Solvability for a class of generalized standard materials with thermomechanical coupling

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 dependence, coupled to the heat-transfer equation. More precisely, the coupling terms are linear with respect to the temperature and the displacement and nonlinear 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 L1L1 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 the global existence is discussed under some additional assumptions on the data.