  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.
