ABSTRACT

This paper considers conservation and balance laws and the constitutive theories for non-classical viscous fluent continua without memory, in which internal rotation rates due to the velocity gradient tensor are incorporated in the thermodynamic framework. The constitutive theories for the deviatoric part of the symmetric Cauchy stress tensor and the Cauchy moment tensor are derived based on integrity. The constitutive theories for the Cauchy moment tensor are considered when the balance of moments of moments 1) is not a balance law and 2) is a balance law. The constitutive theory for heat vector based on integrity is also considered. Restrictions on the material coefficients in the constitutive theories for the stress tensor, moment tensor, and heat vector are established using the conditions resulting from the entropy inequality, keeping in mind that the constitutive theories derived here based on integrity are in fact nonlinear constitutive theories. It is shown that in the case of the simplest linear constitutive theory for stress tensor used predominantly for compressible viscous fluids, Stokes’ hypothesis or Stokes’ assumption has no thermodynamic basis, hence may be viewed incorrect. Thermodynamically consistent derivations of the restrictions on various material coefficients are presented for non-classical as well as classical theories that are applicable to nonlinear constitutive theories, which are inevitable if the constitutive theories are derived based on integrity.

Keywords:

Non-Classical Continua, Polar Continua, Eulerian Description, Viscous Fluids, Material Coefficients

1. Introduction, Literature Review, and Scope of Work

In fluent continua, velocities are observable quantities and the deformation physics is completely contained in the velocities ( $\bar{v}$ ) and the velocity gradient tensor ( $\bar{L}$ ). Thus, the velocities and the velocity gradient tensor in their entirety must form the basis for the thermodynamic framework that describes the behavior of fluent continua. The velocity gradient tensor can be decomposed into symmetric ( $\bar{D}$ ) and antisymmetric tensors ( $\bar{W}$ ). The symmetric part represents strain rates and the antisymmetric part contains rotation rates. Alternatively, the polar decomposition of $\bar{L}$ yields right or left stretch rate tensors ( ${}^t\bar{S}{}_r$ or ${}^t\bar{S}{}_l$ ) and the rotation rate tensor ( ${}^t\bar{R}$ ). The tensors $\bar{D}$ , ${}^t\bar{S}{}_r$ , and ${}^t\bar{S}{}_l$ contain the same physics in different forms related to strain rates. Likewise, $\bar{W}$ contains rotation rates whereas ${}^t\bar{R}$ is a rotation rate matrix. The same physics of rotation rates is contained in both but in different forms. The classical continuum theories for fluent continua are derived using only $\bar{v}$ and $\bar{D}$ ; $\bar{W}$ or ${}^t\bar{R}$ are not considered at all in the derivation of the conservation and balance laws and the constitutive theories.

We note that $\bar{W}$ contains rotation rates that are completely defined by the antisymmetric part of the velocity gradient tensor. We refer to these as internal rotation rates (as these arise due to $\bar{L}$ ), and the associated continuum theory as non-classical continuum theory with internal rotation rates or simply non-classical internal polar continuum theory. Recent papers by Surana, et al. [1] - [10] contain details of the derivations of such non-classical continuum theories and associated constitutive theories for solid and fluent continua. Prior to these works, there have been many published works under the title couple stress theories [11] - [19] , particularly in context with solid continua that contain somewhat similar derivations, but use completely different motivation and rationale. Some concepts similar to those used in References [1] - [10] can also be traced in various different forms in the works of Eringen [20] - [28] related to micro-theories of various types.

In the present work, we consider non-classical continuum theories for fluent continua in which both $\bar{D}$ and $\bar{W}$ , that is, $\bar{L}$ in its entirety, are incorporated in deriving the conservation and balance laws and the constitutive theories for thermoviscous compressible fluids without memory. Constitutive theories are derived using the conditions resulting from entropy inequality and the representation theorem (or theory of generators and invariants). Such constitutive theories, when based on integrity, are nonlinear in most instances. The investigation presented in this paper establishes necessary restrictions on the material coefficients in the constitutive theories that ensure that the resulting constitutive theories satisfy the conditions resulting from the entropy inequality. The work presented in this paper is compared and contrasted with the published works.

In the thermodynamic framework for the non-classical continuum theory used here, we have additional physics of rotation rates due to $\bar{W}$ which, when resisted, result in conjugate moments that lead to Cauchy moment tensor (through Cauchy principle). This physics is absent in the classical continuum theory for fluent continua. Thus, it is natural to ask “are the conservation and balance laws of classical continuum theories sufficient when this new physics of rotation rates is present to ensure equilibrium of the deforming matter?’’ Surana, et al. [1] - [10] [29] [30] and Yang, et al. [31] have shown that in the case of non-classical solid and fluent continua, an additional balance law is required, the balance of moments of moments to ensure equilibrium of the deforming matter. The constitutive theories for the moment tensor are affected by the absence or the presence of this balance law. In the work presented here, we examine the constitutive theories in the presence as well as absence of this balance law for establishing restrictions on the material coefficients.

2. Notations and Definitions of Bases

The notations used in this paper conform to Reference [32] but are different than conventional notations in continuum mechanics writings. These new notations are used to provide more clarity and transparency. $x$ , $A$ , $V$ , $\partial A$ , $\partial V$ refer to material point coordinates (in a fixed Cartesian frame), area, volume, boundary of $A$ , and surface bounding $V$ , all in the reference or undeformed configuration, whereas $\bar{x}$ , $\bar{A}$ , $\bar{V}$ , $\partial \bar{A}$ , $\partial \bar{V}$ are their counterparts in the current configuration. $Q=Q\left(x\mathrm{,}t\right)$ and $\bar{Q}=\bar{Q}\left(\bar{x}\mathrm{,}t\right)$ are Lagrangian and Eulerian descriptions of a quantity $Q$ at a material point $x$ in the reference configuration with its corresponding location $\bar{x}$ in the current configuration.

A tetrahedron in the undeformed configuration (volume $V$ ) with its oblique plane constituting a part of surface $\partial V$ bounding $V$ deforms and rotates in the current configuration. Equilibrium considerations associated with conservation and balance laws require measurement of stress, strain rates, etc. associated with the deformed tetrahedron. Two obvious choices are covariant and contravariant bases. If the edges of the tetrahedron in the undeformed configuration represent material lines, then upon finite deformation the material lines will become curvilinear. The tangent vectors to these deformed lines at a material point (a point from which the material lines emanate) forming the edges of the deformed tetrahedron are covariant base vectors ( ${\tilde{g}}_i$ ). The vectors orthogonal to the faces of the deformed tetrahedron (formed by the covariant base vectors) are called contravariant base vectors ( ${\tilde{g}}^i$ ). ${\tilde{g}}_i$ and ${\tilde{g}}^i$ form nonorthogonal covariant and contravariant bases that are reciprocal to each other. Since the covariant base vectors are tangent to the deformed material lines, the convected time derivative of the covariant strain tensor is a physical measure of the strain rate tensor. Likewise the contravariant directions normal to the faces of the tetrahedron is a natural way to define stress tensor. Thus, we define ${\bar{\sigma }}^{\left(0\right)}$ as contravariant Cauchy stress tensor, ${\bar{\gamma }}_{\left(1\right)}$ as the first convected time derivative of the Green’s strain tensor. These measures are physical as these are related to the faces and edges of the true deformed tetrahedron. Since ${\tilde{g}}_i$ and ${\tilde{g}}^i$ form reciprocal bases, we could also use covariant directions for stress measure and contravarian directions for strain rate measures, i.e., ${\bar{\sigma }}_{\left(0\right)}$ and ${\bar{\gamma }}^{\left(1\right)}$ , covariant Cauchy stress tensor and contravariant strain rate tensor. Mathematically this is justified, however in terms of physics, this description requires ${\tilde{g}}_i$ to be normal to the tetrahedron faces and ${\tilde{g}}^i$ to be the material line tangent vectors. In other words, this description requires a new configuration of the actual deformed tetrahedron that is non-physical. When strain rates are small, the two measures are the same as the deformed and undeformed configurations are virtually the same.

