Explicit Bounds of Effective Stiffness Tensors for Textured Aggregates of Cubic Crystallites

Abstract

For isotropic aggregates of cubic crystallites, Hashin and Shtrikman derived lower and upper bounds for the effective stiffness tensor, which are tighter than the lower and upper bound provided by the Reuss and Voigt model, respectively. In this paper we consider anisotropic aggregates of cubic crystallites with arbitrary texture. We model the elastic polycrystal in question as an assemblage of space-filling spherical grains. Moreover, we assume that every point within one grain has the same crystallographic orientation, whereas the orientations of different grains are uncorrelated. Under this model, we appeal to the variational principles of Hashin and Shtrikman and derive explicit lower and upper bounds for the effective stiffness tensor, which are quadratic in texture coefficients and carry parameters given in terms of the single-crystal elastic constants. For weakly-textured aggregates of cubic crystallites, several examples suggest that our bounds for the effective elastic tensor provide estimates much tighter than those delivered by the Reuss lower bound and the Voigt upper bound.