Homogenizing elastic lattices with mechanisms

We propose an asymptotic method for homogenizing periodic elastic lattices that works in the presence of mechanisms, both of the macroscopic type (strain-producing modes) and of the microscopic type (internal modes). When a microscopic mechanism is present, the unit-cell problem produced by classical homogenization is singular. It can be fixed by including the amplitude θ ( X ) of the mechanism as an additional macroscopic degree of freedom (enrichment variable) contributing to the effective energy via its gradient ∇ θ ( X ) . When a macroscopic mechanism is present, homogenization delivers a degenerate effective energy at leading order, which can be regularized by accounting for the strain gradient. We introduce an asymptotic second-order homogenization scheme that integrates these two features: it delivers an effective energy capturing both the strain-gradient effect ∇ ɛ ( X ) relevant to macroscopic mechanisms, and the ∇ θ ( X ) regularization relevant to microscopic mechanisms, if any is present. The versatility of the 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. Whereas the procedure delivers known effective models for elastic lattices without mechanisms, it can generate novel effective models for lattices possessing mechanisms. • Asymptotic homogenization method suited to lattices possessing mechanisms. • Handles both microscopic (internal) and macroscopic (strain-producing) mechanisms. • Relies on kinematic enrichment, achieves second-order accuracy. • An open-source implementation in the library shoal is provided, allowing automatic homogenization of lattices.