3. Internal Rotation Rates and Their Gradients

Velocities ( $\bar{v}$ ) and velocity gradients ( $L_{ij}=\partial {\bar{v}}_i\partial {\bar{x}}_j$ ) are fundamental measures of deformation physics in fluent continua, hence these in their entirety must form the basis for a complete thermodynamic framework. Polar decomposition of the changing velocity gradient tensor in the deforming fluent continua into stretch rates and pure rotation rates shows that a location and its neighboring locations can experience different rotation rates during deformation. Alternatively, we can also consider decomposition of the velocity gradient tensor into symmetric and antisymmetric tensors. The symmetric tensor is a measure of strain rates whereas the antisymmetric tensor is a measure of pure rotation rates. The measures of internal rotation rates due to deformation in the two approaches describe the same physics but in different forms. Polar decomposition gives rotation rate matrix and not the rotation angle rates whereas the antisymmetric part of the velocity gradient tensor yields rotation angle rates that are explicitly defined in terms of velocity gradients.

If the varying internal rotation rates between the neighboring locations are resisted by the fluent continua, then there must exist conjugate internal moments corresponding to these. The internal rotation rates and the conjugate moments can result in additional energy storage and/or dissipation as well as memory. Since this physics of internal rotation rates arises due to $\bar{L}$ , it exists in all deforming isotropic, homogeneous fluent continua. Incorporating entirety of $\left[\bar{L}\right]$ in the conservation and balance laws implies that we incorporate the additional physics due to internal rotation rates in the existing thermodynamic framework as the physics due to the symmetric part of velocity gradient tensor is already present in it. The internal rotation rates can be visualized as the rotation rates about the axes of a triad located at a material point (or location) whose axes are parallel to the fixed Cartesian x-frame. We present details in the following. The velocity gradient tensor $\left[\bar{L}\right]$ can be decomposed into pure rotation rate tensor $\left[{}^t\bar{R}\right]$ and the right and left stretch rate tensors $\left[{}^t\bar{S}{}_r\right]$ and $\left[{}^t\bar{S}{}_l\right]$ . $\left[{}^t\bar{R}\right]$ is orthogonal and $\left[{}^t\bar{S}{}_r\right]$ and $\left[{}^t\bar{S}{}_l\right]$ are symmetric and positive-definite.

$\left[\bar{L}\right]=\left[{}^t\bar{R}\right]\left[{}^t\bar{S}{}_r\right]=\left[{}^t\bar{S}{}_l\right]\left[{}^t\bar{R}\right]$ (1)

Let ( ${}^t\lambda {}_i\mathrm{,}{\left\{\varphi \right\}}_i$ ); $i=1,2,3$ be the eigenpairs of ${\left[\bar{L}\right]}^T\left[\bar{L}\right]$ in which ${\left\{\varphi \right\}}_i^T{\left\{\varphi \right\}}_j={\delta }_{ij}$ , then

${\left[\bar{L}\right]}^T\left[\bar{L}\right]=\left[\bar{\Phi }\right]\left[{}^t\bar{\lambda }\right]{\left[\bar{\Phi }\right]}^T={\left[{}^{t}S{}_r\right]}^2$ (2)

The columns of $\left[\bar{\Phi }\right]$ are eigenvectors ${\left\{\varphi \right\}}_i$ and $\left[{}^t\bar{\lambda }\right]$ is a diagonal matrix of the eigenvalues ${}^t\lambda {}_i$ ; $i=1,2,3$ . If we choose

$\left[{}^t\bar{S}{}_r\right]=\left[\bar{\Phi }\right]\left[\sqrt{{}^t\bar{\lambda }}\right]{\left[\bar{\Phi }\right]}^T$ (3)

then (2) holds, hence definition of $\left[{}^t\bar{S}{}_r\right]$ in (3) is valid. $\left[{}^t\bar{R}\right]$ can now be defined using (1).

$\left[{}^t\bar{R}\right]=\left[\bar{L}\right]{\left[{}^t\bar{S}{}_r\right]}^{-1}$ (4)

Furthermore, using

$\left[\bar{L}\right]{\left[\bar{L}\right]}^T={\left[{}^t\bar{S}{}_l\right]}^2$ (5)

and following a similar procedure we can establish

$\left[{}^t\bar{S}{}_l\right]=\left[\bar{\Phi }\right]\left[\sqrt{{}^t\bar{\lambda }}\right]{\left[\bar{\Phi }\right]}^T$ (6)

$={\left[{}^t\bar{S}{}_l\right]}^{-1}\left[\bar{L}\right]$ (7)

$\left[{}^t\bar{R}\right]$ defined by (4) and (7) is unique. We note that in this approach $\left[{}^t\bar{R}\right]$ is a rotation rate transformation matrix, hence does not contain rotation angle rates. Alternatively, we can consider decomposition of $\left[\bar{L}\right]$ into symmetric ( $\left[\bar{D}\right]$ ) and antisymmetric ( $\left[\bar{W}\right]$ ) tensors.

$\left[\bar{L}\right]=\left[\frac{\partial \left\{\bar{v}\right\}}{\partial \left\{\bar{x}\right\}}\right]=\left[\bar{D}\right]+\left[\bar{W}\right]$ (8)

$\left[\bar{D}\right]=\frac{1}{2}\left(\left[\bar{L}\right]+{\left[\bar{L}\right]}^T\right);\left[\bar{W}\right]=\frac{1}{2}\left(\left[\bar{L}\right]-{\left[\bar{L}\right]}^T\right)$ (9)

or

${\bar{D}}_{ij}=\frac{1}{2}\left({\bar{v}}_{i,j}+{\bar{v}}_{j,i}\right);{\bar{W}}_{ij}=\frac{1}{2}\left({\bar{v}}_{i,j}-{\bar{v}}_{j,i}\right)$ (10)

Expanded form of $\left[\bar{W}\right]$ can be written as

$\left[\bar{W}\right]=\left[\begin{array}{ccc}0& {}_i{}^t\bar{\Theta }{}_{x3}& -{}_i{}^t\bar{\Theta }{}_{x2}\\ -{}_i{}^t\bar{\Theta }{}_{x3}& 0& {}_i{}^t\bar{\Theta }{}_{x1}\\ {}_i{}^t\bar{\Theta }{}_{x2}& -{}_i{}^t\bar{\Theta }{}_{x1}& 0\end{array}\right]$ (11)

