Transport properties of a three-phase composite material: the square array of coated cylinders

Abstract

The analytic properties of the effective dielectric constant of a class of three-phase composite materials are studied. Specifically, we investigate the effective dielectric constant of a periodic array of coated cylinders, as a function of the core dielectric constant ( ϵ c ) and the shell dielectric constant ( ϵ s ), while keeping the matrix dielectric constant ( ϵ b ) fixed. We show that when ϵ s = – ϵ c , the composite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constant ϵ c and radius equal to the outer radius of the original coated cylinder. We also show that when ϵ s = – 1, the composite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constant ϵ c , and radius exceeding the shell radius. We explore the location of poles and zeros of the three-phase effective dielectric constant in the ( ϵ s , ϵ c ) plane. The lines ϵ s = – 1 and ϵ s + ϵ c = 0 are loci of essential singularities. We also comment on the behaviour of the effective dielectric constant in the neighbourhood of the two special points ( ϵ s , ϵ c ) = (0,0) and ( ϵ s , ϵ c ) = ( - 1 , + 1 ).