1 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

$$ \rho \dot{\eta } + {\text{div}}\frac{{\mathbf{q}}}{\theta } - \rho \frac{r}{\theta } \ge 0, $$
(1)

where \( \rho \) is density, \( \eta \) is the specific entropy density, \( {\mathbf{q}} \) 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 \( {\varvec{\Phi}} \) 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.,

$$ {\varvec{\Phi}}_{\kappa } = \frac{1}{\theta }{\mathbf{q}}_{\kappa } ,\quad s = \frac{1}{\theta }r. $$
(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. Further, Sellitto et al. [2] demonstrated that the proportionality relations (2) do not hold for nonlocal heat transfer at nanoscale and provide an illustrative example of cylindrical nanodevice connected to a graphene layer.

There is an extended formulation of the second law of thermodynamics which has been applied to nonequilibrium thermodynamics by Serrin [3] and Silhavy [4] and summarized by Truesdell and Bharatha [5]. See also Muschik [6], Müller [7] and Domínguez-Cascante and Jou [8]. A comprehensive review of the related literature and detailed derivations can be found in Lebon et al. [9] and Jou et al. [10, 11].

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 Eq. (2) were abandoned and the entropy flux \( {\varvec{\Phi}} \) and the heat flux vector q are treated as independent constitutive quantities and hence leaving the entropy inequality in its general form

$$ \rho \dot{\eta } + {\text{div}}{\varvec{\Phi}} - \rho s \ge 0. $$
(3)

Liu [12] proposed a method, reminiscent of the classical method of Lagrange multipliers, for expanding the inequality (3). Instead of this inequality restricting the solution of field equations, he considered solutions of an extended inequality which should hold for all fields. This can be done if one considers the field equations as constraints on solutions of the energy inequality.

Further, Liu [13] analyzed the thermodynamic theory of viscoelastic bodies and proved that for isotropic viscoelastic materials the results are identical to the classical results given by Eq. (2). In the same paper, he also proved that the body is hyperelastic because of the Lagrange multiplier \( \varLambda^{\varepsilon } = \varLambda (\theta ) \).

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 Liu [14, 15], 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.

Three years later, Bargmann et al. [16] considered the energy influx–entropy influx relation in the Green–Naghdi type III theory of heat conduction and showed that the entropy influx and the energy influx are proportional via the absolute temperature if heat conduction is isotropic. Further, they demonstrated that influx proportionality cannot be postulated in general by giving counterexample of transversely isotropic conduction. Their proof is based on a representation formula for isotropic vector-valued mappings of two vector arguments where Lagrange multiplier is assumed to have the following general form: \( \varLambda = \bar{\varLambda }(\left| {\mathbf{u}} \right|,\left| {\mathbf{v}} \right|,{\mathbf{u}} \cdot {\mathbf{v}}) \). Using Green–Naghdi, set of internal variables \( S\left\{ {\alpha ,\dot{\alpha },\nabla \alpha ,\nabla \dot{\alpha }} \right\} \) where α is thermal displacement they showed, through an algebraic analysis, that the multiplier \( \varLambda = \bar{\varLambda }\left( {\alpha ,\dot{\alpha }} \right) \) is at least proportional to coldness (the inverse of absolute temperature and does not depend on the thermal displacement gradient \( \nabla \alpha \) nor on the temperature gradient \( \nabla \dot{\alpha } \)) for isotropic conduction.

Podio–Guidugli [17] considered energy and entropy inflows in the theory of heat conduction and demonstrated proportional via the absolute temperature for isotropic conduction using similar procedure.

We do not see the possibility for the Bargmann et al. procedure to be generalized and extended to conduction in continuum with other symmetry types.

In this paper, we investigate the functional dependence of the Lagrange multiplier conjugated to the energy balance equation \( \varLambda^{\varepsilon } = \varLambda (\theta ) \), irrespective of whether the classical entropy flux relation is valid. This enables derivation of 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 Sect. 2, the basic ideas and formulas typically used in this field are given as a starting point of our investigation. In Sect. 3, the entropy flux relation for viscoelastic bodies and transverse isotropic elastic materials is reconsidered assuming the Lagrange multiplier dependency on temperature, i.e., \( \varLambda^{\varepsilon } = \varLambda (\theta ) \). In Sect. 4, the procedure introduced in Sect. 3 is extended for the derivation of the entropy flux relation of anisotropic elastic materials defined by the crystal classes listed above. In Sect. 5, the entropy inequality for anisotropic bodies is examined further. Conclusions related to the outcome of this work are given in Sect. 6.

2 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:

$$ \begin{aligned} & \dot{\rho } + \rho {\text{div}}{\dot{\mathbf{x}}} = 0, \\ & \rho {\ddot{\mathbf{x}}} - {\text{div}}{\mathbf{T}} = \rho {\mathbf{b}}, \\ & \rho \dot{\varepsilon } + {\text{div}}{\mathbf{q}} - {\mathbf{T}} \cdot {\text{grad}}\,\dot{x} = \rho r \\ \end{aligned} $$
(4)

where \( {\mathbf{T}} \) is the Cauchy stress tensor, \( {\mathbf{b}} \) 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:

$$ \begin{aligned} & \rho = J^{ - 1} \rho_{\kappa } , \\ & \rho_{\kappa } {\ddot{\mathbf{x}}} - {\text{div}}{\mathbf{T}}_{\kappa } = {\mathbf{0}}, \\ & \rho_{\kappa } \dot{\varepsilon } + {\text{Div}}{\mathbf{q}}_{\kappa } - {\mathbf{T}}_{\kappa } \cdot {\dot{\mathbf{F}}} = 0, \\ \end{aligned} $$
(5)

and the entropy inequality

$$ \rho_{\kappa } \dot{\eta } + {\text{Div}}{\varvec{\Phi}}_{\kappa } \ge 0. $$

Here, the first Piola–Kirchhoff stress tensor \( {\mathbf{T}}_{\kappa } \), the material heat flux vector \( {\mathbf{q}}_{\kappa } \) and the material entropy flux vector \( {\varvec{\Phi}}_{\kappa } \) are related to the Cauchy stress tensor \( {\mathbf{T}} \), the heat flux vector \( {\mathbf{q}} \) and the entropy flux vector \( {\varvec{\Phi}} \) by

$$ {\mathbf{T}}_{\kappa } = {J{\mathbf{T}}{\kern 1pt} {\mathbf{F}}^{ - T},}\quad {\mathbf{q}}_{\kappa } = J{\mathbf{F}}^{{{\mathbf{ - }}1}} {\kern 1pt} {\mathbf{q}},\quad {\varvec{\Phi}}_{\kappa } = J{\mathbf{F}}^{ - 1} {\kern 1pt} {\varvec{\Phi}}, $$
(6)

where \( {\mathbf{F}} \) is the deformation gradient in referential coordinates and \( J = |\det {\mathbf{F}}| \). “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 and is 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 within the framework of continuum mechanics. The thermodynamics of the continua sufficiently close to equilibrium so that the principle of equipresence can be used. Consequently, the local constitutive relations for viscoelastic materials can be written as functions of the state variables

$$ \left( {{\mathbf{F}},{\dot{\mathbf{F}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right) , $$
(7)

i.e.,

$$ \begin{aligned} \begin{array}{*{20}l} {{\mathbf{T}}_{\kappa } = {\hat{\mathbf{T}}}_{\kappa } } \hfill \\ \end{array} \left( {{\mathbf{F}},{\dot{\mathbf{F}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right), \hfill \\ \begin{array}{*{20}l} {{\mathbf{q}}_{\kappa } = {\hat{\mathbf{q}}}_{\kappa } } \hfill \\ \end{array} \left( {{\mathbf{F}},{\dot{\mathbf{F}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right), \hfill \\ \begin{array}{*{20}l} {\varepsilon = \hat{\varepsilon }} \hfill \\ \end{array} \left( {{\mathbf{F}},{\dot{\mathbf{F}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right), \hfill \\ \begin{array}{*{20}l} {\eta = \hat{\eta }} \hfill \\ \end{array} \left( {{\mathbf{F}},{\dot{\mathbf{F}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right), \hfill \\ \begin{array}{*{20}l} {{\varvec{\Phi}}_{\kappa } = {\hat{\mathbf{\varPhi }}}_{\kappa } } \hfill \\ \end{array} \left( {{\mathbf{F}},{\dot{\mathbf{F}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right), \hfill \\ \end{aligned} $$
(8)

where \( {\dot{\mathbf{F}}} \) is time derivative of the deformation gradient, ε is the specific internal energy, \( {\mathbf{g}}_{\kappa } = \nabla \theta \) is temperature gradient, \( \theta \) is an empirical temperature, which is some convenient measure of the hotness (or coldness) of the thermodynamic state. Note that the density field \( \rho \left( {{\mathbf{X}},{\kern 1pt} \,t} \right) \) is completely determined by the motion \( {\mathbf{x}}\left( {{\mathbf{X}},{\kern 1pt} \,t} \right) \) and the density \( \rho_{\kappa } \left( {\mathbf{X}} \right) \) in the reference configuration. Therefore, the thermodynamic process is defined as the solution

$$ \left\{ {{\mathbf{x}}\left( {{\mathbf{X}},\,{\kern 1pt} t} \right),{\kern 1pt} \, \theta \left( {{\mathbf{X}},{\kern 1pt} \,t} \right)} \right\} $$
(9)

of the field equations (the balance laws of the linear momentum and energy) and integration of the constitutive relations for \( {\mathbf{T}}_{\kappa } \), \( {\mathbf{q}}_{\kappa } \) and \( \varepsilon \).

