Preferential addition of cracks

The factor (1-y) on the left hand sides of both (DEMK) and (DEMG) arises from the need to account for the fact that, when an inclusion is placed in a composite, the volume of the inclusion replaces not only host material, but also some of the other inclusion material previously placed in the composite. When y is the inclusion volume fraction, the remaining host volume fraction is (1-y). So random replacement of dy of the composite medium only replaces (1-y)dy of the host material. Replacing instead dy/(1-y) of the composite then gives the correct factor of dy host replacement; thus, the factor of (1-y) is required in (DEMK) and (DEMG) for random inclusion placement at finite values of y.

If we now assume instead that the inclusions are place preferentially in pure host material (and this gets progressively harder to do in practice for larger integrated overall inclusions fractions y), then the DEM equations must be modified to account for this situation.

For example, with preferential addition of inclusions, it is clear from the preceding considerations that DEM equation (DEMK1) is replaced by

ddy(1K^*) 1K_f - 1K_m.   Integrating (prefK) gives

1K^* - 1K(_0) = (1K_f - 1K_m)_crack.   The validity of this result clearly depends on $\phi_0$ being sufficiently small so that it is possible to find enough pure host material to which cracks can be added ``randomly.'' Taking $\phi_0 \to 0$ guarantees satisfaction of the requirement, but the approximation must eventually break down as $\phi_0 \to 1$.

Eqn.(newMJ3) is almost the corresponding result of Mavko and Jizba (1991). Mavko and Jizba use as their comparison state the dry porous material, assuming that no cracks are present or that, when present, they are closed due to applied external pressure. We can also obtain the same result using (prefK), but now $K^*(\phi_0) = K_{dry}(\phi_0)$, so the integration has a different starting value than in the previous paragraph. Then, we find

1K^*_MJ - 1K_dry(_0) = (1K_f - 1K_m)_crack.   Eqn.(newMJ4) is exactly the corresponding result of Mavko and Jizba (1991). Although the right hand sides of (newMJ3) and (newMJ4) are identical, the results differ, i.e., $K^* \ne K^*_{MJ}$, since the assumed host material is fluid saturated in the first case and dry in the second case.