Euler buckling on curved surfaces

TL;DR

This work extends classical Euler buckling to an inextensible elastic line constrained to a curved surface, using both asymptotic analysis for a short line and numerical solutions for longer lines. By contrasting intrinsic (geodesic) and extrinsic (normal-curvature) bending, it shows that curvature can reduce the lowest buckling threshold to $F_* = 0$ when symmetry is broken ($AB\neq0$) and induces higher-mode pairing, while intrinsic buckling shifts the threshold by Gaussian curvature terms. A novel post-buckling snap-through instability is found for long lines on curved surfaces, with the instability region depending on surface parameters and curvature sign. These findings establish a foundation for understanding buckling-driven shape changes on curved biological surfaces and have potential implications for tissue morphogenesis and related biophysical phenomena.

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.

Figures (2)

• Figure 1: Euler buckling in the plane and on a curved surface. (A) Classical Euler buckling in the plane. An inextensible elastic line of length $\ell$, clamped at one end, relaxes into a straight, geodesic shape. Clamping of the other end and compression by a force $F$ along the straight shape leads to buckling. The relative compression is $\delta$. "Up" and "down" buckling (solid and dashed line in the bottom panel) is equivalent by symmetry. (B) Euler buckling on a curved surface $z=h(x,y)$. The relaxed elastic line differs, in general, from the "straight" geodesic, and the curvature of the surface selects either "up" or "down" buckling. (C) Bifurcation diagram of classical Euler buckling for $\ell\ll 1$: Plot of relative compression $\delta$ against compressive force $F$, from asymptotic analysis for $\delta=O(\ell^2)$. The first buckling mode (inset) appears at the critical force $F_\ast=F_1^\pm$ for "up" or "down" buckling. Higher buckling modes (insets) have higher critical forces $F_2^\pm,F_3^\pm,\dots$. (D) Corresponding bifurcation diagram for a general curved surface: "Up" and "down" buckling modes disconnect and the asymptotic analysis at order $O(\ell^2)$ breaks down. Asymptotics for $\delta=O(\ell^4)$ show that $F_\ast=0$ and that higher modes connect in pairs. Asymptotics for $\delta=O(\ell^{4/3})$ show that they undergo a snap-through instability.
• Figure 2: Numerical Euler buckling on curved surfaces. (A) Force-compression diagram for the lowest buckling mode on a surface of Gaussian curvature $K>0$ (${A=B=C=1}$), for short and long elastic lines ($\ell=0.1$, $\ell=1.5$). For $\ell=0.1$, the forces $F_0\neq F_1$ at either end increase monotonically with the compression $\delta$. For $\ell=1.5$, a snap-through instability arises ($\partial F_0/\partial \delta<0$, $\partial F_1/\partial \delta<0$; arrows). (B) Analogous bifurcation diagram for a surface with $K<0$ (${-A=B=C=1}$). (C) Plot of the surface, relaxed elastic line (black), and buckled line (blue) in panel (A). (D) Analogous plot for panel (B). (E) Phase diagram for this snap-through instability for an elastic line of fixed length $\ell=1$ in surface parameter space $(A,B,C)$, for $A,B\geqslant 0$. This is extended to all $A,B$ by the symmetries $(B,y)\mapsto-(B,y)$ and $(A,B,C)\mapsto-(A,B,C)$ of the problem.