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It has been shown elsewhere (Berryman, 2004a,b)
that the Peselnick-Meister-Watt
bounds for bulk modulus of a random polycrystal composed of
hexagonal (or transversely isotropic) grains are given by
| |
(10) |

where () is the uniaxial shear energy per
unit volume for a unit applied shear strain (stress).
The second equality follows directly from the product formula
(9). Parameters are defined by
| |
(11) |

In (11), values of (shear moduli of isotropic
comparison materials) are determined by inequalities
| |
(12) |

and
| |
(13) |

The values of (bulk moduli of isotropic
comparison materials) are then determined by equalities
| |
(14) |

given by Peselnick and Meister (1965) and Watt and Peselnick (1980). Also see Berryman, 2004b).
Bounds on the shear moduli are then given by

| |
(15) |

where and are given by
| |
(16) |

*K*_{V} is the Voigt average of the bulk modulus as defined previously.

** Next:** POROELASTICITY ESTIMATES AND BOUNDS
** Up:** BOUNDS ON ELASTIC CONSTANTS
** Previous:** Voigt and Reuss Bounds
Stanford Exploration Project

5/3/2005