Statics and Inertial Dynamics of a Ruck in a Rug

We consider the familiar problem of a bump, or ruck, in a rug. Under lateral compression, a rug bends to form a ruck—a localized region in which it is no longer in contact with the floor. We show that when the external force that created the ruck is removed, the ruck flattens out unless the initial compression is greater than a critical value, which we determine. We also study the inertial motion of a ruck that is generated when one end of the rug is moved rapidly. We show that the equations of motion admit a traveling ruck solution for which a linear combination of the tension and kinetic energy is determined by the ruck size. We confirm these findings experimentally. We end by discussing the potential implications of our work for the analogous propagation of localized slip pulses in the sliding of two bodies in contact.

Figure 1

The ruck shape observed in a sheet of natural rubber with thickness $h=0.75\mathrm{mm}$ for different imposed end-end compressions $\Delta L\equiv \Delta l/{\ell }_g=0.3$, 2.7 and 4.2.

Figure 2

Regime diagram showing the regions of static friction-length ($\mu L_0$), compression ($\Delta L$) parameter space for which a ruck is observed to stick (i.e., remain) or slip (i.e., flatten out) when the external compressive force is removed. The numerically computed boundary between sticking and slipping (solid curve) compares well with the asymptotic result (6) (dashed curve) for $\Delta L\ll 1$. The unit of length is ${\ell }_g$ (2).

Figure 3

Experimental results for the dynamic propagation of a ruck in a mylar sheet ($h=125\mu \mathrm{m}$). (a) The horizontal position of the peak of the ruck as a function of time (points). The speed in the steady state phase, $c(g{\ell }_g{)}^{1/2}$, is the gradient of the best fit line (dashed line) in the region where the shape is steady. (b) The shape of the traveling ruck at six instants of time. [The time of each profile is given by the position of the corresponding symbol in (a)]. The solution of the heavy elastica equation (3) with $\Delta L=1.05$ is shown by the solid curve.

Figure 4

Main figure: Logarithmic plot of the effective tension, $\sigma +c^2$, (see text) measured experimentally for rucks in mylar sheets as a function of the dimensionless ruck height, $\delta \equiv d/{\ell }_g$. Experimental points are shown for different sheet thicknesses and substrates (hence different values of the friction coefficient $\mu$) as shown in the legend. The values measured of the friction coefficients are $\mu =0.13$ (copper), $\mu =0.41$ (carpet) and $\mu =0.16$ (wood). The dependence of $\sigma +c^2$ on $\delta$ determined from the numerical solution of (9) is plotted as the solid curve while the asymptotic result (10) is plotted as the dashed line. Inset: Linear plot of dimensional raw data showing the dependence of ruck speed $c(g{\ell }_g{)}^{1/2}$ on ruck height $d$.