1. Introduction
The thermodynamic consistency of models in continuum physics is established by the compatibility with the balance equations (mass, linear and angular momentum, energy, and entropy). The balance of entropy is expressed by the Clausius–Duhem (CD) inequality ([
1], § 257). The conceptual role of the CD inequality is now due to a famous paper by Coleman and Noll [
2]. Thermodynamic processes are the sets of pertinent fields that describe a body and satisfy the balance equations and the constitutive assumptions. The CD inequality is assumed to hold for all admissible processes and hence it places restrictions on the constitutive assumptions. Hence, the CD inequality, while being equivalent to the assertion that the entropy production cannot be negative, sets restrictions on physical, real processes.
Let
be the absolute temperature,
the specific entropy,
the heat flux, and
r the energy supply. The physical content of the balance of entropy ([
3], § 6.5),
is made formal in [
2] by letting the entropy transfer consist of the entropy flux
and the entropy supply
. Hence, in local form, the entropy production (per unit mass)
is assumed to be given by
where the superposed dot denotes the total time derivative and
is the mass density. The basic postulate in [
2] is that the entropy production
is non-negative for every admissible process. Next Müller [
4] observed that the entropy flux, say
, need not be
. Hence, the admissible constitutive functions of
and
(or
) are required to satisfy
. Later on, Green and Naghdi [
5] pointed out that also the entropy production
has to be considered as given by an admissible constitutive equation. To our knowledge, this view has not received much attention in the literature; among the few approaches involving
as a constitutive function we mention ref. [
6], recent works of ours [
7,
8,
9], and the systematic procedure developed in [
10].
The purpose of this paper is two-fold. The first fold is to show the general role of the entropy production. Almost always the constitutive function of
is merely inherited by the other constitutive equations or Equation (
1) is an identity. As a simple example, for a rigid heat conductor obeying Fourier’s law
the CD inequality becomes the heat conduction inequality ([
11], § 2.3)
and
follows. Instead, in more involved models a further degree of generality is gained by letting (
1) be an equation, not an identity.
The second fold is to point out that the CD inequality may result in a relation between appropriate rates and the entropy production. A representation formula allows the inequality to be solved with respect to a rate in terms of the remaining rates and of the entropy production. In this way we can establish the thermodynamic consistency of uncommon non-linear rate-type equations. Two models are developed in detail. First we examine a general thermodynamic scheme leading to the modelling of heat conduction and viscoelasticity. The application of the representation formula, for vectors and tensors, yields general relations; as particular cases some classical models of the literature are derived. Next, some models of heat conduction are investigated where the temperature rate is one of the variables. The possible models are framed within different schemes in the literature and the wave propagation properties are established.
We consider a body occupying a time-dependent region . The motion is described by means of the function , providing the position vector . The symbols ∇ and denote the gradient operator with respect to , . The function is assumed to be differentiable; hence, we can define the deformation gradient as or, in suffix notation, . The invertibility of is guaranteed by letting . For any tensor we define as . Throughout . We let be the velocity field. For any function we let be the total time derivative, . A prime denotes the derivative of a function with respect to the argument.
2. Balance of Entropy and Statement of the Second Law
Let
be any sub-region of the body that is convected by the motion. As with any balance equation we may express the balance of entropy in
by letting the rate consist of a volume integral and a surface integral,
where
is the entropy flux. Notice that if we start with a scalar integrand, say
h, in the surface integral then a Cauchy-like theorem will lead to a linear dependence of
h on the unit normal
, say
. Within the physical scheme ([
3], Chapter 6) the variation of entropy is greater than
, where
is the heat transfer at the pertinent region while
is the absolute temperature at that region. In the continuum setting we then let
s comprise
,
r being the energy supply, while
comprises
,
being the heat flux. We then let
where
is the extra-entropy flux and
is the entropy production (per unit mass). Consequently, we have
where, in local form,
Equation (
2) is the general form of the CD inequality. Following is the statement of the second law:
any thermodynamic process is required to satisfy the CD inequality (2).For definiteness we consider a deformable solid. We then express the balance equations for mass, linear momentum, angular momentum and energy in the form
where
is the body force and
is the internal energy density. Substitution of
from (
5) into (
2) results in
Hence, by means of the Helmholtz free energy
we can write the inequality in the form
In the application of the second law, and hence of inequality (
6), we require that the thermodynamic process under consideration satisfies the balance Equations (
3)–(
5) with any functions
and
.
