Buckled Configuration Research Articles

Enhanced Efficiency of Electrostatically Actuated Bistable Microswitches Using Bow-Like Operation

To induce snap-through (ST) switching in an electrostatically actuated bistable latchable curved microbeam, a close gap electrode facing the concave surface of the beam is commonly used. In the framework of an alternative architecture considered here, the electrode is placed in front of the convex (backward), rather than concave (forward) side of a beam. The initially curved stress free beam is activated by an electrostatic force pulling it quasistatically in the backward direction, away from the desired buckled configuration. Once the voltage is turned off , the accumulated strain energy is released, and the beam is catapulted toward the buckled state, similarly to an arrow fired by a bow. In the case where the beam exhibits latching, the device may remain in its buckled state indefinitely under zero voltage. The beam can also be snapped back (released), by applying a voltage to the same electrode. In this article, the efficiency of this bow-like actuation is estimated. By means of a reduced order model, it is shown that due to the nonlinearity of the electrostatic force, the suggested operation scenario may lead to a 54 ${\%}$ reduction in the ST switching voltage, when compared to the traditional forward actuation. As a result, the device allows bidirectional switching by using a single electrode, and can be viewed as a micromechanical realization of a toggle flip-flop element.

Nonlinear bifurcation analysis of stiffener profiles via deflation techniques

When loading experiments are repeated on different samples, qualitatively different results can occur. This is due to factors such as geometric imperfections, load asymmetries, unevenly stressed regions or uneven material distributions created by manufacturing processes. This fact makes designing robust thin-walled structures difficult. One numerical strategy for exploring these different possible responses is to impose various initial imperfections on the geometry before loading, leading to different final solutions. However, this strategy is tedious, error-prone and gives an incomplete picture of the possible buckled configurations of the system. The present study demonstrates how a deflation strategy can be applied to obtain multiple solutions for the more robust design of thin-walled structures under displacement controlled uniaxial compression. We first demonstrate that distinct initial imperfections trigger different sequences of instability events in the Saint Venant–Kirchhoff hyperelastic model. We then employ deflation to investigate multiple bifurcation paths without the use of initial imperfections. A key advantage of this approach is that it can capture disconnected branches that cannot easily be discovered by conventional arc-length continuation and branch switching algorithms. Numerical experiments are given for three types of aircraft stiffener profiles. Our proposed technique is shown to be a powerful tool for exploring multiple disconnected bifurcation paths without requiring detailed insight for designing initial imperfections. We hypothesise that this technique will be very useful in the design process of robust thin-walled structures that must consider a variety of bifurcation paths.

Two-span piezoelectric beam energy harvesting

This paper reports an analytical study of nonlinear vibratory energy harvesting via a two-span piezoelectric beam. The distributed parameter electromechanical model is applied to explore the benefits of nonlinear interaction between buckled beam spans. The Galerkin truncation method and harmonic balance method are applied together to find forced responses of the energy harvesting system. The effects of axial pre-pressure on energy harvesting are also explored. The analytical results are supported by utilizing direct time integration of the equation of motion. The amplitude and voltage output frequency response functions (FRF) of single-span, buckled, and two-span piezoelectric beam configurations reveal some differences. For example, the voltage output FRF of the two-span buckled configuration is larger than that of the other two configurations at high frequencies, whereas at low frequencies, the voltage output FRF of two-span unbuckled configuration is larger than that of the other two configurations. Moreover, two-span buckled configuration has more natural frequencies lying at the concerned frequency range. Finally, a hybrid harvester is proposed, which combines harvested voltage from single-span buckled, two-span, and two-span buckled piezoelectric beam to cover a demand of broadband voltages output.

Water Affects Morphogenesis of Growing Aquatic Plant Leaves.

Lotus leaves floating on water usually experience short-wavelength edge wrinkling that decays toward the center, while the leaves growing above water normally morph into a global bending cone shape with long rippled waves near the edge. Observations suggest that the underlying water (liquid substrate) significantly affects the morphogenesis of leaves. To understand the biophysical mechanism under such phenomena, we develop mathematical models that can effectively account for inhomogeneous differential growth of floating and freestanding leaves to quantitatively predict formation and evolution of their morphology. We find, both theoretically and experimentally, that the short-wavelength buckled configuration is energetically favorable for growing membranes lying on liquid, while the global buckling shape is more preferable for suspended ones. Other influencing factors such as the stem or vein, heterogeneity, and dimension are also investigated. Our results provide a fundamental insight into a variety of plant morphogenesis affected by water foundation and suggest that such surface instabilities can be harnessed for morphology control of biomimetic deployable structures using substrate or edge actuation.

