DISCUSSION

The results obtained so far show that, for the shear modulus $G_{\rm eff}$of uniaxial shear for a transversely isotropic system, we have $2G_{\rm eff}^V = \omega_+$ when the bulk modulus of the system is uniform. In this case, the quasi-shear eigenvector is exactly in the same direction as the uniaxial shear component, so the quantity $2G_{\rm eff}^V$ -- while more generally a strict upper bound on the eigenvalue $\omega_+$ -- is exactly equal to it in this special case. Thus, the uniaxial shear mode is in this instance an eigenvector of this system. This happens in particular when K_n = K is a constant for random polycrystals of laminates. The simplified formula (13) for the bounds is therefore the main new result of this paper. When compared to (8), it is suggestive that some very simple forms for Hashin-Shtrikman bounds on shear can probably be found for many such polycrystalline systems, and especially so for granular laminates. The constant bulk modulus limit is a most convenient place to begin a search for such simplified expressions for the bounds.

Once these HS bounds are known, it is an elementary operational exercise to determine self-consistent (SC) estimates based just on the analytical form of the bounds. Monotonicity of the functional  

$$M(\zeta) = \left[\frac{1}{5}\left(\frac{1}{G_{\rm eff}^V+\ze... ...+\zeta} + \frac{2}{c_{66}+\zeta}\right)\right]^{-1} - \zeta,$$ (15)

appearing in (13), is easy to prove [see Berryman (1982)] for examples of such proofs), and furthermore $\zeta(K,G) = (G/6)(9K+8G)/(K+2G)$ is a monotonic functional of both arguments. These facts guarantee that there is a unique solution to the self-consistency relation  

$$\mu_{SC} \equiv M(\zeta(K,\mu_{SC})),$$ (16)

and, furthermore, this solution always lies between the bounds. To provide an example, consider the case of Figure 4 when the volume fractions are both 50%. Then, $\mu_{44} = 7.2727$,$\mu_{66} = 22.0000$, $G_{\rm eff} = 16.5546$,$\mu_{PM^-} = 13.1164$, $\mu_{PM^+} = 13.8659$,and $\mu_{SC} = 13.5537$. So the self-consistent estimate is not closely correlated with the value of $G_{\rm eff}$, which is itself usually found outside the correlated bounds on $\mu$.Figure 5 illustrates these results for the full range of volume fractions with the same choice of constituents.

 

mucomparison Figure 5 Comparison of the shear modulus estimates over all choices of volume fraction, for the same case considered in Figure 4.

The results in Fig. 5 show very clearly that self-consistent values fall between the bounds as expected, and that the bounds themselves are in any case very close together for this high contrast example. Thus, an exact result for shear modulus has not been found [so the analogy to Hill's formula (1) is not perfect]. Nevertheless, for most practical purposes, the results show that the predictions of the theory using such correlated bounds -- and related self-consistent estimates -- will often be as good as, or perhaps better than, the precision of experimental measurements. (Maximum error incurred by using the self-consistent estimate in the example of Figure 5 is about 2%.) The value of $G_{\rm eff}$, while playing an important role in the analysis, clearly should not be interpreted as the actual value of the effective overall shear modulus for the random polycrystal. $G_{\rm eff}$ does however contribute about 20% of the overall magnitude of the effective shear modulus.

In conclusion, we note that, applications of the analytical results presented here include benchmarking of numerical procedures used for studying elastic behavior of complex composites, as well as estimating coefficients needed in up-scaled equations for elasticity and/or poroelasticity of heterogeneous systems. In particular, up-scaling methods typically determine the form of the effective equations of motion, but most often do not provide any means (or at least any very useful means) of estimating/computing the elastic/poroelastic coefficients. The methods described here are therefore expected to be especially useful for earth sciences and oil reservoir engineering applications, as well as for obvious uses in the practice and theory of elastic composites and heterogeneous media.