \partial x^4} + \frac{\partial^4 W}{\partial x^4} + \frac{\left(\beta\_1 \mathcal{A} + \beta\_2 \mathcal{B} + \beta\_3 + \beta\_4\right)}{4} \frac{\partial^2 W}{\partial t^2} = \frac{\left(a\_1 \mathcal{A} + a\_2 \mathcal{B}\right) \partial P(\mathbf{x}, 0)}{4} + \frac{C\_a}{4} \frac{\partial^2 V\_a}{\partial x^2} \tag{26}$$

where<sup>A</sup> denotes 1 � <sup>v</sup><sup>2</sup> <sup>1</sup> and <sup>β</sup> denotes 1 � <sup>v</sup><sup>2</sup> <sup>2</sup> and the following parameters have been defined viz.:

$$\alpha\_1 = \frac{6\mu}{\mathrm{E\_{eq1}h\_{\mathrm{eq1}}^2}}, \alpha\_2 = \frac{6\mu}{\mathrm{E\_{eq2}h\_{\mathrm{eq2}}^2}} \tag{27}$$

heq1 <sup>¼</sup> ð Þ h1 <sup>þ</sup> ha , heq1 <sup>¼</sup> ð Þ ha <sup>þ</sup> hs , Ca <sup>¼</sup> <sup>1</sup>

ment. (b) An elliptical model representation of a ship/FPSO bottom hull configuration.

quantification can be evaluated from ongoing equation

composite structure.

By utilizing the closed form expression of the interfacial slip motion, the energy dissipation

Figure 4. (a) A mathematical model of ship/Floating Production Storage and Offloading vessel (FPSO) in ocean environ-

Figure 3. (a) Preslip geometry for the composite sandwich structure under dynamic load. (b) Layering cross section of

2

Ead31b hð Þ <sup>a</sup> þ hs (29)

Introductory Chapter: Laminations - Theory and Applications

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$$\beta\_1 = \frac{\rho\_1 \text{bh}\_1}{\text{E}\_1 \text{I}\_1}, \beta\_2 = \frac{\rho\_2 \text{bh}\_2}{\text{E}\_2 \text{I}\_2}, \beta\_3 = \frac{\rho\_3 \text{bh}\_3}{\text{E}\_a \text{I}\_{a1}}, \beta\_4 = \frac{\rho\_s \text{bh}\_s}{\text{E}\_s \text{I}\_s} \tag{28}$$

EI <sup>d</sup><sup>4</sup> W dx<sup>4</sup> þ

subject to the form computed as in [3] as:

8 Lamination - Theory and Application

D ∂4 W ∂x<sup>4</sup> þ

> D ∂4 W ∂x<sup>4</sup> þ

h μ0

> h μ0

<sup>1</sup> � <sup>μ</sup><sup>0</sup> μm � �

<sup>∂</sup>x<sup>4</sup> <sup>þ</sup> <sup>β</sup>1<sup>A</sup> <sup>þ</sup> <sup>β</sup>2<sup>B</sup> <sup>þ</sup> <sup>β</sup><sup>3</sup> <sup>þ</sup> <sup>β</sup><sup>4</sup> � � 4

> <sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>6</sup><sup>μ</sup> Eeq1h<sup>2</sup> eq1

> > , <sup>β</sup><sup>2</sup> <sup>¼</sup> <sup>r</sup>2bh2 E2I2

<sup>1</sup> and <sup>β</sup> denotes 1 � <sup>v</sup><sup>2</sup>

<sup>β</sup><sup>1</sup> <sup>¼</sup> <sup>r</sup>1bh1 E1I1

B0 <sup>2</sup> ∂<sup>2</sup> W <sup>∂</sup>x<sup>2</sup> � <sup>σ</sup>ch<sup>3</sup>

slip damping was hinged on the following pictorial representation in Figure 3.

<sup>1</sup> � <sup>μ</sup><sup>0</sup> μm � �

B0 2 ∂2 W <sup>∂</sup>x<sup>2</sup> � <sup>σ</sup>ch<sup>3</sup>

with literature.

the form:

following:

depicted as follows,

where<sup>A</sup> denotes 1 � <sup>v</sup><sup>2</sup>

W ∂t∂x<sup>4</sup> þ ∂4 W

c<sup>1</sup> þ c<sup>2</sup> 4 � � ∂<sup>5</sup> bh μ0

> ð 1

> > 0

Pav ¼ Po

<sup>1</sup> � <sup>μ</sup><sup>0</sup> μm � �

B2 o d2 W dx<sup>2</sup> <sup>¼</sup> <sup>1</sup> 2 bhμ dP

ð Þ 1 þ εx dx ¼ Po 1 þ

B0 2

∂3 W <sup>∂</sup>t∂x<sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>h</sup>

B0 2

∂3 W <sup>∂</sup>t∂x<sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>h</sup>

<sup>2</sup> <sup>¼</sup> ð Þ <sup>α</sup>1<sup>A</sup> <sup>þ</sup> <sup>α</sup>2<sup>B</sup> 4

12

∂2 W ∂t <sup>2</sup> <sup>¼</sup> <sup>1</sup> 2 hμ ∂P

12 1 � <sup>v</sup><sup>2</sup> ð Þ (24)

∂2 W ∂t

> ∂P xð Þ ; 0 ∂x

<sup>2</sup> and the following parameters have been defined viz.:

, <sup>β</sup><sup>4</sup> <sup>¼</sup> <sup>r</sup>sbhs EsIs

þ Ca 4 ∂2 Va <sup>∂</sup>x<sup>2</sup> (26)

(27)

(28)

12

In the meantime, several derivatives of this analysis can be conjured for comparative analysis

For the case of a layer of sandwich homogenous magnetoelastic beam-plate of thickness h with non-uniform pressure at the interface, the formulated equation governing the vibration takes

<sup>∀</sup><sup>D</sup> <sup>¼</sup> Eh<sup>3</sup>

so that for the case corresponding to uniform interfacial pressure, Eq. (23) reduces to the

Nonetheless, a theoretical investigation by [5] on the proposed uses of lamination theory via

Figure 3 illustrates a slip damping theory and its application in ship hull subject to Eq. (26). Detailed solution methods of this boundary value mathematical physics problem are outlined in [5]. The intent here is to demonstrate possible means to enhance moving ocean craft stability subject to turbulent flow conditions in Figure 4. Meanwhile, the governing equation is

> ∂2 W ∂t

, <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>6</sup><sup>μ</sup> Eeq2h<sup>2</sup> eq2

, <sup>β</sup><sup>3</sup> <sup>¼</sup> <sup>r</sup>abha EaIa1

ε 2 � � dx (21)

(22)

<sup>∂</sup><sup>x</sup> (23)

<sup>2</sup> ¼ 0 (25)

Figure 3. (a) Preslip geometry for the composite sandwich structure under dynamic load. (b) Layering cross section of composite structure.

Figure 4. (a) A mathematical model of ship/Floating Production Storage and Offloading vessel (FPSO) in ocean environment. (b) An elliptical model representation of a ship/FPSO bottom hull configuration.

$$\mathbf{h\_{eq1}} = (\mathbf{h\_1} + \mathbf{h\_a}),\\\mathbf{h\_{eq1}} = (\mathbf{h\_a} + \mathbf{h\_s}),\\\mathbf{C\_a} = \frac{1}{2}\mathbf{E\_a}\mathbf{d\_{31}}\mathbf{b(h\_a + h\_s)}\tag{29}$$

By utilizing the closed form expression of the interfacial slip motion, the energy dissipation quantification can be evaluated from ongoing equation

$$D = 4\mu b \int\_0^{\pi/2a} \int\_0^L P(\mathbf{x}) \Delta u(\mathbf{x}, t) f d\mathbf{x} \, dt \tag{30}$$

and nondimensionalized as:

$$\overline{D} = 4\mu \int\_0^{1/4} \int\_0^1 \overline{P}\_{av} \Delta \overline{u} d\overline{x} d\tau,\tag{31}$$

Figure 6. (a) Sectional view through a modified aircraft engine. (b) Blown-out section of modified new jet engine concept

Introductory Chapter: Laminations - Theory and Applications

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11

Figure 7. (a) Proposed application of lamination for vibration damping and isolating system. (b) Modified axially

Figure 8. (a) 3D section of jet fighter engine showing modified engine. (b) 3D view of jet fighter modified engine.

showing laminated vibration damping mechanism.

compressing turbo jet aircraft engines.

where <sup>D</sup> <sup>¼</sup> <sup>D</sup>ðx, <sup>τ</sup>ÞEbh<sup>3</sup> =L<sup>3</sup> F2 0

For the purpose of clarity, some nomenclatures are described. Readers are advised to read the papers in the references for detailed description of symbols in the equations. Theoretical simulated studies of the foregoing based on the generalized energy dissipation equations and possible scientific applications and engineering design purposes are well highlighted in these papers.

## 2. Proposed applications

A number of technological applications are possible in the areas of machine structures, power plants, ship turbines, military jets and commercial aircrafts with suitable laminated materials in conjunction with intelligent manufacturing systems to produce systems that are more comfortable for use by humans especially in the modern day world with advanced 3D manufacturing techniques. Arising from research results based on lamination theory and applications, the following modern noise dissipation and vibration isolation systems can be incorporated in new aircraft engines and power generating turbo machines to enhance their operational stability are itemized in the following diagrams (Figures 5–11).

Figure 5. (a) A commercial airliner with proposed vibration attenuation devices. (b) Proposed 2D cross section of vibration damping system.

Introductory Chapter: Laminations - Theory and Applications http://dx.doi.org/10.5772/intechopen.74457 11

D ¼ 4μb

and nondimensionalized as:

10 Lamination - Theory and Application

where <sup>D</sup> <sup>¼</sup> <sup>D</sup>ðx, <sup>τ</sup>ÞEbh<sup>3</sup>

2. Proposed applications

vibration damping system.

=L<sup>3</sup> F2 0 ð<sup>π</sup>=2<sup>ω</sup>

ð L

P xð ÞΔu xð Þ ; t fdx dt (30)

PavΔudxdτ, (31)

0

ð 1=4 ð 1

0

For the purpose of clarity, some nomenclatures are described. Readers are advised to read the papers in the references for detailed description of symbols in the equations. Theoretical simulated studies of the foregoing based on the generalized energy dissipation equations and possible scientific applications and engineering design purposes are well highlighted in these papers.

A number of technological applications are possible in the areas of machine structures, power plants, ship turbines, military jets and commercial aircrafts with suitable laminated materials in conjunction with intelligent manufacturing systems to produce systems that are more comfortable for use by humans especially in the modern day world with advanced 3D manufacturing techniques. Arising from research results based on lamination theory and applications, the following modern noise dissipation and vibration isolation systems can be incorporated in new aircraft engines and power generating turbo machines to enhance their

Figure 5. (a) A commercial airliner with proposed vibration attenuation devices. (b) Proposed 2D cross section of

operational stability are itemized in the following diagrams (Figures 5–11).

0

0

D ¼ 4μ

Figure 6. (a) Sectional view through a modified aircraft engine. (b) Blown-out section of modified new jet engine concept showing laminated vibration damping mechanism.

Figure 7. (a) Proposed application of lamination for vibration damping and isolating system. (b) Modified axially compressing turbo jet aircraft engines.

Figure 8. (a) 3D section of jet fighter engine showing modified engine. (b) 3D view of jet fighter modified engine.

In Chapter 1, the focus is on the study of multiscale hierarchical structure and laminated strengthening and toughening mechanisms. Here, the authors emphasize how higher strength can be achieved in titanium matrix composites by adjusting the multiscale hierarchical structure. The chapter further deduced how the elastic properties and yield strength of laminated composites can be modulated through the "rule of average." It is also discussed in this chapter how impossible it is to predict fracture elongation and toughness via this rule. In addition, the authors pointed out that fracture elongations of laminated composites are closely tied to strain

Introductory Chapter: Laminations - Theory and Applications

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13

Figure 11. (a, b) Proposed 3D view of a modified turbojet engine with laminated enclosure.

In Chapter 2, large deflection analysis of laminated composite plates using a higher order zigzag theory is discussed. This proposed refined theory is expected to ease the determination of deflections and stresses in composite sandwich laminate analysis. The presented model which incorporated an efficient C0 plate finite element (FE) was shown to accurately calculate the deflections as well as stresses for different geometries of composite and sandwich laminates. The results obtained agree with existing exact 3D elasticity solutions for thin, moder-

In Chapter 3, the dynamic modeling of a serial link robot laminated with plastic film to improve waterproofing and dustproofing of serial link robots is the main focus. It also discusses how to improve lubrication between the links and the film through an insulating fluid that was encapsulated in the plastic film by employing detailed appropriate mathematical analysis. Dynamic performance of the laminated body was validated for flexural rigidities. Through detailed mathematical analysis, the dynamic performance of the laminated body was

In Chapter 4, advanced technologies in manufacturing 3D layered structures for defense and aerospace systems is the main focus of the study. It emphasizes the importance of additive manufacturing techniques and their robust applications in aeronautical and defense structural systems. This study further reveals that not all advanced materials and alloys can be automatically layered by rapid prototyping system or machine. It also highlighted how efforts are underway to apply automated layering technology in many materials with potential applications in nowadays plastics and reinforced polymers for ease of manufacturing 3D parts. The discussion in this chapter presented a review of additive manufacturing history and the

indexing exponent and strain rate parameter.

ately thick composite and sandwich laminates.

confirmed for different flexural rigidities.

potential advantages the proposed method would offer.

Figure 9. (a, b) Proposed civilian aircraft with modified engines fitted with laminated vibration attenuation devices. (c) Proposed military jet fighter with modified engine fitted with laminated devices. (d) Proposed aircraft engine ducting with laminated devices.

Figure 10. Jet engine vibration isolation system with laminated devices.

The intention of this book is to describe new concepts of producing laminated structures and possible modern engineering applications as demonstrated in the introductory chapter and other chapters. The introductory chapter of this book offers new engine design concepts on how laminated enclosures can be useful in machine, ocean and aerospace structures. The thrust is to showcase how the world can benefit from these innovative concepts.

Figure 11. (a, b) Proposed 3D view of a modified turbojet engine with laminated enclosure.

In Chapter 1, the focus is on the study of multiscale hierarchical structure and laminated strengthening and toughening mechanisms. Here, the authors emphasize how higher strength can be achieved in titanium matrix composites by adjusting the multiscale hierarchical structure. The chapter further deduced how the elastic properties and yield strength of laminated composites can be modulated through the "rule of average." It is also discussed in this chapter how impossible it is to predict fracture elongation and toughness via this rule. In addition, the authors pointed out that fracture elongations of laminated composites are closely tied to strain indexing exponent and strain rate parameter.

In Chapter 2, large deflection analysis of laminated composite plates using a higher order zigzag theory is discussed. This proposed refined theory is expected to ease the determination of deflections and stresses in composite sandwich laminate analysis. The presented model which incorporated an efficient C0 plate finite element (FE) was shown to accurately calculate the deflections as well as stresses for different geometries of composite and sandwich laminates. The results obtained agree with existing exact 3D elasticity solutions for thin, moderately thick composite and sandwich laminates.

In Chapter 3, the dynamic modeling of a serial link robot laminated with plastic film to improve waterproofing and dustproofing of serial link robots is the main focus. It also discusses how to improve lubrication between the links and the film through an insulating fluid that was encapsulated in the plastic film by employing detailed appropriate mathematical analysis. Dynamic performance of the laminated body was validated for flexural rigidities. Through detailed mathematical analysis, the dynamic performance of the laminated body was confirmed for different flexural rigidities.

In Chapter 4, advanced technologies in manufacturing 3D layered structures for defense and aerospace systems is the main focus of the study. It emphasizes the importance of additive manufacturing techniques and their robust applications in aeronautical and defense structural systems. This study further reveals that not all advanced materials and alloys can be automatically layered by rapid prototyping system or machine. It also highlighted how efforts are underway to apply automated layering technology in many materials with potential applications in nowadays plastics and reinforced polymers for ease of manufacturing 3D parts. The discussion in this chapter presented a review of additive manufacturing history and the potential advantages the proposed method would offer.

The intention of this book is to describe new concepts of producing laminated structures and possible modern engineering applications as demonstrated in the introductory chapter and other chapters. The introductory chapter of this book offers new engine design concepts on how laminated enclosures can be useful in machine, ocean and aerospace structures. The

Figure 9. (a, b) Proposed civilian aircraft with modified engines fitted with laminated vibration attenuation devices. (c) Proposed military jet fighter with modified engine fitted with laminated devices. (d) Proposed aircraft engine ducting

with laminated devices.

12 Lamination - Theory and Application

thrust is to showcase how the world can benefit from these innovative concepts.

Figure 10. Jet engine vibration isolation system with laminated devices.

## Nomenclature


εx<sup>2</sup> axial strain in layer-2

r density of laminate material

τxz shear stress at the interface of the laminates

ð Þ σ<sup>x</sup> <sup>1</sup> bending stress at the upper half of the laminates ð Þ σ<sup>x</sup> <sup>2</sup> bending stress at the lower half of the laminates

Address all correspondence to: charlesosheku2002@yahoo.com

Centre for Space Transport and Propulsion, National Space Research and Development Agency, Federal Ministry of Science and Technology, LASU Epe Campus, Epe, Lagos, Nigeria

Introductory Chapter: Laminations - Theory and Applications

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15

[1] Osheku CA, Olunloyo VOS, Damisa O, Akano TT. Acoustic pressure waves in vibrating 3- D laminated beam-plate enclosures. Advances in Acoustics and Vibration. 2009;2009,

[2] Olunloyo VOS, Osheku CA. On vibration and noise dissipation in ship and FPSO structures with smart systems. International Scholarly Research Network: Mechanical Engi-

[3] Osheku CA. Mechanics of static slip and energy dissipation in sandwich structures: Case of homogeneous elastic beams in transverse magnetic fields. International Scholarly

[4] Goodman LE, Klumpp JH. Analysis of slip damping with reference to turbine blade

[5] Olunloyo VOS, Osheku CA. On vibration and noise dissipation in ship and FPSO structures with smart systems. International Scholarly Research Network: Mechanical Engi-

Research Network: Mechanical Engineering. 2012;2012, Article ID 372019:23

γxz angular strain

ξ dummy variable υ Poisson's ratio

Author details

References

Article ID 853407:14

neering. 2012;2012, Article ID 127238:19

neering. 2012;2012, Article ID 127238:19

vibration. Journal of Applied Mechanics. 1956;23:421

Charles Attah Osheku


Nomenclature

14 Lamination - Theory and Application

b width of laminated beam

F applied end force amplitude

h depth of laminated beam H magnetic field intensity

L length of laminated beam

U<sup>1</sup> displacement of the lower laminate U<sup>2</sup> displacement of the upper laminate

W<sup>F</sup> dynamic response in Laplace transform plane

W~ dynamic response in Fourier transform plane

x space coordinate along the beam interface

W~ <sup>F</sup> dynamic response in Fourier-Laplace transform plane

z space coordinate perpendicular to the beam interface

P clamping pressure at the interface of the laminated beams

I moment of inertia

t time coordinate

W dynamic response

ε pressure gradient

μ dry friction coefficient

εx<sup>1</sup> axial strain in layer-1

μ<sup>m</sup> permeability of the medium

μ<sup>r</sup> relative permeability of the medium χ normalized magnetic field intensity

u, v velocity

u, \_ v\_ acceleration

B<sup>0</sup> magnetic flux density d/dx differential operator E electric field intensity E modulus of rigidity


## Author details

Charles Attah Osheku

Address all correspondence to: charlesosheku2002@yahoo.com

Centre for Space Transport and Propulsion, National Space Research and Development Agency, Federal Ministry of Science and Technology, LASU Epe Campus, Epe, Lagos, Nigeria

## References


**Section 2**

**Theory of Lamination**

## **Theory of Lamination**

**Chapter 2**

Provisional chapter

**Multiscale Hierarchical Structure and Laminated**

DOI: 10.5772/intechopen.69976

Metal matrix composites with multiscale hierarchical structure and laminated structure have been developed to provide a novel route to achieve high strength, toughness and ductility. In this chapter, a lot of scientific research has been carried out in the preparation, processing, properties and application of metal matrix composite. Many toughening mechanisms and fracture behavior of composites with multiscale hierarchical structure and laminated structure are overviewed. It is revealed that elastic property and yield strength of laminated composites follow the "rule of average." However, the estimation of fracture elongation and fracture toughness is complex, which is inconsistent with the "rule of average." The fracture elongation of laminated composites is related to the layer thickness size, interface, gradient structure, strain hardening exponent, strain rate parameter and tunnel crack, which are accompanied with crack deflection, crack blunting, crack bridging, stress redistribution, local stress deformation, interfacial delamination crack and so on. The concept of laminated composites can be extended by applying different combination of individual layer, and provides theoretical as well as experimental fundamentals on strengthening and toughening of metal

Keywords: metal matrix composites, multiscale hierarchical structure, laminated

Since the 1980s in the last century, as a result of the significant progress in the material science and technology, many metal matrix composites (MMCs) reinforced with fiber,

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is 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.

Multiscale Hierarchical Structure and Laminated

**Strengthening and Toughening Mechanisms**

Strengthening and Toughening Mechanisms

Baoxi Liu, Lujun Huang, Lin Geng and Fuxing Yin

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69976

Fuxing Yin

Abstract

matrix composites.

1. Introduction

structure, rule of average, tunnel crack

Baoxi Liu, Lujun Huang, Lin Geng and

## **Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms** Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms

DOI: 10.5772/intechopen.69976

Baoxi Liu, Lujun Huang, Lin Geng and Fuxing Yin Baoxi Liu, Lujun Huang, Lin Geng and

Additional information is available at the end of the chapter Fuxing Yin

http://dx.doi.org/10.5772/intechopen.69976 Additional information is available at the end of the chapter

## Abstract

Metal matrix composites with multiscale hierarchical structure and laminated structure have been developed to provide a novel route to achieve high strength, toughness and ductility. In this chapter, a lot of scientific research has been carried out in the preparation, processing, properties and application of metal matrix composite. Many toughening mechanisms and fracture behavior of composites with multiscale hierarchical structure and laminated structure are overviewed. It is revealed that elastic property and yield strength of laminated composites follow the "rule of average." However, the estimation of fracture elongation and fracture toughness is complex, which is inconsistent with the "rule of average." The fracture elongation of laminated composites is related to the layer thickness size, interface, gradient structure, strain hardening exponent, strain rate parameter and tunnel crack, which are accompanied with crack deflection, crack blunting, crack bridging, stress redistribution, local stress deformation, interfacial delamination crack and so on. The concept of laminated composites can be extended by applying different combination of individual layer, and provides theoretical as well as experimental fundamentals on strengthening and toughening of metal matrix composites.

Keywords: metal matrix composites, multiscale hierarchical structure, laminated structure, rule of average, tunnel crack

## 1. Introduction

Since the 1980s in the last century, as a result of the significant progress in the material science and technology, many metal matrix composites (MMCs) reinforced with fiber,

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

whisker and particle reinforcements, especially titanium matrix composites (TMCs), have been developed in many applications in order to improve special stiffness, special strength and thermal stability, such as automobile, nuclear, aerospace, armor and sectors and so on [1]. However, MMCs usually exhibit low fracture elongation and toughness in comparison with monolithic metal and alloys due to the addition of reinforcements, which seriously limits their applications [2]. Herein, control of the microstructure and interface characteristics are the important keys to the development of high strength and toughness MMCs and have been achieved by thermomechanical, heat treatment, alloy composition, reinforcement selection and reinforcement surface modification methods [3]. These favorite methods contain many strengthening effects, such as grain refinement, dispersion, solution, work hardening and phase transition strengthening mechanisms. However, grain refinement strengthening is the only method to improve the strength without decreasing fracture elongation, but it is usually no longer suitable when the grain sizes fall below 1 μm [4]. Therefore, it is believed that innovative design of metal matrix composites should make a guide to improve the comprehensive mechanical properties of superior strength, toughness and fracture elongation [2].

Recently, more and more scientists have realized that the nature has provided numerous smart structures for desired mechanical and functional properties after many billion years of stringent evolution [2]. A good idea was inspired from bio-inspired microstructures in nature. Nature materials always reveal multiscale hierarchical structures, such as nacre, which contains laminated, brick, protrusions, bridge and network structures from macroscale, mesoscale, microscale, nanoscale and even the atom-scale as shown in Figure 1. They can perfectly delicate themselves at different levels and play a synergistic strengthening and toughening effect during the deformation and fracture process, which results in a superior combination of performances, such as the 3000 times higher toughness of mollusk shells than their individual constituents [5].

> In 2010, Lu [6] proposed that the metal matrix composites can obtain superior tensile strength and fracture elongation by designing multiscale hierarchical structures. For example, surface mechanical grinding treatment (SMGT) can induce gradient grain distribution and laminated structure in Cu and Ni matrix, respectively. The inspired hierarchical structure is useful to the enhanced fracture toughness of metal matrix composites apart from the monolithic MMCs as shown in Figure 2 [4, 7, 8]. Palkovic et al. [9] reported that slurry can be designed by the ideal of multiscale hierarchical structures, which can be expected to replace the silicate cement with huge energy consumption. Zheng et al. [10] fabricated multiscale hierarchical metallic metamaterials by three-dimensional (3D) architectures as shown in Figure 3, obtaining supe-

Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms

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21

Figure 2. Strength-ductility synergy of metals with gradient grained structure [4].

On the basis of ideal that metal matrix composites with multiscale hierarchical structures can obtain superior mechanical properties, this proposal aims to synthesize laminated Ti-(TiBw/ Ti) composites by using powder metallurgy, which contains laminated, network and needlelike structures as shown in Figure 4, in order to achieve reinforcing, toughening and plasticizing effect, and to clarify the influence rule of fabrication parameters, laminated, interfacial and network structure parameters on microstructural evolution and mechanical behavior of

rior tensile elasticity of 20%.

Figure 1. The multiscale hierarchical structures of nacre at several length scales [2, 5].

Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms http://dx.doi.org/10.5772/intechopen.69976 21

Figure 2. Strength-ductility synergy of metals with gradient grained structure [4].

whisker and particle reinforcements, especially titanium matrix composites (TMCs), have been developed in many applications in order to improve special stiffness, special strength and thermal stability, such as automobile, nuclear, aerospace, armor and sectors and so on [1]. However, MMCs usually exhibit low fracture elongation and toughness in comparison with monolithic metal and alloys due to the addition of reinforcements, which seriously limits their applications [2]. Herein, control of the microstructure and interface characteristics are the important keys to the development of high strength and toughness MMCs and have been achieved by thermomechanical, heat treatment, alloy composition, reinforcement selection and reinforcement surface modification methods [3]. These favorite methods contain many strengthening effects, such as grain refinement, dispersion, solution, work hardening and phase transition strengthening mechanisms. However, grain refinement strengthening is the only method to improve the strength without decreasing fracture elongation, but it is usually no longer suitable when the grain sizes fall below 1 μm [4]. Therefore, it is believed that innovative design of metal matrix composites should make a guide to improve the comprehensive mechanical properties of superior strength, toughness

Recently, more and more scientists have realized that the nature has provided numerous smart structures for desired mechanical and functional properties after many billion years of stringent evolution [2]. A good idea was inspired from bio-inspired microstructures in nature. Nature materials always reveal multiscale hierarchical structures, such as nacre, which contains laminated, brick, protrusions, bridge and network structures from macroscale, mesoscale, microscale, nanoscale and even the atom-scale as shown in Figure 1. They can perfectly delicate themselves at different levels and play a synergistic strengthening and toughening effect during the deformation and fracture process, which results in a superior combination of performances, such as the 3000 times higher toughness of mollusk shells than their individual

Figure 1. The multiscale hierarchical structures of nacre at several length scales [2, 5].

and fracture elongation [2].

20 Lamination - Theory and Application

constituents [5].

In 2010, Lu [6] proposed that the metal matrix composites can obtain superior tensile strength and fracture elongation by designing multiscale hierarchical structures. For example, surface mechanical grinding treatment (SMGT) can induce gradient grain distribution and laminated structure in Cu and Ni matrix, respectively. The inspired hierarchical structure is useful to the enhanced fracture toughness of metal matrix composites apart from the monolithic MMCs as shown in Figure 2 [4, 7, 8]. Palkovic et al. [9] reported that slurry can be designed by the ideal of multiscale hierarchical structures, which can be expected to replace the silicate cement with huge energy consumption. Zheng et al. [10] fabricated multiscale hierarchical metallic metamaterials by three-dimensional (3D) architectures as shown in Figure 3, obtaining superior tensile elasticity of 20%.

On the basis of ideal that metal matrix composites with multiscale hierarchical structures can obtain superior mechanical properties, this proposal aims to synthesize laminated Ti-(TiBw/ Ti) composites by using powder metallurgy, which contains laminated, network and needlelike structures as shown in Figure 4, in order to achieve reinforcing, toughening and plasticizing effect, and to clarify the influence rule of fabrication parameters, laminated, interfacial and network structure parameters on microstructural evolution and mechanical behavior of

2. Multiscale hierarchical structure

structure and (d) needle-like structure [2].

formation of cracks.

Yin [11] concluded that the one-scale hierarchical distribution structure of reinforcements contains dual matrix, network, laminated, gradient structural and other bi-continuous types. Figure 5 shows the four structures in the titanium matrix composites reinforced with TiBw. Morsi and Patel et al. [12–14] fabricated a so-called "dual matrix" Ti-TiB composites as shown in Figure 5(a), which contains isolated reinforcement-rich phase and continuous Ti matrix phase, exhibiting superior fracture toughness. Huang et al. [2, 15, 16] fabricated one-scale network hierarchical TiBw/Ti composites and two-scale network hierarchical TiBw + Ti5Si3/Ti composites as shown in Figure 5(b). The elastic modulus, passion ratio and yield strength reach to the upper limit of Hashin-Shtrikman theory, as well as high oxidation resistance, superior fracture elongation, toughness and tensile strength. Han et al. [17] fabricated the laminated Ti-(TiB + La2O3)/Ti composites by powder metallurgy and hot rolling process as shown in Figure 5(c), which results in an improvement of 30% in elongation compared with monolithic composites with same volume fraction of reinforcements. In 2003, Panda et al. [18– 21] fabricated TiB/Ti and TiB2/Ti functional gradient composites by powder metallurgy as shown in Figure 5(d), which effectively weakens the thermal residual stress and inhibited the

Figure 4. The multiscale hierarchical Ti-(TiBw/Ti) composites. (a) Macrograph, (b) laminated structure, (c) network

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Figure 6 shows the fracture characteristics of the "dual matrix" Al2O3/Cu composites. The crack propagated along the Cu matrix and Al2O3 ceramic interface or inside the Cu matrix as

Figure 3. Multiscale hierarchical metallic metamaterials (a-f) [10].

titanium matrix composites. By using tensile and bending tests, the influence regular pattern of sintering temperature, layer thickness, layer thickness ratio and reinforcement volume fraction on mechanical properties should be investigated. It may solve the inverted shortcoming between strength and toughness. Through design and optimization of titanium matrix composites with two-scale hierarchical structures, this work will provide theoretical as well as experimental fundamentals on strengthening and toughening of metal matrix composites.

Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms http://dx.doi.org/10.5772/intechopen.69976 23

Figure 4. The multiscale hierarchical Ti-(TiBw/Ti) composites. (a) Macrograph, (b) laminated structure, (c) network structure and (d) needle-like structure [2].

## 2. Multiscale hierarchical structure

titanium matrix composites. By using tensile and bending tests, the influence regular pattern of sintering temperature, layer thickness, layer thickness ratio and reinforcement volume fraction on mechanical properties should be investigated. It may solve the inverted shortcoming between strength and toughness. Through design and optimization of titanium matrix composites with two-scale hierarchical structures, this work will provide theoretical as well as experimental fundamentals on strengthening and toughening of metal matrix

Figure 3. Multiscale hierarchical metallic metamaterials (a-f) [10].

composites.

22 Lamination - Theory and Application

Yin [11] concluded that the one-scale hierarchical distribution structure of reinforcements contains dual matrix, network, laminated, gradient structural and other bi-continuous types. Figure 5 shows the four structures in the titanium matrix composites reinforced with TiBw. Morsi and Patel et al. [12–14] fabricated a so-called "dual matrix" Ti-TiB composites as shown in Figure 5(a), which contains isolated reinforcement-rich phase and continuous Ti matrix phase, exhibiting superior fracture toughness. Huang et al. [2, 15, 16] fabricated one-scale network hierarchical TiBw/Ti composites and two-scale network hierarchical TiBw + Ti5Si3/Ti composites as shown in Figure 5(b). The elastic modulus, passion ratio and yield strength reach to the upper limit of Hashin-Shtrikman theory, as well as high oxidation resistance, superior fracture elongation, toughness and tensile strength. Han et al. [17] fabricated the laminated Ti-(TiB + La2O3)/Ti composites by powder metallurgy and hot rolling process as shown in Figure 5(c), which results in an improvement of 30% in elongation compared with monolithic composites with same volume fraction of reinforcements. In 2003, Panda et al. [18– 21] fabricated TiB/Ti and TiB2/Ti functional gradient composites by powder metallurgy as shown in Figure 5(d), which effectively weakens the thermal residual stress and inhibited the formation of cracks.

Figure 6 shows the fracture characteristics of the "dual matrix" Al2O3/Cu composites. The crack propagated along the Cu matrix and Al2O3 ceramic interface or inside the Cu matrix as

shown in Figure 6(b, c). The two tortuous propagation paths absorbed a lot of fracture energy and plastic deformation work, respectively, which are benefit to the high fracture toughness compared to the popular belief that crack propagation occurs in the brittle ceramic phase. Herein, the interface fracture is attributed to the high thermal tensile stress at the interface

Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms

Huang et al. [2, 23, 24] concluded that the titanium matrix composites with network structure correspond to the upper limit of Hashin-Shtrikman (H-S) bound. Based on H-S bounds proposed by Hashin and Shtrikman in 1963 [25, 26], the reinforcement with network structure can be viewed as "stiffer" phase, which is continuous and encapsulates the titanium matrix as shell structure. The elastic modulus of this structure can be predicted by the H-S upper bounds. However, the titanium matrix composites with homogeneous distribution of reinforcement can be taken as that the soft phase encapsulates the stiffer phase, whose elastic modulus is near

EHS�upper <sup>¼</sup> Ew½EwVw <sup>þ</sup> Emð<sup>2</sup> � VwÞ�

EHS�lower <sup>¼</sup> Em½Emð<sup>1</sup> � VwÞ þ Ewð<sup>1</sup> <sup>þ</sup> VwÞ�

where Ew and Em are the elastic modulus of whisker reinforcement and matrix, respectively, Vw and Vm are the volume fractions of the whisker and matrix, respectively. The network structure can enhance the strengthening and toughening effect at room temperature, which is similar with the grain-boundary strengthening mechanism. At high temperature, it can also overcome the grain-boundary weakening behavior. Therefore, TiBw/Ti composites with network struc-

Figure 8 shows the illustration diagram of oxidation and fracture resistant mechanisms of titanium matrix composites with network structure [2]. After oxidation process of 140 h at 973K, the surface of titanium matrix composites with network structure is still smooth without any spallation and cracks. The in situ reacted TiC particle forms a dense continuous wall as shown in Figure 8(b), which can effectively inhibit the oxidation velocity as shown in Figure 8(c) [2, 27].

EmVw <sup>þ</sup> Ewð<sup>2</sup> � Vw<sup>Þ</sup> (1)

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25

Ewð<sup>1</sup> � VwÞ þ Emð<sup>2</sup> � Vw<sup>Þ</sup> (2)

to the lower limit of H-S bound, and the H-S bounds can be expressed as follows [2]:

ture obtain superior room and high-temperature properties as shown in Figure 7.

Figure 7. It shows the strengthening and toughening mechanisms of TiBw/Ti composites.

developed during the fabrication process [22].

Figure 5. Four structural types of TiBw/Ti composites. (a) Dual matrix [12], (b) network [2], (c) laminated [17] and (d) gradient structure [18].

Figure 6. The fracture morphology of Al2O3/Cu composites [22]. (a) Microstructure, (b) interface fracture and (c) matrix fracture.

shown in Figure 6(b, c). The two tortuous propagation paths absorbed a lot of fracture energy and plastic deformation work, respectively, which are benefit to the high fracture toughness compared to the popular belief that crack propagation occurs in the brittle ceramic phase. Herein, the interface fracture is attributed to the high thermal tensile stress at the interface developed during the fabrication process [22].

Huang et al. [2, 23, 24] concluded that the titanium matrix composites with network structure correspond to the upper limit of Hashin-Shtrikman (H-S) bound. Based on H-S bounds proposed by Hashin and Shtrikman in 1963 [25, 26], the reinforcement with network structure can be viewed as "stiffer" phase, which is continuous and encapsulates the titanium matrix as shell structure. The elastic modulus of this structure can be predicted by the H-S upper bounds. However, the titanium matrix composites with homogeneous distribution of reinforcement can be taken as that the soft phase encapsulates the stiffer phase, whose elastic modulus is near to the lower limit of H-S bound, and the H-S bounds can be expressed as follows [2]:

$$E\_{HS-upper} = \frac{E\_w[E\_w V\_w + E\_m(2 - V\_w)]}{E\_m V\_w + E\_w(2 - V\_w)}\tag{1}$$

$$E\_{HS-lower} = \frac{E\_m[E\_m(1 - V\_w) + E\_w(1 + V\_w)]}{E\_w(1 - V\_w) + E\_m(2 - V\_w)}\tag{2}$$

where Ew and Em are the elastic modulus of whisker reinforcement and matrix, respectively, Vw and Vm are the volume fractions of the whisker and matrix, respectively. The network structure can enhance the strengthening and toughening effect at room temperature, which is similar with the grain-boundary strengthening mechanism. At high temperature, it can also overcome the grain-boundary weakening behavior. Therefore, TiBw/Ti composites with network structure obtain superior room and high-temperature properties as shown in Figure 7.

Figure 8 shows the illustration diagram of oxidation and fracture resistant mechanisms of titanium matrix composites with network structure [2]. After oxidation process of 140 h at 973K, the surface of titanium matrix composites with network structure is still smooth without any spallation and cracks. The in situ reacted TiC particle forms a dense continuous wall as shown in Figure 8(b), which can effectively inhibit the oxidation velocity as shown in Figure 8(c) [2, 27].

Figure 7. It shows the strengthening and toughening mechanisms of TiBw/Ti composites.

Figure 5. Four structural types of TiBw/Ti composites. (a) Dual matrix [12], (b) network [2], (c) laminated [17] and (d)

Figure 6. The fracture morphology of Al2O3/Cu composites [22]. (a) Microstructure, (b) interface fracture and (c) matrix

gradient structure [18].

24 Lamination - Theory and Application

fracture.

Figure 8. Schematic illustration showing the oxidation and fracture resistant mechanisms of titanium matrix composites with network structure [2]. (a–c) Oxidation and (d–f) fracture resistance.

During the fracture process, the TiBw/Ti6Al4V composites are mainly fractured at the network boundary as shown in Figure 8(d, e), and the matrix tearing is also the fracture characteristics with low volume fraction of reinforcement. Figure 8(f) illustrates the crack propagation paths and toughening behaviors of TiBw/Ti6Al4V composites with low volume fraction [2]. Crack A shows that the crack perpendicular to the network boundary cannot propagate through the softer matrix phase because of low crack size, which can play strengthening and toughening effect in the composites. However, the different propagation paths along path I or path II mainly depend on their absorbing energy. The effect reinforcement strength and absorbing energy can be expressed as follows [2, 28]:

$$
\sigma = \sqrt{\frac{\pi E G}{2(1 - \nu^2)d}}\tag{3}
$$

where σ is the effective reinforcement strength and G is the critical strain energy release rate for the dynamic propagation of a crack. ν is the Poisson's ratio, d is the reinforcement size along the crack direction, S is the crack surface area, L is the open crack length and A is the parameter

The absorbing energy of crack propagation along path I (QI) and II (QII) can be deduced as

s

s

The concept of functionally gradient Ti-TiB composites arose as a result of efforts to prevent the cracking in TiB-Ti joints induced by the coefficient of thermal expansion (CTE) mismatch during high-temperature processing. Functionally gradient materials (FGM) effectively decrease the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πEIGI <sup>2</sup>ð<sup>1</sup> � <sup>ν</sup><sup>2</sup> IÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πEIIGII <sup>2</sup>ð<sup>1</sup> � <sup>ν</sup><sup>2</sup> IIÞ

:ð3aÞ 3

Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms

:ð2aÞ 3

<sup>2</sup> (5)

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27

<sup>2</sup> (6)

QI ¼ AL � B

QII ¼ AL � B

Therefore, the crack can propagate along path I when QI ≤ QII.

Figure 9. The crack propagation of functionally gradient TiB-Ti composites [29].

related to the surface and length.

Eqs. (5) and (6), respectively [2].

$$Q = \sigma S.L = AL \sqrt{\frac{\pi E G}{2(1 - \nu^2)d}} d^2 = AL \sqrt{\frac{\pi E G}{2(1 - \nu^2)}} d^2 \tag{4}$$

where σ is the effective reinforcement strength and G is the critical strain energy release rate for the dynamic propagation of a crack. ν is the Poisson's ratio, d is the reinforcement size along the crack direction, S is the crack surface area, L is the open crack length and A is the parameter related to the surface and length.

The absorbing energy of crack propagation along path I (QI) and II (QII) can be deduced as Eqs. (5) and (6), respectively [2].

$$Q\_{\rm I} = AL \cdot B \sqrt{\frac{\pi E\_{\rm I} G\_{\rm I}}{2(1 - \nu\_{\rm I}^2)}} \cdot (3a)^{\frac{3}{2}} \tag{5}$$

$$\mathbf{Q}\_{\rm II} = A \mathbf{L} \cdot \mathbf{B} \sqrt{\frac{\pi E\_{\rm II} \mathbf{G}\_{\rm II}}{2(1 - \nu\_{\rm II}^2)}}. (2a)^{\frac{3}{2}} \tag{6}$$

Therefore, the crack can propagate along path I when QI ≤ QII.

The concept of functionally gradient Ti-TiB composites arose as a result of efforts to prevent the cracking in TiB-Ti joints induced by the coefficient of thermal expansion (CTE) mismatch during high-temperature processing. Functionally gradient materials (FGM) effectively decrease the

Figure 9. The crack propagation of functionally gradient TiB-Ti composites [29].

During the fracture process, the TiBw/Ti6Al4V composites are mainly fractured at the network boundary as shown in Figure 8(d, e), and the matrix tearing is also the fracture characteristics with low volume fraction of reinforcement. Figure 8(f) illustrates the crack propagation paths and toughening behaviors of TiBw/Ti6Al4V composites with low volume fraction [2]. Crack A shows that the crack perpendicular to the network boundary cannot propagate through the softer matrix phase because of low crack size, which can play strengthening and toughening effect in the composites. However, the different propagation paths along path I or path II mainly depend on their absorbing energy. The effect reinforcement strength and absorbing energy can be expressed

Figure 8. Schematic illustration showing the oxidation and fracture resistant mechanisms of titanium matrix composites

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πEG 2ð1 � ν<sup>2</sup>Þd

:d<sup>2</sup> <sup>¼</sup> AL

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πEG 2ð1 � ν<sup>2</sup>Þ

:d 3

s

(3)

<sup>2</sup> (4)

σ ¼

s

Q ¼ σ:S:L ¼ AL

with network structure [2]. (a–c) Oxidation and (d–f) fracture resistance.

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πEG 2ð1 � ν<sup>2</sup>Þd

as follows [2, 28]:

26 Lamination - Theory and Application

thermal stresses and delay the plastic yielding and the fracture failure of the metallic layer by the load shearing of the ceramic [18–21]. Meanwhile, it is a better overall use of the available materials by distributing them in a suitable spatial direction. Ma et al. concluded that the main crack deflected to the layer interface, and it is interesting to note that crack bifurcation was initiated at the interface between layers 5 and 4, resulting into the stepwise fracture process rather than catastrophic failure as shown in Figure 9. Moreover, it has good ballistic properties that make it popular armor application [21, 29].

