Reciprocal relations, bounds, and size effects for composites with highly conducting interface

We provide a reciprocal relation linking the effective conductivity of a composite with highly conducting phase interfaces to that of a composite with the same phase geometry but with an electrical contact resistance at phase interfaces. A field relationship linking the electric field inside a composite with highly conducting phase interfaces to the current in a composite with contact resistance between phases is found. New size effects exhibited by isotropic particulate suspensions with highly conducting interface are obtained. The effective properties of periodic composites are shown to be monotonically increasing as the size of the period cell tends to zero. The role of surface energy for energy minimizing polydisperse suspensions of disks is examined; a necessary condition for isotropic polydisperse suspensions with minimal effective conductivity is found. For monodisperse suspensions of spheres, a critical radius is found for which the electric field is uniform throughout the composite