Static and dynamic stability results for a class of three-dimensional configurations of Kirchhoff elastic rods

Majumdar, Apala; Goriely, Alain

Static and dynamic stability results for a class of three-dimensional configurations of Kirchhoff elastic rods

Type

Article

Authors

Majumdar, Apala
Goriely, Alain

KAUST Grant Number

KUK-C1-013-04

Date

2013-06

Abstract

We analyze the dynamical stability of a naturally straight, inextensible and unshearable elastic rod, under tension and controlled end rotation, within the Kirchhoff model in three dimensions. The cases of clamped boundary conditions and isoperimetric constraints are treated separately. We obtain explicit criteria for the static stability of arbitrary extrema of a general quadratic strain energy. We exploit the equivalence between the total energy and a suitably defined norm to prove that local minimizers of the strain energy, under explicit hypotheses, are stable in the dynamic sense due to Liapounov. We also extend our analysis to damped systems to show that static equilibria are dynamically stable in the Liapounov sense, in the presence of a suitably defined local drag force. © 2013 Elsevier B.V. All rights reserved.

Citation

Majumdar A, Goriely A (2013) Static and dynamic stability results for a class of three-dimensional configurations of Kirchhoff elastic rods. Physica D: Nonlinear Phenomena 253: 91–101. Available: http://dx.doi.org/10.1016/j.physd.2013.03.003.

Acknowledgements

AM is supported by an EPSRC Career Acceleration Fellowship, EP/J001686/1, an OCCAM Visiting Fellowship and a Keble Research Fellowship, University of Oxford (till October 2012). AM would like to thank the Oxford Center for Collaborative Applied Mathematics for its hospitality over the months of August-October 2012, during which this work was completed. This publication is based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). AG is a Wolfson Royal Society Merit Holder and is supported by a Reintegration Grant under EC Framework VII. The authors thank John Maddocks for helpful discussions and for drawing their attention to the crucial role of polar singularities in the second variation analysis. The authors also thank Sebastien Neukirch for helpful discussions on isoperimetric constraints.