New bounds on the elastic moduli of suspensions of spheres

We derive rigorous three-point upper and lower bounds on the effective bulk and shear moduli of a two-phase material composed of equisized spheres randomly distributed throughout a matrix. Our approach is analogous to previously derived three-point cluster bounds on the effective conductivity of suspensions of spheres. Our bounds on the effective elastic moduli are then compared to other known three-point bounds for statistically homogeneous and isotropic random materials. For the case of totally impenetrable spheres, the bulk modulus bounds are shown to be equivalent to the Beran-Molyneux bounds, and the shear modulus bounds are compared to the McCoy and Milton-Phan-Thien bounds. For the case of fully penetrable spheres, our bounds are shown to be simple analytical expressions, in contrast to the numerical quadratures required to evaluate the other three-point bounds.