Curvature screening in draped mechanical metamaterial sheet

TL;DR

This paper develops a continuum elastic theory for mechanical metamaterial sheets on curved surfaces, introducing a gauge-like field to analyze strain modes and predicting their behavior on various curved substrates.

Contribution

It introduces a novel theoretical framework using a gauge-like field to understand the mechanics of curved metamaterial sheets, extending elasticity theory to soft modes on curved surfaces.

Findings

Good agreement between numerical simulations and analytical predictions.

Framework applicable to both dilational and shear metamaterials on curved substrates.

Potential for exploring non-linear soft modes in future work.

Abstract

We develop a framework to understand the mechanics of metamaterial sheets on curved surfaces. Here we have constructed a continuum elastic theory of mechanical metamaterials by introducing an auxiliary, scalar gauge-like field that absorbs the strain along the soft mode and projects out the stiff ones. We propose a general form of the elastic energy of a mechanism based metamaterial sheet and specialize to the cases of dilational metamaterials and shear metamaterials conforming to positively and negatively curved substrates in the F\"{o}ppl-Von K\'{a}rm\'{a}n limit of small strains. We perform numerical simulations of these systems and obtain good agreement with our analytical predictions. This work provides a framework that can be easily extended to explore non-linear soft modes in metamaterial elasticity in future.