Thermoelastic materials with heat flux evolution equation

Abstract

The results obtained in this paper refer to the class of materials for which specific free enetgy ψ, the specific entropy η, and the first Piola-Kirchhoff stress tensor S are, respectively, determined through the constitutive functionals ψ, η and S which are defined in their common domain consisting of quadruples (F, θ, G, Q), called states of the material, and where F is the deformation gradient, θ is the absolute temperature, G is the material gradient of the temperature, and Q is the referential heat flux. The heat flux Q behaves as a "hidden variable" or an "intemal variable" [1] and its evolution in time is described by a differential equation Q = H(F, θ, G; Q), where H is a constitutive functional of the material. Such materials will be called themoelastic material with heat flux evolution equation. To a certain extent, this class of materials may be considered as a limit case of thermomechanical materials with internal state variables examined by COLEMAN and GURTIN [1]. It is for this reason that this fundamental work of modern continuum thermodynamics inspired much of the results in this paper. On the other hand, the above heat flux evolution equation is generalizing Cattaneo's heat conduction equation [2] for isotropic materials. So this theory is convenient for predicting thermal waves propagating at finite speed.