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*Reuss average for shear*

The well-known Reuss result for shear modulus is

but
| |
(31) |

again using the definitions from (19). Combining these
results, we have
| |
(32) |

Since the multiplicity of the shear modulus eigenvalues (*l* and *m*) is properly
accounted for (2 and 2, respectively, out of 5), this result strongly suggests that
one reasonable estimate of the fifth shear modulus for the system is
| |
(33) |

*Voigt average for shear*

The Voigt average for shear modulus is

[Note that (38) corrects an error in equation (69) of Berryman and
Wang (2001).]
This result shows that the combinations 2(*a*-*m*-*f*) and (*c*-*f*) again play the
roles of twice a shear modulus contribution. By analogy to (36) and
(37), we can define another effective constant
| |
(34) |

This constant has a sensible dependence on these parameters and is
consistent with the rest of our analysis (see *Discussion of
**G*_{eff} below), and also is a somewhat simpler form than (37).
Both this constant and *G*_{eff}^{(1)} reappear in the later analyses.
Because these two estimates are related to rigorous bounds, it seems
that estimates not lying in the range from *G*_{eff}^{(1)} to
*G*_{eff}^{(2)} can surely be excluded from further consideration.

** Next:** Estimates based on matrix
** Up:** EFFECTIVE SHEAR MODULUS ESTIMATES
** Previous:** Singular value decomposition
Stanford Exploration Project

7/8/2003