Post-buckling Configurations Research Articles

Observations on the general nonlocal theory applied to axially loaded nanobeams

Static and dynamic analyses for the pre- and post-buckling behavior of an axially loaded nanobeam modeled with a general form of Eringen’s nonlocal theory are conducted and discussed. The general nonlocal theory is a modified form of Eringen’s nonlocal elasticity theory that, contrary to Eringen’s, considers separate attenuation functions to account for the long-range interactions for the two different material moduli and thus introduces an additional nonlocal parameter. The nanobeam is modeled considering Euler–Bernoulli beam theory and the von Karman geometric nonlinearity for midplane stretching in end-constrained beams. Three sets of boundary conditions are studied, namely clamped–clamped, clamped-hinged, and hinged–hinged. The governing equations are derived by virtue of the Hamilton’s principle. Introducing the general nonlocal theory into a beam system increases the order of the resultant transverse equation of motion from fourth to sixth, thus requiring two additional higher-order boundary conditions. These higher-order boundary conditions are derived using a weighted residual approach and are thus variationally consistent. Through this effort, a focus is placed on the effects that these higher-order boundary conditions have on the static and dynamic responses of the system. Specifically, the nonlocal parameters, Poisson ratio, boundary conditions, and axial load are varied. The static critical buckling loads, bifurcation diagrams, and static post-buckling configurations are examined and presented. Additionally, through a linear eigenvalue analysis, the natural frequencies and mode shapes in the pre- and post-buckling regimes are examined. It is shown that the system is very sensitive to the selected nonlocal parameters and higher-order boundary conditions for both the static and dynamic responses. It is the hope that this expanded model can be used by other researchers to accurately model nanobeams composed of materials outside the limits of applicability of Eringen’s nonlocal theory.

Nonlinear buckling and free vibration of curved CNTs by doublet mechanics

In this manuscript, static and dynamic behaviors of geometrically imperfect carbon nanotubes (CNTs) subject to different types of end conditions are investigated. The Doublet Mechanics (DM) theory, which is length scale dependent theory, is used in the analysis. The Euler-Bernoulli kinematic and nonlinear mid-plane stretching effect are considered through analysis. The governing equation of imperfect CNTs is a sixth order nonlinear integro-partial-differential equation. The buckling problem is discretized via the differential-integral-quadrature method (DIQM) and then it is solved using Newton's method. The equation of linear vibration problem is discretized using DIQM and then solved as a linear eigenvalue problem to get natural frequencies and corresponding mode shapes. The DIQM results are compared with analytical ones available in the literature and excellent agreement is obtained. The numerical results are depicted to illustrate the influence of length scale parameter, imperfection amplitude and shear foundation constant on critical buckling load, post-buckling configuration and linear vibration behavior. The current model is effective in designing of NEMS, nano-sensor and nano-actuator manufactured by CNTs.

Semi-analytical solution for post-buckling analysis of simply-supported multi-stepped columns

In this paper, semi-analytical solution is developed to predict post-buckling configurations of simply-supported columns consisting of an arbitrary number of segments. Geometric and equilibrium relations of the deformed column are utilized to derive the nonlinear governing equations and the closed-form solution is obtained. Unknown parameters in the proposed solution are determined by using a numerical method. The column is assumed to be an inextensible and the Euler-Bernoulli beam hypotheses are used. The proposed solution is validated by comparing the results obtained with available data in the literature. Furthermore, by using the proposed solution, the post-buckling behaviour of various columns with geometry or material discontinuities is investigated.

Postbuckling Analysis of a Nonlocal Nanorod Under Self-Weight

This paper presents the postbuckled configurations of simply supported and clamped-pinned nanorods under self-weight based on elastica theory. Numerical solution is considered in this work since closed-form solution of postbuckling analysis under self-weight cannot be obtained. The set of nonlinear differential equations of a nanorod including the effect of nonlocal elasticity are investigated. The constraint equation at boundary condition technique is introduced for the solution of postbuckling analysis. In order to solve the set of nonlinear differential equations, the shooting method is utilized, where the set of these equations along with boundary conditions are integrated by the fourth-order Runge-Kutta algorithm. Numerical results are obtained and the highlighting influences of the nonlocal elasticity on postbuckling behavior of nanorods are discussed. The obtained results indicate that the rotation angle and the postbuckled configurations of nanorods are varied by changing the nonlocal elasticity parameter. The effect of nonlocal elasticity shows the softening behavior in comparison with the Euler beam. The present formulation together with constraint boundary condition technique is an effective solution for postbuckling analysis of a nanorod under self-weight including the effect of nonlocal elasticity.

