1. Introduction
The constitutive properties of materials are often described through evolution equations whereby the time derivative of suitable fields is eventually related to appropriate fields and their derivatives of various order. The dependence on the space and time derivatives is a way of accounting for non-local properties in space and time. Relative to the description of response of the material in terms of space and time integrals, differential evolution equations show the advantage of greater flexibility.
Starting from rheological models [
1,
2], examples of differential equations are settled in the mechanical domain. For example, in connection with viscoelastic materials a simple equation is that associated with the Maxwell model. In the three-dimensional setting, we can write the stress–strain relation in the form,
where
is the stress tensor and
is the (infinitesimal) strain tensor. Likewise, the Kelvin–Voigt model of fluids is expressed by
where
is the viscous stress and
is the stretching tensor. Here,
and
are constants; we might view these coefficients as functions of temperature and mass density. Other models may involve higher-order time differentiation, as seen in the Burgers material, which involves second-order time derivatives (see, e.g., [
3], ch. 4).
As for thermal models, it is worth mentioning the Maxwell–Cattaneo equation [
4,
5,
6,
7]
for the temperature
and the heat flux vector
. In nanoscale systems continuum models might be applicable provided non-local and/or rate properties are properly described. The evolution is often modelled by the Guyer–Krumhansl [
8,
9] equation, namely
where
is the heat flux vector, ∇ is the gradient operator,
is the Laplacian, and the superposed dot denotes the material time derivative. Equation (
4) is then a generalization of (
18) with the dependence of
on the higher-order spatial derivatives
and
.
Within continuum mechanics, the constitutive properties are required to be objective, meaning they remain form-invariant under the set of Euclidean transformations, SO(3). However, Equations (
1)–(
4) are not objective, merely because the time derivative is not objective. Hence, these equations deserve an objective form. Furthermore, the physical admissibility of any constitutive equation, like, e.g., (
1)–(
4), is required to be consistent with the second law of thermodynamics. Consequently, we need a procedure to ascertain the physical admissibility of constitutive equations.
The physical admissibility is by now a well-known topic; however, it is performed in various ways. A formal statement about the thermodynamic consistency traces back to Coleman and Noll [
10]. Next, Müller [
11] generalized the statement by letting the entropy flux be a constitutive quantity to be determined. Lately, a further generalization has been accomplished by regarding also the entropy production as a constitutive quantity per se ([
2], ch. 3).
The purpose of this paper is to establish a systematic procedure to ascertain whether given constitutive equations are thermodynamically consistent. While this is classical in the literature, albeit with technical variants, this paper shows how the outcome is in general a set of constitutive equations that possibly includes a selected equation, such as Equations (
1)–(
4). Mathematically, the procedure is made operative by using a representation formula that determines a wide class of equations once a set of variables is chosen depending on the given particular equation.
This paper revisits the thermodynamic approach and emphasizes the operative character of the second law in connection with physically admissible models. Next, some significant examples of evolution equations are investigated, both in the Lagrangian and in the Eulerian description.
2. Notation and Balance Equations
We denote by , the time-dependent, three-dimensional, region occupied by the body under consideration. The position vector of a point in is denoted by . Hence, and are the mass density and the velocity fields at , at time . The symbol ∇ denotes the gradient, with respect to , while is the divergence operator. For any pair of vectors , or tensors , the notations and denote the inner product. Cartesian coordinates are used and then, in the suffix notation, , , the summation over repeated indices being understood. Also, and denote the symmetric and skew-symmetric parts of , while is the space of symmetric tensors. A superposed dot denotes the total time derivative and hence, for any function on , we have . The symbol denotes the velocity gradient, , while and . Further, is the Cauchy stress tensor, is the specific body force, and ⊗ denotes the dyadic product.
2.1. Balance Equations
The conservation of mass is expressed locally by the continuity equation
The balance of linear momentum leads to the equation of motion
where
. The balance of angular momentum implies that
. To obtain the balance of energy, we consider the specific energy density
(per unit mass) and the power
per unit area, with unit external normal
, and
per unit mass;
are quantities of a non-mechanical character. In light of the equation of motion (
5), it follows that the internal energy
is subject to the local equation
Let
be the specific entropy density,
the entropy flux, and
the entropy supply. The balance of entropy leads to
where
is the (rate of) entropy production. It is assumed that
, which implies the increase of entropy for isolated systems (
).
We assume and r are arbitrarily given time-dependent fields on . A process is the set entering the balance equations. Since the number of components in a process is larger than the number of balance equations, then a corresponding set of constitutive equations are needed to make the two numbers equal. Hence, the balance equations provide ; the assumed arbitrariness of and allows us to view , , and their derivatives as arbitrary.
