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We consider the response of a multicomponent body to $n$ fields, such aselectric fields, magnetic fields, temperature gradients, concentrationgradients, etc., where each component, which is possibly anisotropic, may crosscouple the various fields with different fluxes, such as electrical currents,electrical displacement currents, magnetic induction fields, energy fluxes,particle fluxes, etc. We obtain the form of the perturbation expansions of thefields and response tensor in powers of matrices which measure the differencebetween each component tensor and a homogeneous reference tensor ${\bf L}_0$.For the case of a statistically homogeneous or periodic composite the expansioncoefficients can be expressed in terms of positive semidefinite normalizationmatrices alternating with positive semidefinite weight matrices, which at eachgiven level sum to the identity matrix. In an appropriate basis the projectionoperators onto the relevant subspaces can be expressed in block tridiagonalform, where the blocks are functions of these weight and normalizationmatrices. This leads to continued fraction expansions for the effective tensor,and by truncating the continued fraction at successive levels one obtains anested sequence of bounds on the effective tensor incorporating successivelymore weight and normalization matrices. The weight matrices and normalizationmatrices can be calculated from the series expansions of the fields which solvethe conductivity problem alone, without any couplings to other fields, and thenthey can be used to obtain the solution for the fields and effective tensor incoupled field problems in composites.

The response of linear inhomogeneous systems to coupled fields: Bounds and perturbation expansions