J. Non-Equilib. Thermodyn.
2006 Vol. 31 pp. 319–353
J. Non-Equilib. Thermodyn. 2006 Vol. 31 No. 4
6Copyright 2006 Walter de Gruyter Berlin New York. DOI 10.1515/JNETDY.2006.014
Thermodynamical Frameworks for Higher Grade
Material Theories with Internal Variables or
Additional Degrees of Freedom
Christina Papenfuss
1,
* and Samuel Forest
2
1
Technische Universita
¨t Berlin, Institut fu¨r Mechanik, Straße des 17. Juni 135,
10623 Berlin, Germany
2
Ecole des Mines de Paris, CNRS, Centre des Mate
´riaux, UMR 7633 BP 87,
91003 Evry, France
Communicated by W. Muschik, Berlin, Germany
Abstract
The objective of the present work is to compare several thermomechanical
frameworks, taking into account the influence of strain gradient, internal vari-
ables, gradient of internal variables, and temperature gradient on the consti-
tutive behavior of materials. In particular, the restrictions by the second law
of thermodynamics are derived. The method of exploitation consists of two
steps: an application of the well-known method by Liu and a new method of
exploiting the residual inequality. The first example introduces an enlarged
set of variables for the constitutive functions including in particular the strain
gradient, an internal variable, its gradient, and the temperature gradient. In
the second example, the power of internal forces is enriched to incorporate
generalized stress measures. In the third example, the classical thermome-
chanical setting is complemented by a balance-type di¤erential equation for
an additional variable. Finally, material theories of grade nare envisaged. It
is shown that the free energy density may depend on gradients only in the
case that an additional balance equation is introduced. We also demonstrate
that for isotropic materials the second law of thermodynamics implies for a
large class of state spaces that the entropy flux equals the heat flux divided
by temperature.
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1. Introduction
1.1. Scope of this work
A precise description of the thermoelastoviscoplastic behavior of materials
based on phenomenological constitutive equations requires the introduction
of internal variables [1–4]. An example of such an internal variable is the
alignment tensor in liquid crystals [5–7], and it is known that in this case the
transport of orientational order gives an extra contribution to the entropy
flux [8]. For an overview over the field and for di¤erent applications of inter-
nal variables, see for instance [9–16].
It is also possible within the continuum mechanical framework to account for
the size-dependent material behavior observed in many physical situations
(like grain size e¤ects in polycrystals or particle size e¤ects in composites) by
introducing strain gradients or gradients of internal variables into the consti-
tutive modelling [17–20].
When the constitutive functions like stress and free energy density are as-
sumed to depend on additional variables with respect to the classical frame-
work, such as strain gradients and/or gradients of internal variables, the ther-
modynamical setting of the theory must be reconsidered. In particular, the
restrictions on such dependences induced by the second law of thermodynam-
ics must be examined. Three main trends can be distinguished in the literature
to tackle this problem:
In many cases, the thermomechanical consistency of proposed strain gradi-
ent models is not checked systematically. In particular, the balance equa-
tions (momentum and energy) are assumed to keep their classical form so
that the boundary value problem remains una¤ected. This is the case for
instance for the strain gradient plasticity models in [21–23]. Some limita-
tions of such models are reviewed in [24].
The incorporation of higher order gradient e¤ects is made possible in some
theories by enriching the power of internal forces or by adding contribu-
tions to the energy balance. The mechanical power is enriched in the second
gradient of displacement theories [25–27] by introducing higher order gen-
eralized stress tensors. Such a generalized contribution is added at the level
of the energy balance in [28]. The case of the gradient of damage is tackled
in [29] by extending the power of internal forces through the product of the
gradient of damage rate by a generalized stress measure. Mindlin’s theory
of second gradient media is extended to thermoelasticity in [30, 31] by in-
troducing the gradient of temperature into the thermomechanical setting.
Strain gradient plasticity models are proposed in [32, 33]. These extensions
320 C. Papenfuss and S. Forest
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lead to additional balance equations and additional boundary conditions to
be taken into account.
The introduction of an extra entropy flux khas been considered as an
alternative generalization of the classical theory in [34, 10] to incorporate
internal variables and their gradients. An extra entropy flux in the case of
materials with additional degrees of freedom has been proposed in [9].
In contrast to these approaches, in [35] the problem of thermoviscous fluids is
treated with heat flux and stress tensor as independent variables in the spirit
of extended thermodynamics. Assuming that there is no influence of gra-
dients, the authors find the classical expression for the entropy flux being
heat flux over temperature. Including gradients into the set of variables, they
find an extra entropy flux, i.e., a modified relation between entropy flux and
heat flux.
The objective of the present work is to compare several thermomechanical
frameworks taking the influence of strain gradient, gradient of internal vari-
ables, and temperature gradient on the constitutive behavior of materials into
account. In particular, the restrictions on such constitutive dependences im-
plied by the exploitation of the second law of thermodynamics are derived
systematically and compared for the di¤erent frameworks. The method of
exploiting the second law consists of two steps: The first step is an application
of the well-known methods of Liu [36], or equivalently of the method of Co-
leman and Noll [37]. The second step is a new method of exploiting the resid-
ual inequality.
