A fast algorithm for the simulation of polycrystalline misfits. II. Martensitic transformations in three space dimensions

Abstract

We present a fast numerical method for the simulation of martensitic transformations in three–dimensional polycrystals. To produce the relevant overall elastic energy arising from given boundary conditions, this method proceeds by reducing the corresponding non–convex minimization problem to minimization of a certain quadratic form—over the set of arrays of transformation strains which are compatible with a given distribution of crystallite orientations. The evaluation of this quadratic form for a given array of transformation strains requires solution of certain linear elasticity problems. An acceleration strategy we use, which for a polycrystal containing N grains reduces the complexity of the algorithm from O(N^2) to O(N) operations, results from a formulation of the minimization problem which takes advantage of certain decorrelations present in the minimizing arrays of transformation strains. We illustrate our presentation with a number of examples involving cubic–to–monoclinic and cubic–to–orthorhombic polycrystalline phase transitions, such as those arising in the TiNi and CuAl shape–memory alloys. In particular, our study quantifies the effects of texture on the overall properties of such polycrystalline shape–memory alloys.