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

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 w ith misfitting inclusions. Acta Metall. 21, 571-574) and the self-consisten t model (Hill, R., 1965. A self-consistent mechanics of composite materials . J. Mech. Phys. Solids 13, 213-222; Budiansky, B., 1965. On the elastic mo duli of some heterogeneous material. J. Mech. Phys. Solids, 13, 223-227) fo r aligned inclusions have been established. By choosing the shape and the r elative orientation of the inclusion and the matrix judiciously, the double -inclusion model can produce results for a two-phase composite containing r andomly oriented ellipsoidal inclusions for the Ponte Castaneda-Willis (PCW ) model (Ponte Castaneda, P., Willis, J.R., 1995. The effect of spatial dis tribution on the effective behavior of composite materials and cracked medi a. J. Mech. Phys. Solids 43, 1919-1951), MT model, and Kuster-Toksoz (KT) m odel (Kuster, G.T., Toksoz, M.N., 1974. Velocity and attenuation of seismic waves in two-phase media: I Theoretical formulation. Geophysics, 39, 587-6 06). These connections have also shed some light into the possible microgeo metries for the MT and KT models. The microstructure for the PCW model is a lready known, and it is now established that the outer shape and orientatio n 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. (C) 2000 Elsevier Science Ltd. All rights reserved .