Lifting, Loading, and Buckling in Conical Shells

Liquid crystal elastomer films that morph into cones are strikingly capable lifters. Thus motivated, we combine theory, numerics, and experiments to reexamine the load-bearing capacity of conical shells. We show that a cone squashed between frictionless surfaces buckles at a smaller load, even in scaling, than the classical Seide-Koiter result. Such buckling begins in a region of greatly amplified azimuthal compression generated in an outer boundary layer with oscillatory bend. Experimentally and numerically, buckling then grows subcritically over the full cone. We derive a new thin-limit formula for the critical load, $\propto t^{5/2}$, and validate it numerically. We also investigate deep postbuckling, finding further instabilities producing intricate states with multiple Pogorelov-type curved ridges arranged in concentric circles or Archimedean spirals. Finally, we investigate the forces exerted by such states, which limit lifting performance in active cones.

Figure 1

(a) A $2\times 2$ array of heat-activated LCE cones supporting a load $1100\times$ their own weight [5]. (b) Compressing a conical shell between frictionless slides. (c) Controlled-force compression of experimental (upper images) and numerical (lower images, force-compression plot) LCE cones ($\alpha \approx 60°$, $R=200t$, $\nu =1/2$), both exhibiting subcritical buckling. Using numerical and experimental (Supplemental Material [14], Sec. S9A) modulus values, both buckling thresholds are $\sim 20\%$ of the classical Seide threshold $\sim 3\mathrm{mN}$.

Figure 2

(a) Stress profiles shortly before buckling for the cone in Fig. 1, with (beige) morphoshell and (dashed) boundary-layer (8) results agreeing well. The region of large compressive azimuthal stress near the boundary drives boundary-layer buckling at $f_{*}\propto t^2\sqrt{t/R}$, but is absent from the membrane base state (dotted). (b) Transitory morphoshell topographies just after boundary buckling, labelled by $t/R$, colored by elevation, with smaller thicknesses yielding larger azimuthal mode numbers. (c) Buckling mode shapes $\mathsf{w}(\mathsf{s})$ from numerical linear stability analyses (dashed, $\alpha =60°$) using our boundary-layer base state, converging to our thin-limit theory (solid) as $t/R$ decreases by factors of 10. (d) Buckling force against $t/R$ for $\nu =1/2$ and various $\alpha$, with numerics converging to our thin-limit result (12).

Figure 3

(a) Vertical force against height ratio (initial/squashed) for an LCE cone compressed quasistatically in morphoshell ($\alpha =60°$, $R=100t$), with (un)squashing in (teal) purple. (b) Enlarged view of the small-compression region, where the unsquashing single-ridge force agrees well with our theory (dotted). (c) Height-colored experimental topographies of a square-type LCE cone, with two successive compressions of the same sample (left, middle) exhibiting both concentric-circle and spiral ridges, and (right) a spiral ridge in a similar simulated cone.

Figure 4

(a) Profile sketch of a concentric-ridged lifter. (b) morphoshell confirms that concentric ridges are roughly equispaced while (c) spiral ridges are Archimedean. (d) Final height of a concentric-ridged lifter ($\alpha \approx 60°$, $R\approx 100t$) against weight lifted from a many-ridged state. Our simple theory (line) agrees well with morphoshell (markers) for small weights, but significantly underestimates lifting performance at large weights.