Universality in constrained thin sheets

Suppose you hang a curtain down before your window. Set the x-axis and y-axis in the vertical and horizontal direction, respectively. The upper edge (at x=0) of the curtain is somewhat constrained by the reel bar. Certain kind of creases develop that can be described by the out-of-plane deformation, a function z(x,y). Very generally, this function is largely sinusoidal along y given x, with a wavelength lambda(x). More interestingly, such crumpling pattern is also seen in much smaller thin sheets, such as graphene. Is there any universal manner to all thin sheets with constrained boundaries ? The answer is yes. The work has just been done to demonstrate it [PRL 106, 224301 (2011)]. A key result is that, lambda follows a simple power law, namely, lambda~x^m, where m for light sheet (2/3) is different from that for heavy sheet (1/2), thus showing a hierarchy. The amazing thing is the universality.

We show that thin sheets under boundary confinement spontaneously generate a universal self-similar hierarchy of wrinkles. From simple geometry arguments and energy scalings, we develop a formalism based on wrinklons, the localized transition zone in the merging of two wrinkles, as building blocks of the global pattern. Contrary to the case of crumpled paper where elastic energy is focused, this transition is
described as smooth in agreement with a recent numerical work [R. D. Schroll, E. Katifori, and B. Davidovitch, Phys. Rev. Lett. 106, 074301 (2011)]. This formalism is validated from hundreds of nanometers for graphene sheets to meters for ordinary curtains, which shows the universality of our description. We finally describe the effect of an external tension to the distribution of the wrinkles.