Min et al. [30, 31] fabricated a multilayered Ti-10Mo-1Fe/Ti-10Mo-3Fe composites by hot roll bonding and subsequent heat treatment as shown in Figure 10(a). The layer interface can be detected by Fe concentration as shown in Figure 10(b). It is interesting to note that Ti-10Mo-1Fe layer was deformed by twinning, whereas deformation characteristic of Ti-10Mo-3Fe is dislocation slip as shown in Figure 10(c, d). The artificially multilayered twinning/dislocation slip deformation characteristics can provide a high strength-ductility combination over a

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29

Actually, composite materials with laminated structures have been a design ideal to improve the mechanical and functional properties from antiquity. "The Iliad" of homer has described that Achilles's shield has five layers, and the sequence of laminates is bronze/tin/gold/tin/ bronze. The Aeneas's spear penetrated the first two layers but trapped by gold layer during combat, revealing superior mechanical properties. In 1837, an iron-laminated composite dating as far back as 2750 BC was found in the Great Pyramid in Gizeh, Egypt, and many layers of wrought iron have been inexpertly welded tougher by hammering as shown in Figure 11 [32, 33]. Lamination architect ideal may be attributed to the following three reasons: (1) only thin sheets could be carburized in the earliest form of wrought iron, so lamination was a good way to create bulk composite materials; (2) it is no expensive to sandwich hard rare steel between more cheap common materials and (3) optimizing the combination of strength, toughness and sharpness is the basis for lamination from the mechanical view of materials. Therefore, many weapons were made by lamination design from ancient, such as Adze blade,

Since 1988, American scientists took the sandwich materials into the aircraft door C-17, and the investigations of laminated and functional gradient metal composites enter into the period of rapid development [34]. Jiang et al. [35] fabricated laminated Ti-Al3Ti composites using reaction vacuum sintering in 2003. During the impact testing process, the laminated composites can absorb lots of impact works through forming delamination cracks, and it is expected to be applied in the armor equipment and ballistic system. In 2006, Japan government sponsored a research project named "the layer integrated steel and metals," which aims to fabricate multilayer steel with high strength (>1200 MPa) and superior tensile elongation (>20%) through the composite design of high strength steels and ductile steel layers [36]. In 2008, Inoue et al. [37] fabricated multilayered SS420/SS304 steel with high toughness and reported that fracture elongation is gradually increasing with the decrease of layer thickness, which is attributed to the decreasing size of tunnel cracks. Nambu et al. [38] reported that multilayered steel with strong interfacial bonding strength exhibits superior longitudinal fracture elongation compared with the laminated composites with weak interfaces. Moreover, they found that the laminated structures can change the deformation mode of martensite layer, transiting from

Strength and toughness are the most important performance in traditional structural materials [1–3, 40]. Recently, with the development of national defense and civil sectors, such as aerospace, nuclear, shipbuilding, automobile and so on, a high demand of strength and

wider range to meet industrially applicable strength and formability requirements.

Japanese sword, Merovingian blade and Chinese blade [33].

poor work hardening to improved hardening [39].

3. Laminated structure

Figure 10. (a–d) {332}<113> Twinning and dislocation deformation of laminated Ti composites [30].

Min et al. [30, 31] fabricated a multilayered Ti-10Mo-1Fe/Ti-10Mo-3Fe composites by hot roll bonding and subsequent heat treatment as shown in Figure 10(a). The layer interface can be detected by Fe concentration as shown in Figure 10(b). It is interesting to note that Ti-10Mo-1Fe layer was deformed by twinning, whereas deformation characteristic of Ti-10Mo-3Fe is dislocation slip as shown in Figure 10(c, d). The artificially multilayered twinning/dislocation slip deformation characteristics can provide a high strength-ductility combination over a wider range to meet industrially applicable strength and formability requirements.

## 3. Laminated structure

thermal stresses and delay the plastic yielding and the fracture failure of the metallic layer by the load shearing of the ceramic [18–21]. Meanwhile, it is a better overall use of the available materials by distributing them in a suitable spatial direction. Ma et al. concluded that the main crack deflected to the layer interface, and it is interesting to note that crack bifurcation was initiated at the interface between layers 5 and 4, resulting into the stepwise fracture process rather than catastrophic failure as shown in Figure 9. Moreover, it has good ballistic properties

Figure 10. (a–d) {332}<113> Twinning and dislocation deformation of laminated Ti composites [30].

that make it popular armor application [21, 29].

28 Lamination - Theory and Application

Actually, composite materials with laminated structures have been a design ideal to improve the mechanical and functional properties from antiquity. "The Iliad" of homer has described that Achilles's shield has five layers, and the sequence of laminates is bronze/tin/gold/tin/ bronze. The Aeneas's spear penetrated the first two layers but trapped by gold layer during combat, revealing superior mechanical properties. In 1837, an iron-laminated composite dating as far back as 2750 BC was found in the Great Pyramid in Gizeh, Egypt, and many layers of wrought iron have been inexpertly welded tougher by hammering as shown in Figure 11 [32, 33]. Lamination architect ideal may be attributed to the following three reasons: (1) only thin sheets could be carburized in the earliest form of wrought iron, so lamination was a good way to create bulk composite materials; (2) it is no expensive to sandwich hard rare steel between more cheap common materials and (3) optimizing the combination of strength, toughness and sharpness is the basis for lamination from the mechanical view of materials. Therefore, many weapons were made by lamination design from ancient, such as Adze blade, Japanese sword, Merovingian blade and Chinese blade [33].

Since 1988, American scientists took the sandwich materials into the aircraft door C-17, and the investigations of laminated and functional gradient metal composites enter into the period of rapid development [34]. Jiang et al. [35] fabricated laminated Ti-Al3Ti composites using reaction vacuum sintering in 2003. During the impact testing process, the laminated composites can absorb lots of impact works through forming delamination cracks, and it is expected to be applied in the armor equipment and ballistic system. In 2006, Japan government sponsored a research project named "the layer integrated steel and metals," which aims to fabricate multilayer steel with high strength (>1200 MPa) and superior tensile elongation (>20%) through the composite design of high strength steels and ductile steel layers [36]. In 2008, Inoue et al. [37] fabricated multilayered SS420/SS304 steel with high toughness and reported that fracture elongation is gradually increasing with the decrease of layer thickness, which is attributed to the decreasing size of tunnel cracks. Nambu et al. [38] reported that multilayered steel with strong interfacial bonding strength exhibits superior longitudinal fracture elongation compared with the laminated composites with weak interfaces. Moreover, they found that the laminated structures can change the deformation mode of martensite layer, transiting from poor work hardening to improved hardening [39].

Strength and toughness are the most important performance in traditional structural materials [1–3, 40]. Recently, with the development of national defense and civil sectors, such as aerospace, nuclear, shipbuilding, automobile and so on, a high demand of strength and

Figure 11. The fabrication process of slip blacksmiths [32, 33].

toughness is necessary for structural materials to adapt the increasingly stringent environment and application conditions [4, 6, 41]. For centuries, scientists have been conducting investigations to develop composite materials with laminated structures to improve strength and toughness [32–34, 42]. Typical laminated composites, such as ceramic-ceramic [43], metalmetallic glass [44], metal-ceramic [45], metal-metal [46], metal-metal composites [47] and metal-ceramic-intermetallic [48] systems have shown desirable structural properties as a result of many layers and interfaces, and these laminated composites allow for the possibility of combining the high ductility and toughness of soft layers with the high strength of hard layers. 4. Strengthening and toughening mechanisms of laminated structure

nisms [32, 49].

50% weak (or ductile) component [40].

Typical laminated composites have shown desirable mechanical properties, such as tensile strength, fracture toughness, compact toughness, formability, bending properties and fracture elongation. A number of papers have been published on the reinforcing and toughening of laminated composites and illustrated that laminated metal composites are especially effective in improving fracture-related properties by various intrinsic and extrinsic mecha-

Figure 12. (a) Yield strength and (b) elongation of different metals consisting of 50% strong (of brittle) component and

Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms

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Recently, the strengthening and toughening mechanisms of laminated composites mainly focus on the investigation of plastic deformation, crack propagation and fracture morphology. Herein, Cetin and Hutchinson et al. [50–52] carried out the depth investigation on the tunnel cracks, plastic bifurcations and delamination cracks and found that the fracture elongation is related to the layer thickness, layer thickness ratio, yield strength, ultimate strength, interfacial bonding strength, work hardening index, strain rate sensitivity exponent and interfacial reaction phase of individual layer. Serror [53] reported that multilocalized necking of laminated composites is benefit to relieve the localized stress concentration and improve plastic deformation capability during maximum tensile testing process based on the plastic bifurcation theory and finite element method (FEM) as shown in Figure 13. Zhang et al. [54] fabricated multilayer Cu/Al sheets using rolling bonding method and found that laminated structure can effectively

Though the design concept, strengthening and toughening mechanisms of laminated composites are ancient and attempted for a long time, the relationship between microstructure and mechanical properties are still unclear and ambiguous. Lesuer et al. [32] concluded that the elastic property and yield strength of laminated composites follow the "rule of average." However, the estimation of fracture elongation and fracture strength is complex, which is inconsistent with the "rule of average" as shown in Figure 12 [40]. Therefore, the further investigation of the fabrication and structure factors of laminated composites may prove useful to understand the relationship between structure and performation. The results could be used as guide to design microstructures and improve mechanical properties by multilayer structure, which require a completely different approach from the investigations of conventional monolithic composite material.

Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms http://dx.doi.org/10.5772/intechopen.69976 31

Figure 12. (a) Yield strength and (b) elongation of different metals consisting of 50% strong (of brittle) component and 50% weak (or ductile) component [40].

## 4. Strengthening and toughening mechanisms of laminated structure

toughness is necessary for structural materials to adapt the increasingly stringent environment and application conditions [4, 6, 41]. For centuries, scientists have been conducting investigations to develop composite materials with laminated structures to improve strength and toughness [32–34, 42]. Typical laminated composites, such as ceramic-ceramic [43], metalmetallic glass [44], metal-ceramic [45], metal-metal [46], metal-metal composites [47] and metal-ceramic-intermetallic [48] systems have shown desirable structural properties as a result of many layers and interfaces, and these laminated composites allow for the possibility of combining the high ductility and toughness of soft layers with the high strength of hard layers.

Figure 11. The fabrication process of slip blacksmiths [32, 33].

30 Lamination - Theory and Application

Though the design concept, strengthening and toughening mechanisms of laminated composites are ancient and attempted for a long time, the relationship between microstructure and mechanical properties are still unclear and ambiguous. Lesuer et al. [32] concluded that the elastic property and yield strength of laminated composites follow the "rule of average." However, the estimation of fracture elongation and fracture strength is complex, which is inconsistent with the "rule of average" as shown in Figure 12 [40]. Therefore, the further investigation of the fabrication and structure factors of laminated composites may prove useful to understand the relationship between structure and performation. The results could be used as guide to design microstructures and improve mechanical properties by multilayer structure, which require a completely different approach from the investigations of conventional mono-

lithic composite material.

Typical laminated composites have shown desirable mechanical properties, such as tensile strength, fracture toughness, compact toughness, formability, bending properties and fracture elongation. A number of papers have been published on the reinforcing and toughening of laminated composites and illustrated that laminated metal composites are especially effective in improving fracture-related properties by various intrinsic and extrinsic mechanisms [32, 49].

Recently, the strengthening and toughening mechanisms of laminated composites mainly focus on the investigation of plastic deformation, crack propagation and fracture morphology. Herein, Cetin and Hutchinson et al. [50–52] carried out the depth investigation on the tunnel cracks, plastic bifurcations and delamination cracks and found that the fracture elongation is related to the layer thickness, layer thickness ratio, yield strength, ultimate strength, interfacial bonding strength, work hardening index, strain rate sensitivity exponent and interfacial reaction phase of individual layer. Serror [53] reported that multilocalized necking of laminated composites is benefit to relieve the localized stress concentration and improve plastic deformation capability during maximum tensile testing process based on the plastic bifurcation theory and finite element method (FEM) as shown in Figure 13. Zhang et al. [54] fabricated multilayer Cu/Al sheets using rolling bonding method and found that laminated structure can effectively delay the premature localized necking behavior of Cu sheet. They also investigate that nanometer indentation behavior and found that size effect on the Cu/Cr, Cu/Au and Cu/Ni multilayered composites by magnetron sputtering. The ultimate strength would obviously be enhanced with the decreasing layer thickness in the nanoscale. However, the ductility and fracture toughness of laminated composites are ambiguous due to the complex interaction between layer thickness and interfacial structure [55]. The contribution of this section is to draw the series of deformation and fracture mechanisms of laminated composites and to provide guide for the reinforcing and toughening effect research and design.

Figure 14 shows the mechanical properties and toughening mechanisms of laminated ceramics with different interfacial status and constitute layers [56, 57]. Two kinds of laminated ceramic composites, the laminated composites with strong interface and weak interface, can be divided according to interfacial bonding strength. When the laminated ceramic composites afford bending force, the two kinds of laminated composites always improve the bending properties and fracture toughness through different toughening types. The laminated ceramic composites with strong interfacial strength possess high residual stress during fabrication process, and the main crack may appear to deflect or bifurcation on the strong interface during the fracture process induced by residual compression stress. Therefore, crack propagation path can be extended and the stress concentration at the crack tip can be mitigated, so the laminated ceramic composites with strong interfacial bonding strength can obtain high fracture toughness. The laminated ceramic composites with weak interfacial strength can also obtain high fracture toughness by delamination crack. When the main crack propagates into the weak interfaces forward, it also deflects or forms interfacial crack along the weak interface due to large amount of defects. The stress concentration can be relieved and the stress state can be changed into biaxial stress from complex triaxial stress, which resulted into the shielding

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The improvement in fracture toughness of laminated ceramic composites can be mainly attributed to two categories: intrinsic and extrinsic mechanisms as shown in Figure 15 [59, 60]. Intrinsic toughening mechanism implies in situ resistance of the microstructure (precipitates, particle spacing, grain size, morphology and so on) on crack propagation. For example, toughness can be enhanced by changing the nature, distribution and/or interface properties of second phase particles to suppress damage in the form of microcracking or microvoid

Figure 14. The mechanical properties and toughening mechanisms of laminated ceramics with different interfacial status

phenomenon of main crack tip [58].

and constitute layers [57].

## 4.1. High fracture toughness of laminated composites

The typical natural laminated composite is nacre material with the block-wall structures as shown in Figure 1. The nacre contains mineral layer (CaCO3) with the volume fraction of 99% and protein with the volume fraction of 1%. Nacre reveals high toughening mechanisms as follows: (1) the layer boundary with many projections reveals obvious bridging reinforcing effect; (2) interfacial delamination can play toughening effect when loaded on the nacre; (3) the staggered block-wall microstructures with a high sheet lap length and length/diameter ratio can decrease the relative displacement of fiber sheets during fracture process; (4) the mineral layer can be bonded by protein with three-dimensional network structure and (5) the nacre with convex-curved macroscopic morphology can state biaxial stress state, and the convexcurved plane only affords average compression stress, seldom of tensile stress and bending stress, which is a benefit to the ceramic matrix composites [5].

Figure 13. Three snapshots of the deformation and corresponding plastic strain contours of laminated composites under plane strain uniaxial tension [53].

Figure 14 shows the mechanical properties and toughening mechanisms of laminated ceramics with different interfacial status and constitute layers [56, 57]. Two kinds of laminated ceramic composites, the laminated composites with strong interface and weak interface, can be divided according to interfacial bonding strength. When the laminated ceramic composites afford bending force, the two kinds of laminated composites always improve the bending properties and fracture toughness through different toughening types. The laminated ceramic composites with strong interfacial strength possess high residual stress during fabrication process, and the main crack may appear to deflect or bifurcation on the strong interface during the fracture process induced by residual compression stress. Therefore, crack propagation path can be extended and the stress concentration at the crack tip can be mitigated, so the laminated ceramic composites with strong interfacial bonding strength can obtain high fracture toughness. The laminated ceramic composites with weak interfacial strength can also obtain high fracture toughness by delamination crack. When the main crack propagates into the weak interfaces forward, it also deflects or forms interfacial crack along the weak interface due to large amount of defects. The stress concentration can be relieved and the stress state can be changed into biaxial stress from complex triaxial stress, which resulted into the shielding phenomenon of main crack tip [58].

delay the premature localized necking behavior of Cu sheet. They also investigate that nanometer indentation behavior and found that size effect on the Cu/Cr, Cu/Au and Cu/Ni multilayered composites by magnetron sputtering. The ultimate strength would obviously be enhanced with the decreasing layer thickness in the nanoscale. However, the ductility and fracture toughness of laminated composites are ambiguous due to the complex interaction between layer thickness and interfacial structure [55]. The contribution of this section is to draw the series of deformation and fracture mechanisms of laminated composites and to

The typical natural laminated composite is nacre material with the block-wall structures as shown in Figure 1. The nacre contains mineral layer (CaCO3) with the volume fraction of 99% and protein with the volume fraction of 1%. Nacre reveals high toughening mechanisms as follows: (1) the layer boundary with many projections reveals obvious bridging reinforcing effect; (2) interfacial delamination can play toughening effect when loaded on the nacre; (3) the staggered block-wall microstructures with a high sheet lap length and length/diameter ratio can decrease the relative displacement of fiber sheets during fracture process; (4) the mineral layer can be bonded by protein with three-dimensional network structure and (5) the nacre with convex-curved macroscopic morphology can state biaxial stress state, and the convexcurved plane only affords average compression stress, seldom of tensile stress and bending

Figure 13. Three snapshots of the deformation and corresponding plastic strain contours of laminated composites under

provide guide for the reinforcing and toughening effect research and design.

4.1. High fracture toughness of laminated composites

32 Lamination - Theory and Application

stress, which is a benefit to the ceramic matrix composites [5].

plane strain uniaxial tension [53].

The improvement in fracture toughness of laminated ceramic composites can be mainly attributed to two categories: intrinsic and extrinsic mechanisms as shown in Figure 15 [59, 60]. Intrinsic toughening mechanism implies in situ resistance of the microstructure (precipitates, particle spacing, grain size, morphology and so on) on crack propagation. For example, toughness can be enhanced by changing the nature, distribution and/or interface properties of second phase particles to suppress damage in the form of microcracking or microvoid

Figure 14. The mechanical properties and toughening mechanisms of laminated ceramics with different interfacial status and constitute layers [57].

Figure 15. The intrinsic and extrinsic toughening mechanisms [60].

formation ahead of the crack tip. Extrinsic toughening mechanism indicates the microstructure behind of the crack tip to effectively decrease the crack driving force, which is initiated by the residual stress that reduces local stress intensity at the crack tip, such as delamination at weak interfaces [45], crack bridging [61], stress redistribution [32, 33], crack front convolution [32], local front convolution [33], crack trapping [48], ductile ligament [60], buckling [62], multiple cracking [63], crack bifurcation in the compressive stress layer [60], phase transformation under stress and compositional gradient [59] and crack deflection by mismatch of thermal expansion coefficients between different layers inevitably generates thermal residual stresses during subsequent cooling of laminated composites [64]. These behaviors need a significant amount of energy absorption and long crack propagation paths, resulting in an increase in fracture toughness as shown in Table 1 [33]. Therefore, extrinsic mechanisms, on the other hand, improve fracture properties by providing alternative crack propagation routes or reducing the driving force for crack growth through various processes that shield the crack tip from the applied stress [65].

### 4.2. Gradient laminated distribution of microstructure

It is all known that metal with nanoscale grains can obtain superior compression properties but low tensile ductility, which is due to premature strain localization during the tensile processing [4, 6]. However, metal with coarse grains can obtain high tensile fracture elongation but low tensile strength, which is attributed low strain hardening capacity. What happens if different metals with nanometer grains and with coarse grain are combined? In 2011, Lu et al. [7] introduced nanosize layer on the surface of copper during the serious plastic deformation process by surface mechanical attrition technology. The grain size can be changed from nanoscale into microscale with the increase of depth gradually. The laminated copper with gradient grain size distribution can obtain superior tensile yield strength and high tensile ductility, and the tensile ductility can reach to 100% without crack initiation. In 2014, Liu et al. introduce high-rate shear deformation with strain gradients on bulk nickel, forming nanometer-scale laminated structure with 20 nm, which exhibits a superior hardness of 6.4 GPA [8]. Lu

proposed that the gradient nanometer grained surface layer results in synergy effect of strength and ductility rather than the tradition trade-off between strength and ductility [4].

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Gao et al. [66] reveal that the laminated composites become insensitive to flaw when the layer thickness is lower than the critical flaw size according to the research on nacre. The stress field becomes more and more uniform as the layer thickness decreases. The critical flaw size is

4.3. Size effect in the laminated composites

Table 1. Toughening mechanisms for laminated metal composites [33].

expressed as follows:

Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms http://dx.doi.org/10.5772/intechopen.69976 35


Table 1. Toughening mechanisms for laminated metal composites [33].

proposed that the gradient nanometer grained surface layer results in synergy effect of strength and ductility rather than the tradition trade-off between strength and ductility [4].

## 4.3. Size effect in the laminated composites

formation ahead of the crack tip. Extrinsic toughening mechanism indicates the microstructure behind of the crack tip to effectively decrease the crack driving force, which is initiated by the residual stress that reduces local stress intensity at the crack tip, such as delamination at weak interfaces [45], crack bridging [61], stress redistribution [32, 33], crack front convolution [32], local front convolution [33], crack trapping [48], ductile ligament [60], buckling [62], multiple cracking [63], crack bifurcation in the compressive stress layer [60], phase transformation under stress and compositional gradient [59] and crack deflection by mismatch of thermal expansion coefficients between different layers inevitably generates thermal residual stresses during subsequent cooling of laminated composites [64]. These behaviors need a significant amount of energy absorption and long crack propagation paths, resulting in an increase in fracture toughness as shown in Table 1 [33]. Therefore, extrinsic mechanisms, on the other hand, improve fracture properties by providing alternative crack propagation routes or reducing the driving force for crack growth through various processes that shield the crack tip from

It is all known that metal with nanoscale grains can obtain superior compression properties but low tensile ductility, which is due to premature strain localization during the tensile processing [4, 6]. However, metal with coarse grains can obtain high tensile fracture elongation but low tensile strength, which is attributed low strain hardening capacity. What happens if different metals with nanometer grains and with coarse grain are combined? In 2011, Lu et al. [7] introduced nanosize layer on the surface of copper during the serious plastic deformation process by surface mechanical attrition technology. The grain size can be changed from nanoscale into microscale with the increase of depth gradually. The laminated copper with gradient grain size distribution can obtain superior tensile yield strength and high tensile ductility, and the tensile ductility can reach to 100% without crack initiation. In 2014, Liu et al. introduce high-rate shear deformation with strain gradients on bulk nickel, forming nanometer-scale laminated structure with 20 nm, which exhibits a superior hardness of 6.4 GPA [8]. Lu

the applied stress [65].

34 Lamination - Theory and Application

4.2. Gradient laminated distribution of microstructure

Figure 15. The intrinsic and extrinsic toughening mechanisms [60].

Gao et al. [66] reveal that the laminated composites become insensitive to flaw when the layer thickness is lower than the critical flaw size according to the research on nacre. The stress field becomes more and more uniform as the layer thickness decreases. The critical flaw size is expressed as follows:

$$L\_{critical} = \frac{\alpha \Gamma E}{\sigma^2} \tag{7}$$

where α is crack propagation parameter, about 1.3–1.9. E is the Young's modulus, and Γ is the fracture energy of atomically sharp cracks. Kumar et al. proved that when the layer thickness is small enough compared to the critical flaw size, Lcritical, no matter how high value of applied stress, even reach the ultimate strength of the material and the laminated composites will rupture wholly rather than classical damage mode, which experience nucleation, coalescence, growth and step-by-step crack propagation path. Once layer thickness is beyond the Lcritical and tunnel crack forms, the laminated composites will damage like domino, which does not need a lot of fracture energy. However, the wholly fracture process will spend more fracture energy. Therefore, laminated composites with small layer thickness can obtain more fracture strength and fracture toughness.

However, most experimental results reveal that the ductility of laminated composites is seriously limited with reducing layer thickness to a critical value (<100 nm), although fracture strength has been shifted. Zhang et al. [55] indicated that the fracture mode of the laminated Cu/Au and Cu/Cr composites changed from ductile shearing to brittle shearing with decreasing layer thickness as shown in Figure 16. Laminated composites with thick layer thickness exhibit high plastic deformation behavior, which is attributed to dislocation multiplication and gliding mechanism, until stress concentration at the interface due to the dislocation piling up, and the yield strength is consistent with the Hall-Patch equation in microscale. However, the interfacial bonding strength gradually decreases with decreasing layer thickness, and the dislocation can transmit across the layer interface to accommodate plastic deformation when the shear stress exceeds the interfacial bonding strength; the yield strength can be estimated by Hall-Patch equation. Therefore, the transition from ductile fracture to brittle fracture may be attributed to the layer thickness size, interfacial constraining effect and columnar grain orientation.

According to Hsia et al. [67] theoretical model as shown in Figure 17, the plastic deformation zone size (rp) at the onset of crack tip is expressed as follows:

$$r\_p = \frac{1}{2\pi} \left(\frac{\mathcal{K}\_{Ic}}{\sigma\_s}\right)^2 \tag{8}$$

with decreasing ductile layer thickness by introducing the laminated structure, the confined layer thickness makes dislocations pile up at interface. A low equilibrium number of dislocations can be obtained in the thin ductile layer thickness, which leads to the super-high tensile stress and low fracture toughness at the crack tip, and even cleavage occurs when the tensile stress reaches the theoretical fracture strength. An equilibrium number of dislocations exists at a given load level and ductile layer thickness. Dislocations emitted from the crack tip can not only blunt the crack, but also sent a back stress to the crack tip. Above two effects are mutually competing each other; the former reduces the tensile stress at the crack tip, whereas the latter prevents more dislocation emissions and results into the increase of

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Figure 16. Size effect of laminated composites with layer thickness of (a, b) 200nm and (c, d) 50nm [55].

tensile stress.

where KIc is the fracture toughness and σ<sup>s</sup> is the yield strength of the metal. Therefore, there is no confinement exists, and the fracture energy is independent of layer thickness when the layer thickness (h) is higher than 2rp. However, the plastic deformation behavior becomes confined when h ≤ 2rp, resulting in a lower fracture energy. Meanwhile, the tensile stress at the crack tip increases as h decreases. When h approaches the micro level, the dislocation activities within the thin ductile layer can no longer be treated with continuum plasticity theory, and the confined ductile layer with thin thickness makes dislocations pile up at interface, and even the preexisting dislocations with low number are unlikely to blunt the tunnel crack tip, which results in the transition from ductile to brittle behavior

Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms http://dx.doi.org/10.5772/intechopen.69976 37

Lcritical <sup>¼</sup> αΓ<sup>E</sup>

where α is crack propagation parameter, about 1.3–1.9. E is the Young's modulus, and Γ is the fracture energy of atomically sharp cracks. Kumar et al. proved that when the layer thickness is small enough compared to the critical flaw size, Lcritical, no matter how high value of applied stress, even reach the ultimate strength of the material and the laminated composites will rupture wholly rather than classical damage mode, which experience nucleation, coalescence, growth and step-by-step crack propagation path. Once layer thickness is beyond the Lcritical and tunnel crack forms, the laminated composites will damage like domino, which does not need a lot of fracture energy. However, the wholly fracture process will spend more fracture energy. Therefore, laminated composites with small layer thickness can obtain more fracture

However, most experimental results reveal that the ductility of laminated composites is seriously limited with reducing layer thickness to a critical value (<100 nm), although fracture strength has been shifted. Zhang et al. [55] indicated that the fracture mode of the laminated Cu/Au and Cu/Cr composites changed from ductile shearing to brittle shearing with decreasing layer thickness as shown in Figure 16. Laminated composites with thick layer thickness exhibit high plastic deformation behavior, which is attributed to dislocation multiplication and gliding mechanism, until stress concentration at the interface due to the dislocation piling up, and the yield strength is consistent with the Hall-Patch equation in microscale. However, the interfacial bonding strength gradually decreases with decreasing layer thickness, and the dislocation can transmit across the layer interface to accommodate plastic deformation when the shear stress exceeds the interfacial bonding strength; the yield strength can be estimated by Hall-Patch equation. Therefore, the transition from ductile fracture to brittle fracture may be attributed to the layer thickness size, interfacial constra-

According to Hsia et al. [67] theoretical model as shown in Figure 17, the plastic deformation

where KIc is the fracture toughness and σ<sup>s</sup> is the yield strength of the metal. Therefore, there is no confinement exists, and the fracture energy is independent of layer thickness when the layer thickness (h) is higher than 2rp. However, the plastic deformation behavior becomes confined when h ≤ 2rp, resulting in a lower fracture energy. Meanwhile, the tensile stress at the crack tip increases as h decreases. When h approaches the micro level, the dislocation activities within the thin ductile layer can no longer be treated with continuum plasticity theory, and the confined ductile layer with thin thickness makes dislocations pile up at interface, and even the preexisting dislocations with low number are unlikely to blunt the tunnel crack tip, which results in the transition from ductile to brittle behavior

KIc σs <sup>2</sup>

rp <sup>¼</sup> <sup>1</sup> 2π

strength and fracture toughness.

36 Lamination - Theory and Application

ining effect and columnar grain orientation.

zone size (rp) at the onset of crack tip is expressed as follows:

<sup>σ</sup><sup>2</sup> (7)

(8)

Figure 16. Size effect of laminated composites with layer thickness of (a, b) 200nm and (c, d) 50nm [55].

with decreasing ductile layer thickness by introducing the laminated structure, the confined layer thickness makes dislocations pile up at interface. A low equilibrium number of dislocations can be obtained in the thin ductile layer thickness, which leads to the super-high tensile stress and low fracture toughness at the crack tip, and even cleavage occurs when the tensile stress reaches the theoretical fracture strength. An equilibrium number of dislocations exists at a given load level and ductile layer thickness. Dislocations emitted from the crack tip can not only blunt the crack, but also sent a back stress to the crack tip. Above two effects are mutually competing each other; the former reduces the tensile stress at the crack tip, whereas the latter prevents more dislocation emissions and results into the increase of tensile stress.

Figure 17. Schematic illustration of fracture energy as a function of layer thickness [67].

#### 4.4. Interface layer

#### 4.4.1. Interfacial bonding strength

Nambu et al. [38] reported that laminated composites with strong interfacial bonding strength exhibit superior longitudinal fracture elongation compared with the laminated composites with weak interfaces. For a metal film deposited on an elastic substrate, Li et al. [68] predicted that strong interface can delay strain localization by finite element simulation (FEM). However, if the metal film approaches to form interfacial delamination crack due to weak interfacial bonding strength, the metal film becomes freestanding and is easy to form a localized necking. Koseki et al. [40] proved that delamination crack is related to the low interfacial toughness, while laminated composites with high interfacial toughness reveal high fracture elongation before failure, leading to localized necking and ductile fracture surface, exhibiting interfacial delamination crack and brittle fracture of brittle layer when the interface becomes unstable with the increase of strain. The debonding criterion is expressed as follows:

$$
\Gamma\_{\rm int} \ge 0.26 \frac{\pi t\_B \sigma^2}{2E\_B} \tag{9}
$$

4.4.2. Interfacial reaction phase

The interfacial reaction between two different layers can result in decreasing mechanical properties of laminated composites [50]. There are a series of studies on the mechanical response of laminated composites based on Ti-Al [69], Ti-Cu [70], Ti-Ni [71], Ni-Al [72], Cu-Al [73], Ti-Nb [74], Fe-Al [75, 76] and Fe-Cu [77] systems. Table 2 shows the room temperature tensile properties of these laminated composites. The laminated composites reveal low fracture elongation compared to the constituent metals, which is attributed to the sintering voids and in situ reacted brittle intermetallic layer.

Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms

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39

The size of reacted intermetallic layer can also affect the mechanical properties of laminated composites. Pang et al. [78] and Huang et al. [79] reveal that laminated Ti-(SiCp/Al) and Ti/Al composites exhibit high fracture ductility due to the in situ reacted nanosized TiAl3 intermetallic layer, whose fracture elongation is even higher than that of the constitute layers. Guo et al. [74] reported that the intermetallic reacted layer can influent the fracture mechanism of adjacent layer. The plastic deformation behavior of adjacent Al layer can be constrained by the hard brittle intermetallic reacted Al2Cu layer, leading to brittle fracture morphologies with crack propagating along the grain boundary of Al. Liu et al. [54] proposed that addition of thinner Al layer can enhance the tensile plasticity of cold-rolled Cu without losing strength by delaying onset of premature localized necking. Overall, the constrained interface can effectively improve the strain

For the strong interface without transition phase, the deformation behavior of laminated composites can depend on the strain hardening exponent (n) and strain rate parameter (m) of the constitute layers. Based on the power-law Hollomon equation, the flow stress (σ) of the

<sup>σ</sup> <sup>¼</sup> <sup>K</sup> <sup>ε</sup><sup>n</sup> <sup>þ</sup> <sup>m</sup>ln <sup>ε</sup>\_

where K is the strength constant and ε\_<sup>r</sup> is a reference strain-rate. Herein, n and m exert the most dramatic influence, high value of n indicates the high uniform fracture elongation, and

Laminated systems Brittle intermetallic layer Maximum elongation at room temperature (%)

ε\_r

(10)

delocalization of soft layers and delay the formation of tunnel cracks [79].

4.5. Strain hardening exponent (n) and strain rate parameter (m)

Ti-TiBw/Ti (present work) – 27.5 Ti-Al [69] Al3Ti, Al2Ti, TiAl, TiAl3 3.4 Ti-Cu [70] Ti2Cu, TiCu, TiCu3, TiCu4 4.5 Ti-Ni [71] TiNi3, Ti2Ni, TiNi ≤3 Ti-Nb [72] NbTi 3.6 Ni-Al [73] Ni3Al, Al2Al3, NiAl 12 Cu-Al [74] Al2Cu, AlCu, Al3Cu4, Al2Cu3, Al4Cu9 11 Fe-Al [75, 76] Fe2Al5, Fe4Al13 18 Fe-Cu [77] — 60

Table 2. The elongation of laminated composites at room temperature in reported different system.

individual layer can be expressed as follows [50]:

where Γint is the interfacial toughness and σ is the stress imposed on the brittle layer. This criterion provides a reasonable conditions of sufficient interfacial toughness to inhibit the interfacial delamination phenomenon.

## 4.4.2. Interfacial reaction phase

4.4. Interface layer

38 Lamination - Theory and Application

4.4.1. Interfacial bonding strength

interfacial delamination phenomenon.

Nambu et al. [38] reported that laminated composites with strong interfacial bonding strength exhibit superior longitudinal fracture elongation compared with the laminated composites with weak interfaces. For a metal film deposited on an elastic substrate, Li et al. [68] predicted that strong interface can delay strain localization by finite element simulation (FEM). However, if the metal film approaches to form interfacial delamination crack due to weak interfacial bonding strength, the metal film becomes freestanding and is easy to form a localized necking. Koseki et al. [40] proved that delamination crack is related to the low interfacial toughness, while laminated composites with high interfacial toughness reveal high fracture elongation before failure, leading to localized necking and ductile fracture surface, exhibiting interfacial delamination crack and brittle fracture of brittle layer when the interface becomes unstable

<sup>Γ</sup>int <sup>≥</sup> <sup>0</sup>:<sup>26</sup> <sup>π</sup>tBσ<sup>2</sup>

where Γint is the interfacial toughness and σ is the stress imposed on the brittle layer. This criterion provides a reasonable conditions of sufficient interfacial toughness to inhibit the

2EB

(9)

with the increase of strain. The debonding criterion is expressed as follows:

Figure 17. Schematic illustration of fracture energy as a function of layer thickness [67].

The interfacial reaction between two different layers can result in decreasing mechanical properties of laminated composites [50]. There are a series of studies on the mechanical response of laminated composites based on Ti-Al [69], Ti-Cu [70], Ti-Ni [71], Ni-Al [72], Cu-Al [73], Ti-Nb [74], Fe-Al [75, 76] and Fe-Cu [77] systems. Table 2 shows the room temperature tensile properties of these laminated composites. The laminated composites reveal low fracture elongation compared to the constituent metals, which is attributed to the sintering voids and in situ reacted brittle intermetallic layer.

The size of reacted intermetallic layer can also affect the mechanical properties of laminated composites. Pang et al. [78] and Huang et al. [79] reveal that laminated Ti-(SiCp/Al) and Ti/Al composites exhibit high fracture ductility due to the in situ reacted nanosized TiAl3 intermetallic layer, whose fracture elongation is even higher than that of the constitute layers. Guo et al. [74] reported that the intermetallic reacted layer can influent the fracture mechanism of adjacent layer. The plastic deformation behavior of adjacent Al layer can be constrained by the hard brittle intermetallic reacted Al2Cu layer, leading to brittle fracture morphologies with crack propagating along the grain boundary of Al. Liu et al. [54] proposed that addition of thinner Al layer can enhance the tensile plasticity of cold-rolled Cu without losing strength by delaying onset of premature localized necking. Overall, the constrained interface can effectively improve the strain delocalization of soft layers and delay the formation of tunnel cracks [79].

## 4.5. Strain hardening exponent (n) and strain rate parameter (m)

For the strong interface without transition phase, the deformation behavior of laminated composites can depend on the strain hardening exponent (n) and strain rate parameter (m) of the constitute layers. Based on the power-law Hollomon equation, the flow stress (σ) of the individual layer can be expressed as follows [50]:

$$
\sigma = K \left[ \varepsilon^n + m \ln \left( \frac{\dot{\varepsilon}}{\dot{\varepsilon}\_r} \right) \right] \tag{10}
$$

where K is the strength constant and ε\_<sup>r</sup> is a reference strain-rate. Herein, n and m exert the most dramatic influence, high value of n indicates the high uniform fracture elongation, and


Table 2. The elongation of laminated composites at room temperature in reported different system.

high value of m will even the material superplastic. The strain-rate hardening increases the elongation, which not only delays the onset of instability, but also by retarding the consequences of necking. The laminated composites with superior tensile ductility can be obtained by adjusting the values of n and m of constitute layer [80].

elastic-plastic laminated composites, the following criterion has already been proposed for

where tB, KIC and σ are the thickness, fracture toughness and the ultimate strength of the brittle phase layer, respectively. σY,D is the yielding strength of ductile phase layer, and tcritical is the critical size. The above equation indicated that if the thickness brittle phase layer is lower than the critical thickness size, the tunnel crack in brittle phase layer can be effectively suppressed by the ductile phase layer. Therefore, a notable transition in the fracture morphologies of S420 brittle layer from brittle cleavage to ductile shear and dimples is identified as the layer thick-

Meanwhile, Liu et al. [65, 82] reported that the laminated Ti-TiBw/Ti composites with thinner layer thickness possess higher fracture elongation than the laminated composites with thick layer thickness due to presence of more tunnel cracks. Figure 19 shows the in situ tensile fracture process of laminated Ti-TiBw/Ti composites with layer thickness of 500 and 300 μm. It is revealed that the laminated composites with 500 μm layer thickness sharply fail at the tensile strain of 12.7%, and only a tunnel crack propagates into the overall sample as shown in Figure 19(f). That is to say, when the brittle layer thickness extends beyond the critical thickness size, the ductile layer cannot prevent/blunt the propagation of tunnel crack, and the laminated composites will sharply fracture. However, the laminated composites with layer thickness of 300 μm revealed that one tunnel crack is presented at the strain of 13.4% as shown in Figure 19(k). Finally, the laminated composites fail with two or more tunnel cracks at the strain of 15.4% as shown in Figure 19(l), which is taken as a noncatastrophic fracture stage.

Figure 19. The fracture processes of the laminated Ti-TiBw/Ti composites with different layer thicknesses at the in situ tensile strains of (a, f) 0%; (b, h) 4.5%; (c, d, i) 10.7%; (e, j) 12.6%; (f) 12.7%; (k) 13.4%; (l) 15.3%. (a–f) For the laminated

composites with 500 μm layer thickness; (g–l) for the laminated composites with 300 μm layer thickness [65].

3 <sup>p</sup> <sup>K</sup><sup>2</sup>

ICσY,D

Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms

<sup>σ</sup><sup>3</sup> (11)

http://dx.doi.org/10.5772/intechopen.69976

41

tB <sup>≤</sup> tcritical <sup>¼</sup> <sup>2</sup> ffiffiffi

suppressing crack propagation [40]:

ness is reduced.

Figure 18 shows the fracture elongation map of many power-law hardening monolithic materials [80, 81]. It is interesting to note that a low ductility material laminated with another material that can coax the laminated composites toward higher elongation. For example, austenitic stainless steel and commercial purity iron make them attractive options as the ductility-enhancing component within a laminated metal composites, and this rule can also be indicated in the stainless steel/ultrahigh carbon steel multilayered materials during the high-temperature tensile testing process [32].

### 4.6. Tunnel crack

In 2008, Inoue et al. [37] reported that fracture elongation is gradually increased with the decrease of layer thickness, which is attributed to decreasing size of tunnel crack. For the

Figure 18. The fracture elongation map of many power-law hardening monolithic materials [80].

elastic-plastic laminated composites, the following criterion has already been proposed for suppressing crack propagation [40]:

high value of m will even the material superplastic. The strain-rate hardening increases the elongation, which not only delays the onset of instability, but also by retarding the consequences of necking. The laminated composites with superior tensile ductility can be obtained

Figure 18 shows the fracture elongation map of many power-law hardening monolithic materials [80, 81]. It is interesting to note that a low ductility material laminated with another material that can coax the laminated composites toward higher elongation. For example, austenitic stainless steel and commercial purity iron make them attractive options as the ductility-enhancing component within a laminated metal composites, and this rule can also be indicated in the stainless steel/ultrahigh carbon steel multilayered materials during the

In 2008, Inoue et al. [37] reported that fracture elongation is gradually increased with the decrease of layer thickness, which is attributed to decreasing size of tunnel crack. For the

Figure 18. The fracture elongation map of many power-law hardening monolithic materials [80].

by adjusting the values of n and m of constitute layer [80].

high-temperature tensile testing process [32].

4.6. Tunnel crack

40 Lamination - Theory and Application

$$t\_B \le t\_{critical} = \frac{2\sqrt{3}\mathcal{K}\_{IC}^2 \sigma\_{Y,D}}{\sigma^3} \tag{11}$$

where tB, KIC and σ are the thickness, fracture toughness and the ultimate strength of the brittle phase layer, respectively. σY,D is the yielding strength of ductile phase layer, and tcritical is the critical size. The above equation indicated that if the thickness brittle phase layer is lower than the critical thickness size, the tunnel crack in brittle phase layer can be effectively suppressed by the ductile phase layer. Therefore, a notable transition in the fracture morphologies of S420 brittle layer from brittle cleavage to ductile shear and dimples is identified as the layer thickness is reduced.

Meanwhile, Liu et al. [65, 82] reported that the laminated Ti-TiBw/Ti composites with thinner layer thickness possess higher fracture elongation than the laminated composites with thick layer thickness due to presence of more tunnel cracks. Figure 19 shows the in situ tensile fracture process of laminated Ti-TiBw/Ti composites with layer thickness of 500 and 300 μm. It is revealed that the laminated composites with 500 μm layer thickness sharply fail at the tensile strain of 12.7%, and only a tunnel crack propagates into the overall sample as shown in Figure 19(f). That is to say, when the brittle layer thickness extends beyond the critical thickness size, the ductile layer cannot prevent/blunt the propagation of tunnel crack, and the laminated composites will sharply fracture. However, the laminated composites with layer thickness of 300 μm revealed that one tunnel crack is presented at the strain of 13.4% as shown in Figure 19(k). Finally, the laminated composites fail with two or more tunnel cracks at the strain of 15.4% as shown in Figure 19(l), which is taken as a noncatastrophic fracture stage.

Figure 19. The fracture processes of the laminated Ti-TiBw/Ti composites with different layer thicknesses at the in situ tensile strains of (a, f) 0%; (b, h) 4.5%; (c, d, i) 10.7%; (e, j) 12.6%; (f) 12.7%; (k) 13.4%; (l) 15.3%. (a–f) For the laminated composites with 500 μm layer thickness; (g–l) for the laminated composites with 300 μm layer thickness [65].

Therefore, the laminated composites with 300 μm layer thickness can obtain higher fracture elongation than the composites with 500 μm layer thickness, which is attributed to more tunnel cracks formed in the thin TiBw/Ti layers.

Acknowledgements

Author details

, Lujun Huang<sup>2</sup>

\*Address all correspondence to: huanglujun@hit.edu.cn

Royal Society A. 2007;365:2907-2919

ture in nickel. Science. 2013;342:337-340

[6] Lu K. The future of metals. Science. 2010;328:319-320

gradient nano-grained copper. Science. 2011;331:1587-1590

approach. Construction and Building Materials. 2016;115:13-31

Baoxi Liu<sup>1</sup>

References

168

This work is financially supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51601055, 51471063, 51571071, 51671068, the Hebei Science and Technology program under Grant No. 130000048 and the National Natural Science Founda-

Multiscale Hierarchical Structure and Laminated Strengthening and Toughening Mechanisms

http://dx.doi.org/10.5772/intechopen.69976

43

tion of Hebei Province under Grant Nos. E201620218 and QN2016029.

