Homogenization of polycrystalline aggregates

The effective elastic properties of a polycrystalline material depend on th e single crystal elastic constants of the crystallites comprising the polyc rystal and on the manner in which the crystallites are arranged. In this pa per we apply the techniques of homogenization to put the problem of determi ning effective elastic constants in a precise mathematical framework that p ermits us to derive an expression for the effective elasticity tensor. We a lso study how the homogenized elasticity tensor changes as the probability characterizing the ensemble changes. Under the assumption that the field of orientations of the crystallographic axes of the crystallites is an indepe ndent random field, we show that our theory is compatible with the formulat ion used in texture analysis. In particular, we are able to prove that the physical assumption made by MAN [10] in his study of weakly-textured polycr ystals holds true. In addition, we establish some elementary bounds on the material constants that characterize the effective elasticity tensor of wea kly-textured orthorhombic aggregates of cubic crystallites.