Compression, crumpling and collapse of spherical shells and capsules

The deformation of thin spherical shells by applying an external pressure or by reducing the volume is studied by computer simulations and scaling arguments. The shape of the deformed shells depends on the deformation rate, the reduced volume V/V₀ and the Föppl–von Kármán number γ. For slow deformations the shell attains its ground state, a shell with a single indentation, whereas for large deformation rates the shell appears crumpled with many indentations. The rim of the single indentation undergoes a shape transition from smooth to polygonal for γ≃7000(ΔV/V₀)^{− 3/4}. For the smooth rim the elastic energy scales like γ^1/4 whereas for the polygonal indentation we find a much smaller exponent, even smaller than the exponent 1/6 that is predicted for stretching ridges. The relaxation of a shell with multiple indentations towards the ground state follows an Ostwald ripening type of pathway and depends on the compression rate and on the Föppl–von Kármán number. The number of indentations decreases as a power law with time t following N_ind∼t^{− 0.375} for γ=8×10³ and γ=8×10⁴ whereas for γ=8×10⁵ the relaxation time is longer than the simulation time.