Wrinkling Instability Induced by Geometrical Frustration

Thin elastic sheets are characterized by their deformability and their tendency to undergo mechanical instabilities. One common such instability is wrinkling. Wrinkling is familiar in sheets that adhere to a soft substrate or fluid interface under compression (e.g., skin), and the interplay between the substrate deformation and the bending of the sheet determines their wavelength. Surprisingly, wrinkles can appear also in isolated thin sheets that undergo non-uniform growth, yet their properties are still not well understood.

In this work, we investigate wrinkles of the second type, from a geometrical point of view. We use latex sheets that undergo differential swelling – one half of the sheet is exposed to a solvent that swells it, while the other is not. The two areas meet at a sharp transition. This geometry corresponds to two cylinders of different radii joined by a smooth transition region, like a wine bottle. However, we show that when the radius of one of the cylinders is fixed to a non-equilibrium value, a smooth isometric embedding of the system is no longer possible, and longitudinal wrinkles appear to prevent the stretching of the material. We study the origin of this phenomenon and identify the scaling of the wavelength and size of the wrinkled domain. We find that when the radius grows to infinity, the system behaves like hanging drapes. The study results are relevant to other cases where differential swelling induces wrinkling instability, such as in plants and in synthetic responsive materials.