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For hexagonal symmetry, the nonzero stiffness constants are:
*C*_{11}, *C*_{12}, *C*_{13} = *C*_{23}, *C*_{33}, *C*_{44} = *C*_{55}, and
*C*_{66} = (*C*_{11}-*C*_{12})/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

| |
(22) |

Similarly, for the shear modulus we have
| |
(23) |

where the new term appearing here is essentially defined by
(23) and given explicitly by
| |
(24) |

The quantity is the energy
per unit volume in a grain when a pure ``uniaxial shear'' *strain* of
unit magnitude [*i.e.*,
], 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(*S*_{11} + *S*_{12}) + 4*S*_{13} + *S*_{33}, which can also be
written as

| |
(25) |

in terms of stiffness coefficients. The Reuss average for shear is
| |
(26) |

which again may be taken as the definition of -
*i.e.*, the energy per unit volume in a grain when a pure uniaxial
shear *stress* of unit magnitude [*i.e.*,
],
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 .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.

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Stanford Exploration Project

1/16/2007