Vol. 2, No. 5, 2009

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ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Ineffective perturbations in a planar elastica

Kaitlyn Peterson and Robert Manning

Vol. 2 (2009), No. 5, 559–580
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.

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