TRANSPORT-PROPERTIES OF A 3-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 (epsilon(c)) and the shell dielectric constant (epsilon(s)), while keeping the matrix dielectric constant (epsilon(b)) fixed. We show that when epsilon(s) = - epsilon(c), the composite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constant epsilon(c) and radius equal to the outer radius of the original coated cylinder. We also show that when epsilon(s) = - 1, the composite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constant epsilon(c), and radius exceeding the shell radius. We explore the location of poles and zeros of the three-phase effective dielectric constant in the (epsilon(s), epsilon(c)) plane. The lines epsilon(s) = - 1 and epsilon(s) + epsilon(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 (epsilon(s), epsilon(c)) = (0, 0) and (epsilon(s), epsilon(c)) = (-1, + 1).