From cylindrical to stretching ridges and wrinkles in twisted ribbons

TL;DR

This paper investigates the complex morphologies of twisted ribbons under tension, focusing on tessellated faceted patterns, and introduces an asymptotic isometry framework combining geometry and elasticity to predict these structures.

Contribution

It develops a novel asymptotic isometry approach to describe and predict faceted tessellation patterns in twisted ribbons, linking geometry and elasticity.

Findings

Identifies the morphological phase diagram of twisted ribbons.

Predicts the existence domain of faceted structures.

Provides a geometric-elasticity framework for pattern selection.

Abstract

Twisted ribbons subjected to a tension exhibit a remarkably rich morphology, from smooth and wrinkled helicoids, to cylindrical or faceted patterns. These shapes are intimately related to the instability of the natural, helicoidal symmetry of the system, which generates both longitudinal and transverse stresses, thereby leading to buckling of the ribbon. In this paper, we focus on the tessellation patterns made of triangular facets. Our experimental observations are described within an "asymptotic isometry" approach that brings together geometry and elasticity. The geometry consists of parametrized families of surfaces, isometric to the undeformed ribbon in the singular limit of vanishing thickness and tensile load. The energy, whose minimization selects the favored structure among those families, is governed by the tensile work and bending cost of the pattern. This framework describes the coexistence lines in a morphological phase diagram, and determines the domain of existence of faceted structures.