Bounds on the Effective Elastic Properties of Martensitic Polycrystals

Abstract

We draw attention to the problem of estimation of elastic energies in martensitic polycrystals. In particular we introduce a tensorial parameter η=η_(ijkl) which contains information about the microgeometry and disorder of the polycrystalline structure. Under the assumption of isotropic elasticity and mild hypothesis on the statistics of the polycrystal, this parameter allows for explicit calculation of rigorous and stringent upper bounds on the effective energy. For circular grains in two dimensions η gives the elastic energy resulting from transformation of a single circular inclusion in an elastic matrix and the bounds coincide with those derived recently by Bruno, Reitich and Leo. Consideration of such particular cases shows that our bounds can yield substantial improvements over those obtained under Taylor’s constant strain hypothesis. For arbitrary microgeometries the statistical parameter η can be calculated by means of two-point correlations functions.