It is well-known that an elastic sheet loaded in tension will wrinkle and that the length scale of the wrinkles tends to 0 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., Proc. Natl. Acad. Sci. 108 (2011), 18227]. Our main achievement is identification of the scaling law of the minimum energy as the thickness of the sheet tends to 0. 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.
ASJC Scopus subject areas
- Applied Mathematics