Thermodynamics of Elastic Material with Relaxing Heat Flux

Abstract—

To obtain the heat equation, the first and second laws of thermodynamics and the consequences from them, which are obtained from the requirement that the laws are independent in the sense of choosing the inertial reference frame, are used. To write the second law of thermodynamics, the Clausius-Duhem inequality has been used. In this article, the derivation of the heat equation from the laws of thermodynamics for elastic materials with a relaxing heat flux is carried out. The constituting equations are formulated for a medium operating under conditions of finite deformations. It is shown that the Cauchy stress tensor in an elastic material depends on the heat flux gradient. An approximate version of using the nonlinear heat equation in materials with a very short relaxation time of the heat flux is considered. These are processes in which the relaxing heat flux quickly becomes close to the flux determined by Fourier’s law. We consider a state, in which the temperature gradient is of great importance and the square of the modulus of the heat flux vector cannot be neglected in the heat conduction equation. In this case, it is advisable to talk about using the concept of “nonequilibrium heat capacity” of the material, which depends on the temperature gradient.