1. Introduction
The modelling of materials with memory through functionals on an appropriate set of histories shows interesting questions about the correct assumptions on the constitutive equations. This is the case even for the linear theory of viscoelasticity within the realm of rational thermodynamics.
The classical linear theory traces back to Boltzmann [
1,
2] and assumes that the stress is determined (linearly) by the present value of the strain and the strain history. The consistency of this model with the second law of thermodynamics is well established. Here, we re-examine the thermodynamic restrictions and find that the kernel (Boltzmann function) is required to have a negative half-range sine transform. Well-known forms of the free-energy functional are shown to be consistent with thermodynamics, only if additional conditions on the kernel hold.
There are models in linear viscoelasticity where the kernel is unbounded [
3]. This is particularly the case of kernels in a power law form [
4,
5,
6]. We examine the thermodynamic consistency and show that the power law form is allowed if the exponent of the kernel is within
and is applied to the strain history.
The modelling of viscoelasticity, as developed by Pipkin [
7], involves the strain–rate history rather than the strain history. The strain–rate dependence may be justified as more appropriate to represent the continuity of the stress functional in that small changes in the strain–rate history produce small changes in the stress. In addition, we might think that the viscous character of viscoelasticity is more properly described by the strain–rate history. Mathematical difficulties arise if the strain–rate history is involved along with an unbounded kernel.
The power law form of the kernel is also characteristic of viscoelastic models with derivatives of fractional order. Fractional calculus is a well-established scheme in engineering science, particularly in materials modelling. Despite the extensive literature on the applications of derivatives of fractional order (see, e.g., [
8,
9,
10] and Refs therein), it seems that no definite thermodynamic analysis has been developed so far. Here, the thermodynamic consistency is investigated by requiring the compatibility—with the second law—of a linear dependence of the stress on a fractional derivative of the strain and the existence of a corresponding free-energy functional.
Notation. We consider a solid occupying a time-dependent region . Throughout, is the mass density, is the velocity, is the symmetric stress tensor, is the internal energy, and is the heat flux. The symbol ∇ denotes the gradient operator in , and is the partial time derivative at a point , while a superposed dot stands for the total time derivative, . Cartesian coordinates are used, is the velocity gradient, , and is the Eulerian rate of deformation. We let be a reference configuration; the motion is a function that maps each point vector into a point . The deformation gradient is defined by and , while is the gradient in . The Green–St. Venant strain is , where is the unit second-order tensor. If is a second-order or fourth-order tensor, then the inequality (or ) means that is positive (or negative) semi-definite.
3. Second Law and Free-Energy Functionals
Let
be the absolute temperature,
be the entropy density per unit mass,
be the heat flux in
,
r be the energy supply, and
be the mass density in
. Following the approach of rational thermodynamics, we take the balance of energy in the following form:
and the balance of entropy in the following form:
We assume, as with the statement of the second law of thermodynamics, that inequality (
6) holds for any admissible process; that is, for any set of admissible constitutive equations satisfying the balance equations. By replacing
from the balance of energy and considering the Helmholtz free energy
, we obtain the second-law inequality in the following form:
It is worth remarking that here we follow the scheme of continuum mechanics merely because this is the customary framework of viscoelasticity. Thermoviscous properties might be framed within a more general scheme by enlarging the notion of state with appropriate internal variables, as is the common case in extended irreversible thermodynamics [
11,
12]. Moreover, nonequilibrium properties might be described by means of rate-type equations, as is the case, e.g., in [
13]. We also observe that inequality (
7) shows that occurrence of flux–force pairs, such as
and
. This is not generally the case, as is shown by balances derived within microscopic statistical approaches [
12,
14].
As a further simplifying assumption, we let
be uniform,
, so that some properties of the free energy are conserved while accounting for equilibrium thermal processes. Hence, inequality (
7) simplifies to
Furthermore, with reference to the Boltzmann law, let
, at time
t, be dependent on the present values
, and the history
,
Let
be any time and let
be constant at all times subsequent to
. Formally, given
, with
, consider the static continuation of
,
while
is constant on
. Hence,
and
on
. Consequently, integration of (
8) on
yields
Assume that the functional (
9) satisfies
where
is the constant history
, and then
for brevity, we omit writing the dependence on the present values
.
