EXAMPLES

We now consider some applications of these formulas. We take quartz as the host medium, having K_m = 37.0 GPa and G_m = 44.0 GPa. Poisson's ratio is then found to be $\nu_m = 0.074$.

For liquid saturation, the shear modulus goes to zero as the crack volume fraction increases, while the bulk modulus approaches the bulk modulus of the saturating liquid, which we take as water here (K_f = 2.2 GPa). This means that the effective value of Poisson's ratio increases towards $\nu^* = 0.5$ as the crack volume fraction increases, and thus the approximation that $\nu^*$ is constant clearly does not hold for this case. We therefore expect that the greatest deviations of the analytical approximation should occur for the case of liquid saturation.

In contrast, for the dry case, both shear modulus and bulk modulus tend towards zero as the crack volume fraction increases. Thus, since the trends for both moduli are similar, the approximation of constant Poisson ratio might hold in some cases, depending whether bulk and shear moduli go to zero at similar or very different rates with increasing crack volume fraction.

We consider three cases in Figures 1-6: (1) $\alpha = 0.1$ for Figures 1 and 2. (2) $\alpha = 0.01$ for Figures 3 and 4. (3) $\alpha = 0.001$ for Figures 5 and 6. The first two cases are easily integrated for DEM. We use two Runge-Kutta schemes from Hildebrand (1956): equations (6.13.15) and (6.14.5). When these two schemes give similar results to graphical accuracy, we can be confident that the step size used is small enough. If they differ or if either of them does not converge over the range of crack volume fractions of interest, then it is necessary to choose a smaller step size for integration steps. We found that a step size of $\Delta y = 0.01$ was sufficiently small for both $\alpha = 0.1$ and $\alpha = 0.01$, while it was necessary to decrease this step size to $\Delta y = 0.001$ for the third case, $\alpha = 0.001$.(Still smaller steps were used in some of the calculations to be described later.)

The results show that our expectations for the agreement between the analytical and numerical results are in concert with the results actually obtained in all cases. The analytical approximation gives a remarkably good estimate of the numerical results in nearly all cases, with the largest deviations occurring -- as anticipated -- for the intermediate values of crack volume fraction in the cases of liquid saturation for the bulk modulus estimates. We consider that the results of Figures 1-6 are in sufficiently good agreement that they provide cross-validation of both the numerical and the analytical methods.

For the saturated case, we anticipated little if any deviation for the bulk modulus between the analytical results and the full DEM as is observed for $\alpha = 0.1$ and $\alpha = 0.01$. Larger deviations are found for $\alpha = 0.001$.We also observed the anticipated small deviations for the shear modulus between the analytical formula and the full DEM.

Note that Gassmann's predictions for bulk modulus are in very good agreement with the numerical DEM results for saturated cracks and $\alpha = 0.001$.

For the dry case, we anticipated that the analytical shear modulus formula would be a somewhat better approximation of the full DEM, than that for the bulk modulus. Both approximations were expected to be quite good. These results are also observed in the Figures.