Voigt and Reuss Bounds for Hexagonal Symmetry

For hexagonal symmetry, the nonzero stiffness constants are: C₁₁, C₁₂, C₁₃ = C₂₃, C₃₃, C₄₄ = C₅₅, and C₆₆ = (C₁₁-C₁₂)/2. We assume a vertical (i.e., 3 or z) axis of symmetry. In cases where this was not true of the numerical data, we permuted the axis definitions until it was true.

The Voigt average for bulk modulus of these hexagonal systems is well-known to be  

$$K_V = \left[2(C_{11}+C_{12}) +4C_{13}+C_{33}\right]/9.$$ (22)

Similarly, for the shear modulus we have  

$$G_V = \frac{1}{5}\left(G_{\rm eff}^v + 2C_{44} + 2C_{66}\right),$$ (23)

where the new term appearing here is essentially defined by (23) and given explicitly by  

$$G_{\rm eff}^v = (C_{11} + C_{33} - 2C_{13} - C_{66})/3.$$ (24)

The quantity $G_{\rm eff}^v$ is the energy per unit volume in a grain when a pure ``uniaxial shear'' strain of unit magnitude [i.e., $(e_{11},e_{22},e_{33}) = (1,1,-2)/\sqrt{6}$], whose main compressive strain is applied to the grain along its axis of symmetry Berryman (2004a,b).

The Reuss average for bulk modulus is determined by 1/K_R = 2(S₁₁ + S₁₂) + 4S₁₃ + S₃₃, which can also be written as  

$$\frac{1}{K_R - C_{13}} = \frac{1}{C_{11} - C_{66} - C_{13}} + \frac{1}{C_{33} - C_{13}}$$ (25)

in terms of stiffness coefficients. The Reuss average for shear is  

$$G_R = \left[\frac{1}{5}\left(\frac{1}{G_{\rm eff}^r} + \frac{2}{C_{44}} + \frac{2}{C_{66}}\right)\right]^{-1},$$ (26)

which again may be taken as the definition of $G_{\rm eff}^r$ - i.e., the energy per unit volume in a grain when a pure uniaxial shear stress of unit magnitude [i.e., $(\sigma_{11},\sigma_{22},\sigma_{33}) = (1,1,-2)/\sqrt{6}$], whose main compressive pressure is applied to a grain along its axis of symmetry.

We use the following product formula as the formal definition of $G_{\rm eff}^r$.For each grain having hexagonal symmetry, two product formulas hold (Berryman, 2004b): $3K_RG_{\rm eff}^v = 3K_VG_{\rm eff}^r = \omega_+\omega_-/2 = C_{33}(C_{11}-C_{66})-C_{13}^2$.The symbols $\omega_\pm$ stand for the quasi-compressional and quasi-uniaxial-shear eigenvalues for the crystalline grains. Thus, $G_{\rm eff}^r = K_RG_{\rm eff}^v/K_V$ is a general formula that holds for all crystals having hexagonal symmetry. We can also treat (23) and (26) as the fundamental defining equations for $G_{\rm eff}^v$ and $G_{\rm eff}^r$, respectively.