For finite deformation, finite strain physics in TVE compressible solid matter with rheology, we consider CBL of CCM derived using the contravariant second Piola-Kirchhoff stress tensor ${\sigma }^{\left[0\right]}$ and the covariant Green’s strain tensor ${\epsilon }_{\left[0\right]}$ in Lagrangian description. The dissipation mechanism is described by the convected time derivatives of the Green’s strain tensor up to order n ( ${\epsilon }_{\left[i\right]}\mathrm{<mo>\; ;\; </mo>}i=\mathrm{1,2,}\cdots \mathrm{,}n$). The relaxation or rheology physics is incorporated by considering convected time derivatives ${}_d{\sigma }^{\left[i\right]}\mathrm{<mo>\; ;\; </mo>}i=\mathrm{1,2,}\cdots \mathrm{,}m$ of the deviatoric contravariant second Piola-Kirchhoff stress tensor, each rate results in a characteristic time constant, relaxation time. Thus, in this constitutive theory, we have a spectrum of relaxation times, a more suitable model of the physics of rheology [30] . ${}_d{\sigma }^{\left[m\right]}$ is used as the constitutive tensor with ${\epsilon }_{\left[i\right]}\mathrm{<mo>\; ;\; </mo>}i=\mathrm{0,1,}\cdots \mathrm{,}n$, ${}_d{\sigma }^{\left[j\right]}\mathrm{<mo>\; ;\; </mo>}j=\mathrm{0,1,}\cdots \mathrm{,}m-1$ and $\theta$ as its argument tensors. Conservation of mass (CM), balance of linear momentum (BLM), balance of angular momentum (BAM), and the first and second laws of thermodynamics [31] [32] are given in the following:

${\rho }_0\left(x\right)=\left|J\right|\rho \left(x,t\right)$ (1)

${\rho }_0\frac{{\partial }^2\left\{u\right\}}{\partial t^2}-{\rho }_0\left\{F^b\right\}-\left(\left[J\right]\left[{\sigma }^{\left[0\right]}\right]\right)\{\nabla \}=0$ (2)

${ϵ}_{ijk}{\sigma }_{ij}^{\left(0\right)}=0$ (3)

${\rho }_0{c}_v\frac{De}{Dt}-\nabla \cdot q-{\sigma }^{\left[0\right]}\mathrm{<mo>\; :\; </mo>}{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}=0$ (4)

${\rho }_0\left(\frac{D\varphi }{Dt}-\eta \frac{D\theta }{Dt}\right)-{\sigma }^{\left[0\right]}\mathrm{<mo>\; :\; </mo>}{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}+\frac{q\cdot g}{\theta }\le 0$ (5)

where ${\rho }_0$ is the density at a material point in the reference (or initial) configuration, $\rho \left(x\mathrm{,}t\right)$ is the density at a material point at $x_i$ in the current configuration, $\left[J\right]$ is the deformation gradient tensor, $u$ are displacements in the fixed x-frame, $F^b$ is the body force vector per unit mass, ${\sigma }^{\left[0\right]}$ is the symmetric contravariant second Piola-Kirchhoff stress tensor, $\nabla$ is the gradient operator, ${ϵ}_{ijk}$ is the permutation tensor, $c_v$ is the specific heat, e is specific internal energy density, $q$ is the heat vector, $\varphi$ is the Helmholtz free energy density, $\eta$ is the entropy density, $\theta$ is the absolute temperature, and $g$ is the temperature gradient vector. We consider the additive decomposition of the stress tensor ${\sigma }^{\left[0\right]}$ into equilibrium ( ${}_e{\sigma }^{\left[0\right]}$) and deviatoric ( ${}_d{\sigma }^{\left[0\right]}$) stress tensors:

${\sigma }^{\left[0\right]}={}_e\sigma {}^{\left[0\right]}+{}_d\sigma {}^{\left[0\right]}$ (6)

The constitutive theory for ${}_e{\sigma }^{\left[0\right]}$ describes volumetric deformation and the constitutive theory for ${}_d{\sigma }^{\left[0\right]}$ describes distortional deformation physics. Following references [31] [32] , the constitutive theory for ${}_e{\sigma }^{\left[0\right]}$ can be derived using thermodynamic pressure $p\left(\rho \mathrm{,}\theta \right)$ and contravariant Cauchy stress tensor, and we can write:

$\left[{}_e{\sigma }^{\left[0\right]}\right]=\left|J\right|p\left(\rho \mathrm{,}\theta \right){\left[{\left[J\right]}^{\text{T}}\left[J\right]\right]}^{-1}$ (7)

Thus, we note that ${}_e{\sigma }^{\left[0\right]}$ is not a pressure field. The reduced form of the entropy inequality [32] is given by:

$-{\sigma }^{\left[0\right]}\mathrm{<mo>\; :\; </mo>}{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}+\frac{q\cdot g}{\theta }\le 0$ (8)