Supercritical and Subcritical Hopf Bifurcations in a Delay Differential Equation Model of a Heat-Exchanger Tube Under Cross-Flow

Abstract Nonlinear vibrations of a heat-exchanger tube modeled as a simply supported Euler–Bernoulli beam under axial load and cross-flow have been studied. The compressive axial loads are a consequence of thermal expansion, and tensile axial loads can be induced by design (prestress). The fluid forces are represented using an added mass, damping, and a time-delayed displacement term. Due to the presence of the time-delayed term, the equation governing the dynamics of the tube becomes a partial delay differential equation (PDDE). Using the modal-expansion procedure, the PDDE is converted into a nonlinear delay differential equation (DDE). The fixed points (zero and buckled equilibria) of the nonlinear DDE are found, and their linear stability is analyzed. It is found that stability can be lost via either supercritical or subcritical Hopf bifurcation. Using Galerkin approximations, the characteristic roots (spectrum) of the DDE are found and reported in the parametric space of fluid velocity and axial load. Furthermore, the stability chart obtained from the Galerkin approximations is compared with the critical curves obtained from analytical calculations. Next, the method of multiple scales (MMS) is used to derive the normal-form equations near the supercritical and subcritical Hopf bifurcation points for both zero and buckled equilibrium configurations. The steady-state amplitude response equation, obtained from the MMS, at Hopf bifurcation points is compared with the numerical solution. The coexistence of multiple limit cycles in the parametric space is found, and has implications in the fatigue life calculations of the heat-exchanger tubes.

Interaction Between Thermal Field and Two-Dimensional Functionally Graded Materials: A Structural Mechanical Example

The buckling and free vibration of a Euler–Bernoulli beam composed of two-directional functionally graded materials (FGMs) in thermal environment are analyzed. The material properties and temperature distributions are considered to be continuously varied along both axial and thickness directions. Such two-directional FGMs provide the basis of a promising strategy to tune the dynamic behavior of a structure in a controlled fashion, achieving tunable response as desired. The dynamic equation of the beam and relevant boundary conditions are derived based on Hamilton’s principle. The generalized differential quadrature method is used for determining the exact buckling configuration and the natural frequencies of the beam with different boundary conditions. Numerical results are presented to examine the effects of material gradations on the critical buckling temperature. It is concluded that both temperature change and material properties have significant influences on the natural frequency, which suggests that it is possible to tailor or tune the dynamic behaviors of a beam by using man-made FGMs in a complex environment.

Bistable nonlinear damper based on a buckled beam configuration

This article addresses a particular realization of a compact bistable nonlinear absorber based on the concept of nonlinear energy sink. The article presents both a detailed description of the absorber mechanics and an illustration of the targeted energy transfer between the absorber and a linear system. The experimental results are accompanied with the numerical simulations. Beside practical improvements linked to the features of absorber design, the obtained results stay in line with those found for simpler realizations of a bistable nonlinear energy sinks.

Harnessing energy landscape exploration to control the buckling of cylindrical shells

Even for relatively simple thin shell morphologies, many different buckled configurations can be stable simultaneously. Which state is observed in practice is highly sensitive to both environmental perturbations and shell imperfections. The complexity and unpredictability of postbuckling responses has therefore raised great challenges to emerging technologies exploiting buckling transitions. Here we show how the buckling landscapes can be explored through a comprehensive survey of the stable states and the transition mechanisms between them, which we demonstrate for cylindrical shells. This is achieved by combining a simple and versatile triangulated lattice model with efficient high-dimensional free-energy minimisation and transition path finding algorithms. We then introduce the method of landscape biasing to show how the landscapes can be exploited to exert control over the postbuckling response, and develop structures which are resistant to lateral perturbations. These methods now offer the potential for studying complex buckling phenomena on a range of elastic shells.

