In this paper, we present a nonlinear MPNCCT for finite deformation, finite strain deformation physics, addressing both volumetric and distortional deformation physics of compressible thermoviscoelastic solid matter with memory i.e., polymeric solids. The NCCT presented in this paper consists of modified conservation and balance laws of CCM, additionally, it uses balance of moment of moments, a balance law shown to be essential in 3M NCCT [13] [14] [21] and the constitutive theories that are derived using entropy inequality and the representation theorem. Contravariant second Piola-Kirchhoff stress and moment tensors ${\sigma }^{\left[0\right]}$ and $m^{\left[0\right]}$, in conjunction with finite deformation conjugate strain measures in the entropy inequality that are supported by deformation measures derived for nonlinear 3M NCCT, are used to derive the conservation and balance laws. Balance of moment of moments balance law [13] [14] [44] , shown to be an essential balance law, is also a part of the CBL contained in this paper. Since the details of the derivation of CBL have already been presented, only their final forms are summarized in this paper for the sake of brevity. We must consider the additive decomposition of ${\sigma }^{\left[0\right]}$ to obtain suitable and valid constitutive tensors for describing volumetric and distortional deformation physics and to avoid the constitutive theory for ${}_a{\sigma }^{\left[0\right]}$ as it is defined by BAM.

${\sigma }^{\left[0\right]}={}_s{\sigma }^{\left[0\right]}+{}_a{\sigma }^{\left[0\right]}\mathrm{<mo>\; ;\; </mo>}{}_s{\sigma }^{\left[0\right]}={}_s^e{\sigma }^{\left[0\right]}+{}_s^d{\sigma }^{\left[0\right]}$(1)

The constitutive theory for equilibrium stress tensor ${}_s^e{\sigma }^{\left[0\right]}$ describes volumetric deformation physics that remains the same as in references [57] [58] ; hence, only the final form of the equations for ${}_s^e{\sigma }^{\left[0\right]}$ is presented. Constitutive theories for ${}_s^d{\sigma }^{\left[0\right]}$ and $m^{\left[0\right]}$ address distortional physics as well as dissipation mechanism and rheology. The rate of work conjugate pair ${\sigma }^{\left[0\right]}:{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}$ in the entropy inequality suggests that elasticity must be due to ${\epsilon }_{\left[0\right]}$ and dissipation mechanism is due to ${\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}={\epsilon }_{\left[1\right]}$ i.e., Green’s strain rate of order one. In the present work, we generalized the dissipation mechanism; hence we consider dissipation to be dependent on Green’s strain rates of up to order $n$ i.e., ${\epsilon }_{\left[i\right]}\mathrm{<mo>\; ;\; </mo>}i=\mathrm{1,2,}\cdots \mathrm{,}n$. Thus, at this stage:

${}_s^d{\sigma }^{\left[0\right]}={}_s^d{\sigma }^{\left[0\right]}\left({\epsilon }_{\left[0\right]}\mathrm{,}{\epsilon }_{\left[i\right]}\mathrm{,}\theta \right)\mathrm{<mo>\; ;\; </mo>}i=\mathrm{1,2,}\cdots \mathrm{,}n$(2)

is valid. Since the material has rheology, the constitutive theory for ${}_s^d{\sigma }^{\left[0\right]}$ must be at least a first-order differential equation in time to describe rheology due to viscosity and long chain molecules. This requires that we must use ${}_s^d{\sigma }^{\left[0\right]}$ and ${}_s^d{\sigma }^{\left[1\right]}$, leading to ${}_s^d{\sigma }^{\left[1\right]}$ as a constitutive tensor with ${}_s^d{\sigma }^{\left[0\right]}$ as its argument tensor. Thus, now we have:

${}_s^d{\sigma }^{\left[1\right]}={}_s^d{\sigma }^{\left[1\right]}\left({\epsilon }_{\left[0\right]}\mathrm{,}{\epsilon }_{\left[i\right]}\mathrm{,}{}_s^d{\sigma }^{\left[0\right]}\mathrm{,}\theta \right)\mathrm{<mo>\; ;\; </mo>}i=\mathrm{1,2,}\cdots \mathrm{,}n$(3)

