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A new approach to modeling the thermomechanical, orthotropic, elastic-inelastic response of soft materials

  • M. B. RubinEmail author
Original Paper
  • 24 Downloads

Abstract

This paper generalizes six previously developed nonlinear distortional deformation invariants for general hyperelastic orthotropic materials to model the thermomechanical, orthotropic, elastic-inelastic response of soft materials. These new invariants depend on two independent functions of elastic dilatation and temperature and characterize elastic distortional deformations from Hydrostatic States of Stress (HSS). When the Helmholtz free energy depends on these invariants, elastic dilatation and temperature, the correct response in HSS is automatically satisfied so the determination of the functional form of the Helmholtz free energy is simplified and can focus on modeling the response causing deviatoric stress. In addition, the new invariants are based on an Eulerian formulation of evolution equations for microscructural vectors that describe elastic deformation and directions of anisotropy. In contrast with the standard Lagrangian formulation, the Eulerian formulation is unaffected by arbitrary choices of the reference configuration, an intermediate configuration, a total deformation measure, and an inelastic deformation measure.

Keywords

Elastic-inelastic response Large deformation Orthotropic Soft material Thermomechanical 

1 Introduction

Modeling the thermomechanical, elastic-inelastic response of anisotropic materials is an important challenging problem. One application, with small elastic deformations, is predicting the anisotropic response of metals due to forming processes. Another application, with large elastic deformations, is predicting the response of soft tissues with anisotropy due to fiber bundle orientations.

Most models for elastically anisotropic response of metals are based on the Lagrangian formulation presented in [1, 2, 3]. Within the context of this approach the total deformation gradient F and the plastic deformation gradient Fp are determined by integrating evolution equations of the forms
$$ \dot{{\mathbf{F}}} = {\mathbf{L}} {\mathbf{F}} , \quad \dot{{\mathbf{F}}}_{p} = \boldsymbol{{\Lambda} }_{p} {\mathbf{F}}_{p} , $$
(1)
where L is the velocity gradient and Λp is a second order tensor that controls the rate of plastic dissipation and needs to be specified by a constitutive equation. Also, the elastic deformation gradient Fe is defined by
$$ {\mathbf{F}}_{e} = {\mathbf{F}} {\mathbf{F}}_{p}^{-1} . $$
(2)
For soft tissues with growth, Fp,Λp are replaced by a growth tensor Fg and a rate of growth Λg (e.g. [4]).
Eckart [5] seems to be the first to suggest that since stress is determined by elastic deformations, a theory of elastic-inelastic response can be proposed which introduces an elastic deformation measure directly by an evolution equation. This model was proposed for elastically isotropic material response and the same model was proposed by Leonov [6] for polymeric liquids. Motivated by these ideas, an alternative to the Lagrangian formulation for elastically anisotropic response was proposed in [7]. Specifically, this approach introduces a triad of linearly independent microstructural vectors mi determined by the evolution equations
$$ \dot{{\mathbf{m}}}_{i} = ({\mathbf{L}} - {\mathbf{L}}_{p}) {\mathbf{m}}_{i} , \quad J_{e} = {\mathbf{m}}_{1} \times {\mathbf{m}}_{2} \cdot {\mathbf{m}}_{3} > 0 , $$
(3)
where Je is the elastic dilatation and the second order tensor Lp characterizes the rate of inelasticity and needs to be specified by a constitutive equation. In this formulation mi and the metric
$$ m_{ij} = {\mathbf{m}}_{i} \cdot {\mathbf{m}}_{j} , $$
(4)
characterize directions of anisotropy and elastic deformations. In contrast with the Lagrangian formulation, this Eulerian formulation is unaffected by arbitrary choices of the reference configuration, a stress-free intermediate configuration, a measure of total deformation, and a measure of inelastic deformation. Moreover, in [8], it was shown that when this arbitrariness of the Lagrangian formulation is removed, the resulting formulation reduces to the formulation characterized by Eq. 3.

Using the Lagrangian formulation, physically based orthotropic invariants for purely mechanical hyperelastic response where developed in [9] and modified in [10]. The objective of this paper is to use the Eulerian formulation to develop generalized thermomechanical invariants that can be used for thermomechanical elastic-inelastic orthotropic material response. This yields a new approach to the formulation of orthotropic thermoelastic expansion and rate of inelasticity.

An outline of the paper is as follows. Section 2 introduces basic equations for the kinematics and balance laws, and Section 3 presents the new thermomechanical orthotropic invariants. Section 4 discusses general thermomechanical constitutive equations and Section 5 presents restrictions for metal plasticity. Section 6 proposes specific constitutive equations and Section 7 records equations for small elastic deformations and temperature differences. Section 8 discusses isotropic response and Section 9 presents examples of steady-state viscoplastic flow. Section 10 presents conclusions and the Appendix records details of some approximations.

2 Basic equations

2.1 Kinematics

Within the context of the Lagrangian approach to continuum mechanics it is common to let X denote the location of a material point in a fixed reference configuration and let x be the location of the same material point in the present configuration at time t. The motion of the continuum is specified by the mapping
$$ {\mathbf{x} } = {\mathbf{x} }({\mathbf{X} },t) , \quad {\mathbf{F}} = \partial {\mathbf{x} }/\partial {\mathbf{X} } , \quad J = \det({\mathbf{F}}) > 0 , $$
(5)
where F is the total deformation gradient and J is the total dilatation, both measured from the reference configuration. Also, the right Cauchy-Green deformation tensor C is defined by
$$ {\mathbf{C}} = {\mathbf{F}}^{T} {\mathbf{F}} . $$
(6)
Next, the velocity v, velocity gradient L, rate of deformation tensor D and spin tensor W are defined by
$$ {\mathbf{v}} = \dot{{\mathbf{x} }} , \quad {\mathbf{L}} = \partial {\mathbf{v}}/\partial {\mathbf{x} } = {\mathbf{D}} + {\mathbf{W}} , $$
(7)
$$ {\mathbf{D}} = \frac{1}{2}({\mathbf{L}}+{\mathbf{L}}^{T}) = {\mathbf{D}}^{T} , \quad {\mathbf{W}} = \frac{1}{2}({\mathbf{L}}-{\mathbf{L}}^{T}) = - {\mathbf{W}}^{T} , $$
(8)
where a superposed (⋅) denotes material time differentiation following the material point X.

2.2 Basic thermomechanical balance laws

Within the context of the themomechanical theory proposed in [11, 12], the conservation of mass, balance of linear momentum and balance of entropy are given by
$$ \dot{\rho} + \rho {\mathbf{D}} \cdot {\mathbf{I}} = 0 , \quad \rho \dot{{\mathbf{v}}} = \rho {\mathbf{b}} + \text{div}{\mathbf{T}} , \quad \rho \dot{\eta} = \rho(s + \xi) - \text{div} {\mathbf{p}} , $$
(9)
where ρ is the current mass density, I is the unit second order tensor, AB = tr(ABT) denotes the inner product of two arbitrary second order tensors (A,B), b is the specific (per unit mass) external body force, div() denotes the divergence operator with respect to x, T denotes the Cauchy stress, η denotes the specific entropy, s denotes the specific external rate of supply of entropy, ξ denotes the specific internal rate of production of entropy and p denotes the entropy flux vector per unit present area of a material surface. Also, the reduced forms of the balances of angular momentum and energy are given by
$$ {\mathbf{T}}^{T} = {\mathbf{T}} , \quad \rho \dot{{\varepsilon}} = \rho r - \text{div}{\mathbf{q}} + {\mathbf{T}} \cdot {\mathbf{D}} , $$
(10)
where ε is the specific internal energy, r = 𝜃s is the specific external rate of supply of energy, q = 𝜃p is the energy flux vector per unit present area of a material surface and 𝜃 is the absolute temperature.
Next, introducing the specific Helmholtz free energy
$$ \psi = {\varepsilon} - \theta \eta , $$
(11)
and using the work in [13], the internal rate of production of entropy can be separated additively into thermal and mechanical parts, such that
$$ \rho \theta \xi = - {\mathbf{p}} \cdot {\mathbf{g}} + \rho \theta \xi^{\prime} , \quad {\mathbf{g}} = \partial \theta / \partial {\mathbf{x} } , $$
(12)
where g is the temperature gradient. Using the balance laws (9) and these expessions, the balance of energy (10) can be rewritten in the form
$$ \rho \theta \xi^{\prime} = {\mathbf{T}} \cdot {\mathbf{D}} - \rho (\dot{\psi}+\eta \dot{\theta}) . $$
(13)
The balance laws (9) are used to determine the quantities (ρ,x,𝜃) and the reduced balance of angular momentum in Eq. 10 and balance of energy (13) are satisfied by placing restrictions on the constitutive equations. Moreover, two forms of the second law of thermodynamics require heat to flow from hot to cold regions and the rate of material dissipative to be non-negative
$$ - {\mathbf{p}} \cdot {\mathbf{g}} > 0 \text{for} {\mathbf{g}} \ne 0 , \quad \rho \theta \xi^{\prime} \ge 0 . $$
(14)

