Buckling mediated by mobile localized elastic excitations

TL;DR

Thin elastic sheets undergo buckling transitions that are mediated by mobile localized excitations called crumples or d-cones, which nucleate at regions where surface generators converge and propagate to induce global shape changes. High-speed experiments reveal two stable crumple pair configurations, S-ridge and O-valley, that organize larger-scale patterns, and show that the transient crumple size scales with thickness approximately as the cube root, in line with crescent-like feature theory. A geometry-based origamization model predicts forward snap-through behavior without fitting parameters, while transient dynamics and ridge/valley interactions shape both forward and return paths, highlighting a dynamic mechanism for buckling in thin plates and shells. The findings provide a framework for predicting energy scales and pattern formation in thin-film and shell structures.

Abstract

Experiments reveal that structural transitions in thin sheets are mediated by the passage of transient and stable mobile localized elastic excitations. These ``crumples'' or ``d-cones'' nucleate, propagate, interact, annihilate, and escape. Much of the dynamics occurs on millisecond time scales. Nucleation sites correspond to regions where generators of the ideal unstretched surface converge. Additional stable intermediate states illustrate two forms of quasistatic inter-crumple interaction through ridges or valleys. These interactions create pairs from which extended patterns may be constructed in larger specimens. The onset of localized transient deformation with increasing sheet size is correlated with a characteristic stable crumple size, whose measured scaling with thickness is consistent with prior theory and experiment for localized elastic features in thin sheets. We offer a new theoretical justification of this scaling.

Figures (5)

• Figure 1: (a) Sheets of width $W$ (an overbar will denote normalization by this quantity) and length $L$ are clamped in a reference quarter-cylinder U state. A parallel lateral displacement $u$ is applied to a flat side, leading to formation of a ridge and sharpening of features, then snap-through to the M state. (b) Critical displacement $\bar{u}$ for bifurcation away from the U state (forward) and M state (return) as a function of the aspect ratio $\bar{L}$, for several sheet thicknesses $h$ and widths $W$. Some error bars are horizontally shifted for clarity. The theoretical line $\bar{u}_c= 0.06\bar{L}^{-1}$ is when the sharpening of features in the U state terminates in an "origamized" limit (Appendix \ref{['origami']}); this has no fitting parameters, but uses assumptions valid for large $\bar{L}$. (c) Principal hysteresis loop in an $\bar{L} = 1.2$, $\bar{h} = 5\times 10^{-4}$ sheet: forward shear from U to M via stable S-ridge, return shear from M to U via stable O-valley (see also vidsite). Paths between the two intermediate states involve other states (not shown, see also vidsite). (d) Crumple diameter $\bar{d}_c$ (measured as described in Appendix \ref{['crumplemeasure']}), and minimum aspect ratio $\bar{L}$ for localization during the forward and return bifurcations, as a function of the thickness $\bar{h}$. Some error bars are horizontally shifted for clarity.
• Figure 2: Transient phenomena mediating localized snap-through of narrow sheets of thickness $\bar{h} = 5\times 10^{-4}$. (a) Aspect ratio $\bar{L} = 0.175$, U to M transition: a pair of crumples nucleate near the central ridge of the strip and take a nonlinear path out through the sides (6000 fps, see also vidsite). (b) $\bar{L} = 0.2$, U to M transition: a single crumple nucleates in a highly curved corner, the edge of a ridge, and travels to the other side (2000 fps, see also vidsite). (c) $\bar{L} = 0.6$, M to U transition: shallow crumples nucleate near inflection points on the sides, travel inward and imperfectly annihilate (4000 fps, see also vidsite).
• Figure 3: Complex transient phenomena leading to stable crumple pairs in larger sheets of aspect ratio $\bar{L} = 1.2$, thickness $\bar{h} = 5\times 10^{-4}$. Crumple positions continue to oscillate after the last frames shown here. (a) In the U to M transition, events begin with nucleation of a transient valley near a corner, which triggers a cascade. Two (arrowed) crumples eventually remain to form an S-ridge (4000 fps, see also vidsite). (b) In the M to U transition, stable puckers are initially present. A shallow crumple (arrowed) nucleated near an inflection point begins a series of events leading to a stable O-valley (4000 fps, see also vidsite).
• Figure 4: Geometry (a) of the sheet and (b) of its projection on to the secant plane, when the U state origamizes. Details of the calculation are in the text.
• Figure 5: Example of a crumple measurement (red circles), for a thickness $\bar{h} = 4\times 10^{-4}$ sheet. The dark line is for calibration.