  We investigate buckling of soft elastic capsules under negative pressure or
for reduced capsule volume. Based on nonlinear shell theory and the assumption
of a hyperelastic capsule membrane, shape equations for axisymmetric and
initially spherical capsules are derived and solved numerically. A rich
bifurcation behavior is found, which is presented in terms of bifurcation
diagrams. The energetically preferred stable configuration is deduced from a
least-energy principle both for prescribed volume and prescribed pressure. We
find that buckled shapes are energetically favorable already at smaller
negative pressures and larger critical volumes than predicted by the classical
buckling instability. By preventing self-intersection for strongly reduced
volume, we obtain a complete picture of the buckling process and can follow the
shape from the initial undeformed state through the buckling instability into
the fully collapsed state. Interestingly, the sequences of bifurcations and
stable capsule shapes differ for prescribed volume and prescribed pressure. In
the buckled state, we find a relation between curvatures at the indentation rim
and the bending modulus, which can be used to determine elastic moduli from
experimental shape analysis.
