Case III. $A^*_{33} - A^*_{11}/2 \ne 0$, $A^*_{13} \ne 0$.

This case is the most general one of the three, and the one we will study at greater length in the remainder of this discussion.

We want to understand how the introduction of liquid into the pore space affects the shear modulus. We also want to know how the anisotropy influences, i.e., aids or hinders, the impact of the liquid on the shear behavior. To achieve this understanding, it should be sufficient to consider the case when $(A^*_{13})^2 \ll (A^*_{33}-A^*_{11}/2)^2$, assuming as we do that both factors are nonzero. Then, expanding the square root in (19), we have

$$\begin{eqnarray} \Lambda^*_+ = 6A^*_{33} + \Delta \quad\hbox{and}\quad \Lambda^*_- = 3A^*_{11} - \Delta, \end{eqnarray}$$ (25)

where $\Delta$ is defined consistently by either of the two preceeding expressions or by $2\Delta \equiv \Lambda^*_+ - \Lambda^*_- + 3A_{11} - 6A_{33}$and is also given approximately for cases of interest here by

$$\begin{eqnarray} \Delta \simeq {{3(A^*_{13})^2}\over{A^*_{33}-A^*_{11}/2}}. \end{eqnarray}$$ (26)

In the quasi-isotropic soft anisotropy limit under consideration, we find

$$\begin{eqnarray} \Delta \simeq {{2(\beta_1-\beta_3)^2(2\beta_1+\beta_3)^2/27\gam... ...{ \nu/E +[(2\beta_1+\beta_3)^2 -2(\beta_1-\beta_3)^2]/9\gamma}}. \end{eqnarray}$$ (27)

All of the mechanical effects of the liquid that contribute to this formula appear in the factor $\gamma$. The order at which $\gamma$ appears depends on the relative importance of the two terms in the denominator of this expression. If the second term ever dominates, then one factor of $\gamma$ cancels, and therefore $\Delta \sim O(\gamma^{-1})$, and furthermore $\Delta \sim 2(\beta_1-\beta_3)^2/3\gamma$ if $\vert\beta_1-\beta_3\vert << \vert 2\beta_1+\beta_3\vert$. If instead what seems to be the more likely situation holds and the first term in the denominator dominates, then $\Delta \sim O(\gamma^{-2})$. So in either of these cases, as long as $\beta_1 - \beta_3 \ne 0$ (which is the condition for soft anisotropy), we always have contributions to $\Delta$ from liquid mechanical effects. There do not appear to be any combinations of the parameters for which the fluid effects disappear whenever the material is in the class of anisotropic solids considered here.