next up previous print clean
Next: EFFECTIVE SHEAR MODULUS ESTIMATES Up: Effective bulk modulus estimates Previous: Effective bulk modulus estimates

Voigt average

The Voigt average is obtained with a constant strain assumption, and leads directly to the estimate in terms of stiffnesses
   \begin{eqnarray}
K_V = {{1}\over{9}}\left(2C_{11} + 2C_{12} + C_{33} + 4C_{13}\r...
 ...- f) + (c-f)\right]/9
= \left[9f + 4\mu_1^* + 2\mu_3^*\right]/9,
 \end{eqnarray}
where the final equality makes use of the definitions from (19). It is well-known that $K_V \ge K_{eff} \ge K_R$ [see Hill (1952)].

For an isotropic system, the bulk modulus $K = \lambda + 2\mu/3$.The results (18) and (22) obtained for Keff suggest that f plays the role of $\lambda$and that some combination or combinations of the constants $\mu_1^*$ and $\mu_3^*$ may play the role of the one nontrivial effective shear modulus Geff for both the simple layered system and for the polycrystalline system. The combinations arising here are
   \begin{eqnarray}
G_{KR} \equiv
\left[{{2}\over{3}}\left({{1}\over{\mu_1^*}}+{{1}...
 ...hbox{and}\qquad
G_{KV} \equiv \left(2\mu_1^* + \mu_3^*\right)/3,
 \end{eqnarray} (21)
the harmonic mean and mean, respectively, of $\mu_1^*$ and $\mu_3^*$after having accounted for the duplication of $\mu_1^*$ in the system. We might anticipate (incorrectly!) that these two estimates of the magnitude the remaining shear response will be, respectively, the lowest and the highest that we will find. However, in fact both these estimates usually take lower values than the ones we study more carefully in the next section.


next up previous print clean
Next: EFFECTIVE SHEAR MODULUS ESTIMATES Up: Effective bulk modulus estimates Previous: Effective bulk modulus estimates
Stanford Exploration Project
7/8/2003