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Polytechnica

, Volume 1, Issue 1–2, pp 19–35 | Cite as

Ordered Rate Constitutive Theories for Non-classical Thermoviscoelastic Solids with Dissipation and Memory Incorporating Internal Rotations

  • K. S. SuranaEmail author
  • D. Mysore
  • J. N. Reddy
Original Article
  • 143 Downloads

Abstract

This paper presents constitutive theories for non-classical thermoviscoelastic solids with dissipation and memory using thermodynamic framework based on entirety of the displacement gradient tensor. Thus, the conservation and the balance laws used in this work incorporate symmetric as well as antisymmetric parts of the displacement gradient tensor. In this paper, we only consider small deformation small strain; hence, the constitutive theories are basis independent. The constitutive theories are derived in the presence as well as in the absence of balance of moment of moments balance law. It is shown that the energy storage, dissipation mechanism, and the fading memory in the non-classical thermoviscoelastic solids are due to strain rates, rotation rates, stress tensor, moment tensor, and their rates. Constitutive theories are derived using the conditions resulting from the entropy inequality in conjunction with the representation theorem. The constitutive theories derived using integrity are followed by simplified constitutive theories. Material coefficients are derived and discussed for both cases. Constitutive models parallel to non-classical Maxwell, Oldroyd-B, and Giesekus models for thermoviscoelastic fluids are derived and shown to be a subset of a more generalized simplified constitutive theory presented in the paper. Retardation moduli are derived for stress tensor as well as moment tensor and are compared with those in classical continuum theories for similar solids.

Keywords

Ordered rate constitutive theories Non-classical Thermoviscoelastic solids Internal roatations Polar theory 

1 Introduction

The conservation and the balance laws for non-classical solids and fluent continua incorporating internal rotations and the internal rotation rates due to displacement gradient tensor and velocity gradient tensor and the derivation of the ordered rate constitutive theories for non-classical thermoelastic and thermoviscoelastic solids without memory have been derived and presented by Surana et al. (2015a, b, c, e, f; 2016d). For the benefit of the readers and for the sake of completeness, we briefly describe the motivation behind the non-classical continuum theories incorporating internal rotations and their rates as well as brief description of the theory. In deforming solids and fluids, the displacement gradient tensor and the velocity gradient tensor that are measures of deformation physics vary between a material point (or location) and its neighboring material points (or locations). Polar decomposition of the Jacobian of deformation tensor or the decomposition of the displacement gradient tensor into symmetric and antisymmetric tensors shows that varying deformation between material points results in varying rotations between them. Since thesevarying rotations arise due to varying deformation of the solid continua, hence, they are completely defined by the antisymmetric part of the displacement gradient tensor, thus do not constitute additional degrees of freedom at a material point. We refer to these rotations as internal rotations to distinguish them from Cosserat rotations. If the varying internal rotations are resisted by the deforming solid matter, then there must exist corresponding conjugate moments. This physics is all internal to the deforming continua and is always present in all deforming solids but is completely neglected in currently used classical continuum theories. The continuum theory presented in Surana et al. (2015c, f) considers internally varying rotations in addition to strains and the associated conjugate moments in the derivation of the conservation and the balance laws. This theory has been referred to as internal polar or non-classical continuum theory with internal rotations.

There is much published works on non-classical continuum theories under the titles: micropolar theories, stress couple theories, rotation gradient theories, strain gradient theories, etc. with applications to beams, plates, shells, vibration studies etc. (Voigt 1887, 1894; Cosserat and Cosserat 1909; Surana et al. 2015d; Günther 1958; Grioli 1960; Aero and Kuvshinskii 1961; Schäfer 1962; Truesdell and Toupin 1960; Mindlin and Tiersten 1962; Toupin 1962; Koiter 1964; Eringen 1962, 1964a, b; Eringen and Suhubi 1964a, b; Mindlin 1964; Green and Rivlin 1964; Mindlin 1965; Brand and Rubin 2007; Cao and Tucker 2008; Riahi and Curran 2009; Sansour 1998; Sansour and Skatulla 2009; Sansour et al. 2010; Varygina et al. 2010; Nikabadze 2011; Ieşan 2011; Jung et al. 2010; Alonso-Marroquín 2011; Chiriţă and Ghiba 2012; Cao et al. 2013; Addessi et al. 2013; Cialdea et al. 2013; Skatull and Sansour 2013; Liu 2013; Del Piero 2014; Genovese 2014; Huang et al. 2014; Eringen 1967). A comprehensive review of these works can be found in Surana et al. (2015a, b, c; e, f, 2016d) and others by Surana et al. (2016a, b, c; 2017a, b, c, d, f, g). This is not repeated here for the sake of brevity. In this paper, we utilize the conservation and the balance laws presented in Surana et al. (2015a, c, f; 2016d) for solid continua with small deformation and small strain but incorporating internal rotations due to Jacobian of deformation tensor or displacement gradient tensor to derive constitutive theories for the thermoviscoelastic solids with dissipation and memory. Due to small deformation assumption, the conservation and the balance laws and the constitutive theories contain stress and strain measures and their rates that are not basis dependent (covariant, contravariant, or Jaumann rates). Entropy inequality and the other conservation and balance laws are used to determine the constitutive variables. Their argument tensors are decided based on the conjugate pairs in the entropy as well as the additional desired physics that is not obvious from the conjugate pairs in the entropy inequality. The constitutive variables and their argument tensors are generalized to include the desired rates up to certain orders giving rise to the ordered rate constitutive theories.

The constitutive theories are derived using representation theorem (Prager 1945; Reiner 1945; Todd 1948; Rivlin and Ericksen 1955; Rivlin 1955; Wang 1969a, b, 1970, 1971; Smith 1970, 1971; Spencer and Rivlin 1959, 1960; Spencer 1971; Boehler 1977; Zheng 1993a, b) or the theory of generators and invariants. Material coefficients for each constitutive theory are established using Taylor series expansion of the coefficients in the linear combination about a known configuration. It is shown that in addition to the energy storage, dissipation, and rheology mechanism due to deviatoric part of the symmetric Cauchy stress tensor and its rates and conjugate strain tensor and its rates, such non-classical solids also contain similar mechanisms due to Cauchy moment tensor and its rates and rotation tensor gradient and its rates. The constitutive theories are derived by not considering balance of moment of moments as an additional required balance law as well as by requiring balance of moment of moments as a necessary balance law (Yang et al. 2002, Surana et al. 2017d; g). The constitutive theories are first derived using integrity and then simplified to show that the constitutive models similar to Maxwell, Oldroyd-B, Giesekus, etc. (used for fluids) for thermoviscoelastic solids with memory are a subset of the general but simplified constitutive theories presented in this paper. Retardation moduli are derived for stress tensor as well as moment tensor.

2 Notations, Internal Rotations, Their Gradients and Rates, Stress and Strain Measures, and Decompositions

In the following, we give a brief description of the rotations; this is necessary as some of the notation are new (Surana 2015). If xi and \(\bar {x}_{i}\) denote the position coordinates of a material point in the reference and the current configurations respectively in fixed x-frame, then
$$\begin{array}{@{}rcl@{}} \bar{x}_{i} &=& \bar{x}_{i} (x_{1}, x_{2}, x_{3}, t) \end{array} $$
(1)
$$\begin{array}{@{}rcl@{}} \text{or } \quad x_{i} &=& x_{i} (\bar{x}_{1}, \bar{x}_{2}, \bar{x}_{3}, t) \end{array} $$
(2)
If \(\{dx\} = [dx_{1}, dx_{2}, dx_{3}]\) and \(\{d\bar {x}\} = [d\bar {x}_{1}, d\bar {x}_{2}, d\bar {x}_{3}]\) are components of length ds and \(d\bar {s}\) in the reference and the current configurations, then we have
$$\begin{array}{@{}rcl@{}} \{d\bar{x}\} &=& [J]\{dx\} \end{array} $$
(3)
$$\begin{array}{@{}rcl@{}} \{dx\} &=& [\bar{J}]\{d\bar{x}\} \end{array} $$
(4)
with
$$ [J] \!= [\bar{J}]^{-1}, \quad [\bar{J}] = [J]^{-1}; \quad [J][\bar{J}] = [\bar{J}][J] = [I] $$
(5)
In Murnaghan’s notation
$$ [J] = \left[\frac{\partial\{\bar{x}\}}{\partial\{x\}}\right] = \left[\frac{\bar{x}_{1},\bar{x}_{2},\bar{x}_{3}}{x_{1},x_{2},x_{3}}\right] ; \quad [\bar{J}] = \left[\frac{\partial\{x\}}{\partial\{\bar{x}\}}\right] = \left[\frac{x_{1},x_{2},x_{3}}{\bar{x}_{1},\bar{x}_{2},\bar{x}_{3}}\right] $$
(6)
in which columns of [J] are covariant base vectors \(\tilde { \pmb {\boldsymbol {g }}}_{i}\), whereas the rows of \([\bar {J}]\) are contravariant base vectors \(\underset {\sim }{\pmb {\boldsymbol {g}}}^{i}\) (Surana 2015). [J] and \([\bar {J}]\) are Jacobians of deformation tensors. The basis defined by \([\bar {J}]\) is reciprocal to the basis defined by [J]. The over bar on all dependent quantities refer to their Eulerian description, i.e., they are functions of \(\bar {x}_{i}\) and t whereas quantities without bar are their Lagrangian descriptions, i.e., they are functions of xi and t. Thus, \(\bar {Q}(\bar { \pmb {\boldsymbol {x }}},t)\) and Q(x,t) are Eulerian and Lagrangian descriptions of a quantity Q in the current configuration. Since the deformation and strains are assumed small \(\bar {x}_{i}\simeq x_{i}\), hence, basis dependency of various measures disappears. Thus, we can use σ and m as Cauchy stress and moment tensors. We present details of various measures in the following
$$\begin{array}{@{}rcl@{}} &&[J] = [{~}^d\!J] + [I] = \left[ \frac{ \partial\{u\} }{ \partial\{x\} } \right] + [I] \end{array} $$
(7)
$$\begin{array}{@{}rcl@{}} &&[{~}^d\!J] = [{~}^d_{s}\!J] + [{~}^d_{a }\!J] \end{array} $$
(8)
$$\begin{array}{@{}rcl@{}} &&[{~}^d_{s }\!J] = \frac{1}{2}\left([{~}^d\!J] + [{~}^d\!J]^{T} \right) = [\varepsilon] \end{array} $$
(9)
$$\begin{array}{@{}rcl@{}} [{~}^d_{a }\!J] = \frac{1}{2}\left([{~}^d\!J] - [{~}^d\!J]^{T} \right) = \left[\begin{array}{lll} 0 & {~}_i{\Theta}_{x_{3}} & -{~}_i{\Theta}_{x_{2}} \\ -{~}_i{\Theta}_{x_{3}} & 0 & {~}_i{\Theta}_{x_{1}} \\ {~}_i{\Theta}_{x_{2}} & -{~}_i{\Theta}_{x_{1}} & 0 \end{array}\right] \end{array} $$
(10)
$$ {~}_i{\Theta}_{x_{1}} = \frac{1}{2}\left(\frac{\partial u_{2}}{\partial x_{3}} - \frac{\partial u_{3}}{\partial x_{2}}\right) $$
(11)
$$ {~}_i{\Theta}_{x_{2}} = \frac{1}{2}\left(\frac{\partial u_{3}}{\partial x_{1}} - \frac{\partial u_{1}}{\partial x_{3}}\right) $$
(12)
$$ {~}_i{\Theta}_{x_{3}} = \frac{1}{2}\left(\frac{\partial u_{1}}{\partial x_{2}} - \frac{\partial u_{2}}{\partial x_{1}}\right) $$
(13)
[ d J] is displacement gradient tensor, \([{~}^d_{s }\!J]\) and \([{~}^d_{a }\!J]\) are symmetric and antisymmetric parts of [ d J]. [ε] is strain tensor based on small deformation small strain theory. \({~}_i{\Theta }_{x_{1}}\), \({~}_i{\Theta }_{x_{2}}\), \({~}_i{\Theta }_{x_{3}}\) are rotations about the axes of a triad with axes parallel to x-frame assumed positive when clockwise. Alternatively, we can consider
$$ \boldsymbol\nabla\times\boldsymbol{u} = \boldsymbol{e}_{i}\times\boldsymbol{e}_{j} \frac{\partial u_{j}}{\partial x_{i}} = \epsilon_{ijk}\boldsymbol{e}_{k}\frac{\partial u_{j}}{\partial x_{i}} $$
(14)
$$ \boldsymbol\nabla\times\boldsymbol{u} = \boldsymbol{e}_{1}\left(\frac{\partial u_{3}}{\partial x_{2}} - \frac{\partial u_{2}}{\partial x_{3}}\right) + \boldsymbol{e}_{2}\left(\frac{\partial u_{1}}{\partial x_{3}} - \frac{\partial u_{3}}{\partial x_{1}}\right) + \boldsymbol{e}_{3}\left(\frac{\partial u_{2}}{\partial x_{1}} - \frac{\partial u_{1}}{\partial x_{2}}\right) $$
(15)
$$ = \pmb{\boldsymbol{e }}_{1}\left(-2({~}_i{\Theta}_{x_{1}} )\right) + \pmb{\boldsymbol{e }}_{2}\left(-2({~}_i{\Theta}_{x_{2}} )\right) + \pmb{\boldsymbol{e }}_{3}\left(-2({~}_i{\Theta}_{x_{3}} )\right) $$
(16)
Coefficients of ei in (16) are double the magnitudes of the rotations in (10) and the rotations are positive when in the counterclockwise direction, hence the negative sign in (16) using clockwise definitions of rotation in (11) – (13). In this paper, we use definitions (11)–(13), i.e., clockwise rotations are assumed positive. Since \([{~}^d_{a }\!J]\) is a tensor of rank 2, the gradients of \([{~}^d_{a }\!J]\) would be a tensor of rank 3 (Steinmann 1994; Srinivasa and Reddy 2013; Segerstad et al. 2008; Surana et al. 2017b, c). However, this representation can be simplified if we arrange rotations \({~}_i{\Theta }_{x_{1}}\), \({~}_i{\Theta }_{x_{2}}\), \({~}_i{\Theta }_{x_{3}}\) as a vector, i.e., a tensor of rank 1. Let
$$ \{{~}_{i}{\Theta}\}^{T} = [{~}_i{\Theta}_{x_{1}}, {~}_i{\Theta}_{x_{2}}, {~}_i{\Theta}_{x_{3}}] $$
(17)
Gradient of { iΘ} in (17) and its decomposition into symmetric and antisymmetric tensors are given by the following.
$$\begin{array}{@{}rcl@{}} [ {~}^{{~}_i{\Theta}}\!J ] &=& \left[ \frac{ \partial\{{~}_{i}{\Theta}\}}{ \partial\{x\}} \right] = [ {~}_{ s}^{{~}_i{\Theta}}\!J ] + [ {~}_{ a}^{{~}_i{\Theta}}\!J ] \end{array} $$
(18)
$$\begin{array}{@{}rcl@{}} {[{~}_{ s}^{{~}_i{\Theta}}\!J ]} &=& \frac{1}{2} \left([ {~}^{{~}_i{\Theta}}\!J ] + [ {~}^{{~}_i{\Theta}}\!J ]^{T} \right) \end{array} $$
(19)
$$\begin{array}{@{}rcl@{}} {[ {~}_{ a}^{{~}_i{\Theta}}\!J ]} &=& \frac{1}{2} \left([{~}^{{~}_i{\Theta}}\!J ] - [ {~}^{{~}_i{\Theta}}\!J ]^{T} \right) \end{array} $$
(20)

3 Conservation and Balance Laws

Consider a tetrahedron in the reference configuration (volume V and boundary V) whose planes are parallel to the planes of the fixed x-frame and whose oblique plane is subjected to average stress \(\bar { \pmb {\boldsymbol {P }}}\) and average moment \(\bar { \pmb {\boldsymbol {M }}}\). Upon deformation, it occupies volume \(\bar {V}\) and with boundary \(\partial {\bar {V}}\). Since we only consider small deformation small strain \(\bar {V} \simeq V\) and \(\partial {\bar {V}} \simeq \partial V\), i.e., \(\bar {x}\) and x are interchangeable in the derivation of the conservation and the balance laws. Thus, for this case, covariant base vectors are parallel to the axes of the x-frame; hence, the first and the second Piola-Kirchhoff stress tensors are same as the Cauchy stress tensor. The same is true for the moment tensors. Conservation of mass, balance of linear momenta, balance of angular momenta, and first and the second laws of thermodynamics yield the following (Surana et al. 2015c, f) in Lagrangian description.
$$\begin{array}{@{}rcl@{}} \rho_{_{_{\!\!0}}}(\pmb{\boldsymbol{x }}) = |J|\rho(\pmb{\boldsymbol{x }},t)\\ |J| \simeq 1, \quad \rho_{_{_{\!\!0}}} \simeq \rho \end{array} $$
(21)
$$ \rho_{_{_{\!\!0}}}\frac{D \pmb{\boldsymbol{v }}}{Dt}-\rho_{_{_{\!\!0}}}{ \pmb{\boldsymbol{F }}}^{b} - { \pmb{\boldsymbol{\nabla \cdot }} \pmb{\boldsymbol{\sigma }}}= 0 $$
(22)
$$ \pmb{\boldsymbol{\nabla \cdot }} \pmb{\boldsymbol{m }} - \pmb{\boldsymbol{\epsilon: }} \pmb{\boldsymbol{\sigma }} = 0 $$
(23)
$$ \rho_{_{_{\!\!0}}}\frac{De}{Dt} + \pmb{\boldsymbol{\nabla \cdot q }} - \sigma_{jk} \frac{\partial v_{k}}{\partial x_{j}} - m_{jk} \frac{\partial\left({~}_{i}\overset{ \text{\textbf{.}}}{{\Theta}}_{k} \right)}{\partial x_{j}} - {~}_{i}\overset{ \text{\textbf{.}}}{ \pmb{\boldsymbol{{\Theta} }}} \pmb{\boldsymbol{\cdot }} (\pmb{\boldsymbol{\nabla \cdot }} \pmb{\boldsymbol{m }} ) = 0 $$
(24)
$$ \rho_{_{_{\!\!0}}}\left(\frac{D{\Phi}}{Dt} + \eta\frac{D\theta}{Dt} \right) + \frac{ \pmb{\boldsymbol{{q \cdot g} }}}{\theta} - \sigma_{jk} \frac{\partial v_{k}}{\partial x_{j}} - m_{jk} \frac{\partial\left({~}_{i}\overset{ \text{\textbf{.}}}{{\Theta}}_{k} \right)}{\partial x_{j}} - {~}_{i}\overset{ \text{\textbf{.}}}{ \pmb{\boldsymbol{{\Theta} }}} \pmb{\boldsymbol{\cdot }} (\pmb{\boldsymbol{\nabla \cdot }} \pmb{\boldsymbol{m }} ) \!\leq\! 0 $$
(25)
where v are the velocities, Fb are the body forces per unit mass, 𝜖 is the permutation tensor, e is the specific internal energy, q is the heat flux, Φ is the Helmholtz free energy density, η is the entropy density, 𝜃 is the temperature, and g is temperature gradient. \({~}_{i}\overset { \text {\textbf {.}}}{\boldsymbol {{\Theta }}}\) are the rates of internal rotations that are defined by (11)–(13). The Cauchy stress tensor is non-symmetric and so is the Cauchy moment tensor.