We note that (8) is satisfied if ${\sigma }^{\left[0\right]}\mathrm{<mo>\; :\; </mo>}{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}>0$ (i.e., the rate of work is positive)

and $\frac{q\cdot g}{\theta }\le 0$. From the rate of work conjugate pair in conjunction with the

axioms of constitutive theory [31] [32] , we conclude that the deviatoric contravariant second Piola-Kirchhoff stress tensor is the constitutive variable and the Green’s

strain tensor is its argument tensor. Likewise, the conjugate pair $\frac{q\cdot g}{\theta }$ suggests

that $q$ is constitutive tensor and $g$ and $\theta$ its argument tensors. The presence of memory effects necessitates the inclusion of a memory modulus, which dictates that the constitutive theory for ${}_d{\sigma }^{\left[0\right]}$ must be represented as a differential equation in time. Consequently, it is essential to consider both ${}_d{\sigma }^{\left[0\right]}$ and ${}_d{\sigma }^{\left[1\right]}$, with ${}_d{\sigma }^{\left[1\right]}$ serving as a constitutive tensor that incorporates ${}_d{\sigma }^{\left[0\right]}$ as one of its argument tensors, among others. We generalize this to include ${}_d{\sigma }^{\left[j\right]}\mathrm{<mo>\; ;\; </mo>}j=\mathrm{0,1,}\cdots \mathrm{,}m$ convected time derivatives in place of ${}_d{\sigma }^{\left[0\right]}$ as argument tensors. The constitutive theory for the deviatoric contravariant second Piola-Kirchhoff stress tensor is derived by considering ${}_d{\sigma }^{\left[m\right]}$, convected time derivatives of ${}_e{\sigma }^{\left[0\right]}$ of order m, as the constitutive tensor with the following argument tensors:

${}_d\sigma {}^{\left[m\right]}={}_d\sigma {}^{\left[m\right]}\left({\epsilon }_{\left[i\right]}\mathrm{<mo>\; ;\; </mo>}{}_d\sigma {}^{\left[j\right]}\mathrm{<mo>\; ;\; </mo>}\theta \right)\mathrm{<mo>\; ;\; </mo>}i=\mathrm{0,1,}\cdots \mathrm{,}n\mathrm{<mo>\; ;\; </mo>}j=\mathrm{0,1,}\cdots \mathrm{,}m-1$ (9)

In an insulated system, entropy production and the resulting temperature rise are solely attributed to dissipation. This dissipation is minimal for non-cyclic external stimuli, indicating that non-isothermal effects may not need to be considered in the context of deformation physics. However, we present a derivation of the constitutive theories that includes the variable $\theta$. Using (9) in conjunction with the representation theorem, the constitutive theory for ${}_d{\sigma }^{\left[m\right]}$ can be derived using the derivation presented in [31] [32] . The basic steps are outlined in the following.

Let ${}^{\sigma }{\underset{˜}{\mathcal{G}}}^i\mathrm{<mo>\; ;\; </mo>}i=\mathrm{1,2,}\cdots \mathrm{,}N$ be the combined generators of the argument tensors of ${}_d{\sigma }^{\left[m\right]}$ in (9) that are symmetric tensors of rank two and let ${}^{\sigma }{I}^j\mathrm{<mo>\; ;\; </mo>}j=\mathrm{1,2,}\cdots \mathrm{,}M$ be the combined invariants of the same argument tensors in (9), all in the current configuration. Then $I\mathrm{,}{}^{\sigma }\underset{˜}{\mathcal{G}}{}^i\mathrm{<mo>\; ;\; </mo>}i=\mathrm{1,2,}\cdots \mathrm{,}N$ constitute the complete basis (integrity) of the space containing constitutive tensor ${}_d{\sigma }^{\left[m\right]}$. Hence, ${}_d{\sigma }^{\left[m\right]}$ can be expressed as a linear combination of $I\mathrm{,}{}^{\sigma }\underset{˜}{\mathcal{G}}{}^i\mathrm{<mo>\; ;\; </mo>}i=\mathrm{1,2,}\cdots \mathrm{,}N$ in the current configuration

${}_d{\sigma }^{\left[m\right]}={}^{\sigma }\underset{˜}{\alpha }{}^{0}I+{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{}^{\sigma }\underset{˜}{\alpha }{}^i\left({}^{\sigma }\underset{˜}{\mathcal{G}}{}^i\right)$ (10)

in which

${}^{\sigma }\underset{˜}{\alpha }{}^i={}^{\sigma }\underset{˜}{\alpha }{}^i\left({}^{\sigma }\underset{˜}{I}{}^j,\theta \right);j=1,2,\cdots ,M;i=0,1,\cdots ,N$ (11)

