  We consider the equilibrium shapes of a thin, annular strip cut out in an
elastic sheet. When a central fold is formed by creasing beyond the elastic
limit, the strip has been observed to buckle out-of-plane. Starting from the
theory of elastic plates, we derive a Kirchhoff rod model for the folded strip.
A non-linear effective constitutive law incorporating the underlying
geometrical constraints is derived, in which the angle the ridge appears as an
internal degree of freedom. By contrast with traditional thin- walled beam
models, this constitutive law captures large, non-rigid deformations of the
cross-sections, including finite variations of the dihedral angle at the ridge.
Using this effective rod theory, we identify a buckling instability that
produces the out-of-plane configurations of the folded strip, and show that the
strip behaves as an elastic ring having one frozen mode of curvature. In
addition, we point out two novel buckling patterns: one where the centerline
remains planar and the ridge angle is modulated; another one where the bending
deformation is localized. These patterns are observed experimentally, explained
based on stability analyses, and reproduced in simulations of the post-buckled
configurations.
