CONCLUSIONS

Mori-Tanaka and Kuster-Toksöz give the same results for composites with spherical inclusions as illustrated in Figure 1. In fact, host microgeometry cannot contribute to the results for either Mori-Tanaka and Kuster-Toksöz estimates since $\matT^{hi} = \matI$ for any ellipsoidally shaped particle if i=h. Although this result is reasonable for low concentrations of inclusions in a host medium, it clearly limits the usefulness of these theories. In particular, I have no control (i.e., I cannot specify the limiting host geometry) over the behavior of either theory when the limit of vanishing host volume fraction is approached. Viewing these approximations as interpolation schemes, both theories have no means of adjusting the slope of the elastic constant trajectories as the host volume fraction vanishes. This failure has some serious consequences: For example, Norris (1989) has shown that MT can give results violating the Hashin-Shtrikman bounds for multiphase composites. Similarly, Ferrari (1991) has shown that for anisotropic composites the MT scheme produces the absurd result that the composite can still depend on the host elastic constants even in the limit of vanishing host volume fraction. Berryman (1980) shows that the KT scheme violates the Hashin-Shtrikman bounds when the inclusions are either disks at any finite concentration or needles at volume fractions greater than about $60\%$.

I conclude that MT and KT have limited ranges of usefulness and in particular should not be used when the nominal host material does not have the dominant volume fraction. A reasonable recommendation for the use of these schemes then is to limit calculations to situations where the host occupies at least $50\%$ of the overall volume, as I have done in this paper.