Localization and snaking in axially compressed and internally pressurized thin cylindrical shells

This paper uncovers new manifestations of the homoclinic snaking mechanism in the post-buckling regime of a pressurized thin cylindrical shell under axial load. These new forms tend to propagate either wholly or partially in a direction that is orthogonal to the direction of the applied load and so, unlike earlier forms in Woods & Champneys (1999, Heteroclinic tangles in the unfolding of a degenerate Hamiltonian Hopf bifurcation. Phys. D, 129, 147-170), are fundamentally 2D in nature. The main effect of internal pressurization on the snaking mechanism is firstly to transition the circumferential multiplication of buckles from a one-tier pattern to a three-tier pattern. Secondly, internal pressurization can induce oblique snaking, whereby the sequential multiplication of buckles occurs in a helical pattern across the cylinder domain. For low levels of internal pressure, the single dimple remains - as in the unpressurized case - the unstable edge state that forms the smallest energy barrier around the stable pre-buckling equilibrium. For greater levels of pressure, the edge state changes to a single dimple surrounded by four smaller dimples. By tracing the limit point that denotes the onset of these edge states in the parameter space of internal pressure and axial load, we justify and validate the empirically derived design guideline for buckling of pressurized cylinders proposed by Fung & Sechler (1957, Buckling of thin-walled circular cylinders under axial compression and internal pressure. J. Aeronaut. Sci., 24, 351-356).