Entropy Production Assumption and Objectivity in Continuum Physics Modelling

Abstract

This paper revisits some aspects connected with the methods for the determination of thermodynamically consistent models. While the concepts apply to the general context of continuum physics, the details are developed for the modelling of deformable dielectrics. The symmetry condition arising from the balance of angular momentum is viewed as a constraint for the constitutive equations and is shown to be satisfied by sets of objective fields that account jointly for deformation and electric field. The second law of thermodynamics is considered in a generalized form where the entropy production is given by a constitutive function possibly independent of the other constitutive functions. Furthermore, a representation formula is applied for solving the Clausius–Duhem inequality with respect to the chosen unknown fields.

1. Introduction

This paper is devoted to methods used for the elaboration of models of continuous media on the basis of the principles of continuum physics. The topic is well established, with an enormous literature (see, e.g., [1,2,3]). Yet, some new ideas deserve a careful attention. For definiteness, as well as for the purpose of providing significant examples, the methods are applied to the formulation of constitutive models of deformable dielectrics.

A thermodynamic process is a set of functions on the region $\Omega$ occupied by a body and the pertinent time interval describing the evolution of the body. For each process, the balance equations are required to hold and this results in a set of constraints on the space–time dependence of corresponding functions. The physical properties of the body are described by constitutive equations that provide relations between the quantities pertaining to the process. The body is then described by the whole set of balance equations and constitutive equations.

The constitutive equations are required to be consistent with the second law of thermodynamics, which eventually restricts the set of possible relations characterizing the behaviour of the material. This paper provides a restatement of the postulate of the second law and, although within the view of the Coleman–Noll approach [4], establishes a generalization that allows for an improvement in the modelling of rate-type models and especially dissipative materials.

The generalization involves the (rate of) entropy production or entropy imbalance ([1], Sect. 27). As a general statement, the second law is stated by asserting that the rate of increase in entropy of a convecting region of a body is at least as great as the rate at which entropy flows into the region under consideration. As such, entropy production (or imbalance) might appear to be a purely physical property. Instead, it is known in the (rational) thermodynamics of continua [5] that the expression, and then the value, of the entropy production is a consequence of the chosen constitutive functions. The novelty described in this paper is connected with the view that the entropy production is given by a peculiar constitutive function, thus improving the model through a larger set of constitutive equations.

Electromagnetism in matter also generates problems in connection with balance equations, mainly about the appropriate stress tensor. Here, I start with the balance equations for dielectrics and revisit the constraint placed by the balance of angular momentum. Next, I show that the constraint is satisfied if the dependence on the electric field in matter is described through a class of vector fields that involve the electric field and the deformation gradient. Among these fields is the Lagrangian electric field already considered in the literature (e.g., [6,7]).

Objectivity is a further principle that governs the formulation of constitutive equations in continuum physics. It requires that the constitutive relations be form-invariant under the group of Euclidean transformations referred to as SO(3). The class of vector fields so-established enjoys the property of objectivity, namely invariance, under SO(3).

Following the view that the entropy production is given by its own constitutive function, the Clausius–Duhem inequality is shown to provide a systematic way to express the sought constitutive equation (e.g., for stress and heat flux) when dissipative properties are involved.

Notation 1.

Denote by $\Omega \subset {\mathcal{E}}^3$ the time-dependent region occupied by a body and let $t\in \mathbb{R}$ be the time. The position vector of points in $\Omega$ of the body is denoted by $\mathbf{x}$. Fields $f(\mathbf{x},t)$ are considered on $\Omega \times \mathbb{R}$. The symbol $\nabla ={\partial }_{\mathbf{x}}$ denotes the gradient in $\Omega$ and $\mathbf{v}$ is the velocity. $\mathbf{1}$ is the unit second-order tensor and $\mathsf{I}$ is the unit fourth-order tensor. To avoid confusion, mechanical vectors and tensors are denoted by bold characters; $\mathbf{b}$ is the body force, $\mathbf{T}$ is the Cauchy stress tensor, $\mathbf{F}$ is the deformation gradient, $\mathbf{C}={\mathbf{F}}^T\mathbf{F}$, and $\mathbf{E}=\frac{1}{2}(\mathbf{C}-\mathbf{1})$, $J=det\mathbf{F}$. Instead, electric vectors are denoted by mathsf symbols; $\mathsf{E}$ is the electric field, $\mathsf{P}$ is the polarization, and $\mathsf{D}$ is the electric displacement. $\mathbf{u}·\mathbf{w},\mathbf{u}\times \times \mathbf{w},\mathbf{u}\otimes \mathbf{w}$ denotes the inner, vector, and dyadic product between the vectors. Also, $\mathbf{A}·\mathbf{B}$ is the inner product between the tensors; in components, $\mathbf{A}·\mathbf{B}=A_{ij}{B}_{ij}$. For vectors $\mathbf{u}$ and tensors $\mathbf{A}$, we define ${\mathbf{u}}^2=\mathbf{u}·\mathbf{u}$ and ${\left|\mathbf{A}\right|}^2=\mathbf{A}·\mathbf{A}$.

Throughout, ${\mathcal{P}}_t$ is any subregion, at time t, of the region $\Omega$. Components are referred to an orthonormal right-handed basis $\left\{{\mathbf{e}}_i\right\}$, while $\mathrm{sym}$ and $\mathrm{skw}$ denote the symmetric and skew-symmetric parts and $\mathrm{Sym}$ and $\mathrm{Skw}$ give the sets of symmetric and skew-symmetric tensors. Further, $\mathbf{v}$ is the velocity, $\mathbf{L}$ is the velocity gradient, $L_{ij}={\partial }_{x_j}{v}_i$, and $\mathbf{D}=\mathrm{sym}\mathbf{L}$, $\mathbf{W}=\mathrm{skw}\mathbf{L}$. ${ϵ}_{ijk}$ is the alternating symbol, and then for any two vectors $\mathbf{a},\mathbf{b}$, it is ${\left(\mathbf{a}\times \times \mathbf{b}\right)}_i={ϵ}_{ijk}{a}_j{b}_k$. The Greek symbols $\theta ,\epsilon ,\eta ,$ and ψ denote the absolute temperature, the specific internal energy (density), the specific entropy, and the specific Helmholtz free energy, $\psi =\epsilon -\theta \eta$. The letter $\gamma$ denotes the (rate of) specific entropy production.

2. General Formulation of Balance Equations

The balance equations can be given a common formulation as follows. Let ${\mathcal{P}}_t$ be a convecting sub-region of a body that changes in time through the action of its motion while keeping the same points at any time. Denote by $\mathbf{x}$ the position of a point in $\Omega$ relative to a chosen origin O. Let $\Phi (\mathbf{x},t)$ be a density function defined on ${\mathcal{P}}_t\times \mathbb{R}$. The corresponding rate is governed by a body term density, say $\beta$, and a surface term density, say s, so that

$${\displaystyle \frac{d}{dt}}{\int }_{{\mathcal{P}}_t}\Phi dv={\int }_{{\mathcal{P}}_t}\beta dv+{\int }_{\partial {\mathcal{P}}_t}sda,$$

(1)

where $\partial {\mathcal{P}}_t$ denotes the boundary of ${\mathcal{P}}_t$.

Let $\mathbf{v}(\mathbf{x},t)$ denote the velocity field and ∇ the gradient operator. Hence, $\dot{\Phi }={\partial }_t\Phi +\mathbf{v}·\nabla \Phi$ denotes the total (or Lagrangian) time derivative. If $\Phi$ is differentiable, then, by the Reynolds transport relation ([1], ch. 16; [3], sec. 1.5), we can write Equation (1) in the form

$${\displaystyle \frac{d}{dt}}{\int }_{{\mathcal{P}}_t}\Phi dv={\int }_{{\mathcal{P}}_t}[\dot{\Phi }+\Phi \nabla ·\mathbf{v}]dv.$$

If $\Phi$ is differentiable, then Equation (1) can be written as

$${\int }_{{\mathcal{P}}_t}[\dot{\Phi }+\Phi \nabla ·\mathbf{v}]dv={\int }_{{\mathcal{P}}_t}\beta dv+{\int }_{\partial {\mathcal{P}}_t}sda.$$

(2)

The consistency of (1) and (2) is satisfied if the surface integral on $\partial {\mathcal{P}}_t$ equals a corresponding volume integral on ${\mathcal{P}}_t$. This aspect is now examined.

The functions $\Phi ,\beta ,s$ are all scalar- or vector-valued. The function s may depend on the normal (unit, outward) $\mathbf{n}$ to the surface so that $s=s(\mathbf{x},\mathbf{n},t)$. Cauchy’s theorem on the traction field $s=\mathbf{t}$ proves that

$$\mathbf{t}(\mathbf{x},\mathbf{n},t)=\mathbf{T}(\mathbf{x},t)\mathbf{n}(\mathbf{x},t),\mathbf{T}(\mathbf{x},t)\mathbf{n}(\mathbf{x},t):=\sum _{i=1}^3[\mathbf{t}(\mathbf{x},{\mathbf{e}}_i,t)\otimes {\mathbf{e}}_i];$$

hereafter, $\left\{{\mathbf{e}}_i\right\}$ is a chosen orthonormal right-handed basis.

If $\Phi ,\beta$, and s are scalars, then Cauchy’s theorem can be generalized to any balance equation as follows.

Theorem 1

(Generalized Cauchy’s theorem). If $\dot{\Phi }+\Phi \nabla ·\mathbf{v}-\beta$ is bounded, then there exists a vector, or a tensor, $\mathbf{h}$ defined as

$$\mathbf{h}=h_i{\mathbf{e}}_i,h_i(\mathbf{x},t):=s(\mathbf{x},{\mathbf{e}}_i,t),$$

such that

$$s(\mathbf{x},\mathbf{n},t)=\mathbf{h}(\mathbf{x},t)·\mathbf{n}(\mathbf{x},t).$$

Of course, if s is a scalar, then $\mathbf{h}$ is a vector. If s is a vector, say $s=\mathbf{t}$, then by Cauchy’s theorem, $\mathbf{h}$ is a tensor, namely, $\mathbf{t}(\mathbf{x},\mathbf{n},t)=\mathbf{T}(\mathbf{x},t)\mathbf{n}(\mathbf{x},t)$.

The linear dependence on $\mathbf{n}$—$s=\mathbf{h}·\mathbf{n}$—allows the application of the divergence theorem so that

$${\int }_{\partial {\mathcal{P}}_t}sda={\int }_{{\mathcal{P}}_t}\nabla ·\mathbf{h}dv.$$

(3)

This will be the case directly in dealing with the balance of energy and entropy. Instead, for the balance of linear and angular momentum, s has a known expression in terms of the stress tensor $\mathbf{T}$. We then examine the surface integral associated with linear and angular momentum.