The determination of the restrictions imposed on the constitutive functions by the entropy principle is one of the major objectives in modern continuum thermodynamics.

2.1 Method of Lagrange Multipliers

According to the entropy principle, there exist Lagrange multipliers \( \varLambda^{v} \) conjugated to the momentum balance equation and \( \varLambda^{\varepsilon } \) conjugated to the energy balance equation which depend on the state variables, such that the inequality

$$ \rho_{\kappa } \dot{\eta } + {\text{Div}}{\varvec{\Phi}}_{\kappa } - \varLambda^{v} \cdot \left( {\rho_{\kappa } {\ddot{\mathbf{x}}} - {\text{Div}}{\mathbf{T}}_{\kappa } } \right) - \varLambda^{\varepsilon } \left( {\rho_{\kappa } \dot{\varepsilon } + {\text{Div}}{\mathbf{q}}_{\kappa } - {\mathbf{T}}_{\kappa } \cdot {\dot{\mathbf{F}}}} \right) \ge 0 $$
(10)

is valid under no additional constraints, i.e., valid for any field \( {\mathbf{x}}\left( {{\mathbf{X}},{\kern 1pt} t} \right),{\kern 1pt} \, \theta \left( {{\mathbf{X}},{\kern 1pt} t} \right) \).

Further, we invoke the condition of material objectivity, which implies the following reduced constitutive equations for viscoelastic materials

$$ \begin{aligned} {\mathbf{T}}_{\kappa } & = {\hat{\mathbf{T}}}_{\kappa } \left( {{\mathbf{C}},{\dot{\mathbf{C}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right), \\ {\mathbf{q}}_{\kappa } & = {\hat{\mathbf{q}}}_{\kappa } \left( {{\mathbf{C}},{\dot{\mathbf{C}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right), \\ \varepsilon & = \hat{\varepsilon }\left( {{\mathbf{C}},{\dot{\mathbf{C}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right), \\ \eta & = \hat{\eta }\left( {{\mathbf{C}},{\dot{\mathbf{C}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right), \\ {\varvec{\Phi}}_{\kappa } & = {\hat{\varvec{\Phi }}}_{\kappa } \left( {{\mathbf{C}},{\dot{\mathbf{C}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right), \\ \end{aligned} $$
(11)

where \( {\mathbf{C = F}}^{T} {\kern 1pt} {\mathbf{F}} \) is the right Cauchy–Green tensor.

Since the inequality (10) must hold for any \( {\mathbf{x}}\left( {{\mathbf{X}},{\kern 1pt} t} \right) \) and \( \theta \left( {{\mathbf{X}},{\kern 1pt} t} \right) \), the values of \( \left\{ {\theta ,{\kern 1pt} \,{\mathbf{g}}_{\kappa } ,{\kern 1pt} \,{\mathbf{C}},\,{\kern 1pt} {\dot{\mathbf{C}}}} \right\} \) and \( \left\{ {\dot{\theta },{\kern 1pt} \,{\ddot{\mathbf{x}}},\,\,{\dot{\mathbf{g}}}_{\kappa } ,{\ddot{\mathbf{C}}},{\kern 1pt} \,\nabla {\mathbf{g}}_{\kappa } ,{\kern 1pt} \,\nabla {\mathbf{C}},{\kern 1pt} \,\nabla {\dot{\mathbf{C}}}} \right\} \) in (10) can have arbitrary values at any point and any instant.

First, note that (10) is linear with respect to \( {\ddot{\mathbf{x}}} \). Consequently, \( \rho_{\kappa } \varLambda^{v} \) the coefficient of \( {\ddot{\mathbf{x}}} \) must be equal to zero, i.e.,

$$ \varLambda^{v} = 0. $$
(12)

Thus, (10) becomes

$$ \rho_{\kappa } \dot{\eta } + {\text{Div}}{\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } \rho_{\kappa } \dot{\varepsilon } + \varLambda^{\varepsilon } {\text{Div}}{\mathbf{q}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{T}}_{\kappa } \cdot {\dot{\mathbf{F}}} \ge 0 $$
(13)

Next, we consider terms in

$$ \begin{aligned} \dot{\eta } - \varLambda^{\varepsilon } \dot{\varepsilon } & = \left( {\frac{\partial \eta }{\partial \theta } - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{\partial \theta }} \right)\dot{\theta } + \left( {\frac{\partial \eta }{{\partial {\mathbf{g}}_{\kappa } }} - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{{\partial {\mathbf{g}}_{\kappa } }}} \right) \cdot {\dot{\mathbf{g}}}_{\kappa } \\ & \quad + \left( {\frac{\partial \eta }{{\partial {\mathbf{C}}}} - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{{\partial {\mathbf{C}}}}} \right) \cdot {\dot{\mathbf{C}}} + \left( {\frac{\partial \eta }{{\partial {\dot{\mathbf{C}}}}} - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{{\partial {\dot{\mathbf{C}}}}}} \right) \cdot {\ddot{\mathbf{C}}} \\ \end{aligned} $$
(14)
$$ \begin{aligned} {\text{Div}}{\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\text{Div}}{\mathbf{q}}_{\kappa } & = \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{\partial \theta } - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{\partial \theta }} \right) \cdot {\mathbf{g}}_{\kappa } + \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }}} \right) \cdot \nabla {\mathbf{g}}_{\kappa } \\ & \quad + \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\mathbf{C}}}} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\mathbf{C}}}}} \right) \cdot \nabla {\mathbf{C}} + \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\dot{\mathbf{C}}}}} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\dot{\mathbf{C}}}}}} \right) \cdot \nabla {\dot{\mathbf{C}}} \\ \end{aligned} $$
(15)

Note that the term

$$ \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }}} \right) \cdot \nabla {\mathbf{g}}_{\kappa } $$

in component form reads as

$$ \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }}} \right) \cdot \nabla {\mathbf{g}}_{\kappa } = \left( {\frac{{\partial \varPhi_{\kappa K} }}{{\partial \theta_{,L} }} - \varLambda^{\varepsilon } \frac{{\partial q_{\kappa } }}{{\partial \theta_{,L} }}} \right)\theta_{,LK} . $$

The other terms in the Eq. (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 \( \left\{ {\dot{\theta },{\kern 1pt} \,{\ddot{\mathbf{x}}},\mathop {\,{\mathbf{g}}}\limits^{.}_{\kappa } ,\,{\ddot{\mathbf{C}}},{\kern 1pt} \,\nabla {\mathbf{g}}_{\kappa } ,{\kern 1pt} \,\nabla {\mathbf{C}},{\kern 1pt} \,\nabla {\dot{\mathbf{C}}}} \right\} \).

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

$$ \begin{aligned} & \varLambda^{v} = 0 \\ & \frac{\partial \eta }{\partial \theta } - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{\partial \theta } = 0 \\ & \frac{\partial \eta }{{\partial {\mathbf{g}}_{\kappa } }} - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{{\partial {\mathbf{g}}_{\kappa } }} = 0 \\ & \frac{\partial \eta }{{\partial {\dot{\mathbf{C}}}}} - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{{\partial {\dot{\mathbf{C}}}}} = 0 \\ \end{aligned} $$
(16)
$$ \begin{aligned} & \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }}} \right)_{\text{sym}} = 0 \\ & \frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\mathbf{C}}}} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\mathbf{C}}}} = 0 \\ & \frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\dot{\mathbf{C}}}}} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\dot{\mathbf{C}}}}} = 0 \\ \end{aligned} $$
(17)

Then, the entropy inequality (13) reduces to

$$ \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{\partial \theta } - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{\partial \theta }} \right) \cdot {\mathbf{g}}_{\kappa } + \rho_{\kappa } \left( {\frac{\partial \eta }{{\partial {\mathbf{C}}}} - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{{\partial {\mathbf{C}}}}} \right) \cdot {\dot{\mathbf{C}}} + \varLambda^{\varepsilon } {\mathbf{T}}_{\kappa } \cdot {\dot{\mathbf{F}}} \ge 0 $$
(18)

Making use of second Piola–Kirchhoff tensor \( {\mathbf{S}}_{{\varvec{\upkappa}}} = {\mathbf{F}}^{{{\mathbf{ - }}1}} {\kern 1pt} {\mathbf{T}}_{{\varvec{\upkappa}}} = J{\kern 1pt} {\mathbf{F}}^{ - 1} {\kern 1pt} {\mathbf{T}}{\kern 1pt} {\mathbf{F}}^{ - T} \) and the right Cauchy–Green tensor \( {\dot{\mathbf{C}}} = {\dot{\mathbf{F}}}^{T} {\kern 1pt} {\mathbf{F}} + {\mathbf{F}}^{T} {\kern 1pt} {\dot{\mathbf{F}}} = 2\left( {{\dot{\mathbf{F}}}^{T} {\kern 1pt} {\mathbf{F}}} \right)_{\text{sym}} \) (which are symmetric tensors), the inequality (18) can be written in a more compact form as:

$$ \sigma = \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{\partial \theta } - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{\partial \theta }} \right) \cdot {\mathbf{g}}_{\kappa } + \rho_{\kappa } \left( {\frac{\partial \eta }{{\partial {\mathbf{C}}}} - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{{\partial {\mathbf{C}}}} + \frac{1}{{2\rho_{\kappa } }}\varLambda^{\varepsilon } {\mathbf{S}}_{\kappa } } \right) \cdot {\dot{\mathbf{C}}} \ge 0 $$
(19)

where \( \sigma \) is entropy production density. Moreover, from (16) we obtain

$$ \begin{aligned} & \frac{{{\text{d}}\eta }}{{{\text{d}}t}} - \varLambda^{\varepsilon } \frac{{{\text{d}}\varepsilon }}{{{\text{d}}t}} = \left( {\frac{\partial \eta }{{\partial {\mathbf{C}}}} - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{{\partial {\mathbf{C}}}}} \right) \cdot \frac{{{\text{d}}{\mathbf{C}}}}{{{\text{d}}t}} \\ & {\text{d}}\eta = \varLambda^{\varepsilon } {\kern 1pt} {\text{d}}\varepsilon + \left( {\frac{\partial \eta }{{\partial {\mathbf{C}}}} - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{{\partial {\mathbf{C}}}}} \right) \cdot {\text{d}}{\mathbf{C}} \\ \end{aligned} $$
(20)

which has the form of the thermostatic Gibbs relation.

