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Another invariant of an elasticity matrix is its determinant, which is
given by the product of its eigenvalues. Thus,

| |
(39) |

and so there is only one new estimate available based on this
fact. In particular, if we assume that a reasonable estimate of
*G*_{eff} might be obtained from this condition, then we would again
make use of the four known eigenvalues for shear, and the bulk modulus
for the simple layered medium. Setting
| |
(40) |

comparing to (44), and recalling that *K* = *K*_{R}, we immediately find
| |
(41) |

which is a repeat of an earlier estimate.
It is easy to show that , and, since
*G*_{eff}^{(2)} was derived from the Voigt average for shear, now we should
be able to exclude *G*_{eff}^{(4)} safely from any further consideration.

**Relationship among estimates**

These three shear modulus estimates, *G*_{eff}^{(3)}, *G*_{eff}^{(4)}, and
*G*_{eff}^{(5)}, can be related, by making use of the fact that
.The main result is

| |
(42) |

which shows first of all (since *K* > 0 and *G*_{eff}^{(3)} > 0)
that if any two of these constants are equal,
then they are all equal. Eq.(47) also shows that the
value of *G*_{eff}^{(5)} always lies between the other two estimates,
so in general ,with equality holding only in the case of isotropic composites.

** Next:** Energy estimates
** Up:** EFFECTIVE SHEAR MODULUS ESTIMATES
** Previous:** Estimates based on matrix
Stanford Exploration Project

7/8/2003