Random addition of cracks

Random addition of inclusions is the case considered in the main text, so the DEM equations themselves do not change. We use (DEMK) and (DEMG) as before, but the range of integration changes to the interval starting at $y = \phi_0$ up to $y = \phi = \phi_0 + \phi_{crack}$. The only differences in the resulting formulas are that (a) everywhere the factor $(1-\phi)$appeared before it is now replaced by the ratio $(1-\phi)/(1-\phi_0)$,and (b) everywhere K_m and G_m appeared these material constants are replaced by $K(\phi_0)$ and $G(\phi_0)$.So, for example, equation (DEMKr11) becomes

1K_f - 1K^* = (1K_f - 1K(_0)) (1-1-_0)   When $\phi_{crack} << (1-\phi_0)$, we can rewrite this expression as

1K^* - 1K(_0) = (1K_f - 1K(_0)) (_crack1-_0)   This formula still differs from the Mavko and Jizba (1991) result, but the difference is nevertheless expected because their derivation does not assume random placement of cracks. We can resolve this discrepancy when we make use of an assumption of preferential addition of cracks in the next subsection.