General equations using a reference material

Now consider a reference material r (with stiffness and compliance tensors $\matL_r$ and $\matM_r$) 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 $\sigma_r$ 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., $r \to *$), the new tensors satisfy $\matG^{*i} \equiv \matA^{*i}$ and $\matH^{*i} \equiv \matB^{*i}$.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 $\matL_r$ or $\matM_r$ 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 $\matL^*$, 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 $\matM^*$ can be found using an equation analogous to the one used to find $\matL^*$. 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.