Configurations evolution of a buckled ribbon in response to out-of-plane loading

Classical Euler buckled ribbon has become the cornerstone of many unprecedented applications such as the recently developed stretchable electronics and mechanical-guided 3D assembly. In practical applications, such Euler buckled ribbon may be subjected to out-of-plane loading. However, previous studies mostly concern about the buckling and post-buckling configurations of ribbons under in-plane compression, but the further mechanical behavior of buckled ribbons in response to out-of-plane loading is rarely involved. In this paper, the mechanical behaviors of Euler buckling ribbons under out-of-plane loading are systemically investigated, and distinct configurations evolution paths were observed, which are dependent on both axial in-plane compressive strain of Euler buckling and out-of-plane loading. An analytical model based on geometrical decomposition method and finite deformation beam theory is proposed to clarify the mechanism of the competitions between different configurations evolution paths, to construct phase diagrams to distinguish different deformation modes, and to draw the corresponding deformed profiles, which agree well with the experiments and finite elemental stimulations. Furthermore, the out-of-plane stiffness of Euler buckled ribbons with unusual characteristics (e.g., negative stiffness and stiffness reduction) is also quantified, which has general utility as an engineering design rule for exploiting Euler buckled ribbons in stretchable electronics, 3D bio-interfaces, mechanical-guided 3D assembly and mechanical metamaterials.

Massless Dirac fermions in stable two-dimensional carbon-arsenic monolayer

We predict from DFT based electronic structure calculations that a monolayer made up of Carbon and Arsenic atoms, with a chemical composition (CAs3) forms an energetically and dynamically stable system. The optimized geometry of the monolayer is slightly different from the buckled geometric configuration observed for silicene and germanene. The results of electronic structure calculations predict that it is a semi-metal. Interestingly, the electronic band structure of this material possesses a linear dispersion and a Dirac cone at the Fermi level around the high symmetric K point in the reciprocal lattice. Thus, at low energy excitation (up to 105 meV), the charge carriers in this system behave as massless Dirac-Fermions. Detailed analysis of partial density of state suggests that the 2pz orbital of C atoms plays vital role in determining the nature of the states, which has a linear dispersion and hence the Dirac cone, around the Fermi level. Thus, the electronic properties of CAs3 monolayer are similar to those of graphene and other group IV based monolayers. In addition, we have also investigated the influence of mechanical strain on the properties of CAs3 monolayer. The buckled configuration becomes the planar configuration for a tensile strain beyond 18%. Our results indicate that the monolayer possesses linear dispersion in the electronic band structure for a wide range of mechanical strain from -12 to 20%, though the position of Dirac point may not lie exactly at the Fermi level. The linear dispersion disappears for a compressive strain beyond -12% & it is due to the drastic changes in the geometrical environment around C atom. Finally, we wish to point out that CAs3 monolayer belongs to the class of Dirac materials where the behaviour of particles, at low energy excitations, are characterized by the Dirac-like Hamiltonian rather than the Schrodinger Hamiltonian.

Nonlinear Forced Vibration of Thermally Postbuckled Double-Layered Triangular Graphene Sheet with Clamped Boundary Conditions

In the present research, vibration behavior is presented for a thermally postbuckled double-layered triangular graphene sheet (DLTGS). The DLTGS is modeled as a nonlocal orthotropic plate and contains small-scale effects. The formulations are based on the Kirchhoff’s plate theory, and a nonlinearity of von Karman-type is considered in strain–displacement relations. The thermal effects and van der Waals forces between layers are also included and some of the material properties are assumed to be temperature-dependent. A semi-analytical solution is obtained using the multiple time scales method. The effects of variation of small-scale parameter to the natural frequencies, deflections and response curve of DLTGS are analyzed, and the numerical results are obtained from the nonlocal plate model. Numerical results are compared with those of similar researches. Effects of various parameters on the postbuckled vibration of DLTGS in thermal environments such as scale parameter and thermal load are presented. The stability and occurrence of the internal resonance between vibration modes around a stable buckled configuration is investigated.

Asymmetric compressive stability of rotating annular plates

In the present research, buckling behaviour of an isotropic homogeneous rotating annular plate subjected to uniform compression on both inner and outer edges is analysed. It is further assumed that the plate is rotatingwith a constant angular speed. Formulation is based on the first order shear deformation plate theory, which is valid for thin and moderately thick plates. The complete set of equilibrium quations and the associated boundary conditions are obtained for the plate. Prebuckling loads of the plate are obtained under flatness and axisymmetric deformations. Using the adjacent equilibrium riterion, the linearised stability equations are extracted. An asymmetric stability analysis is performed to obtain the critical buckling loads of the plate and the buckled configurations of the rotating plate. To this end, trigonometric functions through the circumferential direction and the generalised differential quadrature discretization across the radial direction are used which result in an algebraic eigenvalue problem. Benchmark results are given in graphical presentations for combinations of free, simply-supported, sliding supported, and clamped types of boundary conditions. It is shown that rotation enhances the buckling loads of the plate for all types of boundary conditions and alters the buckled shape of the plate.

