Conformal elasticity of mechanism-based metamaterials

Michael Czajkowski,Corentin Coulais,Martin Van Hecke,D Zeb Rocklin

doi:10.1038/s41467-021-27825-0

Abstract

Deformations of conventional solids are described via elasticity, a classical field theory whose form is constrained by translational and rotational symmetries. However, flexible metamaterials often contain an additional approximate symmetry due to the presence of a designer soft strain pathway. Here we show that low energy deformations of designer dilational metamaterials will be governed by a scalar field theory, conformal elasticity, in which the nonuniform, nonlinear deformations observed under generic loads correspond with the well-studied—conformal—maps. We validate this approach using experiments and finite element simulations and further show that such systems obey a holographic bulk-boundary principle, which enables an analytic method to predict and control nonuniform, nonlinear deformations. This work both presents a unique method of precise deformation control and demonstrates a general principle in which mechanisms can generate special classes of soft deformations.

Highlights

Deformations of conventional solids are described via elasticity, a classical field theory whose form is constrained by translational and rotational symmetries
To test the hypothesis of conformal deformation, we investigate the elastic response of rotating square (RS)-based metamaterials at a range of hinge thicknesses using finite element (FEM) simulations which preserve the intricate pore structure
Dilation dominates over shear in simulations of three-point “bending” (Fig. 2a), a local “dipole” dilation (Fig. 2b) and even when the system is subject to global “pure shear” via compression along one axis and expansion along the other (Fig. 2c)

Summary

Introduction

Deformations of conventional solids are described via elasticity, a classical field theory whose form is constrained by translational and rotational symmetries. We show that low energy deformations of designer dilational metamaterials will be governed by a scalar field theory, conformal elasticity, in which the nonuniform, nonlinear deformations observed under generic loads correspond with the wellstudied—conformal—maps. We validate this approach using experiments and finite element simulations and further show that such systems obey a holographic bulk-boundary principle, which enables an analytic method to predict and control nonuniform, nonlinear deformations. We use the bulkboundary correspondence principle obeyed by these metamaterials to introduce a recipe for on-demand activation of each soft configuration

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Conclusion