3.1 Balance of Moment of Moments Balance Law

Yang et al. (2002) and Surana et al. (2017d, g) have shown that when the additional physics of internal rotations is considered for the deforming volume of solid matter, the conservation and the balance laws used in classical continuum theories are not sufficient to ensure equilibrium of the deforming matter. Surana et al. (2017d, g) have shown this to be true in case of non-classical solid as well as fluent continua. In non-classical theories based on internal rotations or rotation rates additional balance law, balance of moment of moments is required (Yang et al. 2002, Surana et al. 2017d, g). Details regarding the need for this balance law in non-classical continuum theories and its derivation can be found in Surana et al. (2017d, g); this material is not repeated here for the sake of brevity. Based on this balance law
$$ \epsilon_{ijk} m_{ij} = 0 $$
(26)
must hold. Hence, \(m_{ij} = m_{ji}\), i.e., Cauchy moment tensor must be symmetric. Obviously in the absence of this balance law, the Cauchy moment tensor m is not symmetric. In the derivation of the constitutive theories for m in this paper, we assume m to be non-symmetric as this is more general case. The constitutive theories for m when it is symmetric are a subset of the constitutive theories when m is not symmetric.

4 Conjugate Pairs in Entropy Inequality, Constitutive Variables, and Their Argument Tensors

From the entropy inequality (25), we note that in each of two trace terms, both tensors are non-symmetric; thus, based on the works of Spencer, Wang, Zheng, and others (Prager 1945; Reiner 1945; Todd 1948; Rivlin and Ericksen 1955; Rivlin 1955; Wang 1969a, b, 1970, 1971; Smith 1970, 1971; Spencer and Rivlin 1959, 1960; Spencer 1971; Boehler 1977; Zheng 1993a, b), the pair of tensors in each trace term do not constitute rate of work conjugate pair. That is neither of the tensors in each pair can be expressed in terms of the other (without the rate) due to lack of existence of integrity for non-symmetric tensors. We consider the following equations, decompositions, substitutions, and simplifications (using back subscript s and a for symmetric and antisymmetric).
$$ { \pmb{\boldsymbol{\nabla \cdot }} \pmb{\boldsymbol{m }} = \pmb{\boldsymbol{\epsilon: }} \pmb{\boldsymbol{\sigma }}} $$
(27)
$$ \pmb{\boldsymbol{\sigma }} = {~}_s\boldsymbol{\sigma} + {~}_a\boldsymbol{\sigma} $$
(28)
$$\begin{array}{@{}rcl@{}} \begin{array}{ll} & \pmb{\boldsymbol{m }} = {~}_s \boldsymbol{m} + {~}_a \boldsymbol{m} \\ &[L] = \left[ \frac{\partial \{v\}}{\partial\{x\}} \right] \end{array} \end{array} $$
(29)
$$ \sigma_{jk} \frac{\partial v_{k}}{\partial x_{j}} = tr\left([\sigma][L] \right) = tr\left([\sigma][\overset{ \text{\textbf{.}}}{J}] \right) = tr\left([\sigma][^{d}\overset{ \text{\textbf{.}}}{J}] \right) $$
(30)
$$ m_{jk} \frac{\partial\left({~}_{i}\overset{ \text{\textbf{.}}}{{\Theta}}_{k} \right)}{\partial x_{j}} = tr\left([m][^{{~}_{i}{\Theta}}\overset{ \text{\textbf{.}}}{J}] \right) $$
(31)
and
$$ [{~}^d\!\overset{ \text{\textbf{.}}}{J}] = [ {~}^d_{s }\!\overset{ \text{\textbf{.}}}{J} ] + [ {~}^d_{a }\!\overset{ \text{\textbf{.}}}{J} ] = [ \overset{ \text{\textbf{.}}}{\varepsilon} ] + [ {~}^d_{a }\!\overset{ \text{\textbf{.}}}{J} ] $$
(32)
$$ [ {~}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} ] = [ {~}_{ s}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} ] + [ {~}_{ a}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} ] $$
(33)
using (32) and (33) in (30) and (31) and then substituting (27)–(31) in (25) and noting that
$$\begin{array}{@{}rcl@{}} &&\text{tr} \left([ {~}_s{\sigma} ] [ {~}^d_{a }\!\overset{ \text{\textbf{.}}}{J} ] \right) = 0\\ &&\text{tr} \left([{~}_{a}\sigma] [ {~}^d_{s }\!\overset{ \text{\textbf{.}}}{J} ] \right) = 0\\ &&\text{tr} \left([ {~}_s {m} ] [ {~}_{ a}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} ] \right) = 0\\ &&\text{tr} \left([ {~}_a {m} ] [ {~}_{ s}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} ] \right) = 0 \end{array} $$
(34)
$$ \text{tr} \left([\sigma][ {~}_{ a}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} ] \right) = - {~}_{i} \overset{ \text{\textbf{.}}}{ \pmb{\boldsymbol{{\Theta} }}} \pmb{\boldsymbol{\cdot }} \pmb{\boldsymbol{\nabla \cdot }} \pmb{\boldsymbol{m }} $$
(35)
we obtain the following from (25)
$$\begin{array}{@{}rcl@{}} \rho_{_{_{\!\!0}}}\left(\frac{D{\Phi}}{Dt} + \eta\frac{D\theta}{Dt} \right) &+& \frac{ \pmb{\boldsymbol{{q \cdot g} }}}{\theta} -\! \text{tr}\left([ {~}_s{\sigma} ][\overset{ \text{\textbf{.}}}{\varepsilon}] \right) - \text{tr}\left([ {~}_s {m} ][ {~}_{ s}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} ] \right)\\& -& \text{tr}\left([ {~}_a {m} ][ {~}_{ a}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} ] \right) \leq 0 \end{array} $$
(36)
The energy equation is modified likewise and we have
$$ \rho_{_{_{\!\!0}}}\frac{De}{Dt} + \pmb{\boldsymbol{\nabla \cdot q }} - \text{tr}\left([ {~}_s{\sigma} ][\overset{ \text{\textbf{.}}}{\varepsilon}] \right) - \text{tr}\left([ {~}_s {m} ][ {~}_{ s}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} ] \right) - \text{tr}\left([ {~}_a {m} ][ {~}_{ a}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} ] \right) = 0 $$
(37)
We note that in (36), in the trace terms, either both tensors are symmetric or antisymmetric; hence, all three trace terms in (36) indeed define rate of work conjugate pairs. From (36), we easily infer that Φ, η, q, \({~}_s\boldsymbol {\sigma } \), \({~}_s \boldsymbol {m}\), and \({~}_a \boldsymbol {m} \) are a possible choice of constitutive variables. This choice is also in agreement with the other balance laws. The argument tensors of these constitutive variables are decided using the conjugate pairs as well as the desired physics to be described by them that may not be entirely obvious from the entropy inequality. For small strain, small deformation \(\bar {x} \simeq x\) hence |J|≅ 1 i.e. \(\rho _{_{_{\!\!0}}} = \rho (\pmb {\boldsymbol {x }},t)\), hence density is constant. Choice of 𝜃 as an argument tensor for all constitutive variables is rather obvious. [ε] as an argument tensor of [ sσ] can be included from conjugate pair in (36). Likewise, choices of \([ {~}_{ s}^{{~}_i{\Theta }}\!J ]\) and \([ {~}_{ a}^{{~}_i{\Theta }}\!J ]\) as arguments of [ sm] and [ am] are straight forward from the conjugate pairs in (36). Choice of g as argument tensor of q is also obvious from the entropy inequality (36). Φ and η at this stage must contain totality of all the argument tensors, some of which may be ruled out later due to some other considerations. Dissipation mechanism due to \({~}_s\boldsymbol {\sigma } \) necessitates that \( \overset { \text {\textbf {.}}}{\boldsymbol {\varepsilon }} \) or ε[1] be an argument tensor of \({~}_s\boldsymbol {\sigma } \). Likewise, \([ {~}_{ s}^{{~}_i{\Theta }}\!\overset { \text {\textbf {.}}}{J} ]\) or \([ {~}_{ s}^{{~}_i{\Theta }}\!J_{[1]} ]\) and \([ {~}_{ a}^{{~}_i{\Theta }}\!\overset { \text {\textbf {.}}}{J} ]\) or \([ {~}_{ a}^{{~}_i{\Theta }}\!J_{[1]} ]\) must also be arguments of \({~}_s \boldsymbol {m} \) and \({~}_a \boldsymbol {m} \) due to dissipation considerations. Furthermore, the presence of rheology or memory in the solid matter due to \({~}_s\boldsymbol {\sigma } \), \({~}_s \boldsymbol {m}\), and \({~}_a \boldsymbol {m} \) necessitates that the constitutive equations for \({~}_s\boldsymbol {\sigma } \), \({~}_s \boldsymbol {m}\), and \({~}_a \boldsymbol {m} \) must contain their time derivatives only then memory mechanism is possible. If \({~}_s\boldsymbol {\sigma }^{[i]} ; i = 0, 1, {\dots } , m_{\! {~}_s\!\sigma } \), \({~}_s \boldsymbol {m}^{[j]} ; j = 0, 1, {\dots } , m_{\! {~}_s\!m } \) and \({~}_a \boldsymbol {m}^{[k]} ; k = 0, 1, {\dots } , m_{\! {~}_a\!m } \) are the rates of \({~}_s\boldsymbol {\sigma } \), \({~}_s \boldsymbol {m} \) and \({~}_a \boldsymbol {m} \) of up to orders \(m_{\! {~}_s\!\sigma }\), \( m_{\! {~}_s\!m } \) and \( m_{\! {~}_a\!m } \) in which \({~}_s\boldsymbol {\sigma }^{[0]} = {~}_s\boldsymbol {\sigma } \), \({~}_s \boldsymbol {m}^{[0]} = {~}_s \boldsymbol {m} \) and \({~}_a \boldsymbol {m}^{[0]} = {~}_a \boldsymbol {m} \), then due to the rheology considerations, at the very minimum, we must consider \({~}_s\boldsymbol {\sigma }^{[1]} \), \({~}_s \boldsymbol {m}^{[1]} \), and \({~}_a \boldsymbol {m}^{[1]} \) as constitutive variable with \({~}_s\boldsymbol {\sigma }^{[0]} \), \({~}_s \boldsymbol {m}^{[0]} \), and \({~}_a \boldsymbol {m}^{[0]} \) as their argument tensors in addition to ε (or ε[0]), \({~}_{ s}^{{~}_i{\Theta }}\!\boldsymbol {J} \) (or \({~}_{ s}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[0]} \) ), and \({~}_{ a}^{{~}_i{\Theta }}\!\boldsymbol {J} \) (or \({~}_{ a}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[0]} \)) respectively. Thus, at this stage, we must at least have the following constitutive variables and their argument tensors.
$$\begin{array}{@{}rcl@{}} {~}_s\boldsymbol{\sigma}^{[1]} &=& {~}_s\boldsymbol{\sigma}^{[1]} (\boldsymbol{\varepsilon}_{[0]} , \boldsymbol{\varepsilon}_{[1]} , {~}_s\boldsymbol{\sigma}^{[0]} , \theta )\\ {~}_s \boldsymbol{m}^{[1]} &=& {~}_s \boldsymbol{m}^{[1]} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} , {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} , {~}_s \boldsymbol{m}^{[0]} , \theta ) \end{array} $$
$$\begin{array}{@{}rcl@{}} {~}_a \boldsymbol{m}^{[1]} &=& {~}_a \boldsymbol{m}^{[1]} ({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} , {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} , {~}_a \boldsymbol{m}^{[0]} , \theta ) \\ {\Phi} &=& {\Phi} (\boldsymbol{\varepsilon}_{[0]} , \boldsymbol{\varepsilon}_{[1]} , {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} , {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} , {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} , {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} , \boldsymbol{g} , \theta )\\ \eta &=& \eta (\boldsymbol{\varepsilon}_{[0]} , \boldsymbol{\varepsilon}_{[1]} , {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} , {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} , {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} , {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} , \boldsymbol{g} , \theta )\\ \boldsymbol{q} &=& \boldsymbol{q} (\boldsymbol{g} , \theta ) \end{array} $$
(38)
We can further modify the argument tensors and the choice of constitutive variables based on the following remarks:
  1. 1.

    If we assume that the dissipation mechanism is not limited due to ε[1], \({~}_{ s}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[1]} \) and \({~}_{ a}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[1]} \) but ε[i];i = 2,3,…,nε, \({~}_{ s}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[j]} ; j = 2, 3, \dots , n_{\! \left ({~}_{ s}^{{~}_i{\Theta }}\!{J} \right )} \), \({~}_{ a}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[k]} ; k = 2, 3, \dots , n_{\! \left ({~}_{ a}^{{~}_i{\Theta }}\!{J} \right )} \) also contribute to dissipation, then, these tensors should also be included in the argument lists of \({~}_s\boldsymbol {\sigma }^{[1]} \), \({~}_s \boldsymbol {m}^{[1]} \), and \({~}_a \boldsymbol {m}^{[1]} \) in (38) and likewise in the arguments of Φ and η.

     
  2. 2.

    If we further assume that the memory mechanism is also related to \({~}_s\boldsymbol {\sigma }^{[i]} ; i = 2, 3, {\dots } , m_{\! {~}_s\!\sigma } \), \({~}_s \boldsymbol {m}^{[j]} ; j = 2, 3, {\dots } , m_{\! {~}_s\!m } \) and \({~}_a \boldsymbol {m}^{[k]} ; k = 2, 3, {\dots } , m_{\! {~}_a\!m } \), then the choice of the constitutive variables \({~}_s\boldsymbol {\sigma }^{[1]} \), \({~}_s \boldsymbol {m}^{[1]} \) and \({~}_a \boldsymbol {m}^{[1]} \) must be replaced by \({~}_s\boldsymbol {\sigma }^{[ m_{\! {~}_s\!\sigma } ]} \), \({~}_s \boldsymbol {m}^{[ m_{\! {~}_s\!m } ]} \) and \({~}_a \boldsymbol {m}^{[ m_{\! {~}_a\!m } ]} \) and \({~}_s\boldsymbol {\sigma }^{[i]} ; i = 0, 1, {\dots } , m_{\! {~}_s\!\sigma } - 1\), \({~}_s \boldsymbol {m}^{[j]} ; j = 0, 1, {\dots } , m_{\! {~}_s\!m } - 1\) and \({~}_a \boldsymbol {m}^{[k]} ; k = 0, 1, {\dots } , m_{\! {~}_a\!m } - 1\) must be included as their argument tensors respectively. These choices will enable ordered rate constitutive theories for \({~}_s\boldsymbol {\sigma } \), \({~}_s \boldsymbol {m} \), and \({~}_a \boldsymbol {m} \) in terms of the rates of ε, \({~}_{ s}^{{~}_i{\Theta }}\!\boldsymbol {J} \) and \({~}_{ a}^{{~}_i{\Theta }}\!\boldsymbol {J} \). Accordingly we also modify the argument tensors of Φ and η.

