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Estimates based on matrix invariants 1.Trace estimates

The trace of the stiffness matrix C is an invariant, and equals the sum of its eigenvalues. Similarly, the trace of the compliance matrix S is also invariant, and equals the sum of its eigenvalues, which are the inverses of the eigenvalues of C. These facts provide two more ways of obtaining estimates of the Geff we seek.



Trace estimate from S

After eliminating the four eigenvalues associated with simple shear, the remainder of the trace of S is just the sum of the inverses of $\omega_+$ and $\omega_-$. To obtain an estimate of Geff, we again make use of the known bulk modulus K and set
   \begin{eqnarray}
{{1}\over{3K}} + {{1}\over{2G_{eff}^{(3)}}} = {{1}\over{\omega_+}} + 
{{1}\over{\omega_-}},
 \end{eqnarray} (35)
which can easily be shown to imply that
   \begin{eqnarray}
G_{eff}^{(3)} = \left(\mu_1^* + 2\mu_3^*\right){{K_R}\over{3K_V}} = G_{eff}^{(1)}.
 \end{eqnarray} (36)
So this compliance estimate again produces the same result found earlier in (37).



Trace estimate from C

Again eliminating the four eigenvalues associated with simple shear, the remainder of the trace of C is just the sum of $\omega_+$ and $\omega_-$. To obtain another estimate of Geff, we make use of the known bulk modulus K as before and set
   \begin{eqnarray}
3K + 2G_{eff}^{(4)} = \omega_+ + \omega_-.
 \end{eqnarray} (37)
After some manipulation, we find
   \begin{eqnarray}
2G_{eff}^{(4)} = {{2(a-m-f)^2+(c-f)^2}\over{(a-m-f)+(c-f)}} = 
2{{(\mu_1^*)^2 + 2(\mu_3^*)^2}\over{\mu_1^* + 2\mu_3^*}}.
 \end{eqnarray} (38)
So this estimate does not agree with any of the others, but it is nevertheless an interesting new combination of the shear modulus measures (a-m-f) and (c-f).


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