Evidence of Topological Edge States in Buckled Antimonene Monolayers.

Two-dimensional topological materials have attracted intense research efforts owing to their promise in applications for low-energy, high-efficiency quantum computations. Group-VA elemental thin films with strong spin-orbit coupling have been predicted to host topologically nontrivial states as excellent two-dimensional topological materials. Herein, we experimentally demonstrated for the first time that the epitaxially grown high-quality antimonene monolayer islands with buckled configurations exhibit significantly robust one-dimensional topological edge states above the Fermi level. We further demonstrated that these topologically nontrivial edge states arise from a single p-orbital manifold as a general consequence of atomic spin-orbit coupling. Thus, our findings establish monolayer antimonene as a new class of topological monolayer materials hosting the topological edge states for future low-power electronic nanodevices and quantum computations.

Exact solutions for the buckling and postbuckling of a shear-deformable cantilever subjected to a follower force

The buckling and postbuckling of a shear-deformable cantilever is studied using Reissner’s geometrically exact relations for the planar deformation of beams. The cantilever is subjected to a compressive follower force whose line of action passes through a spatially fixed point. To study the buckling behavior, a consistent linearization of equilibrium and kinematic relations is introduced. The influence of shear deformation and extensibility on the critical loads is studied. The buckling behavior turns out to crucially depend on the ratio between the shear stiffness and the extensional stiffness of the structure. Closed-form solutions in terms of elliptic integrals for buckled configurations of the cantilever are derived in the present paper.

Nonlinear modeling and experimental verification of Gannet-inspired beam systems during diving

A nonlinear model is proposed to answer at which diving speeds and beak angles will cause injury to Gannet-inspired beam systems during plunge-diving. In doing so, the critical velocities at which buckling occurs with various types of boundary conditions are first obtained for vertical dives and the resulting forces at the point of impact are determined. The Gannet-inspired system is modeled as an Euler–Bernoulli beam to represent the neck and body of the Gannet, while the head of the Gannet is modeled as a cone with varying half-angles. The experimental investigations of Gannet-like diving systems are first introduced to present the varying parameters and assumptions of the simplified model. Next, the resulting forces during impact are investigated and a study is conducted to compare various approximations of the drag coefficient for the cone-shaped head. Considering the mid-plane stretching nonlinearity, the equations of motion for the structural system under various types of boundary conditions are derived using the Hamilton’s principle. The characteristic equations, buckled configurations, and critical velocities are determined for each set of boundary conditions. The results show that the system with the smallest half-beak angle and thus the lowest drag force and beam length delays the critical velocity and is most representative of a Gannet during diving. The obtained results demonstrate great agreement with the conducted experiments. For clamped–clamped boundary conditions, the critical velocity is found to be the greatest because of the increased stability at both ends of the beam. It is also noted that a nonlinear approximation for the coefficient of drag offers the best fit with the provided experimental values when compared to a hyperbolic tangent approximation, which predicts the coefficient of drag to be less than that obtained in experiments, and thus predicts that the systems will buckle at higher velocities.

A Low-Threshold Bistable Device for Energy Scavenging From Wideband Mechanical Vibrations

In this paper, the results of the investigations on a system for energy harvesting from wideband vibrations are presented. The harvester adopts a nonlinear snap-through buckling configuration and two piezoelectric transducers, placed near the inflection points of the potential energy function that underpins the dynamics of the system. The device is capable of scavenging energy from vibration sources in the range of 0.5–7 Hz, but could be exploited up to 10 Hz with an acceptable loss of efficiency. The bandwidth of the device is compatible with applications where the vibrations occur at low frequencies, e.g., in the case of a running human. Powers of about $416~\mu \text{W}$ with root-mean-square accelerations of 13.35 m/s2 at 7 Hz and an efficiency of about 24.6% have been measured in case of a sinusoidal vibration while powers of about $250~\mu \text{W}$ and an efficiency of about 15.7%, have been measured in case of a noise input limited at 15 Hz, with a standard deviation of the accelerations of about 15.1 m/s2. The performance of the harvester is compared with a different configuration where the piezoelectric transducers are placed at the two minima of the (bistable) potential energy function that underpins the dynamics; the minima correspond to the stable states of the beam.

