Effective elasticity of a solid suspension of spheres

We consider the problem of the elastic behaviour of a random medium composed of identical spherically symmetric inclusions, each made of a linear elastic material with varying shear and bulk moduli, distributed randomly in a uniform matrix made of a linear elastic material with moduli μ0, K0. We consider the response of a finite sample to an externally imposed incident displacement field and by eliminating the external field obtain a linear constitutive relation between mean stress and mean strain in the material. Using the cluster expansion method of Felderhof, Ford and Cohen we show that in the thermodynamic limit of an infinite system there is a well defined effective elasticity operator for the medium expressible as a sum of terms which involve successively the problem of 1, 2,...,p,... inclusions in an infinite matrix. The elasticity operator is given in detail through second order terms in the number density of inclusions. By passing to a long wavelength limit we show how the constitutive relation reduces to a local relation expressible in terms of effective moduli μeff, Keff.