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?

5/3/2005