Fourth-order bounds on the effective bulk and shear moduli of random dispersions of penetrable spheres

The structural parameters entering the fourth-order bounds of Milton & Phan-Thien (Proc. R. Soc. Lond. A 380, 305 (1982)) on the effective bulk and shear moduli of two-component materials are expressed in terms of integrals over three- and four-point correlation functions to facilitate numerical calculations on systems for which the correlation functions are known. These integral expressions are then used to numerically calculate the structural parameters of random dispersions of penetrable spheres.