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To place our method using random polycrystal analysis in perspective, we will now give a brief review and discussion of other effective medium theories that have often been applied to the same types of crack-related problems treated here.

A very good review of many (but not all) of the effective medium theories we will mention here has already been given by Zimmerman (1991). To keep the present discussion as brief as possible, we will just provide an overview here, but no equations. In Zimmerman's treatment of the methods, he does include the equations and therefore much of what is missing here can be found in Zimmerman (1991), and the remainder is found in the other references given.

One of the main advantages of the random polycrystals of cracked-grains model is that it provides a definite means of building in a type of microstructure that is usually not possible to obtain using other effective medium theories. This microstructure arises naturally in the polycrystal analysis because we must first construct a grain, using an initial estimate of the effect of the crack (or cluster of cracks) on the grain. Then, we imagine jumbling these cracked grains together to arrive at the overall polycrystalline microstructure. So this approach involves two distinct steps of upscaling: first at the grain level, and then for the overall fractured composite.

The presence of two, or even many, steps of upscaling in effective medium theories is not at all unusual. The best examples of this are the differential schemes (Bruner, 1976; Henyey and Pomfrey, 1982; Norris, 1985; Zimmerman, 1984; 1991; Berryman et al., 2002) in which we take a pristine elastic medium and imbed a very small amount of some inclusion (or a small crack in our case). Then, we use an integration method to deduce what the global effect of this small volume or small crack density will be on the overall medium. The stated procedure is already one level of upscaling. Then, we do this over and over again in the differential scheme, each time starting from the result of the last upscaling step. So small quantities of new inclusions or cracks get imbedded in a medium that already has inclusions or cracks, and so on. Certain of these differential schemes are realizable (Norris, 1985; Avellaneda, 1987), and therefore never violate rigorous bounds.

In contrast, traditional self-consistent schemes (O'Connell and Budiansky, 1974) and also more modern types of self-consistent scheme (Berryman, 1980) that are some times termed the ``coherent potential approximation'' (or CPA) also achieve their final results using multiple steps of upscaling, but this happens at a fixed target volume fraction through an iterative method: we start with coupled equations that depend on constituent properties, volume fractions, and also on the overall properties (usually overall bulk and shear modulus); then these equations are provided with some initial guess of the overall property values, and subsequently iterated until they converge to a definite result. This iteration process itself can be viewed as being very similar to the differential scheme in the sense that each new iteration is using the result of previous (approximate) upscaling steps to generate the next upscaling, until final convergence is achieved. We can also think of this procedure as providing a type of scale-separation at each iteration (Milton, 1985).

One important difference between these two schemes (differential and self-consistent) is that the differential scheme usually starts with one material as the host, and so that host remains connected throughout the integration process. In contrast, the self-consistent scheme treats two or more constituents equally, with no one of them necessarily playing the role of host (but of course if one has significantly higher volume fraction than all the others, then a host-inclusion type of microstructure will naturally arise).

In addition to these implicit schemes (requiring integration or iteration), there are also some explicit schemes: Mori-Tanaka (Benveniste, 1987), Kuster-Toksöz (1974), and the non-interaction approximation (Zimmerman, 1991). Explicit schemes provide formulas: needing numerical evaluation, but not needing either integration or iteration. However, these schemes are known not to be so reliable for very high concentrations of inclusions, and, furthermore, they can lead to incorrect results, such as violations of known rigorous bounds such as the Hashin-Shtrikman bounds when the inclusion shapes are extreme (i.e., differing greatly from spheres). So care must be exercised when using these methods (-- and also the other methods as well). But the CPA (Milton, 1985) and the differential scheme due to Norris (1985) [see also Avellaneda (1987)] are known not to violate the Hashin-Shtrikman bounds, and therefore tend to be fairly reliable estimators, depending on the application and the particular microstructure that one is trying to emulate.

So, among all of the effective medium theories mentioned, only the explicit schemes use just a single upscaling step to arrive at their elastic constant estimates, and this one step may not be trustworthy if the volume fractions of the inclusions, or crack-density of the cracks, is too large (Berryman and Berge, 1996).

In TABLE 2 we show the results found by applying the non-interaction (NI) approximation, the Norris differential scheme (DS), the coherent potential approximation (CPA) and the traditional self-consistent (SC) scheme as applied by O'Connell and Budiansky (1974). We find (as expected) that all these methods give very comparable results for the Sayers and Kachanov (1991) crack-influence decomposition parameters $\eta_1$ and $\eta_2$ at low crack densities. These results suggest that it is entirely appropriate to use the NIA when making the first upscaling step for estimating the properties of a typical cracked grain.

The method proposed here for the random polycrystal of cracked-grains model is then a two-step process: The first step has been chosen to be a non-interaction approximation based on low crack density results from the theory. This step gives us the effective elastic behavior of an average cracked grain. Once we have this compliance in hand, our second upscaling step uses the polycrystal analysis to provide Voigt and Reuss bounds, and also self-consistent estimates. We show further results for Hashin-Shtrikman bounds in the figures in the text, but unfortunately these results show that HS bounds are too tight for these applications, and therefore it is inferred that assumptions implicit in the derivation of HS bounds have been violated. Note that the second step involving Voigt and Reuss bounds is also entirely explicit, while the alternative second step involving the self-consistent estimates for polycrystals is implicit (requiring iteration). So the random polycrystal bounding approach should be very appealing to those users who have a strong preference for explicit methods.

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