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

We present a fast method for the solution of problems of elasticity involvi ng microscopic misfit strains. While the main case we consider is that asso ciated with martensitic transformations in polycrystals, our methods can be applied to a variety of systems whose constituents undergo misfit deformat ions, including polycrystalline magnetostriction, thermal expansion, etc., as well as mathematically analogous phenomena in ferroelectricity and ferro magnetism. The basic component of our method is an explicit solution for Es helby-type problems on square elements. Fast computation of the polycrystal energy results through a rapidly convergent sequence of approximations whi ch can, in fact, be interpreted as a generalization of a class of upper bou nds introduced recently. The overall complexity of the method is O(N) opera tions, where N is the number of component crystallites. We also present a n ew lower bound for the energy, giving additional insights on the microscopi c phenomena leading to the observed structural behaviour. The present work applies to two-dimensional polycrystals; extensions to the three-dimensiona l case have been implemented and will be presented elsewhere.