Voigt and Reuss Bounds

For hexagonal symmetry, the nonzero stiffness constants are: c₁₁, c₁₂, c₁₃ = c₂₃, c₃₃, c₄₄ = c₅₅, and c₆₆ = (c₁₁-c₁₂)/2.

The Voigt average (Voigt, 1928) for bulk modulus of hexagonal systems is well-known to be  

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

Similarly, for the shear modulus we have  

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

where the new term appearing here is essentially defined by (5) and given explicitly by  

$$G_{\rm eff}^v = (c_{11} + c_{33} - 2c_{13} - c_{66})/3.$$ (6)

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 (Reuss, 1929) K_R for bulk modulus can also be written in terms of stiffness coefficients as  

$$\frac{1}{K_R - c_{13}} = \frac{1}{c_{11} - c_{66} - c_{13}} + \frac{1}{c_{33} - c_{13}}.$$ (7)

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 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.

For each grain having hexagonal symmetry, two product formulas hold (Berryman, 2004a): $3K_RG_{\rm eff}^v = 3K_VG_{\rm eff}^r = \omega_+\omega_-/2 = c_{33}(c_{11}-c_{66})-c_{13}^2$.The symbols $\omega_\pm$ stand for the quasi-compressional and quasi-uniaxial-shear eigenvalues for the crystalline grains. Thus, it follows that  

$$G_{\rm eff}^r = K_RG_{\rm eff}^v/K_V$$ (9)

is a general formula, valid for hexagonal symmetry. We can choose to treat (5) and (8) as the fundamental defining equations for $G_{\rm eff}^v$ and $G_{\rm eff}^r$,respectively. Equivalently, we can use (9) as the definition of $G_{\rm eff}^r$.