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## General results

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).

Next: Wu tensors and single Up: BACKGROUND AND GENERAL ANALYSIS Previous: BACKGROUND AND GENERAL ANALYSIS
Stanford Exploration Project
11/17/1997