Material's Models Developing

*Advances in Composite Materials Development*

Mustafizul Karim AN, Dhar NR, Begum S, editors. Proceedings of the International Conference on Manufacturing, ICM

of optimization of wear behaviour of Al-Al2O3 composites using Taguchi technique. Procedia Engineering.

[56] Culliton D, Betts AJ, Kennedy D. Impact of intermetallic precipitates on the tribological and/or corrosion performance of cast aluminium alloys: A short review. International Journal of Cast Metals Research. 2013;**26**(2):65-71

2013;**64**:973-982

Microstructure and some properties of aluminium alloy Al6061 reinforced in situ formed zirconium diboride particulate stir cast composite. International Journal of Cast Metals Research.

[49] Murat Lus H, Ozer G, Altug Guler K, Erzi E, Dispinar D. Wear properties of squeeze cast in situ Mg2Si–A380 alloy. International Journal of Cast Metals

Dhaka. 2002. pp. 21-28

2014;**27**(2):115-121

[48] Dinaharan I, Murugan N.

Research. 2015;**28**(1):59-64

2012;**25**(4):246-250

[50] Unal N, Camurlu HE, Koçak S, Düztepe G. Effect of external ultrasonic

treatment on hypereutectic cast aluminium–silicon alloy. International Journal of Cast Metals Research.

[51] Yamanoglu R, Karakulak E, Zeren M, Koç FG. Effect of nickel on microstructure and wear behaviour of pure aluminium against steel and alumina counterfaces. International Journal of Cast Metals Research. 2013;**26**(5):289-295

[52] Hariprasad T, Varatharajan K, Ravi S. Wear characteristics of B4C and Al2O3 reinforced with Al 5083 metal matrix based hybrid composite. Procedia Engineering. 2014;**97**:925-929

[53] Iacob G, Ghica VG, Buzatu M. Studies on wear rate and micro-hardness of the Al/Al2O3/Gr hybrid composites produced via powder metallurgy. Composites Part B: Engineering. 2014

Chandrasekaran K. Dry sliding wear behavior of powder metallurgy

aluminium matrix composite. Materials Today: Proceedings. 2015;**2**:1441-1449

[55] Baradeswaran A, Elayaperumal A, Franklin Issac R. A statistical analysis

[54] Ganesh R, Subbiah R,

**60**

Chapter 4

Abstract

An Alternative Framework

Ladislav Écsi, Pavel Élesztős, Róbert Jerábek,

Roland Jančo and Branislav Hučko

deformations of elastoplastic media.

1. Introduction

63

plastic-based material models with internal damping

for Developing Material Models

for Finite-Strain Elastoplasticity

Contemporary plasticity theories and their related material models for finite deformations are either based on additive decomposition of a strain-rate tensor or on multiplicative decomposition of a deformation gradient tensor into an elastic part and a plastic part. From the standpoint of the nonlinear continuum mechanics, the former theories, which are used to model hypoelastic-plastic materials, are rather incomplete theories, while the latter theories, which are used to model hyperelastic-plastic materials, are not even continuum-based theories, while none of their related material models are thermodynamically consistent. Recently, a nonlinear continuum theory for finite deformations of elastoplastic media was proposed, which allows for the development of objective and thermodynamically consistent material models. Therefore, the analysis results of the models are

independent of the description and the particularities of their mathematical formulation. Here by the description we mean total or updated Lagrangian description and by the particularities of formulation, the ability to describe the model in various stress spaces using internal mechanical power conjugate stress measures and strain rates. In this chapter, an alternative framework for developing objective and thermodynamically consistent hypoelastic-plastic- and hyperelastic-plastic-based material models is presented using the first nonlinear continuum theory of finite

Keywords: nonlinear continuum theory for finite deformations of elastoplastic media, objective and thermodynamically consistent formulation, J2 generalised plasticity with isotropic hardening, hypoelastic-plastic- and hyperelastic-

There are two types of phenomenological flow plasticity theories and their related material models used at present to model plastic behaviour of deformable bodies within the framework of finite-strain elastoplasticity. The first type of theories are considered to be ad hoc extensions of small-strain flow plasticity theories into the area of finite deformations to describe materials, in which small elastic deformations are accompanied by finite inelastic deformations during the deformation process. They are based on additive decomposition of a strain-rate

## Chapter 4

## An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity

Ladislav Écsi, Pavel Élesztős, Róbert Jerábek, Roland Jančo and Branislav Hučko

## Abstract

Contemporary plasticity theories and their related material models for finite deformations are either based on additive decomposition of a strain-rate tensor or on multiplicative decomposition of a deformation gradient tensor into an elastic part and a plastic part. From the standpoint of the nonlinear continuum mechanics, the former theories, which are used to model hypoelastic-plastic materials, are rather incomplete theories, while the latter theories, which are used to model hyperelastic-plastic materials, are not even continuum-based theories, while none of their related material models are thermodynamically consistent. Recently, a nonlinear continuum theory for finite deformations of elastoplastic media was proposed, which allows for the development of objective and thermodynamically consistent material models. Therefore, the analysis results of the models are independent of the description and the particularities of their mathematical formulation. Here by the description we mean total or updated Lagrangian description and by the particularities of formulation, the ability to describe the model in various stress spaces using internal mechanical power conjugate stress measures and strain rates. In this chapter, an alternative framework for developing objective and thermodynamically consistent hypoelastic-plastic- and hyperelastic-plastic-based material models is presented using the first nonlinear continuum theory of finite deformations of elastoplastic media.

Keywords: nonlinear continuum theory for finite deformations of elastoplastic media, objective and thermodynamically consistent formulation, J2 generalised plasticity with isotropic hardening, hypoelastic-plastic- and hyperelasticplastic-based material models with internal damping

## 1. Introduction

There are two types of phenomenological flow plasticity theories and their related material models used at present to model plastic behaviour of deformable bodies within the framework of finite-strain elastoplasticity. The first type of theories are considered to be ad hoc extensions of small-strain flow plasticity theories into the area of finite deformations to describe materials, in which small elastic deformations are accompanied by finite inelastic deformations during the deformation process. They are based on additive decomposition of a strain-rate

tensor into an elastic part and a plastic part to describe the plastic flow in the material. As an example of such material is ductile metal, which at present is modelled mainly by a kind of a hypoelastic-plastic-based material model, whose constitutive equation does not have a form in terms of a finite-strain measure. Without a need for completeness, let us just mention a few comprehensive studies in technical literature, such as [1–4], where detailed descriptions of the most frequently used contemporary hypoelastic-plastic-based material models are presented.

additive decomposition-based theories are in reality finite-strain theories, but they are constrained when the plastic flow in them is defined in terms of a Cauchy's stress tensor-based yield surface in the current configuration of the body, while contemporary deformation gradient multiplicative split-based theories are not even continuum-based theories. Moreover, none of their related material models is thermodynamically consistent. In addition to this, we will show that all flow plasticity theories are just variants of the nonlinear continuum theory for finite deformations of elastoplastic media presented in this chapter, using the additive decomposition of a Lagrangian displacement field into an elastic part and a plastic part. Eventually, we will demonstrate the theory in numerical experiments using simple hypoelasticplastic- and hyperelastic-plastic-based material models with internal damping and

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity

The Lagrangian description is used to describe the kinematics of motion and constitutive and evolution equations of the material of a deformable body. Though a single form of the constitutive equation of a material is sufficient to describe the material, all forms of the constitutive equation of the material are needed in order to

2.1 A short overview of the nonlinear continuum mechanical theory for finite

The nonlinear continuum theory for finite deformations of elastic media has been developed in an elegant manner in the past decades [15–19]. The theory is particularly suitable for modelling elastic materials, whose constitutive equations are defined either in terms of a finite-strain tensor, as in the case of the St-Venant-Kirchhoff material, or derived from an appropriate strain energy density function, as in the case of the hyperelastic materials [17, 18]. Developing material models for finite-strain elastoplasticity within the framework of thermodynamics with internal variables of state, however, requires a somewhat different approach [20]. The constitutive and evolution equations of these materials either exist in rate forms only, or contain rate equations, which at some point during the solution process have to be integrated. Moreover, in nonlinear continuum mechanics, no kinematics of motion can be described without the mathematical definitions of the motion, the Lagrangian and Eulerian displacement fields and the deformation

Starting with the definitions (see Figure 1), the motion x ¼ Φð Þ X; t from the mathematical pint of view is a vector function or vector field, which maps each material point <sup>0</sup>P ∈<sup>0</sup>Ω with a position vector X in the initial configuration of the

of the body. The function must exist whenever the body moves, and it determines

In Eq. (1) <sup>0</sup>Ω is the domain of the function, which stands for the volume of the

volume of the body in its current configuration; and t is time. The vector field that connects the position vectors of the material particle is the displacement field. It

the position vector of a material particle at each time instant t≥0 [16]:

<sup>x</sup>j<sup>x</sup> <sup>¼</sup> <sup>Φ</sup>ð Þ <sup>X</sup>; <sup>t</sup> ;for <sup>X</sup> <sup>∈</sup><sup>0</sup>Ω; <sup>x</sup>∈<sup>t</sup>

Ω with a position vector x in the current configuration

Ω is the range of the function, which stands for the

Ω and t≥0 : (1)

prove that its formulation is thermodynamically consistent.

then briefly discuss their analysis results.

DOI: http://dx.doi.org/10.5772/intechopen.85112

deformations of elastoplastic media

2. Theory

gradient, respectively.

body, into a spatial pint P∈<sup>t</sup>

body in its initial configuration; <sup>t</sup>

65

The second type of flow plasticity theories, in which multiplicative decomposition of a deformation gradient tensor into an elastic part and a plastic part is used to describe the plastic flow in the material, is based on the theory of single-crystal plasticity [5–7]. The theories and their related material models, which are now considered as 'proper material models' to model plastic behaviour of the deformable body, assume that the intermediate configuration of the body is stress-free [1] or at least locally unstressed [4]. As a result, there cannot exist a deformation or strain tensor field that meets the conditions of compatibility [4]. Therefore these theories treat the kinematics of motion differently between the initial and current or intermediate configurations of the body. This means that the motion, the displacement field and the deformation gradient, all of which have an exact physical meaning in continuum mechanics, are considered in accordance with the continuum theory between the initial and current configurations of the body, but not between the configurations where one is an intermediate configuration. Here the motion and the displacement fields are disregarded, and as a result, the deformation gradient loses its physical meaning. Moreover, it should also be noted that the assumption of an unstressed intermediate configuration is not compatible with the theory of nonlinear continuum mechanics, as it violates proper stress transformations, resulting from the invariance of the internal mechanical power, when switching from one stress space to the other in any configuration of the body. As a result, contemporary multiplicative plasticity theories and their related material models in realty are not continuum-based.

Hypoelastic-plastic- and hyperelastic-plastic-based material models have been the subject of study over recent decades, and there are a few issues to be concerned about when the models are used in numerical analyses. These include energy accumulation and residual stresses along a closed elastic strain path in the case of hypoelastic-plastic-based material models in which the Jaumann rate is used to calculate the Cauchy stress tensor [8], residual stress accumulation up to unacceptable values during multiple loading cycles along a closed elastic strain path using the Jaumann rate and a few other rates [9] or shear oscillation in finite shearing problems [10, 11]. The aforementioned problems however can be eliminated by replacing the Jaumann rate with the Green-Naghdi or Truesdell rate in the formulation of the models [4]. Simo and Pistner showed that employing a constant spatial elasticity tensor in objective stress integration is not compatible with elasticity and that such models in fact fail to define the elastic material [12]. Equivalent rate descriptions of hyperelastic-based models in terms of different strain measures have been thoroughly discussed by Perić, who also showed that the Jaumann rate- and the Green-Naghdi rate-based models provide different levels of approximation to problems governed by the logarithmic strain-based Hencky hyperelastic law [4, 13]. We will show herein that all of the above problems actually resulted from the fact that the related material models use thermodynamically inconsistent formulation.

The aim of this chapter is to present an alternative framework for developing objective and thermodynamically consistent hypoelastic-plastic- and hyperelasticplastic-based material models using the first nonlinear continuum theory for finite deformations of elastoplastic media [14]. We will show that the strain-rate tensor

### An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity DOI: http://dx.doi.org/10.5772/intechopen.85112

additive decomposition-based theories are in reality finite-strain theories, but they are constrained when the plastic flow in them is defined in terms of a Cauchy's stress tensor-based yield surface in the current configuration of the body, while contemporary deformation gradient multiplicative split-based theories are not even continuum-based theories. Moreover, none of their related material models is thermodynamically consistent. In addition to this, we will show that all flow plasticity theories are just variants of the nonlinear continuum theory for finite deformations of elastoplastic media presented in this chapter, using the additive decomposition of a Lagrangian displacement field into an elastic part and a plastic part. Eventually, we will demonstrate the theory in numerical experiments using simple hypoelasticplastic- and hyperelastic-plastic-based material models with internal damping and then briefly discuss their analysis results.

### 2. Theory

tensor into an elastic part and a plastic part to describe the plastic flow in the material. As an example of such material is ductile metal, which at present is modelled mainly by a kind of a hypoelastic-plastic-based material model, whose constitutive equation does not have a form in terms of a finite-strain measure. Without a need for completeness, let us just mention a few comprehensive studies in technical literature, such as [1–4], where detailed descriptions of the most frequently used contemporary hypoelastic-plastic-based material models are

Advances in Composite Materials Development

The second type of flow plasticity theories, in which multiplicative decomposition of a deformation gradient tensor into an elastic part and a plastic part is used to describe the plastic flow in the material, is based on the theory of single-crystal plasticity [5–7]. The theories and their related material models, which are now considered as 'proper material models' to model plastic behaviour of the deformable body, assume that the intermediate configuration of the body is stress-free [1] or at least locally unstressed [4]. As a result, there cannot exist a deformation or strain tensor field that meets the conditions of compatibility [4]. Therefore these theories treat the kinematics of motion differently between the initial and current or intermediate configurations of the body. This means that the motion, the displacement field and the deformation gradient, all of which have an exact physical meaning in continuum mechanics, are considered in accordance with the continuum theory between the initial and current configurations of the body, but not between the configurations where one is an intermediate configuration. Here the motion and the displacement fields are disregarded, and as a result, the deformation gradient loses its physical meaning. Moreover, it should also be noted that the assumption of an unstressed intermediate configuration is not compatible with the theory of nonlinear continuum mechanics, as it violates proper stress transformations, resulting from the invariance of the internal mechanical power, when switching from one stress space to the other in any configuration of the body. As a result, contemporary multiplicative plasticity theories and their related material models in

Hypoelastic-plastic- and hyperelastic-plastic-based material models have been the subject of study over recent decades, and there are a few issues to be concerned about when the models are used in numerical analyses. These include energy accumulation and residual stresses along a closed elastic strain path in the case of hypoelastic-plastic-based material models in which the Jaumann rate is used to calculate the Cauchy stress tensor [8], residual stress accumulation up to unacceptable values during multiple loading cycles along a closed elastic strain path using the Jaumann rate and a few other rates [9] or shear oscillation in finite shearing problems [10, 11]. The aforementioned problems however can be eliminated by replacing the Jaumann rate with the Green-Naghdi or Truesdell rate in the formulation of the models [4]. Simo and Pistner showed that employing a constant spatial elasticity tensor in objective stress integration is not compatible with elasticity and that such models in fact fail to define the elastic material [12]. Equivalent rate descriptions of hyperelastic-based models in terms of different strain measures have been thoroughly discussed by Perić, who also showed that the Jaumann rate- and the Green-Naghdi rate-based models provide different levels of approximation to problems governed by the logarithmic strain-based Hencky hyperelastic law [4, 13]. We will show herein that all of the above problems actually resulted from the fact that the related material models use thermodynamically inconsistent formulation. The aim of this chapter is to present an alternative framework for developing objective and thermodynamically consistent hypoelastic-plastic- and hyperelasticplastic-based material models using the first nonlinear continuum theory for finite deformations of elastoplastic media [14]. We will show that the strain-rate tensor

presented.

realty are not continuum-based.

64

The Lagrangian description is used to describe the kinematics of motion and constitutive and evolution equations of the material of a deformable body. Though a single form of the constitutive equation of a material is sufficient to describe the material, all forms of the constitutive equation of the material are needed in order to prove that its formulation is thermodynamically consistent.

### 2.1 A short overview of the nonlinear continuum mechanical theory for finite deformations of elastoplastic media

The nonlinear continuum theory for finite deformations of elastic media has been developed in an elegant manner in the past decades [15–19]. The theory is particularly suitable for modelling elastic materials, whose constitutive equations are defined either in terms of a finite-strain tensor, as in the case of the St-Venant-Kirchhoff material, or derived from an appropriate strain energy density function, as in the case of the hyperelastic materials [17, 18]. Developing material models for finite-strain elastoplasticity within the framework of thermodynamics with internal variables of state, however, requires a somewhat different approach [20]. The constitutive and evolution equations of these materials either exist in rate forms only, or contain rate equations, which at some point during the solution process have to be integrated. Moreover, in nonlinear continuum mechanics, no kinematics of motion can be described without the mathematical definitions of the motion, the Lagrangian and Eulerian displacement fields and the deformation gradient, respectively.

Starting with the definitions (see Figure 1), the motion x ¼ Φð Þ X; t from the mathematical pint of view is a vector function or vector field, which maps each material point <sup>0</sup>P ∈<sup>0</sup>Ω with a position vector X in the initial configuration of the body, into a spatial pint P∈<sup>t</sup> Ω with a position vector x in the current configuration of the body. The function must exist whenever the body moves, and it determines the position vector of a material particle at each time instant t≥0 [16]:

$$\{\mathbf{x}|\mathbf{x} = \boldsymbol{\Phi}(\mathbf{X}, t), \text{for } \mathbf{X} \in {}^{0}\boldsymbol{\Omega}, \mathbf{x} \in {}^{t}\boldsymbol{\Omega} \text{ and } t \ge 0\}. \tag{1}$$

In Eq. (1) <sup>0</sup>Ω is the domain of the function, which stands for the volume of the body in its initial configuration; <sup>t</sup> Ω is the range of the function, which stands for the volume of the body in its current configuration; and t is time. The vector field that connects the position vectors of the material particle is the displacement field. It

represent spatial vector fields, because the intermediate configuration of the body

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity

<sup>X</sup>; <sup>t</sup> � � � <sup>i</sup>

Moreover, because the plastic motion exists, the vector fields have Lagrangian

Lagrangian elastic displacement field <sup>0</sup>uel <sup>¼</sup> <sup>0</sup>uelð Þ <sup>X</sup>; <sup>t</sup> , defined over the initial vol-

h i; for <sup>X</sup> <sup>∈</sup><sup>0</sup>Ω; <sup>x</sup>∈<sup>t</sup>

pl�elΦplð Þ <sup>X</sup>; <sup>t</sup> ; <sup>t</sup> h i � pl�el<sup>Φ</sup>

Eqs. (1) and (8) then imply the following composite function for the overall

<sup>x</sup> <sup>¼</sup> <sup>Φ</sup>ð Þ¼ <sup>X</sup>; <sup>t</sup> pl�elΦel pl�elΦplð Þ <sup>X</sup>; <sup>t</sup> ; <sup>t</sup>

Moreover, after adding Eqs. (5) and (9) up, the following formula for the overall

<sup>0</sup>uelð Þþ <sup>X</sup>; <sup>t</sup> <sup>0</sup>uplð Þ¼ <sup>X</sup>; <sup>t</sup> <sup>Φ</sup>el <sup>Φ</sup>plð Þ <sup>X</sup>; <sup>t</sup> ; <sup>t</sup> � � � <sup>X</sup> <sup>¼</sup> <sup>x</sup> � <sup>X</sup> <sup>¼</sup> <sup>0</sup>u Xð Þ ; <sup>t</sup> : (11)

Eq. (11) states that the Lagrangian displacement field can additively be decomposed into a Lagrangian elastic part and a Lagrangian plastic part when the kinematics of motion is considered in accordance with the theory of nonlinear

> ∂<sup>0</sup>u <sup>∂</sup><sup>X</sup> <sup>¼</sup> <sup>I</sup> <sup>þ</sup>

It should be noted that Eq. (12) is the simplest form of the deformation gradient, irrespective of whether the additive decomposition of the displacement field in the above or the multiplicative decomposition of the deformation gradient tensor is used as a starting point in its formulation. In the latter case the formulation would

<sup>∂</sup><sup>X</sup> <sup>¼</sup> pl�elFelð Þ� <sup>X</sup>; <sup>t</sup> pl�elFplð Þ¼ <sup>X</sup>; <sup>t</sup> <sup>I</sup> <sup>þ</sup>

∂<sup>0</sup>u el <sup>∂</sup><sup>X</sup> � pl�el<sup>F</sup>

<sup>X</sup> <sup>¼</sup> <sup>I</sup> <sup>þ</sup>

∂<sup>0</sup>upl ∂X þ ∂<sup>0</sup>u el

> ∂<sup>0</sup>u el ∂X þ

plð Þ X; t h i�<sup>1</sup>

<sup>∂</sup><sup>X</sup> : (12)

∂<sup>0</sup>upl

<sup>∂</sup><sup>X</sup> , (13)

, (14)

continuum mechanics. The deformation gradient then takes the form

F ¼ F Xð Þ¼ ; t I þ

n o, (8)

X ∈<sup>i</sup>

n o, (6)

Ω; x∈<sup>t</sup>

Ω; t≥ 0

X ∈<sup>i</sup>

ð Þ <sup>X</sup>; <sup>t</sup> ,for <sup>t</sup>≥0 and <sup>X</sup> <sup>∈</sup><sup>0</sup>Ω:

(9)

h i and the

Ω; t≥0

h i: (10)

Ω: (7)

X, for t ≥0 and <sup>i</sup>

pl

X; t � �; for <sup>i</sup>

<sup>X</sup> <sup>¼</sup> pl�elΦel <sup>i</sup>

forms too. Then the Lagrangian elastic motion pl�elΦel pl�elΦplð Þ <sup>X</sup>; <sup>t</sup> ; <sup>t</sup>

el

changes during the deformation process.

DOI: http://dx.doi.org/10.5772/intechopen.85112

<sup>u</sup>el iX; <sup>t</sup> � � <sup>¼</sup> <sup>x</sup> � <sup>i</sup>

ð Þ¼ <sup>X</sup>; <sup>t</sup> <sup>x</sup> � <sup>i</sup>

i <sup>u</sup>el <sup>¼</sup> <sup>i</sup>

0u el <sup>¼</sup> <sup>0</sup><sup>u</sup> el

motion:

modify as follows:

∂x <sup>∂</sup><sup>X</sup> <sup>¼</sup> <sup>∂</sup><sup>x</sup> ∂i X � ∂i X

pl�elFelð Þ¼ <sup>X</sup>; <sup>t</sup>

∂x ∂i

<sup>X</sup> <sup>¼</sup> <sup>I</sup> <sup>þ</sup>

∂i u el

∂i

F Xð Þ¼ ; t

where

67

<sup>x</sup>j<sup>x</sup> <sup>¼</sup> pl�elΦel <sup>i</sup>

ume of the body, can be expressed as follows:

<sup>x</sup>j<sup>x</sup> <sup>¼</sup> pl�elΦel pl�elΦplð Þ <sup>X</sup>; <sup>t</sup> ; <sup>t</sup>

Lagrangian displacement field can be arrived at

<sup>X</sup> <sup>¼</sup> pl�el<sup>Φ</sup>

Figure 1. The proper kinematics of motion of elastoplastic media.

must have both, Lagrangian <sup>0</sup><sup>u</sup> <sup>¼</sup> <sup>0</sup>u Xð Þ ; <sup>t</sup> and Eulerian <sup>u</sup> <sup>¼</sup> u xð Þ ; <sup>t</sup> forms; otherwise a physical phenomenon expressed in Lagrangian form cannot be re-expressed in Eulerian form and vice versa. The Lagrangian and Eulerian displacement fields are then defined as [16].

$$\mathbf{^0u} = \mathbf{^0u}(\mathbf{X}, t) = \mathbf{x} - \mathbf{X} = \boldsymbol{\Phi}(\mathbf{X}, t) - \mathbf{X}, \text{ for } \ t \ge 0 \quad \text{and} \quad \mathbf{X} \in \prescript{0}{}{\Omega},\tag{2}$$

$$\mathbf{u} = \mathbf{u}(\mathbf{x}, t) = \mathbf{x} - \mathbf{X} = \mathbf{x} - \boldsymbol{\Phi}^{-1}(\mathbf{x}, t), \text{ for } \ t \ge 0 \quad \text{and} \quad \mathbf{x} \in \boldsymbol{\Omega}. \tag{3}$$

The deformation gradient F then becomes the derivative of the position vector of the material point after motion with respect to the position vector of the material point before motion <sup>F</sup> <sup>¼</sup> <sup>∂</sup>x=∂<sup>X</sup> or simply the gradient of the vector function describing the motion.