We note that in the constitutive theory (10), ${}^{\sigma }\underset{˜}{\alpha }{}^i\mathrm{<mo>\; ;\; </mo>}i=\mathrm{0,1,}\cdots \mathrm{,}N$ are not material coefficients. These are simply coefficients in the linear combination in (10) that show dependence on invariants ${}^{\sigma }\underset{˜}{I}{}^j\mathrm{<mo>\; ;\; </mo>}j=\mathrm{1,2,}\cdots \mathrm{,}M$ and $\theta$ (for non-isothermal case). To determine material coefficients, ${}^{\sigma }\underset{˜}{\alpha }{}^i\mathrm{<mo>\; ;\; </mo>}i=\mathrm{0,1,}\cdots \mathrm{,}N$ are expanded in Taylor series in ${}^{\sigma }\underset{˜}{I}{}^j\mathrm{<mo>\; ;\; </mo>}j=\mathrm{1,2,}\cdots \mathrm{,}M$ and $\theta$ about a known configuration $\underset{\_}{\Omega }$. These are then substituted in (10) and terms are rearranged so that each term has quantities in the current configuration multiplied with quantities or coefficients in the known configuration $\underset{\_}{\Omega }$. The quantities in the known configuration are the material coefficients. We can obtain the following [31] [32] .

Consider constitutive theory (10). We expand each ${}^{\sigma }\underset{˜}{\alpha }{}^i\mathrm{<mo>\; ;\; </mo>}i=\mathrm{0,1,}\cdots \mathrm{,}N$ in Taylor series in ${}^{\sigma }\underset{˜}{I}{}^j\mathrm{<mo>\; ;\; </mo>}j=\mathrm{1,2,}\cdots \mathrm{,}M$ and $\theta$ about a known configuration $\Omega$ and retain only up to linear terms in ${}^{\sigma }\underset{˜}{I}{}^j\mathrm{<mo>\; ;\; </mo>}j=\mathrm{1,2,}\cdots \mathrm{,}M$ and $\theta$ (for simplicity).

$\begin{array}{l}{}^{\sigma }\underset{˜}{\alpha }{}^i={{}^{\sigma }\underset{˜}{\alpha }{}^i|}_{\underset{\_}{\Omega }}+{\displaystyle \underset{j=1}{\overset{M}{\sum }}}{\frac{\partial {}^{\sigma }\underset{˜}{\alpha }{}^i}{\partial {}^{\sigma }\underset{˜}{I}{}^j}|}_{\underset{\_}{\Omega }}\left({}^{\sigma }\underset{˜}{I}{}^j-{{}^{\sigma }\underset{˜}{I}{}^j|}_{\underset{\_}{\Omega }}\right)\\ +{\frac{\partial {}^{\sigma }\underset{˜}{\alpha }{}^i}{\partial \theta }|}_{\underset{\_}{\Omega }}\left(\theta -{\theta |}_{\underset{\_}{\Omega }}\right);i=0,1,\cdots ,N\end{array}$ (12)

Substitute ${}^{\sigma }\underset{˜}{\alpha }{}^i\mathrm{<mo>\; ;\; </mo>}i=\mathrm{0,1,}\cdots \mathrm{,}N$ from (12) in (10)

$\begin{array}{c}{}_d{\sigma }^{\left[m\right]}=({{}^{\sigma }\underset{˜}{\alpha }{}^0|}_{\underset{\_}{\Omega }}+{\displaystyle \underset{j=1}{\overset{M}{\sum }}}{\frac{\partial {}^{\sigma }\underset{˜}{\alpha }{}^0}{\partial {}^{\sigma }\underset{˜}{I}{}^j}|}_{\underset{\_}{\Omega }}\left({}^{\sigma }\underset{˜}{I}{}^j-{{}^{\sigma }\underset{˜}{I}{}^j|}_{\Omega }\right)+{\frac{\partial {}^{\sigma }\underset{˜}{\alpha }{}^0}{\partial \theta }|}_{\underset{\_}{\Omega }}\left(\theta -{\theta |}_{\underset{\_}{\Omega }}\right))I\\ +{\displaystyle \underset{i=1}{\overset{N}{\sum }}}({{}^{\sigma }\underset{˜}{\alpha }{}^i|}_{\underset{\_}{\Omega }}+{\displaystyle \underset{j=1}{\overset{M}{\sum }}}{\frac{\partial {}^{\sigma }\underset{˜}{\alpha }{}^i}{\partial {}^{\sigma }\underset{˜}{I}{}^j}|}_{\underset{\_}{\Omega }}\left({}^{\sigma }\underset{˜}{I}{}^j-{{}^{\sigma }\underset{˜}{I}{}^j|}_{\underset{\_}{\Omega }}\right)+{\frac{\partial {}^{\sigma }\underset{˜}{\alpha }{}^i}{\partial \theta }|}_{\underset{\_}{\Omega }}\left(\theta -{\theta |}_{\underset{\_}{\Omega }}\right)){}^{\sigma }\underset{˜}{\mathcal{G}}{}^i\end{array}$ (13)

collecting coefficients (those terms that are defined in known configuration $\underset{\_}{\Omega }$) of $I\mathrm{,}{}^{\sigma }\underset{˜}{I}{}^{j}I\mathrm{,}{}^{\sigma }\underset{˜}{\mathcal{G}}{}^i\mathrm{,}\left(\theta -{\theta |}_{\underset{\_}{\Omega }}\right){}^{\sigma }\underset{˜}{\mathcal{G}}{}^i$ and $\left(\theta -{\theta |}_{\underset{\_}{\Omega }}\right)I$ and defining new coefficients we can write the following.

