On periodic homogenization of highly contrasted elastic structures

While homogenization of periodic linear elastic structures is now a well-known procedure when the stiffness of the material varies inside fixed bounds, no homogenization formula is known which enables us to compute the effective properties of highly contrasted structures. Examples have been given in which the effective energy involves the strain-gradient but no general formula provides this strain-gradient dependence. Some formulas have been proposed which involve such terms and provide a small correction to the classical effective energy still when the stiffness of the material varies inside fixed bounds. The goal of this paper is to check the applicability of these formulas for highly contrasted structures. To that aim we focus on structures whose limit energy is already known and we compare the energies given by (i) the convergence results, (ii) the corrective formulas and (iii) by a direct numerical simulation of the complete structure