When the motion is decomposed into several parts, so that the body moves from its initial configuration into its current configuration through several intermediate configurations, the above definitions apply between any two configurations of the body. Let us now consider a deformation process during which the body first undergoes plastic deformations and then elastic deformations at its each constituent. Then the Lagrangian plastic motion pl�elΦpl <sup>¼</sup> pl�elΦplð Þ <sup>X</sup>; <sup>t</sup> and the Lagrangian plastic displacement field <sup>0</sup>upl <sup>¼</sup> <sup>0</sup>uplð Þ <sup>X</sup>; <sup>t</sup> , defined over the initial volume of the body, take the following forms:

$$\left\{ ^i \mathbf{X} \middle| ^i \mathbf{X} = {}^{pl-el} \boldsymbol{\Phi}^{pl} (\mathbf{X}, t), \text{for} \ \mathbf{X} \in {}^0 \boldsymbol{\Omega}, {}^i \mathbf{X} \in {}^i \boldsymbol{\Omega} \text{ and } t \ge 0 \right\},\tag{4}$$

$$\mathbf{u}^{0}\mathbf{u}^{pl} = {}^{0}\mathbf{u}^{pl}(\mathbf{X},t) = {}^{i}\mathbf{X} - \mathbf{X} = {}^{pl-el}\boldsymbol{\Phi}^{pl}(\mathbf{X},t) - \mathbf{X}, \text{for } t \ge 0, \text{ and } \mathbf{X} \in {}^{0}\boldsymbol{\Omega}.\tag{5}$$

In Eqs. (4) and (5) <sup>i</sup> X stands for the position vector of the spatial point <sup>i</sup> P ∈<sup>i</sup> Ω, at which the material particle is located when the body has undergone plastic deformations only, and the left superscript pl�elð Þ• denotes the order of elastic and plastic deformations. Although the Eulerian elastic motion pl�elΦel <sup>¼</sup> pl�elΦel <sup>i</sup> X; t � � and the Eulerian elastic displacement field <sup>i</sup> <sup>u</sup>el <sup>¼</sup> <sup>i</sup> uel <sup>i</sup> X; t � � defined over the intermediate volume of the body <sup>i</sup> Ω have similar forms to the fields above, these

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity DOI: http://dx.doi.org/10.5772/intechopen.85112

represent spatial vector fields, because the intermediate configuration of the body changes during the deformation process.

$$\left\{ \mathbf{x} | \mathbf{x} = {}^{pl \ -el} \boldsymbol{\Phi}^{el} ({}^i \mathbf{X}, t), \text{ for } {}^i \mathbf{X} \in {}^i \boldsymbol{\Omega}, \ \mathbf{x} \in {}^t \boldsymbol{\Omega}, \ t \ge 0 \right\},\tag{6}$$

$$\mathbf{u}^i \mathbf{u}^d = {}^i \mathbf{u}^d(^i \mathbf{X}, t) = \mathbf{x} - {}^i \mathbf{X} = {}^{pl-d} \boldsymbol{\Phi}^d(^i \mathbf{X}, t) - {}^i \mathbf{X}, \text{ for } \ t \ge 0 \quad \text{and} \ {}^i \mathbf{X} \in {}^i \boldsymbol{\Omega}. \tag{7}$$

Moreover, because the plastic motion exists, the vector fields have Lagrangian forms too. Then the Lagrangian elastic motion pl�elΦel pl�elΦplð Þ <sup>X</sup>; <sup>t</sup> ; <sup>t</sup> h i and the Lagrangian elastic displacement field <sup>0</sup>uel <sup>¼</sup> <sup>0</sup>uelð Þ <sup>X</sup>; <sup>t</sup> , defined over the initial volume of the body, can be expressed as follows:

$$\left\{ \mathbf{x} \middle| \mathbf{x} = {}^{pl\ -el} \boldsymbol{\Phi}^d \left[ {}^{pl\ -el} \boldsymbol{\Phi}^{pl} (\mathbf{X}, t), t \right], \text{ for } \mathbf{X} \in {}^0 \boldsymbol{\Omega}, \ \mathbf{x} \in {}^t \boldsymbol{\Omega}, \ t \ge 0 \right\}, \tag{8}$$

$$\mathbf{u}^{0}\mathbf{u}^{cl} = \mathbf{^0u}^{cl}(\mathbf{X},t) = \mathbf{x} - \,^i\mathbf{X} = \,^{pl-cl}\boldsymbol{\Phi}\left[\prescript{pl-cl}{}{\mathbf{d}}\boldsymbol{\Phi}^{pl}(\mathbf{X},t),t\right] - \,^{pl-cl}\boldsymbol{\Phi}\left(\mathbf{X},t\right), \text{for } t \ge 0 \text{ and } \mathbf{X} \in \prescript{pl}{}{\boldsymbol{\Omega}}\boldsymbol{\Omega}.\tag{9}$$

Eqs. (1) and (8) then imply the following composite function for the overall motion:

$$\mathbf{x} = \boldsymbol{\Phi}(\mathbf{X}, t) = {}^{pl\\_el} \boldsymbol{\Phi}^{el} \left[ {}^{pl\\_el} \boldsymbol{\Phi}^{pl}(\mathbf{X}, t), t \right]. \tag{10}$$

Moreover, after adding Eqs. (5) and (9) up, the following formula for the overall Lagrangian displacement field can be arrived at

$$\mathbf{u}^{0}\mathbf{u}^{cl}(\mathbf{X},t) + {}^{0}\mathbf{u}^{pl}(\mathbf{X},t) = \boldsymbol{\Phi}^{cl}\left[\boldsymbol{\Phi}^{pl}(\mathbf{X},t), t\right] - \mathbf{X} = \mathbf{x} - \mathbf{X} = {}^{0}\mathbf{u}(\mathbf{X},t). \tag{11}$$

Eq. (11) states that the Lagrangian displacement field can additively be decomposed into a Lagrangian elastic part and a Lagrangian plastic part when the kinematics of motion is considered in accordance with the theory of nonlinear continuum mechanics. The deformation gradient then takes the form

$$\mathbf{F} = \mathbf{F}(\mathbf{X}, t) = \mathbf{I} + \frac{\partial^0 \mathbf{u}}{\partial \mathbf{X}} = \mathbf{I} + \frac{\partial^0 \mathbf{u}^{pl}}{\partial \mathbf{X}} + \frac{\partial^0 \mathbf{u}^{el}}{\partial \mathbf{X}}.\tag{12}$$

It should be noted that Eq. (12) is the simplest form of the deformation gradient, irrespective of whether the additive decomposition of the displacement field in the above or the multiplicative decomposition of the deformation gradient tensor is used as a starting point in its formulation. In the latter case the formulation would modify as follows:

$$\mathbf{F}(\mathbf{X},t) = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} \cdot \frac{\partial^{l} \mathbf{X}}{\partial \mathbf{X}} = {}^{pl-d} \mathbf{F}^{cl}(\mathbf{X},t) \cdot {}^{pl-d} \mathbf{F}^{pl}(\mathbf{X},t) = \mathbf{I} + \frac{\partial^{0} \mathbf{u}^{d}}{\partial \mathbf{X}} + \frac{\partial^{0} \mathbf{u}^{pl}}{\partial \mathbf{X}}, \tag{13}$$

where

$$\mathbf{F}^{pl-cl}\mathbf{F}^{cl}(\mathbf{X},t) = \frac{\partial \mathbf{x}}{\partial^{\dagger}\mathbf{X}} = \mathbf{I} + \frac{\partial^{\prime}\mathbf{u}^{cl}}{\partial^{\prime}\mathbf{X}} = \mathbf{I} + \frac{\partial^{0}\mathbf{u}^{cl}}{\partial\mathbf{X}} \cdot \left[\mathbf{f}^{pl-cl}\mathbf{F}^{pl}(\mathbf{X},t)\right]^{-1},\tag{14}$$

must have both, Lagrangian <sup>0</sup><sup>u</sup> <sup>¼</sup> <sup>0</sup>u Xð Þ ; <sup>t</sup> and Eulerian <sup>u</sup> <sup>¼</sup> u xð Þ ; <sup>t</sup> forms; otherwise a physical phenomenon expressed in Lagrangian form cannot be re-expressed in Eulerian form and vice versa. The Lagrangian and Eulerian displacement fields

<sup>0</sup><sup>u</sup> <sup>¼</sup> <sup>0</sup>u Xð Þ¼ ; <sup>t</sup> <sup>x</sup> � <sup>X</sup> <sup>¼</sup> <sup>Φ</sup>ð Þ� <sup>X</sup>; <sup>t</sup> <sup>X</sup>, for <sup>t</sup>≥0 and <sup>X</sup> <sup>∈</sup><sup>0</sup>Ω, (2)

The deformation gradient F then becomes the derivative of the position vector of the material point after motion with respect to the position vector of the material point before motion <sup>F</sup> <sup>¼</sup> <sup>∂</sup>x=∂<sup>X</sup> or simply the gradient of the vector function

When the motion is decomposed into several parts, so that the body moves from its initial configuration into its current configuration through several intermediate configurations, the above definitions apply between any two configurations of the body. Let us now consider a deformation process during which the body first undergoes plastic deformations and then elastic deformations at its each constituent. Then the Lagrangian plastic motion pl�elΦpl <sup>¼</sup> pl�elΦplð Þ <sup>X</sup>; <sup>t</sup> and the Lagrangian plastic displacement field <sup>0</sup>upl <sup>¼</sup> <sup>0</sup>uplð Þ <sup>X</sup>; <sup>t</sup> , defined over the initial volume of the

ð Þ <sup>X</sup>; <sup>t</sup> ;for <sup>X</sup> <sup>∈</sup><sup>0</sup>Ω; <sup>i</sup>

at which the material particle is located when the body has undergone plastic deformations only, and the left superscript pl�elð Þ• denotes the order of elastic and plastic deformations. Although the Eulerian elastic motion pl�elΦel <sup>¼</sup> pl�elΦel <sup>i</sup>

n o

ð Þ <sup>x</sup>; <sup>t</sup> , for <sup>t</sup>≥0 and <sup>x</sup>∈<sup>t</sup>

X ∈<sup>i</sup>

X stands for the position vector of the spatial point <sup>i</sup>

uel <sup>i</sup>

Ω have similar forms to the fields above, these

<sup>u</sup>el <sup>¼</sup> <sup>i</sup>

Ω and t≥ 0

ð Þ� <sup>X</sup>; <sup>t</sup> <sup>X</sup>,for <sup>t</sup>≥0, and <sup>X</sup> <sup>∈</sup> <sup>0</sup>Ω: (5)

X; t � � defined over the inter-

Ω: (3)

, (4)

P ∈<sup>i</sup> Ω,

X; t � �

are then defined as [16].

Figure 1.

describing the motion.

body, take the following forms:

i X � � i

<sup>0</sup>upl <sup>¼</sup> <sup>0</sup>uplð Þ¼ <sup>X</sup>; <sup>t</sup> <sup>i</sup>

In Eqs. (4) and (5) <sup>i</sup>

mediate volume of the body <sup>i</sup>

66

<sup>u</sup> <sup>¼</sup> u xð Þ¼ ; <sup>t</sup> <sup>x</sup> � <sup>X</sup> <sup>¼</sup> <sup>x</sup> � <sup>Φ</sup>�<sup>1</sup>

The proper kinematics of motion of elastoplastic media.

Advances in Composite Materials Development

<sup>X</sup> <sup>¼</sup> pl�elΦpl

and the Eulerian elastic displacement field <sup>i</sup>

<sup>X</sup> � <sup>X</sup> <sup>¼</sup> pl�elΦpl

$$\mathbf{F}^{pl-el}\mathbf{F}^{pl}(\mathbf{X},t) = \frac{\partial^i \mathbf{X}}{\partial \mathbf{X}} = \mathbf{I} + \frac{\partial^0 \mathbf{u}^{pl}}{\partial \mathbf{X}},\tag{15}$$

In the above <sup>E</sup> <sup>¼</sup> <sup>1</sup>=<sup>2</sup> � <sup>F</sup><sup>T</sup> � <sup>F</sup> � <sup>I</sup> denotes the Green-Lagrangian strain tensor

A crucial role in the Lagrangian description plays the invariance of the internal mechanical power. It not only defines conjugate pairs of stress measures and strain or deformation rates, but, being the expression of the conservation of internal mechanical energy (first law of thermodynamics), it plays an inevitable role in making sure that the total Lagrangian description and the updated Lagrangian description are equivalent. As a result, appropriate transformations can be found between various stress measures and strain or deformation rates constituting conjugate pairs, when switching from one stress space in one configuration of the body to the other stress space in the same or any other configuration of the body [15–18]. Unfortunately contemporary continuum theory does not cover materials whose constitutive and evolution equations are defined in rate forms. In order to extend the theory, so that it could cover the materials, Cauchy's stress theorem [16] had to

for all n ¼ 0, 1, …, n ∈ N, where n denotes objective differentiation with respect to time t ≥0 and not an exponent and N the set of natural numbers. In Eq. (24) the

surface traction vectors T ¼ T Xð Þ ; t; N ,t ¼ t xð Þ ; t; n in the initial and current configurations of the body, and N, n are the corresponding unit outwards surface

<sup>P</sup> ð Þ� <sup>P</sup> <sup>N</sup> � dS<sup>0</sup> <sup>¼</sup> <sup>L</sup>ð Þ <sup>n</sup>

<sup>T</sup> ð Þ� <sup>σ</sup> <sup>n</sup> � ds <sup>¼</sup> <sup>L</sup>ð Þ <sup>n</sup>

where <sup>d</sup>S0, d<sup>s</sup> <sup>¼</sup> <sup>J</sup> � <sup>F</sup>�<sup>T</sup> � <sup>d</sup>S<sup>0</sup> denote the infinitesimal surface elements in the initial and current configurations of the body, where the latter is expressed using

objective derivatives of the first Piola-Kirchhoff stress tensor P ¼ P Xð Þ ; t and the Cauchy stress tensor σ ¼ σð Þ x; t , respectively. Then the requirements of thermodynamic consistency of the Lagrangian formulation are ensured by the following

Postulate no. 1. The product of a surface traction vector, including all its higherorder objective time derivatives and the surface of an infinitesimal volume element in the initial and current configurations of the body, on which they act, have to be

λ and an appropriate yield surface normal, ∂<sup>P</sup>Ψ=∂P,

tr ½ �¼ t xð Þ ; <sup>t</sup>; <sup>n</sup> <sup>L</sup>ð Þ <sup>n</sup>

tr ½ � t xð Þ ; <sup>t</sup>; <sup>n</sup> stand for the nth objective derivatives of the

<sup>P</sup> ½ � P Xð Þ ; <sup>t</sup> , <sup>L</sup>ð Þ <sup>n</sup>

<sup>P</sup> ð Þ� <sup>P</sup> <sup>d</sup>S<sup>0</sup> <sup>¼</sup> <sup>L</sup>ð Þ <sup>n</sup>

<sup>T</sup> ð Þ� <sup>σ</sup> <sup>F</sup>�<sup>T</sup> � <sup>d</sup>S0,for all <sup>n</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, <sup>2</sup>, …, n<sup>∈</sup> N, (26)

<sup>T</sup> ½ �� σð Þ x; t n, (24)

<sup>T</sup> ½ � <sup>σ</sup>ð Þ <sup>x</sup>; <sup>t</sup> denote the nth

tr ð Þ� t ds

<sup>T</sup> ð Þ� <sup>σ</sup> <sup>d</sup>s, (25)

and <sup>e</sup> <sup>¼</sup> <sup>1</sup>=<sup>2</sup> � <sup>I</sup> � <sup>F</sup>�<sup>T</sup> � <sup>F</sup>�<sup>1</sup> the Eulerian-Almansi strain tensor, respectively. The symbols E\_ el, E\_ pl=del, dpl stand for the elastic and the plastic material/spatial strain-rate tensors, wherein the latter plastic flow is defined by Eq. (23)1 as a

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity

defined in terms of a first Piola-Kirchhoff stress tensor P: Here the symbol <sup>L</sup>eð Þ¼ <sup>e</sup> <sup>F</sup>�<sup>T</sup> � <sup>∂</sup> <sup>F</sup><sup>T</sup> � ð Þ� <sup>e</sup> <sup>F</sup> <sup>=</sup>∂<sup>t</sup> � <sup>F</sup>�<sup>1</sup> denotes the Lie derivative of the Eulerian-Almansi strain tensor e: It should also be noted that both elastic and the plastic strain-rate tensors have forms similar to the strain-rate tensor itself. Besides, it can be shown that the plastic flow defined by Eq. (23)1 is not constrained, resulting in Eqs. (22)2 and (21)3, respectively, being the only non-degenerated forms of the

<sup>P</sup> ½ �� P Xð Þ ; <sup>t</sup> <sup>N</sup> and <sup>L</sup>ð Þ <sup>n</sup>

product of a plastic multiplier \_

DOI: http://dx.doi.org/10.5772/intechopen.85112

be generalised as follows:

Tr ½ �¼ T Xð Þ ; <sup>t</sup>; <sup>N</sup> <sup>L</sup>ð Þ <sup>n</sup>

Tr ½ � T Xð Þ ; <sup>t</sup>; <sup>N</sup> , <sup>L</sup>ð Þ <sup>n</sup>

normal vectors. Similarly, the quantities Lð Þ <sup>n</sup>

the same during the deformation process, i.e.:

<sup>¼</sup> <sup>L</sup>ð Þ <sup>n</sup>

<sup>T</sup> ð Þ� <sup>σ</sup> <sup>d</sup><sup>s</sup> <sup>¼</sup> <sup>J</sup> � <sup>L</sup>ð Þ <sup>n</sup>

Tr ð Þ� <sup>T</sup> dS<sup>0</sup> <sup>¼</sup>Lð Þ <sup>n</sup>

Lð Þ <sup>n</sup>

variables Lð Þ <sup>n</sup>

postulates:

or

69

Lð Þ <sup>n</sup>

Lð Þ <sup>n</sup>

<sup>P</sup> ð Þ� <sup>P</sup> <sup>d</sup>S<sup>0</sup> <sup>¼</sup> <sup>L</sup>ð Þ <sup>n</sup>

the Nanson's formula [16].

material and spatial plastic strain-rate tensors.

$$\frac{\partial^{\dot{t}}\mathbf{u}^{\prime l}}{\partial^{\dot{t}}\mathbf{X}} = \frac{\partial^{0}\mathbf{u}^{\prime l}}{\partial\mathbf{X}} \cdot \frac{\partial\mathbf{X}}{\partial^{\dot{t}}\mathbf{X}} = \frac{\partial^{0}\mathbf{u}^{\prime l}}{\partial\mathbf{X}} \cdot \left[^{pl-cl}\mathbf{F}^{pl}(\mathbf{X},t)\right]^{-1},\tag{16}$$

and where considering that <sup>0</sup>uel <sup>¼</sup> <sup>i</sup> uel, in the Lagrangian form Eq. (16) of the Eulerian gradient ∂<sup>i</sup> uel=∂<sup>i</sup> <sup>X</sup>, the inverse of pl�elFplð Þ <sup>X</sup>; <sup>t</sup> would be expressed as

$$\mathbf{I} = \frac{\delta^i \mathbf{X}}{\delta^i \mathbf{X}} = \frac{\delta^i \mathbf{X}}{\delta \mathbf{X}} \cdot \frac{\delta \mathbf{X}}{\delta^i \mathbf{X}} = {}^{pl\ -el} \mathbf{F}^{pl}(\mathbf{X}, t) \cdot \frac{\delta \mathbf{X}}{\delta^i \mathbf{X}} \Leftrightarrow \frac{\delta \mathbf{X}}{\delta^i \mathbf{X}} = \left[ {}^{pl\ -el} \mathbf{F}^{pl}(\mathbf{X}, t) \right]^{-1}. \tag{17}$$

Alternatively (see Eq. (10)), the deformation gradient can also be expressed as

$$\begin{split} \mathbf{F}(\mathbf{X},t) &= \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \frac{\partial^{pl-el} \boldsymbol{\Phi}^{el} \left[ {}^{pl-el} \boldsymbol{\Phi}^{pl}(\mathbf{X},t), t \right]}{\partial \mathbf{X}} \\ &= \frac{\partial^{pl-el} \boldsymbol{\Phi}^{el} \left[ {}^{pl-el} \boldsymbol{\Phi}^{pl}(\mathbf{X},t), t \right]}{{}^{pl-el} \boldsymbol{\Phi}^{pl}(\mathbf{X},t)} \cdot \frac{{}^{pl-el} \boldsymbol{\Phi}^{pl}(\mathbf{X},t)}{{}^{pl} \mathbf{X}} = {}^{pl-el} \boldsymbol{\mathsf{F}}^{el}(\mathbf{X},t) \cdot {}^{pl-el} \boldsymbol{\mathsf{F}}^{pl}(\mathbf{X},t). \end{split} \tag{18}$$

It should be noted that by employing the same procedure (Eqs. (13)–(17)), identical formula (Eq. (12)) for the deformation gradient would be arrived at, if the order of elastic and plastic deformations was reversed, although in that case the definitions of the elastic motion el�plΦelð Þ <sup>X</sup>; <sup>t</sup> , the plastic motion el�plΦplð Þ <sup>X</sup>; <sup>t</sup> , the elastic el�plFelð Þ <sup>X</sup>; <sup>t</sup> and plastic el�plFplð Þ <sup>X</sup>; <sup>t</sup> parts of the deformation gradient would be different. The corresponding elastic deformation gradient el�plFelð Þ <sup>X</sup>; <sup>t</sup> , which from now on will be denoted as el�plFel <sup>¼</sup> <sup>F</sup>el, then would take the following form:

$$\mathbf{F}^{el} = \mathbf{I} + \frac{\partial^0 \mathbf{u}^{el}}{\partial \mathbf{X}} = \mathbf{F} - \frac{\partial^0 \mathbf{u}^{pl}}{\partial \mathbf{X}} \,. \tag{19}$$

When the deformation gradient is in the form of Eq. (12), the material E\_ and the spatial d ¼ Leð Þe strain-rate tensors take the forms

$$
\dot{\mathbf{E}} = \frac{1}{2} \cdot \left( \dot{\mathbf{F}}^T \cdot \mathbf{F} + \mathbf{F}^T \cdot \dot{\mathbf{F}} \right) = \dot{\mathbf{E}}^{el} + \dot{\mathbf{E}}^{pl}, \\
\Rightarrow \mathbf{d} = \mathbf{d}^{el} + \mathbf{d}^{pl}, \tag{20}
$$

where

$$\mathbf{d} = \mathbf{F}^{-T} \cdot \dot{\mathbf{E}} \cdot \mathbf{F}^{-1}, \quad \mathbf{d}^{el} = \mathbf{F}^{-T} \cdot \dot{\mathbf{E}}^{el} \cdot \mathbf{F}^{-1}, \quad \mathbf{d}^{pl} = \mathbf{F}^{-T} \cdot \dot{\mathbf{E}}^{pl} \cdot \mathbf{F}^{-1}, \tag{21}$$

$$\dot{\mathbf{E}}^{cl} = \frac{\mathbf{1}}{2} \cdot \left[ \left( \frac{\partial^0 \mathbf{u}^{cl}}{\partial \mathbf{X}} \right)^T \cdot \mathbf{F} + \mathbf{F}^T \cdot \frac{\partial^0 \mathbf{u}^{cl}}{\partial \mathbf{X}} \right], \quad \dot{\mathbf{E}}^{pl} = \frac{\dot{\lambda}}{2} \cdot \left[ \left( \frac{\partial^p \mathbf{V}}{\partial \mathbf{P}} \right)^T \cdot \mathbf{F} + \mathbf{F}^T \cdot \frac{\partial^p \mathbf{V}}{\partial \mathbf{P}} \right], \tag{22}$$

$$\frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} = \dot{\lambda} \cdot \frac{\partial^p \Psi}{\partial \mathbf{P}}, \quad \text{and} \quad \frac{\partial^p \Psi}{\partial \mathbf{P}} \neq \left(\frac{\partial^p \Psi}{\partial \mathbf{P}}\right)^T. \tag{23}$$

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity DOI: http://dx.doi.org/10.5772/intechopen.85112

In the above <sup>E</sup> <sup>¼</sup> <sup>1</sup>=<sup>2</sup> � <sup>F</sup><sup>T</sup> � <sup>F</sup> � <sup>I</sup> denotes the Green-Lagrangian strain tensor and <sup>e</sup> <sup>¼</sup> <sup>1</sup>=<sup>2</sup> � <sup>I</sup> � <sup>F</sup>�<sup>T</sup> � <sup>F</sup>�<sup>1</sup> the Eulerian-Almansi strain tensor, respectively. The symbols E\_ el, E\_ pl=del, dpl stand for the elastic and the plastic material/spatial strain-rate tensors, wherein the latter plastic flow is defined by Eq. (23)1 as a product of a plastic multiplier \_ λ and an appropriate yield surface normal, ∂<sup>P</sup>Ψ=∂P, defined in terms of a first Piola-Kirchhoff stress tensor P: Here the symbol <sup>L</sup>eð Þ¼ <sup>e</sup> <sup>F</sup>�<sup>T</sup> � <sup>∂</sup> <sup>F</sup><sup>T</sup> � ð Þ� <sup>e</sup> <sup>F</sup> <sup>=</sup>∂<sup>t</sup> � <sup>F</sup>�<sup>1</sup> denotes the Lie derivative of the Eulerian-Almansi strain tensor e: It should also be noted that both elastic and the plastic strain-rate tensors have forms similar to the strain-rate tensor itself. Besides, it can be shown that the plastic flow defined by Eq. (23)1 is not constrained, resulting in Eqs. (22)2 and (21)3, respectively, being the only non-degenerated forms of the material and spatial plastic strain-rate tensors.

A crucial role in the Lagrangian description plays the invariance of the internal mechanical power. It not only defines conjugate pairs of stress measures and strain or deformation rates, but, being the expression of the conservation of internal mechanical energy (first law of thermodynamics), it plays an inevitable role in making sure that the total Lagrangian description and the updated Lagrangian description are equivalent. As a result, appropriate transformations can be found between various stress measures and strain or deformation rates constituting conjugate pairs, when switching from one stress space in one configuration of the body to the other stress space in the same or any other configuration of the body [15–18]. Unfortunately contemporary continuum theory does not cover materials whose constitutive and evolution equations are defined in rate forms. In order to extend the theory, so that it could cover the materials, Cauchy's stress theorem [16] had to be generalised as follows:

$$\mathcal{L}\_{Tr}^{(n)}[\mathbf{T}(\mathbf{X},t,\mathbf{N})] = \mathcal{L}\_{P}^{(n)}[\mathbf{P}(\mathbf{X},t)] \cdot \mathbf{N} \text{ and } \mathcal{L}\_{tr}^{(n)}[\mathbf{t}(\mathbf{x},t,\mathbf{n})] = \mathcal{L}\_{T}^{(n)}[\boldsymbol{\sigma}(\mathbf{x},t)] \cdot \mathbf{n}, \tag{24}$$

for all n ¼ 0, 1, …, n ∈ N, where n denotes objective differentiation with respect to time t ≥0 and not an exponent and N the set of natural numbers. In Eq. (24) the variables Lð Þ <sup>n</sup> Tr ½ � T Xð Þ ; <sup>t</sup>; <sup>N</sup> , <sup>L</sup>ð Þ <sup>n</sup> tr ½ � t xð Þ ; <sup>t</sup>; <sup>n</sup> stand for the nth objective derivatives of the surface traction vectors T ¼ T Xð Þ ; t; N ,t ¼ t xð Þ ; t; n in the initial and current configurations of the body, and N, n are the corresponding unit outwards surface normal vectors. Similarly, the quantities Lð Þ <sup>n</sup> <sup>P</sup> ½ � P Xð Þ ; <sup>t</sup> , <sup>L</sup>ð Þ <sup>n</sup> <sup>T</sup> ½ � <sup>σ</sup>ð Þ <sup>x</sup>; <sup>t</sup> denote the nth objective derivatives of the first Piola-Kirchhoff stress tensor P ¼ P Xð Þ ; t and the Cauchy stress tensor σ ¼ σð Þ x; t , respectively. Then the requirements of thermodynamic consistency of the Lagrangian formulation are ensured by the following postulates:

Postulate no. 1. The product of a surface traction vector, including all its higherorder objective time derivatives and the surface of an infinitesimal volume element in the initial and current configurations of the body, on which they act, have to be the same during the deformation process, i.e.:

$$\begin{split} \mathfrak{L}\_{Tr}^{(n)}(\mathbf{T}) \cdot d\mathbf{S}\_0 &= \mathfrak{L}\_p^{(n)}(\mathbf{P}) \cdot \mathbf{N} \cdot d\mathbf{S}\_0 = \mathfrak{L}\_p^{(n)}(\mathbf{P}) \cdot d\mathbf{S}\_0 = \mathfrak{L}\_{tr}^{(n)}(\mathbf{t}) \cdot d\mathbf{s} \\ &= \mathfrak{L}\_T^{(n)}(\mathbf{o}) \cdot \mathbf{n} \cdot d\mathbf{s} = \mathfrak{L}\_T^{(n)}(\mathbf{o}) \cdot d\mathbf{s}, \end{split} \tag{25}$$

or

pl�elFplð Þ¼ <sup>X</sup>; <sup>t</sup>

∂X ∂i

<sup>X</sup> <sup>¼</sup> <sup>∂</sup><sup>0</sup><sup>u</sup>

el ∂X �

<sup>X</sup> <sup>¼</sup> pl�elFplð Þ� <sup>X</sup>; <sup>t</sup>

<sup>∂</sup>pl�elΦel pl�elΦplð Þ <sup>X</sup>; <sup>t</sup> ; <sup>t</sup>

∂X

<sup>F</sup>el <sup>¼</sup> <sup>I</sup> <sup>þ</sup>

, <sup>d</sup>el <sup>¼</sup> <sup>F</sup>�<sup>T</sup> � <sup>E</sup>\_ el � <sup>F</sup>�<sup>1</sup>

