BACKGROUND AND GENERAL ANALYSIS

The constituents of the composites under consideration are linear elastic, isotropic materials whose average stresses $\sigma_i$ 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 $\matL_i$ are defined by

(_i)_mnpq = _i_mn_pq + _i(_mp_nq+_np_mq),   where $\lambda_i$ and $\mu_i$ 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 $K_i = \lambda_i + {2\over3}\mu_i$, while the shear modulus is $\mu_i$. Following Hill (1963) [also see Gubernatis and Krumhansl (1975)], I treat $\matL_i$ as a six-by-six matrix, with $\sigma_i$and e_i being treated as six component vectors. Then, the compliance tensor $\matM_i$, when also viewed as a six-by-six matrix, satisfies

_i_i = = _i_i,   so $\matM_i$ is the matrix inverse of $\matL_i$.

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 $\matL^*$ and $\matM^*$ to the properties of the components contained in the constituents' tensors $\matL_i$ and $\matM_i$.