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The Voigt average is obtained with a constant strain assumption, and
leads
directly to the estimate in terms of stiffnesses

where the final equality makes use of the definitions from
(19).
It is well-known that [see Hill (1952)].
For an isotropic system, the bulk modulus .The results (18) and (22)
obtained for *K*_{eff} suggest that *f* plays the role of and that some combination or combinations of the constants
and may play the role of the one nontrivial
effective shear modulus *G*_{eff} for both the simple layered system
and for the polycrystalline system. The combinations arising here are

| |
(21) |

the harmonic mean and mean, respectively, of and after having accounted for the duplication of 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.

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Stanford Exploration Project

7/8/2003