The Lame` problem: A prototypical model for wrinkling in thin sheets
Wednesday, May 18, 2011 - 10:30am - 11:30am
Wrinkling is a fundamental mechanism for the relief of compressive stress in thin elastic sheets. It is natural to consider wrinkling as a (supercritical) instability of an appropriate flat, highly-symmetric state of the sheet. This talk will address the subtlety of this approach by considering wrinkling in the Lame` geometry: an annular sheet under radial tension. This axi-symmetric system seems to be the most elementary, yet nontrivial extension of Euler buckling (that emerges under uniaxial compression). Nevertheless, despite its apparent simplicity, the Lame` geometry exhibits a dramatic change of the wrinkling pattern beyond the instability threshold. I will address the distinct features of wrinkling patterns in the near-threshold (NT) and far-from-threshold (FFT) regimes, and will show how they emanate from different asymptotic expansions of Foppl-van-Karman (FvK) equations in these two limits. Our systematic theory of the FFT regime unifies the old “membrane limit” approach for the asymptotic stress field (Wagner, Stein&Hedgepeth, Pipkin) with more recent scaling ideas for the wavelength of wrinkles (Cerda&Mahadevan). Combining the analysis of these asymptotic regimes allows us to construct a complete “phase diagram” for wrinkling patterns in the Lame` geometry that sheds new light on experiments in this field. I will discuss general lessons that can be extracted from this analysis, and will conclude with some conjectures on possible universal aspects of this study.