$\begin{array}{ll}{{\underset{˜}{\sigma }}_i^0|}_{\underset{\_}{\Omega }}={{}^{\sigma }\underset{˜}{\alpha }{}^0|}_{\underset{\_}{\Omega }}-{\displaystyle \underset{j=1}{\overset{M}{\sum }}}{\frac{\partial {}^{\sigma }\underset{˜}{\alpha }{}^0}{{\partial }^{\sigma }{I}^j}|}_{\underset{\_}{\Omega }}\left({{}^{\sigma }\underset{˜}{I}{}^j|}_{\underset{\_}{\Omega }}\right);\hfill & {}^{\sigma }\underset{˜}{a}{}_j={\frac{\partial {}^{\sigma }\underset{˜}{\alpha }{}^0}{{\partial }^{\sigma }{I}^j}|}_{\underset{\_}{\Omega }}\hfill \\ {}^{\sigma }\underset{˜}{b}{}_i={{}^{\sigma }\underset{˜}{\alpha }{}^i|}_{\underset{\_}{\Omega }}-{\displaystyle \underset{j=1}{\overset{M}{\sum }}}{\frac{\partial {}^{\sigma }\underset{˜}{\alpha }{}^i}{\partial {}^{\sigma }\underset{˜}{I}{}^j}|}_{\underset{\_}{\Omega }}\left({{}^{\sigma }\underset{˜}{I}{}^j|}_{\underset{\_}{\Omega }}\right);\hfill & {}^{\sigma }\underset{˜}{c}{}_{ij}={\frac{\partial {}^{\sigma }\underset{˜}{\alpha }{}^i}{\partial {}^{\sigma }\underset{˜}{I}{}^j}|}_{\underset{\_}{\Omega }}\hfill \\ i=1,2,\cdots ,N;\hfill & j=1,2,\cdots ,M\hfill \end{array}$ (14)

${}_d{\sigma }^{\left[m\right]}={{\sigma }^0|}_{\underset{\_}{\Omega }}I+{\displaystyle \underset{j=1}{\overset{M}{\sum }}}{}^{\sigma }\underset{˜}{a}{}_j\left({}^{\sigma }\underset{˜}{I}{}^j\right)I+{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle \underset{j=1}{\overset{M}{\sum }}}{}^{\sigma }\underset{˜}{c}{}_{ij}\left({}^{\sigma }\underset{˜}{I}{}^j\right){}^{\sigma }\underset{˜}{\mathcal{G}}{}^i+{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{}^{\sigma }\underset{˜}{b}{}_i\left({}^{\sigma }\underset{˜}{\mathcal{G}}{}^i\right)$ (15)

This constitutive theory (15) is based on integrity (complete basis), the representation theorem, and the conjugate pairs in the entropy inequality, hence it is thermodynamically and mathematically consistent constitutive theory. The coefficients ${}^{\sigma }\underset{˜}{a}{}_j\mathrm{,}{}^{\sigma }\underset{˜}{c}{}_{ij}\mathrm{,}{}^{\sigma }\underset{˜}{b}{}_i$ are the material coefficients that can be functions of

${{}^{\sigma }\underset{˜}{I}{}^j|}_{\underset{\_}{\Omega }}\mathrm{<mo>\; ;\; </mo>}j=\mathrm{1,2,}\cdots \mathrm{,}M$ and ${\theta |}_{\underset{\_}{\Omega }}$. This constitutive theory requires $\left(M+N+MN\right)$

material coefficients.

