DISCUSSION AND CONCLUSIONS

Standard effective medium theories (differential scheme, self-consistent scheme, etc.), when applied to fractured systems, all overpredict the influence of the cracks. Of all the standard effective medium theories, the best one from this point of view is clearly the non-interaction approximation (NIA) because it assumes there is no influence of one crack on any other, while -- with this one exception -- all the other effective medium theories predict that multiple cracks enhance the effects of each other. But numerical experiments show that the actual behavior is the opposite; when cracks interact, they screen or shield each other and reduce the overall effects of the cracks that are present. So NIA is the best one within this restricted class of methods in the sense that, unlike all the others, it does not go in the wrong direction at higher crack densities.

The main question then becomes whether or not there is some simple theoretical method that gives the right correction, i.e., introduces crack shielding and, therefore, stiffer (less compliant) results than all these other methods.

This question has been answered in the affirmative in this paper by introducing and studying the quasistatic behavior of the random polycrystals of cracked-grains model. Each grain has one or more cracks and is therefore anisotropic, having either hexagonal or perhaps more generally orthotropic symmetry. But each of these grains will tend to be weakest in the direction normal to its main crack, and strongest in the two orthogonal directions defined by the plane of the main crack (or cracks). So, when these cracked grains are jumbled together into an isotropic composite medium, the overall effect includes a high proportion (2 to 1 ratio) of strong directions interacting with the strong directions of other grains. This interaction ultimately produces bridging across the entire composite material and is a guaranteed result for bulk modulus (hydrostatic) behavior since the bulk modulus of each grain is determined exactly by its Reuss average (and therefore the weakest possible combination of elastic constants), while the bulk behavior of the composite polycrystal is dominated instead by the strongest connected paths passing throughout the whole composite. From this point of view, for small to moderate values of crack density, we see that fractures/cracks exist as isolated weak patches in an otherwise comparatively strong background framing material. When fractures/cracks intersect only locally, this basic picture does not change qualitatively. But clearly if the crack density increases to the point where cracks start coalescing and spanning the entire composite, then their effect is multiplied and the composite medium can become fragmented thus basically having no strength overall under such circumstances. We have assumed here that all the cases we were studying are far from this extreme limit of the theory, and of course that is also consistent with the types of numerical data used in our comparisons with the theory.

We conclude that the random polycrystals of cracked-grains model is a useful approach to the modeling of small to moderate crack density (i.e., $0 < \rho \le 0.1$) materials. The method involves established formulas from the theory of composites including both bounds and self-consistent estimates for polycrystals. This approach has apparently not been applied to this class of problems previously. Results for shear modulus prediction were found to be especially good, as the simplest method tried actually worked quite well in both of the numerical modeling experiments considered here. Getting the theory to agree simultaneously with the bulk and shear behavior required the introduction of some quadratic corrections, whose coefficients were treated as fitting parameters. This approach is consistent with the crack-influence decomposition method of Sayers and Kachanov (1991). One quadratic fitting parameter was sufficient to produce a good fit when Poisson's ratio $\nu_0 = 0.0$. Only one parameter was also used at first for the case $\nu_0 = 0.4375$. But the results were not an unqualified success in this case. Actual fitting to the numerical data was clearly better for the first model, but fitting was surely adequate for most practical purposes in both cases. Clearly the basic trends have been captured using this approach, involving only one fitting parameter.

In the cases considered, the Hashin-Shtrikman bounds were always too close together to give any useful estimate of the range of scatter in the numerical data for the cracked media. (This fact shows that the assumptions implicit in the derivation of the HS bounds have been violated in the random crack problem.) On the other hand, the differences between Voigt and Reuss bounds gave reasonably accurate estimates of the range of values seen in the numerical data. In one example, the actual highest and lowest values for both bulk and shear modulus were also well-constrained. These results depend to some extent on choosing appropriate fitting parameters for the corrections quadratic in crack density. The self-consistent estimate for the polycrystal analysis lies approximately in the middle of the region bounded by the Voigt and Reuss bounds, and so seems again to be a useful estimate - subject however to the same caveats (of fitting) as for the Voigt and Reuss bounds. A more satisfying result would have been establishing some means of predicting the values of these coefficients (i.e., the $\eta$'s) of the quadratic fitting terms, but doing so in a meaningful way will require more numerical data than are available at the present time and/or some new theoretical analysis of this problem.

This study was clearly limited in its potential scope by the quantity of numerical data available. In particular, it would have been helpful to have some intermediate values of Poisson's ratio $0.0 < \nu_0 < 0.4375$, so that the dependence of the quadratic fitting parameters on Poisson's ratio could be determined. With just two values of $\nu_0$, this goal could not be achieved here as the simplest possible fit (and really the only fit possible with just two data points) is clearly a straight line -- but this fact by itself does not contain any useful information. Work currently in progress will address these, as well as other related, issues concerning quality of fit, and number of parameters needed to achieve good fits across a greater selection of Poisson's ratios.