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 , or equivalently for the Biot-Willis parameter .

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

(^*-^(r))v_i ^ri_r =
v_i(^(i)-^(r))^ri_r,
where is the effective stiffness matrix (inverse of the compliance
matrix ) to be determined, is the stiffness
matrix of some convenient elastic reference material,
*v*_{i} is the volume fraction and
the stiffness matrix of the *i*th constituent of
the elastic composite, is the strain in the reference
material, and is the (exact and generally
unknown) linear coefficient relating strains in material *i* to those in
material *r* according to .

- Coherent potential approximation
- Average t-matrix/Kuster-Toksöz scheme
- Differential effective medium approximation
- Mori-Tanaka approximation

7/5/1998