Mori-Tanaka scheme

The first explicit scheme considered is that due to Mori and Tanaka (1973) as described by Weng (1984) and Benveniste (1987). To obtain this approximation from (Lr) and (Mr), assume that the composite has a host material with inclusions and choose the host to serve as reference material, so $r \to h$. Then, approximating $\matG^{hi} \simeq \matT^{hi}$,I find

v_i (_i-^*_MT)^hi = 0,   in simplest form. The corresponding equation for the compliance is

v_i (_i-^*_MT)^hi = 0,   where $\matW^{hi} \equiv \matL_i\matT^{hi}\matM_h$. Again by multiplying (MTvsSCinverse) on the right by $\matL_h$ and on the left by $\matL^*_{MT}$ to check consistency, I find that (MTvsSC) is recovered, as shown previously by Weng (1984) and Benveniste (1987). The equation for $\matL^*_{MT}$ can also be written as

(^*_MT-_h)v_i ^hi = v_i (_i-_h)^hi,   where I have retained the redundant terms $\matL_h\sum v_i\matT^{hi}$ on both sides of the equation. This form of the equation for $\matL^*_{MT}$is convenient for comparison with the KT explicit scheme to be considered next.