     
$$\begin{array}{@{}rcl@{}} &&{\Phi} = {\Phi} (\boldsymbol{\varepsilon}_{[i]} ; i = 0, 1, \dots, n_{\!\varepsilon} , {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[j]} ; j = 0, 1, \dots, n_{\! \left({~}_{ s}^{{~}_i{\Theta}}\!{J} \right)} , {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[k]} ; k = 0, 1, \dots, n_{\! \left({~}_{ a}^{{~}_i{\Theta}}\!{J} \right)} ,\\ &&\qquad{} {~}_s\boldsymbol{\sigma}^{[i]} ; i = 0, 1, {\dots} , (m_{\! {~}_s\!\sigma } - 1), {~}_s \boldsymbol{m}^{[j]} ; j = 0, 1, {\dots} , (m_{\! {~}_s\!m } - 1), {~}_a \boldsymbol{m}^{[k]} ; k = 0, 1, {\dots} , (m_{\! {~}_a\!m } - 1), \pmb{\boldsymbol{g }}, \theta )\\ &&\eta = \eta (\boldsymbol{\varepsilon}_{[i]} ; i = 0, 1, \dots, n_{\!\varepsilon} , {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[j]} ; j = 0, 1, \dots, n_{\! \left({~}_{ s}^{{~}_i{\Theta}}\!{J} \right)} , {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[k]} ; k = 0, 1, \dots, n_{\! \left({~}_{ a}^{{~}_i{\Theta}}\!{J} \right)} ,\\ &&\qquad{} {~}_s\boldsymbol{\sigma}^{[i]} ; i = 0, 1, {\dots} , (m_{\! {~}_s\!\sigma } - 1), {~}_s \boldsymbol{m}^{[j]} ; j = 0, 1, {\dots} , (m_{\! {~}_s\!m } - 1), {~}_a \boldsymbol{m}^{[k]} ; k = 0, 1, {\dots} , (m_{\! {~}_a\!m } - 1), \pmb{\boldsymbol{g }}, \theta )\\ && {~}_s\boldsymbol{\sigma}^{[ m_{\! {~}_s\!\sigma } ]} = {~}_s\boldsymbol{\sigma}^{[ m_{\! {~}_s\!\sigma } ]} (\boldsymbol{\varepsilon}_{[i]} ; i = 0, 1, \dots, n_{\!\varepsilon} , {~}_s\boldsymbol{\sigma}^{[i]} ; i = 0, 1, {\dots} , (m_{\! {~}_s\!\sigma } - 1), \theta )\\ && {~}_s \boldsymbol{m}^{[ m_{\! {~}_s\!m } ]} = {~}_s \boldsymbol{m}^{[ m_{\! {~}_s\!m } ]} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[j]} ; j = 0, 1, \dots, n_{\! \left({~}_{ s}^{{~}_i{\Theta}}\!{J} \right)} , {~}_s \boldsymbol{m}^{[j]} ; j = 0, 1, {\dots} , (m_{\! {~}_s\!m } - 1), \theta )\\ && {~}_a \boldsymbol{m}^{[ m_{\! {~}_a\!m } ]} = {~}_a \boldsymbol{m}^{[ m_{\! {~}_a\!m } ]} ({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[k]} ; k = 0, 1, \dots, n_{\! \left({~}_{ a}^{{~}_i{\Theta}}\!{J} \right)} , {~}_a \boldsymbol{m}^{[k]} ; k = 0, 1, {\dots} , (m_{\! {~}_a\!m } - 1), \theta )\\ && \pmb{\boldsymbol{q }} = \pmb{\boldsymbol{q }} (\pmb{\boldsymbol{g }}, \theta ) \end{array} $$
(39)
Using Φ(⋅) in (39), we can determine \(\overset { \text {\textbf {.}}}{{\Phi }}(\cdot )\)
$$\begin{array}{@{}rcl@{}} \frac{D{\Phi}}{Dt} &=& \overset{ \text{\textbf{.}}}{{\Phi}} = \sum\limits_{j = 0}^{n_{\!\varepsilon} } \frac{\partial {{\Phi}}}{\partial {({\varepsilon}_{[j]} )_{ik}}} (\overset{ \text{\textbf{.}}}{\varepsilon}_{[j]} )_{ik} + \sum\limits_{j = 0}^{n_{\! \left({~}_{ s}^{{~}_i{\Theta}}\!{J} \right)} } \frac{\partial {{\Phi}}}{\partial {({~}_{ s}^{{~}_i{\Theta}}\!J_{[j]} )_{ik}}} ({~}_{ s}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} {~}_{[j]} )_{ik}\\ &&+\sum\limits_{j = 0}^{n_{\! \left({~}_{ a}^{{~}_i{\Theta}}\!{J} \right)} } \frac{\partial {{\Phi}}}{\partial {({~}_{ a}^{{~}_i{\Theta}}\!J_{[j]} )_{ik}}} ({~}_{ a}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} {~}_{[j]} )_{ik} + \sum\limits_{j = 0}^{(m_{\! {~}_s\!\sigma } -1)} \frac{\partial {{\Phi}}}{\partial {({~}_s {\sigma}^{[j]} )_{ik}}} ({~}_s \overset{ \text{\textbf{.}}}{\sigma} {~}^{[j]} )_{ik}\end{array} $$
$$\begin{array}{@{}rcl@{}} &&+\sum\limits_{j = 0}^{(m_{\! {~}_s\!m } -1)} \frac{\partial {{\Phi}}}{\partial {({~}_s {m}^{[j]} )_{ik}}} ({~}_s \overset{ \text{\textbf{.}}}{m} {~}^{[j]} )_{ik} + \sum\limits_{j = 0}^{(m_{\! {~}_a\!m } -1)} \frac{\partial {{\Phi}}}{\partial {({~}_a {m}^{[j]} )_{ik}}} ({~}_a \overset{ \text{\textbf{.}}}{m} {~}^{[j]} )_{ik} \\ &&+ \frac{\partial {{\Phi}}}{\partial {g_{i}}} \overset{ \text{\textbf{.}}}{g}_{i} + \frac{\partial {{\Phi}}}{\partial {\theta}} \overset{ \text{\textbf{.}}}{\theta} \end{array} $$
(40)
Substituting \(\overset { \text {\textbf {.}}}{{\Phi }}(\cdot )\) from (40) into the entropy inequality and grouping terms
$$\begin{array}{@{}rcl@{}} &&\left(\rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({\varepsilon}_{[0]} )_{ik}}} - ({~}_s {\sigma}^{[0]} )_{ik} \right) (\overset{ \text{\textbf{.}}}{\varepsilon}_{[0]} )_{ik} + \sum\limits_{j = 1}^{n_{\varepsilon}} \rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({\varepsilon}_{[j]} )_{ik}}} (\overset{ \text{\textbf{.}}}{\varepsilon}_{[j]} )_{ik}\\ &&\!+\left(\rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({~}_{ s}^{{~}_i{\Theta}}\!J_{[0]} )_{ik}}} - ({~}_s {m}^{[0]} )_{ik} \right)({~}_{ s}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} {~}_{[0]} )_{ik} + \! \sum\limits_{j = 1}^{n_{\! \left({~}_{ s}^{{~}_i{\Theta}}\!{J} \right)} } \rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({~}_{ s}^{{~}_i{\Theta}}\!J_{[j]} )_{ik}}} ({~}_{ s}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} {~}_{[j]} )_{ik} \\ &&\!+\left(\rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({~}_{ a}^{{~}_i{\Theta}}\!J_{[0]} )_{ik}}} - ({~}_a {m}^{[0]} )_{ik} \right) ({~}_{ a}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} {~}_{[0]} )_{ik} + \sum\limits_{j = 1}^{n_{\! \left({~}_{ a}^{{~}_i{\Theta}}\!{J} \right)} } \rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({~}_{ a}^{{~}_i{\Theta}}\!J_{[j]} )_{ik}}} ({~}_{ a}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} {~}_{[j]} )_{ik} \\ &&\!+\sum\limits_{j = 0}^{(m_{\! {~}_s\!\sigma } -1)} \rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({~}_s {\sigma}^{[j]} )_{ik}}} ({~}_s \overset{ \text{\textbf{.}}}{\sigma} {~}^{[j]} )_{ik} + \sum\limits_{j = 0}^{(m_{\! {~}_s\!m } -1)} \rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({~}_s {m}^{[j]} )_{ik}}} ({~}_s \overset{ \text{\textbf{.}}}{m} {~}^{[j]} )_{ik} \\ &&\!\!+\sum\limits_{j = 0}^{(m_{\! {~}_a\!m } -1)} \rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({~}_a {m}^{[j]} )_{ik}}} ({~}_a \overset{ \text{\textbf{.}}}{m} {~}^{[j]} )_{ik} + \rho_{_{_{\!\!0}}} \left(\frac{\partial {{\Phi}}}{\partial {\theta}} +\! \eta \right) \overset{ \text{\textbf{.}}}{\theta} + \rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {g_{i}}} \overset{ \text{\textbf{.}}}{g}_{i} + \frac{ \pmb{\boldsymbol{q\cdot g }}}{\theta} \!\leq\! 0\\ \end{array} $$
(41)
Entropy inequality in (41) is satisfied for arbitrary but admissible values of \( \overset { \text {\textbf {.}}}{\boldsymbol {\varepsilon }}_{[j]} ; j = 1,2,\dots ,n_{\varepsilon }\), \({~}_{ s}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[j]} ; j = 1,2,\dots , n_{\! \left ({~}_{ s}^{{~}_i{\Theta }}\!{J} \right )} \), \({~}_{ a}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[j]} ; j = 1,2,\dots , n_{\! \left ({~}_{ a}^{{~}_i{\Theta }}\!{J} \right )} \), \({~}_s \overset { \text {\textbf {.}}}{\boldsymbol {\sigma }}^{[j]} ; j = 0,1,\dots , m_{\! {~}_s\!\sigma } -1\), \({~}_s \overset { \text {\textbf {.}}}{\boldsymbol {m}}^{[j]} ; j = 0,1,\dots , m_{\! {~}_s\!m } -1\), \({~}_a \overset { \text {\textbf {.}}}{\boldsymbol {m}}^{[j]} ; j = 0,1,\dots , m_{\! {~}_a\!m } -1\), \(\overset { \text {\textbf {.}}}{\theta }\), \(\overset { \text {\textbf {.}}}{ \pmb {\boldsymbol {g }}}_{i}\) if the following hold (i.e. if their coefficients are zero).
$$\begin{array}{@{}rcl@{}} \rho_{_{_{\!\!0}}} \frac{\partial {{\Phi}}}{\partial {(\boldsymbol{\varepsilon}_{[j]} )}}&=& 0 ; j = 1,2, \dots, n_{\!\varepsilon} \Rightarrow {\Phi} \neq {\Phi}(\boldsymbol{\varepsilon}_{[j]}\\ && ; j = 1,2, \dots, n_{\!\varepsilon} )\\ \rho_{_{_{\!\!0}}} \frac{\partial {{\Phi}}}{\partial {({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[j]} )}} &=& 0 ; j = 1,2, \dots, n_{\! \left({~}_{ s}^{{~}_i{\Theta}}\!{J} \right)} \Rightarrow {\Phi} \neq {\Phi}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[j]}\\ && ; j = 1,2, \dots, n_{\! \left({~}_{ s}^{{~}_i{\Theta}}\!{J} \right)} ) \\ \rho_{_{_{\!\!0}}} \frac{\partial {{\Phi}}}{\partial {({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[j]} )}} &=& 0 ; j = 1,2, \dots, n_{\! \left({~}_{ a}^{{~}_i{\Theta}}\!{J} \right)} \Rightarrow {\Phi} \neq {\Phi}({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[j]}\\ && ; j = 1,2, \dots, n_{\! \left({~}_{ a}^{{~}_i{\Theta}}\!{J} \right)} ) \\ \rho_{_{_{\!\!0}}} \frac{\partial {{\Phi}}}{\partial {({~}_s\boldsymbol{\sigma}^{[j]} )}} &=& 0 ; j = 0,1, \dots, (m_{\! {~}_s\!\sigma } - 1) \Rightarrow {\Phi} \neq {\Phi}({~}_s\boldsymbol{\sigma}^{[j]}\\ && ; j = 0,1, \dots, (m_{\! {~}_s\!\sigma } -1))\\ \rho_{_{_{\!\!0}}} \frac{\partial {{\Phi}}}{\partial {({~}_s \boldsymbol{m}^{[j]} )}} &=& 0 ; j = 0,1, \dots, (m_{\! {~}_s\!m } - 1) \Rightarrow\! {\Phi} \neq {\Phi}({~}_s \boldsymbol{m}^{[j]}\\ && ; j = 0,1, \dots, (m_{\! {~}_s\!m } -1))\\ \rho_{_{_{\!\!0}}} \frac{\partial {{\Phi}}}{\partial {({~}_a \boldsymbol{m}^{[j]} )}} &=& 0 ; j = 0,1, \dots, (m_{\! {~}_a\!m } - 1) \Rightarrow\! {\Phi} \neq\! {\Phi}({~}_a \boldsymbol{m}^{[j]}\\ && ; j = 0,1, \dots, (m_{\! {~}_a\!m } -1)) \end{array} $$
(42)
$$\begin{array}{@{}rcl@{}} \rho_{_{_{\!\!0}}} \left(\frac{\partial {{\Phi}}}{\partial {\theta}} + \eta \right) = 0 \Rightarrow \eta = -\frac{\partial{\Phi}}{\partial \theta} \end{array} $$
(43)
$$\begin{array}{@{}rcl@{}} \rho_{_{_{\!\!0}}} \frac{\partial {{\Phi}}}{\partial { \pmb{\boldsymbol{g }}}} = 0 \Rightarrow {\Phi} \neq {\Phi}(\pmb{\boldsymbol{g }}) \end{array} $$
(44)
Based on (43), η is not a constitutive variable as it can be determined using \(\frac {\partial {\Phi }}{\partial \theta }\). Using (42) and (44), the argument tensors of Φ can be modified as
$$\begin{array}{@{}rcl@{}} {\Phi} = {\Phi}(\boldsymbol{\varepsilon}_{[0]} , {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} , {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} , \theta) \end{array} $$
(45)
using (42) and (44), the entropy inequality reduces to
$$\begin{array}{@{}rcl@{}} &&\left(\rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({\varepsilon}_{[0]} )_{ik}}} - ({~}_s {\sigma}^{[0]} )_{ik} \right) (\overset{ \text{\textbf{.}}}{\varepsilon}_{[0]} )_{ik}\\&+& \left(\rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({~}_{ s}^{{~}_i{\Theta}}\!J_{[0]} )_{ik}}} - ({~}_s {m}^{[0]} )_{ik} \right)({~}_{ s}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} {~}_{[0]} )_{ik}\\ &+&\left(\rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({~}_{ a}^{{~}_i{\Theta}}\!J_{[0]} )_{ik}}} - ({~}_a {m}^{[0]} )_{ik} \right) ({~}_{ a}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} {~}_{[0]} )_{ik} + \frac{ \boldsymbol{q} \pmb{\boldsymbol{\cdot }} \boldsymbol{g} }{\theta} \leq 0 \end{array} $$
(46)
In (46), if we set the coefficients of \({~}_{ s}^{{~}_i{\Theta }}\!\overset { \text {\textbf {.}}}{\boldsymbol {J}} {~}_{[0]} \) and \({~}_{ a}^{{~}_i{\Theta }}\!\overset { \text {\textbf {.}}}{\boldsymbol {J}} {~}_{[0]} \) to zero, then this implies that the constitutive theories for \({~}_s \boldsymbol {m}^{[0]} \) and \({~}_a \boldsymbol {m}^{[0]} \) can be derived using \(\frac {\partial {{\Phi }}}{\partial {({~}_{ s}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[0]} )}}\) and \(\frac {\partial {{\Phi }}}{\partial {({~}_{ a}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[0]} )}}\) which imply that \({~}_s \boldsymbol {m}^{[0]} \) and \({~}_a \boldsymbol {m}^{[0]} \) are only functions of \({~}_{ s}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[0]} \) and \({~}_{ a}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[0]} \) which is not true based on (39); hence, we must have \(\frac {\partial {{\Phi }}}{\partial {({~}_{ s}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[0]} )}} = 0\) and \(\frac {\partial {{\Phi }}}{\partial {({~}_{ a}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[0]} )}} = 0\) and the entropy inequality reduces to
$$\begin{array}{@{}rcl@{}} &&\left(\rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({\varepsilon}_{[0]} )_{ik}}} - ({~}_s {\sigma}^{[0]} )_{ik} \right) (\overset{ \text{\textbf{.}}}{\varepsilon}_{[0]} )_{ik} - \text{tr} \left([ {~}_s {m}^{[0]} ][ {~}_{ s}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} {~}_{[0]} ] \right)\\ && - \text{tr} \left([ {~}_a {m}^{[0]} ][{~}_{ a}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} {~}_{[0]} ] \right) + \frac{ \pmb{\boldsymbol{q\cdot g }}}{\theta} \leq 0 \end{array} $$
(47)
$$\begin{array}{@{}rcl@{}} \text{and } \quad {\Phi} = {\Phi} (\boldsymbol{\varepsilon}_{[0]} ,\theta ) \end{array} $$
(48)
with entropy inequality (47), Φ and its argument tensors in (48) and the other constitutive variables and their argument tensors in (39) we can not proceed any further without some more considerations.