$\begin{array}{l}{}_i{}^t\bar{\Theta }{}_{x1}=\frac{1}{2}\left(\frac{\partial {\bar{v}}_2}{\partial {\bar{x}}_3}-\frac{\partial {\bar{v}}_3}{\partial {\bar{x}}_2}\right);{}_i{}^t\bar{\Theta }{}_{x2}=\frac{1}{2}\left(\frac{\partial {\bar{v}}_3}{\partial {\bar{x}}_1}-\frac{\partial {\bar{v}}_1}{\partial {\bar{x}}_3}\right);\\ {}_i{}^t\bar{\Theta }{}_{x3}=\frac{1}{2}\left(\frac{\partial {\bar{v}}_1}{\partial {\bar{x}}_2}-\frac{\partial {\bar{v}}_2}{\partial {\bar{x}}_1}\right)\end{array}$ (12)

Alternatively, (12) can be derived as

$\bar{\nabla }\times \bar{v}=e_i\times e_j\frac{\partial {\bar{v}}_j}{\partial {\bar{x}}_i}={\epsilon }_{ijk}{e}_k\frac{\partial {\bar{v}}_j}{\partial {\bar{x}}_i}$ (13)

$\bar{\nabla }\times \bar{v}=e_1\left(\frac{\partial {\bar{v}}_3}{\partial {\bar{x}}_2}-\frac{\partial {\bar{v}}_2}{\partial {\bar{x}}_3}\right)+e_2\left(\frac{\partial {\bar{v}}_1}{\partial {\bar{x}}_3}-\frac{\partial {\bar{v}}_3}{\partial {\bar{x}}_1}\right)+e_3\left(\frac{\partial {\bar{v}}_2}{\partial {\bar{x}}_1}-\frac{\partial {\bar{v}}_1}{\partial {\bar{x}}_2}\right)$ (14)

or

$\bar{\nabla }\times \bar{v}=e_1\left(-2\left({}_i^t{\bar{\Theta }}_{x1}\right)\right)+e_2\left(-2\left({}_i^t{\bar{\Theta }}_{x2}\right)\right)+e_3\left(-2\left({}_i^t{\bar{\Theta }}_{x3}\right)\right)$ (15)

The rotation rates in (12) are in clockwise sense, whereas quantities in (15) are twice the magnitude compared to (12) and are in counterclockwise sense. We note that $\left[\bar{W}\right]$ , the antisymmetric part of $\left[\bar{L}\right]$ , has rotation rates whereas $\left[{}^t\bar{R}\right]$ from the polar decomposition of $\left[\bar{L}\right]$ is a transformation matrix related to rotation rates. The details in both are related to rotation rates and are derived using $\left[\bar{L}\right]$ , hence use of $\left[{}^t\bar{R}\right]$ or $\left[\bar{W}\right]$ is interchangeable depending upon the need. Another important point we note is that from (11), $\left[\bar{W}\right]$ is undoubtedly a tensor of rank two. This is also obvious from (13) containing $e_i\times e_j$ term. However, the rotation rates ${}_i^t{\bar{\Theta }}_i$ as in (12) can be viewed as a vector quantity. That is, the three rotations about the axes of a triad at a material point can be arranged in the form of a vector. This form is advantageous when determining gradients of the rotation rates (shown later). We clearly observe that ${}_i^t{\bar{\Theta }}_i$ are completely defined by the components of $\left[\bar{L}\right]$ , i.e., dependent on the components of $\left[\bar{L}\right]$ , therefore are not unknown degrees of freedom at a material point or at a location. Definition of $\bar{W}$ from (9) or (10) clearly shows that it is a tensor of rank two, i.e.,

$\bar{W}=\frac{1}{2}{e}_i\otimes e_j\left(\frac{\partial {\bar{v}}_j}{\partial {\bar{x}}_i}-\frac{\partial {\bar{v}}_i}{\partial {\bar{x}}_j}\right)$ (16)

The gradient of $\bar{W}$ in (16) can be written as

$\bar{\nabla }\bar{W}=e_l\frac{\partial }{\partial {\bar{x}}_l}\otimes \left(\frac{1}{2}{e}_i\otimes e_j\left(\frac{\partial {\bar{v}}_j}{\partial {\bar{x}}_i}-\frac{\partial {\bar{v}}_i}{\partial {\bar{x}}_j}\right)\right)=\frac{1}{2}{e}_l\otimes e_i\otimes e_j\frac{\partial }{\partial {\bar{x}}_l}\left(\frac{\partial {\bar{v}}_j}{\partial {\bar{x}}_i}-\frac{\partial {\bar{v}}_i}{\partial {\bar{x}}_j}\right)$ (17)

Clearly $\bar{\nabla }\bar{W}$ , i.e., gradient of $\bar{W}$ , is a tensor of rank three. An alternative presentation of the gradients of ${}_i^t\bar{\Theta }$ is simple and easier to incorporate in the further developments. Let us represent rotation rates as a vector

${\left\{{}_i^t\bar{\Theta }\right\}}^T=\left[{}_i{}^t\bar{\Theta }{}_{x1}\mathrm{,}{}_i{}^t\bar{\Theta }{}_{x2}\mathrm{,}{}_i{}^t\bar{\Theta }{}_{x3}\right]$ (18)

Gradients of ${}_i^t\bar{\Theta }$ in (18) can be defined using

$\left[{}^{{}_i^t\text{}\bar{\Theta }}\bar{J}\right]=\frac{\partial \left\{{}_i^t\bar{\Theta }\right\}}{\partial \left\{\bar{x}\right\}}\text{or}{}^{{}_i^t\text{}\bar{\Theta }}{\bar{J}}_{ij}=\frac{\partial \left({}_i^t{\bar{\Theta }}_i\right)}{\partial {\bar{x}}_j}$ (19)

The gradient tensor ${}^{{}_i^t\text{}\bar{\Theta }}\bar{J}$ of rotation rates in (19) can be decomposed into symmetric and antisymmetric tensors $\left[{}_s^{{}_i^t\text{}\bar{\Theta }}\bar{J}\right]$ and $\left[{}_a^{{}_i^t\text{}\bar{\Theta }}\bar{J}\right]$ .

$\left[{}^{{}_i^t\text{}\bar{\Theta }}\bar{J}\right]=\left[{}_s^{{}_i^t\text{}\bar{\Theta }}\bar{J}\right]+\left[{}_a^{{}_i^t\text{}\bar{\Theta }}\bar{J}\right]$ (20)

$\begin{array}{l}\left[{}_s^{{}_i^t\text{}\bar{\Theta }}\bar{J}\right]=\frac{1}{2}\left(\left[{}^{{}_i^t\text{}\bar{\Theta }}\bar{J}\right]+{\left[{}^{{}_i^t\text{}\bar{\Theta }}\bar{J}\right]}^T\right)\\ \left[{}_a^{{}_i^t\text{}\bar{\Theta }}\bar{J}\right]=\frac{1}{2}\left(\left[{}^{{}_i^t\text{}\bar{\Theta }}\bar{J}\right]-{\left[{}^{{}_i^t\text{}\bar{\Theta }}\bar{J}\right]}^T\right)\end{array}$ (21)

4. Considerations of Stress, Moment, and Strain Rate Tensors

When the velocity gradient tensor varies between neighboring material points (or locations), so do the internal rotation rates ${}_i{}^t\bar{\Theta }$ . Hence, the rotation rate tensor $\bar{W}$ can vary between the material points. When the rotation rates $\bar{W}$ are resisted by the deforming fluent continua, conjugate moments are created. $\bar{W}$ and conjugate moments can result in additional energy storage, dissipation, and rheology, in addition to dissipation, rheology, etc., which are already present due to Cauchy stress tensor and the strain rate tensor. Thus, in the deforming fluent continua, rotation rates $\bar{W}$ are conjugate to the moment tensor which necessitates that on the boundary of the deformed volume there must exist resultant moment.

