Dynamics of Phase Transitions and Hysteresis in a Viscoelastic Ericksen’s Bar on an Elastic Foundation

This work is a follow-up on the study [32] of interface dynamics and hysteresis in materials undergoing solid-solid phase transitions. We consider the dynamics of a viscoelastic bar with a nonmonotone stress-strain relation and viscous stress linearly proportional to the strain rate. The bar is placed on an elastic foundation with stiffness β mimicking the interaction of phases in higher dimensions. Time-dependent displacement-controlled loading of the bar results in a tilted and serrated hysteresis loop, in qualitative agreement with some experimental observations in shapememory alloys. The model exhibits three phase transition processes: phase nucleation, interface slip and phase annihilation. Between these dynamic processes the system gets stuck in local minimizers of the potential energy. As β increases from zero, a slip-dominated solution behavior transforms to the one where slip and annihilation events are preceded by a step-by-step nucleation process. We show that this transition is caused by an interplay between the slip-favoring inertia term and the nucleation-favoring elastic foundation terms.