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Relation between the pressure and pole buckling deflection for axisymmetric buckling of a perfect spherical shell. The dimple buckle is localized at the poles. The limit of moderate rotation theory as R/t becomes large is f (ξ ), and the results of moderate rotation theory are well approximated by f (ξ ) if ξ < 0.2R/t.

Relation between the pressure and pole buckling deflection for axisymmetric buckling of a perfect spherical shell. The dimple buckle is localized at the poles. The limit of moderate rotation theory as R/t becomes large is f (ξ ), and the results of moderate rotation theory are well approximated by f (ξ ) if ξ < 0.2R/t.

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The nonlinear axisymmetric post-buckling behaviour of perfect, thin, elastic spherical shells subject to external pressure and their asymmetric bifurcations are characterized, providing results for a structure/loading combination with an exceptionally nonlinear buckling response. Immediately after the onset of buckling, the buckling mode localizes...

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... plot of the normalized pressure as a function of the dimensionless pole buckling deflection is presented in figure 1. The dashed curves in this figure have been computed using moderate rotation theory for several values of R/t with ν = 0.3. ...
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... summary, f (ξ ) does not accurately capture the shell behaviour in the range 0 < ξ < 0.5 but is accurate in the range of larger ξ as long as the dimple is shallow. Figure 1 reveals that shallow shell theory provides an accurate approximation for p/p C from moderate rotation theory in a range of ξ = w pole /w R that depends on R/t, which is given approximately by ξ < 0.2R/t. The larger the R/t, the wider the range of ξ for which p/p C = f (ξ ) is accurate. ...
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... already discussed, the localized nature of the buckling when the dimples are shallow essentially decouples behaviour at one pole from the other and thus the results in table 2 apply either to a single dimple at one pole or to symmetric dimples at the two poles. Figure 10. The results of Berke & Carlson [9] showing the changes in dimple shape and size during an unloading sequence, on a plot of pressure versus change of volume, both scaled relative to their critical values. ...
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... their very thin shells, and rigid loading, the authors [9] were able to perform post- buckling tests without the mandrel which nevertheless remained in the elastic range, offering repeatable results. Figure 10 shows a typical pressure-volume result for the unloading of a manually induced dimple in a single shell. This starts at point A, at a value of V/V C just over 0.5 with a five-sided 'pentagonal' dimple as shown in the inset photograph (5). ...
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... black squares represent the sequential results for the pentagonal dimples, from right to left, through which a red line has been fitted. on April 7, 2017 http://rsta.royalsocietypublishing.org/ Downloaded from (a) (b) Figure 11. Final post-buckling configurations of two tests by Thompson [26] under rigid volume control (a) and prescribed pressure loading (b). ...
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... must next think about the theoretical results of the present paper, which are of course for a perfect shell. The axisymmetric post-buckling path shown in deep blue in figure 10 has been computed using the results of §2 assuming that a single dimple forms. On this path are the four bifurcation points into non-symmetric modes with the wavenumbers m = 4, 5, 6 and 7. ...
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... have checked the stability of the axisymmetric path against non-axisymmetric modes for higher deflections than before, and uncovered bifurcations into modes with wavenumbers progressing systematically from 4 to 9. These are shown to tie in well with experimental observations of the square and pentagonal dimples displayed in figure 10. ...
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... important than the Maxwell points are the energy barriers [21] against finite static or dynamic disturbances plotted against the controlled V/V C in figure 7 for R/t = 100. We compare the energy barriers for dead pressure and rigid volume control in figure 12 for the same R/t. The difference between these 'shock sensitivity' barriers is an indication of how much safer against disturbances is rigid volume control than dead pressure loading. ...
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... difference between these 'shock sensitivity' barriers is an indication of how much safer against disturbances is rigid volume control than dead pressure loading. For rigid volume control there exists no possibility of creating a buckle at pressures lower than that of point N, above L in figure 12b, and the curve for this case terminates at that pressure. The rather surprising fact that there is so little difference between the barriers of the perfect shell for the two extreme loadings is consistent with the fact that experiments on thin shells that have explored this issue have found very little dependency on loading compliance [25,27]. ...
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... important step in that direction is made by Lee et al. [6], who have measured buckling pressures of elastomeric spherical shells of R/t = 108 with carefully manufactured dimple imperfections with amplitudes up to 2.5 times the shell thickness. For imperfection Figure 12. Sketches of the system energy barriers for (a) dead pressure, E p , and (b) rigid volume control, E V . ...

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