Buckling of a Nonlinearly Elastic Column: Variational Principles, Bifurcation and Asymptotics

Abstract

We study the equilibrium states of an elastic column subject to an end thrust. The column is modeled as an inextensible rod with a nonlinear moment-curvature equation. We bring various mathematical theories to bear on this problem; the calculus of variations, bifurcation theory, phase plane analysis, and singular perturbation theory for ordinary differential equations. We study the connections between the various approaches.Many of the results presented here are not new. Our object is to show how this problem can serve to illustrate various abstract theories and to show how the various approaches compliment each other.