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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.

Source publication

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...

## Contexts in source publication

**Context 1**

... 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. ...

**Context 2**

... 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. ...

**Context 3**

... 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. ...

**Context 4**

... 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). ...

**Context 5**

... 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). ...

**Context 6**

... 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. ...

**Context 7**

... 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. ...

**Context 8**

... 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. ...

**Context 9**

... 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]. ...

**Context 10**

... 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 . ...

## Citations

... The response of pressurized spherical shell on radial probing force was investigated in [13][14][15]. The concept of energy barrier (perturbation energy) corresponding to local shell buckling and its application to KDF estimation was studied in [5,6,16,17]. Dimple-like initial geometrical imperfections were considered in [18][19][20][21][22][23]. ...

The simple analytical model of spherical shell buckling suggested in the previous publication in the TWS journal is applied to studying the response of the structure to different perturbations given as random variables. Among them probing radial force, energy barrier and geometrical imperfections are considered as factors decreasing uniform buckling pressure. The model is expanded to the case of imperfect shells. The propagation of uncertainty of perturbations is studied by estimation the standard deviation of the buckling load. Monte Carlo method and the propagation of error formula based on Tailor expansion are used. The application of the methods to design buckling pressure estimation is discussed.

... For a historical perspective and a more thorough contextualization of the modern account of single-defect shell buckling, we direct the reader to Refs. [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. ...

We perform finite element simulations to study the impact of defect-defect interactions on the pressure-induced buckling of thin, elastic, spherical shells containing two dimpled imperfections. Throughout, we quantify the critical buckling pressure of these shells using their knockdown factor. We examine cases featuring either identical or different geometric defects and systematically explore the parameter space, including the angular separation between the defects, their widths and amplitudes, and the radius-to-thickness ratio of the shell. As the angular separation between the defects is increased, the buckling strength initially decreases, then increases before reaching a plateau. Our primary finding is that the onset of defect-defect interactions, as quantified by a characteristic length scale associated with the onset of the plateau, is set by the critical buckling wavelength reported in the classic shell-buckling literature. Beyond this threshold, within the plateau regime, the buckling behavior of the shell is dictated by the largest defect.

... The seminal work of Koiter [47] has been recently revisited by Hutchinson [43], in a context of growing interest for the complex instability mechanisms of hollow shells [48][49][50]. To handle the delicate calculation of the stability analysis of such shells, a simplifying hypothesis is typically made regarding the amplitude of the deformations. ...

Collapse of lipidic ultrasound contrast agents under high-frequency compressive load has been historically interpreted by the vanishing of surface tension. By contrast, buckling of elastic shells is known to occur when costly compressible stress is released through bending. Through quasi-static compression experiments on lipidic shells, we analyse the buckling events in the framework of classical elastic buckling theory and deduce the mechanical characteristics of these shells. They are then compared with that obtained through acoustic characterization.
This article is part of the theme issue ‘Probing and dynamics of shock sensitive shells’.

... The notion of lateral probing and its application to the buckling of cylindrical and spherical shells was developed by Thompson and co-workers [26,27]. Prior work involving non-destructive evaluation can also be found in [10,18]. ...

... A very recent publication also suggested using multiple probes in order to enhance the 'ridge-tracking' associated with probing [45], i.e. a (linear) correlation between the peak lateral probe load and the axial buckling load, useful for prediction purposes based on extrapolation. 3D-printing also has scope for producing shells of variable thickness [37,46,47], spherical shells [26,48] and the incorporation of stiffening ribs is straightforward [38,49]. ...

Cylinder buckling is notoriously sensitive to small geometric imperfections. This is an underlying motivation for the use of knock-down factors in the design process, especially in circumstances in which minimum weight is a key design goal, an approach well-established at NASA, for example. Not only does this provide challenges in the practical design of this commonly occurring structural load-bearing configuration, but also in the carefully controlled laboratory setting. The recent development of 3D-printing (additive manufacturing) provides an appealing experimental platform for conducting relatively high-fidelity experiments on the buckling of cylinders. However, in addition to geometric precision, there are a number of shortcomings with this approach, and this article seeks to describe the challenges and opportunities associated with the use of 3D-printing in cylinder buckling in general, and probing the robustness of equilibrium configurations in particular.
This article is part of the theme issue ‘Probing and dynamics of shock sensitive shells’.

