Effective stiffness tensor of composite media - I. Exact series expansions

The problem of determining exact expressions for the effective stiffness tensor macroscopically anisotropic, two-phase composite media of arbitrary microstructure in arbitrary space dimension d is considered. We depart from previous treatments by introducing an integral equation for the "cavity" strain field. This leads to new, exact series expansions for the effective stiffness tensor of macroscopically anisotropic, d-dimensional, two-phase composite media in powers of the "elastic polarizabilities". The nth-order tensor coefficients of these expansions are explicitly expressed as absolutely convergent integrals over products of certain tensor fields and a determinant involving n-point correlation functions that characterize the microstructure. For the special case of macroscopically isotropic media, these series expressions may be regarded as expansions that perturb about the optimal structures that realize the Hashin-Shtrikman bounds (e.g. coated-inclusion assemblages or finite-rank laminates). Similarly, for macroscopically anisotropic media, the series expressions may be regarded as expansions that perturb about optimal structures that realize Willis' bounds. For isotropic multiphase composites, we remark on the behavior of the effective moduli as the space dimension d tends to infinity.