Now consider a reference material *r* (with stiffness and compliance tensors
and ) which may or may not be one of the
actual constituents of the composite under study. If there is one material
in the composite that acts as host to the others, the reference material may
be chosen to be the host material *h*. However, in general, I
want to leave the choice of reference material open for later consideration.

Define tensors relating (possibly fictitious) strains *e*_{r} and stresses
to the actual average strains and stresses in the composite
by the linear relations

e_i = ^rie_r and _i = ^ri_r. The consistency relation between the new tensors is

^ri = _i^ri_r.
Note that, in the case where the reference material is chosen to be the
composite material itself (*i.e.*, ), the new tensors satisfy
and .These new tensors are therefore simply formal generalizations of
and suitable for the wide variety of circumstances
that may arise in my modeling efforts.

Again substituting (newtensors) into (exactLe) and (exactMGs) yields the equations

(^*-_r)v_i ^rie_r = v_i(_i-_r)^rie_r and

(^*-_r)v_i ^ri_r =
v_i(_i-_r)^ri_r.
These equations are not in simplest form (there are redundant terms
proportional to either or on both
side of the equations) for reasons that will become apparent. Nevertheless,
these two equations are exact (*i.e.*, no approximations have been made)
and mutually consistent. Consistency may
be checked by deleting the redundant terms from (Mr), multiplying the
result by , and then including the redundant terms found in
(Lr). The result is exactly (Lr).

It is sometimes claimed that certain theories are superior to others because the compliance tensor can be found using an equation analogous to the one used to find . However, since all the theories considered in this paper may be derived from the pair of equations (Lr) and (Mr) which are themselves consistent in this sense, I have found that all these theories possess this desirable property. Thus, this criterion appears to be too weak to force choices among the theories.

The advantage of the introduction of the reference material will become apparent as I derive various approximate expressions from (Lr) and (Mr) using Eshelby's and Wu's results for single ellipsoidal inclusions amd isotropic averages of ellipsoidal inclusions. These equations also enable me to present a unified approach to derivation of the various approximation schemes considered.

11/17/1997