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

Abstract

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. PACS number(s): 03.40.Dz, 46.30.Cn, 62.20.Dc, 81.40.Jj The calculation of the effective elastic properties of a solid composite is an important problem of material science. In this article, we study a solid suspension consisting of spheres with a spherically symmetric elastic profile embedded in a uniform and isotropic matrix. We derive mean-field expressions for the effective shear and bulk moduli of the suspension by a method analogous to that used by Lorentz [1] for the derivation of the ClausiusMossotti formula in the theory of dielectrics [2,3]. For the special case of uniform spheres the mean-field expressions reduce to the Hashin-Shtrikman bounds [4], except when the shear modulus of the spheres is larger than and the bulk modulus is smaller than that of the matrix, or vice versa. In the statistical theory of dielectrics it is known [5,6] how the Clausius-Mossotti formula may be obtained as an approximation from exact cluster expansions, which have been derived by statistical methods. The cluster expansions provide exact expressions for the effective linear transport properties of suspensions. They allow a systematic calculation of corrections to the mean-field expressions due to correlations in the sphere positions. We show in the following how, in the elastic problem, the mean-field expressions may be obtained from the direct cluster expansion [7), as well as from the renormalized cluster expansion [8]. The latter is the most suitable for the calculation of the correction terms. p(r) = p2( [r — R [ ), «(r) =«2( ~r — RJ [)