Ineffective perturbations in a planar elastica (Q970220)

An elastica is a bendable one-dimensional continuum, or idealized elastic rod. When it is subjected to compression while its ends are constrained to remain tangent to a single straight line, buckling may occur. In this paper, the bifurcation for buckling of a planar elastica under a load \(\lambda\) is investigated. In particular, the perturbation of the bifurcation diagram is analysed due to the addition of a small intrinsic shape in the elastica. It is found that the difference between the buckled solution and the trivial solution is \(O(\varepsilon^{1/3})\) for almost all intrinsic shapes \(\varepsilon f(s)\) but the difference is \(O(\varepsilon)\) for some ineffective \(f\). The important consequences in numerical simulations of these ineffective perturbations are also pointed out.