Existence, numerical convergence and evolutionary relaxation for a rate-independent phase-transformation model

Abstract

We revisit the model for a two-well phase transformation in a linearly elastic body that was introduced and studied in Mielke et al. (2002 Arch. Ration. Mech. Anal. 162 , 137–177). This energetic rate-independent system is posed in terms of the elastic displacement and an internal variable that gives the phase portion of the second phase. We use a new approach based on mutual recovery sequences , which are adjusted to a suitable energy increment plus the associated dissipated energy and, thus, enable us to pass to the limit in the construction of energetic solutions. We give three distinct constructions of mutual recovery sequences which allow us (i) to generalize the existence result in Mielke et al. (2002), (ii) to establish the convergence of suitable numerical approximations via space–time discretization and (iii) to perform the evolutionary relaxation from the pure-state model to the relaxed-mixture model. All these results rely on weak converge and involve the H-measure as an essential tool.