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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
 (4)
Similarly, for the shear modulus we have
 (5)
where the new term appearing here is essentially defined by (5) and given explicitly by
 (6)
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 (Reuss, 1929) KR for bulk modulus can also be written in terms of stiffness coefficients as
 (7)
The Reuss average for shear is
 (8)
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.

For each grain having hexagonal symmetry, two product formulas hold (Berryman, 2004a): .The symbols stand for the quasi-compressional and quasi-uniaxial-shear eigenvalues for the crystalline grains. Thus, it follows that
 (9)
is a general formula, valid for hexagonal symmetry. We can choose to treat (5) and (8) as the fundamental defining equations for and ,respectively. Equivalently, we can use (9) as the definition of .

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Stanford Exploration Project
5/3/2005