Small Geometric Imperfections Research Articles

3D-printing and cylinder buckling: challenges and opportunities.

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'.

New insight into the stability and dynamics of fluid-conveying supported pipes with small geometric imperfections

In several previous studies, it was reported that a supported pipe with small geometric imperfections would lose stability when the internal flow velocity became sufficiently high. Recently, however, it has become clear that this conclusion may be at best incomplete. A reevaluation of the problem is undertaken here by essentially considering the flow-induced static deformation of a pipe. With the aid of the absolute nodal coordinate formulation (ANCF) and the extended Lagrange equations for dynamical systems containing non-material volumes, the nonlinear governing equations of a pipe with three different geometric imperfections are introduced and formulated. Based on extensive numerical calculations, the static equilibrium configuration, the stability, and the nonlinear dynamics of the considered pipe system are determined and analyzed. The results show that for a supported pipe with the geometric imperfection of a half sinusoidal wave, the dynamical system could not lose stability even if the flow velocity reaches an extremely high value of 40. However, for a supported pipe with the geometric imperfection of one or one and a half sinusoidal waves, the first-mode buckling instability would take place at high flow velocity. Moreover, based on a further parametric analysis, the effects of the amplitude of the geometric imperfection and the aspect ratio of the pipe on the static deformation, the critical flow velocity for buckling instability, and the nonlinear responses of the supported pipes with geometric imperfections are analyzed.

EXPERIMENTAL AND NUMERICAL INVESTIGATIONS ON THE STABILITY OF CYLINDRICAL SHELLS

The stability of thin-walled cylindrical shells under axial pressure is investigated. The results of both experiments and numerical simulations are presented. An appropriate finite element model is introduced that accounts not only for geometric imperfections but also for non-linearities. It is found that small geometrical imperfections within a given tolerance range have considerable negative effect on the buckling load compared to perfect geometry. Various post buckling shell shapes are possible, which depend on these imperfections. The experiments and simulations show a very good correlation.

Big influence of small random imperfections in origami-based metamaterials

Origami structures demonstrate great theoretical potential for creating metamaterials with exotic properties. However, there is a lack of understanding of how imperfections influence the mechanical behaviour of origami-based metamaterials, which, in practice, are inevitable. For conventional materials, imperfection plays a profound role in shaping their behaviour. Thus, this paper investigates the influence of small random geometric imperfections on the nonlinear compressive response of the representative Miura-ori, which serves as the basic pattern for many metamaterial designs. Experiments and numerical simulations are used to demonstrate quantitatively how geometric imperfections hinder the foldability of the Miura-ori, but on the other hand, increase its compressive stiffness. This leads to the discovery that the residual of an origami foldability constraint, given by the Kawasaki theorem, correlates with the increase of stiffness of imperfect origami-based metamaterials. This observation might be generalizable to other flat-foldable patterns, in which we address deviations from the zero residual of the perfect pattern; and to non-flat-foldable patterns, in which we would address deviations from a finite residual.

Nonlinear hydrodynamic instability and turbulence in pulsatile flow

Pulsating flows through tubular geometries are laminar provided that velocities are moderate. This in particular is also believed to apply to cardiovascular flows where inertial forces are typically too low to sustain turbulence. On the other hand, flow instabilities and fluctuating shear stresses are held responsible for a variety of cardiovascular diseases. Here we report a nonlinear instability mechanism for pulsating pipe flow that gives rise to bursts of turbulence at low flow rates. Geometrical distortions of small, yet finite, amplitude are found to excite a state consisting of helical vortices during flow deceleration. The resulting flow pattern grows rapidly in magnitude, breaks down into turbulence, and eventually returns to laminar when the flow accelerates. This scenario causes shear stress fluctuations and flow reversal during each pulsation cycle. Such unsteady conditions can adversely affect blood vessels and have been shown to promote inflammation and dysfunction of the shear stress-sensitive endothelial cell layer.

