A fast algorithm for the simulation of polycrystalline misfits: martensitic transformations in two space dimensions

Abstract

We present a fast method for the solution of problems of elasticity involving microscopic misfit strains. While the main case we consider is that associated with martensitic transformations in polycrystals, our methods can be applied to a variety of systems whose constituents undergo misfit deformations, including polycrystalline magnetostriction, thermal expansion, etc., as well as mathematically analogous phenomena in ferroelectricity and ferromagnetism. The basic component of our method is an explicit solution for Eshelby–type problems on square elements. Fast computation of the polycrystal energy results through a rapidly convergent sequence of approximations which can, in fact, be interpreted as a generalization of a class of upper bounds introduced recently. The overall complexity of the method is O(N) operations, where N is the number of component crystallites. We also present a new lower bound for the energy, giving additional insights on the microscopic phenomena leading to the observed structural behaviour. The present work applies to two–dimensional polycrystals; extensions to the three–dimensional case have been implemented and will be presented elsewhere.