A sequence of elastic patterns in a sheared bent sheet

TL;DR

This work investigates the formation of localized crumples in thin elastic sheets subjected to shear, revealing a snaking-like sequence of bifurcations as patterns evolve from a unimodal to multimodal state. The authors perform quasi-static boundary displacement experiments on polyester sheets, capturing force-displacement loops that exhibit nested, overlapping branches corresponding to different numbers of crumple pairs (O-valleys and S-ridges). They identify coarsening and refinement mechanisms, including splitting, merging, and open-boundary escape, and show that while most forces scale with sheet length, the initial nucleation drop decreases with length. The results connect experimental observations to theoretical concepts of snaking in pattern-forming systems and suggest avenues for path-following analyses in buckling PDE models.

Abstract

We document a sequence of bifurcations and elastic patterns in sheared bent sheets of intermediate aspect ratio. The sheets undergo inversion of curvature through the passage of localized features, often in S-shaped pairs. Nested force-displacement hysteresis loops provide experimental evidence for snaking. Several mechanisms for coarsening and refinement of the patterns are observed, including splitting, merging, and escape through open boundaries. While most forces, including that required for full snap-through, scale with the length of the sheet, the initial drop in force upon pattern nucleation decreases rapidly with length.

Figures (15)

• Figure 1: Experimental setup showing a laterally translating, instrumented boundary, and a 12 inch long ($L$), 4 inch wide ($W$), 0.003 inch thick sheet of polyester. Top: sheet is bent into a quarter-circle in the unsheared unimodal U state. Bottom: the lower boundary has moved rightward parallel to the fixed boundary, shearing the sheet into the multimodal M state, in which two small puckers are also visible.
• Figure 2: Unfiltered (red) and filtered (black) force-displacement loops, traversed clockwise. The black curve is the same as that in Figure \ref{['force_displacement_6in_alt']} in Section \ref{['results']} below.
• Figure 3: Spacetime plot corresponding to Video \ref{['vidgreen']}. Horizontal locations of upper (grey) and lower (dark grey) crumples are given in pixels, with a correspondence of approximately 135-138 px/in on the secant plane, with values decreasing upwards in the sheet, and unknown error due to projection of the elevated center of the sheet. Grey lines indicate the spacetime locations of the fixed (solid) and moving (dotted, note slight slopes and jumps at the break) corners of the sheet. Some fast motions are indicated with arrows. Pair positions like the configuration at around 105 seconds correspond to a vertically oriented S-ridge. Details in text.
• Figure 4: Force-displacement loops, traversed clockwise, for a 5 inch long sheet: five full loops in black and two partial loops in orange. Some intermediate stable states are shown in the insets. Details in text.
• Figure 5: Four force-displacement loops, traversed clockwise, for a 6 inch long sheet: increasing shear in red and decreasing shear in blue. Insets show stable states with two pairs of crumples in S-ridge and O-valley arrangements. Details in text.