Analysis of martensitic microstructures in shape-memory-alloys

Abstract

In this work, we study the limiting behavior of a variational model which arises in the analysis of microstructures at austenite-martensite interfaces in shape-memory alloys. We consider an energy containing an elastic bulk term and a surface term, in which the bulk term is geometrically linearized. Two martensitic phases, one of them with a much smaller volume fraction, form microstructures at a straight interface to an austenite phase. From literature we know a scaling law for the behavior of minimizers which depends on the ratio of volume fraction and prefactor in the surface energy.<br /> For a transition regime of this behavior we consider two rescaled energies, one for scalar-valued functions, one for vector-valued functions. The limiting energy in the sense of Gamma-limits for the first functional is derived. This limiting functional is only finite on a subspace of SBV which bears constraints on the direction of the jump set. A key ingredient for the proof of the Gamma-limit is an approximation result for this subspace with respect to the energy.<br /> We conjecture the form of the limiting energy for the vector-valued case that is defined on a subspace of SBD which again does only allow some special directions of the jump set. The Gamma-convergence result for this case is still an open problem since we have not yet been able to provide an appropriate density result. We are, however, able to prove the liminf-inequality, a compactness result and recovery sequences for functions of higher regularity. Additionally, we provide a Korn-Poincare-type inequality for the subspace of SBD.