Elastic Snap-Through Instabilities Are Governed by Geometric Symmetries

Many elastic structures exhibit rapid shape transitions between two possible equilibrium states: umbrellas become inverted in strong wind and hopper popper toys jump when turned inside out. This snap through is a general motif for the storage and rapid release of elastic energy, and it is exploited by many biological and engineered systems from the Venus flytrap to mechanical metamaterials. Shape transitions are known to be related to the type of bifurcation the system undergoes, however, to date, there is no general understanding of the mechanisms that select these bifurcations. Here we analyze numerically and analytically two systems proposed in recent literature in which an elastic strip, initially in a buckled state, is driven through shape transitions by either rotating or translating its boundaries. We show that the two systems are mathematically equivalent, and identify three cases that illustrate the entire range of transitions described by previous authors. Importantly, using reduction order methods, we establish the nature of the underlying bifurcations and explain how these bifurcations can be predicted from geometric symmetries and symmetry-breaking mechanisms, thus providing universal design rules for elastic shape transitions.

Figure 1

(a),(b) Elastic buckled strip with clamped-clamped boundary conditions exhibits two symmetric stable equilibria $U_A$ and $U_B$, and pairs of unstable equilibria of alternating symmetry at increasing energy levels, $S_A$ and $S_B$ denoting the first unstable pair ($S_B$ not shown). [(c)–(f)] Actuation of buckled strip by (quasistatically) translating its left end by a distance $d$ leads to loss of bistability and a shape transition that depends on BCs: (c),(e) CH strip exhibits a violent snap through, (d),(f) the transition in the CC strip is smooth. [(a)–(d)] 3D computer graphics rendering of the Cosserat numerical simulations. (e),(f) Midpoint deflection $w/\sqrt{L\mathrm{\Delta }L}$ versus bifurcation parameter $d/\sqrt{L\mathrm{\Delta }L}$. In all figures, (green) square markers represent data obtained based on the Cosserat rod theory. Solid (green) and dashed (brown) lines represent, respectively, stable and unstable branches obtained from the Euler beam model.

Figure 2

Boundary actuation and bifurcation diagrams from (quasistatically) rotating one or both clamped ends in (a) asymmetric, (b) symmetric, and (c) antisymmetric fashion. [(d)–(f)] Midpoint deflection as a function of bifurcation parameter $\mu =\alpha \sqrt{L/\mathrm{\Delta }L}$. Experimental data from [4] are superimposed. [(g)–(i)] Asymptotic analysis near the bifurcation point ${\mu }^{*}$ gives access to normal forms describing the amplitude $A(t)$ of the leading order mode. Bifurcation diagrams of the normal forms (black lines) agree quantitatively with data obtained from the Euler beam equations (green and brown lines) and Cosserat simulations (green and brown square markers), and experimental data.

Figure 3

(a) Energy landscape at $\mu =0$: two potential wells at the two stable equilibria $U_{A,B}$ separated by lowest energy barriers at the first pair of unstable equilibria $S_{A,B}$. The two paths connecting $U_A$ to $U_B$ via either $S_A$ or $S_B$ constitutes the energetically cheapest routes to pass from $U_A$ to $U_B$. (b) 1D periodic representation of energy landscape. [(c)–(e)] Rotating one or both of the boundaries reshapes the energy landscape: breaking both $U$- and $S$-twin symmetries leads to a saddle-node bifurcation; breaking either $U$- or $S$-twin symmetry leads to a pitchfork bifurcation.

Figure 4

(a) Tapered elastic strip under antisymmetric boundary rotation. (b) Midpoint deflection (green symbols) versus bifurcation parameter exhibits snap through as opposed to the graceful merging of a homogeneous strip subject to the same actuation (black lines). (c) Critical slowing down near the bifurcation scales as $(\mathrm{\Delta }\mu {)}^{1/4}$ as in the case of a saddle node.