3 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 \( \left\{ {\varepsilon ,{\kern 1pt} \,{\mathbf{T}}_{\kappa } ,{\kern 1pt} \,{\mathbf{q}}_{\kappa } ,{\kern 1pt} \,{\varvec{\Phi}}_{\kappa } ,{\kern 1pt} \,\eta } \right\} \) for isotropic viscoelastic bodies. For instance, this condition for heat flux can be expressed as:

$$ {\hat{\mathbf{q}}}_{\kappa } \left( {{\mathbf{QCQ}}^{T} ,{\kern 1pt} {\mathbf{Q}}{\dot{\mathbf{C}}}{\mathbf{Q}}^{T} ,{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{Qg}}_{\kappa } } \right) = {\mathbf{Q}}\hat{\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},{\kern 1pt} {\dot{\mathbf{C}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right), \, \forall {\mathbf{Q}} \in O $$
(21)

where \( O \) is the full orthogonal group. Note that \( {\hat{\mathbf{q}}}_{\kappa } \) is an isotropic vector-valued function of \( \left\{ {{\mathbf{C}},{\kern 1pt} \,{\dot{\mathbf{C}}},{\kern 1pt} \,\theta ,{\kern 1pt} \,{\mathbf{g}}_{\kappa } } \right\} \) and that (21) imposes restriction on its form. After a lengthy calculation starting from (15), Liu [13] proved that the following entropy flux relation holds

$$ {\varvec{\Phi}}_{\kappa } = \varLambda^{\varepsilon } \left( {{\mathbf{C}},{\kern 1pt} {\dot{\mathbf{C}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right){\mathbf{q}}_{\kappa } $$
(22)

Further, based on (22) and (17), I-Shih Liu concluded that \( \varLambda^{\varepsilon } \) must be independent of \( {\mathbf{C}} \), \( {\dot{\mathbf{C}}} \) and \( {\mathbf{g}}_{\kappa } \). Thus,

$$ \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ). $$
(23)

Accordingly, (20) becomes

$$ {\text{d}}{\kern 1pt} \eta = \varLambda^{\varepsilon } {\kern 1pt} {\text{d}}\varepsilon + \frac{\partial }{{\partial {\mathbf{C}}}}\left( {\eta - \varLambda^{\varepsilon } \varepsilon } \right) \cdot {\text{d}}{\mathbf{C}} $$

or

$$ {\text{d}}\eta = \varLambda^{\varepsilon } {\kern 1pt} \left( {{\text{d}}\varepsilon - \frac{\partial \psi }{{\partial {\mathbf{C}}}} \cdot {\text{d}}{\mathbf{C}}} \right) $$

where

$$ \psi = \varepsilon - \eta /\varLambda^{\varepsilon } $$
(24)

By comparison with the classical Gibbs relation in thermostatics, the function \( \varLambda^{\varepsilon } \) can be identified as the reciprocal of the absolute temperature \( \theta \), i.e.,

$$ \varLambda^{\varepsilon } = \frac{1}{\theta } $$
(25)

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 \( \left( {{\mathbf{F}},{\kern 1pt} \,\theta ,{\kern 1pt} \,{\mathbf{g}}_{\kappa } } \right) \) the relation (25) holds.

3.1 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 \( \left( {{\mathbf{F}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right) \) is considered. In this case, the requirements (16) and (17) reduce to (26) and (27), respectively.

$$ \begin{aligned} & \frac{\partial \eta }{\partial \theta } - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{\partial \theta } = 0, \\ & \frac{\partial \eta }{{\partial {\mathbf{g}}_{\kappa } }} - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{{\partial {\mathbf{g}}_{\kappa } }} = 0, \\ \end{aligned} $$
(26)
$$ \begin{aligned} & \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }}} \right)_{\text{sym}} = 0, \\ & \frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\mathbf{C}}}} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\mathbf{C}}}} = 0. \\ \end{aligned} $$
(27)

In addition, the material symmetry condition for anisotropic elastic bodies has to be satisfied by \( \left\{ {\varepsilon ,{\kern 1pt} {\mathbf{T}}_{\kappa } ,{\kern 1pt} {\mathbf{q}}_{\kappa } ,{\kern 1pt} {\varvec{\Phi}}_{\kappa } ,{\kern 1pt} \eta } \right\} \).

The first paper considering entropy flux for transversely isotropic elastic bodies, was published by Liu [14], which relies on his paper [18]. In this, anisotropic materials properties in preferential directions were characterized by a number of unit vectors \( {\mathbf{m}}_{1} , \ldots ,{\mathbf{m}}_{a} \) and tensors \( {\mathbf{M}}_{1} , \ldots ,{\mathbf{M}}_{b} \). Ig \( g \) is a group of transformations which preserve these characteristics, i.e.,

$$ g = \left\{ {{\mathbf{Q}} \in G;\quad {\mathbf{Q}}{\mathfrak{m}} = {\mathfrak{m}},\quad {\mathbf{Q}}{\mathfrak{M}}{\mathbf{Q}}^{T} = {\mathfrak{M}}} \right\}, $$

where \( G \) is a subgroup of \( O \) the full orthogonal group, \( {\mathfrak{m}} = \left( {{\mathbf{m}}_{1} , \ldots ,{\mathbf{m}}_{a} } \right) \) and \( {\mathfrak{M}} = \left( {{\mathbf{M}}_{1} , \ldots ,{\mathbf{M}}_{b} } \right) \).

In other words, \( g \) is characterized by the set \( \left( {{\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}} \right) \) and the group \( G \in O \), i.e.,

$$ g = (G;{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}). $$
(28)

Theorem 1

A function \( f({\mathbf{v}},{\kern 1pt} {\mathbf{A}}) \) is invariant to \( g \) if and only if it can be represented by

$$ f({\mathbf{v}},{\kern 1pt} {\mathbf{A}}) = \hat{f}({\mathbf{v}},{\kern 1pt} {\mathbf{A}},{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}), $$
(29)

where \( \hat{f}({\mathbf{v}},{\kern 1pt} {\mathbf{A}},{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}) \) is invariant relative to \( G \).

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 \( \hat{f}({\mathbf{v}},{\kern 1pt} {\mathbf{A}},{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}) \) is an isotropic function then:

  • for a scalar-valued function

    $$ \hat{f}({\mathbf{Qv}},{\kern 1pt} {\mathbf{QAQ}}^{T} ,{\mathbf{Q}}{\mathfrak{m}},{\kern 1pt} {\mathbf{Q}}{\mathfrak{M}}{\mathbf{Q}}^{T} ) = \hat{f}({\mathbf{v}},{\kern 1pt} {\mathbf{A}},{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}), $$

  • for a vector-valued function

    $$ {\mathbf{Q}}\hat{f}({\mathbf{v}},{\kern 1pt} {\mathbf{A}},{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}) = \hat{f}({\mathbf{Qv}},{\kern 1pt} {\mathbf{QAQ}}^{T} ,{\kern 1pt} {\mathbf{Q}}{\mathfrak{m}},{\kern 1pt} {\mathbf{Q}}{\mathfrak{M}}{\mathbf{Q}}^{T} ), $$

  • and for a tensor-valued function

    $$ {\mathbf{Q}}\hat{f}({\mathbf{v}},{\kern 1pt} {\mathbf{A}},{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}){\mathbf{Q}}^{T} = \hat{f}({\mathbf{Qv}},{\kern 1pt} {\mathbf{QAQ}}^{T} ,{\kern 1pt} {\mathbf{Q}}{\mathfrak{m}},{\kern 1pt} {\mathbf{Q}}{\mathfrak{M}}{\mathbf{Q}}^{T} ). $$

In the same paper, I-Shih Liu gives a list of 14 such groups \( g \) for some crystal classes. He uses the following notation:

\( {\mathbf{n}}_{i} \), \( i = 1,2,3 \) where \( {\mathbf{n}}_{i} \) are orthonormal vectors, i.e.,

$$ {\mathbf{n}}_{i} \cdot {\mathbf{n}}_{j} = \delta_{ij} . $$
(30)

\( {\mathbf{N}}_{i} \) are skew-symmetric tensors defined by

$$ {\mathbf{N}}_{i} = e_{ijk} {\mathbf{n}}_{j} \otimes {\mathbf{n}}_{k} ,\quad i,j,k = 1,2,3. $$
(31)

In order to write the exact form of the constitutive functions used in our further investigation, we need several functional relations particularly among \( {\mathbf{n}}_{i} \) and \( {\mathbf{N}}_{i} \). These basic relations are given in “Appendix.”

