Effective elasticity of a solid suspension of spheres

We evaluate explicitly through terms of second order in volume fraction the effective elastic moduli derived formally in a preceeding article for a random suspension of spherically symmetric inclusions in a background matrix. By interpreting the earlier formulae in terms of induced stresses on an inclusion we reduce the calculation to the treatment of a one-inclusion and a two-inclusion problem only. To treat the one-inclusion problem we derive a Faxén type theorem for the total stress induced on the inclusion by an incident displacement field. For the two-inclusion problem we use a method of reflections whereby we obtain the contribution of lowest order in the separation of inclusion centres. By introducing a point stress model of the inclusions we sum a subset of reflections to all orders in inclusion separation. We find that in the point stress model the effective moduli are given in terms of only two of the scattering coefficients introduced elsewhere for the one-inclusion problem. For the special case of uniform inclusions we give numerical results for the effective moduli which are compared with exact upper and lower bounds.