Figure 1 - uploaded by John Michael T Thompson
Content may be subject to copyright.
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.

Source publication
Article
Full-text available
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

... However, while these modes give rise to a meta-stable fundamental path above the minimum post-buckling pressure, there may exist additional meta-stable paths in the immediate vicinity of the fundamental path that are also associated with a very low energy barrier, and which would yield an earlier transition into the post-buckling regime. Such broken away paths were recently studied for the case of the cylindrical shell under axial compression and for the spherical shell under pressure [33][34][35]. In the case of the cylinder the corresponding mode shapes take the form of a localized dimple. ...
Article
Full-text available
This paper analyzes the buckling and postbuckling behavior of ultralight ladder-type coilable structures, called strips, composed of thin-shell longerons connected by thin rods. Based on recent research on the stability of cylindrical and spherical shells, the stability of strip structures loaded by normal pressure is studied by applying controlled perturbations through localized probing. A plot of these disturbances for increasing pressure is the stability landscape for the structure, which gives insight into the structure’s buckling, postbuckling, and sensitivity to disturbances. The probing technique is generalized to higher-order bifurcations along the postbuckling path, and low-energy escape paths into buckling that cannot be predicted by a classical eigenvalue formulation are identified. It is shown that the stability landscape for a pressure-loaded strip is similar to the landscape for classical shells, such as the axially loaded cylinder and the pressure-loaded sphere. Similarly to classical shells, the stability landscape for the strip shows that an early transition into buckling can be triggered by small disturbances; however, while classical shell structures buckle catastrophically, strip structures feature a large stable postbuckling range.
... Some 56 years later, I added the experimental results onto Fig. 9 of an article on shock sensitivity [Thompson & Sieber, 2016], and photographs showing the pentagonal post-buckling dimples are in [Hutchinson & Thompson, 2017a]. Theoretically, I made a four-degrees-of-freedom analysis of the post-buckling behavior of a complete, uniformly compressed spherical shell, solving the nonlinear algebraic equations on the Cambridge EDSAC II computer, built in 1958 with its host of vacuum tubes and paper-tape input. ...
... The result was a set of papers about energy barriers and probing of a complete spherical shell, which stimulated some new experimental studies. First, in [Hutchinson & Thompson, 2017a] we explored energy barriers and symmetry-breaking dimples. Accurate recently established shell equations [Hutchinson, 2016] allowed an evaluation of criteria such as the Maxwell condition for which the energies in the unbuckled and buckled states are the same. ...
Article
Full-text available
This article is an informal auto-biographical memoir by Mike Thompson, reflecting in retirement on his scientific researches in nonlinear phenomena, wandering pictorially from shell buckling, through bifurcations and chaos to climate tipping points. Some ideas and advice to young researchers are offered whenever it seems appropriate. Two research groups at University College London, and their two IUTAM Symposia are given some prominence, as are the ten years editing the Philosophical Transactions of the Royal Society.
... Refs. [13,[19][20][21]). These developments afford engineers the opportunity to design sturdy structures with more specific -and permissive [2] -lower bounds on the load carrying capacity. ...
Article
Full-text available
We performed dynamic pressure buckling experiments on defect-seeded spherical shells made of a common silicone elastomer. Unlike in quasi-static experiments, shells buckled at ostensibly subcritical pressures, i.e. below the experimentally determined critical load at which buckling occurs elastically, often following a significant delay period from the time of load application. While emphasizing the close connections to elastic shell buckling, we rely on viscoelasticity to explain our observations. In particular, we demonstrate that the lower critical load may be determined from the material properties, which is rationalized by a simple analogy to elastic spherical shell buckling. We then introduce a model centred on empirical quantities to show that viscoelastic creep deformation lowers the critical load in the same predictable, quantifiable way that a growing defect would in an elastic shell. This allows us to capture how both the deflection at instability and the time delay depend on the applied pressure, material properties and defect geometry. These quantities are straightforward to measure in experiments. Thus, our work not only provides intuition for viscoelastic behaviour from an elastic shell buckling perspective but also offers an accessible pathway to introduce tunable, time-controlled actuation to existing mechanical actuators, e.g. pneumatic grippers.
