Shape and Motion of a Ruck in a Rug

The motion of a ruck in a rug is used as an analogy to explain the role of dislocations in crystalline solids. We take literally one side of this analogy and study the shape and motion of a bump, wrinkle or ruck in a thin sheet in partial contact with a rough substrate in a gravitational field. Using a combination of experiments, scaling analysis and numerical solutions of the governing equations, we quantify the static shape of a ruck on a horizontal plane. When the plane is inclined, the ruck becomes asymmetric and moves by rolling only when the inclination of the plane reaches a critical angle, at a speed determined by a simple power balance. We find that the angle at which rolling starts is larger than the angle at which the ruck stops; i.e., static rolling friction is larger than dynamic rolling friction. We conclude with a generalization of our results to wrinkles in soft adherent extensible films.

Figure 1

(a) A ruck in a thin latex rug on a flat substrate, of height $\Delta$, length measured along the arc $S$ and horizontal extent $S-ϵ$. Here $ϵ$ is analogous to the Burgers vector. (b) A schematic shows the ruck on a plane at an angle of inclination $\theta$. The upper right hand corner shows the local stress and moment resultants for a segment of length $ds$, where $\varphi$ is the angle between the local ruck tangent and the plane.

Figure 2

(a) The scaled ruck height $\Delta$ as a function of the excess length $ϵ$, for different thicknesses [$h=0.25\mathrm{mm}$ (+), 0.5 mm (⋄), 0.75 mm (△), 1.0 mm (*), 1.5 mm (○)] measured experimentally and calculated numerically by solving (1, 2, 3) (solid line) collapse onto a single curve. All lengths are normalized by the elastic gravity length, ${\ell }_g=(\frac{E{h}^2}{\rho g}{)}^{1/3}$. The inset shows the numerically calculated values (○) of $\Delta$ as a function of $ϵ$ on a log-log plot and yields a power law $\Delta ={ϵ}^{0.54}$ which compares well with the scaling law $\Delta \sim {ϵ}^{4/7}$ given by (4). (b) Ruck shape as a function of the excess length, $ϵ$; the solid lines correspond to experimental measurements, while the dashed lines correspond to numerical solutions of (1, 2, 3). (c) The ruck shape on an inclined plane with $\sin \theta =0.3$ shows an increasing asymmetry as $ϵ$ increases. The solid lines correspond to experiments and the dashed lines correspond to numerical solutions of (1, 2, 3).

Figure 3

(a) The path of a point on the rug as a ruck (in white *) and time-lapse images of the ruck for $ϵ=1.25$, $\theta =20°$, $V=0.7\mathrm{m}/\mathrm{s}$. The inset shows the path of a point on the ruck as it approaches its zenith (*), compared with the curve $\eta \sim {\xi }^{2/3}$ that characterizes the trajectory of a point on a cycloidal curve characterizing the rolling of a rigid circle. (b) The numerically calculated shape of a static ruck ($ϵ=3.5$, $\theta =17°$—dashed line), and the path of a point on the rug (inset, solid line) as the ruck moves through it.

Figure 4

(a) The tilt for the onset of rolling ${\theta }_g$ and its arrest ${\theta }_s$ as a function of the shape of the ruck, characterized by the ratio of the excess length to the contour length $ϵ/S$. Since ${\theta }_s>{\theta }_g$ dynamic rolling friction is less than static rolling friction. (b) Ruck speed $V$ vs $\sqrt{\sin \theta }$ when $\theta >{\theta }_s$ for $ϵ=0.5$, 0.8 and 1.25, is consistent with $V\sim \sqrt{\sin \theta }$ (see text). The inset shows the dynamic distortion of the shape of the ruck due to air drag ($\sin \theta =0.6$, $ϵ=1.25$). Velocity experiments used a latex sheet of thickness $h=0.75\mathrm{mm}$.