Asymptotic strain-gradient theory for one-dimensional continua

The goal of periodic homogenization is to identify an effective model specified by an energy functional Φ_ɛ[u] depending on the macroscopic displacement u. We consider second-order homogenization, a case where the effective energy depends not only on the strain u^′ but also on its gradients u^′′ and u^′′′. Functionals Φ_ɛ[u] obtained in prior work are typically made stationary order by order in the expansion parameter, and are not positive when truncated: they are not proper strain-gradient theories. Starting from a functional Φ_ɛ[u] produced by linear, second-order homogenization of a periodic elastic lattice in dimension 1, we propose a systematic method to upgrade it to a positive strain-gradient energy Ψ_ɛ[u]. This enables us to formulate second-order homogenization as a variational problem. Boundary layers are represented in an effective and asymptotically correct way by boundary terms in the energy Ψ_ɛ[u].