EFFECTIVE MEDIUM THEORIES

The analysis to follow requires two main steps for each of the examples to be presented. The first step involves recovering the elastic result for the case when the pore pressure vanishes, i.e., for the drained porous frame. Then, Eqs.(strain) and (effectivestress) imply, when p_f = 0, that

_pq = S_pqrs_rs.   Therefore, this step is completely equivalent to the analysis already presented in Berryman and Berge (1996). I will present these results (along with quick derivations for the sake of completeness) because the results are needed to understand the second step of the analysis in each case. The second step is to derive the equivalent effective medium theory expression for $K^{\prime *}_s$, or equivalently for the Biot-Willis parameter $\alpha^*$.

The general result I use for the drained analysis takes the form [see Eq.(19) of Berryman and Berge (1996)]

(C

^*-C

^(r))v_i ^ri_r =                  v_i(C

^(i)-C

^(r))^ri_r,   where ${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}^*$ is the effective stiffness matrix (inverse of the compliance matrix $\matS^*$) to be determined, ${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}^{(r)}$ is the stiffness matrix of some convenient elastic reference material, v_i is the volume fraction and ${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}^{(i)}$ the stiffness matrix of the ith constituent of the elastic composite, $\varepsilon_r$ is the strain in the reference material, and $\matG^{ri}$ is the (exact and generally unknown) linear coefficient relating strains in material i to those in material r according to $\varepsilon_i = \matG^{ri}\varepsilon_r$.