Existence result for a class of generalized standard materials with thermomechanical coupling

TL;DR

This paper establishes the existence of solutions for a complex three-dimensional thermomechanical model of viscous solids with hysteresis, coupling heat transfer and stress-strain behavior within a generalized standard materials framework.

Contribution

It provides the first local existence proof for a thermodynamically consistent, coupled thermomechanical model with nonlinear coupling terms and L1 data in the heat equation.

Findings

Local existence of solutions using fixed-point methods

Solutions are physically admissible

Global existence under additional data assumptions

Abstract

This paper deals with the study of a three-dimensional model of thermomechanical coupling for viscous solids exhibiting hysteresis effects. This model is written in accordance with the formalism of generalized standard materials and it is composed of the momentum equilibrium equation combined with the flow rule, which describes some stress-strain dependance, coupled to the heat-transfer equation. More precisely, the coupling terms are linear with respect to the temperature and the displacement and non linear with respect to the internal variable. The main mathematical difficulty lies in the fact that the natural framework for the right-hand side of the heat equation is the space of L1 functions. A local existence result for this thermodynamically consistent problem is obtained by using a fixed-point argument. Then the solutions are proved to be physically admissible and global existence is discussed under some additional assumptions on the data.