Classifying soft elastic lattices using higher-order homogenization

TL;DR

This paper introduces a homogenization method for periodic elastic lattices, including unstable ones with mechanisms, using second-order asymptotic analysis to classify their effective behaviors.

Contribution

It develops a unified homogenization framework that handles unstable lattices with mechanisms by treating mechanism amplitude as an enrichment variable.

Findings

Effective energy capturing strain-gradient effects

Classification of lattice behaviors based on homogenization

Application to various lattice structures

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.