5 Constitutive Theories

In this section, we begin with (47), (48), and the other constitutive variables and their argument tensors in (39). Consider decomposition of symmetric Cauchy stress tensor \({~}_s\boldsymbol {\sigma }^{[0]} \) into equilibrium and deviatoric stress tensors
$$ {~}_s\boldsymbol{\sigma}^{[0]} = {~}_e({~}_s\boldsymbol{\sigma}^{[0]}) + {~}_d({~}_s\boldsymbol{\sigma}^{[0]}) $$
(49)
Substituting (49) into (47)
$$\begin{array}{@{}rcl@{}} &&\left(\rho_{_{_{\!\!0}}}\frac{\partial {{\Phi}}}{\partial {({\varepsilon}_{[0]} )_{ik}}} - {~}_e({~}_s{\sigma}^{[0]})_{ik} \right) (\overset{ \text{\textbf{.}}}{\varepsilon}_{[0]} )_{ik} - \text{tr} \left([ {~}_d({~}_s{\sigma}^{[0]}) ][ \overset{ \text{\textbf{.}}}{\varepsilon}_{[0]} ] \right)\\ &&- \text{tr} \left([ {~}_s {m}^{[0]} ][ {~}_{ s}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} {~}_{[0]} ] \right) - \text{tr} \left([ {~}_a {m}^{[0]} ][{~}_{ a}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} {~}_{[0]} ] \right) + \frac{ \pmb{\boldsymbol{q\cdot g }}}{\theta} \leq 0 \end{array} $$
(50)

5.1 Constitutive Theory for \({~}_e({~}_s\boldsymbol {\sigma }^{[0]}) \)

For small strain, small deformation |J| = 1, hence
$$ \frac{\partial {{\Phi}}}{\partial { \boldsymbol{\varepsilon}_{[k]} }} = \frac{\partial {{\Phi}}}{\partial {|J|}} \frac{\partial {|J|}}{\partial { \boldsymbol{\varepsilon}_{[k]} }} = 0 \quad \text{as} \quad \frac{\partial {{\Phi}}}{\partial {|J|}} = 0 $$
(51)
Thus, the first term in (50) cannot be used to derive constitutive theory for \({~}_e({~}_s\boldsymbol {\sigma }^{[0]}) \). Note that \({~}_e({~}_s\boldsymbol {\sigma }^{[0]}) \) in (50) is only valid for compressible matter if the coefficient if \((\overset { \text {\textbf {.}}}{\varepsilon }_{[0]} )_{ik}\) is set to zero. The incompressibility condition must be introduced in (50).
$$ \pmb{\boldsymbol{\bar{\nabla} \cdot \bar{v} }} = \text{tr} [\bar{D}] = \text{tr} [\bar{L}] = \text{tr} [\overset{ \text{\textbf{.}}}{ \pmb{\boldsymbol{J }}} \pmb{\boldsymbol{J }}^{-1}] = \overset{ \text{\textbf{.}}}{J}_{kl} (J^{-1})_{lk} = \overset{ \text{\textbf{.}}}{J}_{kl} \delta_{lk} = 0 $$
(52)
$$ \text{Also } \quad \text{tr} [\bar{L}]^{T} = \text{tr} \left(\pmb{\boldsymbol{J }}^{-1} (\overset{ \text{\textbf{.}}}{ \pmb{\boldsymbol{J }}})^{T} \right) $$
(53)
$$ \text{Since } \quad \text{tr} [\bar{L}] = \text{tr} [\bar{L}]^{T} $$
(54)
we can write
$$ \frac{1}{2} \left(\text{tr} [\bar{L}] + \text{tr} [\bar{L}]^{T} \right) = \frac{1}{2} \left(\overset{ \text{\textbf{.}}}{J}_{kl} \delta_{lk} + \overset{ \text{\textbf{.}}}{J}_{lk} \delta_{lk} \right) = (\overset{ \text{\textbf{.}}}{\boldsymbol{\varepsilon}}_{[0]} )_{kl} \delta_{kl} = 0 $$
(55)
Let p(𝜃) be the arbitrary Lagrange multiplier. Then, the incompressibility condition based on (55) becomes
$$ p(\theta)(\overset{ \text{\textbf{.}}}{\varepsilon}_{[0]} )_{lk} \delta_{lk} = 0 $$
(56)
Adding (56) to the left side of (50) and using \(\frac {\partial {{\Phi }}}{\partial { \boldsymbol {\varepsilon }_{[0]} }} = 0\)
$$\begin{array}{@{}rcl@{}} &&\left(p(\theta)\delta_{lk} - {~}_e({~}_s{\sigma}^{[0]})_{lk} \right)(\overset{ \text{\textbf{.}}}{\varepsilon}_{[0]} )_{lk} -\text{tr} \left([ {~}_d({~}_s{\sigma}^{[0]}) ] [ \overset{ \text{\textbf{.}}}{\varepsilon}_{[0]} ] \right) \\&-& \text{tr} \left([ {~}_s {m}^{[0]} ][ {~}_{ s}^{{~}_i{\Theta}}\!J_{[0]} ] \right)\\ &-& \text{tr} \left([ {~}_a {m}^{[0]} ][ {~}_{ a}^{{~}_i{\Theta}}\!J_{[0]} ] \right) + \frac{ \boldsymbol{q} \pmb{\boldsymbol{\cdot }} \boldsymbol{g} }{\theta} \leq 0 \end{array} $$
(57)
for arbitrary but admissible \( \overset { \text {\textbf {.}}}{\boldsymbol {\varepsilon }}_{[0]} \) (57) holds if
$$ \left(p(\theta)\delta_{lk} - {~}_e({~}_s\boldsymbol{\sigma}^{[0]})_{lk} \right) = 0 $$
(58)
or
$$ {~}_e({~}_s\boldsymbol{\sigma}^{[0]}) = p(\theta) \pmb{\boldsymbol{I }} $$
(59)
This is the constitutive theory for \({~}_e({~}_s\boldsymbol {\sigma }^{[0]}) \) for incompressible solid matter. p(𝜃) is called mechanical pressure. If compressive pressure is assumed positive, then p(𝜃) in (59) can be replaced by − p(𝜃). The entropy inequality (50) now reduces to the following.
$$\begin{array}{@{}rcl@{}} &-&\text{tr} \left([ {~}_d({~}_s{\sigma}^{[0]}) ] [ \overset{ \text{\textbf{.}}}{\varepsilon}_{[0]} ] \right) - \text{tr} \left([ {~}_s {m}^{[0]} ][ {~}_{ s}^{{~}_i{\Theta}}\!J_{[0]} ] \right)\\ &-& \text{tr} \left([ {~}_a {m}^{[0]} ][ {~}_{ a}^{{~}_i{\Theta}}\!J_{[0]} ] \right) + \frac{ \boldsymbol{q} \pmb{\boldsymbol{\cdot }} \boldsymbol{g} }{\theta} \leq 0 \end{array} $$
(60)
The entropy inequality is satisfied if
$$\begin{array}{@{}rcl@{}} &&{~}^{{~}_s{\sigma} }{\Psi} = \text{tr} \left([ {~}_d({~}_s{\sigma}^{[0]}) ] [ \overset{ \text{\textbf{.}}}{\varepsilon}_{[0]} ] \right) > 0\\ &&{~}^{{~}_s {m} }{\Psi} = \text{tr} \left([ {~}_s {m}^{[0]} ][ {~}_{ s}^{{~}_i{\Theta}}\!J_{[0]} ] \right) > 0\\ &&{~}^{{~}_a {m} }{\Psi} = \text{tr} \left([ {~}_a {m}^{[0]} ][ {~}_{ a}^{{~}_i{\Theta}}\!J_{[0]} ] \right) > 0 \end{array} $$
(61)
$$ \frac{ \pmb{\boldsymbol{q\cdot g }}}{\theta} \leq 0 $$
(62)
Condition (61) implies that rate of work due to \({~}_s\boldsymbol {\sigma }^{[0]} \), \({~}_s \boldsymbol {m}^{[0]}\), and \({~}_a \boldsymbol {m}^{[0]} \) must be positive. The final set of constitutive variables and their argument tensors are from (39) and (48), keeping in mind that \({~}_s\boldsymbol {\sigma }^{[ m_{\! {~}_s\!\sigma } ]} \) and its argument tensors \({~}_s\boldsymbol {\sigma }^{[i]} ; i = 0,1, \dots , (m_{\! {~}_s\!\sigma } -1)\) need to be replaced with the corresponding deviatoric stress tensor and its rates.
$$\begin{array}{@{}rcl@{}} &&{\Phi} = {\Phi} (\theta )\\ && {~}_d({~}_s\boldsymbol{\sigma}^{[ m_{\! {~}_s\!\sigma } ]}) = {~}_d({~}_s\boldsymbol{\sigma}^{[ m_{\! {~}_s\!\sigma } ]}) (\boldsymbol{\varepsilon}_{[i]} ; i = 0, 1, {\dots} , n_{\!\varepsilon} , {~}_s\boldsymbol{\sigma}^{[j]}\\ &&; j = 0, 1, {\dots} , (m_{\! {~}_s\!\sigma } - 1), \theta )\\ && {~}_s \boldsymbol{m}^{[ m_{\! {~}_s\!m } ]} = {~}_s \boldsymbol{m}^{[ m_{\! {~}_s\!m } ]} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[i]} ; i = 0, \dots, n_{\! \left({~}_{ s}^{{~}_i{\Theta}}\!{J} \right)} , {~}_s \boldsymbol{m}^{[j]}\\ && ; j = 0, 1, {\dots} , (m_{\! {~}_s\!m } - 1), \theta )\\ && {~}_a \boldsymbol{m}^{[ m_{\! {~}_a\!m } ]} = {~}_a \boldsymbol{m}^{[ m_{\! {~}_a\!m } ]} ({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[i]} ; i = 0, \dots, n_{\! \left({~}_{ a}^{{~}_i{\Theta}}\!{J} \right)} , {~}_a \boldsymbol{m}^{[j]}\\ &&; j = 0, 1, {\dots} , (m_{\! {~}_a\!m } - 1), \theta ) \end{array} $$
(63)
$$ \pmb{\boldsymbol{q }} = \pmb{\boldsymbol{q }} (\pmb{\boldsymbol{g }}, \theta ) $$
(64)

5.2 Theory of Generators and Invariants (Representation Theorem)

In the following sections, we present derivations of the constitutive theories for \({~}_d({~}_s\boldsymbol {\sigma }^{[0]}) \), \({~}_s \boldsymbol {m}^{[0]} \), \({~}_a \boldsymbol {m}^{[0]} \), and q using theory of generators and invariants (representation theorem) based on pioneering works of Spencer, Wang, Zheng, etc. (Prager 1945; Reiner 1945; Todd 1948; Rivlin and Ericksen 1955; Rivlin 1955; Wang 1969a, b, 1970, 1971; Smith 1970, 1971; Spencer and Rivlin 1959, 1960; Spencer 1971; Boehler 1977; Zheng 1993a, b). To illustrate the basic concept of representation theorem, consider a symmetric tensor T(A1,A2,…,Ak) of rank r with A1,A2,…,Ak as its arguments that could be a mix of tensors of rank r or lower. If tensor T belongs to a space then the space must have a basis, referred to as integrity. Spencer, Wang, Zheng, etc. (Prager 1945; Reiner 1945; Todd 1948; Rivlin and Ericksen 1955; Rivlin 1955; Wang 1969a, b, 1970, 1971; Smith 1970, 1971; Spencer and Rivlin 1959, 1960; Spencer 1971; Boecher 1977; Zheng 1993a, b) have shown that for a symmetric tensor T of rank r, the basis consists of all possible tensors of rank r that are derived using its arguments Ai,i = 1,2,…,k, referred to as combined generators of the argument tensors. If \(\boldsymbol {I},\underset {\sim }{\boldsymbol {G}}^{i} , i = 1,2,\dots ,N\) are the combined generators constituting the basis of the space of the tensor T, then we can represent T by a linear combination of \(\boldsymbol {I},\underset {\sim }{\boldsymbol {G}}^{i} , i = 1,2,\dots ,N\), i.e.
$$ \pmb{\boldsymbol{T }} = {\alpha}^{0} \boldsymbol{I} + \sum\limits_{i = 1}^{N} {\alpha}^{i} \underset{\sim}{\boldsymbol{G}}^{i} $$
(65)
$$ {\alpha}^{i} = {\alpha}^{i} (\!\underset{\sim}{I}{~}^{j} ; j = 1,2, \dots, M); \quad i = 0,1,\dots,N $$
(66)
in which \((\!\underset {\sim }{I}{~}^{j} ; j = 1,2, \dots , M)\) are the combined invariants of the argument tensors of T().

Remarks 1

  1. (1)

    When T is an antisymmetric tensor of rank r then the same representation theorem concept applies except that in this case the combined generators \(\underset {\sim }{\boldsymbol {G}}^{i}\) will all be antisymmetric tensors of rank r and I will not be a generator.

     
  2. (2)

    If T is a non-symmetric tensor of some rank with non-symmetric tensors as its arguments, then based on Prager (1945), Reiner (1945), Todd (1948), Rivlin and Ericksen (1955), Rivlin (1955), Wang (1969a, b, 1970, 1971), Smith (1970, 1971), Spencer and Rivlin (1959, 1960), Spencer (1971), Boehler (1977), Zheng (1993a, b), and other published works, the representation theorem does not hold.

     
  3. (3)

    Material coefficients are derived from αi(⋅);i = 0,1,…,N using their Taylor series expansion in the invariants and others (like temperature 𝜃).

     

5.3 Constitutive Theory for Deviatoric Part of the Symmetric Cauchy Stress Tensor

We consider \({~}_d({~}_s\boldsymbol {\sigma }^{[ m_{\! {~}_s\!\sigma } ]}) \) and its argument tensors in (63). This constitutive theory is a rate theory in stress and strain rate tensors in order to incorporate dissipation and memory. Let \({~}^{{~}_s\!{\sigma }}\!\underset {\sim }{\boldsymbol {G}}^{i} ; i = 1,2,\dots ,N_{\!{~}_s\!{\sigma }}\) be the combined generators of the argument tensors of \({~}_d({~}_s\boldsymbol {\sigma }^{[ m_{\! {~}_s\!\sigma } ]}) \) that are symmetric tensors of rank two and let \({~}^{{~}_s\!\sigma }\!\underset {\sim }{I}{~}^{j} ; j = 1,2,\dots ,M_{\!{~}_s\!{\sigma }}\) be the combined invariants of the same argument tensors, then using representation theorem, we can express \({~}_d({~}_s\boldsymbol {\sigma }^{[m_{s^{\sigma }}]}) \) as a linear combination of \({~}^{{~}_s\!{\sigma }}\!\underset {\sim }{\boldsymbol {G}}^{i} ; i = 1,2,\dots ,N_{\!{~}_s\!{\sigma }}\) and I in the current configuration.
$$ {~}_d({~}_s\boldsymbol{\sigma}^{[ m_{\! {~}_s\!\sigma } ]}) = {~}^{{~}_s\!\sigma}\!\!\underset{\sim}{\alpha}^{0} \boldsymbol{I} + \sum\limits_{i = 1}^{N_{\!{~}_s\!{\sigma}}} {~}^{{~}_s\!\sigma}\!\!\underset{\sim}{\alpha}^{i} ({~}^{{~}_s\!{\sigma}}\!\underset{\sim}{\boldsymbol{G}}^{i} ) $$
(67)
in which
$$ {~}^{{~}_s\!\sigma}\!\!\underset{\sim}{\alpha}^{i} = {~}^{{~}_s\!\sigma}\!\!\underset{\sim}{\alpha}^{i} ({~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{j} ; j = 1,2,\dots,M_{\!{~}_s\!{\sigma}},\theta ) $$
(68)
To determine the material coefficients in (67), we expand each \({~}^{{~}_s\!\sigma }\!\!\underset {\sim }{\alpha }^{i}\) in Taylor series in \({~}^{{~}_s\!\sigma }\!\underset {\sim }{I}{~}^{j} ; j = 1,2,\dots ,M_{\!{~}_s\!{\sigma }}\) and 𝜃 about a known configuration \(\underline {\Omega }\), retaining only up to linear terms in \({~}^{{~}_s\!\sigma }\!\underset {\sim }{I}{~}^{j} ; j = 1,2,\dots ,M_{\!{~}_s\!{\sigma }}\) and 𝜃 (for simplicity) and then we substitute these expansions of \({~}^{{~}_s\!\sigma }\!\!\underset {\sim }{\alpha }^{i}\) back in (67). After collecting coefficients of those terms that are defined in the current configuration, we obtain the following
$$\begin{array}{@{}rcl@{}} {~}_d({~}_s\boldsymbol{\sigma}^{[ m_{\! {~}_s\!\sigma } ]}) &= & \left.{ {{~}_{ {~}_s\!\sigma}\!{~}^0}\!\sigma }\right|_{\underline{\Omega}} \boldsymbol{I} + \sum\limits_{j = 1}^{M_{\!{~}_s\!{\sigma}}} {~}^{{~}_s\!\sigma}\!\underline{a}_{j} ({~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{j})\boldsymbol{I} - {~}^{{~}_s\!\sigma}\!\underset{\sim}{\alpha}_{\text{tm}}(\theta-\theta_{\!\underline{\scriptscriptstyle{\Omega}}})\boldsymbol{I}\\ &&+\sum\limits_{i = 1}^{N_{\!{~}_s\!{\sigma}}} {~}^{{~}_s\!\sigma}\!\underline{b}_{i} ({~}^{{~}_s\!{\sigma}}\!\underset{\sim}{\boldsymbol{G}}^{i}) + \sum\limits_{i = 1}^{N_{\!{~}_s\!{\sigma}}}\sum\limits_{j = 1}^{M_{\!{~}_s\!{\sigma}}} {~}^{{~}_s\!\sigma}\!\underset{\sim}{c}_{{i}{j}} ({~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{j})({~}^{{~}_s\!{\sigma}}\!\underset{\sim}{\boldsymbol{G}}^{i})\\ &&+\sum\limits_{i = 1}^{N_{\!{~}_s\!{\sigma}}} {~}^{{~}_s\!\sigma}\!\underset{\sim}{d}_{i} (\theta - \theta_{\!\underline{\scriptscriptstyle{\Omega}}})({~}^{{~}_s\!{\sigma}}\!\underset{\sim}{\boldsymbol{G}}^{i}) \end{array} $$
(69)
in which
$$\begin{array}{@{}rcl@{}} && \left.{ {{~}_{ {~}_s\!\sigma}\!{~}^0}\!\sigma }\right|_{\underline{\Omega}} = {~}^{{~}_s\!\sigma}\!\!\underset{\sim}{\alpha}^{0} - \sum\limits_{j = 1}^{M_{\!{~}_s\!{\sigma}}} \left.{ \frac{\partial {({~}^{{~}_s\!\sigma}\!\!\underset{\sim}{\alpha}^{0})}}{\partial {{~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{j}}} }\right|_{\underline{\Omega}} ({~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{j})_{\!\underline{\scriptscriptstyle{\Omega}}}\\ && {~}^{{~}_s\!\sigma}\!\underline{a}_{j} = \left.{ \frac{\partial {({~}^{{~}_s\!\sigma}\!\!\underset{\sim}{\alpha}^{0})}}{\partial {({~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{j})}} }\right|_{\underline{\Omega}} ; \ j = 1,2,\dots,M_{\!{~}_s\!{\sigma}} \\ && {~}^{{~}_s\!\sigma}\!\underline{b}_{i} = \left.{ {~}^{{~}_s\!\sigma}\!\!\underset{\sim}{\alpha}^{i} }\right|_{\underline{\Omega}} - \sum\limits_{j = 1}^{M_{\!{~}_s\!{\sigma}}} \left.{ \frac{\partial {({~}^{{~}_s\!\sigma}\!\!\underset{\sim}{\alpha}^{i})}}{\partial {({~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{j})}} }\right|_{\underline{\Omega}}({~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{j})_{\!\underline{\scriptscriptstyle{\Omega}}} ; \ i = 1,2,\dots,N_{\!{~}_s\!{\sigma}}\\ && {~}^{{~}_s\!\sigma}\!\underset{\sim}{c}_{{i}{j}} = \left.{ \frac{\partial {({~}^{{~}_s\!\sigma}\!\!\underset{\sim}{\alpha}^{i})}}{\partial {({~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{j})}} }\right|_{\underline{\Omega}} ; \ i = 1,2,\dots,N_{\!{~}_s\!{\sigma}}\\ && \quad \qquad{} \qquad{} \qquad{} \qquad{} j = 1,2,\dots,M_{\!{~}_s\!{\sigma}} \end{array} $$
(70)
$$\begin{array}{@{}rcl@{}} &&{~}^{{~}_s\!\sigma}\!\underset{\sim}{\alpha}_{\text{tm}} = - \left.{ \frac{\partial {({~}^{{~}_s\!\sigma}\!\!\underset{\sim}{\alpha}^{0})}}{\partial {\theta}} }\right|_{\underline{\Omega}} \\ && {~}^{{~}_s\!\sigma}\!\underset{\sim}{d}_{i} = \left.{ \frac{\partial {({~}^{{~}_s\!\sigma}\!\!\underset{\sim}{\alpha}^{i})}}{\partial {\theta}} }\right|_{\underline{\Omega}} ; \ i = 1,2,\dots,N_{\!{~}_s\!{\sigma}} \end{array} $$
in which \({~}^{{~}_s\!\sigma }\!\underline {a}_{j} \), \({~}^{{~}_s\!\sigma }\!\underline {b}_{i} \), \({~}^{{~}_s\!\sigma }\!\underset {\sim }{c}_{{i}{j}} \), \({~}^{{~}_s\!\sigma }\!\underset {\sim }{d}_{i} \), and \({~}^{{~}_s\!\sigma }\!\underset {\sim }{\alpha }_{\text {tm}}\) are material coefficients defined in a known configuration \(\underline {\Omega }\). This constitutive theory requires \((M_{\!{~}_s\!{\sigma }} + N_{\!{~}_s\!{\sigma }} + M_{\!{~}_s\!{\sigma }}N_{\!{~}_s\!{\sigma }} + N_{\!{~}_s\!{\sigma }} + 1)\) material coefficients. The material coefficients defined in (70) can be functions of \({({~}^{{~}_s\!\sigma }\!\underset {\sim }{I}{~}^{j})}_{\!\underline {\scriptscriptstyle {\Omega }}} ; j = 1,2,\dots ,M_{\!{~}_s\!{\sigma }} \) and \(\theta _{\!\underline {\scriptscriptstyle {\Omega }}}\). This constitutive theory is based on integrity (complete basis), the only approximation being truncation of the Taylor series expansions of \({~}^{{~}_s\!\sigma }\!\!\underset {\sim }{\alpha }^{i} ; i = 0,1,\dots ,N_{\!{~}_s\!{\sigma }}\). We consider simplified form of this constitutive theory in later sections.

