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For a periodically perforated structure, for which homogenization takes placein the linear theory of elasticity, the components of the effective elasticitytensor depend in general on the geometry of the holes as well as on the localelastic properties. These dependencies were shown by Vigdergauz in [7] to beseparated in an elementary way for one particular class of structures. Theoriginal proof of this relation made use of the lattice approach to describeperiodic functions using complex variables. In this paper we present a proof ofthe so-called Vigdergauz equations for a related class of square symmetricstructures. Our proof relies solely on the fundamental theorem of real variablecalculus. The differences between the two mentioned classes of structures arenontrivial which makes our result a partial generalization as well.

An elementary proof of the Vigdergauz equations for a class of square symmetric structures