The constituents of the composites under consideration are linear elastic, isotropic materials whose average stresses and average strains ei are related by the constitutive relations
_i = _i e_i and e_i = _i _i. The subscript i refers to the i-th constituent, of which I assume there are N. The components of the fourth ranked stiffness tensor are defined by
(_i)_mnpq = _i_mn_pq + _i(_mp_nq+_np_mq), where and are the Lamé parameters of the i-th constituent. The indices m,n,p,q take the values 1,2,3, corresponding to Cartesian axes x1 = x, x2 = y, and x3 = z. The bulk modulus , while the shear modulus is . Following Hill (1963) [also see Gubernatis and Krumhansl (1975)], I treat as a six-by-six matrix, with and ei being treated as six component vectors. Then, the compliance tensor , when also viewed as a six-by-six matrix, satisfies
_i_i = = _i_i, so is the matrix inverse of .
I assume for simplicity that the overall behavior of the composite is also linear elastic and isotropic, and that the effective constitutive laws are given by
= ^* e and e = ^*, with
^*^*= = ^*^*. My problem is to find ways of relating the effective tensors and to the properties of the components contained in the constituents' tensors and .