Average t-matrix/Kuster-Toksöz scheme

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

(C

^*-C

_h)= v_i(C

_i-C

_h)^hi_h.   I obtained this form from (general) by noting that $\varepsilon= \sum v_i \varepsilon_i = \sum v_i \matG^{ri}\varepsilon_r$.The Kuster-Toksöz approximation includes the assumptions that $\varepsilon= \matG^{h*}\varepsilon_h \simeq \matT^{h*}\varepsilon_h$and that $\matG^{ri} \simeq \matT^{hi}$. Then, the resulting formula for the approximation is

(C

^*_KT-C

_h)^h* = v_i(C

_i-C

_h)^hi.   The further assumption is normally made that the tensor $\matT^{h*}$ is always the one for spherical inclusions, while $\matT^{hi}$ can be for arbitrary shapes of inclusions.

To derive a formula within ATA/KT for the Biot-Willis constant $\alpha^*$,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 $\varepsilon$ 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 ${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}^*_{KT}$ and $\alpha^*_{KT}$ 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 $\sum v_i \varepsilon^{(i)} = \varepsilon^*$ 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 $\ldots$ are of the form $\sum v_i (\matT^{hi} - \matT^{h*})\varepsilon$ 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 $\matG^{h*} = \sum v_i\matG^{hi}$ as a rigorous result of the theory. Thus, the KT formula for the Biot-Willis parameter $\alpha^*$ 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 $\sum v_i(\matT^{hi}-\matT^{h*}) \simeq 0$that should be checked for approximate satisfaction by ${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}^*_{KT}$. 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).