Equipment Materials, Hebei University of Technology, Tianjin, China

composites. Materials Science and Engineering R. 2000;29:49-113

\*, Lin Geng<sup>2</sup> and Fuxing Yin<sup>1</sup>

1 Tianjin Key Laboratory of Materials Laminating Fabrication and Interfacial Controlling Technology, School of Materials Science and Engineering, Research Institute for Energy

2 School of Materials Science and Engineering, Harbin Institute of Technology, Harbin, China

[1] Tjong C, Ma ZY. Microstructural and mechanical characteristics of in situ metal matrix

[2] Huang LJ, Geng L, Peng HX. Microstructurally inhomogeneous composites: Is a homogeneous reinforcement distribution optimal? Progress in Materials Science. 2005;71:93-

[3] Tjong SC, Mai YW. Processing-structure-property aspects of particulate- and whiskerreinforced titanium matrix composites. Composites Science and Technology. 2008;68:583-601

[4] Lu K. Making strong nanomaterials ductile with gradients. Science. 2014;345:1455-1456 [5] Barthelat F. Biomimetics for next generation materials. Philosophical Transactions of the

[7] Fang TH, Li WL, Tao NR, Lu K. Revealing extraordinary intrinsic tensile plasticity in

[8] Liu XC, Zhang HW, Lu K. Strain-induced ultrahard and ultrastable nanolaminated struc-

[9] Palkovic SD, Brommer DB, Kupwade-Patil K, Masic A, Buehler MJ, Buyukozturk O. Roadmap across the mesoscale for durable and sustainable cement paste: A bioinspired

In the laminated Ti-TiBw/Ti systems, Liu et al. reported [47, 83–87] that the formation and number of tunnel crack are also related to interfacial bonding strength and residual stress, and the ductile Ti layer exhibits ductile-brittle transition at the tunnel crack with the decreasing Ti layer thickness and the increasing volume fraction of TiB reinforcement, which is attributed to that the plastic deformation of thin Ti layer is constrained by TiBw/Ti layer, resulting in high triaxial tensile stress at the tunnel crack. Therefore, the laminated composites with thin Ti layer thickness reveal brittle fracture characteristics. With increasing volume fraction from 5 to 12%, the fracture manner in the TiBw/Ti layer changes from quasi-cleavage fracture to flat brittle fracture. Moreover, Ti layer reveals obviously ductile-brittle transition manner: the ductile dimples, crack bifurcation and lath-like intergranular fracture are presented in the laminated composites with volume fraction of 5%, 8% and 10%, respectively, and a typical brittle fracture manner is existed in the Ti layer. Huang et al. [85] reported that the plastic deformation zone can effectively delay the propagating behavior of tunnel crack. However, the size of plastic zone is decreased with the increasing volume fraction of TiB. Therefore, the tunnel crack is difficult to be blunted and easy to propagate in a brittle manner in the laminated composites with high volume fraction of reinforcements.

## 5. Conclusions

A background to the development of toughening mechanisms of metal matrix composites with multiscale hierarchical structure and laminated structure was overviewed in this article as follows:


## Acknowledgements

Therefore, the laminated composites with 300 μm layer thickness can obtain higher fracture elongation than the composites with 500 μm layer thickness, which is attributed to more

In the laminated Ti-TiBw/Ti systems, Liu et al. reported [47, 83–87] that the formation and number of tunnel crack are also related to interfacial bonding strength and residual stress, and the ductile Ti layer exhibits ductile-brittle transition at the tunnel crack with the decreasing Ti layer thickness and the increasing volume fraction of TiB reinforcement, which is attributed to that the plastic deformation of thin Ti layer is constrained by TiBw/Ti layer, resulting in high triaxial tensile stress at the tunnel crack. Therefore, the laminated composites with thin Ti layer thickness reveal brittle fracture characteristics. With increasing volume fraction from 5 to 12%, the fracture manner in the TiBw/Ti layer changes from quasi-cleavage fracture to flat brittle fracture. Moreover, Ti layer reveals obviously ductile-brittle transition manner: the ductile dimples, crack bifurcation and lath-like intergranular fracture are presented in the laminated composites with volume fraction of 5%, 8% and 10%, respectively, and a typical brittle fracture manner is existed in the Ti layer. Huang et al. [85] reported that the plastic deformation zone can effectively delay the propagating behavior of tunnel crack. However, the size of plastic zone is decreased with the increasing volume fraction of TiB. Therefore, the tunnel crack is difficult to be blunted and easy to propagate in a brittle manner in the laminated composites

A background to the development of toughening mechanisms of metal matrix composites with multiscale hierarchical structure and laminated structure was overviewed in this article

1. The development of metal matrix composites approaches to the microstructure configuration design, and titanium matrix composites can achieve both high strength and tough-

2. The design ideal of metal matrix composites with laminated structure is aiming to achieve high strength, toughness and ductility. It is revealed that elastic property, and yield strength of laminated composites follow the "rule of average." However, the estimation of fracture elongation and fracture toughness is complex, which is inconsistent with the

3. The fracture elongation and toughness of laminated composites are related to interfacial bonding strength and tunnel crack number, and the fracture elongation of laminated composites with strong interface can be controlled by adjusting the strain hardening

4. The tensile properties and fracture characteristics of laminated composites reveal obvious size effect, with the decrease of layer thickness; the tensile strength is gradually improved, whereas the fracture elongation is first increased in microscale and then sharply decreased

tunnel cracks formed in the thin TiBw/Ti layers.

42 Lamination - Theory and Application

with high volume fraction of reinforcements.

ness by adjusting the multiscale hierarchical structure.

exponent n and strain rate parameter m.

in the submicroscale and nanoscale.

5. Conclusions

"rule of average."

as follows:

This work is financially supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51601055, 51471063, 51571071, 51671068, the Hebei Science and Technology program under Grant No. 130000048 and the National Natural Science Foundation of Hebei Province under Grant Nos. E201620218 and QN2016029.

## Author details

Baoxi Liu<sup>1</sup> , Lujun Huang<sup>2</sup> \*, Lin Geng<sup>2</sup> and Fuxing Yin<sup>1</sup>

\*Address all correspondence to: huanglujun@hit.edu.cn

1 Tianjin Key Laboratory of Materials Laminating Fabrication and Interfacial Controlling Technology, School of Materials Science and Engineering, Research Institute for Energy Equipment Materials, Hebei University of Technology, Tianjin, China

2 School of Materials Science and Engineering, Harbin Institute of Technology, Harbin, China

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**Chapter 3**

Provisional chapter

**Bending of Laminated Composite Plates in Layerwise**

DOI: 10.5772/intechopen.69975

Bending of Laminated Composite Plates in Layerwise

Determination of stress-strain state in contemporary laminated composite plates containing layers with continuous unidirectional fibers requires the application of refined plate theories, which include layerwise theory. In contrast to homogeneous isotropic plates, heterogeneity of the anisotropic structure of laminated composite plates often leads to the appearance of imperfections in the connection between the layers. Mathematical models, which are formed on the assumption that the plate is homogeneous and isotropic, cannot properly include irregularities that can occur at the level of the layer in the process of manufacture, transportation, installation, or exploitation. Mathematical models of layerwise theory allow defining a more realistic stress-strain state through the thickness of the plate, where consideration is carried out at the level of the layer. Additionally, this model makes possible to include delaminations that might occur on the connection between the individual layers. In this chapter, Reddy's layerwise theory is applied in order to determine equations for the problem of bending of laminated composite plates. The bending equations are solved by applying analytical method by means of double trigonometric series, as well as by using numerical methods based on the finite elements. This chapter presents

Keywords: laminated plate, bending, layerwise theory, analytical solution, finite

Composite materials are obtained by a combination of two or more materials which together have characteristics that separately usually cannot be achieved with reasonable costs. During the last decades, a sudden increase of the use of modern composite materials is based on the achieved advanced properties which include their lower weight and higher strength and resistance compared to the classical and conventional composite materials. It is clear that the

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is 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.

**Theory**

Theory

Marina Rakočević

Marina Rakočević

Abstract

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69975

examples for both applied approaches.

element method

1. Introduction

Provisional chapter

## **Bending of Laminated Composite Plates in Layerwise Theory** Bending of Laminated Composite Plates in Layerwise Theory

DOI: 10.5772/intechopen.69975

## Marina Rakočević

Additional information is available at the end of the chapter Marina Rakočević

http://dx.doi.org/10.5772/intechopen.69975 Additional information is available at the end of the chapter

## Abstract

Determination of stress-strain state in contemporary laminated composite plates containing layers with continuous unidirectional fibers requires the application of refined plate theories, which include layerwise theory. In contrast to homogeneous isotropic plates, heterogeneity of the anisotropic structure of laminated composite plates often leads to the appearance of imperfections in the connection between the layers. Mathematical models, which are formed on the assumption that the plate is homogeneous and isotropic, cannot properly include irregularities that can occur at the level of the layer in the process of manufacture, transportation, installation, or exploitation. Mathematical models of layerwise theory allow defining a more realistic stress-strain state through the thickness of the plate, where consideration is carried out at the level of the layer. Additionally, this model makes possible to include delaminations that might occur on the connection between the individual layers. In this chapter, Reddy's layerwise theory is applied in order to determine equations for the problem of bending of laminated composite plates. The bending equations are solved by applying analytical method by means of double trigonometric series, as well as by using numerical methods based on the finite elements. This chapter presents examples for both applied approaches.

Keywords: laminated plate, bending, layerwise theory, analytical solution, finite element method

## 1. Introduction

Composite materials are obtained by a combination of two or more materials which together have characteristics that separately usually cannot be achieved with reasonable costs. During the last decades, a sudden increase of the use of modern composite materials is based on the achieved advanced properties which include their lower weight and higher strength and resistance compared to the classical and conventional composite materials. It is clear that the

© 2018 The Author(s). Licensee InTech. This chapter is 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 Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

modern composite materials are materials of the future, because they allow the designers to choose the appropriate characteristics of the elements structure depending on the applicable problem. Due to its complexity, the researches of modern materials involve several scientific fields, and each of them, in their own way, contributes to their development.

deformation in the calculation which increases accuracy compared to the classical laminated theory. Layerwise theories consider laminated composite plate at the level of the layer. LW theories are recommended for moderately thick and thick laminated plates and for plates with a pronounced anisotropic behavior for which it is necessary to define a more realistic internal stresses distribution, as well as where it is necessary to include delamination and the other geometrical and material imperfections in calculation. On the basis of LW theories, many calculation procedures, analytical, and numerical methods were created in order to accurately

Bending of Laminated Composite Plates in Layerwise Theory

http://dx.doi.org/10.5772/intechopen.69975

53

This chapter describes the application of layerwise theories in solving problems of bending of laminated plates, whereby the three-dimensional (3D) problem of elasticity is reduced by solving differential equations with two-dimensional (2D) variables in the plane plate. Additionally, the approximation of the displacement through the thickness of the plate is conducted with one-dimensional interpolation functions of coordinate perpendicular to the plane of the layer. Mathematical model is based on the Generalized Laminated Plate Theory (GLPT) devel-

Based on the adopted assumptions of the layerwise theory, bending equations are obtained by applying the principle of virtual displacements. In this chapter, the governing differential equations of LW theory are solved by applying analytical and numerical methods. Closed analytical solution of bending equations in LW theory is done by using the double trigonometric series of the simple supported rectangular plate with the angle of the layers orientation

Analytical solution for this mechanical problem is difficult to determine in case of complex plate geometry, arbitrary angles of layers orientation, arbitrary boundary conditions, as well as for various forms of nonlinearity. These are the reasons why the applied Finite Element Method (FEM) as one of the most used numerical methods was used, see Refs. [1, 10–14, 21]. The idea of the FEM is a representation of the domain as the sets of appropriate simple geometrical shapes called the finite elements. Discretization of the domain with finite elements makes this method a valuable tool in solving a large number of complex engineering

Since the FEM is a method of discretization, errors can be classified into two categories: geometric discretization errors and interpolation errors. Accuracy of the results obtained by applying FEM depends on the choice of interpolation function in the plane (x, y) and interpolation function through the thickness of the plate. It also depends on the number of elements and the number of nodes of the adopted network in the plane plate and perpendicular to the plane plate. The increase of the number of finite element nodes gives rise to the increase of the number of degrees of freedom, making this calculation uneconomic. Regardless of the pointed disadvantages, FEM is much more general and more acceptable for the analysis of laminated composite plates than analytical methods. In this chapter, the variable kinematic finite element (with C0 continuity through the plate thickness) based on the assumptions of the partial LW theory is presented. The element is hierarchical and can include delamination. This finite element can be expanded in order to obtain a more complex finite element, by simply adding

analyze mentioned stresses of the laminated composite plates.

neglected elongation perpendicular to the plane of the plate.

oped by Reddy and Reddy et al. [10–14].

0 and 90, see Refs. [10, 18, 22–26].

problems.

Composite plates, made of layers of the various material and geometric characteristics, are called laminated plates. Ply or layer is the basic element of the laminated plate and it is made of fibers installed into the matrix. The fibers can be discontinuous, continuous unidirectional, bidirectional, woven unidirectional, or woven bidirectional. In the unidirectional laminated composites, the fibers are load-bearing elements of the layer, while the matrix has a role to protect the fibers from external influences, to hold the fibers together and to perform uniform distribution of the influences to each of the fibers. The materials used for fibers have better material property and greater capacity compared to the matrix, and the geometrical characteristics of the fiber cross section are significantly smaller than its length. Materials used for fibers can be aluminum, copper, iron, nickel, steel, titanium, or organic materials such as glass, carbon, and graphite. A layer with unidirectional fibers has significantly better characteristics in fiber direction than in a direction perpendicular to the fiber.

Heterogeneity of anisotropic laminated composite plates often causes the appearance of a large number of imperfections that can occur at the level of the laminated plate or at the level of the layer, as well as at the local level of the fiber/matrix. General deformation of laminated plates is often defined by complex coupling between the axial deformation, bending, and shear deformation. In laminated composite plates for smaller aspect ratio, the importance of shear deformation is higher than in the corresponding homogeneous isotropic plates.

At the level of the layer, composites often contain concentrations of transverse stresses near the geometrical and material imperfections which lead to damage. Discontinuity between adjacent layers can occur in the stages of production, transportation, installation, or exploitation. This phenomenon in the literature is known as delamination. Delamination and its increase during exploitation is one of the most important parameters that affect the bearing capacity of the plate. Plates with delamination have reduced bearing capacity to bending. In the analysis of these plates, it is necessary to choose the mathematical models which can successfully incorporate delamination in the calculations.

Mathematical models for these particular problems need to determine the real stress-strain state in the laminated plate, which requires the application of more accurate theories. In addition, it is important to find a balance between the desired accuracy and calculation costs.

The literature that studies the problems of laminated composite plates is significant and extensive. Available mathematical models for the laminated plates are based on assumptions of Equivalent Single Layer (ESL) theories and assumptions of LayerWise (LW) theories, see Refs. [1–26]. ESL theories consider plate as one layer with equivalent stiffnesses. The most commonly applied ESL theories of laminated composite plates are Classical Laminated Plate Theory (CLPT), First-Order Shear Deformation Theory (FSDT), Third-Order Shear Deformation Theory (TSDT), and other high-order shear theories. ESL theories are recommended to apply for thin and very thin laminated plates. More accurate ESL plate theories introduce shear deformation in the calculation which increases accuracy compared to the classical laminated theory. Layerwise theories consider laminated composite plate at the level of the layer. LW theories are recommended for moderately thick and thick laminated plates and for plates with a pronounced anisotropic behavior for which it is necessary to define a more realistic internal stresses distribution, as well as where it is necessary to include delamination and the other geometrical and material imperfections in calculation. On the basis of LW theories, many calculation procedures, analytical, and numerical methods were created in order to accurately analyze mentioned stresses of the laminated composite plates.

modern composite materials are materials of the future, because they allow the designers to choose the appropriate characteristics of the elements structure depending on the applicable problem. Due to its complexity, the researches of modern materials involve several scientific

Composite plates, made of layers of the various material and geometric characteristics, are called laminated plates. Ply or layer is the basic element of the laminated plate and it is made of fibers installed into the matrix. The fibers can be discontinuous, continuous unidirectional, bidirectional, woven unidirectional, or woven bidirectional. In the unidirectional laminated composites, the fibers are load-bearing elements of the layer, while the matrix has a role to protect the fibers from external influences, to hold the fibers together and to perform uniform distribution of the influences to each of the fibers. The materials used for fibers have better material property and greater capacity compared to the matrix, and the geometrical characteristics of the fiber cross section are significantly smaller than its length. Materials used for fibers can be aluminum, copper, iron, nickel, steel, titanium, or organic materials such as glass, carbon, and graphite. A layer with unidirectional fibers has significantly better characteristics

Heterogeneity of anisotropic laminated composite plates often causes the appearance of a large number of imperfections that can occur at the level of the laminated plate or at the level of the layer, as well as at the local level of the fiber/matrix. General deformation of laminated plates is often defined by complex coupling between the axial deformation, bending, and shear deformation. In laminated composite plates for smaller aspect ratio, the importance of shear defor-

At the level of the layer, composites often contain concentrations of transverse stresses near the geometrical and material imperfections which lead to damage. Discontinuity between adjacent layers can occur in the stages of production, transportation, installation, or exploitation. This phenomenon in the literature is known as delamination. Delamination and its increase during exploitation is one of the most important parameters that affect the bearing capacity of the plate. Plates with delamination have reduced bearing capacity to bending. In the analysis of these plates, it is necessary to choose the mathematical models which can successfully incor-

Mathematical models for these particular problems need to determine the real stress-strain state in the laminated plate, which requires the application of more accurate theories. In addition, it is important to find a balance between the desired accuracy and calculation costs.

The literature that studies the problems of laminated composite plates is significant and extensive. Available mathematical models for the laminated plates are based on assumptions of Equivalent Single Layer (ESL) theories and assumptions of LayerWise (LW) theories, see Refs. [1–26]. ESL theories consider plate as one layer with equivalent stiffnesses. The most commonly applied ESL theories of laminated composite plates are Classical Laminated Plate Theory (CLPT), First-Order Shear Deformation Theory (FSDT), Third-Order Shear Deformation Theory (TSDT), and other high-order shear theories. ESL theories are recommended to apply for thin and very thin laminated plates. More accurate ESL plate theories introduce shear

fields, and each of them, in their own way, contributes to their development.

in fiber direction than in a direction perpendicular to the fiber.

porate delamination in the calculations.

52 Lamination - Theory and Application

mation is higher than in the corresponding homogeneous isotropic plates.

This chapter describes the application of layerwise theories in solving problems of bending of laminated plates, whereby the three-dimensional (3D) problem of elasticity is reduced by solving differential equations with two-dimensional (2D) variables in the plane plate. Additionally, the approximation of the displacement through the thickness of the plate is conducted with one-dimensional interpolation functions of coordinate perpendicular to the plane of the layer. Mathematical model is based on the Generalized Laminated Plate Theory (GLPT) developed by Reddy and Reddy et al. [10–14].

Based on the adopted assumptions of the layerwise theory, bending equations are obtained by applying the principle of virtual displacements. In this chapter, the governing differential equations of LW theory are solved by applying analytical and numerical methods. Closed analytical solution of bending equations in LW theory is done by using the double trigonometric series of the simple supported rectangular plate with the angle of the layers orientation 0 and 90, see Refs. [10, 18, 22–26].

Analytical solution for this mechanical problem is difficult to determine in case of complex plate geometry, arbitrary angles of layers orientation, arbitrary boundary conditions, as well as for various forms of nonlinearity. These are the reasons why the applied Finite Element Method (FEM) as one of the most used numerical methods was used, see Refs. [1, 10–14, 21]. The idea of the FEM is a representation of the domain as the sets of appropriate simple geometrical shapes called the finite elements. Discretization of the domain with finite elements makes this method a valuable tool in solving a large number of complex engineering problems.

Since the FEM is a method of discretization, errors can be classified into two categories: geometric discretization errors and interpolation errors. Accuracy of the results obtained by applying FEM depends on the choice of interpolation function in the plane (x, y) and interpolation function through the thickness of the plate. It also depends on the number of elements and the number of nodes of the adopted network in the plane plate and perpendicular to the plane plate. The increase of the number of finite element nodes gives rise to the increase of the number of degrees of freedom, making this calculation uneconomic. Regardless of the pointed disadvantages, FEM is much more general and more acceptable for the analysis of laminated composite plates than analytical methods. In this chapter, the variable kinematic finite element (with C0 continuity through the plate thickness) based on the assumptions of the partial LW theory is presented. The element is hierarchical and can include delamination. This finite element can be expanded in order to obtain a more complex finite element, by simply adding neglected elongation perpendicular to the plane of the plate.

## 2. Theoretical backgrounds

Laminated composite plate, loaded by transversal load, is considered in Cartesian coordinate system (x, y, z). The plate includes N orthotropic layers wherein each layer carries a predetermined direction in the plane of the plate (x, y). Layer orientation angle is defined by angle θ, angle between continuous fibers, and axis x of the global coordinate system of the plate.

It is assumed that the dilation ε<sup>z</sup> perpendicular to the middle plane of the plate is neglected which results in a constant deflection through the plate thickness w(x, y, z) ¼ w(x, y). A displacement field is determined by components:

$$\begin{array}{l} \mu(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \mu(\mathbf{x}, \mathbf{y}) + \mathcal{U}(\mathbf{x}, \mathbf{y}, \mathbf{z})\\ \nu(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \nu(\mathbf{x}, \mathbf{y}) + \mathcal{V}(\mathbf{x}, \mathbf{y}, \mathbf{z})\\ \nu(\mathbf{x}, \mathbf{y}, \mathbf{z}) = w(\mathbf{x}, \mathbf{y}) \end{array} \tag{1}$$

For linear interpolation p ¼ 1, the number of nodes through the plate thickness is one larger than the number of layers n ¼ Nþ1, and the linear interpolation functions are determined by

> z � zJ�<sup>1</sup> zJ � zJ�<sup>1</sup>

zJþ<sup>1</sup> � z zJ � zJ�<sup>1</sup>

where zJ is the coordinate of the Jth node which is located between the layers of J and J-1.

of the layered plates has an effect on the previous and the next layer of the laminate.

Based on Eqs. (1) and (2), strain components can be expressed as follows:

u,x v, <sup>y</sup> u, <sup>y</sup> þ v,x w,x w,y

9 >>>>>>>=

Figure 1. 3D geometry of the laminated composite plates and discretization through the plate thickness with linear

>>>>>>>;

{ε} <sup>¼</sup> ½ � <sup>H</sup> {Δ}, {ε<sup>J</sup>

where the members of the vector ε determine the deformation of the middle plane of the plate,

where members of matrix H and H are the first derivative of the interpolation functions with

For an orthotropic material, the constitutive equations of the kth layer are determined by

<sup>ε</sup><sup>J</sup> � � <sup>¼</sup>

UJ , x VJ ,y 9 >>>>>>>=

>>>>>>>;

} (7)

UJ VJ

determine the deformation through the thickness of the plate.

UJ ,y <sup>þ</sup> <sup>V</sup><sup>J</sup> ,x

8 >>>>>>><

>>>>>>>:

} ¼ H � �{Δ<sup>J</sup>

f gε ¼

and the members of the vector ε<sup>J</sup>

interpolation.

The short form of Eq. (3) is defined by

respect to the global coordinates x and y.

8 >>>>>>><

>>>>>>>:

Graphical display of the laminated composite plates with linear interpolation through the plate thickness is shown in Figure 1. By analyzing separate layers, this LW theory can be understood as a first-order shear deformation theory for each layer of the observed laminated plate. A layer

, zJ�<sup>1</sup> < z < zJ

9 >>=

Bending of Laminated Composite Plates in Layerwise Theory

http://dx.doi.org/10.5772/intechopen.69975

>>;

(5)

55

(6)

, zJ < z < zJþ<sup>1</sup>

ψ<sup>J</sup>�<sup>1</sup> <sup>1</sup> ð Þ¼ z

8 >><

>>:

ψJ <sup>2</sup>ð Þ¼ z

ϕJ ð Þ¼ z

Displacement components u(x, y, z) and v(x, y, z) are obtained as a sum of displacements of the middle plane of the plate u(x, y), v(x, y) and the additional displacements U(x, y, z) and V(x, y, z) through the plate thickness. Reduction of a 3D problem to a 2D problem is carried out by introducing the following relations:

$$\begin{aligned} \mathcal{U}(\mathbf{x}, \boldsymbol{y}, \boldsymbol{z}) &= \sum\_{l=1}^{n} u^{l}(\mathbf{x}, \boldsymbol{y}) \boldsymbol{\psi}^{l}(\boldsymbol{z}) \\ \mathcal{V}(\mathbf{x}, \boldsymbol{y}, \boldsymbol{z}) &= \sum\_{l=1}^{n} \boldsymbol{\upsilon}^{l}(\mathbf{x}, \boldsymbol{y}) \boldsymbol{\psi}^{l}(\boldsymbol{z}) \end{aligned} \tag{2}$$

where uJ and vJ are the displacement components of Jth node through the plate thickness in the directions of x- and y-axes, respectively; ψ<sup>J</sup> (z) is the one-dimensional (1D) interpolation function.

The governing equations of the laminated composite plates in case of bending are solved by displacement method of analysis, so that the primary variables are displacement components defined by Eqs. (1) and (2). The primary variables can be conveniently displayed in a vector form:

$$\{\!\!\{\Delta\} = \begin{Bmatrix} u \\ v \\ w \end{Bmatrix} \quad \{\!\!\Delta^{\!\!\{\}} \} = \begin{Bmatrix} u^{\!\!\!\} \\ v^{\!\!\!\/)} \end{Bmatrix} \tag{3}$$

where Δ are the components of the displacement vector of the middle plane plate and Δ<sup>J</sup> are the components of the displacement vector of the Jth node through the plate thickness.

By relations (Eq. (2)), the discretization through the plate thickness is performed, wherein the uJ and vJ are unknown nodal displacements and ψ<sup>J</sup> (z) are adopted general 1D interpolation functions that can be polynomials. The degree of a polynomial is related to the number of nodes in which determined unknown nodal displacements occur, where

$$m = pN + 1\tag{4}$$

where p is the polynomial degree; n is the number of nodes through the plate thickness; N is the number of layers.

For linear interpolation p ¼ 1, the number of nodes through the plate thickness is one larger than the number of layers n ¼ Nþ1, and the linear interpolation functions are determined by

2. Theoretical backgrounds

54 Lamination - Theory and Application

introducing the following relations:

directions of x- and y-axes, respectively; ψ<sup>J</sup>

A displacement field is determined by components:

Laminated composite plate, loaded by transversal load, is considered in Cartesian coordinate system (x, y, z). The plate includes N orthotropic layers wherein each layer carries a predetermined direction in the plane of the plate (x, y). Layer orientation angle is defined by angle θ, angle

It is assumed that the dilation ε<sup>z</sup> perpendicular to the middle plane of the plate is neglected which results in a constant deflection through the plate thickness w(x, y, z) ¼ w(x, y).

> u x, y, z ð Þ¼ u x, y ð Þþ U x, y, z ð Þ νð Þ¼ x, y, z νð Þþ x, y V x, y, z ð Þ

Displacement components u(x, y, z) and v(x, y, z) are obtained as a sum of displacements of the middle plane of the plate u(x, y), v(x, y) and the additional displacements U(x, y, z) and V(x, y, z) through the plate thickness. Reduction of a 3D problem to a 2D problem is carried out by

> J¼1 uJ ð Þ x, y <sup>ψ</sup><sup>J</sup>

J¼1 vJ ð Þ x, y <sup>ψ</sup><sup>J</sup>

where uJ and vJ are the displacement components of Jth node through the plate thickness in the

The governing equations of the laminated composite plates in case of bending are solved by displacement method of analysis, so that the primary variables are displacement components defined by Eqs. (1) and (2). The primary variables can be conveniently displayed in a vector form:

where Δ are the components of the displacement vector of the middle plane plate and Δ<sup>J</sup> are

By relations (Eq. (2)), the discretization through the plate thickness is performed, wherein the

functions that can be polynomials. The degree of a polynomial is related to the number of

where p is the polynomial degree; n is the number of nodes through the plate thickness; N is

<sup>Δ</sup><sup>J</sup> � � <sup>¼</sup> uJ

vJ � �

ð Þz

ð Þz

(z) is the one-dimensional (1D) interpolation function.

(z) are adopted general 1D interpolation

n ¼ pN þ 1 (4)

(1)

(2)

(3)

between continuous fibers, and axis x of the global coordinate system of the plate.

w x, y, z ð Þ¼ w x, y ð Þ

U x, y, z ð Þ¼ <sup>X</sup><sup>n</sup>

V x, y, z ð Þ¼ <sup>X</sup><sup>n</sup>

u v w

the components of the displacement vector of the Jth node through the plate thickness.

9 = ;

8 < :

f g Δ ¼

nodes in which determined unknown nodal displacements occur, where

uJ and vJ are unknown nodal displacements and ψ<sup>J</sup>

the number of layers.

$$\phi^l(z) = \left\{ \begin{aligned} \psi\_1^{l-1}(z) &= \frac{z - z\_{l-1}}{z\_l - z\_{l-1}}, z\_{l-1} < z < z\_l \\ \psi\_2^l(z) &= \frac{z\_{l+1} - z}{z\_l - z\_{l-1}}, z\_l < z < z\_{l+1} \end{aligned} \right\} \tag{5}$$

where zJ is the coordinate of the Jth node which is located between the layers of J and J-1.

Graphical display of the laminated composite plates with linear interpolation through the plate thickness is shown in Figure 1. By analyzing separate layers, this LW theory can be understood as a first-order shear deformation theory for each layer of the observed laminated plate. A layer of the layered plates has an effect on the previous and the next layer of the laminate.

Figure 1. 3D geometry of the laminated composite plates and discretization through the plate thickness with linear interpolation.

Based on Eqs. (1) and (2), strain components can be expressed as follows:

$$\{\varepsilon\} = \left\{ \begin{array}{c} \mathsf{u}\_{,x} \\ \upsilon\_{,y} \\ \mathsf{u}\_{,y} + \upsilon\_{,x} \\ \mathsf{w}\_{,x} \\ \mathsf{w}\_{,y} \end{array} \right\} \quad \{\varepsilon^{l}\} = \left\{ \begin{array}{c} \mathsf{U}^{l}\_{,x} \\ \mathsf{V}^{l}\_{,y} \\ \mathsf{U}^{l}\_{,y} + \mathsf{V}^{l}\_{,x} \\ \mathsf{U}^{l}\_{,} \\ \mathsf{V}^{l}\_{} \end{array} \right\} \tag{6}$$

where the members of the vector ε determine the deformation of the middle plane of the plate, and the members of the vector ε<sup>J</sup> determine the deformation through the thickness of the plate. The short form of Eq. (3) is defined by

$$\{\varepsilon\} = [H]\{\Delta\}, \quad \{\varepsilon^{\dagger}\} = \left[\overline{H}\right] \{\Delta^{\dagger}\} \tag{7}$$

where members of matrix H and H are the first derivative of the interpolation functions with respect to the global coordinates x and y.

For an orthotropic material, the constitutive equations of the kth layer are determined by

$$\{\sigma\}^{(k)} = \begin{bmatrix} \overline{\mathbf{Q}}\_{11} & \overline{\mathbf{Q}}\_{12} & \overline{\mathbf{Q}}\_{16} & 0 & 0\\ \overline{\mathbf{Q}}\_{12} & \overline{\mathbf{Q}}\_{22} & \overline{\mathbf{Q}}\_{26} & 0 & 0\\ \overline{\mathbf{Q}}\_{16} & \overline{\mathbf{Q}}\_{26} & \overline{\mathbf{Q}}\_{66} & 0 & 0\\ 0 & 0 & 0 & \overline{\mathbf{Q}}\_{44} & \overline{\mathbf{Q}}\_{45}\\ 0 & 0 & 0 & \overline{\mathbf{Q}}\_{45} & \overline{\mathbf{Q}}\_{55} \end{bmatrix}^{(k)} \{\varepsilon\} \tag{8}$$

f g <sup>N</sup> <sup>¼</sup> ½ � <sup>A</sup> f g<sup>ε</sup> <sup>þ</sup>X<sup>n</sup>

<sup>N</sup><sup>J</sup> � � <sup>¼</sup> BJ � �f g<sup>ε</sup> <sup>þ</sup>X<sup>n</sup>

where ½ � <sup>A</sup> , <sup>B</sup><sup>J</sup> � �, and <sup>D</sup>IJ � � are stiffnesses of a laminated plate

Aij <sup>¼</sup> <sup>X</sup> N

> BJ ij <sup>¼</sup> <sup>X</sup> N

BJ ij <sup>¼</sup> <sup>X</sup> N

DJI ij <sup>¼</sup> <sup>X</sup> N

DJI ij <sup>¼</sup> <sup>X</sup> N

Aij <sup>¼</sup> <sup>X</sup> N

BJ ij <sup>¼</sup> <sup>Q</sup><sup>J</sup>�<sup>1</sup> ij

BJ ij <sup>¼</sup> <sup>Q</sup><sup>J</sup>�<sup>1</sup>

DIJ ij <sup>¼</sup> <sup>Q</sup><sup>J</sup>�<sup>1</sup> ij

DIJ

DIJ ij <sup>¼</sup> <sup>D</sup>JI

DIJ ij <sup>¼</sup> <sup>D</sup>JI

k¼1 QK

f or J ¼ I

ij <sup>¼</sup> <sup>Q</sup><sup>J</sup>�<sup>1</sup> ij ðzJ � zJ�<sup>1</sup>Þ

f or J ¼ I � 1

ij <sup>¼</sup> <sup>Q</sup><sup>J</sup> ij

ij ¼ � <sup>Q</sup><sup>J</sup>

J¼1

ðzkþ1

Q<sup>ð</sup>k<sup>Þ</sup>

Q<sup>ð</sup>k<sup>Þ</sup> ij ψ<sup>J</sup>

> Q<sup>ð</sup>k<sup>Þ</sup> ij ψ<sup>J</sup>

zk

ðzkþ1

zk

ðzkþ1

zk

Q<sup>ð</sup>k<sup>Þ</sup> ij ψ<sup>J</sup> ψI

Q<sup>ð</sup>k<sup>Þ</sup> ij ψ<sup>J</sup> , <sup>z</sup>ψ<sup>I</sup>

For the adopted linear interpolation functions (Eq. (5)) from Eqs. (12)–(14) can be obtained

ij

ij

QJ ij

ðzkþ1

zk

ðzkþ1

zk

<sup>2</sup> <sup>þ</sup> <sup>Q</sup><sup>J</sup>

<sup>3</sup> <sup>þ</sup> <sup>Q</sup><sup>J</sup>

ðzJþ<sup>1</sup> � zJÞ

ij

do not depend on the adopted discretization through the thickness of the plate.

k¼1

k¼1

k¼

k¼1

ðzJ � zJ�<sup>1</sup>Þ

ðzJ � zJ�<sup>1</sup>Þ

þ

ij � <sup>Q</sup><sup>J</sup>

J¼1

J¼1

B<sup>J</sup> � � ε<sup>J</sup> � �

dz ði, j ¼ 1, 2, 6Þ

, <sup>z</sup>dz ði, j ¼ 4, 5Þ

dz ði, j ¼ 1, 2, 6Þ

, <sup>z</sup>dz ði, j ¼ 4, 5Þ

<sup>2</sup> <sup>ð</sup>i, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>6</sup><sup>Þ</sup>

<sup>3</sup> <sup>ð</sup>i, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>6</sup><sup>Þ</sup>

ijðzk�<sup>1</sup> � zkÞ, ði, j ¼ 1, 2, 6, 4, 5Þ

ðzJþ<sup>1</sup> � zJÞ

ij ði, j ¼ 4, 5Þ

ðzJþ<sup>1</sup> � zJÞ

<sup>ð</sup>zJþ<sup>1</sup> � zJ<sup>Þ</sup> <sup>ð</sup>i, j <sup>¼</sup> <sup>4</sup>, <sup>5</sup><sup>Þ</sup>

<sup>6</sup> <sup>ð</sup>i, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>6</sup><sup>Þ</sup>

<sup>ð</sup>zJþ<sup>1</sup> � zJ<sup>Þ</sup> <sup>ð</sup>i, j <sup>¼</sup> <sup>4</sup>, <sup>5</sup><sup>Þ</sup>

It is noted that the elements of matrix A are determined for the laminated plate as a whole and

DIJ � � ε<sup>J</sup> � �

ij dz, ði, j ¼ 1, 2, 6, 4, 5Þ (12)

Bending of Laminated Composite Plates in Layerwise Theory

http://dx.doi.org/10.5772/intechopen.69975

(11)

57

(13)

(14)

(15)

In the previous equation, Qij are members of the transformed plane stress-reduced stiffnesses of kth layer which are obtained as follows; see Ref. [1]:

$$\begin{aligned} \overline{Q}\_{11} &= Q\_{11}\cos^4\theta + 2(Q\_{12} + 2Q\_{66})\sin^2\theta\cos^2\theta + Q\_{22}\sin^4\theta \\ \overline{Q}\_{12} &= (Q\_{11} + Q\_{22} - 4Q\_{66})\sin^2\theta\cos^2\theta + Q\_{12}(\sin^4\theta + \cos^4\theta) \\ \overline{Q}\_{22} &= Q\_{11}\sin^4\theta + 2(Q\_{12} + 2Q\_{66})\sin^2\theta\cos^2\theta + Q\_{22}\cos^4\theta \\ \overline{Q}\_{16} &= (Q\_{11} - Q\_{12} - 2Q\_{66})\cos^3\theta\sin\theta + (Q\_{12} - Q\_{22} + 2Q\_{66})\sin^3\theta\cos\theta \\ \overline{Q}\_{26} &= (Q\_{11} - Q\_{12} - 2Q\_{66})\sin^3\theta\cos\theta + (Q\_{12} - Q\_{22} + 2Q\_{66})\cos^3\theta\sin\theta \\ \overline{Q}\_{66} &= (Q\_{11} + Q\_{22} - 2Q\_{12} - 2Q\_{66})\sin^2\theta\cos^2\theta + Q\_{66}(\sin^4\theta + \cos^4\theta) \\ \overline{Q}\_{44} &= Q\_{44}\cos^2\theta + Q\_{55}\sin^2\theta \\ \overline{Q}\_{45} &= (Q\_{55} - Q\_{44})\cos\theta\sin\theta \\ \overline{Q}\_{56} &= Q\_{55}\cos^2\theta + Q\_{44}\sin^2\theta \end{aligned} \tag{9}$$

where θ is the angle between the fibers and the x-axis of the global coordinate system of the plate and Qij is the plane stress-reduced stiffnesses of kth layer (including materials constants).

Transverse stresses σ<sup>k</sup> xz and σ<sup>k</sup> yz obtained using Eq. (8) have a constant value along the kth layer. In order to determine a more realistic distribution of transverse shear stresses through the thickness approximate method that is given in Refs. [23, 26] can be used.

The internal forces in a cross section of the laminated plate are defined as integrals of stresses

$$\left(N\_{\mathbf{x}\prime}N\_{y\prime}N\_{\mathbf{x}y\prime}Q\_{\mathbf{x}\prime}Q\_{\mathbf{y}}\right) = \int\_{0}^{h} \left(\sigma\_{\mathbf{x}\prime}\sigma\_{yy\prime}\sigma\_{\mathbf{x}y\prime}\sigma\_{\mathbf{x}z\prime}\sigma\_{yz}\right)dz$$

$$\left(N\_{\mathbf{x}\prime}^{l}N\_{y\prime}^{l}N\_{\mathbf{x}y}^{l}\right) = \int\_{o}^{h} \left(\sigma\_{\mathbf{x}\prime}\sigma\_{yy\prime}\sigma\_{\mathbf{x}y}\right)\phi^{l}(z)dz\tag{10}$$

$$\left(Q\_{\mathbf{x}\prime}^{l}Q\_{\mathbf{y}}^{l}\right) = \int\_{0}^{h} \left(\sigma\_{\mathbf{x}z\prime}\sigma\_{yz}\right)\phi^{l}\_{,z}dz$$

where σx, σy, σxy, σxz, σyz are stress components.

The internal forces can be expressed in terms of the displacements by inserting Eq. (6) into Eq. (7) and then in the integrals in Eq. (10):

#### Bending of Laminated Composite Plates in Layerwise Theory http://dx.doi.org/10.5772/intechopen.69975 57

$$\begin{aligned} \{N\} &= [A] \{\varepsilon\} + \sum\_{J=1}^{n} [\mathcal{B}^{J}] \{\varepsilon^{J}\} \\ \{N^{J}\} &= [\mathcal{B}^{J}] \{\varepsilon\} + \sum\_{J=1}^{n} [\mathcal{D}^{J}] \{\varepsilon^{J}\} \end{aligned} \tag{11}$$

where ½ � <sup>A</sup> , <sup>B</sup><sup>J</sup> � �, and <sup>D</sup>IJ � � are stiffnesses of a laminated plate

f g<sup>σ</sup> <sup>ð</sup>k<sup>Þ</sup> <sup>¼</sup>

of kth layer which are obtained as follows; see Ref. [1]:

<sup>Q</sup><sup>12</sup> ¼ ðQ<sup>11</sup> <sup>þ</sup> <sup>Q</sup><sup>22</sup> � <sup>4</sup>Q66Þsin<sup>2</sup>

<sup>Q</sup><sup>26</sup> ¼ ðQ<sup>11</sup> � <sup>Q</sup><sup>12</sup> � <sup>2</sup>Q66Þsin<sup>3</sup>

<sup>Q</sup><sup>44</sup> <sup>¼</sup> <sup>Q</sup>44cos<sup>2</sup><sup>θ</sup> <sup>þ</sup> <sup>Q</sup>55sin<sup>2</sup>

xz and σ<sup>k</sup>

NJ x, N<sup>J</sup> y, N<sup>J</sup> xy

QJ x, Q<sup>J</sup> y � �

where σx, σy, σxy, σxz, σyz are stress components.

Eq. (7) and then in the integrals in Eq. (10):

� �

Q<sup>45</sup> ¼ ðQ<sup>55</sup> � Q44Þcosθsinθ <sup>Q</sup><sup>55</sup> <sup>¼</sup> <sup>Q</sup>55cos2<sup>θ</sup> <sup>þ</sup> <sup>Q</sup>44sin<sup>2</sup>

<sup>Q</sup><sup>66</sup> ¼ ðQ<sup>11</sup> <sup>þ</sup> <sup>Q</sup><sup>22</sup> � <sup>2</sup>Q<sup>12</sup> � <sup>2</sup>Q66Þsin<sup>2</sup>

<sup>Q</sup><sup>22</sup> <sup>¼</sup> <sup>Q</sup>11sin<sup>4</sup>

56 Lamination - Theory and Application

Transverse stresses σ<sup>k</sup>

<sup>Q</sup><sup>11</sup> <sup>¼</sup> <sup>Q</sup>11cos<sup>4</sup><sup>θ</sup> <sup>þ</sup> <sup>2</sup>ðQ<sup>12</sup> <sup>þ</sup> <sup>2</sup>Q66Þsin<sup>2</sup>

<sup>θ</sup> <sup>þ</sup> <sup>2</sup>ðQ<sup>12</sup> <sup>þ</sup> <sup>2</sup>Q66Þsin<sup>2</sup>

θ

θ

thickness approximate method that is given in Refs. [23, 26] can be used.

¼ ðh o

σxz, σyz � �ϕ<sup>J</sup>

Nx, Ny, Nxy, Qx, Qy � �

> ¼ ðh 0

where θ is the angle between the fibers and the x-axis of the global coordinate system of the plate and Qij is the plane stress-reduced stiffnesses of kth layer (including materials constants).

In order to determine a more realistic distribution of transverse shear stresses through the

The internal forces in a cross section of the laminated plate are defined as integrals of stresses

¼ ðh 0

σxx, σyy, σxy � �ϕ<sup>J</sup>

, <sup>z</sup>dz

The internal forces can be expressed in terms of the displacements by inserting Eq. (6) into

<sup>Q</sup><sup>16</sup> ¼ ðQ<sup>11</sup> � <sup>Q</sup><sup>12</sup> � <sup>2</sup>Q66Þcos<sup>3</sup>θsin<sup>θ</sup> þ ðQ<sup>12</sup> � <sup>Q</sup><sup>22</sup> <sup>þ</sup> <sup>2</sup>Q66Þsin<sup>3</sup>

Q<sup>11</sup> Q<sup>12</sup> Q<sup>16</sup> 0 0 Q<sup>12</sup> Q<sup>22</sup> Q<sup>26</sup> 0 0 Q<sup>16</sup> Q<sup>26</sup> Q<sup>66</sup> 0 0 000 Q<sup>44</sup> Q<sup>45</sup> 000 Q<sup>45</sup> Q<sup>55</sup>

In the previous equation, Qij are members of the transformed plane stress-reduced stiffnesses

<sup>θ</sup>cos2<sup>θ</sup> <sup>þ</sup> <sup>Q</sup>22sin<sup>4</sup>

<sup>θ</sup>cos<sup>2</sup><sup>θ</sup> <sup>þ</sup> <sup>Q</sup>22cos<sup>4</sup><sup>θ</sup>

<sup>θ</sup>cos<sup>θ</sup> þ ðQ<sup>12</sup> � <sup>Q</sup><sup>22</sup> <sup>þ</sup> <sup>2</sup>Q66Þcos<sup>3</sup>θsin<sup>θ</sup>

yz obtained using Eq. (8) have a constant value along the kth layer.

