It is known (Grechka and Kachanov, 2006a,b,c) that the quasistatic behavior of cracked or fractured systems cannot be successfully modeled using traditional effective medium theory methods: (a) in part because most of those methods are based on energy stored in the volume of the inclusions, whereas the effects of cracks are known to be nearly independent of their volume fraction, and (b) in part because addition of more cracks results in shielding effects, thereby reducing the softening influence of all the cracks on each other and on the overall system. This reduced softening effect for multiple cracks is usually missed entirely by the traditional volume-fraction-based effective medium theories, typically based on Eshelby's analysis (Eshelby, 1957) of ellipsoidal inclusions.
To circumvent the volume fraction issue, we consider herein a model based on grains containing cracks. The grains are assembled into (for example) an isotropic polycrystal of cracked grains. (The assumed overall isotropy is not a requirement of the approach, but it does greatly simplify presentation of the modeling results.) The analysis of polycrystal behavior then proceeds using ensemble averaging and, therefore, is not limited by the lack of crack-volume sensitivity of such cracked systems. This model of polycrystals composed of cracked grains also contains within it an effect similar to the shielding effect observed in high crack-density systems. In particular, it is not difficult to show that the natural definition of the bulk modulus of an anisotropic grain is always given precisely by the Reuss average of the bulk modulus. (Imagine immersing a grain in a water bath, and then measuring total grain strain as a function of fluid pressure.) But since the same Reuss average is also the rigorous lower bound of the bulk modulus of a polycrystal composed of like grains, it is certain that a polycrystal of such grains will be hydrostatically stiffer than the grains themselves (Berryman, 2004a). We can attribute this effect to grain-to-grain bridging of the strongest components (i.e., the large volume of solid that is not cracked in the present study). The effect just described will always be present in true polycrystals, and may be contributing part of the observed ``shielding'' in cracked systems. But, we do not expect that this is the only type of crack-crack shielding present in real systems. In particular, the assumed granular structure of polycrystals also prevents various long-range connections among cracks from occurring, and thus limits the range of behaviors that can be present in the model; by assumption, cracks never intersect grain boundaries in these models, so these systems are thereby inherently constrained never to fail (either locally or globally) in the elastic regime to which the analysis is restricted.
We first provide an overview of the technical approach based on random polycrystals of cracked grains in the next section; the details of the method are contained in Appendix A. Then, two sets of numerical examples are analyzed using the polycrystal approach. To motivate the method used to fit the polycrystal analysis output to these numerical data, we also make use of the crack-influence decomposition method of Sayers and Kachanov (1991). The pertinent details of this method are presented in Appendix B. Our results and conclusions are discussed in the final section. Some additional review of pertinent effective medium theories is found in Appendix C.