Rate-type equations are framed naturally in the Lagrangian description. In this connection quantities related to the reference configuration are denote by the index
R. The referential mass density
, the second Piola stress
, and the referential vectors
are defined by
while
Hence, the multiplication of (
6) by
J yields
For formal convenience hereafter we let .
In the next section we investigate the restrictions placed by (
6) and (
7) in connection with generalized models for thermoelastic solids.
3. Rate Equations and Euclidean Invariance
Rate-type models are often based on rheological analogues (e.g., [
12], Chapter 8). Mathematically rate-type models are characterized by setting the time derivative of appropriate fields among the constitutive functions of the model. The interest in rate-type models is well motivated by a comment on hypo-elastic materials, described by
versus materials with memory in that the entire kinematical history of a body can rarely be known ([
13], § 99). The use of rate equations is standard in the extended irreversible thermodynamics [
14,
15]. Yet, as with any constitutive equation, the rate-type form is also required to comply with Euclidean invariance.
A change of frame
given by a Euclidean transformation, such that
, is expressed by
Under the transformation (
11), the deformation gradient
changes as a vector,
and hence it is not invariant. Yet invariant scalars, vectors, and tensors occur in connection with
.
We first look at invariants of mechanical character. The right Cauchy–Green tensor
and the Green–Lagrange (or Green–St. Venant) strain tensor
, defined as
are invariant in that
Consequently, the scalar
is invariant too. Since
then is apparently non-invariant. Decompose
in the classical form
where
is the stretching tensor and
is the spin; we have
The second Piola stress is invariant; this is checked by observing that
We observe that since
then
The referential heat flux and temperature gradient
are invariant and so is the power
4. Thermodynamic Consistency of Thermo-Viscoelastic Solids
Here, we look for rate-type models of thermoelastic materials in that rate equations are considered for the stress tensor and the heat flux in deformable solids.
From the mechanical viewpoint we look for a scheme that accounts for a persistent rate of the response under a constant action (viscoelastic behaviour). For heat conduction the model is thought to describe both a non-instantaneous approach to stationarity and a higher-order spatial interaction. This suggests that we allow for rate equations of
and
and let
be a variable. Thus, we might take
,
as the set of independent variables. Yet, invariance requirements demand that the dependence on the derivatives occurs in an objective way. Moreover, the Euclidean invariance of the free energy
implies that the dependence of
be through a function of Euclidean invariants. Hence, we let
and the same for
and
. The constitutive assumptions are completed by letting the rates
and
be given by constitutive functions of
.
Computing the time derivative of
and substituting it into (
7), one obtains
where
. The (linearity and) arbitrariness of
, and
implies that
Owing to the dependence of
on
it follows that
where the dots denote possible terms which are independent of
. Hence, the arbitrariness of
in (
14) results in
For the present purposes no significant generality is lost by letting
. Thus, we have
Substitution into (
14) yields
where
Notice that, since depends on , then may depend on . Only and can depend on .
The unknown functions
, and
can be related by common dependencies so that cross-coupling terms are allowed. For simplicity we examine a sufficient condition for the validity of (
15) namely that the three inequalities
are satisfied while
. For definiteness we now let
4.1. Consequences of (18); Heat Conduction
As for Equation (
18), we observe that if
then it follows the heat equation
which is satisfied by any function
where
and
is positive definite and hence
As a particular example, let
. Then
which is Fourier’s law.
If instead
then we can apply (
10), with
and
, to obtain
The choice
results in
Equation (
19) has the form of a Maxwell–Cattaneo (MC for short) Equation (see, e.g., [
16]) with relaxation time
and conductivity
given by
As expected, the positiveness of results in the positiveness of the conductivity .
4.2. Consequences of (16)
The continuity of
implies that
as
. Consequently, for sufficiently small
we have
The arbitrary sign of
implies
and hence
As an example, we may have
The dependence on
has also been considered in an attempt to establish a model allowing for wave propagation at finite speed. For simplicity, assume the body is undeformable (
) and
. The evolution of
is governed by the balance of energy,
If
,
then (
20) yields
Since
then (
21) is an elliptic differential equation, not a hyperbolic one.
