Post-buckling Configurations Research Articles

Thermal Loading on the Natural Frequencies of Postbuckled Clamped Circular Panels

Given a constrained slender structure and subject to thermal loading, that is, relative to ambient conditions and any support structure, a reduction in stiffness not only leads to buckling, but also has a strong effect on the natural frequencies of the system. One of the simplest structural forms is a thin circular panel, clamped around its perimeter, and the current paper focuses attention on such a system. A key aspect of this work is the extraction of sets of natural frequencies throughout the buckling process, with a primary focus on experimentally measured frequencies. Given the cyclically symmetric nature of the system, there are also some interesting observations concerning the changing vibration mode shapes as a function of thermal loading, and more clearly in the postbuckling regime. Another novel aspect of this research is an additional focus on the characterization of snap-through between postbuckled equilibrium configurations and its influence on the natural frequencies. That is, thermally buckled panels often possess multiple equilibrium configurations (bistability), with some characteristic transitions (both naturally occurring and forced) between them. Three panels are tested, consisting of two different materials (aluminum and brass) and two different thicknesses.

Investigating the bending and buckling behaviors of composite porous beams reinforced with carbon nanotubes and graphene platelets using a TRPIM path following mesh-free approach

The aim of the present work consists in investigating the nonlinear behavior of porous beams reinforced with graphene platelets (GPL) and supported carbon nanotubes (CNT), termed functionally graded graphene platelets reinforced composite beam (FG-GPLRC) and functionally graded nanotube carbon reinforced composite beam (FG-CNTRC), respectively. Notably, the distribution of GPL/CNT is explored in both uniform and non-uniform patterns across the beam's thickness. What sets this research apart is its utilization of a refined beam model as enhanced FSDT incorporating nonlinear shear terms which is a crucial advancement in accurately capturing the post-buckling response in certain boundary conditions, a feature lacking in the existing FSDT literature. Innovatively, the post-buckling load-deflection relationship is derived through the solution of governing equations incorporating cubic nonlinearity. This is achieved by employing Galerkin's method alongside a non-iterative high-order continuation technique based on the asymptotic numerical method coupled with the Tchebychev-radial point interpolation method (TRPIM), using a path-following where the solutions are obtained branch-by-branch by eliminating the need for iterative processes. In essence, this research underscores the pivotal role of porosity and GPL/CNT reinforcement in shaping the post-buckling configuration of both perfect and imperfect nanocomposite beams, thereby advancing our understanding of structural behavior in porous nanocomposite materials. The findings of this study illuminate the significant influence of parameters such as porosity coefficient, porosity distribution, GPL/CNT distribution, and GPL-weight/CNT-volume fraction on the nonlinear buckling behavior of porous beams.

Sound generation mechanisms in a collapsible tube.

Collapsible tubes can be employed to study the sound generation mechanism in the human respiratory system. The goals of this work are (a) to determine the airflow characteristics connected to three different collapse states of a physiological tube and (b) to find a relation between the sound power radiated by the tube and its collapse state. The methodology is based on the implementation of computational fluid dynamics simulation on experimentally validated geometries. The flow is characterized by a radical change of behavior before and after the contact of the lumen. The maximum of the sound power radiated corresponds to the post-buckling configuration. The idea of an acoustic tube law is proposed. The presented results are relevant to the study of self-excited oscillations and wheezing sounds in the lungs.

Low-velocity impact response of the post-buckled FG-MEE plate resting on visco-Pasternak foundation: Magneto-electro-mechanical effects-based interaction analysis

Revealing the coupling of nonlinear behavior and the magneto-electro-mechanical effects in the impact responses of post-bucked magneto-electro-elastic structures contributes to the intelligent development of aircraft structural systems. The low-velocity impact responses of the post-buckled functional gradient magneto-electro-elastic(FG-MEE) plate resting on visco-Pasternak foundation are investigated. In the framework of the von Kármán-type nonlinear model considering post-buckling configurations, the low-velocity impact dynamics model of the post-buckled FG-MEE plate is constructed. The two-step perturbation method is developed to obtain the post-buckling equilibrium path induced by the magneto-electro-mechanical effects. Further, the higher-order form of the two-step perturbation method-Galerkin integral method is proposed for the post-buckled FG-MEE plate resting on visco-Pasternak foundation, acquiring the high-order truncated solutions of displacement, electric potential, and magnetic potential. Ultimately, the snap-through phenomenon of the post-buckled FG-MEE plate under low-velocity impact is captured, and the variation in the degree of coupling between nonlinear behavior and the magneto-electro-mechanical effects is systematically revealed.