σxx, σyy, σxy, σxz, σyz � �dz

ð Þz dz

<sup>θ</sup>cos<sup>2</sup><sup>θ</sup> <sup>þ</sup> <sup>Q</sup><sup>66</sup> sin<sup>4</sup>

<sup>θ</sup>cos<sup>2</sup><sup>θ</sup> <sup>þ</sup> <sup>Q</sup><sup>12</sup> sin<sup>4</sup>

ðkÞ

θ

<sup>θ</sup> <sup>þ</sup> cos<sup>4</sup> θ � �

f gε (8)

θcosθ

(9)

(10)

<sup>θ</sup> <sup>þ</sup> cos4 θ � �

$$A\_{ij} = \sum\_{l=1}^{N} \int\_{z\_k}^{z\_{k+1}} \overline{Q}\_{ij}^{(k)} dz \qquad \qquad (i, j = 1, 2, 6, 4, 5) \tag{12}$$

$$\begin{aligned} \boldsymbol{B}\_{ij}^{l} &= \sum\_{k=1}^{N} \sum\_{z\_{k}}^{z\_{k+1}} \overline{\boldsymbol{\nabla}}\_{ij}^{(k)} \boldsymbol{\psi}^{l} dz & (i, j = 1, 2, 6) \\ \boldsymbol{B}\_{ij}^{l} &= \sum\_{k=1}^{N} \int\_{z\_{k}}^{z\_{k+1}} \overline{\boldsymbol{\nabla}}\_{ij}^{(k)} \boldsymbol{\psi}\_{,z}^{l} dz & (i, j = 4, 5) \\ \boldsymbol{D}\_{ij}^{l} &= \sum\_{k=1}^{N} \int\_{z\_{k}}^{z\_{k+1}} \overline{\boldsymbol{\nabla}}\_{ij}^{(k)} \boldsymbol{\psi}^{l} \boldsymbol{\psi}^{l} dz & (i, j = 1, 2, 6) \\ \boldsymbol{D}\_{ij}^{l} &= \sum\_{k=1}^{N} \int\_{z\_{k}}^{z\_{k+1}} \overline{\boldsymbol{\nabla}}\_{ij}^{(k)} \boldsymbol{\psi}\_{,z}^{l} \boldsymbol{\psi}\_{,z}^{l} dz & (i, j = 4, 5) \end{aligned} \tag{14}$$

For the adopted linear interpolation functions (Eq. (5)) from Eqs. (12)–(14) can be obtained

Aij <sup>¼</sup> <sup>X</sup> N k¼1 QK ijðzk�<sup>1</sup> � zkÞ, ði, j ¼ 1, 2, 6, 4, 5Þ BJ ij <sup>¼</sup> <sup>Q</sup><sup>J</sup>�<sup>1</sup> ij ðzJ � zJ�<sup>1</sup>Þ <sup>2</sup> <sup>þ</sup> <sup>Q</sup><sup>J</sup> ij ðzJþ<sup>1</sup> � zJÞ <sup>2</sup> <sup>ð</sup>i, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>6</sup><sup>Þ</sup> BJ ij <sup>¼</sup> <sup>Q</sup><sup>J</sup>�<sup>1</sup> ij � <sup>Q</sup><sup>J</sup> ij ði, j ¼ 4, 5Þ f or J ¼ I DIJ ij <sup>¼</sup> <sup>Q</sup><sup>J</sup>�<sup>1</sup> ij ðzJ � zJ�<sup>1</sup>Þ <sup>3</sup> <sup>þ</sup> <sup>Q</sup><sup>J</sup> ij ðzJþ<sup>1</sup> � zJÞ <sup>3</sup> <sup>ð</sup>i, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>6</sup><sup>Þ</sup> DIJ ij <sup>¼</sup> <sup>Q</sup><sup>J</sup>�<sup>1</sup> ij ðzJ � zJ�<sup>1</sup>Þ þ QJ ij <sup>ð</sup>zJþ<sup>1</sup> � zJ<sup>Þ</sup> <sup>ð</sup>i, j <sup>¼</sup> <sup>4</sup>, <sup>5</sup><sup>Þ</sup> f or J ¼ I � 1 DIJ ij <sup>¼</sup> <sup>D</sup>JI ij <sup>¼</sup> <sup>Q</sup><sup>J</sup> ij ðzJþ<sup>1</sup> � zJÞ <sup>6</sup> <sup>ð</sup>i, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>6</sup><sup>Þ</sup> DIJ ij <sup>¼</sup> <sup>D</sup>JI ij ¼ � <sup>Q</sup><sup>J</sup> ij <sup>ð</sup>zJþ<sup>1</sup> � zJ<sup>Þ</sup> <sup>ð</sup>i, j <sup>¼</sup> <sup>4</sup>, <sup>5</sup><sup>Þ</sup> (15)

It is noted that the elements of matrix A are determined for the laminated plate as a whole and do not depend on the adopted discretization through the thickness of the plate.

The governing equations of motion can be derived by using the principle of displacements. For zero values of stiffnesses A16 ¼ A26 ¼ A45 ¼ B16 J ¼ B26 J ¼ B45 J ¼ D16 JI <sup>¼</sup> D26 JI <sup>¼</sup> D45 JI, the equations take the form:

Boundary conditions for simply supported plate are met:

n), a system of equations is obtained:

Elements of the submatrices K, KJ

displacements are determined.

m, <sup>n</sup>

m, <sup>n</sup>

m, <sup>n</sup>

<sup>σ</sup>xxðx, y, z޼�X<sup>∞</sup>

<sup>σ</sup>yyðx, y, z޼�X<sup>∞</sup>

<sup>σ</sup>ðx, y, z޼�Q66X<sup>∞</sup>

σj

σj

xzð Þ¼ x, y, z <sup>X</sup><sup>∞</sup>

xzð Þ¼ x, y, z <sup>X</sup><sup>∞</sup>

m, <sup>n</sup> Qj 45 1 hk S<sup>j</sup>þ<sup>1</sup> mn � <sup>S</sup><sup>j</sup>

m, <sup>n</sup> Qj 44 1 hk S<sup>j</sup>þ<sup>1</sup> mn � <sup>S</sup><sup>j</sup>

laminated plate (Eq. (15)) and variables α and β [26].

plane of the plate are determined by the expressions:

<sup>Q</sup>11<sup>α</sup> Xmn <sup>þ</sup>X<sup>n</sup>

<sup>Q</sup>12<sup>α</sup> Xmn <sup>þ</sup>X<sup>n</sup>

<sup>β</sup> Xmn <sup>þ</sup>X<sup>n</sup>

J¼1 RJ mnϕ<sup>J</sup> ðzÞ

!

J¼1 RJ mnϕ<sup>J</sup> ðzÞ

Shear stresses σxz and σ<sup>y</sup> are constant through the layer J and can be obtained with

!

J¼1 RJ mnϕ<sup>J</sup> ðzÞ

mn � � <sup>þ</sup> <sup>β</sup>Wmn � �sinα<sup>x</sup> cosβ<sup>y</sup> � �

mn � � <sup>þ</sup> <sup>β</sup>Wmn � �sinα<sup>x</sup> cosβ<sup>y</sup> � �

!

<sup>u</sup> <sup>¼</sup> <sup>w</sup> <sup>¼</sup> <sup>V</sup><sup>J</sup> <sup>¼</sup> Nx <sup>¼</sup> <sup>N</sup><sup>J</sup>

<sup>v</sup> <sup>¼</sup> <sup>w</sup> <sup>¼</sup> <sup>U</sup><sup>J</sup> <sup>¼</sup> Ny <sup>¼</sup> <sup>N</sup><sup>J</sup>

Transverse load is displayed in the form of double Fourier series:

<sup>q</sup>ðx, yÞ ¼ <sup>X</sup><sup>∞</sup>

½ � <sup>K</sup> <sup>K</sup><sup>J</sup> � � <sup>K</sup><sup>J</sup> � �<sup>T</sup> CJI � � " #

m, <sup>n</sup>

where Qmn is the Furie's coefficients which depend on the type of transverse load.

Qmn sin <sup>m</sup>π<sup>x</sup> a

When derivatives of the displacement (Eq. (17)) are substituted into Eq. (16), for each pair (m,

8 >>>>>><

>>>>>>:

By solving Eq. (20), the unknown coefficients Xmn, Ymn, Wmn, Rmn, and Smn for each pair (m, n) are determined. Additionally, the application of Eq. (17) and the summing of the unknown

The strains are determined using Eq. (6) and then the stresses using Eq. (8). Stresses in the

" # !

" # !

" # !

Xmn Ymn Wmn RJ mn SJ mn

9 >>>>>>=

>>>>>>; ¼

<sup>x</sup> ¼ 0 x ¼ 0,a J ¼ 1, n

(18)

59

(20)

sinαx sinβy

sinαx sinβy

(21)

(22)

cosαx sinβy

mn � � <sup>þ</sup> <sup>α</sup>Wmn � �cosα<sup>x</sup> sinβ<sup>y</sup>

mn � � <sup>þ</sup> <sup>α</sup>Wmn � �cosα<sup>x</sup> sinβ<sup>y</sup>

<sup>b</sup> (19)

Bending of Laminated Composite Plates in Layerwise Theory

http://dx.doi.org/10.5772/intechopen.69975

<sup>y</sup> ¼ 0 y ¼ 0,b J ¼ 1, n

sin <sup>n</sup>π<sup>y</sup>

9 >>>>>>=

>>>>>>;

, and KJI are given in the function of the stiffnesses of

J¼1 SJ mnϕ<sup>J</sup> ðzÞ

J¼1 SJ mnϕ<sup>J</sup> ðzÞ

Rjþ<sup>1</sup> mn � <sup>R</sup><sup>j</sup>

Rjþ<sup>1</sup> mn � <sup>R</sup><sup>j</sup>

8 >>>>>><

>>>>>>:

<sup>þ</sup> <sup>Q</sup>12<sup>β</sup> Ymn <sup>þ</sup>X<sup>n</sup>

<sup>þ</sup> <sup>Q</sup>22<sup>β</sup> Ymn <sup>þ</sup>X<sup>n</sup>

J¼1 SJ mnϕ<sup>J</sup> ðzÞ

( )

( )

<sup>þ</sup> <sup>α</sup> Ymn <sup>þ</sup>X<sup>∞</sup>

<sup>þ</sup> Qj 55 1 hk

<sup>þ</sup> Qj 45 1 hk

� � ��

� � ��

( )

<sup>A</sup>11u, xx <sup>þ</sup> <sup>A</sup>12v, yx <sup>þ</sup> <sup>A</sup><sup>66</sup> u, yy <sup>þ</sup> v, xy � � <sup>þ</sup>X<sup>n</sup> J¼1 BJ 11U<sup>J</sup> , xx <sup>þ</sup> BJ 12V<sup>J</sup> , yx <sup>þ</sup> BJ <sup>66</sup> <sup>U</sup><sup>J</sup> ,yy <sup>þ</sup> <sup>V</sup><sup>J</sup> , xy h i � � <sup>¼</sup> <sup>0</sup> <sup>A</sup>12u, xy <sup>þ</sup> <sup>A</sup>22v, yy <sup>þ</sup> <sup>A</sup><sup>66</sup> u, yx <sup>þ</sup> v, xx � � <sup>þ</sup>X<sup>n</sup> J¼1 BJ 12U<sup>J</sup> , xy <sup>þ</sup> BJ 22V<sup>J</sup> , yy <sup>þ</sup> BJ <sup>66</sup> <sup>U</sup><sup>J</sup> ,yx <sup>þ</sup> <sup>V</sup><sup>J</sup> , xx h i � � <sup>¼</sup> <sup>0</sup> <sup>A</sup>55w, xx <sup>þ</sup> <sup>A</sup>44w, yy <sup>þ</sup>X<sup>n</sup> J¼1 BJ 55U<sup>J</sup> , <sup>x</sup> <sup>þ</sup> BJ 44V<sup>J</sup> , y h i <sup>þ</sup> <sup>q</sup> <sup>¼</sup> <sup>0</sup> BJ <sup>11</sup>u, xx <sup>þ</sup> BJ <sup>12</sup>v, xy <sup>þ</sup> BJ <sup>66</sup> u, yy <sup>þ</sup> v, xy � � � BJ <sup>55</sup>w, <sup>x</sup> <sup>þ</sup>X<sup>n</sup> J¼1 DJI 11U<sup>J</sup> , xx <sup>þ</sup> <sup>D</sup>JI 12V<sup>J</sup> ,yx <sup>þ</sup> DJI <sup>66</sup> <sup>U</sup><sup>J</sup> , yy <sup>þ</sup> <sup>V</sup><sup>J</sup> ,xy � � � DJI <sup>55</sup>U<sup>I</sup> h i <sup>¼</sup> <sup>0</sup> BJ <sup>11</sup>u, xy <sup>þ</sup> BJ <sup>21</sup>v, yy <sup>þ</sup> BJ <sup>66</sup> u, yx <sup>þ</sup> v, xx � � � BJ <sup>44</sup>w, <sup>y</sup> <sup>þ</sup>X<sup>n</sup> J¼1 DJI 12U<sup>J</sup> , xy <sup>þ</sup> <sup>D</sup>JI 22V<sup>J</sup> ,yy <sup>þ</sup> DJI <sup>66</sup> <sup>U</sup><sup>J</sup> , yx <sup>þ</sup> <sup>V</sup><sup>J</sup> ,xx � � � DJI <sup>44</sup>V<sup>I</sup> h i <sup>¼</sup> <sup>0</sup> (16)

where q is load perpendicular to the middle plane of the plate.

The system of Eq. (16) contains (3þ2n) differential equations with the unknown displacements of the middle plane of the plate (u, v, w), and the displacements through the plate thickness U<sup>J</sup> and V<sup>J</sup> , where J¼1, …, n. These equations can be solved by using analytical or numerical and approximate methods.

This chapter shows the analytical solution that is obtained by using the double trigonometric series for simply supported plate, as well as the numerical solution obtained by using the finite element method as one of the currently most applied numerical methods.

## 3. Analytical solution

#### 3.1. Navier's solution in layerwise theory

Simply supported rectangular laminated plate axb with N orthotropic layers with angles of orientation θ¼0� or 90� is considered. Displacements (Eqs. (1) and (2)) are given in the form of double trigonometric series

$$\begin{aligned} \mu &= \sum\_{m\_r n}^{\infty} X\_{mn} \cos \frac{m \pi x}{a} \sin \frac{n \pi y}{b} \\ \sigma &= \sum\_{m\_r n}^{\infty} Y\_{mn} \sin \frac{m \pi x}{a} \cos \frac{n \pi y}{b} \\ \sigma &= \sum\_{m\_r n}^{\infty} W\_{mn} \sin \frac{m \pi x}{a} \sin \frac{n \pi y}{b} \\ \sigma^l &= \sum\_{m\_r n}^{\infty} R\_{mn}^l \cos \frac{m \pi x}{a} \sin \frac{n \pi y}{b} \\ V^l &= \sum\_{m\_r n}^{\infty} S\_{mn}^l \sin \frac{m \pi x}{a} \cos \frac{n \pi y}{b} \end{aligned} \tag{17}$$

where <sup>α</sup> <sup>¼</sup> <sup>m</sup><sup>π</sup> <sup>a</sup> ; <sup>β</sup> <sup>¼</sup> <sup>n</sup><sup>π</sup> <sup>b</sup> J ¼ 1, …, n Boundary conditions for simply supported plate are met:

The governing equations of motion can be derived by using the principle of displacements. For zero

JI <sup>¼</sup> D26

,yy <sup>þ</sup> <sup>V</sup><sup>J</sup> , xy

,yx <sup>þ</sup> <sup>V</sup><sup>J</sup> , xx

,yx <sup>þ</sup> DJI

,yy <sup>þ</sup> DJI

<sup>66</sup> <sup>U</sup><sup>J</sup>

<sup>66</sup> <sup>U</sup><sup>J</sup>

55U<sup>I</sup> h i

44V<sup>I</sup> h i

<sup>66</sup> <sup>U</sup><sup>J</sup>

<sup>66</sup> <sup>U</sup><sup>J</sup>

h i � �

h i � �

, xx <sup>þ</sup> <sup>D</sup>JI 12V<sup>J</sup>

, xy <sup>þ</sup> <sup>D</sup>JI 22V<sup>J</sup> JI <sup>¼</sup> D45 JI

¼ 0

¼ 0

, yy <sup>þ</sup> <sup>V</sup><sup>J</sup> ,xy � �

, yx <sup>þ</sup> <sup>V</sup><sup>J</sup> ,xx � �

, the equations take

� DJI

� DJI

¼ 0

¼ 0

(16)

(17)

J ¼ B26 J ¼ B45 J ¼ D16

þ q ¼ 0

J¼1

J¼1

DJI 11U<sup>J</sup>

DJI 12U<sup>J</sup>

The system of Eq. (16) contains (3þ2n) differential equations with the unknown displacements of the middle plane of the plate (u, v, w), and the displacements through the plate thickness U<sup>J</sup>

This chapter shows the analytical solution that is obtained by using the double trigonometric series for simply supported plate, as well as the numerical solution obtained by using the finite

Simply supported rectangular laminated plate axb with N orthotropic layers with angles of orientation θ¼0� or 90� is considered. Displacements (Eqs. (1) and (2)) are given in the form of

> mπx a

> > mπx a

sin <sup>n</sup>π<sup>y</sup> b

> sin <sup>n</sup>π<sup>y</sup> b

sin <sup>n</sup>π<sup>y</sup> b

Xmncos

Ymnsin <sup>m</sup>π<sup>x</sup> a cos nπy b

Wmnsin <sup>m</sup>π<sup>x</sup> a

> mnsin <sup>m</sup>π<sup>x</sup> a cos nπy b

, where J¼1, …, n. These equations can be solved by using analytical or numerical and

<sup>55</sup>w, <sup>x</sup> <sup>þ</sup>X<sup>n</sup>

<sup>44</sup>w, <sup>y</sup> <sup>þ</sup>X<sup>n</sup>

J¼1 BJ 11U<sup>J</sup> , xx <sup>þ</sup> BJ 12V<sup>J</sup> , yx <sup>þ</sup> BJ

J¼1 BJ 12U<sup>J</sup> , xy <sup>þ</sup> BJ 22V<sup>J</sup> , yy <sup>þ</sup> BJ

values of stiffnesses A16 ¼ A26 ¼ A45 ¼ B16

J¼1 BJ 55U<sup>J</sup> , <sup>x</sup> <sup>þ</sup> BJ 44V<sup>J</sup> , y

<sup>66</sup> u, yy þ v, xy � � � BJ

<sup>66</sup> u, yx þ v, xx � � � BJ

� � þX<sup>n</sup>

� � þX<sup>n</sup>

h i

where q is load perpendicular to the middle plane of the plate.

element method as one of the currently most applied numerical methods.

<sup>u</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup> m, <sup>n</sup>

<sup>v</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup> m, <sup>n</sup>

<sup>w</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup> m, <sup>n</sup>

<sup>U</sup><sup>J</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup> m, <sup>n</sup> RJ mncos

<sup>V</sup><sup>J</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup> m, <sup>n</sup> SJ

<sup>b</sup> J ¼ 1, …, n

A11u, xx þ A12v, yx þ A<sup>66</sup> u, yy þ v, xy

58 Lamination - Theory and Application

A12u, xy þ A22v, yy þ A<sup>66</sup> u, yx þ v, xx

<sup>12</sup>v, xy <sup>þ</sup> BJ

<sup>21</sup>v, yy <sup>þ</sup> BJ

<sup>A</sup>55w, xx <sup>þ</sup> <sup>A</sup>44w, yy <sup>þ</sup>X<sup>n</sup>

approximate methods.

3. Analytical solution

double trigonometric series

where <sup>α</sup> <sup>¼</sup> <sup>m</sup><sup>π</sup>

<sup>a</sup> ; <sup>β</sup> <sup>¼</sup> <sup>n</sup><sup>π</sup>

3.1. Navier's solution in layerwise theory

the form:

BJ

BJ

and V<sup>J</sup>

<sup>11</sup>u, xx <sup>þ</sup> BJ

<sup>11</sup>u, xy <sup>þ</sup> BJ

$$\begin{aligned} \mu = w = V^l = \mathbf{N}\_x = \mathbf{N}\_x^l = \mathbf{0} & \quad \mathbf{x} = \mathbf{0}, a & \quad J = 1, n\\ w = w = \mathbf{U}^l = \mathbf{N}\_y = \mathbf{N}\_y^l = \mathbf{0} & \quad y = \mathbf{0}, b & \quad J = 1, n \end{aligned} \tag{18}$$

Transverse load is displayed in the form of double Fourier series:

$$q(\mathbf{x}, y) = \sum\_{m,n}^{\prime} Q\_{mn} \sin \frac{m \pi x}{a} \sin \frac{n \pi y}{b} \tag{19}$$

where Qmn is the Furie's coefficients which depend on the type of transverse load.

When derivatives of the displacement (Eq. (17)) are substituted into Eq. (16), for each pair (m, n), a system of equations is obtained:

$$\begin{Bmatrix} \begin{bmatrix} \mathbf{K} \end{bmatrix} & \begin{bmatrix} \mathbf{K}^{l} \end{bmatrix} \\\\ \begin{bmatrix} \mathbf{K}^{l} \end{bmatrix}^{T} & \begin{bmatrix} \mathbf{C}^{l} \end{bmatrix} \end{Bmatrix} \begin{Bmatrix} \mathbf{Y}\_{mn} \\\\ \mathbf{W}\_{mn} \\\\ \mathbf{R}\_{mn}^{l} \\\\ \mathbf{S}\_{mn}^{l} \end{Bmatrix} = \begin{Bmatrix} \mathbf{0} \\\\ \mathbf{0} \\\\ \mathbf{0} \\\\ \mathbf{0} \end{Bmatrix} \end{Bmatrix} = \begin{Bmatrix} \mathbf{0} \\\\ \mathbf{0} \\\\ \mathbf{0} \\\\ \mathbf{0} \end{Bmatrix} \tag{20}$$

Elements of the submatrices K, KJ , and KJI are given in the function of the stiffnesses of laminated plate (Eq. (15)) and variables α and β [26].

By solving Eq. (20), the unknown coefficients Xmn, Ymn, Wmn, Rmn, and Smn for each pair (m, n) are determined. Additionally, the application of Eq. (17) and the summing of the unknown displacements are determined.

The strains are determined using Eq. (6) and then the stresses using Eq. (8). Stresses in the plane of the plate are determined by the expressions:

$$\begin{aligned} \sigma\_{xx}(x,y,z) &= -\sum\_{m,n}^{m} \left\{ \left[ \overline{\mathbf{Q}}\_{11} a \left( \mathbf{X}\_{mn} + \sum\_{l=1}^{n} \mathbf{R}\_{mn}^{l} \phi^{l}(z) \right) + \overline{\mathbf{Q}}\_{12} \theta \left( Y\_{mn} + \sum\_{l=1}^{n} \mathbf{S}\_{mn}^{l} \phi^{l}(z) \right) \right] \text{sinc} x \sin \theta y \right\} \\ \sigma\_{yy}(x,y,z) &= -\sum\_{m,n}^{m} \left\{ \left[ \overline{\mathbf{Q}}\_{12} a \left( \mathbf{X}\_{mn} + \sum\_{l=1}^{n} \mathbf{R}\_{mn}^{l} \phi^{l}(z) \right) + \overline{\mathbf{Q}}\_{22} \theta \left( Y\_{mn} + \sum\_{l=1}^{n} \mathbf{S}\_{mn}^{l} \phi^{l}(z) \right) \right] \text{sinc} x \sin \theta y \right\} \\ \sigma(x,y,z) &= -\overline{\mathbf{Q}}\_{66} \sum\_{m,n}^{m} \left\{ \left[ \beta \left( \mathbf{X}\_{mn} + \sum\_{l=1}^{n} \mathbf{R}\_{mn}^{l} \phi^{l}(z) \right) + a \left( Y\_{mn} + \sum\_{l=1}^{n} \mathbf{S}\_{mn}^{l} \phi^{l}(z) \right) \right] \text{cos} x \sin \theta y \right\} \end{aligned} \tag{21}$$

Shear stresses σxz and σ<sup>y</sup> are constant through the layer J and can be obtained with

$$\begin{aligned} \boldsymbol{\sigma}\_{xx}^{j}(\mathbf{x},y,z) &= \sum\_{m,\mu}^{\bullet} \left\{ \overline{Q}\_{45} \left\{ \frac{1}{\hbar\_{k}} \left[ \left( S\_{uu}^{j+1} - S\_{mu}^{j} \right) + \beta \mathcal{W}\_{mu} \right] \text{sinc} \mathbf{x} \cos \theta y \right\} + \overline{Q}\_{45}^{j} \left\{ \frac{1}{\hbar\_{k}} \left[ \left( R\_{uu}^{j+1} - R\_{mu}^{j} \right) + a \mathcal{W}\_{mu} \right] \cos \mathbf{x} \sin \theta y \right\} \right\} \\ \boldsymbol{\sigma}\_{xx}^{j}(\mathbf{x},y,z) &= \sum\_{m,\mu}^{\bullet} \left\{ \overline{Q}\_{44} \left\{ \frac{1}{\hbar\_{k}} \left[ \left( S\_{uu}^{j+1} - S\_{mu}^{j} \right) + \beta \mathcal{W}\_{mu} \right] \sin \mathbf{x} \cos \theta y \right\} + \overline{Q}\_{45}^{j} \left\{ \frac{1}{\hbar\_{k}} \left[ \left( R\_{uu}^{j+1} - R\_{mu}^{j} \right) + a \mathcal{W}\_{mu} \right] \cos \mathbf{x} \sin \theta y \right\} \right\} \end{aligned}$$

(22)

A more realistic change of shear stresses σxz and σ<sup>y</sup> through the plate thickness can be obtained using approximate procedures. In this chapter, approximate procedure as described in Ref. [23] was used.

#### 3.2. Numerical examples

For a simply supported rectangular plate, stresses are determined in dimensionless form:

$$
\overline{\sigma}\_{xx} = \frac{1}{q\overline{s}^2} \sigma\_{xx} \qquad \overline{\sigma}\_{yy} = \frac{1}{q\overline{s}^2} \sigma\_{yy} \qquad \overline{\sigma}\_{xz} = \frac{1}{q\overline{s}} \sigma\_{xz} \qquad \overline{\sigma}\_{yz} = \frac{1}{q\overline{s}} \sigma\_{yz} \tag{23}
$$

Example 2: The square laminated composite plate with six layers 0<sup>o</sup>

the plate.

4. Finite element method

Figure 4. σxx and σyy through the thickness of the plate 0o

Figure 5. σyz and σxz through the thickness of the plate 0o

4.1. Stiffness matrix of the kinematic variable finite element

way as the primary variables (displacements, see Eq. (25)):

dimensional interpolation function ϕ<sup>i</sup> and nodal displacements ui, vi, wi, ui

u, v, w, uj, vj � � <sup>¼</sup> <sup>X</sup><sup>m</sup>

The components of displacement of finite element can be written as a combination of two-

/90<sup>o</sup> /0o /90o /0o /90o .

i¼1

Interpolation functions ϕ<sup>i</sup> are defined in the natural coordinates (ξ,η). The isoparametric formulation of the finite element is adopted, so that the geometry is interpolated the same

ui, vi, wi, u<sup>j</sup>

i , vj i

the aspect ratio a/h¼10 loaded by uniformly distributed load of intensity q is considered. The material properties of the layers are as in Example 1. Figures 4 and 5 graphically show the change of dimensionless stresses σxx, σyy, σyz, and σxz through the thickness of

> /90<sup>o</sup> /0o /90<sup>o</sup> /0o /90�.

/90<sup>o</sup> /0o /90<sup>o</sup> /0o

http://dx.doi.org/10.5772/intechopen.69975

Bending of Laminated Composite Plates in Layerwise Theory

j , vi j

� �ϕ<sup>i</sup> (25)

/90<sup>o</sup> and

61

where s¼a/h, a¼b are plate dimensions in the plane of the plate; h is the plate thickness

All layers of the laminated plate have the same thickness and material properties. For predefined material characteristics of layers E1, E2, G12, G21, G23, ν12, and ν13, changes of dimensionless stresses σxx, σyy, σyy, and σxz are displayed in the points with coordinates A¼1,105662(a/2) and B¼1,894338(a/2)

$$
\overline{\sigma}\_{xx} = \overline{\sigma}\_{xx}(A, A, z), \quad \overline{\sigma}\_{yy} = \overline{\sigma}\_{yy}(A, A, z), \quad \overline{\sigma}\_{xz} = \overline{\sigma}\_{xz}(B, A, z), \quad \overline{\sigma}\_{yz} = \overline{\sigma}\_{yz}(A, B, z) \tag{24}
$$

Example 1: The square laminated composite plate with five layers 0�/90�/0�/90�/0� and the aspect ratio a/h¼4 loaded by a bi-sinusoidal is considered. Characteristics of the layers are E1/E2¼25, E2¼1, G12¼G13¼0.5, G23¼0.2, ν12¼ν13¼0.25. The change of dimensionless normal stresses σxx, σyy and shear stresses σyz, σxz are displayed in Figures 2 and 3.

Figure 2. σxx and σyy through the thickness of the plate 0o /90<sup>o</sup> /0o /90<sup>o</sup> /0o , bi-sinusoidal load.

Figure 3. σyz and σxz through the thickness of the plate 0o /90<sup>o</sup> /0o /90o /0o , bi-sinusoidal load.

Example 2: The square laminated composite plate with six layers 0<sup>o</sup> /90<sup>o</sup> /0o /90<sup>o</sup> /0o /90<sup>o</sup> and the aspect ratio a/h¼10 loaded by uniformly distributed load of intensity q is considered. The material properties of the layers are as in Example 1. Figures 4 and 5 graphically show the change of dimensionless stresses σxx, σyy, σyz, and σxz through the thickness of the plate.

Figure 4. σxx and σyy through the thickness of the plate 0o /90<sup>o</sup> /0o /90<sup>o</sup> /0o /90�.

Figure 5. σyz and σxz through the thickness of the plate 0o /90<sup>o</sup> /0o /90o /0o /90o .

## 4. Finite element method

A more realistic change of shear stresses σxz and σ<sup>y</sup> through the plate thickness can be obtained using approximate procedures. In this chapter, approximate procedure as described in Ref. [23] was used.

For a simply supported rectangular plate, stresses are determined in dimensionless form:

where s¼a/h, a¼b are plate dimensions in the plane of the plate; h is the plate thickness

qs<sup>2</sup> <sup>σ</sup>yy, <sup>σ</sup>xz <sup>¼</sup> <sup>1</sup>

All layers of the laminated plate have the same thickness and material properties. For predefined material characteristics of layers E1, E2, G12, G21, G23, ν12, and ν13, changes of dimensionless stresses σxx, σyy, σyy, and σxz are displayed in the points with coordinates A¼1,105662(a/2) and

Example 1: The square laminated composite plate with five layers 0�/90�/0�/90�/0� and the aspect ratio a/h¼4 loaded by a bi-sinusoidal is considered. Characteristics of the layers are E1/E2¼25, E2¼1, G12¼G13¼0.5, G23¼0.2, ν12¼ν13¼0.25. The change of dimensionless normal stresses σxx, σyy

> /90<sup>o</sup> /0o /90<sup>o</sup> /0o

/90<sup>o</sup> /0o /90o /0o

, bi-sinusoidal load.

, bi-sinusoidal load.

σxx ¼ σxxðA, A, zÞ, σyy ¼ σyyðA, A, zÞ, σxz ¼ σxzðB, A, zÞ, σyz ¼ σyzðA, B, zÞ (24)

qs

<sup>σ</sup>xz, <sup>σ</sup>yz <sup>¼</sup> <sup>1</sup>

qs

σyz (23)

3.2. Numerical examples

60 Lamination - Theory and Application

B¼1,894338(a/2)

<sup>σ</sup>xx <sup>¼</sup> <sup>1</sup>

qs<sup>2</sup> <sup>σ</sup>xx, <sup>σ</sup>yy <sup>¼</sup> <sup>1</sup>

and shear stresses σyz, σxz are displayed in Figures 2 and 3.

Figure 2. σxx and σyy through the thickness of the plate 0o

Figure 3. σyz and σxz through the thickness of the plate 0o

#### 4.1. Stiffness matrix of the kinematic variable finite element

The components of displacement of finite element can be written as a combination of twodimensional interpolation function ϕ<sup>i</sup> and nodal displacements ui, vi, wi, ui j , vi j

$$\left(\left(u,v,w,u\_{\flat},v\_{\flat}\right) = \sum\_{i=1}^{m} \left(\mu\_{i\star}v\_{i\star}w\_{i\star}u\_{i\star}^{j}v\_{i}^{j}\right)\phi\_{i} \tag{25}$$

Interpolation functions ϕ<sup>i</sup> are defined in the natural coordinates (ξ,η). The isoparametric formulation of the finite element is adopted, so that the geometry is interpolated the same way as the primary variables (displacements, see Eq. (25)):

$$\begin{aligned} x &= \sum\_{i=1}^{m} x\_i \phi\_i(\xi, \eta) \\ y &= \sum\_{i=1}^{m} y\_i \phi\_i(\xi, \eta) \end{aligned} \tag{26}$$

ϕ, xx ϕ, yy ϕ, xy 9 = ; <sup>¼</sup> <sup>J</sup><sup>1</sup> ½ ��<sup>1</sup>

J<sup>1</sup> ½ �¼

By using Eq. (7), the strains can be written in the following form:

ΔTHTH<sup>T</sup>X<sup>n</sup>

J¼1 ϕJ <sup>Δ</sup>JT ! � <sup>Q</sup> � <sup>H</sup>δΔ <sup>þ</sup> <sup>H</sup>

After multiplication, the stiffness matrix of the finite element takes the form:

<sup>H</sup> <sup>þ</sup>X<sup>n</sup> I¼1 HT Qϕ<sup>I</sup>

<sup>H</sup> <sup>þ</sup>X<sup>n</sup> I¼1 HT BI

!

!

HTQϕ<sup>J</sup>

HTBJ

The stiffness matrix of the finite element can be written in a matrix from as

ϕi , ξξ ϕi , ηη ϕi , ξη 9 =

� � <sup>0</sup>

; � <sup>J</sup><sup>2</sup> ½ � <sup>ϕ</sup><sup>i</sup>

2 ξx, 2 ξ

2 ηx, 2 η

3

x, ηξ y, ηξ x, ηξ þ y, ηξ

x, ξξ y, ξξ x, ηη y, ηη x, ηξ y, ηξ

According to the principle of stationary potential energy, the first variation of the potential energy must be zero for equilibrium conditions. The first variation of the potential energy of

ð

Ve

ð Þ¼ <sup>z</sup> <sup>ψ</sup><sup>j</sup>

Inserting Eq. (35) into Eq. (34), the first variation of the potential energy of the finite element

� �X<sup>n</sup> j¼1

are the interpolation functions perpendicular to the plane of the plate or its derivatives:

ðzÞ ψj ðzÞ, <sup>z</sup>

, x ϕi , y

3

1

Bending of Laminated Composite Plates in Layerwise Theory

http://dx.doi.org/10.5772/intechopen.69975

A (31)

63

5 (32)

5 (33)

ϕ<sup>J</sup> Δ<sup>J</sup> � � (35)

dV (37)

dV ¼

(38)

dA

εTQδε dV (34)

� � (36)

Xn I¼1 ϕI δΔ<sup>I</sup>

!

Xn J¼1 HT Qϕ<sup>I</sup> ϕJ H

Xn J¼1 HT DIJH

<sup>H</sup> <sup>þ</sup>X<sup>n</sup> I¼1

<sup>H</sup> <sup>þ</sup>X<sup>n</sup> I¼1

8 < :

@

x, 2 <sup>ξ</sup> y, 2 <sup>ξ</sup> 2y,

2 4

x, 2 <sup>η</sup> y, 2 <sup>η</sup> 2y,

J<sup>2</sup> ½ �¼

δΠ<sup>e</sup> ¼

f g¼ ε ½ � H f gþ Δ H

ϕJ

2 4

8 < :

where

the finite element is

where ϕ <sup>J</sup>

becomes

δΠ<sup>e</sup> ¼

ke ¼ ððð

Ve

ðð

Ae

ð

Ve

<sup>H</sup>TQH <sup>þ</sup>X<sup>n</sup>

<sup>H</sup>TAH <sup>þ</sup>X<sup>n</sup>

J¼1

J¼1

where x, y are point coordinates and xi, yi are coordinates of the finite element. According to Eqs. (6) and (25), matrices H and H are defined as follows:

½ � <sup>H</sup> ð Þ <sup>5</sup>x3<sup>m</sup> <sup>¼</sup> ∂ϕ<sup>1</sup> ∂x 0 0 ∂ϕ<sup>2</sup> ∂x 0 0 … <sup>∂</sup>ϕ<sup>m</sup> ∂x 0 0 0 ∂ϕ<sup>1</sup> ∂y 0 0 ∂ϕ<sup>2</sup> ∂y 0 … 0 ∂ϕ<sup>m</sup> ∂y 0 ∂ϕ<sup>1</sup> ∂y ∂ϕ<sup>1</sup> ∂x 0 ∂ϕ<sup>2</sup> ∂y ∂ϕ<sup>2</sup> ∂x <sup>0</sup> … <sup>∂</sup>ϕ<sup>m</sup> ∂y ∂ϕ<sup>m</sup> ∂x 0 0 0 ∂ϕ<sup>1</sup> ∂x 0 0 ∂ϕ<sup>2</sup> <sup>∂</sup><sup>x</sup> … 0 0 ∂ϕ<sup>m</sup> ∂x 0 0 ∂ϕ<sup>1</sup> ∂y 0 0 ∂ϕ<sup>2</sup> <sup>∂</sup><sup>y</sup> … 0 0 ∂ϕ<sup>m</sup> ∂y 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (27) H � � ð Þ 5x2m ¼ ∂ϕ<sup>1</sup> ∂x 0 ∂ϕ<sup>2</sup> ∂x <sup>0</sup> … <sup>∂</sup>ϕ<sup>m</sup> ∂x 0 0 ∂ϕ<sup>1</sup> ∂y 0 ∂ϕ<sup>2</sup> <sup>∂</sup><sup>y</sup> … <sup>0</sup> ∂ϕ<sup>m</sup> ∂y ∂ϕ<sup>1</sup> ∂y ∂ϕ<sup>1</sup> ∂x ∂ϕ<sup>2</sup> ∂y ∂ϕ<sup>2</sup> <sup>∂</sup><sup>x</sup> … <sup>∂</sup>ϕ<sup>m</sup> ∂y ∂ϕ<sup>m</sup> ∂x ϕ<sup>1</sup> 0 ϕ<sup>2</sup> 0 … ϕ<sup>m</sup> 0 0 ϕ<sup>1</sup> 0 ϕ<sup>2</sup> … 0 ϕ<sup>m</sup> 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 (28)

where m is the number of nodes in the plane of the plate.

The first derivatives of the ϕ<sup>i</sup> with respect to the global coordinates x, y can be obtained by using the local coordinates ξ, η as follows:

$$\begin{Bmatrix} \frac{\partial \phi\_i}{\partial x} \\ \frac{\partial \phi\_i}{\partial y} \end{Bmatrix} = \begin{bmatrix} \mathbf{x}\_{,\boldsymbol{\xi}} & \mathbf{y}\_{,\boldsymbol{\xi}} \\ \mathbf{x}\_{,\boldsymbol{\eta}} & \mathbf{y}\_{,\boldsymbol{\eta}} \end{bmatrix}^{-1} \begin{Bmatrix} \phi\_{,\boldsymbol{\xi}}^{\dagger} \\ \phi\_{,\boldsymbol{\eta}}^{\dagger} \end{Bmatrix} = [\mathbf{J}]^{-1} \begin{Bmatrix} \phi\_{,\boldsymbol{\xi}}^{\dagger} \\ \phi\_{,\boldsymbol{\eta}}^{\dagger} \end{Bmatrix} \tag{29}$$

where J is the Jacobian matrix of the transformation in the natural coordinates:

$$[f] = \begin{bmatrix} \frac{\partial x}{\partial \xi} & \frac{\partial y}{\partial \xi} \\ \frac{\partial x}{\partial \eta} & \frac{\partial y}{\partial \eta} \end{bmatrix} \tag{30}$$

The second derivatives of the interpolation functions, which are used during determinations of the shear stresses perpendicular to the plane plate, are

Bending of Laminated Composite Plates in Layerwise Theory http://dx.doi.org/10.5772/intechopen.69975 63

$$\begin{Bmatrix} \phi\_{\land \text{xx}}\\ \phi\_{\land \text{yy}}\\ \phi\_{\land \text{xy}} \end{Bmatrix} = [I\_1]^{-1} \left( \left\{ \begin{matrix} \phi\_{i^\* \cdot \xi \xi}\\ \phi\_{i^\* \cdot \eta \eta}\\ \phi\_{i^\* \cdot \xi \eta} \end{matrix} - [I\_2] \begin{matrix} \phi\_{i^\* \cdot \text{x}}\\ \phi\_{i^\* \cdot \text{y}} \end{matrix} \right\} \right) \tag{31}$$

where

(26)

(27)

(28)

(29)

(30)

<sup>x</sup> <sup>¼</sup> <sup>X</sup><sup>m</sup> i¼1 xiϕ<sup>i</sup> ð Þ ξ, η

<sup>y</sup> <sup>¼</sup> <sup>X</sup><sup>m</sup> i¼1 yi ϕi ð Þ ξ, η

> ∂ϕ<sup>2</sup> ∂x

> ∂ϕ<sup>2</sup> ∂y

∂ϕ<sup>2</sup> ∂y

∂ϕ<sup>2</sup> ∂x

0 0

0 0

∂ϕ<sup>2</sup> ∂x

0

∂ϕ<sup>2</sup> ∂y

The first derivatives of the ϕ<sup>i</sup> with respect to the global coordinates x, y can be obtained by

∂x ∂ξ

∂x ∂η

The second derivatives of the interpolation functions, which are used during determinations of

0 0

0

∂ϕ<sup>1</sup> ∂x

∂ϕ<sup>1</sup> ∂y

0

∂ϕ<sup>1</sup> ∂y

∂ϕ<sup>1</sup> ∂x

∂ϕ<sup>1</sup> ∂x

0

∂ϕ<sup>1</sup> ∂y

<sup>¼</sup> x, <sup>ξ</sup> y, <sup>ξ</sup> x, <sup>η</sup> y, <sup>η</sup> � ��<sup>1</sup> ϕ,

where J is the Jacobian matrix of the transformation in the natural coordinates:

½�¼J

0 0 … <sup>∂</sup>ϕ<sup>m</sup>

∂ϕ<sup>2</sup>

∂ϕ<sup>2</sup>

∂ϕ<sup>2</sup>

∂ϕ<sup>2</sup>

ϕ<sup>1</sup> 0 ϕ<sup>2</sup> 0 … ϕ<sup>m</sup> 0 0 ϕ<sup>1</sup> 0 ϕ<sup>2</sup> … 0 ϕ<sup>m</sup>

> i ξ ϕ, i η

( )

∂y ∂ξ

∂y ∂η <sup>0</sup> … <sup>∂</sup>ϕ<sup>m</sup>

<sup>∂</sup><sup>y</sup> … <sup>0</sup>

<sup>∂</sup><sup>x</sup> … <sup>∂</sup>ϕ<sup>m</sup>

<sup>¼</sup> ½ �<sup>J</sup> �<sup>1</sup> <sup>ϕ</sup>,

i ξ ϕ, i η

( )

0 … 0

<sup>0</sup> … <sup>∂</sup>ϕ<sup>m</sup>

∂x

∂y

<sup>∂</sup><sup>x</sup> … 0 0

<sup>∂</sup><sup>y</sup> … 0 0

∂x

∂y

0 0

0

0

∂ϕ<sup>m</sup> ∂x

∂ϕ<sup>m</sup> ∂y

∂ϕ<sup>m</sup> ∂y

∂ϕ<sup>m</sup> ∂x

0

∂ϕ<sup>m</sup> ∂y

∂ϕ<sup>m</sup> ∂x

where x, y are point coordinates and xi, yi are coordinates of the finite element.

According to Eqs. (6) and (25), matrices H and H are defined as follows:

0 0

∂ϕ<sup>1</sup> ∂y

∂ϕ<sup>1</sup> ∂x

0 0

0 0

ð Þ 5x2m ¼

where m is the number of nodes in the plane of the plate.

