Evolution of martensitic phase boundaries in heterogeneous media

We study the role of defects in the quasistatic evolution of a martensitic phase boundary. Martensitic phase transformations involve a change in shape of the underlying crystal, and thus the propagation of the phase boundary is accompanied by an evolving mechanical stress and strain field. This gives rise to a nonlocal free boundary problem, one where the evolution of the free boundary is coupled to an elliptic partial differential equation. Often, real materials contain defects which can potentially pin the interface and contribute to the hysteresis, and understanding this is the goal of this work. We present a mathematical model and a proof of existence in the sense of sets of finite perimeter. We then present numerical simulations and analysis of both the fully nonlocal problem as well as a linearized version, and draw conclusions on the role of defects on hysteresis.