INTRODUCTION

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, $6K_R G_{\rm eff}^{V} = \omega_+\omega_-$,relating the Reuss estimate K_R of the bulk modulus times the Voigt estimate $G_{\rm eff}^V$ of the uniaxial part of the shear modulus to the product of the two system eigenvalues $\omega_\pm$ 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  

$$K^*= \left[\sum_{n=1}^{N} \frac{f_n}{K_n+4\mu/3}\right]^{-1} - 4\mu/3.$$ (1)

Here the bulk modulus of the n-th constituent is K_n, the shear modulus takes the same value $\mu_n = \mu$ for all $n = 1,\ldots,N$,and the overall effective bulk modulus is K^*. The volume fractions f_n are all nonnegative, and add up to unity. In general there is in fact no corresponding relationship for the overall shear modulus $\mu^*$, when instead the system has constant bulk modulus K_n = K for all N constituents. But, nevertheless, the existence of formulas quite analogous to (1) for shear will be demonstrated for a model random polycrystal composed of laminated grains.

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.