J Elast (2011) 105:137–170 DOI 10.1007/s10659-010-9290-5 Wrinkling of a Stretched Thin Sheet Eric Puntel · Luca Deseri · Eliot Fried Received: 8 October 2010 / Published online: 23 December 2010 © Springer Science+Business Media B.V. 2010 Abstract 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 for- mation 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. Moti- vated 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 consid- eration, 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 Dedicated to the memory of Donald E. Carlson, whose insight and clarity of thought were exceeded only by his modesty and generosity. E. Puntel Dipartimento di Georisorse e Territorio, Università di Udine, via Cotonificio 114, 33100 Udine, Italy L. Deseri Dipartimento di Ingegneria Meccanica e Strutturale, Università di Trento, via Mesiano 77, 38023 Trento, Italy E. Fried ( ) Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montréal, Québec H3A 2K6, Canada e-mail: eliot.fried@mcgill.ca