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Topological Rigidity in Twisted, Elastic Ribbons.

Carlos E Moguel-Lehmer1, Christian D Santangelo1

  • 1Syracuse University, Department of Physics, Syracuse, New York 13244, USA.

Physical Review Letters
|October 5, 2025
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Summary
This summary is machine-generated.

Topology influences condensed-matter systems and elasticity. This study reveals how ribbon geometry and nonorientability create topological obstructions, leading to geometric phases and soft deformation modes in twisted sheets.

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Area of Science:

  • Condensed Matter Physics
  • Theoretical Mechanics
  • Materials Science

Background:

  • Topology significantly impacts condensed-matter systems.
  • The role of topology in elasticity has been historically underexplored.
  • Twisted non-Euclidean sheets present unique geometric properties.

Purpose of the Study:

  • To develop a theory for the deformations of twisted non-Euclidean sheets.
  • To investigate the influence of topology, specifically nonorientability and ribbon geometry, on symmetry realization.
  • To identify conditions for constructing soft modes of deformation.

Main Methods:

  • Development of a theoretical framework for ribbon deformations.
  • Analysis of symmetry based on the Bonnet isometry.
  • Investigation of geometric phases and topological obstructions.
  • Examination of the winding of the unit normal vector around the ribbon centerline.

Main Results:

  • Nonorientability obstructs the global realization of the Bonnet isometry symmetry.
  • A geometric phase is induced, analogous to memory effects in 2D metamaterials.
  • Orientable ribbons can also obstruct symmetry realization through normal vector winding.
  • Conditions for constructing soft modes of deformation in multiply twisted ribbons are established.

Conclusions:

  • Topology plays a crucial role in the elastic behavior of twisted sheets.
  • Geometric phases and topological obstructions are key features of these systems.
  • The study provides a new understanding of how ribbon geometry dictates deformation modes.