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* = *v*_{i} 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 *K*_{DEM}(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 ^{*}*

7/5/1998