Influence of initial geometric imperfections on the 1:1:1:1 internal resonances and nonlinear vibrations of thin-walled cylindrical shells

The analysis of internal resonances in continuous structural systems is one of the main research areas in the field of nonlinear dynamics. Internal resonances enable the transfer of energy between the related vibration modes, usually leading to new phenomena that have a profound influence on nonlinear oscillations, bifurcations and dynamic instabilities. Shells of revolution usually exhibit internal resonances due to their inherent circumferential symmetry and dense frequency spectrum in the lower frequency range, which may lead not only to m:n internal resonances but also to multiple internal resonances. In this work, the nonlinear response of an imperfect circular cylindrical shell, simply supported at the edges, to harmonic excitation is studied. Geometries that have two modes with n and n+1 circumferential waves, corresponding to the lowest natural frequency, are identified. These two modes are driven to resonance, with each being in one-to-one internal resonance with its companion mode, thus leading to a possible 1:1:1:1 internal resonance, a topic rarely investigated in the technical literature. The investigation of internal resonances in continuous systems is usually conducted using low-dimensional discrete models. Here, using a perturbation procedure, a consistent modal expansion is derived for an arbitrary number of interacting modes, leading to reliable low-dimensional models. Using the discrete models derived in this way, the shell nonlinear dynamics is explored by using bifurcation diagrams of the Poincaré map, continuation techniques and the Floquet stability criterion. The importance of internal resonances to the nonlinear vibrations and instabilities of the shell is clarified. It is well known that small geometric imperfections in the order of the shell thickness have a strong influence on the buckling and post buckling behavior of a thin-walled shell. However, their influence on internal resonances, dynamic instability and energy transfer is largely unknown. Thus, a detailed parametric analysis that considers different types of modal imperfection is conducted in the present work, and the influence of such imperfections on the activation of energy exchanges between the modes involved is analyzed. The results confirm that the form and magnitude of initial geometric imperfections have a profound influence on the results, enabling or preventing the transfer of energy among the resonant modes being considered. • Derivation of a consistent modal expansion by a perturbation procedure considering arbitrary number of interacting modes. • The influence of geometric imperfections on internal resonances, dynamic instability and energy transfer between modes. • Analysis of 1:1:1:1 internal resonance considering four vibration modes with the same natural frequency. • The dynamic bifurcations are sensitive to the form and magnitude of initial geometric imperfections.

Buckling and collapse of pseudoelastic NiTi tubes under bending

Pseudoelastic NiTi structural components have been shown to exhibit tension/compression asymmetry. Under tension, phase transformation induces localized strains of nearly 7% that spread under nearly constant stress, while under compression the transformation leads to hardening at significantly higher stress, the strain is smaller and grows homogeneously. Bending of tubes ovalizes their cross-section, a nonlinearity that reduces the bending rigidity and precipitates instabilities that can lead to collapse. This paper uses experiment and analysis to examine the interaction of these geometric nonlinearities with the complex material behavior of pseudoelastic NiTi tubes of moderate diameter-to-thickness ratios under bending. An experiment on a tube with a diameter-to-thickness ratio of 18.7 shows that transformation causes the nucleation of several dispersed high strain banded patterns of martensite on the tensioned side. The zone where the bands coalesce into intersecting diamonds undergoes excessive local ovalization causing buckling and collapse at a relatively low overall tube curvature. The bending experiment is simulated with a finite element model coupled with a custom constitutive model of pseudoelastic NiTi that allows for plastic deformation of the martensitic phase. The analysis reproduces the moment-end rotation response, and the nucleation and evolution of the high strain diamond shaped patterns. Zones that develop such patterns exhibit accelerated growth of ovalization that degrades the local bending rigidity. In the presence of small geometric imperfections this degradation evolves into a buckle in the form of diffuse local ovalization that precipitates collapse that can negate the recoverable property of NiTi. Parametric studies show that as the tube D/t increases, the interaction between the material and geometric nonlinearities becomes stronger making buckling more imperfection sensitive.

Pulse transmission and acoustic non-reciprocity in a granular channel with symmetry-breaking clearances

Granular media composed of ordered arrays of discrete spherical granules have attracted considerable attention due to their highly nonlinear and discontinuous dynamics and acoustics that enable passively adaptive and tailorable properties. In this work we numerically study nonlinear pulse propagation in a two-dimensional (2D) granular channel composed of a main homogeneous lattice with several pairs of side granules. Depending on the direction of pulse transmission in the main lattice, a periodic series of symmetry-breaking clearances alter the topology of this 2D granular network. The rotational dynamics of the individual granules, as well as dissipative effects due to friction between granules (and their boundaries) and inherent material damping, are proven to play a significant role in the acoustics of the granular channel. This is demonstrated by comparing the theoretical predictions of this work to experimental measurements of the same system reported earlier. Moreover, the strong nonlinearity of this system in synergy with the topological asymmetry introduced by the clearances passively breaks acoustic reciprocity. To this end, a detailed study of pulse transmission in the 2D granular channel is performed, and it is shown that by simply changing the direction of pulse transmission it is possible to switch the nonlinear acoustics from Nesterenko solitary pulses to strongly decaying propagating pulses. In the latter case there is continuous and irreversible transfer of energy from the main propagating pulse to the side granules, which act, in essence, as nonlinear energy absorbers. This work highlights the important role that (even small) geometric imperfections (asymmetries) may play on the acoustics of 2D granular media.

