Euler Buckling on Curved Surfaces

Euler buckling epitomizes mechanical instabilities: an inextensible straight elastic line in the plane buckles under compression when the compressive force F reaches a critical value F_{*}>0. But how does an elastic line buckle within a general curved surface? Here, we reveal that the classical instability changes fundamentally: by weakly nonlinear analysis of the buckling of an asymptotically short elastic line, we show that the critical force for the lowest buckling mode is F_{*}=0 and discover a new bifurcation structure in which the modes of classical Euler buckling split into pairs. For long elastic lines, we numerically find an additional bifurcation by which the second of these new modes becomes the lowest mode and show that, at sufficiently large F, they snap discontinuously to higher end-to-end compression. Our results constitute the foundations for a class of buckling instabilities that arise within curved surfaces, for example when biological shape emerges in development.