If we generalize and assume that rheology depends on stress rates of up to order $m$ i.e., ${}_s^d{\sigma }^{\left[j\right]}\mathrm{<mo>\; ;\; </mo>}j=\mathrm{1,2,}\cdots \mathrm{,}m$, then ${}_s^d{\sigma }^{\left[m\right]}$ is the constitutive tensor, and ${}_s^d{\sigma }^{\left[j\right]}\mathrm{<mo>\; ;\; </mo>}j=\mathrm{0,1,}\cdots \mathrm{,}m-1$ are its argument tensors. Thus, we finally have:

${}_s^d{\sigma }^{\left[m\right]}={}_s^d{\sigma }^{\left[m\right]}\left({\epsilon }_{\left[i\right]}\mathrm{,}{}_s^d{\sigma }^{\left[j\right]}\mathrm{,}\theta \right)\mathrm{<mo>\; ;\; </mo>}i=\mathrm{0,1,}\cdots \mathrm{,}n\mathrm{<mo>\; ;\; </mo>}j=\mathrm{1,2,}\cdots \mathrm{,}m-1$(4)

We use (4) and representation theorem to derive constitutive theory for ${}_s^d{\sigma }^{\left[m\right]}$. The other conjugate pair in entropy inequality, $m^{\left[0\right]}$ and ${}^{{}_c\Theta }{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}$, suggests that $m^{\left[0\right]}$ and ${}^{{}_c\Theta }{\epsilon }_{\left[0\right]}$ are work conjugate. Microconstituent rotations in a viscous medium result in secondary dissipation dependent on rotation rates of the microconstituents. If we generalize this to consider dissipation dependent on rotation rates up to order $\underset{˜}{n}$, then we can write the following for the constitutive tensor $m^{\left[0\right]}$.

$m^{\left[0\right]}=m^{\left[0\right]}\left({}^{{}_c\Theta }{\epsilon }_{\left[0\right]}\mathrm{,}{}^{{}_c\Theta }{\epsilon }_{\left[i\right]}\mathrm{,}\theta \right)\mathrm{<mo>\; ;\; </mo>}i=\mathrm{1,2,}\cdots \mathrm{,}n$(5)

The interaction of the microconstituents with long chain molecules in a viscous medium results in another mechanism of rheology. Upon cessation of an external stimulus, the presence of long chain molecules inhibits microconstituents from resuming their original state immediately. As long chain molecules relax, the microconstituents progressively approach their original undeformed position. This mechanism of rheology is referred to as micropolar rheology, with a corresponding micropolar relaxation time. This describes the relaxation phenomenon of the second Piola-Kirchhoff moment tensor. This mechanism requires that the constitutive theory for $m^{\left[0\right]}$ be a differential equation in time, which necessitates that at the very minimum, $m^{\left[0\right]}$ and $m^{\left[1\right]}$ must be considered in deriving the constitutive theory for the moment tensor. We generalize and assume that micropolar rheology depends on rate of $m^{\left[0\right]}$ up to order $\underset{˜}{m}$ i.e., $m^{\left[k\right]}\mathrm{<mo>\; ;\; </mo>}k=\mathrm{1,2,}\cdots \mathrm{,}\underset{˜}{m}$. Now, $m^{\left[\underset{˜}{m}\right]}$ can be considered constitutive tensor and $m^{\left[k\right]}\mathrm{<mo>\; ;\; </mo>}k=\mathrm{0,1,}\cdots \mathrm{,}\underset{˜}{m}-1$ as its argument tensor. Thus, finally we have:

$m^{\left[\underset{˜}{m}\right]}=m^{\left[\underset{˜}{m}\right]}\left({}^{{}_c\Theta }{\epsilon }_{\left[i\right]}\mathrm{,}{m}^{\left[j\right]}\mathrm{,}\theta \right)\mathrm{<mo>\; ;\; </mo>}i=\mathrm{0,1,}\cdots \mathrm{,}\underset{˜}{n}\mathrm{<mo>\; ;\; </mo>}j=\mathrm{0,1,}\cdots \mathrm{,}\underset{˜}{m}-1$(6)

We use (6) in conjunction with representation theorem to derive constitutive theory for $m^{\left[\underset{˜}{m}\right]}$.