Nonlinear analysis of axially loaded piezoelectric energy harvesters with flexoelectricity

In this work, nonlinear electromechanical models for energy harvesters based on axially preloaded piezoelectric beams incorporating flexoelectric effect are presented. Depending on the amplitude of the applied axial load, the proposed energy harvester could operate in either prebuckling or postbuckling configuration. According to the theory of flexoelectricity and the Hamilton's principle, the nonlinear electromechanical coupling equations of the proposed energy harvesters under base excitations are derived. For a simply-supported piezoelectric beam, the expression of the static buckling load is analytically determined. Then, for energy harvesters in the prebuckling and postbuckling configurations, we obtain the discrete nonlinear governing equations by employing the Galerkin's method. These coupling equations are solved numerically by the Runge–Kutta method. Case studies are provided to show the steady-state output voltage and power of the energy harvesters. Results indicate that the frequency response curves show a typical hardening nonlinear behavior in the prebuckling configuration and a softening nonlinear behavior in the postbuckling configuration. Such nonlinear behaviors of energy harvesters imply a wider frequency operation bandwidth of the proposed energy harvesters. We also find that the energy harvesters utilizing flexoelectricity have a better performance than those based on piezoelectricity in the prebuckling configuration. Moreover, we examine the influences of resistive load, mechanical damping coefficient and the amplitude of the base excitation on the performance of the energy harvesters as well as the size effect due to flexoelectricity. It is also interesting to observe that both intrawell and interwell oscillations of the energy harvesters could occur in the postbuckling configuration. This work provides an efficient route to design energy harvesting systems at micro- and nano-scales with enhanced performance.

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.

Lateral buckling of pipelines due to internal pressure: A geometrically nonlinear finite element analysis

This work presents an analysis of the lateral buckling of pipelines triggered by internal pressure. Geometrical imperfection amplitudes and friction of pipeline-soil are studied concerning their influence on the critical load and on the post-buckling configuration. To perform the analyses, numerical models are developed using geometrically-exact 3D Timoshenko beam finite elements, which may undergo large displacements and finite rotations. The interaction between the pipeline and the soil is simulated through a contact model. Geometrical imperfections are addressed by imposing prescribed displacements and considering uneven surfaces to represent the soil. The latter, by the way, allow to input concomitant vertical and lateral imperfections to the models. Varying vertical and lateral imperfection amplitudes and soil friction coefficients, the results show that all the parameters together influence the lateral buckling concerning critical loads and shape, order and amplitude of post-buckling configurations of the pipeline. Comparisons with analytical models are also presented.

Effects of thermal loads representations on the dynamics and characteristics of carbon nanotubes-based mass sensors

Uniform, linear, and nonlinear temperature gradients are considered to perform buckling and vibration analyses of carbon nanotube (CNT)-based mass sensors modeled as a clamped-clamped Euler–Bernoulli nanobeam with a deposited atomic-scale particle. Size dependent effects are taken into account using the Eringen’s nonlocal elasticity theory. Employing the Hamilton’s principle, the governing equations of motion considering the three types of temperature distribution are derived. To investigate the effects of various thermal loadings on the mass detection sensitivity of clamped CNT-mass detectors, four distinct noble gas atoms, namely Helium (He), Neon (Ne), Argon (Ar), and Krypton (Kr) with very low chemical reactivity are considered as attached atomic-scale masses. The influence of important parameters, such as the length and diameter of the CNT, the deposited mass and its location, the nonlocal parameter, the surface temperature difference, the temperature rise, and the type of temperature distribution on the errors in thermal buckling loads and frequency shifts are also studied. Assessing the impacts of the inaccuracies of uniform and linear temperature distributions on the critical thermal buckling load and the frequency shift of the CNT-based mechanical resonator is the primary contribution of the work. The numerical results indicate that in the pre-buckling region, the assumptions of uniform and linear temperature distributions through the thickness of the CNT estimate higher values of the natural frequency and the frequency shift compared to the nonlinear temperature gradient. On the contrary, the nonlinear thermal gradient across the radius of the CNT-based mass detector yields the largest values of the frequency shift and hence the highest mass detection sensitivities of the CNT-based mass sensor in the post-buckling configuration.