3 Thermomechanical orthotropic invariants

3.1 Additional vectors and metrics

To develop the Eulerian forms of the thermomechanical othotropic invariants it is convenient to introduce the reciprocal vectors mi defined by
$$\begin{array}{@{}rcl@{}} {\mathbf{m}}^{1} &=& J_{e}^{-1} {\mathbf{m}}_{2} \times {\mathbf{m}}_{3}, \quad {\mathbf{m}}^{2} = J_{e}^{-1} {\mathbf{m}}_{3} \times {\mathbf{m}}_{1} ,\\ {\mathbf{m}}^{3} &=& J_{e}^{-1} {\mathbf{m}}_{1} \times {\mathbf{m}}_{2} , \quad {\mathbf{m}}^{i} \cdot {\mathbf{m}}_{j} = {\delta^{i}_{j}} ,\end{array} $$
(15)
where \({\delta ^{i}_{j}}\) is the Kronecker delta symbol. Also, the reciprocal metric mij is defined by
$$ m^{ij} = {\mathbf{m}}^{i} \cdot {\mathbf{m}}^{j} , \quad m^{im} m_{mj} = {\delta^{i}_{j}} , $$
(16)
where the usual summation convention is used for repeated indices. Then, using the evolution (3), it can be shown that
$$ \dot{{\mathbf{m}}}^{i} =- ({\mathbf{L}} - {\mathbf{L}}_{p})^{T} {\mathbf{m}}^{i} , $$
(17)
and that the metrics mij,mij satisfy the evolution equations
$$\dot{m}_{ij} = 2 ({\mathbf{m}}_{i} \otimes {\mathbf{m}}_{j}) \cdot ({\mathbf{D}} - {\mathbf{D}}_{p}) , $$
$$ \dot{m}^{ij} = -2 ({\mathbf{m}}^{i} \otimes {\mathbf{m}}^{j}) \cdot ({\mathbf{D}} - {\mathbf{D}}_{p}), $$
(18)
where Lp has been expressed in terms of the plastic deformation rate Dp and the plastic spin Wp, such that
$$\begin{array}{@{}rcl@{}} &&{\mathbf{L}}_{p} = {\mathbf{D}}_{p} + {\mathbf{W}}_{p} , \quad {\mathbf{D}}_{p} = \frac{1}{2} ({\mathbf{L}}_{p} + {{\mathbf{L}}_{p}^{T}}) ,\\ &&{\mathbf{W}}_{p} = \frac{1}{2} ({\mathbf{L}}_{p} - {{\mathbf{L}}_{p}^{T}}) .\end{array} $$
(19)
For definiteness, the vectors mi are defined so that they form an orthonomal triad in any Reference Lattice State (RLS), which is a stress-free state at reference temperature 𝜃0
$$ m_{ij} = \delta_{ij} \text{for any RLS with} {\mathbf{T}} = 0 \text{and} \theta = \theta_{0} . $$
(20)
Also, using Eqs. 3 and 19, it can be shown that
$$ \dot{J}_{e} = J_{e} \dot{{\mathbf{m}}}_{i} \cdot {\mathbf{m}}^{i} = J_{e} ({\mathbf{D}} - {\mathbf{D}}_{p}) \cdot {\mathbf{I}} . $$
(21)
Furthermore, for the orthotropic material considered here, mi define the principle directions of orthotropy in any RLS.
Next, using the work in [14] it is possible to define the vectors \({\mathbf {m}}_{i}^{\prime }, {\mathbf {m}}^{i\prime }\) and the elastic distortional deformation metrics \(m_{ij}^{\prime },m^{ij\prime }\) by the equations
$$\begin{array}{@{}rcl@{}} {\mathbf{m}}_{i}^{\prime} &=& J_{e}^{-1/3} {\mathbf{m}}_{i} , \quad {\mathbf{m}}^{i\prime} = J_{e}^{1/3} {\mathbf{m}}^{i} , \\ m_{ij}^{\prime} &=& {\mathbf{m}}_{i}^{\prime} \cdot {\mathbf{m}}_{j}^{\prime} , \quad m^{ij \prime} = {\mathbf{m}}^{i \prime} \otimes {\mathbf{m}}^{j \prime} . \end{array} $$
(22)
Then, it can be shown that these elastic distortional deformation metrics satisfy the evolution equations
$$\begin{array}{@{}rcl@{}} {\dot{m}_{ij}^{\prime}} &=& 2 ({\mathbf{m}}_{i}^{\prime} \otimes {\mathbf{m}}_{j}^{\prime} - \frac{1}{3} m_{ij}^{\prime} {\mathbf{I}} ) \cdot ({\mathbf{D}} - {\mathbf{D}}_{p}) ,\\ {\dot{m}^{ij \prime}} &=& - 2 ({\mathbf{m}}^{i\prime} \otimes {\mathbf{m}}^{j\prime} - \frac{1}{3} m^{ij\prime} {\mathbf{I}} ) \cdot ({\mathbf{D}} - {\mathbf{D}}_{p}) . \end{array} $$
(23)
Moreover, it is convenient to write Lp,Dp,Wp in the forms
$$\begin{array}{@{}rcl@{}} {\mathbf{L}}_{p} &=& {\Gamma} \overline{{\mathbf{L}}}_{p} , \quad {\mathbf{D}}_{p} = {\Gamma} \overline{{\mathbf{D}}}_{p} ,\\ {\mathbf{W}}_{p} &=& {\Gamma} \overline{{\mathbf{W}}}_{p} , \quad {\Gamma} \ge 0 \end{array} $$
(24)
where Γ controls the rate of inelastic deformation and \(\overline {{\mathbf {L}}}_{p}, \overline {{\mathbf {D}}}_{p}, \overline {{\mathbf {W}}}_{p}\) control the directions of Lp,Dp,Wp.

3.2 Hardening

To model anisotropic hardening it is convenient to introduce a symmetric hardening matrix Hij by an evolution equation of the form
$$ \dot{H}_{ij} = {\Gamma} H_{ijkl} ({\mathbf{m}}_{k} \otimes {\mathbf{m}}_{l}) \cdot \overline{{\mathbf{D}}}_{p} , $$
(25)
where Hijkl are functions having the symmetries
$$ H_{ijkl} = H_{jikl} = H_{ijlk} = H_{klij} . $$
(26)

3.3 Rate-independent and rate-dependent response

Assuming that \(\overline {{\mathbf {L}}}_{p}, H_{ijkl}\) are functions of the state of the material mi,𝜃,Hij and not the rates \({\mathbf {D}},\dot {\theta }\), the evolution (3) for mi and Eq. 25 for Hij predict rate-independent response if Γ is a homogeneous function of order one in \(\mathbf {D},\dot {\theta }\). Otherwise, these evolution equations predict rate-dependent response.

Often the inelastic deformation rate is controlled by a yield function of the form
$$ g = g(m_{ij}, \theta, H_{ij}) . $$
(27)

3.3.1 Rate-independent response

Motivated by the rate-independent strain-space formulation of plasticity [15], g is required to remain non-positive (g ≤ 0) and the material derivative of g can be expressed as
$$\begin{array}{@{}rcl@{}} \dot{g} &=& \hat{g} - {\Gamma} \overline{g} , \\ \hat{g} &=& 2 \frac{\partial g}{\partial m_{ij}} ({\mathbf{m}}_{i} \otimes {\mathbf{m}}_{j}) \cdot {\mathbf{D}} + \frac{\partial g}{\partial \theta} \dot{\theta} , \\ \overline{g} &=& - \left[2 \frac{\partial g}{\partial m_{ij}} + \frac{\partial g}{\partial H_{kl}} H_{klij}\right] ({\mathbf{m}}_{i} \otimes {\mathbf{m}}_{j}) \cdot \overline{{\mathbf{D}}}_{p} . \end{array} $$
(28)
Then, the loading and consistency conditions are given by
$$\begin{array}{@{}rcl@{}} {\Gamma} &=&0 {\kern30pt} \text{for} (g <0) \text{or} (g = 0, \hat{g} \le 0) , \\ {\Gamma} &=& \frac{\hat{g}}{\overline{g}} > 0{\kern8pt} \text{for} (g = 0, \hat{g}> 0) . \end{array} $$
(29)
Since Γ is a homogeneous function of order one in \({\mathbf {D}},\dot {\theta }\), the material response is rate-independent.

3.3.2 Rate-dependent response

A unified model of dynamic viscoplastic response combining creep and relaxation has been developed (e.g., [16, 17, 18, 19, 20, 21]), which is equivalent to proposing a functional form for Γ that is independent of the rates \(\mathbf {D},\dot {\theta }\). Alternatively, the overstress model developed in [22, 23] eliminates the consistency condition which restricts g to be non-positive and takes Γ to be a function of g with non-zero rate of inelasticity when g is positive. Also, an overstress model with a smooth elastic-inelastic transition for both rate-independent and rate-dependent response has been developed in [24, 25].