The CBL and the constitutive theories for nonlinear MPNCCT expressed in terms of ${}_s^d{\sigma }^{\left[j\right]}\mathrm{<mo>\; ;\; </mo>}j=\mathrm{0,1,}\cdots \mathrm{,}m$, $m^{\left[k\right]}\mathrm{<mo>\; ;\; </mo>}k=\mathrm{0,1,}\cdots \mathrm{,}\underset{˜}{m}$, ${\epsilon }_{\left[i\right]}\mathrm{<mo>\; ;\; </mo>}i=\mathrm{0,1,}\cdots \mathrm{,}n$ and ${}^{{}_c\Theta }{\epsilon }_{\left[j\right]}\mathrm{<mo>\; ;\; </mo>}j=\mathrm{0,1,}\cdots \mathrm{,}\underset{˜}{n}$ are a system of nonlinear partial differential equations in spatial coordinates and time that have closure. The linear MPNCCT for TVES with memory is a complete subset of the nonlinear MPNCCT for compressible TVES with memory presented in this paper.

Surana et al. have presented details of the derivation of conservation and balance laws for nonlinear MPNCCT for compressible solid matter using ${\sigma }^{\mathrm{<mo>\; *\; </mo>}}$ and $m^{\mathrm{<mo>\; *\; </mo>}}$ as stress and moment tensors. These CBL in their final form were expressed in terms of ${\sigma }^{\left[0\right]}$ and $m^{\left[0\right]}$. We begin with CBL (Equations (47)-(52)) expressed in terms of ${\sigma }^{\mathrm{<mo>\; *\; </mo>}}$ and $m^{\mathrm{<mo>\; *\; </mo>}}$, which include: Conservation of mass (CM), balance of linear momenta (BLM), balance of angular momenta (BAM), first law of thermodynamics (FLT) and the second law of thermodynamics (SLT).

${\rho }_0\left(x\right)=\left|J\right|\rho \left(x\mathrm{,}t\right)$(15)

${\rho }_0\frac{Dv}{Dt}-{\rho }_0{F}^b-\nabla \cdot {\sigma }^{\mathrm{<mo>\; *\; </mo>}}=0$(16)

$\nabla \cdot m^{\mathrm{<mo>\; *\; </mo>}}+ϵ:{\sigma }^{\mathrm{<mo>\; *\; </mo>}}+{\rho }_0{m}^b=0$(17)

${ϵ}_{ijk}{m}_{ij}=0$(18)

${\rho }_0\frac{De}{Dt}+\nabla \cdot q-{\sigma }^{\mathrm{<mo>\; *\; </mo>}}:\stackrel{\dot{}}{J}-m^{\mathrm{<mo>\; *\; </mo>}}:{}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}-{}_c\stackrel{\dot{}}{\Theta }\cdot \left(\nabla \cdot m^{\mathrm{<mo>\; *\; </mo>}}+{\rho }_0{m}^b\right)=0$(19)

${\rho }_0\left(\frac{D\Phi }{Dt}+\eta \frac{D\theta }{Dt}\right)+\frac{q\cdot g}{\theta }-{\sigma }^{\mathrm{<mo>\; *\; </mo>}}:\stackrel{\dot{}}{J}-m^{\mathrm{<mo>\; *\; </mo>}}:{}^{{}_c\Theta }\stackrel{\dot{}}{J}-{}_c\stackrel{\dot{}}{\Theta }\cdot \left(\nabla \cdot m^{\mathrm{<mo>\; *\; </mo>}}+{\rho }_0{m}^b\right)\le 0$(20)

This mathematical model contains: $u\left(3\right)\mathrm{,}\sigma \left(9\right)\mathrm{,}m\left(6\right)\mathrm{,}q\left(3\right)\mathrm{,}\theta \left(1\right)$, a total of 22 dependent variables, but has only balance of linear momenta (3), balance of angular momenta (3), first law of thermodynamics (1), seven partial differential equations. Thus, additional 15 equations are needed for closure. These are provided by the constitutive theories for stress tensor (6), moment tensor (6) and heat vector (3).

