Mean-field approximation to the effective elastic moduli of a solid suspension of spheres

We study the effective shear and bulk moduli of a solid suspension of spheres with a spherically symmetric elastic profile. A mean-field approximation is derived which corresponds to the Lorentz local field in the theory of dielectrics. Thus the approximate expressions for the effective shear and bulk moduli are the analogs of the Clausius-Mossotti equation for the effective dielectric constant. For the case of uniform spheres the expressions are closely related to the Hashin-Shtrikman bounds. We show that the mean-field expression may be corrected systematically for correlations in the sphere positions on the basis of cluster expansions derived by statistical methods.