INEQUALITIES FOR ELECTRIC AND ELASTIC POLARIZATION TENSORS WITH APPLICATIONS TO RANDOM COMPOSITES

NEW BOUNDS on the elastic and electric polarization tensors are found for grains of arbitrary shape or connectivity. For a grain shape specified by the characteristic function chi(x), the bounds are given explicitly in terms of the geometric function \chi(k)\2. For electric polarizations one of the bounds may be interpreted as the polarization of a homogeneous ellipsoidal inclusion with axes determined by \chi(k)\2. The other bound corresponds to a convex sum of polarization tensors for plate-like inclusions. Here the plate normals and weights are specified by \chi(k)\2. These bounds are used to predict the range of effective transport properties for hierarchical random suspensions and aggregates that realize the Effective Medium Approximation. The inequalities also provide rigorous bounds for the effective properties of dilute statistically anisotropic random suspensions.