On the energy-minimizing strains in martensitic microstructures—Part 1: Geometrically nonlinear theory

This paper addresses the theoretical prediction of the quasiconvex hull of energy-minimizing strains that can be realized by martensitic microstructures. Polyconvexification and related notions are used to derive some upper bounds (in the sense of inclusion) on the quasiconvex hull. Lower bounds are constructed by lamination techniques. The geometrically nonlinear theory (finite strains) is considered in the present Part 1. Analytical expressions are obtained for a three-well problem which encompasses the cubic to tetragonal transformation as a special case. Twelve-well problems related to cubic to monoclinic transformations are also studied. In that case, sufficient conditions are derived for the microstructure to be restricted to only two of the 12 wells.