For elasticity, we considered various improvements on the Hashin-Shtrikman bounds such as the Beran-Molyneux bounds, the McCoy-Silnutzer bounds, and the Milton-Phan-Thien bounds. We found that knowledge of microstructure can be used very effectively in these improved bounds. New estimates (not themselves rigorous bounds) can also be formulated based on the analytical structure of the bounds, and the microstructure parameters can then be incorporated directly into these estimates in a way so the resulting estimates always satisfy the bounds. When making comparisons between models based on disk-like inclusions in a host medium, and the random polycrystals of laminates model, we found that these models predict very similar results when there is a relatively small volume fraction of disks present. But when the volume fraction of disks is large, the bounds do not constrain the results as well, and so there is still more work to be done relating constants to microstructure in the mid-range of volume fractions, and generally for high contrast problems.

For electrical conductivity and other related physical constants such
as thermal conductivity and dielectric constant (and in some cases fluid
permeability), the microstructure can be introduced not only through
the volume fractions and
microstructural parameters as was done in the case of elasticity, but also
through the use of more global measures of microstructure such as the
formation factors. Global measures like the *F*_{i}'s that determine the
long-range spatial correlations and connectivity
(within our material object of study) -- by means of two fairly
common and relatively simple measurements -- are
very advantageous and clearly more information of this type
is desirable. The case of high contrast composites is always very important
for all types of conductivity estimation and so formation factor
bounds and Padé approximant schemes both provide convenient
means of addressing this problem. The formation factor bounds are
elementary in mathematical structure, but nevertheless provide
very useful lower bounds on conductivity
and permittivity for high contrast problems.

One general observation is that behavior of high contrast mixtures
and composites remains poorly
constrained by most of the methods presented, and more work in that direction
is therefore still needed. A typical example that always generates
high contrast situations is porous and/or granular media, where the
pores may be filled with air; then, both the mechanical and the
transport properties can have a very wide range of variation depending
on the details of the microstructure. Some of the same types of
information (such as formation factors) used here
for studies of transport properties can also be applied
to elasticity estimates in porous media as has been pointed out previously in
studies of ``cross-property'' relationships and bounding methods
(Berryman and Milton, 1988; Gibiansky and Torquato, 1995)
- *i.e.*, estimating one physical quantity after measuring another.
So, one possibility for future progress
that has yet to be explored in very great detail is
how the formation factor bounds as well as other improved bounds
on electrical or thermal conductivity may provide useful information
about microstructure that can then be used to constrain further
the elastic behavior of the same system.

5/3/2005