Geometric approach to mechanical design principles in continuous elastic sheets

TL;DR

This paper introduces a geometric formalism for designing and controlling the mechanical properties of slender elastic sheets, enabling the encoding of extreme behaviors through geometric frustration.

Contribution

It develops a systematic theoretical method for inverse design of mechanical properties in non-euclidean thin sheets using geometric formalism.

Findings

Derived geometries with tunable mechanical responses

Demonstrated anomalous behaviors like vanishing rigidity

Provided a discretizable formalism for design of solids

Abstract

Using a geometric formalism of elasticity theory we develop a systematic theoretical method for controlling and manipulating the mechanical response of slender solids to external loads. We formally express global mechanical properties associated with non-euclidean thin sheets, and interpret the expressions as inverse problem for designing desired mechanical properties. We show that by wisely designing geometric frustration, extreme mechanical properties can be encoded into a material using accessible experimental techniques. To test the methodology we derive a family of geometries that result with anomalous mechanical behavior such as tunable, an-harmonic, and even vanishing rigidities. The presented formalism can be discretized and thus opens a new pathway for the design of both continuum and discrete solids and structures.