Hierarchy of Geometrical Frustration in Elastic Ribbons: shape-transitions and energy scaling obtained from a general asymptotic theory

TL;DR

This paper develops an asymptotic theory to analyze shape transitions and energy scaling in geometrically frustrated elastic ribbons, explaining how their configurations depend on width and thickness with experimental validation.

Contribution

It introduces a general asymptotic approach to predict shape transitions and energy scaling in frustrated ribbons, covering various morphological behaviors.

Findings

Identifies critical widths for shape transitions in different ribbon types.

Predicts energy scaling laws for frustrated ribbons.

Experimental results confirm theoretical predictions across systems.

Abstract

Geometrically frustrated elastic ribbons exhibit, in many cases, significant changes in configuration depending on the relation between their width and thickness. We show that the existence of such a transition, and the scaling at which it occurs, strongly depend on the system considered. Using an asymptotic approach, treating the width as a small parameter, we find the leading energy terms resulting from the frustration and predict the existence and scaling of the shape transition. We study in detail 5 different types of frustrated ribbons with a different morphological dependence on ribbon's width: a sharp shape-transition at a critical width, a moderate transition with an intermediate regime, and no transition at all. We show that the predictions of our approach match experimental results from two different experimental systems: prestressed rubber bilayers and 4D printed thermoplastics, in a wide variety of geometric settings.