On Entropy Flux of Anisotropic Elastic Bodies

The framework of irreversible thermodynamics is fundamental in development of constitutive models. One of the important aspects of the extended irreversible thermodynamics is the relationship between the entropy flux and the heat flux, especially for phenomena far from equilibrium. In this paper, we demonstrate that the assumption that Lagrange multiplier conjugated to the energy balance equation (in the expression for the second law of thermodynamics) is a function of temperature $$ \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ) $$Λε=Λε(θ) is a sufficient condition to derive the entropy flux–heat flux relation for all isotropic materials as well as for a number of crystal classes including transverse isotropy, orthotropy, triclinic systems and rhombic systems. For all considered crystal classes, the entropy flux–heat flux relation was derived explicitly. Further, we demonstrate that for some crystal classes heat flux is nonzero even when temperature gradient vanishes (as stated by Eringen). The anisotropic functions, with respect to the symmetry groups of the crystal classes, were expressed in terms of isotropic functions. The proposed procedure is very general in the sense that it can be used with nonlinear constitutive relations as demonstrated here. The presented results confirm that the all crystal elastic bodies considered are hyperelastic.


Introduction
One of the important problems in thermodynamics is the relationship between the entropy flux and the heat flux for phenomena far from equilibrium.
The entropy principle based on the Clausius-Duhem inequality d 0, r iv where  is density,  is the specific entropy density, q the heat flux and r the heat supply, has been widely adopted in the development of modern rational thermodynamics after the fundamental work of Coleman and Noll [1]. The main assumptions, motivated by the result of classical thermostatics, are that the entropy flux Φ and the entropy supply s are proportional to the heat flux and the heat supply, respectively. Moreover, both constants of proportionality. are assumed to be the reciprocal of the absolute temperature, i.e. 11 = , = . sr   Φq (2) These main assumptions, while tacit in the classical theory of continuum mechanics, do not hold particularly well for materials in general. In fact, it is known that they are inconsistent with the kinetic theory of ideal gases and are also found to be inadequate to account for the thermodynamics of diffusion. There is an extended formulation of the second law of thermodynamics which has been applied to nonequilibrium thermodynamics by Serrin [2] and Silhavy [3] and summarized by Truesdell and Bharatha [4]. See also Muschik [5] and Müller Error! Reference source not found.. A comprehensive review of related literature and detailed derivations can be found in Lebon at.al. [20] and Jou at. al. [21]. The extended form of the second law, usually called the entropy inequality, seems to be the most general formulation of the continuous second law of thermodynamics proposed so far . In this theory, the assumptions given by equation (2) were abandoned and the entropy flux Φ and the heat flux vector q are treated as independent constitutive quantities and hence leaving the entropy inequality in its general form div 0. Further, I-Shih Liu [7], analysed the thermodynamic theory of viscoelastic bodies and proved that for isotropic viscoelastic materials the results are identical to the classical results given by equation (2). In the same paper, he also proved that the body is hyperelastic because of the Lagrange multiplier = ( )   .
However, for anisotropic elastic materials in general, the validity of the classical entropy flux relation is yet to be demonstrated.
The first contribution in this direction has been given by I-Shih Liu [8] [9], who proved by considering transversely isotropic elastic bodies and transversely isotropic rigid heat conductors that the classical entropy flux relation (2) need not be valid in general.
In this paper we investigate the functional dependence of the Lagrange multiplier = ( ) irrespective of whether the classical entropy flux relation is valid. This enable us to derive the entropy flux -heat flux relations for a number of crystal classes including transverse isotropy, orthotropy, triclinic systems, monoclinic systems and rhombic systems. The paper is organized as follows: In Section 2, the basic ideas and formulas typically used in this field are given as a starting point of our investigation. In Section 3 the entropy flux relation for viscoelastic bodies and transverse isotropic elastic materials is reconsidered assuming the Lagrange multiplier dependency on temperature, i.e. = ( )   . In Section 4 the procedure introduced in Section 3 is extended for the derivation of the entropy flux relation of anisotropic elastic materials defined by the crystal classes listed above. In Section 5 the entropy inequality for anisotropic bodies is examined further. Conclusions related to the outcome of this work are given in Section 6.

The entropy principle
In this section the basic framework of the entropy principle for viscoelastic materials is presented. The balance laws of mass, linear momentum and energy can be stated in current configuration as div = 0, div = , div grad = xr where T is the Cauchy stress tensor, b is external body force and r is external heat supply. Note that for solid bodies it is more convenient to use a referential description. Also, since constitutive relations do not depend on external supplies, it suffices to consider only supply-free bodies. Consequently, the balance laws can be rewritten as and the entropy inequality Div 0.

  Φ
Here the first Piola-Kirchhoff stress tensor  T , the material heat flux vector  q and the material entropy flux vector  Φ are related to the Cauchy stress tensor T , the heat flux vector q and the entropy flux vector Φ by 11 = , = , = , where F is the deformation gradient in referential coordinates and = | det | J F . "Div" is the divergence operator with respect to the referential coordinates.
It is well known that the entropy principle imposes severe restrictions on constitutive functions and the exploitation of such restrictions based on the Clausius Duhem inequality are relatively easy. For elastic materials, in general, the thermodynamic restrictions can be easily obtained by the well-known Coleman-Noll procedure [1].
The derivation of the relation between the entropy flux and the heat flux based on the entropy principle, referred to as entropy flux relation, is a typical problem in this new theory.
Here we outline the consideration for isotropic viscoelastic materials with isotropic elasticity as a special case. Using the principle of equipresence, the constitutive relations for viscoelastic materials can be written as functions of the state variables   , , , i.e.
where F is time derivative of the deformation gradient, ε is the specific internal energy, =    g is temperature gradient,  is an empirical temperature, which is some convenient measure of the hotness (or coldness) of the thermodynamic state. Note that the density field   Therefore, the thermodynamic process is defined as the solution tt  x X X (9) of the field equations (the balance laws of the linear momentum and energy) and integration of the constitutive relations for  T ,  q , and  .
The determination of the restrictions imposed on the constitutive functions by the entropy principle is one of the major objectives in modern continuum thermodynamics.

Method of Lagrange multipliers
According to the entropy principle, there exist Lagrange multipliers v  and   which depend on the state variables, such that the inequality is valid under no additional constraints, i.e. valid for any field     , , , tt  x X X .
Further, we invoke the condition of material objectivity, which implies the following reduced constitutive equations for viscoelastic materials where T C = F F is the right Cauchy-Green tensor.
Since the inequality (10) must hold for any   Div Div The other terms in the equation (15) have equivalent component forms.
After substituting (14) and (15) into (13), by inspection, we conclude that this inequality is also linear with respect to the following derivatives   . , , , , , , As the inequality must hold for arbitrary fields we have eliminated the constraints imposed by the field equations. The coefficients of the above derivatives must vanish identically. Otherwise, we could choose the fields in such a way that one negative term would dominate all others and the inequality would be violated. Hence, we obtain the following equations =0

Φq gg
Φq CC Φq CC (17) Then the entropy inequality (13) reduces to Making use of second Piola-Kirchhoff tensor F T F TF and the right Cauchy-Green tensor (18) can be written in a more compact form as Where  is entropy production density. Moreover, from (16) we obtain (20) which has the form of the thermostatic Gibbs relation.

Entropy flux relation for viscoelastic materials
For further evaluation of the consequences of the entropy principle, particularly in connection with relations (17), we invoke the material symmetry condition that has to be satisfied by for isotropic viscoelastic bodies. For instance, this condition for heat flux can be expressed as O is the full orthogonal group. Note that ˆ q is an isotropic vector-valued function of   , , ,   C C g and that (21) imposes restriction on its form. After a lengthy calculation starting from (15), I-Shih Liu [7] proved that the following entropy flux relation holds Further, based on (22) and (17) By comparison with the classical Gibbs relation in thermostatics, the function   can be identified as the reciprocal of the absolute temperature Which leads to the classical entropy flux relation (2). Consequently, one can come to the conclusion that, following I-Shih Liu's procedure for viscoelastic bodies outlined above, that for isotropic elastic materials with state variables Fg the relation (25) holds.