5.4 Constitutive Theory for Symmetric Cauchy Moment Tensor

Consider the constitutive variable \({~}_s \boldsymbol {m}^{[ m_{\! {~}_s\!m } ]} \) and its argument tensors defined by (63). As in case of \({~}_d({~}_s\boldsymbol {\sigma }^{[ m_{\! {~}_s\!\sigma } ]}) \) in Section 5.3, here also the constitutive theory should be a rate theory in the rate of the symmetric moment tensor and the rates of the symmetric part of the rotation gradient tensor. Let \({~}^{{~}_s\!m}\!\underset {\sim }{\boldsymbol {G}}^{i} ; i = 1,2,\dots ,N_{\!{~}_s\!m}\) be the combined generators of the argument tensors of \({~}_s \boldsymbol {m}^{[ m_{\! {~}_s\!m } ]} \) that are symmetric tensors of rank two and let \({~}^{{~}_s\!m}\!\underset {\sim }{I}{~}^{j} ; j = 1,2,\dots ,M_{\!{~}_s\!m}\) be the combined invariants of the same argument tensors; then, using the representation theorem, we can express \({~}_s \boldsymbol {m}^{[ m_{\! {~}_s\!m } ]} \) as linear combination of \({~}^{{~}_s\!m}\!\underset {\sim }{\boldsymbol {G}}^{i} ; i = 1,2,\dots ,N_{\!{~}_s\!m}\) and I in the current configuration.
$$ {~}_s \boldsymbol{m}^{[ m_{\! {~}_s\!m } ]} = {~}^{{~}_s\!m}\!\underset{\sim}{\alpha}^{0} \boldsymbol{I} + \sum\limits_{i = 1}^{N_{\!{~}_s\!m}} {~}^{{~}_s\!m}\!\underset{\sim}{\alpha}^{i} ({~}^{{~}_s\!m}\!\underset{\sim}{\boldsymbol{G}}^{i} ) $$
(71)
in which
$$ {~}^{{~}_s\!m}\!\underset{\sim}{\alpha}^{i} = {~}^{{~}_s\!m}\!\underset{\sim}{\alpha}^{i} ({~}^{{~}_s\!m}\!\underset{\sim}{I}{~}^{j} ; j = 1,2,\dots,M_{\!{~}_s\!m},\theta ) $$
(72)
To determine the material coefficients in (71), we expand each \({~}^{{~}_s\!m}\!\underset {\sim }{\alpha }^{i}\) in Taylor series in \({~}^{{~}_s\!m}\!\underset {\sim }{I}{~}^{j} ; j = 1,2,\dots ,M_{\!{~}_s\!m}\) and 𝜃 about a known configuration \(\underline {\Omega }\), retaining only up to linear terms in \({~}^{{~}_s\!m}\!\underset {\sim }{I}{~}^{j} ; j = 1,2,\dots ,M_{\!{~}_s\!m}\) and 𝜃 (for simplicity) and then we substitute these in (71). After collecting coefficients of those terms that are defined in the current configuration, we obtain the following
$$\begin{array}{@{}rcl@{}} {~}_s \boldsymbol{m}^{[ m_{\! {~}_s\!m } ]} &= & \left.{ {{~}_{ {~}_s\!m}\!{~}^0}\!m }\right|_{\underline{\Omega}} \boldsymbol{I} + \sum\limits_{j = 1}^{M_{\!{~}_s\!m}} {~}^{{~}_s\!m}\!\underline{a}_{j} ({~}^{{~}_s\!m}\!\underset{\sim}{I}{~}^{j})\boldsymbol{I} - {~}^{{~}_s\!m}\!\underset{\sim}{\alpha}_{\text{tm}}(\theta-\theta_{\!\underline{\scriptscriptstyle{\Omega}}})\boldsymbol{I}\\ &&+\sum\limits_{i = 1}^{N_{\!{~}_s\!m}} {~}^{{~}_s\!m}\!\underline{b}_{i} ({~}^{{~}_s\!m}\!\underset{\sim}{\boldsymbol{G}}^{i}) + \sum\limits_{i = 1}^{N_{\!{~}_s\!m}}\sum\limits_{j = 1}^{M_{\!{~}_s\!m}} {~}^{{~}_s\!m}\!\underset{\sim}{c}_{{i}{j}} ({~}^{{~}_s\!m}\!\underset{\sim}{I}{~}^{j})({~}^{{~}_s\!m}\!\underset{\sim}{\boldsymbol{G}}^{i})\\ &&+\sum\limits_{i = 1}^{N_{\!{~}_s\!m}} {~}^{{~}_s\!m}\!\underset{\sim}{d}_{i} (\theta - \theta_{\!\underline{\scriptscriptstyle{\Omega}}})({~}^{{~}_s\!m}\!\underset{\sim}{\boldsymbol{G}}^{i}) \end{array} $$
(73)
in which \({~}^{{~}_s\!m}\!\underline {a}_{j} \), \({~}^{{~}_s\!m}\!\underline {b}_{i} \), \({~}^{{~}_s\!m}\!\underset {\sim }{c}_{{i}{j}} \), \({~}^{{~}_s\!m}\!\underset {\sim }{d}_{i} \), and \({~}^{{~}_s\!m}\!\underset {\sim }{\alpha }_{\text {tm}}\) are material coefficients defined in a known configuration \(\underline {\Omega }\). Explicit expressions for these can be obtained from (70) by replacing subscript and the superscript of \({~}_s\boldsymbol {\sigma } \) by \({~}_s \boldsymbol {m} \). This constitutive theory requires \((M_{\!{~}_s\!m} + N_{\!{~}_s\!m} + M_{\!{~}_s\!m}N_{\!{~}_s\!m} + N_{\!{~}_s\!m} + 1)\) material coefficients. This constitutive theory is based on integrity. The only approximation being truncation of the Taylor series expansions of \({~}^{{~}_s\!m}\!\underset {\sim }{\alpha }^{i} ; i = 0,1,\dots ,N_{\!{~}_s\!m}\). Simplified form of the constitutive theory (73) is considered in later sections.

5.5 Constitutive Theory for Antisymmetric Cauchy Moment Tensor

Consider \({~}_a \boldsymbol {m}^{[ m_{\! {~}_a\!m } ]} \) and its argument tensors defined by (63). \({~}_a \boldsymbol {m}^{[ m_{\! {~}_a\!m } ]} \) is an antisymmetric tensor of rank 2 and so are its argument tensors except 𝜃, a tensor of rank 0. Let \({~}^{{~}_a\!m}\!\underset {\sim }{\boldsymbol {G}}^{i} ; i = 1,2,\dots ,N_{\!{~}_a\!m}\) be the combined generators of the argument tensors of \({~}_a \boldsymbol {m}^{[ m_{\! {~}_a\!m } ]} \) that are antisymmetric tensors of rank 2 and let \({~}^{{~}_a\!m}\!\underset {\sim }{I}{~}^{j} ; j = 1,2,\dots ,M_{\!{~}_a\!m}\) be the combined invariants of the same argument tensors. Similar to Sections 5.3 and 5.4 in this case, also, we consider possible energy storage, dissipation and theology physics. Based on representation theorem, we can represent \({~}_a \boldsymbol {m}^{[ m_{\! {~}_a\!m } ]} \) as a linear combination of \({~}^{{~}_a\!m}\!\underset {\sim }{\boldsymbol {G}}^{i} ; i = 1,2,\dots ,N_{\!{~}_a\!m}\) in the current configuration.
$$ {~}_a \boldsymbol{m}^{[ m_{\! {~}_a\!m } ]} = \sum\limits_{i = 1}^{N_{\!{~}_a\!m}} {~}^{{~}_a\!m}\!\underset{\sim}{\alpha}^{i} ({~}^{{~}_a\!m}\!\underset{\sim}{\boldsymbol{G}}^{i} ) $$
(74)
in which
$$ {~}^{{~}_a\!m}\!\underset{\sim}{\alpha}^{i} = {~}^{{~}_a\!m}\!\underset{\sim}{\alpha}^{i} ({~}^{{~}_a\!m}\!\underset{\sim}{I}{~}^{j}; j = 1,2,\dots,M_{\!{~}_a\!m},\theta ) $$
(75)
To determine the material coefficients in (74), we expand each \({~}^{{~}_a\!m}\!\underset {\sim }{\alpha }^{i} ; i = 1,2,\dots ,N_{\!{~}_a\!m}\) in Taylor series in \({~}^{{~}_a\!m}\!\underset {\sim }{I}{~}^{j}; j = 1,2,\dots ,M_{\!{~}_a\!m}\) and 𝜃 about a known configuration \(\underline {\Omega }\), retaining only up to linear terms in \({~}^{{~}_a\!m}\!\underset {\sim }{I}{~}^{j}; j = 1,2,\dots ,M_{\!{~}_s\!m}\) and 𝜃 (for simplicity) and then we substitute them in (74). After collecting coefficients of those terms that are defined in the current configuration, we obtain the following
$$\begin{array}{@{}rcl@{}} {~}_a \boldsymbol{m}^{[ m_{\! {~}_a\!m } ]} &= &\sum\limits_{i = 1}^{N_{\!{~}_a\!m}} {~}^{{~}_a\!m}\!\underline{b}_{i} ({~}^{{~}_a\!m}\!\underset{\sim}{\boldsymbol{G}}^{i}) + \sum\limits_{i = 1}^{N_{\!{~}_a\!m}}\sum\limits_{j = 1}^{M_{\!{~}_a\!m}} {~}^{{~}_a\!m}\!\underset{\sim}{c}_{{i}{j}} ({~}^{{~}_a\!m}\!\underset{\sim}{I}{~}^{j})({~}^{{~}_a\!m}\!\underset{\sim}{\boldsymbol{G}}^{i})\\ &&\sum\limits_{i = 1}^{N_{\!{~}_a\!m}} {~}^{{~}_a\!m}\!\underset{\sim}{d}_{i} (\theta - \theta_{\!\underline{\scriptscriptstyle{\Omega}}})({~}^{{~}_a\!m}\!\underset{\sim}{\boldsymbol{G}}^{i}) \end{array} $$
(76)
In which \({~}^{{~}_a\!m}\!\underline {b}_{i} \), \({~}^{{~}_a\!m}\!\underset {\sim }{c}_{{i}{j}} \), and \({~}^{{~}_a\!m}\!\underset {\sim }{d}_{i} \) are material coefficients defined in a known configuration \(\underline {\Omega }\). These can be functions of \(\left .{{~}^{{~}_a\!m}\!\underset {\sim }{I}{~}^{j}}\right |_{\underline {\Omega }} ; j = 1,2,\dots ,M_{\!{~}_a\!m}\) and \(\theta _{\!\underline {\scriptscriptstyle {\Omega }}}\). Explicit expressions for these can be obtained from (70) by replacing subscript and the superscript of \({~}_s\boldsymbol {\sigma } \) by \({~}_a \boldsymbol {m} \). This constitutive theory requires \((N_{\!{~}_a\!m} + M_{\!{~}_a\!m}N_{\!{~}_a\!m} + N_{\!{~}_a\!m})\) material coefficients. This constitutive theory is also based on integrity and has the same approximation in Taylor series expansion as the theories in Sections 5.3 and 5.4. Simplified form of this constitutive theory will also be considered in a later section. This constitutive theory is only needed when the balance of moment of moments is not used as a balance law.

5.6 Constitutive Theory for q

Recall the inequality (62) resulting from entropy inequality
$$ \pmb{\boldsymbol{q \cdot g }} \leq 0 \quad (\text{as } \theta > 0) $$
(77)
In (77), q and g are conjugate. The simplest possible constitutive theory for q can be derived by assuming that q is proportional to −g which leads to the following for q (Surana 2015)
$$ \boldsymbol{q} = - k(\theta) \boldsymbol{g} $$
(78)
This is standard Fourier heat conduction law with temperature-dependent thermal conductivity. Alternatively, if we assume
$$ \boldsymbol{q} = \boldsymbol{q} (\boldsymbol{g} ,\theta) $$
(79)
Then, based on the representation theorem, since g is the only combined generator of rank one of g and 𝜃, we can write the following in the current configuration
$$ \boldsymbol{q} = - {~}^{q}\!{\alpha} \boldsymbol{g} $$
(80)
$$ \text{in which } {~}^{q}\!{\alpha} = {~}^{q}\!{\alpha} ({~}^{q}\!{I} ,\theta) ; \quad {~}^{q}\!{I} = \pmb{\boldsymbol{g \cdot g }} $$
(81)
\({~}^{q}\!{I} \) is the only invariant of the argument tensors of q and 𝜃. We expand \({~}^{q}\!{\alpha } \) in Taylor series of \({~}^{q}\!{I} \) and 𝜃 about a known configuration \(\underline {\Omega }\) and retain only up to linear terms in \({~}^{q}\!{I} \) and 𝜃. Collecting coefficients of the terms defined in the current configuration, we can obtain the following
$$ \boldsymbol{q} = - \left.{ k }\right|_{\underline{\Omega}} \boldsymbol{g} - \left.{ k_{1} }\right|_{\underline{\Omega}} (\pmb{\boldsymbol{g \cdot g }}) \boldsymbol{g} - \left.{ k_{2} }\right|_{\underline{\Omega}} (\theta - \theta_{\!\underline{\scriptscriptstyle{\Omega}}}) \boldsymbol{g} $$
(82)
where
$$\begin{array}{@{}rcl@{}} && \left.{ k }\right|_{\underline{\Omega}} = \left.{ {~}^{q}\!{\alpha} }\right|_{\underline{\Omega}} + \left.{\frac{\partial { {~}^{q}\!{\alpha} }}{\partial {({~}^{q}\!{I} )}}}\right|_{\underline{\Omega}} (\pmb{\boldsymbol{g\cdot g }})_{\!\underline{\scriptscriptstyle{\Omega}}}\\ && \left.{ k_{1} }\right|_{\underline{\Omega}} = \left.{\frac{\partial { {~}^{q}\!{\alpha} }}{\partial {({~}^{q}\!{I} )}}}\right|_{\underline{\Omega}} \\ && \left.{ k_{2} }\right|_{\underline{\Omega}} = \left.{\frac{\partial { {~}^{q}\!{\alpha} }}{\partial {(\theta)}}}\right|_{\underline{\Omega}} \end{array} $$
(83)
The constitutive theory (82) is the simplest possible constitutive theory based on representation theorem \( \left .{ k }\right |_{\underline {\Omega }} \), \( \left .{ k_{1} }\right |_{\underline {\Omega }} \) and \( \left .{ k_{2} }\right |_{\underline {\Omega }} \) can be functions of \(\left .{ {~}^{q}\!{I} }\right |_{\underline {\Omega }}\) and \(\theta _{\!\underline {\scriptscriptstyle {\Omega }}}\). Clearly constitutive theory (78) is a subset of (82). Constitutive theory (82) is cubic in q.

