Deformation of loops in 2D packing of flexible rods

TL;DR

This study investigates how the shape of loops in 2D flexible rod packings relates to local deformations and forces, using experiments and elastic theory to model and analyze the system.

Contribution

It introduces a model linking loop shape to deformation and force in 2D flexible rod packings, validated by experiments and image analysis.

Findings

Loop dimensions follow the same deformation dependence regardless of confinement.

The elastic model based on Euler's elasticas accurately predicts forces and stored energy.

Experimental compression forces agree with theoretical predictions.

Abstract

The injection of a long flexible rod into a two-dimensional domain yields a complex pattern commonly studied through elasticity theory, packing analysis, and fractal geometries. "Loop" is a one-vertex entity that is naturally formed in this system. The role of the elastic features of each loop in 2D packing has not yet been discussed. In this work, we point out how the shape of a given loop in the complex structure allows estimating local deformations and forces. First, we build sets of symmetric free loops and performed compression experiments. Then, tight packing configurations are analyzed by using image processing. We found that the dimensions of the loops, confined or not, obey the same dependence on the deformation. The result is consistent with a simple model based on 2D elastic theory for filaments, where the rod adopts the shape of Euler's elasticas between its contact points. The force and the stored energy are obtained from numerical integration of the analytic expressions. In an additional experiment, we obtain that the compression force for deformed loops corroborates the theoretical findings. The importance of the shape of the loop is discussed and we hope that the theoretical curves may allow statistical considerations in future investigations.