Geometry and Physics of Wrinkling

The wrinkling of thin elastic sheets occurs over a range of length scales, from the fine scale patterns in substrates on which cells crawl to the coarse wrinkles seen in clothes. Motivated by the wrinkling of a stretched elastic sheet, we deduce a general theory of wrinkling, valid far from the onset of the instability, using elementary geometry and the physics of bending and stretching. Our main result is a set of simple scaling laws; the wavelength of the wrinkles lambda approximately K(-1/4), where K is the stiffness due to an "elastic substrate" effect with a multitude of origins, and the amplitude of the wrinkle A approximately lambda. These could form the basis of a highly sensitive quantitative wrinkling assay for the mechanical characterization of thin solid membranes.