Pore pressure and porosity effects on bending and thermal postbuckling behavior of FG saturated porous circular plates

Postbuckling and nonlinear bending of a geometrically perfect circular plate arisen by the temperature of one side of a axisymmetric circular plate made from a poroelastic solid whose matrix pores have been saturated by fluid have been numerically analyzed. The plate porosity varies continuously through the plate thickness according to some given specific functions. The postbuckling and nonlinear bending configurations of respectively clamped and simply-supported plates, as well as the critical postbuckling temperature, have been obtained under the influence of the plate one side surface temperature, thickness, type of pore distribution, porosity and compressibility of fluid confined by the pores. Equilibrium equations of the plate have been derived on the basis of the classical plate theory, Love–Kirchhoff hypothesis and the Sander’s nonlinear strain–displacement relationship. They have been discretized via differential quadrature method. The numerical results appropriately coincide with the relevant literature, namely the results of saturated porous and mathematically equivalent plates made from functionally grated materials.

Size-dependent secondary resonance of a piezoelectrically laminated bistable MEMS arch resonator

This paper deals with the size-dependent nonlinear dynamics of a bistable fully clamped resonant shallow microarch under subharmonic resonance of order 1/2. The microarch is laminated between thin piezoelectric layers and subjected to a combination of piezoelectric and electrostatic actuations. Nonlinearities associated with clamped ends, curvature and electrostatic actuation are taken into account in the formulation of the problem. Size effect is considered using the so-called strain gradient theory, and the result is compared with the classical theory. The frequency response is numerically obtained using the shooting technique and in some cases analytically using multiple time scales method. The effect of piezoelectric actuation and the initial rise on the resonant behavior of the system are investigated in the pre and post-buckling configurations as well as the bistability band of the microarch. Applying a DC voltage to the piezoelectric layers imposes a longitudinal compression or tension force along the microbeam depending on the polarity of the voltage. As a result, the initial rise and bending stiffness of the microarch changes. This property is used as a frequency tuning tool in this paper. It is shown that the resonance frequency of the microarch resonator can be tuned in both forward and backward directions thanks to appropriate piezoelectric actuation either in the pre-buckling or the post-buckling configuration. It is also revealed that taking advantage of piezoelectric actuation, the bandwidth of the static bistable band can be adjusted effectively. The results of this paper can be used for designing MEMS resonators.

A component coupling approach to dynamic analysis of a buckled, bistable vibration energy harvester structure

Over the past ten years, the energy harvesting community has focused on bistable structures as a means of broadening the working frequency range and, by extension, the effective efficiency of vibration-based power scavenging systems. In the current study, a new method is implemented to statically and dynamically analyze a bistable buckled, multi-component coupled structure designed specifically for low-frequency (< 30 Hz) vibration energy harvesting. First, the system is divided into its individual components and the governing equations for each part are developed based on Euler–Bernoulli beam theory. These governing equations are then solved in one single system of equations by applying geometrical and force-moment boundary conditions at the connections. Solving the nonlinear static equations gives the critical buckling loads, as well as the exact static, post-buckled configuration of the system about which the dynamic response is formulated. The natural frequencies and mode shapes of the system are obtained by solving the free vibration case of the linearized buckled structure, which are then used as spatial functions in a Galerkin approach to discretize the nonlinear partial differential equations of the coupled system. To validate the modeling approach, the obtained results are compared with the ones captured from both finite element analysis model and the experimental setup, which shows good agreement between them. Furthermore, the amplitude-frequency response of the system and snap-through regime with the variation of various parameters, including exciting frequency, base vibration and buckling loads are investigated based on the developed model. It is shown that for a weakly buckled configuration, bistable motion can be captured for a wide range of frequencies, which is crucial for the performance of energy harvesting devices.