∂<sup>0</sup>u\_ el ∂X

3

<sup>∂</sup><sup>P</sup> , and <sup>∂</sup><sup>P</sup><sup>Ψ</sup>

<sup>5</sup>, <sup>E</sup>\_ pl <sup>¼</sup> \_

� <sup>F</sup> <sup>þ</sup> <sup>F</sup><sup>T</sup> �

spatial d ¼ Leð Þe strain-rate tensors take the forms

<sup>E</sup>\_ <sup>¼</sup> <sup>1</sup>

<sup>d</sup> <sup>¼</sup> <sup>F</sup>�<sup>T</sup> � <sup>E</sup>\_ � <sup>F</sup>�<sup>1</sup>

el ∂X !<sup>T</sup>

> ∂<sup>0</sup>u\_ pl <sup>∂</sup><sup>X</sup> <sup>¼</sup> \_ λ � ∂<sup>P</sup>Ψ

<sup>2</sup> � <sup>∂</sup><sup>0</sup>u\_

2 4 h i

pl�elΦplð Þ <sup>X</sup>; <sup>t</sup>

It should be noted that by employing the same procedure (Eqs. (13)–(17)), identical formula (Eq. (12)) for the deformation gradient would be arrived at, if the order of elastic and plastic deformations was reversed, although in that case the definitions of the elastic motion el�plΦelð Þ <sup>X</sup>; <sup>t</sup> , the plastic motion el�plΦplð Þ <sup>X</sup>; <sup>t</sup> , the elastic el�plFelð Þ <sup>X</sup>; <sup>t</sup> and plastic el�plFplð Þ <sup>X</sup>; <sup>t</sup> parts of the deformation gradient would be different. The corresponding elastic deformation gradient el�plFelð Þ <sup>X</sup>; <sup>t</sup> , which from now on will be denoted as el�plFel <sup>¼</sup> <sup>F</sup>el, then would take the following form:

> ∂<sup>0</sup>u el

<sup>∂</sup><sup>X</sup> <sup>¼</sup> <sup>F</sup> � <sup>∂</sup><sup>0</sup>upl

<sup>2</sup> � <sup>F</sup>\_ <sup>T</sup> � <sup>F</sup> <sup>þ</sup> <sup>F</sup><sup>T</sup> � <sup>F</sup>\_ � � <sup>¼</sup> <sup>E</sup>\_ el <sup>þ</sup> <sup>E</sup>\_ pl, ) <sup>d</sup> <sup>¼</sup> <sup>d</sup>el <sup>þ</sup> <sup>d</sup>pl, (20)

λ <sup>2</sup> � <sup>∂</sup><sup>P</sup><sup>Ψ</sup> ∂P � �<sup>T</sup>

<sup>∂</sup><sup>P</sup> 6¼ <sup>∂</sup><sup>P</sup><sup>Ψ</sup> ∂P � �<sup>T</sup>

, <sup>d</sup>pl <sup>¼</sup> <sup>F</sup>�<sup>T</sup> � <sup>E</sup>\_ pl � <sup>F</sup>�<sup>1</sup>

When the deformation gradient is in the form of Eq. (12), the material E\_ and the

∂i u el

Advances in Composite Materials Development

∂i

and where considering that <sup>0</sup>uel <sup>¼</sup> <sup>i</sup>

uel=∂<sup>i</sup>

∂X ∂i

Eulerian gradient ∂<sup>i</sup>

<sup>I</sup> <sup>¼</sup> <sup>∂</sup><sup>i</sup> X ∂i <sup>X</sup> <sup>¼</sup> <sup>∂</sup><sup>i</sup> X ∂X �

F Xð Þ¼ ; t

¼

where

68

<sup>E</sup>\_ el <sup>¼</sup> <sup>1</sup>

∂x ∂X ¼

<sup>∂</sup>pl�elΦel pl�elΦplð Þ <sup>X</sup>; <sup>t</sup> ; <sup>t</sup>

h i

pl�elΦplð Þ <sup>X</sup>; <sup>t</sup> �

<sup>X</sup> <sup>¼</sup> <sup>∂</sup><sup>0</sup><sup>u</sup>

∂i X <sup>∂</sup><sup>X</sup> <sup>¼</sup> <sup>I</sup> <sup>þ</sup>

el <sup>∂</sup><sup>X</sup> � pl�el<sup>F</sup>

∂X ∂i X ⇔ ∂X ∂i

Alternatively (see Eq. (10)), the deformation gradient can also be expressed as

<sup>∂</sup><sup>X</sup> <sup>¼</sup> pl�el<sup>F</sup>

∂<sup>0</sup>upl

pl ð Þ X; t h i�<sup>1</sup>

<sup>X</sup>, the inverse of pl�elFplð Þ <sup>X</sup>; <sup>t</sup> would be expressed as

<sup>X</sup> <sup>¼</sup> pl�el<sup>F</sup>

el

uel, in the Lagrangian form Eq. (16) of the

pl ð Þ X; t h i�<sup>1</sup>

ð Þ� <sup>X</sup>; <sup>t</sup> pl�el<sup>F</sup>

pl ð Þ X; t :

<sup>∂</sup><sup>X</sup> : (19)

, (21)

∂<sup>P</sup>Ψ ∂P

: (23)

,

(22)

� <sup>F</sup> <sup>þ</sup> <sup>F</sup><sup>T</sup> �

" #

<sup>∂</sup><sup>X</sup> , (15)

, (16)

: (17)

(18)

$$d\mathfrak{L}\_{\mathbb{P}}^{(n)}(\mathbf{P}) \cdot d\mathbf{S}\_{0} = \mathfrak{L}\_{\mathbb{T}}^{(n)}(\boldsymbol{\sigma}) \cdot d\mathbf{s} = \boldsymbol{J} \cdot \mathfrak{L}\_{\mathbb{T}}^{(n)}(\boldsymbol{\sigma}) \cdot \mathbf{F}^{-T} \cdot d\mathbf{S}\_{0}, \text{for all } n = 0, 1, 2, \dots, n \in \mathbb{N}, \tag{26}$$

where <sup>d</sup>S0, d<sup>s</sup> <sup>¼</sup> <sup>J</sup> � <sup>F</sup>�<sup>T</sup> � <sup>d</sup>S<sup>0</sup> denote the infinitesimal surface elements in the initial and current configurations of the body, where the latter is expressed using the Nanson's formula [16].

Postulate no. 2. The rate of change of the internal mechanical energy accumulated in the infinitesimal volume element in the initial and current configurations of the body and all its higher-order time derivatives have to be the same during the deformation process, i.e.:

Lð Þ <sup>n</sup>

DOI: http://dx.doi.org/10.5772/intechopen.85112

Lð Þ <sup>n</sup>

Lð Þ <sup>n</sup>

Lð Þ <sup>n</sup>

Lð Þ <sup>n</sup> <sup>T</sup> ð Þ¼ σ J

S <sup>Ψ</sup> <sup>¼</sup> <sup>S</sup>

σσeqð Þ <sup>σ</sup> ; <sup>q</sup>

∂u\_ pl <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_

71

<sup>Ψ</sup> <sup>S</sup>σeqð Þ <sup>S</sup> ; <sup>q</sup>

pl

<sup>∂</sup><sup>X</sup> � <sup>F</sup>�<sup>1</sup> <sup>¼</sup> \_

<sup>P</sup> ð Þ¼ <sup>P</sup> <sup>F</sup> � <sup>∂</sup><sup>n</sup> <sup>F</sup>�<sup>1</sup> � <sup>P</sup> � �

<sup>F</sup> <sup>F</sup>\_ � � <sup>¼</sup> <sup>F</sup>�<sup>T</sup> � <sup>∂</sup><sup>n</sup> <sup>F</sup><sup>T</sup> � <sup>F</sup>\_ � �

<sup>O</sup> ð Þ¼ <sup>τ</sup> <sup>F</sup> � <sup>∂</sup><sup>n</sup> <sup>F</sup>�<sup>1</sup> � <sup>τ</sup> � <sup>F</sup>�<sup>T</sup> � �

∂tn " #

�<sup>1</sup> � <sup>F</sup> � <sup>∂</sup><sup>n</sup> <sup>J</sup> � <sup>F</sup>�<sup>1</sup> � <sup>σ</sup> � <sup>F</sup>�<sup>T</sup> � � ∂tn " #

<sup>e</sup> ð Þ¼ <sup>d</sup> <sup>F</sup>�<sup>T</sup> � <sup>∂</sup><sup>n</sup> <sup>F</sup><sup>T</sup> � <sup>d</sup> � <sup>F</sup> � �

2.2 Modelling of the plastic flow in the material

h i, <sup>P</sup><sup>Ψ</sup> <sup>¼</sup> <sup>P</sup><sup>Ψ</sup> <sup>P</sup>σeqð Þ <sup>P</sup> ; <sup>q</sup>

λ � ∂<sup>σ</sup>Ψ <sup>∂</sup><sup>σ</sup> ,

∂tn " #

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity

∂tn " #

∂tn " # <sup>¼</sup> <sup>F</sup> � <sup>∂</sup><sup>n</sup><sup>S</sup> ∂tn

� <sup>F</sup><sup>T</sup> <sup>¼</sup> <sup>F</sup> � <sup>∂</sup><sup>n</sup><sup>S</sup>

<sup>∂</sup><sup>n</sup> <sup>F</sup><sup>T</sup> �

∂tn � �

� <sup>F</sup>�<sup>1</sup>

∂tn � �

�<sup>1</sup> � <sup>F</sup> � <sup>∂</sup><sup>n</sup><sup>S</sup> ∂tn � �

∂0u\_ ∂X � �

<sup>¼</sup> <sup>F</sup>�<sup>T</sup> �

� <sup>F</sup>�<sup>1</sup> <sup>¼</sup> <sup>F</sup>�<sup>T</sup> � <sup>∂</sup><sup>n</sup>E\_

� <sup>F</sup><sup>T</sup> <sup>¼</sup> <sup>J</sup>

It should also be noted here that Eq. (27) is the result of straightforward manipulation of the nth time derivative of the internal mechanical power (see Eq. (28)), whose last terms formally define the formulas for evaluating the nth objective derivative of the power with respect to time in the first Piola-Kirchhoff, Kirchhoff and Cauchy stress spaces. Then Eq. (27) defines not only conjugate pairs of stress measures and strain or deformations rates but also conjugate pairs of objective differentiation operators and derivatives. Moreover, we used intentionally the term 'requirements of thermodynamic consistency' for the transformations Eq. (29), because without the invariance of the internal mechanical power (first law of thermodynamics) and its higher-order time derivatives (Eq. (27) or (28)), no formulation is thermodynamically consistent, in spite of the fact that in thermodynamics the term is associated with the second law of thermodynamics to show that the constitutive equation of a material is compatible with the second law.

In order to modify the nonlinear continuum theory for finite deformations of elastoplastic media, it is assumed that the yield surface of the material has definitions

<sup>Ψ</sup> <sup>¼</sup> <sup>τ</sup>

Ψ <sup>τ</sup>

σeqð Þτ ; q

h i, σΨ <sup>¼</sup> σΨ

∂<sup>0</sup>u\_ pl <sup>∂</sup><sup>X</sup> <sup>¼</sup> \_ λ � ∂<sup>S</sup>Ψ <sup>∂</sup><sup>S</sup> , (35)

h i, <sup>τ</sup>

h i in terms of the second Piola-Kirchhoff stress tensor <sup>S</sup>, the first

to the material gradient of the plastic velocity field, Eq. (23)1 is as follows:

∂u\_ pl <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_

Piola-Kirchhoff stress tensor P, the Kirchhoff stress tensor τ, the Cauchy stress tensor σ and a vector of internal variables q in the second Piola-Kirchhoff, first

Piola-Kirchhoff, Kirchhoff and Cauchy stress spaces, where <sup>S</sup>σeqð Þ <sup>S</sup> , <sup>P</sup>σeqð Þ <sup>P</sup> , <sup>σ</sup>σeqð Þ<sup>τ</sup> , <sup>σ</sup>σeqð Þ <sup>σ</sup> are the corresponding equivalent stresses. After changing the physical interpretation of the plastic flow and applying push-forward and pull-back operations

pl

<sup>∂</sup><sup>X</sup> � <sup>F</sup>�<sup>1</sup> <sup>¼</sup> \_

λ � ∂τ Ψ <sup>∂</sup><sup>τ</sup> , <sup>F</sup><sup>T</sup> �

� �, (30)

<sup>∂</sup>tn , (31)

� <sup>F</sup>T, (32)

, (33)

� <sup>F</sup><sup>T</sup>: (34)

$$\begin{split} &\frac{\partial^{n}dW}{\partial t^{n}} = \left[\sum\_{k=0}^{n} \binom{n}{k} \cdot \frac{\partial^{n-k}\mathbf{S}}{\partial t^{n-k}} : \frac{\partial^{k}\dot{\mathbf{E}}}{\partial t^{k}}\right] \cdot dV\_{0} = \left[\sum\_{k=0}^{n} \binom{n}{k} \cdot \mathfrak{L}\_{p}^{(n-k)}(\mathbf{P}) : \mathfrak{L}\_{F}^{(k)}\left(\frac{\partial^{0}\dot{\mathbf{u}}}{\partial \mathbf{X}}\right)\right] \\ &\cdot dV\_{0} = \left[\sum\_{k=0}^{n} \binom{n}{k} \cdot \mathfrak{L}\_{O}^{(n-k)}(\mathbf{r}) : \mathfrak{L}\_{\epsilon}^{(k)}(\mathbf{d})\right] \cdot dV\_{0} = \left[\sum\_{k=0}^{n} \binom{n}{k} \cdot \mathfrak{L}\_{T}^{(n-k)}(\mathbf{r}) : \mathfrak{L}\_{\epsilon}^{(k)}(\mathbf{d})\right] \\ &\cdot dv, \quad \text{for all } \ n = 0, 1, 2, \dots, n \in \mathbb{N}, \end{split}$$

(27) or

$$\frac{d^n d\mathbf{W}}{dt^n} = \left[\sum\_{k=0}^n \binom{n}{k} \cdot \frac{\partial^{n-k} \mathbf{S}}{\partial t^{n-k}} \cdot \frac{\partial^k \dot{\mathbf{E}}}{\partial t^k} \right] \cdot dV\_0 = \left[\sum\_{k=0}^n \binom{n}{k} \cdot \mathbf{F} \cdot \left(\frac{\partial^{n-k} \mathbf{S}}{\partial t^{n-k}}\right) \cdot \mathbf{F}^{-T} \cdot \frac{\partial^k \left(\mathbf{F}^T \cdot \dot{\mathbf{F}}\right)}{\partial t^k} \right]$$

$$\cdot dV\_0 = \left[\sum\_{k=0}^n \binom{n}{k} \cdot \mathbf{F} \cdot \left(\frac{\partial^{n-k} \mathbf{S}}{\partial t^{n-k}}\right) \cdot \mathbf{F}^T \cdot \mathbf{F}^{-T} \cdot \left(\frac{\partial^k \dot{\mathbf{E}}}{\partial t^k}\right) \cdot \mathbf{F}^{-1}\right] \cdot dV\_0 =$$

$$\frac{1}{n} = \left[\sum\_{k=0}^n \binom{n}{k} \cdot \frac{\mathbf{F}}{\dot{f}} \cdot \left(\frac{\partial^{n-k} \mathbf{S}}{\partial t^{n-k}}\right) \cdot \mathbf{F}^T \cdot \mathbf{F}^{-T} \cdot \left(\frac{\partial^k \dot{\mathbf{E}}}{\partial t^k}\right) \cdot \mathbf{F}^{-1}\right] \cdot dv \quad \text{for all } n = 0, 1, 2, \dots, n \in \mathcal{N}. \tag{28}$$

where dV0, dv ¼ J � dV<sup>0</sup> stand for the infinitesimal volume elements in the initial and current configurations of the body and J ¼ detð Þ F : Then Eqs. (26)–(28) define the following transformations:

Lð Þ <sup>n</sup> <sup>P</sup> ð Þ¼ <sup>P</sup> <sup>F</sup> � <sup>∂</sup><sup>n</sup><sup>S</sup> ∂tn � �, <sup>L</sup>ð Þ <sup>n</sup> <sup>O</sup> ð Þ¼ <sup>τ</sup> <sup>F</sup> � <sup>∂</sup><sup>n</sup><sup>S</sup> ∂tn � � � <sup>F</sup>T, <sup>L</sup>ð Þ <sup>n</sup> <sup>T</sup> ð Þ¼ σ J �<sup>1</sup> � <sup>F</sup> � <sup>∂</sup><sup>n</sup><sup>S</sup> ∂tn � � � <sup>F</sup>T, Lð Þ <sup>n</sup> <sup>e</sup> ð Þ¼ <sup>d</sup> <sup>F</sup>�<sup>T</sup> � <sup>∂</sup><sup>n</sup>E\_ ∂tn � � � <sup>F</sup>�<sup>1</sup> , <sup>∂</sup><sup>n</sup>E\_ <sup>∂</sup>tn <sup>¼</sup> <sup>F</sup><sup>T</sup> � <sup>L</sup>ð Þ <sup>n</sup> <sup>F</sup> <sup>F</sup>\_ � � h isym ¼ <sup>∂</sup><sup>n</sup> <sup>F</sup><sup>T</sup> � ∂0u\_ ∂X � � ∂tn 2 4 3 5 sym , Lð Þ <sup>n</sup> <sup>e</sup> ð Þ¼ <sup>d</sup> <sup>L</sup>ð Þ <sup>n</sup> <sup>F</sup> <sup>F</sup>\_ � � � <sup>F</sup>�<sup>1</sup> h isym <sup>¼</sup> <sup>F</sup>�<sup>T</sup> � <sup>∂</sup><sup>n</sup> <sup>F</sup>T� ∂0u\_ ∂X � � ∂tn � � � <sup>F</sup>�<sup>1</sup> � �sym for all n ¼ 0, 1, ::, n ∈ N, (29)

as the sufficient conditions of thermodynamic consistency, because they ensure that the two postulates above are met. It should also be noted that for n ¼ 0 the transformations define the necessary conditions of thermodynamic consistency. In that case the generalised Cauchy's stress theorem Eq. (24) reduces to its well-known form, T ¼ P � N and t ¼ σ � n, , while the transformations Eq. (29) reduce to the already well-known transformations in nonlinear continuum mechanics, defining the relationship between various stress measures and strain or deformation rates constituting the conjugate pairs.

The objective rates, which meet the sufficient conditions of thermodynamic consistency defined by Eq. (29), are already known in nonlinear continuum mechanics as the nth Lie derivative of the first Piola-Kirchhoff stress tensor P (Eq. (30)), the nth Lie derivative of the rate of deformation gradient tensor F\_ (Eq. (31)), the nth Oldroyd derivative of the Kirchhoff stress τ tensor (Eq. (32)), the nth Lie derivative of the spatial strain-rate tensor d (Eq. (33)) and the nth Truesdell derivative of the Cauchy stress tensor Eq. (34), respectively:

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity DOI: http://dx.doi.org/10.5772/intechopen.85112

$$\mathcal{L}\_P^{(n)}(\mathbf{P}) = \mathbf{F} \cdot \left[ \frac{\partial^n \left( \mathbf{F}^{-1} \cdot \mathbf{P} \right)}{\partial t^n} \right] = \mathbf{F} \cdot \left( \frac{\partial^n \mathbf{S}}{\partial t^n} \right), \tag{30}$$

$$\mathcal{L}\_F^{(n)}\left(\dot{\mathbf{F}}\right) = \mathbf{F}^{-T} \cdot \left[\frac{\partial^n \left(\mathbf{F}^T \cdot \dot{\mathbf{F}}\right)}{\partial t^n}\right] = \mathbf{F}^{-T} \cdot \frac{\partial^n \left(\mathbf{F}^T \cdot \frac{\partial^p \dot{\mathbf{u}}}{\partial \mathbf{X}}\right)}{\partial t^n},\tag{31}$$

$$\mathfrak{L}\_{O}^{(n)}(\mathbf{r}) = \mathbf{F} \cdot \left[ \frac{\partial^{n} \left( \mathbf{F}^{-1} \cdot \mathbf{r} \cdot \mathbf{F}^{-T} \right)}{\partial t^{n}} \right] \cdot \mathbf{F}^{T} = \mathbf{F} \cdot \left( \frac{\partial^{n} \mathbf{S}}{\partial t^{n}} \right) \cdot \mathbf{F}^{T}, \tag{32}$$

$$\mathcal{L}\_{\epsilon}^{(n)}(\mathbf{d}) = \mathbf{F}^{-T} \cdot \left[ \frac{\partial^{n} \left( \mathbf{F}^{T} \cdot \mathbf{d} \cdot \mathbf{F} \right)}{\partial t^{n}} \right] \cdot \mathbf{F}^{-1} = \mathbf{F}^{-T} \cdot \left( \frac{\partial^{n} \dot{\mathbf{E}}}{\partial t^{n}} \right) \cdot \mathbf{F}^{-1}, \tag{33}$$

$$\mathcal{L}\_T^{(n)}(\boldsymbol{\sigma}) = \boldsymbol{J}^{-1} \cdot \mathbf{F} \cdot \left[ \frac{\partial^n \left( \boldsymbol{J} \cdot \mathbf{F}^{-1} \cdot \boldsymbol{\sigma} \cdot \mathbf{F}^{-T} \right)}{\partial t^n} \right] \cdot \mathbf{F}^T = \boldsymbol{J}^{-1} \cdot \mathbf{F} \cdot \left( \frac{\partial^n \mathbf{S}}{\partial t^n} \right) \cdot \mathbf{F}^T. \tag{34}$$

It should also be noted here that Eq. (27) is the result of straightforward manipulation of the nth time derivative of the internal mechanical power (see Eq. (28)), whose last terms formally define the formulas for evaluating the nth objective derivative of the power with respect to time in the first Piola-Kirchhoff, Kirchhoff and Cauchy stress spaces. Then Eq. (27) defines not only conjugate pairs of stress measures and strain or deformations rates but also conjugate pairs of objective differentiation operators and derivatives. Moreover, we used intentionally the term 'requirements of thermodynamic consistency' for the transformations Eq. (29), because without the invariance of the internal mechanical power (first law of thermodynamics) and its higher-order time derivatives (Eq. (27) or (28)), no formulation is thermodynamically consistent, in spite of the fact that in thermodynamics the term is associated with the second law of thermodynamics to show that the constitutive equation of a material is compatible with the second law.

#### 2.2 Modelling of the plastic flow in the material

In order to modify the nonlinear continuum theory for finite deformations of elastoplastic media, it is assumed that the yield surface of the material has definitions S <sup>Ψ</sup> <sup>¼</sup> <sup>S</sup> <sup>Ψ</sup> <sup>S</sup>σeqð Þ <sup>S</sup> ; <sup>q</sup> h i, <sup>P</sup><sup>Ψ</sup> <sup>¼</sup> <sup>P</sup><sup>Ψ</sup> <sup>P</sup>σeqð Þ <sup>P</sup> ; <sup>q</sup> h i, <sup>τ</sup> <sup>Ψ</sup> <sup>¼</sup> <sup>τ</sup> Ψ <sup>τ</sup> σeqð Þτ ; q h i, σΨ <sup>¼</sup> σΨ σσeqð Þ <sup>σ</sup> ; <sup>q</sup> h i in terms of the second Piola-Kirchhoff stress tensor <sup>S</sup>, the first Piola-Kirchhoff stress tensor P, the Kirchhoff stress tensor τ, the Cauchy stress tensor σ and a vector of internal variables q in the second Piola-Kirchhoff, first Piola-Kirchhoff, Kirchhoff and Cauchy stress spaces, where <sup>S</sup>σeqð Þ <sup>S</sup> , <sup>P</sup>σeqð Þ <sup>P</sup> , <sup>σ</sup>σeqð Þ<sup>τ</sup> , <sup>σ</sup>σeqð Þ <sup>σ</sup> are the corresponding equivalent stresses. After changing the physical interpretation of the plastic flow and applying push-forward and pull-back operations to the material gradient of the plastic velocity field, Eq. (23)1 is as follows:

$$\frac{\partial \dot{\mathbf{u}}^{pl}}{\partial \mathbf{x}} = \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} \cdot \mathbf{F}^{-1} = \dot{\boldsymbol{\lambda}} \cdot \frac{\partial^r \mathbf{V}}{\partial \mathbf{\sigma}},\\\frac{\partial \dot{\mathbf{u}}^{pl}}{\partial \mathbf{x}} = \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} \cdot \mathbf{F}^{-1} = \dot{\boldsymbol{\lambda}} \cdot \frac{\partial^r \mathbf{V}}{\partial \mathbf{\tau}},\\\mathbf{F}^T \cdot \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} = \dot{\boldsymbol{\lambda}} \cdot \frac{\partial^3 \mathbf{V}}{\partial \mathbf{S}},\tag{35}$$

Postulate no. 2. The rate of change of the internal mechanical energy accumulated in the infinitesimal volume element in the initial and current configurations of the body and all its higher-order time derivatives have to be the same during the

� dV<sup>0</sup> ¼ ∑

<sup>e</sup> ð Þ d

(27) or

n k¼0

∂tk � �

� <sup>F</sup>�<sup>1</sup>

� <sup>F</sup>T, <sup>L</sup>ð Þ <sup>n</sup>

<sup>F</sup> <sup>F</sup>\_ � � h isym

∂0u\_ ∂X � � ∂tn � �

� �sym

as the sufficient conditions of thermodynamic consistency, because they ensure that the two postulates above are met. It should also be noted that for n ¼ 0 the transformations define the necessary conditions of thermodynamic consistency. In that case the generalised Cauchy's stress theorem Eq. (24) reduces to its well-known form, T ¼ P � N and t ¼ σ � n, , while the transformations Eq. (29) reduce to the already well-known transformations in nonlinear continuum mechanics, defining the relationship between various stress measures and strain or deformation rates

The objective rates, which meet the sufficient conditions of thermodynamic consistency defined by Eq. (29), are already known in nonlinear continuum mechanics as the nth Lie derivative of the first Piola-Kirchhoff stress tensor P (Eq. (30)), the nth Lie derivative of the rate of deformation gradient tensor F\_ (Eq. (31)), the nth Oldroyd derivative of the Kirchhoff stress τ tensor (Eq. (32)), the nth Lie derivative of the spatial strain-rate tensor d (Eq. (33)) and the nth Truesdell

<sup>T</sup> ð Þ¼ σ J

� <sup>F</sup>�<sup>1</sup>

¼

2 4

� dV<sup>0</sup> ¼ ∑

� <sup>F</sup><sup>T</sup> : <sup>F</sup>�<sup>T</sup> � <sup>∂</sup><sup>k</sup>E\_

∂tk � �

where dV0, dv ¼ J � dV<sup>0</sup> stand for the infinitesimal volume elements in the initial and current configurations of the body and J ¼ detð Þ F : Then Eqs. (26)–(28) define

n k¼0

� dV<sup>0</sup> ¼ ∑

n k !

� <sup>F</sup>�<sup>1</sup>

n k

> n k¼0

� <sup>L</sup>ð Þ <sup>n</sup>�<sup>k</sup>

n k

� <sup>F</sup> � <sup>∂</sup><sup>n</sup>�<sup>k</sup><sup>S</sup> ∂tn�<sup>k</sup> � �

" #

� dV<sup>0</sup> ¼

!