Mechanics and topology of twisted hyperelastic filaments under prescribed elongations: Experiment, theory, and simulation

Soft filaments can be stretched, bent, and twisted, exhibiting complex configurations. When a filament undergoes large torsional deformation, it can display instabilities, and its post-buckling behavior and configuration evolution differs significantly from that observed under small deformation. We study the mechanics and topologically complex morphologies of twisted rubber filaments under prescribed elongation by combining experiment and finite strain theory. Based on the Mooney-Rivlin model, a finite strain theory of the hyperelastic filament under combined tension, bending, and torsion has been established, accounting for both geometrical and material nonlinearities. An experimental and theoretical morphological phase diagram is constructed as a function of the twist density and the initial elongation. The buckling and post-buckling behaviors of the twisted rubber filaments under prescribed elongation are well captured by the theory that considers geometrical nonlinearity and self-contact. By tracking the interconversion of link, twist, and writhe, we accurately determine the configuration and the critical points of phase transitions. The theoretical predictions agree closely with the measurements. This work sheds light on understanding the morphological complexity of the loaded hyperelastic rod.

Influence of surface effect on post-buckling behavior of piezoelectric nanobeams

Piezoelectric nanobeams with excellent mechanical, thermal and electrical properties are important components in micro-nano electromechanical systems, which are widely used as sensors, brakes and resonators. Based on the Euler–Bernoulli beam model, the influence of surface effect on the post-buckling behaviour of piezoelectric nanobeams is analysed. According to the surface elasticity theory and the ‘core–shell’ model, the surface energy model is used to introduce the influence of surface effect. The governing equations and boundary conditions of the post-buckling of piezoelectric nanobeams under the influence of surface effect are derived by the principle of minimum potential energy. The analytical solution of post-buckling is obtained by the eigenvalue method. The influence of surface effect on the post-buckling configuration, post-buckling path, amount of induced charge and critical load of piezoelectric nanobeams with different external constraints and cross-sectional dimensions are discussed. The results show that surface effect has a significant influence on the post-buckling of piezoelectric nanobeams. Considering surface effect, the effective elastic modulus and critical load of piezoelectric nanobeams are increased, and the post-buckling configuration, post-buckling path and amount of induced charge are reduced. These findings contribute to the study of micro-nano electromechanical systems based on nanobeam structures and provide a theoretical basis for the design and manufacture of nanodevices.

Non-contact actuated snap-through buckling of a pre-buckled bistable hard-magnetic elastica

Snap-through buckling of bistable structures is a classic topic in mechanics which has been widely studied and applied in various fields such as mechanical meta-materials and soft robotics. Obstacles that hinder broader applications of conventional bistable structures include the requirement of contact actuation to trigger instability and difficulty to control post-buckling configurations. In contrast, hard magnetic elastica (HME), a composite made of hard ferromagnetic particles and soft elastomer that deforms in response to an externally applied magnetic field, exhibits great potential to bring major advances in this field by allowing non-contact actuation and programmable control of snap-through buckling via magnetization distribution (M−distribution). Here, we develop a theoretical framework to trace the instability and post-buckling evolution process of snap-through buckling of a bistable HME. In contrast to the conventional snapping through end-end shortening, the design space for bistable HME includes two key parameters: the remanent magnetization density after pre-magnetization and the external magnetic field. We focus on two simple yet practical cases: a fixed amplitude of magnetization density along the HME with direction reversed at the magnetization interface (M−interface), and a uniform magnetic field with varied direction. We identify an optimal position for the single M−interface and direction for the uniform actuation field for pre-buckled beams with two-ends fixed, which can reduce the required actuation field for snapping to nearly half in comparison with the symmetric cases. Experiments and finite element analysis are performed to validate the model predictions. Our work may stimulate further studies on utilizing snap-through buckling in applications where fast and large shape transitions from one stable state to another can be actuated in a low-energy, non-contact mode through a remotely applied stimulus field.