Section 2 considers the introduction of an enlarged state space including in
particular the strain gradient, an internal variable, its gradient, and the tem-
perature gradient, without considering any additional di¤erential equation.
The restrictions on the constitutive functional dependences are derived. In
Section 3, the power of internal forces is enriched to incorporate generalized
stress measures. In Section 4, the classical thermomechanical setting is com-
plemented by a balance equation for the internal degree of freedom instead
of the usual evolution equation. Finally, material theories of grade nare en-
visaged in Section 5.
In this work, we reserve the term ‘‘internal variable’’ for a quantity which is
neither observable nor controllable. The example of molecule or lattice rota-
tion in liquid or solid crystal shows that a quantity can be observable (by X-
ray analysis for example) but not easily controllable because it is experimen-
tally di‰cult to monitor it. The question then arises whether it should be
treated as an internal variable or as an actual degree of freedom. When the gra-
dient of such a variable plays a role in the material theory, we show especially
in Section 3 that the variable can be treated as an additional degree of freedom.
Thermodynamical Frameworks for Higher Grade Material Theories 321
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The analysis is confined to the case of solid bodies within the context of small
perturbations, although straightforward extensions of most results are pos-
sible in the context of finite deformations or in the case of fluids. Accord-
ingly, the privileged constitutive function in this work is the free energy den-
sity C.
The notation used throughout this work is the following: scalar quantities,
vectors, second- and third-rank tensors are denoted respectively, by a,a,aî,
aî
ô
. Simple, double, and triple contraction read ,:,.
.
.
.‘is the nabla operator.
The symmetric strain tensor is denoted by eîwithin the context of small defor-
mations. _
aa is the time derivative of a.
1.2. Method of exploiting the dissipation inequality
According to the second law of thermodynamics, all processes are connected
with a non-negative entropy production sb0. In principle, this could be
guaranteed either by ‘‘ruling out’’ the (mathematical) solutions of the balance
equations, which contradict this dissipation inequality, or by restrictions on
constitutive functions, such that there exist only solutions of the balance equa-
tions together with constitutive equations in agreement with the dissipation
inequality. With a very reasonable amendment to the second law, it can be
shown that the second possibility is the case [38]: the second law of thermody-
namics imposes restrictions on constitutive functions. The most general form
of these restrictions is derived by the method of Liu [36] after having chosen
the state space (the set of variables the constitutive functions depend upon).
Let us give here a short sketch of the method:
The balance equations, after the chain rule is applied to the constitutive func-
tions, form a system of equations linear in the so-called higher derivatives,
i.e., the derivatives are not included in the state space but one order higher.
These higher derivatives will be put together in a row y. Then the system of
balance equations and the dissipation inequality can be written symbolically
as:
Ay¼C;ð1Þ
BybD;ð2Þ
with a matrix A, rows Band C, and a scalar function D. All these quantities
are functions of the state space variables.
LIU’s proposition [39, 36, 40]: Constitutive equations satisfy the relations
BðZÞ¼lðZÞAðZÞ;ð3Þ
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lðZÞCðZÞbDðZÞ;ð4Þ
with state space functions lðZÞ. The entropy production density
s:¼lCDb0ð5Þ
is independent of the process direction, i.e., it depends only on the state space
elements and not on higher derivatives.
The set of equations (3) will be denoted as ‘‘LIU equations’’, and Eq. (4) is
the residual inequality. The LIU equations are as many equations as there
are higher derivatives (elements in the row y). These are more equations
than there are unknown factors l, which is the same number as the number
of balance equations. The equations remaining after eliminating the unknowns
lfrom the LIU equations are the restrictions on constitutive functions. After
eliminating the multipliers lfrom the residual inequality, an expression for
the entropy production is obtained. In the examples, it turns out that this ex-
pression is of the form of a sum of products where one factor can be denoted
as a thermodynamic force and the other one as a thermodynamic flux, i.e., it
is of the form
s¼X
i
vifiðvjÞb0:ð6Þ
One can show that for continuous functions fiðvjÞit follows [41] that fiis a
homogeneous function,
vi¼0)fi¼0:ð7Þ
This fact as well as the LIU equations will be exploited in the following.
Another way of exploiting the inequality of the second law of thermodynam-
ics is the Coleman–Noll procedure [42]. Both methods have been used for the
results presented in this work and can be shown to lead to the same conclu-
sions under conditions stated in Section 3.
2. Application to a gradient theory with an internal variable
A higher order theory involving the first and second gradients of the displace-
ment field is envisaged in this section using the method proposed in [25]. The
influence of an internal variable aand of its first gradient is also taken into
account. A full thermomechanical framework is the aim of this part.
Thermodynamical Frameworks for Higher Grade Material Theories 323
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