EXACT SOLUTION OF A BOUNDARY-VALUE PROBLEM FOR A RECTANGULAR CHECKERBOARD FIELD

A class of two-phase composite materials with a biperiodic structure i s investigated by the methods of complex analysis. Two interface condi tions - continuity of normal component of a desired vector omega and t angential component of <(rho)over cap omega> at the contact boundary a s well as the double-periodicity condition - are involved in rigorous form. The exact analytic solution of the corresponding generalized Rie mann boundary-value problem is obtained. The explicit values of the ef fective parameters, namely effective resistivity and dissipation of en ergy of an elementary cell and resistivities along the symmetry axes a re calculated in closed analytic form. The coincidence of our formulae with the well-known effective resistivity (conductivity) formula of K eller (1964), Dykhne (1970) and Mendelson (1975) and the dissipation f ormula of Dykhne (1970) is shown in the case of square checkerboard he ld. The Keller (1963) identity is generalized for the heterogeneous st ructure studied.