<sup>P</sup> ð Þ <sup>P</sup> : <sup>L</sup>ð Þ<sup>k</sup>

� <sup>L</sup>ð Þ <sup>n</sup>�<sup>k</sup>

" #

: <sup>F</sup>�<sup>T</sup> �

� dv for all n ¼ 0, 1, 2, …, n∈ N:

�<sup>1</sup> � <sup>F</sup> � <sup>∂</sup><sup>n</sup><sup>S</sup> ∂tn � �

∂tn

∂0u\_ ∂X � �

3 5

for all n ¼ 0, 1, ::, n ∈ N,

sym ,

<sup>∂</sup><sup>n</sup> <sup>F</sup><sup>T</sup> �

" # � �

F

<sup>T</sup> ð Þ <sup>σ</sup> : <sup>L</sup>ð Þ<sup>k</sup>

∂<sup>0</sup>u\_ ∂X

<sup>∂</sup><sup>k</sup> <sup>F</sup><sup>T</sup> � <sup>F</sup>\_ � � ∂tk

<sup>e</sup> ð Þ d

(28)

(29)

� <sup>F</sup>T,

!

deformation process, i.e.:

n k¼0

n k¼0

> n k¼0

n k¼0

n k ! � F <sup>J</sup> � <sup>∂</sup><sup>n</sup>�<sup>k</sup><sup>S</sup> ∂tn�<sup>k</sup> � �

the following transformations:

∂tn � �

> ∂tn � �

<sup>F</sup> <sup>F</sup>\_ � � � <sup>F</sup>�<sup>1</sup> h isym

constituting the conjugate pairs.

, Lð Þ <sup>n</sup>

� <sup>F</sup>�<sup>1</sup>

<sup>P</sup> ð Þ¼ <sup>P</sup> <sup>F</sup> � <sup>∂</sup><sup>n</sup><sup>S</sup>

<sup>e</sup> ð Þ¼ <sup>d</sup> <sup>F</sup>�<sup>T</sup> � <sup>∂</sup><sup>n</sup>E\_

<sup>e</sup> ð Þ¼ <sup>d</sup> <sup>L</sup>ð Þ <sup>n</sup>

n k

n k

� dv, for all n ¼ 0, 1, 2, …, n∈ N,

n k ! � ∂<sup>n</sup>�<sup>k</sup>S <sup>∂</sup>tn�<sup>k</sup> :

n k !

" #

� <sup>F</sup> � <sup>∂</sup><sup>n</sup>�<sup>k</sup><sup>S</sup> ∂tn�<sup>k</sup> � �

" #

<sup>O</sup> ð Þ¼ <sup>τ</sup> <sup>F</sup> � <sup>∂</sup><sup>n</sup><sup>S</sup>

, <sup>∂</sup><sup>n</sup>E\_

derivative of the Cauchy stress tensor Eq. (34), respectively:

∂tn � �

<sup>∂</sup>tn <sup>¼</sup> <sup>F</sup><sup>T</sup> � <sup>L</sup>ð Þ <sup>n</sup>

<sup>¼</sup> <sup>F</sup>�<sup>T</sup> � <sup>∂</sup><sup>n</sup> <sup>F</sup>T�

!

� ∂<sup>n</sup>�<sup>k</sup>S <sup>∂</sup>tn�<sup>k</sup> :

" #

� <sup>L</sup>ð Þ <sup>n</sup>�<sup>k</sup>

" #

∂<sup>k</sup>E\_ ∂tk

<sup>O</sup> ð Þ<sup>τ</sup> : <sup>L</sup>ð Þ<sup>k</sup>

∂<sup>k</sup>E\_ ∂tk

" #

� <sup>F</sup><sup>T</sup> : <sup>F</sup>�<sup>T</sup> � <sup>∂</sup><sup>k</sup>E\_

!

Advances in Composite Materials Development

∂ndW

<sup>∂</sup>tn <sup>¼</sup> <sup>∑</sup>

� dV<sup>0</sup> ¼ ∑

∂ndW <sup>∂</sup>tn <sup>¼</sup> <sup>∑</sup>

¼ ∑ n k¼0

Lð Þ <sup>n</sup>

Lð Þ <sup>n</sup>

Lð Þ <sup>n</sup>

70

�dV<sup>0</sup> ¼ ∑

where it can be shown that the definitions of the yield surface are not independent of each other but are related, and the following formulas hold true:

$$\frac{\partial^{\mathbb{P}}\Psi}{\partial\mathbf{P}}\cdot\mathbf{F}^{-1}=\frac{\partial^{\mathbb{P}}\Psi}{\partial\mathbf{\sigma}},\quad\frac{\partial^{\mathbb{P}}\Psi}{\partial\mathbf{P}}\cdot\mathbf{F}^{-1}=\frac{\partial^{\mathbb{P}}\Psi}{\partial\mathbf{\sigma}},\quad\mathbf{F}^{T}\cdot\frac{\partial^{\mathbb{P}}\Psi}{\partial\mathbf{P}}=\frac{\partial^{\mathbb{Q}}\Psi}{\partial\mathbf{S}}.\tag{36}$$

<sup>L</sup>Pð Þ¼ <sup>P</sup> <sup>F</sup> � <sup>S</sup>\_ <sup>¼</sup> mixCel : <sup>F</sup>\_ � xx �

DOI: http://dx.doi.org/10.5772/intechopen.85112

LTð Þ¼ σ J

mixCel

ijkl ¼ J

spatCel

73

∂<sup>0</sup>u\_ pl ∂X

<sup>L</sup>Oð Þ¼ <sup>τ</sup> <sup>F</sup> � <sup>S</sup>\_ � <sup>F</sup><sup>T</sup> <sup>¼</sup> <sup>J</sup> � spatCel : <sup>d</sup> � xx � <sup>d</sup>pl � � <sup>þ</sup> <sup>J</sup> � spatCvis : <sup>L</sup>eð Þ� <sup>d</sup> ð Þ� <sup>1</sup> � xx <sup>L</sup><sup>e</sup> <sup>d</sup>pl h i � � ,

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity

where (45)

<sup>2</sup> � ð Þ <sup>1</sup> <sup>þ</sup> <sup>ν</sup> , <sup>λ</sup>el <sup>¼</sup> <sup>ν</sup> � <sup>E</sup>

<sup>2</sup> � <sup>1</sup> <sup>þ</sup> <sup>ν</sup>vis ð Þ, <sup>λ</sup>vis <sup>¼</sup> <sup>ν</sup>vis � Evis

mnop, mixCvis

mnop, spatCvis

In Eqs. (42)–(50) the symbols <sup>S</sup>\_, <sup>L</sup>Pð Þ <sup>P</sup> , <sup>L</sup>Oð Þ<sup>τ</sup> , <sup>L</sup>Tð Þ <sup>σ</sup> denote the time derivative of the second Piola-Kirchhoff stress tensor, the Lie derivative of the first Piola-Kirchhoff stress tensor, the Oldroyd rate of the Kirchhoff stress and the Truesdell rate of the Cauchy stress, respectively. Here the fourth-order material elasticity tensor matCel and the fourth-order material viscosity tensor matCvis have similar forms as the fourth-order elasticity tensor of the St. Venant-Kirchhoff material [17] using two independent material parameters E, ν and Evis, νvis, respectively, where I

denotes the symmetric fourth-order identity tensor and I the second-order identity tensor. The fourth-order mixed spatial-material elasticity and viscosity tensors matCel, matCvis then can be determined in accordance with Eq. (49), where δij is the Kronecker delta, and the fourth-order spatial elasticity and viscosity tensors in accordance with Eq. (50). The variable xx denotes the ratio of ductile and total damage increment [21]. It should be noted that the objective rates

and Cauchy (Eq. (54)) stress spaces are modified as follows:

<sup>S</sup>\_,LPð Þ <sup>P</sup> , <sup>L</sup>Oð Þ<sup>τ</sup> , <sup>L</sup>Tð Þ <sup>σ</sup> transform in the same way from one form into another as do the stress tensors S, P, τ, σ, which ensure that the formulation is thermodynamically consistent (see also Eq. (29)). Then, the corresponding rate forms of loading/ unloading discrete Kuhn-Tucker plastic optimization conditions in the second Piola-Kirchhoff (Eq. (51)), first Piola-Kirchhoff (Eq. (52)), Kirchhoff (Eq. (53))

ijkl ¼ J

�<sup>1</sup> � <sup>F</sup> � <sup>S</sup>\_ � <sup>F</sup><sup>T</sup> <sup>¼</sup> spatCel : <sup>d</sup> � xx � <sup>d</sup>pl � � <sup>þ</sup> spatCvis : <sup>L</sup>eð Þ� <sup>d</sup> ð Þ� <sup>1</sup> � xx <sup>L</sup><sup>e</sup> <sup>d</sup>pl h i � � ,

matCel <sup>¼</sup> <sup>2</sup> � <sup>G</sup> � <sup>I</sup> <sup>þ</sup> <sup>λ</sup>el � <sup>I</sup>⊗I, matCvis <sup>¼</sup> <sup>2</sup> � <sup>G</sup>vis � <sup>I</sup> <sup>þ</sup> <sup>λ</sup>vis � <sup>I</sup>⊗I, (46)

<sup>þ</sup> mixCvis : <sup>L</sup><sup>F</sup> <sup>F</sup>\_ � � � ð Þ� <sup>1</sup> � xx <sup>L</sup><sup>F</sup>

∂<sup>0</sup>u\_ pl ∂X

(43)

(44)

mnop, (49)

> mnop: (50)

,

" # !

ð Þ� <sup>1</sup> <sup>þ</sup> <sup>ν</sup> ð Þ <sup>1</sup> � <sup>2</sup> � <sup>ν</sup> , (47)

<sup>1</sup> <sup>þ</sup> <sup>ν</sup>vis ð Þ� <sup>1</sup> � <sup>2</sup> � <sup>ν</sup>vis ð Þ, (48)

ijkl <sup>¼</sup> Fim � <sup>δ</sup>jn � Fko � <sup>δ</sup>lp � matCvis

�<sup>1</sup> � Fim � Fjn � Fko � Flp � matCvis

!

<sup>G</sup> <sup>¼</sup> <sup>E</sup>

<sup>G</sup>vis <sup>¼</sup> Evis

ijkl <sup>¼</sup> Fim � <sup>δ</sup>jn � Fko � <sup>δ</sup>lp � matCel

�<sup>1</sup> � Fim � Fjn � Fko � Flp � matCel

As a result, one of the definitions of the yield surfaces has to be chosen as a reference to define the material model, and the rest of them can be calculated by solving the differential equations in Eq. (36). Moreover, when σΨ or <sup>τ</sup> Ψ is used as the reference definition of the yield surface in the current configuration of the body, the contemporary flow plasticity models will be recovered. It also can be verified that the various definitions of the yield surface and their equivalent stresses <sup>P</sup>σeq, <sup>S</sup>σeq, <sup>τ</sup>σeq, <sup>σ</sup>σeq, which also meet the transformations defined by Eq. (36), have the following properties:

$$\prescript{P}{}{\sigma}\_{eq} = \prescript{S}{}{\sigma}\_{eq} = \prescript{\tau}{}{\sigma}\_{eq} = \prescript{}{f}{\cdot} \prescript{\sigma}{}{\sigma}\_{eq},\tag{37}$$

$$\frac{\partial^{\mathbb{S}}\Psi}{\partial\mathbf{S}} : \mathbf{S} = \frac{\partial^{\mathbb{P}}\Psi}{\partial\mathbf{P}} : \mathbf{P} = \frac{\partial^{\mathbb{s}}\Psi}{\partial\mathbf{\sigma}} : \mathbf{\sigma} = \boldsymbol{J} \cdot \frac{\partial^{\sigma}\Psi}{\partial\mathbf{\sigma}} : \mathbf{\sigma},\tag{38}$$

$$\frac{\partial^3 \Psi}{\partial \mathbf{S}} : \dot{\mathbf{S}} = \frac{\partial^3 \Psi}{\partial \mathbf{P}} : \mathfrak{L}\_P(\mathbf{P}) = \frac{\partial^r \Psi}{\partial \mathbf{r}} : \mathfrak{L}\_O(\mathbf{r}) = \boldsymbol{f} \cdot \frac{\partial^r \Psi}{\partial \mathbf{r}} : \mathfrak{L}\_T(\mathbf{r}), \tag{39}$$

where Eqs. (38) and (39) represent 'normality rules', which from the physical point of view are equivalent with the following equations:

$$d\!\!\!W^{pl} = \dot{\mathbf{E}}^{pl} : \mathbf{S} \cdot \text{d}V\_0 = \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} : \mathbf{P} \cdot \text{d}V\_0 = \mathbf{d}^{pl} : \mathbf{r} \cdot \text{d}V\_0 = \mathbf{d}^{pl} : \boldsymbol{\sigma} \cdot \text{d}v,\tag{40}$$

$$\dot{\mathbf{E}}^{pl} : \dot{\mathbf{S}} \cdot d\mathbf{V}\_0 = \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} : \mathfrak{L}\_P(\mathbf{P}) \cdot d\mathbf{V}\_0 = \mathbf{d}^{pl} : \mathfrak{L}\_O(\mathbf{r}) \cdot d\mathbf{V}\_0 = \mathbf{d}^{pl} : \mathfrak{L}\_T(\mathbf{o}) \cdot d\mathbf{v}. \tag{41}$$

where Wpl is the internal plastic power.

#### 2.3 The constitutive equations of the material

Proper formulation of a material model for finite-strain elastoplasticity allows for the definition of the model in all stress spaces in any configuration of the body. These, however, have to comply with the principles of material modelling, particularly to meet the requirements of material objectivity and be thermodynamically consistent in order that they would define the same material. Finite-strain computational plasticity distinguishes between two major types of material models known as hypoelastic-plastic-based material models and hyperelastic-plastic-based material models. Moreover, hypoelastic-plastic-based material models exist in rate forms only, because the additive decomposition of the strain-rate tensor Eqs. (20)–(23) and (36) exists either in rate forms only. In this research we have modified our former material model with internal damping, capable of imitating even ductile-tobrittle failure mode transition at high strain rates, to model our hypoelastic-plasticbased material [21]. The rate form of the constitutive equation of the material then can take any of the following forms:

$$\dot{\mathbf{S}} = {}^{mat}\mathbb{C}^{l} : \left(\dot{\mathbf{E}} - \boldsymbol{\infty} \cdot \dot{\mathbf{E}}^{pl}\right) + {}^{mat}\mathbb{C}^{vis} : \left[\ddot{\mathbf{E}} - (\mathbf{1} - \boldsymbol{\infty}) \cdot \ddot{\mathbf{E}}^{pl}\right],\tag{42}$$

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity DOI: http://dx.doi.org/10.5772/intechopen.85112

$$\mathfrak{L}\_P(\mathbf{P}) = \mathbf{F} \cdot \dot{\mathbf{S}} = {}^{\text{mix}}\mathbb{C}^d : \left(\dot{\mathbf{F}} - \mathbf{x}\mathbf{x} \cdot \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}}\right) + {}^{\text{mix}}\mathbb{C}^{vis} : \left[\mathfrak{L}\_F(\dot{\mathbf{F}}) - (\mathbf{1} - \mathbf{x}\mathbf{x}) \cdot \mathfrak{L}\_F\left(\frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}}\right)\right],\tag{43}$$

$$\mathfrak{L}\_{O}(\mathfrak{r}) = \mathbf{F} \cdot \dot{\mathbf{S}} \cdot \mathbf{F}^{T} = \boldsymbol{f} \cdot \mathbf{^{spt}} \mathbf{C}^{d} : \left(\mathbf{d} - \boldsymbol{\infty} \cdot \mathbf{d}^{pl}\right) + \boldsymbol{f} \cdot \mathbf{^{spt}} \mathbf{C}^{\dot{\boldsymbol{\alpha}}i} : \left[\mathfrak{L}\_{\mathbf{e}}(\mathbf{d}) - (\mathbf{1} - \boldsymbol{\infty}) \cdot \mathfrak{L}\_{\mathbf{e}}\left(\mathbf{d}^{pl}\right)\right],\tag{44}$$

$$\mathfrak{L}\_{T}(\boldsymbol{\sigma}) = \boldsymbol{f}^{-1} \cdot \mathbf{F} \cdot \dot{\mathbf{S}} \cdot \mathbf{F}^{T} = {}^{\text{part}}\mathbb{C}^{d} : \left(\mathbf{d} - \boldsymbol{\text{x}}\boldsymbol{\text{x}} \cdot \mathbf{d}^{pl}\right) + {}^{\text{spat}}\mathbb{C}^{\text{vir}} : \left[\mathfrak{L}\_{\text{e}}(\mathbf{d}) - (\mathbb{1} - \boldsymbol{\text{x}}\boldsymbol{\text{x}}) \cdot \mathfrak{L}\_{\text{e}}\left(\mathbf{d}^{pl}\right)\right],\tag{45}$$
  $\text{where}$ 

$$\mathbf{C}^{mat}\mathbf{C}^{el} = \mathbf{2} \cdot \mathbf{G} \cdot \mathbb{I} + \lambda^{el} \cdot \mathbf{I} \otimes \mathbf{I}, \quad \mathbf{^{mat}C}^{vis} = \mathbf{2} \cdot \mathbf{G}^{vis} \cdot \mathbb{I} + \lambda^{vis} \cdot \mathbf{I} \otimes \mathbf{I}, \tag{46}$$

$$G = \frac{E}{2 \cdot (1 + \nu)},\\ \lambda^{cl} = \frac{\nu \cdot E}{(1 + \nu) \cdot (1 - 2 \cdot \nu)},\tag{47}$$

$$G^{vis} = \frac{E^{vis}}{2 \cdot (1 + \nu^{vis})}, \; \lambda^{vis} = \frac{\nu^{vis} \cdot E^{vis}}{(1 + \nu^{vis}) \cdot (1 - 2 \cdot \nu^{vis})},\tag{48}$$

$${}^{m\dot{x}i}\mathbb{C}^{el}\_{ijkl} = F\_{im} \cdot \delta\_{jn} \cdot F\_{ko} \cdot \delta\_{lp} \cdot {}^{mat}\mathbb{C}^{el}\_{mnpp} \cdot \mathbb{C}^{oi}\_{ijkl} = F\_{im} \cdot \delta\_{jn} \cdot F\_{ko} \cdot \delta\_{lp} \cdot {}^{mat}\mathbb{C}^{oi}\_{mnpp} \tag{49}$$

$${}^{\rm spat}\mathbb{C}\_{ijkl}^{\rm cl} = \boldsymbol{J}^{-1} \cdot \boldsymbol{F}\_{im} \cdot \boldsymbol{F}\_{jn} \cdot \boldsymbol{F}\_{ko} \cdot \boldsymbol{F}\_{lp} \cdot \boldsymbol{J}^{mat}\_{mnpp} \cdot \boldsymbol{J}^{mat}\_{ijkl} = \boldsymbol{J}^{-1} \cdot \boldsymbol{F}\_{im} \cdot \boldsymbol{F}\_{jn} \cdot \boldsymbol{F}\_{ko} \cdot \boldsymbol{F}\_{lp} \cdot \boldsymbol{J}^{mat}\_{mnpp} \cdot \boldsymbol{J}^{rot}\_{mnpq} \tag{50}$$

In Eqs. (42)–(50) the symbols <sup>S</sup>\_,LPð Þ <sup>P</sup> , <sup>L</sup>Oð Þ<sup>τ</sup> , <sup>L</sup>Tð Þ <sup>σ</sup> denote the time derivative of the second Piola-Kirchhoff stress tensor, the Lie derivative of the first Piola-Kirchhoff stress tensor, the Oldroyd rate of the Kirchhoff stress and the Truesdell rate of the Cauchy stress, respectively. Here the fourth-order material elasticity tensor matCel and the fourth-order material viscosity tensor matCvis have similar forms as the fourth-order elasticity tensor of the St. Venant-Kirchhoff material [17] using two independent material parameters E, ν and Evis, νvis, respectively, where I denotes the symmetric fourth-order identity tensor and I the second-order identity tensor. The fourth-order mixed spatial-material elasticity and viscosity tensors matCel, matCvis then can be determined in accordance with Eq. (49), where δij is the Kronecker delta, and the fourth-order spatial elasticity and viscosity tensors in accordance with Eq. (50). The variable xx denotes the ratio of ductile and total damage increment [21]. It should be noted that the objective rates <sup>S</sup>\_,LPð Þ <sup>P</sup> ,LOð Þ<sup>τ</sup> , <sup>L</sup>Tð Þ <sup>σ</sup> transform in the same way from one form into another as do the stress tensors S, P, τ, σ, which ensure that the formulation is thermodynamically consistent (see also Eq. (29)). Then, the corresponding rate forms of loading/ unloading discrete Kuhn-Tucker plastic optimization conditions in the second Piola-Kirchhoff (Eq. (51)), first Piola-Kirchhoff (Eq. (52)), Kirchhoff (Eq. (53)) and Cauchy (Eq. (54)) stress spaces are modified as follows:

where it can be shown that the definitions of the yield surface are not independent of each other but are related, and the following formulas hold true:

<sup>∂</sup><sup>P</sup> � <sup>F</sup>�<sup>1</sup> <sup>¼</sup> <sup>∂</sup><sup>τ</sup>

As a result, one of the definitions of the yield surfaces has to be chosen as a reference to define the material model, and the rest of them can be calculated by

the reference definition of the yield surface in the current configuration of the body, the contemporary flow plasticity models will be recovered. It also can be verified that the various definitions of the yield surface and their equivalent stresses <sup>P</sup>σeq, <sup>S</sup>σeq, <sup>τ</sup>σeq, <sup>σ</sup>σeq, which also meet the transformations defined by Eq. (36), have

<sup>σ</sup>eq <sup>¼</sup> <sup>τ</sup>

Ψ <sup>∂</sup><sup>τ</sup> : <sup>τ</sup> <sup>¼</sup> <sup>J</sup> �

<sup>∂</sup><sup>τ</sup> : <sup>L</sup>Oð Þ¼ <sup>τ</sup> <sup>J</sup> �

Ψ

where Eqs. (38) and (39) represent 'normality rules', which from the physical

Proper formulation of a material model for finite-strain elastoplasticity allows for the definition of the model in all stress spaces in any configuration of the body. These, however, have to comply with the principles of material modelling, particularly to meet the requirements of material objectivity and be thermodynamically consistent in order that they would define the same material. Finite-strain computational plasticity distinguishes between two major types of material models known as hypoelastic-plastic-based material models and hyperelastic-plastic-based material models. Moreover, hypoelastic-plastic-based material models exist in rate forms only, because the additive decomposition of the strain-rate tensor Eqs. (20)–(23) and (36) exists either in rate forms only. In this research we have modified our former material model with internal damping, capable of imitating even ductile-tobrittle failure mode transition at high strain rates, to model our hypoelastic-plasticbased material [21]. The rate form of the constitutive equation of the material then

<sup>S</sup>\_ <sup>¼</sup> matCel : <sup>E</sup>\_ � xx � <sup>E</sup>\_ pl <sup>þ</sup> matCvis : <sup>E</sup>€ � ð Þ� <sup>1</sup> � xx <sup>E</sup>€pl , (42)

<sup>∂</sup><sup>P</sup> : <sup>P</sup> <sup>¼</sup> <sup>∂</sup><sup>τ</sup>

Ψ <sup>∂</sup><sup>τ</sup> , <sup>F</sup><sup>T</sup> � ∂<sup>P</sup>Ψ <sup>∂</sup><sup>P</sup> <sup>¼</sup> <sup>∂</sup><sup>S</sup><sup>Ψ</sup>

<sup>σ</sup>eq <sup>¼</sup> <sup>J</sup> � <sup>σ</sup>σeq, (37)

<sup>∂</sup><sup>σ</sup> : <sup>σ</sup>, (38)

<sup>∂</sup><sup>σ</sup> : <sup>L</sup>Tð Þ <sup>σ</sup> , (39)

∂σΨ

<sup>∂</sup><sup>X</sup> : <sup>P</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>d</sup>pl : <sup>τ</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>d</sup>pl : <sup>σ</sup> � dv, (40)

<sup>∂</sup><sup>X</sup> : <sup>L</sup>Pð Þ� <sup>P</sup> dV<sup>0</sup> <sup>¼</sup> <sup>d</sup>pl : <sup>L</sup>Oð Þ� <sup>τ</sup> dV<sup>0</sup> <sup>¼</sup> <sup>d</sup>pl : <sup>L</sup>Tð Þ� <sup>σ</sup> dv: (41)

∂σΨ

<sup>∂</sup><sup>S</sup> : (36)

Ψ is used as

<sup>∂</sup><sup>σ</sup> , <sup>∂</sup><sup>P</sup><sup>Ψ</sup>

solving the differential equations in Eq. (36). Moreover, when σΨ or <sup>τ</sup>

<sup>P</sup>σeq <sup>¼</sup> <sup>S</sup>

<sup>∂</sup><sup>P</sup> : <sup>L</sup>Pð Þ¼ <sup>P</sup> <sup>∂</sup><sup>τ</sup>

pl

<sup>∂</sup><sup>S</sup> : <sup>S</sup> <sup>¼</sup> <sup>∂</sup><sup>P</sup><sup>Ψ</sup>

point of view are equivalent with the following equations:

∂<sup>P</sup>Ψ

the following properties:

∂<sup>S</sup>Ψ

dWpl <sup>¼</sup> <sup>E</sup>\_ pl : <sup>S</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_

can take any of the following forms:

72

<sup>E</sup>\_ pl : <sup>S</sup>\_ � dV<sup>0</sup> <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_

<sup>∂</sup><sup>P</sup> � <sup>F</sup>�<sup>1</sup> <sup>¼</sup> <sup>∂</sup><sup>σ</sup><sup>Ψ</sup>

Advances in Composite Materials Development

∂<sup>S</sup>Ψ

<sup>∂</sup><sup>S</sup> : <sup>S</sup>\_ <sup>¼</sup> <sup>∂</sup><sup>P</sup><sup>Ψ</sup>

pl

where Wpl is the internal plastic power.

