Folding pathways to crumpling a sheet with cyclic wringing

We will discuss an experimental study in which a cylindrically rolled inextensible sheet is repeatedly wrung about its axis by slowly twisting and untwisting the clamped ends through a prescribed angle. For sufficiently large twist angles and aspect ratios, the sheet buckles, folds, and gives rise to a disordered pattern of intersecting creases as the sheet crushes onto itself by contracting both along its length and width. We measure the end-to-end distance between the clamped ends as a function of twist angle and find that it is hysteretic as the sheet follows a slightly different path during the twisting phase compared with untwisting, with the area enclosed decreasingly with the number of cycles. Monitoring a cross section, we find that the creased features can repeat over hundreds of cycles but then undergo sudden rearrangements. The frequency of these sudden increases with prescribed twist angles. By scanning the sheet surface, we find that the mean curvature of a crease increases logarithmically as a function of the number of the folding and unfolding cycles. We show that this crease-weakening leads the structure of the crumpled sheet to evolve slowly till a snap-through event occurs leading to rapid reconfiguration of its structure. Thus, we demonstrate that a repeatedly crumpled sheet can follow a complex seemly reversible folding and unfolding pathway over long periods without converging.