Continuum modeling of crushing of low density foams

Under compression deformation in low-density foams localizes into narrow bands of crushed cells. Crushing spreads at nearly constant stress with crushed and relatively undeformed material coexisting. The material returns to homogeneous deformation with increasing stress when the crushing has spread over the whole specimen. This paper presents a first attempt at representing the inhomogeneous behavior exhibited by open-cell aluminum alloy foams with relative density of 8%. A plasticity model is presented with a compressible yield function calibrated to a set of multiaxial foam crushing tests coupled with a non-associated flow rule. An essential component of the modeling effort is the introduction of a softening branch to the material stress-strain response. A cubical finite element model with an irregular mesh of solid elements is used to simulate a set of numerical crushing tests on micromechanically accurate foam models performed in a true triaxial apparatus in Yang and Kyriakides (2019). Small geometric imperfections are used to trigger localized deformation in the form of planar bands of high strain normal to the loading directions of compression. The bands broaden with the stresses tracing plateaus that mimic the random foam test results. The predicted mean stress–change in volume responses perform equally well up to a volume reduction of about 50%. At higher compressions, the stresses exhibit the gradual increase characteristic of foams in the densification regime. These parts of the stress–displacement trajectories, and mean stress–change in volume responses under-predict to some degree those recorded in the random foams. The material hardening in the densification regime appears to be deformation-dependent causing this difference.

Programmable higher-order Euler buckling modes in hierarchical beams

We present a numerical-aided experimental study on the buckling of hierarchical beams comprising multiple self-similar modules. Each module consists of multiple elemental beams and is arranged in series to form the hierarchical beam. We show, through a combination of experiments and computations, that these beams exhibit stable and realizable higher-order buckling modes. By contrast to the canonical Euler buckling problem, such modes emerge naturally in the proposed self-similar beams since they correspond to almost identical critical loads. By harnessing the imperfection sensitivity of the hierarchical structures, we 3D-print weakly imperfect polymer samples with a small geometric imperfection corresponding to the desired eigenmode. We subsequently carry out uniaxial compression experiments and show in practice that higher-order patterns can be triggered selectively upon buckling. Moreover, these patterns are preserved in the post-bifurcation regime in many cases and are reversible upon load release. The ability to trigger higher-order buckling modes is found to depend on two main geometrical parameters which lead to scale coupling. Those are the slenderness of the macroscopic hierarchical beam and the slenderness of the lower-scale elemental beam. With increasing slenderness of the hierarchical beam, we observe a significant softening in the overall stress-strain response and patterns exhibiting curvature localization in the post-bifurcation regime. The numerical finite-strain simulations carried out in the present study are found to be in very good agreement with the experiments and are used to quantify further the observed curvature localization in the hierarchical beams. The present study and the obtained results are geometric in nature and thus can be extended to different scales and hierarchies ad infinitum.

Experiments on the thermal post-buckling of panels, including localized heating

The suppression of expansion in thin clamped panels subjected to elevated thermal loading often results in buckling. However, a number of possible post-buckled equilibrium configurations typically exist, and which shape ensues depends on a number of factors including the role of symmetry, boundary conditions, aspect ratio, and the effect of small geometric imperfections associated with the initial shape. It is possible to force the panel to go between different buckled shapes, given a sufficiently large perturbation. Sometimes the panel will spontaneously jump, or snap, when the temperature is gradually increased or decreased (mode jumping). The extent to which these features occur when the thermal loading is applied locally is also investigated. This paper describes some interesting nonlinear (essentially buckling) behavior in thermally-loaded panels from a primarily experimental perspective, with an additional focus on non-uniform heating. The full force of stereo 3D digital image correlation and forward-looking infrared cameras are exploited to provide a relatively complete picture of this behavior.