2.2. Second Law of Thermodynamics
The second law of thermodynamics is assumed as follows.
Postulate.For every process
admissible in a body, the inequality
is valid at any internal point.
Henceforth, we refer to Equation (
7) as the Clausius–Duhem (CD) inequality. It is convenient to let
is said to be the extra-entropy flux and
the Helmholtz free energy. Substitution of
from (
6) makes (
7) in the form
This is the thermodynamic scheme for fields in the Eulerian description, where is the space–time domain of the process. For later use, we now establish the corresponding relations in the space–time domain , where is the region at some convenient time. To fix ideas, let .
2.3. Lagrangian Description
Let
be the position vector of a point in
and hence,
is the motion. We denote by
the gradient operator in
. We let
be the deformation gradient and hence, for any differentiable function
, we have
Define
and let
be the Green–Lagrange strain tensor. Hence,
Let
is the second Piola (or Piola–Kirchhoff) stress while
is the heat flux in
. We can prove that ([
2], ch. 1; [
12], §25)
Furthermore, letting
m, we have
Since
is the mass density in the reference configuration
, then
J times the CD inequality (
8) yields
Two points characterize the present approach. Firstly, the entropy production is given by a constitutive equation per se as is the case for the flux . Secondly, the constitutive equation for is not assumed a priori but is derived as a consequence of the other constitutive equations and the CD inequality. In particular, this shows how and influence the equation for .
2.4. Representation Formula
Depending on the set of variables, the CD inequality (
8) or (
9) has the form
in the unknown
with
as vectors or tensors. For example, if
depends on
, then (
9) eventually has the form (
10) for the unknown
, with
Notice that, for any unit vector
, a vector
can be represented in the form
where
is the longitudinal part (in the direction
) while
is the transverse part,
. If only
is known, say
, then
Any vector
perpendicular to
can be represented in the form
for any vector
, where
is the identity tensor. This is so in that
Back to (
10), we have
If
and
are tensors, say
then we repeat the formal steps to conclude that
where
is the fourth-order identity tensor and
is any second-order tensor.
3. Thermodynamic Restrictions
Our purpose is to determine evolution equations of
consistent with thermodynamics and hence, we look for a constitutive equation of
. Possible nonlocal effects are described by a suitable dependence on the spatial derivatives. For definiteness, we direct our attention to a dependence on
and
. A dependence on the strain
is allowed to account for the elastic properties of the solid. Hence, we let
be the set of independent variables and assume
, and
are continuous functions of
, while
is continuously differentiable.
Compute
and substitute in (
9) to obtain
The linearity and arbitrariness of
, imply that
Hence, (
13) simplifies to
Inequality (
15) requires the compatibility among the constitutive functions
along with the non-negative valuedness of
. For the present purpose, we now use Equation (
15) as an equation in the unknown
.
The representation formula (
11) is now applied to obtain
from (
10). Assume that
and let
. By (
11), it follows that
Equation (
16) is the general thermodynamic requirement on
under the assumption
with variables
.
The representation (
16) and the arbitrariness of the field
show that any transverse term relative to
is allowed, irrespective of its connection with the entropy production
or the extra-entropy flux
. This property is apparent from the CD inequality (
9), where
occurs through the component along
. Physical examples are given in the next section.
4. Remarkable Models of Heat Conduction
Models are now derived by taking different assumptions on .
4.1. Non-Dissipative Heat Conduction
Let
. Hence, Equation (
16) simplifies to
The non-dissipative character is characterized by the assumption
. Now, as
and
are constants, we have
The classical entropy production
times
, which enters the heat conduction inequality, produces a corresponding rate of the free energy in this model.
4.2. Fourier Model
Let
and
. If, further, we let
, then we find
, which means that
is stationary. Now, if we let
where
is positive definite, then it follows that
Thus, the Fourier law follows with heat conductivity tensor
.
4.3. Some Evolutionary Models
We now look for some equations involving
and relate them to models that appear in the literature. Hence, for definiteness, hereafter, we let
where
is possibly dependent on
and
. Equation (
16) then is written in the form
Maxwell–Cattaneo Equation
Let
. Hence, Equation (
17) simplifies to
Assume
. If
it follows
Equation (
18) is just the Maxwell–Cattaneo (MC for short) equation with relaxation time
and heat conductivity
, given by
The positive value of the entropy dissipation
implies the positive value of
. Instead, the sign of
is left undetermined by thermodynamics; it is customary to assume
, which follows from the assumption
.
