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
The Reuss average for bulk modulus is determined by 1/KR = 2(S11 + S12) + 4S13 + S33, which can also be written as
We use the following product formula as the formal definition of .For each grain having hexagonal symmetry, two product formulas hold (Berryman, 2004b): .The symbols stand for the quasi-compressional and quasi-uniaxial-shear eigenvalues for the crystalline grains. Thus, 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 and , respectively.