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).