4.4. Cross-Couplings between Deformation and Heat Conduction
Cross-coupling terms are modelled by the joint dependence on
and
. For simplicity, let
where
is any scalar function of
. Hence, Equation (
16) takes the form
The selection of appropriate functions
and
completes the model.
4.5. Second-Order Non-Local Models
Non-local models are established in terms of the dependence on the spatial derivatives of
. A model often considered in the literature is named after Guyer and Krumhansl and involves second-order derivatives [
8,
9]. Indeed, we consider the generalized (Lagrangian) form
the coefficients
are expected to be positive-valued functions of temperature.
With the purpose of obtaining an evolution equation like (
20) and recalling the assumption
made in connection with the MC Equation (
18), we let
where
are functions of
and
. Letting
, we can write (
17) in the form
We can get Equation (
20) by the identifications
and
The form (
20) is then obtained, albeit with
as a function of the second-order derivatives
. Owing to the occurrence of
and
, we ask whether we can eliminate the dependence of
on the derivatives of
and
.
Let
We can then write Equation (
21) in the form
Notice that
Hence, we select
to find
If
h and
l are constants, then
provided only that
and hence,
.
The generalized form (
20) is thermodynamically consistent if
h and
l are positive constants. In that case the extra-entropy flux has the form (
22), while the entropy production is
5. Higher-Order Differential Equations
Experiments on heat propagation in graphene [
13] show that the results are best fitted with the differential equation
for the temperature
. There are various physical models leading to (
23). The simplest one is to assume the conductor is rigid and let
be the specific heat so that the balance of energy simplifies to
with a zero energy supply. The evolution of the heat flux is assumed to be governed by the MC equation with a higher-order correction so that
Assuming
c is constant, upon some rearrangements, we find Equation (
23) with
. Owing to the formal occurrence of two times
, Equation (
23) is also referred to as a heat equation with two phase lags. Other physical motivations of (
23) are given in [
14].
We now examine the thermodynamic consistency of heat equations of the form (
24). For generality, we let the body be deformable and hence, we follow again the Lagrangian description. We then consider equations of the form
Owing to the term
in Equation (
25), we let both
and
be among the variables. Hence, we let
be the set of variables and
and
be given by constitutive functions. Computation of
and substitution in the CD inequality yields
The linearity and arbitrariness of
implies that
The occurrence of
affects the properties of the entropy
. Hence, we define
In (
25), it is
; consistent with (
25), we assume
, and hence,
, are independent of
. Substitution of
in (
26) and division by
result in
where
Now, observe that
Hence, we let
Consequently, we can write Equation (
27) in the form
where
If
and
are uniform, that is
, then
If, further,
, then
equals the variational derivative of
with respect to
. Accordingly, we can say that
is a generalized variational derivative of
.
While
is independent of
, both
and
can depend on
. If we let
then we find that
where
is the value of the entropy production
at
.
Equation (
28) reduces to
Hence, using the representation (
11) with
, we can determine
in the form
We assume
whence
The selection
leads to
The particular case
results in
This in turn leads to the rate equation for
in the form
whence
In stationary conditions, we have
which ascribes to
the meaning of heat conductivity. Since
the conductivity
K proves to be positive, as expected.
6. Rate Equations in the Eulerian Description
In accordance with the objectivity principle ([
15], p. 35; [
16], §41), the constitutive equations must be form-invariant under Euclidean transformations, namely under the map
such that
where the vector
and the tensor
are time-dependent and
is orthogonal,
. Consequently, if a constitutive equation is expressed by a rate equation then the time derivatives must be objective in that the objective time derivative of a vector transforms as a vector, and the same is true for tensors.
A natural question arises since there are infinitely many objective derivatives and the non-uniqueness remains, even though we require the thermodynamic consistency to hold. We now examine rate-type models. For formal simplicity, we consider separately the occurrence of rate effects for the stress and the heat flux.
6.1. The Navier–Stokes–Voigt Fluid
Roughly, the NSV (Navier–Stokes–Voigt) fluid model is the counterpart of the Kelvin–Voigt solid, whereby the stress is a linear combination of the strain and the strain-rate. Consistently, the NSV system of equations modifies the Navier–Stokes equations by addition of a term of form to the acceleration . Really, this is the approximation induced by replacing the Navier–Stokes stress with the linear combination . However, the total time derivative, along with the partial time derivative, is not objective and hence, the idea of the linear combination has to be revisited.