The result,
means that among the free energies of the histories
with a given present value, that associated with the constant history has the minimum value. This conclusion holds, irrespective of the linearity of the model.
In light of the dependence on
, inequality (
7) becomes
where
denotes the Fréchet derivative at
in the direction of
. The linearity and arbitrariness of
and
imply
These relations hold for any functional ; to save writing we let and stand for and .
If
is independent of
, then
is the standard relation characterizing hyperelasticity. In such a case, it is often assumed that the reference configuration is natural ([
15], §48.2.3) in that
and
at
. The requirement (
15) is viewed as the convexity condition and allows for the invertibility of
. A property of
is related to wave propagation. If we look for jump discontinuities
, in the direction
, by the equation of motion
, at
, then we find that
where
U is the speed and
is the acoustic tensor,
. To guarantee wave propagation the tensor
is assumed to be strongly elliptic ([
16], ch. 11),
for all
. In the linear case,
Lin(Sym, Sym) and
satisfies the propagation condition (
17).
We now go back to linear viscoelasticity and specify
in the Boltzmann form (
2). By (
14) we find
whence, if
,
where
and
possibly are parameterized by the temperature
. The functional
is subject to two conditions. First, by (
14)
, we have
Secondly, by (
13) it follows that
Inequalities (
19) and (
20) are necessary conditions on the free energy for the validity of the Boltzmann law (
2).
Consider the functional
and investigate the consistency with the requirements (
19) and (
20). We first observe that
is in the form (
18). Splitting the dependence on
and that on
we have
Since
, it follows that
has the form (
18) with
At constant histories, namely when
, we have
Hence, it follows that
has a minimum value at constant histories if—and only if— the following holds
where
Now, assuming
we have
Observe that
. Moreover, letting
we have
Hence, an integration by parts yields the following:
Since
and
, the boundary terms (at
) vanish. Hence, inequality (
14)
holds for any history
if—and only if—the following holds:
Thus, the conditions (
26) and (
28) are necessary and sufficient for the consistency of the free-energy functional
.
The functional
for the free energy traces back to Volterra [
17,
18]. The thermodynamic consistency of
has been investigated by Graffi [
19,
20].
The results (
26) and (
28) have been obtained in the literature through various approaches; see, e.g., [
21], where scalar-valued relaxation functions are considered. It is worth emphasizing that these restrictions follow up on the selection of a (nonunique) free-energy functional, as is the case also for the next example.
As with the previous scheme, let
. A further free-energy functional satisfying (
18) is
where
To verify the thermodynamic consistency of the functional (
29), we first observe that the minimum property of
at constant histories is apparent. Next, we note that
Hence, the form (
18) holds with
Furthermore, we verify the requirement
. Since
an integration by parts yields
The vanishing of the boundary terms implies that
Consequently,
for any history
if—and only if—the following holds:
Remark 1. It is worth emphasizing that the restrictions (14) hold for possibly nonlinear models. Inequalities for are related to linear models. 4. Restrictions Induced by Periodic Histories
We now examine the restrictions on the Boltzmann function
induced by a particular set of functions of
and
. Consider functions
and
, such that
and
for any time
t. Consequently,
and hence the history
is periodic, with period
d. This in turn implies that
where
is also periodic. Moreover, let
and
N be the integral of
, so that
Now,
is periodic too, with period
d. Hence, the integration of (
8) over
results in
and the same conclusion follows for any function
, if attention is restricted to isothermal processes, then
. In view of (
32) and (
2), we have
for any periodic functions
with period
d. Here, we do not assume the symmetry of
and
.