Furthermore, the more involved model with the dependence on would not result in a hyperbolic equation.
Though it is unusual in the literature, we might consider constitutive equations where both
and
are in the set of variables. In this case, inequality (
16) would be in the form
thus resulting in the rate equation
To establish the physical relevance of Equation (
22), a careful investigation is required.
4.3. Consequences of (17); Viscoelasticity
If
and
are independent of each other then we have
and
. Hence,
depends only on
. Furthermore,
is no longer an independent variable but is equal to
. This relation can be viewed as a model of a thermo-hyperelastic material.
We now consider
and
, with the assumption
. Equation (
17), i.e.,
yields
as a function of
. Using the representative Formula (
9), with
we find
where
is any second-order tensor function of
. Among the possible forms of
, we consider
where
is a fourth-order tensor, which maximizes the (linear) dependence on
. Hence, it follows
where
The result (
24) gives a general representation of viscoelastic modelling.
Of course, the entropy production
may depend on
. Indeed,
has to be non-negative and a dependence of
on
has been shown to be the natural modelling of the hysteretic behaviour in plastic materials [
9].
The constitutive relation (
23) has the form
and is linear in
if
and
. This equation can be viewed as characterizing hypo-elastic materials in that they experience a stress increase arising in response to the rate of strain
from the immediately preceding state ([
1], page 731). In fact, Truesdell [
17] restricted hypo-elasticity to
with
depending on the stress; in elasticity
also depends on the strain.
We now return to (
24) and look for a simple example of
induced by the free energy
, while
and
. Let
be a non-singular, fully-symmetric, fourth-order tensor and
a smooth function from
to
. Hence, we consider the free energy
where
. Substitution of
results in
In the linear case,
, with
being a fourth-order tensor, we have
Now we show that the general form (
23) of the representation of
also allows us to find very simple models of viscoelasticity. Consider again the free energy (
25) and look for an entropy production given by
to save writing we let
stand for
. Notice that
Hence, since
we obtain
where
. Further, by (
26) and (
25) we find
Substitution into (
23) yields
Whenever the rate equation has the form
for any tensor
the choice
results in
Consequently, letting
we have
For definiteness, let
be linear, namely
, and define
[
18]. Hence, Equation (
28) becomes
which may be viewed as a three-dimensional version of the standard linear solid.
5. A Rate-Type Approach to the Equation for the Heat Flux
Equation (
18) is an implicit relation for the heat flux
. This is consistent with the fact that Equation (
19) is derived by assuming
. Hence, if
, the consistency follows by requiring that
. This might appear as a restriction on the consistency of the MC Equation (
19). Furthermore, in the investigation of wave propagation properties the requirement
implies that we cannot account for discontinuity waves propagating in a region with
. Hence, we re-examine the consistency of heat-flux equations by starting with a rate-type form of the sought equation. For simplicity we let the body be rigid and at rest in the chosen frame of reference. Consequently,
. The CD inequality then simplifies to
Let
and
be the variables. Hence,
and
are given by functions of
and
is taken in the form
subject to
. The scalars
and
g are assumed to depend on
and
through the magnitudes
and
. Computation of
and substitution in (
29) yield
The arbitrariness of
and
imply that
Likewise, we find that
might only depend on
; we loose no generality by letting
. The remaining inequality is
Since
and
are independent vectors then the arbitrariness of
implies
it then follows that
g is independent of
. Hence, we find the entropy production in the form
Furthermore, the dependence of
on
through
q leads to
An analogous model for anisotropic solids is obtained by letting
where
while
is a positive definite. Hence,
plays the role of a conductivity tensor under stationary conditions. We assume that
and
have a common basis of eigenvectors. By paralleling the previous derivation it follows from (
29) that
Thus, and this in turn implies the positive definiteness of the conductivity tensor . Since then .
We now check the wave properties associated with (
30) and the balance of energy for solids,
We consider discontinuity waves and denote the difference between the limit value behind (
) and ahead (
) of the wave by
. We investigate weak discontinuities in that we assume the jump conditions
Let
U be the wave speed. By means of geometrical and kinematical conditions ([
1], § 172) we have
where
is the unit normal to the wave and
denotes the normal derivative. Notice that
where
is the specific heat and
if
. By (
30) it follows
Hence, by (
30) and (
31), and some algebraic manipulations we obtain
Non-zero discontinuities can propagate with speeds
In particular, waves entering a region where (and hence ) propagate with the speed . Hence, greater tensor increases the speed.