On the failure configuration of suspended tubular strings when helical buckling just occurs

When the sinusoidal buckling of suspended tubular strings is transforming to helical buckling, the deformation changes drastically, which makes the tubular strings extremely unstable. The quantitative description of this buckling configuration in previous theoretical studies is not precise enough. In our current study, dynamic relaxation method is applied to carry out numerical simulation of buckling transition of suspended tubular strings. To obtain stable buckling configurations, larger Rayleigh damping is used to attenuate the dynamic effect. With the increase of axial compression load, the lateral buckling of the suspended strings suddenly becomes spiral shape with two contact points. As the string length increases, the position of the lower contact point remains almost at the same position. While the upper contact point gradually moves upward, and the helix angle between the two points increases and gradually tends to be constant. The maximum contact force is located at the lower contact point, the maximum shear force is located at the bottom of the strings, and the dimensionless distance between the location of the maximum bending moment and the well bottom is about 1.0. These loads, including the maximum contact force, maximum shear force and maximum bending moment, gradually decrease with the increasing string length and tend to be constant. The dimensionless critical load is reduced from 4.41 to 3.81 with the effect of string length. The critical load calculated in this paper is the minimum value when helical buckling just occurs. The results obtained in the current study have wide engineering applications, particularly in failure analysis and prevention of helical buckling.

Buckling configurations of stiff thin films tuned by micro-patterns on soft substrate

Wrinkling and buckling of stiff film/soft substrate system has great potential applications in many areas, especially in stretchable electronics. Inspired by the micro-patterns to tune the interfacial adhesion strength, a novel way to tune three buckling configurations (i.e., wrinkling, upward and downward buckling) by using the micro-patterns on the soft substrate was found experimentally. The established governing equations to distinguish the three buckling configurations are consistent with the experimental observations. Results show that the geometries of the micro-patterns, the material properties and the interfacial adhesion between the stiff film and the micro-patterned soft substrate can tremendously influence the final buckling configurations. The proposed analytical governing equations can not only clarify the mechanism of the micro-patterns on the tuning of different buckling configurations, but also be used to design the buckling configurations as desired.

Modeling Floppy Iris Syndrome and the Impact of Phenylephrine on Iris Buckling

Abnormal iris movement (floppy iris syndrome) during intraocular surgery is associated with an increased risk of intraoperative complications. We have previously investigated this scenario with respect to intracameral air in corneal endothelial transplantation, and described the concept of iris buckling. As a number of clinical interventions are recommended for addressing floppy iris syndrome, we wished to evaluate the impact of intracameral phenylephrine on iris buckling and so refine our mathematical model. We considered the stability of an iris structure under a uniform pressure loading. We performed mathematical and computational simulations to demonstrate iris buckling, and then altered the parameters to assess the impact of phenylephrine on the model. We elucidated a number of buckled iris configurations which become unstable as the intraocular pressure increased, for transversely isotropic iris material properties, and identified a positive correlation between the critical pressure and the iris stiffness. A mechanical analysis with a dilated pupil (mimicking phenylephrine use) was also conducted, and demonstrated a significant increase in the critical pressure required to induce iris buckling. We have shown that iris buckling can arise at lower pressures when the iris stiffness is reduced, as in floppy iris syndrome. The use of phenylephrine was shown to prevent iris movement (buckling) by increasing the required critical pressures. This refined model demonstrates the positive effectiveness of phenylephrine in the management of floppy iris syndrome and gives evidence to the clinical practice of using this as a preventative measure.

Imperfection Sensitivity of Post-Buckling Behavior and Vibration Response in Pre- and Post-Buckled Regions of Nanoscale Plates Considering Surface Effects

This paper aims to investigate the imperfection sensitivity of the post-buckling behavior and the free vibration response under pre- and post-buckling of nanoplates with various edge supports in the thermal environment. Formulation is based on the higher-order shear deformation plate theory, von Kármán kinematic hypothesis including an initial geometrical imperfection and Gurtin–Murdoch surface stress elasticity theory. The discretized nonlinear coupled in-plane and out-of-plane equations of motion are simultaneously obtained using the variational differential quadrature (VDQ) method and Hamilton’s principle. To this end, the displacement vector and nonlinear strain–displacement relations corresponding to the bulk and surface layers are matricized. Also, the variations of potential strain energies, kinetic energies and external work are obtained in matrix form. Then, the VDQ method is employed to discretize the obtained energy functional on space domain. By Hamilton’s principle, the discretized quadratic form of nonlinear governing equations is derived. The resulting equations are solved employing the pseudo-arc-length technique for the post-buckling problem. Moreover, considering a time-dependent small disturbance around the buckled configuration, the vibrational characteristics of pre- and post-buckled nanoplates are determined. The influences of initial imperfection, thickness, surface residual stress and temperature rise are examined in the numerical results.