Consider a volume of matter $\underset{˜}{V}$ in the reference configuration with closed boundary $\partial \underset{˜}{V}$ . The volume $V$ is isolated from $\underset{˜}{V}$ by a hypothetical surface $\partial V$ as in the cut principle of Cauchy. Consider a tetrahedron $T_1$ such that its oblique plane is part of $\partial V$ and its other three planes are orthogonal to each other and parallel to the planes of the x-frame. Upon deformation, $\underset{˜}{V}$ and $\partial \underset{˜}{V}$ occupy $\underset{˜}{\bar{V}}$ and $\partial \underset{˜}{\bar{V}}$ and likewise $V$ and $\partial V$ deform into $\bar{V}$ and $\partial \bar{V}$ . The tetrahedron $T_1$ deforms into ${\bar{T}}_1$ whose edges (under finite deformation) are non-orthogonal covariant base vectors ${\tilde{g}}_i$ . The planes of the tetrahedron formed by the covariant base vectors are flat but obviously non-orthogonal to each other. We assume the tetrahedron to be the small neighborhood of material point $\bar{o}$ so that the assumption of the oblique plane $\bar{A}\bar{B}\bar{C}$ being flat but still part of $\partial \bar{V}$ is valid. When the deformed tetrahedron is isolated from volume $\bar{V}$ it must be in equilibrium under the action of disturbance on surface $\bar{A}\bar{B}\bar{C}$ from the volume surrounding $\bar{V}$ and the internal fields that act on the flat faces which equilibrate with the mating faces in volume $\bar{V}$ when the tetrahedron $T_2$ is placed back in the volume $\bar{V}$ .

Consider the deformed tetrahedron ${\bar{T}}_1$ . Let $\bar{P}$ be the average stress per unit area on plane $\bar{A}\bar{B}\bar{C}$ , $\bar{M}$ be the average moment per unit area on plane $\bar{A}\bar{B}\bar{C}$ (henceforth referred to as moment for short), and $\bar{n}$ be the unit exterior normal to the face $\bar{A}\bar{B}\bar{C}$ . $\bar{P}$ , $\bar{M}$ , and $\bar{n}$ all have different directions when the deformation is finite [32] .

As mentioned earlier, the edges of the deformed tetrahedron are covariant base vectors ${\tilde{g}}_i$ that are tangent to deformed curvilinear material lines:

${\tilde{g}}_i=e_k\frac{\partial {\bar{x}}_k}{\partial x_i}$ (22)

and

$J_{ij}=\frac{\partial {\bar{x}}_i}{\partial x_j}$ (23)

The columns of $J$ are covariant base vectors ${\tilde{g}}_i$ that form non-orthogonal covariant basis. Contravariant base vectors ${\tilde{g}}^i$ are normal to the faces of the deformed tetrahedron formed by the covariant base vectors:

${\tilde{g}}^j=e_l\frac{\partial x_j}{\partial {\bar{x}}_l}$ (24)

and

${\bar{J}}_{ij}=\frac{\partial x_i}{\partial {\bar{x}}_j}$ (25)

The rows of $\bar{J}$ are contravariant base vectors ${\tilde{g}}^j$ . These form a non-orthogonal contravariant basis. Covariant and contravariant bases are reciprocal to each other [32] .

4.1. Contravariant Cauchy Stress Tensor

The definition of the stresses on the non-oblique faces of the deformed tetrahedron formed by the covariant base vectors ${\tilde{g}}_i$ in the contravariant directions orthogonal to the faces of the deformed tetrahedron is most natural. Let ${\overline{\underset{˜}{\sigma }}}^{\left(0\right)}$ or ${\underset{˜}{\sigma }}^{\left(0\right)}$ be the contravariant stress tensor with components ${\overline{\underset{˜}{\sigma }}}_{ij}^{\left(0\right)}$ or ${\underset{˜}{\sigma }}_{ij}^{\left(0\right)}$ with dyads ${\tilde{g}}_i\otimes {\tilde{g}}_j$ . Component ${\overline{\underset{˜}{\sigma }}}_{11}^{\left(0\right)}$ or ${\underset{˜}{\sigma }}_{11}^{\left(0\right)}$ is in the ${\tilde{g}}^1$ direction on a face of the tetrahedron with unit exterior normal ${\tilde{g}}^1$ , i.e., on the ${\tilde{g}}^1$ face. Likewise ${\overline{\underset{˜}{\sigma }}}_{12}^{\left(0\right)}$ or ${\underset{˜}{\sigma }}_{12}^{\left(0\right)}$ and ${\overline{\underset{˜}{\sigma }}}_{31}^{\left(0\right)}$ or ${\overline{\underset{˜}{\sigma }}}_{31}^{\left(0\right)}$ act on ${\tilde{g}}^1$ and ${\tilde{g}}^3$ faces in the ${\tilde{g}}^2$ and ${\tilde{g}}^1$ directions. Using dyads ${\tilde{g}}_i\otimes {\tilde{g}}_j$ or contravariant law of transformation, we can write [32]

${\sigma }^{\left(0\right)}={\tilde{g}}_i\otimes {\tilde{g}}_j{\underset{˜}{\sigma }}_{ij}^{\left(0\right)}$ (26)

Using (22) in (26), we can write

$\begin{array}{l}{\sigma }^{\left(0\right)}=e_i\otimes e_j{\sigma }_{ij}^{\left(0\right)}\\ {\sigma }_{ij}^{\left(0\right)}=J_{ik}{\underset{˜}{\sigma }}_{kl}^{\left(0\right)}{J}_{jl}\\ \left[{\sigma }^{\left(0\right)}\right]=\left[J\right]\left[{\underset{˜}{\sigma }}^{\left(0\right)}\right]{\left[J\right]}^T\end{array}$ (27)

${\sigma }^{\left(0\right)}$ is a contravariant Cauchy stress tensor (Lagrangian description) from which ${\bar{\sigma }}^{\left(0\right)}$ can be easily obtained by replacing $\left[J\right]$ by ${\left[\bar{J}\right]}^{-1}$ and ${\sigma }^{\left(0\right)}$ by ${\bar{\sigma }}^{\left(0\right)}$ in (27). Since the dyads of ${\sigma }^{\left(0\right)}$ or ${\bar{\sigma }}^{\left(0\right)}$ are $e_i\otimes e_j$ , the Cauchy principle holds between $\bar{P}$ and ${\bar{\sigma }}^{\left(0\right)}$ .

