Frictional sliding strength of knotted and capstan configurations along the axis of a cylinder

TL;DR

This study analyzes the frictional sliding strength of knotted and capstan configurations on cylinders, combining experiments, simulations, and analytical modeling to understand nonlinear behaviors and transitions to linearity.

Contribution

It introduces a comprehensive mechanical framework explaining superlinear frictional behavior in knot and capstan systems across various materials and geometries.

Findings

Both FEM and DER simulations replicate experimental superlinear scaling.

Capstan angles influence the transition from superlinear to linear behavior.

The analytical model captures the coupling of forces and contact length during tightening.

Abstract

We investigate the sliding strength of thin filaments in frictional contact with a translating cylinder, perpendicular to the filaments' axes, in knotted (clove hitch) and unknotted (capstan) configurations. Recent work reported superlinear scaling for surgical knots with elasto-plastic filaments [1]. Testing the clove hitch with various materials (elastomeric rods, metallic wires, braided ropes) reveals similar nonlinear behavior, ruling out plasticity. To explore the source of the previously reported nonlinear behavior, we perform three-dimensional FEM simulations (resolving full 3D mechanics) and reduced-order DER simulations (isolating geometric effects by neglecting cross-sectional deformation). Both FEM and DER simulations reproduce the experimental scaling. Simplifying the knot topology by studying capstan angles from $\pi/4$ to $4\pi$ yields comparable superlinear behavior, transitioning to linearity at smaller angles. We rationalize the results by developing an analytical model based on planar elastica theory for the capstan configuration (which exhibits behavior similar to the clove hitch but with a simpler topology). The model reproduces the observed superlinear behavior and rationalizes it by coupling the evolution of normal forces and contact arclength during tightening. The analysis further predicts transition to linearity when full contact between the filament and the cylinder is established, providing a mechanical framework applicable across materials, geometries, and topologies.