Nonlinear dynamics of spherical shells buckling under step pressure

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.

Delayed buckling of spherical shells due to viscoelastic knockdown of the critical load

“…a time delay between the application of the pressure load and the buckling event. We observe singular thresholds and a delay time which increases monotonically as pressure decreases—this contrasts the findings of a recent numerical study on dynamic step loading of spherical shells which are much thinner than our own [35].…”

“…Following e.g. [35,62,63], we expect that the elastic timescale t∗∼false(2Rfalse)2/false(chfalse), where c=E0/ρ=23.37 m s −1 is the speed of sound within the material and ρ = 1080 kg m −3 is the material density according to the manufacturer. For our shells in order of increasing thickness, this gives t∗∼falsefalse{0.1027,0.0493,0.0264,0.0173falsefalse} s. We have indicated the elastic snap-through time for an arch tinertial=23t∗ [62], with horizontal dashed lines in figure 6.…”

Post-buckling Dynamics of Spherical Shells

“…While the picture in terms of equilibrium states is now quite clear, at least for simple geometries (rods, half-spheres, closed spheres), full control of soft structures by external signals requires to know more about their dynamics. Recent papers have shed light on the complexity of the first-stage dynamics, close to the buckling threshold, where the response time of the material depends on classical dissipative mechanisms coupled to intrinsic slowing down observed in such a critical phenomenon [13][14][15]. The goal of the present paper is to explore the second-stage dynamics, around the buckled state, where the geometry is often more complex than that of the initial state.…”

Post-buckling dynamics of spherical shells

“…While the picture in terms of equilibrium states is now quite clear, at least for simple geometries (rods, half-spheres, closed spheres), full control of soft structures by external signals requires to know more about their dynamics. Recent papers have shed light on the complexity of the first-stage dynamics, close to the buckling threshold, where the response time of the material depends on classical dissipative mechanisms coupled to intrinsic slowing down observed in such a critical phenomenon [13–15]. The goal of the present paper is to explore the second-stage dynamics, around the buckled state, where the geometry is often more complex than that of the initial state.…”

“…More recently, shells made of non-isotropic material have also attracted some attention [31–33]. In terms of dynamics, the reaction of shells to a steep increase of pressure have been recently studied [15,34], while an experimental study has highlighted the role of dissipation within the shell while reaching the stable buckled state [29].…”