3.4 Orthotropic invariants

In general, the stress T can be expressed in terms of its pressure p and its deviatoric part T, such that
$$ {\mathbf{T}} = - p {\mathbf{I}} + {\mathbf{T}^{\prime\prime}} . $$
(30)
As discussed in [9, 10] an orthotropic material distorts when it is subjected to a Hydrostatic State of Stress (HSS) with T = 0. For the thermomechanical theory developed here this HSS can also allow distortions due to temperature changes. The orthotropic invariants proposed here measure additional thermoelastic distortions which cause deviatoric stress. Specifically, the metrics \(m_{ij}^{\prime }, m^{ij\prime }\) in a HSS take the forms
$$\begin{array}{@{}rcl@{}} m_{11}^{\prime} &=& {\eta_{1}^{2}} {\kern15pt} m_{22}^{\prime} = {\eta_{2}^{2}} {\kern15pt} m_{33}^{\prime} = {\eta_{3}^{2}} , \\ m_{12}^{\prime} &=& 0 {\kern19pt} m_{13}^{\prime}= 0 {\kern20pt}m_{23}^{\prime} = 0 , \\ m^{11^{\prime}} &=& \frac{1}{{\eta_{1}^{2}}} {\kern15pt} m^{22^{\prime}} = \frac{1}{{\eta_{2}^{2}}} {\kern8pt} m^{22^{\prime}} = \frac{1}{{\eta_{3}^{2}}} , \\ m^{12^{\prime}} &=& 0 {\kern23pt} m^{13^{\prime}} = 0{\kern17pt} m^{23^{\prime}} = 0 ,\\ &&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{for any HSS} , \end{array} $$
(31)
where ηi are positive constitutive functions of the elastic dilatation Je and temperature 𝜃 satisfying the restrictions
$$ \eta_{i} = \eta_{i}(J_{e},\theta) > 0 , \quad \eta_{1} \eta_{2} \eta_{3} = 1 , \quad \eta_{i}(1,\theta_{0}) = 1 , $$
(32)
with ηi being unity in any RLS. It also follows that in any HSS
$$\begin{array}{@{}rcl@{}} \frac{1}{\eta_{1}} {\mathbf{m}}_{1}^{\prime} = \eta_{1} {\mathbf{m}}^{1\prime} , \quad \frac{1}{\eta_{2}} {\mathbf{m}}_{2}^{\prime} = \eta_{2} {\mathbf{m}}^{2\prime} , \quad \frac{1}{\eta_{3}} {\mathbf{m}}_{3}^{\prime} = \eta_{3} {\mathbf{m}}^{3\prime} ,\\ &&{} \text{for any HSS} .\end{array} $$
(33)
The three physically based invariants β1,β2,β3 defined in [9] are rewritten in the forms
$$\begin{array}{@{}rcl@{}} \beta_{1} &=& \frac{m_{11}^{\prime}}{{\eta_{1}^{2}}} + {\eta_{1}^{2}} m^{11 \prime} , \quad \beta_{2} = \frac{m_{22}^{\prime}}{{\eta_{2}^{2}}} + {\eta_{2}^{2}} m^{22 \prime} ,\\ \beta_{3} &=& \frac{m_{33}^{\prime}}{{\eta_{3}^{2}}} + {\eta_{3}^{2}} m^{33 \prime} , \quad \beta_{i} \ge 2 \text{for} i = 1,2,3 , \end{array} $$
(34)
and the three additional modified invariants defined in [10] are rewritten in the forms
$$ \beta_{4} = \frac{m_{12}^{\prime 2}} {m_{11}^{\prime} m_{22}^{\prime}} , \quad \beta_{5} = \frac{m_{13}^{\prime 2}} {m_{11}^{\prime} m_{33}^{\prime}} , \quad \beta_{6} = \frac{m_{23}^{\prime 2}} {m_{22}^{\prime} m_{33}^{\prime}} , $$
(35)
where the lower bounds for β1,β2,β3 were proved in [9]. In non-hydrostatic states of stress, the invariants β1,β2,β3 mainly control the components of stress in the principle directions of orthotropy and the invariants β4,β5,β6 mainly control the shear stresses. In addition, using Eqs. 21 and 23 it can be shown that
$$\begin{array}{@{}rcl@{}} \dot{\eta}_{i} &=& J_{e} \frac{\partial \eta_{i}}{\partial J_{e}} {\mathbf{I}} \cdot ({\mathbf{D}} - {\mathbf{D}}_{p}) + \frac{\partial \eta_{i}}{\partial \theta} \dot{\theta} ,\\ \dot{\beta}_{i} &=& 2(- N_{i} {\mathbf{I}} + {\mathbf{B}}_{i}^{\prime \prime}) \cdot ({\mathbf{D}} - {\mathbf{D}}_{p}) + 2A_{i} \dot{\theta} ,\\ N_{1} &=& \frac{J_{e}}{\eta_{1}} (\frac{m_{11}^{\prime}}{{\eta_{1}^{2}}} - {\eta_{1}^{2}} m^{11 \prime}) \frac{\partial \eta_{1}}{\partial J_{e}} ,\\ N_{2} &=& \frac{J_{e}}{\eta_{2}} (\frac{m_{22}^{\prime}}{{\eta_{2}^{2}}} - {\eta_{2}^{2}} m^{22 \prime}) \frac{\partial \eta_{2}}{\partial J_{e}} ,\\ N_{3} &=& \frac{J_{e}}{\eta_{3}} (\frac{m_{33}^{\prime}}{{\eta_{3}^{2}}} - {\eta_{3}^{2}} m^{33 \prime}) \frac{\partial \eta_{3}}{\partial J_{e}} ,\\ N_{4} &=& N_{5} = N_{6} = 0 ,\\ A_{1} &=& - \frac{1}{\eta_{1}} (\frac{m_{11}^{\prime}}{{\eta_{1}^{2}}} - {\eta_{1}^{2}} m^{11 \prime})\frac{\partial \eta_{1}}{\partial \theta} ,\\ A_{2} &=& - \frac{1}{\eta_{2}} (\frac{m_{22}^{\prime}}{{\eta_{2}^{2}}} - {\eta_{2}^{2}} m^{22 \prime}) \frac{\partial \eta_{2}}{\partial \theta} ,\\ A_{3} &=& - \frac{1}{\eta_{3}} (\frac{m_{33}^{\prime}}{{\eta_{3}^{2}}} - {\eta_{3}^{2}} m^{33 \prime}) \frac{\partial \eta_{3}}{\partial \theta} ,\\ A_{4} &=& A_{5} = A_{6} = 0 , \end{array} $$
(36)
where the deviatoric tensors \({\mathbf {B}}_{i}^{\prime \prime }\) are defined by
$$\begin{array}{@{}rcl@{}} {\mathbf{B}}_{1}^{\prime \prime} &=& \frac{1}{{\eta_{1}^{2}}} {\mathbf{m}}_{1}^{\prime} \otimes {\mathbf{m}}_{1}^{\prime} - {\eta_{1}^{2}} {\mathbf{m}}^{1\prime} \otimes {\mathbf{m}}^{1\prime}\\ &-& \frac{1}{3} (\frac{m_{11}^{\prime}}{{\eta_{1}^{2}}} - {\eta_{1}^{2}} m^{11 \prime}) {\mathbf{I}} ,\\ {\mathbf{B}}_{2}^{\prime \prime} &=& \frac{1}{{\eta_{2}^{2}}} {\mathbf{m}}_{2}^{\prime} \otimes {\mathbf{m}}_{2}^{\prime} - {\eta_{2}^{2}} {\mathbf{m}}^{2\prime} \otimes {\mathbf{m}}^{2\prime}\\ &-& \frac{1}{3} (\frac{m_{22}^{\prime}}{{\eta_{2}^{2}}} - {\eta_{2}^{2}} m^{22 \prime}) {\mathbf{I}} ,\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} {\mathbf{B}}_{3}^{\prime \prime} &=& \frac{1}{{\eta_{3}^{2}}} {\mathbf{m}}_{3}^{\prime} \otimes {\mathbf{m}}_{3}^{\prime} - {\eta_{3}^{2}} {\mathbf{m}}^{3\prime} \otimes {\mathbf{m}}^{3\prime}\\ &-& \frac{1}{3} (\frac{m_{33}^{\prime}}{{\eta_{3}^{2}}} - {\eta_{3}^{2}} m^{33 \prime}) {\mathbf{I}} ,\\ {\mathbf{B}}_{4}^{\prime \prime} &=& \frac{m_{12}^{\prime}}{m_{11}^{\prime} m_{22}^{\prime}} [({\mathbf{m}}_{1}^{\prime} \otimes {\mathbf{m}}_{2}^{\prime} + {\mathbf{m}}_{2}^{\prime} \otimes {\mathbf{m}}_{1}^{\prime})\\ &-& \frac{m_{12}^{\prime}}{m_{11}^{\prime}} ({\mathbf{m}}_{1}^{\prime} \otimes {\mathbf{m}}_{1}^{\prime}) - \frac{m_{12}^{\prime}}{m_{22}^{\prime}} ({\mathbf{m}}_{2}^{\prime} \otimes {\mathbf{m}}_{2}^{\prime})] ,\\ {\mathbf{B}}_{5}^{\prime \prime} &=& \frac{m_{13}^{\prime}}{m_{11}^{\prime} m_{33}^{\prime}} [({\mathbf{m}}_{1}^{\prime} \otimes {\mathbf{m}}_{3}^{\prime} + {\mathbf{m}}_{3} \otimes {\mathbf{m}}_{1}^{\prime})\\ &-& \frac{m_{13}^{\prime}}{m_{11}^{\prime}} ({\mathbf{m}}_{1}^{\prime} \otimes {\mathbf{m}}_{1}^{\prime}) - \frac{m_{13}^{\prime}}{m_{33}^{\prime}} ({\mathbf{m}}_{3}^{\prime} \otimes {\mathbf{m}}_{3}^{\prime})] ,\\ {\mathbf{B}}_{6}^{\prime \prime} &=& \frac{m_{23}^{\prime}}{m_{22}^{\prime} m_{33}} [({\mathbf{m}}_{2}^{\prime} \otimes {\mathbf{m}}_{3}^{\prime} + {\mathbf{m}}_{3}^{\prime} \otimes {\mathbf{m}}_{2}^{\prime})\\ &-& \frac{m_{23}^{\prime}}{m_{22}^{\prime}} ({\mathbf{m}}_{2}^{\prime} \otimes {\mathbf{m}}_{2}^{\prime}) - \frac{m_{23}^{\prime}}{m_{33}^{\prime}} ({\mathbf{m}}_{3}^{\prime} \otimes {\mathbf{m}}_{3}^{\prime})] . \end{array} $$
(37)

