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Estimates based on matrix invariants 2.Determinant estimates

Another invariant of an elasticity matrix is its determinant, which is given by the product of its eigenvalues. Thus,
   \begin{eqnarray}
\det{C} = \omega_+\omega_-(2l)^2(2m)^2 \qquad\hbox{and}\qquad
\det{S} = [\det{C}]^{-1},
 \end{eqnarray} (39)
and so there is only one new estimate available based on this fact. In particular, if we assume that a reasonable estimate of Geff 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
   \begin{eqnarray}
\det{C} = (3K)(2G_{eff})(2l)^2(2m)^2,
 \end{eqnarray} (40)
comparing to (44), and recalling that K = KR, we immediately find
   \begin{eqnarray}
2G_{eff}^{(5)} \equiv {{\omega_+\omega_-}\over{3K}} = 2G_{eff}^{(2)},
 \end{eqnarray} (41)
which is a repeat of an earlier estimate.

It is easy to show that $G_{eff}^{(5)} \le G_{eff}^{(4)}$, and, since Geff(2) was derived from the Voigt average for shear, now we should be able to exclude Geff(4) safely from any further consideration.





Relationship among estimates

These three shear modulus estimates, Geff(3), Geff(4), and Geff(5), can be related, by making use of the fact that $\omega_+ + \omega_- = \omega_+\omega_-(1/\omega_+ + 1/\omega_-)$.The main result is
   \begin{eqnarray}
{{G_{eff}^{(5)}-G_{eff}^{(3)}}\over{2G_{eff}^{(3)}}} =
{{G_{eff}^{(4)}-G_{eff}^{(5)}}\over{3K}},
 \end{eqnarray} (42)
which shows first of all (since K > 0 and Geff(3) > 0) that if any two of these constants are equal, then they are all equal. Eq.(47) also shows that the value of Geff(5) always lies between the other two estimates, so in general $G_{eff}^{(4)} \ge G_{eff}^{(5)} \ge G_{eff}^{(3)}$,with equality holding only in the case of isotropic composites.


next up previous print clean
Next: Energy estimates Up: EFFECTIVE SHEAR MODULUS ESTIMATES Previous: Estimates based on matrix
Stanford Exploration Project
7/8/2003