  When a spherical elastic capsule is deflated, it first buckles
axisymmetrically and subsequently loses its axisymmetry in a secondary
instability, where the dimple acquires a polygonal shape. We explain this
secondary polygonal buckling in terms of wrinkles developing at the inner side
of the dimple edge in response to compressive hoop stress. Analyzing the
axisymmetric buckled shape, we find a compressive hoop stress with parabolic
stress profile at the dimple edge. We further show that there exists a critical
value for this hoop stress, where it becomes favorable for the membrane to
buckle out of its axisymmetric shape, thus releasing the compression. The
instability mechanism is analogous to the formation of wrinkles under
compressive stress. A simplified stability analysis allows us to quantify the
critical stress for secondary buckling. Applying this secondary buckling
criterion to the axisymmetric shapes, we can determine the critical volume for
secondary buckling. Our analytical result is in close agreement with existing
numerical data.