4 General thermomechanical constitutive equations

For general thermomechanical orthotropic response, the Helmholtz free energy is a function of the form
$$ \psi = \psi(J_{e},\beta_{i},\theta) , $$
(38)
and the pressure p, deviatoric stress T and entropy η are specified by
$$\begin{array}{@{}rcl@{}} p &=& - \rho J_{e} \frac{\partial \psi}{\partial J_{e}} + 2 \sum\limits_{i = 1}^{3} \rho \frac{\partial \psi}{\partial \beta_{i}} N_{i} , \\ {\mathbf{T}^{\prime\prime}} &=& 2 \sum\limits_{i = 1}^{6} \rho \frac{\partial \psi}{\partial \beta_{i}} {\mathbf{B}}_{i}^{\prime \prime} , \\ \eta &=& - \frac{\partial \psi}{\partial \theta} - 2 \sum\limits_{i = 1}^{3} \frac{\partial \psi}{\partial \beta_{i}} A_{i} .\end{array} $$
(39)
Then, the rate of material dissipation (14) requires
$$ \rho \theta \xi^{\prime} = - p ({\mathbf{D}}_{p} \cdot {\mathbf{I}}) + {\mathbf{T}^{\prime\prime}} \cdot {\mathbf{D}}_{p} \ge 0 . $$
(40)
In addition, the second law (14) associated the entropy flux p can be satisfied by specifying
$$\begin{array}{@{}rcl@{}} {\mathbf{p}} &=& - \frac{{\mathbf{K}}}{\theta} {\mathbf{g}} , \quad {\mathbf{K}} = {\mathbf{K}}^{T} = K^{ij} ({\mathbf{m}}_{i} \otimes {\mathbf{m}}_{j}) , \\ K^{ji} &=& K^{ij} , \end{array} $$
(41)
where Kij is a positive definite symmetric matrix characterizing anisotropic heat conduction coefficients.
Using Eqs. 3438, it can be seen that the invariants βi are constants and the tensors \({\mathbf {B}}_{i}^{\prime \prime }\) and scalars Ni,Ai vanish for all HSS
$$\begin{array}{@{}rcl@{}} \beta_{1} &=& \beta_{2} = \beta_{3} = 2 , \quad \beta_{4} = \beta_{5} = \beta_{6} = 0 ,\\ {\mathbf{B}}_{i}^{\prime \prime} &=& 0 , \quad N_{i} = A_{i} = 0 (i = 1,2,..,6) \text{for all HSS} .\end{array} $$
(42)
Moreover, it can be seen from Eq. 39 that the pressure p and entropy η are given by
$$ p \!= - \rho J_{e} \frac{\partial \psi}{\partial J_{e}} , \quad \eta = - \frac{\partial \psi}{\partial \theta} (i = 1,2,..,6) \text{for all HSS} . $$
(43)
Consequently, the distortional deformations (31) due to Je,𝜃 in all HSS are automatically satisfied for any functional dependence of ψ on its arguments. This means that the determination of the functional form for ψ only requires additional modeling of non-hydrostatic states of stress.

5 Restrictions for metal plasticity

For metal plasticity it is assumed that plastic deformation rate is isochoric with Dp satisfying the restriction
$$ {\mathbf{D}}_{p} \cdot {\mathbf{I}} = 0 , $$
(44)
so that with the help of the conservation of mass (9) and Eq. 21 it can be shown that
$$ \frac{d(\rho J_{e})}{dt} = 0 . $$
(45)
Now, using the fact the Je = 1 for any RLS this equation can be integrated to deduce that
$$ J_{e} = \frac{\rho_{0}}{\rho} , $$
(46)
where ρ0 is the density of the material in any RLS. Moreover, with the help of Eq. 44 the rate of material dissipation (40) for metals requires
$$ \rho \theta \xi^{\prime} = {\mathbf{T}^{\prime\prime}} \cdot {\mathbf{D}}_{p} \ge 0 . $$
(47)
In the remainder of this paper attention will be limited to this case when inelastic deformation rate is isochoric (44).

6 Specific constitutive equations

Motivated by the work in [9, 10, 21], the Helmholtz free energy is specified in the form
$$\begin{array}{@{}rcl@{}} \psi &=& \psi_{1}(J_{e},\theta) + \psi_{2}(J_{e}, \theta, m_{ij}) ,\\ \rho_{0} \psi_{1} &=& \rho_{0} C_{v} [\theta-\theta_{0} - \theta \ln(\frac{\theta}{\theta_{0}})]\\ &&- (\theta-\theta_{0}) f_{1}(J_{e}) + f_{2}(J_{e}) ,\\ \rho_{0} \psi_{2} &=& \frac{1}{2} \sum\limits_{i = 1}^{3} K_{i} (\beta_{i}- 2) + \frac{1}{2} \sum\limits_{i = 4}^{6} K_{i} \beta_{i} , \quad K_{i} \ge 0, \end{array} $$
(48)
where ψ1 controls the isotropic thermomechancal response to changes in Je,𝜃, and ψ2 controls the response to non-hydrostatic distortional deformations and temperature changes. Also, Cv is the constant specific heat at constant volume, f1(Je) controls isotropic thermomechanical coupling, f2(Je) controls the isotropic response to nonlinear compression and Ki are non-negative material constants. The relationship between the mechanical constants Ki and standard orthotropic constants for the purely mechanical model can be found in [9]. Then, using Eq. 39, it follows that
$$\begin{array}{@{}rcl@{}} p &=& p_{1}(J_{e},\theta) + p_{2}(J_{e},\theta, m_{ij}) ,\\ p_{1} &=& (\theta-\theta_{0}) \frac{df_{1}}{dJ_{e}} - \frac{df_{2}}{dJ_{e}} ,\\ p_{2} &=& J_{e}^{-1} \sum\limits_{i = 1}^{3} K_{i} N_{i} ,\\ {\mathbf{T}^{\prime\prime}} &=& J_{e}^{-1} \sum\limits_{i = 1}^{6} K_{i} {\mathbf{B}}_{i}^{\prime \prime} ,\\ \eta &=& \hat{\eta}_{1}(J_{e},\theta) + \hat{\eta}_{2}(J_{e},\theta,m_{ij}) ,\\ \rho_{0} \hat{\eta}_{1} &=& \rho_{0} C_{v} \ln(\frac{\theta}{\theta_{0}}) + f_{1} ,\\ \rho_{0} \hat{\eta}_{2} &=& - \sum\limits_{i = 1}^{3} K_{i} A_{i} . \end{array} $$
(49)
In these expressions, \(p_{1}, \hat {\eta }_{1}\) control the isotropic thermomechancal response to changes in Je,𝜃 and \(p_{2}, \hat {\eta }_{2}\) include dependence on mij. The functions \(\hat {\eta }_{1}, \hat {\eta }_{2}\) in Eq. 32 should not be confused with the functions η1,η2, which control elastic distortions in HSS. Also, the internal energy can be expressed in the form
$$\begin{array}{@{}rcl@{}} &&{\varepsilon} = {\varepsilon}_{1}(J_{e},\theta) + {\varepsilon}_{2}(J_{e}, \theta, m_{ij}),\\ &&\rho_{0} {\varepsilon}_{1} = \rho_{0} C_{v} (\theta-\theta_{0}) + \theta_{0} f_{1}(J_{e}) + f_{2}(J_{e}) ,\\ &&\rho_{0} {\varepsilon}_{2} = \frac{1}{2} \sum\limits_{i = 1}^{3} K_{i} (\beta_{i}- 2) + \frac{1}{2}\sum\limits_{i = 4}^{6} K_{i} \beta_{i} - \sum\limits_{i = 1}^{3} K_{i} \theta A_{i} .\end{array} $$
(50)
Specifically, it was shown in [21, 26] that the functions f1,f2 can be specified to model a Mie-Gruneisen equation for the pressure p1 of the form
$$ p_{1} = p_{H}(J_{e}) + \rho_{0} \gamma_{0} [{\varepsilon}_{1}-{\varepsilon}_{H}(J_{e})] , $$
(51)
where pH,εH are the Hugoniot values of pressure and energy, respectively, and γ0 is a constant Gruneisen parameter that controls thermomechanical coupling.
Motivated by the work in [9, 10], the functions ηi, which control distortional deformations in any HSS, are specified by
$$\begin{array}{@{}rcl@{}} \eta_{i} &=& J_{e}^{n_{i}/3} (\frac{\theta}{\theta_{0}})^{(\alpha_{i} \theta_{0}/3)} , \quad n_{i} = n_{i} (J_{e}, \theta) ,\\ \alpha_{i} &=& \alpha_{i}(J_{e}, \theta) ,\end{array} $$
(52)
where in view of Eq. 32 the functions ni,αi must satisfy the restrictions
$$ n_{1}+n_{2}+n_{3} = 0 , \quad \alpha_{1}+\alpha_{2}+\alpha_{3} = 0 . $$
(53)
Moreover, using Eq. 49 it can be seen that the dissipation inequality (47) will be satisfied when \(\overline {{\mathbf {D}}}_{p}\) is specified by
$$\begin{array}{@{}rcl@{}} \overline{{\mathbf{D}}}_{p} &=& \sum\limits_{i = 1}^{6} d_{i} \text{Sign}({\mathbf{T}^{\prime\prime}} \cdot {\mathbf{B}}_{i}^{\prime\prime}) {\mathbf{B}}_{i}^{\prime\prime} , \quad d_{i} \ge 0 ,\\ &&\text{Sign}(x) = -1 \text{for} x < 0 ,\\ &&\text{Sign}(x) = 1 \text{for} x \ge 0 , \end{array} $$
(54)
where di are non-negative functions. In addition, motivated by the work in [26], the plastic spin is proposed in the forms
$$\begin{array}{@{}rcl@{}} \overline{{\mathbf{W}}}_{p} &=& {\Omega}_{12} m_{12}^{\prime} ({\mathbf{m}}^{1\prime} \otimes {\mathbf{m}}^{2\prime} -{\mathbf{m}}^{2\prime} \otimes {\mathbf{m}}^{1\prime}) \\ &+& {\Omega}_{13} m_{13}^{\prime} ({\mathbf{m}}^{3\prime} \otimes {\mathbf{m}}^{1\prime} -{\mathbf{m}}^{1\prime} \otimes {\mathbf{m}}^{3\prime}) \\ &+& {\Omega}_{23} m_{23}^{\prime} ({\mathbf{m}}^{2\prime} \otimes {\mathbf{m}}^{3\prime} -{\mathbf{m}}^{3\prime} \otimes {\mathbf{m}}^{2\prime}) , \end{array} $$
(55)
where Ω121323 are material constants to be specified.

