Stiffness formulation

The dual to the problem just studied replaces compliances everywhere with stiffnesses, and then proceeds as before. Equations (15)-(18) are replaced by

$$\begin{eqnarray} \left(\begin{array} {c} v_1^T \\ v_3^T\end{array}\right) C^* \... ...*_{11} & 18B^*_{13} \\ 18B^*_{13} & 36B^*_{33}\end{array}\right) \end{eqnarray}$$ (28)

(in all cases the * superscripts indicate that the pore-fluid effects are included) and the reduced matrix

$$\begin{eqnarray} \left(\Sigma^*\right)^{-1} = B^*_{11}v_1v_1^T + B^*_{13}(v_1v_3^T + v_3v_1^T) + B^*_{33} v_3v_3^T, \end{eqnarray}$$ (29)

where

$$\begin{eqnarray} B^*_{11} = [2(c^*_{11}+c^*_{12}+2c^*_{13})+c^*_{33}]/9, \nonumb... ...B^*_{33} = (c^*_{11}+c^*_{12}-4c^*_{13}+2c^*_{33})/18. \nonumber \end{eqnarray}$$ (30)

It is a straightforward exercise to check that the two reduced problems are in fact inverses of each other. We will not repeat this analysis here, as it is wholly repetitive of what has gone before. The main difference in the details is that the expressions for the B's in terms of the $\beta$'s are rather more complicated than those for the compliance version, which is also why we chose to display the compliance formulation instead.