A new look at Hill's arithmetic and geometric means for a two-phase, isotropic composite

In this paper we take a fresh look at the classical problem of Hill's arithmetic and geometric means. It is shown that both types of means can be derived from a dual class of polarization fields that lead to an "ordered" structure. Hill's arithmetic mean of Voigt's and Reuss' moduli is derived from the displacement-prescribed boundary condition, whereas the arithmetic mean of the compliances is derived from the traction-prescribed condition. These means are found to correspond to 1/4-th of the maximum elastic energy from the perturbed elastic field, the maximum values being associated with the Reuss and Voigt approaches, respectively. Hill's geometric mean, on the other hand, always lies between this pair of moduli. It also represents the limiting case of the arithmetic means for this class of ordered materials that can be constructed with a hierarchical scheme. Despite their simplicity, both the arithmetic and the geometric means are found to fully meet the rigorous requirement for the shift property of the effective moduli recently derived by Hu and Weng [10].