2.3 The constitutive equations of the material

$$
\dot{\lambda} \ge 0, \qquad \prescript{S}{}{\Psi} \le 0, \qquad \dot{\lambda} \cdot \prescript{S}{}{\Psi} = 0,\tag{51}
$$

2.4 The reference definition of the yield surface

DOI: http://dx.doi.org/10.5772/intechopen.85112

longer a cylinder but a sphere:

q

<sup>¼</sup> springe\_

spring e\_ pl � dt ¼

0

pl

<sup>P</sup>σ<sup>y</sup> <sup>¼</sup> FUT<sup>11</sup> �

spring e\_ pl

spring e pl <sup>¼</sup> ðt

<sup>¼</sup> dampere\_

dampere\_ pl � dt ¼ ðt

0

dampere\_ pl

dampere

75

pl ¼ ðt

<sup>P</sup><sup>Ψ</sup> <sup>¼</sup> <sup>P</sup>σeq�<sup>P</sup>σ<sup>y</sup> <sup>≤</sup>0, where <sup>P</sup>σeq <sup>¼</sup> <sup>P</sup>σeqð Þ¼ <sup>P</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>r</sup><sup>2</sup> � <sup>a</sup> � <sup>e</sup> ½ � pl � center <sup>2</sup>

pl xx �

ð Þ� <sup>1</sup> � xx <sup>∂</sup><sup>0</sup>u\_ pl

0

� �

∂<sup>0</sup>u\_ pl ∂X � �

ðt

xx � \_

¼

r

0

∂X

ð Þ� <sup>1</sup> � xx \_

¼

r

Objective and thermodynamically consistent formulation of the plastic flow allows for the development of consistent material models. As a result, the material model can by formulated in any stress space and in whatever configuration of the body, though we intentionally omitted to give an example of the constitutive equation of our hypoelastic-plastic-based and hyperelastic-plastic-based material models in the intermediate configuration of the body, as it is just a matter of proper stress transformation using the multiplicative split of the deformation. Moreover, one of the formulations of the yield surface has to be a reference, from which other definitions of the yield surface in the second Piola-Kirchhoff, first Piola-Kirchhoff, Kirchhoff and Cauchy stress spaces can be calculated by solving Eq. (36). The various definitions of the yields surface then have the properties Eqs. (37)–(39), from which the 'normality rules' (Eqs. (38) and (39)) (whose physical meaning is defined by Eqs. (40) and (41)) are used in the return mapping/rate form of the return mapping algorithms to calculate the plastic multiplier. In this study the reference yield surface was defined in the first Piola-Kirchhoff stress space, because the corresponding plastic flow Eq. (23)1 is the only not constrained. Then the generalised J2 flow plasticity theory with isotropic hardening is defined by

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity

Eqs. (63)–(67). It should be noted that the PJ2ð Þ¼ <sup>P</sup> <sup>P</sup> : <sup>P</sup> invariant in the definition of the equivalent stress no longer bases on the deviatoric part of the first Piola-Kirchhoff stress tensor. The change was implied by the objectivity requirements, since the first Piola-Kirchhoff stress tensor transforms under the change of the observer as <sup>P</sup><sup>þ</sup> <sup>¼</sup> <sup>Q</sup><sup>R</sup> � <sup>P</sup> and PJ2ð Þ <sup>P</sup> is the only invariant, which is not affected by the change, i.e. PJ2ð Þ¼ <sup>P</sup> PJ<sup>2</sup> <sup>P</sup><sup>þ</sup> ð Þ, where <sup>Q</sup><sup>R</sup> is a rotating tensor expressing the relative rotation of the coordinate systems of an arbitrarily moving observer with respect to the reference coordinate system. The resulting yield surface is then no

, r ¼ σ<sup>y</sup> þ Q, center ¼

xx �

ffiffiffiffiffiffiffiffiffiffiffiffi PJ2ð Þ P q

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r<sup>2</sup> � σ<sup>2</sup> y

∂<sup>0</sup>u\_ pl ∂X

<sup>∂</sup><sup>X</sup> : ð Þ� <sup>1</sup> � xx <sup>∂</sup><sup>0</sup>u\_ pl

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λ � dt, with xx∈h i 0; 1 ,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

∂<sup>0</sup>u\_ pl <sup>∂</sup><sup>X</sup> : xx �

ð Þ� <sup>1</sup> � xx <sup>∂</sup><sup>0</sup>u\_ pl

λ � dt, with xx∈h i 0; 1 ,

<sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi

<sup>¼</sup> xx � \_ λ,

∂X

<sup>P</sup> : <sup>P</sup> <sup>p</sup> , (63)

and <sup>a</sup> <sup>¼</sup> center <sup>þ</sup> <sup>r</sup>

<sup>b</sup> , (64)

(65)

<sup>¼</sup> ð Þ� <sup>1</sup> � xx \_

(66)

λ,

$$
\dot{\lambda} \ge \mathbf{0}, \qquad \prescript{p}{}{\Psi} \le \mathbf{0}, \qquad \dot{\lambda} \cdot \mathfrak{L}\_P(^{p}\Psi) = \mathbf{0}, \tag{52}
$$

$$
\dot{\lambda} \ge 0, \qquad \tau \Psi \le 0, \qquad \dot{\lambda} \cdot \mathfrak{L}\_O^{\epsilon} (\tau \Psi) = 0,\tag{53}
$$

$$
\dot{\lambda} \ge 0, \qquad \prescript{\sigma}{}{\Psi} \le 0, \qquad \dot{\lambda} \cdot \mathfrak{L}\_T \Big( \prescript{\sigma}{}{\Psi} \Big) = 0. \tag{54}
$$

It should be noted here that the above conditions resulted from the invariance of the internal mechanical power and its first time derivative (see Eq. (27)), which now also define conjugate pairs of differentiation operators and derivatives in all stress spaces, and that all yield surface is the expression of the conservation of internal plastic power, which will be shown later.

Hyperelastic-plastic-based material models are essentially elastic material models. The starting point in their development is Eq. (19), wherein the constitutive equation of the material the elastic Green-Lagrangian strain tensor and its time derivative are modified as follows:

$$\mathbf{E}^{\*}\mathbf{E}^{el} = \frac{1}{2} \cdot \left[ \left( \mathbf{F}^{el} \right)^{T} \cdot \mathbf{F}^{el} - \mathbf{I} \right], \qquad \prescript{\*}{}{\mathbf{E}}^{el} = \frac{1}{2} \cdot \left[ \left( \dot{\mathbf{F}}^{el} \right)^{T} \cdot \mathbf{F}^{el} + \left( \mathbf{F}^{el} \right)^{T} \cdot \dot{\mathbf{F}}^{el} \right]. \tag{55}$$

In our research, the St. Venant hyperelastic material with internal damping was used, whose constitutive equation then takes any of the following forms:

$$\mathbf{S} = {}^{mat}\mathbf{C}^{el} : {}^{\*}\mathbf{E}^{el} + {}^{mat}\mathbf{C}^{vis} : {}^{\*}\dot{\mathbf{E}}^{el},\tag{56}$$

$$\mathbf{P} = \mathbf{F} \cdot \mathbf{S}, \qquad \boldsymbol{\sigma} = \mathbf{F} \cdot \mathbf{S} \cdot \mathbf{F}^T, \qquad \boldsymbol{\sigma} = \boldsymbol{J}^{-1} \cdot \mathbf{F} \cdot \mathbf{S} \cdot \mathbf{F}^T. \tag{57}$$

In the above equations and also in (19), the incremental form of the material gradient of the plastic displacement field is determined as follows:

$$\frac{\partial^{n+1,0}\mathbf{u}^{pl}}{\partial \mathbf{X}} = \Delta t \cdot \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} + \frac{\partial^{n,0} \mathbf{u}^{pl}}{\partial \mathbf{X}},\tag{58}$$

where the material gradient of the plastic velocity field ∂<sup>0</sup>u\_ pl=∂X is calculated in accordance with Eq. (23)1 or Eq. (36), depending on the reference definition of the yield surface. The corresponding loading/unloading discrete Kuhn-Tucker plastic optimization conditions then take the forms:


An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity DOI: http://dx.doi.org/10.5772/intechopen.85112

## 2.4 The reference definition of the yield surface

\_

\_

Advances in Composite Materials Development

\_

\_

internal plastic power, which will be shown later.

derivative are modified as follows:

<sup>2</sup> � <sup>F</sup>el � �<sup>T</sup> � <sup>F</sup>el � <sup>I</sup> h i

<sup>∗</sup> <sup>E</sup>el <sup>¼</sup> <sup>1</sup>

74

λ ≥0, <sup>τ</sup>

λ ≥0, <sup>S</sup>

λ ≥0, <sup>P</sup>Ψ ≤0, \_

λ ≥0, σΨ ≤0, \_

Ψ ≤0, \_

Ψ ≤0, \_

Hyperelastic-plastic-based material models are essentially elastic material models. The starting point in their development is Eq. (19), wherein the constitutive equation of the material the elastic Green-Lagrangian strain tensor and its time

, <sup>∗</sup> <sup>E</sup>\_ el <sup>¼</sup> <sup>1</sup>

used, whose constitutive equation then takes any of the following forms:

<sup>P</sup> <sup>¼</sup> <sup>F</sup> � <sup>S</sup>, <sup>τ</sup> <sup>¼</sup> <sup>F</sup> � <sup>S</sup> � <sup>F</sup>T, <sup>σ</sup> <sup>¼</sup> <sup>J</sup>

<sup>∂</sup><sup>X</sup> <sup>¼</sup> <sup>Δ</sup><sup>t</sup> �

λ ≥ 0, <sup>P</sup>Ψ ≤0, \_

λ ≥0, σΨ ≤0, \_

gradient of the plastic displacement field is determined as follows:

∂<sup>n</sup>þ1;<sup>0</sup>upl

λ ≥ 0, <sup>S</sup>

λ ≥0, <sup>τ</sup>

optimization conditions then take the forms:

\_

\_

\_

\_

In our research, the St. Venant hyperelastic material with internal damping was

In the above equations and also in (19), the incremental form of the material

∂<sup>0</sup>u\_ pl ∂X þ

where the material gradient of the plastic velocity field ∂<sup>0</sup>u\_ pl=∂X is calculated in accordance with Eq. (23)1 or Eq. (36), depending on the reference definition of the yield surface. The corresponding loading/unloading discrete Kuhn-Tucker plastic

Ψ ≤0, \_

Ψ ≤0, \_

<sup>λ</sup> � <sup>S</sup>

<sup>λ</sup> � <sup>τ</sup>

It should be noted here that the above conditions resulted from the invariance of the internal mechanical power and its first time derivative (see Eq. (27)), which now also define conjugate pairs of differentiation operators and derivatives in all stress spaces, and that all yield surface is the expression of the conservation of

<sup>λ</sup> � <sup>S</sup>

λ � L<sup>P</sup>

<sup>λ</sup> � <sup>L</sup><sup>O</sup> <sup>τ</sup>

Ψ

<sup>Ψ</sup>\_ <sup>¼</sup> <sup>0</sup>, (51)

<sup>P</sup><sup>Ψ</sup> � � <sup>¼</sup> <sup>0</sup>, (52)

� � <sup>¼</sup> <sup>0</sup>, (53)

<sup>λ</sup> � <sup>L</sup><sup>T</sup> <sup>σ</sup>Ψ� � <sup>¼</sup> <sup>0</sup>: (54)

<sup>2</sup> � <sup>F</sup>\_ el � �<sup>T</sup> � <sup>F</sup>el <sup>þ</sup> <sup>F</sup>el � �<sup>T</sup> � <sup>F</sup>\_ el h i

<sup>S</sup> <sup>¼</sup> matCel : <sup>∗</sup> <sup>E</sup>el <sup>þ</sup> matCvis : <sup>∗</sup> <sup>E</sup>\_ el, (56)

∂<sup>n</sup>;<sup>0</sup>upl

: (55)

�<sup>1</sup> � <sup>F</sup> � <sup>S</sup> � <sup>F</sup><sup>T</sup>: (57)

<sup>∂</sup><sup>X</sup> , (58)

Ψ ¼ 0, (59)

Ψ ¼ 0, (61)

<sup>λ</sup> � σΨ <sup>¼</sup> <sup>0</sup>: (62)

<sup>λ</sup> � <sup>P</sup><sup>Ψ</sup> <sup>¼</sup> <sup>0</sup>, (60)

Objective and thermodynamically consistent formulation of the plastic flow allows for the development of consistent material models. As a result, the material model can by formulated in any stress space and in whatever configuration of the body, though we intentionally omitted to give an example of the constitutive equation of our hypoelastic-plastic-based and hyperelastic-plastic-based material models in the intermediate configuration of the body, as it is just a matter of proper stress transformation using the multiplicative split of the deformation. Moreover, one of the formulations of the yield surface has to be a reference, from which other definitions of the yield surface in the second Piola-Kirchhoff, first Piola-Kirchhoff, Kirchhoff and Cauchy stress spaces can be calculated by solving Eq. (36). The various definitions of the yields surface then have the properties Eqs. (37)–(39), from which the 'normality rules' (Eqs. (38) and (39)) (whose physical meaning is defined by Eqs. (40) and (41)) are used in the return mapping/rate form of the return mapping algorithms to calculate the plastic multiplier. In this study the reference yield surface was defined in the first Piola-Kirchhoff stress space, because the corresponding plastic flow Eq. (23)1 is the only not constrained. Then the generalised J2 flow plasticity theory with isotropic hardening is defined by Eqs. (63)–(67). It should be noted that the PJ2ð Þ¼ <sup>P</sup> <sup>P</sup> : <sup>P</sup> invariant in the definition of the equivalent stress no longer bases on the deviatoric part of the first Piola-Kirchhoff stress tensor. The change was implied by the objectivity requirements, since the first Piola-Kirchhoff stress tensor transforms under the change of the observer as <sup>P</sup><sup>þ</sup> <sup>¼</sup> <sup>Q</sup><sup>R</sup> � <sup>P</sup> and PJ2ð Þ <sup>P</sup> is the only invariant, which is not affected by the change, i.e. PJ2ð Þ¼ <sup>P</sup> PJ<sup>2</sup> <sup>P</sup><sup>þ</sup> ð Þ, where <sup>Q</sup><sup>R</sup> is a rotating tensor expressing the relative rotation of the coordinate systems of an arbitrarily moving observer with respect to the reference coordinate system. The resulting yield surface is then no longer a cylinder but a sphere:

$${}^{P}\Psi = {}^{P}\sigma\_{eq} - {}^{P}\sigma\_{\mathcal{V}} \leq \mathbf{0}, \quad \text{where} \quad {}^{P}\sigma\_{eq} = {}^{P}\sigma\_{eq}(\mathbf{P}) = \sqrt{{}^{P}I\_{2}(\mathbf{P})} = \sqrt{\mathbf{P} : \mathbf{P}}, \tag{63}$$

$$\sigma^P \sigma\_\eta = F\_{UT1} \cdot \sqrt{r^2 - \left[a \cdot e^{\eta l} - \text{center}\right]^2},\\ r = \sigma\_\eta + Q, \quad \text{center} = \sqrt{r^2 - \sigma\_\eta^2} \text{ and } \quad a = \frac{\text{center} + r}{b},\tag{64}$$

$$\begin{aligned} \stackrel{spring}{e} \dot{\mathbf{e}}^{pl} &= \,^{spring} \dot{\mathbf{e}}^{pl} \left( \mathbf{x} \mathbf{x} \cdot \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} \right) = \sqrt{\mathbf{x} \mathbf{x} \cdot \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} \, : \mathbf{x} \mathbf{x} \cdot \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}}} = \mathbf{x} \mathbf{x} \cdot \dot{\boldsymbol{\lambda}}, \\\ ^{spring} \boldsymbol{e}^{pl} &= \int\_0^t \mathbf{s} \mathbf{r} \frac{\partial^l}{\partial \mathbf{x}} \cdot \mathbf{d}t = \int\_0^t \mathbf{x} \mathbf{x} \cdot \dot{\boldsymbol{\lambda}} \cdot d\mathbf{t}, \qquad \text{with} \qquad \boldsymbol{\text{xc}} \in \langle \mathbf{0}; \mathbf{1} \rangle, \end{aligned} \tag{65}$$

$${}^{dampr}\boldsymbol{\varepsilon}^{pl} = {}^{dampr}\boldsymbol{\varepsilon}^{pl} \left( (\mathbf{1} - \boldsymbol{\varepsilon}\mathbf{x}) \cdot \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} \right) = \sqrt{(\mathbf{1} - \boldsymbol{\varepsilon}\mathbf{x}) \cdot \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} \cdot (\mathbf{1} - \boldsymbol{\varepsilon}\mathbf{x}) \cdot \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}}} = (\mathbf{1} - \boldsymbol{\varepsilon}\mathbf{x}) \cdot \dot{\boldsymbol{\lambda}},\tag{6.18}$$
 
$${}^{dampr}\boldsymbol{\varepsilon}^{pl} = \int\_0^t {}^{dampr}\boldsymbol{\varepsilon}^{pl} \cdot \boldsymbol{dt} = \int\_0^t (\mathbf{1} - \boldsymbol{\varepsilon}\mathbf{x}) \cdot \dot{\boldsymbol{\lambda}} \cdot \boldsymbol{dt},\qquad \text{with} \qquad {}^{d\infty}\boldsymbol{\varepsilon}\mathbf{x} \in (0; 1),\tag{6.19}$$

$$\dot{\boldsymbol{\sigma}}^{pl} = \dot{\boldsymbol{\sigma}}^{pl} \left( \dot{\mathbf{F}}^{pl} \right) = \sqrt{\frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} : \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} = \dot{\boldsymbol{\lambda}}, \quad \boldsymbol{\varepsilon}^{pl} = \int\_0^t \dot{\boldsymbol{\varepsilon}}^{pl} \cdot d\mathbf{t} = \int\_0^t \dot{\boldsymbol{\lambda}} \cdot d\mathbf{t}, \quad \frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} = \dot{\boldsymbol{\lambda}} \cdot \frac{\partial^0 \mathbf{V}}{\partial \mathbf{X}}. \tag{67}$$

<sup>P</sup>σeq�Pσ<sup>y</sup> <sup>¼</sup> <sup>0</sup>: (68)

<sup>P</sup>σ<sup>y</sup> � <sup>S</sup><sup>0</sup> � dX <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_

<sup>∂</sup><sup>P</sup> : <sup>P</sup>: (69)

λ and the infinitesimal

pl UTX=∂X�

> � e\_ pl. It

<sup>P</sup>σy�

In order to show that the return mapping is thermodynamically consistent, the

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity

ffiffiffiffiffiffiffiffiffiffi <sup>P</sup> : <sup>P</sup> <sup>p</sup> : <sup>P</sup> <sup>¼</sup> <sup>∂</sup><sup>P</sup><sup>Ψ</sup>

<sup>λ</sup> � <sup>∂</sup><sup>P</sup>Ψ=∂<sup>P</sup> : <sup>P</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_ pl=∂<sup>X</sup> : <sup>P</sup> � dV<sup>0</sup> <sup>¼</sup> dWpl: Similarly, the

pl�

<sup>∂</sup><sup>P</sup> : <sup>L</sup>Pð Þ� <sup>P</sup> ½ � <sup>L</sup>Pð Þ <sup>P</sup>UT <sup>11</sup> <sup>¼</sup> <sup>0</sup>, (70)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>r</sup><sup>2</sup> � <sup>a</sup> � <sup>e</sup> ½ � pl � center <sup>2</sup>

<sup>∂</sup><sup>X</sup> � ½ � <sup>L</sup>Pð Þ <sup>P</sup>UT <sup>11</sup> � <sup>S</sup><sup>0</sup> � dX <sup>¼</sup> <sup>0</sup>: (71)

λ and the infinitesimal volume ele-

equivalent stress in the material model is manipulated as follows:

Then after multiplying Eq. (68) by the plastic multiplier \_

λ�

<sup>P</sup> : <sup>P</sup> <sup>p</sup> <sup>¼</sup> <sup>P</sup>

volume element dV<sup>0</sup> in the initial configuration of the body, the first term of Eq. (68) becomes the internal plastic power at a material constituent of the model

The second type of return mapping procedure, which is best suited for hypoelastic-plastic-based materials, is the rate form of the return mapping procedure. Its thermodynamically consistent form in terms of the normality rules is defined by Eq. (39). Here the objective time derivative of the yield surface is used for the plastic multiplier calculation, which in the case of our hypoelastic-plastic

<sup>P</sup>σ<sup>y</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>e</sup>\_

<sup>S</sup><sup>0</sup> � dX <sup>¼</sup> dWpl, which is just the internal plastic power at a material constituent of the specimen coming from the tensile test of the material, where S<sup>0</sup> ¼ S0ð Þ X is the

where LPð Þ P is then replaced by the rate form of the constitutive equation of the

should be noted that the first term of Eq. (70) can be replaced by any other term of Eq. (39), because the formulation is thermodynamically consistent. Moreover,

ment dV<sup>0</sup> in the initial configuration of the body, it is easy to see that by solving

pl UTX

Therefore, both return mapping algorithms in the above result in such a plastic multiplier calculation, during which the internal plastic power density of the models becomes the same as the internal plastic power density of the specimen, coming from the uniaxial tensile test of the material. It should also be noted that Eq. (70) is just the objective and thermodynamically consistent rate form of the Eq. (68) resulting from the invariance of the internal mechanical power and its first derivative Eq. (27), which now defines not only conjugate pairs of stress measures and strain or deformation rates but also conjugate differentiation operators and derivatives. Moreover, using Eqs. (20)–(23) and (40) and considering the fact that the physical meaning of Eqs. (68)–(71) is the conservation of plastic energy, the

� � q

material Eq. (43) and the second term on the LHS of Eq. (70) by the form

<sup>P</sup>σeq <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi

∂<sup>P</sup>Ψ

½ � <sup>L</sup>Pð Þ <sup>P</sup>UT <sup>11</sup> <sup>¼</sup> FUT<sup>11</sup> � �<sup>a</sup> � <sup>a</sup> � epl � center � � � � <sup>=</sup>

multiplying Eq. (70) by the plastic multiplier \_

Eq. (70), the following conservation equation is enforced:

<sup>∂</sup><sup>X</sup> : <sup>L</sup>Pð Þ� <sup>P</sup> dV<sup>0</sup> � <sup>∂</sup><sup>0</sup>u\_

\_

77

<sup>λ</sup> � <sup>P</sup>σeq � dV<sup>0</sup> <sup>¼</sup> \_

second term of Eq. (68) becomes \_

DOI: http://dx.doi.org/10.5772/intechopen.85112

cross-sectional area of the specimen.

material model takes the form

∂<sup>0</sup>u\_ pl

The actual yield stress <sup>P</sup>σy, which is a first Piola-Kirchhoff stress measure, determines the radius of the yield surface and is defined by Eq. (64)1. It is the only nonzero component of a stress tensor <sup>P</sup>UT (i.e. <sup>P</sup>σ<sup>y</sup> <sup>¼</sup> <sup>P</sup>UT<sup>11</sup> <sup>¼</sup> ½ � <sup>P</sup>UT <sup>11</sup>) coming from a uniaxial tensile test of the modelled material, where the operator ½ � ð Þ• <sup>11</sup> extracts the element in the first row and the first column of a second-order tensor, ð Þ• , is written as a 3 � 3 square matrix. The corresponding deformation gradient and the Jacobian of deformation are denoted as FUT, JUT, where FUT<sup>11</sup> ¼ ½ � FUT <sup>11</sup> and JUT ¼ detð Þ FUT . Please also note that the only nonzero element of the related second Piola-Kirchhoff stress tensor SUT, coming from the tensile test of the material, is SUT<sup>11</sup> <sup>¼</sup> ½ � <sup>S</sup>UT <sup>11</sup>¼<sup>S</sup>σ<sup>y</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>r</sup><sup>2</sup> � <sup>a</sup> � <sup>e</sup> ½ � pl � center <sup>2</sup> q . The equation defines an arc of a circle using three material parameters: the constant yield stress of the material σy, the maximum hardening stress Q by which the material can harden and the maximum accumulated plastic strain value b ¼ e pl max, at which the material loses its integrity, i.e. <sup>S</sup>σ<sup>y</sup> <sup>¼</sup> 0. The relationship between the corresponding stress measures then can be written in tensor form as PUT ¼ FUT � SUT, in which the parameters <sup>σ</sup>y, Q are second Piola-Kirchhoff stresses and <sup>e</sup>pl <sup>⊂</sup> h i <sup>0</sup>; <sup>b</sup> . It should also be noted that the definitions of the accumulated plastic strain rate e\_ pl (an equivalent strain rate defined by Eq. (67)1); the accumulated plastic strain epl (Eq. (67)2), which controls the hardening and denotes the plastic damage; and the equivalent stress <sup>P</sup>σ eq (Eq. (63)2), respectively, which have the following physical meaning in the uniaxial tensile test of the material, e\_ pl <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_ pl UTX=∂X, epl <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_ pl UTX=∂X, <sup>P</sup><sup>σ</sup> eq <sup>¼</sup> <sup>P</sup>UT<sup>11</sup> <sup>¼</sup> FUT<sup>11</sup> � SUT11, have changed. The changes were required by consistency conditions, so that the model could work properly in either a one-dimensional (1D) stress state or a three-dimensional (3D) stress state, which may occur at a material particle during the analysis. In the above equations, the variables spring epl (see Eq. (65)) and damperepl (see Eq. (66)) denote the ductile and brittle damage defined in terms of the ratio of ductile and total damage increment xx [21].

#### 2.5 Calculation of the plastic multiplier

A thermodynamically consistent formulation of the plastic flow allows for the calculation of the plastic multiplier in whatever (second Piola-Kirchhoff, first Piola-Kirchhoff, Kirchhoff and Cauchy) stress spaces using the corresponding definition of the yield surface. There are altogether two types of return mapping procedures for plastic multiplier calculation, which result in a thermodynamically consistent material model.

The first type of return mapping procedure, which is best suited for hyperelastic-plastic-based materials, is the ordinary return mapping procedure. Its thermodynamically consistent form in terms of the normality rules is defined by Eq. (38). Here the equation of the yield surface is solved directly for the plastic multiplier value, which in the case of our hyperelastic-plastic material takes the form (see Eq. (63)1)

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity DOI: http://dx.doi.org/10.5772/intechopen.85112

$$\prescript{P}{}{\sigma}\_{eq} - \prescript{P}{}{\sigma}\_{\mathcal{Y}} = \mathbf{0}.\tag{68}$$

In order to show that the return mapping is thermodynamically consistent, the equivalent stress in the material model is manipulated as follows:

$${}^{P}\sigma\_{eq} = \sqrt{\mathbf{P} : \mathbf{P}} = \frac{\mathbf{P}}{\sqrt{\mathbf{P} : \mathbf{P}}} : \mathbf{P} = \frac{\delta^{P} \Psi}{\partial \mathbf{P}} : \mathbf{P}. \tag{69}$$

Then after multiplying Eq. (68) by the plastic multiplier \_ λ and the infinitesimal volume element dV<sup>0</sup> in the initial configuration of the body, the first term of Eq. (68) becomes the internal plastic power at a material constituent of the model \_ <sup>λ</sup> � <sup>P</sup>σeq � dV<sup>0</sup> <sup>¼</sup> \_ <sup>λ</sup> � <sup>∂</sup><sup>P</sup>Ψ=∂<sup>P</sup> : <sup>P</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_ pl=∂<sup>X</sup> : <sup>P</sup> � dV<sup>0</sup> <sup>¼</sup> dWpl: Similarly, the second term of Eq. (68) becomes \_ λ� <sup>P</sup>σ<sup>y</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>e</sup>\_ pl� <sup>P</sup>σ<sup>y</sup> � <sup>S</sup><sup>0</sup> � dX <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_ pl UTX=∂X� <sup>P</sup>σy� <sup>S</sup><sup>0</sup> � dX <sup>¼</sup> dWpl, which is just the internal plastic power at a material constituent of the specimen coming from the tensile test of the material, where S<sup>0</sup> ¼ S0ð Þ X is the cross-sectional area of the specimen.