A simplified linear constitutive theory

We consider an ordered rate constitutive theory of up to orders m and n that is linear in the components of ${\epsilon }^{\left[0\right]}$, ${\epsilon }^{\left[i\right]}$; $i=\mathrm{1,2,}\cdots \mathrm{,}n$ and ${}_d{\sigma }^{\left[j\right]}$; $j=\mathrm{0,1,}\cdots \mathrm{,}m-1$ tensors. This constitutive theory will contain generators $I$, ${\epsilon }_{\left[0\right]}$, ${\epsilon }_{\left[i\right]}$; $i=\mathrm{1,2,}\cdots \mathrm{,}n$, ${}_d{\sigma }^{\left[j\right]}$; $j=\mathrm{0,1,}\cdots \mathrm{,}m-1$ and invariants $\text{tr}\left({\epsilon }_{\left[0\right]}\right)$, $\text{tr}\left({\epsilon }_{\left[i\right]}\right)$; $i=\mathrm{1,2,}\cdots \mathrm{,}n$, $\text{tr}\left({}_d{\sigma }^{\left[j\right]}\right)$; $j=\mathrm{0,1,}\cdots \mathrm{,}m-1$. Using (15) and only retaining these generators and invariants, we can write (after introducing new notation for material coefficients):

$\begin{array}{c}{}_d{\sigma }^{\left[m\right]}={{\underset{˜}{\sigma }}^0|}_{\underset{\_}{\Omega }}I+a_1{\epsilon }_{\left[0\right]}+a_2\text{tr}\left({\epsilon }_{\left[0\right]}\right)I+{\displaystyle \underset{i=1}{\overset{n}{\sum }}}{b}_i^1{\epsilon }_{[i]}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}}{b}_i^2\text{tr}\left({\epsilon }_{\left[i\right]}\right)I\\ +{\displaystyle \underset{j=0}{\overset{m-1}{\sum }}}{c}_j^1\left({}_d{\sigma }^{\left[j\right]}\right)+{\displaystyle \underset{j=0}{\overset{m-1}{\sum }}}{c}_j^2\text{tr}\left({}_d{\sigma }^{\left[j\right]}\right)I\end{array}$ (16)

We can rewrite (16) in the following form using more familiar notation for the material coefficients and neglecting the last term (as done usually in the published works).

${}_d{\sigma }^{\left[0\right]}+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\lambda }_i\left({}_d{\sigma }^{\left[i\right]}\right)={{\underset{˜}{\sigma }}^0|}_{\underset{\_}{\Omega }}I+{\displaystyle \underset{i=0}{\overset{n}{\sum }}}2{\eta }_i\left({\epsilon }_{\left[i\right]}\right)+{\displaystyle \underset{i=0}{\overset{n}{\sum }}}{\kappa }_i\text{tr}\left({\epsilon }_{\left[i\right]}\right)I$ (17)

${\lambda }_i\mathrm{,}i=\mathrm{1,2,}\cdots \mathrm{,}m$, in the spectrum of relaxation times. ${\eta }_i$ and ${\kappa }_i$ are the first and second viscosities associated with strain rates ${\epsilon }_{\left[i\right]}$ ( $i=\mathrm{1,2,}\cdots \mathrm{,}n$), where ${\eta }_0$ and ${\kappa }_0$ are similar to Lamé’s constants in linear elasticity. For heat flux $q$ the following simple heat conduction law is commonly used (see [31] [32] for a more comprehensive constitutive theory for $q$).

$q=-kg$ (18)

The simplest possible constitutive theory can be obtained from (17) by considering $n=1$ and $m=1$ (neglecting ${\sigma }_0$):

${}_d{\sigma }^{\left[0\right]}+{\lambda }_1\left({}_d{\sigma }^{\left[1\right]}\right)=2{\eta }_0\mathrm{<mo\; stretchy="false">\; (\; </mo>}{\epsilon }_{\left[0\right]}\mathrm{<mo\; stretchy="false">\; )\; </mo>}+{\kappa }_0\text{tr}\left({\epsilon }_{\left[0\right]}\right)I+2{\eta }_1\left({\epsilon }_{\left[1\right]}\right)+{\kappa }_1\text{tr}\left({\epsilon }_{\left[1\right]}\right)I$ (19)

In which:

${\epsilon }^{\left[0\right]}=\frac{1}{2}\left({\left[J\right]}^{\text{T}}\left[J\right]-\left[I\right]\right)\mathrm{,}\left[J\right]=\left[{}^{d}J\right]+\left[I\right]\mathrm{,}\left[{}^{d}J\right]=\left[\frac{\partial \left\{u\right\}}{\partial \left\{x\right\}}\right]$

with $\left[{}^{d}J\right]$ being the displacement gradient tensor and

${\epsilon }_{\left[1\right]}=\frac{D}{Dt}\left[{\epsilon }_{\left[0\right]}\right]=\frac{\partial }{\partial t}\left[{\epsilon }_{\left[0\right]}\right]=\left({\left[\stackrel{\dot{}}{J}\right]}^{\text{T}}\left[J\right]+{\left[J\right]}^{\text{T}}\left[\stackrel{\dot{}}{J}\right]\right)$

$\left[\stackrel{\dot{}}{J}\right]$ being the velocity gradient tensor in Lagrangian description.

This mathematical model ((2), (4), (18), and (19)) consists of 13 equations: BLM (3), FLT (1), constitutive theories for ${}_d{\sigma }^{\left[0\right]}$ (6) and $q$ (3) in thirteen variables: $u$ (3), $\theta$ (1), ${}_d{\sigma }^{\left[0\right]}$ (6) and $q$ (3), hence the mathematical model has closure.

