The third scheme I consider is the Differential Effective Medium (DEM) Approximation (Cleary et al., 1980; Norris, 1985; Avellaneda, 1987). I limit the treatment here to the two-component case, as that is the easiest to explain in a small space. This method is derived by assuming the composite is formed by successively mixing very small (infinitesimal) fractions dy of one inclusion material i in another host material. The host medium changes gradually during this process from material h at y=0 into the desired composite material * at some finite y value. Starting with (general), the resulting formula for the stiffness is the differential equation
(1-y)ddy^*_DEM(y) = [^(i)-^*_DEM(y)]^*i, where the initial value of the stiffness tensor is . The Eshelby-Wu tensor is the one corresponding to inclusions of stiffness imbedded in host material of stiffness . The resulting system of coupled equations may be integrated to any desired value of total inclusion volume fraction y = vi easily using (for example) a Runge-Kutta scheme.
The formula for the Biot-Willis parameter is obtained in this scheme most easily by starting from (KTalpha*), noting first that the sum on the right is reduced to a single term for the phase that is not the initial host phase, replacing the parameters for the host medium by their values evaluated at concentration y and the * parameters by their values evaluated at concentration y+dy. The volume fraction is replaced by to account for the fact that more than the amount dy of the composite host material must be replaced in order to achieve the new desired volume fraction y + dy. Finally, taking the limit as gives the desired formula. For spherical inclusions, the result is
(1-y)ddy^*_DEM(y) = [^(i)-^*_DEM(y)]P^*i, where .The corresponding result for the bulk modulus obtained directly from (DEM) is
(1-y)ddyK^*_DEM(y) = [K^(i)-K^*_DEM(y)]P^*i, where KDEM(0) = K(h). Both results were obtained previously for spherical inclusions (Berryman, 1992), but the present derivation is much more compact. The generalization to nonspherical inclusions is now straightforward.
Since this version of DEM is only valid for two component composites, I can take (DEMBW) as the correct generalization formula for nonspherical inclusions with P*i evaluated for arbitrary ellipsoidal inclusions. The motivation for this choice is that, when (DEMBW) and (DEMK) are taken together, they guarantee satisfaction of the known exact results for two-component materials, since their ratio gives
ddy^*_DEM^*_DEM-^(i) = ddyK^*_DEMK^*_DEM-K^(i). Upon integration, (ratio) gives
^*_DEM-^(i)^(h)-^(i) = K^*_DEM-K^(i)K^(h)-K^(i), as required (see the discussion in the next section). We could alternatively (and more simply) take (DEMexact) as the formula for when K*DEM has been previously computed.