The second type of return mapping procedure, which is best suited for hypoelastic-plastic-based materials, is the rate form of the return mapping procedure. Its thermodynamically consistent form in terms of the normality rules is defined by Eq. (39). Here the objective time derivative of the yield surface is used for the plastic multiplier calculation, which in the case of our hypoelastic-plastic material model takes the form

$$\frac{\partial^{\mathbb{P}}\Psi}{\partial\mathbf{P}} \colon \mathfrak{L}\_{\mathbb{P}}(\mathbf{P}) - [\mathfrak{L}\_{\mathbb{P}}(\mathbf{P}\_{UT})]\_{\mathfrak{1}\mathfrak{1}} = \mathbf{0},\tag{70}$$

where LPð Þ P is then replaced by the rate form of the constitutive equation of the material Eq. (43) and the second term on the LHS of Eq. (70) by the form

$$[\mathfrak{L}\_{\mathbf{P}}(\mathbf{P}\_{UT})]\_{11} = F\_{UT11} \cdot \left\{ \left[ -a \cdot \left( a \cdot e^{pl} - \text{center} \right) \right] / \sqrt{r^2 - \left[ a \cdot e^{pl} - \text{center} \right]^2} \right\} \cdot \dot{e}^{pl} \text{ Jt}$$

should be noted that the first term of Eq. (70) can be replaced by any other term of Eq. (39), because the formulation is thermodynamically consistent. Moreover, multiplying Eq. (70) by the plastic multiplier \_ λ and the infinitesimal volume element dV<sup>0</sup> in the initial configuration of the body, it is easy to see that by solving Eq. (70), the following conservation equation is enforced:

$$\frac{\partial^0 \dot{\mathbf{u}}^{pl}}{\partial \mathbf{X}} : \mathfrak{L}\_{\mathbb{P}}(\mathbf{P}) \cdot dV\_0 - \frac{\partial^0 \dot{u}\_{\mathrm{UTX}}^{pl}}{\partial \mathbf{X}} \cdot [\mathfrak{L}\_{\mathbb{P}}(\mathbf{P}\_{\mathrm{UT}})]\_{\mathbf{11}} \cdot \mathfrak{S}\_0 \cdot d\mathbf{X} = \mathbf{0}. \tag{71}$$

Therefore, both return mapping algorithms in the above result in such a plastic multiplier calculation, during which the internal plastic power density of the models becomes the same as the internal plastic power density of the specimen, coming from the uniaxial tensile test of the material. It should also be noted that Eq. (70) is just the objective and thermodynamically consistent rate form of the Eq. (68) resulting from the invariance of the internal mechanical power and its first derivative Eq. (27), which now defines not only conjugate pairs of stress measures and strain or deformation rates but also conjugate differentiation operators and derivatives. Moreover, using Eqs. (20)–(23) and (40) and considering the fact that the physical meaning of Eqs. (68)–(71) is the conservation of plastic energy, the

e\_ pl <sup>¼</sup> <sup>e</sup>\_

pl <sup>F</sup>\_ pl � � <sup>¼</sup>

SUT<sup>11</sup> <sup>¼</sup> ½ � <sup>S</sup>UT <sup>11</sup>¼<sup>S</sup>σ<sup>y</sup> <sup>¼</sup>

tensile test of the material, e\_

material model.

form (see Eq. (63)1)

76

s

Advances in Composite Materials Development

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂<sup>0</sup>u\_ pl <sup>∂</sup><sup>X</sup> :

q

mum accumulated plastic strain value b ¼ e

the definitions of the accumulated plastic strain rate e\_

the ratio of ductile and total damage increment xx [21].

2.5 Calculation of the plastic multiplier

pl <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_

pl

∂<sup>0</sup>u\_ pl ∂X

¼ \_

<sup>λ</sup>, epl <sup>¼</sup>

The actual yield stress <sup>P</sup>σy, which is a first Piola-Kirchhoff stress measure, determines the radius of the yield surface and is defined by Eq. (64)1. It is the only nonzero component of a stress tensor <sup>P</sup>UT (i.e. <sup>P</sup>σ<sup>y</sup> <sup>¼</sup> <sup>P</sup>UT<sup>11</sup> <sup>¼</sup> ½ � <sup>P</sup>UT <sup>11</sup>) coming from a uniaxial tensile test of the modelled material, where the operator ½ � ð Þ• <sup>11</sup> extracts the element in the first row and the first column of a second-order tensor, ð Þ• , is written as a 3 � 3 square matrix. The corresponding deformation gradient and the Jacobian of deformation are denoted as FUT, JUT, where FUT<sup>11</sup> ¼ ½ � FUT <sup>11</sup> and

JUT ¼ detð Þ FUT . Please also note that the only nonzero element of the related second Piola-Kirchhoff stress tensor SUT, coming from the tensile test of the material, is

circle using three material parameters: the constant yield stress of the material σy, the maximum hardening stress Q by which the material can harden and the maxi-

integrity, i.e. <sup>S</sup>σ<sup>y</sup> <sup>¼</sup> 0. The relationship between the corresponding stress measures then can be written in tensor form as PUT ¼ FUT � SUT, in which the parameters <sup>σ</sup>y, Q are second Piola-Kirchhoff stresses and <sup>e</sup>pl <sup>⊂</sup> h i <sup>0</sup>; <sup>b</sup> . It should also be noted that

defined by Eq. (67)1); the accumulated plastic strain epl (Eq. (67)2), which controls the hardening and denotes the plastic damage; and the equivalent stress <sup>P</sup>σ eq

(Eq. (63)2), respectively, which have the following physical meaning in the uniaxial

FUT<sup>11</sup> � SUT11, have changed. The changes were required by consistency conditions, so that the model could work properly in either a one-dimensional (1D) stress state or a three-dimensional (3D) stress state, which may occur at a material particle during the analysis. In the above equations, the variables spring epl (see Eq. (65)) and damperepl (see Eq. (66)) denote the ductile and brittle damage defined in terms of

A thermodynamically consistent formulation of the plastic flow allows for the calculation of the plastic multiplier in whatever (second Piola-Kirchhoff, first Piola-Kirchhoff, Kirchhoff and Cauchy) stress spaces using the corresponding definition of the yield surface. There are altogether two types of return mapping procedures for plastic multiplier calculation, which result in a thermodynamically consistent

The first type of return mapping procedure, which is best suited for

hyperelastic-plastic-based materials, is the ordinary return mapping procedure. Its thermodynamically consistent form in terms of the normality rules is defined by Eq. (38). Here the equation of the yield surface is solved directly for the plastic multiplier value, which in the case of our hyperelastic-plastic material takes the

UTX=∂X, epl <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_

pl

pl

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>r</sup><sup>2</sup> � <sup>a</sup> � <sup>e</sup> ½ � pl � center <sup>2</sup>

ðt

ðt

<sup>λ</sup> � dt, <sup>∂</sup><sup>0</sup>u\_ pl

. The equation defines an arc of a

max, at which the material loses its

pl (an equivalent strain rate

UTX=∂X, <sup>P</sup><sup>σ</sup> eq <sup>¼</sup> <sup>P</sup>UT<sup>11</sup> <sup>¼</sup>

<sup>∂</sup><sup>X</sup> <sup>¼</sup> \_ λ � ∂<sup>P</sup>Ψ <sup>∂</sup><sup>X</sup> :

(67)

0 \_

0 e\_ pl � dt <sup>¼</sup> overall power at a material particle of the body dW can be decomposed into an elastic part dWel and a plastic part dWpl as follows:

$$dW = \dot{\mathbf{E}} : \mathbf{S} \cdot dV\_0 = \frac{\partial^0 \dot{\mathbf{u}}}{\partial \mathbf{X}} : \mathbf{P} \cdot dV\_0 = \mathbf{d} : \mathbf{r} \cdot dV\_0 = \mathbf{d} : \boldsymbol{\sigma} \cdot dv = dW^d + dW^l,\tag{72}$$

where

$$d\mathcal{W}^{cl} = \dot{\mathbf{E}}^{cl} : \mathbf{S} \cdot d\boldsymbol{V}\_0 = \frac{\partial^0 \dot{\mathbf{u}}^{cl}}{\partial \mathbf{X}} : \mathbf{P} \cdot d\boldsymbol{V}\_0 = \mathbf{d}^{cl} : \boldsymbol{\sigma} \cdot d\boldsymbol{V}\_0 = \mathbf{d}^{cl} : \boldsymbol{\sigma} \cdot d\boldsymbol{v},\tag{73}$$

$$d\mathscr{W}^{pl} = \dot{\lambda} \cdot \frac{\partial^{\mathbb{S}} \Psi}{\partial \mathbf{S}} : \mathbf{S} \cdot d\mathscr{V}\_{0} = \dot{\lambda} \cdot \frac{\partial^{p} \Psi}{\partial \mathbf{P}} : \mathbf{P} \cdot d\mathscr{V}\_{0} = \dot{\lambda} \cdot \frac{\partial^{r} \Psi}{\partial \mathbf{r}} : \mathbf{r} \cdot d\mathscr{V}\_{0} = \dot{\lambda} \cdot \frac{\partial^{r} \Psi}{\partial \mathbf{r}} : \boldsymbol{\sigma} \cdot d\mathbf{v},\tag{74}$$

which also proves that the formulation of the material model is thermodynamically consistent.

#### 2.6 The ratio of ductile and total damage increment

The idea of the ratio of ductile and total damage increment xx was first introduced by Écsi and Élesztős in order to take into account the internal damping properly during plastic deformation of the hypoelastic-plastic material, where xx allowed for the redistribution of the plastic flow between the spring and the damper of a 1D frictional device representing the rheological model of the material [21]. The ratio is determined in an elastic corrector phase during return mapping, and its value is then kept constant. Since the return mapping procedure in our material model is carried out in the first Piola-Kirchhoff stress space, we had to modify the definition of the ratio as follows:

$$\text{acc} = \frac{\left\langle \mathbf{N} : \mathbf{F} \cdot \left( ^{mat}\mathbb{C}^{el} : \dot{\mathbf{E}} \right) \right\rangle}{\left\langle \mathbf{N} : \mathbf{F} \cdot \left( ^{mat}\mathbb{C}^{el} : \dot{\mathbf{E}} \right) \right\rangle + \left\langle \mathbf{N} : \mathbf{F} \cdot \left( ^{mat}\mathbb{C}^{vis} : \ddot{\mathbf{E}} \right) \right\rangle},\tag{75}$$

In order to assess the value of the axial component of the deformation gradient coming from the tensile stress of the material FUT11, we solved the one-dimensional (1D) rate form of the constitutive equation of the material (Eq. (42)) for the unknown component of the derivative of the axial elastic displacement field with

equation of this specific 1D stress analysis, after neglecting the internal damping in

∂<sup>0</sup>uel x ∂X þ

S<sup>11</sup> is the axial component of the second Piola-Kirchhoff stress rate

" # r

" # !

el x <sup>∂</sup><sup>X</sup> � <sup>1</sup> <sup>þ</sup>

tensor from the tensile test of the material and E is the Young's modulus. Furthermore considering that the accumulated plastic strain rate Eq. (67) <sup>1</sup> in this 1D stress

<sup>λ</sup>, and that its integral is <sup>e</sup>pl <sup>¼</sup> <sup>∂</sup>0upl

pl � � <sup>þ</sup>

the material, can be expressed in the following finite-strain form:

Hypoelastic-plastic material with damping

DOI: http://dx.doi.org/10.5772/intechopen.85112

<sup>E</sup> [Pa] 7.31 � 1010 7.31 � <sup>10</sup><sup>10</sup> <sup>E</sup>vis ½ � Pa � <sup>s</sup> 7.31 � <sup>10</sup><sup>2</sup> 7.31 � <sup>10</sup><sup>2</sup>

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity

<sup>ν</sup> <sup>¼</sup> <sup>ν</sup>vis ½ �‐ 0.33 0.33 <sup>σ</sup><sup>y</sup> ½ � Pa 220.0 � <sup>10</sup><sup>6</sup> 220.0 � 106 <sup>Q</sup> ½ � Pa 110.0 � 106 110.0 � <sup>10</sup><sup>6</sup> <sup>b</sup> ½ �‐ <sup>10</sup><sup>64</sup> <sup>10</sup><sup>64</sup> <sup>ρ</sup><sup>0</sup> kg � <sup>m</sup>�<sup>3</sup> ½ � 2770.0 2770.0

<sup>S</sup><sup>11</sup> <sup>¼</sup> <sup>E</sup> � <sup>∂</sup><sup>0</sup>u\_

can find FUT<sup>11</sup> as a function of the accumulated plastic strain e

<sup>∂</sup><sup>X</sup> <sup>¼</sup> <sup>1</sup> þ � <sup>1</sup> <sup>þ</sup> <sup>e</sup>

<sup>x</sup> <sup>=</sup>∂X. The rate form of the constitutive

Hyperelastic-plastic material with/ without damping

/0.0

, (77)

<sup>x</sup> =∂X (Eq. (67)2), one

<sup>S</sup>σy E

pl only in the follow-

þ e

pl, (78)

∂<sup>0</sup>upl x ∂X

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> epl ð Þ<sup>2</sup> <sup>þ</sup> <sup>2</sup> �

respect to the axial material coordinate ∂<sup>0</sup>uel

S <sup>σ</sup>\_ <sup>y</sup> <sup>¼</sup> \_

where <sup>S</sup>σ\_ <sup>y</sup> <sup>¼</sup> \_

pl <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_

pl <sup>x</sup> <sup>=</sup>∂<sup>X</sup> <sup>¼</sup> \_

∂<sup>0</sup>uel x ∂X þ ∂0upl x

where <sup>S</sup>σ<sup>y</sup> <sup>¼</sup> <sup>S</sup>σy epl � � see also Eq. (64)1.

state is e\_

Table 1. Material properties.

Figure 2.

The spatially discretized model.

ing form:

79

FUT<sup>11</sup> ¼ 1 þ

where

$$\frac{\partial^p \Psi}{\partial \mathbf{P}} = \mathbf{N}, \qquad \mathbf{N} = \frac{\mathbf{P}}{\sqrt{\mathbf{P} : \mathbf{P}}} = \frac{\mathbf{P}}{||\mathbf{P}||}, \qquad \langle \mathbf{y} \rangle = \frac{y + |\mathbf{y}|}{2} \ge \mathbf{0}. \tag{76}$$

Eqs. (75) and (76)h iy denote the McCauly's brackets, which return zero if y < 0 and where we also used the transformation <sup>L</sup>Pð Þ¼ <sup>P</sup> <sup>F</sup> � <sup>S</sup>\_ . Please also note that all terms of Eq. (75) are objective stress rates, so that the value of xx is not affected by the change of the observer.

### 3. Numerical experiment

In our numerical experiment a cantilever, size 50 mm � 50 mm � 600 mm was studied applying dynamic pressure on 1/3 of its upper surface near the cantilever free end. The loading was defined as a product of a constant p = 3 MPa pressure and the Heaviside step function. The analysis was run as dynamic using <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>5</sup> � <sup>10</sup>�<sup>6</sup> <sup>s</sup> time step size. Table 1 lists the used material properties, and Figure 2 depicts the spatially discretized model of the cantilever.


An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity DOI: http://dx.doi.org/10.5772/intechopen.85112

Table 1. Material properties.

overall power at a material particle of the body dW can be decomposed into an

<sup>∂</sup><sup>P</sup> : <sup>P</sup> � dV<sup>0</sup> <sup>¼</sup> \_

which also proves that the formulation of the material model is thermodynami-

The idea of the ratio of ductile and total damage increment xx was first introduced by Écsi and Élesztős in order to take into account the internal damping properly during plastic deformation of the hypoelastic-plastic material, where xx allowed for the redistribution of the plastic flow between the spring and the damper of a 1D frictional device representing the rheological model of the material [21]. The ratio is determined in an elastic corrector phase during return mapping, and its value is then kept constant. Since the return mapping procedure in our material model is carried out in the first Piola-Kirchhoff stress space, we had to modify the

<sup>N</sup> : <sup>F</sup> � matCel : <sup>E</sup>\_ D E � �

el

λ � ∂<sup>P</sup>Ψ

2.6 The ratio of ductile and total damage increment

<sup>N</sup> : <sup>F</sup> � matCel : <sup>E</sup>\_ D E � �

ffiffiffiffiffiffiffiffiffiffi <sup>P</sup> : <sup>P</sup> <sup>p</sup> <sup>¼</sup> <sup>P</sup>

Eqs. (75) and (76)h iy denote the McCauly's brackets, which return zero if y < 0

terms of Eq. (75) are objective stress rates, so that the value of xx is not affected by

In our numerical experiment a cantilever, size 50 mm � 50 mm � 600 mm was studied applying dynamic pressure on 1/3 of its upper surface near the cantilever free end. The loading was defined as a product of a constant p = 3 MPa pressure and the Heaviside step function. The analysis was run as dynamic using <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>5</sup> � <sup>10</sup>�<sup>6</sup> <sup>s</sup> time step size. Table 1 lists the used material properties, and Figure 2 depicts the

<sup>∂</sup><sup>P</sup> <sup>¼</sup> <sup>N</sup>, <sup>N</sup> <sup>¼</sup> <sup>P</sup>

and where we also used the transformation <sup>L</sup>Pð Þ¼ <sup>P</sup> <sup>F</sup> � <sup>S</sup>\_

<sup>∂</sup><sup>X</sup> : <sup>P</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>d</sup> : <sup>τ</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>d</sup> : <sup>σ</sup> � dv <sup>¼</sup> dWel <sup>þ</sup> dWpl, (72)

λ � ∂τ Ψ

<sup>∂</sup><sup>X</sup> : <sup>P</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>d</sup>el : <sup>τ</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>d</sup>el : <sup>σ</sup> � dv, (73)

<sup>∂</sup><sup>τ</sup> : <sup>τ</sup> � dV<sup>0</sup> <sup>¼</sup> \_

<sup>þ</sup> <sup>N</sup> : <sup>F</sup> � matCvis : <sup>E</sup>€ D E � � , (75)

<sup>2</sup> <sup>≥</sup>0: (76)

. Please also note that all

k k<sup>P</sup> , yh i <sup>¼</sup> <sup>y</sup> <sup>þ</sup> j j <sup>y</sup>

λ � ∂<sup>σ</sup>Ψ

<sup>∂</sup><sup>σ</sup> : <sup>σ</sup> � dv,

(74)

elastic part dWel and a plastic part dWpl as follows:

dW <sup>¼</sup> <sup>E</sup>\_ : <sup>S</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_

dWel <sup>¼</sup> <sup>E</sup>\_ el : <sup>S</sup> � dV<sup>0</sup> <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_

Advances in Composite Materials Development

definition of the ratio as follows:

xx ¼

∂<sup>P</sup>Ψ

the change of the observer.

3. Numerical experiment

spatially discretized model of the cantilever.

<sup>∂</sup><sup>S</sup> : <sup>S</sup> � dV<sup>0</sup> <sup>¼</sup> \_

where

dWpl <sup>¼</sup> \_

cally consistent.

where

78

λ � ∂<sup>S</sup>Ψ

Figure 2. The spatially discretized model.

In order to assess the value of the axial component of the deformation gradient coming from the tensile stress of the material FUT11, we solved the one-dimensional (1D) rate form of the constitutive equation of the material (Eq. (42)) for the unknown component of the derivative of the axial elastic displacement field with respect to the axial material coordinate ∂<sup>0</sup>uel <sup>x</sup> <sup>=</sup>∂X. The rate form of the constitutive equation of this specific 1D stress analysis, after neglecting the internal damping in the material, can be expressed in the following finite-strain form:

$$\prescript{S}{}{\dot{\sigma}}\_{\mathcal{Y}} = \dot{\mathcal{S}}\_{11} = E \cdot \left[ \frac{\partial^0 \dot{u}\_{\mathcal{X}}^{cl}}{\partial X} \cdot \left( \mathbf{1} + \frac{\partial^0 u\_{\mathcal{X}}^{cl}}{\partial X} + \frac{\partial^0 u\_{\mathcal{X}}^{pl}}{\partial X} \right) \right], \tag{77}$$

where <sup>S</sup>σ\_ <sup>y</sup> <sup>¼</sup> \_ S<sup>11</sup> is the axial component of the second Piola-Kirchhoff stress rate tensor from the tensile test of the material and E is the Young's modulus. Furthermore considering that the accumulated plastic strain rate Eq. (67) <sup>1</sup> in this 1D stress state is e\_ pl <sup>¼</sup> <sup>∂</sup><sup>0</sup>u\_ pl <sup>x</sup> <sup>=</sup>∂<sup>X</sup> <sup>¼</sup> \_ <sup>λ</sup>, and that its integral is <sup>e</sup>pl <sup>¼</sup> <sup>∂</sup>0upl <sup>x</sup> =∂X (Eq. (67)2), one can find FUT<sup>11</sup> as a function of the accumulated plastic strain e pl only in the following form:

$$F\_{UT11} = \mathbf{1} + \frac{\partial^0 u\_{\mathbf{x}}^{cl}}{\partial \mathbf{X}} + \frac{\partial^0 u\_{\mathbf{x}}^{pl}}{\partial \mathbf{X}} = \mathbf{1} + \left[ -\left(\mathbf{1} + \mathbf{e}^{pl}\right) + \sqrt{\left(\mathbf{1} + \mathbf{e}^{pl}\right)^2 + 2 \cdot \frac{\mathbf{S}\boldsymbol{\sigma}\mathbf{y}}{E}} \right] + \mathbf{e}^{pl}, \quad \text{(78)}$$

where <sup>S</sup>σ<sup>y</sup> <sup>¼</sup> <sup>S</sup>σy epl � � see also Eq. (64)1.

## 4. Numerical results

Figure 3 depicts a few selected analysis results. These are the first principal Cauchy stress and the accumulated plastic strain distributions over the body, which is similar in the case of both materials, the vertical displacement time history curves and the accumulated plastic strain time history curves for the used material at selected nodes (see Figure 2 for the location of the nodes).

numerical experiments. The most important implication of the presented theory is that the analysis results of the related models are no longer affected by the description and the particularities of the mathematical formulation. The nonlinear continuum theory was also briefly presented, while the thermodynamic consistency of the formulation was in detail discussed. Another important implication of the theory is that the dissipated plastic power density of the model can directly be related to the dissipated plastic power density of the specimen coming from the uniaxial tensile stress of the modelled material. Moreover, contemporary tensile testing for material parameter determination will also have to be extended by determination of the deformation gradient of the specimen of the modelled material, as it is an important

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity

Funding from the VEGA 1/0740/16 grant, the KEGA 017STU-4/2018 grant and

Ladislav Écsi\*, Pavel Élesztős, Róbert Jerábek, Roland Jančo and Branislav Hučko Department of Applied Mechanics and Mechatronics, Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, Bratislava, Slovakia

© 2019 The Author(s). Licensee IntechOpen. 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,

\*Address all correspondence to: ladislav.ecsi@stuba.sk

provided the original work is properly cited.

entry for the presented material models.

DOI: http://dx.doi.org/10.5772/intechopen.85112

the APVV-15-0757 grant resources are greatly appreciated.

Acknowledgements

Author details

81

It should be noted that in order to avoid problems with convergence with the used hypoelastic-plastic material, the value of the b parameter had to be set extraordinarily high. As a result, the isotropic hardening curve becomes flat in the range of the accumulated plastic strain value that occurred in the analysis, i.e. no isotropic hardening took place in the analysis. Moreover, for the same reasons, a very small viscous damping parameter has to be used with the model. The viscous damping, however, did not affect the analysis result at this kind of intensive loading, resulting in the plastic collapse of the beam, which can be seen in the time history curves. The analysis results otherwise seem to be reasonable.

Figure 3. Selected results from the analyses.

## 5. Conclusions

In this chapter an alternative framework for developing objective and thermodynamically consistent hypoelastic-plastic- and hyperelastic-plastic-based material models was presented using the first nonlinear continuum theory for finite deformations of elastoplastic media. The related material models were demonstrated in

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity DOI: http://dx.doi.org/10.5772/intechopen.85112

numerical experiments. The most important implication of the presented theory is that the analysis results of the related models are no longer affected by the description and the particularities of the mathematical formulation. The nonlinear continuum theory was also briefly presented, while the thermodynamic consistency of the formulation was in detail discussed. Another important implication of the theory is that the dissipated plastic power density of the model can directly be related to the dissipated plastic power density of the specimen coming from the uniaxial tensile stress of the modelled material. Moreover, contemporary tensile testing for material parameter determination will also have to be extended by determination of the deformation gradient of the specimen of the modelled material, as it is an important entry for the presented material models.

## Acknowledgements

4. Numerical results

Advances in Composite Materials Development

5. Conclusions

Selected results from the analyses.

Figure 3.

80

Figure 3 depicts a few selected analysis results. These are the first principal Cauchy stress and the accumulated plastic strain distributions over the body, which is similar in the case of both materials, the vertical displacement time history curves and the accumulated plastic strain time history curves for the used material at

It should be noted that in order to avoid problems with convergence with the

In this chapter an alternative framework for developing objective and thermodynamically consistent hypoelastic-plastic- and hyperelastic-plastic-based material models was presented using the first nonlinear continuum theory for finite deformations of elastoplastic media. The related material models were demonstrated in

used hypoelastic-plastic material, the value of the b parameter had to be set extraordinarily high. As a result, the isotropic hardening curve becomes flat in the range of the accumulated plastic strain value that occurred in the analysis, i.e. no isotropic hardening took place in the analysis. Moreover, for the same reasons, a very small viscous damping parameter has to be used with the model. The viscous damping, however, did not affect the analysis result at this kind of intensive loading, resulting in the plastic collapse of the beam, which can be seen in the time

history curves. The analysis results otherwise seem to be reasonable.

selected nodes (see Figure 2 for the location of the nodes).

Funding from the VEGA 1/0740/16 grant, the KEGA 017STU-4/2018 grant and the APVV-15-0757 grant resources are greatly appreciated.

## Author details

Ladislav Écsi\*, Pavel Élesztős, Róbert Jerábek, Roland Jančo and Branislav Hučko Department of Applied Mechanics and Mechatronics, Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, Bratislava, Slovakia

\*Address all correspondence to: ladislav.ecsi@stuba.sk

© 2019 The Author(s). Licensee IntechOpen. 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.

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An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity

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[14] Écsi L, Élesztős P. An alternative material model using a generalized J2 finite-strain flow plasticity theory with isotropic hardening. International Journal of Applied Mechanics and Engineering. 2018;2(23):351-365. DOI: 10.2478/ijame-2018-0019

[15] Gurtin ME. An Introduction to Continuum Mechanics. Orlando: Academic Press; 1981. p. 265. ISBN-10: 9780123097507

[16] Holzapfel GA. Nonlinear Solid Mechanics. A Continuum Approach for Engineering. Chichester: Wiley; 2001. p. 455. ISBN-10: 0471823198

[17] Bonet J, Wood RD. Nonlinear Continuum Mechanics for Finite

An Alternative Framework for Developing Material Models for Finite-Strain Elastoplasticity DOI: http://dx.doi.org/10.5772/intechopen.85112

Element Analysis. 2nd ed. Cambridge: Cambridge University Press; 2008. p. 340. ISBN-10: 0521838703

References

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1118632702

ISBN-10: 0470694521

10: 0120020238

90047-9

82

[1] Simo JC, Hughes TJR. Computational Inelasticity. New York: Springer; 2000.

Advances in Composite Materials Development

[9] Meyers A, Xiao H, Bruhns OT. Choice of objective rate in single parameter hypoelastic deformation cycles. Computers and Structures. 2006;

84:1134-1140. DOI: 10.1016/j. compstruc.2006.01.012

10.1007/BF02096162

10.1007/BF01379008

[10] Lehmann T. Anisotrope plastishe Formänderungen. Romanian Journal of Technical Sciences. Series: Applied Mechanics. 1972;17:1077-1086. DOI:

[11] Diens JK. On the analysis of rotation and stress rate in deforming bodies. Acta Mechanica. 1979;32:217-232. DOI:

[12] Simo JC, Pistner KS. Remark on the rate of constitutive equations for finite deformation problems: Computational implications. Computer Methods in Applied Mechanics. 1984;46:201-215

[13] Perić D. On constitutive stress rates in solid mechanics: Computational implications. The International Journal for Numerical Methods in Engineering.

1992;33:799-817. DOI: 10.1002/

10.2478/ijame-2018-0019

9780123097507

[14] Écsi L, Élesztős P. An alternative material model using a generalized J2 finite-strain flow plasticity theory with isotropic hardening. International Journal of Applied Mechanics and Engineering. 2018;2(23):351-365. DOI:

[15] Gurtin ME. An Introduction to Continuum Mechanics. Orlando: Academic Press; 1981. p. 265. ISBN-10:

[16] Holzapfel GA. Nonlinear Solid Mechanics. A Continuum Approach for Engineering. Chichester: Wiley; 2001.

