Pierluigi Cesana

Mathematical models and ideas for disclinations

In this talk, I will describe a few recent results on the modeling of disclinations (rotational mismatches at the level of a crystal lattice) and on the modeling of self-similar martensitic microstructure, two phenomena which appear to be strongly interconnected.

First, we introduce an energy functional defined over a triangular lattice accounting for nearest-neighbor interactions. We design special rotational-type boundary value problems on the lattice so that the minimizers necessarily exhibit non-homogeneous rotations. We are interested in the asymptotics of the energy minima and minimizers as the lattice spacing vanishes which we characterize with Gamma-convergence. We perform some numerical calculations for the discrete model and show that both the shape of the solutions as well as the values of the energies are in agreement with classical results for positive and negative disclinations. This is a collaboration with P. van Meurs (Kanazawa).

Second. We present a probabilistic model for the description of the martensitic microstructure as an avalanche process. A martensitic phase-transformation is a first-order diffusionless transition occurring in elastic crystals and characterized by an abrupt change of shape of the underlying crystal lattice. It is the basic activation mechanism for the Shape-Memory effect. Our approach to the analysis of the model is based on an associated general branching random walk process. Comparisons are reported for numerical and analytical solutions and experimental observations. This is a joint project with John M. Ball and Ben Hambly (Oxford).