This mathematical model can be used to study 3D compressible deformation physics of TVES with memory. To illustrate the shock physics of deviatoric stress and density waves more clearly, it is necessary to consider a model problem in which dispersion and boundary effects causing transmission and reflection do not contaminate the solution. For this reason, we consider 1D stress wave propagation in compressible thermoviscoelastic medium with memory. Furthermore, if we consider an insulated system, then the entropy generation is only due to viscous dissipation, which leads to extremely small temperature changes (for non-cyclic loads) that have virtually no effect on mechanical deformation. Thus, with this assumption, we can eliminate the energy Equation (4) and temperature $\theta$ as a dependent variable in the mathematical model.

As pointed earlier, in the Lagrangian description, CM is not part of the mathematical model as $\rho \left(x\mathrm{,}t\right)$ in the current configuration is deterministic using CM (1). Thus, for the 1D case, we have displacement $u_1$ and deviatoric contravariant second Piola-Kirchhoff stress ${}_d{\sigma }_{11}^{\left[0\right]}$ as the only two dependent variables in the balance of linear momenta in the $x_1$ direction and the constitutive theory for ${}_d{\sigma }_{11}^{\left[0\right]}$ defined using (19). Thermodynamic pressure is only a function of $\rho$ and is known when $\left[J\right]$ is known (Equation (7)).

For 1D finite deformation, finite strain deformation we have:

$\left[J\right]=J=\left|J\right|=\left(1+\frac{\partial u_1}{\partial x_1}\right)$ (20)

${\epsilon }_{\left[0\right]}{}_{{}_{11}}=\frac{\partial u_1}{\partial x_1}+\frac{1}{2}{\left(\frac{\partial u_1}{\partial x_1}\right)}^2$ (21)

${\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}={\epsilon }_{\left[1\right]}{}_{{}_{11}}=\frac{\partial {\left[{\epsilon }_{\left[0\right]}\right]}_{11}}{\partial t}=\frac{\partial v_1}{\partial x_1}+\frac{\partial u_1}{\partial x_1}\frac{\partial v_1}{\partial x_1};{}_d{\sigma }_{11}^{\left[1\right]}=\frac{\partial }{\partial t}\left({}_d{\sigma }_{11}^{\left[0\right]}\right)$ (22)

where

$\frac{\partial v_1}{\partial x_1}=\frac{\partial }{\partial t}\left(\frac{\partial u_1}{\partial x_1}\right)$ (23)

Using (20)-(23) and (6), the mathematical model for 1D wave propagation in a compressible TVE medium with memory can be written as:

${\rho }_0\frac{{\partial }^2{u}_1}{\partial t^2}-{\rho }_0{F}_1^b-\frac{\partial }{\partial x_1}\left[\left(1+\frac{\partial u_1}{\partial x_1}\right){}_e\sigma {}_{11}^{\left[0\right]}\right]-\frac{\partial }{\partial x_1}\left[\left(1+\frac{\partial u_1}{\partial x_1}\right){}_d\sigma {}_{11}^{\left[0\right]}\right]=0$ (24)

${}_e{\sigma }_{11}^{\left[0\right]}=\frac{p\left(\rho \right)}{1+\frac{\partial u_1}{\partial x_1}}$ (25)

${}_d{\sigma }_{11}^{\left[0\right]}+{\lambda }_1\left({}_d{\sigma }_{11}^{\left[1\right]}\right)=C_1\left(\frac{\partial u_1}{\partial x_1}+\frac{1}{2}{\left(\frac{\partial u_1}{\partial x_1}\right)}^2\right)+C_2\left(\frac{\partial v_1}{\partial x_1}+\frac{\partial u_1}{\partial x_1}\frac{\partial v_1}{\partial x_1}\right)$ (26)

and

$v_1=\frac{\partial u_1}{\partial t}\forall \left(x,t\right)\in {\Omega }_{xt}={\Omega }_x\times {\Omega }_t$ (27)

in which $C_1=2{\eta }_0+{\kappa }_0$ and $C_2=2{\eta }_1+{\kappa }_1$. $C_1$ and $C_2$ are material coefficients for elasticity and dissipation. Further details on the equilibrium contravariant second Piola-Kirchhoff stress tensor are given in the following.

In (25), we choose a simple equation of state $p\left(\rho \right)$ given in the following:

$p\left(\rho \right)=C\left(\frac{\rho }{{\rho }_0}-1\right)$ (28)

in which C is the bulk modulus. From (28), we note that increasing $\rho$ implies $\frac{\rho }{{\rho }_0}>1$, yielding positive p, confirming that (28) has correct physics.