Conjugate pairs $\frac{q\cdot g}{\theta },{\sigma }^{\mathrm{<mo>\; *\; </mo>}}:\stackrel{\dot{}}{J}$ and $m^{\mathrm{<mo>\; *\; </mo>}}:{}^{{}_c\Theta }\stackrel{\dot{}}{J}$ in conjunction with axioms of constitutive theory [57] [58] suggest the choice of $q\mathrm{,}{\sigma }^{\mathrm{<mo>\; *\; </mo>}}$ and $m^{\mathrm{<mo>\; *\; </mo>}}$ as constitutive tensors with $g\mathrm{,}J$ and ${}^{{}_c\Theta }J$ as their argument tensors in addition to temperature $\theta$. ${\sigma }^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{m}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}J$ and ${}^{{}_c\Theta }J$ are all nonsymmetric tensors and $J$ is not objective. Thus, the following do not hold, i.e.:

${\sigma }^{\mathrm{<mo>\; *\; </mo>}}\ne {\sigma }^{\mathrm{<mo>\; *\; </mo>}}\left(J\mathrm{,}\theta \right)$(21)

$m^{\mathrm{<mo>\; *\; </mo>}}\ne m^{\mathrm{<mo>\; *\; </mo>}}\left({}^{{}_c\Theta }J\mathrm{,}\theta \right)$(22)

but

$q=q\left(g\mathrm{,}\theta \right)$(23)

is a valid choice. Intentionally, in the derivation of energy equation and entropy inequality, we have used ${\sigma }^{\mathrm{<mo>\; *\; </mo>}}$ and $m^{\mathrm{<mo>\; *\; </mo>}}$ for simplicity. At this stage, we transform these to ${\sigma }^{\left[0\right]}$ and $m^{\left[0\right]}$.

From BAM, we have:

$\nabla \cdot m^{\mathrm{<mo>\; *\; </mo>}}+{\rho }_0{m}^b=-ϵ:{\sigma }^{\mathrm{<mo>\; *\; </mo>}}$(24)

Using (24), the last term in (20) can be written as:

${}_c\stackrel{\dot{}}{\Theta }\cdot \left(\nabla \cdot m^{\mathrm{<mo>\; *\; </mo>}}+{\rho }_0{m}^b\right)={-}_c\stackrel{\dot{}}{\Theta }\cdot \left(ϵ:{\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)$(25)

A simple calculation shows that

${}_c\stackrel{\dot{}}{\Theta }\cdot \left(ϵ:{\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)={}_a{\sigma }^{\mathrm{<mo>\; *\; </mo>}}:\stackrel{\dot{}}{J}={\sigma }^{\mathrm{<mo>\; *\; </mo>}}:{}_a\stackrel{\dot{}}{J}$(26)

Therefore,

${}_c\stackrel{\dot{}}{\Theta }\cdot \left(\nabla \cdot m^{\mathrm{<mo>\; *\; </mo>}}+{\rho }_0{m}^b\right)=-{\sigma }^{\mathrm{<mo>\; *\; </mo>}}:{}_a\stackrel{\dot{}}{J}$(27)

Using (27), the entropy inequality (20) can be written as:

${\rho }_0\left(\frac{D\theta }{Dt}+\eta \frac{D\theta }{Dt}\right)+\frac{q\cdot g}{\theta }-{\sigma }^{\mathrm{<mo>\; *\; </mo>}}:\stackrel{\dot{}}{J}-m^{\mathrm{<mo>\; *\; </mo>}}:{}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}+{\sigma }^{\mathrm{<mo>\; *\; </mo>}}:{}_a\stackrel{\dot{}}{J}\le 0$(28)

Consider ${\sigma }^{\mathrm{<mo>\; *\; </mo>}}:\stackrel{\dot{}}{J}$ term in (28)