On the influence of edge damage on thermally induced buckling of stepped circular bi-laminates

The influence of edge damage on thermo-mechanical buckling of stepped circular bi-laminates is examined analytically. The nonlinear equations of equilibrium, the associated boundary conditions and the transversality conditions for the propagating interior boundaries of a region of sliding contact are derived via the Principal of Stationary Potential Energy as a propagating boundary problem in the calculus of variations. The problem is ultimately recast in a mixed formulation expressed in terms of the membrane forces and the transverse displacements of the composite structure. A loading parameter is identified and closed form analytical solutions to the resulting non-linear problem are determined. In addition, a stability criterion is established based on the second variation of the total potential energy of the structure, and qualitative behavior concerning contact is deduced analytically. Results of numerical simulations based on the analytical solutions established herein are generated and the phenomenon of “sling-shot buckling” is observed, whereby the structure first deflects in one direction upon heating (or cooling) and then, when a critical temperature is achieved, dynamically “sling-shots” from the current equilibrium configuration to an alternate equilibrium configuration possessing deflections in the opposite sense. In addition, the phenomenon of “buckle-trapping” is observed for stepped circular bi-laminates with hinged-fixed supports, as well as for clamped-fixed structures with relatively large damaged area. A robust mixture of contact behavior, including propagating sliding contact, edge contact and no contact, and the transition from one form of contact to another, is exhibited for both pre-buckling and post-buckling configurations. The influence of edge damage on critical load and characteristic behavior is assessed.

SECONDARY BUCKLING OF A HEATED CIRCULAR PLATE

Upon heating, a circular plate with immovable edge exhibits buckling, which results in ax-isymmetric postcritical bending. Progressive axisymmetric bending due to increasing thermal load in the postbuckling range leads to substantial redistribution of the stresses in the plate and occurrence of high compressive circumferential stresses in a narrow zone adjacent to the plate edge. In this case, secondary buckling can occur resulting in unsymmetric stress-strain state. The aim of the present work is to study stability behavior of the postbuckling axisymmetric equilibrium configurations of a circular plate subjected to uniform temperature rise. The plate edge is assumed to be immovable in the radial and transverse directions and elastically re-strained against bending rotation, which allows one to model the boundary conditions between two extremes of clamped and simply supported. The stability analysis is performed by the semi-analytical finite element method, where unknown displacements are approximated by truncated Fourier series in the circumferential coordinate. The geometrical nonlinearity is taken into ac-count in a quadratic approximation using the Fӧppl – von Karman nonlinear plate theory. To find equilibrium states of the plate, an iterative algorithm is proposed which requires determination of the coefficients of the first and second variations of the total potential energy. Stability of the equilibrium states is examined using the criterion of positive defineteness of the Hess matrix of the plate finite-element model. The critical temperature rise at which secondary buckling occurs is determined. The post-buckling nonlinear deformation characterized by local winkling near the plate edge is studied. The effect of boundary conditions and temperature-dependent material properties on the critical thermal load and secondary buckling modes is discussed.

Buckling Induced Strongly Nonlinear Vibration of Supercritical Axially Moving Beam

In this paper, the postbuckling problem and the free linear and nonlinear vibration surrounding postbuckling configuration are accurately analyzed. The placement of translating beams that undergo significant overall buckling deformations in this high-speed region, leading to strongly nonlinear vibration, is studied. Using the Galerkin method, we found that the partial differential equation is reduced to an ordinary differential equation with cubic and quadratic terms, which differs from the previous literature. The supercritical natural frequency is analytically obtained, the strongly nonlinear vibration problem is solved using the homotopy analysis method (HAM), and the solution is examined using numerical results. Results show that the obtained analytical solution is valid for both weakly and strongly nonlinear cases.

