Methods for computing estimates of elastic constants for composites are
either implicit or explicit. Implicit methods are generally computed
either by iteration [for example, self-consistent methods such as those
described by Gubernatis and Krumhansl (1975) or Berryman (1980)] or
by integration [for example, differential effective medium methods such
as those discussed by Cleary *et al.* (1980) or Norris (1985)]. On the
other hand, explicit methods are easier to evaluate because they
supply *formulas* for estimates of constants, not requiring any further
computation except direct substitution.

Two of the more common implicit methods -- the self-consistent
(SC) method and the differential effective medium (DEM) method
mentioned in the preceeding paragraph -- are known to be realizable
(Milton, 1985; Norris, 1985; Avellaneda, 1987), *i.e.*, they compute
constants that correspond to values realized for specific microgeometries
(Berryman and Berge, 1993). Realizability of an estimator implies that
the estimator will *always* satisfy rigorous bounds on the constants
[such as Hashin-Shtrikman (1963) and Walpole (1969) bounds]. On the contrary,
the two most common explicit methods and the ones I intend to study here --
Mori-Tanaka (1973) and Kuster-Toksöz (1974) -- are known not to be
realizable, and are in fact both known to violate some of the rigorous bounds
(Norris, 1989; Ferrari, 1991; Berryman, 1980) in limiting cases
with extreme microgeometries (such as high concentrations of thin disk
inclusions).

Nevertheless there are circumstances in which explicit schemes are useful.
In one situation frequently encountered in geophysical/rock mechanics
studies, it is desirable to invert the formulas for constants to try
to estimate other physical properties of the composite/rock, such
as the crack aspect ratio distribution (Toksöz *et al.*, 1976;
Cheng and Toksöz, 1979) or the porosity (Berge *et al.*, 1992),
from measurements of seismic velocities or elastic constants.
Such an inversion procedure is practically intractable with an implicit scheme
but quite feasible with any explicit estimation scheme. The main limitations
on the use of explicit schemes for inversion are the nonuniqueness of the
results obtained (a common difficulty in geophysical inversion problems) and
the inherent inaccuracies of the approximate estimators themselves. My
present purpose is not to study the inverse problem, but rather to analyze two
of the most commonly used explicit schemes: Mori-Tanaka (MT) and
Kuster-Toksöz (KT). I do this with a view to
determining relations between them and also which of them might be
more accurate and therefore more useful for such inversion studies.
Range of validity of the two methods will also be discussed.

To achieve these goals, I first introduce a unified approach for obtaining these approximation schemes and use this method to present different derivations of the SC, MT, and KT estimates. For example, I show how to obtain the KT approximation within a quasistatic analysis, whereas this approximation has generally been derived using a long-wavelength approximation within scattering theory (Kuster-Toksöz, 1974; Berryman, 1980).

11/17/1997