I have demonstrated that the generalized Eshelby formula (genEshelby) derived earlier (Berryman, 1997) can be successfully used in various well-known effective medium theories to estimate the Biot-Willis parameter when the inclusions are of arbitrary ellipsoidal shape. This generalizes other work of the author (Berryman, 1985; 1992) that provided means of computing these same constants but only for the case of spherical inclusions. The new formulas are no more difficult to compute that the corresponding formulas for the bulk and shear (empty porous) frame moduli of these materials.
The two most robust theories for the examples shown are clearly the CPA and the Mori-Tanaka theory. Neither of these theories has any problem computing estimates for any of the extreme cases considered. Both DEM and Kuster-Toksöz have problems with the disk or penny-shaped inclusions when the filling material has low shear modulus. This result shows that, although the Kuster-Toksöz and Mori-Tanaka methods are identical for spherical inclusions, they are easily distinguished for nonspherical inclusions and it appears from these examples that Mori-Tanaka may be the preferred explicit scheme of these two.
The work presented is incomplete because it does not yet show how to compute the remaining parameter B (Skempton's coefficient) for a general ellipsoidal inclusion within these various effective medium theories. Nevertheless, the procedure for doing so is a straightforward extension of work published earlier by Berryman and Milton (1991) based on an analysis of (porositychange) and will be presented in a future publication.