Wrinkling of a stretched thin sheet

When a thin rectangular sheet is clamped along two opposing edges and stretched, its inability to accommodate the Poisson contraction near the clamps may lead to the formation of wrinkles with crests and troughs parallel to the axis of stretch. A variational model for this phenomenon is proposed. The relevant energy functional includes bending and membranal contributions, the latter depending explicitly on the applied stretch. Motivated by work of Cerda, Ravi-Chandar, and Mahadevan, the functional is minimized subject to a global kinematical constraint on the area of the mid-surface of the sheet. Analysis of a boundary-value problem for the ensuing Euler-Lagrange equation shows that wrinkled solutions exist only above a threshold of the applied stretch. A sequence of critical values of the applied stretch, each element of which corresponds to a discrete number of wrinkles, is determined. Whenever the applied stretch is sufficiently large to induce more than three wrinkles, previously proposed scaling relations for the wrinkle wavelength and, modulo a multiplicative factor that depends on the Poisson ratio of the sheet and the applied stretch and arises from the more general and weaker nature of geometric constraint under consideration, root-mean-square amplitude are confirmed. In contrast to the scaling relations for the wrinkle wavelength and amplitude, the applied stretch required to induce any number of wrinkles depends on the in-plane aspect ratio of the sheet. When the sheet is significantly longer than it is wide, the critical stretch scales with the fourth power of the length-to-width ratio but, when the sheet is significantly wider than it is long, the critical stretch scales with the square of that same ratio.