Here, for generality, we let the fluid be compressible, though usually in the literature the fluid is incompressible (see, e.g., [
17,
18,
19]). Hence, we let
where
p is the thermodynamic pressure, assumed to be a function of the temperature
and the mass density
. The stress
is expected to be related to
and the rate of
. Furthermore, we keep modelling heat conduction through a rate equation. Hence, we let
be the set of variables for the constitutive equations of
, and the rates of
and
. Indeed, the rate equations for
and
have to be established via objective time derivatives.
Compute the time derivative of
and substitute in the CD inequality (
8) to obtain
The linearity and arbitrariness of
imply that
By the thermodynamic character of the pressure, we conclude that
Thus, Equation (
30) simplifies to
where, for technical purposes, we have divided throughout by
.
Notice that
and then
Consequently, we let
Hence,
times the remaining part of (
31) results in
where
denotes the variational derivative,
By Equation (
32), the set
of variables allows for a rich variety of properties. Formally, one of the rates, e.g.,
, is a function of
and the remaining quantities. For definiteness, we look for separate effects or, mathematically, sufficient conditions to satisfy Equation (
32).
Firstly, let
, as well as
be independent of
. Hence, Equation (
32) holds only if
and
To proceed with the analysis of (
33), it is worth observing that a rate equation of the form
does not comply with the objectivity principle in that
is not form invariant. We then look for a recourse to objective derivatives. To fix ideas, consider the objective derivative
If
, then
are the Truesdell derivatives of
and
. If, instead,
, then
and
coincide with the corotational derivatives,
The other objective derivatives have as common part the corotational derivative [
2]. Replacing
we can write Equation (
33) in the form
The linearity and arbitrariness of the spin
in (
34) imply that
whence
Hence, (
34) simplifies to
Furthermore, in light of (
35), we have
Two significant instances are taken from the literature.
6.2. Non-Heat Conducting NSV Fluid
This approximation traces back to Oskolkov, though in the particular case of incompressible fluids [
20]. We let
and notice that
Consequently, Equation (
36) simplifies to
We now apply the representation formula (
11) for the unknown
with
Hence,
where
is any tensor function of
. If
, then
reduces to the corotational derivative; thus, we have
If
, then
The choice
results in
Borrowing from the Navier–Stokes theory we identify the coefficient of
with the shear viscosity
and hence, we can write
which gives
where
represents the relaxation time.
In light of (
35), the free energy has the form
Hence, also
and
have additive terms proportional to
.
Remark 1. Two theories of NSV fluids have been developed with different schemes; however, both of them use a kinetic energy density proportional to [21,22]. We then see a similarity with the free energy ψ, comprising a term proportional to . 6.3. NSV Fluid with Non-Fourier Conduction
Back to Equation (
34), we now focus our attention on the corotational derivative as the objective derivative (and hence, we let
). Furthermore, the model of heat conduction is sought by looking for a rate-type equation; thus, we obtain, in particular, a Maxwell-Cattaneo equation. In this way, a generalization is established of the model applied by Straughan [
23].
By (
36) and (
37), it follows
For simplicity, cross couplings (e.g.,
terms) are set aside and hence, the entropy production is assumed in the separate form
Both
and
are non-negative, in that
is the value of
when
, and
is the value of
when
. Hence, (
39) splits into two equations,
Equation (
40) has just been considered in the previous subsection. As for (
41), we apply the representation formula with
to obtain
If we assume
, then it follows
If no action on the system happens (
), then
is directed along
. Furthermore, in stationary conditions,
, it follows
that is, the standard thermodynamic condition on the heat flux when only the temperature gradient is considered to produce the heat flux.
If we assume
, then it follows
Hence, letting
, we have
Equation (
43) has the form of the MC equation where the time derivative is in the corotational (and hence objective) form. Indeed, we can view
as the relaxation time and the heat conductivity. The positive value of the entropy production
yields
. Any assumption
makes
a function of
and
. The assumption
implies
and hence, guarantees the boundedness of the solution
in time. This procedure in turn shows how the MC equation is merely a particular case of the family of thermodynamically consistent evolution equations
with
.
7. Evolution Equations, Second Law, and Objective Derivatives
By analogy with the MC equation for the heat flux, in fluid-dynamics, evolution equations for the stress tensor are considered in the Eulerian description. In this context, we contemplate the selection of objective derivatives and the restrictions placed by the second law, as well as the objectivity of thermodynamically consistent evolution equations.
The stress
is represented in the form
, with
p representing the thermodynamic pressure. Describe a thermo-viscous fluid through the variables
for the constitutive functions
. The Clausius–Duhem inequality (with
) reads
The linearity and arbitrariness of
and
imply
Substitute
with
and define the thermodynamic pressure by
The remaining condition is
For simplicity, we neglect cross-coupling terms (i.e.,
is independent of
and
, while
is independent of
and
) and
both
and
are non-negative. Hence, we have
By applying the representation formula to (
44), it follows
where
is any tensor function of
. Equation (
46) is not objective because
is not. Thus, the second law per se does not provide objective evolution equations. Yet, we show that thermodynamic consistency and objectivity are compatible.
