A local-effective relation in planar linear elasticity

We present a relation between local and effective properties for a class of two-dimensional elastic structures. The model considered is a periodic structure, which is locally isotropic and homogeneous. The corresponding physical model is a flat two-dimensional body with traction-free holes, such as a perforated plate. We show how the effective properties of the structure depend on the local properties in a way that separates the dependence on the holes. Our result extends a result of Vigdergauz [1], which describes how effective properties depend on local properties for effectively square symmetric structures, to the case when the
effective elasticity tensor is allowed to be anisotropic.