6 Simplified Constitutive Theories: Non-classical and Classical Simplified Constitutive Theories

Polymeric fluids are generally classified as dilute polymeric fluids or dense polymeric fluids. dilute polymers behave much like thermoviscous fluids with some elasticity due to relatively low concentration of polymer molecules in the solvent. Dense polymers on the other hand have significantly higher concentration of polymer molecules in the solvent, hence exhibit pronounced elastic behavior. Surana et al. (2012, 2014a, b, c) have presented derivations of Maxwell, Oldroyd-B, and Gisekus constitutive models for classical continuum theories strictly using entropy inequality and representation theorem. The work presented in Surana (2015) and Surana et al. (2012, 2014a, b, c) highlights limitations of currently used Maxwell, Oldryod-B, and Giesekus constitutive models for polymers and provides derivations of the ordered rate theories based on integrity that permit much more comprehensive constitutive models. In a recent paper, Surana et al. (2017e) have presented constitutive theories for non-classical polymers in which the internal rotation rates due to velocity gradient tensor are incorporated in the conservation and the balance laws. The authors show that a single non-classical constitutive model for polymer can be derived that contains corresponding Maxwell, Oldroyd-B, and Giesekus constitutive models for non-classical as well for classical cases as subset. The authors show that in non-classical polymers, the dissipation and rheology are due to deviatoric Cauchy stress tensor as well as the Cauchy moment tensor.

Surana (2015) and Surana et al. (2012, 2014a, b, c) have presented constitutive theories for classical thermoviscoelastic solids based on integrity. The constitutive models in Surana (2015) and Surana et al. (2012, 2014a, b, c) were compared with phenomenological models such as Kelvin Voigt model. One-dimensional wave propagation model problem studies were presented. The simplified constitutive theories presented in this section for non-classical polymers parallel those for classical thermoviscoelastic fluids such as Maxwell, Oldroyd-B, and Giesekus. The intent here is to show that a large variety of diverse models are possible based on the constitutive theories presented here using integrity. The constitutive theories for classical thermoviscoelastic solids are always a subset of the constitutive theories for the non-classical polymeric solids presented in this paper. We make some remarks.

Remarks 2

  1. (1)

    The constitutive theories for classical theories are easily obtained by removing the internal rotation physics that leads to \({~}_s \boldsymbol {m} = 0\), \({~}_a \boldsymbol {m} = 0\) and the Cauchy stress tensor becomes symmetric and with this the balance of angular momenta holds. The resulting constitutive theories are same as those by Surana (2015) for classical thermoviscoelastic solids.

     
  2. (2)

    We reiterate that since the constitutive theories presented here are based on integrity, the specific forms of the constitutive theories are all subset of these. Hence, it would be possible to derive a single simplified constitutive model using the general theory that should contain various possible models related to specific physics.

     
  3. (3)
    The simplest way to proceed is to assume that our objective is to derive constitutive models parallel to Maxwell, Oldroyd-B, and Giesekus for polymeric liquids are desired for non-classical thermoviscoelastic solids. Furthermore in order for the non-classical constitutive models to contain classical models as subset we need to choose the following ordered rates.
    $$ n_{\!\varepsilon} = 2, m_{\! {~}_s\!\sigma } = 1, m_{\! {~}_s\!m } = 1, m_{\! {~}_a\!m } = 1, n_{\! \left({~}_{ s}^{{~}_i{\Theta}}\!{J} \right)} = 2, n_{\! \left({~}_{ a}^{{~}_i{\Theta}}\!{J} \right)} = 2 $$
    (84)
     
For this choice, the constitutive variables and their argument tensors are
$$\begin{array}{@{}rcl@{}} {~}_d({~}_s\boldsymbol{\sigma}^{[1]}) &=& {~}_d({~}_s\boldsymbol{\sigma}^{[1]}) (\boldsymbol{\varepsilon}_{[0]} , \boldsymbol{\varepsilon}_{[1]} , \boldsymbol{\varepsilon}_{[2]} , {~}_d({~}_s\boldsymbol{\sigma}^{[0]}) , \theta )\\ {~}_s \boldsymbol{m}^{[1]} &=& {~}_s \boldsymbol{m}^{[1]} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} , {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} , {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} , {~}_s \boldsymbol{m}^{[0]} , \theta )\\ {~}_a \boldsymbol{m}^{[1]} &=& {~}_a \boldsymbol{m}^{[1]} ({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} , {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} , {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} , {~}_a \boldsymbol{m}^{[0]} , \theta ) \end{array} $$
(85)
The constitutive theories using (85) when based on integrity still requires too many material coefficients. We consider the following simplification. The motivation being that their further simplification for classical case will yield standard, well known and familiar constitutive theories.
  1. (i)

    Consider the constitutive theories to be linear in ε[0], ε[1], ε[2], \({~}_{ s}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[0]} \), \({~}_{ s}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[1]} \), \({~}_{ s}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[2]} \), \({~}_{ a}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[0]} \), \({~}_{ a}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[1]} \), \({~}_{ a}^{{~}_i{\Theta }}\!\boldsymbol {J}_{[2]} \)

     
  2. (ii)

    Neglect product terms of the terms in (6). Also neglect products of the invariants as well as their products with generators.

     
  3. (iii)

    Neglect \((\theta - \theta _{\!\underline {\scriptscriptstyle {\Omega }}})\) terms (to conform to the currently used models).

     
  4. (iv)

    Neglect the first term in each constitutive theory containing the influence of initial stress or moment tensor.

     
  5. (v)

    Consider \(({~}_d({~}_s\boldsymbol {\sigma }^{[0]}) )^{2}\), \(({~}_s \boldsymbol {m}^{[0]} )^{2}\) and \(({~}_a \boldsymbol {m}^{[0]} )^{2}\) as generators but neglect quadratic and cubic trace terms in the invariants as well as their products. Include tr\(({~}_d({~}_s\boldsymbol {\sigma }^{[0]}) )\), tr\(({~}_s \boldsymbol {m}^{[0]} )\) and tr\(({~}_a \boldsymbol {m}^{[0]} )\) terms in the constitutive theories.

     

6.1 Constitutive Theory for Deviatoric Part of the Symmetric Cauchy Stress Tensor

If we consider
$$\begin{array}{@{}rcl@{}} {~}^{{~}_s\!{\sigma}}\!\underset{\sim}{\boldsymbol{G}}{~}^{1} &=& \boldsymbol{\varepsilon}_{[0]} , {~}^{{~}_s\!{\sigma}}\!\underset{\sim}{\boldsymbol{G}}{~}^{2} = \boldsymbol{\varepsilon}_{[1]} , {~}^{{~}_s\!{\sigma}}\!\underset{\sim}{\boldsymbol{G}}{~}^{3} = \boldsymbol{\varepsilon}_{[2]} ,\\ {~}^{{~}_s\!{\sigma}}\!\underset{\sim}{\boldsymbol{G}}{~}^{4} &=& {~}_d({~}_s\boldsymbol{\sigma}^{[0]}) \text{ and } {~}^{{~}_s\!{\sigma}}\!\underset{\sim}{\boldsymbol{G}}{~}^{5} = ({~}_d({~}_s\boldsymbol{\sigma}^{[0]}) )^{2} \\ {~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{1} &=& \text{tr}(\boldsymbol{\varepsilon}_{[0]} ) , {~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{2} = \text{tr}(\boldsymbol{\varepsilon}_{[1]} ) , {~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{3} = \text{tr}(\boldsymbol{\varepsilon}_{[2]} ) ,\\ {~}^{{~}_s\!\sigma}\!\underset{\sim}{I}{~}^{4} &=& \text{tr}({~}_d({~}_s\boldsymbol{\sigma}^{[0]}) ) \end{array} $$
then we obtain the following from (69) (using \(n_{\!\varepsilon } = 2\), \(m_{\! {~}_s\!\sigma } = 1\))
$$\begin{array}{@{}rcl@{}} {~}_d({~}_s\boldsymbol{\sigma}^{[1]}) &=& {~}^{{~}_s\!\sigma}\!\underline{a}_{1} \text{tr}(\boldsymbol{\varepsilon}_{[0]} )\boldsymbol{I} + {~}^{{~}_s\!\sigma}\!\underline{a}_{2} \text{tr}(\boldsymbol{\varepsilon}_{[1]} )\boldsymbol{I} + {~}^{{~}_s\!\sigma}\!\underline{a}_{3} \text{tr}(\boldsymbol{\varepsilon}_{[2]} )\boldsymbol{I}\\ &&+ {~}^{{~}_s\!\sigma}\!\underline{a}_{4} \text{tr}({~}_d({~}_s\boldsymbol{\sigma}^{[0]}) )\boldsymbol{I} + {~}^{{~}_s\!\sigma}\!\underline{b}_{1} (\boldsymbol{\varepsilon}_{[0]} ) + {~}^{{~}_s\!\sigma}\!\underline{b}_{2} (\boldsymbol{\varepsilon}_{[1]}) \\ &&+ {~}^{{~}_s\!\sigma}\!\underline{b}_{3} (\boldsymbol{\varepsilon}_{[2]}) + {~}^{{~}_s\!\sigma}\!\underline{b}_{4} ({~}_d({~}_s\boldsymbol{\sigma}^{[0]}) ) + {~}^{{~}_s\!\sigma}\!\underline{b}_{5} ({~}_d({~}_s\boldsymbol{\sigma}^{[0]}) )^{2}\\ \end{array} $$
(86)
In order to obtain standard form from (86), we transfer \({~}^{{~}_s\!\sigma }\!\underline {b}_{4} ({~}_d({~}_s\boldsymbol {\sigma }^{[0]}) )\) term on the left-hand side and divide the whole equation by \({~}^{{~}_s\!\sigma }\!\underline {b}_{4} \) and define
$$\begin{array}{@{}rcl@{}} {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} &=& - \frac{1}{ {~}^{{~}_s\!\sigma}\!\underline{b}_{4} } , {{~}^{{~}_s\!\sigma} \! \mu} = \left(- \frac{ {~}^{{~}_s\!\sigma}\!\underline{b}_{1} }{ {~}^{{~}_s\!\sigma}\!\underline{b}_{4} } \right) , {~}^{{~}_s\!\sigma}\!\lambda = \left(- \frac{ {~}^{{~}_s\!\sigma}\!\underline{a}_{1} }{ {~}^{{~}_s\!\sigma}\!\underline{b}_{4} } \right) , \\ {~}^{{~}_s\!\sigma}\!\eta_{1} &=& \left(- \frac{ {~}^{{~}_s\!\sigma}\!\underline{b}_{2} }{ {~}^{{~}_s\!\sigma}\!\underline{b}_{4} } \right) , {~}^{{~}_s\!\sigma} \! k_{1} = \left(- \frac{ {~}^{{~}_s\!\sigma}\!\underline{a}_{2} }{ {~}^{{~}_s\!\sigma}\!\underline{b}_{4} } \right) , {~}^{{~}_s\!\sigma}\!\eta_{2} = \left(- \frac{ {~}^{{~}_s\!\sigma}\!\underline{b}_{3} }{ {~}^{{~}_s\!\sigma}\!\underline{b}_{4} } \right) , \\ {~}^{{~}_s\!\sigma} \! k_{2} &=& \left(- \frac{ {~}^{{~}_s\!\sigma}\!\underline{a}_{3} }{ {~}^{{~}_s\!\sigma}\!\underline{b}_{4} } \right) , {~}^{{~}_s\!\sigma} \! k_{3} = \left(- \frac{ {~}^{{~}_s\!\sigma}\!\underline{a}_{4} }{ {~}^{{~}_s\!\sigma}\!\underline{b}_{4} } \right)\\ {~}^{{~}_s\!\sigma}\!\eta_{4} &=& \left(- \frac{ {~}^{{~}_s\!\sigma}\!\underline{b}_{5} }{ {~}^{{~}_s\!\sigma}\!\underline{b}_{4} } \right) \end{array} $$
(87)
Then, (86) can be written as
$$\begin{array}{@{}rcl@{}} {~}_d({~}_s\boldsymbol{\sigma}^{[0]}) + {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} {~}_d({~}_s\boldsymbol{\sigma}^{[1]}) &=& \ {{~}^{{~}_s\!\sigma} \! \mu} (\boldsymbol{\varepsilon}_{[0]} ) + {~}^{{~}_s\!\sigma}\!\lambda \text{tr}(\boldsymbol{\varepsilon}_{[0]} )\boldsymbol{I} + {~}^{{~}_s\!\sigma}\!\eta_{1} (\boldsymbol{\varepsilon}_{[1]} )\\ &&+ {~}^{{~}_s\!\sigma} \! k_{1} \text{tr}(\boldsymbol{\varepsilon}_{[1]} )\boldsymbol{I} + {~}^{{~}_s\!\sigma}\!\eta_{2} (\boldsymbol{\varepsilon}_{[2]} )\\ &&+ {~}^{{~}_s\!\sigma} \! k_{2} \text{tr}(\boldsymbol{\varepsilon}_{[2]} )\boldsymbol{I} + {~}^{{~}_s\!\sigma} \! k_{3} \text{tr}({~}_d({~}_s\boldsymbol{\sigma}^{[0]}) )\boldsymbol{I}\\ &&+ {~}^{{~}_s\!\sigma}\!\eta_{4} ({~}_d({~}_s\boldsymbol{\sigma}^{[0]}) )^{2}\\ \end{array} $$
(88)
In which \({~}^{{~}_s\!\sigma }\!\underset {\sim }{\lambda } \) is the relaxation time, \({{~}^{{~}_s\!\sigma } \! \mu } \) and \({~}^{{~}_s\!\sigma }\!\lambda \) are material coefficients similar to Lame’s constants associated with strains, \({~}^{{~}_s\!\sigma }\!\eta _{1} \) and \({~}^{{~}_s\!\sigma } \! k_{1} \) are material coefficients associated with strain rates (dissipating mechanism). For incompressible matter (as the case in here), approximate coefficient(s) can be set to 0. The constitutive model (88) remains the same regardless of the consideration of classical or non-classical continuum theory. Constitutive theory (88) contains models similar to Maxwell, Oldroyd-B, and Giesekus polymeric fluids as subset. We present explicit form of these in the following.

6.1.1 Constitutive Theory Similar to Maxwell Polymeric Fluid

In order to obtain a constitutive theory similar to Maxwell model for fluids, we use (88) with
$${~}^{{~}_s\!\sigma}\!\eta_{2} = 0 , {~}^{{~}_s\!\sigma} \! k_{2} = 0 , {~}^{{~}_s\!\sigma} \! k_{3} = 0 \quad \text{ and } {~}^{{~}_s\!\sigma}\!\eta_{4} = 0 $$
which gives us
$$\begin{array}{@{}rcl@{}} {~}_d({~}_s\boldsymbol{\sigma}^{[0]}) &+& {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} {~}_d({~}_s\boldsymbol{\sigma}^{[1]}) = \ {{~}^{{~}_s\!\sigma} \! \mu} (\boldsymbol{\varepsilon}_{[0]} ) + {~}^{{~}_s\!\sigma}\!\lambda \text{tr}(\boldsymbol{\varepsilon}_{[0]} )\boldsymbol{I}\\& +& {~}^{{~}_s\!\sigma}\!\eta_{1} (\boldsymbol{\varepsilon}_{[1]} ) + {~}^{{~}_s\!\sigma} \! k_{1} \text{tr}(\boldsymbol{\varepsilon}_{[1]} )\boldsymbol{I} \end{array} $$
(89)
When (89) is compared with Maxwell model for fluids, strain terms containing material coefficeints \( {{~}^{{~}_s\!\sigma } \! \mu } \) and \({~}^{{~}_s\!\sigma }\!\lambda \) are additional terms due to elasticity. tr\((\boldsymbol {\varepsilon }_{[1]} )\) can be set to 0 for incompressible case, then \({{~}^{{~}_s\!\sigma } \! \mu } \neq 0\), \({~}^{{~}_s\!\sigma }\!\lambda \neq 0\), \({~}^{{~}_s\!\sigma }\!\eta _{1} \neq 0\) but \({~}^{{~}_s\!\sigma } \! k_{4} = 0\) is the Maxwell model for solids parallel to the Maxwell model for incompressible fluids.

6.1.2 Constitutive Theory Similar to Oldroyd-B Model for Fluids

For this constitutive model, we set \({~}^{{~}_s\!\sigma } \! k_{3} \) and \({~}^{{~}_s\!\sigma }\!\eta _{4} = 0\) in (88) to obtain
$$\begin{array}{@{}rcl@{}} {~}_d({~}_s\boldsymbol{\sigma}^{[0]})\! +\! {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} {~}_d({~}_s\boldsymbol{\sigma}^{[1]}) &=& \ {{~}^{{~}_s\!\sigma} \! \mu} (\boldsymbol{\varepsilon}_{[0]} ) + {~}^{{~}_s\!\sigma}\!\lambda \text{tr}(\boldsymbol{\varepsilon}_{[0]} )\boldsymbol{I} + {~}^{{~}_s\!\sigma}\!\eta_{1} (\boldsymbol{\varepsilon}_{[1]} )\\ &&+ {~}^{{~}_s\!\sigma} \! k_{1} \text{tr}(\boldsymbol{\varepsilon}_{[1]} )\boldsymbol{I} + {~}^{{~}_s\!\sigma}\!\eta_{2} (\boldsymbol{\varepsilon}_{[2]} )\\ &&+ {~}^{{~}_s\!\sigma} \! k_{2} \text{tr}(\boldsymbol{\varepsilon}_{[2]} )\boldsymbol{I} \end{array} $$
(90)
In this model, it is generally assumed that tr(ε[1]) = 0 (due to incompressibility) and tr(ε[2]) = 0.