[17] Bonet J, Wood RD. Nonlinear Continuum Mechanics for Finite

p. 455. ISBN-10: 0471823198

nme.1620330409

392 p. ISBN-10: 0387975209

[2] Nemat-Nasser S. Plasticity: A Treatise on Finite Deformation of Heterogeneous Inelastic Materials. Cambridge: Cambridge University Press; 2004. 760 p. ISBN-10:

[3] Belytschko T, Liu WK, Moran B. Nonlinear Finite Elements for Continua and Structures. 2nd ed. Chichester: Wiley; 2014. 830 p. ISBN-10:

[4] De Souza Neto EA, Perić D, Owen DRJ. Computational Methods for Plasticity: Theory and Applications. 1st ed. Singapore: Wiley; 2008. 814 p.

[5] Asaro RJ. Micromechanics of crystals and polycrystals. In: Hutchinson JW, Wu TY, editors. Advances in Applied Mechanics. Vol. 23. New York:

Academic Press; 1983. pp. 1-115. ISBN-

[6] Peirce D, Asaro RJ, Needlemann A. An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metallurgica. 1982;30:1087-1119. DOI: 10.1016/0001-6160(82)90005-0

[7] Peirce D. Shear band bifurcation in ductile single crystals. Journal of the Mechanics and Physics of Solids. 1983; 31:133-153. DOI: 10.1016/0022-5096(83)

[8] Kojić M, Bathe KJ. Studies of finite element procedures–Stress solution of a

formulation. Computers and Structures. 1987;28(1/2):175-179. DOI: 10.1016/

closed elastic strain paths with stretching and shearing using the updated Lagrangian Jaumann

0045-7949(87)90247–1

[18] Spencer AJM. Continuum Mechanics. 1st ed. New York: Longman; 2012. p. 208. ISBN-10: 0486435946

[19] Borisenko AI, Tarapov IE. Vector and Tensor Analysis with Applications. New York: Dover; 2012. p. 288. ISBN-10: 0486638332

[20] Maugin GA. The saga of internal variables of state in continuum thermomechanics. Research Communications. 2015;69:79-86. DOI: 10.1016/j. mechrescom.2015.06.009

[21] Écsi L, Élesztős P. Moving toward a more realistic material model of a ductile material with failure mode transition. Materialwissenschaft und Werkstofftechnik. 2012;5(43):379-387. DOI: 10.1002/mawe.201200969

Section 4

Applications for Nuclear

Fuel Elements

85

Section 4

## Applications for Nuclear Fuel Elements

Chapter 5

Abstract

CERMET fuels.

1. Introduction

density, uranium distribution

fuel systems under consideration.

87

Dennis S. Tucker

CERMETS for Use in Nuclear

NASA is currently investigating nuclear thermal propulsion as an alternative to chemical propulsion for manned missions to the outer planets. There are a number of materials being considered for use as fuel elements. These materials include tricarbides and CERMETS such as W/UO2, Mo/UO2, W/UN and Mo/UN. All of these materials require high temperature processing to achieve the required densities. It has been found that Spark Plasma Sintering is a good choice for sintering these materials to the required densities while maintaining a uniform grain size. In this chapter a brief history of NASA's research into nuclear thermal propulsion will be given, followed by specific research by this author and others to produce

Keywords: nuclear thermal propulsion, fuel elements, spark plasma sintering,

Long duration spaceflight can expose astronauts to two major problems. These

necessary to develop alternate means of propulsion to that of chemical propulsion. A reasonable option being studied by NASA is nuclear thermal propulsion (NTP), whereby nuclear fuel elements made from metal/UO2, metal/UN or tricarbides are being considered. These fuel elements must be capable of operating in excess of

From previous studies it was shown that W/UO2 as a fuel element can be used both in power and propulsion at temperatures as high as 3000 K [2–4]. Two factors need to be considered when producing W/UO2 fuel elements. The first is the importance of a uniform distribution of UO2 particles in the tungsten matrix. If one has segregation of UO2 particles it can lead to hot spots and ultimately failure of the fuel element. The second factor which needs to be considered is the stoichiometry of the UO2 particles. Maintaining stoichiometry is vital to ensure stability and proper operation of the fuel element. This chapter will detail a brief history of NTP by NASA including fuel element work, followed by more recent research on various

are extended periods of weightlessness and radiation exposure. Thus, it is

2700 K while being compatible with the propellant, typically hydrogen [1].

Thermal Propulsion

## Chapter 5

## CERMETS for Use in Nuclear Thermal Propulsion

Dennis S. Tucker

## Abstract

NASA is currently investigating nuclear thermal propulsion as an alternative to chemical propulsion for manned missions to the outer planets. There are a number of materials being considered for use as fuel elements. These materials include tricarbides and CERMETS such as W/UO2, Mo/UO2, W/UN and Mo/UN. All of these materials require high temperature processing to achieve the required densities. It has been found that Spark Plasma Sintering is a good choice for sintering these materials to the required densities while maintaining a uniform grain size. In this chapter a brief history of NASA's research into nuclear thermal propulsion will be given, followed by specific research by this author and others to produce CERMET fuels.

Keywords: nuclear thermal propulsion, fuel elements, spark plasma sintering, density, uranium distribution

## 1. Introduction

Long duration spaceflight can expose astronauts to two major problems. These are extended periods of weightlessness and radiation exposure. Thus, it is necessary to develop alternate means of propulsion to that of chemical propulsion. A reasonable option being studied by NASA is nuclear thermal propulsion (NTP), whereby nuclear fuel elements made from metal/UO2, metal/UN or tricarbides are being considered. These fuel elements must be capable of operating in excess of 2700 K while being compatible with the propellant, typically hydrogen [1]. From previous studies it was shown that W/UO2 as a fuel element can be used both in power and propulsion at temperatures as high as 3000 K [2–4]. Two factors need to be considered when producing W/UO2 fuel elements. The first is the importance of a uniform distribution of UO2 particles in the tungsten matrix. If one has segregation of UO2 particles it can lead to hot spots and ultimately failure of the fuel element. The second factor which needs to be considered is the stoichiometry of the UO2 particles. Maintaining stoichiometry is vital to ensure stability and proper operation of the fuel element. This chapter will detail a brief history of NTP by NASA including fuel element work, followed by more recent research on various fuel systems under consideration.

## 2. NTP history

As far back as the 1940s, it was recognized that energy from nuclear fission could be used to power spacecraft by heating a working fluid such as hydrogen and provide thrust to the rocket via expansion of the propellant through a rocket nozzle. A simple drawing of an NTP rocket is shown in Figure 1 below.

Due to the high specific impulse, NTP is considered to be the preferred propulsion method for future manned flights to Mars. Specific impulse (Isp) is a method to measure and compare the efficiency of different propulsion systems. It is determined by the ratio of thrust to the propellant mass flow rate through the engine. The typical NTP engine would have and Isp = 900 s, which is twice that of chemical propulsion systems. During a Mars manned mission the engine would be run for a total of 4 hours. One hour to accelerate from Earth to Mars, followed by a 1 hour deceleration burn. The same burn cycles would occur on a return trip. Thus it is quite critical for the fuel elements to retain their stability during these burns.

The United States was involved in the production and testing of NTP engines during the period of 1955–1972. This was the Rover/NERVA program which tested 20 prototype reactors during this period. These prototypes included fuel test reactors, a safety reactor and prototype engines. Figure 2 below shows a test of one of the NERVA engines. This engine reached an Isp of 850 s during a 2 hour burn. Twenty prototype reactors were ground tested. Fuel forms evolved over the duration of the program [2–9].

The fuel elements used during testing were of varying compositions. These were coated graphite-matrix elements followed by advanced fuel elements consisting of UC-ZrC-C and all carbide elements ((U, Zr)C) [5–8]. Most of the testing was performed using the coated graphite fuel elements. These elements were full length (52″) with a hexagonal cross section (0.75″ flat-to-flat) and 19 axial holes for propellant flow. These elements were arranged to create a cylindrical reactor core. The NERVA/Rover program proved NTP to be a viable technology [9]. Several prototype reactors were produced which survived multiple restarts and power levels over 4000 MW, thrust levels of 250 klbf, maximum propellant outlet temperatures of 2550 K, a maximum net specific impulse of 850 s and over an hour of

continuous operation [9]. The fuel elements had to perform various tasks. The elements contained fissile material (UO2 or UC2) and graphite as a moderator. The

Presently there are a number of fuels under consideration for NTP. These include graphite composites, tricarbides (U-Zr-Nb)C and CERMETS (MUO2 and MUN). At Marshall Space Flight Center, we are concentrating on CERMETS (W/UO2, Mo/UO2, W/UN, Mo/UN) and tricarbides. Consolidation of these

fuel elements also functioned as structural components.

3. Present-day NTP fuels research

CERMETS for Use in Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.85220

Figure 2.

89

Test of NERVA engine.

Figure 1. Schematic of NTP rocket.

CERMETS for Use in Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.85220

2. NTP history

Advances in Composite Materials Development

tion of the program [2–9].

Figure 1.

88

Schematic of NTP rocket.

As far back as the 1940s, it was recognized that energy from nuclear fission could be used to power spacecraft by heating a working fluid such as hydrogen and provide thrust to the rocket via expansion of the propellant through a rocket nozzle.

Due to the high specific impulse, NTP is considered to be the preferred propulsion method for future manned flights to Mars. Specific impulse (Isp) is a method to measure and compare the efficiency of different propulsion systems. It is determined by the ratio of thrust to the propellant mass flow rate through the engine. The typical NTP engine would have and Isp = 900 s, which is twice that of chemical propulsion systems. During a Mars manned mission the engine would be run for a total of 4 hours. One hour to accelerate from Earth to Mars, followed by a 1 hour deceleration burn. The same burn cycles would occur on a return trip. Thus it is quite critical for the fuel elements to retain their stability during these burns. The United States was involved in the production and testing of NTP engines during the period of 1955–1972. This was the Rover/NERVA program which tested 20 prototype reactors during this period. These prototypes included fuel test reactors, a safety reactor and prototype engines. Figure 2 below shows a test of one of the NERVA engines. This engine reached an Isp of 850 s during a 2 hour burn. Twenty prototype reactors were ground tested. Fuel forms evolved over the dura-

The fuel elements used during testing were of varying compositions. These were coated graphite-matrix elements followed by advanced fuel elements consisting of UC-ZrC-C and all carbide elements ((U, Zr)C) [5–8]. Most of the testing was performed using the coated graphite fuel elements. These elements were full length

(52″) with a hexagonal cross section (0.75″ flat-to-flat) and 19 axial holes for propellant flow. These elements were arranged to create a cylindrical reactor core. The NERVA/Rover program proved NTP to be a viable technology [9]. Several prototype reactors were produced which survived multiple restarts and power levels over 4000 MW, thrust levels of 250 klbf, maximum propellant outlet temperatures of 2550 K, a maximum net specific impulse of 850 s and over an hour of

A simple drawing of an NTP rocket is shown in Figure 1 below.

Figure 2. Test of NERVA engine.

continuous operation [9]. The fuel elements had to perform various tasks. The elements contained fissile material (UO2 or UC2) and graphite as a moderator. The fuel elements also functioned as structural components.

## 3. Present-day NTP fuels research

Presently there are a number of fuels under consideration for NTP. These include graphite composites, tricarbides (U-Zr-Nb)C and CERMETS (MUO2 and MUN). At Marshall Space Flight Center, we are concentrating on CERMETS (W/UO2, Mo/UO2, W/UN, Mo/UN) and tricarbides. Consolidation of these

CERMETS and tricarbides has been performed using RF induction furnaces, hot pressing and spark plasma sintering (SPS).

With respect to CERMET processing a number of approaches have been utilized in order to obtain a uniform distribution of the fissile material within the metal matrix. These include traditional powder processing techniques and coating the fissile material with metal using chemical vapor deposition (CVD) [10]. Recently, a fluidized bed reactor to coat UO2 with tungsten was tested [11]. CVD shows great promise, however, there are technical issues due to the complexity of the experimental CVD apparatus and the CVD process. These issues lead to expense and long reaction times to apply the appropriate thickness coatings. Thus, a technique for obtaining a uniformly coated spherical UO2 particles was developed [12]. In this technique a small amount of high density polyethylene binder (0.25 w/o) is added to a mixture of tungsten and UO2 particles. The powders are then mixed thoroughly using a turbula, then heated and mixed on a magnetic stir plate.

Traditional sintering methods can be used to densify W/UO2. Both hot pressing and hot isostatic pressing have been used. There are drawbacks to these two methods including incomplete sintering and dissociation of UO2 at high temperatures, pressures and long sintering times. Another issue is a problem of exaggerated grain growth which can occur under these processing conditions.

mixed for 45 minutes in a turbula. This powder was placed in a 400 ml Pyrex beaker and then stirred on a hot plate for 10 minutes above the drop point of the polyethylene (101°C). This process was repeated until 500 g was produced. The powder was shipped to the Center for Space Nuclear Research in Idaho Falls, Idaho for sintering in the SPS. Thirty one grams was placed in a graphene die for sintering. Samples were densified at 1600, 1700, 1750, 1800 and 1850°C. Samples were heated at a rate of 100°C/minute to the sintering temperature. The pressure was increased by 10 Mpa/minute to 50 Mpa. After soaking at the maximum temperature for 20 minutes, the pressure was decreased by 10 Mpa/minute to 5 Mpa and the

Scanning electron micrographs of hot isostatic pressed W/UO2. Dark area on left is UO2 while on right shows

Density was obtained using the Archimedes method. Carbon content was analyzed using instrumental gas analysis (EAG, NY). Scanning electron microscopy with energy dispersive x-ray analysis was performed on all samples. Microstructural and chemical analyses were carried out by using transmission electron microscopy (TEM) and atom probe tomography (APT) techniques. TEM and APT specimens were prepared at phase boundaries using lift-out methods with a focused ion beam (FIB). The size of each TEM lamella was 10 10 μm. TEM characterization was carried out using a FEI Tecnai G2 F30 STEM FEG equipped with energy dispersive x-ray spectrometry (EDS). The EDS analyses were done in Scanning TEM (STEM) mode with a beam size of 1 nm. APT was carried out using a CAMECA LEAP 4000X HR. APT data reconstruction was done using the CAMECA IVAS software. Figure 4 shows a scanning electron micrograph of UO2 particles coated with

In Figure 4 one can note that the UO2 particles are almost completely covered with the tungsten powder. The polyethylene binder is viscous above its drop point (101°C) and coats both the UO2 particle and tungsten particles which when subsequently mixed together results in the image in Figure 4. The mixing temperature was 140°C which led to a binder viscosity of 140 cP. As the mixture was stirred, the nearly spherical UO2 particles rolled around the bottom of the beaker and were

As can be seen in Table 1 below, the density is relatively high at a sintering temperature of 1600°C and gradually increases up to 1800°C and jumps to 99.46%

The lower sintering temperature densities align well with what was previously reported for which the density was reported as 97.9% of theoretical for W-Re/UO2 at 1500°C and 40 Mpa applied pressure using SPS [20]. The higher sintering

temperature was decreased by 20°C/minute to room temperature.

tungsten powder.

91

Figure 3.

higher magnification image of UO2.

CERMETS for Use in Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.85220

coated with the tungsten particles.

of theoretical density at 1850°C.

A sintering method which has been shown to be a reasonable alternative to these traditional method is Spark Plasma Sintering (SPS) [13–15]. SPS leads to higher densities at lower temperatures and processing times while minimizing grain growth. Grain growth is detrimental to densification during the sintering process. In one study, UO2 was produced by hot pressing, however it was found that a large number of pores were present on the grain faces which led to a smaller grain boundary contact area [16]. In this same study it was observed that grains which had not undergone exaggerated grain growth had pores at the grain corners. It has been observed that pores located on grain faces have greater mobility than those at grain corners and ultimately end up within the grains [17, 18]. Joule heating is utilized in SPS which results in passing a current through the powder during sintering [19]. A pulsed current is utilized in SPS which leads to two different operating temperatures: the average temperature and the maximum temperature. The average temperature is lower than the melting point of the materials. During current discharge, material is transported by a plasma across pores of the matrix. While the pulse is off, the matrix cools rapidly, and this lead to condensation of the material vapor within regions where there is mechanical contact between grains. This mechanism leads to necking between grains. There have been a number of studies have using SPS to consolidate tungsten and a surrogate or tungsten and UO2. In one study W/CeO2 was sintered using SPS [19]. In a second study W/UO2 was densified using SPS [20]. A shortcoming in both of these studies was the segregation of the tungsten and the oxides. In the W/UO2 study, the materials were mixed in a turbula for 1 hour then hot isostatic pressed [21]. This result was a segregated CERMET due to the differences in powder sizes (W-15 μm, UO2-200 μm) and density differences where size differences made the largest difference. In Figure 3 on can plainly see segregation in the sintered CERMET.

Studies [22, 23] were undertaken to eliminate this segregation using an inexpensive, simple technique. Depleted UO2 particles were obtained from Oak Ridge National Laboratory. These particles had an average size of 200 μm. Tungsten powder with a particle size of 5/15 μm was purchased and used as the matrix material. In order to coat the UO2 particles with tungsten, a powder processing technique was developed. In this technique, high molecular weight polyethylene powder was milled to approximately 1 μm in diameter. Next a mixture of 50 g of 60 vol% UO2, 40 vol% W and 0.25 wt% polyethylene powder were thoroughly

Figure 3.

CERMETS and tricarbides has been performed using RF induction furnaces, hot

in order to obtain a uniform distribution of the fissile material within the metal matrix. These include traditional powder processing techniques and coating the fissile material with metal using chemical vapor deposition (CVD) [10]. Recently, a fluidized bed reactor to coat UO2 with tungsten was tested [11]. CVD shows great promise, however, there are technical issues due to the complexity of the experimental CVD apparatus and the CVD process. These issues lead to expense and long reaction times to apply the appropriate thickness coatings. Thus, a technique for obtaining a uniformly coated spherical UO2 particles was developed [12]. In this technique a small amount of high density polyethylene binder (0.25 w/o) is added to a mixture of tungsten and UO2 particles. The powders are then mixed thoroughly

using a turbula, then heated and mixed on a magnetic stir plate.

grain growth which can occur under these processing conditions.

on can plainly see segregation in the sintered CERMET.

90

Studies [22, 23] were undertaken to eliminate this segregation using an inexpensive, simple technique. Depleted UO2 particles were obtained from Oak Ridge National Laboratory. These particles had an average size of 200 μm. Tungsten powder with a particle size of 5/15 μm was purchased and used as the matrix material. In order to coat the UO2 particles with tungsten, a powder processing technique was developed. In this technique, high molecular weight polyethylene powder was milled to approximately 1 μm in diameter. Next a mixture of 50 g of 60 vol% UO2, 40 vol% W and 0.25 wt% polyethylene powder were thoroughly

With respect to CERMET processing a number of approaches have been utilized

Traditional sintering methods can be used to densify W/UO2. Both hot pressing

A sintering method which has been shown to be a reasonable alternative to these

traditional method is Spark Plasma Sintering (SPS) [13–15]. SPS leads to higher densities at lower temperatures and processing times while minimizing grain growth. Grain growth is detrimental to densification during the sintering process. In one study, UO2 was produced by hot pressing, however it was found that a large number of pores were present on the grain faces which led to a smaller grain boundary contact area [16]. In this same study it was observed that grains which had not undergone exaggerated grain growth had pores at the grain corners. It has been observed that pores located on grain faces have greater mobility than those at grain corners and ultimately end up within the grains [17, 18]. Joule heating is utilized in SPS which results in passing a current through the powder during sintering [19]. A pulsed current is utilized in SPS which leads to two different operating temperatures: the average temperature and the maximum temperature. The average temperature is lower than the melting point of the materials. During current discharge, material is transported by a plasma across pores of the matrix. While the pulse is off, the matrix cools rapidly, and this lead to condensation of the material vapor within regions where there is mechanical contact between grains. This mechanism leads to necking between grains. There have been a number of studies have using SPS to consolidate tungsten and a surrogate or tungsten and UO2. In one study W/CeO2 was sintered using SPS [19]. In a second study W/UO2 was densified using SPS [20]. A shortcoming in both of these studies was the segregation of the tungsten and the oxides. In the W/UO2 study, the materials were mixed in a turbula for 1 hour then hot isostatic pressed [21]. This result was a segregated CERMET due to the differences in powder sizes (W-15 μm, UO2-200 μm) and density differences where size differences made the largest difference. In Figure 3

and hot isostatic pressing have been used. There are drawbacks to these two methods including incomplete sintering and dissociation of UO2 at high temperatures, pressures and long sintering times. Another issue is a problem of exaggerated

pressing and spark plasma sintering (SPS).

Advances in Composite Materials Development

Scanning electron micrographs of hot isostatic pressed W/UO2. Dark area on left is UO2 while on right shows higher magnification image of UO2.

mixed for 45 minutes in a turbula. This powder was placed in a 400 ml Pyrex beaker and then stirred on a hot plate for 10 minutes above the drop point of the polyethylene (101°C). This process was repeated until 500 g was produced. The powder was shipped to the Center for Space Nuclear Research in Idaho Falls, Idaho for sintering in the SPS. Thirty one grams was placed in a graphene die for sintering. Samples were densified at 1600, 1700, 1750, 1800 and 1850°C. Samples were heated at a rate of 100°C/minute to the sintering temperature. The pressure was increased by 10 Mpa/minute to 50 Mpa. After soaking at the maximum temperature for 20 minutes, the pressure was decreased by 10 Mpa/minute to 5 Mpa and the temperature was decreased by 20°C/minute to room temperature.

Density was obtained using the Archimedes method. Carbon content was analyzed using instrumental gas analysis (EAG, NY). Scanning electron microscopy with energy dispersive x-ray analysis was performed on all samples. Microstructural and chemical analyses were carried out by using transmission electron microscopy (TEM) and atom probe tomography (APT) techniques. TEM and APT specimens were prepared at phase boundaries using lift-out methods with a focused ion beam (FIB). The size of each TEM lamella was 10 10 μm. TEM characterization was carried out using a FEI Tecnai G2 F30 STEM FEG equipped with energy dispersive x-ray spectrometry (EDS). The EDS analyses were done in Scanning TEM (STEM) mode with a beam size of 1 nm. APT was carried out using a CAMECA LEAP 4000X HR. APT data reconstruction was done using the CAMECA IVAS software.

Figure 4 shows a scanning electron micrograph of UO2 particles coated with tungsten powder.

In Figure 4 one can note that the UO2 particles are almost completely covered with the tungsten powder. The polyethylene binder is viscous above its drop point (101°C) and coats both the UO2 particle and tungsten particles which when subsequently mixed together results in the image in Figure 4. The mixing temperature was 140°C which led to a binder viscosity of 140 cP. As the mixture was stirred, the nearly spherical UO2 particles rolled around the bottom of the beaker and were coated with the tungsten particles.

As can be seen in Table 1 below, the density is relatively high at a sintering temperature of 1600°C and gradually increases up to 1800°C and jumps to 99.46% of theoretical density at 1850°C.

The lower sintering temperature densities align well with what was previously reported for which the density was reported as 97.9% of theoretical for W-Re/UO2 at 1500°C and 40 Mpa applied pressure using SPS [20]. The higher sintering

This figure shows the distribution of UO2 particles within the tungsten matrix. The UO2 particles are the darker near spherical particles in the lighter gray tungsten matrix. As can be seen, the distribution of UO2 particles is nearly uniform with the

extremely difficult using traditional powder mixing techniques without the aid of a binder. There four properties which give rise to segregation in powder mixes are: particle size difference, variations in particle density, along with shape and resilience [24]. Particle size difference has been shown to be the most important factor [21]. Segregation of powders causes fluctuations in the size distributions of particles and this leads to variations in bulk density which can affect the desired properties. There are three mechanisms of segregation which can occur during mixing and vibration. Vibration is often used to increase packing density in the powder which in leads to higher sintered densities. Segregation can occur during mixing when fine particles travel further than coarse particles during the mixing operation. If a mass of particles is disturbed so that individual particles move, a rearrangement can take place. This is termed percolation. Over time, gaps between particles occur, which allows particles from above to move down, while a particles from some other location replaces them. When the powder mass contains different size particles, small particles will fall through the large particle interstices leading to segregation. Percolation occurs when the mass of particles is disturbed due to a shear stress within the particle mass. This phenomenon is explained in which a large particle causes an increase in pressure in the region below it which compacts the material and stops the particle from moving downward. Upward movement allows fines to run in under the coarse particle and these in turn lock in position. If the vibration intensity is large enough the larger particles will migrate to the powder surface. The powder process described above which uses polyethylene binder, overcomes this difficulty using a minimum amount of binder which burns out during the SPS process and is drawn away from the sample by the vacuum system. In all the sintering temperatures listed in Table 1 the vacuum was <sup>2</sup> <sup>10</sup><sup>3</sup> Torr. The binder acts as an adhesive for the W/UO2 powders eliminating the disparate particle

tungsten matrix. Obtaining a uniform mixture of disparate particle sizes is

size effect. It was found that the carbon content for the mixed powders was 0.025 wt%, while the sintered samples were below the detectable limit which is in

Figure 6 below shows EDS maps for the sample sintered at 1850°C.

In this figure one can see the SEM in the figure on the left and the x-ray maps on the center and right figure. The UO2 particles are blue and the tungsten matrix is orange. One can also note some fracturing of the UO2 particles. This is most likely due to the pressure during sintering but could also be caused during the grinding

parts per million.

93

Figure 5.

SEM of SPS sintered W/UO2 at 1850°C.

CERMETS for Use in Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.85220

#### Figure 4.

Scanning electron micrograph showing UO2 particles coated with tungsten powder.


Table 1.

Sintering temperature versus % theoretical density for SPS W/UO2.

temperatures and increased applied pressure used in this study can account for an increase in density seen in Table 1. It is also thought that decreasing the cooling rate to 20°C/minute could also have contributed to further densification. At lower cooling rated the insulated die will retain heat which will allow more densification.

Figure 5 below is a low magnification SEM micrograph of the sample sintered at 1850°C.

CERMETS for Use in Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.85220

Figure 5. SEM of SPS sintered W/UO2 at 1850°C.

This figure shows the distribution of UO2 particles within the tungsten matrix. The UO2 particles are the darker near spherical particles in the lighter gray tungsten matrix. As can be seen, the distribution of UO2 particles is nearly uniform with the tungsten matrix. Obtaining a uniform mixture of disparate particle sizes is extremely difficult using traditional powder mixing techniques without the aid of a binder. There four properties which give rise to segregation in powder mixes are: particle size difference, variations in particle density, along with shape and resilience [24]. Particle size difference has been shown to be the most important factor [21]. Segregation of powders causes fluctuations in the size distributions of particles and this leads to variations in bulk density which can affect the desired properties. There are three mechanisms of segregation which can occur during mixing and vibration. Vibration is often used to increase packing density in the powder which in leads to higher sintered densities. Segregation can occur during mixing when fine particles travel further than coarse particles during the mixing operation. If a mass of particles is disturbed so that individual particles move, a rearrangement can take place. This is termed percolation. Over time, gaps between particles occur, which allows particles from above to move down, while a particles from some other location replaces them. When the powder mass contains different size particles, small particles will fall through the large particle interstices leading to segregation. Percolation occurs when the mass of particles is disturbed due to a shear stress within the particle mass. This phenomenon is explained in which a large particle causes an increase in pressure in the region below it which compacts the material and stops the particle from moving downward. Upward movement allows fines to run in under the coarse particle and these in turn lock in position. If the vibration intensity is large enough the larger particles will migrate to the powder surface. The powder process described above which uses polyethylene binder, overcomes this difficulty using a minimum amount of binder which burns out during the SPS process and is drawn away from the sample by the vacuum system. In all the sintering temperatures listed in Table 1 the vacuum was <sup>2</sup> <sup>10</sup><sup>3</sup> Torr. The binder acts as an adhesive for the W/UO2 powders eliminating the disparate particle size effect. It was found that the carbon content for the mixed powders was 0.025 wt%, while the sintered samples were below the detectable limit which is in parts per million.

