Ineffective perturbations in a planar elastica

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 $\lambda$ is made up of a trivial branch of unbuckled configurations for all $\lambda$ 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 $ϵf\left(s\right)$, the difference between the buckled solution and the trivial solution is $O\left({ϵ}^{1∕3}\right)$, but for some ineffective $f$, this difference is $O\left(ϵ\right)$, and we find functions $u_j\left(s\right)$ so that $f$ is ineffective at bifurcation point number $j$ when $\langle f,u_j\rangle =0$. 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.