Substituting (28) in (25) and then (25) in the third term of BLM (24), we can obtain the following (assuming C to be constant):

$\frac{\partial }{\partial x_1}\left[\left(1+\frac{\partial u_1}{\partial x_1}\right){}_e\sigma {}^{\left[0\right]}\right]=\frac{\partial p\left(\rho \right)}{\partial x_1}$ (29)

in which $p\left(\rho \right)$ can be written as follows (using CM):

$p\left(\rho \right)=C\left(\frac{\rho }{{\rho }_0}-1\right)=C\left[{\left(1+\frac{\partial u_1}{\partial x_1}\right)}^{-1}-1\right]$ (30)

We substitute (30) in (29) to obtain

$\frac{\partial }{\partial x_1}\left[{\left(1+\frac{\partial u_1}{\partial x_1}\right)}_e{\sigma }_{11}^{\left[0\right]}\right]=C\left(-\frac{1}{{\left(1+\frac{\partial u_1}{\partial x_1}\right)}^2}\frac{{\partial }^2{u}_1}{\partial x^2}\right)$ (31)

Equation (31) can be used in the balance of linear momentum (Equation (24)) to obtain the final form of the BLM. If we consider thermodynamic pressure to be positive when compressive (28) then we should change the sign of the third term in (24). We do so in (24) and then write (32). The final form of the CBL and the constitutive theory are given in the following for 1D wave propagation in a TVES medium with memory.

${\rho }_0\frac{\partial v_1}{\partial t}-{\rho }_0{F}_1^b+C\left(-\frac{1}{{\left(1+\frac{\partial u_1}{\partial x_1}\right)}^2}\frac{{\partial }^2{u}_1}{\partial x_1^2}\right)-\frac{\partial }{\partial x_1}\left[\left(1+\frac{\partial u_1}{\partial x_1}\right)\text{}{\sigma }_{11}^{\left[0\right]}\right]=0$ (32)

${\sigma }_{11}^{\left[0\right]}+{\lambda }_1\frac{\partial {\sigma }_{11}^{\left[0\right]}}{\partial t}=C_1\left(\frac{\partial u_1}{\partial x_1}+\frac{1}{2}{\left(\frac{\partial u_1}{\partial x_1}\right)}^2\right)+C_2\left(\frac{\partial v_1}{\partial x_1}+\frac{\partial u_1}{\partial x_1}\frac{\partial v_1}{\partial x_1}\right)$ (33)

$v_1=\frac{\partial u_1}{\partial t}\forall \left(x,t\right)\in {\Omega }_{xt}={\Omega }_x\times {\Omega }_t$ (34)

in which $C_1$ is the elastic coefficient and $C_2$ is the damping coefficient.

In linear TVES with rheology, due to small strain and small deformation physics, the material is nearly incompressible, hence density remains approximately constant during the evolution. If the elasticity, i.e., stiffness does not change during deformation, which is the case in linear viscoelasticity, then the wave speed remains the same for TES as well as for TVES with and without memory, regardless of the extent of dissipation and memory. However, the stresses will differ in TES, TVES without memory, and TVES with memory. We expect TVES without memory to have lower stress compared to TES due to viscous dissipation that causes amplitude decay and base elongation, and TVES with memory to have higher stresses compared to TVES without memory due to additional elasticity of the long-chain molecules.

Figure 4(a) and Figure 4(b) show the plots of ${}_d{\sigma }_{11}^{\left[0\right]}$ versus $x_1$ at $t=7\Delta t$ and $18\Delta t$ (after reflection from the boundary at $x_1=0$). We clearly observe larger peak values of ${}_d{\sigma }_{11}^{\left[0\right]}$ for TVES with memory compared to TVES without memory ( $De=0$). Since the energy imparted to the rod by velocity pulse is fixed, larger peak values of ${}_d{\sigma }_{11}^{\left[0\right]}$ must be accompanied by a reduction in the base of the wave so that energy content of the wave remains the same as for the imparted velocity pulse. We clearly note progressive peak increase and base reduction from the results in Figure 4(a) and Figure 4(b) as $De$ is increased.

Even though for infinitesimal deformation, the material is assumed incompressible

we know that $\frac{\partial u_1}{\partial x_1}\ne 0$, which must be the case to have strain and stress. Thus,

a change in density due to

$\rho \left(x\mathrm{,}t\right)=\frac{{\rho }_0\left(x\mathrm{,}t\right)}{1+\frac{\partial u_1}{\partial x_1}}$

is inevitable. Figure 5(a) and Figure 5(b) show plots of $\rho$ versus $x_1$ at $t=7\Delta t$ and $18\Delta t$. Density values follow stress ${}_d{\sigma }_{11}^{\left[0\right]}$, i.e., an increase in tensile ${}_d{\sigma }_{11}^{\left[0\right]}$ results in an reduction in $\rho$. By reducing $v^{\mathrm{<mo>\; *\; </mo>}}$, we can obviously reduce the change in density, suggesting that we are approaching the incompressible case. We clearly observe the same wave speed for TES, TVES with and without memory. As expected, an increased Deborah number results in increased peak stress value but further reduction in the base of the wave. We shall see in the study presented in section 5.2 that finite deformation, finite strain, deformation physics for compressible TVES with rheology is drastically different from what we have observed here for infinitesimal deformation (linear viscoelasticity with memory).