$\bar{P}={\left({\bar{\sigma }}^{\left(0\right)}\right)}^T\cdot \bar{n}$ (28)

4.2. Covariant Cauchy Stress Tensor

Instead of using contravariant directions and stress components ${\underset{˜}{\sigma }}^{\left(0\right)}$ and covariant basis ${\tilde{g}}_i$ , we could use covariant stress components ${\left({\underset{˜}{\sigma }}_{\left(0\right)}\right)}_{ij}$ or ${\left({\overline{\underset{˜}{\sigma }}}_{\left(0\right)}\right)}_{ij}$ and contravariant basis ${\tilde{g}}^i$ . Consideration of ${\left({\underset{˜}{\sigma }}_{\left(0\right)}\right)}_{ij}$ of course will require a different deformed tetrahedron such that covariant base vectors ${\tilde{g}}_i$ are normal to its oblique faces. The adverse consequences of choosing this measure of stress for finite deformation are discussed by Surana, et al. [32] [33] . Here we proceed with this measure as an alternative to the contravariant stress measure. Using dyads ${\tilde{g}}^i\otimes {\tilde{g}}^j$ and components ${\left({\underset{˜}{\sigma }}_{\left(0\right)}\right)}_{ij}$ , we can write [32]

${\bar{\sigma }}_{\left(0\right)}={\tilde{g}}^i\otimes {\tilde{g}}^j{\left({\underset{˜}{\sigma }}_{\left(0\right)}\right)}_{ij}$ (29)

And using (24)

$\begin{array}{l}{\bar{\sigma }}_{\left(0\right)}=e_i\otimes e_j{\left({\bar{\sigma }}_{\left(0\right)}\right)}_{ij}\\ {\left({\bar{\sigma }}_{\left(0\right)}\right)}_{ij}={\bar{J}}_{ki}{\left({\underset{˜}{\sigma }}_{\left(0\right)}\right)}_{kl}{\bar{J}}_{lj}\\ \left[{\bar{\sigma }}_{\left(0\right)}\right]={\left[\bar{J}\right]}^T\left[{\sigma }_{\left(0\right)}\right]\left[\bar{J}\right]\end{array}$ (30)

${\bar{\sigma }}_{\left(0\right)}$ is the covariant Cauchy stress tensor (Eulerian description) from which ${\sigma }_{\left(0\right)}$ can be obtained by replacing $\left[\bar{J}\right]$ with ${\left[J\right]}^{-1}$ and ${\bar{\sigma }}_{\left(0\right)}$ with ${\sigma }_{\left(0\right)}$ in (30). Since the dyads of ${\bar{\sigma }}_{\left(0\right)}$ are $e_i\otimes e_j$ , the Cauchy principle holds between $\bar{P}$ and ${\bar{\sigma }}_{\left(0\right)}$ .

$\bar{P}={\left({\bar{\sigma }}_{\left(0\right)}\right)}^T\cdot \bar{n}$ (31)

Remarks.

The Cauchy stress tensors ${\sigma }^{\left(0\right)}$ or ${\bar{\sigma }}^{\left(0\right)}$ and ${\sigma }_{\left(0\right)}$ or ${\bar{\sigma }}_{\left(0\right)}$ are non-symmetric at this stage and so are the stress tensors ${\underset{˜}{\sigma }}^{\left(0\right)}$ and ${\underset{˜}{\sigma }}_{\left(0\right)}$ . Following the details given in Reference [32] we can also define Jaumann stress tensor ${}^{\left(0\right)}{\sigma }^J$ using ${\bar{\sigma }}^{\left(0\right)}$ and ${\bar{\sigma }}_{\left(0\right)}$ stress measures.

4.3. Contravariant and Covariant Cauchy Moment Tensor

When the deformed tetrahedron with moment $\bar{M}$ on its oblique face $\bar{A}\bar{B}\bar{C}$ is isolated from the volume $\bar{V}$ , its faces will have existence of moments (per unit area) on them. As in case of stress measure, contravariant basis is the most natural way to define these. Following the notations parallel to those used in case of Cauchy stress tensors, we can write the following using contravariant measures of moment tensor:

$m={\tilde{g}}_i\otimes {\tilde{g}}_j{m}_{ij}^{\left(0\right)}$ (32)

Using (22) in (32) we obtain

$\begin{array}{l}m=e_i\otimes e_j{m}_{ij}^{\left(0\right)}\\ m_{ij}^{\left(0\right)}=J_{ik}{\underset{˜}{m}}_{kl}^{\left(0\right)}{J}_{jl}\\ \left[m^{\left(0\right)}\right]=\left[J\right]\left[{\underset{˜}{m}}^{\left(0\right)}\right]{\left[J\right]}^T\\ \left[{\bar{m}}^{\left(0\right)}\right]={\left[\bar{J}\right]}^{-1}\left[{\underset{˜}{m}}^{\left(0\right)}\right]{\left[{\left[\bar{J}\right]}^{-1}\right]}^T\end{array}$ (33)

and the Cauchy principle

$\bar{M}={\left({\bar{m}}^{\left(0\right)}\right)}^T\cdot \bar{n}$ (34)

Likewise when using covariant measure of moment tensor we have

${\bar{m}}_{\left(0\right)}={\tilde{g}}^i\otimes {\tilde{g}}^j{\left({\underset{˜}{m}}_{\left(0\right)}\right)}_{ij}$ (35)

And using (24) in (35) we obtain

$\begin{array}{l}{\bar{m}}_{\left(0\right)}=e_i\otimes e_j{\left({\bar{m}}_{\left(0\right)}\right)}_{ij}\\ {\left({\bar{m}}_{\left(0\right)}\right)}_{ij}={\bar{J}}_{ki}{\left({\underset{˜}{m}}_{\left(0\right)}\right)}_{kl}{\bar{J}}_{lj}\\ \left[{\bar{m}}_{\left(0\right)}\right]={\left[\bar{J}\right]}^T\left[{\underset{˜}{m}}_{\left(0\right)}\right]\left[\bar{J}\right]\\ \left[m_{\left(0\right)}\right]={\left[{\left[J\right]}^{-1}\right]}^T\left[{\underset{˜}{m}}_{\left(0\right)}\right]{\left[J\right]}^{-1}\end{array}$ (36)

and the Cauchy principle

$\bar{M}={\left({\bar{m}}_{\left(0\right)}\right)}^T\cdot \bar{n}$ (37)

As in case of stress tensors ${\bar{\sigma }}^{\left(0\right)}$ and ${\bar{\sigma }}_{\left(0\right)}$ , the moment tensors ${\bar{m}}^{\left(0\right)}$ and ${\bar{m}}_{\left(0\right)}$ are also non-symmetric at this stage.

4.4. Convected Time Derivatives of the Stress and Strain Tensors

Convected time derivatives of strain and stress tensors in covariant and contravariant bases play an important role in the Eulerian description, especially in

constitutive theories. If we define $\left[{\bar{T}}_{\left(0\right)}\right]$ and $\left[{\bar{T}}^{\left(0\right)}\right]$ covariant and contrava-

riant Cauchy stress tensors and $\left[{\bar{T}}_{\left(0\right)}\right]$ and $\left[{\bar{T}}^{\left(0\right)}\right]$ as corresponding second

Piola-Kirchhoff stress tensors, then the convected time derivatives of $\left[{\bar{T}}_{\left(0\right)}\right]$

and $\left[{\bar{T}}^{\left(0\right)}\right]$ in co- and contravariant bases are defined by $\left[{\bar{T}}_{\left(k\right)}\right]$ and $\left[{\bar{T}}^{\left(k\right)}\right]$ ;

$k=1,2,\cdots ,n$ and are given by the following for compressible matter [32] .