Liu [14] considered only two different classes of transversally isotropic bodies

$$ \begin{aligned} g_{2} & = \left( {O;{\kern 1pt} \, {\mathbf{Qn}}_{1} = {\mathbf{n}}_{1} } \right), \\ g_{5} & = \left( {O;{\kern 1pt} \, {\mathbf{Qn}}_{1} \otimes {\mathbf{n}}_{1} {\mathbf{Q}}^{T} = {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} } \right), \\ \end{aligned} $$

where \( {\mathbf{n}}_{1} \) is the preferred direction of transverse isotropy.

In applying the isotropic representation of constitutive functions, instead of the Cauchy–Green strain tensor C, he used the Green–St. Venant strain tensor E, i.e.,

$$ {\mathbf{E}} = \frac{1}{2}({\mathbf{C}} - {\mathbf{I}}) $$

which vanishes when there is no deformation and considers constitutive functions \( {\mathbf{q}}_{\kappa } \) and \( {\varvec{\Phi}}_{\kappa } \) of \( \left( {{\mathbf{E}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right) \) up to bilinear terms in E and \( {\mathbf{g}}_{\kappa } \), i.e.,

$$ \begin{aligned} {\mathbf{q}}_{\kappa } & = \left( {a_{1} + a_{2} {\kern 1pt} {\text{tr}}{\mathbf{E}} + a_{3} {\mathbf{n}} \cdot {\mathbf{En}}} \right){\mathbf{g}}_{\kappa } + a_{4} {\mathbf{Eg}}_{\kappa } \\ & \quad + \left( {b_{1} + b_{2} {\kern 1pt} {\text{tr}}{\mathbf{E}} + b_{3} {\mathbf{n}} \cdot {\mathbf{En}}} \right)\left( {{\mathbf{n}} \otimes {\mathbf{n}}} \right){\mathbf{g}}_{\kappa } + b_{4} \left( {{\mathbf{n}} \otimes {\mathbf{En}}} \right){\mathbf{g}}_{\kappa } + b_{5} \left( {{\mathbf{En}} \otimes {\mathbf{n}}} \right){\mathbf{g}}_{\kappa } \\ & \quad + \left( {c_{1} + c_{2} {\kern 1pt} {\text{tr}}{\mathbf{E}} + c_{3} {\mathbf{n}} \cdot {\mathbf{En}}} \right){\mathbf{n}} + c_{4} {\mathbf{En}}, \\ {\varvec{\Phi}}_{\kappa } & = \left( {\alpha_{1} + \alpha_{2} {\kern 1pt} {\text{tr}}{\mathbf{E}} + \alpha_{3} {\mathbf{n}} \cdot {\mathbf{En}}} \right){\mathbf{g}}_{\kappa } + \alpha_{4} {\mathbf{Eg}}_{\kappa } \\ & \quad + \left( {\beta_{1} + \beta_{2} {\kern 1pt} {\text{tr}}{\mathbf{E}} + \beta_{3} {\mathbf{n}} \cdot {\mathbf{En}}} \right)\left( {{\mathbf{n}} \otimes {\mathbf{n}}} \right){\mathbf{g}}_{\kappa } + \beta_{4} \left( {{\mathbf{n}} \otimes {\mathbf{En}}} \right){\mathbf{g}}_{\kappa } + \beta_{5} \left( {{\mathbf{En}} \otimes {\mathbf{n}}} \right){\mathbf{g}}_{\kappa } \\ & \quad + \left( {\gamma_{1} + \gamma_{2} {\kern 1pt} {\text{tr}}{\mathbf{E}} + \gamma_{3} {\mathbf{n}} \cdot {\mathbf{En}}} \right){\mathbf{n}} + \gamma_{4} {\mathbf{En}}, \\ \end{aligned} $$

where all the material coefficients are functions of the temperature \( \theta \) only.

For the class of transversally isotropic bodies defined by

$$ g_{2} = \left( {O;{\kern 1pt} \, {\mathbf{Qn}}_{1} = {\mathbf{n}}_{1} } \right), $$

he was able to prove that \( \varLambda^{\varepsilon } \) is a function of the temperature only, i.e.,

$$ \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ) = \frac{1}{\theta }. $$

Further, he obtained the entropy flux and the heat flux relation as:

$$ {\varvec{\Phi}}_{\kappa } = \frac{1}{\theta }{\mathbf{q}}_{\kappa } + k(\theta ){\mathbf{n}}_{1} . $$
(32)

Therefore, for this class of transversally isotropic bodies, the classical result does not hold in general.

For the class of transversally isotropic bodies defined by

$$ g_{5} = \left( {O;{\kern 1pt} \, {\mathbf{Qn}}_{1} \otimes {\mathbf{n}}_{1} {\mathbf{Q}}^{T} = {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} } \right), $$

he obtained

$$ \varLambda^{\varepsilon } = \frac{1}{\theta },\quad {\varvec{\Phi}}_{\kappa } = \frac{1}{\theta }{\mathbf{q}}_{\kappa } , $$
(33)

which is identical to the classical result (2).

For these cases, a functional form of \( \varLambda^{\varepsilon } \) had to be found first to determine the relation between entropy flux and heat flux. It appears that \( \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ) \) holds in all these cases, i.e., \( \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ) \) is a necessary condition for the determination of a relation between the entropy flux and the heat flux.

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.

This assumption differs substantially from Green and Laws [19], as well as from Hutter [20] and Bargmann and Steinmann [21]. In Green and Laws [19], the entropy flux and heat flux relationships are defined as \( {\varvec{\Phi}}_{\kappa } = \frac{1}{\varphi }{\mathbf{q}}_{\kappa } \), where \( \varphi \) is a constitutive function which reduces to the absolute temperature \( \theta \) in equilibrium. Hutter [20] postulated the classical entropy flux heat flux relation. In their contribution, Bargmann and Steinmann [21] adopted the Green and Naghdi approach for non-classical theory of thermos-elasticity for isotropic materials and to obtain the entropy flux–heat flux relation (4).The consequence of the assumption that Lagrange multiplier conjugated to the energy balance equation is function of temperature only \( \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ) \).

Eringen stated [22] that “it is always possible to express the entropy change as a sum of entropy flux and entropy source as”

$$ {\varvec{\Phi}} = \frac{1}{\theta }{\kern 1pt} {\mathbf{q}} + {\varvec{\Phi}}_{1} $$
(34)

where \( {\varvec{\Phi}}_{1} \) 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 \( \varLambda^{\varepsilon } \) is a function of temperature \( \theta \) only, i.e., \( \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ) \). The implication of this assumption for isotropic viscoelastic bodies is considered first. In this, the starting point is the set of Eq. (17) which are restated below

$$ \begin{aligned} & \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }}} \right)_{\text{sym}} = 0 \\ & \frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\mathbf{C}}}} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\mathbf{C}}}} = 0 \\ & \frac{{\partial {\varvec{\Phi}}_{\kappa } }}{{\partial {\dot{\mathbf{C}}}}} - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\dot{\mathbf{C}}}}} = 0 \\ \end{aligned} $$
(35)

Let us introduce a new variable \( {\mathbf{k}} = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } . \) Recall that \( \begin{array}{*{20}l} {{\varvec{\Phi}}_{\kappa } = {\hat{\mathbf{\varPhi }}}_{\kappa } } \hfill \\ \end{array} \left( {{\mathbf{C}},{\dot{\mathbf{C}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right) \) and \( \begin{array}{*{20}l} {{\mathbf{q}}_{\kappa } = {\hat{\mathbf{q}}}_{\kappa } } \hfill \\ \end{array} \left( {{\mathbf{C}},{\dot{\mathbf{C}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right) \), consequently \( {\mathbf{k}} = {\hat{\mathbf{k}}}\left( {{\mathbf{C}},{\kern 1pt} {\dot{\mathbf{C}}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right) \). Using \( {\mathbf{k}} , \) Eq. (35) can be written as:

$$ \begin{aligned} & \left( {\frac{{\partial {\mathbf{k}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }}} \right)_{\text{sym}} = 0 \\ & \frac{{\partial {\mathbf{k}}_{\kappa } }}{{\partial {\mathbf{C}}}} = 0 \\ & \frac{{\partial {\mathbf{k}}_{\kappa } }}{{\partial {\dot{\mathbf{C}}}}} = 0 \\ \end{aligned} $$
(36)

Thus, \( {\mathbf{k}} = {\hat{\mathbf{k}}}(\theta ,\,{\kern 1pt} {\mathbf{g}}_{\kappa } ) \) and the set of Eq. (36) reduces to

$$ \left( {\frac{{\partial {\mathbf{k}}}}{{\partial {\mathbf{g}}_{\kappa } }}} \right)_{\text{sym}} = 0 $$
(37)

For clarity reasons, the following few steps are written in index notation or component notation. Equation (37) when written in component form is \( \frac{{\partial k_{i} }}{{\partial g_{j} }} + \frac{{\partial k_{j} }}{{\partial g_{i} }} = 0, \) where \( k_{i} \) and \( g_{i} \) are components of vectors \( {\mathbf{k}} \) and \( {\mathbf{g}}_{\kappa } \), respectively. Differentiation of (37) with respect to \( {\mathbf{g}}_{\kappa } \) yields

$$ \frac{{\partial^{2} k_{p} }}{{\partial g_{r} \partial g_{q} }} + \frac{{\partial^{2} k_{q} }}{{\partial g_{r} \partial g_{p} }} = 0 $$

