The connections between the double-inclusion model and the Ponte Castaneda–Willis, Mori–Tanaka, and Kuster–Toksoz models

Abstract

In this paper, it is shown that the double-inclusion model (Hori, M., Nemat-Nasser, S., 1993. Double-inclusion model and overall moduli of multi-phase composites. Mech. Mater. 14, 189–206) carries more theoretical connections with other micromechanical models than what is presently realized. In the past, only connections with the Mori–Tanaka (MT) model (Mori, T., Tanaka, K., 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21, 571–574) and the self-consistent model (Hill, R., 1965. A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222; Budiansky, B., 1965. On the elastic moduli of some heterogeneous material. J. Mech. Phys. Solids, 13, 223–227) for aligned inclusions have been established. By choosing the shape and the relative orientation of the inclusion and the matrix judiciously, the double-inclusion model can produce results for a two-phase composite containing randomly oriented ellipsoidal inclusions for the Ponte Castaneda–Willis (PCW) model (Ponte Castaneda, P., Willis, J.R., 1995. The effect of spatial distribution on the effective behavior of composite materials and cracked media. J. Mech. Phys. Solids 43, 1919–1951), MT model, and Kuster–Toksoz (KT) model (Kuster, G.T., Toksoz, M.N., 1974. Velocity and attenuation of seismic waves in two-phase media: I Theoretical formulation. Geophysics, 39, 587–606). These connections have also shed some light into the possible microgeometries for the MT and KT models. The microstructure for the PCW model is already known, and it is now established that the outer shape and orientation of the double inclusion is exactly the spatial distribution ellipsoid of the PCW model. The result also proves that the KT model, widely used in the geophysics community, actually provides a result that is identical to the PCW model and, thus, has a well-defined microstructure that was previously said to be non-existent.