Euler buckling in a potential field

P. Holmes, G. Domokos, G. Hek

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

We consider elastic buckling of an inextensible rod with free ends, confined to the plane, and in the presence of distributed body forces derived from a potential. We formulate the geometrically nonlinear (Euler) problem with nonzero preferred curvature, and show that it may be written as a three-degree-of-freedom Hamiltonian system. We focus on the special case of an initially straight rod subject to body forces derived from a quadratic potential uniform in one direction; in this case the system reduces to two degrees of freedom. We find two classes of trivial (straight) solutions and study the primary non-trivial branches bifurcating from one of these classes, as a load parameter, or the rod's length, increases. We show that the primary branches may be followed to large loads (lengths) and that segments derived from primary solutions may be concatenated to create secondary solutions, including closed loops, implying the existence of disconnected branches. At large loads all finite energy solutions approach homoclinic and heteroclinic orbits to the other class of straight states, and we prove the existence of an infinite set of such 'spatially chaotic' solutions, corresponding to arbitrary concatenations of 'simple' homoclinic and heteroclinic orbits. We illustrate our results with numerically computed equilibria and global bifurcation diagrams.

Original languageEnglish
Pages (from-to)477-505
Number of pages29
JournalJournal of Nonlinear Science
Volume10
Issue number4
Publication statusPublished - Jul 2000

Fingerprint

Euler buckling
Potential Field
potential fields
Buckling
Straight
Euler
Heteroclinic Orbit
Branch
Homoclinic Orbit
Degree of freedom
rods
Inextensible
Global Bifurcation
Concatenation
Bifurcation Diagram
Orbits
elastic buckling
degrees of freedom
Closed-loop
Hamiltonian Systems

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mathematics(all)
  • Applied Mathematics
  • Mathematical Physics
  • Statistical and Nonlinear Physics

Cite this

Euler buckling in a potential field. / Holmes, P.; Domokos, G.; Hek, G.

In: Journal of Nonlinear Science, Vol. 10, No. 4, 07.2000, p. 477-505.

Research output: Contribution to journalArticle

Holmes, P, Domokos, G & Hek, G 2000, 'Euler buckling in a potential field', Journal of Nonlinear Science, vol. 10, no. 4, pp. 477-505.
Holmes, P. ; Domokos, G. ; Hek, G. / Euler buckling in a potential field. In: Journal of Nonlinear Science. 2000 ; Vol. 10, No. 4. pp. 477-505.
@article{8bf8b045e5bb4235a21e5e22eec7c15d,
title = "Euler buckling in a potential field",
abstract = "We consider elastic buckling of an inextensible rod with free ends, confined to the plane, and in the presence of distributed body forces derived from a potential. We formulate the geometrically nonlinear (Euler) problem with nonzero preferred curvature, and show that it may be written as a three-degree-of-freedom Hamiltonian system. We focus on the special case of an initially straight rod subject to body forces derived from a quadratic potential uniform in one direction; in this case the system reduces to two degrees of freedom. We find two classes of trivial (straight) solutions and study the primary non-trivial branches bifurcating from one of these classes, as a load parameter, or the rod's length, increases. We show that the primary branches may be followed to large loads (lengths) and that segments derived from primary solutions may be concatenated to create secondary solutions, including closed loops, implying the existence of disconnected branches. At large loads all finite energy solutions approach homoclinic and heteroclinic orbits to the other class of straight states, and we prove the existence of an infinite set of such 'spatially chaotic' solutions, corresponding to arbitrary concatenations of 'simple' homoclinic and heteroclinic orbits. We illustrate our results with numerically computed equilibria and global bifurcation diagrams.",
author = "P. Holmes and G. Domokos and G. Hek",
year = "2000",
month = "7",
language = "English",
volume = "10",
pages = "477--505",
journal = "Journal of Nonlinear Science",
issn = "0938-8974",
publisher = "Springer New York",
number = "4",

}

TY - JOUR

T1 - Euler buckling in a potential field

AU - Holmes, P.

AU - Domokos, G.

AU - Hek, G.

PY - 2000/7

Y1 - 2000/7

N2 - We consider elastic buckling of an inextensible rod with free ends, confined to the plane, and in the presence of distributed body forces derived from a potential. We formulate the geometrically nonlinear (Euler) problem with nonzero preferred curvature, and show that it may be written as a three-degree-of-freedom Hamiltonian system. We focus on the special case of an initially straight rod subject to body forces derived from a quadratic potential uniform in one direction; in this case the system reduces to two degrees of freedom. We find two classes of trivial (straight) solutions and study the primary non-trivial branches bifurcating from one of these classes, as a load parameter, or the rod's length, increases. We show that the primary branches may be followed to large loads (lengths) and that segments derived from primary solutions may be concatenated to create secondary solutions, including closed loops, implying the existence of disconnected branches. At large loads all finite energy solutions approach homoclinic and heteroclinic orbits to the other class of straight states, and we prove the existence of an infinite set of such 'spatially chaotic' solutions, corresponding to arbitrary concatenations of 'simple' homoclinic and heteroclinic orbits. We illustrate our results with numerically computed equilibria and global bifurcation diagrams.

AB - We consider elastic buckling of an inextensible rod with free ends, confined to the plane, and in the presence of distributed body forces derived from a potential. We formulate the geometrically nonlinear (Euler) problem with nonzero preferred curvature, and show that it may be written as a three-degree-of-freedom Hamiltonian system. We focus on the special case of an initially straight rod subject to body forces derived from a quadratic potential uniform in one direction; in this case the system reduces to two degrees of freedom. We find two classes of trivial (straight) solutions and study the primary non-trivial branches bifurcating from one of these classes, as a load parameter, or the rod's length, increases. We show that the primary branches may be followed to large loads (lengths) and that segments derived from primary solutions may be concatenated to create secondary solutions, including closed loops, implying the existence of disconnected branches. At large loads all finite energy solutions approach homoclinic and heteroclinic orbits to the other class of straight states, and we prove the existence of an infinite set of such 'spatially chaotic' solutions, corresponding to arbitrary concatenations of 'simple' homoclinic and heteroclinic orbits. We illustrate our results with numerically computed equilibria and global bifurcation diagrams.

UR - http://www.scopus.com/inward/record.url?scp=0004507501&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0004507501&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0004507501

VL - 10

SP - 477

EP - 505

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 4

ER -