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 (ϵc) and the shell dielectric constant (ϵs), while keeping the matrix dielectric constant (ϵb) fixed. We show that whenϵs= –ϵc, the composite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constantϵcand radius equal to the outer radius of the original coated cylinder. We also show that whenϵs= – 1, the composite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constantϵc, and radius exceeding the shell radius. We explore the location of poles and zeros of the three-phase effective dielectric constant in the (ϵs,ϵc) plane. The linesϵs= – 1 andϵs+ϵc= 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 (ϵs,ϵc) = (0,0) and (ϵs,ϵc) = ( - 1 , + 1 ).

This publication has 10 references indexed in Scilit: