Kuster-Toksöz scheme

The second explicit scheme considered is that due to Kuster and Toksöz (1974). To obtain this approximation, first recall, from the definitions (newtensors) of the tensors relating the reference material stresses and strains to those of the constituents, that the average strain and average stress are formally related to reference material quantities by

e = v_i ^rie_r   and  = v_i ^ri_r.   So (Lr) and (Mr) may be rewritten as

(^*-_r)e = v_i (_i-_r)^rie_r   and

(^*-_r)= v_i (_i-_r)^ri_r.   Now, consider the relationship between the average strain e in the composite and the strain e_r in the reference material, supposing in particular that the composite is imbedded in the reference material so

e = ^r*e_r   and  = ^r*_r.   Note that it follows from (averages) and (etoer) that

^r* = v_i ^ri   and   ^r* = v_i ^ri.  

Substituting (etoer) into (LrKT) and (MrKT), taking the reference material to be the host $r \to h$, and making the usual approximation that $\matG \simeq \matT$ gives the KT approximation

(^*_KT-_h)^h* = v_i (_i-_h)^hi.   The corresponding equation for the compliance is

(^*_KT-_h)^h* = v_i (_i-_h)^hi,   where $\matW^{h*} \equiv \matL^*_{MT}\matT^{h*}\matM_h$and $\matW^{hi} \equiv \matL_i\matT^{hi}\matM_h$. Again multiplying (KTinverse) on both the left and right by $\matL_h$, (KT) is recovered.

Equations (KT) and (KTinverse) may at first appear to be implicit rather than explicit formulas. However, Kuster and Toksöz (1974) made the further assumption that tensor $\matT^{h*}$ on the left had side of (KT) is always the tensor for spherical inclusions. Then, (KT) reduces to the two equations

(K^*_KT-K_h)K_h+43_hK^*_KT+43_h= v_i (K_i-K_h)P^hi   and (note that $\zeta$ is defined in Table 1)

(^*_KT-_h)_h+_h^*_KT+_h = v_i (_i-_h)Q^hi,   which are easily rearranged into explicit formulas for K^*_KT and $\mu^*_{KT}$.

Kuster and Toksöz (1974) consider their approach to be a low concentration approximation based on a ``noninteraction'' assumption. They translate this assumption into a quantitative condition stating that the aspect ratio of spheroidal inclusions should not exceed the corresponding volume fraction. Such a condition clearly places severe constraints on models containing oblate spheroids and disks.