Post-buckling and vibration of post-buckled rotating pre-twisted FG microbeams in thermal environment

In this paper, comprehensive studies on the size-dependent thermal post-buckling and coupled bi-directional transverse-longitudinal free vibration behavior of post-buckled rotating pre-twisted functionally graded (FG) microbeams in thermal environment are presented. The material properties are assumed to be temperature-dependent and graded in the thickness direction with mid-plane symmetric distribution, i.e., symmetric functionally graded (S-FG). At first, the discretized nonlinear post-buckling governing equations are derived based on the modified strain gradient theory (MSGT) in conjunction with the first-order shear deformation theory (FSDT) of beams under the von Kármán geometric nonlinearity assumptions using Hamilton's principle and Chebyshev-Ritz method. The resulting nonlinear equations under different boundary conditions are solved iteratively by employing Newton-Raphson's method. Then, using the same theories and solution procedure, the free vibration equations around the post-buckled configuration are derived and solved. After validating the approach, the effects of angular velocity, twist angle, length scale parameters, thickness-to-length ratio, hub radius, material gradient index and boundary conditions on the post-buckling equilibrium path and linear free vibration behavior of rotating pre-twisted S-FG microbeams are investigated. The results are prepared for both linear and nonlinear variations of the twist angle along the microbeam axis.

Stability and nonplanar buckling analysis of a current-carrying mircowire in three-dimensional magnetic field

This paper investigates the stability and nonplanar buckling of a current-carrying microwire immersed in a three-dimensional (3D) magnetic field based on modified coupled stress theory and Euler–Bernoulli beam assumptions. Utilizing Hamilton’s principle, the partial differential governing equations for 3D vibrations of the current-carrying microwire are derived. These governing equations are then discretized via the Galerkin’s approach and solved by a fourth-order Runge–Kutta method. Numerical results show that the slenderness ratio and magnetic field have obvious effect on the stability of the microwire. When the applied magnetic field is along the axial direction of the undeformed microwire, the shape of the postbuckling configuration of the microwire is very sensitive to initial conditions employed. If, however, the magnetic field is deviated from the axial direction of the undeformed microwire, it is interesting that the shape of the postbuckling configuration is independent of initial conditions even if the y- and/or z-component of the 3D magnetic field strength is sufficiently small. Another attractive feature of the results is that the lateral deflection of the microwire can be adjusted by changing the magnetic field strength in each direction, which may have potential application to the microelectromechanical systems (MEMS). Finally, the influence of a 3D magnetic field on the axial displacement of the microwire is analyzed, and it is demonstrated that in practice the axial motion of the microwire may be considered negligible.

A robust Bézier based solution for nonlinear vibration and post-buckling of random checkerboard graphene nano-platelets reinforced composite beams

In the present study, an accurate Bézier based multi-step method is developed and implemented to find the nonlinear vibration and post-buckling configurations of Euler-Bernoulli composite beams reinforced with graphene nano-platelets (GnP). The GnP is assumed to be randomly and uniformly dispersed in the composite mix-proportion, with a random checkerboard configuration. Therefore, a probabilistic model together with an efficient simulation technique is proposed to find the effective moduli of a matrix reinforced GnP. It is worth noting that the presented micro-mechanics model found by the employed Monte-Carlo simulation matches exactly the experimental data and predicts the composite elastic constants more accurate than that found from other common methods, including the Halpin-Tsai theory. Also, for mathematical simplification, the composite beam in-plane inertia is neglected. The presented multi-step method is based on Burnstein polynomial basis functions while shows interesting potential to provide robust solutions for various initial and boundary value problems. It is found that adding a relatively low content of GnP would drastically increase the composite elastic constants, particularly in the transverse direction to fiber. In addition, the numerical results are compared with those provided by exact analytical solutions, where the stability of results suggests the effectiveness of the presented methodology.

Static and dynamic analysis of the postbuckling of bi-directional functionally graded material microbeams

In this paper, the static and dynamic responses of bi-directional functionally graded (BDFG) microbeams are investigated. The material properties vary along both thickness and axial directions. Employing Hamilton’s principle, the differential equations are derived based on von-Karman geometric nonlinearity and third-order shear deformation beam theory. The modified couple stress theory is adopted to capture the size effects. The material length scale parameter of microbeam is considered as a function of spatial coordinates and varies with the material gradient parameters. The differential equations and boundary conditions are discretized using differential quadrature method. Subsequently, static bifurcation of microbeams is calculated utilizing pseudo-arc length continuation technique. The free vibration around the postbuckling configuration of microbeams is studied by solving the associated linear eigenvalue problem. The influences of the functional gradient parameters, dimensionless length scale parameter and aspect ratio on the static response and vibration characteristics of buckled BDFG microbeams with various boundary conditions are investigated. It is demonstrated that the buckling of BDFG microbeam occurs through a transcritical bifurcation. The static stable responses of BDFG microbeam are asymmetric due to the stretching-bending coupling that results from the asymmetry of the material distribution in thickness direction of beam. The mode veering phenomenon is detected in postbuckling domain of microbeams. It is examined that the longitudinal dynamic displacement plays a pivotal role in the mode veering phenomenon.

