A quaternion-based continuation method to follow the equilibria and stability of slender elastic rods

TL;DR

This paper introduces a quaternion-based numerical continuation framework for analyzing the equilibrium configurations and stability of slender elastic rods, enabling efficient bifurcation tracking and comparison with experimental results.

Contribution

The authors develop a novel quaternion-based finite element method combined with asymptotic continuation to efficiently compute equilibrium branches and stability of elastic rods.

Findings

Accurate prediction of buckling instabilities in curved rods

Quantitative agreement between simulations and experiments

Efficient computation of complex equilibrium configurations

Abstract

We present a theoretical and numerical framework to compute bifurcations of equilibria and stability of slender elastic rods. The 3D kinematics of the rod is treated in a geometrically exact way by parameterizing the position of the centerline and making use of quaternions to represent the orientation of the material frame. The equilibrium equations and the stability of their solutions are derived from the mechanical energy which takes into account the contributions due to internal moments (bending and twist), external forces and torques. Our use of quaternions allows for the equilibrium equations to be written in a simple quadratic form and solved efficiently with an asymptotic numerical continuation method. This finite element perturbation method gives interactive access to semi-analytical equilibrium branches, in contrast with the individual solution points obtained from classical minimization or predictor-corrector techniques. By way of example, we apply our numerics to address the specific problem of a naturally curved rod under extreme twisting and perform a detailed comparison against our own precision model experiments of this system. Excellent quantitative agreement is found between experiments and simulations for the underlying 3D buckling instabilities and the characterization of the resulting complex configurations.