The remaining two relations obtained by cyclic index permutation are

$$ \frac{{\partial^{2} k_{q} }}{{\partial g_{p} \partial g_{r} }} + \frac{{\partial^{2} k_{r} }}{{\partial g_{p} \partial g_{q} }} = 0 $$
$$ \frac{{\partial^{2} k_{r} }}{{\partial g_{q} \partial g_{p} }} + \frac{{\partial^{2} k_{p} }}{{\partial g_{q} \partial g_{r} }} = 0 $$

From them, we have

$$ \frac{{\partial^{2} k_{p} }}{{\partial g_{r} \partial g_{q} }} = 0 $$

The solution of this simple set of differential equations is

$$ k_{p} = A_{pq} (\theta )g_{q} + a_{p} (\theta ),\quad A_{(pq)} = 0 $$

This solution can be rewritten, using symbolic notation, as:

$$ {\mathbf{k}}(\theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } ) = {\mathbf{A}}\left( \theta \right){\mathbf{g}}_{\kappa } + {\mathbf{a}}(\theta ) $$
(38)

where \( {\mathbf{A}}(\theta ) \) is skew symmetric. Since we are dealing with isotropic viscoelastic bodies, \( {\mathbf{k}}(\theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } ) \) must be a vector-valued isotropic function. Thus

$$ {\mathbf{Qk}}(\theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } ) = {\mathbf{k}}(\theta ,{\mathbf{Q}}{\kern 1pt} {\mathbf{g}}_{\kappa } ) $$
(39)

must hold for all \( {\mathbf{Q}} \in O \) and an arbitrary \( {\mathbf{g}}_{\kappa } \). Equivalently, (39) can be restated as:

$$ {\mathbf{QA}}\left( \theta \right){\mathbf{g}}_{\kappa } + {\mathbf{Qa}}(\theta ) = {\mathbf{A}}\left( \theta \right){\mathbf{Qg}}_{\kappa } + {\mathbf{a}}(\theta ) $$
(40)

Particularly, for the case of \( {\mathbf{Q}} = - {\mathbf{I}} , \) it follows that \( {\mathbf{a}}(\theta ) = 0 \) and consequently (40) reduces to

$$ {\mathbf{QA}}\left( \theta \right){\mathbf{g}}_{\kappa } = {\mathbf{A}}\left( \theta \right){\mathbf{Qg}}_{\kappa } $$
(41)

which can be rewritten as:

$$ {\mathbf{QA}}(\theta ){\mathbf{Q}}^{T} {\mathbf{Qg}}_{\kappa } = {\mathbf{A}}(\theta ){\mathbf{Qg}}_{\kappa } $$
(42)

Since (42) must hold for all \( {\mathbf{Q}} \in O \) and an arbitrary \( {\mathbf{g}}_{\kappa } , \) we have that

$$ {\mathbf{QA}}(\theta ){\mathbf{Q}}^{T} = {\mathbf{A}}(\theta ) $$
(43)

since \( {\mathbf{A}}(\theta ) \) is skew symmetric, only \( {\mathbf{A}}(\theta ) = 0 \) satisfies (43), and

$$ {\mathbf{k}} = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = {\mathbf{0}} $$
(44)

This result is identical to Liu’s [13] 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 \( \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ) \) significantly simplifies and shortens the procedure.

3.2 Entropy Flux Relation for Anisotropic Elastic Materials in General

This section considers anisotropic bodies characterized with

$$ g = \left\{ {{\mathbf{Q}} \in O;\quad {\mathbf{Q}}{\mathfrak{m}} = {\mathfrak{m}},\quad {\mathbf{Q}}{\mathfrak{M}}{\mathbf{Q}}^{T} = {\mathfrak{M}}} \right\}. $$
(45)

In other words, \( g \) comprises the set \( \left( {{\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}} \right) \) and the group \( O \), i.e., \( g = (O;{\kern 1pt} \, {\mathfrak{m}},{\kern 1pt} \, {\mathfrak{M}}). \) Consequently,

$$ {\mathbf{k}} = {\hat{\mathbf{k}}}(\theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } ,{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}) = {\mathbf{A}}(\theta ,{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}){\mathbf{g}}_{\kappa } + {\mathbf{a}}(\theta ,{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}) $$
(46)

is isotropic vector-valued function, i.e.,

$$ {\mathbf{Qk}}(\theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } ,{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}) = {\mathbf{k}}(\theta ,{\kern 1pt} {\mathbf{Qg}}_{\kappa } ,{\kern 1pt} {\mathbf{Q}}{\mathfrak{m}},{\kern 1pt} {\mathbf{Q}}{\mathfrak{M}}{\mathbf{Q}}^{T} ), $$
(47)

or

$$ {\mathbf{QA}}(\theta ,{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}){\mathbf{g}}_{\kappa } + {\mathbf{Qa}}(\theta ,{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}) = {\mathbf{A}}(\theta ,{\kern 1pt} {\mathbf{Q}}{\mathfrak{m}},{\kern 1pt} {\mathbf{Q}}{\mathfrak{M}}{\mathbf{Q}}^{T} ){\mathbf{Qg}}_{\kappa } + {\mathbf{a}}(\theta ,{\kern 1pt} {\mathbf{Q}}{\mathfrak{m}},{\kern 1pt} {\mathbf{Q}}{\mathfrak{M}}{\mathbf{Q}}^{T} ) $$
(48)

which must hold for all \( {\mathbf{Q}} \in O \) and arbitrary \( {\mathbf{g}}_{\kappa } \). Particularly, for \( {\mathbf{g}}_{\kappa } = 0 \), we have

$$ {\mathbf{Qa}}(\theta ,{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}) = {\mathbf{a}}(\theta ,{\kern 1pt} {\mathbf{Q}}{\mathfrak{m}},{\kern 1pt} {\mathbf{Q}}{\mathfrak{M}}{\mathbf{Q}}^{T} ), $$
(49)

i.e., \( {\mathbf{a}}(\theta ,{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}) \) is vector-valued isotropic function of its arguments. Moreover,

$$ {\mathbf{QA}}(\theta ,{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}){\mathbf{g}}_{\kappa } = {\mathbf{A}}(\theta ,{\kern 1pt} {\mathbf{Q}}{\mathfrak{m}},{\kern 1pt} {\mathbf{Q}}{\mathfrak{M}}{\mathbf{Q}}^{T} ){\mathbf{Qg}}_{\kappa } $$
(50)

or

$$ {\mathbf{QA}}(\theta ,{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}){\mathbf{Q}}^{T} {\mathbf{Qg}}_{\kappa } = {\mathbf{A}}(\theta ,{\kern 1pt} {\mathbf{Q}}{\mathfrak{m}},{\kern 1pt} {\mathbf{Q}}{\mathfrak{M}}{\mathbf{Q}}^{T} ){\mathbf{Qg}}_{\kappa } . $$

This must hold for all \( {\mathbf{Q}} \in O \), and hence

$$ {\mathbf{QA}}\left( {\theta ,{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}} \right){\mathbf{Q}}^{T} = {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{Q}}{\mathfrak{m}},{\kern 1pt} {\mathbf{Q}}{\mathfrak{M}}{\mathbf{Q}}^{T} } \right) , $$
(51)

i.e., \( {\mathbf{A}}(\theta ,{\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}) \) is skew-symmetric tensor-valued isotropic function of its arguments.

For vector-valued and skew-symmetric tensor-valued isotropic functions in R3 see Smith [23] and Spencer [24]. All groups of crystal classes given by Liu [18] are considered below.

3.3 Transversally Isotropic Material Bodies

In this section, we consider transversely isotropic material bodies divided into four cases. The cases (a) characterized by \( g_{2} = (O;{\kern 1pt} \,{\mathbf{n}}_{1} ) \) and the case (b) characterized by \( g_{5} = (O;{\kern 1pt} \,{\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} ) \) were selected in order to validate the proposed procedure against the results obtained by Liu [14] for the same materials. The results for the cases (c) characterized by \( g_{1} = (O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} ) \) and d) characterized \( g_{1} = (O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} ) \) are new and, to the best of our knowledge, not available in the published literature.

  1. (a)

    transversally isotropic bodies with group symmetry \( g_{2} = (O;{\kern 1pt} \,{\mathbf{n}}_{1} ) \).

    In this case,

    $$ {\mathbf{k}} = {\hat{\mathbf{k}}}\left( {\theta ,{\mathbf{g}}_{\kappa } ,{\kern 1pt} {\mathbf{n}}_{1} } \right) = {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} } \right){\kern 1pt} {\mathbf{g}}_{\kappa } + {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} } \right) $$
    (52)

    where the following must hold

    $$ {\mathbf{Qa}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} } \right) = {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{Qn}}_{1} } \right) , $$
    (53)

    i.e., \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} } \right) \) is vector-valued function which can be expressed as

    $$ {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} } \right) = \lambda \left( \theta \right){\mathbf{n}}_{1} $$
    (54)

    where \( \lambda (\theta ) \) is an arbitrary scalar function. Further,

    $$ {\mathbf{QA}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} } \right){\mathbf{Q}}^{T} = {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{Qn}}_{1} } \right) $$
    (55)

    Thus, \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} } \right) \) is skew-symmetric tensor-valued isotropic function. Consequently, \( {\mathbf{A = 0}} \) (see Smith [23]). In this case,

    $$ {\mathbf{k}} = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = \lambda (\theta ){\mathbf{n}}_{1} . $$
    (56)

    Moreover,

    $$ \lambda (\theta ) = \left. {\left( {{\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } } \right)} \right|_{{{\mathbf{C}} = {\mathbf{I}},{\mathbf{g}}_{\kappa } = {\mathbf{0}}}} \cdot {\mathbf{n}}_{{\mathbf{1}}} $$
    (57)

    where C = I implies that there is no deformation and \( {\mathbf{g}}_{\kappa } = {\mathbf{0}} \) implies that there is no temperature gradient.

