THE OVERALL ELASTIC ENERGY OF POLYCRYSTALLINE MARTENSITIC SOLIDS

We are concerned with the overall elastic energy in martensitic polycr ystals. These are polycrystals whose constituent crystallites can unde rgo shape-deforming phase transitions as a result of changes in their stress or temperature. We approach the problem of calculation of the n onlinear overall energy via a statistical optimization method which in volves solution of a sequence of linear elasticity problems. As a case study we consider simulations on a two-dimensional model in which cir cular randomly-oriented crystallites are arranged in a square pattern within an elastic matrix. The performance of our present code suggests that this approach can be used to compute the overall energies in rea listic three-dimensional polycrystals containing grains of arbitrary s hape. In addition to numerical results we present upper bounds on the overall energy. Some of these bounds apply to the square array mention ed above. Others apply to polycrystals containing circular, randomly-o riented crystallites with sizes ranging to infinitesimal, and no inter grain matrix. The square-array bounds are consistent with our numerica l results. In some regimes they approximate them closely, thus providi ng an insight on the convergence of the numerical method. On the other hand, in the case of the random array the bounds carry substantial pr actical significance, since in this case the energy contains no artifi cial contributions from an elastic matrix. In all the cases we have co nsidered our bounds compare favorably with those obtained under the we ll-known Taylor hypothesis; they show that, as far as polycrystalline martensite is concerned, calculations of the elastic energy based on t he Taylor assumption may lead to substantial overestimates of this qua ntity. Copyright (C) 1996 Elsevier Science Ltd