Involve: A Journal of Mathematics

  • Involve
  • Volume 2, Number 5 (2009), 559-580.

Ineffective perturbations in a planar elastica

Kaitlyn Peterson and Robert Manning

Full-text: Open access

Abstract

An elastica is a bendable one-dimensional continuum, or idealized elastic rod. If such a rod is subjected to compression while its ends are constrained to remain tangent to a single straight line, buckling can occur: the elastic material gives way at a certain point, snapping to a lower-energy configuration.

The bifurcation diagram for the buckling of a planar elastica under a load λ is made up of a trivial branch of unbuckled configurations for all λ and a sequence of branches of buckled configurations that are connected to the trivial branch at pitchfork bifurcation points. We use several perturbation expansions to determine how this diagram perturbs with the addition of a small intrinsic shape in the elastica, focusing in particular on the effect near the bifurcation points.

We find that for almost all intrinsic shapes ϵf(s), the difference between the buckled solution and the trivial solution is O(ϵ13), but for some ineffective f, this difference is O(ϵ), and we find functions uj(s) so that f is ineffective at bifurcation point number j when f,uj=0. These ineffective perturbations have important consequences in numerical simulations, in that the perturbed bifurcation diagram has sharper corners near the former bifurcation points, and there is a higher risk of a numerical simulation inadvertently hopping between branches near these corners.

Article information

Source
Involve, Volume 2, Number 5 (2009), 559-580.

Dates
Received: 12 February 2009
Accepted: 2 May 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513799211

Digital Object Identifier
doi:10.2140/involve.2009.2.559

Mathematical Reviews number (MathSciNet)
MR2601577

Zentralblatt MATH identifier
1197.34079

Subjects
Primary: 34B15: Nonlinear boundary value problems 34E10: Perturbations, asymptotics 34G99: None of the above, but in this section 74K10: Rods (beams, columns, shafts, arches, rings, etc.)

Keywords
elastic rod intrinsic shape undetermined-gauges perturbation expansion pitchfork bifurcations

Citation

Peterson, Kaitlyn; Manning, Robert. Ineffective perturbations in a planar elastica. Involve 2 (2009), no. 5, 559--580. doi:10.2140/involve.2009.2.559. https://projecteuclid.org/euclid.involve/1513799211


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