    The result (57) agrees with I-Shih Liu’s result for transversally isotropic bodies with group symmetry group g2 [14].

  2. (b)

    transversally isotropic bodies with group symmetry \( g_{5} = (O;{\kern 1pt} {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} ). \)

    In this case,

    $$ {\mathbf{k}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{g}}_{\kappa } } \right) = {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} } \right){\kern 1pt} {\mathbf{g}}_{\kappa } + {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} } \right) $$
    (58)

    where \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} } \right) \) and \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} } \right) \), as isotropic functions, must vanish, resulting in

    $$ {\mathbf{k}} = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = {\mathbf{0}}, $$
    (59)

    The result (59) agrees with I-Shih Liu’s result for transversally isotropic bodies with group symmetry group g5 [14].

    Therefore, for the above cases, we demonstrated that \( \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ) \) 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 Liu [18]. Note, in all these cases, representations of anisotropic invariant function are obtained using the tables for isotropic functions. This more general entropy principle delivers relations which are the same as those obtained by the Clausius–Duhem inequality and hold for all materials in classical thermodynamics. Heuristically, we do not see any physically based objection which would contradict the possibility that \( \varLambda^{\varepsilon } \) could be different to \( \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ) \) for crystal classes from G3 to G14. It is also in accordance with assumption given by Hutter [25], p. 211, who demonstrated that \( \varLambda^{\varepsilon } (\theta ) \) is independent of the material properties for the heat conducting compressible fluids.

    Further, this could be related to the statement by Ingo Muller and Tommaso Ruggeri in relation to extended thermodynamics: “Physicists firmly believe that the differential equations of nature should be hyperbolic so as to exclude action at a distance. This incompatibility between the expectation of the physicist and the classical laws of thermodynamics has prompted the formulation of extended thermodynamics”.

    In addition, we do not see any physical reason that \( \varLambda^{\varepsilon } \) would have a different form to \( \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ) \) for these crystal classes.

    The result for the following cases is new (to the best of our knowledge not available in the published literature).

  3. (c)

    transversally isotropic bodies with group symmetry \( g_{1} = (O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} ) \).

    In this case,

    $$ {\mathbf{k}} = {\hat{\mathbf{k}}}\left( {\theta ,{\kern 1pt} {\mathbf{g}}_{\kappa } ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) = {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} } \right){\kern 1pt} {\mathbf{g}}_{\kappa } + {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) $$
    (60)

    Here and in what follows, we use particularly the representation formulae [(2.41), (4.2), (4.3), (4.6), (4.7)] given by Smith [23] (see also Liu [18], Wilmanski [26]) for vector-valued isotropic function and skew-symmetric tensor-valued function of their arguments. We strictly apply these formulae for the vector function \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \) and a skew-symmetric tensor-valued function \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \). Their scalar invariant functions are always functions of \( \theta \). Therefore, we must find the basis invariants of the set \( \left( {{\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \). They are

    $$ \begin{aligned} & {\mathbf{n}}_{1} \cdot {\mathbf{n}}_{1} \\ & {\text{tr}}{\mathbf{N}}_{1}^{2} \\ & {\mathbf{n}}_{1} \cdot {\mathbf{N}}_{1}^{2} {\mathbf{n}}_{1} \\ \end{aligned} $$
    (61)

    The generator of the set \( \left( {{\mathbf{n}}_{1} ,{\kern 1pt} \,{\mathbf{N}}_{1} } \right) \) for \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} \,{\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} } \right) \) is \( {\mathbf{n}}_{1} , \) and consequently, \( {\mathbf{a}} = \lambda (\theta ){\mathbf{n}}_{1} \). The generator of the set \( \left( {{\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \) for \( A\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \) is \( {\mathbf{N}}_{1} \) and, accordingly, \( {\mathbf{A}}(\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} ) = \mu (\theta ){\mathbf{N}}_{1} \).

    Thus,

    $$ \begin{aligned} {\mathbf{k}} & = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = - \mu (\theta ){\mathbf{N}}_{1} {\kern 1pt} {\mathbf{g}}_{\kappa } + \lambda (\theta ){\mathbf{n}}_{1} \\ & = \mu (\theta ){\mathbf{n}}_{1} \times {\mathbf{g}}_{\kappa } + \lambda (\theta ){\mathbf{n}}_{1} . \\ \end{aligned} $$
    (62)
  4. (d)

    transversally isotropic bodies with group symmetry \( g_{3} = \left( {O;{\kern 1pt} {\mathbf{N}}_{1} } \right) \).

    In this case,

    $$ {\mathbf{k}} = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = \mu (\theta ){\mathbf{n}}_{1} \times {\mathbf{g}}_{\kappa } $$
    (63)

3.3.1 Orthotropic Material Bodies

This section considers anisotropic bodies characterized with group symmetry \( g_{6} = \left( {O;{\kern 1pt} {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right) \). The basis of invariants for \( \left( {\theta ,{\kern 1pt} \,{\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} ,{\kern 1pt} \,{\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right) \) are functions only of \( \theta \) (see Smith [23] and the “Appendix”). There are no generators for \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right) \), i.e., \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right) = {\mathbf{0}} \) and \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right) = {\mathbf{0}} \).

Thus

$$ {\mathbf{k}} = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = {\mathbf{0}} $$
(64)

3.3.2 Triclinic System

  1. (a)

    Predial class characterized with group symmetry \( g_{7} = \left( {O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} } \right) \). For this class, there are no invariants of \( \left( {O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} } \right) \). Generators for \( {\mathbf{a}}\left( {\theta ,{\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} } \right) \) are \( {\mathbf{n}}_{1} ,{\kern 1pt} \,{\mathbf{n}}_{2} \) and \( {\mathbf{n}}_{3} \). Thus, \( a\left( {{\kern 1pt} \theta ,{\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} } \right) = \lambda (\theta ){\mathbf{n}}_{1} + \mu (\theta ){\mathbf{n}}_{2} + \nu (\theta ){\mathbf{n}}_{3} \). Generators for \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} } \right) \) are \( {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{2} - {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{1} \), \( {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{3} - {\mathbf{n}}_{3} \otimes {\mathbf{n}}_{2} \) and \( {\mathbf{n}}_{3} \otimes {\mathbf{n}}_{1} - {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{3} \). Thus,

    $$ \begin{aligned} {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} } \right) & = - p(\theta )\left( {{\mathbf{n}}_{1} \otimes {\mathbf{n}}_{2} - {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{1} } \right) \\ & \quad - q(\theta )\left( {{\mathbf{n}}_{2} \otimes {\mathbf{n}}_{3} - {\mathbf{n}}_{3} \otimes {\mathbf{n}}_{2} } \right) - r(\theta )\left( {{\mathbf{n}}_{3} \otimes {\mathbf{n}}_{1} - {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{3} } \right) \\ & = - p(\theta ){\mathbf{N}}_{3} - q(\theta ){\mathbf{N}}_{1} - r(\theta ){\mathbf{N}}_{2} \\ \end{aligned} $$

    and consequently

    $$ \begin{aligned} {\mathbf{k}} & = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = \left[ {p\left( \theta \right){\mathbf{n}}_{3} + q\left( \theta \right){\mathbf{n}}_{1} + r\left( \theta \right){\mathbf{n}}_{2} } \right] \times {\mathbf{g}}_{\kappa } \\ & \quad + \lambda \left( \theta \right){\mathbf{n}}_{3} + \mu \left( \theta \right){\mathbf{n}}_{1} + \nu \left( \theta \right){\mathbf{n}}_{2} \\ \end{aligned} $$
    (65)
  2. (b)

    Pinacoidal class characterized with group symmetry \( g_{8} = \left( {O;{\kern 1pt} {\mathbf{N}}_{1} ,{\kern 1pt} {\mathbf{N}}_{2} } \right) \). For this class, there are no invariants and no generators for \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{N}}_{1} ,{\kern 1pt} {\mathbf{N}}_{2} } \right) \), i.e., \( {\mathbf{a}} = {\mathbf{0}} \). Generators for \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{N}}_{1} ,{\kern 1pt} {\mathbf{N}}_{2} } \right) \) are \( {\mathbf{N}}_{1} ,{\mathbf{N}}_{2} \) and \( {\kern 1pt} {\mathbf{N}}_{1} {\mathbf{N}}_{2} - {\mathbf{N}}_{2} {\mathbf{N}}_{1} = - {\mathbf{N}}_{3} \). Thus

    $$ \begin{aligned} {\mathbf{k}} & = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = \left[ { - p\left( \theta \right){\mathbf{N}}_{3} - q\left( \theta \right){\mathbf{N}}_{1} - r\left( \theta \right){\mathbf{N}}_{2} } \right]{\mathbf{g}}_{\kappa } \\ & = \left[ {p\left( \theta \right){\mathbf{n}}_{3} + q\left( \theta \right){\mathbf{n}}_{1} + r\left( \theta \right){\mathbf{n}}_{2} } \right] \times {\mathbf{g}}_{\kappa } \\ \end{aligned} $$
    (66)

