Quasiconvex hulls and recoverable strains in shape-memory alloys

We present work in progress on determining the set of recoverable strains in shape-memory single crystals from lattice parameters in a full 3D setting. More precisely, we seek the set of macroscopic deformation gradients which can be accomodated below the transition temperature by a zero energy martensitic microstructure. Mathematically, this amounts to determining the quasiconvex hull (it turns out typically to be non convex) of the set of pure martensitic phases, and is a prototype case of a homogeneization problem for a system of nonlinear partial differential equations. Via a novel, purely geometric approach it is possible to determine the desired set for cubic-to-tetragonal transformations, subject to an as yet unproved conjecture (that the quasiconvex hull in this case equals the polyconvex hull). This givex explicit predictions, e.g. for Ni-36at%Al: recoverable tensile strain of 12.7% in 100> direction, 1.4% in 110>, 0.74% in 111>. It would be interesting to check these predictions experimentally, especially as geometrically linear theory (mis?)predicts the cubic-to-tetragonal 111> recovery to be zero.