The constituents of the composites under consideration are linear elastic,
isotropic materials whose average stresses and average strains
*e*_{i} 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 *x _{1}* =

_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

^*^*= = ^*^*. My problem is to find ways of relating the effective tensors and to the properties of the components contained in the constituents' tensors and .

- General results
- Wu tensors and single inclusion problems
- General equations using a reference material

11/17/1997