Consider the objective time derivative
with generic coefficients
, and examine the thermodynamic consistency of the sought constitutive function
. Substitution of
in (
44) results in
Since
then the linearity and arbitrariness of
in (
47) imply
the condition (
48) holds if
depends on
through the invariants
or
. Hence, Equation (
47) reduces to
For definiteness, let
The representation formula for
yields
Equation (
49) shows various terms for the objective derivative
, namely the term induced by the entropy production, the terms due to the chosen derivative, the term induced by the power
and the term associated with the transverse tensor
. Among the particular cases, we may consider the tensor
. Letting
, we find that
If, further,
, then we have
Equation (
50) can be viewed as the tensor form of the Maxwell–Wiechert fluid model ([
2], sec. 6.1.4). Accordingly,
can be viewed as the relaxation time. In stationary conditions (
), we have
thus ascribing to
the role of viscosity coefficient. The assumption
implies that
.
Remarks about Rate Equations and Thermodynamic Restrictions
Depending on the set of variables, it may happen that the CD inequality has the form (as, e.g., in (
44))
If
, then by the representation formula, we can derive
, as in (
46), but this rate equation would be non-objective. Hence, objectivity is an additional requirement. In fact, this requirement is realized by replacing the time derivative with an objective derivative, say
, and then deriving the (objective) equation for
. Owing to the non-uniqueness of objective derivatives, this shows that thermodynamics and objectivity lead to a class of rate-type models rather than to a single equation.
The need of objective constitutive equations and the non-uniqueness of the objective variables are well established in the literature. For instance, in connection with fluids of differential type [
24,
25], the dependence on the variables
is replaced from the start with that on the Rivlin–Ericksen tensors
, e.g.,
The literature also shows another way of modelling through objective derivatives. The starting assertion in [
26] is that a scalar subjected to an objective rate is equal to its conventional time rate. Hence, one might expect the validity of the equation
for the two vector functions
and
; in [
26]
is the logarithmic derivative. Yet, this equation in general is not true. For simplicity, let
We have
Since
, then
, but
need not be zero. Accordingly, in general, the most convenient procedure is to start from the time derivatives and then to replace them through the selected objective derivatives.
8. Conclusions
This paper addresses the modelling of material behaviour in continuous bodies. In general, non-locality effects, in space and time, have to be described by involving time and space dependence of the pertinent fields. In this connection, the paper looks at the modelling of constitutive properties in terms of differential equations as relations among space and time derivatives of various orders.
The occurrence of time derivatives has to comply with the objectivity principle. Objective equations are obtained in two ways. In the Lagrangian description some pertinent fields (e.g., , as well as the densities ) are invariant under the set of Euclidean transformations. Hence, their time derivatives are invariant and objective. Furthermore, the recourse to the Lagrangian description naturally allows models with finite deformations. Instead, in the Eulerian description, the standard fields (e.g., ) are not invariant and their derivatives are not objective. Objectivity is obtained by using objective time derivatives; the corotational derivative is the simplest one and seemingly the most appropriate.
Physically admissible models are required to comply with the second law of thermodynamics. Here, the postulate of the second law is eventually expressed in terms of the Clausius–Duhem inequality. Two features have to be emphasized with respect to the tradition. Firstly, the (non-negative) entropy production is viewed as a constitutive function per se and not merely deduced from the other constitutive properties. Secondly, the Clausius–Duhem inequality is viewed as a constraint on the pertinent fields and it is solved with respect to the interested vector or tensor by using a representation formula (
Section 2; see also [
2]). This allows for qualitatively different constitutive equations, as is emphasized in
Section 4,
Section 5 and
Section 6. Rather than confining to the admissibility of a single equation, the procedure leads to the admissibility of a class of models characterized by the entropy production
and the transverse vector or tensor (determined by
or
).
Section 3,
Section 4 and
Section 5 are devoted to models of heat conduction, within Lagrangian descriptions, while
Section 6 develops some models of the Navier–Stokes–Voigt fluid, within Eulerian descriptions. In connection with thermo-viscous fluids,
Section 7 gives a systematic approach to the modelling of evolution equations in the Eulerian description. It is shown that the thermodynamic consistency is compatible with both objective and non-objective evolution equations, which emphasizes that objectivity is an independent principle of continuum mechanics.