We apply a tensile velocity pulse of maximum amplitude $v^{\mathrm{<mo>\; *\; </mo>}}=0.1$ and of base $2\Delta t$ at the right end of the rod ( $x_1=1.0$). Figures 6(a)-(d) show plots of ${}_d{\sigma }_{11}^{\left[0\right]}$ versus $x_1$ for $t=2\Delta t\mathrm{,5}\Delta t\mathrm{,10}\Delta t$, and $15\Delta t$. We have chosen a damping coefficient of 0.0009 and Deborah numbers of 0.0004 and 0.0006 for this study. At $t=2\Delta t$, the tensile stress pulse has just entered the rod ( Figure 6(a) ). Even at this early stage in the evolution, the stress wave in TVES without rheology is moving

slower than the stress wave in TVES with rheology. In Figure 6(b) , we observe the formation of a stress front or shock front ahead the peak of the wave. The shock front is more pronounced in the case of TVES with rheology due to higher stresses (obvious from the peak values in Figure 6(a) and Figure 6(b) ). Increased stiffness due to the tensile stress field [1] [33] and reduced density result in higher wave speed in the case of TVES with memory.

In our earlier paper [3] on tensile shock physics, we showed that in the case of compressible TES, there are oscillations in the evolution of ${}_d{\sigma }_{11}^{\left[0\right]}$ behind the

shock front. We explained that their presence is due to vibrations of material points in the neighborhood of the shock front. In the absence of dissipation, there is no mechanism of dampening these except when the shock moves further from these locations, resulting in these material points receiving less energy for excitation. In reference [3] , we also showed that the addition of even very mild dissipation completely eliminates these oscillations, as the vibrational energy of these material points is now converted into entropy. From Figures 6(a)-6(d) , we note that for $De=0$, the ${}_d{\sigma }_{11}^{\left[0\right]}$ versus $x_1$ plot has no oscillations behind or ahead of the peak due to dissipation converting vibrational energy into entropy. However, when $De\ne 0$, we observe the appearance of oscillations ahead of the peak as well as behind it but more pronounced ahead of the peak. These are stresses that have not relaxed yet. We see more intense stress and density oscillations close to the shock front. These diminish as we move away from the shock front, as at these locations stresses have had time to relax some. Furthermore, for $De=0.0006$, the magnitude of the oscillation is higher compared to $De=0.0004$, as expected, because at $De=0.0006$, longer time is required for relaxation compared to $De=0.0004$.

Figure 6(c) at $t=10\Delta t$ shows the shock front at the impermeable boundary ( $x_1=0$). The shock front for $De=0$ is moving slower and hence has just reached the boundary $x_1=0$ at $t=10\Delta t$, whereas for $De\ne 0$, the shock front has already reflected and we see the reflected waves. Reflected shock fronts propagating towards $x_1=1.0$ are shown in Figure 6(d) for $t\mathrm{<mo>\; =\; </mo>15}\Delta t$. We observe the shock front is always ahead of the wave with oscillation in ${}_d{\sigma }_{11}^{\left[0\right]}$ in the vicinity of the shock front, with the highest peak immediately adjacent to the shock front. We note that a higher $De$ results in a higher magnitude of oscillations. Lower wave speed of ${}_d{\sigma }_{11}^{\left[0\right]}$ versus $x_1$ for $De=0$ compared to $De\ne 0$ can be observed. Since $De=0.0004$ and $De=0.0006$ are fairly close, we don’t observe appreciable increase in wave speed for $De=0.0006$ compared to $De=0.0004$. Figures 7(a)-(d) show the density $\rho$ versus $x_1$ graphs for $t=2\Delta t\mathrm{,5}\Delta t\mathrm{,10}\Delta t$, and $15\Delta t$. These follow exactly the same pattern as the ${}_d{\sigma }_{11}^{\left[0\right]}$ versus $x_1$ graphs in Figures 6(a)-(d) .

In this study, we consider a rectangular tensile velocity pulse of duration $4\Delta t$ with a peak value of $v^{\mathrm{<mo>\; *\; </mo>}}=0.1$, as shown in Figure 8 . For $0\le t\le \Delta t$ and $3\Delta t\le t\le 4\Delta t$, $v\left(t\right)$ is a cubic distribution with the values 0, $v^{\mathrm{<mo>\; *\; </mo>}}$ and $v^{\mathrm{<mo>\; *\; </mo>}}$, 0, and