In the balance of linear momentum, $s=\mathbf{t}$ is a vector and $\mathbf{t}=\mathbf{T}\mathbf{n}$. Then, upon inner-multiplying by a constant vector $\mathbf{a}$ and using the divergence theorem, we have

$$\mathbf{a}·{\int }_{\partial {\mathcal{P}}_t}sda={\int }_{\partial {\mathcal{P}}_t}\mathbf{a}·\mathbf{T}\mathbf{n}da={\int }_{\partial {\mathcal{P}}_t}\left({\mathbf{T}}^T\mathbf{a}\right)·\mathbf{n}da={\int }_{{\mathcal{P}}_t}\nabla ·\left({\mathbf{T}}^T\mathbf{a}\right)dv=\mathbf{a}·{\int }_{{\mathcal{P}}_t}(\nabla ·\mathbf{T})dv,$$

where

$$\nabla ·\mathbf{T}:={\partial }_{x_j}{T}_{ij}{\mathbf{e}}_i.$$

(4)

The arbitrariness of $\mathbf{a}$ implies that

$${\int }_{\partial {\mathcal{P}}_t}sda={\int }_{\partial {\mathcal{P}}_t}\mathbf{T}\mathbf{n}da={\int }_{{\mathcal{P}}_t}\nabla ·\mathbf{T}dv.$$

Hence, Equation (3) holds even if s is a vector and $\mathbf{h}$ is a tensor, provided the divergence is defined as in Equation (4).

In the balance of angular momentum, we consider the torque relative to a chosen, fixed, base point $O_B$. Let $\mathbf{d}$ be the constant position vector of O relative to $O_B$. Hence, $\mathbf{r}=\mathbf{d}+\mathbf{x}$ is the position vector relative to $O_B$. Hence,

$$s=\mathbf{r}\times \times \mathbf{t}=\mathbf{r}\times \times \left(\mathbf{T}\mathbf{n}\right).$$

Now, inner-multiplying by the constant vector $\mathbf{a}$, we have

$$\mathbf{a}·\left[\mathbf{r}\times \times \left(\mathbf{T}\mathbf{n}\right)\right]=\left(\mathbf{T}\mathbf{n}\right)·\mathbf{a}\times \times \mathbf{r}=\mathbf{n}·\left[{\mathbf{T}}^T\left(\mathbf{a}\times \times \mathbf{r}\right)\right]$$

and

$$\begin{array}{c}\hfill \nabla ·\left[{\mathbf{T}}^T\left(\mathbf{a}\times \times \mathbf{r}\right)\right]=\left({\partial }_{x_j}{T}_{jk}^T\right){ϵ}_{kpq}{a}_p{r}_q+T_{jk}^T{ϵ}_{kpq}{a}_p{\partial }_{x_j}{r}_q\\ \hfill =(\nabla ·\mathbf{T})·\mathbf{a}\times \times \mathbf{r}+\mathbf{a}·\mathbf{{\rm Y}}=\mathbf{a}·\left[\mathbf{r}\times \times \right(\nabla ·\mathbf{T})+\mathbf{{\rm Y}}],\end{array}$$

where, in components,

$${{\rm Y}}_p={ϵ}_{pjk}{T}_{kj}.$$

Hence, it follows that

$${\int }_{\partial {\mathcal{P}}_t}\mathbf{r}\times \times \left(\mathbf{T}\mathbf{n}\right)da={\int }_{{\mathcal{P}}_t}[\mathbf{r}\times \times (\nabla ·\mathbf{T})+\mathbf{{\rm Y}}]dv.$$

(5)

In the balance of energy, s comprises the power

$$\mathbf{t}·\mathbf{v}=\left(\mathbf{T}\mathbf{n}\right)·\mathbf{v}=\mathbf{n}·\left({\mathbf{T}}^T\mathbf{v}\right);$$

the divergence of the vector ${\mathbf{T}}^T\mathbf{v}$ gives

$$\nabla ·\left({\mathbf{T}}^T\mathbf{v}\right)=(\nabla ·\mathbf{T})·\mathbf{v}+\mathbf{T}·\mathbf{L},$$

(6)

where $\mathbf{T}·\mathbf{L}=T_{ij}{L}_{ij}=T_{ij}{\partial }_{x_j}{v}_i$.

If $\Phi$ and s are continuously differentiable, then Equation (1) can be written as

$${\int }_{{\mathcal{P}}_t}[\dot{\Phi }+\Phi \nabla ·\mathbf{v}-\beta -\nabla ·\mathbf{h}]dv=0.$$

The assumed continuity of the integrand and the arbitrariness of ${\mathcal{P}}_t$ imply the local form of the balance as

$$\dot{\Phi }+\Phi \nabla ·\mathbf{v}-\beta -\nabla ·\mathbf{h}=0.$$

(7)

With the global and local balance laws (1) and (7), we can now revisit a wide set of balance equations.

• The balance of mass is characterized by

  $$\Phi =\rho ,\beta =0,s=0,$$

  where $\rho$ is the mass density. Consequently, Equation (7) becomes the standard continuity equation

  $$\dot{\rho }+\rho \nabla ·\mathbf{v}=0,$$

  (8)

  or

  $${\partial }_t\rho +\nabla ·\left(\rho \mathbf{v}\right)=0.$$
  Furthermore, letting $\phi =\Phi /\rho$ be the specific density per unit mass, Equation (7) can be written in the form

  $$\rho \dot{\phi }-\beta -\nabla ·\mathbf{h}=0.$$

  (9)
• In the balance of linear momentum, it is

  $$\phi =\mathbf{v},\beta =\rho \mathbf{b},\mathbf{h}=\mathbf{T},$$

  with $\mathbf{T}$ being the stress tensor. Equation (9) results in

  $$\rho \dot{\mathbf{v}}-\rho \mathbf{b}-\nabla ·\mathbf{T}=0.$$

  (10)
• In the balance of angular momentum, with respect to a fixed base point $O_B$, it is

  $$\phi =\mathbf{r}\times \times \mathbf{v},\beta =\rho \mathbf{r}\times \times \mathbf{b}+\mathbf{l},s=\mathbf{r}\times \times \mathbf{T}\mathbf{n},$$

  (11)

  where $\mathbf{l}$ represents a possible couple density or torque per unit mass. In view of (5), the densities (11) lead to

  $$\mathbf{r}\times \times (\rho \dot{\mathbf{v}}-\rho \mathbf{b}-\nabla ·\mathbf{T})-(\mathbf{{\rm Y}}+\mathbf{l})=\mathbf{0}.$$
  In light of (10), it follows that

  $$\mathbf{{\rm Y}}+\mathbf{l}=\mathbf{0}.$$

  (12)

Since ${{\rm Y}}_p={ϵ}_{pjk}{T}_{kj}$, we look for a tensor form of (12). Notice that any tensor $\mathbf{A}$ can be given the additive decomposition

$$\mathbf{A}={\mathbf{A}}^{\mathrm{sym}}+{\mathbf{A}}^{\mathrm{skw}},{\mathbf{A}}^{\mathrm{sym}}=\frac{1}{2}(\mathbf{A}+{\mathbf{A}}^T),{\mathbf{A}}^{\mathrm{skw}}=\frac{1}{2}(\mathbf{A}-{\mathbf{A}}^T).$$

Since

$${ϵ}_{pjk}{A}_{jk}^{\mathrm{sym}}=0$$

then

$${ϵ}_{pjk}{A}_{jk}={ϵ}_{pjk}{A}_{jk}^{\mathrm{skw}}.$$

Consequently,

$${ϵ}_{pjk}{A}_{jk}=0⟹{\mathbf{A}}^{\mathrm{skw}}=\mathrm{skw}\mathbf{A}=\mathbf{0}.$$

If $\mathbf{l}$ is also given by a tensor $\Lambda$ in the form

$$l_p={ϵ}_{pjk}{\Lambda }_{kj}$$

then (12) implies that

$$\mathrm{skw}(\mathbf{T}+\Lambda )=0.$$

(13)

If, in particular, $\mathbf{l}$ is given by a vector product, say $\mathbf{l}=\mathbf{u}\times \times \mathbf{a}$, then we have

$$\Lambda =\mathbf{a}\otimes \mathbf{u}.$$

• The balance of energy involves mechanical and non-mechanical terms; the mechanical terms are

  $${\phi }_m=\rho \frac{1}{2}{\mathbf{v}}^2,{\beta }_m=\rho \mathbf{b}·\mathbf{v},s_m=\mathbf{t}·\mathbf{v}.$$
  The non-mechanical terms ${\varphi }_{nm},{\beta }_{nm},s_{nm}$ are

  $${\phi }_{nm}=\rho \epsilon ,{\beta }_{nm}=\rho r+\rho \mathsf{E}·\dot{\mathsf{p}},s_{nm}=\lambda ,$$

  where $\epsilon$ is viewed as the internal energy density, r is an energy supply, $\rho \mathsf{E}·\dot{\mathsf{p}}$ is the power density of electrical origin, and $\lambda$ is a power per unit area. Using Equation (6), we can write the global balance of energy in the form

  $${\int }_{{\mathcal{P}}_t}[\mathbf{v}·(\rho \dot{\mathbf{v}}-\rho \mathbf{b}-\nabla ·\mathbf{T})+\rho \dot{\epsilon }-\mathbf{T}·\mathbf{L}-\rho r-\rho \mathbf{E}·\dot{\mathsf{p}}]dv={\int }_{\partial {\mathcal{P}}_t}\lambda da$$

  and hence, in light of (10),

  $${\int }_{{\mathcal{P}}_t}[\rho \dot{\epsilon }-\mathbf{T}·\mathbf{L}-\rho r-\rho \mathsf{E}·\dot{\mathsf{p}}]dv={\int }_{\partial {\mathcal{P}}_t}\lambda da.$$
  Again, a Cauchy-like theorem allows us to show that there is a vector, say $\mathbf{q}$, such that

  $$\lambda =-\mathbf{q}·\mathbf{n}.$$
  Accordingly, we obtain the local balance of energy in the form

  $$\rho \dot{\epsilon }=\mathbf{T}·\mathbf{L}+\rho r+\rho \mathsf{E}·\dot{\mathsf{p}}-\nabla ·\mathbf{q}.$$

  (14)
• Let $\eta$ be the specific entropy. The balance of entropy is written in the general form in (1) with $\Phi =\rho \eta$ and the corresponding terms $\beta$ and h. By the Cauchy-like theorem, we prove the existence of a vector field, say $\mathbf{j}$, such that $h=-\mathbf{j}·\mathbf{n}$. Hence, we obtain the balance equation

  $$\rho \dot{\eta }=\beta -\nabla ·\mathbf{j}.$$
  Letting $\theta$ be the absolute temperature, we define

  $$\beta =\rho ({\displaystyle \frac{r}{\theta }}+\gamma ).$$
  Hence,

  $$\rho \dot{\eta }-{\displaystyle \frac{\rho r}{\theta }}+\nabla ·\mathbf{j}=\rho \gamma .$$

  (15)
• Further balance laws can occur. Here, we restrict our attention to dielectrics and then we need the laws for the electric field $\mathsf{E}$ and the electric displacement $\mathsf{D}$ while

  $$\mathsf{D}={ϵ}_0\mathsf{E}+\mathsf{P},$$

  where ${ϵ}_0=8.854·{10}^{-12}$ C²/N·m² is the permittivity of free space. We assume that neither free charges nor electric current occur. Hence, the pertinent Maxwell equations are

  $$\nabla ·\mathsf{D}=0,\nabla \times \times \mathsf{E}=-{\partial }_t\mathsf{B}=\mathbf{0},$$

  (16)

  with the vanishing of ${\partial }_t\mathsf{B}$ being valid in stationary conditions.