$\begin{array}{c}{\sigma }^{\mathrm{<mo>\; *\; </mo>}}:\stackrel{\dot{}}{J}={\left({\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\text{T}}:{\stackrel{\dot{}}{J}}^{\text{T}}={\left({\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\text{T}}\cdot \stackrel{\dot{}}{J}\\ ={\left({\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\text{T}}:\left({}_s\stackrel{\dot{}}{J}+{}_a\stackrel{\dot{}}{J}\right)\\ ={\left({\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\text{T}}:{}_s\stackrel{\dot{}}{J}+{\left({\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\text{T}}:{}_a\stackrel{\dot{}}{J}\\ ={\left({\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\text{T}}:{}_s\stackrel{\dot{}}{J}+{\sigma }^{\mathrm{<mo>\; *\; </mo>}}:{}_a\stackrel{\dot{}}{J}\end{array}$(29)

Consider ${\left({\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\text{T}}:{}_s\stackrel{\dot{}}{J}$ term

$\begin{array}{l}{\left({\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\text{T}}:{}_s\stackrel{\dot{}}{J}={\left({\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\text{T}}\cdot \frac{1}{2}\left(\bar{L}\cdot J+J^{\text{T}}\cdot {\bar{L}}^{\text{T}}\right)\\ =\frac{1}{2}J\cdot {\left({\sigma }^{\left[0\right]}\right)}^{\text{T}}:\left(\left(\bar{D}+\bar{W}\right)\cdot J+J^{\text{T}}\cdot \left(\bar{D}-\bar{W}\right)\right)\\ =\frac{1}{2}J\cdot \left({}_s{\sigma }^{\left[0\right]}-{}_a{\sigma }^{\left[0\right]}\right):\left(\left(\bar{D}+\bar{W}\right)\cdot J+J^{\text{T}}\cdot \left(\bar{D}-\bar{W}\right)\right)\\ =\frac{1}{2}J\cdot \left({}_s{\sigma }^{\left[0\right]}\right):\bar{D}\cdot J+\frac{1}{2}J\cdot \left({}_s{\sigma }^{\left[0\right]}\right):J^{\text{T}}\cdot \bar{D}\\ +\frac{1}{2}J\cdot \left({}_s{\sigma }^{\left[0\right]}\right):\bar{W}\cdot J-\frac{1}{2}J\cdot \left({}_s{\sigma }^{\left[0\right]}\right):J^{\text{T}}\cdot \bar{W}\\ -\frac{1}{2}J\cdot \left({}_a{\sigma }^{\left[0\right]}\right):\bar{D}\cdot J-\frac{1}{2}J\cdot \left({}_a{\sigma }^{\left[0\right]}\right):J^{\text{T}}\cdot \bar{D}\\ -\frac{1}{2}J\cdot \left({}_a{\sigma }^{\left[0\right]}\right):\bar{W}\cdot J+\frac{1}{2}J\cdot \left({}_a{\sigma }^{\left[0\right]}\right):J^{\text{T}}\cdot \bar{W}\end{array}$(30)

We consider each term in (30)

$\frac{1}{2}\left(J\cdot \left({}_s{\sigma }^{\left[0\right]}\right)\right):\left(\bar{D}\cdot J\right)=\frac{1}{2}{}_s{\sigma }^{\left[0\right]}:\left(J^{\text{T}}\cdot \bar{D}\cdot J\right)=\frac{1}{2}{}_s{\sigma }^{\left[0\right]}:{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}$(31)

$\frac{1}{2}J\cdot \left({}_s{\sigma }^{\left[0\right]}\right):J\cdot \bar{D}=\frac{1}{2}{}_s{\sigma }^{\left[0\right]}:\left(J^{\text{T}}\cdot \bar{D}\cdot J\right)=\frac{1}{2}{}_s{\sigma }^{\left[0\right]}:{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}$(32)

