  We present a combined analytical approach and numerical study on the
stability of a ring bound to an annular elastic substrate, which contains a
circular cavity. The system is loaded by depressurizing the inner cavity. The
ring is modeled as an Euler-Bernoulli beam and its equilibrium equations are
derived from the mechanical energy which takes into account both stretching and
bending contributions. The curvature of the substrate is considered explicitly
to model the work done by its reaction force on the ring. We distinguish two
different instabilities: periodic wrinkling of the ring or global buckling of
the structure. Our model provides an expression for the critical pressure, as
well as a phase diagram that rationalizes the transition between instability
modes. Towards assessing the role of curvature, we compare our results for the
critical stress and the wrinkling wavelength to their planar counterparts. We
show that the critical stress is insensitive to the curvature of the substrate,
while the wavelength is only affected due to the permissible discrete values of
the azimuthal wavenumber imposed by the geometry of the problem. Throughout, we
contrast our analytical predictions against finite element simulations.
