The Wrinkling of a Twisted Ribbon

Recent experiments by Chopin and Kudrolli (Phys Rev Lett 111:174302, 2013) showed that a thin elastic ribbon, when twisted into a helicoid, may wrinkle in the center. We study this from the perspective of elastic energy minimization, building on recent work by Chopin et al. (J Elast 119(1–2):137–189, 2015) in which they derive a modified von Kármán functional and solve the relaxed problem. Our main contribution is to show matching upper and lower bounds for the minimum energy in the small-thickness limit. Along the way, we show that the displacements must be small where we expect that the ribbon is helicoidal, and we estimate the wavelength of the wrinkles.