An incremental variational formulation of dissipative and non-dissipative coupled thermoelasticity for solids

Abstract

In the early 1990s, Green and Naghdi introduced a theory attracting interest as heat propagates as thermal waves at finite speed and does not necessarily involve energy dissipation. Another outstanding property of the so-called theory of type II is the fact that the entropy flux vector is determined by means of the same potential as the mechanical stress tensor. Motivated by the procedure of [9, 15], we formulate a variational formulation within the incremental framework for coupled thermoelasticity for type I mimicing type II. The entropy flux of type II is determined via the free energy acting as a potential. This is not possible for the classical Fourier case in the continuous setting. Therefore, we target a derivation of an incremental entropy flux following Fourier’s law by means of incremental potentials similar to the spirit of the theory of the non-dissipative Green–Naghdi-type II. Subsequently, we show that the resulting update algorithm is a convenient fully coupled finite element formulation of the proposed thermoelastic problem.