Euler buckling on curved surfaces

Abstract

Euler buckling epitomises mechanical instabilities: An inextensible straight elastic line buckles under compression when the compressive force reaches a critical value $F_{\ast }>0$. Here, we extend this classical, planar instability to the buckling under compression of an inextensible relaxed elastic line on a curved surface. By weakly nonlinear analysis of an asymptotically short elastic line, we reveal that the buckling bifurcation changes fundamentally: The critical force for the lowest buckling mode is $F_{\ast }=0$ and higher buckling modes disconnect from the undeformed branch to connect in pairs. Solving the buckling problem numerically, we additionally find a new post-buckling instability: A long elastic line on a curved surface snaps through under sufficient compression. Our results thus set the foundations for understanding the buckling instabilities on curved surfaces that pervade the emergence of shape in biology.

18 pages, 2 figures