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

Handle URI:
http://hdl.handle.net/10754/599726
Title:
Static and dynamic stability results for a class of three-dimensional configurations of Kirchhoff elastic rods
Authors:
Majumdar, Apala; Goriely, Alain
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.
Publisher:
Elsevier BV
Journal:
Physica D: Nonlinear Phenomena
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
Jun-2013
DOI:
10.1016/j.physd.2013.03.003
Type:
Article
ISSN:
0167-2789
Sponsors:
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.
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Full metadata record

DC FieldValue Language
dc.contributor.authorMajumdar, Apalaen
dc.contributor.authorGoriely, Alainen
dc.date.accessioned2016-02-28T06:08:24Zen
dc.date.available2016-02-28T06:08:24Zen
dc.date.issued2013-06en
dc.identifier.citationMajumdar 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.en
dc.identifier.issn0167-2789en
dc.identifier.doi10.1016/j.physd.2013.03.003en
dc.identifier.urihttp://hdl.handle.net/10754/599726en
dc.description.abstractWe 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.en
dc.description.sponsorshipAM 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.en
dc.publisherElsevier BVen
dc.subjectDynamic stabilityen
dc.subjectElastic rodsen
dc.subjectEnergy minimizersen
dc.subjectEuler bucklingen
dc.subjectLocal drag modelsen
dc.subjectStatic stabilityen
dc.titleStatic and dynamic stability results for a class of three-dimensional configurations of Kirchhoff elastic rodsen
dc.typeArticleen
dc.identifier.journalPhysica D: Nonlinear Phenomenaen
dc.contributor.institutionUniversity of Bath, Bath, United Kingdomen
dc.contributor.institutionUniversity of Oxford, Oxford, United Kingdomen
kaust.grant.numberKUK-C1-013-04en
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