A06 – Hysteresis and microstructure in shape memory alloys
Our aim is to develop mathematical tools for the study of microstructure and hysteresis in the martensitic phase transition within the framework of nonlinear elasticity. We focus on a variational formulation, in which the different phases are represented by different eigenstrains, and include the interfacial energy as a singular perturbation to the functional. The key modeling assumption is that hysteresis is largely determined by the height of the energy barrier, which in turn can be estimated by the minimal energy required to generate a nucleus of the critical size. Therefore we shall consider a variational model with boundary conditions which correspond to (part of) a martensitic nucleus in an austenitic matrix, and parameters related to the sample geometry, the interfacial energy, the elastic constants of the material and the eigenstrains.
We shall start from the study of the phase diagram of our model in dependence on the various parameters, in order to identify the different microstructure regimes with two martensite variants near an austenite-martensite interface. We shall then consider in more detail three specific issues. Firstly, the regime with very small volume fraction of one martensite variant, in which we expect this variant to concentrate on a lower-dimensional manifold. Secondly, the regime of needle-like microstructure, and the precise geometry of the needles. Thirdly, the possibility of low-energy martensitic inclusions in an austenitic matrix.