Technical Brief: Finite Element Modeling of Tight Elastic Knots

TL;DR

This paper introduces a comprehensive 3D finite element modeling approach to analyze the complex mechanics of tight elastic knots, validated by experiments, revealing significant 3D deformation effects overlooked by simpler models.

Contribution

The paper presents a novel 3D finite element methodology for simulating tight elastic knots, capturing detailed deformation and contact mechanics beyond previous simplified theories.

Findings

3D deformation is crucial for understanding tight knot mechanics.

Finite element simulations match experimental results with high accuracy.

Tight knots exhibit complex behaviors not predicted by 1D models.

Abstract

We present a methodology to simulate the mechanics of knots in elastic rods using geometrically nonlinear, full three-dimensional (3D) finite element analysis. We focus on the mechanical behavior of knots in tight configurations, for which the full 3D deformation must be taken into account. To set up the topology of our knotted structures, we apply a sequence of prescribed displacement steps to the centerline of an initially straight rod that is meshed with 3D solid elements. Self-contact is enforced with a normal penalty force combined with Coulomb friction. As test cases, we investigate both overhand and figure-of-eight knots. Our simulations are validated with precision model experiments, combining rod fabrication and X-ray tomography. Even if the focus is given to the methods, our results reveal that 3D deformation of tight elastic knots is central to their mechanical response. These findings contrast to a previous analysis of loose knots, for which 1D centerline-based rod theories sufficed for a predictive understanding. Our method serves as a robust framework to access complex mechanical behavior of tightly knotted structures that are not readily available through experiments nor existing reduced-order theories.