∂ϕ<sup>i</sup> ∂x ∂ϕ<sup>i</sup> ∂y

the shear stresses perpendicular to the plane plate, are

9 >>=

>>;

8 >><

>>:

∂ϕ<sup>1</sup> ∂x

0

∂ϕ<sup>1</sup> ∂y

½ � <sup>H</sup> ð Þ <sup>5</sup>x3<sup>m</sup> <sup>¼</sup>

62 Lamination - Theory and Application

H � �

using the local coordinates ξ, η as follows:

$$[f\_1] = \begin{bmatrix} \mathbf{x}.\mathbf{\hat{z}}\_{\boldsymbol{\xi}} & \mathbf{y}.\mathbf{\hat{z}}\_{\boldsymbol{\xi}} & 2\mathbf{y}.\mathbf{\hat{z}}.\mathbf{\hat{z}}\_{\boldsymbol{\xi}} \\ \mathbf{x}.\mathbf{\hat{z}}\_{\boldsymbol{\eta}} & \mathbf{y}.\mathbf{\hat{\eta}}\_{\boldsymbol{\eta}} & 2\mathbf{y}.\mathbf{\hat{\eta}}\_{\boldsymbol{\eta}} \mathbf{x}.\mathbf{\hat{\eta}}\_{\boldsymbol{\eta}} \\ \mathbf{x}.\mathbf{\hat{\eta}} & \mathbf{y}.\mathbf{\hat{\eta}} & \mathbf{x}.\mathbf{\hat{\eta}}\_{\boldsymbol{\eta}} + \mathbf{y}.\mathbf{\hat{\eta}}\_{\boldsymbol{\eta}} \end{bmatrix} \tag{32}$$

$$[J\_2] = \begin{bmatrix} \propto\_{\times,\xi\xi} & y\_{\cdot,\xi\xi} \\ \propto\_{\eta\eta} & y\_{\cdot,\eta\eta} \\ \propto\_{\cdot,\eta\xi} & y\_{\cdot,\eta\xi} \end{bmatrix} \tag{33}$$

According to the principle of stationary potential energy, the first variation of the potential energy must be zero for equilibrium conditions. The first variation of the potential energy of the finite element is

$$
\delta \Pi\_{\epsilon} = \int\_{V\_{\epsilon}} \varepsilon^{T} \overline{\mathbf{Q}} \delta \varepsilon \, dV \tag{34}
$$

By using Eq. (7), the strains can be written in the following form:

$$\{\varepsilon\} = [H]\{\Delta\} + [\overline{H}]\sum\_{j=1}^{n} \phi^{J} \{\Delta^{J}\} \tag{35}$$

where ϕ <sup>J</sup> are the interpolation functions perpendicular to the plane of the plate or its derivatives:

$$\phi^{\jmath}(\mathbf{z}) = \begin{bmatrix} \psi^{\jmath}(\mathbf{z}) \\ \psi^{\jmath}(\mathbf{z})\_{\cdot z} \end{bmatrix} \tag{36}$$

Inserting Eq. (35) into Eq. (34), the first variation of the potential energy of the finite element becomes

$$\delta \Pi\_{\epsilon} = \int\_{V\_{\epsilon}} \left( \Delta^{T} H^{T} \overline{H}^{T} \sum\_{l=1}^{n} \phi^{l} \Delta^{lT} \right) \cdot \overline{\mathcal{Q}} \cdot \left( H \delta \Delta + \overline{H} \sum\_{l=1}^{n} \phi^{l} \delta \Delta^{l} \right) dV \tag{37}$$

After multiplication, the stiffness matrix of the finite element takes the form:

$$\begin{aligned} k\_{\epsilon} &= \iiint\_{V\_{\epsilon}} \left( H^T \overline{\mathbf{Q}} H + \sum\_{l=1}^{n} H^T \overline{\mathbf{Q}} \phi^l \overline{H} + \sum\_{l=1}^{n} \overline{H}^T \overline{\mathbf{Q}} \phi^l H + \sum\_{l=1}^{n} \sum\_{l=1}^{n} \overline{H}^T \overline{\mathbf{Q}} \phi^l \phi^l \overline{H} \right) dV = \\ &\int\_{A\_{\epsilon}} \left( H^T A H + \sum\_{l=1}^{n} H^T B^l \overline{H} + \sum\_{l=1}^{n} \overline{H}^T B^l H + \sum\_{l=1}^{n} \sum\_{l=1}^{n} \overline{H}^T D^l \overline{H} \right) dA \end{aligned} \tag{38}$$

The stiffness matrix of the finite element can be written in a matrix from as

$$k\_{\epsilon} = \begin{bmatrix} k^{11} & k\_1^{12} & k\_2^{12} & \dots & k\_n^{12} \\ \\ k\_1^{21} & k\_{11}^{22} & k\_{12}^{22} & \dots & k\_{1n}^{22} \\ \\ k\_2^{21} & k\_{12}^{22} & k\_{22}^{22} & \dots & k\_{2n}^{22} \\ \\ \dots & \dots & \dots & \dots & \dots \\ \\ k\_n^{21} & k\_{n1}^{22} & k\_{n2}^{22} & \dots & k\_{nn}^{22} \end{bmatrix} \tag{39}$$

k <sup>11</sup> <sup>¼</sup> þ ð 1

k12 <sup>i</sup> ¼

k 21 <sup>i</sup> ¼

k22 ij ¼

4.2. Interpolation functions and numerical integration

through the thickness of the plate.

�1

þ ð 1

�1

þ ð 1

�1

þ ð 1

�1

þ ð 1

�1

þ ð 1

HTBi

HT

HT

�1

þ ð 1

�1

þ ð 1

�1

variable (z) and can be linear, quadratic, or higher order functions.

<sup>H</sup>TAH detJ d<sup>ξ</sup> <sup>d</sup><sup>η</sup> <sup>i</sup> <sup>¼</sup> <sup>1</sup>, …, n

H detJ dξ dη i ¼ 1, …, n

Bending of Laminated Composite Plates in Layerwise Theory

http://dx.doi.org/10.5772/intechopen.69975

(44)

65

BiTH detJ d<sup>ξ</sup> <sup>d</sup><sup>η</sup> <sup>i</sup> <sup>¼</sup> <sup>1</sup>, …, n

<sup>D</sup>ijH detJ d<sup>ξ</sup> <sup>d</sup><sup>η</sup> i, j <sup>¼</sup> <sup>1</sup>, …, n

Interpolation or shape functions in finite element analysis depend on the dimensionality of the problem and type of elements used for discretization of the problem. Layerwise theory introduces two types of interpolation functions. The first type is the interpolation functions in the plane (x,y) that can be selected from the family of well-known two-dimensional shape functions. The degree and type of interpolation functions depend on the type of finite element and the number of nodes in which variables are defined. The second type is the interpolation functions through the plate thickness. These interpolation functions are the functions of one

For the models of rectangular finite element, it is possible to implement a wide range of standard 2D interpolation functions in the plane of the plate, as well as 1D interpolation functions through the thickness of the layer. At our disposal are 2D finite elements such as E4—four-node Lagrange rectangular element, E8—eight-node Serendipity rectangular element, E9—nine-node Lagrange rectangular element, E12—12-node Serendipity rectangular element, and E16—16-node Lagrange rectangular element. Each of these elements can be combined with one or more 1D Lagrange elements: L—linear element, Q—square element, C—cubic element, in order to create a wide spectrum of a variety of finite elements based on the layerwise theory. We use notation E16-L4 to denote 2D element with 16 nodes combined with four linear 1D elements

For laminated plate containing N layers with adopted linear interpolation through the thickness (number of nodes is n¼N þ 1) and the interpolation with m nodes in the plane of the plate, the total number of degrees of freedom of one finite element is (3 þ 2n)m. For the finite element displayed in Figure 6, m¼8 and N¼3, which leads to the number of degrees of freedom (3þ2 � 4) � 8¼88. It is clear that the number of degrees of freedom of the finite

To determine the stiffness matrix of elements, it is necessary to perform numerical integration.

element is rapidly growing with the increasing degree of interpolation functions.

If Gauss-Legendre integration is used, then the integral is calculated by

From the structure of the matrix (Eq. (39)), four submatrices can be noticed, where elements (see Eq. (38)) are determined as follows:

$$\begin{aligned} k^{11} &= \iint\_{A\_{\epsilon}} H^T A H dA & i &= 1, \ldots, n\\ k\_i^{12} &= \iint\_{A\_{\epsilon}} H^T B^i \overline{H} dA & i &= 1, \ldots, n\\ k\_i^{21} &= \iint\_{A\_{\epsilon}} \overline{H}^T B^i H dA & i &= 1, \ldots, n \end{aligned} \tag{40}$$

$$\begin{aligned} k\_{ij}^{22} &= \iint\_{A\_{\epsilon}} \overline{H}^T D^{ij} \overline{H} dA & i, j &= 1, \ldots, n \end{aligned}$$

The area integral can be written in the form of the natural coordinates:

$$dA = d\mathbf{x} \, dy = \det I \, d\xi \, d\eta \tag{41}$$

as follows:

$$\iint\_{A\_{\epsilon}} (\ldots)dA = \int\_{-1}^{+1} \int (\ldots) \det I \, d\xi \, d\eta \tag{42}$$

where det J is the determinant of the Jacobian matrix of the transformation according to Eqs. (26) and (30):

$$|f| = \begin{vmatrix} \sum\_{i=1}^{m} \mathbf{x}\_i \frac{\partial \phi}{\partial \xi} & \sum\_{i=1}^{m} \mathbf{y}\_i \frac{\partial \phi}{\partial \xi} \\\\ \sum\_{i=1}^{m} \mathbf{x}\_i \frac{\partial \phi}{\partial \eta} & \sum\_{i=1}^{m} \mathbf{y}\_i \frac{\partial \phi}{\partial \eta} \end{vmatrix} \tag{43}$$

Thereafter, the submatrices of the stiffness matrix (Eq. (39)) are given by

$$\begin{aligned} k^{1} &= \int \bigcap\_{i=1}^{i+1} H^{T} A H \, \text{det} f \, d\xi \quad d\eta \qquad i = 1, \ldots, n\\ k\_{i}^{1} &= \int \bigcap\_{i=1}^{i+1} H^{T} B \overline{H} \, \text{det} f \, d\xi \, d\eta \qquad i = 1, \ldots, n\\ k\_{i}^{2} &= \int \bigcap\_{i=1}^{i+1} H^{T} B^{T} H \, \text{det} f \, d\xi \, d\eta \qquad i = 1, \ldots, n \end{aligned} \tag{44}$$
 
$$\begin{aligned} k\_{ij}^{2} &= \int \bigcap\_{i=1}^{i+1} H^{T} D^{j} \overline{H} \, \text{det} f \, d\xi \, d\eta \qquad i, j = 1, \ldots, n \end{aligned}$$

## 4.2. Interpolation functions and numerical integration

ke ¼

<sup>k</sup><sup>11</sup> <sup>¼</sup> ðð

k 12 <sup>i</sup> ¼ ðð

k 21 <sup>i</sup> ¼ ðð

k 22 ij ¼ ðð

Ae

Ae

Ae

Ae

The area integral can be written in the form of the natural coordinates:

ðð

ð…ÞdA ¼

jJj ¼

Thereafter, the submatrices of the stiffness matrix (Eq. (39)) are given by

þ ð 1 þ ð 1

�1

where det J is the determinant of the Jacobian matrix of the transformation according to

Xm i¼1 yi ∂ϕ ∂ξ

� � � � � � � � � �

Xm i¼1 yi ∂ϕ ∂η

¼1

Xm i¼1 xi ∂ϕ ∂ξ

� � � � � � � � � �

Xm i¼1 xi ∂ϕ ∂η

Ae

HTBi

HT Bi

HT

(see Eq. (38)) are determined as follows:

64 Lamination - Theory and Application

as follows:

Eqs. (26) and (30):

k <sup>11</sup> k 12 <sup>1</sup> <sup>k</sup><sup>12</sup>

k 21 <sup>1</sup> k 22 <sup>11</sup> <sup>k</sup><sup>22</sup>

k 21 <sup>2</sup> k 22 <sup>12</sup> k 22 <sup>22</sup> … k

k 21 <sup>n</sup> k 22 <sup>n</sup><sup>1</sup> <sup>k</sup><sup>22</sup>

<sup>2</sup> … k

<sup>12</sup> … <sup>k</sup><sup>22</sup>

<sup>n</sup><sup>2</sup> … <sup>k</sup><sup>22</sup>

……………

<sup>H</sup>TAHdA i <sup>¼</sup> <sup>1</sup>, …, n

HdA i ¼ 1, …, n

HdA i ¼ 1, …, n

<sup>D</sup>ijHdA i, j <sup>¼</sup> <sup>1</sup>,…, n

dA ¼ dx dy ¼ det J dξ dη (41)

ð…Þ det J dξ dη (42)

From the structure of the matrix (Eq. (39)), four submatrices can be noticed, where elements

12 n

(39)

(40)

(43)

1n

22 2n

nn

Interpolation or shape functions in finite element analysis depend on the dimensionality of the problem and type of elements used for discretization of the problem. Layerwise theory introduces two types of interpolation functions. The first type is the interpolation functions in the plane (x,y) that can be selected from the family of well-known two-dimensional shape functions. The degree and type of interpolation functions depend on the type of finite element and the number of nodes in which variables are defined. The second type is the interpolation functions through the plate thickness. These interpolation functions are the functions of one variable (z) and can be linear, quadratic, or higher order functions.

For the models of rectangular finite element, it is possible to implement a wide range of standard 2D interpolation functions in the plane of the plate, as well as 1D interpolation functions through the thickness of the layer. At our disposal are 2D finite elements such as E4—four-node Lagrange rectangular element, E8—eight-node Serendipity rectangular element, E9—nine-node Lagrange rectangular element, E12—12-node Serendipity rectangular element, and E16—16-node Lagrange rectangular element. Each of these elements can be combined with one or more 1D Lagrange elements: L—linear element, Q—square element, C—cubic element, in order to create a wide spectrum of a variety of finite elements based on the layerwise theory. We use notation E16-L4 to denote 2D element with 16 nodes combined with four linear 1D elements through the thickness of the plate.

For laminated plate containing N layers with adopted linear interpolation through the thickness (number of nodes is n¼N þ 1) and the interpolation with m nodes in the plane of the plate, the total number of degrees of freedom of one finite element is (3 þ 2n)m. For the finite element displayed in Figure 6, m¼8 and N¼3, which leads to the number of degrees of freedom (3þ2 � 4) � 8¼88. It is clear that the number of degrees of freedom of the finite element is rapidly growing with the increasing degree of interpolation functions.

To determine the stiffness matrix of elements, it is necessary to perform numerical integration. If Gauss-Legendre integration is used, then the integral is calculated by

$$I(\xi,\eta) = \int\_{-1}^{+1} \int f(\xi,\eta)d\xi d\eta = \sum\_{i=1}^{m} \sum\_{j=1}^{m} \omega\_i \omega\_j f(\overline{\xi}, \overline{\eta}) \tag{45}$$

where ω<sup>i</sup> and ω<sup>j</sup> are the weights of the integration, and ðξ, ηÞ coordinates of the points at which the integration is performed. The weights and the position of integration points can be found in the wide literature for the finite element method.

Figure 6. Rectangular finite element E8–L3.

## 4.3. Governing equations

Governing equations in the finite element method are calculated from the principle of virtual displacements. The first variation of the potential of the system of finite elements is

$$
\delta\Omega \mathbf{U} = \delta\mathbf{U}^T \mathbf{K} \mathbf{U} - \delta\mathbf{U}^T \mathbf{F} \tag{46}
$$

By applying the principle of virtual displacements, the system of algebraic equations of the finite element method is obtained:

$$\mathbf{K}\mathbf{U} = \mathbf{F} \tag{47}$$

very extensive process of calculation with a large number of unknowns, a relatively rough

Bending of Laminated Composite Plates in Layerwise Theory

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67

Material characteristics of the layers are E1¼25, E2¼1, G12¼G13¼0.5, G23¼0.2, ν12¼ν13¼0.25. Stresses are determined in a dimensionless form according to Eq. (23) described in Section 2 of this chapter, wherein the shear stress in the plane of the plate is also given in the same dimensionless form as the normal stresses. Stresses are calculated in points with coordinates:

mesh of finite elements with 4�4 finite elements is adopted.

Figure 7. σxx and σyy through the thickness of the plate.

Figure 8. σxy and σxz through the thickness of the plate.

Figure 9. σyz through the thickness of the plate.

where U is the vector of the nodal displacements, F is the vector of the nodal forces, and K is the stiffness matrix, all for system of assembly of finite elements formed in the global coordinate system.

Methods of forming the stiffness matrix K and the vectors U and F are generally known and can be found in the expanded literature that deal with the finite element method. Solutions of the system of algebraic equations (Eq. (46)) are the nodal displacements.

#### 4.4. Numerical examples

A square four-layer simply supported laminated plate with the arrangement of layers 60�/- 45�/-45�/60� with boundary conditions u ¼ v ¼ w ¼ 0 for x ¼ 0, a and y ¼ 0, b, loaded by the uniformly distributed load intensity q, was considered. The adopted finite element is E4–L1 with layers of the same thickness and the same material properties. Due to the demanding and Bending of Laminated Composite Plates in Layerwise Theory http://dx.doi.org/10.5772/intechopen.69975 67

Figure 7. σxx and σyy through the thickness of the plate.

Iðξ, ηÞ ¼

in the wide literature for the finite element method.

4.3. Governing equations

66 Lamination - Theory and Application

Figure 6. Rectangular finite element E8–L3.

finite element method is obtained:

4.4. Numerical examples

þ ð 1 þ ð 1

<sup>f</sup>ðξ, <sup>η</sup>Þdξd<sup>η</sup> <sup>¼</sup> <sup>X</sup><sup>m</sup>

where ω<sup>i</sup> and ω<sup>j</sup> are the weights of the integration, and ðξ, ηÞ coordinates of the points at which the integration is performed. The weights and the position of integration points can be found

Governing equations in the finite element method are calculated from the principle of virtual

By applying the principle of virtual displacements, the system of algebraic equations of the

where U is the vector of the nodal displacements, F is the vector of the nodal forces, and K is the stiffness matrix, all for system of assembly of finite elements formed in the global coordinate system. Methods of forming the stiffness matrix K and the vectors U and F are generally known and can be found in the expanded literature that deal with the finite element method. Solutions of

A square four-layer simply supported laminated plate with the arrangement of layers 60�/- 45�/-45�/60� with boundary conditions u ¼ v ¼ w ¼ 0 for x ¼ 0, a and y ¼ 0, b, loaded by the uniformly distributed load intensity q, was considered. The adopted finite element is E4–L1 with layers of the same thickness and the same material properties. Due to the demanding and

<sup>∂</sup><sup>Π</sup> <sup>¼</sup> <sup>δ</sup>UTKU � <sup>δ</sup>UTF (46)

KU ¼ F (47)

displacements. The first variation of the potential of the system of finite elements is

the system of algebraic equations (Eq. (46)) are the nodal displacements.

i¼1

Xm j¼1

ωiωjfðξ, ηÞ (45)

�1

�1

Figure 8. σxy and σxz through the thickness of the plate.

Figure 9. σyz through the thickness of the plate.

very extensive process of calculation with a large number of unknowns, a relatively rough mesh of finite elements with 4�4 finite elements is adopted.

Material characteristics of the layers are E1¼25, E2¼1, G12¼G13¼0.5, G23¼0.2, ν12¼ν13¼0.25.

Stresses are determined in a dimensionless form according to Eq. (23) described in Section 2 of this chapter, wherein the shear stress in the plane of the plate is also given in the same dimensionless form as the normal stresses. Stresses are calculated in points with coordinates:

σxxð Þ 0:4471688a; 0:4471688a σyyð Þ 0:4471688a; 0:4471688a σxyð Þ 0:05283122a; 0:05283122a σyzð Þ 0:4471688a; 0:9471688a σxzð Þ 0:9471688a; 0:4471688a

Author details

Marina Rakočević

References

716-718

Wiley; 1984

Address all correspondence to: marinara@ac.me

NY: CRC Press; 1997. ISBN 0-8493-3101-3

plates. Journal of Composite Materials. 2004;3:534-547

Faculty of Civil Engineering, University of Montenegro, Podgorica, Montenegro

[1] Reddy JN. Mechanics of Laminated Composite Plate: Theory and Analysis. New York,

Bending of Laminated Composite Plates in Layerwise Theory

http://dx.doi.org/10.5772/intechopen.69975

69

[2] Whitney JM. The effect of transverse shear deformation in the bending of laminates

[3] Reissner E. A consistent treatment of transverse shear deformations in laminated anisotropic plates. The American Institute of Aeronautics and Astronautics Journal. 1972;10(5):

[4] Reddy JN. Energy and Variational Methods in Applied Mechanics. New York, NY: John

[5] Christensen RM, Lo KH, Wu EM. A high-order theory of plate deformation, part1:

[6] Christensen RM, Lo KH, Wu EML. A high-order theory of plate deformation, part 2:

[7] Vuksanović DJ. Linear analysis of laminated composite plates using single layer higher-

[8] Reddy JN. A simple higher-order theory for laminated composite plates. Journal of

[9] Reddy JN. A refined nonlinear theory of plates with transverse shear deformation. Inter-

[10] Reddy JN, Barbero EJ, Teply JL. A plate bending element based on a generalized laminated plate theory. International Journal for Numerical Methods in Engineering. 1989;28:2275-2292

[11] Reddy JN, Barbero EJ, Teply JL. An accurate determination of stresses in thick laminates using a generalized plate theory. International Journal for Numerical Methods in Engi-

[12] Redyy JN, Barbero EJ. Modelling of thick composites using a layerwise laminate theory. International Journal for Numerical Methods in Engineering. 1993;36:655-677

[13] Reddy JN, Robbins DH. Theories and computational models for composite laminates.

Homogeneous plates. Journal of Applied Mechanics. 1977;44(7):636-668

Laminated plates. Journal of Applied Mechanics. 1977;44(4):669-676

order discrete models. Composite Structures. 2000;48:205-211

national Journal of Solids and Structure. 1984;20(9):881-906

Applied Mechanics Review. 1994;47(6):147-165

Applied Mechanics. 1984;51:745-752

neering. 1990;29:1-14

Changing of dimensionless stresses through the thickness of the laminated plate is presented in Figures 7–9.

## 5. Conclusion

This chapter refers to the application of Reddy's layerwise theory, based on the assumption that dilatation perpendicular to the middle plane of the laminated plate is neglected. In the bending equations for laminated plates, the primary variables of node J are displacements through the thickness of the plate uJ (x,y,z) and vJ (x,y,z) and deflections wJ (x,y,z)¼w(x,y).

Analytical solution of the bending equations in the layerwise theory is presented for simply supported rectangular laminated plate which contains orthogonal layers. A graphical representation of the distribution of dimensionless stresses σxx, σyy, σyz, and σxz through the thickness for the square plate with five layers 0�/90�/0�/90�/0� and the aspect ratio a/h¼4, which is loaded by bi-sinusoidal load, as well as for six-layer plate 0�/90�/0�/90�/0�/90� with the aspect ratio a/h¼10, which is loaded with uniformly distributed load, is presented. Obtained results indicate that the analytical solution provides determination of more realistic change of interlayer stresses, and it also allows the incorporation of delamination in the calculation procedures. However, the analytical solution is difficult to define for complex geometry of the plates, for arbitrary angle of orientation layers, for arbitrary boundary conditions, as well as for various forms of nonlinearity. These are the reasons why in layerwise theory the use of numerical method such as finite element method is usually recommended.

In Chapter 4, a graphical representation of the distribution of dimensionless stresses σxx, σyy, σxy, σyz, and σxz through the thickness of the square, simply supported rectangular laminated plate with four layers 60�/-45�/-45�/60� and the aspect ratio a/h¼10, which is loaded by uniformly distributed load, is presented. The accuracy of the solution, obtained by using the finite element method, is highly influenced by the selection of interpolation functions in the plane (2D), by the selection of interpolation functions through the thickness (1D) of the plate. Additionally, as in general for this calculation method, a number of finite elements of the analyzed domain highly influence the accuracy of obtained solution. However, arbitrary increase of the number of finite element nodes in the plane of the plate and through the thickness of the plate suddenly drastically increases the number of degrees of freedom, which gives rise to the uneconomical nature of the finite element method for more complex mechanical and engineering problems. Regardless of this deficiency, the finite element method is a more general and acceptable for the analysis of laminated composite plates than the proposed analytical method.

Aside the drawbacks and disadvantages of the analytical solution, the same should not be excluded from the engineering and scientific practice, since it represents the ultimate convergence and accuracy criteria for any other numerical or approximate methods, including the finite element method presented in this chapter.

## Author details

σxxð Þ 0:4471688a; 0:4471688a σyyð Þ 0:4471688a; 0:4471688a σxyð Þ 0:05283122a; 0:05283122a

Changing of dimensionless stresses through the thickness of the laminated plate is presented in

This chapter refers to the application of Reddy's layerwise theory, based on the assumption that dilatation perpendicular to the middle plane of the laminated plate is neglected. In the bending equations for laminated plates, the primary variables of node J are displacements

Analytical solution of the bending equations in the layerwise theory is presented for simply supported rectangular laminated plate which contains orthogonal layers. A graphical representation of the distribution of dimensionless stresses σxx, σyy, σyz, and σxz through the thickness for the square plate with five layers 0�/90�/0�/90�/0� and the aspect ratio a/h¼4, which is loaded by bi-sinusoidal load, as well as for six-layer plate 0�/90�/0�/90�/0�/90� with the aspect ratio a/h¼10, which is loaded with uniformly distributed load, is presented. Obtained results indicate that the analytical solution provides determination of more realistic change of interlayer stresses, and it also allows the incorporation of delamination in the calculation procedures. However, the analytical solution is difficult to define for complex geometry of the plates, for arbitrary angle of orientation layers, for arbitrary boundary conditions, as well as for various forms of nonlinearity. These are the reasons why in layerwise theory the use of

In Chapter 4, a graphical representation of the distribution of dimensionless stresses σxx, σyy, σxy, σyz, and σxz through the thickness of the square, simply supported rectangular laminated plate with four layers 60�/-45�/-45�/60� and the aspect ratio a/h¼10, which is loaded by uniformly distributed load, is presented. The accuracy of the solution, obtained by using the finite element method, is highly influenced by the selection of interpolation functions in the plane (2D), by the selection of interpolation functions through the thickness (1D) of the plate. Additionally, as in general for this calculation method, a number of finite elements of the analyzed domain highly influence the accuracy of obtained solution. However, arbitrary increase of the number of finite element nodes in the plane of the plate and through the thickness of the plate suddenly drastically increases the number of degrees of freedom, which gives rise to the uneconomical nature of the finite element method for more complex mechanical and engineering problems. Regardless of this deficiency, the finite element method is a more general and acceptable for the analysis of laminated composite plates than

Aside the drawbacks and disadvantages of the analytical solution, the same should not be excluded from the engineering and scientific practice, since it represents the ultimate convergence and accuracy criteria for any other numerical or approximate methods, including the

(x,y,z) and deflections wJ

(x,y,z)¼w(x,y).

(x,y,z) and vJ

numerical method such as finite element method is usually recommended.

σyzð Þ 0:4471688a; 0:9471688a σxzð Þ 0:9471688a; 0:4471688a

Figures 7–9.

5. Conclusion

68 Lamination - Theory and Application

through the thickness of the plate uJ

the proposed analytical method.

finite element method presented in this chapter.

Marina Rakočević

Address all correspondence to: marinara@ac.me

Faculty of Civil Engineering, University of Montenegro, Podgorica, Montenegro

## References


[14] Reddy JN. Theory and analysis of laminated composite plates. Mechanics of Composite Materials and Structures. 1999;361:1-79

**Section 3**

**Application of Lamination**


## **Application of Lamination**

[14] Reddy JN. Theory and analysis of laminated composite plates. Mechanics of Composite

[15] Ferreira AJM, Fasshauer GE, Batra RC, Rodrigues JD. Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and RBF-PS discretizations with optimal shape parameter. Composite Structures. 2008;86:328-343 [16] Vuksanović Ð, Rakočević M. Kompozitne ploče sa delaminacijama-analitičko rješenje za slobodno oslonjene ploče. In: Simpozijum o istraživanjima i primjeni savremenih dostignuća u našem gradevinarstvu u oblasti materijala i konstrukcija (JUDIMK); Oktobar

[17] Vuksanović Ð, Rakočević M. Opšta teorija laminarnih ploča–analitičko rješenje za slobodno oslonjene ploće, In: Simpozijum o istraživanjima i primjeni savremenih dostignuća u našem graevinarstvu u oblasti materijala i konstrukcija, JUDIMK; Oktobar 2002. pp. 125-130 [18] Vuksanović Ð, Ćetković M. Analytical solution for multilayer plates using general layerwise plate theory. Facta Universitates Series. Architecture and Civil Engineering. 2005;3(2):121-136

[19] Cetković M, Vuksanović D. Bending, free vibrations and buckling of laminated composite and sandwich plates using a layerwise displacement model. Composite Structures.

[20] Cetković M, Vuksanović D. Large deflection analysis of laminated composite plates using layerwise displacement model. Structural Engineering & Mechanics. 2011;40(2):257-277

[21] Rakočević M. Stress in multi-layered composite plates. Gradevinar. 2005;57(7):503-509 [22] Rakočević M. Analysis of laminated composite plate. Gradevinar. 2011;63(9/10):819-825

[23] Rakocevic M. Approximate procedure for calculation of shear stresses σxz and σyz. Journal

[24] Rakočević M. Analysis of simply supported laminated composite plates using double trigonometric series. In: Proceedings of the 16th International Symposium of Macedonian Association of Structural Engineers (MASE 2016); October 2015; Ohrid, Macedonia [25] Rakocevic M. Analytical solution for simply supported laminated composite plate based on partial layerwise theory. Journal of Applied Engineering Science. 2016;14(1):102-108

[26] Rakočević M, Popović S, Ivanišević N. A computational method for laminated composite

plates based on layerwise theory. Composites Part B. 2017;122:202-218

of Applied Engineering Science. 2012;10(1):37-42

Materials and Structures. 1999;361:1-79

2002. pp. 131-136

70 Lamination - Theory and Application

2009;88(2):219-227

**Chapter 4**

Provisional chapter

**Dynamic Modeling of a Serial Link Robot Laminated**

DOI: 10.5772/intechopen.72441

Dynamic Modeling of a Serial Link Robot Laminated

This chapter presents serial link robots laminated with a plastic film, a derivation of the equations of motion of the laminated robots, and numerical simulation. Recently, to become capable of wide application for several serial link robots that work outside, waterproofing and dustproofing techniques are required. We have proposed a robot packaging method to improve waterproof and dustproof properties of serial link robots. Using the proposed packaging method, rigid links with some active joints are loosely laminated with plastic film to protect the links from dust and water. In the next step of our research, we must derive the equations of motion of the laminated robots for the design and performance improvement from the viewpoint of high speed and high energy efficiency. We assume a plastic film as a closed-loop link structure with passive joints in this chapter. A rigid serial link (fin) connected with a motor-actuated joint moves a closed-loop link structure with passive joints. We numerically investigate the influence of the flexural rigidity of a plastic film on the motion of the rigid fin. This research not only contributes to the lamination techniques but also develops a novel

application of waterproofing and dustproofing techniques in robotics.

mechanism, vacuum packaging, fish-like robots

Keywords: closed-loop mechanism, equations of motion, serial link robot, flexible

This chapter presents a description of equations of motion of serial link robots laminated with plastic film. Serial link robots are designed as a series of links connected by motor-actuated joints. Typical applications of serial link robots are serial manipulators, which are the most common industrial robots, such as pick-and-place assembly robots [1, 2] and welding robots [3, 4]. A salient feature of serial link robots is their large workspace in comparison with the robot size [5]. Additionally, serial link robots consist of simple structures. For that reason, they

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is 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.

**with Plastic Film**

with Plastic Film

Abstract

1. Introduction

Norimitsu Sakagami and Mizuho Shibata

Norimitsu Sakagami and Mizuho Shibata

http://dx.doi.org/10.5772/intechopen.72441

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

## **Dynamic Modeling of a Serial Link Robot Laminated with Plastic Film** Dynamic Modeling of a Serial Link Robot Laminated with Plastic Film

DOI: 10.5772/intechopen.72441

Norimitsu Sakagami and Mizuho Shibata

Additional information is available at the end of the chapter Norimitsu Sakagami and Mizuho Shibata

http://dx.doi.org/10.5772/intechopen.72441 Additional information is available at the end of the chapter

## Abstract

This chapter presents serial link robots laminated with a plastic film, a derivation of the equations of motion of the laminated robots, and numerical simulation. Recently, to become capable of wide application for several serial link robots that work outside, waterproofing and dustproofing techniques are required. We have proposed a robot packaging method to improve waterproof and dustproof properties of serial link robots. Using the proposed packaging method, rigid links with some active joints are loosely laminated with plastic film to protect the links from dust and water. In the next step of our research, we must derive the equations of motion of the laminated robots for the design and performance improvement from the viewpoint of high speed and high energy efficiency. We assume a plastic film as a closed-loop link structure with passive joints in this chapter. A rigid serial link (fin) connected with a motor-actuated joint moves a closed-loop link structure with passive joints. We numerically investigate the influence of the flexural rigidity of a plastic film on the motion of the rigid fin. This research not only contributes to the lamination techniques but also develops a novel application of waterproofing and dustproofing techniques in robotics.

Keywords: closed-loop mechanism, equations of motion, serial link robot, flexible mechanism, vacuum packaging, fish-like robots

## 1. Introduction

This chapter presents a description of equations of motion of serial link robots laminated with plastic film. Serial link robots are designed as a series of links connected by motor-actuated joints. Typical applications of serial link robots are serial manipulators, which are the most common industrial robots, such as pick-and-place assembly robots [1, 2] and welding robots [3, 4]. A salient feature of serial link robots is their large workspace in comparison with the robot size [5]. Additionally, serial link robots consist of simple structures. For that reason, they

© 2018 The Author(s). Licensee InTech. This chapter is 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 Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

have also been used as various applications as humanoids [6, 7], robotic hands [8, 9], biped robots [10, 11], robotic legs [12, 13], snake robots [14, 15], fish-like robots [16, 17], and a jumping robot [18]. Kinematics of a serial link robot itself has remained a hot topic in robotics and has been studied for the last few decades [19].

mechanism but also to a prismatic joint mechanism. A ball-screw mechanism was packaged in

Dynamic Modeling of a Serial Link Robot Laminated with Plastic Film

http://dx.doi.org/10.5772/intechopen.72441

75

In the robot fabricated using the packaging method, the rigid links and the plastic film are loosely laminated. The serial link in the plastic film is packaged in the chamber of the vacuum packaging machine after the decompression process. No adhesion exists between the rigid link and the plastic film. However, the rigid link and the plastic film contact mutually with little slippage between them after depressurization during the robot packaging method. To move the serial link with an actuator in the plastic film, the actuator torque must overcome the static frictional force between the links and the film. We encapsulate an insulating fluid to improve lubricity between the links and the film. Therefore, we must consider not only the material properties of the plastic film but also the effects of the insulating fluid to improve the perfor-

In the next step of our research, we must derive equations of motion to achieve performance improvements such as thrust force and energy efficiencies of the laminated robot system. Over the past few decades, several researchers have studied models of laminated structures. In magnetics, modeling of hysteresis losses [26–28], eddy current losses [29, 30], and temperature effects [31] for magnetic laminations have been proposed under some conditions. Depending on those conditions, a 1-D model [32], a 2-D finite-element model [33], and a 3-D finite-element model [34] have been selected to analyze the laminated structure performance. However, we must derive the equations of motion and analyze the motions of laminated structures including a serial link robot from the viewpoint of robotics. In this chapter, we propose equations of motion of the laminated robot underwater. In addition, based on the equations of motion, we numerically estimate the influence of flexural rigidity of plastic films for the motion perfor-

This research not only contributes to lamination techniques but also develops a novel application of waterproofing and dustproofing techniques for application to robotics. Several techniques using an elastic material such as silicone have been applied to robots for waterproofing [35], impact absorption [36, 37], and decoration [38]. In terms of most of the robots that are covered with an elastic material, the actuator force and torque must overcome the elastic force during motions of the robot body. Thin plastic films can be flexible, but they have lower elasticity for bending than other elastic materials such as silicone. Therefore, we can select a low-force and torque actuator to inflect the robot body laminated with a thin plastic film.

This chapter is organized as follows: Section 2 briefly outlines the concept of a serial link robot laminated with a plastic film. This plastic film has flexible and nonextendable properties, which are useful for vacuum packaging in food industry. Also Section 2 takes applications of two serial link robots, such as fish-like robots we have proposed. Section 3 discusses derivation of the equations of motion of the serial link robots including a plastic film that laminates the robot body. Here, we assume that the plastic film has a closed-loop link structure with passive joints. A serial link as a fin connected by a motor-actuated joint drives the closed-loop link structure (the plastic fin) with passive joints. We consider equations of motion to estimate the motion characteristics of the proposed robot system. Section 4 presents validation of the equations of motion through several simulations. Section 5 presents a summary our future

plastic film to fabricate an attitude control system for underwater robots [25].

mance of the serial link robots.

mance of the serial link robot.

work and conclusions.

Recently, to become capable of wide applications for several serial link robots that must be used outdoors, waterproofing and dustproofing techniques must be improved. These techniques have been developed for ocean engineering and field engineering for the past few decades. For example, most underwater robots have waterproof and pressure-tight housings made of metal such as stainless steel or titanium alloy [20–22]. In another waterproofing method used in ocean engineering, called the pressure equalization method, waterproof housings are filled with an insulating fluid such as an industrial oil or a cleaning fluid used for semiconductors. This method has been applied to underwater equipment of several types such as undersea batteries of a submarine [23] and light devices.

We have proposed a robot packaging method to improve waterproofing of a serial link robot [24]. In the robot packaging method, a serial link mechanism connected with motor-actuated joints is packaged in plastic film in a chamber of a vacuum packaging machine, so that the serial link robot is laminated with a plastic film (Figure 1). We applied the packaging method to fabrication of a fish-like robot (Figure 2(a)). The body (the float in Figure 2(b)) and fin (the oscillation plate in Figure 2(b)) are connected via a servo motor in series. The outer plastic film is inflected by the motion of the inside fin. The fish-like robot generates thrust using the body inflection underwater. This packaging method is applicable not only to a rotary joint

Figure 1. Concept of a serial link robot laminated with plastic film.

Figure 2. Concept of a fish-like robot laminated with plastic film.

mechanism but also to a prismatic joint mechanism. A ball-screw mechanism was packaged in plastic film to fabricate an attitude control system for underwater robots [25].

have also been used as various applications as humanoids [6, 7], robotic hands [8, 9], biped robots [10, 11], robotic legs [12, 13], snake robots [14, 15], fish-like robots [16, 17], and a jumping robot [18]. Kinematics of a serial link robot itself has remained a hot topic in robotics

Recently, to become capable of wide applications for several serial link robots that must be used outdoors, waterproofing and dustproofing techniques must be improved. These techniques have been developed for ocean engineering and field engineering for the past few decades. For example, most underwater robots have waterproof and pressure-tight housings made of metal such as stainless steel or titanium alloy [20–22]. In another waterproofing method used in ocean engineering, called the pressure equalization method, waterproof housings are filled with an insulating fluid such as an industrial oil or a cleaning fluid used for semiconductors. This method has been applied to underwater equipment of several types such

We have proposed a robot packaging method to improve waterproofing of a serial link robot [24]. In the robot packaging method, a serial link mechanism connected with motor-actuated joints is packaged in plastic film in a chamber of a vacuum packaging machine, so that the serial link robot is laminated with a plastic film (Figure 1). We applied the packaging method to fabrication of a fish-like robot (Figure 2(a)). The body (the float in Figure 2(b)) and fin (the oscillation plate in Figure 2(b)) are connected via a servo motor in series. The outer plastic film is inflected by the motion of the inside fin. The fish-like robot generates thrust using the body inflection underwater. This packaging method is applicable not only to a rotary joint

and has been studied for the last few decades [19].

74 Lamination - Theory and Application

as undersea batteries of a submarine [23] and light devices.

Figure 1. Concept of a serial link robot laminated with plastic film.

Figure 2. Concept of a fish-like robot laminated with plastic film.

In the robot fabricated using the packaging method, the rigid links and the plastic film are loosely laminated. The serial link in the plastic film is packaged in the chamber of the vacuum packaging machine after the decompression process. No adhesion exists between the rigid link and the plastic film. However, the rigid link and the plastic film contact mutually with little slippage between them after depressurization during the robot packaging method. To move the serial link with an actuator in the plastic film, the actuator torque must overcome the static frictional force between the links and the film. We encapsulate an insulating fluid to improve lubricity between the links and the film. Therefore, we must consider not only the material properties of the plastic film but also the effects of the insulating fluid to improve the performance of the serial link robots.

In the next step of our research, we must derive equations of motion to achieve performance improvements such as thrust force and energy efficiencies of the laminated robot system. Over the past few decades, several researchers have studied models of laminated structures. In magnetics, modeling of hysteresis losses [26–28], eddy current losses [29, 30], and temperature effects [31] for magnetic laminations have been proposed under some conditions. Depending on those conditions, a 1-D model [32], a 2-D finite-element model [33], and a 3-D finite-element model [34] have been selected to analyze the laminated structure performance. However, we must derive the equations of motion and analyze the motions of laminated structures including a serial link robot from the viewpoint of robotics. In this chapter, we propose equations of motion of the laminated robot underwater. In addition, based on the equations of motion, we numerically estimate the influence of flexural rigidity of plastic films for the motion performance of the serial link robot.

This research not only contributes to lamination techniques but also develops a novel application of waterproofing and dustproofing techniques for application to robotics. Several techniques using an elastic material such as silicone have been applied to robots for waterproofing [35], impact absorption [36, 37], and decoration [38]. In terms of most of the robots that are covered with an elastic material, the actuator force and torque must overcome the elastic force during motions of the robot body. Thin plastic films can be flexible, but they have lower elasticity for bending than other elastic materials such as silicone. Therefore, we can select a low-force and torque actuator to inflect the robot body laminated with a thin plastic film.

This chapter is organized as follows: Section 2 briefly outlines the concept of a serial link robot laminated with a plastic film. This plastic film has flexible and nonextendable properties, which are useful for vacuum packaging in food industry. Also Section 2 takes applications of two serial link robots, such as fish-like robots we have proposed. Section 3 discusses derivation of the equations of motion of the serial link robots including a plastic film that laminates the robot body. Here, we assume that the plastic film has a closed-loop link structure with passive joints. A serial link as a fin connected by a motor-actuated joint drives the closed-loop link structure (the plastic fin) with passive joints. We consider equations of motion to estimate the motion characteristics of the proposed robot system. Section 4 presents validation of the equations of motion through several simulations. Section 5 presents a summary our future work and conclusions.

## 2. Application

In this section, we briefly describe the concept of lamination of a serial link robot for waterproofing and pressure tightness. In robotics, piezoelectric actuators have been used to drive several robots as a useful application of lamination techniques [39, 40]. We used a vacuum packaging machine to laminate a serial link robot with a plastic film. We designated this fabrication as "robot packaging."

## 2.1. Robot packaging method

Figure 3 presents the "robot packaging" process as an example of fabricating a fish-like robot. The entire robot body was covered by a flexible plastic film. The process is divisible into four steps: (a) encapsulation of the internal components, including a microcontroller, a drive circuit, a battery, a servomotor, and an oscillation plate, in a plastic film bag; (b) pouring of an insulating fluid, specifically industrial oil or cleaning fluid for semiconductors, into the plastic bag; (c) depressurization of the inside of the robot using a vacuum packaging machine. This process reduces the quantity of air in the film bag; and (d) sealing of the plastic film by a sealer within the chamber of the vacuum packaging machine after depressurizing.

readily fabricate the entire body of a serial link robot laminated with a plastic film at low cost

Dynamic Modeling of a Serial Link Robot Laminated with Plastic Film

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77

As shown in Figure 2, we proposed a fish-like robot as a prototype hardware laminated using a plastic film, which was fabricated using a vacuum packaging machine. In the prototype robot, we applied an insulating fluid (Fluorinert FC-3283; 3 M Corp.) filled in the body. The insulating fluid is generally used as a cleaning fluid for semiconductors. The specific gravity of the insulating fluid is approximately 1.83, which is heavier than water. Therefore, we used a copolymer foam (NiGK Corp.) not only for the frame structure of the body but also for generating buoyancy. The specific gravity of the foam is approximately 0.2, which is a much

We also applied a servomotor (SG51R; Tower Pro) to the prototype robot actuator. In this prototype, we used a microcontroller (Arduino Nano ver. 3.1). A styrene board of 80 mm height, 40 mm width, and 2 mm thickness was used as an oscillation plate to generate the thrust force of the prototype robot in water. The plastic film (poly bag TL12-38; Fukusuke Kogyo Co. Ltd.) covering the internal contents of the robot was multilayered for use in food packaging. This film is sealed by thermal adhesion of a vacuum packaging machine. We made use of a TM-HV made by Furukawa Mfg. Co., Ltd. as a vacuum packaging machine. In this prototype, the depressurization time is 15 s. The sealing time is 4.0 s. A battery (9 V) is used as

After design, the prototype size without the film was approximately 185 mm long, 80 mm high, and 18 mm wide. After fabrication, the prototype including the film in Figure 2 was 230 mm long, 90 mm high, and 18 mm wide. Additionally, it weighed approximately 290 g, including 80 ml of the insulating fluid, thereby realizing almost neutral buoyancy of the body. Under these conditions, the oscillation plate driven by the servo motor was moved 10 deg. in

This section presents a discussion of the derivation of the equations of motion of a serial link robot laminated with a plastic film. The laminated robot consists of a rigid link fin, a plastic film, and an enclosed insulating fluid as illustrated in Figure 5. In this section, we assume that

the power source for driving the electrical circuits and servomotor.

the left and right directions at a frequency of approximately 2.0 Hz (Figure 4).

and in a short time. The body can also be lighter than other underwater systems.