3. Second Law of Thermodynamics

The entropy supply $\rho r/\theta$ is reminiscent of the physical scheme of heat divided by the absolute temperature at which heat is transferred. Quite naturally, $\mathbf{j}$ is viewed as the entropy flux. In the Coleman–Noll paper given in [4], $\mathbf{j}$ is identified with $\mathbf{q}/\theta$ and $\gamma$ is viewed as the (rate of) entropy production (density); Equation (15) takes the form

$$\rho \dot{\eta }-{\displaystyle \frac{\rho r}{\theta }}+\nabla ·{\displaystyle \frac{\mathbf{q}}{\theta }}=\rho \gamma .$$

(17)

A thermodynamic process is the set of functions $\rho ,\mathbf{v},\mathbf{T},\mathbf{l},\epsilon ,\mathsf{E},\mathsf{p},\eta ,$ and $\mathbf{q}$ on $\Omega \times \mathbb{R}$ satisfying the balance Equations (8), (10), (12), (14), and (15) and Maxwell’s equations (16). The functions $\mathbf{b}$ and r are assumed to be arbitrary functions. Indeed, in rational thermodynamics, they are required to be assigned arbitrarily to ensure satisfaction of the balance Equations (10) and (14). Hence, in addition to Maxwell’s equations, only (8), (12), and (15) are constraints on the admissible functions. Yet, Equation (15) involves the function $\gamma$ that is unknown and cannot be arbitrarily assigned.

As everyone knows, the number of unknown functions exceeds that of the balance equations. Consistently, we need to add constitutive assumptions, which then limit the class of processes that bodies comprising a given material may undergo. The whole set of balance equations, up to (15), is a restriction on the admissible constitutive assumptions.

The balance of entropy, (15) or (17), contains the right-hand side $\rho \gamma$ as an unknown function. Consistently, borrowing from the principle of the increase of entropy [8], Coleman and Noll stated the following:

Postulate 1.

For every admissible process, the inequality $\gamma \ge 0$ is valid.

With the condition $\gamma \ge 0$, Equation (17), or Equation (15), is usually referred to as the CD (Clausius–Duhem) inequality. Müller [9] observed that $\mathbf{j}$ should also be unknown and given by a constitutive equation. Accordingly, letting

$$\mathbf{j}={\displaystyle \frac{\mathbf{q}}{\theta }}+\mathbf{k},$$

we can write (15) in the form

$$\rho \dot{\eta }-{\displaystyle \frac{\rho r}{\theta }}+\nabla ·({\displaystyle \frac{\mathbf{q}}{\theta }}+\mathbf{k})=\rho \gamma ;$$

(18)

the vector $\mathbf{k}$ is denoted as extra-entropy flux and is an unknown function within the thermodynamic process. Much research has been undertaken with the purpose of characterizing restrictions on the models of material behaviour, at least along the lines of rational thermodynamics [5].

Relation (15) can be viewed, and is currently so, as the definition of the value of $\rho \gamma$, the entropy production per unit volume. Indeed, by the character of the definition, we conclude that the left-hand side is eventually the expression of $\rho \gamma$. For definiteness, if we model a rigid heat conductor through $\mathbf{q}(\theta ,\nabla \theta )$, we arrive at

$$-{\displaystyle \frac{1}{{\theta }^2}}\mathbf{q}·\nabla \theta =\rho \gamma .$$

By using Fourier’s law

$$\mathbf{q}=-\kappa \left(\theta \right)\nabla \theta$$

we have

$$\rho \gamma ={\displaystyle \frac{\kappa \left(\theta \right)}{{\theta }^2}}{|\nabla \theta |}^2.$$

As a second example, if we describe a viscous fluid through the Navier–Stokes stress function

$$\mathbf{T}=-p(\rho ,\theta )\mathbf{1}+2\mu \mathbf{D}+\lambda \left(\mathrm{tr}\mathbf{D}\right)\mathbf{1}$$

we arrive at

$$\rho \theta \gamma ={2\mu \left|\mathbf{D}\right|}^2+\lambda {\left(\mathrm{tr}\mathbf{D}\right)}^2.$$

The positive character of $\gamma$ implies $\kappa \ge 0$ and $\mu \ge 0,2\mu +3\lambda \ge 0$ [3] (§2.6).

Still, with the formulation (15) and the conceptual role stated in the Coleman–Noll postulate, a generalization has been stated letting $\gamma$ be a constitutive function per se [3,10]. Hence, the thermodynamic process comprises the entropy flux $\mathbf{j}$ and the entropy production $\gamma$.

Generalized second law of thermodynamics.

The entropy production $\gamma$ satisfies (18) and is a non-negative function for every admissible thermodynamic process for all times t and points $\mathbf{x}$ of the body.

As a comment, in the current literature, the equality in (17) is a definition of $\gamma$, and hence the equality holds identically. Instead, if $\gamma$ is a peculiar constitutive function, then $\gamma$ satisfies the equality for certain processes and not identically. This conceptually new approach is commented upon in the next sections.

Remark 1.

In principle, a further (couple) term, say $\widehat{\mathbf{l}}$, might occur in (12) and be assigned arbitrarily so that Equation (12) is not a constraint. However, if the material does not involve any extra couple, then we have to satisfy (12). This is why, hereafter, we shall regard (12) as a constraint.

Remark 2.

The view that $\rho \gamma$ in Equation (18), or in Equation (15), is the entropy production traces back to Clausius [11]. Indeed, the Clausius inequality for closed systems, $dS\ge dQ/\theta$, (equality for reversible heat transfers) shows that $dQ/\theta$ does not account for all contributions to the entropy change $dS$. The Clausius inequality is a consequence of applying the second law of thermodynamics (for cycles) at each infinitesimal stage of heat transfer.

For later application, we multiply (18) by $\theta$ to obtain

$$\rho \theta \dot{\eta }-(\rho r-\nabla ·\mathbf{q})-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta +\theta \nabla ·\mathbf{k}=\rho \theta \gamma .$$

Replacing $\rho r-\nabla ·\mathbf{q}$ from the balance of energy (14), it follows that

$$-\rho (\dot{\epsilon }-\theta \dot{\eta })+\mathbf{T}·\mathbf{L}+\rho \mathsf{E}·\dot{\mathsf{p}}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta +\theta \nabla ·\mathbf{k}=\rho \theta \gamma .$$

(19)

For formal convenience, we consider the specific free energy

$$\varphi =\epsilon -\theta \eta -\mathsf{E}·\mathsf{p}.$$

Hence, the CD inequality (19) can be written in the form

$$-\rho (\dot{\varphi }+\eta \dot{\theta })+\mathbf{T}·\mathbf{L}-\mathsf{P}·\dot{\mathsf{E}}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta +\theta \nabla ·\mathbf{k}=\rho \theta \gamma ,$$

(20)

where $\mathsf{P}=\rho \mathsf{p}$ is the polarization (per unit volume).

We now proceed by examining three aspects of materials modelling in connection with the thermodynamic restrictions. First, possible consequences of the entropy production $\gamma$ as a constitutive function instead of being given by the CD inequality are considered. Second, a method of finding models satisfying constraints like (12) is discussed. Third, the possible restrictions induced by the objectivity principle are discussed. For definiteness, these aspects are investigated for deformable dielectric solids.

Relation to Other Approaches to Entropy Production

Within non-equilibrium thermodynamics, the entropy production $\rho \gamma$ (usually denoted by $\sigma$) is expressed in terms of fluxes $\left\{J_i\right\}$ and forces $\left\{X_i\right\}$ in the form

$$\rho \gamma ={\sum }_{i=1}^n{J}_i{X}_i,$$

where n is the number of independent fluxes (see, e.g., [12], ch. 1, [13], and refs therein). This statement is based on a precise characterization of fluxes and forces, which would not be the case in all continuum models.

A more general approach is developed by recourse to dissipation potentials. By analogy with Rayleigh’s function in Lagrangian mechanics, there are approaches where entropy is replaced with a dissipation potential, say D, as a function of the time derivative $\dot{q}$ of the state variables q. The corresponding variational principle leads to the system of n equations

$${\displaystyle \frac{\delta F}{\delta q}}={\displaystyle \frac{\delta D}{\delta \dot{q}}}$$

where $\delta$ denotes the variational derivative and F is the free energy [14].

Among the approaches to entropy production in dissipative materials, it is worth mentioning the multiscale thermodynamics where the dynamics of macroscopic systems is developed jointly on various different scales [15]. The multiscale scheme has some similarities to the approach of modeling using internal variables in continuum mechanics, where additional (internal) variables are introduced to describe complex dissipative behaviors [16].

