Euclidean Frustrated Ribbons

TL;DR

This paper introduces a new type of geometrical frustration in Euclidean sheets caused by violation of compatibility equations, leading to shape transitions and stress focusing, with implications for natural and engineered thin structures.

Contribution

It identifies and characterizes a novel form of geometrical frustration in Euclidean sheets arising from compatibility violations, expanding understanding beyond Gauss frustration.

Findings

Shape transitions and symmetry breaking observed experimentally.

Spontaneous stress focusing demonstrated in Euclidean ribbons.

Analytic solutions explain the geometric phenomena.

Abstract

Geometrical frustration in thin sheets is ubiquitous across scales in biology and becomes increasingly relevant in technology. Previous research identified the origin of the frustration as the violation of Gauss's \emph{Theorema Egregium}. Such "Gauss frustration" exhibits rich phenomenology; it may lead to mechanical instabilities, anomalous mechanics and shape-morphing abilities that can be harnessed in engineering systems. Here we report a new type of geometrical frustration, one that is as general as Gauss frustration. We show that its origin is the violation of Mainardi-Codazzi-Peterson compatibility equations and that it appears in Euclidean sheets. Combining experiments, simulations and theory, we study the specific case of a Euclidean ribbon with radial and geodesic curvatures. Experiments, conducted using different materials and techniques, reveal shape transitions, symmetry breaking and spontaneous stress focusing. These observations are quantitatively rationalized using analytic solutions and geometrical arguments. We expect this frustration to play a significant role in natural and engineering systems, specifically in slender 3D printed sheets.