Ordered buckling structures in a twisted crimped tube

Abstract

When a ribbon or tube is twisted far enough it forms buckles and wrinkles. Its new geometry can be strikingly ordered, or hopelessly disordered. Here we study this process in a tube with hybrid boundary conditions: one end a cylinder, and the other end crimped flat like a ribbon, so that the sample resembles a toothpaste tube. The resulting irregular structures and mechanical responses can be dramatically different from those of a ribbon. However, when we form two creases in the tube prior to twisting, we obtain an ordered structure composed of repeating triangular facets oriented at varying angles, and a more elastic torque response, reminiscent of the creased helicoid structure of a twisted ribbon. We measure how the torque and structural evolution depend on parameters such as material thickness and the twist angle. When only part of the tube is pre-creased, the ordered structures are confined to this segment. Surprisingly, in some tubes made from thicker sheets, an ordered structure forms without pre-creasing. This study provides insights into controlling the buckling of thin shells, offering a potential pathway for designing ordered structures in soft materials.

Figures (8)

• Figure 1: Buckling morphologies of three slender structures under tension and twist. All samples have $t=38$$\mu$m, $W=13.5$ mm, $L=108$ mm. (a) A crimped tube evolves into a disordered twisted structure, under constant tension $T=0.5$ N. (b) A pre-creased crimped tube develops an ordered creased helicoid structure, emanating from the crimped end, for $T=0.5$ N. The structure is reminiscent of a buckling pattern in ribbons Korte2010. (c) Twisted ribbon, for comparison with the crimped tube. We set $T=0.25$ N for the ribbon as the emergent structure in panel (b) is effectively a double-layered ribbon. Figure \ref{['fig:sm1']} shows the same sequences with increments of $30^\circ$.
• Figure 2: Mechanical response of the three samples in Fig. \ref{['fig:360_twist']}. Samples are twisted from $0^\circ$ to $360^\circ$ and back to $0^\circ$. Because the ribbon width is half the circumference of the tubes, we plot twice the value of the measured torque for the ribbon. The data show that pre-creasing a crimped tube brings its mechanical response close to that of a ribbon.
• Figure 3: How the morphology varies with twist, tension, and thickness. (a--c) Each panel shows a different sheet thickness. The vertical axis is the twist angle $\theta$, scaled by $W/L$ to convert to the pitch of an equivalent helix. Results are robust under a variation of the aspect ratio, indicated by the symbol shape. Open circles in (b) indicate the 60$^\circ$ and 360$^\circ$ images in Fig. \ref{['fig:360_twist']}(b). (d) Qualitative phase diagram that labels the colors used in (a--c), and that captures the results for all three thicknesses.
• Figure 4: Buckling of four twisted crimped tubes that are partially pre-creased. (a) Schematic of a crimped tube with a fraction $\ell/L$ pre-creased (red lines along its sides), which leads to ordered facets in a fraction $H/L$ (shaded region). (b) Photos of samples twisted to $360^\circ$ with pre-creased fractions $\ell/L=0.125$, $0.25$, $0.5$, and $0.75$. The samples have $t=38~\mu$m, $W=25.5$ mm, and $L=408$ mm, and the crimped end is at the top. (c) We measure the length $H$ of the region that develops ordered facets when each tube is twisted. Error bars represent the uncertainty in assessing the end of the ordered faceted region.
• Figure 5: Buckling morphology of thick, crimped tubes with no pre-creasing. All samples have $W=13.5$ mm and $L=108$ mm. The thicknesses are: (a) $t=76~\mu$m, (b) $t=102~\mu$m, (c) $t=127~\mu$m, and the normal forces are $T = 1$ N, 1.33 N, and 1.67 N, in proportion to thickness. The $76~\mu$m tube develops ordered facets, despite the absence of pre-creasing to direct the buckling pathway.