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We study third-order upper and lower bounds on the shear modulus of a model composite made up of equisized, impenetrable spherical inclusions randomly distributed throughout a matrix phase. We determine greatly simplified expressions for the two key multidimensional cluster integrals (involving the three-point distribution function for one of the phases) arising in these bounds. These expressions are obtained by expanding the orientation-dependent terms in the integrand in spherical harmonics and employing the orthogonality property of this basis set. The resulting simplified integrals are in a form that makes them much easier to compute. The approach described here is quite general in the sense that it has application in cases where the spheres are permeable to one another (models of consolidated media such as sandstones and sintered materials) and to the determination of other bulk properties, such as the bulk modulus, thermal/electrical conductivity, and fluid permeability.

Bulk properties of composite media. I. Simplification of bounds on the shear modulus of suspensions of impenetrable spheres