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Bounds for polycrystals

Reuss average for shear

The well-known Reuss result for shear modulus is
   \begin{eqnarray}
G_R^{-1} = {{1}\over{15}}\left(8S_{11} - 4S_{12} + 4S_{33} - 8S...
 ...\over{m}} + 
{{9f + 4(a-m-f) + (c-f)}\over{(a-m)c-f^2}}
\right),
 \end{eqnarray}
but
   \begin{eqnarray}
(a-m)c-f^2 = \left[(a-m-f) + (c-f)\right]K_R =
[\mu_1^* + 2\mu_3^*]K_R,
 \end{eqnarray} (31)
again using the definitions from (19). Combining these results, we have
   \begin{eqnarray}
G_R^{-1} = {{1}\over{5}}\left({{2}\over{l}} + {{2}\over{m}}
+ \left[\mu_1^* + 2\mu_3^*\right]^{-1}{{3K_V}\over{K_R}}\right)
 \end{eqnarray} (32)
Since the multiplicity of the shear modulus eigenvalues (l and m) is properly accounted for (2 and 2, respectively, out of 5), this result strongly suggests that one reasonable estimate of the fifth shear modulus for the system is
   \begin{eqnarray}
G_{eff}^{(1)} \equiv \left(\mu_1^* +
2\mu_3^*\right){{K_R}\over{3K_V}}.
 \end{eqnarray} (33)

Voigt average for shear

The Voigt average for shear modulus is
   \begin{eqnarray}
G_V = {{1}\over{15}}\left[2C_{11} - C_{12} + C_{33} - 2C_{13}
 ...
 ...number \
= {{1}\over{15}}\left[(a-m-f) + (c-f) + 6l + 6m\right].
 \end{eqnarray}
[Note that (38) corrects an error in equation (69) of Berryman and Wang (2001).] This result shows that the combinations 2(a-m-f) and (c-f) again play the roles of twice a shear modulus contribution. By analogy to (36) and (37), we can define another effective constant
   \begin{eqnarray}
G_{eff}^{(2)} = [(a-m-f) + (c-f)]/3 =
\left(\mu_1^* + 2\mu_3^*\right)/3.
 \end{eqnarray} (34)
This constant has a sensible dependence on these parameters and is consistent with the rest of our analysis (see Discussion of Geff below), and also is a somewhat simpler form than (37). Both this constant and Geff(1) reappear in the later analyses. Because these two estimates are related to rigorous bounds, it seems that estimates not lying in the range from Geff(1) to Geff(2) can surely be excluded from further consideration.


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