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.

Role and significance of thermal loading on the performance of carbon nanotube-based mass sensors

Insights on the sensitivity of carbon nanotube-based mass sensors subject to a uniform thermal environment are provided. The thermal loads are introduced as a means of buckling nonlocal carbon nanotube-based mass sensors in an effort to obtain high frequency shifts before and after the deposition of a nanoparticle on a carbon nanotube (CNT), leading to improved mass detection. In introducing thermal loads, the critical buckling temperature, pre-buckling, and post-buckling behaviors of CNTs with an arbitrary deposited mass are investigated. The CNT is modeled as an Euler-Bernoulli beam and the nonlinear governing equations of motion are derived by virtue of the Hamilton's principle. Eringen's nonlocal theory is employed to account for the interatomic long-range interactions, i.e., the size dependent phenomena occurring at the nano-scale. The proposed model allows for an arbitrarily positioned atomic-mass particle and accounts for the effects of the thermal field and the nonlocal parameter on the pre-buckling and post-buckling dynamics of clamped-clamped nanobeams. It is demonstrated that the temperature change has a remarkable influence on the frequency shift and hence on the sensitivity of the CNT-based mass sensor. The results show that augmenting the temperature in the pre-buckling regime leads to a decrease in the frequency shift, while the temperature rise in the post-buckling configuration has an increasing effect on the sensitivity of the CNT-based mass sensor. Also, it is found that when the temperature exceeds a certain point in the post-buckling region, higher frequency shifts can be obtained compared to the pre-buckling regime. In varying parameters, such as the length, diameter, number of walls (single-walled CNTs or multi-walled CNTs), location of the deposited mass, and nonlocal parameter, greater frequency shifts can be obtained in the post-buckling configurations of the CNT, leading to enhanced sensitivity in the system.

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.

The filament-bundle elastica

Filament-bundles are ubiquitous in nature. They are composed by an assembly of flexible rods held together by elastic springs, such as found in ciliary systems and flagella. We study the static, post-transient, post-buckled configurations of a generalised filament-bundle elastica or flagella. We recur to linear and weakly non-linear analysis, as well as geometrically exact numerical solutions. The bundle cross-linking mechanics is characterised by non-local moments affecting distant parts of the bundle. This induces a bimodal post-buckling response sensitive to the interfilament sliding at the base. We report the occurrence of a novel reversed cusp catastrophe, reminiscent of the counterbend phenomenon, that folds and suppresses the saddle-node bifurcation back a pitchfork bistability landscape, found in classical elastica systems. The filament-bundle elastica can thus prevent violent jumps, non-uniqueness and hysteresis. This non-trivial folding of the imperfection-sensitivity diagram may impact bundle systems with naturally occurring buckling phenomena.

Three-dimensional vortex-induced vibrations of supported pipes conveying fluid based on wake oscillator models

The present study develops a new three-dimensional nonlinear model for investigating vortex-induced vibrations (VIV) of flexible pipes conveying internal fluid flow. The unsteady hydrodynamic forces associated with the wake dynamics are modeled by two distributed van der Pol wake oscillators. In particular, the nonlinear partial differential equations of motion of the pipe and the wake are derived, taking into account the coupling between the structure and the fluid. The nonlinear equations of motion for the coupled system are then discretized by means of the Galerkin technique, resulting in a high-dimensional reduced-order model of the system. It is shown that the natural frequencies for in-plane and out-of-plane motions of the pipe may be different at high internal flow velocities beyond the threshold of buckling instability. The orientation angle of the postbuckling configuration is time-varying due to the disturbance of hydrodynamic forces, thus yielding sometimes unexpected results. For a buckled pipe with relatively low cross-flow velocity, interestingly, examining the nonlinear dynamics of the pipe indicates that the combined effects of the cross-flow-induced resonance of the in-plane first mode and the internal-flow-induced buckling on the IL and CF oscillation amplitudes may be significant. For higher cross-flow velocities, however, the effect of internal fluid flow on the nonlinear VIV responses of the pipe is not pronounced.