CONSTANT BULK MODULUS

As a first result, consider a laminated grain composed of isotropic constituents, all having the same bulk modulus K in each layer, but differing shear moduli. Then, if we define the function [compare (1)]  

$$g(\zeta) = \left[\sum_{n=1}^{N} \frac{f_n}{\mu_n+\zeta}\right]^{-1} - \zeta,$$ (6)

we find from (3) that $G_{\rm eff}^{V} = g(\zeta)$ with $\zeta = 3K/4$.This function $g(\zeta)$ has the interesting and useful properties that  

$$c_{44} = \left<1/\mu\right\gt^{-1}\equiv g_- \le g(\zeta) \le g_+ \equiv \left<\mu\right\gt = c_{66}.$$ (7)

Furthermore, $g(\zeta)$ is a monotonic function, achieving its lower bound when $\zeta = 0$ and approaching its upper bound as $\zeta \to \infty$.This formula shows in an elementary way how $G_{\rm eff}^V = g(3K/4)$ -- and therefore $\mu_V$ -- depends on the constant bulk modulus of the system, and also that this component of the Voigt bound on the overall shear modulus increases with increasing magnitude of the bulk modulus. The overall Voigt bound/estimate (5) for shear therefore has very similar character, but the magnitude of the effect is reduced by a factor of 5, since this is only one of the five distinct contributors to the overall shear behavior of the system. So the largest change in the Voigt shear modulus that variations in bulk modulus can ever induce are expected to be on the order of 20% (or less) of the difference c₆₆-c₄₄.

Similarly, the Reuss average for shear is  

$$\mu_R = \left[\frac{1}{5}\left(\frac{1}{G_{\rm eff}^R} + \frac{2}{c_{44}} + \frac{2}{c_{66}}\right)\right]^{-1},$$ (8)

which is also a rigorous lower bound on the overall shear modulus of the polycrystal (Hill, 1952). For each hexagonal grain, the product formulas $3K_RG_{\rm eff}^V = 3K_VG_{\rm eff}^R = \omega_+\omega_-/2 = c_{33}(c_{11}-c_{66})-c_{13}^2$ are valid. The symbols $\omega_\pm$stand for the quasi-compressional and quasi-uniaxial shear eigenvalues for all the grains (Berryman, 2004). The product formulas show immediately that $G_{\rm eff}^R = G_{\rm eff}^VK_R/K_V = G_{\rm eff}^V$, since K_R = K_V = K. Thus, for this relatively simple system, pure compression or tension (e₁₁ = e₂₂ = e₃₃) is an eigenvector corresponding to stiffness eigenvalue 3K. Uniaxial shear strain (e₃₃ = -2e₁₁ = -2e₂₂) is also an eigenvector and $2G_{\rm eff}^V = 2G_{\rm eff}^R$ is the corresponding eigenvalue.