Variational methods, bounds, and size effects for composites with highly conducting interface

New variational principles are developed for the effective conductivity tensor for anisotropic two-phase electric conductors. Here the interface between phases is assumed to be highly conducting. Extra geometric information is encoded into the principles through the solution operators of simpler transport problems. These operators can be expressed as gradients of simple layer potentials with densities supported on phase interfaces or in terms of simple Dirichlet problems inside each phase region. New upper bounds on the effective conductivity are found that depend upon component volume fractions, a surface energy tensor and a scale-free matrix of parameters. This matrix corresponds to the effective conductivity tensor associated with the same geometry but with perfectly conducting inclusions. New lower bounds are given in terms of two-point correlation functions, component volume fractions, and interfacial geometric parameters. Both upper and lower bounds are found to be optimal for certain choices of interfacial parameters. For isotropic polydisperse suspensions of spheres we are able to estimate the effective conductivity based on measured values of the size distribution of the spheres. Conversely, we are able to characterize the size distribution of the spherical inclusions based on measured values of the effective conductivity. (C) 1997 Elsevier Science Ltd.