$\begin{array}{l}\frac{D}{Dt}\left[{\bar{T}}_{\left[k-1\right]}\right]=\left[{\bar{T}}_{\left[k\right]}\right]\hfill \\ \left[{\bar{T}}_{\left[k\right]}\right]=\left|J\right|{\left[J\right]}^T\left[{\bar{T}}_{\left(k\right)}\right]\left[J\right]\hfill \\ \left[{\bar{T}}_{\left(k\right)}\right]=\frac{D}{Dt}\left[{\bar{T}}_{\left(k-1\right)}\right]+{\left[\bar{L}\right]}^T\left[{\bar{T}}_{\left(k-1\right)}\right]+\left[{\bar{T}}_{\left(k-1\right)}\right]\left[\bar{L}\right]+\left[{\bar{T}}_{\left(k-1\right)}\right]\text{tr}\left[\bar{L}\right]\hfill \end{array}\}k=\mathrm{1,2,}\cdots \mathrm{,}n$ (38)

and

$\begin{array}{l}\frac{D}{Dt}\left[{\bar{T}}^{\left[k-1\right]}\right]=\left[{\bar{T}}^{\left[k\right]}\right]\hfill \\ \left[{\bar{T}}^{\left[k\right]}\right]=\left|J\right|\left[\bar{J}\right]\left[{\bar{T}}^{\left(k\right)}\right]{\left[\bar{J}\right]}^T\hfill \\ \left[{\bar{T}}^{\left(k\right)}\right]=\frac{D}{Dt}\left[{\bar{T}}^{\left(k-1\right)}\right]-\left[\bar{L}\right]\left[{\bar{T}}^{\left(k-1\right)}\right]-\left[{\bar{T}}^{\left(k-1\right)}\right]{\left[\bar{L}\right]}^T+\left[{\bar{T}}^{\left(k-1\right)}\right]\text{tr}\left[\bar{L}\right]\hfill \end{array}\}k=\mathrm{1,2,}\cdots \mathrm{,}n$ (39)

and Jaumann rates are defined as

$\left[{}^{\left(k\right)}{\bar{T}}^J\right]=\frac{1}{2}\left(\left[{\bar{T}}^{\left(k\right)}\right]+\left[{\bar{T}}_{\left(k\right)}\right]\right)\mathrm{<mo>\; ;\; </mo>}k=\mathrm{1,2,}\cdots \mathrm{,}n$ (40)

If $\left[{\epsilon }_{\left[0\right]}\right]$ and $\left[{\bar{\epsilon }}^{\left[0\right]}\right]$ are Green’s and Almansi strain tensors in co- and contravariant bases, then their convected time derivatives $\left[{\bar{\gamma }}_{\left(k\right)}\right]$ and $\left[{\bar{\gamma }}^{\left(k\right)}\right]$ ; $k=1,2,\cdots ,n$ are defined as [32]

$\begin{array}{l}\frac{D}{Dt}\left[{\bar{\gamma }}_{\left[k-1\right]}\right]=\left[{\bar{\gamma }}_{\left[k\right]}\right]\hfill \\ \left[{\bar{\gamma }}_{\left[k\right]}\right]=\left|J\right|{\left[J\right]}^T\left[{\bar{\gamma }}_{\left(k\right)}\right]\left[J\right]\hfill \\ \left[{\bar{\gamma }}_{\left(k\right)}\right]=\frac{D}{Dt}\left[{\bar{\gamma }}_{\left(k-1\right)}\right]+{\left[\bar{L}\right]}^T\left[{\bar{\gamma }}_{\left(k-1\right)}\right]+\left[{\bar{\gamma }}_{\left(k-1\right)}\right]\left[\bar{L}\right]+\left[{\bar{\gamma }}_{\left(k-1\right)}\right]\text{tr}\left[\bar{L}\right]\hfill \\ \left[{\bar{\gamma }}_{\left[1\right]}\right]=\frac{D}{Dt}\left[{\epsilon }_{\left[0\right]}\right]=\left|J\right|{\left[J\right]}^T\left[{\bar{\gamma }}_{\left(1\right)}\right]\left[J\right]\hfill \\ \left[{\bar{\gamma }}_{\left(1\right)}\right]=\frac{1}{2}\left(\left[\bar{L}\right]+{\left[\bar{L}\right]}^T\right)=\left[\bar{D}\right]\hfill \end{array}\}k=\mathrm{1,2,}\cdots \mathrm{,}n$ (41)

and

$\begin{array}{l}\frac{D}{Dt}\left[{\bar{\gamma }}^{\left[k-1\right]}\right]=\left[{\bar{\gamma }}^{\left[k\right]}\right]\hfill \\ \left[{\bar{\gamma }}^{\left[k\right]}\right]=\left|J\right|\left[\bar{J}\right]\left[{\bar{\gamma }}^{\left(k\right)}\right]{\left[\bar{J}\right]}^T\hfill \\ \left[{\bar{\gamma }}^{\left(k\right)}\right]=\frac{D}{Dt}\left[{\bar{\gamma }}^{\left(k-1\right)}\right]+\left[\bar{L}\right]\left[{\bar{\gamma }}^{\left(k-1\right)}\right]+\left[{\bar{\gamma }}^{\left(k-1\right)}\right]{\left[\bar{L}\right]}^T+\left[{\bar{\gamma }}^{\left(k-1\right)}\right]\text{tr}\left[\bar{L}\right]\hfill \\ \left[{\bar{\gamma }}^{\left[1\right]}\right]=\frac{D}{Dt}\left[{\epsilon }^{\left[0\right]}\right]=\left|J\right|\left[\bar{J}\right]\left[{\bar{\gamma }}^{\left(1\right)}\right]{\left[\bar{J}\right]}^T\hfill \\ \left[{\bar{\gamma }}^{\left(1\right)}\right]=\frac{1}{2}\left(\left[\bar{L}\right]+{\left[\bar{L}\right]}^T\right)=\left[\bar{D}\right]\hfill \end{array}\}k=\mathrm{1,2,}\cdots \mathrm{,}n$ (42)

and Jaumann rates $\left[{}^{\left(k\right)}{\gamma }^J\right]$ ; $k=1,2,\cdots ,n$ are defined as

$\left[{}^{\left(k\right)}{\gamma }^J\right]=\frac{1}{2}\left(\left[{\bar{\gamma }}^{\left(k\right)}\right]+\left[{\bar{\gamma }}_{\left(k\right)}\right]\right)\mathrm{<mo>\; ;\; </mo>}k=\mathrm{1,2,}\cdots \mathrm{,}n$ (43)