... Because of their ubiquitousness in nature but also of their simplicity in terms of fabrication and of modelling, spherical closed shells enclosing a compressible fluid are particularly in the spotlight [3,[16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Existing studies are essentially focused on understanding the scenario of the buckling instability that occurs beyond a certain threshold of compression or deflation, and on characterizing the stability branches [22,24,25,29]. ...
... Both in numerical simulations [22,24,26] or in experiments [29], it is now well established that at equilibrium, the pressure difference P ext − P quasi-plateaus to a constant P pl as a function of equilibrium volumes V e . Therefore, varying the fourth parameterP ext strictly amounts to varying the equilibrium pressureP e inside the shell, which we already set here by varying the initial pressure. ...
... Therefore, varying the fourth parameterP ext strictly amounts to varying the equilibrium pressureP e inside the shell, which we already set here by varying the initial pressure. In our experiments and simulations,P ext is typically chosen such that the pressure difference is right above the buckling threshold P b = 4d/3 [22,26]. Increasing too much the external pressure, or starting with very low pressure inside the shell, leads to a full collapse of the shell, with the two opposite poles being in contact. ...
Article
Full-text available
We explore the intrinsic dynamics of spherical shells immersed in a fluid in the vicinity of their buckled state, through experiments and three-dimensional axisymmetric simulations. The results are supported by a theoretical model that accurately describes the buckled shell as a two-variable-only oscillator. We quantify the effective ‘softening’ of shells above the buckling threshold, as observed in recent experiments on interactions between encapsulated microbubbles and acoustic waves. The main dissipation mechanism in the neighbouring fluid is also evidenced.
... Two of the buckled specimens are shown below in figure 1. Some 56 years later, I added the experimental results onto figure 9 of an article on shock sensitivity [Thompson & Sieber, 2016], and photographs showing the pentagonal post-buckling dimples are in [Hutchinson & Thompson, 2017a]. Theoretically, I made a four-degrees-of-freedom analysis of the post-buckling behaviour of a complete, uniformly compressed spherical shell, solving the nonlinear algebraic equations on the Cambridge EDSAC II computer, built in 1958 with its host of vacuum tubes and paper-tape input. ...
... The result was a set of papers about energy barriers and probing of a complete spherical shell, which stimulated some new experimental studies. First, in [Hutchinson & Thompson, 2017a] we explored energy barriers and symmetry-breaking dimples. Accurate recently established shell equations [Hutchinson, 2016] allowed an evaluation of criteria such as the Maxwell condition for which the energies in the unbuckled and buckled states are the same. ...
... A breakthrough came only recently, from experiments, when the combination of a rapid prototyping technique for spherical shells 18 and its adaptation to seed precisely designed defects led to a quantitative relationship between the knockdown factor and defect geometry 17 . This study demonstrated that if imperfections can be measured precisely, then the knockdown factor can be precisely predicted using an appropriate shell theory [19][20][21][22][23][24][25] , thus opening the door for less conservative designs of shell structures. In parallel, nondestructive probing methods have recently been proposed to experimentally access the stability landscape of shells [26][27][28] , while accepting the inevitable presence of multiple imperfections to set the knockdown factor. ...
... The full 3D configuration and the corresponding cross-section profile of the shell are visualized through X-ray micro-computed tomography (μCT, 100 Scanco Medical AG), a representative example of which is presented in Fig. 1c. To measure the buckling strength, we depressurize the shell using a pneumatic-loading system under imposed-volume conditions and measure the associated pressure sustained by the shell (Methods section and Supplementary Note 2). Figure 1d presents the load-carrying behavior of the shell, characterized by the pressure (p) as a function of the volume change (ΔV), both of which are normalized, respectively, by the classic buckling prediction, p c , and the corresponding volume change immediately prior to buckling 19,20 , ΔV c ¼ 2πð1 À νÞR 2 h= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3ð1 À ν 2 Þ p . The onset of buckling corresponds to the maximum of each curve, p max ¼ p max =p c , and the accompanying pressure drop indicates the loss of load-carrying capacity of the shell. ...
... We also note that, since the generation of magnetic torque is not constrained to a specific defect geometry, we anticipate our mechanism would also work for imperfections with other geometric profiles or arrangements. The localized nature of buckling in spherical shells 19,20 is also underlined in our system by the localized distribution of shell rotation and the associated magnetic torque in the vicinity of the defect (see Supplementary Fig. 6 in Supplementary Note 3). This localization ensures that the tunability would not be affected by the shell opening angle and boundary conditions (except for extremely shallow shells). ...
