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h = shell thickness
delta = initial amplitude of a polar dimple
p = uniform external pressure
p-sub-c = classical buckling pressure of a perfect spherical shell
This and the next 2 images are from:
Jan Sieber, John W. Hutchinson and J. Michael T. Thompson, “Nonlinear dynamics of spherical shells buckling under step pressure”, Proceedings of the Royal Society Series A, Vol. 475, No. 2223, 6 March 2019, https://doi.org/10.1098/rspa.2018.0884
ABSTRACT: Dynamic buckling is addressed for complete elastic spherical shells subject to a rapidly applied step in external pressure. Insights from the perspective of nonlinear dynamics reveal essential mathematical features of the buckling phenomena. To capture the strong buckling imperfection-sensitivity, initial geometric imperfections in the form of an axisymmetric dimple at each pole are introduced. Dynamic buckling under the step pressure is related to the quasi-static buckling pressure. Both loadings produce catastrophic collapse of the shell for conditions in which the pressure is prescribed. Damping plays an important role in dynamic buckling because of the time-dependent nonlinear interaction among modes, particularly the interaction between the spherically symmetric ‘breathing’ mode and the buckling mode. In general, there is not a unique step pressure threshold separating responses associated with buckling from those that do not buckle. Instead, there exists a cascade of buckling thresholds, dependent on the damping and level of imperfection, separating pressures for which buckling occurs from those for which it does not occur. For shells with small and moderately small imperfections, the dynamic step buckling pressure can be substantially below the quasi-static buckling pressure.
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