The volume fraction of the i-th constituent is vi 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-^*)e_i = 0 and
v_i (_i-^*)_i = 0. Following Hill (1963), then recall that, since ei 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
^*i = _i^*i^*.
Substituting (defAandB) into (exactLe) and (exactMGs), I find (since e and may be considered arbitrary) that
v_i (_i-^*)^*i = 0 and
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).