Article
Full-text available
Shell buckling is central in many biological structures and advanced functional materials, even if, traditionally, this elastic instability has been regarded as a catastrophic phenomenon to be avoided for engineering structures. Either way, predicting critical buckling conditions remains a long-standing challenge. The subcritical nature of shell buckling imparts extreme sensitivity to material and geometric imperfections. Consequently, measured critical loads are inevitably lower than classic theoretical predictions. Here, we present a robust mechanism to dynamically tune the buckling strength of shells, exploiting the coupling between mechanics and magnetism. Our experiments on pressurized spherical shells made of a hard-magnetic elastomer demonstrate the tunability of their buckling pressure via magnetic actuation. We develop a theoretical model for thin magnetic elastic shells, which rationalizes the underlying mechanism, in excellent agreement with experiments. A dimensionless magneto-elastic buckling number is recognized as the key governing parameter, combining the geometric, mechanical, and magnetic properties of the system. Predicting and controlling the (in)stability of thin shells are important tasks in many practical applications but hampered by the presence of imperfections. Here Yan et al. show that when the material composition is magnetic the conditions for collapse can simply be readjusted with external fields.
... Refs. [13,[19][20][21]). These developments afford engineers the opportunity to design sturdy structures with more specific -and permissive [2] -lower bounds on the load carrying capacity. ...
Preprint
Full-text available
We performed dynamic pressure buckling experiments on defect-seeded spherical shells made of a common silicone elastomer. Unlike in quasi-static experiments, shells buckled at ostensibly subcritical pressures (i.e. below the experimentally-determined load at which buckling occurs elastically), often following a significant time delay. While emphasizing the close connections to elastic shell buckling, we rely on viscoelasticity to explain our observations. In particular, we demonstrate that the lower critical load may be determined from the material properties, which is rationalized by a simple analogy to elastic spherical shell buckling. We then introduce a model centered on empirical quantities to show that viscoelastic creep deformation lowers the critical load in the same predictable, quantifiable way that a growing defect would in an elastic shell. This allows us to capture how both the critical deflection and the delay time depend on the applied pressure, material properties, and defect geometry. These quantities are straightforward to measure in experiments. Thus, our work not only provides intuition for viscoelastic behavior from an elastic shell buckling perspective, but also offers an accessible pathway to introduce tunable, time-controlled actuation to existing mechanical actuators, e.g. pneumatic grippers.
... Recently, the quest for high-fidelity estimates of the buckling capacity has regained significant attention due to the renewed interest in space-flight and in thin soft material [22][23][24][25][26][27][28][29]. Indeed, a promising new framework based on the probing of axially compressed cylinders has emerged for the evaluation of the buckling capacity of thin cylindrical shells without complete knowledge of the shell's underlying imperfections: the stability landscape [7,[30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. However, this framework is still in the infant state, and many issues have to be resolved, e.g., the role of probing location, the influence of the imperfection's size and shape, and the impact of the interaction among imperfections. ...
Article
Full-text available
The axial buckling capacity of a thin cylindrical shell depends on the shape and the size of the imperfections that are present in it. Therefore, the prediction of the shells' buckling capacity is difficult, expensive, and time-consuming, if not impossible, because the prediction requires a priori knowledge about the imperfections. As a result, thin cylindrical shells are designed conservatively using the knockdown factor approach that accommodates the uncertainties associated with the imperfections that are present in the shells; almost all the design codes follow this approach explicitly or implicitly. A novel procedure is proposed for the accurate prediction of the axial buckling capacity of thin cylindrical shells without measuring the imperfections and is based on the probing of the axially loaded shells. Computational and experimental implementation of the procedure yields accurate results when the probing is done in location of highest imperfection amplitude. However, the procedure over-predicts the capacity when the probing is done away from that point. This study demonstrates the crucial role played by the probing location and shows that the prediction of imperfect cylinders is possible if the probing is done at the proper location.
... There exist a number of contributions on the buckling of spherical shells ( Hutchinson, 1967;2016;Hutchinson and Thompson, 2017;Koiter, 1969;Thompson, 1964 ), yet only few on spherical film/substrate systems ( Breid and Crosby, 2013;Cao et al., 2008;Li et al., 2011b;Stoop et al., 2015 ). The latter works reported similar scenario of pattern transitions, where periodic hexagonal patterns appear at the onset of instability and then transform into disordered patterns such as labyrinthlike or mixed shapes combining 1D and 2D modes. ...