$\frac{1}{2}\left(J\cdot \left({}_s{\sigma }^{\left[0\right]}\right)\right):\bar{W}\cdot J=\frac{1}{2}{}_s{\sigma }^{\left[0\right]}\cdot J:\bar{W}\cdot J=\frac{1}{2}{}_s{\sigma }^{\left[0\right]}:\left(J^{\text{T}}\cdot \bar{W}\cdot J\right)=0$(33)

$\frac{1}{2}\left(J\cdot \left({}_s{\sigma }^{\left[0\right]}\right)\right):J^{\text{T}}\cdot \bar{W}=\frac{1}{2}\left({}_s{\sigma }^{\left[0\right]}\right):\left(J^{\text{T}}\cdot \bar{W}\cdot J\right)=0$(34)

$-\frac{1}{2}\left(J\cdot \left({}_a{\sigma }^{\left[0\right]}\right)\right):\bar{D}\cdot J=-\frac{1}{2}{}_a{\sigma }^{\left[0\right]}\cdot J^{\text{T}}:\bar{D}\cdot J=-\frac{1}{2}{}_a{\sigma }^{\left[0\right]}:\left(J^{\text{T}}\cdot \bar{D}\cdot J\right)=0$(35)

$-\frac{1}{2}J\cdot {}_s{\sigma }^{\left[0\right]}:J^{\text{T}}\cdot \bar{D}=\frac{1}{2}\left({}_s{\sigma }^{\left[0\right]}\right):\left(J^{\text{T}}\cdot \bar{D}\cdot J\right)=0$(36)

$-\frac{1}{2}J\cdot \left({}_a{\sigma }^{\left[0\right]}\right):\left(\bar{W}\cdot J\right)=-\frac{1}{2}J\cdot {}_a{\sigma }^{\left[0\right]}:\left(J^{\text{T}}\cdot {\bar{W}}^{\text{T}}\right)=-\frac{1}{2}{}_a{\sigma }^{\left[0\right]}:\left(J^{\text{T}}\cdot {\bar{W}}^{\text{T}}\cdot J\right)$(37)

$\begin{array}{l}\frac{1}{2}J\cdot {}_a{\sigma }^{\left[0\right]}:\left(J^{\text{T}}\cdot \bar{W}\right)=\frac{1}{2}{}_a{\sigma }^{\left[0\right]}:\left(J^{\text{T}}\cdot \bar{W}\cdot J\right)=\frac{1}{2}{}_a{\sigma }^{\left[0\right]}:{\left(J^{\text{T}}\cdot \bar{W}\cdot J\right)}^{\text{T}}\\ =\frac{1}{2}{}_a{\sigma }^{\left[0\right]}:\left(J^{\text{T}}\cdot {\bar{W}}^{\text{T}}\cdot J\right)\end{array}$(38)

Substituting (31)-(38) in (30), we obtain

${\left({\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\text{T}}:{}_s\stackrel{\dot{}}{J}={}_s{\sigma }^{\left[0\right]}:{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}$(39)

Substituting from (39) into (29) we obtain

${\sigma }^{\mathrm{<mo>\; *\; </mo>}}:\stackrel{\dot{}}{J}={}_s{\sigma }^{\left[0\right]}:{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}+{\sigma }^{\mathrm{<mo>\; *\; </mo>}}:{}_a\stackrel{\dot{}}{J}$(40)

Substituting from (40) into (28), and noting that ${\sigma }^{\mathrm{<mo>\; *\; </mo>}}:{}_a\stackrel{\dot{}}{J}$ term cancel, we obtain the following form of entropy inequality:

${\rho }_0\left(\frac{D\Phi }{Dt}+\eta \frac{D\theta }{Dt}\right)+\frac{q\cdot g}{\theta }-{}_s{\sigma }^{\left[0\right]}:{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}-m^{\mathrm{<mo>\; *\; </mo>}}:{}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\le 0$(41)

We substitute

${\left(m^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\text{T}}=J\cdot {\left(m^{\left[0\right]}\right)}^{\text{T}}$(42)

in the last term of (41).

