Session X55: Shell Buckling I

Non-destructive prediction of the buckling load of soda cans and space rockets

What is the critical load required to crush a soda can or a space rocket shell? Surprisingly, there is no good way to estimate it, because of the high defect-sensitivity of the buckling instability. Here we measure the response of (imperfect) soda cans to lateral poking and identify a generic stability landscape, which fully characterizes the stability of real imperfect shells in the case where one single defect dominates. We show that the landscape of stability is independent of the loading protocol and the poker geometry.
Our results suggest that the complex stability of shells reduces to a low dimensional description and that tracking the ridge and the valley of the landscape of stability defines natural coordinates for describing the stability of shells. By using this new paradigm, we show that we can accurately and non-destructively predict the buckling load of real imperfect shell structures, thereby promising drastic reductions of the costs of structural engineering experimental tests.   

How localized imperfections modify the buckling threshold of cylindrical shells

Shells buckle and collapse at load levels below the thresholds predicted by linear stability analysis. The reduction of the load-carrying capacity has been linked to extreme sensitivity to geometric imperfections. Historically, spatially extended periodic imperfections have been analyzed using linear and weakly nonlinear theory. Recent fully nonlinear approaches suggest however the importance of spatially localized imperfections. We thus characterize experimentally and numerically the effect of spatially localized imperfections in an axially loaded cylindrical shell. For different types of imperfections, the buckling thresholds as a function of its size shows similar behaviour. This suggests a universal failure mechanism due to local buckling.   

Localized edge state equilibria control when a soda can buckles

Thin-walled cylindrical shells such as rocket walls (or soda cans) offer exceptional strength-to-weight ratios yet predicting at which load the structure becomes unstable and fails remains an unsolved problem. Shells buckle and collapse at loading conditions much below those predicted by linear stability theory. We thus propose a fully nonlinear approach and show that fully nonlinear equilibrium states located on the boundary of the unbuckled state's basin of attraction define critical perturbation amplitudes and guide the nonlinear initiation of catastrophic buckling. For a clamped thin cylindrical shell under axial compression a fully localized single dimple deformation is identified as the edge state—the attractor for the dynamics restricted to the stability boundary. Under variation of the axial load, the single dimple undergoes homoclinic snaking in the azimuthal direction, creating states with multiple dimples arranged around the central circumference. Once the circumference is completely filled with a ring of dimples, snaking in the axial direction leads to further growth of the dimple pattern. The bifurcation structure of the equilibria closely resembles that observed in the Swift-Hohenberg equation with quadratic-cubic nonlinearity.   

Interacting defects affect the buckling of imperfect spherical shells

The presence, distribution, and interaction of defects dictate the buckling strength of shell structures. Even if the sensitivity of shell buckling to imperfections has long been recognized, to date, the successful prediction of critical loads is restricted to cases with a single defect, of known geometry. However, in reality, shells typically contain a distribution of defects and their interaction has not been well studied. In this talk, we will focus on spherical shells with multiple defects and study the role of defect interactions in dictating the buckling pressure. In the experiments, we fabricate polymeric shells containing two precisely engineered geometric defects through a customizable coating technique. We vary the relative size and location of these two defects and quantify the relationship between buckling pressure and defect distribution. The experimental results are then contrasted against finite element modeling (FEM) simulations. Upon validation of the numerics, we use FEM to perform a broader and more systematic exploration of the parameter space. Our results provide a better understanding of defect interactions, which we hope will put us on a step to better predict the buckling pressure of practical shell structures.   

On Establishing Buckling Knockdowns for Imperfection-Sensitive Shell Structures

This presentation contributes to recent efforts aiming to revise long-standing knock-down factors for elastic shell buckling, which are widely regarded as overly conservative for well-constructed shells. The presented work focuses on cylindrical shells under axial compression with emphasis on the role of local geometric dimple imperfections and the use of lateral force probes as surrogate imperfections. Two buckling thresholds are identified (local and global buckling) and related for the two kinds of imperfections. Four sets of relevant boundary conditions are accounted revealing a strong dependence of the global buckling load on overall end-rotation constraint when local buckling precedes global buckling. A reasonably complete picture emerges, which should be useful for informing decisions on establishing knockdown factors.   

Quasi-static experimental path-following

Our work aims to exploit structural nonlinearity in engineering, with a particular focus on aerospace applications, to develop well-behaved nonlinear structures [1].

In addition to improved numerical methods, experimental validation of nonlinear structures is critical in ensuring their use in engineering applications, especially in conservative industries such as commercial aviation.

Existing experimental techniques are unable to fully characterise the nonlinear response of even simple nonlinear structures, as they cannot measure structures with force-displacement responses that include limit points and snapping behaviour. An experimental method has been developed to extend our ability to measure the structural response of nonlinear structures [2]. In this presentation we will present our recent developments in experimental path-following on nonlinear structures.