2.2. Fish-like robot laminated with a plastic film

Figure 4. Swimming test of our prototype at narrow space.

lighter fluid than water.

3. Modeling

This plastic film has flexible and nonextendable properties, which is used for vacuum packaging in food industry. This packaging method corresponds with the method used in the food industry [41]. The internal components including electrical circuits in the body of the robot are not shortened by the insulating fluid surrounding the circuits. Using this method, we can

Figure 3. Fabrication process by robot packaging.

Figure 4. Swimming test of our prototype at narrow space.

2. Application

76 Lamination - Theory and Application

fabrication as "robot packaging."

Figure 3. Fabrication process by robot packaging.

2.1. Robot packaging method

In this section, we briefly describe the concept of lamination of a serial link robot for waterproofing and pressure tightness. In robotics, piezoelectric actuators have been used to drive several robots as a useful application of lamination techniques [39, 40]. We used a vacuum packaging machine to laminate a serial link robot with a plastic film. We designated this

Figure 3 presents the "robot packaging" process as an example of fabricating a fish-like robot. The entire robot body was covered by a flexible plastic film. The process is divisible into four steps: (a) encapsulation of the internal components, including a microcontroller, a drive circuit, a battery, a servomotor, and an oscillation plate, in a plastic film bag; (b) pouring of an insulating fluid, specifically industrial oil or cleaning fluid for semiconductors, into the plastic bag; (c) depressurization of the inside of the robot using a vacuum packaging machine. This process reduces the quantity of air in the film bag; and (d) sealing of the plastic film by a sealer

This plastic film has flexible and nonextendable properties, which is used for vacuum packaging in food industry. This packaging method corresponds with the method used in the food industry [41]. The internal components including electrical circuits in the body of the robot are not shortened by the insulating fluid surrounding the circuits. Using this method, we can

within the chamber of the vacuum packaging machine after depressurizing.

readily fabricate the entire body of a serial link robot laminated with a plastic film at low cost and in a short time. The body can also be lighter than other underwater systems.

## 2.2. Fish-like robot laminated with a plastic film

As shown in Figure 2, we proposed a fish-like robot as a prototype hardware laminated using a plastic film, which was fabricated using a vacuum packaging machine. In the prototype robot, we applied an insulating fluid (Fluorinert FC-3283; 3 M Corp.) filled in the body. The insulating fluid is generally used as a cleaning fluid for semiconductors. The specific gravity of the insulating fluid is approximately 1.83, which is heavier than water. Therefore, we used a copolymer foam (NiGK Corp.) not only for the frame structure of the body but also for generating buoyancy. The specific gravity of the foam is approximately 0.2, which is a much lighter fluid than water.

We also applied a servomotor (SG51R; Tower Pro) to the prototype robot actuator. In this prototype, we used a microcontroller (Arduino Nano ver. 3.1). A styrene board of 80 mm height, 40 mm width, and 2 mm thickness was used as an oscillation plate to generate the thrust force of the prototype robot in water. The plastic film (poly bag TL12-38; Fukusuke Kogyo Co. Ltd.) covering the internal contents of the robot was multilayered for use in food packaging. This film is sealed by thermal adhesion of a vacuum packaging machine. We made use of a TM-HV made by Furukawa Mfg. Co., Ltd. as a vacuum packaging machine. In this prototype, the depressurization time is 15 s. The sealing time is 4.0 s. A battery (9 V) is used as the power source for driving the electrical circuits and servomotor.

After design, the prototype size without the film was approximately 185 mm long, 80 mm high, and 18 mm wide. After fabrication, the prototype including the film in Figure 2 was 230 mm long, 90 mm high, and 18 mm wide. Additionally, it weighed approximately 290 g, including 80 ml of the insulating fluid, thereby realizing almost neutral buoyancy of the body. Under these conditions, the oscillation plate driven by the servo motor was moved 10 deg. in the left and right directions at a frequency of approximately 2.0 Hz (Figure 4).

## 3. Modeling

This section presents a discussion of the derivation of the equations of motion of a serial link robot laminated with a plastic film. The laminated robot consists of a rigid link fin, a plastic film, and an enclosed insulating fluid as illustrated in Figure 5. In this section, we assume that

Figure 5. Modeling of a serial link robot laminated with a plastic film: (a) serial link robot laminated with a plastic film; (b) plastic film is modeled mathematically as a closed-loop link structure with multi-DOFs consisting of passive joints.

the plastic film is modeled as a multi-DOF closed-loop structure with passive joints that are actuated indirectly by a motor mounted on the base of the fin.

#### 3.1. Rigid fin

The equation of fin motion in Figure 5 is expressed simply as shown below.

$$I\ddot{\theta} = \tau \tag{1}$$

As illustrated in Figure 6(b), a small area Ai of a trapezoid is determined by four points (or a triangle determined by three points at the tip of the fin,) and the center of the mass is used to estimate the fluid inertia. Mass MFi and inertia IFi of the fluid in a small area are obtained

Figure 6. (a) Contact force between the fin and the plastic film based on the penalty method, and (b) calculation of the

MFi ¼ Aidr, IFi ¼ MFil

where d, r, and lFi represent the depth of area Ai, the density of the insulating fluid, and the moment arm for the mass MFi, respectively. Consequently, the equations of the fin motion can

In this chapter, the plastic file is modeled as multi-DOF links with passive joints and with a closed kinematic loop constraint as illustrated in Figure 7. We use a method based on Lagrange-D'Alembert formulation on reduced system described in an earlier report [43, 44] to derive

First, a link of one of the closed-loop links, that is, is cut and a tree system (two serial link structures) is formed, as portrayed in Figure 7. Equations of motion of the serial link structures can be obtained easily using Newton-Euler method or Direct Lagrangian method. Because a plastic film is generally very light, the mass of the links can be negligible. Therefore, the equations of the motion of the multi-DOF serial link structures are only represented by passive

BRθ\_

2

j

ð Þ IR <sup>þ</sup> IF <sup>θ</sup>€ <sup>¼</sup> <sup>τ</sup><sup>A</sup> <sup>þ</sup> <sup>τ</sup><sup>p</sup> (5)

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<sup>R</sup> þ KRΔθ<sup>R</sup> ¼ τ<sup>R</sup> (6)

BLθ\_ <sup>L</sup> <sup>þ</sup> KLΔθ<sup>L</sup> <sup>¼</sup> <sup>τ</sup><sup>L</sup> (7)

Fi (4)

IFi, τ<sup>A</sup> stands for the actuator torque.

quantitatively as

be rewritten as follows.

3.2. Plastic film

Therein, IR signifies the inertia of the rigid fin. IF <sup>¼</sup> <sup>P</sup>

equations of motion of the closed-loop links.

additional inertia of the enclosed fluid in the plastic film.

joints with elasticity and damping effects:

In this equation, scalar I represents the inertia of the rigid fin and the enclosed fluid, θ stands for the active joint angle of the fin, and the scalar τ includes an actuator torque and external torque related to the contact force from the film and additional inertia of the enclosed fluid.

To simplify the mathematical model of the contact force between the fin and the plastic film, we introduce a penalty method [42]. The penalty method treats elasticity and damping force for slight penetration between two objects. Here, we assume a small penetration between the fin and the joints of the links, as shown in Figure 6(a). Contact force f pi based on the penalty method is calculated as

$$f\_{\,\,pi} = K\_{\,\,pi} D\_i - B\_{\,\,pi} \dot{D}\_i \tag{2}$$

where Di denotes the small penetration between the fin and the plastic film at a contact point. Kpi and Bpi represent the elasticity and damping constant coefficients, respectively. As a result, the external torque τ<sup>p</sup> generated by the contact forces f pi can be calculated as

$$
\pi\_p = \sum\_i \left( l\_i \times f\_{pi} \right) \tag{3}
$$

where vector li represents the position for contact force f pi.

In addition, the inertia of the enclosed fluid is included as well as the rigid inertia in the mathematical model because the rigid link fin carries the enclosed fluid during the fin motion. Dynamic Modeling of a Serial Link Robot Laminated with Plastic Film http://dx.doi.org/10.5772/intechopen.72441 79

Figure 6. (a) Contact force between the fin and the plastic film based on the penalty method, and (b) calculation of the additional inertia of the enclosed fluid in the plastic film.

As illustrated in Figure 6(b), a small area Ai of a trapezoid is determined by four points (or a triangle determined by three points at the tip of the fin,) and the center of the mass is used to estimate the fluid inertia. Mass MFi and inertia IFi of the fluid in a small area are obtained quantitatively as

$$M\_{\rm Fi} = A\_i d\rho\_\prime \, I\_{\rm Fi} = M\_{\rm Fi} l\_{\rm Fi}^2 \tag{4}$$

where d, r, and lFi represent the depth of area Ai, the density of the insulating fluid, and the moment arm for the mass MFi, respectively. Consequently, the equations of the fin motion can be rewritten as follows.

$$(I\_R + I\_F)\ddot{\bar{\Theta}} = \pi\_A + \pi\_p \tag{5}$$

Therein, IR signifies the inertia of the rigid fin. IF <sup>¼</sup> <sup>P</sup> j IFi, τ<sup>A</sup> stands for the actuator torque.

#### 3.2. Plastic film

the plastic film is modeled as a multi-DOF closed-loop structure with passive joints that are

Figure 5. Modeling of a serial link robot laminated with a plastic film: (a) serial link robot laminated with a plastic film; (b) plastic film is modeled mathematically as a closed-loop link structure with multi-DOFs consisting of passive joints.

In this equation, scalar I represents the inertia of the rigid fin and the enclosed fluid, θ stands for the active joint angle of the fin, and the scalar τ includes an actuator torque and external torque related to the contact force from the film and additional inertia of the enclosed fluid.

To simplify the mathematical model of the contact force between the fin and the plastic film, we introduce a penalty method [42]. The penalty method treats elasticity and damping force for slight penetration between two objects. Here, we assume a small penetration between the fin and the joints of the links, as shown in Figure 6(a). Contact force f pi based on the penalty

where Di denotes the small penetration between the fin and the plastic film at a contact point. Kpi and Bpi represent the elasticity and damping constant coefficients, respectively. As a result,

In addition, the inertia of the enclosed fluid is included as well as the rigid inertia in the mathematical model because the rigid link fin carries the enclosed fluid during the fin motion.

<sup>i</sup> li � f pi � �

the external torque τ<sup>p</sup> generated by the contact forces f pi can be calculated as

where vector li represents the position for contact force f pi.

<sup>τ</sup><sup>p</sup> <sup>¼</sup> <sup>X</sup>

<sup>I</sup>θ€ <sup>¼</sup> <sup>τ</sup> (1)

<sup>f</sup> pi <sup>¼</sup> KpiDi � BpiD\_ <sup>i</sup> (2)

(3)

actuated indirectly by a motor mounted on the base of the fin.

The equation of fin motion in Figure 5 is expressed simply as shown below.

3.1. Rigid fin

78 Lamination - Theory and Application

method is calculated as

In this chapter, the plastic file is modeled as multi-DOF links with passive joints and with a closed kinematic loop constraint as illustrated in Figure 7. We use a method based on Lagrange-D'Alembert formulation on reduced system described in an earlier report [43, 44] to derive equations of motion of the closed-loop links.

First, a link of one of the closed-loop links, that is, is cut and a tree system (two serial link structures) is formed, as portrayed in Figure 7. Equations of motion of the serial link structures can be obtained easily using Newton-Euler method or Direct Lagrangian method. Because a plastic film is generally very light, the mass of the links can be negligible. Therefore, the equations of the motion of the multi-DOF serial link structures are only represented by passive joints with elasticity and damping effects:

$$
\hbar \mathcal{B}\_{\mathcal{R}} \dot{\Theta}\_{\mathcal{R}} + \mathcal{K}\_{\mathcal{R}} \Delta \Theta\_{\mathcal{R}} = \tau\_{\mathcal{R}} \tag{6}
$$

$$
\hbar \dot{B}\_L \dot{\Theta}\_L + K\_L \Delta \Theta\_L = \tau\_L \tag{7}
$$

Figure 7. Mathematical model of the closed-loop link structure for a plastic film.

Therein, vectors θ<sup>R</sup> ¼ ½ � θR1; ⋯; θRnþ<sup>1</sup> <sup>T</sup> and <sup>θ</sup><sup>L</sup> <sup>¼</sup> ½ � <sup>θ</sup>L1; <sup>⋯</sup>; <sup>θ</sup>Ln <sup>T</sup> denote the passive joint angle vectors of the film, the vectors <sup>Δ</sup>θ<sup>R</sup> <sup>¼</sup> tan <sup>Δ</sup>θR<sup>1</sup> <sup>2</sup> ; <sup>⋯</sup>; tan <sup>Δ</sup>θRnþ<sup>1</sup> 2 h i<sup>T</sup> and <sup>Δ</sup>θ<sup>L</sup> <sup>¼</sup> tan <sup>Δ</sup>θL<sup>1</sup> <sup>2</sup> ; <sup>⋯</sup>; tan � ΔθLn <sup>2</sup> � <sup>T</sup>, where ΔθRi and ΔθLi represent the rotational displacement for a joint. The matrices BR ¼ diagð Þ bR1; ⋯; bRnþ<sup>1</sup> and BL ¼ diagð Þ bL1; ⋯; bLn represent the damping matrices, KR ¼ diagð Þ kR1; ⋯; kRnþ<sup>1</sup> and KL ¼ diagð Þ kL1; ⋯; kLn are constant matrices related to the flexural rigidity and curvature of the plastic film. τ<sup>R</sup> and τ<sup>L</sup> represent the external torque related to the hydrodynamic effect acting on the film and the contact force between the rigid fin and the film, respectively.

Based on reports of the literature [43, 44], the equations of motion of the closed-loop structure is obtainable as

$$\mathbf{Q}^{T} \begin{bmatrix} B\_R \dot{\theta}\_R + K\_R \Delta \theta\_R \\ B\_L \dot{\theta}\_L + K\_L \Delta \theta\_L \end{bmatrix} = \mathbf{Q}^{T} \begin{bmatrix} \tau\_R \\ \tau\_L \end{bmatrix} \tag{8}$$

f 0

τpR <sup>τ</sup>pL � � <sup>¼</sup>

> <sup>f</sup> Di <sup>¼</sup> <sup>1</sup> 2

interference between the film and the fin is

velocity of a link can be approximated by the following.

τR τL � �

¼

configuration singularity does not occur, angular velocities θ\_

P i J T Rið Þ θRi f <sup>0</sup>

P j J T Lj <sup>θ</sup>Lj � �<sup>f</sup> <sup>0</sup>

θ\_ i; θ\_ j; θ\_ k

Jð Þ θR; θ<sup>L</sup> is the 3 � ð Þ 2n � 2 matrix derived from the constraint equations.

For vector <sup>θ</sup><sup>i</sup> <sup>¼</sup> <sup>θ</sup>R1; <sup>⋯</sup>; <sup>θ</sup>Ri ½ �<sup>T</sup>

summarized as shown below.

almost zero.

where f <sup>0</sup>

pi ¼ �<sup>f</sup> pi ¼ � KpD � BpD\_ � � (10)

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81

pjL " # (11)

CdrwSi v⊥<sup>i</sup> j jv⊥<sup>i</sup> (12)

<sup>i</sup> ð Þ <sup>θ</sup><sup>i</sup> <sup>θ</sup>\_ <sup>i</sup> (13)

� �<sup>T</sup> and

i, θ\_ j, θ\_ <sup>k</sup> and

� �<sup>T</sup> of any three joints of

or <sup>θ</sup><sup>i</sup> <sup>¼</sup> <sup>θ</sup>L1; <sup>⋯</sup>; <sup>θ</sup>Li ½ �<sup>T</sup>, matrix Ri is the rotational matrix to calcu-

Rkð Þ θRk f DkR

Rkð Þ <sup>θ</sup>Rk <sup>f</sup> DkR " # (14)

i; θ\_ j; θ\_ k

� �<sup>T</sup> <sup>¼</sup> <sup>J</sup>ð Þ <sup>θ</sup>R; <sup>θ</sup><sup>L</sup> <sup>q</sup>\_ (15)

pi is the reaction force of f pi in Figure 6(a). Using Eq. (9), the torque related to the

piR

P i J T Rið Þ θRi f <sup>0</sup>

P j J T Lj <sup>θ</sup>Lj � �<sup>f</sup> <sup>0</sup>

Finally, we assume that the hydrodynamic damping force acts as an external force on each link of the film. The hydrodynamic force acting on a link is modeled as shown below (Figure 8):

In this equation, Cd represents the drag force coefficient, r<sup>w</sup> stands for the surrounding water density outside the film, Si is the representative area of the ith link, and v⊥<sup>i</sup> is the relative flow velocity that is perpendicular to the ith link. We presume that the film moves in still water. The

late the velocity perpendicular to the link. Ji is the Jacobian matrix of the ith link. However, we assume that <sup>v</sup>⊥<sup>i</sup> <sup>¼</sup> <sup>0</sup> <sup>f</sup> Di <sup>¼</sup> <sup>0</sup> � � when the <sup>i</sup>th link of the plastic film moves toward the rigid fin in the insulating fluid because the encapsulated insulating fluid in the plastic film moves with the fin and the film. Therefore, the relative velocity v⊥<sup>i</sup> between the film and the insulating fluid is

Consequently, the external torque related to the contact forces and the hydrodynamic forces is

In the model shown in Figure 5, three constraint equations related to the position xE; yE

the closed-loop structure can be expressed by angular velocities of the 2n � 2 other joints.

Therein, <sup>q</sup>\_ is the 2<sup>n</sup> � 2 vector composed of all joint angular velocities except for <sup>θ</sup>\_

orientation θ<sup>E</sup> of the two serial link structures should be considered. The constraint reduces the degrees of freedom of the serial link structures from 2n þ 1 to 2n � 2. In other words, when

piR <sup>þ</sup> <sup>P</sup> kJ T

pjL <sup>þ</sup> <sup>P</sup> kJ T

T

v⊥<sup>i</sup> ¼ Rið Þ θ<sup>i</sup> J

where <sup>Φ</sup><sup>T</sup> is the 2ð Þ� <sup>n</sup> � <sup>2</sup> ð Þ <sup>2</sup><sup>n</sup> <sup>þ</sup> <sup>1</sup> matrix which comprises the Jacobian matrix of the constraint equations. Matrix <sup>Φ</sup><sup>T</sup> reduces the number of the equations from (2<sup>n</sup> <sup>þ</sup> 1) to (2<sup>n</sup> � <sup>2</sup>Þ.

The following part of this discussion explains the external torque τ<sup>R</sup> and τL. When forces f <sup>i</sup> apply to their links, the torque is given as the following.

$$
\begin{bmatrix} \tau\_R \\ \tau\_L \end{bmatrix} = \begin{bmatrix} \sum\_{i} I\_{Ri}^T(\theta\_{Ri}) f\_{Ri} \\ \sum\_{j} I\_{Lj}^T(\theta\_{Lj}) f\_{Lj} \end{bmatrix} \tag{9}
$$

In that equation, JRi and JLi are the Jacobian matrices of a contact point where a force is applied. Here, we consider forces related to the hydrodynamic effects and the contact forces between the plastic film and the rigid fin.

As described before, the contact force between the rigid fin and the film is derived based on the penalty method. The force applied to the film is expressed as

$$f\_{\,\,\dot{p}i}^{\prime} = -f\_{\,\,\dot{p}i} = -\left(\mathcal{K}\_{\mathcal{p}}\mathcal{D} - B\_{\mathcal{p}}\dot{\mathcal{D}}\right) \tag{10}$$

where f <sup>0</sup> pi is the reaction force of f pi in Figure 6(a). Using Eq. (9), the torque related to the interference between the film and the fin is

$$
\begin{bmatrix} \tau\_{p\mathcal{R}} \\ \tau\_{p\mathcal{L}} \end{bmatrix} = \begin{bmatrix} \sum\_{i} l\_{Ri}^{T} (\Theta\_{Ri}) f'\_{p\mathcal{R}} \\ \sum\_{j} l\_{Lj}^{T} (\Theta\_{Lj}) f'\_{p\mathcal{L}} \end{bmatrix} \tag{11}
$$

Finally, we assume that the hydrodynamic damping force acts as an external force on each link of the film. The hydrodynamic force acting on a link is modeled as shown below (Figure 8):

$$f\_{Di} = \frac{1}{2} \mathbf{C}\_d \rho\_w \mathbf{S}\_i |v\_{\perp i}| v\_{\perp i} \tag{12}$$

In this equation, Cd represents the drag force coefficient, r<sup>w</sup> stands for the surrounding water density outside the film, Si is the representative area of the ith link, and v⊥<sup>i</sup> is the relative flow velocity that is perpendicular to the ith link. We presume that the film moves in still water. The velocity of a link can be approximated by the following.

Therein, vectors θ<sup>R</sup> ¼ ½ � θR1; ⋯; θRnþ<sup>1</sup>

80 Lamination - Theory and Application

ΔθLn <sup>2</sup> �

the film, respectively.

is obtainable as

vectors of the film, the vectors <sup>Δ</sup>θ<sup>R</sup> <sup>¼</sup> tan <sup>Δ</sup>θR<sup>1</sup>

Figure 7. Mathematical model of the closed-loop link structure for a plastic film.

apply to their links, the torque is given as the following.

penalty method. The force applied to the film is expressed as

the plastic film and the rigid fin.

<sup>T</sup> and <sup>θ</sup><sup>L</sup> <sup>¼</sup> ½ � <sup>θ</sup>L1; <sup>⋯</sup>; <sup>θ</sup>Ln

<sup>T</sup>, where ΔθRi and ΔθLi represent the rotational displacement for a joint. The matrices BR ¼ diagð Þ bR1; ⋯; bRnþ<sup>1</sup> and BL ¼ diagð Þ bL1; ⋯; bLn represent the damping matrices, KR ¼ diagð Þ kR1; ⋯; kRnþ<sup>1</sup> and KL ¼ diagð Þ kL1; ⋯; kLn are constant matrices related to the flexural rigidity and curvature of the plastic film. τ<sup>R</sup> and τ<sup>L</sup> represent the external torque related to the hydrodynamic effect acting on the film and the contact force between the rigid fin and

Based on reports of the literature [43, 44], the equations of motion of the closed-loop structure

where <sup>Φ</sup><sup>T</sup> is the 2ð Þ� <sup>n</sup> � <sup>2</sup> ð Þ <sup>2</sup><sup>n</sup> <sup>þ</sup> <sup>1</sup> matrix which comprises the Jacobian matrix of the constraint equations. Matrix <sup>Φ</sup><sup>T</sup> reduces the number of the equations from (2<sup>n</sup> <sup>þ</sup> 1) to (2<sup>n</sup> � <sup>2</sup>Þ.

The following part of this discussion explains the external torque τ<sup>R</sup> and τL. When forces f <sup>i</sup>

P i J T Rið Þ θRi f Ri

" #

P j J T Lj θLj � �<sup>f</sup> Lj

In that equation, JRi and JLi are the Jacobian matrices of a contact point where a force is applied. Here, we consider forces related to the hydrodynamic effects and the contact forces between

As described before, the contact force between the rigid fin and the film is derived based on the

<sup>Φ</sup><sup>T</sup> BRθ\_ <sup>R</sup> <sup>þ</sup> KRΔθ<sup>R</sup> BLθ\_ <sup>L</sup> <sup>þ</sup> KLΔθ<sup>L</sup> " #

> τR τL � �

¼

<sup>2</sup> ; <sup>⋯</sup>; tan <sup>Δ</sup>θRnþ<sup>1</sup>

<sup>¼</sup> <sup>Φ</sup><sup>T</sup> <sup>τ</sup><sup>R</sup> τL � �

h i<sup>T</sup>

2

<sup>T</sup> denote the passive joint angle

<sup>2</sup> ; <sup>⋯</sup>; tan �

(8)

(9)

and <sup>Δ</sup>θ<sup>L</sup> <sup>¼</sup> tan <sup>Δ</sup>θL<sup>1</sup>

$$\boldsymbol{\sigma}\_{\perp i} = \mathcal{R}\_i(\boldsymbol{\Theta}\_i) \boldsymbol{I}\_i^T(\boldsymbol{\Theta}\_i) \dot{\boldsymbol{\theta}}\_i \tag{13}$$

For vector <sup>θ</sup><sup>i</sup> <sup>¼</sup> <sup>θ</sup>R1; <sup>⋯</sup>; <sup>θ</sup>Ri ½ �<sup>T</sup> or <sup>θ</sup><sup>i</sup> <sup>¼</sup> <sup>θ</sup>L1; <sup>⋯</sup>; <sup>θ</sup>Li ½ �<sup>T</sup>, matrix Ri is the rotational matrix to calculate the velocity perpendicular to the link. Ji is the Jacobian matrix of the ith link. However, we assume that <sup>v</sup>⊥<sup>i</sup> <sup>¼</sup> <sup>0</sup> <sup>f</sup> Di <sup>¼</sup> <sup>0</sup> � � when the <sup>i</sup>th link of the plastic film moves toward the rigid fin in the insulating fluid because the encapsulated insulating fluid in the plastic film moves with the fin and the film. Therefore, the relative velocity v⊥<sup>i</sup> between the film and the insulating fluid is almost zero.

Consequently, the external torque related to the contact forces and the hydrodynamic forces is summarized as shown below.

$$
\begin{bmatrix} \tau\_R \\ \tau\_L \end{bmatrix} = \begin{bmatrix} \sum\_i I\_{Ri}^T(\Theta\_{Ri}) f'\_{p\bar{i}R} + \sum\_k I\_{Rk}^T(\Theta\_{Rk}) f\_{DkR} \\ \sum\_j I\_{Lj}^T(\Theta\_{lj}) f'\_{p\bar{j}L} + \sum\_k I\_{Rk}^T(\Theta\_{Rk}) f\_{DkR} \end{bmatrix} \tag{14}$$

In the model shown in Figure 5, three constraint equations related to the position xE; yE � �<sup>T</sup> and orientation θ<sup>E</sup> of the two serial link structures should be considered. The constraint reduces the degrees of freedom of the serial link structures from 2n þ 1 to 2n � 2. In other words, when configuration singularity does not occur, angular velocities θ\_ i; θ\_ j; θ\_ k � �<sup>T</sup> of any three joints of the closed-loop structure can be expressed by angular velocities of the 2n � 2 other joints.

$$\left[\dot{\Theta}\_i, \dot{\Theta}\_\rangle, \dot{\Theta}\_k\right]^T = \mathbf{J}(\theta\_R, \theta\_L)\dot{\boldsymbol{q}}\tag{15}$$

Therein, <sup>q</sup>\_ is the 2<sup>n</sup> � 2 vector composed of all joint angular velocities except for <sup>θ</sup>\_ i, θ\_ j, θ\_ <sup>k</sup> and Jð Þ θR; θ<sup>L</sup> is the 3 � ð Þ 2n � 2 matrix derived from the constraint equations.

Figure 8. Model of the hydrodynamic damping force acting on the ith link.

Consequently, the equations of film motion are rewritten as shown below.

$$\left[\mathbf{B}^{\prime} + \mathbf{J}^{T}(\boldsymbol{\Theta}\_{\mathcal{R}}, \boldsymbol{\Theta}\_{L})\mathbf{B}^{\prime}\mathbf{J}(\boldsymbol{\Theta}\_{\mathcal{R}}, \boldsymbol{\Theta}\_{L})\right]\dot{\boldsymbol{q}} + \boldsymbol{\Phi}^{T}\begin{bmatrix} \boldsymbol{K}\_{\mathcal{R}}\boldsymbol{\Delta}\boldsymbol{\Theta}\_{\mathcal{R}}\\ \boldsymbol{K}\_{L}\boldsymbol{\Delta}\boldsymbol{\Theta}\_{L} \end{bmatrix} = \boldsymbol{\Phi}^{T}\begin{bmatrix} \boldsymbol{\tau}\_{\mathcal{R}}\\ \boldsymbol{\tau}\_{L} \end{bmatrix} \tag{16}$$

Therein,

$$\mathbf{B}' = \begin{bmatrix} B\_R' & 0 \\ 0 & B\_L' \end{bmatrix}, \mathbf{B}'' = \text{diag}\{b\_i, b\_j, b\_k\} \tag{17}$$

times and 100 times the value of 3.14 (31.4 gf cm2 and 314 gf cm<sup>2</sup>

The desired angle θdð Þt [rad] for the rigid fin motion was given as

θdðÞ¼ t

π <sup>2</sup> � <sup>π</sup>

except for the following.

Table 1. Parameters used for simulation.

Moment of inertia of the fin [kg m<sup>2</sup>

Density of insulating fluid [kg/m3

Water density [kg/m3

Representative area [m<sup>2</sup>

4.2. Numerical results

<sup>180</sup> , <sup>θ</sup>R<sup>17</sup> <sup>¼</sup> <sup>14</sup>:59520<sup>π</sup>

upper torque limits: �0:0588 < τ < 0:0588 [Nm].

the cases of (a) 3.14, (b) 31.4, and (c) 314 gf cm2

can be expressed approximately by the angular velocities θ\_

θL. There is no disturbance of water flow in the simulation.

<sup>θ</sup>R<sup>1</sup> <sup>¼</sup> <sup>92</sup>:86241<sup>π</sup>

simulation.

passive joints to imitate a flexible film. The initial angles of the entire joints were set at 0 rad

Drag coefficient CD 1.3

bRi and bLi in Eqs. (6) and (7) bRi, bLi 0.004 kRi and kLi in Eqs. (6) and (7) kRi, kLi 0.001231 Position of the right 1st joint ð Þ lR; 0 ð Þ 0:001; 0 Position of the left 1st joint ð Þ lL; 0 ð Þ �0:001; 0 Link length of the fin [m] dl 0.005 Fin depth [m] d 0.1

Based on the penalty method, the contact forces f pi between the rigid fin and the entire joints of the plastic film were calculated using <sup>f</sup> pi <sup>¼</sup> KpiDi � BpiD\_ <sup>i</sup> where Kpi <sup>¼</sup> 300 and Bpi <sup>¼</sup> <sup>0</sup>:1 in the

Frequency f of the fin motion was set at 2 Hz. For this desired angle, we made use of a conventional Proportional Derivative (PD) feedback controller <sup>τ</sup><sup>A</sup> <sup>¼</sup> KP½ �� <sup>θ</sup>dð Þ� <sup>t</sup> <sup>θ</sup>ð Þ<sup>t</sup> KDθ\_ where KP ¼ 30 and KD ¼ 0:1. We assumed that a commercial waterproof servomotor (SG51R; Tower Pro) was used for the fin motion and assumed that the actuator had small lower and

We presume that the relative flow velocity v⊥<sup>i</sup> for the calculation of the hydrodynamic force

We conducted a numerical simulation to investigate the fin motion of different flexural rigidities of the plastic film. Figure 9 presents an illustration of how the fin changes in time series in

<sup>180</sup> , <sup>θ</sup>L<sup>1</sup> <sup>¼</sup> <sup>87</sup>:13760<sup>π</sup>

] r 1830

] r<sup>W</sup> 1000

] Si ¼ d � dl 0.0005

] IR 0.0004033

Dynamic Modeling of a Serial Link Robot Laminated with Plastic Film

http://dx.doi.org/10.5772/intechopen.72441

83

<sup>180</sup> , <sup>θ</sup>R<sup>21</sup> <sup>¼</sup> <sup>145</sup>:08480<sup>π</sup>

). The plastic film had 41

<sup>180</sup> (18)

<sup>180</sup> , <sup>θ</sup>L<sup>17</sup> ¼ � <sup>14</sup>:59520<sup>π</sup>

<sup>R</sup> and θ\_ <sup>L</sup> and joint angles θ<sup>R</sup> and

<sup>18</sup> sin 2ð Þ <sup>π</sup>ft (19)

. Figures 10 and 11 portray plots of the tracking

where B<sup>0</sup> <sup>R</sup> and B<sup>0</sup> <sup>L</sup> denote the 2ð Þ� n � 2 ð Þ 2n � 2 matrices excluded the i, j, kth rows and columns from BR and BL,respectively . The characters bi, bj, bk are the damping coefficients for the i, j, kth joints.

We use Eqs. (5) and (16) of the rigid fin and film motion for numerical simulation of the motion of the fin robot laminated by the plastic film.

## 4. Simulation

This section presents numerical simulation of the two-dimensional motion of the fin robot laminated by the plastic film. The purpose of the simulator is to design the laminated robots and to estimate their motion performance. A graphical simulator was developed for this purpose using software (Visual Studio 2010; Microsoft Corp.) and OpenGL library. The code was written in C language. The derived equations of the motion were solved numerically using the Runge-Kutta-Gill method by which the time step size was 0.001 s.

#### 4.1. Numerical conditions

Table 1 presents the physical parameters of the laminated robot. We investigated the fin motion in three flexural rigidities of the plastic film. One was the value of the flexural rigidity of the film (3.14 gf cm2 ) used to fabricate the fish robot (Figure 2). The other two values were 10


Table 1. Parameters used for simulation.

Consequently, the equations of film motion are rewritten as shown below.

00

Jð Þ θR; θ<sup>L</sup>

<sup>R</sup> 0 0 B<sup>0</sup> L

, B 00

umns from BR and BL,respectively . The characters bi, bj, bk are the damping coefficients for the

We use Eqs. (5) and (16) of the rigid fin and film motion for numerical simulation of the motion

This section presents numerical simulation of the two-dimensional motion of the fin robot laminated by the plastic film. The purpose of the simulator is to design the laminated robots and to estimate their motion performance. A graphical simulator was developed for this purpose using software (Visual Studio 2010; Microsoft Corp.) and OpenGL library. The code was written in C language. The derived equations of the motion were solved numerically using the Runge-

Table 1 presents the physical parameters of the laminated robot. We investigated the fin motion in three flexural rigidities of the plastic film. One was the value of the flexural rigidity

) used to fabricate the fish robot (Figure 2). The other two values were 10

� �

<sup>q</sup>\_ <sup>þ</sup> <sup>Φ</sup><sup>T</sup> KRΔθ<sup>R</sup> KLΔθ<sup>L</sup> � �

¼ diag bi; bj; bk

<sup>L</sup> denote the 2ð Þ� n � 2 ð Þ 2n � 2 matrices excluded the i, j, kth rows and col-

<sup>¼</sup> <sup>Φ</sup><sup>T</sup> <sup>τ</sup><sup>R</sup> τL � �

� � (17)

(16)

<sup>T</sup>ð Þ <sup>θ</sup>R; <sup>θ</sup><sup>L</sup> <sup>B</sup>

Figure 8. Model of the hydrodynamic damping force acting on the ith link.

h i

<sup>B</sup><sup>0</sup> <sup>¼</sup> <sup>B</sup><sup>0</sup>

B<sup>0</sup> þ J

of the fin robot laminated by the plastic film.

Kutta-Gill method by which the time step size was 0.001 s.

Therein,

where B<sup>0</sup>

i, j, kth joints.

4. Simulation

4.1. Numerical conditions

of the film (3.14 gf cm2

<sup>R</sup> and B<sup>0</sup>

82 Lamination - Theory and Application

times and 100 times the value of 3.14 (31.4 gf cm2 and 314 gf cm<sup>2</sup> ). The plastic film had 41 passive joints to imitate a flexible film. The initial angles of the entire joints were set at 0 rad except for the following.

$$\theta\_{R1} = \frac{92.86241\pi}{180}, \theta\_{R17} = \frac{14.59520\pi}{180}, \theta\_{R21} = \frac{145.08480\pi}{180}, \theta\_{L1} = \frac{87.13760\pi}{180}, \theta\_{L17} = -\frac{14.59520\pi}{180} \tag{18}$$

Based on the penalty method, the contact forces f pi between the rigid fin and the entire joints of the plastic film were calculated using <sup>f</sup> pi <sup>¼</sup> KpiDi � BpiD\_ <sup>i</sup> where Kpi <sup>¼</sup> 300 and Bpi <sup>¼</sup> <sup>0</sup>:1 in the simulation.

The desired angle θdð Þt [rad] for the rigid fin motion was given as

$$
\Theta\_d(t) = \frac{\pi}{2} - \frac{\pi}{18} \sin\left(2\pi ft\right) \tag{19}
$$

Frequency f of the fin motion was set at 2 Hz. For this desired angle, we made use of a conventional Proportional Derivative (PD) feedback controller <sup>τ</sup><sup>A</sup> <sup>¼</sup> KP½ �� <sup>θ</sup>dð Þ� <sup>t</sup> <sup>θ</sup>ð Þ<sup>t</sup> KDθ\_ where KP ¼ 30 and KD ¼ 0:1. We assumed that a commercial waterproof servomotor (SG51R; Tower Pro) was used for the fin motion and assumed that the actuator had small lower and upper torque limits: �0:0588 < τ < 0:0588 [Nm].

We presume that the relative flow velocity v⊥<sup>i</sup> for the calculation of the hydrodynamic force can be expressed approximately by the angular velocities θ\_ <sup>R</sup> and θ\_ <sup>L</sup> and joint angles θ<sup>R</sup> and θL. There is no disturbance of water flow in the simulation.

#### 4.2. Numerical results

We conducted a numerical simulation to investigate the fin motion of different flexural rigidities of the plastic film. Figure 9 presents an illustration of how the fin changes in time series in the cases of (a) 3.14, (b) 31.4, and (c) 314 gf cm2 . Figures 10 and 11 portray plots of the tracking

5. Conclusion

Acknowledgements

Author details

Norimitsu Sakagami<sup>1</sup>

Technology, Tokai University, Japan

This chapter described the equations of motion of serial link robots laminated with a plastic film. We have proposed robot packaging method to improve waterproofing and dustproofing of serial link robots. To improve lubrication between the links and the film, we encapsulated an insulating fluid in the plastic film. Considering these conditions, we derived the equations of motion of the laminated robot to be useful for hardware design, motion analysis, and performance improvement such as thrust force and energy efficiency. In the derivation of the equations of motion, we assumed the plastic film as a closed-loop link structure with passive joints. Through numerical simulation based on the derived mathematical model of the fish-like robot, we estimated the motion performance of the fin in different flexural rigidities. We confirmed that the low torque actuator can inflect the laminated body because of the thin plastic film with lower flexural rigidity. Future work

.

Dynamic Modeling of a Serial Link Robot Laminated with Plastic Film

http://dx.doi.org/10.5772/intechopen.72441

85

includes design of robots laminated with a plastic film using our mathematical model.

1 Department of Navigation and Ocean Engineering, School of Marine Science and

2 Department of Robotics, Faculty of Engineering, Kindai University, Japan

partially supported by JSPS KAKENHI (Grant Number 15 K18011).

Figure 11. Torque pattern during the fin motion. (a) 3.14, (b) 31.4, and (c) 314 gf cm<sup>2</sup>

\* and Mizuho Shibata<sup>2</sup>

\*Address all correspondence to: sakagami@scc.u-tokai.ac.jp

This work was partially supported by Furukawa Mfg. Co. Ltd. and partially by the Center of Innovation Program from Japan Science and Technology (JST) Agency. This work was also

Figure 9. Two-dimensional simulation in three flexural rigidities of the plastic film. (a) 3.14, (b) 31.4, and (c) 314 gf cm<sup>2</sup> .

data for the desired trajectory in Eq. (19) and the torque patterns in each case. For the value of 3.14 gf cm<sup>2</sup> in (a) that was the actual value for the prototype robot, the fin achieved smoothly reciprocating motion in the range of 8 degrees at 2 Hz in the simulation. The actual prototype robot in Figure 4 also achieved approximately 10-deg. reciprocating motion at 2 Hz in the tank test. In (b), the flexural rigidity was 10 times different, no great difference in performance was found between the case (a) and the case (b). Result in (c) shows that higher flexural rigidity tended to prevent the fin motion. These numerical results demonstrate that the low torque actuator can inflect the robot body laminated by the thin plastic film with lower flexural rigidity.

Figure 10. Tracking performance for the desired fin angle. (a) 3.14, (b) 31.4, and (c) 314 gf cm2 .

Figure 11. Torque pattern during the fin motion. (a) 3.14, (b) 31.4, and (c) 314 gf cm<sup>2</sup> .

## 5. Conclusion

.

.

data for the desired trajectory in Eq. (19) and the torque patterns in each case. For the value of 3.14 gf cm<sup>2</sup> in (a) that was the actual value for the prototype robot, the fin achieved smoothly reciprocating motion in the range of 8 degrees at 2 Hz in the simulation. The actual prototype robot in Figure 4 also achieved approximately 10-deg. reciprocating motion at 2 Hz in the tank test. In (b), the flexural rigidity was 10 times different, no great difference in performance was found between the case (a) and the case (b). Result in (c) shows that higher flexural rigidity tended to prevent the fin motion. These numerical results demonstrate that the low torque actuator can inflect the robot body laminated by the thin plastic film with lower

Figure 10. Tracking performance for the desired fin angle. (a) 3.14, (b) 31.4, and (c) 314 gf cm2

Figure 9. Two-dimensional simulation in three flexural rigidities of the plastic film. (a) 3.14, (b) 31.4, and (c) 314 gf cm<sup>2</sup>

flexural rigidity.

84 Lamination - Theory and Application

This chapter described the equations of motion of serial link robots laminated with a plastic film. We have proposed robot packaging method to improve waterproofing and dustproofing of serial link robots. To improve lubrication between the links and the film, we encapsulated an insulating fluid in the plastic film. Considering these conditions, we derived the equations of motion of the laminated robot to be useful for hardware design, motion analysis, and performance improvement such as thrust force and energy efficiency. In the derivation of the equations of motion, we assumed the plastic film as a closed-loop link structure with passive joints. Through numerical simulation based on the derived mathematical model of the fish-like robot, we estimated the motion performance of the fin in different flexural rigidities. We confirmed that the low torque actuator can inflect the laminated body because of the thin plastic film with lower flexural rigidity. Future work includes design of robots laminated with a plastic film using our mathematical model.

## Acknowledgements

This work was partially supported by Furukawa Mfg. Co. Ltd. and partially by the Center of Innovation Program from Japan Science and Technology (JST) Agency. This work was also partially supported by JSPS KAKENHI (Grant Number 15 K18011).

## Author details

Norimitsu Sakagami<sup>1</sup> \* and Mizuho Shibata<sup>2</sup>

\*Address all correspondence to: sakagami@scc.u-tokai.ac.jp

1 Department of Navigation and Ocean Engineering, School of Marine Science and Technology, Tokai University, Japan

2 Department of Robotics, Faculty of Engineering, Kindai University, Japan

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**Chapter 5**

**Provisional chapter**

**Advanced Technologies in Manufacturing 3D-Layered**

In the past 20 years, a great progress has been made in additive manufacturing techniques, which has led to numerous applications in aeronautical and defense structures. Though not all advanced materials and alloys, can be automatically layered by a rapid prototyping system or machine, several interesting application have seen the light of publicity in many sectors. Efforts are underway to apply the automated layering technologies in as many materials as possible, mostly nowadays plastics, reinforced-polymers, and metals can be processed by such systems in order to produce three-dimensional parts. The work is underway internationally in order to promote more and more applications of additive manufacturing or automated layering and to lower the costs in such systems. This paper aims at presenting a review of the additive manufacturing history presenting the major steps that lead to the explosion of this technology, and with a special focus on advanced 3D structures in aerospace and defense applications. An insight is also given on the four

**Keywords:** laminated manufacturing, 3D printing, rapid prototyping, additive

It was back in the late 1960s, when the creation of solid objects using photopolymerized resins was attempted, by employing a laser. The attempt took place at the Battelle Memorial Institute in Columbus, Ohio, USA [1]. During this pioneering experiment, two laser beams with non-similar wavelengths were pointed to intersect in the middle of a transparent container filled with of resin. Inside the vat of resin (usual term nowadays), the polymer crosslinked and solidified

**1. Early research and development of making 3D objects**

**Advanced Technologies in Manufacturing 3D-Layered** 

© 2016 The Author(s). Licensee InTech. This chapter is 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.

