A ‘non-local’ variational approach to the elastic energy minimalization of martensitic polycrystals

Abstract

Solid-solid phase transformations in polycrystals are considered in the context of energy minimization. The energy of a single crystal is specified by a non-convex multi-well energy function, in the approximation of infinitesimal deformations. The number of wells corresponds to the number of distinct phases and each is assumed to have the same isotropic elastic modulus. The polycrystal's energy is defined a priori via minimization of the energy functional with proper account of the orientation 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 involves a non-local functional with a Green's function-related kernel operating on trial 'transformation fields' which are appropriately constrained to accommodate both the single crystal's constitutive law and the polycrystal's texture. For a statistically uniform polycrystal, the variational principle is reformulated to require minimization with respect to all possible two-point correlation functions of 'submicrostructure', compatible with the texture.This variational principle is applied to derive upper bounds for a statistically uniform polycrystal by employing a 'separation of scales', i.e. by constraining the set of trial fields to those with the property that the scale of the trial submicrostructure is much finer than the scale of the polycrystal's texture. Subsequent optimization with respect to this submicrostructure for each particular orientation reveals a connection with relaxation of a single crystal 'with fixed volume fractions' and associated H-measures as discussed by Kohn. The resulting upper bound is developed and compared with a bound derived by Bruno et al. For some examples the new bound is demonstrated to be sharper than the latter, as a result of an improved optimization procedure. The present approach also extends that of Bruno et al. to more general orientation distribution statistics and clarifies the effect of incompatibility of transformation strains in the single crystals for the overall performance of polycrystals.