$m^{\mathrm{<mo>\; *\; </mo>}}:{}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}=m^{\left[0\right]}\cdot J^{\text{T}}:{}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}=m^{\left[0\right]}:\left(J^{\text{T}}\cdot \left({}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\right)\right)$(43)

Substituting (43) in (41), we obtain the final form of entropy inequality

${\rho }_0\left(\frac{D\Phi }{Dt}+\eta \frac{D\theta }{Dt}\right)+\frac{q\cdot g}{\theta }-{}_s{\sigma }^{\left[0\right]}:{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}-m^{\left[0\right]}:\left(J^{\text{T}}\cdot \left({}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\right)\right)\le 0$(44)

Thus, ${\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}$ and $J^{\text{T}}\cdot \left({}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\right)$ are rate of work conjugate to ${}_s{\sigma }^{\left[0\right]}$ and $m^{\left[0\right]}$ (symmetric). We should confirm the validity of the rate terms in (44) by comparing them with the nonlinear deformation measures derived.

The nonlinear deformation measures for MPNCCT are:

${\left[J\right]}^{\text{T}}\left[J\right]-\left[I\right]\mathrm{<mo>\; ;\; </mo>}\left[{}_a{J}^{\left(\alpha \right)}\right]\mathrm{<mo>\; ;\; </mo>}2{\left[J\right]}^{\text{T}}\left[\frac{{\partial }_a^d\left[J^{\left(\alpha \right)}\right]}{\partial x}\right]$(45)

or in the notations of MPNCCT based on classical rotations ${}_c\Theta$, we have

${\left[J\right]}^{\text{T}}\left[J\right]-\left[I\right]\mathrm{<mo>\; ;\; </mo>}\left[{}_{a}J\right]\mathrm{<mo>\; ;\; </mo>}2{\left[J\right]}^{\text{T}}\left[{}^{c^{\Theta }}\text{}\text{}J\right]$(46)

Based on (46) and rates used in (41), the choice of $q\mathrm{,}{\sigma }^{\left[0\right]}$ and $m^{\left[0\right]}$ as constitutive tensors is justified. Last term in (41) requires further considerations.

$m^{\left[0\right]}:\left(J^{\text{T}}\cdot \left({}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\right)\right)=m^{\left[0\right]}:\frac{1}{2}\left(J^{\text{T}}\cdot \left({}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\right)+{\left({}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\right)}^{\text{T}}\cdot J\right)$(47)

Thus, now we can write entropy inequality (41) as:

${\rho }_0\left(\frac{D\Phi }{Dt}+\eta \frac{D\theta }{Dt}\right)+\frac{q\cdot g}{\theta }-{}_s{\sigma }^{\left[0\right]}:{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}-m^{\left[0\right]}:\frac{1}{2}\left(J^{\text{T}}\cdot \left({}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\right)+{\left({}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\right)}^{\text{T}}\cdot J\right)\le 0$(48)

Similar changes can also be made in the energy equation.

If we use

${\left({\sigma }^{\mathrm{<mo>\; *\; </mo>}}\right)}^{\text{T}}=J\cdot {\left({\sigma }^{\left[0\right]}\right)}^{\text{T}}$(49)

and define

${}^{{}_c\Theta }{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}={}^{{}_c\Theta }{\epsilon }_{\mathrm{<mo\; stretchy="false">\; [\; </mo>1\; <mo\; stretchy="false">\; ]\; </mo>}}=\frac{1}{2}\left(J^{\text{T}}\cdot \left({}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\right)+{\left({}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\right)}^{\text{T}}\cdot J\right)$(50)

Thus, the conservation and the balance laws (15)-(20) can be written as:

${\rho }_0\left(x\right)=\left|J\right|{\rho }_0\left(x\mathrm{,}t\right)$(51)

${\rho }_0\frac{Dv}{Dt}-{\rho }_0{F}^b-\nabla \cdot \left({\sigma }^{\left[0\right]}\cdot J^{\text{T}}\right)=0$(52)

$\nabla \cdot \left(m^{\left[0\right]}\cdot J^{\text{T}}\right)+ϵ:\left({\sigma }^{\left[0\right]}\cdot J^{\text{T}}\right)+{\rho }_0{m}^b=0$(53)