6. Models Involving the Temperature Rate
Section 4 shows the difficulties of modelling heat conduction with finite speed in terms of the dependence on
for rigid bodies. Here, we look for a more general model where the dependence on
is considered for thermo-elastic bodies. Due to deformation, the model is simpler within the reference configuration.
Our purpose is to let
and
be independent variables. Yet, the occurrence of
and the equipresence principle also show that the dependence on
is in order. Hence, we let
depend on
and, by analogy with
Section 5, we consider the equation
for the evolution of
. Hence, the referential CD inequality (
7) takes the form
Hence, with no significant loss of generality, we let
The arbitrariness of
implies
Hence, the inequality (
32) simplifies to
Since
is independent of
then it follows
If instead we let
then we have
where
at
. The arbitrariness of
implies
6.1. Structure of and
To understand the possible consequences of the dependence on
and
we determine the explicit form of the internal energy
and the entropy
. Indeed, for simplicity and definiteness we specify the free energy
in (
35) in the form
where
is constant. Hence, we have
where
and
We then compute
to obtain
Relative to the variables
and
, by (
36) we see that
encloses first- and second-order derivatieves and furthermore a third-order term,
.
6.2. Discontinuity Waves
To investigate the existence of thermal-mechanical waves we consider the balance equations and the rate equation for
in the form
where
is the first Piola stress. We look for discontinuity waves ([
1,
19], Chapter 2) where
Hence, by (
37) we can write the system
Let
U be the (normal) speed and
the unit normal of the wave. Using the geometrical–kinematical conditions of compatibility we can write the system (
38) in the form
We notice that, except for , all the jumps are linear in the sought discontinuities. Instead involves the term , a single third-order derivative of . Consequently this scheme is not consistent with propagation at a finite speed. To overcome this drawback we assume , which means that . Hence, accounting for a dependence of the free energy on is not consistent with the propagation at a finite wave speed.
The dependence of the entropy on results in a single second-order term in , i.e., . Hence, the compatibility with a finite wave speed also requires that . We are then confined to the thermoelastic solid with a MC-like equation for the heat flux.
6.3. Waves in Thermoelastic Solids
Letting
we have
For simplicity we assume
, which is true if, e.g.,
. Upon substitution of
, the second equation in the system (
39)
is given the explicit form
where
We notice that
is the classical (positive) specific heat, and hence
is an effective specific heat. Substitution of
from (
39)
yields
Hence, we find two possible speeds,
for the propagation of the discontinuity
.
Meanwhile, Equation (
39)
results in the classical equation for the mechanical discontinuity
of acceleration waves [
19].
Some comments are in order. The independent behaviour of
and
is a consequence of the model; here it follows from the neglect of
in (
39)
. For thermodynamic consistency, the modelling of heat conduction through an MC-like equation affects the rate
in that the free energy is required to depend on the heat flux. A family of models, with different physical properties, is obtained by letting
depend on temperature, while here
is constant for simplicity.
8. Conclusions
This paper investigates the techniques associated with the exploitation of the second law of thermodynamics as a restriction on physically admissible processes. As is standard in the literature, the exploitation consists of the use of arbitrariness occurring in the CD inequality. The present approach emphasizes two uncommon features within the thermodynamic analysis: the representation formula and the entropy production.
There are cases where more terms of the CD inequality are not independent. The representation formulae, for vectors or tensors, allow us to derive a direct dependence between appropriate unknowns. As an example, Equation (
17) has the form
and the representation formula allows the unknown
to be determined. The solution is widely non-unique and this results in a variety of models characterized by the free energy
and the right-hand side
. Among the examples developed in this paper, we obtained the constitutive equation for hypo-elastic solids and for MC-like equations of heat conduction. Further models can be established by using the techniques developed in this paper, as was performed in [
10].
Concerning the entropy production
, we let it be given by a constitutive function per se and not merely by the expression inherited by other constitutive functions. This property results in more general expressions of the representation formulae and, as shown in [
7,
8,
9]), is crucial for the description of hysteretic phenomena.
These features are highlighted in this paper through models of viscoelastic solids and heat conductors. In particular, the models of heat conduction were also investigated in detail in connection with wave propagation properties.