On effective resistivity and related parameters for periodic checkerboard composites

An approach for finding the effective properties of two-dimensional periodic rectangular checkerboard media is presented; it is demonstrated that one can take a relatively simple model problem, an appropriate system of joined quarter planes, and that after appropriate conformal transformation one can use this to solve for either periodic or doubly periodic structures. Illustrative calculations involving three-phase media are presented that, in limiting cases, recover earlier analyses. Useful asymptotic relations are also given.