Continuation of equilibria and stability of slender elastic rods using an asymptotic numerical method

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 orientati...

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Bibliographic Details
Published in:Journal of the mechanics and physics of solids 2013-08, Vol.61 (8), p.1712-1736
Main Authors: Lazarus, A., Miller, J.T., Reis, P.M.
Format: Article
Language:English
Subjects:
Asymptotic properties
Elastic rods
Engineering Sciences
Equilibrium
Equilibrium equations
Mathematical models
Mechanics
Mechanics of materials
Path-following techniques
Quadratic forms
Quaternions
Rods
Stability
Three dimensional
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Summary: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 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 and heavy 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. We believe that our framework is a powerful alternative to other methods for the computation of nonlinear equilibrium 3D shapes of rods in practical scenarios.
ISSN:0022-5096
DOI:10.1016/j.jmps.2013.04.002