Euler buckling on curved surfaces

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.