Classifying soft elastic lattices using higher-order homogenization

Abstract: We propose a methodology for the homogenization of periodic elastic lattices that covers the case of unstable lattices, having affine (macroscopic) or periodic (microscopic) mechanisms. The singular cell problems that are encountered when a periodic mechanism is present are naturally solved by treating the amplitude $\theta(X)$ of the mechanism as an enrichment variable. We use asymptotic second-order homogenization to derive an effective energy capturing both the strain-gradient effect $\nabla \varepsilon$ relevant to affine mechanisms, and the $\nabla \theta$ regularization relevant to periodic mechanisms, if any is present. The proposed approach is illustrated with a selection of lattices displaying a variety of effective behaviors. It follows a unified pattern that leads to a classification of these effective behaviors.