7 Small elastic deformations and temperature differences

With reference to fixed rectangular Cartesian base vectors ei, for small elastic deformations mi are specified by
$$\begin{array}{@{}rcl@{}} {\mathbf{m}}_{1} &=& (1+e_{11}) {\mathbf{e}}_{1} + (e_{12}+\omega_{12}) {\mathbf{e}}_{2} + (e_{13}+\omega_{13}) {\mathbf{e}}_{3} , \\ {\mathbf{m}}_{2} &=& (e_{12}-\omega_{12}) {\mathbf{e}}_{1} + (1+e_{22}) {\mathbf{e}}_{2} + (e_{23}+\omega_{23}) {\mathbf{e}}_{3} , \\ {\mathbf{m}}_{3} &=& (e_{13}-\omega_{13}) {\mathbf{e}}_{1} + (e_{23}-\omega_{23}) {\mathbf{e}}_{2} + (1+e_{33}) {\mathbf{e}}_{3} , \\ {\mathbf{m}}^{1} &=& (1-e_{11}) {\mathbf{e}}_{1} + (-e_{12}+\omega_{12}) {\mathbf{e}}_{2} + (-e_{13}+\omega_{13}) {\mathbf{e}}_{3} , \\ {\mathbf{m}}^{2} &=& (-e_{12}-\omega_{12}) {\mathbf{e}}_{1} + (1-e_{22}) {\mathbf{e}}_{2} + (-e_{23}+\omega_{23}) {\mathbf{e}}_{3} , \\ {\mathbf{m}}^{3} &=& (- e_{13}-\omega_{13}) {\mathbf{e}}_{1} + (-e_{23}-\omega_{23}) {\mathbf{e}}_{2} + (1-e_{33}) {\mathbf{e}}_{3} ,\\ \end{array} $$
(56)
where eij is the symmetric elastic strain and ωij is the skew-symmetric rotation. Moreover, eij can be expressed in terms of the elastic volumetric strain ev and the elastic deviatoric strain \(e_{ij}^{\prime \prime }\), such that
$$\begin{array}{@{}rcl@{}} &&e_{ij} = \frac{1}{3} e_{v} \delta_{ij} + e_{ij}^{\prime \prime} , \quad e_{v}= J_{e}-1= e_{mm} ,\\ &&e_{mm}^{\prime \prime} = 0. \end{array} $$
(57)
Using these expressions, it can be shown that neglecting quadratic terms in small quantities
$$\begin{array}{@{}rcl@{}} m_{ij} = \delta_{ij} + 2 e_{ij} , \quad m^{ij} = \delta_{ij} - 2 e_{ij},\\ m_{ij}^{\prime} = \delta_{ij} + 2 e_{ij}^{\prime \prime} , \quad m^{ij \prime} = \delta_{ij} - 2 e_{ij}^{\prime \prime}, \end{array} $$
$$\begin{array}{@{}rcl@{}} {\mathbf{m}}_{1}^{\prime} &=& (1+e_{11}^{\prime \prime}) {\mathbf{e}}_{1} + (e_{12}^{\prime \prime}+\omega_{12}) {\mathbf{e}}_{2} \\ &+& (e_{13}^{\prime \prime}+\omega_{13}) {\mathbf{e}}_{3} , \\ {\mathbf{m}}_{2}^{\prime} &=& (e_{12}^{\prime \prime}-\omega_{12}) {\mathbf{e}}_{1} + (1+e_{22}^{\prime \prime}) {\mathbf{e}}_{2} \\ &+& (e_{23}^{\prime \prime}+\omega_{23}) {\mathbf{e}}_{3} , \\ {\mathbf{m}}_{3}^{\prime} &=& (e_{13}^{\prime \prime}-\omega_{13}) {\mathbf{e}}_{1} + (e_{23}^{\prime \prime}-\omega_{23}) {\mathbf{e}}_{2} \\ &+& (1+e_{33}^{\prime \prime}) {\mathbf{e}}_{3} , \\ {\mathbf{m}}^{1 \prime} &=& (1-e_{11}^{\prime \prime}) {\mathbf{e}}_{1} + (-e_{12}^{\prime \prime}+\omega_{12}) {\mathbf{e}}_{2} \\ &+& (-e_{13}^{\prime \prime}+\omega_{13}) {\mathbf{e}}_{3} , \\ {\mathbf{m}}^{2 \prime} &=& (-e_{12}^{\prime \prime}-\omega_{12}) {\mathbf{e}}_{1} + (1-e_{22}^{\prime \prime}) {\mathbf{e}}_{2} \\ &+& (-e_{23}^{\prime}+\omega_{23}) {\mathbf{e}}_{3} , \\ {\mathbf{m}}^{3 \prime} &=& (- e_{13}^{\prime \prime}-\omega_{13}) {\mathbf{e}}_{1} + (-e_{23}^{\prime \prime}-\omega_{23}) {\mathbf{e}}_{2} \\ &+& (1-e_{33}^{\prime \prime}) {\mathbf{e}}_{3} . \end{array} $$
(58)
Next, take ni,αi in Eq. 32 to be constants so that ηi can be approximated by
$$ \eta_{i} = 1+(\frac{n_{i}}{3}) e_{v}+ (\frac{\alpha_{i}}{3}) (\theta-\theta_{0}) , $$
(59)
where ni,αi satisfy the restrictions (53). Then, using the approximations in the Appendix it can be shown that the deviatoric stress in Eq. 49 is approximated by
$$\begin{array}{@{}rcl@{}} {\mathbf{T}^{\prime\prime}} &=& \frac{4}{9} [2K_{1}\{3e_{11}^{\prime \prime} -n_{1} e_{v} - \alpha_{1} (\theta-\theta_{0})\} \\ &-& K_{2}\{3e_{22}^{\prime \prime} -n_{2} e_{v} - \alpha_{2} (\theta-\theta_{0})\} \\ &-& K_{3}\{3e_{33}^{\prime \prime} -n_{3} e_{v} - \alpha_{3} (\theta-\theta_{0})\} ] ({\mathbf{e}}_{1} \otimes {\mathbf{e}}_{1}) \\ &+& \frac{4}{9} [-K_{1}\{3e_{11}^{\prime \prime} -n_{1} e_{v} - \alpha_{1} (\theta-\theta_{0})\} \\ &+&2K_{2}\{3e_{22}^{\prime \prime} -n_{2} e_{v} - \alpha_{2} (\theta-\theta_{0})\} \\ &-&K_{3}\{ 3e_{33}^{\prime \prime} -n_{3} e_{v} - \alpha_{3} (\theta-\theta_{0})\} ]({\mathbf{e}}_{2} \otimes {\mathbf{e}}_{2}) \\ &+& \frac{4}{9} [-K_{1}\{3e_{11}^{\prime \prime} -n_{1} e_{v} - \alpha_{1} (\theta-\theta_{0})\} \\ &-&K_{2}\{ 3e_{22}^{\prime \prime} -n_{2} e_{v} - \alpha_{2} (\theta-\theta_{0})\} \\ &+&2K_{3}\{ 3e_{33}^{\prime \prime} -n_{3} e_{v} - \alpha_{3} (\theta-\theta_{0})\} ]({\mathbf{e}}_{3} \otimes {\mathbf{e}}_{3}) \\ &+& 2 (K_{1}+K_{2}+K_{4})e_{12}^{\prime \prime} ({\mathbf{e}}_{1} \otimes {\mathbf{e}}_{2} + {\mathbf{e}}_{2} \otimes {\mathbf{e}}_{1}) \\ &+& 2 (K_{1}+K_{3}+K_{5})e_{13}^{\prime \prime} ({\mathbf{e}}_{1} \otimes {\mathbf{e}}_{3} + {\mathbf{e}}_{3} \otimes {\mathbf{e}}_{1}) \\ &+& 2 (K_{2}+ K_{3}+K_{6})e_{23}^{\prime \prime}({\mathbf{e}}_{2} \otimes {\mathbf{e}}_{3} + {\mathbf{e}}_{3} \otimes {\mathbf{e}}_{2}) .\end{array} $$
(60)
Also, from the expressions in [26], it can be shown that the functions f1,f2 can be approximated by
$$\begin{array}{@{}rcl@{}} f_{1} &=& - \rho_{0} C_{v} \gamma_{0} (1-J_{e}) , \quad f_{2} = \frac{1}{2} K (J_{e}-1)^{2} , \\ K &=& \rho_{0} ({C_{0}^{2}}-{\gamma_{0}^{2}} C_{v} \theta_{0}) , \end{array} $$
(61)
where C0 is the zero-stress shock velocity. Then, the pressure in Eq. 49 is given by
$$\begin{array}{@{}rcl@{}} p &=& \rho_{0} C_{v} \gamma_{0} (\theta-\theta_{0}) - K {\varepsilon}_{v} \\ &+& (\frac{4n_{1}}{9}) K_{1} [3e_{11}^{\prime\prime} -n_{1} e_{v} - \alpha_{1} (\theta-\theta_{0})] \\ &+& (\frac{4n_{2}}{9}) K_{2} [3e_{22}^{\prime\prime} -n_{2} e_{v} - \alpha_{2} (\theta-\theta_{0})] \\ &+&(\frac{4n_{3}}{9}) K_{3} [3e_{33}^{\prime\prime} -n_{3} e_{v} - \alpha_{3} (\theta-\theta_{0})] ,\end{array} $$
(62)
and the entropy is approximated by
$$\begin{array}{@{}rcl@{}} \rho_{0} \eta &=& \rho_{0} C_{v} (\frac{\theta-\theta_{0}}{\theta_{0}}) + \rho_{0} C_{v} \gamma_{0} e_{v} \\ &+& (\frac{4\alpha_{1}}{9}) K_{1} [3e_{11}^{\prime\prime} -n_{1} e_{v} - \alpha_{1} (\theta-\theta_{0})] \\ &+& (\frac{4\alpha_{2}}{9}) K_{2} [3e_{22}^{\prime\prime} -n_{2} e_{v} - \alpha_{2} (\theta-\theta_{0})] \\ &+& (\frac{4\alpha_{3}}{9}) K_{3} [3e_{33}^{\prime\prime} -n_{3} e_{v} - \alpha_{3} (\theta-\theta_{0})] .\end{array} $$
(63)
It then follows that the material will be in a HSS when the elastic distortional strain \(e_{ij}^{\prime \prime }\) is given by
$$\begin{array}{@{}rcl@{}} e_{11}^{\prime \prime} &=& \frac{1}{3} [n_{1} e_{v} + \alpha_{1} (\theta-\theta_{0})] , \quad e_{22}^{\prime \prime} = \frac{1}{3} [n_{2} e_{v} + \alpha_{2} (\theta-\theta_{0})] ,\\ e_{33}^{\prime \prime} &=& \frac{1}{3} [n_{3} e_{v} + \alpha_{3} (\theta-\theta_{0})] , \quad e_{12}^{\prime \prime} = e_{13}^{\prime \prime} = e_{23}^{\prime \prime} = 0 .\end{array} $$
(64)

