APPENDIX

Wu (1966) developed another type of self-consistent estimator, distinct from the approximation (SC) and (SCinverse) that I call the SC theory. It can also be derived from (Lr) and (Mr). First, remove the redundant terms from these equations and then recall (averages). The resulting formulas are

^*e = v_i _i ^hie_h   and

^*= v_i _i ^hi_h,   where I have taken the reference material to be the host material $r \to h$.

Now the strains and stresses of the host material are related to those of the composite by

e_h = ^*he   and  _h = ^*h.   Then, substituting (ehe) into (Wuone) and (Wutwo) and recalling the identities

^*i = ^hi^*h   and   ^*i = ^hi^*h,   together with

v_i ^*i = = v_i ^*i,   I find

(^*-_h)e = v_i (_i-_h) ^*ie   and

(^*-_h)= v_i (_i-_h) ^*i.   So far no approximations have been made in arriving at (WuL) and (WuM).

Now, to obtain Wu's scheme, I make the usual substitution $\matG \to \matT$.Then, for arbitrary e and $\sigma$, I have

^*_Wu-_h = v_i (_i-_h)^*i   and

^*_Wu-_h = v_i (_i-_h)^*i.   These two formulas are exactly the multiphase generalizations of Wu's formulas (Berryman, 1980), and they are consistent in the sense that either one can be derived from the other using the uniqueness of the matrix inverses.