Robustly Tuning Bandgaps in Two-Dimensional Soft Phononic Crystals with Criss-Crossed Elliptical Holes

Tuning band gaps in soft materials by post-buckling deformation is becoming an appealing means to manipulate elastic waves. As one of the most favorable topologies, two-dimensional soft structures with circular holes have been extensively studied. Based on the contrarian thinking, this paper starts from the two-dimensional soft structures with criss-crossed elliptical holes, which is close to the post-buckling configuration of soft structures with circular holes, and then proposes to tune the band gaps through elongating or stretching rather than compressing. Influences of the loading magnitude and loading pattern (i.e., uniaxial and biaxial elongations) on the band gaps are studied via the nonlinear finite element simulations. Effects of the geometric parameters (the major-to-minor half-axis ratio and the porosity of the structure) are also discussed. It is shown that, compared with the traditional circular hole case, the band gaps of the unloaded structure with criss-crossed elliptical holes are much richer, and they could be reversely and continuously tuned by tensile loadings. In particular, the deformation is very robust and is insensitive to small geometric imperfections, which is always necessary for triggering the post-buckling deformations. The present work provides a useful reference to the manipulation of elastic waves in periodic structures as well as the design of soft phononic crystals/acoustic devices.

Nonlinear Dynamics of Imperfect FGM Conical Panel

Structures composed of functionally graded materials (FGM) can satisfy many rigorous requisitions in engineering application. In this paper, the nonlinear dynamics of a simply supported FGM conical panel with different forms of initial imperfections are investigated. The conical panel is subjected to the simple harmonic excitation along the radial direction and the parametric excitation in the meridian direction. The small initial geometric imperfection of the conical panel is expressed by the form of the Cosine functions. According to a power-law distribution, the effective material properties are assumed to be graded along the thickness direction. Based on the first-order shear deformation theory and von Karman type nonlinear geometric relationship, the nonlinear equations of motion are established by using the Hamilton principle. The nonlinear partial differential governing equations are truncated by Galerkin method to obtain the ordinary differential equations along the radial displacement. The effects of imperfection types, half-wave numbers of the imperfection, amplitudes of the imperfection, and damping on the dynamic behaviors are studied by numerical simulation. Maximum Lyapunov exponents, bifurcation diagrams, time histories, phase portraits, and Poincare maps are obtained to show the dynamic responses of the system.

Parametric vibrations of imperfect Timoshenko microbeams

This paper examines the size-dependent nonlinear parametric vibrations of imperfect Timoshenko microbeams subject to a time-dependent axial load. Taking into account rotary inertia and shear deformation, the modified couple stress theory is employed so as to derive a continuous expression for the elastic potential energy of the system; the geometric imperfection is modelled via a small initial deflection in the transverse direction. The continuous model of the system is obtained via an energy balance method. A reduced-order model is obtained by means of a weighted residual method together with an assumed-mode technique. The double-dimensionalised form of the reduced-order model is solved by means of different numerical techniques and stability is analysed. It is shown that even small geometric imperfections may change the parametric response of the system not only quantitatively, but also qualitatively.

Nonlinear Vibrations of Fluid-Filled Cylindrical Shells with Initial Imperfections

Theoretical approaches to studying the free vibrations of imperfect (with small geometrical imperfections) orthotropic cylindrical shells filled with a fluid are proposed. A geometrically nonlinear problem is solved. The emphasis is on the interaction and energy exchange between the “conjugate” flexural modes in the dynamic radial deflection. The effect of initial deflections on the interaction of the initial deflection modes is analyzed

Focusing properties of a spherical perfect lens with eccentric deformations

A spherical perfect lens made of a shell of negative refractive index material is known not only for its subwavelength resolution of images but also for the magnification of the near-field image due to its curved geometry. We investigate the effects on the image resolution arising from geometric deformations of the spherical perfect lens, whereby an eccentricity in the spherical shell is introduced. Using a first-order perturbative method valid for small deformations in the near-field (quasi-static) limit, we find that small deformations (∼1% of the inner radius of the shell) appear to affect the strength of the multipoles in the image by only few percent and do not have any drastic effects. Our results indicate that near-field imaging can be robust against small geometric imperfections in these imaging devices, which is expected to be highly encouraging to nanoscale/microscale design in optical engineering.

Imperfection sensitivity of locally supported cylindrical silos subjected to uniform axial compression

For the prediction of the real failure load of shell structures, such as locally supported cylindrical steel silos under axial compression, it is convenient to take into account imperfections. It is assumed that such silos are very sensitive to a wide range of (even small) geometric imperfections, and that they lower the failure load significantly. Furthermore, these imperfections caused by the fabrication or the manufacturing process, are the dominant factor in the discrepancy between the theoretical/numerical predictions based on a perfect geometry and the experimental results of an imperfect geometry. In other words, it is important to make a well-considered choice for an imperfection when predicting the real failure load. However, the imperfection sensitivity depends, among other things, on the shape of the shell, the stiffening configuration, the boundary and loading conditions, etc. Before proceeding to the calculation of interaction curves and the development of new design rules for imperfect barrels, it is essential to perform an extensive study to examine the influence of imperfections to the failure behaviour and to choose a sufficiently detrimental imperfection shape. In this study, different imperfection forms are numerically investigated: the linear bifurcation mode, the non-linear buckling mode, several post-buckling deformed shapes of the perfect shell, and a weld depression type A and B. Additional aspects, such as the orientation, the amplitude of the equivalent imperfection, and the position of the influence of the weld depression are also investigated. The present study takes into account the European normative documents and the guidelines of the recommendations of the ECCS.