[1] G. Arena, R.M.J. Groh, A. Brinkmeyer, R. Theunissen, P.M. Weaver and A. Pirrera, Adaptive compliant structures for flow regulation, Proceedings of Royal Society A 473, 20170334 (2017)
[2] R.M. Neville, R.M.J. Groh, A. Pirrera, M. Schenk (2018), Shape Control for Experimental Continuation, Physical Review Letters, 120(25), 254101 (2018)   

On the role of localised post-buckling equilibria in axially compressed cylinders

We revisit buckling of axially compressed cylinders by considering fully localised post-buckling states in the form of one or multiple dimples. Using a combination of nonlinear quasi-static finite element methods and numerical continuation algorithms, we trace the evolution of odd and even dimples into one ring of circumferential diamond waves. The growth of the post-buckling pattern with varying compression is driven by a homoclinic snaking sequence, with even and odd dimple solutions intertwined. The initially stable and axially localised ring of circumferential diamonds destabilises at a pitchfork bifurcation to produce a second circumferential snaking sequence that results in the Yoshimura pattern. Localised dimple solutions represent saddle points in the energy landscape providing an exponentially decreasing energy barrier between the stable pre-buckling and re-stabilised post-buckling wells. The significance of the Maxwell load as a measure for quantifying the onset of mountain-pass solutions and the reduced resilience of the pre-buckling state is assessed. Finally, conservative buckling loads for design are inferred by tracing critical boundaries of the snaking set.   

Powering the Renaissance: Methods to Reveal the Energy Landscapes in Thin Shell Buckling

The extreme non-linearity of thin shell buckling introduces significant challenges to studying even the simplest geometries. Here, we develop a new approach by coupling efficient and powerful energy landscape methods to a conceptually simple discretized elastic mesh model. This highly versatile approach can probe the buckling energy landscape of any thin shell, or composite of thin shells, without the need for a priori assumptions about deformation morphologies or symmetries.

As an example, we investigate the landscape of an axially compressed cylinder, revealed to be remarkably rich and highly variable as a function of aspect ratio and compressive strain. Firstly, we observe that a relatively small, discrete set of sub-critical buckled states balloons in size as the aspect ratio is increased. We then use a string method to obtain the minimum energy pathways between any two minima. We show how experimental local probing techniques can access the true initial buckling transition for centrally-located dimples. By recursively connecting the local minima, we reveal the landscape to be dominated by a small number of multiply-dimpled states at small aspect ratios, becoming glassy and highly-connected at large aspect ratios, and featuring many multi-step pathways between minimum pairs.   

Buckling of thermalized cylindrical shells

We explore how thermal fluctuations affect the buckling of thin cylindrical shells. It is known that for flat solid sheets thermal fluctuations effectively increase the bending rigidity and reduce the bulk and shear moduli. As a consequence, thermal fluctuations increase the critical buckling load. In cylindrical shells, thermal fluctuations also increase the bending rigidity and reduce the in-plane elastic constants. However, the additional coupling between the shell curvature, the in-plane stretching modes and the out-of-plane undulations leads to novel phenomena. In shells thermal fluctuations effectively generate compressive load. As a consequence, the critical axial buckling load for cylindrical shells is reduced due to thermal fluctuations, which is similar to the reduced buckling pressure for spherical shells, but different from the enhanced buckling load for flat sheets. Similar to spherical shells, we find that for cylindrical shells with a sufficiently large radius the thermally generated compression can be large enough that shells become unstable even in the absence of external load. Furthermore, we find that the critical radius also depends on the aspect ratio (length/perimeter) of cylindrical shells.   

Imperfection-insensitive thin wavy cylindrical shells under bending: Effect of local radius of curvature on buckling and imperfection-sensitivity.

The imperfection-sensitivity of thin cylindrical shells has long been an obstacle for their optimal applications. To nullify this behavior, a conservative knock-down factor method is utilized for the design of thin cylindrical shells. Alternatively, stiffeners are also used to increase their capacity and to reduce the imperfection-sensitivity. We explore wavy cross-sectional thin cylindrical shells under bending to investigate the impact of imperfections on their load carrying capacity. We found that thin wavy cylindrical shells are insensitive to imperfections under bending in contrast to thin circular cylinders. This insensitivity is achieved by reducing the local radius of curvature and consequently, the effective radius of the cylindrical shell is reduced. This way cylindrical shells become less sensitive to imperfections and increase their load carrying capacity.   

Out-of-plane buckling of architected sheets with non-periodic cut patterns

We investigate the out-of-plane shape morphing capability of elastic sheets with architected cut patterns. These cuts result in arrays of tiles connected by flexible hinges. We demonstrate that a non-periodic cut pattern can cause a sheet to buckle into three-dimensional shapes, such as domes or patterns of wrinkles, when pulled at specific boundary points. This phenomenon stems from the geometric incompatibilities between regions that are designed to undergo different amounts of strain. Global buckling modes observed in experiments are rationalized by an in-plane kinematic analysis, and are reproduced in simulations of homogenized shell models implemented in the open-source finite-element platform FEniCS. Our work illustrates a scalable route towards the fabrication of three-dimensional objects with nonzero Gaussian curvature from initially-flat sheets.