4. Tentative Models of Dielectric Solids

As a preliminary approach, we let the constitutive functions $\varphi ,\eta ,\mathbf{T},\mathsf{P},\mathbf{q}\mathbf{k},$ and $\gamma$ depend on the variables

$$\theta ,\mathbf{F},\mathsf{E},\nabla \theta .$$

We assume that $\varphi$ and $\mathbf{k}$ are continuously differentiable while $\eta ,\mathbf{T},\mathsf{P},\mathbf{q},\mathbf{k},$ and $\gamma$ are continuous. Computing $\dot{\varphi }$ and substituting in (20), we have

$$-\rho ({\partial }_{\theta }\varphi +\eta )\dot{\theta }+\mathbf{T}·\mathbf{L}-\rho {\partial }_{\mathbf{F}}·\dot{\mathbf{F}}-(\mathsf{P}+\rho {\partial }_{\mathsf{E}}\varphi )·\dot{\mathsf{E}}-\rho {\partial }_{\nabla \theta }\varphi ·(\nabla \theta {{\displaystyle )}}^{\dot{}}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta +\theta \nabla ·\mathbf{k}=\rho \theta \gamma .$$

The linearity and arbitrariness of $\dot{\theta },(\nabla \theta {{\displaystyle )}}^{\dot{}}$, and $\dot{\mathsf{E}}$ imply that

$${\partial }_{\nabla \theta }\varphi =\mathbf{0},\eta =-{\partial }_{\theta }\varphi ,$$

$$\mathsf{P}=-\rho {\partial }_{\mathsf{E}}\varphi .$$

(21)

Since $\dot{\mathbf{F}}=\mathbf{L}\mathbf{F}$, the remaining relation reads

$$(\mathbf{T}-\rho {\partial }_{\mathbf{F}}\varphi {\mathbf{F}}^T)·\mathbf{L}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta +\theta \nabla ·\mathbf{k}=\rho \theta \gamma .$$

The linearity and arbitrariness of $\mathbf{L}$ imply that

$$\mathbf{T}=\rho {\partial }_{\mathbf{F}}\varphi {\mathbf{F}}^T.$$

(22)

As for $\nabla ·\mathbf{k}$, the term ${\partial }_{\theta }\mathbf{k}·\nabla \theta$ is allowed and this would imply a joint contribution $(\theta {\partial }_{\theta }\mathbf{k}-\mathbf{q}/\theta )·\nabla \theta$. Yet, a dependence of the vector $\mathbf{k}$ on the scalar $\theta$ is allowed only if ${\partial }_{\theta }\mathbf{k}$ is a vector, say $\mathbf{d}$, so that $\mathbf{k}=\theta \mathbf{d}$. No such vector occur, and we let $\mathbf{k}=\mathbf{0}$. The remaining condition is the heat conduction inequality

$$\mathbf{q}·\nabla \theta =\rho {\theta }^2\gamma ,$$

where, as always, $\gamma \ge 0$.

In polarizable media, the polarization $\mathsf{P}$ produces a couple density $\mathbf{l}=\mathsf{P}\times \times \mathsf{E}$. Hence, the balance condition (13) results in

$$\mathrm{skw}(\mathbf{T}+\mathsf{E}\otimes \mathsf{P})=\mathbf{0}.$$

(23)

In view of (21) and (22), the requirement (23) can be written in the form

$$\mathrm{skw}({\partial }_{\mathbf{F}}\varphi {\mathbf{F}}^T-\mathsf{E}\otimes {\partial }_{\mathsf{E}}\varphi )=\mathbf{0}.$$

(24)

Equation (24) is a constraint on the function $\varphi$ and denotes that $\varphi$ is required to depend on $\mathbf{F}$ and $\mathsf{E}$ in a suitable joint form. For definiteness, we look for a dependence of the form

$$\varphi =\varphi (\theta ,\mathbf{w}(\mathbf{F},\mathsf{E}\left)\right),$$

(25)

where $\mathbf{w}$ is a vector function to be determined. We prove that there is a double infinity of functions $\mathbf{w}$ that satisfy (24). First, we observe that replacing the function (25) in (24), we find

$$\mathrm{sym}\left[{\partial }_{w_Q}\varphi ({\partial }_{F_{iK}}{w}_Q{F}_{jK}-{\mathsf{E}}_i{\partial }_{{\mathsf{E}}_j}{w}_Q)\right]=0,$$

(26)

with the symmetric part being relative to the indices $i,j$. Let

$$\mathbf{w}=f\left(J\right){\mathbf{F}}^T\mathsf{E},w_Q=f\left(J\right)F_{pQ}{\mathsf{E}}_p,$$

(27)

where $J=det\mathbf{F}>0$ is the Jacobian of the deformation. By the derivative of a determinant, we have

$${\mathit{\partial }}_{F_{jK}}J=J{F}_{Ki}^{-1}$$

and then

$${\mathit{\partial }}_{F_{iK}}f\left(J\right)=f^{\prime }J{F}_{Ki}^{-1},$$

where $f^{\prime }$ is the derivative of f. Furthermore,

$${\mathit{\partial }}_{F_{iK}}{w}_Q{F}_{jK}={\mathsf{E}}_i{F}_{jQ}={\mathsf{E}}_i{\partial }_{{\mathsf{E}}_j}{w}_Q.$$

Hence, a direct computation yields

$${\partial }_{w_Q}\varphi ({\partial }_{F_{iK}}{w}_Q{F}_{jK}-{\mathsf{E}}_i{\partial }_{{\mathsf{E}}_j}{w}_Q)={\partial }_{w_Q}\varphi f^{\prime }J{F}_{pQ}{\mathsf{E}}_p{\delta }_{ij}.$$

Thus, the function (27) satisfies Equation (26) for any function f. Furthermore, if f is constant, then $\mathbf{w}=f{\mathbf{F}}^T\mathsf{E}$ makes $\varphi (\theta ,\mathbf{w})$ satisfy the stronger condition

$${\partial }_{\mathbf{F}}\varphi {\mathbf{F}}^T-\mathsf{E}\otimes {\partial }_{\mathsf{E}}\varphi =\mathbf{0}.$$

(28)

Another function $\mathbf{w}$ is now considered in the form

$$\mathbf{w}=f\left(J\right){\mathbf{F}}^{-1}\mathsf{E}.$$

(29)

Notice that

$${\partial }_{F_{iK}}{F}_{Qp}^{-1}=-F_{Qi}^{-1}{F}_{Kp}^{-1}.$$

Hence, a direct calculation yields

$${\partial }_{w_Q}\varphi ({\partial }_{F_{iK}}{w}_Q{F}_{jK}-{\mathsf{E}}_i{\partial }_{{\mathsf{E}}_j}{w}_Q)={\partial }_{w_Q}\varphi [{\displaystyle \frac{f^{\prime }J}{f}}{w}_Q{\delta }_{ij}-f(F_{Qi}^{-1}{\mathsf{E}}_j+{\mathsf{E}}_i{F}_{Qj}^{-1})].$$

Thus, Function (29) also satisfies Equation (26) for any function f. Instead, (29) does not satisfy Equation (26) even though f is constant.

Some comments are in order about the functions $\mathbf{w}$ so determined. The function $\mathbf{w}={\mathbf{F}}^T\mathsf{E}$ is often selected in the literature [6,7,17,18] and the selection is motivated by the fact that ${\mathbf{F}}^T\mathsf{E}$ is a Lagrangian field, namely the electric field in the reference configuration. Both ${\mathbf{F}}^T\mathsf{E}$ and ${\mathbf{F}}^{-1}\mathsf{E}$ are considered in [19] and ([3], ch. 12), on the basis of their invariance under Euclidean transformations; $f\left(J\right){\mathbf{F}}^T\mathsf{E}$ and $f\left(J\right){\mathbf{F}}^{-1}\mathsf{E}$ are invariant too. The connection between Euclidean transformations and the formulation of constitutive equations is revisited in the next section.

The arguments of this section indicate that there can be various behaviours of materials upon deformation. The simplest case involves materials where $\mathsf{P}$ and $\mathsf{E}$ are parallel in any configuration. Hence, $\mathsf{E}\otimes \mathsf{P}\in \mathrm{Sym}$ and (23) degenerates to the standard condition $\mathbf{T}\in \mathrm{Sym}$ of elasticity.

While $\mathbf{T}$ in the formulae above is the stress, viz., the total stress, there are several approaches in the literature where the stress is the sum of a mechanical stress, say $\mathbf{\sigma }$, and a Maxwell stress, say ${\mathbf{\tau }}_M$, so that $\mathbf{T}=\mathbf{\sigma }+{\mathbf{\tau }}_M$. This is so, e.g., in [6,7], while in [20], ${\mathbf{\tau }}_M$ is split in a polarization term ${\mathbf{\sigma }}^{pol}$ and a symmetric Maxwell stress ${\mathbf{\sigma }}^{max}$.

A direct motivation for the splitting $\mathbf{T}=\mathbf{\sigma }+{\mathbf{\tau }}_M$ with the derivation of ${\mathbf{\tau }}_M$ is given in [7] as follows. Consider the electric body force $(\mathsf{P}·\nabla )\mathsf{E}$. Using (16), we find

$$(\mathsf{P}·\nabla ){\mathsf{E}}_j={\mathsf{P}}_i{\partial }_{x_i}{\mathsf{E}}_j={\mathsf{D}}_i{\partial }_{x_i}{\mathsf{E}}_j-{ϵ}_0{\mathsf{E}}_i{\partial }_{x_i}{\mathsf{E}}_j={\partial }_{x_i}\left({\mathsf{D}}_i{\mathsf{E}}_j\right)-{ϵ}_0{\mathsf{E}}_i{\partial }_{x_j}{\mathsf{E}}_i$$

whence

$$(\mathsf{P}·\nabla )\mathsf{E}=\nabla ·{\mathbf{\tau }}_M,{\mathbf{\tau }}_M=\mathsf{E}\otimes \mathsf{D}-\frac{1}{2}{ϵ}_0{\left|\mathsf{E}\right|}^2\mathbf{1}.$$

(30)

5. Constitutive Equations for Deformable Dielectric

Any formulation of constitutive equations is required to obey the objectivity (or material frame indifference) principle. In this connection, we start with the notation concerning Euclidean transformations.

Let $\mathcal{F}$ and ${\mathcal{F}}^{*}$ be two observers. The corresponding position vectors $\mathbf{x}$ and ${\mathbf{x}}^{*}$ of a point are related by the (Euclidean) transformation

$${\mathbf{x}}^{*}=\mathbf{y}\left(t\right)+\mathbf{Q}\left(t\right)\mathbf{x},$$

(31)

where $\mathbf{y}$ is a vector and $\mathbf{Q}$ is a rotation tensor, and $det\mathbf{Q}=1$. Under the transformation, it is

$${\mathbf{F}}^{*}=\mathbf{Q}\mathbf{F},J^{*}=J,$$

while $\mathsf{E}$, as well as $\mathsf{P}$ and $\mathsf{D}$, is assumed to transform as a vector,

$${\mathsf{E}}^{*}=\mathbf{Q}\mathsf{E}.$$

Consequently, ${\mathbf{F}}^T\mathsf{E}$ and ${\mathbf{F}}^{-1}\mathsf{E}$ are invariant in that

$${\left({\mathbf{F}}^T\mathsf{E}\right)}^{*}={\mathbf{F}*}^T{\mathsf{E}}^{*}={\mathbf{F}}^T{\mathbf{Q}}^T\mathbf{Q}\mathsf{E}={\mathbf{F}}^T\mathsf{E}$$

and the like for ${\mathbf{F}}^{-1}\mathsf{E}$ because ${\mathbf{Q}}^{-1}={\mathbf{Q}}^T$. The stress $\mathbf{T}$ transforms as a tensor, ${\mathbf{T}}^{*}=\mathbf{Q}\mathbf{T}{\mathbf{Q}}^T$, and the heat flux $\mathbf{q}$ as a vector, ${\mathbf{q}}^{*}=\mathbf{Q}\mathbf{q}$.

