Stability Landscape of Shell Buckling

We measure the response of cylindrical shells to poking and identify a stability landscape, which fully characterizes the stability of perfect shells and imperfect ones in the case where a single defect dominates. We show that the landscape of stability is independent of the loading protocol and the poker geometry. Our results suggest that the complex stability of shells reduces to a low dimensional description. Tracking ridges and valleys of this landscape defines a natural phase-space coordinates for describing the stability of shells.

Figure 1

(a) Schematics of the experimental setup. (b) The lateral poker is a steel marble glued to a screw; two different marble diameters were tested. (c) Axial load $F_A$ and poker force $F_p$ for a typical experiment: (i) the shell is compressed to $F_A^0$, and then the gap is fixed, (ii) the poker is advanced until buckling occurs. (iii) The probe catches up with the buckled shell (iv).

Figure 2

(a) Force-displacement curves for seven cans. Insets: axial load measurements. (b) Three-dimensional representation of independent poker force measurements (1 in Table 1). The surface generated by the curves defines a landscape, which characterizes the stability of the thin shell. The color indicates $F_p$. (c) Three-dimensional representation of consecutive non-destructive poker force measurements on the same can, probing reversible elastic deformations without reaching buckling (2 in Table 1).

Figure 3

All individual cans share universal stability features. (a) Definitions of the main features. (b) $D_p^{\max }$ at $F_p^{\max }$ vs $F_A^0$. Small dots, large dots and circles correspond to the protocols 1, 3 and 4 respectively, indicated in Table 1. Inset: single can tests. (c) $F_p^{\max }$ vs $F_A^0$. (d) Critical poker displacement $D_p^c$ required to trigger buckling (magenta) and displacement $D_p^s$ to catch up with the post-buckled surface (green) vs $F_A^0$. Inset, hollow squares: the shell is first poked and then loaded (5 in Table 1). The poker displacement $D_p^{\min }$ at the minimum of poking force is plotted with blue stars, and the minimum poker force $F_p^{\min }$ is reported in the second inset. (e) The elastic energy barrier, $E_p$ vs $F_A^0$.

Figure 4

A schematic demonstration of the stability landscape of shell buckling. The ridge is defined as the trajectory of the maxima of poker force $[D_p^{\max }(F_A^0),F_p^{\max }(F_A^0)]$. Following the ridge down to zero poker force leads to the spontaneous buckling (s.b.). The valley is defined as the trajectory of the local force minima $[D_p^{\min }(F_A^0),F_p^{\min }(F_A^0)]$. Following the valley down to zero poker force leads to a minimally buckled state (m.b.), below which no buckling is possible. The magenta and the green dashed lines corresponds to the unstable ($D_p^c$) and stable ($D_p^s$) fixed points, respectively.