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

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 when ϵs = – ϵc, the composite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constant ϵc and radius equal to the outer radius of the original coated cylinder. We also show that when ϵs = – 1, the composite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constant ϵc, 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 lines ϵs = – 1 and ϵs + 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 ).