3.3.3 Monoclinic System

  1. (a)

    Domatic class characterized with group symmetry \( g_{9} = (O;{\kern 1pt} {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} ) \). In the same way as in the case of the predial class, we obtain

    $$ {\mathbf{k}} = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = \left[ {q(\theta ){\mathbf{n}}_{2} + r(\theta ){\mathbf{n}}_{3} } \right] \times {\mathbf{g}}_{\kappa } + \lambda (\theta ){\mathbf{n}}_{2} + \mu (\theta ){\mathbf{n}}_{3} $$
    (67)
  2. (b)

    Sphenoidal class characterized with group symmetry \( g_{10} = (O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} ) \). For this class, there are no invariants for \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \) and \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \). Generator of \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \) is \( {\mathbf{n}}_{1} \) and \( {\mathbf{a}} = \lambda (\theta ){\mathbf{n}}_{1} . \) Generator of \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \) is \( {\mathbf{N}}_{1} \) and \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) = - \mu (\theta ){\mathbf{N}}_{1} \). Therefore,

    $$ {\mathbf{k}} = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = \mu (\theta ){\mathbf{n}}_{1} \times {\mathbf{g}}_{\kappa } + \lambda (\theta ){\mathbf{n}}_{1} . $$
    (68)
  3. (c)

    Prismatic class characterized with group symmetry \( g_{11} = \left( {O;{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \). For this class, there are no invariants for \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \) and \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \). There are no generators of \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \), thus, \( {\mathbf{a}} = {\mathbf{0}} \). Generator of \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) \) is \( {\mathbf{N}}_{1} \) and \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} } \right) = - \mu (\theta ){\mathbf{N}}_{1} \). Therefore,

    $$ {\mathbf{k}} = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = \mu (\theta ){\mathbf{n}}_{1} \times {\mathbf{g}}_{\kappa } . $$
    (69)

3.3.4 Rhombic Systems

  1. (a)

    Pyramidal class characterized with group symmetry \( g_{12} = \left( {O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right) \). For this class, there are no invariants of \( {\mathbf{a}}\left( {O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right) \) and \( {\mathbf{A}}\left( {O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right) \). Generator of \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right) \) is \( {\mathbf{n}}_{1} \) and \( {\mathbf{a}} = \lambda (\theta ){\mathbf{n}}_{1} \). There are no generators of \( {\mathbf{A}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right) \), i.e., \( {\mathbf{A}}\left( {O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right) = {\mathbf{0}} \).

    Hence,

    $$ {\mathbf{k}} = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = \lambda (\theta ){\mathbf{n}}_{1} . $$
    (70)
  2. (b)

    Dipyramidal class characterized with group symmetry \( g_{14} = g_{6} = \left( {O;{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} \otimes {\mathbf{n}}_{3} } \right) \) and consequently

    $$ {\mathbf{k}} = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } = {\mathbf{0}}. $$
    (71)

4 The Entropy Inequality for Anisotropic Bodies

So far, we did not investigate the entropy inequality

$$ \sigma = \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{\partial \theta } - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{\partial \theta }} \right) \cdot {\mathbf{g}}_{\kappa } + \rho_{\kappa } \left( {\frac{\partial \eta }{{\partial {\mathbf{C}}}} - \varLambda^{\varepsilon } \frac{\partial \varepsilon }{{\partial {\mathbf{C}}}} + \frac{1}{{2\rho_{\kappa } }}\varLambda^{\varepsilon } {\mathbf{S}}_{\kappa } } \right){\dot{\mathbf{C}}} \ge 0 $$
(72)

for general anisotropic materials under the assumption that \( \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ) \). Making use of the free energy function \( \psi = \varepsilon - \theta \eta , \) the expression for σ reduces to

$$ \sigma = \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{\partial \theta } - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{\partial \theta }} \right) \cdot {\mathbf{g}}_{\kappa } + \frac{{\varLambda^{\varepsilon } }}{2}\left( {{\mathbf{S}}_{\kappa } - 2\rho_{\kappa } \frac{\partial \psi }{{\partial {\mathbf{C}}}}} \right){\dot{\mathbf{C}}} \ge 0 $$
(73)

which is linear in \( {\dot{\mathbf{C}}} \). Thus,

$$ {\mathbf{S}}_{\kappa } = 2\rho_{\kappa } \frac{\partial \psi }{{\partial {\mathbf{C}}}} $$
(74)

Equation (74) leads to the conclusion that all anisotropic elastic materials are hyperelastic as a consequence of \( \varLambda^{\varepsilon } (\theta ) \), irrespective of whether the classical entropy flux relation is valid. The remaining inequality now reads

$$ \sigma = \left( {\frac{{\partial {\varvec{\Phi}}_{\kappa } }}{\partial \theta } - \varLambda^{\varepsilon } \frac{{\partial {\mathbf{q}}_{\kappa } }}{\partial \theta }} \right) \cdot {\mathbf{g}}_{\kappa } \ge 0 $$
(75)

or

$$ \sigma = \left( {\frac{{\partial {\mathbf{k}}}}{\partial \theta } + \frac{{\partial \varLambda^{\varepsilon } }}{\partial \theta }{\mathbf{q}}_{\kappa } } \right) \cdot {\mathbf{g}}_{\kappa } \ge 0 $$
(76)

where \( {\mathbf{k}} = {\varvec{\Phi}}_{\kappa } - \varLambda^{\varepsilon } {\mathbf{q}}_{\kappa } \). Now, from (46), we have

$$ \frac{{\partial {\mathbf{k}}}}{\partial \theta } = \frac{{\partial {\mathbf{A}}\left( {\theta ,{\mathfrak{m}},{\mathfrak{M}}} \right)}}{\partial \theta }{\mathbf{g}}_{\kappa } + \frac{{\partial {\mathbf{a}}\left( {\theta ,{\mathfrak{m}},{\mathfrak{M}}} \right)}}{\partial \theta } $$
(77)

Since \( {\mathbf{A}}\left( {\theta ,{\mathfrak{m}},{\mathfrak{M}}} \right) \) is skew-symmetric \( {\mathbf{g}}_{\kappa } \cdot \frac{{\partial {\mathbf{A}}\left( {\theta ,{\mathfrak{m}},{\mathfrak{M}}} \right)}}{\partial \theta }{\mathbf{g}}_{\kappa } = 0 \), hence

$$ \frac{{\partial {\mathbf{k}}}}{\partial \theta } \cdot {\mathbf{g}}_{\kappa } = \frac{{\partial {\mathbf{a}}\left( {\theta ,{\mathfrak{m}},{\mathfrak{M}}} \right)}}{\partial \theta } \cdot {\mathbf{g}}_{\kappa } $$
(78)

Therefore (75) becomes

$$ \sigma = \left( {\frac{{\partial {\mathbf{a}}\left( {\theta ,{\mathfrak{m}},{\mathfrak{M}}} \right)}}{\partial \theta } - \frac{1}{{\theta^{2} }}{\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{g}}_{\kappa } ,{\mathfrak{m}},{\mathfrak{M}}} \right)} \right) \cdot {\mathbf{g}}_{\kappa } \ge 0 $$
(79)

The nonnegative entropy production density \( \sigma \) attains its minimum, which is in fact zero, when \( {\mathbf{g}}_{\kappa } = {\mathbf{0}} \). A necessary condition for an extremum at \( {\mathbf{g}}_{\kappa } = {\mathbf{0}} \) is

$$ \left. {\frac{\partial \sigma }{{\partial {\mathbf{g}}_{\kappa } }}} \right|_{{{\mathbf{g}}_{\kappa } {\mathbf{ = 0}}}} = {\mathbf{0}} $$
(80)

or

$$ {\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{g}}_{\kappa } = {\mathbf{0}},{\mathfrak{m}},{\mathfrak{M}}} \right) = \theta^{2} \frac{{\partial {\mathbf{a}}\left( {\theta ,{\mathfrak{m}},{\mathfrak{M}}} \right)}}{\partial \theta }, $$
(81)

at equilibrium.

Since Eq. (81) holds for all anisotropic elastic bodies, it can be used to define the heat flux vector at equilibrium for specific crystal classes. For instance, for transversally isotropic material bodies, considered by Liu [14], with the following group symmetries:

  1. (a)

    \( g_{2} = \left( {O;{\kern 1pt} \,{\mathbf{n}}_{1} } \right) \) and \( {\mathbf{a}}\left( {\theta ,{\kern 1pt} {\mathbf{n}}_{1} } \right) = \lambda \left( \theta \right){\mathbf{n}}_{1} \), where \( \lambda (\theta ) \) is an arbitrary scalar function, we have

    $$ {\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{0}},{\mathfrak{m}},{\mathfrak{M}}} \right) = \theta^{2} \frac{{{\text{d}}\lambda }}{{{\text{d}}\theta }}{\mathbf{n}}_{1} $$
    (82)
  2. (b)
    $$ g_{5} = \left( {O;{\kern 1pt} {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} } \right) $$
    $$ {\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{0}},{\mathfrak{m}},{\mathfrak{M}}} \right) = {\mathbf{0}} $$
    (83)

And similarly, for other group symmetries investigated here, the following results hold:

$$ \begin{array}{*{20}l} {g_{1} = (O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{N}}_{1} ):} \hfill & {{\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{0}},{\mathbf{n}}_{1} ,{\mathbf{N}}_{1} } \right)} \hfill & { = \theta^{2} \frac{{{\text{d}}\lambda }}{{{\text{d}}\theta }}{\kern 1pt} {\mathbf{n}}_{{\mathbf{1}}} } \hfill \\ {g_{2} = (O;{\kern 1pt} {\mathbf{n}}_{1} ):} \hfill & {{\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{0}},{\mathbf{n}}_{1} } \right)} \hfill & { = \theta^{2} \frac{{{\text{d}}\lambda }}{{{\text{d}}\theta }}{\kern 1pt} {\mathbf{n}}_{{\mathbf{1}}} } \hfill \\ {g_{10} = (O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} ):} \hfill & {{\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{0}},{\mathbf{n}}_{1} ,{\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\mathbf{N}}_{1} } \right)} \hfill & { = \theta^{2} \frac{{{\text{d}}\lambda }}{{{\text{d}}\theta }}{\kern 1pt} {\mathbf{n}}_{{\mathbf{1}}} } \hfill \\ {g_{12} = \left( {O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right):} \hfill & {{\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{0}},{\mathbf{n}}_{1} ,{\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right)} \hfill & { = \theta^{2} \frac{{{\text{d}}\lambda }}{{{\text{d}}\theta }}{\kern 1pt} {\mathbf{n}}_{{\mathbf{1}}} } \hfill \\ {g_{3} = \left( {O;{\kern 1pt} {\mathbf{N}}_{1} } \right):} \hfill & {{\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{0}},{\mathbf{N}}_{1} } \right)} \hfill & { = {\mathbf{0}}} \hfill \\ {g_{5} = (O;{\kern 1pt} n_{1} \otimes n_{1} ):} \hfill & {{\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{0}},{\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} } \right)} \hfill & { = {\mathbf{0}}} \hfill \\ {g_{6} = \left( {O;{\kern 1pt} {\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right):} \hfill & {{\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{0}},{\mathbf{n}}_{1} \otimes {\mathbf{n}}_{1} ,{\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} } \right)} \hfill & { = {\mathbf{0}}} \hfill \\ {g_{8} = \left( {O;{\kern 1pt} {\mathbf{N}}_{1} ,{\kern 1pt} {\mathbf{N}}_{2} } \right):} \hfill & {{\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{0}},{\mathbf{N}}_{1} ,{\mathbf{N}}_{2} } \right)} \hfill & { = {\mathbf{0}}} \hfill \\ {g_{11} = \left( {O;{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{N}}_{1} } \right):} \hfill & {{\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{0}},{\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\mathbf{N}}_{1} } \right)} \hfill & { = {\mathbf{0}}} \hfill \\ {g_{14} = \left( {O;{\kern 1pt} {\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} \otimes {\mathbf{n}}_{3} } \right):} \hfill & {{\mathbf{q}}_{\kappa } \left( {{\mathbf{C}},\theta ,{\mathbf{0}},{\mathbf{n}}_{2} \otimes {\mathbf{n}}_{2} ,{\mathbf{n}}_{3} \otimes {\mathbf{n}}_{3} } \right)} \hfill & { = {\mathbf{0}}} \hfill \\ \end{array} $$
(84)

Note, \( g_{14} = g_{6} \).

$$ \begin{array}{*{20}l} {g_{7} = \left( {O;{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} } \right):} \hfill & {{\mathbf{q}}_{\kappa } ({\mathbf{C}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{0}},{\kern 1pt} {\mathbf{n}}_{1} ,{\kern 1pt} {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} )} \hfill & { = \frac{{{\text{d}}\lambda }}{{{\text{d}}\theta }}{\kern 1pt} {\mathbf{n}}_{1} + \frac{{{\text{d}}\mu }}{{{\text{d}}\theta }}{\kern 1pt} {\mathbf{n}}_{2} + \frac{{{\text{d}}\nu }}{{{\text{d}}\theta }}{\kern 1pt} {\mathbf{n}}_{3} } \hfill \\ {g_{9} = (O;{\kern 1pt} {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} ):} \hfill & {{\mathbf{q}}_{\kappa } ({\mathbf{C}},{\kern 1pt} \theta ,{\kern 1pt} {\mathbf{0}},{\kern 1pt} {\mathbf{n}}_{2} ,{\kern 1pt} {\mathbf{n}}_{3} )} \hfill & { = \frac{{{\text{d}}\mu }}{{{\text{d}}\theta }}{\kern 1pt} {\mathbf{n}}_{2} + \frac{{{\text{d}}\nu }}{{{\text{d}}\theta }}{\kern 1pt} {\mathbf{n}}_{3} } \hfill \\ \end{array} $$
(85)

A necessary condition that entropy production \( \sigma \) has minimum at \( {\mathbf{g}}_{\kappa } = 0 \) is that the second derivative of \( \sigma \) with respect to \( {\mathbf{g}}_{\kappa } \) be semi-positive, i.e.,

$$ \left. {\frac{{\partial^{2} \sigma }}{{\partial g_{i} \partial g_{j} }}} \right|_{{{\mathbf{g}}_{\kappa } = {\mathbf{0}}}} \ge 0 $$
(86)

Note that \( \frac{{\partial^{2} \sigma }}{{\partial g_{i} \partial g_{j} }} \) is a symmetric tensor. Using (76) and (79), it is easy to show that

$$ \left. {\frac{{\partial^{2} \sigma }}{{\partial g_{i} \partial g_{j} }}} \right|_{{{\mathbf{g}}_{\kappa } = {\mathbf{0}}}} = 2\left. {\frac{{\partial \varLambda^{\varepsilon } }}{\partial \theta }\frac{{\partial q_{i} }}{{\partial g_{j} }}} \right|_{{(i,j)|{\mathbf{g}}_{\kappa } = {\mathbf{0}}}} = - 2\frac{1}{{\theta^{2} }}\left. {\left( {\frac{{\partial q_{i} }}{{\partial g_{j} }}} \right)} \right|_{{(i,j)|{\mathbf{g}}_{\kappa } = {\mathbf{0}}}} \ge 0 $$

having in mind that \( \varLambda^{\varepsilon } = 1/\theta \).

Finally, (86) can be written as:

$$ \left. {\left( {\frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }}} \right)} \right|_{{\left. {\text{sym}} \right|{\mathbf{g}}_{\kappa } = {\mathbf{0}}}} \le 0 $$
(87)

which holds for all anisotropic elastic materials. The constraints that (87) imposes must be investigated for particular anisotropic materials. For example, for the transversely isotropic elastic materials where

$$ {\mathbf{q}}_{\kappa } = a_{0} {\kern 1pt} {\mathbf{g}}_{\kappa } + \left( {b_{0} + b_{1} \left( {{\mathbf{n}} \cdot {\mathbf{g}}_{\kappa } } \right)} \right){\mathbf{n}} + c_{0} {\mathbf{n}} \times {\mathbf{g}}_{\kappa } $$

we have

$$ \left. {\left( {\frac{{\partial {\mathbf{q}}_{\kappa } }}{{\partial {\mathbf{g}}_{\kappa } }}} \right)} \right|_{{\left. {\text{sym}} \right|{\mathbf{g}}_{\kappa } = {\mathbf{0}}}} = a_{0} {\mathbf{I}} + b_{1} {\mathbf{n}} \otimes {\mathbf{n}} \le 0 $$

with the constraints

$$ a_{0} \le 0,\quad a_{0} + b_{1} \le 0 $$

5 Conclusion

This paper revisits entropy flux and heat flux relations for isotropic and several anisotropic elastic materials. More specifically, we investigated the consequence of the assumption that Lagrange multiplier conjugated to the energy balance equation \( \varLambda^{\varepsilon } = \varLambda (\theta ) \) 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, we re-examined the entropy flux–heat flux relation for viscoelastic materials, isotropic elastic materials and transversely isotropic elastic bodies and demonstrated that our results agree with the results obtained by Liu [12,13,14,15, 27]. All these cases confirm that \( \varLambda^{\varepsilon } = \varLambda (\theta ) \) 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, based on the assumption that \( \varLambda^{\varepsilon } = \varLambda (\theta ) \), 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 nonlinear. 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 analyzed.

We would like to draw attention to the following three points:

  1. (i)

    The vector function \( {\mathbf{a}} \) and skew-symmetric function \( {\mathbf{A}} \) are isotropic functions depending only on the set \( \left( {\theta ,{\kern 1pt} {\kern 1pt} {\mathfrak{m}},{\kern 1pt} {\mathfrak{M}}} \right) \) which simplifies the procedure.

  2. (ii)

    Generally, the classical entropy flux–heat flux relation does not hold; it is true for all crystal classes investigated here except for \( g_{2} ,g_{6} = g_{14} \).

  3. (iii)

    The heat flux in the absence of a temperature gradient is not zero for all crystal classes. This confirms Eringen’s statement that \( {\varvec{\Phi}} = \frac{1}{\theta }{\kern 1pt} {\mathbf{q}} + {\varvec{\Phi}}_{1} \) where \( {\varvec{\Phi}}_{1} \) is the entropy change due to all other effects except heat input.

We repeat that heuristically, we do not see any physically based objection which would contradict the possibility that \( \varLambda^{\varepsilon } \) could be different to \( \varLambda^{\varepsilon } = \varLambda^{\varepsilon } (\theta ) \) for crystal classes we considered. Of course, the assumption \( \varLambda^{\varepsilon } = \varLambda (\theta ) \) and all our predictions have to be verified experimentally. This is a task for future research in collaboration with some experimentalists.