• Objectivity principle. The constitutive equations must be form-invariant under Euclidean transformations.

Invariance means that a function $\mathbf{g}\left(\mathbf{u}\right)$ is subject to

$${\mathbf{g}}^{*}\left(\mathbf{u}\right)=\mathbf{g}\left({\mathbf{u}}^{*}\right).$$

As an important example, consider the possible dependence of the stress $\mathbf{T}$ on the velocity gradient $\mathbf{L}$. As shown in Appendix A, ${\mathbf{L}}^{*}=\mathbf{Q}\mathbf{L}{\mathbf{Q}}^T+\dot{\mathbf{Q}}{\mathbf{Q}}^T$ and $\dot{\mathbf{Q}}{\mathbf{Q}}^T\in \mathrm{Skw}$. Hence, invariance requires that

$$\mathbf{Q}\mathbf{T}{\mathbf{Q}}^T\left(\mathbf{L}\right)=\mathbf{T}(\mathbf{Q}\mathbf{L}{\mathbf{Q}}^T+\dot{\mathbf{Q}}{\mathbf{Q}}^T).$$

Choosing $\mathbf{Q}\left(t\right)$ such that $\mathbf{Q}\left(0\right)=\mathbf{1}$ and $\dot{\mathbf{Q}}=-\mathbf{W}$, it follows that

$$\mathbf{T}\left(\mathbf{L}\right)=\mathbf{T}\left(\mathbf{D}\right).$$

The same argument applies to any constitutive function. Hence, constitutive functions can depend on $\mathbf{L}$ only through the stretching $\mathbf{D}$.

To account for the elastic properties of a material, we allow for a dependence on the deformation gradient $\mathbf{F}$. However, a scalar-valued function can depend only through (Euclidean) invariants. The Green–Lagrange strain $\mathbf{E}=\frac{1}{2}({\mathbf{F}}^T\mathbf{F}-\mathbf{1})$ is invariant and so is any function of $\mathbf{E}$. In particular,

$$J=det\mathbf{F}={[det(2\mathbf{E}+\mathbf{1})]}^{1/2},{\left|\mathbf{F}\right|}^2=2\mathrm{tr}\mathbf{E}+3.$$

5.1. Constitutive Assumptions and Thermodynamic Restrictions

We look for models of dielectrics where $\mathsf{P}$ and $\mathsf{E}$ need not be parallel in deformed configurations. The arguments of the previous section indicate that, in a three-dimensional setting, the symmetry condition (23) is satisfied by considering the field ${\mathbf{F}}^T\mathsf{E}$ (or ${\mathbf{F}}^{-1}\mathsf{E}$) as an effective electric field in the material. For definiteness, let $\mathfrak{E}={\mathbf{F}}^T\mathsf{E}$ and consider the set of variables

$$\theta ,\mathbf{F},\mathfrak{E},\nabla \theta ,\mathbf{D},\mathbf{q}$$

for the constitutive functions $\varphi ,\eta ,\mathsf{P},\mathbf{T},\dot{\mathbf{q}},$ and $\gamma$. As we shall show, the extra-entropy flux $\mathbf{k}$ turns out to be zero; to save writing, we let $\mathbf{k}=\mathbf{0}$ from scratch. The scalars $\varphi ,\eta ,$ and $\gamma$ should depend on $\mathbf{F},\mathfrak{E},\nabla \theta ,\mathbf{D},$ and $\mathbf{q}$ through their invariant scalars.

Computing $\dot{\varphi }$ and substituting in (20) yields

$$\begin{array}{c}\hfill -\rho ({\partial }_{\theta }\varphi +\eta )\dot{\theta }-\rho {\partial }_{\mathbf{F}}\varphi ·\dot{\mathbf{F}}-\rho {\partial }_{\mathfrak{E}}\varphi ·\dot{\mathfrak{E}}-\rho {\partial }_{\nabla \theta }\varphi ·(\nabla \theta {{\displaystyle )}}^{\dot{}}-\rho {\partial }_{\mathbf{D}}\varphi ·\dot{\mathbf{D}}-\rho {\partial }_{\mathbf{q}}\varphi ·\dot{\mathbf{q}}\\ \hfill +\mathbf{T}·\mathbf{L}-\mathsf{P}·\dot{\mathsf{E}}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta \gamma .\end{array}$$

(32)

Notice that $\dot{\mathbf{F}}=\mathbf{L}\mathbf{F}$ and $\mathsf{E}={\mathbf{F}}^{-T}\mathfrak{E}$. Hence, we find that

$${\partial }_{\mathbf{F}}\varphi ·\dot{\mathbf{F}}={\partial }_{\mathbf{F}}\varphi ·\left(\mathbf{L}\mathbf{F}\right)=\left({\partial }_{\mathbf{F}}\varphi {\mathbf{F}}^T\right)·\mathbf{L},$$

$$\dot{\mathsf{E}}=-{\mathbf{L}}^T\mathsf{E}+{\mathbf{F}}^{-T}\dot{\mathfrak{E}},$$

$$\mathsf{P}·\dot{\mathsf{E}}=-(\mathsf{E}\otimes \mathsf{P})·\mathbf{L}+\left({\mathbf{F}}^{-1}\mathsf{P}\right)·\dot{\mathfrak{E}}.$$

Upon substitution in (32) and using the decomposition $\mathbf{L}=\mathbf{D}+\mathbf{W}$, it follows that

$$\begin{array}{c}\hfill -\rho ({\partial }_{\theta }\varphi +\eta )\dot{\theta }+(\mathbf{T}+\mathsf{E}\otimes \mathsf{P}-\rho {\partial }_{\mathbf{F}}\varphi {\mathbf{F}}^T)·\mathbf{D}+(\mathbf{T}+\mathsf{E}\otimes \mathsf{P}-\rho {\partial }_{\mathbf{F}}\varphi {\mathbf{F}}^T)·\mathbf{W}\\ \hfill -(\rho {\partial }_{\mathfrak{E}}\varphi +{\mathbf{F}}^{-1}\mathsf{P})·\dot{\mathfrak{E}}-\rho {\partial }_{\nabla \theta }\varphi ·(\nabla \theta {{\displaystyle )}}^{\dot{}}-\rho {\partial }_{\mathbf{D}}\varphi ·\dot{\mathbf{D}}-\rho {\partial }_{\mathbf{q}}\varphi ·\dot{\mathbf{q}}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta \gamma .\end{array}$$

(33)

The linearity and arbitrariness of $(\nabla \theta {{\displaystyle )}}^{\dot{}},\dot{\mathbf{D}},\mathbf{W},\dot{\theta },$ and $\dot{\mathfrak{E}}$ imply that

$${\partial }_{\nabla \theta }\varphi =\mathbf{0},{\partial }_{\mathbf{D}}\varphi =\mathbf{0},$$

$$\mathbf{T}+\mathsf{E}\otimes \mathsf{P}-\rho {\partial }_{\mathbf{F}}\varphi {\mathbf{F}}^T\in \mathrm{Sym},$$

(34)

$${\mathbf{F}}^{-1}\mathsf{P}=-\rho {\partial }_{\mathfrak{E}}\varphi .$$

(35)

Comparing (34) and (23), we find the necessary condition

$${\partial }_{\mathbf{F}}\varphi {\mathbf{F}}^T\in \mathrm{Sym}.$$

This condition holds if $\varphi$ depends on $\mathbf{F}$ through $\mathbf{E}$ in that

$${\partial }_{\mathbf{F}}\varphi {\mathbf{F}}^T=\mathbf{F}{\partial }_{\mathbf{E}}\varphi {\mathbf{F}}^T.$$

Thus, we let $\varphi$ depend on $\mathbf{F}$ through $\mathbf{E}$ and hence the CD inequality (33) simplifies to

$$(\mathbf{T}+\mathsf{E}\otimes \mathsf{P}-\rho \mathbf{F}{\partial }_{\mathbf{E}}\varphi {\mathbf{F}}^T)·\mathbf{D}-\rho {\partial }_{\mathbf{q}}\varphi ·\dot{\mathbf{q}}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta \gamma .$$

(36)

A further simplification of (36) follows by assuming that

$$\mathbf{T}+\mathsf{E}\otimes \mathsf{P}-\rho \mathbf{F}{\partial }_{\mathbf{E}}\varphi {\mathbf{F}}^T=:\mathcal{T}(\theta ,\mathbf{F},\mathbf{D})=O\left(\right|\mathbf{D}\left|\right),{\partial }_{\mathbf{D}}\dot{\mathbf{q}}=\mathbf{0}$$

(37)

and

$$\dot{\mathbf{q}}(\theta ,\mathbf{F},\mathsf{E},\nabla \theta ,\mathbf{q})=O\left(\right|\mathsf{E}|+|\nabla \theta |+\left|\mathbf{q}\right|).$$

Hence, $-\rho {\partial }_{\mathbf{q}}\varphi ·\dot{\mathbf{q}}-(1/\theta )\mathbf{q}·\nabla \theta$ is independent of $\mathbf{D}$ while $\mathcal{T}·\mathbf{D}$ is independent of $\nabla \theta$. Correspondingly, we assume

$$\gamma ={\gamma }_D(\theta ,\mathbf{F},\mathbf{D})+{\gamma }_q(\theta ,\mathbf{F},\mathsf{E},\nabla \theta ,\mathbf{q}),{\gamma }_D(\theta ,\mathbf{F},\mathbf{0})=0,{\gamma }_q(\theta ,\mathbf{F},\mathsf{E},\mathbf{0},\mathbf{q})=0.$$

Consequently, the CD inequality (36) splits into

$$\mathcal{T}(\theta ,\mathbf{F},\mathbf{D})·\mathbf{D}=\rho \theta {\gamma }_D(\theta ,\mathbf{F},\mathbf{D}),$$

(38)

$$-\rho {\partial }_{\mathbf{q}}\varphi ·\dot{\mathbf{q}}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta {\gamma }_q(\theta ,\mathbf{F},\mathsf{E},\nabla \theta ,\mathbf{q})$$

(39)

where ${\gamma }_D\ge 0,{\gamma }_q\ge 0$.

5.2. Representation Formula

As is exemplified by (38) and (39), the second law, through the CD inequality, leads to equations of the form

$$\mathbf{u}·\mathbf{w}=g$$

(40)

where $\mathbf{u}$ and $\mathbf{w}$ are vectors or tensors and g is a scalar; $\mathbf{u}$ is an unknown vector (or tensor) function; $\mathbf{w}$ is a vector (or tensor) variable; and g is a scalar function. As shown in the next sub-sections, g might be (proportional to) the entropy production, but the problem applies for more general scalar functions.

