The 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 the deformation of crystalline solids. We take the analogy literally 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 first 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 the inclination of the plane reaches a critical angle. We find that the angle at which this first occurs is larger than the angle at which the ruck stops, i.e. static rolling friction is larger than dynamic rolling friction. Once the ruck is in motion, it travels at a constant speed proportional to the sine of the angle of inclination, a result that we rationalize in terms of a simple power balance. We conclude with a simple implication of our study for the onset of rolling motion at soft interfaces.