A variational model for reconstructive phase transformations in crystals, and their relation to dislocations and plasticity

We study the reconstructive martensitic transformations in crystalline solids (i.e., martensitic transformations in which the parent and product lattices have arithmetic symmetry groups admitting no finite supergroup), the best known example of which is the bcc-fcc transformation in iron. We first describe the maximal Ericksen-Pitteri neighborhoods in the space of lattice metrics, thereby obtaining a quantitative characterization of the weak transformations, which occur within these domains. Then, focusing for simplicity on a two-dimensional setting, we construct a class of strain-energy functions admitting large strains in their domain, and which are invariant under the full symmetry group of the lattice. In particular, we exhibit an explicit energy suitable for the square-to-hexagonal reconstructive transformation in planar lattices. We present a numerical scheme based on atomic-scale finite elements and, by means of our constitutive function, we use it to analyze the effects of transformation cycling on a planar crystal. This example illustrates the main phenomena related to the reconstructive martensitic phase changes in crystals: in particular, the formation of dislocations, vacancies and interstitials in the lattice.