The second approximation scheme I will consider is sometimes called the Average T-Matrix Approximation (ATA) (Berryman, 1992) and sometimes the Kuster-Toksöz (KT) Scheme (Kuster and Toksöz, 1974).
In the absence of a pore fluid, the poroelastic problem reduces again precisely to the elastic composite problem. Following the analysis of Berryman and Berge (1996), I find that the general result (general) is conveniently written as
(^*-_h)= v_i(_i-_h)^hi_h. I obtained this form from (general) by noting that .The Kuster-Toksöz approximation includes the assumptions that and that . Then, the resulting formula for the approximation is
(^*_KT-_h)^h* = v_i(_i-_h)^hi. The further assumption is normally made that the tensor is always the one for spherical inclusions, while can be for arbitrary shapes of inclusions.
To derive a formula within ATA/KT for the Biot-Willis constant ,I need to make use of the Eshelby generalization again and make appropriate substitutions into the formula (averagestrain). The thought experiment for KT is a little more complex than that for CPA, however, so I actually need to do this in two steps. First, note that if I view the composite as a finite sphere and imbed this sphere in a host material (that may be and usually is chosen to be the same as one of the constituent materials), then the appropriate generalized Eshelby formula for the poroelastic case is
^(i) = e^hi(p_f) + ^hi(- e^hi(p_f)), where is the applied strain at infinity. Equation (KTEshelby) can then be averaged to give
v_i^(i) = v_i (- ^hi)e^hi(p_f) + v_i ^hi. But now if I consider that the composite has the effective properties and in the composite sphere imbedded in the host material, then I can also write
^* = e^h*(p_f) + ^h*(- e^h*(p_f)), and, since by construction, (KTstrain2) should be equated to (KTstrain1). The final result is
(- ^h*)e^h*(p_f) = v_i (- ^hi)e^hi(p_f) + ..., where the terms indicated by the ellipsis are of the form and should vanish for similar reasons to those discussed in the case of a corresponding term in the derivation for CPA, since in this case we have as a rigorous result of the theory. Thus, the KT formula for the Biot-Willis parameter is
(1 - P^h*) ^*_KT-^(h)K^*_KT - K^(h) = v_i (1 - P^hi) ^(i)-^(h)K^(i) - K^(h).
As in the CPA, I now have a subsidiary condition that should be checked for approximate satisfaction by . Again, we find this condition is satisfied exactly for spherical inclusions.
For nonspherical inclusions, we can again simplify the result (KTalpha*) by considering formulas such as
1-P^h*K^(h)-K^* = - P^h*K^(h)+y^h* and 1-P^hiK^(h)-K^(i) = - P^hiK^(h)+y^hi, where the y's again depend on the shape of the inclusion. Substituting into (KTalpha*) and neglecting the differences in the y's, I find that
(^*_KT - ^(h))P^h* = v_i (^(i)-^(h))P^hi, which should then be compared to
(K^*_KT - K^(h))P^h* = v_i (K^(i)-K^(h))P^hi, which follows directly from (KT2).