A hysteresis approach to phase-field models

Phase-field systems as mathematical models to forecast the evolution of processes involving phase transitions have drawn a considerable interest in recent years. However, while they are capable of capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occuring during phase transition processes. To overcome this shortcoming, a new approach to phase-field models is proposed in this paper which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. The approach taken here leads to highly nonlinearly coupled systems of differential equations containing hysteretic nonlinearities at different places. For such a system, well-posedness and thermodynamic consistency are proved. Due to the lack of smoothness (hysteresis operators are, as a rule, non-differentiable) in the system, the method of proof has to be different from those usually employed for classical phase-field systems