In classical continuum theories for thermoviscous fluids without memory [32] , only stress rates of order zero, i.e., ${\bar{\sigma }}^{\left(0\right)}$ , ${\bar{\sigma }}_{\left(0\right)}$ , and ${}^{\left(0\right)}{\bar{\sigma }}^J$ , are used in the ordered rate constitutive theories of up to order n [32] . In such theories, $\left[{\bar{\gamma }}_{\left(k\right)}\right]$ , $\left[{\bar{\gamma }}^{\left(k\right)}\right]$ , and $\left[{}^{\left(k\right)}{\gamma }^J\right]$ ; $k=1,2,\cdots ,n$ are considered as argument tensors (in addition to some others) of $\left[{\bar{\sigma }}^{\left(0\right)}\right]$ , $\left[{\bar{\sigma }}_{\left(0\right)}\right]$ , and $\left[{}^{\left(0\right)}{\bar{\sigma }}^J\right]$ , respectively. Thus, the Cauchy stress tensor and the constitutive theory for it are basis dependent. In the derivations of the conservation and balance laws and the constitutive theories, we choose basis independent ${}^{\left(0\right)}\bar{\sigma }$ as Cauchy stress tensor which could be ${\bar{\sigma }}^{\left(0\right)}$ , ${\bar{\sigma }}_{\left(0\right)}$ , or ${}^{\left(0\right)}{\bar{\sigma }}^J$ ; $k=1,2,\cdots ,n$ . ${}^{\left(k\right)}\gamma$ ; $k=1,2,\cdots ,n$ convected time derivatives are considered as arguments of ${}^{\left(0\right)}\bar{\sigma }$ which could be ${\gamma }_{\left(k\right)}$ , ${\gamma }^{\left(k\right)}$ , or ${}^{\left(k\right)}{\gamma }^J$ . These choices of ${}^{\left(0\right)}\bar{\sigma }$ and ${}^{\left(k\right)}\gamma$ ; $k=1,2,\cdots ,n$ make all derivations hold for any desired choice of stress measure and the corresponding convected time derivatives of the strain measures. In case of non-classical theories considered here, exactly the same notation is used except that the Cauchy stress tensor is not symmetric, thus ${}^{\left(k\right)}\gamma$ ; $k=1,2,\cdots ,n$ cannot be the argument tensors of ${}^{\left(0\right)}\bar{\sigma }$ , but can be paired with it. Similarly, the Cauchy moment tensor choice can be ${\bar{m}}^{\left(0\right)}$ , ${\bar{m}}_{\left(0\right)}$ , or ${}^{\left(0\right)}{\bar{m}}^J$ , depending upon the choice of basis. In the derivations and the details that follow, we consider ${}^{\left(0\right)}\bar{\sigma }$ and ${}^{\left(0\right)}\bar{m}$ as Cauchy stress and moment tensors. With these choices the details that follow hold for any desired choice of basis.

5. Conservation and Balance Laws

The non-classical continuum theory used in this paper for fluent continua incorporates new physics due to internal rotation rates that are defined by $\bar{L}$ , hence known. This new physics is absent in the currently used thermodynamic framework for fluent continua. Introduction of this new physics may influence some or all conservation and balance laws which can only be determined by initiating their derivations from the most fundamental stage as we do in classical continuum theories [32] [34] . In this process of deriving conservation and balance laws with the new rotation rate physics we may very well find that some conservation and balance laws are not affected; however, such conclusions without rigorous derivations are not possible. In the non-classical continuum theory for fluent continua with velocities, velocity gradients, strain rate tensor, internal rotation rates, and their gradients describing the kinematics of deformation, we must at least consider the following conservation and balance laws based on the assumption of thermodynamic equilibrium that are used for classical continuum theories during the evolution of the deforming matter: 1) conservation of mass, 2) balance of linear momenta, 3) balance of angular momenta, 4) first law of thermodynamics (i.e., balance of energy), and 5) second law of thermodynamics (i.e., entropy inequality).

The use of conservation and balance laws that are necessary for classical continuum theories for non-classical continuum theories with additional physics due to internal rotation rates raises a fundamental concern: are these sufficient to ensure equilibrium of deforming non-classical fluent continua? It is pointed out by Yang, et al. [31] that an additional balance law is required in non-classical continuum theories for solids incorporating internal rotations (also see [29] [30] ) arising from the Jacobian of deformation. In case of fluent continua, the existence of internal rotation rates due to the velocity gradient tensor necessitates an additional balance law to ensure that in the presence of this physics the entire volume of fluid will remain in equilibrium. In recent papers by Surana, et al. [29] [30] , comprehensive discussion of the work of Yang, et al. [31] as well as authors’ own view regarding the need for this additional balance law in non-classical continuum theories for solid and fluid continua have been presented. This is not repeated here for the sake of brevity. The readers can refer to References [29] [30] [31] .

Balance of moments of moments (similar to balance of moments of forces in classical continuum theory) is an additional balance law needed due to the presence of Cauchy moment tensor that is independent of forces. In the derivation presented subsequently, one notes that this balance law yields the Cauchy moment tensor to be symmetric, just like the balance of angular momenta in classical continuum theory gives rise to the symmetry of the Cauchy stress tensor. One can use inductive reasoning to extend this concept of the need for additional balance laws when additional kinematic variables (over and beyond velocities and rotation rates) and their conjugates appear in the theory. One notes that each additional kinematic variable introduces its conjugate that requires two balance laws, out of which the balance law that requires their sum to balance with others already exists from the consideration of prior kinematic variables; hence the new conjugate quantities can be incorporated in it, but the other balance law that requires balance of their moments is an additional balance law. In other words, only one balance law is needed for each conjugate quantity corresponding to each kinematic variable.

In the non-classical continuum theory considered here for fluent continua, we need only one additional balance law, namely the balance of moments of moments, due to the fact that balance of moments balance law already exists from the classical continuum theory. Whether we consider balance of moments of moments as an additional balance law in non-classical continuum theories for fluent continua influences the derivation of the constitutive theories for the Cauchy moment tensor. In the present work we consider both cases and the associated constitutive theories to establish restrictions on the material coefficients appearing in them.

5.1. Conservation of Mass, Balance of Linear and Angular Momenta

We consider compressible fluent non-classical continua with internal rotation rates to present conservation and balance laws. For incompressible fluent continua, $\bar{\rho }={\rho }_0=\text{constant}$ and $\text{tr}\left(\bar{D}\right)=0$ , hence the conservation and balance laws presented here can be easily modified.

Conservation of mass in a deforming volume of fluid leads to continuity equation that remains the same in the present work as it is for the classical continuum theory [32] [34] and is given in the following for compressible fluent continua in Eulerian description.

$\frac{\partial \bar{\rho }}{\partial t}+\bar{\nabla }\cdot \left(\bar{\rho }\bar{v}\right)=0$ (44)

or

$\frac{D\bar{\rho }}{Dt}+\bar{\rho }\text{div}\left(\bar{v}\right)=0$ (45)

in which $\bar{\rho }\left(\bar{x}\mathrm{,}t\right)$ is the density at a material point at $\bar{x}$ in the current configuration.

For a deforming volume of matter, the rate of change of linear momenta must be equal to the sum of all other forces acting on it. This is Newton’s second law applied to a volume of matter. The derivation is exactly same as that for classical continuum theory. Following Reference [32] and using Cauchy stress tensor ${}^{\left(0\right)}\bar{\sigma }$ , we can write the following.

$\bar{\rho }\frac{D\bar{v}}{Dt}-\bar{\rho }{\bar{F}}^b-\bar{\nabla }\cdot {}^{\left(0\right)}\bar{\sigma }=0$ (46)

or

$\bar{\rho }\frac{\partial {\bar{v}}_i}{\partial t}+\bar{\rho }{\bar{v}}_j\frac{\partial {\bar{v}}_i}{\partial {\bar{x}}_j}-\bar{\rho }{\bar{F}}_i^b-\frac{{\partial }^{\left(0\right)}{\bar{\sigma }}_{ji}}{\partial {\bar{x}}_j}=0$ (47)

in which ${\bar{F}}^b$ are body forces per unit mass and ${}^{\left(0\right)}\bar{\sigma }$ is basis independent Cauchy stress tensor. Equations (46) or (47) are momentum equations in $x_1$ , $x_2$ , and $x_3$ directions.

The principle of balance of angular momenta for a non-classical continuum can be stated as: The material derivative (time rate of change) of moments of momenta must be equal to the vector sum of the moments of forces and the moments. Thus, due to the surface stress $\bar{P}$ , total surface moment $\bar{M}$ (per unit area), body force ${\bar{F}}^b$ (per unit mass), and the momentum $\bar{\rho }\bar{v}\text{d}\bar{V}$ for an elemental mass $\bar{\rho }\text{d}\bar{V}$ in the current configuration we can write the following in Eulerian description.

