Conductivity of a periodic two-component system of rhombic type

The conductivity and the distribution of electric field, current, and charge density in a periodic two-component system composed of rhombs with an arbitrary vertex angle of 2alpha are investigated. The effective conductivity of such a medium is represented by a tensor with components sigma(eff)(11)(alpha) and sigma(eff)(22)(alpha) in the principal axes that satisfy the Dykhne relation sigma(eff)(11)(alpha) sigma(eff)(22)(alpha)= sigma(1)sigma(2), where sigma(1), sigma(2) are the isotropic conductivities of media 1 and 2. In addition, the relation sigma(eff)(22)(alpha) = sigma(eff)(11)(pi/2 - alpha) is satisfied. The principal axes are directed along the diagonals of the rhombs. It is shown that there are three lines in the rectangle 0 < alpha less than or equal to pi/2, -1 < Z < 1 (Z = (sigma(1) - sigma(2))/(sigma(1) + sigma(2))) on which the charge density is expressed in terms elliptic functions. An explicit expression is obtained for all physical quantities on these lines. (C) 2004 MAIK "Nauka/Interperiodica".