${ϵ}_{ijk}{m}_{ij}^{\left(0\right)}=0$(54)

${\rho }_0\frac{De}{Dt}+\nabla \cdot q-{}_s{\sigma }^{\left[0\right]}:{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}-m^{\left[0\right]}:{}^{{}_c\Theta }{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}=0$(55)

${\rho }_0\left(\frac{D\Phi }{Dt}+\eta \frac{D\theta }{Dt}\right)+\frac{q\cdot g}{\theta }-{}_s{\sigma }^{\left[0\right]}:{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}-m^{\left[0\right]}:\left({}^{{}_c\Theta }{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}\right)\le 0$(56)

in which

${}^{{}_c\Theta }{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}={}^{{}_c\Theta }{\epsilon }_{\left[1\right]}=\frac{1}{2}\left(J^{\text{T}}\cdot \left({}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\right)+{\left({}^{{}_c\Theta }\text{}\stackrel{\dot{}}{J}\right)}^{\text{T}}\cdot J\right)$(57)

Remarks

(1) The linear MPNCCT for infinitesimal deformation is a complete subset of the nonlinear MPNCCT described by CBL (51)-(57). In this case:

$\begin{array}{l}J\approx I\mathrm{<mo>\; ;\; </mo>}{\sigma }^{\left[0\right]}={\sigma }^{\left(0\right)}\mathrm{<mo>\; ;\; </mo>}{m}^{\left[0\right]}=m^{\left(0\right)}\\ {\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}=\stackrel{\dot{}}{\epsilon }\mathrm{<mo>\; ;\; </mo>}{}^{{}_c\Theta }{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}={}^{{}_c\Theta }\stackrel{\dot{}}{\epsilon }\end{array}$(58)

in which $\epsilon$ is the linear strain measure and ${}^{{}_c\Theta }J$ is the symmetric part of the classical rotation gradient tensor. Thus, (51)-(57) reduce to linear MPNCCT for infinitesimal deformation.

(2) When ${}_c\Theta$ is not considered, the CBL (51)-(57) reduce to finite deformation, finite strain CCM physics.

(3) When ${}_c\Theta$ is a free field (i.e., ${}_c\Theta$ is not described) and when the deformation is infinitesimal, then $\left[J\right]=\left[I\right]$ and $\left|J\right|=1$, and the CBL presented here automatically reduce to CBL for small deformation, small strain infinitesimal physics.

(4) All conjugate pairs in the entropy inequality define constitutive tensors and their argument tensors that are supported by the representation theorem, hence would yield thermodynamically and mathematically consistent constitutive theories.

Additive decomposition of ${}_s{\sigma }^{\left[0\right]}$ into ${}_s^e{\sigma }^{\left[0\right]}$ equilibrium and ${}_s^d{\sigma }^{\left[0\right]}$, deviatoric tensors is necessary to address volumetric deformation (constitutive theory for ${}_s^e{\sigma }^{\left[0\right]}$) distortional deformation (constitutive theory for ${}_s^d{\sigma }^{\left[0\right]}$) physics that are mutually exclusive.

${}_s{\sigma }^{\left[0\right]}={}_s^e{\sigma }^{\left[0\right]}+{}_s^d{\sigma }^{\left[0\right]}$(59)

Substituting (59) into entropy inequality (56)

${\rho }_0\left(\frac{D\Phi }{Dt}+\eta \frac{D\theta }{Dt}\right)+\frac{q\cdot g}{\theta }-{}_s^e{\sigma }^{\left[0\right]}:{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}-{}_s^d{\sigma }^{\left[0\right]}:{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}-m^{\left[0\right]}:\left({}^{{}_c\Theta }{\stackrel{\dot{}}{\epsilon }}_{\left[0\right]}\right)\le 0$(60)

Based on (60), choice of $q\mathrm{,}{}_s^e{\sigma }^{\left[0\right]}\mathrm{,}{}_s^d{\sigma }^{\left[0\right]}$ and $m^{\left[0\right]}$ as constitutive tensor is supported by the axioms of constitutive theory [31] [32] .