Growth of a flexible fibre in a deformable ring

TL;DR

This paper investigates how an elastic fibre grows within a deformable ring, leading to buckling and folding phenomena, with theoretical, numerical, and experimental validation of the equilibrium configurations based on key dimensionless parameters.

Contribution

It introduces a simplified model for the growth-induced buckling and folding of an elastic fibre in a deformable ring, supported by finite element simulations and experiments.

Findings

Fibre buckles beyond a critical length depending on stiffness ratio.

Fibre folding distorts the ring and breaks mirror symmetry.

Theoretical predictions match experimental observations well.

Abstract

We study the equilibrium configurations related to the growth of an elastic fibre in a confining flexible ring. This system represents a paradigm for a variety of biological, medical, and engineering problems. We consider a simplified geometry in which initially the container is a circular ring of radius $R$. Quasi-static growth is then studied by solving the equilibrium equations as the fibre length $l$ increases, starting from $l = 2R$. Considering both the fibre and the ring as inextensible and unshearable, we find that beyond a critical length, which depends on the relative bending stiffness, the fibre buckles. Furthermore, as the fibre grows further it folds, distorting the ring until it induces a break in mirror symmetry at $l>2 \pi R$. We get that the equilibrium shapes depend only on two dimensionless parameters: the length ratio $\mu = l/R$ and the bending stiffnesses ratio $\kappa$. These findings are also supported by finite element simulation. Finally we experimentally validate the theoretical results showing a very good quantitative prediction of the observed buckling and folding regimes at variable geometrical parameters.