8 Isotropic response

The specific constitutive model proposed in Section 6 models isotropic thermoelastic response when
$$\begin{array}{@{}rcl@{}} n_{i} &=& 0 , \quad \alpha_{i} = 0 , \quad K_{1} = K_{2} = K_{3} = \frac{\mu}{2} ,\\ K_{4} &=& K_{5} = K_{6} = 0 ,\end{array} $$
(65)
where μ is the small deformation shear modulus. Since for isotropic response, loading in any material direction produces the same response, it is impossible to distinguish differences between the directions mi in any RLS. Consequently, it is convenient to introduce the elastic distortional deformation unimodular tensor \(\mathbf {B}_{e}^{\prime }\), such that
$$ {\mathbf{B}_e^{\prime}} = {\mathbf{m}}_{i}^{\prime} \otimes {\mathbf{m}}_{i}^{\prime} , \quad {\mathbf{B}}_{e}^{\prime -1} = {\mathbf{m}}^{i \prime} \otimes {\mathbf{m}}^{i \prime} . $$
(66)
Moreover, since for isotropic response (ηi = 1) in Eq. 32 and it can be shown that
$$ \beta_{1}+\beta_{2}+\beta_{3} = ({\mathbf{B}_e^{\prime}} + {\mathbf{B}}_{e}^{\prime-1}) \cdot {\mathbf{I}} , $$
(67)
so that ψ2 in Eq. 48 can be written in the form
$$ \rho_{0} \psi_{2} = \frac{1}{2} \mu [\frac{1}{2} ({\mathbf{B}_e^{\prime}} + {\mathbf{B}}_{e}^{\prime-1}) \cdot {\mathbf{I}} -3] , $$
(68)
which is a function of isotropic invariants of \({\mathbf {B}_e^{\prime }}\) only.
Next, using Eqs. 321222444 and taking the plastic spin \(\overline {{\mathbf {W}}}_{p}\) in Eq. 55 equal to zero for isotropic response, it can be shown that \(\mathbf {B}_{e}^{\prime }\) satisfies the evolution equation
$$\begin{array}{@{}rcl@{}} \dot{{\mathbf{B}}}_{e}^{\prime} &=& {\mathbf{L}} {\mathbf{B}_e^{\prime}} + {\mathbf{B}_e^{\prime}} {\mathbf{L}}^{T} - \frac{2}{3} ({\mathbf{D}} \cdot {\mathbf{I}}) {\mathbf{B}_e^{\prime}} - {\Gamma} \overline{{\mathbf{A}}}_{p} , \\ \overline{{\mathbf{A}}}_{p} &=& \overline{{\mathbf{D}}}_{p} {\mathbf{B}_e^{\prime}} + {\mathbf{B}_e^{\prime}} \overline{{\mathbf{D}}}_{p} . \end{array} $$
(69)
Moreover, using Eqs. 38 and 65, it can be shown that
$$ {\mathbf{B}}_{1} + {\mathbf{B}}_{2} + {\mathbf{B}}_{3} = {\mathbf{B}_e^{\prime}} - {\mathbf{B}}_{e}^{\prime-1} - \frac{1}{3} [({\mathbf{B}_e^{\prime}} - {\mathbf{B}}_{e}^{\prime-1}) \cdot {\mathbf{I}}] {\mathbf{I}} , $$
(70)
so with the help of Eq. 54, \(\overline {{\mathbf {A}}}_{p}\) will characterize isotropic inelasticity when
$$\begin{array}{@{}rcl@{}} &&d_{1} \text{Sign}({\mathbf{T}^{\prime\prime}} \cdot {\mathbf{B}}_{1}^{\prime\prime}) = \frac{1}{4} , \quad d_{2} \text{Sign}({\mathbf{T}^{\prime\prime}} \cdot {\mathbf{B}}_{2}^{\prime\prime}) = \frac{1}{4} , \\ &&d_{3} \text{Sign}({\mathbf{T}^{\prime\prime}} \cdot {\mathbf{B}}_{3}^{\prime\prime}) = \frac{1}{4} , \quad d_{4} = d_{5} = d_{6} = 0 , \\ &&\overline{{\mathbf{A}}}_{p} = \frac{1}{2} [{\mathbf{B}}_{e}^{\prime 2} - {\mathbf{I}} - \frac{1}{3} \{({\mathbf{B}}_{e}^{\prime} - {\mathbf{B}}_{e}^{\prime-1}) \cdot {\mathbf{I}}\} {\mathbf{B}_e^{\prime}}] .\end{array} $$
(71)
Although, these specifications yield isotropic response, it is noted that the evolution Eq. 69 would be identical to that in [27, 28, 29] if \(\overline {\mathbf {A}}_{p}\) were replaced by
$$ {\mathbf{A}}_{p} = {\mathbf{B}}_{e}^{\prime} - (\frac{3}{{\mathbf{B}}_{e}^{\prime -1} \cdot {\mathbf{I}}}) {\mathbf{I}} , $$
(72)
which is preferable to \(\overline {{\mathbf {A}}}_{p}\) since it yields a simple, strongly objective numerical integration scheme for Eq. 69. Also, the values for di in Eq. 71 were specified so that \(\overline {\mathbf {A}}_{p}\) in Eq. 71 would have the same form as Ap in Eq. 72 for small elastic distortional deformations.

9 Examples of steady-state isochoric deformations

For the examples in this section, consider an orthotropic viscoplastic material with constant Γ experiencing isochoric deformation at constant reference temperature 𝜃 = 𝜃0. The values of ηi in Eq. 32 are specified to be constants so that
$$ \eta_{i} = J_{e}^{n_{i}/3} , \quad n_{1} + n_{2} + n_{3} = 0 , $$
(73)
and the functions (f1,f2) are specified by Eq. 61, so the pressure the pressure in Eq. 49 is given by
$$ p = K (1-J_{e}) + J_{e}^{-1} \sum\limits_{i = 1}^{3} K_{i} N_{i} . $$
(74)
Also, the initial state of the material is taken to be an RLS with
$$ {\mathbf{m}}_{i}(0) = {\mathbf{e}}_{i} . $$
(75)

