CAM Colloquium - Timothy Healey: Wrinkling of Highly Stretched Elastic Sheets

Friday, April 8, 2016

Wrinkling of Highly Stretched Elastic Sheets: Modeling, Computation and Analysis

When a finely thin, rectangular, elastic sheet is highly stretched in the longer direction, transverse wrinkles often develop – think of a sheet of sandwich wrap. Much like the famous Euler buckling of a compressed thin rod, this can be treated as a bifurcation problem: the flat, unwrinkled state bifurcates to the wrinkled state as the longitudinal 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, Föppl von Kármán (FvK) plate theory, is woefully inadequate for the problem at hand:

We propose a new model incorporating finite nonlinear elasticity for the membrane behavior, accompanied by very 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 from the FvK model.
We consider the true test for our model – a mathematical existence theorem. This raises some new and interesting questions in the calculus of variations, some of which we can answer while others remain open.