TRANSPORT-PROPERTIES OF A 3-PHASE COMPOSITE-MATERIAL - THE SQUARE ARRAY OF COATED CYLINDERS

The analytic properties of the effective dielectric constant of a clas s of three-phase composite materials are studied. Specifically, we inv estigate the effective dielectric constant of a periodic array of coat ed cylinders, as a function of the core dielectric constant (epsilon(c )) and the shell dielectric constant (epsilon(s)), while keeping the m atrix dielectric constant (epsilon(b)) fixed. We show that when epsilo n(s) = - epsilon(c), the composite has exactly the same effective diel ectric constant as a periodic array of solid cylinders with dielectric constant epsilon(c) and radius equal to the outer radius of the origi nal coated cylinder. We also show that when epsilon(s) = - 1, the comp osite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constant epsilon(c), and rad ius exceeding the shell radius. We explore the location of poles and z eros of the three-phase effective dielectric constant in the (epsilon( s), epsilon(c)) plane. The lines epsilon(s) = - 1 and epsilon(s) + eps ilon(c) = 0 are loci of essential singularities. We also comment on th e behaviour of the effective dielectric constant in the neighbourhood of the two special points (epsilon(s), epsilon(c)) = (0, 0) and (epsil on(s), epsilon(c)) = (-1, + 1).