To exploit the inequality (
33), we consider harmonic strain tensor functions
being arbitrary symmetric tensors; Hence,
. Substituting in (
33), and integrating, we obtain
where
and
are the
-dependent cosine and sine transforms of
; they are defined on
by
Let
. By Riemann’s lemma, it follows that
Inequality (
34) then reduces to
First, inequality (
35) holds, e.g., for any
with
and
, only if
for any pair
. Next, the arbitrariness of
implies that
, namely
We now let
. As we show in a while,
Hence, taking the limit of (
34) as
we find
Finally, let
. Inequality (
34) reduces to
or
The requirements (
36)–(
39) are necessary for the consistency of (
2) with the second law of thermodynamics. By having recourse to Fourier series, we can prove that they are also sufficient [
2]. We now derive some consequences of (
39).
By the inversion formula, we have
Integration of (
40) with respect to
u yields
Inequality (
40) then implies
Inequality (
42) means that
has a maximum at
or that the instantaneous elastic modulus
is the maximum value of
. However, this need not imply that
is monotone, decreasing as we might expect. It follows from (
40) that, if
and
g is monotone decreasing, then
is negative, and
is monotone decreasing.
If
, then an integration by parts yields
Hence, we have
where
stands for
. Consequently,
Equation (
43) in turn implies that
By the same token, we have
and then
4.1. Proof of (37)
By (
3), it follows that for any arbitrarily small
there is
, such that
Now, since
, we have
as
. Consequently,
.
For any
, there is
, such that
Since
then
as
.
4.2. Remarks about the Half-Range Sine Transform
It seems reasonable to assume that
enjoys the same tensor properties for any
. Hence, we let
and (
2) becomes
The restriction to periodic histories implies
and hence
A free energy satisfies the second law, if .
We now show that by means of the following:
Lemma 1. Let . If f is continuous and monotone decreasing on and as , then To prove (Lemma 1) we first observe that, for any
, we can partition
into subintervals
,
For any subinterval, we have
and
Inequalities (
48) and (
49) imply
Now, for any
let
and observe that
Consequently, for any
, we have
For any
, taking the limit of (
50) as
we obtain
while, by definition,
. This concludes the proof.
Applying the Lemma to
, we obtain
, and hence (
46).
6. Viscoelastic Models with Strain–Rate Histories
Based on the observation that the viscoelastic behaviour is a combination of elastic and viscous effects, Pipkin [
7] suggested that the strain–rate history should be involved rather than the strain history [
22]. Hence the constitutive equation might be written in the form
If
is bounded, then an integration by parts and the assumption
yield
Indeed,
would be the instantaneous modulus and
the Boltzmann function. In that case, the thermodynamic requirement is
Yet, it seems that the effect of
on the stress
is significant if we cannot pass to (
54) because
is unbounded as
. Suppose that we cannot integrate by parts and observe that in connection with the time-harmonic dependence
we can repeat the procedure of §4 and require that the functional (
53) satisfy
This requirement results in
where it follows that
Letting
, we find again
. Now, we let
, and obtain
Inequalities (
55) and (
56) are consistent in that, integrating by parts with
bounded, we have
If we let
,
, then
as
and
is integrable. Hence,
also holds if
.
As we show in the next section, in connection with the general context of models of fractional order, consistency with thermodynamics requires also that there exists a free-energy functional
of the form
with
The existence of such a functional is investigated.
7. Viscoelastic Models of Fractional Order
Still with attention to both a power law form of the kernel and dependence of the stress on the strain–rate we may consider the following constitutive equation:
being a dimensional quantity and most likely
. Differently from (
53), we neglect the dependence on the present value
. Since we have in mind the standard notation for models with derivatives of fractional order, in this section,
and
denote the (total) time derivative at constant reference position
.
While the power law of data may be the physical motivation for assuming constitutive equations of the form (
57), we observe that, if
, then
is not integrable on
. We then might restrict attention to the set of strain histories with compact support,
a being a suitable reference time (called base point in the literature) or to histories, such that (
57) converges. This is the case for time-harmonic histories.
This view leads naturally to the modelling via fractional derivatives. In light of the Caputo fractional derivative [
23], for any
, we let
where
is the Gamma function,
By a change of variable, we have
Since
,
as
. Instead of restricting the set of histories, we may replace the integral on
with the integral on
[
24].
Still, for functions in
, we define the fractional derivative of any order. Let
and
. Let
denote the floor function of
, here
. We define the derivative
in the form
For any
we have
and
. Consequently,
We let . For any the fractional derivative is a linear functional of the history . If, instead, , then the fractional derivative coincides with the corresponding time derivative.
A simple example of constitutive equation of fractional order might be considered in the following form:
where
Lin(Sym, Sym). Hence, we have
For formal convenience, let
Hence, the constitutive Equation (
58) can be written in the form
thus ascribing to
the meaning of kernel, parametrized by the order
and possibly the temperature
. If
, then (
58) becomes
Let
and hence
. Equation (
59) simplifies to
Observe
and then, integrating by parts, we find
Equation (
62) would be the classical form of the Boltzmann law. However
diverges and hence a different approach is in order.
Remark 2. We may wonder about the thermodynamic consistency of constitutive equations of the form (58) with the second law of thermodynamics. Friedrich [25] considered the stress–strain relation in the form of the generalized Maxwell modelwhere σ is the stress, E the spring constant, and ε the strain. It emerged that thermodynamic compatibility holds, or the solution is thermodynamically reasonable [26], if . The consistency with thermodynamics is now investigated in detail. Fractional Models and Thermodynamic Requirements
Consider models where the constitutive functionals
, and
, at time
t, depend on the set of variables
. The Clausius–Duhem inequality (
7) yields
where
is the Fréchet derivative of
with respect to
along
. The linearity and arbitrariness of
,
,
imply that
Since
is independent of
, the remaining inequality implies that
The heat conduction inequality
is consistent, though non-necessary, if
depends on
.
For any given history
, we can select a history
, such that
is arbitrary, while
is as small as we please; this is obtained by letting
be continuously differentiable and
as
with
a arbitrarily small. Hence, it follows
If we assume
then we have
where
is positive definite. This makes the relation for
as a Kelvin–Voigt constitutive model, but leaves
free from derivatives of fractional order.
A constitutive equation of fractional order
has the form
By generality and analogy with the linear viscoelasticity we investigate the thermodynamic consistency of the constitutive equation
where
is the kernel possibly of the fractional-order form. By (
66), it follows
Hence, we look for the free energy in the form
being a constant introduced for dimensional reasons. The functional
is required to satisfy the inequality (
66)
,
Consequently,
takes the form
where
Hence, an integration by parts and the assumption
yield
Since
is unbounded as
, we cannot write the limit value
. In case of
bounded, the negative definiteness of
would imply
Two aspects are crucial. First, bounded is inherently in contrast with the kernels related to the derivatives of fractional order. Furthermore, if is bounded, then are both positive or negative definite. This contradicts the view that the influence on the stress of previous strains (or strain–rates) is weaker for those strains that occurred long ago.
8. Conclusions
This paper investigates the thermodynamic consistency of three models of linear viscoelasticity. The classical model due to Boltzmann is consistent, only if the Boltzmann function has a negative half-range sine transform, . Moreover, consistent free-energy functionals are subject to the inequality . This in turn is consistent with the proof that a function that is decreasing, as is shown by , has a positive sine transform, .
The model involving in a power law form, , satisfies the required condition . If, instead, the stress depends on the strain–rate history , rather than on , then the required consistency condition should be , and this is satisfied. However, gives an unbounded response to constant histories.
The idea of the dependence on
traces back to Pipkin [
7] and is of interest in connection with the viscoelastic model of fractional order. For definiteness, a viscoelastic-like constitutive equation is considered in the form (
68). Both the unboundedness of the kernel and the conditions (
69) show that a free-energy functional has still to be determined. For models with derivatives of fractional order, as well as with constitutive equations involving strain–rate histories, finding a free-energy functional consistent with the second law of thermodynamics seems to be an interesting open problem.