... In this case, the buckling mode initially triggers a global, but very small deformation that gradually becomes more and more localized. This type of localization is for instance observed in beams on an elastic foundation [10] and in spherical shells under external pressure [11][12][13]. Another localization scenario is observed when a global post-buckling mode is created through the sequential formation of localized buckles. ...

This paper studies the stability of space structures consisting of longitudinal, open-section thin-shells transversely connected by thin rods subjected to a pure bending moment. Localization of deformation, which plays a paramount role in the nonlinear post-buckling regime of these structures and is extremely sensitive to imperfections, is investigated through probing experiments. As the structures are bent, a probe locally displaces the edge of the thin shells, creating local dimple imperfections. The range of moments for which the early buckling of the structures can be triggered by this perturbation is determined, as well as the energy barrier separating the pre-buckling and post-buckling states. The stability of the local buckling mode is then illustrated by a stability landscape, and probing is extended to the entire structure to reveal alternate buckling modes disconnected from the structure’s fundamental path. These results can be used to formulate efficient buckling criteria and pave the way to operating these structures close to their buckling limits, and even in their post-buckling regime, therefore significantly reducing their mass.
This article is part of the theme issue ‘Probing and dynamics of shock sensitive shells’.

... Fig. 10 by dashed curve. We used notation from [25] for comparison with numerical results (solid line) obtained in [25] (Table 1) for complete shell. Asymptotic formula for volume change in this notation is ℎ ( ) = 0.01∕ 4 . ...

... Fig. 10 by dashed curve. We used notation from [25] for comparison with numerical results (solid line) obtained in [25] (Table 1) for complete shell. Asymptotic formula for volume change in this notation is ℎ ( ) = 0.01∕ 4 . ...

... We suggest to calculate derivative ∕ numerically as the ratio of finite differences for given value of load parameter . The result is shown in Fig. 10 by dotted line which nearly coincides with the numerical solution [25]. There is also good agreement between other results obtained by our new model and numerical data published in [25]. ...

An important property of localization in buckling of spherical shells under external pressure is discussed. It is shown that the localization is possible for structures with nonlinear softening. An analytical model of local buckling of spherical shell is developed. Rayleigh-Ritz method is used at small and moderate deflections. At large deflections corresponding to relatively small pressure (less than 20% of classical bucking load) it is based on asymptotic method. The asymptotic model is then expanded to the practically important range of the load. The response of the structure to local perturbations of different types (including radial probing force, prescribed deflection at the shell pole, and energy barrier) is studied. Special attention is paid to the energy barrier which is required for structure transition from initial equilibrium state to the post-buckling dimple-like state. Energy barrier criterion is used as a measure of metastability of the structure and applied for estimation of load level separating high and low sensitivity of the shell to local perturbations. Based on this pressure value, formulae for design buckling load are proposed and deliberated. Similarities and differences of local buckling of spherical shells under external pressure and axially compressed cylindrical shells are discussed.

... Many studies have been conducted on the effects of local defects and geometrically imperfect shapes on the crushing and buckling characteristics of spherical pressure hulls, and many research results have been published [18][19][20][21][22]. The thickness distribution of a spherical pressure hull has been adjusted to improve the crushing and buckling property, although these issues are yet to be resolved [23][24][25][26]. ...

Spherical shell structures are the most suitable shape for deep-sea pressure hulls because they have ideal mechanical properties for handling symmetrical pressure. However, the shape accuracy requirement for a hull in a spherical shell structure subjected to deep-sea pressure is extremely high. Even minor asymmetry can significantly degrade its mechanical properties. In this study, a new type of spherical deep-sea pressure hull structure and its integral hydro-bulge-forming (IHBF) method are proposed. First, 32 flat metal plate parts are prepared and welded along their straight sides to form a regular polygonally shaped box. Next, water pressure is applied inside the preformed box to create a spherical pressure vessel. We performed a forming experiment using a spherical pressure vessel with a design radius of 250 mm as a verification research object. The radius of the spherical pressure vessel obtained from the forming experiment is 249.32 mm, the error from the design radius is 0.27%, and the roundness of the spherical surface is 2.36 mm. We performed a crushing analysis using uniform external pressure to confirm the crushing and buckling characteristics of the formed spherical pressure vessel. The results show that the work-hardening increased the crushing and buckling load of the spherical pressure vessel, above that of the conventional spherical shell structure. Additionally, it is established that local defects and the size of the weld line significantly and slightly affected the crushing and buckling load of the spherical pressure hull, respectively.

