Snapping and Switching of Elastic Arches with Patterned Preferred Curvature

Abstract

An elastic arch is an archetypal bistable system. Here, we combine elastica theory and photo-mechanical experiments to elucidate the mechanics of an active arch with a spatio-temporally varying preferred curvature $\overline κ(s)$. Our shallow-arch theory completely describes any such system via the decomposition of its $\overline κ(s)$ into Euler-buckling modes. Intuitively, if $\overline κ(s)$ overlaps with the fundamental mode, it snaps the arch up/down. Conversely, non-overlapping $\overline κ(s)$ drives a second-order transition to a higher-order shape. Furthermore, the form of $\overline κ(s)$ enables control over the instability's character; we find the forms for snapping with maximum energy release and at the lowest stimulation (both binary patterns) and design forms for symmetric and asymmetric switching pathways. Analogous control can also be achieved in boundary-driven instabilities of passive arches by fabricating them with suitable $\overline κ(s)$. We thus anticipate our results will improve switchable/snapping elements in MEMS, robotics, and mechanical meta-materials.

Figures (4)

• Figure 1: Snapping of arches with active curvature. (a) A beam of length $L$ is first compressed by $\epsilon L$ to form an arch, which snaps down upon uniform curvature stimulation of the central $70\%$. (b) Theoretical (blue) and experimental (black) hysteresis loop of center height vs curvature magnitude showing both way snapping control. (c) Theoretical force-compression curve showing transition from bi- to mono-stability at high stimulation.
• Figure 2: Snapping condition and subcritical deformation. (a) Compression spectrum and (inset) center heights of an arch stimulated with $\overline \kappa_1$. (b) Snapping at $f_m<f_1$ by stimulating with $\overline \kappa= \alpha_0 \kappa_0+\alpha_1 \kappa_1$ and (c) the corresponding phase diagram showing bistability in the dashed region and monostability outside.
• Figure 3: Optimized stimulus profiles. (a) Elastic energy released during the snap and (b) snap-through thresholds for offset rectangular $\overline \kappa$, inset: optimal profiles. Comparison of center heights for (c) energy optimized $\overline \kappa$ and (d) threshold optimized $\overline \kappa$, theoretical design (blue) and experiment (black).
• Figure 4: Controlling symmetry of snap-through transition. Stimulation profiles $\overline \kappa_\mathrm{rect}|_{\gamma=0.7}$ and $\overline \kappa_\mathrm{sym}$, and the corresponding mid-snap images.