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.