Effective properties of two-phase disordered composite Media. I. Simplification of bounds on the conductivity and bulk modulus of dispersions of impenetrable spheres

We consider the problem of evaluating upper and lower bounds on the effective conductivity and bulk modulus derived, respectively, by Beran and by Beran and Molyneux, for the model of impen- etrable spherical inclusions randomly distributed throughout a matrix. The key multidimensional cluster integral is simplified by expanding the appropriate terms of its integrand in spherical harmonics and employing the orthogonality properties of this basis set. The resulting simplified integrals are in a form that makes them easier to compute. The approach described here can be readily and systematically extended to cases in which the inclusions are permeable to one another and to the determination of other bulk properties of composite media, such as the effective shear modulus.