Homogenized Tensor on Networks

The homogenized tensor that arises in problems of elasticity theory on periodic networks is studied. On the basis of the relaxation formula, optimal networks are described for which the homogenized tensor can be determined in an explicit form, exact calculations for some nonoptimal networks are performed, and the nondegeneracy properties of the homogenized tensor are investigated. Scalar problems are handled similarly; the class of optimal networks for them proves to be larger than that for problems of elasticity theory.