Input parameter and thermal load representation uncertainties effects on the buckling and dynamic characteristics of cylindrical tubes

This research considers different methods in sensitivity analysis and uncertainty quantification applied to cylindrical tubes subject to three types of thermal load representation, namely, uniform, linear, and nonlinear in the radial direction. Sensitivity analysis techniques are used to quantify and rank the parameters that are most influential in altering the critical buckling temperatures, the post-buckled configuration amplitudes, and the natural frequencies in the pre- and post-buckling regimes. The uncertainty quantification and sensitivity analysis strategies show a great potential and usefulness in terms of determining the most influential input parameters for cylindrical tubes subject to thermal loads. Based on the set of nominal parameters in this study, it is shown that the length is the most sensitive parameter in altering the critical thermal buckling load. Additionally, it is demonstrated how uncertainty in the thermal load representation can lead to over or underestimation in both sensitivity analysis and uncertainty quantification findings. The outcome of this analysis can be utilized by other researchers in the design and optimization of cylindrical tubes in thermal environment.

Pattern transformation induced waisted post-buckling of perforated cylindrical shells

A comprehensive investigation on the extremely large post-buckling deformation of perforated cylindrical shells is conducted using experiments and verified with an analytical shell model and nonlinear finite element simulations. A “waisted” post-buckling configuration, which is characterized by uniform shrinking in the middle section of the perforated cylindrical shell, is identified. The waisted behavior is attributed to the triggering of a pattern transformation under compressive load that shows special hyperelastic metamaterial characteristics. The load-carrying capacity of waisted post-buckling suffers a sudden drop and then recovers when the holes are completely collapsed and closed. Plenty of design parameters can be utilized to enrich variations of the waisted post-buckling responses. The negative Poisson's ratio induced by pattern transformation plays a key role in forming the waisted post-buckling modes. This special hyperelastic metamaterial behavior can be easily achieved by fixing the boundaries and adjusting the geometric parameters of the shell. By comparing the characteristics of a porous cylindrical shell with those appeared for an equivalent porous panel, it is highlighted that pattern transformation can occur in a thinner porous cylindrical shell without lateral support. The waisted post-buckling modes of a perforated cylindrical shell are stable, and the shell is invulnerable to the progressively increasing applied loads. In comparison, an ordinary cylindrical shell may snap from one mode to another in the post-buckling process. Moreover, we find that some non-closed cylindrical panels can also buckle into the waisted-like modes. These findings can be applied to the construction of functional devices for soft robotics, actuators, and structural protection for facilities, etc.

Natural frequencies of pre-buckled rods and gridshells

A discrete differential geometry (DDG)-based method is proposed to numerically study the natural frequencies of elastic rods and gridshells in their post-buckling configurations. A fully implicit numerical framework is developed based on Discrete Elastic Rods (DER) algorithm, in order to characterize the mechanical behaviors of an elastic gridshell comprised of multiple rods. When their footprints are constrained along a shrinking trajectory, the slender structures as well as their constructed network would experience a geometrically nonlinear instability and deform into an out-of-plane configuration. By checking the eigenvalues and eigenvectors of the mass and stiffness matrix, the linear vibration near the post-buckling equilibria are characterized through a numerical approach. Exploiting the efficiency and the robustness of the developed discrete method, a systematic parameter sweep is performed to quantify the vibration frequency of pre-buckled gridshells with respect to the number of rods and the pre-compressed distance. It is found that the natural frequency for both pre-deformed rods and gridshells would linearly decrease as the enlargement of compressive distance, even though the geometrically nonlinear deformations have been taken into account. Moreover, the vibration frequency almost linearly rises when the number of rods in a gridshell becomes larger. These findings could provide a fundamental insight in revealing more complex structural dynamics and facilitating the designs of buckling-induced assembly in some man-made systems, e.g., avoidance of the resonance in soft electronics.

