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The results obtained for self-consistent estimates
can be written in many different ways (Berryman, 2005). We take the self-consistent estimate for bulk modulus to be

| |
(34) |

where
| |
(35) |

In (35), *K*^{*} is determined by (34), depending
also on *G*^{*};
*G*^{*} is determined by the self-consistent expression for the
shear modulus to follow, also depending on *K*^{*};
and is then determined by (35).
The final result for *G*^{*} = *G*^{*}_{hex} in polycrystals having grains with
hexagonal symmetry is
| |
(36) |

These formulas can be successfully solved by iteration, starting for
example by using values corresponding to upper or lower bounds for the
values of *K*^{*} and *G*^{*}.
Some details of the derivation of these formulas can be found in
Willis (1977, 1981) and Berryman (2005).

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** Up:** Appendix A: Bounds and
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Stanford Exploration Project

1/16/2007