2012/02/14 by Peter Bella, Robert V. Kohn, Bella, Peter +1
Engineering · Mathematics · Physics and Astronomy · #74K35 #Advanced Materials and Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Materials Science (cond-mat.mtrl-sci) #Mathematical Physics (math-ph) #Pattern Formation and Solitons (nlin.PS) #Structural Analysis and Optimization #Vibration and Dynamic Analysis #cond-mat.mtrl-sci #math-ph #math.AP #math.MP #msc:74K35 #nlin.PS
paper · pdf · doi:10.48550/arxiv.1202.3160
41 pages
arxiv created 2012/02/14 · openalex publication_date 2012/02/14 · arxiv updated 2012/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well known that an elastic sheet loaded in tension will wrinkle and that the length scale of the wrinkles tends to zero with vanishing thickness of the sheet [Cerda and Mahadevan, Phys. Rev. Lett. 90, 074302 (2003)]. We give the first mathematically rigorous analysis of such a problem. Since our methods require an explicit understanding of the underlying (convex) relaxed problem, we focus on the wrinkling of an annular sheet loaded in the radial direction [Davidovitch et al., PNAS 108 (2011), no. 45]. Our main achievement is identification of the scaling law of the minimum energy as the thickness of the sheet tends to zero. This requires proving an upper bound and a lower bound that scale the same way. We prove both bounds first in a simplified Kirchhoff-Love setting and then in the nonlinear three-dimensional setting. To obtain the optimal upper bound, we need to adjust a naive construction (one family of wrinkles superimposed on a planar deformation) by introducing a cascade of wrinkles. The lower bound is more subtle, since it must be ansatz-free.