Entropy flux of anisotropic elastic materials
In this subsection the derivation of the relation between the entropy flux and the heat flux for anisotropic materials with state variables Fg is considered. In this case the requirements (16) and (17)  In addition, the material symmetry condition for anisotropic elastic bodies has to be satisfied by The first paper considering entropy flux for transversely isotropic elastic bodies, was published by I-Shih Liu in 2009 [8], which relies on his paper [11]. In this, anisotropic materials properties in preferential directions were characterized by a number of unit vectors 1 Here v is a vector, A is a second order tensor and f is either scalar-valued, vector-valued or tensor valued function. Particularly, if ˆ( , , , ) f vAmM is an isotropic function then: • and for a tensor-valued function ˆ( , , , ) = ( , , , ).
In the same paper I-Shih Liu gives a list of 14 such groups g for some crystal classes. He uses the following notation:    This is where we pose the question whether this is sufficient condition? Having this in mind we proceed first by re-examining the above cases.


Eringen stated [19] that "it is always possible to express the entropy change as a sum of entropy flux and entropy source as" where 1 Φ is the entropy change due to all other effects except heat input. In our consideration we do not use Eringen's postulate and do not make any assumption about the entropy flux relation. The only assumption used in our derivation is that Lagrange multiplier For clarity reasons the following few steps are written in index notation or component notation.
This result is identical to I-Shih Liu's [7] for isotropic viscoelastic bodies, which validates the new procedure and led to its application to the anisotropic materials considered below. It is important to observe that the assumption = ( )    significantly simplifies and shortens the procedure.

Entropy flux relation for anisotropic elastic materials in general
This i.e. ( , , )  A mM is skew symmetric tensor-valued isotropic function of its arguments. For vector-valued and skew-symmetric tensor-valued isotropic functions in R 3 see Smith [12] and Spencer [18]. All groups of crystal classes given by I-Shih Liu [11] are considered below.

Transversally isotropic material bodies
In this section we consider transversely isotropic material bodies divided into four cases. The An is skew-symmetric tensor-valued isotropic function. Consequently, A = 0 (see Smith [12] (59) The result (59) agrees with I-Shih Liu's result for transversally isotropic bodies with group symmetry group g5 [8].
Therefore, for the above cases we demonstrated that = ( )    is not only a necessary but also a sufficient condition to determine the entropy flux-heat flux relation.
To demonstrate the generality of the proposed procedure we consider the other crystal classes for which representation of anisotropic invariants is given by I-Shih Liu [11]. Note, in all these cases representations of anisotropic invariant function are obtained using the tables for isotropic functions. In addition, we do not see any physical reason that   would have a different form to = ( )    for these crystal classes.
The result for the following cases are new (to the best of our knowledge not available in published literature).

The entropy inequality for anisotropic bodies
So far we did not investigate the entropy inequality 1 =0 2 n N q C 0 n N n n q C 0 n n n n n N q C 0 n n n N n n n n q C 0 n n n n N q C 0 N 0 q C 0 n n 0

Conclusion
This paper revisits entropy flux and heath flux relations for isotropic and several anisotropic elastic materials. More specifically we investigated the consequence of the assumption that = ( )    irrespective of validity of the classical entropy flux relation. This assumption is used to derive the relationship between the entropy flux and heat flux for all isotropic elastic materials as well as for some crystal classes including transverse isotropy, orthotropy, triclinic systems and rhombic systems. First re-examined the entropy flux-heat flux relation for viscoelastic materials, isotropic elastic materials and transversely isotropic elastic bodies and demonstrate that our results agree with the results obtained by I-Shih Liu [6] to [10]. All these cases confirm that = ( )    is a necessary and sufficient condition for the determination of the entropy flux-heat flux relation.
Furthermore, we derived the entropy flux-heat flux relations for all the following crystal classes: transverse isotropy, orthotropy, triclinic systems, monoclinic systems and rhombic systems for which representations of anisotropic functions with respect to their symmetry groups can be expressed in terms of isotropic functions. Our derivation is very general in the sense that the constitutive relations are non-linear. One of our main results is the proof that all crystal elastic bodies, we considered, are hyperelastic. This represents a generalization of I-Shih Liu's finding for transversely isotropic bodies, the only case he analysed.
We would like to draw attention to the following three points: i. The vector function a and skew-symmetric function A are isotropic functions depending only on the set is also a necessary condition, at least for all crystal classes investigated above. This is a task for future investigation. Where v and w are arbitrary vectors