6.1.3 Constitutive Theory Similar to Giesekus Polymeric Fluids

In this case, we set \({~}^{{~}_s\!\sigma }\!\eta _{2} = 0\), \({~}^{{~}_s\!\sigma } \! k_{2} = 0\), \({~}^{{~}_s\!\sigma } \! k_{3} = 0\) in (88).
$$\begin{array}{@{}rcl@{}} {~}_d({~}_s\boldsymbol{\sigma}^{[0]}) + {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} {~}_d({~}_s\boldsymbol{\sigma}^{[1]}) &=& \ {{~}^{{~}_s\!\sigma} \! \mu} (\boldsymbol{\varepsilon}_{[0]} ) + {~}^{{~}_s\!\sigma}\!\lambda \text{tr}(\boldsymbol{\varepsilon}_{[0]} )\boldsymbol{I} + {~}^{{~}_s\!\sigma}\!\eta_{1} (\boldsymbol{\varepsilon}_{[1]} )\\ &&+ {~}^{{~}_s\!\sigma} \! k_{1} \text{tr}(\boldsymbol{\varepsilon}_{[1]} )\boldsymbol{I} \\ &&+ {~}^{{~}_s\!\sigma}\!\eta_{4} ({~}_d({~}_s\boldsymbol{\sigma}^{[0]}) )^{2} \end{array} $$
(91)
For incompressible case, tr(ε[1]) is set to 0 in (91).

6.2 Constitutive Theories for Symmetric Part of the Cauchy Moment Tensor

If we consider
$$\begin{array}{@{}rcl@{}} {~}^{{~}_s\!m}\!\underset{\sim}{\boldsymbol{G}}{~}^{1} &=& {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} , {~}^{{~}_s\!m}\!\underset{\sim}{\boldsymbol{G}}{~}^{2} = {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} , {~}^{{~}_s\!m}\!\underset{\sim}{\boldsymbol{G}}{~}^{3} = {~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} ,\\ {~}^{{~}_s\!m}\!\underset{\sim}{\boldsymbol{G}}{~}^{4} &=& {~}_s \boldsymbol{m}^{[0]} \text{ and } {~}^{{~}_s\!m}\!\underset{\sim}{\boldsymbol{G}}{~}^{5} = ({~}_s \boldsymbol{m}^{[0]} )^{2} \\ {~}^{{~}_s\!m}\!\underset{\sim}{I}{~}^{1} &=& \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} ) , {~}^{{~}_s\!m}\!\underset{\sim}{I}{~}^{2} = \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} ) , {~}^{{~}_s\!m}\!\underset{\sim}{I}{~}^{3} = \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} ) ,\\ {~}^{{~}_s\!m}\!\underset{\sim}{I}{~}^{4} &=& \text{tr}({~}_s \boldsymbol{m}^{[0]} ) \end{array} $$
then we can obtain the following from (73) (using \(n_{\! \left ({~}_{ s}^{{~}_i{\Theta }}\!{J} \right )} = 2\), \(m_{\! {~}_s\!m } = 1\))
$$\begin{array}{@{}rcl@{}} {~}_s \boldsymbol{m}^{[1]} &= & {~}^{{~}_s\!m}\!\underline{a}_{1} \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} )\boldsymbol{I} + {~}^{{~}_s\!m}\!\underline{a}_{2} \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} )\boldsymbol{I}+ {~}^{{~}_s\!m}\!\underline{a}_{3} \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} )\boldsymbol{I}\\ &&+ {~}^{{~}_s\!m}\!\underline{a}_{4} \text{tr}({~}_s \boldsymbol{m} )\boldsymbol{I}+ {~}^{{~}_s\!m}\!\underline{b}_{1} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} ) + {~}^{{~}_s\!m}\!\underline{b}_{2} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} )\\ && + {~}^{{~}_s\!m}\!\underline{b}_{3} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} )+ {~}^{{~}_s\!m}\!\underline{b}_{4} ({~}_s \boldsymbol{m}^{[0]} ) + {~}^{{~}_s\!m}\!\underline{b}_{5} ({~}_s \boldsymbol{m}^{[0]} )^{2} \end{array} $$
(92)
We transfer \({~}^{{~}_s\!m}\!\underline {b}_{4} ({~}_s \boldsymbol {m}^{[0]} )\) term on the left side of (92) and divide the whole equation by \({~}^{{~}_s\!m}\!\underline {b}_{4} \) and define
$$\begin{array}{@{}rcl@{}} {~}^{{~}_s\!m}\!{\underset{\sim}{\lambda}} &=& - \frac{1}{ {~}^{{~}_s\!m}\!\underline{b}_{4} } , {{~}^{{~}_s\!m}\mu} = \left(- \frac{ {~}^{{~}_s\!m}\!\underline{b}_{1} }{ {~}^{{~}_s\!m}\!\underline{b}_{4} } \right) , {~}^{{~}_s\!m} \! \lambda = \left(- \frac{ {~}^{{~}_s\!m}\!\underline{a}_{1} }{ {~}^{{~}_s\!m}\!\underline{b}_{4} } \right) ,\\ {~}^{{~}_s\!m}\eta_{1} &=& \left(- \frac{ {~}^{{~}_s\!m}\!\underline{b}_{2} }{ {~}^{{~}_s\!m}\!\underline{b}_{4} } \right) ,{~}^{{~}_s\!m}k_{1} = \left(- \frac{ {~}^{{~}_s\!m}\!\underline{a}_{2} }{ {~}^{{~}_s\!m}\!\underline{b}_{4} } \right) , {~}^{{~}_s\!m}\eta_{2} = \left(- \frac{ {~}^{{~}_s\!m}\!\underline{b}_{3} }{ {~}^{{~}_s\!m}\!\underline{b}_{4} } \right) ,\\ {~}^{{~}_s\!m}k_{2} &=& \left(- \frac{ {~}^{{~}_s\!m}\!\underline{a}_{3} }{ {~}^{{~}_s\!m}\!\underline{b}_{4} } \right) , {~}^{{~}_s\!m}k_{3} = \left(- \frac{ {~}^{{~}_s\!m}\!\underline{a}_{4} }{ {~}^{{~}_s\!m}\!\underline{b}_{4} } \right)\\ {~}^{{~}_s\!m}\eta_{4} &=& \left(- \frac{ {~}^{{~}_s\!m}\!\underline{b}_{5} }{ {~}^{{~}_s\!m}\!\underline{b}_{4} } \right) \end{array} $$
(93)
Then, (92) can be written as
$$\begin{array}{@{}rcl@{}} {~}_s \boldsymbol{m}^{[0]} + {~}^{{~}_s\!m}\!{\underset{\sim}{\lambda}} {~}_s \boldsymbol{m}^{[1]} &=& \ {{~}^{{~}_s\!m}\mu} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} ) + {~}^{{~}_s\!m} \! \lambda \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} )\boldsymbol{I}\\ &&+ {~}^{{~}_s\!m}\eta_{1} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} ) + {~}^{{~}_s\!m}k_{1} \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} )\boldsymbol{I} \\ &&+ {~}^{{~}_s\!m}\eta_{2} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} ) + {~}^{{~}_s\!m}k_{2} \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} )\boldsymbol{I}\\ &&+ {~}^{{~}_s\!m}k_{3} \text{tr}({~}_s \boldsymbol{m}^{[0]} )\boldsymbol{I} + {~}^{{~}_s\!m}\eta_{4} ({~}_s \boldsymbol{m}^{[0]} )^{2} \end{array} $$
(94)
In which \({~}^{{~}_s\!m}\!{\underset {\sim }{\lambda }} \) is the relaxation time associated with \({~}_s \boldsymbol {m}^{[0]} \). Other material coefficients have similar meaning as those defined in Section 6.1. Constitutive theories for symmetric part of the moment tensor in non-classical thermoviscoelastic solids similar to Maxwell, Oldroyd-B, and Giesekus polymeric fluids can be obtained by setting \({~}^{{~}_s\!m}\eta _{2} = 0\), \({~}^{{~}_s\!m}k_{2} = 0\), \({~}^{{~}_s\!m}k_{3} = 0\) and \({~}^{{~}_s\!m}\eta _{4} = 0\); \({~}^{{~}_s\!m}k_{3} = 0\), \({~}^{{~}_s\!m}\eta _{4} = 0\); and \({~}^{{~}_s\!m}\eta _{2} = 0\), \({~}^{{~}_s\!m}k_{2} = 0\) and \({~}^{{~}_s\!m}k_{3} = 0\) respectively. Details are straight forward.

6.3 Constitutive Theories for Antisymmetric Part of the Cauchy Moment Tensor

Based on the simplification in Section 6 for constitutive variable \({~}_a \boldsymbol {m}^{[1]} \) with argument tensors in (74), we have the following generators and invariants (based on purely linear constitutive theory)
$$\begin{array}{@{}rcl@{}} {~}^{{~}_a\!m}\!\underset{\sim}{\boldsymbol{G}}{~}^{1} = {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} , {~}^{{~}_a\!m}\!\underset{\sim}{\boldsymbol{G}}{~}^{2} = {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} , {~}^{{~}_a\!m}\!\underset{\sim}{\boldsymbol{G}}{~}^{3} = {~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} , {~}^{{~}_a\!m}\!\underset{\sim}{\boldsymbol{G}}{~}^{4} = {~}_a \boldsymbol{m}^{[0]} \end{array} $$
(95)
There are no invariants of these argument tensors that are linear in their components and do not contain the product terms of the components of argument tensor. Invariants containing quadratic term in the the argument tensors are not considered either. That is the following invariants
$$\begin{array}{@{}rcl@{}} &&\text{tr}({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} )({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} ) ,\text{tr}({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} )({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} ) , \text{tr}({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} )({~}_a \boldsymbol{m}^{[0]} ) , {\dots} \text{ etc}\\ &&\text{tr}({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} )^{2} , \text{tr}({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} )^{2} , etc \end{array} $$
(96)
are not considered. Likewise, the generators containing the product and quadratic terms in the argument tensors are not considered either. We remark that if there is a need for any of these invariants and generators to be considered, they can be easily incorporated in the derivation of the constitutive theory. We can write the following (neglecting \((\theta - \theta _{\!\underline {\scriptscriptstyle {\Omega }}})\) terms and the initial moment term)
$$\begin{array}{@{}rcl@{}} {~}_a \boldsymbol{m}^{[1]} &=& {~}^{{~}_a\!m}\!\underline{b}_{1} ({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} ) + {~}^{{~}_a\!m}\!\underline{b}_{2} ({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} ) + {~}^{{~}_a\!m}\!\underline{b}_{3} ({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} )\\&& + {~}^{{~}_a\!m}\!\underline{b}_{4} ({~}_a \boldsymbol{m}^{[0]} ) \end{array} $$
(97)
transferring \({~}^{{~}_a\!m}\!\underline {b}_{4} ({~}_a \boldsymbol {m}^{[0]} )\) term on the left side of (97) and dividing throughout by \({~}^{{~}_a\!m}\!\underline {b}_{4} \) and defining
$$\begin{array}{@{}rcl@{}} {~}^{{~}_a\!m}\!{\underset{\sim}{\lambda}} &=& - \frac{1}{ {~}^{{~}_a\!m}\!\underline{b}_{4} } , {~}^{{~}_a\!m}\eta = \left(- \frac{ {~}^{{~}_a\!m}\!\underline{b}_{1} }{ {~}^{{~}_a\!m}\!\underline{b}_{4} } \right) , {~}^{{~}_a\!m}\eta_{1} = \left(- \frac{ {~}^{{~}_a\!m}\!\underline{b}_{2} }{ {~}^{{~}_a\!m}\!\underline{b}_{4} } \right) , {~}^{{~}_a\!m}\eta_{2}\\& =& \left(- \frac{ {~}^{{~}_a\!m}\!\underline{b}_{3} }{ {~}^{{~}_a\!m}\!\underline{b}_{4} } \right) \end{array} $$
(98)
We obtain
$$\begin{array}{@{}rcl@{}} {~}_a \boldsymbol{m}^{[0]} &+& ({~}^{{~}_a\!m}\!{\underset{\sim}{\lambda}} ) {~}_a \boldsymbol{m}^{[1]} = ({~}^{{~}_a\!m}\eta )({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} )\\ &+& ({~}^{{~}_a\!m}\eta_{1} )({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} ) + ({~}^{{~}_a\!m}\eta_{2} )({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} ) \end{array} $$
(99)
This constitutive theory is obviously linear. As pointed out earlier, by including generators and invariants that are nonlinear functions of the argument tensors, more elaborate form of (99) can be obtained depending upon the need.

7 Complete Mathematical Model

In the following, we present complete mathematical model including the constitutive theories for non-classical thermoviscoelastic solid with memory (but small deformation and small strain) in simplified forms that contain the classical constitutive theories as subset. In the following, we assume that balance of moment of moments is not a balance law; hence, the moment tensor is not symmetric. Conservation and balance laws ((21)–(23), (37), (60))
$$\begin{array}{@{}rcl@{}} \rho_{_{_{\!\!0}}}(\pmb{\boldsymbol{x }}) = |J|\rho(\pmb{\boldsymbol{x }},t)\\ |J| \simeq 1, \quad \rho_{_{_{\!\!0}}} \simeq \rho \end{array} $$
(100)
$$ \rho_{_{_{\!\!0}}}\frac{D \pmb{\boldsymbol{v }}}{Dt}-\rho_{_{_{\!\!0}}}{ \pmb{\boldsymbol{F }}}^{b} - { \pmb{\boldsymbol{\nabla \cdot }} \pmb{\boldsymbol{\sigma }}}= 0 $$
(101)
$$ { \pmb{\boldsymbol{\nabla \cdot }} \pmb{\boldsymbol{m }} - \pmb{\boldsymbol{\epsilon: }} \pmb{\boldsymbol{\sigma }}} = 0 $$
(102)
$$ \rho_{_{_{\!\!0}}}\frac{De}{Dt} + \pmb{\boldsymbol{\nabla \cdot q }} - \text{tr}\left([ {~}_s{\sigma} ][\overset{ \text{\textbf{.}}}{\varepsilon}] \right) - \text{tr}\left([ {~}_s {m} ][ {~}_{ s}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} ] \right) - \text{tr}\left([ {~}_a {m} ][ {~}_{ a}^{{~}_i{\Theta}}\!\overset{ \text{\textbf{.}}}{J} ] \right) = 0 $$
(103)
$$\begin{array}{@{}rcl@{}} &&-\text{tr} \left([ {~}_d({~}_s{\sigma}^{[0]}) ] [ \overset{ \text{\textbf{.}}}{\varepsilon}_{[0]} ] \right) - \text{tr} \left([ {~}_s {m}^{[0]}][ {~}_{ s}^{{~}_i{\Theta}}\!J_{[0]} ] \right)\\ &&- \text{tr} \left([ {~}_a {m}^{[0]}][ {~}_{ a}^{{~}_i{\Theta}}\!J_{[0]} ] \right) + \frac{\{q\}^{T} \{g\}}{\theta} \leq 0 \end{array} $$
(104)
$$\begin{array}{@{}rcl@{}} && \pmb{\boldsymbol{\sigma }} = {~}_s\boldsymbol{\sigma} + {~}_a\boldsymbol{\sigma} , \quad {~}_s\boldsymbol{\sigma} = {~}_d({~}_s\boldsymbol{\sigma}) + {~}_e({~}_s\boldsymbol{\sigma})\\ && \pmb{\boldsymbol{m }} = {~}_s \boldsymbol{m} + {~}_a \boldsymbol{m} \end{array} $$
(105)

7.1 Constitutive Theories

Using general but simplified constitutive theories (59), (88), (94), (99), (82) (for non-classical case), we have
$$ {~}_e({~}_s\boldsymbol{\sigma}^{[0]}) = p(\theta) \pmb{\boldsymbol{I }} $$
(106)
$$\begin{array}{@{}rcl@{}} {~}_d({~}_s\boldsymbol{\sigma}^{[0]}) + {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} {~}_d({~}_s\boldsymbol{\sigma}^{[1]}) \!&=&\! {{~}^{{~}_s\!\sigma} \! \mu} (\boldsymbol{\varepsilon}_{[0]} ) + {~}^{{~}_s\!\sigma}\!\lambda \text{tr}(\boldsymbol{\varepsilon}_{[0]} )\boldsymbol{I}\\ && \!+ {~}^{{~}_s\!\sigma}\!\eta_{1} (\boldsymbol{\varepsilon}_{[1]} ) + {~}^{{~}_s\!\sigma} \! k_{1} \text{tr}(\boldsymbol{\varepsilon}_{[1]} )\boldsymbol{I}\\ && \!+ {~}^{{~}_s\!\sigma}\!\eta_{2} (\boldsymbol{\varepsilon}_{[2]} ) + {~}^{{~}_s\!\sigma} \! k_{2} \text{tr}(\boldsymbol{\varepsilon}_{[2]} )\boldsymbol{I}\\ && \!+ {~}^{{~}_s\!\sigma} \! k_{3} \text{tr}({~}_d({~}_s\boldsymbol{\sigma}^{[0]}) )\boldsymbol{I} + {~}^{{~}_s\!\sigma}\!\eta_{4} ({~}_d({~}_s\boldsymbol{\sigma}^{[0]}) )^{2}\\ \end{array} $$
(107)
$$\begin{array}{@{}rcl@{}} {~}_s \boldsymbol{m}^{[0]} + {~}^{{~}_s\!m}\!{\underset{\sim}{\lambda}} {~}_s \boldsymbol{m}^{[1]} &=& \ {{~}^{{~}_s\!m}\mu} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} ) + {~}^{{~}_s\!m} \! \lambda \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} )\boldsymbol{I} {~}^{{~}_s\!m}\eta_{1} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]}\\ &&+ {~}^{{~}_s\!m}k_{1} \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} )\boldsymbol{I} + {~}^{{~}_s\!m}\eta_{2} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} )\\ &&+ {~}^{{~}_s\!m}k_{2} \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} )\boldsymbol{I} + {~}^{{~}_s\!m}k_{3} \text{tr}({~}_s \boldsymbol{m}^{[0]} )\boldsymbol{I}\\ &&+ {~}^{{~}_s\!m}\eta_{4} ({~}_s \boldsymbol{m}^{[0]} )^{2} \end{array} $$
(108)
$$\begin{array}{@{}rcl@{}} {~}_a \boldsymbol{m}^{[0]} + ({~}^{{~}_a\!m}\!{\underset{\sim}{\lambda}} ) {~}_a \boldsymbol{m}^{[1]} \!&=& \!({~}^{{~}_a\!m}\eta )({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} ) + ({~}^{{~}_a\!m}\eta_{1} )({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} )\\ && + ({~}^{{~}_a\!m}\eta_{2} )({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]}) \end{array} $$
(109)
$$ \boldsymbol{q} = - \left.{ k }\right|_{\underline{\Omega}} \boldsymbol{g} - \left.{ k_{1} }\right|_{\underline{\Omega}} (\pmb{\boldsymbol{g \cdot g }}) \boldsymbol{g} - \left.{ k_{2} }\right|_{\underline{\Omega}} (\theta - \theta_{\!\underline{\scriptscriptstyle{\Omega}}}) \boldsymbol{g} $$
(110)
This mathematical model has closure. Twenty-five dependent variables (number in the parenthesis is the number of variables): v or u(3), σ(9), m(9), q(3), 𝜃(1), total of 25 in 25 equations: balance of linear momenta (3), balance of angular moment (3), constitutive theories for: σ(6), m(9), q(3), energy equation (1).