Exact Solution of Nonlinear Behaviors of Imperfect Bioinspired Helicoidal Composite Beams Resting on Elastic Foundations

This paper presents exact solutions for the nonlinear bending problem, the buckling loads, and postbuckling configurations of a perfect and an imperfect bioinspired helicoidal composite beam with a linear rotation angle. The beam is embedded on an elastic medium, which is modeled by two elastic foundation parameters. The nonlinear integro-differential governing equation of the system is derived based on the Euler–Bernoulli beam hypothesis, von Kármán nonlinear strain, and initial curvature. The Laplace transform and its inversion are directly applied to solve the nonlinear integro-differential governing equations. The nonlinear bending deflections under point and uniform loads are derived. Closed-form formulas of critical buckling loads, as well as nonlinear postbuckling responses of perfect and imperfect beams are deduced in detail. The proposed model is validated with previous works. In the numerical results section, the effects of the rotation angle, amplitude of initial imperfection, elastic foundation constants, and boundary conditions on the nonlinear bending, critical buckling loads, and postbuckling configurations are discussed. The proposed model can be utilized in the analysis of bio-inspired beam structures that are used in many energy-absorption applications.

Effect of ocean load on axial force of coiled tubing with different helical post-buckling degrees

Coiled tubing (CT) is prone to helical post-buckling with variable pitch when being operated in deep water. The influence of ocean load on the axial force of a variable pitch pipe is complex and lacks relevant research. In this study, a “pipe-in-pipe” system test-bed was established for simulation experiments. Based on the experimental results, a mechanical analysis of the CT's buckling model is carried out. The effects of different ocean loads and helical post-buckling configurations on the axial force are studied. The results show that the greater the amplitude of ocean load and the degree of helical post-buckling, the greater is the axial force fluctuation of the CT. Because the buckling configuration is of variable pitch, the decrease in the axial force is different when ocean loads with the same size and opposite direction are applied. Besides, with the increase in the helical post-buckling, the pitch gradually changes from variable pitch to equal pitch, and the pitch number increases. The research results are helpful to the supplement of CT's buckling system.

Post-buckling analysis of two-layer contact concentric tubular strings in pipe

Buckling of concentric tubular strings in pipe is an essential problem in the fields of medical, biological, and nanomaterials, especially in petroleum engineering. Completely different from the buckling of the classic Euler compression columns, the buckling of concentric tubular strings involves geometric nonlinearity and contact nonlinearity, and has more complicated geometric deformations and buckling configurations. In this paper, a finite element method combining the spatial beam element and the two-layer contact gap element is proposed, and a virtual transient dynamic method is introduced to solve the static buckling of concentric tubular strings. The internal concentric tubular strings are discretized into spatial beam elements, and the contact relationships between tubular strings are described by two-layer contact gap element. Meanwhile, the larger damping is used to attenuate the dynamic response of the system, so as to find the static part of the transient dynamic solution. Appropriate damping and element length are determined by numerical simulation. The structural characteristics and contact states of eight post-buckling configurations are studied. The helix pitch values of specific configurations are in good agreement with the theoretical solutions. In addition, the distribution of bending moment, shear force and contact force in various post-buckling configurations is studied. For the full helical buckling, it is found that there is only one non-contact section in the transition section of inner tube and two non-contact sections in the transition section of middle tube. The research results of this paper provide a theoretical basis and reference for further study on the buckling of multi-layer concentric tubular strings.

A generalized differential quadrature-based computational model for describing free vibrations behavior of functionally graded circular plates around buckled configuration

In the present article, the vibrational behavior of buckled functionally graded (FG) circular plates with clamped and simply-supported edge conditions is described. Considering von Karman’s assumptions, the geometric nonlinearity is incorporated into the Kirchhoff plate theory and the nonlinear governing equations of motion are then derived using Hamilton’s principle. Critical buckling load and linear natural frequencies are first calculated using the generalized differential quadrature (GDQ) method. Afterward, the postbuckling characteristics of the circular plate are obtained via solving the nonlinear governing equations, directly. By several comparative studies, the reliability of the presented model is revealed. Finally, the fundamental natural frequency of the plate is evaluated for prebuckled and postbuckled configurations. The effects of material property and boundary conditions on the static bifurcation diagram and the natural frequency of the initial undeflected and bucked plate are studied. It is found that the trend of the fundamental natural frequency changes with the applied radial load around the prebuckled configuration is unlike the one around the buckled configuration.

