Periodic Heterogeneous Structures: New Explicit Solutions and Effective Characteristics of Refraction of an Imposed Field

Abstract

Complex variable methods are applied to study three double-periodic, two-phase, planar heterogeneous structures. Explicit expressions of functionals of energy dissipation and of effective resistivities (conductivities) are found on the basis of strict analytical solutions of a boundary-value problem for an arbitrary direction of an imposed external field. Comparisons with the results of Bakhvalov and Panasenko [ Homogenization of Processes in Periodic Media, Nauka, Moscow, 1984]; Dykhne [Zh. Eksp. Teor. Fiz., 59 (1970), pp. 110--113 ( Soviet. Phys. JETP, 32 (1971), pp. 63--65)]; Emets [Zh. Eksp. Teor. Fiz., 96 (1989), pp. 701--711 ( Soviet. Phys. JETP, 96 (1989))]; Keller [ J. Appl. Phys., 34 (1963), pp. 991--993]; and Mendelson [ J. Appl. Phys., 46 (1975), pp. 4740--4741] are made. In particular, the dissipation in the general case is shown essentially to depend on the direction of the external field. For a rectangular checkerboard, dissipation within the two phases does not coincide. It is rigorously proved that the conductivity curve is an ellipse for all studied structures.