A one dimensional variational model of superelasticity for shape memory alloys

Abstract

In this paper we propose a variational framework for the modeling of superelasticity in shape memory alloys with softening behavior. This model is valid for a class of standard rate-independent materials with a single internal variable. The quasi-static evolution is based on two physical principles: a stability criterion which selects the local minima of the total energy and an energy balance condition which ensures the absolute continuity of the total energy. The stability criterion allows to bypass non-uniqueness issues associated to softening behaviour while the energy balance condition accounts for brutal evolutions at the local levels. We investigate properties of homogeneous and non-homogenous solutions towards this variational evolution problem. Specifically, we show how softening behaviour can lead to instability of the homogeneous states. In this latter case, we show that a stable solution would consist in following the Maxwell line given by the softening behaviour, then resulting in a non-homogeneous evolution.