Buckling and post-buckling of an elastica under a lateral restraining force

Buckling of structures under lateral load restraint has been found to differ from the conventional bifurcation problems and cannot be handled with conventional analysis. Towards a better understanding of this type of unconventional buckling problems, this work investigates the upheaval buckling of an elastica resting on a smooth surface and restrained by a lateral constant force. Analytical solutions for the Mode-1 and Mode-2 post-buckling responses and configurations of the elastica are established. Stability of the post-buckling configurations under force-control are investigated via the energy method. An energy barrier is found to exist and the buckling load depends on the restraining force and perturbation energy. The imperfection sensitivity of the system is different from the conventional buckling ones, and a characteristic imperfection is found to exist that can transform a snap-through instability into a bifurcation buckling. As a first attempt to address the problem of flange wrinkling under a constant blank-holding force during the cup-drawing process, the fundamental solutions are extended to study the wrinkling of a shrinking elastic annulus under a constant lateral restraining force. The post-wrinkling responses are established and an energy barrier is found to exist for each wrinkling mode. A characteristic imperfection is also found to exist for each mode, which eliminates the energy barrier and enables bifurcation wrinkling. The linear relationship between the critical restraining force that prevents the wrinkling of the elastic annulus and the imperfection amplitude is established, which is also verified by numerical simulations. This study contributes to a better understanding of load-restrained buckling problems and is expected to shed light on the elastoplastic buckling of plates and shells under lateral load restraints in general, and on the wrinkling of a thin sheet during cup-drawing in particular.

A higher-order size-dependent beam model for nonlinear mechanics of fluid-conveying FG nanotubes incorporating surface energy

Fluid-conveying tubes in nano scale present different mechanical behaviors from the macro tubes. A higher-order size-dependent beam model is developed coupling with the nonlocal stress, strain gradient effects, surface energy effects for flow-inducing post-buckling and free vibration analysis of the functionally graded nanotubes. Also, the slip flow effect is considered to modify the kinetic energy of the fluid. The governing equations of post-buckling and vibration are derived by the Hamilton variational principle in which the obtained post-buckling tube configuration is also the initial configuration for vibration analysis. The two-step perturbation method is extended to the nonlinear analysis of the fluid-conveying tubes, and a perturbation scheme is proposed. Subsequently, the influences of nano effects on post buckling, natural frequency and nonlinear vibration of the fluid-conveying nanotubes are studied. The imperfection sensitivity of the static and dynamic behaviors is also discussed. Numerical results reveal the dual influences of nano effects on the nonlinear behaviors and indicate the significance to consider the effects of surface energy, small scale and slip flow in nonlinear analyses of the fluid-conveying nanotubes in a comprehensive way.

Thermoelastic modeling and comparative analysis of biomass sensors under rippling deformation and magnetic field

This study is intended for investigating the linear and nonlinear responses of carbon nanotube-based mass sensors in thermal and magnetic environment. For accurate modeling, the carbon nanotube rippling deformation effect is considered and various representations for the thermal expansion are examined and compared to each other. By utilizing Euler-Bernoulli beam theory assumptions and Eringen's nonlocal elasticity theory, the nonlinear reduced-order model is developed on the basis of the extended Hamilton's principle. The results show that the natural and resonant frequencies and frequency shifts of the system are strongly dependent on the magnetic field and thermal expansion representation. The method of multiple scale is used to determine the modulation equation including the von Karman and rippling nonlinearities. The results show a very good agreement between the perturbation solution and the numerical integration results for specific conditions of the forcing, temperature difference, and quality factor. A comparative study between the linear and nonlinear mass sensing approaches is performed to show their limits of applicability. It is demonstrated that the linear approach may result in erroneous detection of the deposited mass. The obtained results indicate that the longitudinal magnetic field enhances the dynamic stability of the carbon nanotube mechanical resonator in the pre-buckling oscillation regime, while the dynamic stability of the nanoscale resonator is decreased in the presence of the magnetic field for the post-buckling configuration. Also, the ripple-based nonlinearity is accompanied by an increase in the mass responsivity of the resonator. On the contrary, mass sensitivity of the carbon nanotube resonator is diminished by considering the von Kármán geometric nonlinearity. This study shows the importance of considering the nonlinear effects on the system's sensitivity from frequency and amplitude sensing techniques.