9.1 Steady-state uniaxial stress

For steady-state uniaxial stress in the e1 direction, the velocity gradient is specified by
$$\begin{array}{@{}rcl@{}} {\mathbf{L}} = {\mathbf{D}} &=& {\Gamma} [D_{11} {\mathbf{e}}_{1} \otimes {\mathbf{e}}_{1} + D_{22} {\mathbf{e}}_{2} \otimes {\mathbf{e}}_{2} \\ &-&(D_{11} + D_{22}) {\mathbf{e}}_{3} \otimes {\mathbf{e}}_{3}] ,\end{array} $$
(76)
where Γ,D11,D22 are constants. Although the steady-state motion is isochoric with constant Je, the transient response is allowed to change volume so the steady-state value of Je need not be unity. Then, using Eqs. 5455, and 75, it can be shown that the solutions of the evolution (3) take the forms
$$\begin{array}{@{}rcl@{}} {\mathbf{m}}_{1}^{\prime} &=& \lambda_{1} {\mathbf{e}}_{1} , \quad {\mathbf{m}}_{2}^{\prime} = \lambda_{2} {\mathbf{e}}_{2} , \quad {\mathbf{m}}_{3}^{\prime} = \frac{1}{\lambda_{1} \lambda_{2}} {\mathbf{e}}_{3} , \\ {\mathbf{m}}_{1} &=& J_{e}^{1/3} \lambda_{1} {\mathbf{e}}_{1} , \quad {\mathbf{m}}_{2} = J_{e}^{1/3} \lambda_{2} {\mathbf{e}}_{2} ,\\ {\mathbf{m}}_{3} &=& \frac{J_{e}^{1/3}}{\lambda_{1} \lambda_{2}} {\mathbf{e}}_{3} ,\end{array} $$
(77)
where Je is a constant elastic dilatation and λ1,λ2 are elastic distortional stretches. Using Eq. 36 and these expressions it can also be shown that
$$\begin{array}{@{}rcl@{}} N_{1} &=& \frac{n_{1}}{3} ({{\Lambda}_{1}^{2}} - \frac{1}{{{\Lambda}_{1}^{2}}}) , \quad N_{2} = \frac{n_{2}}{3} ({{\Lambda}_{2}^{2}} - \frac{1}{{{\Lambda}_{2}^{2}}}) , \\ N_{3} &=& \frac{n_{3}}{3} (\frac{1}{{{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}} - {{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}) , \\ {\Lambda}_{i} &=& \frac{\lambda_{i}}{J_{e}^{n_{i}/3}} , \quad {\Lambda}_{3} = \frac{1}{{\Lambda}_{1} {\Lambda}_{2}} ,\end{array} $$
(78)
and
$$\begin{array}{@{}rcl@{}} {\mathbf{B}}_{1}^{\prime \prime} &=& \frac{1}{3} ({{\Lambda}_{1}^{2}} - \frac{1}{{{\Lambda}_{1}^{2}}}) [2({\mathbf{e}}_{1} \otimes {\mathbf{e}}_{1}) - {\mathbf{e}}_{2} \otimes {\mathbf{e}}_{2} - {\mathbf{e}}_{3} \otimes {\mathbf{e}}_{3}] , \\ {\mathbf{B}}_{2}^{\prime \prime} &=& \frac{1}{3} ({{\Lambda}_{2}^{2}} - \frac{1}{{{\Lambda}_{2}^{2}}}) [- {\mathbf{e}}_{1} \otimes {\mathbf{e}}_{1} + 2({\mathbf{e}}_{2} \otimes {\mathbf{e}}_{2}) - {\mathbf{e}}_{3} \otimes {\mathbf{e}}_{3}] , \\ {\mathbf{B}}_{3}^{\prime \prime} &=& \frac{1}{3} (\frac{1}{{{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}} - {{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}) [- {\mathbf{e}}_{1} \otimes {\mathbf{e}}_{1} - {\mathbf{e}}_{2} \otimes {\mathbf{e}}_{2} \\ &+& 2({\mathbf{e}}_{3} \otimes {\mathbf{e}}_{3}) ] .\end{array} $$
(79)
Specifically, the steady-state values of these stretches are determined by using Eqs. 54 and 55 and setting the right-hand side of the evolution (3) equal zero to deduce the total deformation rates D11,D22 yields
$$\begin{array}{@{}rcl@{}} D_{11} &=& \frac{2}{3} d_{1} \text{Sign}({\mathbf{T}^{\prime\prime}} \cdot {\mathbf{B}}_{1}^{\prime\prime}) ({{\Lambda}_{1}^{2}} - \frac{1}{{{\Lambda}_{1}^{2}}})\\ &&- \frac{1}{3} d_2 \text{Sign}(\mathbf{T}^{\prime\prime} \cdot \mathbf{B}_2^{\prime\prime}) ({\Lambda}_2^2 - \frac{1}{{\Lambda}_2^2}) \end{array} $$
$$\begin{array}{@{}rcl@{}} &-& \frac{1}{3} d_{3} \text{Sign}({\mathbf{T}^{\prime\prime}} \cdot {\mathbf{B}}_{3}^{\prime\prime}) (\frac{1}{{{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}} - {{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}) , \\ D_{22} &=& -\frac{1}{3} d_{1} \text{Sign}({\mathbf{T}^{\prime\prime}} \cdot {\mathbf{B}}_{1}^{\prime\prime}) ({{\Lambda}_{1}^{2}} - \frac{1}{{{\Lambda}_{1}^{2}}}) \\ &+& \frac{2}{3} d_{2} \text{Sign}({\mathbf{T}^{\prime\prime}} \cdot {\mathbf{B}}_{2}^{\prime\prime}) ({{\Lambda}_{2}^{2}} - \frac{1}{{{\Lambda}_{2}^{2}}}) \\ &-& \frac{1}{3} d_{3} \text{Sign}({\mathbf{T}^{\prime\prime}} \cdot {\mathbf{B}}_{3}^{\prime\prime}) (\frac{1}{{{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}} - {{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}) .\end{array} $$
(80)
Next, using the functions (61) and Eq. 78, the pressure in Eq. 49 is given by
$$\begin{array}{@{}rcl@{}} p &=& K(1-J_{e}) + \frac{1}{3} J_{e}^{-1} [n_{1} K_{1} ({{\Lambda}_{1}^{2}} - \frac{1}{{{\Lambda}_{1}^{2}}}) \\ &+& n_{2} K_{2} ({{\Lambda}_{2}^{2}} - \frac{1}{{{\Lambda}_{2}^{2}}}) + n_{3} K_{3} (\frac{1}{{{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}} - {{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}) ] ,\end{array} $$
(81)
and the deviatoric stress is given by
$$\begin{array}{@{}rcl@{}} {\mathbf{T}^{\prime\prime}} &=& \frac{1}{3} J_{e}^{-1} K_{1}({{\Lambda}_{1}^{2}} - \frac{1}{{{\Lambda}_{1}^{2}}}) [2({\mathbf{e}}_{1} \otimes {\mathbf{e}}_{1}) - ({\mathbf{e}}_{2} \otimes {\mathbf{e}}_{2} \\ &+& {\mathbf{e}}_{3} \otimes {\mathbf{e}}_{3})] \\ &+& \frac{1}{3} J_{e}^{-1} K_{2} ({{\Lambda}_{2}^{2}} - \frac{1}{{{\Lambda}_{2}^{2}}}) [- {\mathbf{e}}_{1} \otimes {\mathbf{e}}_{1} + 2({\mathbf{e}}_{2} \otimes {\mathbf{e}}_{2}) \\ &-& {\mathbf{e}}_{3} \otimes {\mathbf{e}}_{3}] \\ &+& \frac{1}{3} J_{e}^{-1} K_{3} (\frac{1}{{{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}} - {{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}) [- {\mathbf{e}}_{1} \otimes {\mathbf{e}}_{1} - {\mathbf{e}}_{2} \otimes {\mathbf{e}}_{2} \\ &+& 2({\mathbf{e}}_{3} \otimes {\mathbf{e}}_{3}) ] .\end{array} $$
(82)
Now, consider uniaxial stress σ in the e1 direction for which
$$\begin{array}{@{}rcl@{}} \sigma &=& - p + \frac{1}{3} J_{e}^{-1} [2K_{1}({{\Lambda}_{1}^{2}} - \frac{1}{{{\Lambda}_{1}^{2}}}) - K_{2} ({{\Lambda}_{2}^{2}} - \frac{1}{{{\Lambda}_{2}^{2}}}) \\ &-& K_{3} (\frac{1}{{{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}} - {{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}})] , \\ 0 &=& - p + \frac{1}{3} J_{e}^{-1} [-K_{1}({{\Lambda}_{1}^{2}} - \frac{1}{{{\Lambda}_{1}^{2}}}) + 2 K_{2} ({{\Lambda}_{2}^{2}} - \frac{1}{{{\Lambda}_{2}^{2}}}) \\ &-& K_{3} (\frac{1}{{{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}} - {{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}})] , \\ 0 &=& - p + \frac{1}{3} J_{e}^{-1} [-K_{1}({{\Lambda}_{1}^{2}} - \frac{1}{{{\Lambda}_{1}^{2}}}) - K_{2} ({{\Lambda}_{2}^{2}} - \frac{1}{{{\Lambda}_{2}^{2}}}) \\ &+& 2 K_{3} (\frac{1}{{{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}} - {{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}})] .\end{array} $$
(83)
Adding these three equations yields the standard result that
$$ p = - \frac{\sigma}{3} , $$
(84)
Moreover, taking the difference of the last two equations in Eq. 83 yields
$$ K_{2} ({{\Lambda}_{2}^{2}} - \frac{1}{{{\Lambda}_{2}^{2}}}) = K_{3} (\frac{1}{{{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}} - {{\Lambda}_{1}^{2}}{{\Lambda}_{2}^{2}}) , $$
(85)
which can be solved to obtain
$$ {\Lambda}_{2} = [ \frac{1 + (\frac{K_{3}}{K_{2}}) \frac{1}{{{\Lambda}_{1}^{2}}}} {1 + (\frac{K_{3}}{K_{2}}) {{\Lambda}_{1}^{2}}} ]^{1/4} . $$
(86)
Then, with the help of Eqs. 84 and 85, the first of Eq. 83 can be rewritten in the form
$$ \frac{\sigma J_{e}}{K} = \frac{K_{1}}{K} ({{\Lambda}_{1}^{2}} - \frac{1}{{{\Lambda}_{1}^{2}}}) - \frac{K_{2}}{K} ({{\Lambda}_{2}^{2}} - \frac{1}{{{\Lambda}_{2}^{2}}}) . $$
(87)
and Eq. 81 yields an equation for Je of the form
$$\begin{array}{@{}rcl@{}} &&{J_{e}^{2}} - J_{e} - C = 0 ,\\ &&C = \frac{1}{3}(1+n_{1}) [\frac{K_{1}}{K} ({{\Lambda}_{1}^{2}} - \frac{1}{{{\Lambda}_{1}^{2}}}) - \frac{K_{2}}{K} ({{\Lambda}_{2}^{2}} - \frac{1}{{{\Lambda}_{2}^{2}}})] ,\end{array} $$
(88)
which can be solved to obtain
$$ J_{e} = \frac{1}{2} [1 + \sqrt{1 + 4 C}] . $$
(89)
This solution is parameterized by the value of Λ1 which is used in Eq. 86 to determine Λ2. These values are used to determine Je in Eq. 89 and σ in Eq. 87. Moreover, the physical distortional stretches λ1,λ2 are determined by Eq. 78, the normalized rates D11,D22 are determined by Eq. 80 and the total rate of deformation tensor D is determined by Eq. 76. In this regard, it is noted that with the specification (86) it can be shown that
$$ \text{Sign}({\mathbf{T}^{\prime\prime}} \cdot {\mathbf{B}}_{1}^{\prime\prime}) = \text{Sign}({\mathbf{T}^{\prime\prime}} \cdot {\mathbf{B}}_{2}^{\prime\prime}) = \text{Sign}({\mathbf{T}^{\prime\prime}} \cdot {\mathbf{B}}_{3}^{\prime\prime}) = 1 , $$
(90)
for any positive values of K1,K2,K31. Although the normalized variables Λ12,Je,D11,D22,σm/K,σp/K are independent of n2, the physical distortional stretches λ2,λ3 in Eq. 78 depend on n2 (with n3 = −n1n2).
To examine the response of this steady-state solution for different values of n1 consider the case with the isotropic constants
$$ K_{1} = K_{2} = K_{3} = K , $$
(91)
and the case with the anisotropic constants
$$ K_{1} = K , \quad K_{2} = 2 K , \quad K_{3} = 3 K . $$
(92)
Figure 1 shows that the response of Λ2 as a function of Λ1 is nearly the same for these two cases. In this regard, it should be noted that even though the forms of Λ2 as a function of Λ1 for isotropic and orthotropic response are similar, Λ12 are normalized quantities so the actual distortional stretches λ1,λ2 are influenced by the orthotropic constants n1,n2. Figures 2 and 3 show that changing n1 has a significant effect on the response for both cases and that the response for the isotropic constants (91) and for the anisotropic constants (92) are similar but quantitatively different. Figure 4 shows a direct comparison of these responses for (n1 = 0). The numerical simulations indicate that for n1 = 0,1 the steady-state uniaixal stress solution ceases to exist for critical values of compression. Consequently, the plots in Figs. 23, and 4 for these values of n1 are limited to their ranges of existence.
Fig. 1

