Tim J. Healey (Department of Mathematics, Cornell University, USA). Wrinkling of Highly Stretched Elastic Sheets: Modeling, Computation and Analysis.

Transverse wrinkles often develop when a finely thin, rectangular, elastic sheet is highly stretched in the longer direction – think of a sheet of sandwich wrap.   Much like the famous Euler buckling of a compressed thin rod, this can be analyzed as a bifurcation problem:  The flat, unwrinkled state bifurcates to the wrinkled state as the applied macroscopic stretch is slowly increased.  This idea is well known, and the problem has been widely popularized in recent years.  Here we argue that the commonly employed model, namely, the Föppl von Kármán (FvK) theory of plates, is woefully inadequate for the problem at hand:  
 
  •    We propose a new model incorporating finite nonlinear elasticity for the membrane behavior, accompanied by small, non-zero bending stiffness. 
  •    We numerically analyze our model via bifurcation/path-following techniques in multiple parameters to uncover realistic wrinkling behavior, while    revealing erroneous predictions by the FvK model.
  •    We consider the true test of our model - a mathematical existence theorem.  Based on the minimization of potential energy, this raises new questions in the calculus of variations, which we address.