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

We consider the response of a multicomponent body to $n$ fields, such as electric fields, magnetic fields, temperature gradients, concentration gradients, etc., where each component, which is possibly anisotropic, may cross couple 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 the fields and response tensor in powers of matrices which measure the difference between each component tensor and a homogeneous reference tensor ${\bf L}_0$. For the case of a statistically homogeneous or periodic composite the expansion coefficients can be expressed in terms of positive semidefinite normalization matrices alternating with positive semidefinite weight matrices, which at each given level sum to the identity matrix. In an appropriate basis the projection operators onto the relevant subspaces can be expressed in block tridiagonal form, where the blocks are functions of these weight and normalization matrices. This leads to continued fraction expansions for the effective tensor, and by truncating the continued fraction at successive levels one obtains a nested sequence of bounds on the effective tensor incorporating successively more weight and normalization matrices. The weight matrices and normalization matrices can be calculated from the series expansions of the fields which solve the conductivity problem alone, without any couplings to other fields, and then they can be used to obtain the solution for the fields and effective tensor in coupled field problems in composites.