Steady-state uniaxial stress: Predictions of the normalized distortional stretch Λ2 for the isotropic constants (I) with K1 = K2 = K3 = K and the anisotropic constants (A) with K1 = K,K2 = 2K,K3 = 3K

Fig. 2

Steady-state uniaxial stress: Predictions of the dilatation Je and the normalized axial stress σ/K for the isotropic constants (K1 = K2 = K3 = K) for different values of n1

Fig. 3

Steady-state uniaxial stress: Predictions of the dilatation Je and the normalized axial stress σ/K for the anisotropic constants (K1 = K,K2 = 2K,K3 = 3K) for different values of n1

Fig. 4

Steady-state uniaxial stress: Comparison of the predictions of the dilatation Je and the normalized axial stress σ/K for the isotropic constants (I) with K1 = K2 = K3 = K and the anisotropic constants (A) with K1 = K,K2 = 2K,K3 = 3K, both with n1 = 0

9.2 Steady-state simple shear

For steady-state simple shear the velocity gradient L and rate of deformation D are specified by
$$ {\mathbf{L}} = {\Gamma} L_{12} {\mathbf{e}}_{1} \otimes {\mathbf{e}}_{2} , \quad {\mathbf{D}} = \frac{1}{2} {\Gamma} L_{12} ({\mathbf{e}}_{1} \otimes {\mathbf{e}}_{2} + {\mathbf{e}}_{2} \otimes {\mathbf{e}}_{1}) , $$
(93)
where Γ,L12 are positive constants. Then, using Eqs. 5455 and 75 it can be shown that the solution of the evolution (3) takes the form
$$\begin{array}{@{}rcl@{}} {\mathbf{m}}_{1}&=& {\mathbf{m}}_{1}^{\prime} = \lambda_{1} [\cos(\theta_{1}) {\mathbf{e}}_{1} + \sin(\theta_{1}) {\mathbf{e}}_{2})] ,\\ {\mathbf{m}}_{2}&=& {\mathbf{m}}_{2}^{\prime} = \lambda_{2} [-\sin(\theta_{2}) {\mathbf{e}}_{1} + \cos(\theta_{2}) {\mathbf{e}}_{2})] ,\\ {\mathbf{m}}_{3}&=&{\mathbf{m}}_{3}^{\prime} = \frac{1}{\lambda_{1} \lambda_{2} \cos(\theta_{2}-\theta_{1})} {\mathbf{e}}_{3} , \quad J_{e} = 1 ,\end{array} $$
(94)
where λ1,λ2 are positive stretches. Also, it can be shown that
$$\begin{array}{@{}rcl@{}} {\mathbf{m}}^{1}&=& {\mathbf{m}}^{1\prime} = \frac{\cos(\theta_{2}) {\mathbf{e}}_{1} + \sin(\theta_{2}) {\mathbf{e}}_{2}}{\lambda_{1} \cos(\theta_{2}-\theta_{1})} , \\ {\mathbf{m}}^{2}&=& {\mathbf{m}}^{2\prime} = \frac{-\sin(\theta_{1}) {\mathbf{e}}_{1} + \cos(\theta_{1}) {\mathbf{e}}_{2}}{\lambda_{2} \cos(\theta_{2}-\theta_{1})} ,\\ {\mathbf{m}}^{3}&=&{\mathbf{m}}^{3 \prime} = \lambda_{1} \lambda_{2} \cos(\theta_{2}-\theta_{1}) {\mathbf{e}}_{3} .\end{array} $$
(95)
The steady-state solution is determined by substituting these expressions into the evolution (3) and setting the right-hand sides equal to zero. This procedure yields the following four non-trivial scalar equations
$$\begin{array}{@{}rcl@{}} ({\mathbf{L}}-{\mathbf{L}}_{p}) \cdot ({\mathbf{m}}^{1} \otimes {\mathbf{m}}_{1}) = 0 , \quad ({\mathbf{L}}-{\mathbf{L}}_{p}) \cdot ({\mathbf{m}}^{2} \otimes {\mathbf{m}}_{1}) = 0 , \\ ({\mathbf{L}}-{\mathbf{L}}_{p}) \cdot ({\mathbf{m}}^{1} \otimes {\mathbf{m}}_{2})= 0 , \quad ({\mathbf{L}}-{\mathbf{L}}_{p}) \cdot ({\mathbf{m}}^{2} \otimes {\mathbf{m}}_{2}) = 0 ,\end{array} $$
(96)
with the fifth scalar equation
$$ ({\mathbf{L}}-{\mathbf{L}}_{p}) \cdot ({\mathbf{m}}^{3} \otimes {\mathbf{m}}_{3}) = 0 , $$
(97)
being automatically satisfied since
$$ {\mathbf{m}}^{i} \otimes {\mathbf{m}}_{i} = {\mathbf{I}} , \quad ({\mathbf{L}} - {\mathbf{L}}_{p}) \cdot {\mathbf{I}} = {\mathbf{D}} \cdot {\mathbf{I}} = 0 . $$
(98)
To examine the influence of the loading rate L12 and the plastic spin parameter Ω12 on the solution, consider the anisotropic inelastic constants
$$\begin{array}{@{}rcl@{}} K_{1} = K , \quad K_{2} = 2 K , \quad K_{3} = 3 K , \quad K_{4} = 4 K , \\ d_{1} = 1 , \quad d_{2} = 2 , \quad d_{3} = 3 , \quad d_{4} = d_{5} = d_{6} = 0 .\end{array} $$
(99)
Using these values, Eqs. 96 were solved numerically for λ1,λ2,𝜃1,𝜃2. Figure 5 plots these quantities as functions of the loading rate L12 for different values of the plastic spin parameter Ω12 and Fig. 6 plots these quantities as functions of the plastic spin parameter Ω12 for different values of the loading rate L12. From these figures it can be seen that both L12 and Ω12 have significant influences on the response. The value Ω12 = 4 is near the smallest value for which a steady-state solution was obtained. Moreover, it is noted that as Ω12 increases the angles 𝜃1,𝜃2 approach the value 45 which causes mi to be nearly parallel to the principle directions of the rate of deformation tensor D.
Fig. 5

Steady-state simple shear: Predictions of λ1,λ2,𝜃1,𝜃2 as functions of the loading rate L12 for different values of the plastic spin parameter Ω12 using the anisotropic constants (99)

Fig. 6

Steady-state simple shear: Predictions of λ1,λ2,𝜃1,𝜃2 as functions of the plastic spin parameter Ω12 for different values of the loading rate L12 using the anisotropic constants (99)

10 Conclusions

This paper introduces a new Eulerian formulation of thermomechanical, orthotropic, elastic-inelastic constitutive equations for soft materials. The Eulerian formulation is based on the evolution equations for a triad of microstructural vectors mi proposed in [7] to model general anisotropic elastic and inelastic response. Motivated by the work in [9], [10], the functions ηi in Eq. 32 are generalized to include dependence on both elastic dilatation Je and temperature 𝜃. These functions characterize thermoelastic distortional deformations in hydrostatic states of stress (HSS) (31) and need to be determined by experiments. Here, the six nonlinear orthotropic invariants βi in Eqs. 34 and 35 are proposed as functions of the components \(m_{ij}^{\prime }\) of the elastic distortional metric and of ηi.

When the Helmholtz free energy ψ is taken to be a function of Je,βi,𝜃, as in Eq. 38, the stress response (39) automatically reproduces the measured thermoelastic distortions for all HSS. This means that βi are measures of the thermoelastic distortions which cause deviatoric stress. It also means that the search for specific functional forms for ψ is simplified since attention need only be focused on modeling the response that causes deviatoric stress and its influence on the entropy η.

Since mi model both elastic response and directions of anisotropy, it is possible to model inelastic anisotropy with the proposed model. Specific constitutive equations have been proposed for ψ in Eq. 48, the functions ηi in Eq. 32, the rate of elastic deformation \(\overline {\mathbf {D}}_{p}\) in Eq. 54 and the plastic spin \(\overline {\mathbf {W}}_{p}\) in Eq. 55, which automatically satisfy the dissipation inequality (47). Also, the formulation allows for modeling anisotropic heat conduction (41).

Examples of steady-state uniaxial tension and simple shear have been considered to demonstrate the influence of various anisotropic material constants in the specific constitutive equations. Since these constitutive equations are valid for large elastic deformations they can be used to model elastic and inelastic effects in soft materials. Additional research is needed to develop a strongly objective, robust numerical integration algorithm (e.g., [28]) for the evolution (3) in the presence of inelastic deformation rate.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Faculty of Mechanical EngineeringTechnion-Israel Institute of TechnologyHaifaIsrael

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