Ineffective perturbations in a planar elastica

Peterson, Kaitlyn; Manning, Robert

doi:10.2140/involve.2009.2.559

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Ineffective perturbations in a planar elastica

Kaitlyn Peterson and Robert Manning

Vol. 2 (2009), No. 5, 559–580

DOI: 10.2140/involve.2009.2.559

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 $ϵ f (s)$ , the difference between the buckled solution and the trivial solution is $O (ϵ^{1 ∕ 3})$ , but for some ineffective $f$ , this difference is $O (ϵ)$ , and we find functions $u_{j} (s)$ so that $f$ is ineffective at bifurcation point number $j$ when $〈 f, u_{j} 〉 = 0$ . 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.

Keywords

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

Mathematical Subject Classification 2000

Primary: 34B15, 34E10, 34G99, 74K10

Milestones

Received: 12 February 2009

Accepted: 2 May 2009

Published: 13 January 2010

Communicated by Natalia Hritonenko

Authors

Kaitlyn Peterson
Mathematics Department Haverford College 370 Lancaster Ave. Haverford, PA 19041 United States

Robert Manning
Mathematics Department Haverford College 370 Lancaster Ave. Haverford, PA 19041 United States

Vol. 2, No. 5, 2009