A dynamical theory for the Mori–Tanaka and Ponte Castañeda–Willis estimates

Most micromechanics models for composite materials have been derived under the static condition, and most of them are not known to possess a dynamical basis. In this article, we seek to establish such a basis for the Mori–Tanaka (Mori and Tanaka, 1973) and Ponte Castañeda–Willis (Ponte Castañeda and Willis, 1995) models. We adopt a low frequency, long wavelength scattering approach to match the far fields between two configurations: a cluster of original inclusions in the matrix and an effective homogeneous domain. We demonstrate that, with aligned ellipsoidal inclusions, the MT moduli can be obtained by choosing the domain shape of the effective medium to coincide with the inclusion shape, and that, by leaving it free from such a restriction, the PCW estimates can be realized. It turns out that the effective domain shape is the shape of the outer cell of the double inclusion (Hori and Nemat-Nasser, 1993) and it also is a reflection of the spatial distribution of inclusions in the composite. We further demonstrate that, in the randomly oriented isotropic case, the Kuster–Toksöz (Kuster and Toksöz, 1974)–Berryman (Berryman, 1980b) model is exactly the dynamical counterpart of the PCW estimate.