Nonlinear dynamics of initially imperfect functionally graded circular cylindrical shell under complex loads

The nonlinear vibration of a simply supported FGM cylindrical shell with small initial geometric imperfection under complex loads is studied. The effects of radial harmonic excitation, compressive in-plane force combined with supersonic aerodynamic and thermal loads are considered. The small initial geometric imperfection of the cylindrical shell is characterized in the form of the sine-type trigonometric functions. The effective material properties of this FGM cylindrical shell are graded in the radial direction according to a simple power law in terms of the volume fractions. Based on Reddy’s third-order shear deformation theory, von Karman-type nonlinear kinematics and Hamilton’s principle, the nonlinear partial differential equation that controls the shell dynamics is derived. Both axial symmetric and driven modes of the cylindrical shell deflection pattern are included. Furthermore, the equations of motion can be reduced into a set of coupled nonlinear ordinary differential equations by applying Galerkin’s method. In the study of the nonlinear dynamics responses of small initial geometric imperfect FGM cylindrical shell under complex loads, the 4th order Runge−Kutta method is used to obtain time history, phase portraits, bifurcation diagrams and Poincare maps with different parameters. The effects of external loads, geometric imperfections and volume fractions on the nonlinear dynamics of the system are discussed.

Liner wrinkling and collapse of girth-welded bi-material pipe under bending

Pipelines and flowlines that carry corrosive hydrocarbons are often protected by lining them internally with a thin layer of a corrosion resistant material. In the most economic method, the liner is brought in contact with a carbon steel carrier pipe by mechanical expansion. In applications involving severe plastic bending, such as winding onto a large diameter drum as is done in the reeling installation method, such a liner can wrinkle and collapse while the carrier pipe remains intact. Collapse has been shown to be sensitive to small initial geometric imperfections in the liner. A numerical framework for establishing the extent to which lined pipe can be bent before liner collapse was presented in [14–16]. This framework, suitably extended is used here to examine the effect of girth welds on liner collapse. The modeling starts by simulating the expansion process that plastically deforms the two tubes bringing them into contact. Bending plastically the composite structure leads to differential ovalization of the two tubes and detachment of the liner. The girth weld locally prevents this detachment creating a periodic boundary disturbance in the liner. With increasing bending the periodic disturbance grows and eventually yields to buckling into a shell-type diamond-shaped mode that causes the liner to collapse inside the intact outer pipe. The problem is investigated using a 12-inch carrier pipe base case. Comparing the collapse curvature of a girth-welded liner with imperfect liners free of welds that have the same collapse curvature, it is concluded that girth welds constitute a “weak” spot on the line. Results from a parametric study of factors that influence the collapse of a girth-welded are presented followed by recommendations.

Liner wrinkling and collapse of bi-material pipe under axial compression

Lined pipe describes a product where a carbon steel pipe is lined internally with a thin layer of a corrosion resistant material in order to protect it from corrosive constituents in the hydrocarbons it carries. Most commonly the liner is brought into contact with the carrier pipe by mechanical expansion. Full-scale tests on this product have demonstrated that, under mechanical loads that plastically deform the composite structure, the thin liner can buckle and collapse inside an intact outer pipe making the structure unserviceable. This paper investigates the extent to which typical lined pipes can be axially compressed before liner collapse. Demonstration experiments on model lined systems illustrate that the liner, although supported by contact with the outer pipe, first buckles unilaterally into an axisymmetric wrinkling mode at a relatively low strain. The wrinkles grow stably with compression but yield to a non-axisymmetric diamond-type mode that results in the collapse of the liner at a higher strain. This process has been modeled numerically starting with simulation of the mechanical expansion through which the composite structure is manufactured. The sensitivity of the collapse strain to the various parameters of the problem is studied and amongst other findings it is shown to be very sensitive to small geometric imperfections in the liner. It is also demonstrated that even modest amounts of internal pressure can delay liner collapse up to strains at which the outer pipe collapses.