Topological rigidity in twisted, elastic ribbons

TL;DR

This paper develops a theoretical framework for understanding how topology influences the deformation behavior of twisted elastic ribbons, revealing new topological obstructions and conditions for soft modes.

Contribution

It introduces a novel theory linking topology and elasticity in twisted ribbons, highlighting the role of non-orientability and normal vector winding in deformation constraints.

Findings

Non-orientability causes a geometric phase affecting global symmetry realization.

Orientable ribbons can also have topological obstructions related to normal vector winding.

Conditions for soft deformation modes are derived based on ribbon topology.

Abstract

Topology is an important determinant of the behavior of a great number of condensed-matter systems, but until recently has played a minor role in elasticity. We develop a theory for the deformations of a class of twisted non-Euclidean sheets which have a symmetry based on the celebrated Bonnet isometry. We show that non-orientability is an obstruction to realizing the symmetry globally, and induces a geometric phase that captures a memory analogous to a previously identified one in 2D metamaterials. However, we show that orientable ribbons can also obstruct realizing the symmetry globally. This new obstruction is mediated by how the unit normal vector winds around the centerline of the ribbon, and provides conditions for constructing soft modes of deformation compatible with the topology of multiply-twisted connected ribbons.