Helicoids, Wrinkles, and Loops in Twisted Ribbons

Julien Chopin and Arshad Kudrolli

Phys. Rev. Lett. 111, 174302 – Published 25 October 2013

Abstract

We investigate the instabilities of a flat elastic ribbon subject to twist under tension and develop an integrated phase diagram of the observed shapes and transitions. We find that the primary buckling mode switches from being localized longitudinally along the length of the ribbon to transverse above a triple point characterized by a crossover tension that scales with ribbon elasticity and aspect ratio. Far from threshold, the longitudinally buckled ribbon evolves continuously into a self-creased helicoid with focusing of the curvature along the triangular edges. Further twist causes an anomalous transition to loops compared with rods due to the self-rigidity induced by the creases. When the ribbon is twisted under high tension, transverse wrinkles are observed due to the development of compressive stresses with higher harmonics for greater width-to-length ratios. Our results can be used to develop functional structures using a wide range of elastic materials and length scales.

Received 17 April 2013

DOI:https://doi.org/10.1103/PhysRevLett.111.174302

Authors & Affiliations

Julien Chopin^* and Arshad Kudrolli^†

Department of Physics, Clark University, Worcester, Massachusetts 01610, USA

^*jchopin@coc.ufrj.br
^†akudrolli@clarku.edu

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Vol. 111, Iss. 17 — 25 October 2013

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Images

Figure 1
(a) A schematic of the ribbon illustrating the applied tension and twist at the boundaries. Images of a helicoid (b), a longitudinally buckled helicoid (c), a transversely buckled helicoid (d), a creased helicoid (e), and a ribbon with a localized loop (f). The images all correspond to a uniform green biaxially oriented polyethylene terephthalate ribbon ( $W = 12.7 mm$ and $h = 76 μ m$ ). (g) The phase diagram of the shapes observed along with the critical twist angle for the longitudinal buckling ( $θ_{L}$ , solid circle) and the transverse buckling ( $θ_{T}$ , ▿) as a function of the tension ( $L = 45 cm$ , $W = 25.4 mm$ , and $h = 127 μ m$ ). Below a crossover tension $T^{⋆}$ , the buckling is longitudinal. A secondary discontinuous buckling transition leading to a loop with self-contact is observed (solid triangle) upon further twist. A transverse buckled mode is observed for $T > T^{⋆}$ . Upon further twisting, a loop with self-contact develops (open circle).Reuse & Permissions
Figure 2
(a),(b) Ribbon shapes before and after longitudinal buckling. (c) Creasing regime at higher twist. The mean curvature is measured from 3D x-ray scans of the ribbon and superimposed on the extracted shape according to the color map shown. (d) $θ_{L}$ as a function of $T$ . All critical angles lie above $θ_{0} = \sqrt{24 T}$ (dashed line), which corresponds to the theoretical critical twist angle for ribbons of vanishing thicknesses. (e) $θ_{L} - θ_{0}$ as a function of $T$ . The symbol corresponds to the mean and the vertical bar to the standard deviation. For ribbons with finite thicknesses, $θ_{L}$ is offset by $9.3 h / W$ . Same symbols as in (d). (f) The longitudinal buckling wavelength $λ_{L}$ normalized by $\sqrt{h W}$ decreases with $T$ in good agreement with Eq. (6).Reuse & Permissions
Figure 3
(a)–(c) Measured transverse cross section of ribbons ( $x = 0$ ) twisted above $θ_{T}$ illustrating different buckling harmonics depending on $h / W$ and $L / W$ ( $E \approx 1 MPa$ ). (d) Evolution of the critical angle $θ_{T}$ for the transverse buckling with the aspect ratio $L / W$ of the ribbons. A given symbol corresponds to a fixed length and thickness and varying width. Four different lengths in the range 10–90 cm are investigated with thicknesses $h = 76 μ m$ (★), $254 μ m$ (diamond), and $127 μ m$ (square, down triangle, right triangle). A dashed line drawn as a guide for the eyes separates the ribbon regime (large $L / W$ ) with a plate regime (small $L / W$ ). (e) Evolution of $θ_{T}$ with $h / W$ in the ribbon regime. All the data [same symbols as in (d)] collapse on a single curve $θ_{T} = 4.4 (h / W)^{1 / 2}$ independent to $L$ as further shown in the corresponding log-log plot in the inset.Reuse & Permissions
Figure 4
(a) Evolution of the critical angle $θ_{T}$ for the transverse buckling with tension [black (blue) solid triangle, gray (red) solid triangle, open triangle, solid lines] using a ribbon with $W = 25.4 mm$ , $h = 127 μ m$ , and $L = 45 cm$ . Twist angles where self-contact is first observed while increasing $θ$ [open circle, black (blue) dashed line], and where self-contact is no longer observed while decreasing $θ$ [solid circle, gray (red) dashed line]. Above $T^{⋆}$ , the transverse buckling (open triangle) and the formation of a self-contact (open circle) are successive and no significant hysteresis is observed. Below $T^{⋆}$ , when increasing $θ$ , a loop with a self-contact is formed and $θ_{T}$ decreases linearly with $T$ [black (blue) solid triangle]. When decreasing $θ$ in the striped area, the loss of self-contact (solid circle) and the transition to a creased helicoid [gray (red) solid triangle] are successive. (b) Mean curvature $H$ of ribbon with loop ( $W = 15 mm$ , $L = 10 cm$ , and $h = 127 μ m$ ). (c) Gaussian curvature $K$ of the same ribbon. Above the transition, the Gaussian curvature vanishes whereas the mean curvature is localized in the loop region.Reuse & Permissions

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