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Energy estimates

The energy W of the elastic system can be written conveniently in either of two ways, based on strain and stress respectively:
   \begin{eqnarray}
2W = a(e_{11}^2 + e_{22}^2) + ce_{33}^2 + 2be_{11}e_{22} +
2fe_...
 ...a_{11}\sigma_{22} +
2S_{13}\sigma_{33}(\sigma_{11}+\sigma_{22}).
 \end{eqnarray}
We can use these energy formulas to help decide whether any of the estimates obtained so far are actually fundamental quantities, by which we mean that they actually provide useful measures of the energy stored in the system. For example, it is well-known (as we have already discussed) that, if we set $\sigma_{11} = \sigma_{22} = \sigma_{33} = \sigma$, then
   \begin{eqnarray}
W = \sigma^2/2K,
 \end{eqnarray} (43)
where K is given by (20). Thus, even though K is not simply related to the eigenvalues of the system in general, it is still the fundamental measure of compressional energy in the simple layered system.

To check to see if any of the shear constants studied so far might play a similar role for shear, we can set $e_{11} = e_{22} = -e_{33}/2 = e/\sqrt{6}$.Then, we find that

 
<I>WI> = 2[(<I>aI> - <I>mI> - <I>fI>) + (<I>cI>-<I>fI>)]<I>e11I>2 = <I>GI><I>effI>(2)<I>e2I>.      (44)
So there is no ambiguity in the result for the shear energy. Clearly, Geff(2) plays the same role for shear energy that K plays for bulk energy in this system, again regardless of the fact that it is not simply related to the eigenvalues.


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