A wide array of results is available for practical studies of the linear elastic constants of composite solid and/or granular materials, fluid suspensions, and emulsions. These results range from rigorous bounds such as the Voigt (1928), Reuss (1929), Hill (1952), and Hashin-Shtrikman (1962; 1963) bounds to the fairly popular and mostly well-justified [for sufficiently small concentrations of inclusions (Berryman and Berge, 1996)] approximate methods such as the explicit approximations of Kuster and Toksöz (1974) and Mori and Tanaka (Benveniste, 1987; Ferrari and Filiponni, 1991) and the implicit methods such as the differential effective medium (DEM) method (Cleary et al., 1980; Norris, 1985) and the self-consistent (Hill, 1965; Budiansky, 1965) or the coherent potential approximation for elastic composites (Gubernatis and Krumhansl, 1975; Korringa et al., 1979; Berryman, 1980; 1982). Older reviews (Watt et al., 1976) and both early (Beran, 1968; Christensen, 1979) and more recent textbooks and research monographs (Nemat-Nasser and Hori, 1993; Cherkaev, 2000; Milton, 2002; Torquato, 2002) survey the state of the art. So it might seem that there is little left to be done in this area of research. However, continuing problems with applications of these methods have included lack of sufficient information [such as the required spatial correlation functions (Torquato, 1980; Torquato and Stell, 1982; Berryman, 1985a)] needed to compute the most accurate bounds known and the failure of some of the explicit methods to satisfy the rigorous bounds in some limiting cases such as three or more constituents (Norris, 1989) or extreme geometries such as disk-like inclusions (Berryman, 1980). The best implicit schemes, even though they are known to be realizable and therefore cannot ever violate the bounds, are often criticized by some workers (Christensen, 1990) because the microgeometry generated implicitly by these methods does not represent the true microgeometry with any obvious fidelity. Nevertheless, it has been shown (Berge et al., 1993; 1995) that knowing general features of the microgeometry such as whether one constituent can be classified as the host medium and others the inclusions, or whether in fact there is no one constituent that serves as the host can be sufficient information to decide on a model that can then be used successfully to study a class of appropriate composites (Berge et al., 1993; 1995; Garboczi and Berryman, 2000; 2001). Some critics also point out that the iteration or integration schemes required to compute the estimates for implicit schemes are sufficiently more difficult to implement than those of the explicit methods that workers are often discouraged from trying these approaches for this reason alone.
Virtually all of the improved bounds (i.e., improved beyond the bounds of Hashin and Shtrikman, which do not make direct use of microstructural information except for the volume fractions) require some information about the microstructure. But it has not been very clear just how precisely this information needs to be known in order for it to be useful. The present work will show for several examples how some general knowledge of microstructure can be used in more than one way to generate estimates. And since the predicted properties (at least in some cases) do not seem to depend too strongly on details beyond those readily incorporated, it gives some confidence that the methods can be successfully applied to real materials. One comparison we can make is between predictions and bounds on elastic constants for random polycrystals of laminates and the predictions of improved bounds based on spatial correlation functions for disks. It is clear that these models should both apply at least approximately to the same types of random composites, yet the microstructure is assumed to be organized rather differently. The random polycrystal is an aggegrate of grains, each of which is a laminate material. These laminated grains are then jumbled together with random orientations so the overall composite is isotropic, even though the individual grains act like crystals having hexagonal symmetry. The improved bounds for composites with disk-shaped inclusions must have a microstructure that is at least crudely the same as the random polycrystal, since each layer of an individual grain could be seen as approximately disk-like. So one quantitative question we can ask is: How closely do these two models agree with each other, and if they are indeed close in value, what do we learn about the sensitivity of elastic constants to microstructure? Also, how does this information affect engineering efforts to design (Cherkaev, 2000; Torquato, 2002) new materials? Or, how does general knowledge of the geology of a given region help us to choose good models of the rocks when we need to interpret our seismic data?