The first approximation scheme considered is an implicit method called the self-consistent (SC) scheme (Gubernatis and Krumhansl, 1975; Berryman, 1980). This scheme should be distinguished from another scheme by Wu (1966) that is also called a self-consistent scheme, but generally differs from the present one. For completeness, Wu's scheme is also derived in the Appendix.
My approximation may be obtained by supposing the reference material is a medium with the same stiffnesses and compliances as the composite of interest. Thus, in (Lr) and (Mr), so the left hand side of each equation vanishes. To obtain a useful approximation, suppose that is well approximated by Wu's tensor .Then, considering er and arbitrary, the self-consistent approximation is given by
v_i (_i-^*_SC) ^*i = 0. The corresponding equation for the compliance tensor is
v_i (_i-^*_SC) ^*i = 0, where . Both of these equations have the same general form as the exact equations (exact) and (exactinverse), but nevertheless the results obtained are necessarily approximate since the exact factors and are only approximated by the factors and , respectively.
I find easily that, by multiplying (SCinverse) on both the left and right by , it reduces to (SC) and is therefore consistent with it in the sense that it can be derived from it. This shows that it is sufficient to solve either equation and then use the uniqueness of the matrix inverse to find the solution of the other. I will continue to point out the existence of this property for the other theories as they are derived, but will not dwell on the proofs, since these proofs are completely analogous to the one just given in all cases.