© 2018 The Author(s). Licensee InTech. This chapter is 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.

DOI: 10.5772/intechopen.74331

**Structures for Defense and Aerospace**

**Structures for Defense and Aerospace**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74331

dimensions of manufacturing concept.

Dionysios E. Mouzakis

**Abstract**

manufacturing

Dionysios E. Mouzakis


## **Advanced Technologies in Manufacturing 3D-Layered Structures for Defense and Aerospace Advanced Technologies in Manufacturing 3D-Layered Structures for Defense and Aerospace**

DOI: 10.5772/intechopen.74331

Dionysios E. Mouzakis Dionysios E. Mouzakis

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Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74331

## **Abstract**

In the past 20 years, a great progress has been made in additive manufacturing techniques, which has led to numerous applications in aeronautical and defense structures. Though not all advanced materials and alloys, can be automatically layered by a rapid prototyping system or machine, several interesting application have seen the light of publicity in many sectors. Efforts are underway to apply the automated layering technologies in as many materials as possible, mostly nowadays plastics, reinforced-polymers, and metals can be processed by such systems in order to produce three-dimensional parts. The work is underway internationally in order to promote more and more applications of additive manufacturing or automated layering and to lower the costs in such systems. This paper aims at presenting a review of the additive manufacturing history presenting the major steps that lead to the explosion of this technology, and with a special focus on advanced 3D structures in aerospace and defense applications. An insight is also given on the four dimensions of manufacturing concept.

**Keywords:** laminated manufacturing, 3D printing, rapid prototyping, additive manufacturing

## **1. Early research and development of making 3D objects**

It was back in the late 1960s, when the creation of solid objects using photopolymerized resins was attempted, by employing a laser. The attempt took place at the Battelle Memorial Institute in Columbus, Ohio, USA [1]. During this pioneering experiment, two laser beams with non-similar wavelengths were pointed to intersect in the middle of a transparent container filled with of resin. Inside the vat of resin (usual term nowadays), the polymer crosslinked and solidified

© 2016 The Author(s). Licensee InTech. This chapter is 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. © 2018 The Author(s). Licensee InTech. This chapter is 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.

at the point where the laser beams intersected. DuPont had already made available the photopolymerization of resins as a technology in the 1950s. Quite similar to this approach, Swainson filed a patent titled "*Method of Producing a 3D Figure by Holography*" in Denmark, in 1967. His procedure was also based on dual-laser beam scheme. From 1967 to 1994, the laser-induced photopolymerization was being developed, but until then no functional 3D printer was offered to the market. In the early 1970s, a company called Formigraphic Engine Co., applied the dual-laser intersection paten, aiming at the first commercially available laser-prototyping machine. At that time, they coded the methodology under the term "photochemical machining." That company, however, succeeded to present the creation of a 3D object in 1974. Later, Formigraphic became Omtec Replication, and led the development of similar techniques with the help of DARPA funding. As the decade expired, a number of patents regarding solid photography were claimed by Dynell Electronics Corp. Their invention described the cutting of cross sections, in any soft material available, usually carton or plastic, using either a milling machine or laser, guided by computer control and stacking the sections sequentially, to form a 3D object. Nowadays, the technique is called laminated object manufacturing (LOM). Dynell merged with United Technologies Corp. in 1977. Their merger was independent and was called Solid Photography. This company opened an affiliated retail outlet named Sculpture, which by mid-1981 had changed its name to Robotic Vision.

the first commercially available AM machine worldwide. This system was the precursor of the SLA 250 machine, which became a commercial success (SLA acronym means Stereo-Lithography-Apparatus). SLA 250 was replaced by Viper SLA, and nowadays replaced by the

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A year later, in 1988, 3D Systems and Ciba-Geigy formed a partnership aiming at the development of stereolithography materials and marketed their first-generation acrylate resins. DuPont developed its stereolithography system named Somos along with proper resins in the same year. Loctite had also attempted to enter the SL resins market in the late 1980s, but

Japan's NTT Data CMET and Sony/D-MEC commercialized versions of stereolithography in 1988 and 1989, as an answer to 3D Systems SL in the U.S. NTT Data CMET (now a part of Teijin Seiki, a subsidiary of Nabtesco) called its machine Solid Object Ultraviolet Plotter (SOUP) and Sony/D-MEC (now D-MEC) called its device Solid Creation System (SCS). Sony closed its production for SL systems for D-MEC in 2007. In 1988, Asahi Denka Kogyo introduced the first epoxy resin for the CMET SL machine. In 1989, Japan Synthetic Rubber (now JSR Corp.) and DSM Desotech started to supply polymers for the Sony/D-MEC stereolithography machines. In 1990, Electro Optical Systems (EOS) based in Germany sold its first Stereos stereolithography system. The same year, Quadrax introduced the Mark 1000 SL system, which employed visible light resin. The following year, Imperial Chemical Industries introduced a visible light resin product for the Mark 1000. ICI stopped selling its resin about 1 year later when Quadrax

Back in 1991, three AM technologies were made available to the market, e.g., fused deposition modeling (FDM) as coined by Stratasys, solid ground curing (SGC) marketed by Cubital, and as mentioned above, laminated object manufacturing (LOM) by Helisys. FDM utilizes molten thermoplastic polymers in the form of a filament to structure an object in a layer-by-layer fashion. In SGC, a UV-sensitive liquid polymer is used, and each complete layer solidifies in each laser scan by ample UV light which passes through masks created using an electrostatic toner on a glass plate. LOM cut sheet material using a digitally guided laser and bonds the stacked layers together into a 3D object. Cubital and Helisys are not in the market anymore. Selective laser sintering (SLS) from DTM (now a part of 3D Systems) and the Soliform stereolithography system from Teijin Seiki became available in 1992. Using laser as an extreme heat source, SLS sinters powder materials by local fusion. DuPont had developed the Soliform technology originally, under the name Somos and licensed it to Teijin Seiki. The latter company had the exclusive distribution rights in parts of East Asia. Also, in the same year, a company called Allied Signal marketed vinylether Exactomer resin polymers for SL applications. In the next year, Soligen of Germany presented a device under the name direct shell production casting (DSPC). DSPC employed an inkjet printer mechanism, which deposited liquid binder onto ceramic powder. In this way, shells were formed for later use in the investment-casting procedure. The patent Soligen used was filed by the Massachusetts Institute of Technology (MIT). In January 2006, Solingen stopped its production of DSPC systems. That year, Denken

ProJet series of SLA Printers.

closed the corresponding department in 1993.

dissolved due legal issues with 3D Systems.

**1.2. Layering systems-non SL**

## **1.1. Stereolithography emerges**

According to many sources, Hideo Kodama, working at the Nagoya Municipal Industrial Research Institute (Nagoya, Japan), was one of the first to invent the single-beam laser curing approach, In mid-1980, he filed a patent in Japan, which unfortunately expired without proceeding to the examination stage, this being a requirement of the Japanese patent application procedure. Kodama apparently had obstacles in securing funds for additional research and development. In 1981, he published a paper titled "*A Scheme for Three Dimensional Display by Automatic Fabrication of Three Dimensional Model*" [2] and another paper titled "*Automatic Method for Fabricating a Three-Dimensional Plastic Model With Photo-Hardening Polymer*" that outlined his work in detail and offering thus the technology openly to the public [3]. During his experiments, UV rays were projected, using a mercury lamp, (Toshiba) into a photosensitivephotopolymerizing resin called Tevistar® produced by Teijin. In Kodama's innovating technique, black and white film was used to mask and accurately determine the region of exposure to UV light, and thus to define each cross section. His first paper also described the application of an x-y plotter device and the use of optical fibers as carriers of UV light. His approach was used by CMET Company in the SOUP 530, 600, and 850 stereolithographic systems. In his paper, Kodama quotes, "*If the solidified layer is immersed into the liquid with the top at a depth equal to the thickness of the layer to be solidified, its top surface is covered with unsolidified liquid polymer*,….", actually giving the description of the main core procedure of the stereolithography technology. Kodama's experiments, and papers, are nowadays considered to be the first successful application of working additive manufacturing (AM) techniques worldwide.

The real market-available additive manufacturing, however, first saw light in 1987 with stereolithography (SL) from 3D Systems. Their process involved solidification of thin layers of light-sensitive liquid resin polymer using a UV-laser beam. They marketed their SLA-1, being the first commercially available AM machine worldwide. This system was the precursor of the SLA 250 machine, which became a commercial success (SLA acronym means Stereo-Lithography-Apparatus). SLA 250 was replaced by Viper SLA, and nowadays replaced by the ProJet series of SLA Printers.

A year later, in 1988, 3D Systems and Ciba-Geigy formed a partnership aiming at the development of stereolithography materials and marketed their first-generation acrylate resins. DuPont developed its stereolithography system named Somos along with proper resins in the same year. Loctite had also attempted to enter the SL resins market in the late 1980s, but closed the corresponding department in 1993.

Japan's NTT Data CMET and Sony/D-MEC commercialized versions of stereolithography in 1988 and 1989, as an answer to 3D Systems SL in the U.S. NTT Data CMET (now a part of Teijin Seiki, a subsidiary of Nabtesco) called its machine Solid Object Ultraviolet Plotter (SOUP) and Sony/D-MEC (now D-MEC) called its device Solid Creation System (SCS). Sony closed its production for SL systems for D-MEC in 2007. In 1988, Asahi Denka Kogyo introduced the first epoxy resin for the CMET SL machine. In 1989, Japan Synthetic Rubber (now JSR Corp.) and DSM Desotech started to supply polymers for the Sony/D-MEC stereolithography machines.

In 1990, Electro Optical Systems (EOS) based in Germany sold its first Stereos stereolithography system. The same year, Quadrax introduced the Mark 1000 SL system, which employed visible light resin. The following year, Imperial Chemical Industries introduced a visible light resin product for the Mark 1000. ICI stopped selling its resin about 1 year later when Quadrax dissolved due legal issues with 3D Systems.

## **1.2. Layering systems-non SL**

at the point where the laser beams intersected. DuPont had already made available the photopolymerization of resins as a technology in the 1950s. Quite similar to this approach, Swainson filed a patent titled "*Method of Producing a 3D Figure by Holography*" in Denmark, in 1967. His procedure was also based on dual-laser beam scheme. From 1967 to 1994, the laser-induced photopolymerization was being developed, but until then no functional 3D printer was offered to the market. In the early 1970s, a company called Formigraphic Engine Co., applied the dual-laser intersection paten, aiming at the first commercially available laser-prototyping machine. At that time, they coded the methodology under the term "photochemical machining." That company, however, succeeded to present the creation of a 3D object in 1974. Later, Formigraphic became Omtec Replication, and led the development of similar techniques with the help of DARPA funding. As the decade expired, a number of patents regarding solid photography were claimed by Dynell Electronics Corp. Their invention described the cutting of cross sections, in any soft material available, usually carton or plastic, using either a milling machine or laser, guided by computer control and stacking the sections sequentially, to form a 3D object. Nowadays, the technique is called laminated object manufacturing (LOM). Dynell merged with United Technologies Corp. in 1977. Their merger was independent and was called Solid Photography. This company opened an affiliated retail outlet named Sculpture, which by

According to many sources, Hideo Kodama, working at the Nagoya Municipal Industrial Research Institute (Nagoya, Japan), was one of the first to invent the single-beam laser curing approach, In mid-1980, he filed a patent in Japan, which unfortunately expired without proceeding to the examination stage, this being a requirement of the Japanese patent application procedure. Kodama apparently had obstacles in securing funds for additional research and development. In 1981, he published a paper titled "*A Scheme for Three Dimensional Display by Automatic Fabrication of Three Dimensional Model*" [2] and another paper titled "*Automatic Method for Fabricating a Three-Dimensional Plastic Model With Photo-Hardening Polymer*" that outlined his work in detail and offering thus the technology openly to the public [3]. During his experiments, UV rays were projected, using a mercury lamp, (Toshiba) into a photosensitivephotopolymerizing resin called Tevistar® produced by Teijin. In Kodama's innovating technique, black and white film was used to mask and accurately determine the region of exposure to UV light, and thus to define each cross section. His first paper also described the application of an x-y plotter device and the use of optical fibers as carriers of UV light. His approach was used by CMET Company in the SOUP 530, 600, and 850 stereolithographic systems. In his paper, Kodama quotes, "*If the solidified layer is immersed into the liquid with the top at a depth equal to the thickness of the layer to be solidified, its top surface is covered with unsolidified liquid polymer*,….", actually giving the description of the main core procedure of the stereolithography technology. Kodama's experiments, and papers, are nowadays considered to be the first suc-

cessful application of working additive manufacturing (AM) techniques worldwide.

The real market-available additive manufacturing, however, first saw light in 1987 with stereolithography (SL) from 3D Systems. Their process involved solidification of thin layers of light-sensitive liquid resin polymer using a UV-laser beam. They marketed their SLA-1, being

mid-1981 had changed its name to Robotic Vision.

**1.1. Stereolithography emerges**

90 Lamination - Theory and Application

Back in 1991, three AM technologies were made available to the market, e.g., fused deposition modeling (FDM) as coined by Stratasys, solid ground curing (SGC) marketed by Cubital, and as mentioned above, laminated object manufacturing (LOM) by Helisys. FDM utilizes molten thermoplastic polymers in the form of a filament to structure an object in a layer-by-layer fashion. In SGC, a UV-sensitive liquid polymer is used, and each complete layer solidifies in each laser scan by ample UV light which passes through masks created using an electrostatic toner on a glass plate. LOM cut sheet material using a digitally guided laser and bonds the stacked layers together into a 3D object. Cubital and Helisys are not in the market anymore. Selective laser sintering (SLS) from DTM (now a part of 3D Systems) and the Soliform stereolithography system from Teijin Seiki became available in 1992. Using laser as an extreme heat source, SLS sinters powder materials by local fusion. DuPont had developed the Soliform technology originally, under the name Somos and licensed it to Teijin Seiki. The latter company had the exclusive distribution rights in parts of East Asia. Also, in the same year, a company called Allied Signal marketed vinylether Exactomer resin polymers for SL applications. In the next year, Soligen of Germany presented a device under the name direct shell production casting (DSPC). DSPC employed an inkjet printer mechanism, which deposited liquid binder onto ceramic powder. In this way, shells were formed for later use in the investment-casting procedure. The patent Soligen used was filed by the Massachusetts Institute of Technology (MIT). In January 2006, Solingen stopped its production of DSPC systems. That year, Denken marketed an SL system that featured a solid-state laser. It should be mentioned here that Denken's SL system was presumably, compact enough, to sit on a bench top. Moreover, it was also given away at a relatively low price, compared to other SL systems available in the market. In 1993, 3D Systems and Ciba made their first epoxy resin product for SLA, commercially available. In the same period, the QuickCast structuring scheme was presented. QuickCast, still used to-date, is a process in which hollow investment-casting molds are produced. After casting, the polymer mold would burn out, without damaging the fragile ceramic shell. In 1994, many new additive manufacturing systems found their way into the markets. ModelMaker from Solidscape (then called Sanders Prototype) was delivered, as many new systems from Japanese and European firms did. By making use of an inkjet print head, ModelMaker had the ability to deposit waxy materials layer, by layer. Aiming at a new market sector, namely jewelry makers, Meiko in Japan produced a novel small stereolithography device. Meiko is no more in the SL sector since 2006. Also, in Japan, KiraCorp. marketed its first non-stereolithographic device. The device called Solid Center was actually a complete LOM system featuring a typical laserprinter engine, toner, an x-y plotter, and knife; it was able to produce wood-like models by paper lamination. Kira referred to Solid Center as the first plain-paper 3D printer. In the same year 1994, Fockele & Schwarze (F&S) in Germany introduced a stereolithography machine, but on a limited basis, and a German company named EOS commercialized a printer called EOSINT based on laser-sintering technology. Ushio from Japan (Unirapid Inc. nowadays) sold its first stereolithography machine in 1995.

has developed processes much alike to FDM and other additive manufacturing technologies, since 1996. Autostrade started to market a stereolithography system called E-DARTS to firms in Japan at prices no higher than \$25,000, in the same year. Also in 1998, Optomec made available to the industry its laser-engineered net shaping (LENS) metal powder system. The machine was based on the technology developed at Sandia National Labs. From that point, the markets begin to open even more and also the international demand for such systems. In March 1999, 3D Systems introduced a machine called ThermoJet, a much faster and less costly version of Actua 2100. At that time, 3D Systems was selling its SLA 7000 system for \$800,000, which was the most expensive AM system for plastic materials, available worldwide. In April 1999, on demand by Motorola, a company under the name Extrude Hone AM business (nowadays named Ex One) installed its first ProMetal RTS-300 system, for building metal parts. The machine was utilizing MIT's 3DP inkjet-printing technology. In 1999, Fockele & Schwarze based in Germany, revealed its selective laser-melting system for steel-based powders. This system was developed in cooperation with the Fraunhofer Institute for Laser Technology, in Aachen. The same Institute provided know-how for Röders, who developed and sold its controlled metal buildup (CMB) machine. Also, in 1999, DSM acquired the Somos branch from DuPont. In January 2000, Helisys announced that Toyoda Machine Works of Japan would manufacture and sell LOM systems in Japan. In June 2000, Toyoda exhibited its proprietary machine based on LOM technology, at a technology fair in Tokyo. Sanders Design International proclaimed the development of a system named Rapid Tool Maker (RTM) in January 2000, and they also announced that it had licensed the RTM technology to a German company called Buss Modeling Technology (BMT). BMT, which was formerly Buss Müller Technology, had the strategic plan to manufacture RTM systems and provide it to the European markets. At that time, BMT announced manufacturing and marketing a color 3D printer based on powder and binder technology, developed by Aad van der Geest of the Netherlands. The process was quite similar to the 3DP process from Z Corp. Since the beginning of the first decade of the new century, many systems are available in the market for producing layered structures, now under the name 3D printed structures utilizing the prevailing technologies as described above in summary and in the following chapters in detail. Nowadays, many affordable systems are available in the open market, even as DIY kits, for applications in almost all industrial sectors as well as for hobby and recreation as it

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Up to year 2009, layering technology of 3D printing was largely employed for industrial applications, as reported above; however, exactly then, the patent protecting fused deposition modeling (FDM)—one of the most simple and commonly used 3D printing technologies nowadays—expired. Thanks to the RepRap people's project's mission and vision, to build a self-replicating 3D printer, the first desktop 3D printer was born. This caused an avalanche effect and many manufacturers followed, and the cost which was initially \$200,000 back in early 2000, suddenly sunk below \$2000, and the consumer 3D printing market took off in

2009. Nowadays, a simple DIY system costs about \$150 for hobbyists.

will be shown.

**2. Market placement**

## **1.3. Low-cost 3D printers**

In 1996, Stratasys introduced the Genisys machine. This type of printer utilized an extrusion process similar to FDM, however, based on technology developed at IBM's Watson Research Center. Being a decade almost in the market of stereolithography systems, 3D Systems offered to the market its first 3D printer (Actua 2100) in 1996, using a technology that deposits waxy materials in a layer after layer fashion, utilizing an inkjet-printing mechanism. In the same year, a company called Z Corp. marketed its Z402 modeling 3D printer, aiming at conceptual use. The Z402 machine profited from MIT's inkjet-printing (3DP) technology, and models were manufactured using starch- or plaster-based powders and a water-soluble liquid binder. Also in 1996, Schroff Development started to offer to the market its semi-automated paper lamination system below the threshold of \$10,000. BPM Technology begun to sell its Personal Modeler 2100 model in 1996 too. By employing a process named ballistic particle manufacturing (BPM), the machine could deposit waxy material layers by use of an inkjet-printing head. The company ceased operations in October 1997. Kinergy based in Singapore started selling its Zippy paper lamination systems, which worked much alike as the LOM process. AeroMet founded in 1997 was a subsidiary of MTS Systems Corp. This company developed a procedure called laser additive manufacturing (LAM) that employed a high-power laser and titanium alloys in the form of powder. Until it stopped operations in December 2005, AeroMet was a 3D printed parts subcontractor for the aerospace industry. That year, Ciba acquired the Exactomer resins business from Allied Signal. In 1998, Yinhua Laser Rapid Prototypes Making & Mould Technology Co., Ltd. based in Beijing-China, intensively pursued the marketing of its products. Not by chance, Tsinghua University in Beijing, being the original developer of these systems, has developed processes much alike to FDM and other additive manufacturing technologies, since 1996. Autostrade started to market a stereolithography system called E-DARTS to firms in Japan at prices no higher than \$25,000, in the same year. Also in 1998, Optomec made available to the industry its laser-engineered net shaping (LENS) metal powder system. The machine was based on the technology developed at Sandia National Labs. From that point, the markets begin to open even more and also the international demand for such systems. In March 1999, 3D Systems introduced a machine called ThermoJet, a much faster and less costly version of Actua 2100. At that time, 3D Systems was selling its SLA 7000 system for \$800,000, which was the most expensive AM system for plastic materials, available worldwide. In April 1999, on demand by Motorola, a company under the name Extrude Hone AM business (nowadays named Ex One) installed its first ProMetal RTS-300 system, for building metal parts. The machine was utilizing MIT's 3DP inkjet-printing technology. In 1999, Fockele & Schwarze based in Germany, revealed its selective laser-melting system for steel-based powders. This system was developed in cooperation with the Fraunhofer Institute for Laser Technology, in Aachen. The same Institute provided know-how for Röders, who developed and sold its controlled metal buildup (CMB) machine. Also, in 1999, DSM acquired the Somos branch from DuPont. In January 2000, Helisys announced that Toyoda Machine Works of Japan would manufacture and sell LOM systems in Japan. In June 2000, Toyoda exhibited its proprietary machine based on LOM technology, at a technology fair in Tokyo. Sanders Design International proclaimed the development of a system named Rapid Tool Maker (RTM) in January 2000, and they also announced that it had licensed the RTM technology to a German company called Buss Modeling Technology (BMT). BMT, which was formerly Buss Müller Technology, had the strategic plan to manufacture RTM systems and provide it to the European markets. At that time, BMT announced manufacturing and marketing a color 3D printer based on powder and binder technology, developed by Aad van der Geest of the Netherlands. The process was quite similar to the 3DP process from Z Corp.

Since the beginning of the first decade of the new century, many systems are available in the market for producing layered structures, now under the name 3D printed structures utilizing the prevailing technologies as described above in summary and in the following chapters in detail. Nowadays, many affordable systems are available in the open market, even as DIY kits, for applications in almost all industrial sectors as well as for hobby and recreation as it will be shown.

## **2. Market placement**

marketed an SL system that featured a solid-state laser. It should be mentioned here that Denken's SL system was presumably, compact enough, to sit on a bench top. Moreover, it was also given away at a relatively low price, compared to other SL systems available in the market. In 1993, 3D Systems and Ciba made their first epoxy resin product for SLA, commercially available. In the same period, the QuickCast structuring scheme was presented. QuickCast, still used to-date, is a process in which hollow investment-casting molds are produced. After casting, the polymer mold would burn out, without damaging the fragile ceramic shell. In 1994, many new additive manufacturing systems found their way into the markets. ModelMaker from Solidscape (then called Sanders Prototype) was delivered, as many new systems from Japanese and European firms did. By making use of an inkjet print head, ModelMaker had the ability to deposit waxy materials layer, by layer. Aiming at a new market sector, namely jewelry makers, Meiko in Japan produced a novel small stereolithography device. Meiko is no more in the SL sector since 2006. Also, in Japan, KiraCorp. marketed its first non-stereolithographic device. The device called Solid Center was actually a complete LOM system featuring a typical laserprinter engine, toner, an x-y plotter, and knife; it was able to produce wood-like models by paper lamination. Kira referred to Solid Center as the first plain-paper 3D printer. In the same year 1994, Fockele & Schwarze (F&S) in Germany introduced a stereolithography machine, but on a limited basis, and a German company named EOS commercialized a printer called EOSINT based on laser-sintering technology. Ushio from Japan (Unirapid Inc. nowa-

In 1996, Stratasys introduced the Genisys machine. This type of printer utilized an extrusion process similar to FDM, however, based on technology developed at IBM's Watson Research Center. Being a decade almost in the market of stereolithography systems, 3D Systems offered to the market its first 3D printer (Actua 2100) in 1996, using a technology that deposits waxy materials in a layer after layer fashion, utilizing an inkjet-printing mechanism. In the same year, a company called Z Corp. marketed its Z402 modeling 3D printer, aiming at conceptual use. The Z402 machine profited from MIT's inkjet-printing (3DP) technology, and models were manufactured using starch- or plaster-based powders and a water-soluble liquid binder. Also in 1996, Schroff Development started to offer to the market its semi-automated paper lamination system below the threshold of \$10,000. BPM Technology begun to sell its Personal Modeler 2100 model in 1996 too. By employing a process named ballistic particle manufacturing (BPM), the machine could deposit waxy material layers by use of an inkjet-printing head. The company ceased operations in October 1997. Kinergy based in Singapore started selling its Zippy paper lamination systems, which worked much alike as the LOM process. AeroMet founded in 1997 was a subsidiary of MTS Systems Corp. This company developed a procedure called laser additive manufacturing (LAM) that employed a high-power laser and titanium alloys in the form of powder. Until it stopped operations in December 2005, AeroMet was a 3D printed parts subcontractor for the aerospace industry. That year, Ciba acquired the Exactomer resins business from Allied Signal. In 1998, Yinhua Laser Rapid Prototypes Making & Mould Technology Co., Ltd. based in Beijing-China, intensively pursued the marketing of its products. Not by chance, Tsinghua University in Beijing, being the original developer of these systems,

days) sold its first stereolithography machine in 1995.

**1.3. Low-cost 3D printers**

92 Lamination - Theory and Application

Up to year 2009, layering technology of 3D printing was largely employed for industrial applications, as reported above; however, exactly then, the patent protecting fused deposition modeling (FDM)—one of the most simple and commonly used 3D printing technologies nowadays—expired. Thanks to the RepRap people's project's mission and vision, to build a self-replicating 3D printer, the first desktop 3D printer was born. This caused an avalanche effect and many manufacturers followed, and the cost which was initially \$200,000 back in early 2000, suddenly sunk below \$2000, and the consumer 3D printing market took off in 2009. Nowadays, a simple DIY system costs about \$150 for hobbyists.

The sales and market of 3D printing have been skyrocketing ever since. Of course, important patents on additive manufacturing expired and technology is made available to more bright minds, hence more innovations are foreseen in the near future as well as enormous revenues in the main and niece markets. There are, nowadays, almost 300,000 enterprise users and more than 1,000,000 private owners of 3D printers in the world. These numbers are doubling every other year. 3D printing industry is still in its childhood phase and its growing bigger and bigger. The number of companies that manufacture 3D printers has doubled in the last 2 years. According to Wohler's Report for 2017, 97 manufacturers produced and sold additive manufacturing machines and devices in 2016. A year later, they were 62. The industry achieved worldwide revenues of \$6.063 billion in 2016 about three times the maximum forecasted value as seen in **Figure 1**.

manufacturing processes required to produce the final product. This results in an item that can be customized to meet a user's specific needs without additional manufacturing costs,

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In metal casting or injection molding, each specific part of each product requires a mold—a factor that can skyrocket manufacturing costs rapidly. To counteract these permanent manufacturing costs, most companies anticipate thousands of the same items being sold. On the other hand, 3D printing is *per se*, a "single tool" or maybe a zero-tool process. In this way, no tool is required and no need to change any condition of the process exists. Additionally, no hidden costs or lead times are involved in making an object complex or simple. Consequently, this paves the way for substantial decreases in production costs. At the cost of manufacturing time of course, needless to mention, as situation stands nowadays, because 3D printing is by

Strategically thinking, once its manifested that there is no expensive tooling required to produce objects through 3D printing technologies, designers or entrepreneurs, might consider it, and they usually do so, as a cost effective method to produce items for a market test run or small production series. Possibly best, exploit the internet, through crowdfunding sites like Kickstarter, in order to launch their products. At the early stages of product development, it appears both also practical and wise to make design changes, in the product, without compromising their name in more formal—and expensive—manufacturing orders. Concluding, 3D printing technology opens a much less dire route to the market for those who want to

Most conventional manufacturing processes are subtractive especially the second grade ones: you start with a block of material (or a cast item), and usually through cutting, milling, drilling or similar, it is being processed at the intended final design. For many products—such as

On the other hand, 3D printing belongs to the additive processes; object is created from the raw material layer-by-layer. According to the laws of Mother Nature (she creates things exactly this way we found), when an object is manufactured this way, it only uses as much material and energy—that is needed to create that particular object and no more. Additionally, most of

Alas, having all of the benefits of manufacturing through additive techniques, 3D printing is not yet competitive with conventional manufacturing processes, when it comes to large production volumes. The critical turning point lies between 1000 and 10,000 units, the numbers being a function on the material and the design. Of course, as the prices of printers and raw materials continue to decrease, the range of efficient production is expected to increase above

Nowadays, there are more than 600 3D printing raw materials marketed, most of which are plastics and metals, but the choices are still limited compared to conventional product materials, available colors and finishes. The pseudo-lack of materials, however, is increasing, the number of new materials added to the 3D printing palette is growing rapidly now including

a bracket for an airplane—90% of the raw material is lost during processing.

these materials can be recycled and repurposed into more 3D-printed objects.

only the design labor remaining.

no means considered a mass production technology.

materialize a novel product or an idea.

the reported numbers in the following years.

wood, metals, alloys, composites, ceramics, and even chocolates.

## **2.1. 3D printing: advantages and disadvantages**

It is of paramount significance to understand that 3D printing is a rapidly developing technology, rather immature, which comes with its family of inherent benefits, but also lags behind traditional manufacturing processes in many aspects. Examples from all aspects will enable the reader, get a grasp of these factors and foresee where the technology is headed in the near future.

3D printing allows designers and engineers create complex shapes and parts—many of which cannot be produced by traditional manufacturing techniques. Of course, evidently, manufacturing through additive methods implies that complexity comes at a price; elaborate product designs with complicated design features now cost just as much to produce as simple product designs that follow all the traditional rules of conventional manufacturing.

Utilizing traditional production methods, at a high volume or numbers of products, it is simply cheaper to make and sell products at reasonable prices to the consumer. Alternatively, 3D printing allows easy customization; one only needs to change the design digitally in CAD software to make changes with no additional tooling or other expensive or high efficiency

**Figure 1.** Estimated 3D printer sales per year from 2013 to 2018 [4].

manufacturing processes required to produce the final product. This results in an item that can be customized to meet a user's specific needs without additional manufacturing costs, only the design labor remaining.

The sales and market of 3D printing have been skyrocketing ever since. Of course, important patents on additive manufacturing expired and technology is made available to more bright minds, hence more innovations are foreseen in the near future as well as enormous revenues in the main and niece markets. There are, nowadays, almost 300,000 enterprise users and more than 1,000,000 private owners of 3D printers in the world. These numbers are doubling every other year. 3D printing industry is still in its childhood phase and its growing bigger and bigger. The number of companies that manufacture 3D printers has doubled in the last 2 years. According to Wohler's Report for 2017, 97 manufacturers produced and sold additive manufacturing machines and devices in 2016. A year later, they were 62. The industry achieved worldwide revenues of \$6.063 billion in 2016 about three times the maximum fore-

It is of paramount significance to understand that 3D printing is a rapidly developing technology, rather immature, which comes with its family of inherent benefits, but also lags behind traditional manufacturing processes in many aspects. Examples from all aspects will enable the reader, get a grasp of these factors and foresee where the technology is headed in the near future. 3D printing allows designers and engineers create complex shapes and parts—many of which cannot be produced by traditional manufacturing techniques. Of course, evidently, manufacturing through additive methods implies that complexity comes at a price; elaborate product designs with complicated design features now cost just as much to produce as simple product

Utilizing traditional production methods, at a high volume or numbers of products, it is simply cheaper to make and sell products at reasonable prices to the consumer. Alternatively, 3D printing allows easy customization; one only needs to change the design digitally in CAD software to make changes with no additional tooling or other expensive or high efficiency

designs that follow all the traditional rules of conventional manufacturing.

casted value as seen in **Figure 1**.

94 Lamination - Theory and Application

**2.1. 3D printing: advantages and disadvantages**

**Figure 1.** Estimated 3D printer sales per year from 2013 to 2018 [4].

In metal casting or injection molding, each specific part of each product requires a mold—a factor that can skyrocket manufacturing costs rapidly. To counteract these permanent manufacturing costs, most companies anticipate thousands of the same items being sold. On the other hand, 3D printing is *per se*, a "single tool" or maybe a zero-tool process. In this way, no tool is required and no need to change any condition of the process exists. Additionally, no hidden costs or lead times are involved in making an object complex or simple. Consequently, this paves the way for substantial decreases in production costs. At the cost of manufacturing time of course, needless to mention, as situation stands nowadays, because 3D printing is by no means considered a mass production technology.

Strategically thinking, once its manifested that there is no expensive tooling required to produce objects through 3D printing technologies, designers or entrepreneurs, might consider it, and they usually do so, as a cost effective method to produce items for a market test run or small production series. Possibly best, exploit the internet, through crowdfunding sites like Kickstarter, in order to launch their products. At the early stages of product development, it appears both also practical and wise to make design changes, in the product, without compromising their name in more formal—and expensive—manufacturing orders. Concluding, 3D printing technology opens a much less dire route to the market for those who want to materialize a novel product or an idea.

Most conventional manufacturing processes are subtractive especially the second grade ones: you start with a block of material (or a cast item), and usually through cutting, milling, drilling or similar, it is being processed at the intended final design. For many products—such as a bracket for an airplane—90% of the raw material is lost during processing.

On the other hand, 3D printing belongs to the additive processes; object is created from the raw material layer-by-layer. According to the laws of Mother Nature (she creates things exactly this way we found), when an object is manufactured this way, it only uses as much material and energy—that is needed to create that particular object and no more. Additionally, most of these materials can be recycled and repurposed into more 3D-printed objects.

Alas, having all of the benefits of manufacturing through additive techniques, 3D printing is not yet competitive with conventional manufacturing processes, when it comes to large production volumes. The critical turning point lies between 1000 and 10,000 units, the numbers being a function on the material and the design. Of course, as the prices of printers and raw materials continue to decrease, the range of efficient production is expected to increase above the reported numbers in the following years.

Nowadays, there are more than 600 3D printing raw materials marketed, most of which are plastics and metals, but the choices are still limited compared to conventional product materials, available colors and finishes. The pseudo-lack of materials, however, is increasing, the number of new materials added to the 3D printing palette is growing rapidly now including wood, metals, alloys, composites, ceramics, and even chocolates.

A key aspect, of course, is the mechanical properties. In most 3D printing technologies, the part strength is neither uniform nor high, due to the layer-by-layer fabrication process. Practically, parts that have been 3D printed are usually weaker than their traditionally manufactured counterparts. Repeatability is also an open question; parts made on different 3D printers might have varying properties. However, as technical improvements are rapidly achieved and as novel continuous 3D printing processes like Carbon3D are made available, these disadvantages are prone to extinct in coming years.

**3. 3D technologies review**

**3.1. Fused deposition modeling (FDM)**

innovative ones are shown in the applications chapter.

**Figure 3.** FDM technique sketch. Source: Caliskan and Durgun [5, 6].

for the next layer of material until the entire object is manufactured.

This part refers to the main 3D printing technologies which enable us in printing layered structures in some detail. Only the principal ones in use nowadays are reported, and some

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The FDM printing process begins with a string of solid material called the filament. This line of filament is pulled from a reel attached to the 3D printer to a heated nozzle inside of the 3D printer that heats the material above its melting point. Once in a melted state, the material pushed out of a nozzle is extruded on a specific and predetermined path guided by the software on the computer usually instructed in G-code language. As the material is extruded, as a layer of the object on this path, it instantly cools down and solidifies—providing the base

Considered nowadays as the cheapest 3D printing technology commercially available, FDM also offers a wide variety of plastic-matrix materials in a rainbow of colors including ABS, PLA, nylon, and blends with more exotic materials, including carbon, bronze, or wood (**Figure 3**). FDM is a considered to be the most practical choice for quick and low-cost prototyping. It can be used for a wide range of applications and objects with a typically wide palette of polymers as filaments in pure or reinforced form. Recently, FDM 3D printing has become very famous among hobbyists for enabling them to design and produce functional products, with embedded electronics and mechanical parts such as drones. FDM 3D printing is hampered by

Despite, the fact that we are not still able to manufacture 3D-printed objects with submicron tolerances like an iPhone, 3D printing technology is considered as a very straightforward and practical procedure of layering objects. These parts feature precision within the scale 20–100 microns, which correspond to a natural scale from the diameter of a human hair to the height of a single sheet of paper. 3D printing enables designers and engineers, who are creating objects with few tolerances and design details, to make products and bright ideas real. As known, many high-tech objects demand fine working parts and even finer details—such as the silent switch on the iPhone—it is still difficult to compete with the high precision capabilities of certain manufacturing processes, but time will prove this technology in every case. In every case, 3D printing is changing business model innovation in a very rapid manner (**Figure 2**) [7].

**Figure 2.** Available technologies and materials for 3D printing and lamination techniques. Source: www.aniwaa.com.

## **3. 3D technologies review**

A key aspect, of course, is the mechanical properties. In most 3D printing technologies, the part strength is neither uniform nor high, due to the layer-by-layer fabrication process. Practically, parts that have been 3D printed are usually weaker than their traditionally manufactured counterparts. Repeatability is also an open question; parts made on different 3D printers might have varying properties. However, as technical improvements are rapidly achieved and as novel continuous 3D printing processes like Carbon3D are made available, these dis-

Despite, the fact that we are not still able to manufacture 3D-printed objects with submicron tolerances like an iPhone, 3D printing technology is considered as a very straightforward and practical procedure of layering objects. These parts feature precision within the scale 20–100 microns, which correspond to a natural scale from the diameter of a human hair to the height of a single sheet of paper. 3D printing enables designers and engineers, who are creating objects with few tolerances and design details, to make products and bright ideas real. As known, many high-tech objects demand fine working parts and even finer details—such as the silent switch on the iPhone—it is still difficult to compete with the high precision capabilities of certain manufacturing processes, but time will prove this technology in every case. In every case, 3D printing is changing business model innovation

**Figure 2.** Available technologies and materials for 3D printing and lamination techniques. Source: www.aniwaa.com.

advantages are prone to extinct in coming years.

96 Lamination - Theory and Application

in a very rapid manner (**Figure 2**) [7].

This part refers to the main 3D printing technologies which enable us in printing layered structures in some detail. Only the principal ones in use nowadays are reported, and some innovative ones are shown in the applications chapter.

## **3.1. Fused deposition modeling (FDM)**

The FDM printing process begins with a string of solid material called the filament. This line of filament is pulled from a reel attached to the 3D printer to a heated nozzle inside of the 3D printer that heats the material above its melting point. Once in a melted state, the material pushed out of a nozzle is extruded on a specific and predetermined path guided by the software on the computer usually instructed in G-code language. As the material is extruded, as a layer of the object on this path, it instantly cools down and solidifies—providing the base for the next layer of material until the entire object is manufactured.

Considered nowadays as the cheapest 3D printing technology commercially available, FDM also offers a wide variety of plastic-matrix materials in a rainbow of colors including ABS, PLA, nylon, and blends with more exotic materials, including carbon, bronze, or wood (**Figure 3**).

FDM is a considered to be the most practical choice for quick and low-cost prototyping. It can be used for a wide range of applications and objects with a typically wide palette of polymers as filaments in pure or reinforced form. Recently, FDM 3D printing has become very famous among hobbyists for enabling them to design and produce functional products, with embedded electronics and mechanical parts such as drones. FDM 3D printing is hampered by

**Figure 3.** FDM technique sketch. Source: Caliskan and Durgun [5, 6].

design and material limitations, although improvements appear almost continuously nowadays. The technology generally is not considered suitable for more intricate designs or where high strength is required [8]. Usually, the parts manufactured with this technique can exhibit some internal anisotropy due to layering procedure [9].

## **3.2. Stereolithography and digital light processing (SLA & DLP)**

These techniques are reported together, due to the fact that both technologies produce 3D-printed parts using a photo polymerizing polymer resin, featuring a UV light source to cross-link the liquid material [10].

Analytically, the procedure is as follows: a building platform submerges into a translucent tank (vat) filled with the liquid photo polymerizing resin. After submerging the tank, the UV light source located inside the machine, focuses through the bottom of the tank, scans each layer of the object, effectively solidifies-crosslinks, or polymerizes the material in other words. Consequently, the platform is lifted upward by a few microns, thus allowing a fresh layer of resin to flow beneath the object. The UV light source shall map and solidify the new layer onto the previous one. Micron-by-micron step, the process is repeated in a layer-by-layer fashion, top to bottom until the whole part is finished. The methods are differentiating only by the light source used: In SLA, a UV-laser is used; whereas in DLP, a UV-projector lamp is employed.

The progress made in the past decades delivered enabled 3D printing processes to be applied in desktop 3D printers. Needless to mention here, materials selection is limited to UV-crosslinked polymers. The materials selection, however, broadens each year, new resins with enhanced strength or flexibility are available on the market.

One of the most favorable advantages of SLA & DLP 3D printers is the high accuracy in the produced objects characterized by very smooth surface finishes. This makes them especially famous among artists, for manufacturing sculptures, jewelry molds, and other prototypes. On the other hand, the SLA-DLP technologies are not suitable for printing relatively large or high strength objects. The technology has been accessed as a useful tool in biomedical engineering too [11] (**Figure 4**).

## **3.3. Selective laser sintering (SLS)**

In the process called selective laser sintering (SLS), a high-power laser is required. The laser is employed in order to melt and solidify layers of powder and produce, again layer-bylayer, 3D objects. The SLS printers are commonly equipped with two plates called pistons. First, a first layer of powder is laid onto the fabrication piston. The high-power laser maps/ scans the first layer in the powder, thus selectively melting and sintering—the powder material [12]. In this way, the first layer is fabricated. After solidification of the first layer, the fabrication piston is slightly lowered, and the powder delivery bed, in which the power is contained, is raised by some microns. Then, a roller forces another layer of powder on top of the previous solidified layer. The aforementioned procedure is repeated, allowing the laser to melt and solidify all successive layers one by one, until the designed part has been finished bottom to top (**Figure 5**).

**Figure 4.** (A) Stereolithography (SLA) vs. (B) digital light processing (DLP) techniques.

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design and material limitations, although improvements appear almost continuously nowadays. The technology generally is not considered suitable for more intricate designs or where high strength is required [8]. Usually, the parts manufactured with this technique can exhibit

These techniques are reported together, due to the fact that both technologies produce 3D-printed parts using a photo polymerizing polymer resin, featuring a UV light source to

Analytically, the procedure is as follows: a building platform submerges into a translucent tank (vat) filled with the liquid photo polymerizing resin. After submerging the tank, the UV light source located inside the machine, focuses through the bottom of the tank, scans each layer of the object, effectively solidifies-crosslinks, or polymerizes the material in other words. Consequently, the platform is lifted upward by a few microns, thus allowing a fresh layer of resin to flow beneath the object. The UV light source shall map and solidify the new layer onto the previous one. Micron-by-micron step, the process is repeated in a layer-by-layer fashion, top to bottom until the whole part is finished. The methods are differentiating only by the light source used: In SLA, a UV-laser is used; whereas in DLP, a UV-projector lamp is employed. The progress made in the past decades delivered enabled 3D printing processes to be applied in desktop 3D printers. Needless to mention here, materials selection is limited to UV-crosslinked polymers. The materials selection, however, broadens each year, new resins

One of the most favorable advantages of SLA & DLP 3D printers is the high accuracy in the produced objects characterized by very smooth surface finishes. This makes them especially famous among artists, for manufacturing sculptures, jewelry molds, and other prototypes. On the other hand, the SLA-DLP technologies are not suitable for printing relatively large or high strength objects. The technology has been accessed as a useful tool in biomedical engineering

In the process called selective laser sintering (SLS), a high-power laser is required. The laser is employed in order to melt and solidify layers of powder and produce, again layer-bylayer, 3D objects. The SLS printers are commonly equipped with two plates called pistons. First, a first layer of powder is laid onto the fabrication piston. The high-power laser maps/ scans the first layer in the powder, thus selectively melting and sintering—the powder material [12]. In this way, the first layer is fabricated. After solidification of the first layer, the fabrication piston is slightly lowered, and the powder delivery bed, in which the power is contained, is raised by some microns. Then, a roller forces another layer of powder on top of the previous solidified layer. The aforementioned procedure is repeated, allowing the laser to melt and solidify all successive layers one by one, until the designed part has been

some internal anisotropy due to layering procedure [9].

cross-link the liquid material [10].

98 Lamination - Theory and Application

too [11] (**Figure 4**).