... The study of the critical buckling conditions of spherical shells has been reinvigorated by recent advances in experiments and computation [16,[21][22][23][24][25][26][27][28]. For a contemporary perspective and overview of the recent activity in the field, we point the reader to the following recent studies [16,[22][23][24][29][30][31][32][33][34][35][36]. Even if similar results are also found for cylindrical shells [7,17,20,25,27,[37][38][39][40], the present study will focus on spherical shells exclusively. ...

... Most of the recent investigations on spherical-shell buckling mentioned in the previous paragraph [33][34][35][41][42][43] have considered standardized dimpled (Gaussian) defects. Other types of imperfections (e.g., throughthickness defects [23,44,45], and dent imperfections [46]) have also been considered, but such cases are sparser. ...

We investigate the effect of defect geometry in dictating the sensitivity of the critical buckling conditions of spherical shells under external pressure loading. Specifically, we perform a comparative study between shells containing dimpled (inward) versus bumpy (outward) Gaussian defects. The former has become the standard shape in many recent shell-buckling studies, whereas the latter has remained mostly unexplored. We employ finite-element simulations, which were validated previously against experiments, to compute the knockdown factors for the two cases while systematically exploring the parameter space of the defect geometry. For the same magnitudes of the amplitude and angular width of the defect, we find that shells containing bumpy defects consistently exhibit significantly higher knockdown factors than shells with the more classic dimpled defects. Furthermore, the relationship of the knockdown as a function of the amplitude and width of the defect is qualitatively different between the two cases, which also exhibit distinct post-buckling behavior. A speculative interpretation of the results is provided based on the qualitative differences in the mean-curvature profiles of the two cases.

... The seminal work of Koiter [49] has been recently revisited by Hutchinson [44], in a context of growing interest for the complex instability mechanisms of hollow shells [50][51][52]. In order to handle the delicate calculation of the stability analysis of such shells, a simplifying hypothesis is typically made regarding the amplitude of the deformations. ...

Collapse of lipidic ultrasound contrast agents under high-frequency compressive load has been historically interpreted by the vanishing of surface tension. By contrast, buckling of elastic shells is known to occur when costly compressible stress is released through bending. Through quasi-static compression experiments on lipidic shells, we analyze the buckling events in the framework of classical elastic buckling theory and deduce the mechanical characteristics of these shells. They are then compared to that obtained through acoustic characterization.

... This is referred to as spatial chaos (Thompson and Virgin, 1988). Localization can arise on postbuckling branches determined by the buckling modes, as observed in the spherical shell under pressure (Audoly and Hutchinson, 2020;Hutchinson and Thompson, 2017a). In addition, localization can also appear on post-buckling paths disconnected from the fundamental path while running asymptotically close to it (Groh and Pirrera, 2019). ...

... It depends on the particular structure under study, and also on whether the experiment/simulation is load controlled or displacement controlled. For example, a spherical shell under external pressure will exhibit stable buckled states when loaded under volume-control but has no stable buckled states (other than complete collapse) under pressure-control (Hutchinson and Thompson, 2017a). For the SSPP strip structures described in the Introduction, it has been observed that the stable buckled equilibrium contour can extend much farther than the first buckling load . ...

The stability of lightweight space structures composed of longitudinal thin-shell elements connected transversely by thin rods is investigated, extending recent work on the stability of cylindrical and spherical shells. The role of localization in the buckling of these structures is investigated and early transitions into the post-buckling regime are unveiled using a probe that locally displaces the structure. Multiple probe locations are studied and the probe force versus probe displacement curves are analyzed and plotted to assess the structure’s stability. The probing method enables the computation of the energy input needed to transition early into a post-buckling state, which is central to determining the critical buckling mechanism for the structure. A stability landscape is finally plotted for the critical buckling mechanism. It gives insight into the post-buckling stability of the structure and the existence of localized post-buckling states in the close vicinity of the fundamental equilibrium path.