8 Retardation and Memory Moduli

Using constitutive theories (107)–(109) and discarding \({~}_d({~}_s\boldsymbol {\sigma }^{[0]}) \) and m[0] terms on the right side of these and defining
$$\begin{array}{@{}rcl@{}} {~}^{\sigma}\!{\boldsymbol{Q}}_s &=& \ {{~}^{{~}_s\!\sigma} \! \mu} (\boldsymbol{\varepsilon}_{[0]} ) + {~}^{{~}_s\!\sigma}\!\lambda \text{tr}(\boldsymbol{\varepsilon}_{[0]} )\boldsymbol{I} + {~}^{{~}_s\!\sigma}\!\eta_{1} (\boldsymbol{\varepsilon}_{[1]} ) + {~}^{{~}_s\!\sigma} \! k_{1} \text{tr}(\boldsymbol{\varepsilon}_{[1]} )\boldsymbol{I}\\&& + {~}^{{~}_s\!\sigma}\!\eta_{2} (\boldsymbol{\varepsilon}_{[2]} )\\ &&+ {~}^{{~}_s\!\sigma} \! k_{2} \text{tr}(\boldsymbol{\varepsilon}_{[2]} )\boldsymbol{I} \end{array} $$
(111)
$$\begin{array}{@{}rcl@{}} {~}^m\!{\boldsymbol{Q}}_s &=& \ {{~}^{{~}_s\!m}\mu} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} ) + {~}^{{~}_s\!m} \! \lambda \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} )\boldsymbol{I} + {~}^{{~}_s\!m}\eta_{1} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} )\\&& + {~}^{{~}_s\!m}k_{1} \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} )\boldsymbol{I} + {~}^{{~}_s\!m}\eta_{2} ({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} )\\ &&+ {~}^{{~}_s\!m}k_{2} \text{tr}({~}_{ s}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} )\boldsymbol{I} \end{array} $$
(112)
$$\begin{array}{@{}rcl@{}} {~}^m\!{\boldsymbol{Q}}_a = ({~}^{{~}_a\!m}\eta )({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[0]} ) + ({~}^{{~}_a\!m}\eta_{1} )({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[1]} ) + ({~}^{{~}_a\!m}\eta_{2} )({~}_{ a}^{{~}_i{\Theta}}\!\boldsymbol{J}_{[2]} )\\ \end{array} $$
(113)
We can write (107)–(109) as
$$\begin{array}{@{}rcl@{}} {~}_d({~}_s\boldsymbol{\sigma}^{[0]}) + ({~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} ) {~}_d({~}_s\boldsymbol{\sigma}^{[1]}) = {~}^{\sigma}\!{\boldsymbol{Q}}_s \end{array} $$
(114)
$$\begin{array}{@{}rcl@{}} {~}_s \boldsymbol{m}^{[0]} + ({~}^{{~}_s\!m}\!{\underset{\sim}{\lambda}} ) {~}_s \boldsymbol{m}^{[1]} = {~}^m\!{\boldsymbol{Q}}_s \end{array} $$
(115)
$$\begin{array}{@{}rcl@{}} {~}_a \boldsymbol{m}^{[0]} + ({~}^{{~}_a\!m}\!{\underset{\sim}{\lambda}} ) {~}_a \boldsymbol{m}^{[1]} = {~}^m\!{\boldsymbol{Q}}_a \end{array} $$
(116)
Equation 114116 are first-order differential equations time in \({~}_d({~}_s\boldsymbol {\sigma }) \), \({~}_s \boldsymbol {m}\), and \({~}_a \boldsymbol {m} \), hence can be integrated using the following The differential equation
$$ \frac{d\phi}{d x} + P(x)\phi = Q(x) $$
(117)
has the solution
$$ \phi = e^{-\int P(x)dx} \left[ \int Q(x) e^{\int P(x)dx}_{~~~~~~~~ dx} + K \right] $$
(118)
where K is a constant of integrationWe consider (114) and rewrite
$$ {~}_d({~}_s\boldsymbol{\sigma}^{[1]}) + \frac{1}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }({~}_d({~}_s\boldsymbol{\sigma}^{[0]}) ) = \left(\frac{1}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} } \right) {~}^{\sigma}\!{\boldsymbol{Q}}_s $$
(119)
Hence, using (117) and (118), we can write
$$\begin{array}{@{}rcl@{}} {~}_d({~}_s\boldsymbol{\sigma}^{[0]}) &=& e^{-\int \frac{1}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }dt} \left[ \int \frac{1}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }({~}^{\sigma}\!{\boldsymbol{Q}}_s) e^{\int \frac{1}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }dt}dt + \boldsymbol{K} \right]\\ &=& e^{-\frac{t}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }} \left[ \int \frac{1}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }({~}^{\sigma}\!{\boldsymbol{Q}}_s) e^{\frac{t^{\prime}}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }}dt^{\prime} + \boldsymbol{K} \right]\\ &=& \frac{ {\int}_{-\infty}^{t} \frac{1}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }({~}^{\sigma}\!{\boldsymbol{Q}}_s) e^{\frac{t^{\prime}}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }}dt^{\prime} }{e^{\frac{t}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }}} + \boldsymbol{K} e^{-\frac{t}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }} \end{array} $$
(120)
Based on Bird et al. (1987), the choice of − is arbitrary. Some other value could result in a different value of K. If we prescribe that the stress at t = − is finite, we must choose K to be 0. We must check the first term in (120), since both numerator and denominator go to 0 as t goes to −. Using L’Hospital’s rule, we get
$$\begin{array}{@{}rcl@{}} \lim\limits_{t \rightarrow-\infty} {~}_d({~}_s\boldsymbol{\sigma}^{[0]}) = \lim\limits_{t \rightarrow-\infty} \!\frac{ \frac{1}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }({~}^{\sigma}\!{\boldsymbol{Q}}_s) e^{\frac{t^{\prime}}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }}dt^{\prime} }{ \frac{1}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} } e^{\frac{t}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }}} = {~}^{\sigma}\!{\boldsymbol{Q}}_s(- \infty) \end{array} $$
(121)
Thus, if \({~}^{\sigma }\!{\boldsymbol {Q}}_s(-\infty )\) is finite, the stress is finite at t = −. Hence, (120) reduces to
$$ {~}_d({~}_s\boldsymbol{\sigma}^{[0]}) = {\int}_{-\infty}^{t} \left(\frac{1}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} } e^{-\frac{(t - t^{\prime})}{ {~}^{{~}_s\!\sigma}\!\underset{\sim}{\lambda} }} \right) {~}^{\sigma}\!{\boldsymbol{Q}}_s(t^{\prime})dt^{\prime} $$
(122)
The quantity in bracket in the integrand in (122) is called retardation modulus, when \({~}^{\sigma }\!{\boldsymbol {Q}}_s\) only contains \( \overset { \text {\textbf {.}}}{\boldsymbol {\varepsilon }}_{[0]} \) as in case of fluids, we can obtain relaxation modulus from (122). We omit this here as this requires approximating \({~}^{\sigma }\!{\boldsymbol {Q}}_s\). Retardation modulus is as good of a measure of rheology as relaxation modulus. Using similar derivations, we can also obtain the following from (115) and (116), first by rewriting then by dividing by \({~}^{{~}_s\!m}\!{\underset {\sim }{\lambda }} \) and \({~}^{{~}_a\!m}\!{\underset {\sim }{\lambda }} \) respectively
$$\begin{array}{@{}rcl@{}} && {~}_s \boldsymbol{m}^{[1]} + \frac{1}{ {~}^{{~}_s\!m}\!{\underset{\sim}{\lambda}} }({~}_s \boldsymbol{m}^{[0]} ) = \left(\frac{1}{ {~}^{{~}_s\!m}\!{\underset{\sim}{\lambda}} } \right) {~}^m\!{\boldsymbol{Q}}_s \end{array} $$
(123)
$$\begin{array}{@{}rcl@{}} && {~}_a \boldsymbol{m}^{[1]} + \frac{1}{ {~}^{{~}_a\!m}\!{\underset{\sim}{\lambda}} }({~}_a \boldsymbol{m}^{[0]} ) = \left(\frac{1}{ {~}^{{~}_a\!m}\!{\underset{\sim}{\lambda}} } \right) {~}^m\!{\boldsymbol{Q}}_a \end{array} $$
(124)
and then following the derivation for \({~}_d({~}_s\boldsymbol {\sigma }) \)
$$ {~}_s \boldsymbol{m}^{[0]} = {\int}_{-\infty}^{t} \left(\frac{1}{ {~}^{{~}_s\!m}\!{\underset{\sim}{\lambda}} } e^{-\frac{(t - t^{\prime})}{ {~}^{{~}_s\!m}\!{\underset{\sim}{\lambda}} }} \right) {~}^m\!{\boldsymbol{Q}}_s(t^{\prime})dt^{\prime} $$
(125)
$$ {~}_a \boldsymbol{m}^{[0]} = {\int}_{-\infty}^{t} \left(\frac{1}{ {~}^{{~}_a\!m}\!{\underset{\sim}{\lambda}} } e^{-\frac{(t - t^{\prime})}{ {~}^{{~}_a\!m}\!{\underset{\sim}{\lambda}} }} \right) {~}^m\!{\boldsymbol{Q}}_a(t^{\prime})dt^{\prime} $$
(126)
The terms in the brackets in the integrand in (125) and (126) are retardation moduli for \({~}_s \boldsymbol {m}^{[0]} \) and \({~}_a \boldsymbol {m}^{[0]} \).

Remarks 3

  1. 1.

    We observe that in non-classical thermoviscoelastic solids (with memory), we have retardation mechanics due to \({~}_d({~}_s\boldsymbol {\sigma }^{[0]}) \) (as in classical case) as well as due to \({~}_s \boldsymbol {m}^{[0]} \) and \({~}_a \boldsymbol {m}^{[0]} \) when the balance of moments of moment is not used as a balance law.

     
  2. 2.

    When the balance of moment of moments is used as a balance law, Cauchy moment tensor is symmetric, i.e., \({~}_a \boldsymbol {m} = 0\) and m[0] = sm[0]; hence, in this case, the retardation mechanism is only due to \({~}_d({~}_s\boldsymbol {\sigma }^{[0]}) \) and \({~}_s \boldsymbol {m}^{[0]} = \pmb {\boldsymbol {m }}^{[0]}\).

     

9 Summary and Conclusions

In this paper, we have given conservation and balance laws for non-classical solid continua with small deformation and small strain incorporating internal rotations arising from the Jacobian of deformation tensor and have presented derivations of constitutive theories for thermoviscoelastic solids with dissipation and memory mechanisms. The internal rotations resulting from the antisymmetric part of the Jacobian of deformation are defined about the axes of a triad located at each material point, the axes being parallel to x-frame. Thus, these internal rotations are completely defined by the Jacobian of deformation tensor and therefore do not constitute additional degrees of freedom at a material point. Constitutive theories are derived using the conditions resulting from the entropy inequality in conjunction with representation theorem, hence are thermodynamically consistent. In the following, we summarize the significant aspects of the work presented in this paper.
  1. 1.

    The rate of work due to internal rotation rates and the conjugate moment are incorporated in the conservation and the balance laws.

     
  2. 2.

    As shown in Yang et al. (2002) and Surana et al. (2017d, g), the balance of moment of moments is a necessary balance law in the non-classical continuum theories to ensure equilibrium of the deforming volume of matter. In the presence of this balance law, the Cauchy moment tensor becomes symmetric. In this paper, we have presented derivation of the constitutive theories in the absence of this balance law, a more general case. When this balance law is considered, the Cauchy moment tensor becomes symmetric, i.e., \({~}_a \boldsymbol {m}^{[0]} = 0\) and \({~}_s \boldsymbol {m}^{[0]} = \pmb {\boldsymbol {m }}^{[0]}\).

     
  3. 3.

    In non-classical thermoviscoelastic solid with internal rotations and their rates, the energy storage, dissipation, and rheology mechanisms are due to \({~}_d({~}_s\boldsymbol {\sigma }^{[0]}) \), \({~}_s \boldsymbol {m}^{[0]} \), and \({~}_a \boldsymbol {m}^{[0]} \) as opposed to only due to \({~}_d({~}_s\boldsymbol {\sigma }^{[0]}) = {~}_{d}(\pmb {\boldsymbol {\sigma }}^{[0]})\) for such solids described using classical continuum theories.

     
  4. 4.

    When the balance of moment of moments is not considered as a balance law, the Cauchy moment tensor is not symmetric, which necessitates constitutive theories for \({~}_s \boldsymbol {m}^{[0]} \) as well as \({~}_a \boldsymbol {m}^{[0]} \), hence brings in the mechanisms of energy storage, dissipation and rheology due to both moment tensors but considered individually.

     
  5. 5.

    The constitutive theories for deviatoric part of the symmetric Cauchy stress tensor, symmetric Cauchy moment tensor, and the antisymmetric moment tensors are ordered rate theories that utilize rates of constitutive variables as well as the rates of the argument tensors upto certain desired orders. Rates of constitutive variables allow us to incorporate rheological behavior. Rates of argument tensors permit more comprehensive energy storage and dissipation physics.

     
  6. 6.

    General constitutive theories presented in this paper based on integrity are the most general and consistent description of the constitutive behavior of the deforming matter, from which any desired simplified and specific constitutive theory can be extracted as shown in the section containing simplified constitutive theories.

     
  7. 7.

    As shown in the paper, the constitutive theories based on integrity are almost always nonlinear in their argument tensors. Their linearization is perfectly valid if only limited physics is of interest; however, the conclusions that may be drawn from the superposition of the linear constitutive theories obviously cannot be applied to the constitutive theories based on integrity as these constitutive theories are almost always nonlinear. For example, linear constitutive theory for \({~}_s \boldsymbol {m}^{[0]} \) and \({~}_a \boldsymbol {m}^{[0]} \) suggesting a constitutive theory for \(\pmb {\boldsymbol {m }}^{[0]} = {~}_s \boldsymbol {m}^{[0]} + {~}_a \boldsymbol {m}^{[0]} \), a non-symmetric tensor in terms of non-symmetric argument tensor(s) is obviously a false conclusion based on the works of Spencer, Wang, Zheng, and others.

     
  8. 8.

    Derivations of memory moduli due to the \({~}_d({~}_s\boldsymbol {\sigma }^{[0]}) \), \({~}_s \boldsymbol {m}^{[0]}\), and \({~}_a \boldsymbol {m}^{[0]} \) are presented.

     

In conclusion, the work presented in this paper utilizes a consistent thermodynamic framework for non-classical solid continua incorporating internal rotations due to Jacobian of deformation tensor in the derivations of the constitutive theories for the thermoviscoelastic solids with mechanisms of energy storage, dissipation, and rheology. It is shown that Cauchy stress tensor as well as Cauchy moment tensor influence this physics. The derivations of the constitutive theories are based on the conditions resulting from the entropy inequality in conjunction with representation theorem. Simplification of the general constitute theories based on integrity is demonstrated to show that the resulting theories that resemble Maxwell, Oldroyd-B, and Giesekus constitute models in classical polymer science can be easily derived.

Notes

Acknowledgments

The first and the third authors are grateful for the support provided by their endowed professorships during the course of this research. Many facilities provided by the Department of mechanical engineering of the University of Kansas are gratefully acknowledged.

Funding Information

The second author received financial support from the Mechanical Engineering Department of the University of Kansas is also acknowledged.

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© Escola Politécnica - Universidade de São Paulo 2018

Authors and Affiliations

  1. 1.Mechanical EngineeringUniversity of KansasLawrenceUSA
  2. 2.Mechanical EngineeringTexas A&M UniversityCollege StationUSA

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