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 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 .

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 of the overall volume, as I have done in this paper.

11/17/1997