**3.3. Selective laser sintering (SLS)**

finished bottom to top (**Figure 5**).

**3.2. Stereolithography and digital light processing (SLA & DLP)**

with enhanced strength or flexibility are available on the market.

**Figure 4.** (A) Stereolithography (SLA) vs. (B) digital light processing (DLP) techniques.

rapidly coated with new layer of metal powder on top of the solidified layer. The process is repeated until all layers have been solidified resulting in the finished part. Contrary to SLS, the SLM and EBM techniques require support structures, in order to stabilize the object to the build platform and enable manufacturing of overhanging parts. As a side effect, these enable heat transfer away from the solidified powder. Moreover, SLM is performed in a low-oxygen environment and EBM in vacuum. These conditions enable thermal stresses reduction and warping prevention, and they also allow reactive metals and alloys to be used as raw material. Due to their high accuracy and costs, SLM and EBM are mainly applied in industrial 3D printing. Materials include various metals and alloys including steel, titanium, aluminum, cobalt-

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Metal printing is considered to be as the "holy grail" of additive manufacturing and 3D printing; it has found its path in the aerospace, aircraft, automotive, and healthcare industries for a range of high-tech, low-volume part production, from prototyping to final production. 3D printed metal parts allow for monolithic structuring (reducing the quantity of components), miniaturization, and mass reduction combined with design optimization, as shown in **Figure 6**. SLM and EBM have evolved to a stage where these prints are directly comparable to traditionally manufactured parts in terms of chemical composition, mechanical properties (static and fatigue), as well as microstructure. In the year 2017, for the first time, direct metal laser sintering (DMLS) devices, as the latest generation of SLM printers are referred to, were presented in the market at a cost lower than \$100,000. Prices for such

**Figure 6.** A stainless-steel bracket optimized for weight reduction (front) and the traditional cast bracket in the back.

systems in the previous years were over half and near 1 M\$.

Source: European Space Agency events via Flickr.

chrome, and nickel.

**Figure 5.** Selective laser sintering (SLS) method. © Materialgeeza—own work, CC BY-SA 3.0, https://commons. wikimedia.org/w/index.php?curid=4032088.

SLS is a highly efficient method, though rather expensive, but has established itself into the industrial 3D printing applications. Desktop SLS printers are widely available on the open market, and prices are already quite affordable. In 2017, a small SLS system could be acquired for as much as \$15,000. The usual materials available nowadays as powders for SLS include most thermoplastics such as polyamides (nylon), polystyrene, thermoplastic elastomers, etc.

Due to its high accuracy and production fidelity, SLS machines are widely used for manufacturing end products as well as functional prototypes. Complete design freedom is its most important advantage. There is no need for support of structure. The surrounding unmolten powder acts as a support for the structure as it is layered, which allows for complex, sophisticated, and delicate shapes to be manufactured. Finished objects, as a side effect, take a bit more time to cool, and thus longer lead times are expected. Excess and attached object powder is removed by blowing air or using high pressure water or liquids, and is recycled after filtering; hence, economy of raw materials is *in situ* achieved.

## **3.4. Metal printing (selective laser melting and electron beam melting)**

Selective laser melting and electron beam melting (SLM and EBM) are two of the most common metal 3D printing technologies. They are considered as offspring of the SLS technique described above. Just like SLS, these processes create objects from thin layers material. Raw material is the form of powder and it is selectively melted using an intense heat source. As metals and also ceramics, are characterized by higher melting points, consequently much more power is required; this is provided by a high-power laser for SLM or even an electron beam in the EBM technology.

The printing process begins by distribution of a thin layer of metal powder onto a build plate. The powder is selectively melted by a laser (SLM) [13] or an electron beam (EBM) [14] which maps the object layer. The platform or build plate is afterwards lowered by some microns and rapidly coated with new layer of metal powder on top of the solidified layer. The process is repeated until all layers have been solidified resulting in the finished part. Contrary to SLS, the SLM and EBM techniques require support structures, in order to stabilize the object to the build platform and enable manufacturing of overhanging parts. As a side effect, these enable heat transfer away from the solidified powder. Moreover, SLM is performed in a low-oxygen environment and EBM in vacuum. These conditions enable thermal stresses reduction and warping prevention, and they also allow reactive metals and alloys to be used as raw material.

Due to their high accuracy and costs, SLM and EBM are mainly applied in industrial 3D printing. Materials include various metals and alloys including steel, titanium, aluminum, cobaltchrome, and nickel.

Metal printing is considered to be as the "holy grail" of additive manufacturing and 3D printing; it has found its path in the aerospace, aircraft, automotive, and healthcare industries for a range of high-tech, low-volume part production, from prototyping to final production. 3D printed metal parts allow for monolithic structuring (reducing the quantity of components), miniaturization, and mass reduction combined with design optimization, as shown in **Figure 6**. SLM and EBM have evolved to a stage where these prints are directly comparable to traditionally manufactured parts in terms of chemical composition, mechanical properties (static and fatigue), as well as microstructure. In the year 2017, for the first time, direct metal laser sintering (DMLS) devices, as the latest generation of SLM printers are referred to, were presented in the market at a cost lower than \$100,000. Prices for such systems in the previous years were over half and near 1 M\$.

SLS is a highly efficient method, though rather expensive, but has established itself into the industrial 3D printing applications. Desktop SLS printers are widely available on the open market, and prices are already quite affordable. In 2017, a small SLS system could be acquired for as much as \$15,000. The usual materials available nowadays as powders for SLS include most thermoplastics such as polyamides (nylon), polystyrene, thermoplastic elastomers, etc. Due to its high accuracy and production fidelity, SLS machines are widely used for manufacturing end products as well as functional prototypes. Complete design freedom is its most important advantage. There is no need for support of structure. The surrounding unmolten powder acts as a support for the structure as it is layered, which allows for complex, sophisticated, and delicate shapes to be manufactured. Finished objects, as a side effect, take a bit more time to cool, and thus longer lead times are expected. Excess and attached object powder is removed by blowing air or using high pressure water or liquids, and is recycled after filter-

**Figure 5.** Selective laser sintering (SLS) method. © Materialgeeza—own work, CC BY-SA 3.0, https://commons.

Selective laser melting and electron beam melting (SLM and EBM) are two of the most common metal 3D printing technologies. They are considered as offspring of the SLS technique described above. Just like SLS, these processes create objects from thin layers material. Raw material is the form of powder and it is selectively melted using an intense heat source. As metals and also ceramics, are characterized by higher melting points, consequently much more power is required; this is provided by a high-power laser for SLM or even an electron

The printing process begins by distribution of a thin layer of metal powder onto a build plate. The powder is selectively melted by a laser (SLM) [13] or an electron beam (EBM) [14] which maps the object layer. The platform or build plate is afterwards lowered by some microns and

ing; hence, economy of raw materials is *in situ* achieved.

beam in the EBM technology.

wikimedia.org/w/index.php?curid=4032088.

100 Lamination - Theory and Application

**3.4. Metal printing (selective laser melting and electron beam melting)**

**Figure 6.** A stainless-steel bracket optimized for weight reduction (front) and the traditional cast bracket in the back. Source: European Space Agency events via Flickr.

## **4. Advanced applications of the technology of 3D manufacturing**

## **4.1. Automotive industry**

3D printing is becoming more and more a familiar technology to the automotive industry, enabling manufacturing of not only prototypes but also finished parts as well. In February 2017, BMW iVentures, the automaker's venture capital arm, announced an investment in Desktop Metal, a startup devoted to 3D printing metal objects. BMW wants to help accelerate the rollout of this technology in both its design and manufacturing departments. Ιn Formula 1, quite a few racing teams have been testing and ultimately creating custom car parts, using 3D printing for prototyping that are used in high speed races. In the same spirit, Swedish car manufacturer Koenigsegg employed 3D printing to manufacture the variable turbocharger for their One:1 model—a car that has an astonishing 1:1 HP-to-Kg curb weight ratio. Although the 100% metal part is not only very lightweight, more importantly, it can also endure the high forces of supercar combustion and demanding racetrack conditions. Other high-tech examples include Ai Design company's bespoke interior items for high-end cars with popular items including housings for radar detectors, iPhones, and aftermarket SatNav units that blend in with the car's interior. The company often services customers with Lamborghinis, Ferraris, and classic Bentleys, so the fit and finish on the OEM-grade thermoplastics has to be perfect. Ai's experienced engineering was reluctant to abandon the highly efficient CNC manufacturing behind. It took a lot of expert consulting from Stratasys to make Ai Design's people to comprehend the potential in manufacturing its models with a 3D printer and fused deposition modeling.

Prosthetics and other assistive medical devices, braces, and retainers are tailored specifically for the needs of the patient. This has totally reversed an inherent problem that of time and energy required to manually produce each product. As a natural consequence, introduction of 3D printing in the dental and orthodontics fields was an inevitable event. With today's technology, a dental surgeon or orthodontist can use an intraoral 3D camera to scan a client's oral cavity and teeth, use afterward a specialized software and digitally design dental prosthetics, braces, crowns, bridges, etc. Then, he can send the files to a dental technician to 3D print the required molds or directly print the prosthetic itself. As if it was meant to be invented for them the dental industry fully adopted 3D printing technologies. Nowadays, there are dedicated 3D printer models produced specifically for manufacturing dental aids and molds. Alone 3D printer company Stratasys offers two wax 3D printers available to the dental industry. The Stratasys CrownWorx and FrameWorx 3D Printers are supposed to provide the highest precision in wax 3D printing, allowing dental laboratories to produce wax-ups for crowns, bridges, and denture frameworks. Imagine economy in time and costly silicon imprinting materials and gypsum molding. 3D Printers for dental applications such as Stratasys CrownWorx and FrameWorx use wax deposition modeling (WDM) technology. Mainly based on jetting technology and waxy polymers, they allow production of wax-ups characterized by smooth surface finishes and minimal post-processing effort and time requirements. Stratasys claims that the waxy materials burn out leaving no residue, no material shrinkage, neither invoking

**Figure 7.** BMW has turned to 3D printing to augment its workers and stop strain on limbs frequently found on manu-

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facturing lines. Photograph: BMW (republished from the Guardian).

Nowadays, many types of 3D printers are used also in other areas of biomedical applications, such as manufacturing scaffolds for tissue engineering [17, 18] and many other areas of bio-

SpaceX designed and built its famous SuperDraco hypergolic propellant liquid rocket engine. It is a member of SpaceX's Draco rocket engines family. Dragon V2 passenger-carrying space

cracking, nor expansion (**Figure 8**).

materials engineering [19].

**4.3. Aerospace**

BMW, as mentioned above, went for 3D printing technologies quite early. The company has its own Rapid Manufacturing Facility at the HQ in Munich. BMW is considered to be of the founding fathers of stereolithography having recently revealed plans and approach for a fully 3D printed car. One can understand how deeply 3D printing has become incorporated into the company culture by the fact that even a thumb cast for assembly line workers was produced in that way. Evidently, the workers have to push by thumb, a huge number of rubber plugs into chassis holes on the assembly line. This repetitive work causes a repetitive strain type injury in many of them. Confronting the issue, BMW engineers came up with a bright idea: a cast of the thumb and hand that relieved all the strain out the process. Quite simple, very brilliant, and it just proves just how deep 3D Printing has gone into the corporate culture at BMW. It also shows that 3D printing goes beyond the actual manufacturing process itself into biomechanics and ergonomic concepts (**Figure 7**) [15, 16].

## **4.2. Medical and dental industry**

Being always at the cutting edge of technology, biomedical and prosthetics fields has largely benefited from the introduction of 3D printing in these sectors. Custom-shaped personalizedhearing aids no longer require manual labor to manufacture; with 3D printing, they can be made with the click of a button in a very short time. This of course implies substantially lower costs and shorter production times. Even orthopedic implants manufacturing at custom dimensions from CT or MRI scans from the patients is nowadays feasible.

**Figure 7.** BMW has turned to 3D printing to augment its workers and stop strain on limbs frequently found on manufacturing lines. Photograph: BMW (republished from the Guardian).

Prosthetics and other assistive medical devices, braces, and retainers are tailored specifically for the needs of the patient. This has totally reversed an inherent problem that of time and energy required to manually produce each product. As a natural consequence, introduction of 3D printing in the dental and orthodontics fields was an inevitable event. With today's technology, a dental surgeon or orthodontist can use an intraoral 3D camera to scan a client's oral cavity and teeth, use afterward a specialized software and digitally design dental prosthetics, braces, crowns, bridges, etc. Then, he can send the files to a dental technician to 3D print the required molds or directly print the prosthetic itself. As if it was meant to be invented for them the dental industry fully adopted 3D printing technologies. Nowadays, there are dedicated 3D printer models produced specifically for manufacturing dental aids and molds. Alone 3D printer company Stratasys offers two wax 3D printers available to the dental industry. The Stratasys CrownWorx and FrameWorx 3D Printers are supposed to provide the highest precision in wax 3D printing, allowing dental laboratories to produce wax-ups for crowns, bridges, and denture frameworks. Imagine economy in time and costly silicon imprinting materials and gypsum molding. 3D Printers for dental applications such as Stratasys CrownWorx and FrameWorx use wax deposition modeling (WDM) technology. Mainly based on jetting technology and waxy polymers, they allow production of wax-ups characterized by smooth surface finishes and minimal post-processing effort and time requirements. Stratasys claims that the waxy materials burn out leaving no residue, no material shrinkage, neither invoking cracking, nor expansion (**Figure 8**).

Nowadays, many types of 3D printers are used also in other areas of biomedical applications, such as manufacturing scaffolds for tissue engineering [17, 18] and many other areas of biomaterials engineering [19].

## **4.3. Aerospace**

**4. Advanced applications of the technology of 3D manufacturing**

3D printing is becoming more and more a familiar technology to the automotive industry, enabling manufacturing of not only prototypes but also finished parts as well. In February 2017, BMW iVentures, the automaker's venture capital arm, announced an investment in Desktop Metal, a startup devoted to 3D printing metal objects. BMW wants to help accelerate the rollout of this technology in both its design and manufacturing departments. Ιn Formula 1, quite a few racing teams have been testing and ultimately creating custom car parts, using 3D printing for prototyping that are used in high speed races. In the same spirit, Swedish car manufacturer Koenigsegg employed 3D printing to manufacture the variable turbocharger for their One:1 model—a car that has an astonishing 1:1 HP-to-Kg curb weight ratio. Although the 100% metal part is not only very lightweight, more importantly, it can also endure the high forces of supercar combustion and demanding racetrack conditions. Other high-tech examples include Ai Design company's bespoke interior items for high-end cars with popular items including housings for radar detectors, iPhones, and aftermarket SatNav units that blend in with the car's interior. The company often services customers with Lamborghinis, Ferraris, and classic Bentleys, so the fit and finish on the OEM-grade thermoplastics has to be perfect. Ai's experienced engineering was reluctant to abandon the highly efficient CNC manufacturing behind. It took a lot of expert consulting from Stratasys to make Ai Design's people to comprehend the potential in manufacturing its models with a 3D printer and fused deposition modeling.

BMW, as mentioned above, went for 3D printing technologies quite early. The company has its own Rapid Manufacturing Facility at the HQ in Munich. BMW is considered to be of the founding fathers of stereolithography having recently revealed plans and approach for a fully 3D printed car. One can understand how deeply 3D printing has become incorporated into the company culture by the fact that even a thumb cast for assembly line workers was produced in that way. Evidently, the workers have to push by thumb, a huge number of rubber plugs into chassis holes on the assembly line. This repetitive work causes a repetitive strain type injury in many of them. Confronting the issue, BMW engineers came up with a bright idea: a cast of the thumb and hand that relieved all the strain out the process. Quite simple, very brilliant, and it just proves just how deep 3D Printing has gone into the corporate culture at BMW. It also shows that 3D printing goes beyond the actual manufacturing process itself

Being always at the cutting edge of technology, biomedical and prosthetics fields has largely benefited from the introduction of 3D printing in these sectors. Custom-shaped personalizedhearing aids no longer require manual labor to manufacture; with 3D printing, they can be made with the click of a button in a very short time. This of course implies substantially lower costs and shorter production times. Even orthopedic implants manufacturing at custom

into biomechanics and ergonomic concepts (**Figure 7**) [15, 16].

dimensions from CT or MRI scans from the patients is nowadays feasible.

**4.2. Medical and dental industry**

**4.1. Automotive industry**

102 Lamination - Theory and Application

SpaceX designed and built its famous SuperDraco hypergolic propellant liquid rocket engine. It is a member of SpaceX's Draco rocket engines family. Dragon V2 passenger-carrying space

On the other hand, in early 2016, rocket and missile propulsion manufacturer Aerojet Rocketdyne, a renowned aerospace and defense leader, was the recipient of a \$6 million contract from the US Air Force to define 3D-printed rocket engine component standards. The standards will be used to qualify the 3D printed components used in liquid-fueled rocket engine applications, in order to follow through with a mandate set down by US Congress: that the Department of Defense will stop using Russian-made RD-180 engines to launch US satellites and national security payloads into space and begin using domestically produced options instead. Shortly after, Aerojet signed its own contract with Sigma Labs, to non-exclusively license its PrintRite 3D software system to evaluate and redefine the 3D-printed components used in Air Force manufacturing (**Figure 10**). Aerojet, being certainly the right company for the Air Force contract, had already successfully completed hot-fire testing of the 3D printed rocket engine injectors for its liquid-fueled AR1 booster rocket engine in 2015; selective laser melting was used to manufacture the components. It has been known that Aerojet has also successfully completed its Critical Design Review (CDR) for the 500,000 foot-pound thrust-class AR1 engine. This achievement will keep the AR1 on track for flight certification in 2019, as a replacement for the Russian RD-180 engine. Twenty-two incremental CDRs came before the recent system-level CDR, along with full-scale testing of critical subsystem components, like the staged combustion system.

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In the aviation sector, GE Aviation and Safran companies have successfully launched a method for 3D printing of jet engine fuel nozzles. The remarkable technology allows engineers to replace complex assemblies with a single part. The 3D-printed nozzles are lighter than previous designs, and boost a jet engine's fuel efficiency by up to 15%. LEAP engines of GE equipped with 3D-printed fuel nozzles which will power new generation narrow-body planes, e.g., Boeing 737MAX and Airbus A320neo. Especially, the A320neo Airbus passenger airplane is powered by twin LEAP jet engines with 3D-printed parts based on new advanced materials. LEAP ("Leading Edge Aviation Propulsion") is a high-bypass turbofan engine which, due to its advanced space age materials operates at higher pressures than the previous CFM56 machine. The LEAP is the first engine equipped with actually 19 (!) 3D-printed fuel nozzles and parts

**Figure 10.** The Aerojet Rocketdyne AR1 booster engine can be configured as a single engine or a twin booster.

**Figure 8.** An example of a dental frame built using wax deposition modeling. Source: www.Stratasys.com.

capsule shall be powered by a redundant array of eight SuperDraco engines. These provide fault-tolerant propulsion in the launch escape system and propulsive-landing thrust (**Figure 9**).

The combustion chamber of the SuperDraco space engine is created with direct metal laser sintering (DMLS), using Inconel powder. This super-strong nickel-chromium-based "superalloy" is quite difficult to machine in the traditional way with CNC's. The use of 3D printing DMLS technology "*resulted in an order of magnitude reduction in lead-time compared with traditional machining – the path from the initial concept to the first hotfire was just over three months*," according to the company's website. Moreover, the combustion chamber is regeneratively cooled. This method allows cryogenic propellant to pass through a jacket covering the combustion chamber, cooling thus the engine, a trusted solution in the rocker motor design technology.

**Figure 9.** SPACEX, Superdraco engine [20].

On the other hand, in early 2016, rocket and missile propulsion manufacturer Aerojet Rocketdyne, a renowned aerospace and defense leader, was the recipient of a \$6 million contract from the US Air Force to define 3D-printed rocket engine component standards. The standards will be used to qualify the 3D printed components used in liquid-fueled rocket engine applications, in order to follow through with a mandate set down by US Congress: that the Department of Defense will stop using Russian-made RD-180 engines to launch US satellites and national security payloads into space and begin using domestically produced options instead. Shortly after, Aerojet signed its own contract with Sigma Labs, to non-exclusively license its PrintRite 3D software system to evaluate and redefine the 3D-printed components used in Air Force manufacturing (**Figure 10**).

Aerojet, being certainly the right company for the Air Force contract, had already successfully completed hot-fire testing of the 3D printed rocket engine injectors for its liquid-fueled AR1 booster rocket engine in 2015; selective laser melting was used to manufacture the components. It has been known that Aerojet has also successfully completed its Critical Design Review (CDR) for the 500,000 foot-pound thrust-class AR1 engine. This achievement will keep the AR1 on track for flight certification in 2019, as a replacement for the Russian RD-180 engine. Twenty-two incremental CDRs came before the recent system-level CDR, along with full-scale testing of critical subsystem components, like the staged combustion system.

In the aviation sector, GE Aviation and Safran companies have successfully launched a method for 3D printing of jet engine fuel nozzles. The remarkable technology allows engineers to replace complex assemblies with a single part. The 3D-printed nozzles are lighter than previous designs, and boost a jet engine's fuel efficiency by up to 15%. LEAP engines of GE equipped with 3D-printed fuel nozzles which will power new generation narrow-body planes, e.g., Boeing 737MAX and Airbus A320neo. Especially, the A320neo Airbus passenger airplane is powered by twin LEAP jet engines with 3D-printed parts based on new advanced materials. LEAP ("Leading Edge Aviation Propulsion") is a high-bypass turbofan engine which, due to its advanced space age materials operates at higher pressures than the previous CFM56 machine. The LEAP is the first engine equipped with actually 19 (!) 3D-printed fuel nozzles and parts

**Figure 10.** The Aerojet Rocketdyne AR1 booster engine can be configured as a single engine or a twin booster.

**Figure 9.** SPACEX, Superdraco engine [20].

104 Lamination - Theory and Application

capsule shall be powered by a redundant array of eight SuperDraco engines. These provide fault-tolerant propulsion in the launch escape system and propulsive-landing thrust (**Figure 9**). The combustion chamber of the SuperDraco space engine is created with direct metal laser sintering (DMLS), using Inconel powder. This super-strong nickel-chromium-based "superalloy" is quite difficult to machine in the traditional way with CNC's. The use of 3D printing DMLS technology "*resulted in an order of magnitude reduction in lead-time compared with traditional machining – the path from the initial concept to the first hotfire was just over three months*," according to the company's website. Moreover, the combustion chamber is regeneratively cooled. This method allows cryogenic propellant to pass through a jacket covering the combustion chamber, cooling thus the engine, a trusted solution in the rocker motor design technology.

**Figure 8.** An example of a dental frame built using wax deposition modeling. Source: www.Stratasys.com.

from space age, super-strong ceramics that make it 15% more fuel efficient than the previous CFM56 airplane jet engines built by CFM International. CFM is the 50/50 joint-venture between GE Aviation and France's Safran (Snecma) who designed the engine. Airbus picked the LEAP for the A320neo in 2010. Since then, CFM has received more than 2500 orders and commitments for the LEAP-1A engine, representing 55% of A320neo orders to-date (**Figures 11** and **12**).

**4.4. Fuel tanks**

In the totally innovative additive manufacturing process technology invented by Sciaky [22, 23], it is possible to build the two hemispherical halves of fuel tanks. Layer-by-layer, spools of titanium wire is spun, providing the material which melts and deposits, forming thus the tank walls. Lockheed Martin Space Systems, after carefully reviewing and applying the process, stated that they plan to re-think the way they produce satellite propellant tanks. Moreover, eventually the construction of those tanks will be shifted in-house, and thus results in significant capital savings. Up to this point, Lockheed Martin bought those critical titanium tanks from Orbital ATK. The three Mars orbiters use the Orbital ATK tanks as will NASA's OSIRIS-Rex asteroid-probing space vehicle upon its completion and launch. Lockheed Martin official Ambrose stated that it has become critical to reduce the lead times for building satellites, and

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Sciaky Inc. had already begun developing the wire-feed electron beam process back in the mid-1960s [24]. The process was further developed in the 1990s allowing for production of jet engine knife edge seals [25]. The EBAM process was more advanced early in the 2000s, allowing manufacturers' significant savings in time and money on the production of large, high-value metal parts. Sciaky formally launched the EBAM process, in 2009, which was then marketed as Electron Beam Direct Manufacturing, as a service option. Lockheed Martin Aeronautics selected Sciaky for the Department of Defense (DOD) Mentor-Protégé Program in 2011. The special focus of this agreement was the application of additive manufacturing for titanium structural components for Lockheed Martin's F-35 Lightning II fighter project. Further on, in 2012, Sciaky signed a partnership with Penn State University, via DARPA (Defense Advanced Research Projects Agency) funding. Their goal was to advance Direct Digital Manufacturing (DDM) technologies for highly engineered and critical metallic systems and components in applications for the Department of Defense (DOD) and US industry. Finally, in 2014, Sciaky started delivering fully operational commercially available EBAM systems. Needless to mention, Lockheed Martin Space Systems was one of the first customers to acquire an EBAM system for developing and producing "3D-printed" titanium propellant tanks. The EBAM system which Lockheed bought last year from Sciaky Inc. costs \$4 million and is capable of "printing" out fuel tanks of nearly 150 cm in diameter. There is even better news: the method cuts down the cost of manufacturing propellant tanks by as much as 50%.

**Figure 13.** The manufacturing procedure of one half of a satellite fuel tank according to the Sciaky method (top left) and

finished half-product (top right). Source: Sciaky Inc. Video capture.

he said that 3D printing holds the keys to reaching that goal (**Figure 13**).

In August 2016, the LEAP engine was installed on the Airbus A320neo with Pegasus Airlines and CFM delivered 77 machines. Following the introduction of Boeing 737 MAX, CFM delivered 257 LEAPS in the first 9 months of 2017, including 110 in the last three: 49 to Airbus and 61 to Boeing, and targets 450 in the year. The predictions for production at CFM are 1200 engines in 2018, 1900 in 2019, and 2100 in 2020, respectively. This is compared to the 1700 CFM56 produced in 2016. FAA has recently certified the first 3D-printed part for a GE jet engine—a casing that houses the compressor inlet temperature sensor inside the GE90 jet engine.

**Figure 11.** An Airbus A320neo powered by a pair of LEAP-1A engines took a maiden flight on May 19 in Toulouse, France. The two engines used for the four-and-a-half-hour flight were the LEAP-1A, developed specifically for the Airbus jet [21].

**Figure 12.** The LEAP engine (to left) has 19 3D-printed fuel nozzles (top right) and static turbine shrouds made from ceramic matrix composites (CMCs) (above left). Image credit: CFM.

## **4.4. Fuel tanks**

**Figure 12.** The LEAP engine (to left) has 19 3D-printed fuel nozzles (top right) and static turbine shrouds made from

**Figure 11.** An Airbus A320neo powered by a pair of LEAP-1A engines took a maiden flight on May 19 in Toulouse, France. The two engines used for the four-and-a-half-hour flight were the LEAP-1A, developed specifically for the

from space age, super-strong ceramics that make it 15% more fuel efficient than the previous CFM56 airplane jet engines built by CFM International. CFM is the 50/50 joint-venture between GE Aviation and France's Safran (Snecma) who designed the engine. Airbus picked the LEAP for the A320neo in 2010. Since then, CFM has received more than 2500 orders and commitments for the LEAP-1A engine, representing 55% of A320neo orders to-date (**Figures 11** and **12**).

In August 2016, the LEAP engine was installed on the Airbus A320neo with Pegasus Airlines and CFM delivered 77 machines. Following the introduction of Boeing 737 MAX, CFM delivered 257 LEAPS in the first 9 months of 2017, including 110 in the last three: 49 to Airbus and 61 to Boeing, and targets 450 in the year. The predictions for production at CFM are 1200 engines in 2018, 1900 in 2019, and 2100 in 2020, respectively. This is compared to the 1700 CFM56 produced in 2016. FAA has recently certified the first 3D-printed part for a GE jet engine—a casing

that houses the compressor inlet temperature sensor inside the GE90 jet engine.

ceramic matrix composites (CMCs) (above left). Image credit: CFM.

Airbus jet [21].

106 Lamination - Theory and Application

In the totally innovative additive manufacturing process technology invented by Sciaky [22, 23], it is possible to build the two hemispherical halves of fuel tanks. Layer-by-layer, spools of titanium wire is spun, providing the material which melts and deposits, forming thus the tank walls. Lockheed Martin Space Systems, after carefully reviewing and applying the process, stated that they plan to re-think the way they produce satellite propellant tanks. Moreover, eventually the construction of those tanks will be shifted in-house, and thus results in significant capital savings. Up to this point, Lockheed Martin bought those critical titanium tanks from Orbital ATK. The three Mars orbiters use the Orbital ATK tanks as will NASA's OSIRIS-Rex asteroid-probing space vehicle upon its completion and launch. Lockheed Martin official Ambrose stated that it has become critical to reduce the lead times for building satellites, and he said that 3D printing holds the keys to reaching that goal (**Figure 13**).

Sciaky Inc. had already begun developing the wire-feed electron beam process back in the mid-1960s [24]. The process was further developed in the 1990s allowing for production of jet engine knife edge seals [25]. The EBAM process was more advanced early in the 2000s, allowing manufacturers' significant savings in time and money on the production of large, high-value metal parts. Sciaky formally launched the EBAM process, in 2009, which was then marketed as Electron Beam Direct Manufacturing, as a service option. Lockheed Martin Aeronautics selected Sciaky for the Department of Defense (DOD) Mentor-Protégé Program in 2011. The special focus of this agreement was the application of additive manufacturing for titanium structural components for Lockheed Martin's F-35 Lightning II fighter project. Further on, in 2012, Sciaky signed a partnership with Penn State University, via DARPA (Defense Advanced Research Projects Agency) funding. Their goal was to advance Direct Digital Manufacturing (DDM) technologies for highly engineered and critical metallic systems and components in applications for the Department of Defense (DOD) and US industry.

Finally, in 2014, Sciaky started delivering fully operational commercially available EBAM systems. Needless to mention, Lockheed Martin Space Systems was one of the first customers to acquire an EBAM system for developing and producing "3D-printed" titanium propellant tanks. The EBAM system which Lockheed bought last year from Sciaky Inc. costs \$4 million and is capable of "printing" out fuel tanks of nearly 150 cm in diameter. There is even better news: the method cuts down the cost of manufacturing propellant tanks by as much as 50%.

**Figure 13.** The manufacturing procedure of one half of a satellite fuel tank according to the Sciaky method (top left) and finished half-product (top right). Source: Sciaky Inc. Video capture.

The process is also much faster than casting those tanks in molds. With around 20 months being the lead time of the casting technique, and the time spent procuring the bulk, forged titanium billets also considered, EBAM additive manufacturing is considerably quicker. Material costs are also reduced with respect to structural titanium parts that are machined from a billet or forged. The EBAM spare also all the required involved machining time by as much as 80%. Dennis Little, Lockheed's vice president of production for space systems, stated that, those manufacturing advantages involved in the 3D EBAM-printed titanium tanks, will be in use on spacecraft before the decade is out, provided that internal evaluation of the process meets certification requirements from the US Air Force and NASA.

including GPS waypoints for navigation and mile-distance control. Even camera hoisting and phone linking capabilities that extend the distance it can be controlled are present. Being 3D printed, it has even greater advantage that it can be modified and reprinted by desire and need. It is fully customizable, as it can be made smaller or bigger, geared to carry a sensor

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Solid Concepts uses additive manufacturing to produce fixed-wing UAS airframes (such as the PTERA shown in **Figure 14**) that are used to test high-risk circulation control systems, conformal fuel tank concepts, and other advanced aerospace concepts with SLS 3D-printed parts.

Additive manufacturing—or 3D printing—is almost 30 years old. Today, it is not only just found in industry but also in households, as the price of high accuracy 3D printers has fallen below US\$1000. Understanding of this new power enabling us to design and print almost anything in three dimensions, not just scribes and symbols on paper, opens up unlimited opportunities for everyday people to manufacture from toys and household appliances, to

Well, guess again, there is even more that can be done with 3D-printed materials. We can make them more flexible and more useful. Smart structures can be also printed, featuring embedded sensors and actuators. These objects, also 3D-printed, can transform in a pre-programmed way in response to an external stimulus. Such a technology enabling objects, parts, or even complete systems to be manufactured was baptized by the popular science name of "4D printing." Perhaps, it leads us to a better or easier way to think about that the object transforms or

Of course, such a behavior of structural deformations is not at all quite new. Researchers have been for decades researching and demonstrating "memory" and "smart material" effects and properties. In their recent Nature article, Raviv et al. propose a new design of complex self-evolving structures that vary over time due to environmental interactions [28]. Logically, in conventional 3D printing systems, materials are expected to exhibit a stable response rather than an active one and fabricated objects are designed and printed as static items or functional parts at the most. In this paper, a novel approach is introduced for simulating and fabricating self-evolving structures. These "smart" structures transform

instead of a camera, or fly slower or faster as each different operation requires.

**4.6. 4D printing**

reacts over time.

jewels and tools, in our homes and work places.

**Figure 14.** 3D-printed military drone-PTERA of solid concepts.

## **4.5. 3D printed drones**

Drone fashion hobby, also used in serious applications, has dramatically increased in the past several years. Global unit sales grew 60% (to 2.2 million) and revenue increased 36% (to \$4.5 billion) in 2016 [26]. American consumers in the United States alone, bought 2.4 million hobbyist drones, compared to 1.1 million in the previous year. In 2017, revenue is anticipated to reach \$6 billion, while units should rise up to 3 million. The latter figures substantiate a revenue growth of 39% and unit growth 34% within 2 years time, according to a study conducted by the research firm Gartner Inc. [27].

Impressively, engineers, designers, and other individual end-users become more and more involved in current research and development efforts to manufacture 3D-printed drones are now eligible for R&D federal and state tax credits.

Military sectors are intensively exploring novel paths to make cheaper, lighter, and more energy-efficient drones. A Marine Corp named Rhet McNeal created Scout, a drone composed of 3D-printed components. Scout only costs \$600 to build. In comparison, a similar military-grade drone costs hundreds of thousands of dollars to build from advanced materials with conventional methods. Another benefit which arises from the technology, being a 3D-printed drone, should it receive any damage, any part or parts can be directly printed out and installed within hours. On the other side of the fence, a standard-issue drone would require weeks, sometimes months, to get a replacement through the Marine Corps' supply chain. Scout has been delivered to Mitre Corp., a USMC drone supplier, for certification testing. Prior to Scout, Mitre Corp. experimented with a 3D-printed drone called Nibbler. In the following period of time, the USMC is planning to test Nibbler into a real combat zone in order to supply troops with required resources. At the same time, they are investing resources in R&D on how to manufacture 3D-printed drones for surveillance purposes.

The University of Virginia, in its term, designed and created a 3D-printed UAV drone for the Department of Defense. The Drone can be printed in less than 24 h at an end-user price of \$2500, electronics development included. The fuselage of the drone costs only \$800. Resembling to one long wing, it was nicknamed the Razor. The Razor can fly at maximum speed of 40 mph for up to 45 min, having a gross weight at 6 pounds with all the equipment installed.

None of its features and capabilities is compromised by the fact that it is 3D-printed; in fact, it features all the same functions and operational capabilities as a typical military-grade drone, including GPS waypoints for navigation and mile-distance control. Even camera hoisting and phone linking capabilities that extend the distance it can be controlled are present. Being 3D printed, it has even greater advantage that it can be modified and reprinted by desire and need. It is fully customizable, as it can be made smaller or bigger, geared to carry a sensor instead of a camera, or fly slower or faster as each different operation requires.

Solid Concepts uses additive manufacturing to produce fixed-wing UAS airframes (such as the PTERA shown in **Figure 14**) that are used to test high-risk circulation control systems, conformal fuel tank concepts, and other advanced aerospace concepts with SLS 3D-printed parts.

## **4.6. 4D printing**

The process is also much faster than casting those tanks in molds. With around 20 months being the lead time of the casting technique, and the time spent procuring the bulk, forged titanium billets also considered, EBAM additive manufacturing is considerably quicker. Material costs are also reduced with respect to structural titanium parts that are machined from a billet or forged. The EBAM spare also all the required involved machining time by as much as 80%. Dennis Little, Lockheed's vice president of production for space systems, stated that, those manufacturing advantages involved in the 3D EBAM-printed titanium tanks, will be in use on spacecraft before the decade is out, provided that internal evaluation of the pro-

Drone fashion hobby, also used in serious applications, has dramatically increased in the past several years. Global unit sales grew 60% (to 2.2 million) and revenue increased 36% (to \$4.5 billion) in 2016 [26]. American consumers in the United States alone, bought 2.4 million hobbyist drones, compared to 1.1 million in the previous year. In 2017, revenue is anticipated to reach \$6 billion, while units should rise up to 3 million. The latter figures substantiate a revenue growth of 39% and unit growth 34% within 2 years time, according to a study conducted

Impressively, engineers, designers, and other individual end-users become more and more involved in current research and development efforts to manufacture 3D-printed drones are

Military sectors are intensively exploring novel paths to make cheaper, lighter, and more energy-efficient drones. A Marine Corp named Rhet McNeal created Scout, a drone composed of 3D-printed components. Scout only costs \$600 to build. In comparison, a similar military-grade drone costs hundreds of thousands of dollars to build from advanced materials with conventional methods. Another benefit which arises from the technology, being a 3D-printed drone, should it receive any damage, any part or parts can be directly printed out and installed within hours. On the other side of the fence, a standard-issue drone would require weeks, sometimes months, to get a replacement through the Marine Corps' supply chain. Scout has been delivered to Mitre Corp., a USMC drone supplier, for certification testing. Prior to Scout, Mitre Corp. experimented with a 3D-printed drone called Nibbler. In the following period of time, the USMC is planning to test Nibbler into a real combat zone in order to supply troops with required resources. At the same time, they are investing resources

The University of Virginia, in its term, designed and created a 3D-printed UAV drone for the Department of Defense. The Drone can be printed in less than 24 h at an end-user price of \$2500, electronics development included. The fuselage of the drone costs only \$800. Resembling to one long wing, it was nicknamed the Razor. The Razor can fly at maximum speed of 40 mph

None of its features and capabilities is compromised by the fact that it is 3D-printed; in fact, it features all the same functions and operational capabilities as a typical military-grade drone,

for up to 45 min, having a gross weight at 6 pounds with all the equipment installed.

in R&D on how to manufacture 3D-printed drones for surveillance purposes.

cess meets certification requirements from the US Air Force and NASA.

**4.5. 3D printed drones**

108 Lamination - Theory and Application

by the research firm Gartner Inc. [27].

now eligible for R&D federal and state tax credits.

Additive manufacturing—or 3D printing—is almost 30 years old. Today, it is not only just found in industry but also in households, as the price of high accuracy 3D printers has fallen below US\$1000. Understanding of this new power enabling us to design and print almost anything in three dimensions, not just scribes and symbols on paper, opens up unlimited opportunities for everyday people to manufacture from toys and household appliances, to jewels and tools, in our homes and work places.

Well, guess again, there is even more that can be done with 3D-printed materials. We can make them more flexible and more useful. Smart structures can be also printed, featuring embedded sensors and actuators. These objects, also 3D-printed, can transform in a pre-programmed way in response to an external stimulus. Such a technology enabling objects, parts, or even complete systems to be manufactured was baptized by the popular science name of "4D printing." Perhaps, it leads us to a better or easier way to think about that the object transforms or reacts over time.

Of course, such a behavior of structural deformations is not at all quite new. Researchers have been for decades researching and demonstrating "memory" and "smart material" effects and properties. In their recent Nature article, Raviv et al. propose a new design of complex self-evolving structures that vary over time due to environmental interactions [28]. Logically, in conventional 3D printing systems, materials are expected to exhibit a stable response rather than an active one and fabricated objects are designed and printed as static items or functional parts at the most. In this paper, a novel approach is introduced for simulating and fabricating self-evolving structures. These "smart" structures transform

**Figure 14.** 3D-printed military drone-PTERA of solid concepts.

into a predetermined shape, by changing properties and functions after fabrication has been completed. The new locally coordinated bending primitives combine into a single system, allowing for a global deformation which can stretch, fold, and bend in the given environmental stimulus. The physics of 4D printing often requires multiple materials to be embedded into a single 3D structure. Much work is anticipated to be invested on such 4D printers and structures over the next decades.

**References**

[1] Wohlers T, Gorne T. History of Additive Manufacturing. Wohlers Report 2014. Colorado,

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2016.01.207

## **5. Conclusions**

In our days, it is apparent that we experience a very large revolution as far as novel production techniques is concerned, in manufacturing structures by advanced layering techniques, processes widely known under the simplistic name of 3D printing, for the wide public. The truth is that this new kind of industrial revolution regarding manufacturing has not been completed yet. Although techniques for glass and ceramics similar to 3D printing are developed as these lines are written, no mature methods have been proposed yet. Moreover, upon completion of this revolution, which owes a lot to electronics, software (CAD), and computer technology, we will be able to manufacture almost all our products utilizing 3D printing or even 4D printing technologies for smart structures.

It is of course common sense that this will influence our way of thinking and designing items and parts or even whole devices at once, since results of simulation and design optimization are nowadays directly printable. Mankind will benefit, as already does, from applications ranging from medical equipment and implants, biomaterials, or biomedical devices, to possible automated orbital factories fully equipped with 3D printers for space exploration equipment. The limits for this new technology for manufacturing are far from being set as yet. Transport and automotive sectors are also supposed to profit from the impact of 3D printing technologies.

Since these techniques are saving energy and CO2 emissions, and their products or by-products are recyclable too; it is also a revolution of green manufacturing. Furthermore, as scientists are moving into MEMS and NEMS device manufacturing techniques, it is certainly implied that in some future era, mankind will possess technology to perform atom by atom structuring. This has already been shown. From atom scale to nano-, micro-, and macroscale, we are almost able to manipulate matter totally.

## **Author details**

Dionysios E. Mouzakis

Address all correspondence to: demouzakis@sse.gr

Department of Military Sciences, Hellenic Army Academy, Sector of Mathematics and Engineering Applications, Greece

## **References**

into a predetermined shape, by changing properties and functions after fabrication has been completed. The new locally coordinated bending primitives combine into a single system, allowing for a global deformation which can stretch, fold, and bend in the given environmental stimulus. The physics of 4D printing often requires multiple materials to be embedded into a single 3D structure. Much work is anticipated to be invested on such 4D printers

In our days, it is apparent that we experience a very large revolution as far as novel production techniques is concerned, in manufacturing structures by advanced layering techniques, processes widely known under the simplistic name of 3D printing, for the wide public. The truth is that this new kind of industrial revolution regarding manufacturing has not been completed yet. Although techniques for glass and ceramics similar to 3D printing are developed as these lines are written, no mature methods have been proposed yet. Moreover, upon completion of this revolution, which owes a lot to electronics, software (CAD), and computer technology, we will be able to manufacture almost all our products utilizing 3D printing or

It is of course common sense that this will influence our way of thinking and designing items and parts or even whole devices at once, since results of simulation and design optimization are nowadays directly printable. Mankind will benefit, as already does, from applications ranging from medical equipment and implants, biomaterials, or biomedical devices, to possible automated orbital factories fully equipped with 3D printers for space exploration equipment. The limits for this new technology for manufacturing are far from being set as yet. Transport and automotive sectors are also supposed to profit from the impact of 3D printing

ucts are recyclable too; it is also a revolution of green manufacturing. Furthermore, as scientists are moving into MEMS and NEMS device manufacturing techniques, it is certainly implied that in some future era, mankind will possess technology to perform atom by atom structuring. This has already been shown. From atom scale to nano-, micro-, and macroscale,

Department of Military Sciences, Hellenic Army Academy, Sector of Mathematics and

emissions, and their products or by-prod-

and structures over the next decades.

even 4D printing technologies for smart structures.

Since these techniques are saving energy and CO2

we are almost able to manipulate matter totally.

Address all correspondence to: demouzakis@sse.gr

**5. Conclusions**

110 Lamination - Theory and Application

technologies.

**Author details**

Dionysios E. Mouzakis

Engineering Applications, Greece


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112 Lamination - Theory and Application


## *Edited by Charles A. Osheku*

The field of lamination has developed significantly over the past 5000 years. Nowadays, we have a humongous array of structures and technological systems where composite laminates are applied. From the viewpoint of structural mechanics, an interface slip motion between two laminated structures, such as beam plate and plate in the presence of dry friction, can be utilized for slip damping systems. By scientific definition, slip damping is a mechanism exploited for dissipating noise and vibration energy in machine structures and systems. Researchers have developed several mathematical models for noise dissipation, minimization and complete vibration isolation laminated mechanisms. The purpose of this book is to describe new concepts of producing laminated structures and possible modern engineering applications.

Lamination - Theory and Application

Lamination

Theory and Application

*Edited by Charles A. Osheku*

Photo by getanov / iStock