The effective energy of a lattice metamaterial

TL;DR

This paper explores how the continuum limit of certain periodic lattice materials can be modeled as nonlinear elastic media, with implications for understanding mechanism-based mechanical metamaterials and their soft modes.

Contribution

It provides a more general and physical framework for the continuum limit of discrete materials, including those with mechanisms, by addressing degeneracy issues through orientation penalization.

Findings

Broad class of systems analyzed, including those with mechanisms

Degeneracy issues can be mitigated by penalizing orientation changes

Defines soft modes as deformations with vanishing effective energy in the continuum limit

Abstract

We study the sense in which the continuum limit of a broad class of discrete materials with periodic structures can be viewed as a nonlinear elastic material. While we are not the first to consider this question, our treatment is more general and more physical than those in the literature. Indeed, it applies to a broad class of systems, including ones that possess mechanisms; and we discuss how the degeneracy that plagues prior work in this area can be avoided by penalizing change of orientation. A key motivation for this work is its relevance to mechanism-based mechanical metamaterials. Such systems often have ``soft modes'', achieved in typical examples by modulating mechanisms. Our results permit the following more general definition of a soft mode: it is a macroscopic deformation whose effective energy vanishes -- in other words, one whose spatially-averaged elastic energy tends to zero in the continuum limit.