If $\mathbf{n}$ is a unit vector, then $\mathbf{u}$ can be represented as

$$\mathbf{u}=(\mathbf{u}·\mathbf{n})\mathbf{n}+(\mathbf{1}-\mathbf{n}\otimes \mathbf{n})\mathbf{u};$$

(41)

geometrically, this means that $\mathbf{u}$ is the sum of the longitudinal part $(\mathbf{u}·\mathbf{n})\mathbf{n}$, along $\mathbf{n}$, and the transverse part, $(\mathbf{1}-\mathbf{n}\otimes \mathbf{n})\mathbf{u}$. Letting $\mathbf{n}=\mathbf{w}/\left|\mathbf{w}\right|$, we can write Equation (41) in the form

$$\mathbf{u}={\displaystyle \frac{\mathbf{u}·\mathbf{w}}{{\mathbf{w}}^2}}\mathbf{w}+(\mathbf{1}-{\displaystyle \frac{\mathbf{w}\otimes \mathbf{w}}{{\mathbf{w}}^2}})\mathbf{u}.$$

(42)

Equation (40) indicates the vector (or tensor) $\mathbf{w}$ and expresses $\mathbf{u}·\mathbf{w}$ as the scalar g. The transverse character of the second term in (42) is verified in that

$$\left[(\mathbf{1}-{\displaystyle \frac{\mathbf{w}\otimes \mathbf{w}}{{\mathbf{w}}^2}})\mathbf{u}\right]·\mathbf{w}=[\mathbf{u}-{\displaystyle \frac{(\mathbf{u}·\mathbf{w})\mathbf{w}}{{\mathbf{w}}^2}}]·\mathbf{w}=0.$$

Now, Equation (40) specifies the longitudinal part in terms of g but leaves the transverse part undetermined. Furthermore, for any vector $\mathbf{f}$, we have that

$$\left[(\mathbf{1}-{\displaystyle \frac{\mathbf{w}\otimes \mathbf{w}}{{\mathbf{w}}^2}})\mathbf{f}\right]·\mathbf{w}=0$$

(identically) and then that $\left[\right(\mathbf{1}-\mathbf{w}\otimes \mathbf{w}/{\left|\mathbf{w}\right|}^2\left)\mathbf{f}\right]$ is transverse, too. Hence, the general representation of $\mathbf{u}$ subject to (40) is

$$\mathbf{u}={\displaystyle \frac{g}{{\mathbf{w}}^2}}\mathbf{w}+(\mathbf{1}-{\displaystyle \frac{\mathbf{w}\otimes \mathbf{w}}{{\mathbf{w}}^2}})\mathbf{f}.$$

(43)

Equation (43) provides the vector (or tensor) representation of $\mathbf{u}$. If $\mathbf{u}$ and g depend on appropriate variables, in addition to $\mathbf{w}$, then these variables are allowed to occur in $\mathbf{f}$ too.

5.3. Representation of $\mathcal{T}$

We now apply the representation formula (43) to the stress tensor $\mathcal{T}$ in (38). We then let $\mathbf{u}=\mathcal{T}$ and $\mathbf{w}=\mathbf{D}$. For definiteness, let

$$\mathbf{f}=\Xi \mathbf{D},$$

where $\Xi$ is a fourth-order symmetric tensor. Hence, we have

$$\mathcal{T}={\displaystyle \frac{g-\mathbf{D}·\Xi \mathbf{D}}{{\left|\mathbf{D}\right|}^2}}\mathbf{D}+\Xi \mathbf{D}.$$

The tensor $\Xi$ need not be positive definite. In any case,

$$\mathcal{T}·\mathbf{D}=g\ge 0$$

as is required by the CD inequality. Instead if, as is customary, we let $\Xi$ be positive definite, then the choice $g=\mathbf{D}·\Xi \mathbf{D}$ gives

$$\mathcal{T}·\mathbf{D}=\mathbf{D}·\Xi \mathbf{D}\ge 0$$

as expected.

Using the definition (37) of $\mathcal{T}$, we can write, e.g.,

$$\mathbf{T}=-\mathsf{E}\otimes \mathsf{P}+\rho \mathbf{F}{\partial }_{\mathbf{E}}\varphi {\mathbf{F}}^T+{\displaystyle \frac{g-\mathbf{D}·\Xi \mathbf{D}}{{\left|\mathbf{D}\right|}^2}}\mathbf{D}+\Xi \mathbf{D}.$$

The choice $g=\mathbf{D}·\Xi \mathbf{D}$ with $\Xi$ positive definite gives a classical representation of the dissipative part.

5.4. Representation of $\dot{\mathbf{q}}$

The thermodynamic requirement given in (39) on the model of heat conduction can be viewed in the form given in (40) via the identifications

$$\mathbf{u}=\dot{\mathbf{q}},\mathbf{w}={\partial }_{\mathbf{q}}\varphi ,g=-(\theta {\gamma }_q+{\displaystyle \frac{1}{\rho \theta }}\mathbf{q}·\nabla \theta );$$

this model shows that g need not coincide with, or merely be proportional to, the pertinent entropy production $\gamma$. However, constitutive equations are required to obey the objectivity principle. Now, $\mathbf{q},{\partial }_{\mathbf{q}}\varphi ,\nabla \theta$ are objective (vectors) and ${\gamma }_q$ has to be an invariant (scalar). Instead, $\dot{\mathbf{q}}$ is not objective. To overcome this drawback, we let $\varphi$ depend quadratically on $\mathbf{q}$ so that

$${\partial }_{\mathbf{q}}\varphi =\alpha (\theta ,\mathbf{E},\mathsf{E})\mathbf{q},$$

with $\alpha$ being invariant under SO(3). Hence,

$${\partial }_{\mathbf{q}}\varphi ·\dot{\mathbf{q}}=\alpha \mathbf{q}·\dot{\mathbf{q}}=\alpha \mathbf{q}·\stackrel{\circ }{\mathbf{q}},$$

where $\stackrel{\circ }{\mathbf{q}}$ denotes the co-rotational derivative,

$$\stackrel{\circ }{\mathbf{q}}:=\dot{\mathbf{q}}-\mathbf{W}\mathbf{q}.$$

Consequently, the identifications are

$$\mathbf{u}=\stackrel{\circ }{\mathbf{q}},\mathbf{w}=\alpha \mathbf{q},g=-(\theta {\gamma }_q+{\displaystyle \frac{1}{\rho \theta }}\mathbf{q}·\nabla \theta ).$$

Substitution in (43) results in

$$\stackrel{\circ }{\mathbf{q}}=-{\displaystyle \frac{\theta {\gamma }_q}{\alpha {\mathbf{q}}^2}}\mathbf{q}-{\displaystyle \frac{1}{\rho \theta \alpha {\mathbf{q}}^2}}(\mathbf{q}·\nabla \theta )\mathbf{q}+(\mathbf{1}-{\displaystyle \frac{\mathbf{q}\otimes \mathbf{q}}{{\mathbf{q}}^2}})\mathbf{f}.$$

In addition to terms collinear with $\mathbf{q}$, we look for terms collinear with $\nabla \theta$ and $\mathsf{E}$. Accordingly, we let

$$\mathbf{f}=\beta \nabla \theta +\nu \mathsf{E}.$$

It follows that

$$\stackrel{\circ }{\mathbf{q}}=-{\displaystyle \frac{\theta {\gamma }_q}{\alpha {\mathbf{q}}^2}}\mathbf{q}-{\displaystyle \frac{1}{\rho \theta \alpha {\mathbf{q}}^2}}(\mathbf{q}·\nabla \theta )\mathbf{q}+\beta (\nabla \theta -{\displaystyle \frac{\mathbf{q}·\nabla \theta }{{\mathbf{q}}^2}}\mathbf{q})+\nu (\mathsf{E}-{\displaystyle \frac{\mathbf{q}·\mathsf{E}}{{\mathbf{q}}^2}}\mathbf{q}).$$

The particular choice

$$\beta =-{\displaystyle \frac{1}{\rho \theta \alpha }}$$

simplifies the representation to

$$\stackrel{\circ }{\mathbf{q}}=-{\displaystyle \frac{\theta {\gamma }_q}{\alpha {\mathbf{q}}^2}}\mathbf{q}-{\displaystyle \frac{1}{\rho \theta \alpha }}\nabla \theta +\nu (\mathsf{E}-{\displaystyle \frac{\mathbf{q}·\mathsf{E}}{{\mathbf{q}}^2}}\mathbf{q}).$$

Thus, $\stackrel{\circ }{\mathbf{q}}$ can have a pure term along $\nabla \theta$; the same is not true for $\mathsf{E}$. In the particular case where $\nu =0$, we obtain the Maxwell–Cattaneo equation

$$\stackrel{\circ }{\mathbf{q}}=-{\displaystyle \frac{\theta {\gamma }_q}{\alpha {\mathbf{q}}^2}}\mathbf{q}-{\displaystyle \frac{1}{\rho \theta \alpha }}\nabla \theta .$$

Hence,

$${\tau }_q={\displaystyle \frac{\alpha {\mathbf{q}}^2}{\theta {\gamma }_q}}$$

plays the role of relaxation time. Furthermore, as $\alpha \to 0$, we obtain Fourier’s law with heat conductivity

$$\kappa ={\displaystyle \frac{{\mathbf{q}}^2}{\rho {\theta }^2{\gamma }_q}}>0.$$

If ${\partial }_{\mathbf{q}}\varphi =\mathbf{0}$, then Equation (40) follows with

$$\mathbf{u}=\mathbf{q}.\mathbf{w}=\nabla \theta ,g=-\rho {\theta }^2{\gamma }_q.$$

Accordingly, we use (43) with $\mathbf{f}=\nu \mathsf{E}$ to obtain

$$\mathbf{q}=-{\displaystyle \frac{\rho {\theta }^2{\gamma }_q}{{|\nabla \theta |}^2}}\nabla \theta +\nu (\mathsf{E}-{\displaystyle \frac{\mathsf{E}·\nabla \theta }{{|\nabla \theta |}^2}}\nabla \theta ).$$

Notice that if $\nu =0$, then we have Fourier’s law with conductivity

$$\kappa =\rho {\theta }^2{\gamma }_q/{|\nabla \theta |}^2$$

or entropy production

$${\gamma }_q={\displaystyle \frac{1}{\rho {\theta }^2\kappa }}{|\nabla \theta |}^2.$$

Again, a purely longitudinal term along the electric field $\mathsf{E}$ cannot occur. Indeed, the transverse character of the term in $\mathsf{E}$ implies that no term in the electric field cannot occur in a one-dimensional setting. A term in $\mathsf{E}$, sometimes motivated by the Seebeck effect [21], can hold in models with appropriate cross-coupled terms. Instead, ${\gamma }_q$ and $\alpha$ can depend on the electric field. Indeed, measures on the conductivity $\kappa$ have shown that the application of an electric field on, e.g., CUO/water nanofluid yields an increase in $\kappa$ by up to 30% [22,23].

