A NONLOCAL VARIATIONAL APPROACH TO THE ELASTIC ENERGY MINIMIZATION OFMARTENSITIC POLYCRYSTALS

Solid-solid phase transformations in polycrystals are considered in th e context of energy minimization. The energy of a single crystal is sp ecified by a non-convex multi-well energy function, in the approximati on of infinitesimal deformations. The number of wells corresponds to t he number of distinct phases and each is assumed to have the same isot ropic elastic modulus. The polycrystal's energy is defined a priori vi a minimization of the energy functional with proper account of the ori entation distribution, with respect to all 'kinematically admissible' displacement fields. A variational principle of Hashin-Shtrikman type is derived for the polycrystal's energy by developing and generalizing the approach of Bruno and co-workers. The variational principle invol ves a non-local functional with a Green's function-related kernel oper ating on trial 'transformation fields' which are appropriately constra ined to accommodate both the single crystal's constitutive law and the polycrystal's texture. For a statistically uniform polycrystal, the v ariational principle is reformulated to require minimization with resp ect to all possible two-point correlation functions of 'submicrostruct ure', compatible with the texture. This variational principle is appli ed to derive upper bounds for a statistically uniform polycrystal by e mploying a 'separation of scales', i.e. by constraining the set of tri al fields to those with the property that the scale of the trial submi crostructure is much finer than the scale of the polycrystal's texture . Subsequent optimization with respect to this submicrostructure for e ach particular orientation reveals a connection with relaxation of a s ingle crystal 'with fixed volume fractions' and associated H-measures as discussed by Kohn. The resulting upper bound is developed and compa red with a bound derived by Bruno et al. For some examples the new bou nd is demonstrated to be sharper than the latter, as a result of an im proved optimization procedure. The present approach also extends that of Bruno et al. to more general orientation distribution statistics an d clarifies the effect of incompatibility of transformation strains in the single crystals for the overall performance of polycrystals.