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The authors write:
“In this paper we investigate the different shapes that appear when a spherical shell is collapsed either under an external pressure or by a volume constraint. To illustrate this we present in [this slide] two representative examples of configurations that we find in the simulations. The reduced volume of the shells is in both cases the same but the Föppl–von Kármán numbers γ are different. The localization of the elastic energy in the highly deformed regions is displayed by the darker shades of grey. The shell with small γ (figure (A)) is characterized by a single indentation and looks rather regular whereas the shell with large γ (figure (B)) has several indentations and appears crumpled.”
(The Föppl–von Kármán number γ is given by: γ = 12(1-nu^2)(R/h)^2, in which R is the radius of the spherical shell and h is the thickness.)
This and the next image are from:
G A Vliegenthart (1) and G Gompper (1 and 2)
(1) Theoretical Soft Matter and Biophysics, Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich 52425, Germany
(2) Theoretical Soft Matter and Biophysics, Institute for Complex Systems, Forschungszentrum Jülich, Jülich 52425, Germany
“Compression, crumpling and collapse of spherical shells and capsules”, New J. Phys. Vol. 13, April 2011, 045020 doi: 10.1088/1367-2630/13/4/045020
ABSTRACT: 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 Foppl - von Karman number gamma. 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 gamma approximately 7000(deltaV/V0)^(-3/4). For the smooth rim the elastic energy scales like gamma^(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 Foppl - von Karman number. The number of indentations decreases as a power law with time t following ….(formula with lots of symbols and powers).
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