Cylinder morphology of a stretched and twisted ribbon

A rich zoology of morphologies emerges from a simple stretched and twisted elastic ribbon. Despite a lot of interest, all the observed shapes are not quantitatively described. This is the case of the cylindrical shape that prevails at large tension and twist, which emerges from a transverse buckling instability of the helicoid. Here, we propose a simple description of this cylindrical shape. By comparing its energy to the energy of other configurations, helicoidal and facetted, we are able to determine its location on the tension-twist phase diagram. The theoretical predictions are in good quantitative agreement with the experimental results and complement previous results from linear stability analysis.

Figure 1

Pictures of the different morphologies: (a) helicoid, (b) facets with isometric ridges (FIR), and (c) cylinder; from [13].

Figure 2

Phase diagram with transitions between helicoid, cylinder, and facetted shapes at (a) moderate tension and (b) low tension [expansion of the gray domain in (a)]. Experimental data for $t=6.5\times {10}^{-3}$ (filled symbols, PET ribbons), data of Ref. [7] (open symbols) and theoretical curves. Thin solid line, longitudinal linear instability of the helicoid [11]; dashed-dotted line, transverse linear instability [8]; thick solid line, helicoid-cylinder transition from energy comparison [Eq. (B2)]; dashed line, helicoid facets with isometric ridges transition from energy comparison [Eq. (18)]; thick dash-dotted line, helicoid-cylinder transition at low tension from the energy comparison [Eq. (17)]; thin dotted line, low tension limit of definition of the facets with isometric ridges.

Figure 3

Cylindrical configurations determined theoretically for $\eta =0.5$, $t=6.5\times {10}^{-3}$, and different values of the tension $T$. The parameters of the cylinder, the radius $R$ and the contraction $\chi$, are represented for $T=7\times {10}^{-6}$. The dark green line represents a line parallel to the center line of the ribbon.

Figure 4

Evolution of the measured cylinder radius for twisted PET ribbons ($t\simeq 6.5\times {10}^{-3}$) in the three regimes: (a) fixed length, (b) vanishing tension, and (c) moderate tension ($T=0.005$, 0.007). The solid lines correspond to the values predicted by the theoretical model. There are no fitting parameters.

Figure 5

(a) Evolution of the tension with $\eta$ at constant length showing the helicoid-cylinder transition [$T={\eta }^2/24$ for the helicoid and $T\simeq 0.28{\left(\eta t\right)}^{2/3}$ for the cylinder]. Squares correspond to helicoidal configurations and circles to cylindrical configurations; the color indicates the thickness, from 0.3 [red (gray)] to $0.9\mathrm{mm}$ (black). (b) Helicoid-cylinder transition at fixed length for various ribbon thicknesses.

Figure 6

Radius $R$ (a), contraction $\chi$ (b), and energy $U_{\text{cyl}}$ (c) of a cylinder as a function of the tension $T$ for $t=6.5\times {10}^{-3}$ and $\eta =0.5$. The numerical minimization (solid lines) of the energy is compared to the analytical expressions obtained in the small tension (dotted lines, $R=1/\eta ,\chi =1,U_{\text{cyl}}={\eta }^2{t}^2/24$) and moderate tension (dashed lines) regimes. When values of the contraction or the energy are negative, their absolute value is plotted in gray. The scaling laws for ${\left(\eta t\right)}^2\ll T\ll {\left(\eta t\right)}^{2/3}$ are indicated.