Figure 6 below shows EDS maps for the sample sintered at 1850°C.

In this figure one can see the SEM in the figure on the left and the x-ray maps on the center and right figure. The UO2 particles are blue and the tungsten matrix is orange. One can also note some fracturing of the UO2 particles. This is most likely due to the pressure during sintering but could also be caused during the grinding

temperatures and increased applied pressure used in this study can account for an increase in density seen in Table 1. It is also thought that decreasing the cooling rate to 20°C/minute could also have contributed to further densification. At lower cooling rated the insulated die will retain heat which will allow more densification. Figure 5 below is a low magnification SEM micrograph of the sample sintered at

Sintering temperature versus % theoretical density for SPS W/UO2.

Scanning electron micrograph showing UO2 particles coated with tungsten powder.

Advances in Composite Materials Development

1850°C.

92

Table 1.

Figure 4.

and polishing operation. This should be avoided to lessen the probability that uranium can escape and diffuse to the tungsten matrix and to the fuel surface. One can also note some UO2 particle-particle contact which can lead to hot spots in the fuel during operation. This in turn could lead to disruption of the fuel element.

Using high resolution transmission electron microscopy (HRTEM) it is possible to image the boundaries between UO2 particles and the tungsten matrix. Figure 7

This type of boundary is typical for all samples except for the one sintered

There was an anomalous phase which was identified as U0.1WO3, space group Pm-3m (221) which is a cubic structure. This was based on its electron diffraction pattern and by considering the atomic ratio of U:W = 1:10 which is consistent with EDS results. Tungsten trioxide, WO3, has a monoclinic structure with a space group mP32. It is possible that this anomalous phase is WO3 with uranium contamination, since U0.1WO3 phase has not been previously identified in the literature. Since the space group and crystal structure are different for these two phases, this could be a new phase. High resolution, high intensity x-ray diffraction could be performed on all sintered samples to make a definite determination. It could be that the U0.1WO3 phase forms due to the availability of oxygen vacancies from the UO2 reduction due to sintering in vacuum. The EDS line scan across the W/UO2 boundary for the

Figure 9a shows the length of the line scan across the boundary. In Figure 9b it can be seen that the uranium has diffused approximately 15 nm into the tungsten matrix. The green line is the uranium curve and one measures where it crosses over the blue curve (tungsten). For all other sintered samples, it was seen that the uranium diffused approximately 10 nm into the tungsten matrix. The atom imaging probe analysis for the sample sintered at 1850°C is shown below in Figure 10.

It can be seen from Figure 10 that the uranium and oxygen were present in the form of UO, UO2 and UO3. The nitrogen present is most likely from the nitrogen gas backfill used during SPS at room temperature after cooling. Silicon was observed for all samples except the one processed at 1750°C. Its origin is unknown but most likely is an impurity picked up during grinding and polishing. The carbon present in the sample which is from the polyethylene binder used during powder processing. The above data led to a formula given as UO1.95 which is slightly sub-stoichiometric. The samples sintered at 1600, 1650 and 1700°C were also calculated to have this same formula. The only difference was for the sample processed at 1750°C which

at 1600°C which also showed an anomalous third phase. This is shown in

below shows a typical region in the sample sintered at 1850°C.

CERMETS for Use in Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.85220

sample sintered at 1850°C is shown in Figure 9a.

Figure 8 below.

had the formula UO2.

Figure 8.

95

TEM of sample sintered at 1600°C showing phase U0.1WO3.

Figure 6. SEM (left figure), EDS maps (UO2—blue and W—orange) for sample sintered at 1850°C.

HRTEM image of boundary of W/UO2 for sample sintered at 1850°C: (a) is the low resolution boundary and (b) is the high resolution image of this boundary.

### CERMETS for Use in Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.85220

and polishing operation. This should be avoided to lessen the probability that uranium can escape and diffuse to the tungsten matrix and to the fuel surface. One can also note some UO2 particle-particle contact which can lead to hot spots in the fuel during operation. This in turn could lead to disruption of the fuel element.

Advances in Composite Materials Development

SEM (left figure), EDS maps (UO2—blue and W—orange) for sample sintered at 1850°C.

HRTEM image of boundary of W/UO2 for sample sintered at 1850°C: (a) is the low resolution boundary and

Figure 6.

Figure 7.

94

(b) is the high resolution image of this boundary.

Using high resolution transmission electron microscopy (HRTEM) it is possible to image the boundaries between UO2 particles and the tungsten matrix. Figure 7 below shows a typical region in the sample sintered at 1850°C.

This type of boundary is typical for all samples except for the one sintered at 1600°C which also showed an anomalous third phase. This is shown in Figure 8 below.

There was an anomalous phase which was identified as U0.1WO3, space group Pm-3m (221) which is a cubic structure. This was based on its electron diffraction pattern and by considering the atomic ratio of U:W = 1:10 which is consistent with EDS results. Tungsten trioxide, WO3, has a monoclinic structure with a space group mP32. It is possible that this anomalous phase is WO3 with uranium contamination, since U0.1WO3 phase has not been previously identified in the literature. Since the space group and crystal structure are different for these two phases, this could be a new phase. High resolution, high intensity x-ray diffraction could be performed on all sintered samples to make a definite determination. It could be that the U0.1WO3 phase forms due to the availability of oxygen vacancies from the UO2 reduction due to sintering in vacuum. The EDS line scan across the W/UO2 boundary for the sample sintered at 1850°C is shown in Figure 9a.

Figure 9a shows the length of the line scan across the boundary. In Figure 9b it can be seen that the uranium has diffused approximately 15 nm into the tungsten matrix. The green line is the uranium curve and one measures where it crosses over the blue curve (tungsten). For all other sintered samples, it was seen that the uranium diffused approximately 10 nm into the tungsten matrix. The atom imaging probe analysis for the sample sintered at 1850°C is shown below in Figure 10.

It can be seen from Figure 10 that the uranium and oxygen were present in the form of UO, UO2 and UO3. The nitrogen present is most likely from the nitrogen gas backfill used during SPS at room temperature after cooling. Silicon was observed for all samples except the one processed at 1750°C. Its origin is unknown but most likely is an impurity picked up during grinding and polishing. The carbon present in the sample which is from the polyethylene binder used during powder processing. The above data led to a formula given as UO1.95 which is slightly sub-stoichiometric. The samples sintered at 1600, 1650 and 1700°C were also calculated to have this same formula. The only difference was for the sample processed at 1750°C which had the formula UO2.

Figure 8. TEM of sample sintered at 1600°C showing phase U0.1WO3.

Oo <sup>¼</sup> <sup>V</sup>o€ <sup>þ</sup> <sup>1</sup>=2O2 <sup>þ</sup> 2 M'

and free uranium as shown in Eq. (4).

CERMETS for Use in Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.85220

particle. This is shown in Eqs. (5) and (6) below.

will be a rare earth oxide addition.

4. Conclusions

97

The result of either of these reactions will be a sub-stoichiometric uranium oxide

The free uranium from this reaction is then available to diffuse into the tungsten matrix. This mechanism occurs due to Fick's law of diffusion. The importance of the presence of free uranium in sintered W/UO2 samples cannot be overstated. These materials will be exposed to hydrogen gas in a thermal cycling environment during engine operation. When thermal cycling takes place in a hydrogen environment, hydrogen will penetrate into the tungsten matrix by both grain boundary and bulk diffusion. The hydrogen can then combine with the free uranium leading to uranium hydride. Uranium hydride can also be formed by reaction with the UO2

The free uranium has a melting point of �1130°C and will rapidly diffuse along the tungsten grain boundaries and form UH3 at �225°C. The formation of UH3 leads to large increases in volume and which can result in tungsten grain separation. This grain separation creates avenues for migration of UO2 to the CERMET surface. This results in the loss of UO2 and can lead to mechanical failure. The free uranium not only forms UH3, but can also reoxidize to form UO2. Both mechanisms result in a large volume expansion and loss of mechanical integrity. There is also a difference between isothermal and cyclic heating. It has been shown that cycling heating

It has been found that oxides such as ThO2, Ce2O3 and Y2O3 reduce fuel loss when added to the CERMET powder [25]. The observation was that the oxide additives did not increase the solubility of uranium in UO2, but stabilized UO2 against oxygen loss. Two mechanisms were proposed to explain the stabilization against oxygen loss: (1) oxide additives lower the partial molar free energy of oxygen in the UO2. This precludes the possibility of forming free uranium upon cooling and (2) when metal oxide is added to the CERMET powder, uranium is transformed to a hexavalent state. This hexavalent state precludes the formation of uranium metal. The UO2 maintains an oxygen-to-metal ration of 2.0–2.1 by forming a defect lattice structure. To maintain electrical neutrality, the uranium ions will be in the hexavalent state. U4<sup>+</sup> cannot be reduced to the metal in the presence of U6<sup>+</sup>

Thus, the initial loss of oxygen from the CERMET will be accompanied by oxygen vacancies rather than the formation of free uranium. The use of hyperstoi-

chiometric uranium oxide (UO2+x). UO2 CERMETS in which the O/U ratio of the starting composition was varied between 1.93 and 2.05 was studied. It was shown in this study that there was minimal effect of varying this ratio. Thus the most likely candidate to stabilize UO2 during sintering and thermal cycling in hydrogen

In this chapter a brief history of the nuclear thermal propulsion program was

given. A present-day research into processing and properties of nuclear fuel

results in more fuel loss than isothermal heating in hydrogen [25].

2UO2�<sup>x</sup> ! ð Þ 2 � x UO2 þ xU (4)

UO2 þ H2 ! UO2�<sup>x</sup> þ xH2O (5)

2UO2�<sup>x</sup> ! ð Þ 2 � x UO2 þ xU (6)

<sup>M</sup> (3)

.

#### Figure 9.

EDS line scan across the W/UO2 boundary for sample sintered at 1850°C in (a) and the results are in (b).

Figure 10.

3-D element maps from UO2 particles and atomic percent from sample sintered at 1850°C.

The loss of uranium from the UO2 particles and uranium migration into the tungsten matrix can be understood in terms of the generation of oxygen vacancies during sintering in a vacuum environment. An reaction for UO2 if oxygen vacancies are abundant is given by Eq. (1).

$$\mathsf{U}\mathsf{O}\_{2} = \mathsf{U}\mathsf{O}\_{2-\mathsf{x}} + \mathsf{x}/2\mathsf{O}\_{2} \tag{1}$$

With the loss of oxygen there are two possible defect reactions that can occur. The first reaction is electronic compensation leading to the creation of oxygen vacancies and electrons. This is shown in Eq. (2).

$$\mathbf{O\_{0}} = \mathbb{V}\ddot{o} + \mathbb{V}\mathbf{O\_{2}} + \mathbf{2e} \tag{2}$$

Ionic substitution can lead to the formation of oxygen vacancies and reduction of the metal oxide on their sites as shown below in Eq. (3).

CERMETS for Use in Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.85220

$$\mathbf{O}\_{\bullet} = \mathsf{V}\ddot{o} + \mathsf{I}/2\mathsf{O}\_{2} + \mathsf{Z}\,\mathsf{M}\_{\bullet} \tag{3}$$

The result of either of these reactions will be a sub-stoichiometric uranium oxide and free uranium as shown in Eq. (4).

$$\mathsf{2U}\mathsf{O}\_{2-\mathsf{x}} \to (\mathsf{2}-\mathsf{x})\mathsf{U}\mathsf{O}\_{2} + \mathsf{x}\mathsf{U} \tag{4}$$

The free uranium from this reaction is then available to diffuse into the tungsten matrix. This mechanism occurs due to Fick's law of diffusion. The importance of the presence of free uranium in sintered W/UO2 samples cannot be overstated. These materials will be exposed to hydrogen gas in a thermal cycling environment during engine operation. When thermal cycling takes place in a hydrogen environment, hydrogen will penetrate into the tungsten matrix by both grain boundary and bulk diffusion. The hydrogen can then combine with the free uranium leading to uranium hydride. Uranium hydride can also be formed by reaction with the UO2 particle. This is shown in Eqs. (5) and (6) below.

$$\mathsf{U}\mathsf{O}\_{2} + \mathsf{H}\_{2} \to \mathsf{U}\mathsf{O}\_{2-\mathsf{x}} + \mathsf{x}\mathsf{H}\_{2}\mathsf{O}\tag{5}$$

$$\mathsf{2U}\mathsf{O}\_{2-\mathsf{x}} \to (\mathsf{2}-\mathsf{x})\mathsf{U}\mathsf{O}\_{2} + \mathsf{x}\mathsf{U} \tag{6}$$

The free uranium has a melting point of �1130°C and will rapidly diffuse along the tungsten grain boundaries and form UH3 at �225°C. The formation of UH3 leads to large increases in volume and which can result in tungsten grain separation. This grain separation creates avenues for migration of UO2 to the CERMET surface. This results in the loss of UO2 and can lead to mechanical failure. The free uranium not only forms UH3, but can also reoxidize to form UO2. Both mechanisms result in a large volume expansion and loss of mechanical integrity. There is also a difference between isothermal and cyclic heating. It has been shown that cycling heating results in more fuel loss than isothermal heating in hydrogen [25].

It has been found that oxides such as ThO2, Ce2O3 and Y2O3 reduce fuel loss when added to the CERMET powder [25]. The observation was that the oxide additives did not increase the solubility of uranium in UO2, but stabilized UO2 against oxygen loss. Two mechanisms were proposed to explain the stabilization against oxygen loss: (1) oxide additives lower the partial molar free energy of oxygen in the UO2. This precludes the possibility of forming free uranium upon cooling and (2) when metal oxide is added to the CERMET powder, uranium is transformed to a hexavalent state. This hexavalent state precludes the formation of uranium metal. The UO2 maintains an oxygen-to-metal ration of 2.0–2.1 by forming a defect lattice structure. To maintain electrical neutrality, the uranium ions will be in the hexavalent state. U4<sup>+</sup> cannot be reduced to the metal in the presence of U6<sup>+</sup> . Thus, the initial loss of oxygen from the CERMET will be accompanied by oxygen vacancies rather than the formation of free uranium. The use of hyperstoichiometric uranium oxide (UO2+x). UO2 CERMETS in which the O/U ratio of the starting composition was varied between 1.93 and 2.05 was studied. It was shown in this study that there was minimal effect of varying this ratio. Thus the most likely candidate to stabilize UO2 during sintering and thermal cycling in hydrogen will be a rare earth oxide addition.

### 4. Conclusions

In this chapter a brief history of the nuclear thermal propulsion program was given. A present-day research into processing and properties of nuclear fuel

The loss of uranium from the UO2 particles and uranium migration into the tungsten matrix can be understood in terms of the generation of oxygen vacancies during sintering in a vacuum environment. An reaction for UO2 if oxygen vacancies

EDS line scan across the W/UO2 boundary for sample sintered at 1850°C in (a) and the results are in (b).

3-D element maps from UO2 particles and atomic percent from sample sintered at 1850°C.

With the loss of oxygen there are two possible defect reactions that can occur. The first reaction is electronic compensation leading to the creation of oxygen

Ionic substitution can lead to the formation of oxygen vacancies and reduction

UO2 ¼ UO2�<sup>x</sup> þ x=2O2 (1)

Oo ¼ Vo€ þ ½O2 þ 2e (2)

are abundant is given by Eq. (1).

Figure 10.

96

Figure 9.

Advances in Composite Materials Development

vacancies and electrons. This is shown in Eq. (2).

of the metal oxide on their sites as shown below in Eq. (3).

elements was discussed. In particular W/UO2 which was spark plasma sintered was discussed. Uranium migration into the tungsten matrix was observed for all samples. The presence of uranium was explained in terms of oxygen vacancy generation due to processing in vacuum and the migration of the uranium by Fick's law of diffusion. Possible solutions to this problem were also discussed.

References

2007

[1] Burkes DE, Wachs DM, Werner JE, Howe SD. An Overview of Current and Past W-UO2 CERMET Fuel Fabrication Technology. Space Nuclear Conference;

CERMETS for Use in Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.85220

> [11] Hickman RR, Broadway JW, Mirales OR. Fabrication and testing of CERMET fuel materials for nuclear thermal propulsion. In: AIAA Joint Propulsion Conference; 13 July-1 August 2012;

[12] Tucker DS, O'Connor A, Hickman R. A methodology for producing a uniform mixture of UO2 in a tungsten matrix. Journal of Physical Science and Applications. 2015;5:255-262. DOI: 10.17265/2159-5348/2015.04.002

[13] O'Brien RC, Ambrosi RM, Bannister NP, Howe SD, Atkinson HV. Spark plasma sintering of simulated

radioisotope materials within tungsten CERMETS. Journal of Nuclear Materials. 2009;393:108-113. DOI: 10.1016/j.

[14] Wang X, Xie Y, Fuo H, Van der Blest O, Vieugels J. Sintering of WC-Co powder with nanocrystalline WC by spark plasma sintering. Rare Metals.

Fundamental of hot isostatic pressing: An overview. Metallurgical and Materials Transactions. 2000;A31:2981

[16] Xu A, Soloman AA. The effects of grain growth on the intergranular porosity distribution in hot pressed and swelled UO2. In: Proceedings of Ceramic Microstructures '86; July 1986; Plenum, Berkley, CA: University of California. 1986. pp. 28-31. ISBN: 0306426811

[17] Carpay FMA. The effect of pore drag on ceramic microstructures. In: Proceedings of Ceramic Microstructures '76; Westview, Boulder, CO. 1977.

[18] German RM. Sintering Theory and Practice. New York: John Wiley; 1996.

p. 171. ISBN: 0891583076

ISBN: 0-471-05786-X

[15] Atkinson HV, Davies S.

Atlanta, Georgia. 2012

jnucmat.2009.05.012

2006;25:246

[2] Bhattacharyya SK. An Assessment of Fuels for Nuclear Thermal Propulsion,

Expansion W/UO2 Nuclear Fuel, NASA-

[5] Koenig DR. Experience Gained from the Space Nuclear Rocket Program (ROVER), Rep. No. LA-10062-H; Los Alamos National Laboratory; 1986

[6] Lyon LL. Performance of (U,Zr)C-Graphite (Composite) and (U,Zr)C (Carbide) Fuel Elements in the Nuclear Furnace 1 Test Reactor, Rep. No. LA-5398-MS; Los Alamos National

[7] Wallace TC. Review of Rover fuel element protective coating development at Los Alamos. In: Proceedings of the Eight Symposium on Space Nuclear Power Systems. 1991. pp. 1024-1036

[8] Taub JM. A Review of Fuel Element Development for Nuclear Rocket Engines, Rep. No. LA-5931; Los Alamos

[9] Qualls L. Graphite Fuel Development for Nuclear Thermal Propulsion; Oak Ridge National Laboratory; 2013

[10] Benensky K. Summary of historical solid Core nuclear thermal propulsion

Pennsylvania, Penn State University;

National Laboratory; 1975

fuels [thesis]. State College

2013

99

ANL/TD/TM01-22; 2001

CR-72711; 1970

Laboratory; 1973

[3] Marlowe MO, Kaznoff AI. Development of Low Thermal

[4] Baker RJ. Basic Behavior and Properties of W/UO2 CERMETS,

NASA-CR-54840; 1965

## Acknowledgements

The author would like to thank the Nuclear Thermal Propulsion office at Marshall Space Flight Center for funding this work. The author would also like to thank the Center for Space Nuclear Research for performing the spark plasma experiments and the Center for Advanced Energy Studies for performing TEM and atom probe measurements. Both institutes are located in Idaho Falls, Idaho, USA.

## Conflict of interest

There is no conflict of interest represented by this work.

## Author details

Dennis S. Tucker NASA Marshall Space Flight Center, Alabama, United States

\*Address all correspondence to: dr.dennis.tucker@nasa.gov

© 2019 The Author(s). Licensee IntechOpen. 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.

CERMETS for Use in Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.85220

## References

elements was discussed. In particular W/UO2 which was spark plasma sintered was discussed. Uranium migration into the tungsten matrix was observed for all samples. The presence of uranium was explained in terms of oxygen vacancy generation due to processing in vacuum and the migration of the uranium by Fick's law of

The author would like to thank the Nuclear Thermal Propulsion office at Marshall Space Flight Center for funding this work. The author would also like to thank the Center for Space Nuclear Research for performing the spark plasma experiments and the Center for Advanced Energy Studies for performing TEM and atom probe measurements. Both institutes are located in Idaho Falls,

diffusion. Possible solutions to this problem were also discussed.

There is no conflict of interest represented by this work.

NASA Marshall Space Flight Center, Alabama, United States

\*Address all correspondence to: dr.dennis.tucker@nasa.gov

provided the original work is properly cited.

© 2019 The Author(s). Licensee IntechOpen. 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,

Acknowledgements

Advances in Composite Materials Development

Conflict of interest

Author details

Dennis S. Tucker

98

Idaho, USA.

[1] Burkes DE, Wachs DM, Werner JE, Howe SD. An Overview of Current and Past W-UO2 CERMET Fuel Fabrication Technology. Space Nuclear Conference; 2007

[2] Bhattacharyya SK. An Assessment of Fuels for Nuclear Thermal Propulsion, ANL/TD/TM01-22; 2001

[3] Marlowe MO, Kaznoff AI. Development of Low Thermal Expansion W/UO2 Nuclear Fuel, NASA-CR-72711; 1970

[4] Baker RJ. Basic Behavior and Properties of W/UO2 CERMETS, NASA-CR-54840; 1965

[5] Koenig DR. Experience Gained from the Space Nuclear Rocket Program (ROVER), Rep. No. LA-10062-H; Los Alamos National Laboratory; 1986

[6] Lyon LL. Performance of (U,Zr)C-Graphite (Composite) and (U,Zr)C (Carbide) Fuel Elements in the Nuclear Furnace 1 Test Reactor, Rep. No. LA-5398-MS; Los Alamos National Laboratory; 1973

[7] Wallace TC. Review of Rover fuel element protective coating development at Los Alamos. In: Proceedings of the Eight Symposium on Space Nuclear Power Systems. 1991. pp. 1024-1036

[8] Taub JM. A Review of Fuel Element Development for Nuclear Rocket Engines, Rep. No. LA-5931; Los Alamos National Laboratory; 1975

[9] Qualls L. Graphite Fuel Development for Nuclear Thermal Propulsion; Oak Ridge National Laboratory; 2013

[10] Benensky K. Summary of historical solid Core nuclear thermal propulsion fuels [thesis]. State College Pennsylvania, Penn State University; 2013

[11] Hickman RR, Broadway JW, Mirales OR. Fabrication and testing of CERMET fuel materials for nuclear thermal propulsion. In: AIAA Joint Propulsion Conference; 13 July-1 August 2012; Atlanta, Georgia. 2012

[12] Tucker DS, O'Connor A, Hickman R. A methodology for producing a uniform mixture of UO2 in a tungsten matrix. Journal of Physical Science and Applications. 2015;5:255-262. DOI: 10.17265/2159-5348/2015.04.002

[13] O'Brien RC, Ambrosi RM, Bannister NP, Howe SD, Atkinson HV. Spark plasma sintering of simulated radioisotope materials within tungsten CERMETS. Journal of Nuclear Materials. 2009;393:108-113. DOI: 10.1016/j. jnucmat.2009.05.012

[14] Wang X, Xie Y, Fuo H, Van der Blest O, Vieugels J. Sintering of WC-Co powder with nanocrystalline WC by spark plasma sintering. Rare Metals. 2006;25:246

[15] Atkinson HV, Davies S. Fundamental of hot isostatic pressing: An overview. Metallurgical and Materials Transactions. 2000;A31:2981

[16] Xu A, Soloman AA. The effects of grain growth on the intergranular porosity distribution in hot pressed and swelled UO2. In: Proceedings of Ceramic Microstructures '86; July 1986; Plenum, Berkley, CA: University of California. 1986. pp. 28-31. ISBN: 0306426811

[17] Carpay FMA. The effect of pore drag on ceramic microstructures. In: Proceedings of Ceramic Microstructures '76; Westview, Boulder, CO. 1977. p. 171. ISBN: 0891583076

[18] German RM. Sintering Theory and Practice. New York: John Wiley; 1996. ISBN: 0-471-05786-X

[19] O'Brien RC, Ambrosi RM, Bannister NP, Howe SD, Atkinson HV. Safe thermoelectric generators and heat sources for space applications. Journal of Nuclear Materials. 2008;377:506

[20] O'Brien RC, Jerred ND. Spark Plasma Sintering of W-UO2 CERMETS; 2013. pp. 433-450

[21] Williams JC. The segregation of particulate materials: A review. Powder Technology. 1976;15:245

[22] Tucker DS, Barnes MW, Hone L, Cook S. High density, uniformly distributed W/UO2 for use in nuclear thermal propulsion. Journal of Nuclear Materials. 2017;486:246-249. DOI: 10.1016/jnucmat.2017.01.033

[23] Tucker DS, Wu Y, Burns J. Uranium migration in spark plasma sintered W/ UO2 CERMETS. Journal of Nuclear Materials. 2018;500:141-144. DOI: 10.1016/jnucmat.2017.12.029

[24] Haertling C, Hanrahan RJ. Literature review of thermal and radiation performance parameters for high-temperature uranium dioxide fueled CERMET materials. Journal of Nuclear Materials. 2007;366:317-335. DOI: 10.1016/jnucmat.2007.03.024

[25] Saunders NT, Gluyas RE, Watson GK. Feasibility of a Tungsten-Water Mediated Rocket II. NASA Lewis Research Center Report, NASA-TM-X-1421; 1968

[19] O'Brien RC, Ambrosi RM, Bannister NP, Howe SD, Atkinson HV. Safe thermoelectric generators and heat sources for space applications. Journal of

Advances in Composite Materials Development

Nuclear Materials. 2008;377:506

[20] O'Brien RC, Jerred ND. Spark Plasma Sintering of W-UO2 CERMETS;

[21] Williams JC. The segregation of particulate materials: A review. Powder

[22] Tucker DS, Barnes MW, Hone L, Cook S. High density, uniformly distributed W/UO2 for use in nuclear thermal propulsion. Journal of Nuclear Materials. 2017;486:246-249. DOI: 10.1016/jnucmat.2017.01.033

[23] Tucker DS, Wu Y, Burns J. Uranium migration in spark plasma sintered W/ UO2 CERMETS. Journal of Nuclear Materials. 2018;500:141-144. DOI: 10.1016/jnucmat.2017.12.029

[24] Haertling C, Hanrahan RJ. Literature review of thermal and radiation performance parameters for high-temperature uranium dioxide fueled CERMET materials. Journal of Nuclear Materials. 2007;366:317-335. DOI: 10.1016/jnucmat.2007.03.024

[25] Saunders NT, Gluyas RE, Watson GK. Feasibility of a Tungsten-Water Mediated Rocket II. NASA Lewis Research Center Report, NASA-TM-X-

1421; 1968

100

2013. pp. 433-450

Technology. 1976;15:245

## *Edited by Dumitra Lucan*

The progress of technology is a permanent challenge in developing new materials with superior properties in terms of quality and reliability. The demand for increased performance continues to focus materials development efforts on exploring new concepts or new generations of composite materials. The chapters contained in this book represent many examples of research results dedicated to obtaining, characterizing, and mathematically modeling composite materials, especially metal matrix composites, with superior properties having a wide range of applications. The book is addressed to researchers and materials specialists, teachers and students at science and material engineering faculties, and all those interested in advances in science and technology of new materials.

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Advances in Composite Materials Development

Advances in Composite

Materials Development

*Edited by Dumitra Lucan*