$\frac{D}{Dt}{\displaystyle \underset{\bar{V}\left(t\right)}{\int }}\bar{x}\times \bar{\rho }\bar{v}\text{d}\bar{V}={\displaystyle \underset{\partial \bar{V}\left(t\right)}{\int }}\left(\bar{x}\times \bar{P}-\bar{M}\right)\text{d}\bar{A}+{\displaystyle \underset{\bar{V}\left(t\right)}{\int }}\bar{x}\times \bar{\rho }{\bar{F}}^b\text{d}\bar{V}$ (48)

The negative sign for $\bar{M}$ is due to the fact that clockwise rotation rates are considered positive. The moments created by these must also be considered positive when clockwise. Following the derivation given by Surana, et al. [1] - [6] , we obtain

${\displaystyle \underset{\bar{V}\left(t\right)}{\int }}{e}_k\left({}^{\left(0\right)}{\bar{m}}_{mk\mathrm{,}m}-{ϵ}_{ijk}{}^{\left(0\right)}{\bar{\sigma }}_{ij}\right)\text{d}\bar{V}=0$ (49)

Since volume $\bar{V}$ is arbitrary, we have

${}^{\left(0\right)}{\bar{m}}_{mk\mathrm{,}m}-{ϵ}_{ijk}{}^{\left(0\right)}{\bar{\sigma }}_{ij}=0$ (50)

or

$\bar{\nabla }\cdot {}^{\left(0\right)}\bar{m}-ϵ:{}^{\left(0\right)}\bar{\sigma }=0$ (51)

Equation (51) represents balance of angular momenta. The basis independent Cauchy stress tensor ${}^{\left(0\right)}\bar{\sigma }$ is non-symmetric and so is the basis independent Cauchy moment tensor ${}^{\left(0\right)}\bar{m}$ .

5.2. First Law of Thermodynamics

The sum of work and heat added to a deforming volume of matter must result in increase of the energy of the system. This is expressed as a rate equation in Eulerian description in the following.

$\frac{D{\bar{E}}_t}{Dt}=\frac{D\bar{Q}}{Dt}+\frac{D\bar{W}}{Dt}$ (52)

${\bar{E}}_t$ , $\bar{Q}$ , and $\bar{W}$ are total energy, heat added, and work done. These can be written as [1] - [6] [32]

$\frac{D{\bar{E}}_t}{Dt}=\frac{D}{Dt}{\displaystyle \underset{\bar{V}\left(t\right)}{\int }}\bar{\rho }\left(\bar{e}+\frac{1}{2}\bar{v}\cdot \bar{v}-{\bar{F}}^b\cdot \bar{u}\right)\text{d}\bar{V}$ (53)

$\frac{D\bar{Q}}{Dt}=-{\displaystyle \underset{\partial \bar{V}\left(t\right)}{\int }}\bar{q}\cdot \bar{n}\text{d}\bar{A}$ (54)

$\frac{D\bar{W}}{Dt}={\displaystyle \underset{\partial \bar{V}\left(t\right)}{\int }}\left(\bar{P}\cdot \bar{v}+\bar{M}\cdot {}_i{}^t\bar{\Theta }\right)\text{d}\bar{A}$ (55)

where $\bar{e}$ is specific internal energy, ${\bar{F}}^b$ is body force vector per unit mass, and $\bar{q}$ is rate of heat. Note that the additional term $\bar{M}\cdot {}_i{}^t\bar{\Theta }$ in $D\bar{W}/Dt$ contributes additional rate of work due to rates of internal rotations ${}_i{}^t\bar{\Theta }$ . Expanding integrals and following Reference [32] , one can show the following.

${\displaystyle \underset{\bar{V}\left(t\right)}{\int }}\left(\bar{\rho }\frac{D\bar{e}}{Dt}+\bar{\nabla }\cdot \bar{q}-{}^{\left(0\right)}\bar{\sigma }{}_{ji}\frac{\partial {\bar{v}}_i}{\partial {\bar{x}}_j}-{}^{\left(0\right)}m{}_{ji}\frac{\partial {}_i{}^t\bar{\Theta }{}_i}{\partial {\bar{x}}_j}-{}_i{}^t\bar{\Theta }\cdot \left(\bar{\nabla }\cdot {}^{\left(0\right)}\bar{m}\right)\right)\text{d}V=0$ (56)

Since volume $\bar{V}$ is arbitrary, the following holds:

$\bar{\rho }\frac{D\bar{e}}{Dt}+\bar{\nabla }\cdot \bar{q}-{}^{\left(0\right)}\bar{\sigma }{}_{ji}\frac{\partial {\bar{v}}_i}{\partial {\bar{x}}_j}-\left({}^{\left(0\right)}m{}_{ji}\frac{{\partial }_i^t{\bar{\Theta }}_i}{\partial {\bar{x}}_j}+{}_i{}^t\bar{\Theta }\cdot \left(\bar{\nabla }\cdot {}^{\left(0\right)}\bar{m}\right)\right)=0$ (57)

We note that in the term ${}_i{}^t\bar{\Theta }\cdot \left(\bar{\nabla }\cdot {}^{\left(0\right)}\bar{m}\right)$ we can substitute $\bar{\nabla }\cdot {}^{\left(0\right)}\bar{m}$ from the balance of angular momenta (51), thereby eliminating gradients of ${}^{\left(0\right)}\bar{m}$ but instead introducing Cauchy stress tensor ${}^{\left(0\right)}\bar{\sigma }$ .

5.3. Second Law of Thermodynamics

Let $\bar{\eta }$ be entropy density in deformed volume $\bar{V}\left(t\right)$ , $\bar{h}$ be the entropy flux between $\bar{V}\left(t\right)$ and the volume of matter surrounding it (i.e., contacting sources), and $\bar{s}$ be the source of entropy in $\bar{V}\left(t\right)$ due to non-contacting bodies, then the rate of increase of entropy in volume $\bar{V}\left(t\right)$ is at least equal to that supplied to $\bar{V}\left(t\right)$ from all contacting and non-contacting sources [32] . Thus

$\frac{D}{Dt}{\displaystyle \underset{\bar{V}\left(t\right)}{\int }}\bar{\eta }\bar{\rho }\text{d}\bar{V}\ge {\displaystyle \underset{\partial \bar{V}\left(t\right)}{\int }}\bar{h}\text{d}\bar{A}+{\displaystyle \underset{\bar{V}\left(t\right)}{\int }}\bar{s}\bar{\rho }\text{d}\bar{V}$ (58)

Using Cauchy’s postulate for $\bar{h}$ , we have

$\bar{h}=-\bar{\psi }\cdot \bar{n}$ (59)

${\displaystyle \underset{\bar{V}\left(t\right)}{\int }}\left(\bar{\rho }\frac{D\bar{\eta }}{Dt}+\bar{\nabla }\cdot \bar{\psi }-\bar{s}\bar{\rho }\right)\text{d}\bar{V}\ge 0$ (60)

Since the volume $\bar{V}$ is arbitrary, the following holds:

$\bar{\rho }\frac{D\bar{\eta }}{Dt}+\bar{\nabla }\cdot \bar{\psi }-\bar{s}\bar{\rho }\ge 0$ (61)