The e&ective energy and laminated microstructures in martensitic phase transformations

We study the behavior of crystals that undergo martensitic transformations. On cooling, the high-temperature phase (austenite) transforms to the martensite phase changing its crystalline symmetry. The lower crystalline symmetry of the martensite gives rise to several variants of martensite. Each variant has an associated transformation strain. These variants accommodate themselves (according to the boundary conditions) forming a microstructure that minimizes the elastic energy. This minimum value of the energy is called the e ective energy. We assume that all the material is in the martensite phase (i.e. the material is at low temperatures). We show that, assuming the geometrically linear approximation, the maximum of the e&ective energy restricted to applied strains in the convex hull of the transformation strains is attained by an applied strain that is a convex combination of only two transformation strains. We derive a recurrence relation to compute the energy corresponding to laminated microstructures of arbitrary rank, under the assumption that the variants of martensite are linearly elastic and their elastic moduli are isotropic. We use this recurrence relation to develop an algorithm that minimizes the energy over microstructures that belong to the class of rank-r laminates. We apply our methods to the case in which the transformation is cubic to monoclinic (corresponding to TiNi). We conclude with some comments on the possible implications of our calculations on the behavior of this shape-memory alloy. c © 2001 Elsevier Science Ltd. All rights reserved.