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Appendix A: Bounds and estimates for random polycrystals of grains having known symmetry

Although the numerical results for stiffness matrices of randomly cracked systems in the experiments of Grechka and Kachanov (2006a,b,c) generally show orthorhombic symmetry, in fact the symmetry has also been observed to be very close to hexagonal in all cases. The hexagonal symmetry is especially pertinent for these numerical experiments when the averaging approach of Huet (1990) is used as in (1) where it was found that the results obtained from sampling and then averaging these numerical stiffness matrices tend to be very nearly hexagonal stiffnesses. For the present purposes, after averaging to obtain CijSMP according to (1), the next step has been to find the closest hexagonal stiffness matrix for these averaged matrices by using a least-squares fitting method. The resulting hexagonal matrix is the one used to compute KSMP and GSMP in the examples. These results were also used in the fitting methods to obtain various bounds and estimates when introducing quadratic corrections to the very low crack density results.

Also, note that the step taken to reduce the matrices to hexagonal symmetry was not necessary for computing the VRH estimates for the individual stiffness matrices originally obtained from the numerical experiments. KVRH and GVRH can both be computed easily for these stiffnesses using the formulas given by Watt (1979) for orthorhombic symmetry. But, after the averaged matrices CijSMP were computed as in (1), these matrices were slightly ``smoothed'' (in addition to the averaging) so the results had exactly hexagonal symmetry. The least-squares fitting method producing these smoothed hexagonal results typically amounted to a change by just a single unit in the third significant digit of only one of the nine distinct -- for orthotropy -- matrix components.

Figure 1 illustrates the type of geometry envisioned for the random polycrystals of cracked-grains model.