5.5. Entropy Production and Non-Equilibrium Thermodynamics

There are similarities and differences regarding the conceptual view of entropy production between rational thermodynamics and mesoscopic non-equilibrium thermodynamics. This point is emphasized using an example involving the diffusion process.

Consider a mixture of non-reacting fluid constituents and label the fields pertaining to the single constituents by the index $\alpha =1,2,...,n$. Hence, ${\rho }_{\alpha }$ and ${\mathbf{v}}_{\alpha }$ are the mass density and the velocity; the continuity equation reads

$${\partial }_t{\rho }_{\alpha }+\nabla ·\left({\rho }_{\alpha }{\mathbf{v}}_{\alpha }\right)=0.$$

(44)

Let

$$\rho ={\sum }_{\alpha }{\rho }_{\alpha },c_{\alpha }={\displaystyle \frac{{\rho }_{\alpha }}{\rho }},\mathbf{v}={\sum }_{\alpha }{c}_{\alpha }{\mathbf{v}}_{\alpha },{\mathbf{u}}_{\alpha }={\mathbf{v}}_{\alpha }-\mathbf{v};$$

$\rho$ is the mass density of the mixture, $c_{\alpha }$ the $\alpha$-th concentration, $\mathbf{v}$ the barycentric velocity, and ${\mathbf{u}}_{\alpha }$ the diffusion velocity. Summing (44) on $\alpha$, we find

$${\partial }_t\rho +\nabla ·\left(\rho \mathbf{v}\right)=0.$$

(45)

Substitution of ${\rho }_{\alpha }=\rho c_{\alpha }$ and ${\rho }_{\alpha }{\mathbf{v}}_{\alpha }=\rho c_{\alpha }(\mathbf{v}+{\mathbf{u}}_{\alpha })$ in (44) and using (45) yields

$$\rho {\dot{c}}_{\alpha }+\nabla ·{\mathbf{h}}_{\alpha }=0,$$

(46)

where ${\dot{c}}_{\alpha }={\partial }_t{c}_{\alpha }+\mathbf{v}·\nabla c_{\alpha }$ and ${\mathbf{h}}_{\alpha }={\rho }_{\alpha }{\mathbf{u}}_{\alpha }$. The diffusion fluxes $\left\{{\mathbf{h}}_{\alpha }\right\}$ satisfy

$${\sum }_{\alpha }{\mathbf{h}}_{\alpha }=\mathbf{0}.$$

In light of (20), we let $\mathsf{P}=\mathbf{0}$, $\mathbf{q}=\mathbf{0}$, $\mathbf{T}=-p\mathbf{1}$, $\mathbf{T}·\mathbf{L}=-p\nabla ·\mathbf{v}$, and $\varphi =\psi =\epsilon -\theta \eta$ to obtain the CD inequality in the form

$$-\rho (\dot{\psi }+\eta \dot{\theta })-p\nabla ·\mathbf{v}+\theta \nabla ·\mathbf{k}=\rho \theta \gamma .$$

(47)

Let $\psi ,\eta ,p,\mathbf{k},{\mathbf{h}}_{\alpha }$, and $\gamma$ be functions of $\theta ,\rho ,\left\{c_{\alpha }\right\},\nabla \theta ,\nabla \rho ,$ and $\left\{c_{\alpha }\right\}$. Notice that, by (45), it follows that

$$\dot{\rho }=-\rho \nabla ·\mathbf{v}.$$

Compute $\dot{\psi }$ and substitute in (47) to obtain

$$\begin{array}{c}\hfill -\rho ({\partial }_{\theta }\psi +\eta )\dot{\theta }+({\rho }^2{\partial }_{\rho }\psi -p)\nabla ·\mathbf{v}-\rho ({\partial }_{\nabla \theta }\psi ·(\nabla \theta {{\displaystyle )}}^{\dot{}}-\rho {\partial }_{\nabla \rho }\psi ·(\nabla \rho {{\displaystyle )}}^{\dot{}}\\ \hfill -\rho {\sum }_{\alpha }{\partial }_{\nabla c_{\alpha }}\psi ·(\nabla c_{\alpha }{{\displaystyle )}}^{\dot{}}+{\sum }_{\alpha }({\partial }_{c_{\alpha }}\psi \nabla ·{\mathbf{h}}_{\alpha })+\theta \nabla ·\mathbf{k}=\rho \theta \gamma ,\end{array}$$

where Equation (46) is used. The linearity and arbitrariness of $\dot{\theta }$, $\nabla ·\mathbf{v}$, $(\nabla \theta {{\displaystyle )}}^{\dot{}}$, $(\nabla \rho ){{\displaystyle )}}^{\dot{}}$, and $(\nabla c_{\alpha }{{\displaystyle )}}^{\dot{}}$ imply

$$\eta =-{\partial }_{\theta }\psi ,p={\rho }^2{\partial }_{\rho }\psi ,{\partial }_{\nabla \theta }\psi =\mathbf{0},{\partial }_{\nabla \rho }\psi =\mathbf{0},{\partial }_{\nabla c_{\alpha }}\psi =\mathbf{0}.$$

Now, we divide the remaining equation by $\theta$ to find

$${\displaystyle \frac{1}{\theta }}{\sum }_{\alpha }({\partial }_{c_{\alpha }}\psi \nabla ·{\mathbf{h}}_{\alpha })+\nabla ·\mathbf{k}=\rho \gamma .$$

(48)

In light of the identity

$${\displaystyle \frac{1}{\theta }}{\sum }_{\alpha }({\partial }_{c_{\alpha }}\psi \nabla ·{\mathbf{h}}_{\alpha })=\nabla ·\left({\displaystyle \frac{1}{\theta }}{\sum }_{\alpha }{\partial }_{c_{\alpha }}\psi {\mathbf{h}}_{\alpha }\right)-{\mathbf{h}}_{\alpha }·\nabla \left({\displaystyle \frac{1}{\theta }}{\sum }_{\alpha }{\partial }_{c_{\alpha }}\psi \right)$$

we can write (48) in the form

$$\nabla ·(\mathbf{k}+{\displaystyle \frac{1}{\theta }}{\sum }_{\alpha }{\partial }_{c_{\alpha }}\psi {\mathbf{h}}_{\alpha })-{\sum }_{\alpha }[{\mathbf{h}}_{\alpha }·\nabla \left({\displaystyle \frac{1}{\theta }}{\partial }_{c_{\alpha }}\psi \right)]=\rho \gamma ,$$

whence we have the extra-entropy flux

$$\mathbf{k}=-{\displaystyle \frac{1}{\theta }}{\sum }_{\alpha }{\partial }_{c_{\alpha }}\psi {\mathbf{h}}_{\alpha },$$

and the entropy production

$$-{\sum }_{\alpha }[{\mathbf{h}}_{\alpha }·\nabla \left({\displaystyle \frac{1}{\theta }}{\partial }_{c_{\alpha }}\psi \right)]=\rho \gamma .$$

(49)

We represent $\psi$ in terms of the peculiar free energies $\left\{{\psi }_{\alpha }\right\}$,

$$\psi ={\sum }_{\alpha }{c}_{\alpha }{\psi }_{\alpha }(\theta ,\rho c_{\alpha }),$$

and compute

$${\partial }_{c_{\alpha }}\psi ={\psi }_{\alpha }+c_{\alpha }{\partial }_{{\rho }_{\alpha }}{\psi }_{\alpha }\rho ={\psi }_{\alpha }+{\displaystyle \frac{p_{\alpha }}{{\rho }_{\alpha }}}=:{\mu }_{\alpha },$$

with ${\mu }_{\alpha }$ being the classical $\alpha$-th chemical potential. Hence, the reduced Equation (49) reads

$${\sum }_{\alpha }({\mathbf{h}}_{\alpha }·{\displaystyle \frac{{\mu }_{\alpha }}{\theta }})=-\rho \gamma .$$

(50)

We now restrict our attention to the simple case of $n=2$ constituents, i.e., solvent and solute. Since ${\mathbf{h}}_1={\mathbf{h}}_2=\mathbf{0}$,

$${\sum }_{\alpha }({\mathbf{h}}_{\alpha }·\nabla {\displaystyle \frac{{\mu }_{\alpha }}{\theta }})=\mathbf{h}·\nabla {\displaystyle \frac{\widehat{\mu }}{\theta }},$$

where $\mathbf{h}={\mathbf{h}}_2,\widehat{\mu }={\mu }_2-{\mu }_1$. Consequently, Equation (50) simplifies to

$$\mathbf{h}·\nabla {\displaystyle \frac{\widehat{\mu }}{\theta }}=-\rho \gamma .$$

(51)

If we let

$$\mathbf{h}=-\kappa \nabla {\displaystyle \frac{\widehat{\mu }}{\theta }},\kappa \ge 0,$$

(52)

then (51) is satisfied and

$$\rho \gamma =\kappa |\nabla (\widehat{\mu }/\theta ){|}^2\ge 0.$$

In non-equilibrium thermodynamics (see, e.g., [24,25]), the chemical potential is given by the probability density $P(y,t)$ in the form

$$\mu (y,t)=k_B\theta ln{\displaystyle \frac{P(y,t)}{P_{eq}\left(y\right)}}+{\mu }_{eq},$$

where y denotes the set of variables, $P_{eq}$ and ${\mu }_{eq}$ denote the values at equilibrium, and $k_B$ is Boltzmann’s constant. The so-called thermodynamic force driving the diffusion process is ${\theta }^{-1}{\partial }_y\mu$ and the entropy production is taken as

$$\gamma =-{\displaystyle \frac{1}{\theta }}\int \kappa \nabla \mu ·d\mathbf{x},\nabla \mu ={\partial }_y\mu \nabla y.$$

This definition is consistent with (51) upon the identification $\mu =\widehat{\mu }$ and the assumption that $\theta$ is constant.

Returning to the reduced Equation (51), we notice that by using the representation Formula (43), we can write

$$\mathbf{h}=-{\displaystyle \frac{\rho \gamma }{|\nabla (\widehat{\mu }/\theta ){|}^2}}\nabla (\widehat{\mu }/\theta )+(\mathbf{1}-{\displaystyle \frac{\nabla (\widehat{\mu }/\theta )\otimes \nabla (\widehat{\mu }/\theta )}{|\nabla (\widehat{\mu }/\theta ){|}^2}})\mathbf{f}.$$

(53)

If

$$\rho \gamma =\kappa |\nabla (\widehat{\mu }/\theta ){|}^2,\mathbf{f}=\mathbf{0},$$

then we obtain (52). However, Equation (53) is thermodynamically consistent for any non-negative function $\gamma$ and vector $\mathbf{f}$. This exemplifies the generalization of having $\gamma$ as a constitutive function (of the variables under consideration).

