It is inherent in the mathematical form of all DEM schemes that they always give correct values and slopes of the curves for small values of the inclusion volume fraction, and that they always give the right values (but not necessarily correct slopes) at high volume fractions. We see that these expectations are fulfilled in all the examples shown here.
The approximations made in the text to arrive at analytical results were chosen as a convenient means to decouple the equations for bulk and shear moduli, which are normally coupled in the DEM scheme. For the liquid saturated case, the approximations for bulk modulus are very good for all values of aspect ratio, but for shear modulus the exponent determined by (cdef) can deviate as much as a factor of 2/3. The value chosen is the maximum value possible, guaranteeing that the analytical approximation will always be a lower bound for this case.
In contrast, for the case of dry cracks, the approximations for the shear modulus are expected to be somewhat better than those for the bulk modulus. The analytical approximation is again expected to be a lower bound for the full DEM result for the shear modulus. Analysis for the bulk modulus is more difficult in this limit as it requires checking that the ratio G*/K* remains finite as the porosity , and this would be difficult to establish if Poisson's ratio were going to , as it does for the liquid saturated case. But, Appendix B shows that Poisson's ratio actually tends to a value of about , so there is no singularity in the K* behavior for this case. This feature is also confirmed by the numerical results.