  We theoretically explain the complete sequence of shapes of deflated
spherical shells. Decreasing the volume, the shell remains spherical initially,
then undergoes the classical buckling instability, where an axisymmetric dimple
appears, and, finally, loses its axisymmetry by wrinkles developing in the
vicinity of the dimple edge in a secondary buckling transition. We describe the
first axisymmetric buckling transition by numerical integration of the complete
set of shape equations and an approximate analytic model due to Pogorelov. In
the buckled shape, both approaches exhibit a locally compressive hoop stress in
a region where experiments and simulations show the development of polygonal
wrinkles, along the dimple edge. In a simplified model based on the stability
equations of shallow shells, a critical value for the compressive hoop stress
is derived, for which the compressed circumferential fibres will buckle out of
their circular shape in order to release the compression. By applying this
wrinkling criterion to the solutions of the axisymmetric models, we can
calculate the critical volume for the secondary buckling transition. Using the
Pogorelov approach, we also obtain an analytical expression for the critical
volume at the secondary buckling transition: The critical volume difference
scales linearly with the bending stiffness, whereas the critical volume
reduction at the classical axisymmetric buckling transition scales with the
square root of the bending stiffness. These results are confirmed by another
stability analysis in the framework of Donnel, Mushtari and Vlasov (DMV) shell
theory, and by numerical simulations available in the literature.
