A proof of the Flaherty-Keller formula on the effective property of densely packed elastic composites

We prove in a mathematically rigorous way the asymptotic formula of Flaherty and Keller on the effective property of densely packed periodic elastic composites with hard inclusions. The proof is based on the primal-dual variational principle, where the upper bound is derived by using the Keller-type test functions and the lower bound by singular functions made of nuclei of strain. Singular functions are solutions of the Lam\'e system and capture precisely singular behavior of the stress in the narrow region between two adjacent hard inclusions.