Clausius-Mossotti-type approximation for elastic moduli of a cubic array of spheres

The method of elastostatic resonances is applied to the three-dimensional problem of nonoverlapping spherical inclusions arranged in a cubic array in order to calculate the effective elastic moduli. Explicit expressions, exact at least to order p(3) (where p is the volume fraction of the inclusions), are obtained for the bulk modulus and for the two shear moduli. The approximation used, which is the leading order in a systematic perturbation expansion of the appropriate modulus, is related to the Clausius-Mossotti approximation of electrostatics. Comparison with numerical calculations of the moduli and with previous work reveals that this approximation provides accurate results at low volume fractions of the inclusions and is a good estimate to the effective moduli at moderate volume fractions even when the contrast is high. Some of the expressions turn out to be identical to the Hashin-Shtrikman (HS) bounds.