Periodic folding of a falling viscoelastic sheet

A viscoelastic solid sheet fed from a certain height towards a rigid horizontal plane folds on itself provided that there is no slip. This phenomenon commonly occurs in the manufacturing process of textile and paper products. In this paper we apply a particle dynamics model to investigate this phenomenon. At a low feeding velocity and low viscosity, the inertial effect and the viscous dissipation within the sheet are negligible, and our model successfully reproduces the existing quasistatic results in the gravitational regime. As the feeding velocity and the viscosity of the sheet increase, the folding process changes significantly. The length of the folds decrease and the "rolling back" motion of the sheet vanishes. In the inertial regime, a scaling law between the fold length and the feeding velocity is derived by balancing the kinetic energy and the elastic bending energy involved in folding, which is verified by the simulation. It is found that above a critical feeding velocity, the folding morphology transforms from line contact into point contact with the sheet exhibiting a lemniscate-like pattern. Finally, a phase diagram for the folding morphology is constructed. The results presented in this work may offer some insights into the high-speed manufacturing of paper and fabric sheets.