General results

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 $\sum v_i = 1$.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 e_i must be a linear function of e and since $\sigma_i$ must be a linear function of $\sigma$,I can write

e_i = ^*i e   and  _i = ^*i ,   where $\matA^{*i}$ and $\matB^{*i}$ 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 $\sigma$ 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 $\matA^{*i}$ and $\matB^{*i}$ are unknown. If I could find approximations to and based on their definitions (defAandB), then such approximations would lead immediately to approximations for $\matL^*$ and $\matM^*$ using (exact) and (exactinverse).