Article
Curvature-induced symmetry-breaking pattern formation and transition are widely observed in curved film/substrate systems across different length scales such as embryogenesis, heterogeneous micro-particles, dehydrated fruits, growing tumors and planetary surfaces. Here, we find, both experimentally and theoretically, that morphological pattern selection of core-shell spheres, upon shrinkage of core or expansion of surface layer, is primarily determined by a single dimensionless parameter Cs which characterizes the stiffness ratio of core/shell and geometric curvature of the system. When the core remains relatively soft (Cs<1.3), the core-shell sphere usually experiences subcritical buckling behavior with local dimples at the critical threshold. With a stiffer substrate (1.3<Cs<15), the system morphs into periodic buckyball patterns. With the continuous increase of the core stiffness (Cs>15), symmetry-breaking disordered patterns involving polygon and labyrinth modes appear to be energetically favorable. With extremely large Cs~1000, the core-shell sphere approximates to a planar film/substrate system and thus checkerboard patterns with grain boundaries are observed. Moreover, we find that the transition from subcritical to supercritical bifurcations can be quantitatively characterized by this parameter. Pattern selection based on this single key factor remarkably agrees with our experimental observations on oxidized polydimethylsiloxane (PDMS) microspheres in the entire validity range. Our results not only provide fundamental understanding of pattern selection in spherical film/substrate systems, but also pave a promising way to facilitate the design of morphology-related functional surfaces by quantitatively harnessing such curvature-modulus co-determined pattern formation.
... While this approach is intended to ensure that a shell will not buckle for loads below the knockeddown value, it does not give insight into how robust a loaded shell will be to accidental disturbances or ancillary loads. For assessing the robustness of imperfection-sensitive shells against buckling, recent research [1][2][3][4][5][6] has focused on the energy barrier that exists at loads below the buckling load. For a shell with unstable post-buckling behavior (which is, thus, imperfection-sensitive), the energy barrier to buckling at a given static load is the difference between energy of the shell/load system in the quasi-static buckled state from that in the unbuckled state. ...
... Except at applied pressures just below p fold , published simulations and experimental observations indicate that the buckling behavior is dominantly axially symmetry about the center of the emerging dimple buckle. Non-axisymmetric features only develop deep into the post-buckling response, well beyond the range of relevance to the considerations in this paper [2]. For these reasons, an axisymmetric dynamic analysis of the hemispherical shell captures the essential aspects of the dynamic buckling process. ...
... While all the results in this paper have assumed a pre-load pressure that is held fixed during the impact event, the energy barrier is almost the same for a spherical shell that is loaded to the same pre-load pressure and then buckled with no subsequent change in volume, except in the range of low pressures, as discussed in detail in Refs. [2,5]. We fully expect that the findings in this paper relating threshold buckling energy to the energy barrier will carry over to spherical shells impacted under conditions where the internal volume of the shell remains unchanged, the so-called rigid volume constraint. ...
Article
Full-text available
This paper investigates the robustness against localized impacts of elastic spherical shells pre-loaded under uniform external pressure. We subjected a pre-loaded spherical shell that is clamped at its equator to axisymmetric blast-like impacts applied to its polar region. The resulting axisymmetric dynamic response is computed for increasing amplitudes of the blast. Both perfect shells and shells with axisymmetric geometric imperfections are analyzed. The impact energy threshold causing buckling is identified and compared with the energy barrier that exists between the buckled and un-buckled static equilibrium states of the energy landscape associated with the pre-loaded pressure. The extent to which the impact energy of the threshold blast exceeds the energy barrier depends on the details of its shape and width. Targeted blasts that approximately replicate the size and shape of the energy barrier buckling mode defined in the paper have an energy threshold that is only modestly larger than the energy barrier. An extensive study is carried out for more realistic Gaussian-shaped blasts revealing that the buckling threshold energy for these blasts is typically in the range of at least ten to forty percent above the energy barrier, depending on the pressure pre-load and the blast width. The energy discrepancy between the buckling threshold and energy barrier is due to elastic waves spreading outward from the impact and dissipation associated with the numerical integration scheme..