The final approximation I consider is the Mori-Tanaka (MT) Scheme of Mori and Tanaka (1973), as described by Weng (1984), Benveniste (1987), and others [see, for example, Berryman and Berge (1996)].
For the drained frame, the Mori-Tanaka approximation is obtained by assuming the composite has a host material with imbedded inclusions and then choosing the host to serve as the reference material, so r = h. Making this choice in (general) and then substituting , I obtain
v_i (^(i) - ^*_MT)^hi = 0. The Mori-Tanaka result for the bulk modulus with arbitrary ellipsoidal inclusion shapes is
v_i (K^(i) - K^*_MT)P^hi = 0. Because the Mori-Tanaka scheme can not be derived using any analogy to scattering theory (unlike the other three schemes considered so far), there is some ambiguity about how to apply the present method within Mori-Tanaka and different choices of formulas for the Biot-Willis parameter may result. One of the more straightforward approaches can be shown to lead to the formula
v_i (^(i) - ^*_MT)P^hi = 0, when the inclusions are all spherical in shape. I stress however that (MTBW) is not the only possible formula that could be obtained or that could be considered to be fully consistent with the Mori-Tanaka scheme.
Note that it is easy to show that both (MTBW) and (DEMBW) have the advantage that they reproduce the known exact results (Berryman and Milton, 1991) for two component poroelastic media. This fact provides a useful criterion for choosing among various possibilities that arise when trying to identify the proper generalizations of these theories for the poroelastic case.