An energetic formulation of a one-dimensional model of superelastic SMA

This paper presents an energetic framework for the study of the macroscopic evolution of shape memory alloys (SMA) with softening behavior. It is written for a class of standard rate-independent materials with an internal variable derived from the Drucker–Ilyushin work property. This one-dimensional model is defined by three material functions of the internal variable and one material parameter. The quasi-static evolution is formulated for a one-dimensional bar under traction and is based on two physical principles: a stability criterion which consists in selecting the local minima of the total energy and an energy balance condition which requires the absolute continuity of the total energy. The stability criterion aims to overcome the non-uniqueness issue associated with the intrinsic softening character of SMA while the energy balance condition accounts for evolutions even with possible time discontinuities. While being consistent with the classical Kuhn–Tucker formulation of the phase transformations, such energetic formulation proved to be more suitable than this latter for the study of stress-softening SMA. Both homogeneous and non-homogenous solutions are investigated with respect to this variational evolution problem. Specifically, we show the instability of the homogeneous states for softening materials and construct, in this latter case, a non-homogeneous stable evolution that follows a transformation stress line which corresponds to the Maxwell line of the softening intrinsic behavior.