The volume fraction of the *i*-th constituent is *v*_{i} and I assume that
the constituents of the composite are space filling so that .Then, the average strain and average stress are given by the volume averages

e = v_i e_i and = v_i _i. It is an elementary exercise to show that (aveeandGs), (effectivelaws), and (basicelasticity) imply

v_i (_i-^*)_i = 0.
Following Hill (1963), then recall that, since *e*_{i} must be a linear
function of *e* and since must be a linear function of ,I can write

e_i = ^*i e and _i = ^*i , where and are unknown dimensionless fourth ranked tensors depending on the properties and microgeometry of the composite and its constituents. From (aveeandGs), these tensors must also satisfy the normalization conditions

v_i ^*i = = v_i ^*i. From (defAandB), (effectivelaws), and (basicelasticity) follow the consistency conditions

Substituting (defAandB) into (exactLe) and (exactMGs),
I find (since *e* and may be considered arbitrary) that

v_i (_i-^*)^*i = 0. There has been no approximation made in arriving at these equations, but they are of only limited usefulness since the tensors and are unknown. If I could find approximations to and based on their definitions (defAandB), then such approximations would lead immediately to approximations for and using (exact) and (exactinverse).

11/17/1997