As an example of the degree of freedom given by $\mathbf{f}$, we can take $\mathbf{f}=\beta \nabla \theta$ to obtain

$$\mathbf{h}=-{\displaystyle \frac{\rho \gamma +\beta \nabla (\widehat{\mu }/\theta )·\nabla \theta }{|\nabla (\widehat{\mu }/\theta ){|}^2}}\nabla (\widehat{\mu }/\theta )+\beta \nabla \theta ,$$

with an arbitrary function $\beta$. In general, any function $\mathbf{f}$ in (43) results in zero entropy production.

6. Stress Tensor and Electrostriction

In Section 4, we found that both

$$f\left(J\right){\mathbf{F}}^T\mathsf{E}\mathrm{and}f\left(J\right){\mathbf{F}}^{-1}\mathsf{E}$$

(54)

satisfy the constraint (24). Now, we examine the symmetry constraint (23) and look for the stress $\mathbf{T}$ depending on the choice of the two variables (54). For simplicity, we let $f=1$ and, to save writing, we consider only the pertinent terms of the CD inequality.

• $\mathfrak{E}={\mathbf{F}}^T\mathsf{E}$. With the variable $\mathfrak{E}$, we find

$$\dot{\mathsf{E}}=-{\mathbf{L}}^T\mathsf{E}+{\mathbf{F}}^{-T}\dot{\mathfrak{E}}$$

and then

$$\begin{array}{c}\hfill -\rho {\partial }_{\mathbf{E}}\varphi ·\dot{\mathbf{E}}+\mathbf{T}·\mathbf{L}-\rho {\partial }_{\mathfrak{E}}\varphi ·\dot{\mathfrak{E}}-\mathsf{P}·\dot{\mathsf{E}}=(\mathbf{T}+\mathsf{E}\otimes \mathsf{P}-\rho \mathbf{F}{\partial }_{\mathbf{E}}\varphi {\mathbf{F}}^T)·\mathbf{D}\\ \hfill +(\mathbf{T}+\mathsf{E}\otimes \mathsf{P})·\mathbf{W}-({\mathbf{F}}^{-1}\mathsf{P}+\rho {\partial }_{\mathfrak{E}}\varphi )·\dot{\mathfrak{E}}.\end{array}$$

Consequently, it follows

$$\mathbf{T}+\mathsf{E}\otimes \mathsf{P}\in \mathrm{Sym},\mathbf{T}+\mathsf{E}\otimes \mathsf{P}=\rho \mathbf{F}{\partial }_{\mathbf{E}}\varphi {\mathbf{F}}^T$$

(55)

and

$${\mathbf{F}}^{-1}\mathsf{P}=-\rho {\partial }_{\mathfrak{E}}\varphi .$$

For definiteness, let

$${\partial }_{\mathbf{E}}\varphi =\mathbb{C}\mathbf{E},$$

(56)

thus ascribing to $\mathbb{C}$ the meaning of elasticity tensor. Hence, by left-multiplying (55) by $J{\mathbf{F}}^{-1}$ and right-multiplying it by ${\mathbf{F}}^{-T}$, we obtain

$${\rho }_R\mathbb{C}\mathbf{E}={\widehat{\mathbf{T}}}_{RR}^{+}$$

(57)

where

$${\widehat{\mathbf{T}}}_{RR}^{+}=J{\mathbf{F}}^{-1}(\mathbf{T}+\mathsf{E}\otimes \mathsf{P}){\mathbf{F}}^{-T}$$

is the second Piola stress of $\mathbf{T}+\mathsf{E}\otimes \mathsf{P}$.

It is worth remarking that the literature develops arguments in terms of mechanical and Maxwell stresses. E.g., in [7],

$$\mathbf{\sigma }=\rho \mathbf{F}{\partial }_{\mathbf{E}}{\mathbf{F}}^T-\mathsf{E}\otimes \mathsf{P}$$

is viewed as the mechanical stress; here, instead, $\mathbf{\sigma }=\mathbf{T}$ is the total stress. Next, also in agreement with the Maxwell stress ${\mathbf{\tau }}_M$ in (30), the total stress tensor $\mathbf{\tau }$ is found to be

$$\mathbf{\tau }=\mathbf{\sigma }+{\mathbf{\tau }}_M=\rho \mathbf{F}{\partial }_{\mathbf{E}}{\mathbf{F}}^T+{ϵ}_0[\mathsf{E}\otimes \mathsf{E}-\frac{1}{2}(\mathsf{E}·\mathsf{E})\mathbf{1}].$$

• $\mathcal{E}={\mathbf{F}}^{-1}\mathsf{E}$. With the variable $\mathcal{E}$, we find

$$\dot{\mathcal{E}}=\mathbf{L}\mathsf{E}+\mathbf{F}\dot{\mathcal{E}}$$

and then

$$\begin{array}{c}\hfill -\rho {\partial }_{\mathbf{E}}\varphi ·\dot{\mathbf{E}}+\mathbf{T}·\mathbf{L}-\rho {\partial }_{\mathcal{E}}\varphi ·\dot{\mathcal{E}}-\mathsf{P}·\dot{\mathsf{E}}=(\mathbf{T}-\mathsf{P}\otimes \mathsf{E}-\rho \mathbf{F}{\partial }_{\mathbf{E}}\varphi {\mathbf{F}}^T)·\mathbf{D}\\ \hfill +(\mathbf{T}-\mathsf{P}\otimes \mathsf{E})·\mathbf{W}-({\mathbf{F}}^T\mathsf{P}+\rho {\partial }_{\mathcal{E}}\varphi )·\dot{\mathcal{E}}.\end{array}$$

Consequently, it follows that

$$\mathbf{T}-\mathsf{P}\otimes \mathsf{E}\in \mathrm{Sym},\mathbf{T}-\mathsf{P}\otimes \mathsf{E}=\rho \mathbf{F}{\partial }_{\mathbf{E}}\varphi {\mathbf{F}}^T$$

(58)

and

$${\mathbf{F}}^T\mathsf{P}=-\rho {\partial }_{\mathcal{E}}\varphi .$$

Again, with the assumption (56), we find that

$${\rho }_R\mathbb{C}\mathbf{E}={\widehat{\mathbf{T}}}_{RR}^{-}$$

(59)

where

$${\widehat{\mathbf{T}}}_{RR}^{-}=J{\mathbf{F}}^{-1}(\mathbf{T}-\mathsf{P}\otimes \mathsf{E}){\mathbf{F}}^{-T}.$$

Since

$$\mathrm{skw}(\mathbf{T}-\mathsf{P}\otimes \mathsf{E})=\mathrm{skw}(\mathbf{T}+\mathsf{E}\otimes \mathsf{P})$$

the symmetry conditions (55)₁ and (58)₁ are equivalent. Instead, the stress functions (55)₂ and (58)₂ are different. Indeed, in the case of (57), the application of an electric field to a sample produces an elongation; in the case of (59), the sample contracts. The two behaviours occur in PZT/PU depending on the volume fraction of PZT (see [26]). This may indicate that different models apply depending on the fraction of PZT.

Electrostriction in Isotropic Deformations

As a particular case, we now consider isotropic deformations so that

$$\mathbf{F}=F\mathbf{1},\mathfrak{E}=F\mathsf{E}.$$

Using $\dot{\mathbf{F}}=\mathbf{L}\mathbf{F}$, it follows that

$$\dot{F}=\frac{1}{3}F\nabla ·\mathbf{v}.$$

Hence, we have

$$\dot{\mathsf{E}}=({\displaystyle \frac{1}{F}}\mathfrak{E}{)}^{\dot{}}=-{\displaystyle \frac{1}{3F}}(\nabla ·\mathbf{v})\mathfrak{E}+{\displaystyle \frac{1}{F}}\dot{\mathfrak{E}}=-\frac{1}{3}(\nabla ·\mathbf{v})\mathsf{E}+{\displaystyle \frac{1}{F}}\dot{\mathfrak{E}}.$$

A direct substitution gives

$$-\rho {\partial }_{\mathbf{E}}\varphi ·\dot{\mathbf{E}}+\mathbf{T}·\mathbf{L}-\rho {\partial }_{\mathfrak{E}}\varphi ·\dot{\mathfrak{E}}-\mathsf{P}·\dot{\mathsf{E}}=(\mathbf{T}+\frac{1}{3}(\mathsf{E}·\mathsf{P})\mathbf{1}-\rho F^2{\partial }_{\mathbf{E}}\varphi )·\mathbf{D}+\mathbf{T}·\mathbf{W}-({\displaystyle \frac{1}{F}}\mathsf{P}+\rho {\partial }_{\mathfrak{E}}\varphi )·\dot{\mathfrak{E}}.$$

Hence, it follows that

$$\mathbf{T}\in \mathrm{Sym},\mathsf{P}=-F\rho {\partial }_{\mathfrak{E}}\varphi ,$$

$$\mathbf{T}=-\frac{1}{3}(\mathsf{E}·\mathsf{P})\mathbf{1}+\rho F^2{\partial }_{\mathbf{E}}\varphi .$$

7. Conclusions

This paper revisits some aspects connected with the methods used for the determination of thermodynamically consistent models. For definiteness, the methods are applied to the modelling of deformable dielectrics.

The symmetry condition given in (23) is derived as a balance equation. Section 4 shows that this constraint is satisfied identically if the dependence of the free energy on the electric field is modelled through the fields $f\left(J\right){\mathbf{F}}^T\mathsf{E}$ or $f\left(J\right){\mathbf{F}}^{-1}\mathsf{E}$. All these fields are also proved to be objective and invariant within the Euclidean transformations. Furthermore, ${\mathbf{F}}^T\mathsf{E}$ is the Lagrangian electric field in that it enters the Maxwell equations in the reference configuration.

The second law of thermodynamics is considered in a generalized form where the entropy production is given by a constitutive function possibly independent of the other constitutive functions. Furthermore, a representation formula is applied where the Clausius–Duhem inequality allows for additional, non-dissipative terms. Both views allow remarkable generalizations, mainly in connection with rate-type models. Some examples are given where the constitutive equations themselves are significantly influenced by the chosen function for the entropy production (Section 5). Indeed, through the example of fluid mixtures, it is shown how a general, thermodynamically consistent representation follows for the diffusion flux. Furthermore, the connection is established with the definition of entropy production in non-equilibrium thermodynamics.

Using the vector $\mathfrak{E}={\mathbf{F}}^T\mathsf{E}$, or $\mathcal{E}={\mathbf{F}}^{-1}\mathsf{E}$, to account for the electric field effects allows for the description of electrostrictive properties naturally via the relations (57) and (59).