We began the paper by pursuing the differential effective medium predictions for the bulk and shear moduli in a cracked material in which the cracks can be either gas-saturated (dry) or liquid-saturated. The DEM equations can be integrated numerically without serious difficulty for the exact model of oblate spheroids of arbitrary aspect ratio, but the full formulas for oblate spheroids are rather involved. In order to make progress on analytical expressions, part of the effort was directed towards study of the penny-shaped crack model of Walsh (1969). This model is not too difficult to analyze if an additional approximation is entertained. The problem for analysis is that the ordinary differential equations for bulk and shear moduli are coupled. If they can be decoupled either rigorously or approximately, then they can be integrated analytically. We accomplished the decoupling for the penny crack model by assuming that changes in Poisson's ratio occurring in those terms proportional to the aspect ratio are negligible to first order. This permits the decoupling to occur and the integration to proceed. We could subsequently check the analytical results against the full DEM integration for penny-shaped cracks, which showed that the analytical results were in quite good agreement with the numerical results.
To attempt to understand why the analytical results worked so well, we studied the behavior of Poisson's ratio for the same system, and found that, as the porosity increases, for the dry systems Poisson's ratio tends to a small positive value on the order of , where is the aspect ratio, and for liquid saturated systems it tends towards 1/2 in all cases. These results permit error estimates for the analytical formulas showing that errors will always be less than about 5%-20%, depending on the aspect ratio and the porosity value.
We have also shown that the Mavko and Jizba (1991) proportionality factor of 4/15 relating the differences in shear compliances to the differences in bulk compliances for cracked systems is an upper bound and that this upper bound is approximately achieved for . The proportionality factor decreases monotonically with increasing aspect ratio of oblate spheroids, and vanishes identically for spheres at .