In the course of analyzing a problem on fluid-dependence of shear modulus
in poroelastic systems, the author (Berryman, 2004a)
uncovered an unanticipated identity in elasticity that appears to have
wider implications for many elastic systems and/or composites.
The basic result states that for *any* hexagonal
(or transversely isotropic) elastic system there is an exact product
formula, namely, ,relating the Reuss estimate *K*_{R} of the bulk modulus
times the Voigt estimate of the uniaxial part
of the shear modulus to the product of the two system eigenvalues
for quasi-compressional and quasi-shear modes.
There is also a second product formula with the roles of the
Reuss and Voigt averages reversed, but this second identity is
somewhat less important as we shall see.

Our goal here will be to show how these facts help to remove in part (although only in one special, but nevertheless interesting, case) the asymmetry in the analysis of elastic composites resulting from the existence of Hill's well-known formula (Hill, 1963, 1964; Milton, 2002) for arbitrary elastic composites, showing that

(1) |

As always in the theory of composites,
there are several clear limitations to the use of the analysis in
practice: (a) the continuum hypothesis, (b) the implicit assumption of
adequate separation of scales between sizes of grains and of the overall
composite, and (c) an assumption of negligible porosity. The continuum
hypothesis will clearly be violated if the grain sizes are too small,
approaching nanometer sizes and below. The deviations expected in
our case are similar to those observed in deviations from the
Hall-Petch effect (Hall, 1951; Petch, 1953; Schiötz, 1998),
*i.e.*,
a softening of the composite as a function of decreasing grain size
once the size is below some threshold. This effect is caused in part
by a significant increase in grain-to-grain interface area (which
is not accounted for by the present theory) in composites when the
grains become too small. At still smaller grain sizes, atomic scale effects
become important and the continuum theory must clearly fail. At the
other extreme, if the grains are too large, then there may not be
sufficient numbers of particles in the sample for the separation of
scales
between composite and grains to be adequate. This issue is related to
the question of what is an adequate REV (representative elementary
volume) (Bear, 1972; Bourbié, 1987; Drugan and Willis, 1996).
If the grains are too large and,
therefore, too few, the entire sample may not be large enough to serve
as an adequate REV. Finally, when a polycrystal is constructed by
assembling many crystalline grains, it is also important that very
little porosity remain in the resulting polycrystal. It has been
estimated
(Berryman, 1994) that as little as 0.5% porosity in a composite is
sufficient to make it important to include the porosity in the model.
But, except to exclude it thus from consideration, porosity is not
discussed here.

The next section introduces the notation and basic results used in the rest of the paper. The third section considers the case of constant bulk modulus, and shows that the Voigt and Reuss averages for shear modulus, although differing in their numerical values, nevertheless depend on simple averages of the shear modulus plus another average comparable to (1). The fourth section considers the general problem for bounds on the moduli of random polycrystals of laminates, with special emphasis on the Peselnick-Meister-Watt bounds (Peselnick and Meister, 1965; Watt and Peselnick, 1980). The discussion of the fifth section summarizes some practical conclusions about the analysis and also makes a comparison with a ``self-consistent'' estimate related to the bounds. Two technical Appendices summarize results used in the main text.

10/23/2004