A thermodynamic framework for non-isothermal phenomenological models of isotropic Mullins effect

The Mullins effect is a common name for a family of intriguing inelastic responses of various solid materials, in particular filled rubbers. Given the importance of the Mullins effect, there have been many attempts to develop mathematical models describing the effect. However, most of available models focus exclusively on the mechanical response, and are restricted to the idealised isothermal setting. We lift the restriction to isothermal processes, and we propose a full thermodynamic framework for a class of phenomenological models of the Mullins effect. In particular, we identify energy storage mechanisms (Helmholtz free energy) and entropy production mechanisms that on the level of stress--strain relation lead to the idealised Mullins effect or to the Mullins effect with permanent strain. The models constructed within the proposed framework can be used in the modelling of fully coupled thermo-mechanical processes, and the models are guaranteed to be consistent with the laws of thermodynamics.