Buckling of double functionally-graded nanobeam system under axial load based on nonlocal theory: an analytical approach

Buckling treatment of a bonded compressed double-FG nanobeam system (DFGNBS) is studied in this paper based on Eringen’s nonlocal elasticity theory and Euler–Bernoulli beam model. Differential equations and boundary conditions are obtained using Hamilton’s principle, and the nonlocal theory is employed to derive differential equations in small scale. The material properties are assumed to be functionally graded (FG) along the thickness direction. The synchronous, asynchronous and stationary-type buckling are considered in detail. Results reveal that the small-scale effects are higher with increasing values of nonlocal parameter for the case of in-phase (synchronous) buckling modes in compare to the out-of phase (asynchronous) buckling modes. Increasing the stiffness of the coupling elastic medium double-FG nanobeam system decreases the small-scale effects during the out-of-phase (asynchronous) buckling modes. A detailed parametric study is conducted to investigate the influences of nonlocal parameter, higher modes, spring constant and distributed coefficient of DFGNBS. Some illustrative examples are also stated to verify the present formulation and solutions which showed an excellent agreement.

     

Nonlocal divergence and flutter instability analysis of embedded fluid-conveying carbon nanotube under magnetic field

This paper deals with nonlocal divergence and flutter instability analysis of carbon nanotubes (CNTs) conveying fluid embedded in an elastic foundation under magnetic field. Nonlocal constitutive equations of Eringen and Euler–Bernoulli beam theory are used in the formulations. Also, the foundation is described by the Winkler and Pasternak models. The governing equation of motion and boundary conditions are derived using extended Hamilton’s variational principle. The extended Galerkin’s approach is adopted to reduce the partial differential equation governing the dynamics of the CNTs to a system of coupled ordinary differential equations. In the present study, four different boundary conditions are considered, namely the pined–pined (P–P), clamped–pined (C–P), clamped–clamped (C–C) and clamped–free (C–F). A detailed parametric study is conducted to elucidate the effects of the nonlocal effect, longitudinal magnetic field, elastic Winkler and Pasternak foundations and geometrically boundary conditions on the instability characteristic of CNTs. It was observed that the only instability type for the investigated CNT with clamped–free boundary condition (cantilever) is flutter, while CNT conveying fluid with both ends supported loses its stability by divergence first and then by flutter with increase in fluid velocity. It was also found that the magnetic field and the Winkler and Pasternak foundations increase the stiffness of the system. Therefore, flutter instability region is enlarged significantly due to the existence of springs, shear foundations and magnetic field. Also, results show that the nonlocal parameter has a prominent effect on the stability behavior of CNTs, in which increasing nonlocal parameter results in the decrease in stability region. Furthermore, it was shown that the stability behavior of CNT is strongly affected by different boundary conditions. Finally, the validity of the present analysis is confirmed by comparing the results with those obtained from the literature.     

Chaotic dynamics and forced harmonic vibration analysis of magneto-electro-viscoelastic multiscale composite nanobeam

In this article, the damping forced harmonic vibration characteristics of magneto-electro-viscoelastic (MEV) nanobeam embedded in viscoelastic foundation is evaluated based on nonlocal strain gradient elasticity theory. The viscoelastic foundation consists of Winkler–Pasternak layer. The governing equations of nonlocal strain gradient viscoelastic nanobeam in the framework of refined shear deformable beam theory are obtained using Hamilton’s principle and solved implementing an analytical solution. In addition, a parametric study is presented to examine the effect of the nonlocal strain gradient parameter, magneto-electro-mechanical loadings, and aspect ratio on the vibration characteristics of nanobeam. From the numerical evaluation, it is revealed that the effect of electric and magnetic loading on the natural frequency has a predominant influence.

     

Size-dependent effects on critical flow velocity of a SWCNT conveying viscous fluid based on nonlocal strain gradient cylindrical shell model

This article investigates vibration and instability analysis of a single-walled carbon nanotube (SWCNT) conveying viscous fluid flow. For this purpose, the first-order shear deformation shell model is developed in the framework of nonlocal strain gradient theory (NSGT) for the first time. The proposed model is a conveying viscous fluid in which the external force of fluid flow is applied by the modified Navier–Stokes relation and considering slip boundary condition and Knudsen number. The NSGT can be reduced to the nonlocal elasticity theory, strain gradient theory or the classical elasticity theory by inserting their specific nonlocal parameters and material length scale parameters into the governing equations. Comparison of above-mentioned theories suggests that the NSGT predicts the greatest critical fluid flow velocity and stability region. The governing equations of motion and corresponding boundary conditions are discretized using the generalized differential quadrature method. Furthermore, the effects of the material length scale, nonlocal parameter, Winkler elastic foundation and Pasternak elastic foundation on vibration behavior and instability of a SWCNT conveying viscous fluid flow with simply supported and clamped–clamped boundary conditions are investigated.     

Nonlinear vibration of nanobeam with attached mass at the free end via nonlocal elasticity theory

In the present study, nonlinear free and forced vibration of Euler–Bernoulli nanobeam with attached nanoparticle at the free end is investigated based on nonlocal elasticity theory. The effects of the different nonlocal parameters (γ) and mass ratios (α) as well as effects of fixed-free boundary conditions on the vibrations are determined. To obtain the equation of motion of the system, the Hamilton’s principle is employed. The stretching of neutral axis which introduces cubic nonlinearity is included into the equation for deriving nonlinear equation. And also effects of damping and forcing are included into the equations. The approximate solutions of the equations are derived by using the multiple scale method. Fundamental frequencies, frequency shift and mode shapes for the linear problem are estimated for a nonlocal Euler–Bernoulli nanobeam with an attached nanoparticle and graphically represented the frequency shift and mode shapes. Nonlinear frequencies are derived depending on amplitude and phase modulation. Frequency–response curves are drawn for different nonlocal parameters and different modes.

     

Vibration analysis of rotating rods based on the nonlocal elasticity theory and coupled displacement field

Free longitudinal vibration analysis of a rotating rod based on the Eringen’s nonlocal elasticity is studied in this paper. Rod is supposed to rotate around a fixed axis with a constant angular velocity. To capture the effect of the rotational motion into analysis of the continuous system, a linear proportional relation is introduced between axial and angular velocities. For the first time the mentioned relation is presented based on the internal motions of the infinitesimal element. This novelty makes the rotational displacement as a dependent function of axial displacement playing a significant role through the analysis. Variational approach is adopted to derive the equations of motion for clamped–clamped and clamped-free boundary conditions. For verification of the results obtained from the Galerkin approach, comparison with technical literature is reported. Finally current results illustrate the dependency of the dynamic-vibration analysis of the presented system on the nonlocality and the rotational velocity parameter. This dependency shows the decrement of the frequency with increment in both the angular velocity and the nonlocal parameter. As a result, the mentioned parameters are key factors in the design and analysis of such systems.

     

Nonlinear analysis of size-dependent frequencies in porous FG curved nanotubes based on nonlocal strain gradient theory

A nonlocal strain gradient model is developed in this research to analyse the nonlinear frequencies of functionally graded porous curved nanotubes. It is assumed that the curved nanotube is in contact with a two-parameter nonlinear elastic foundation and is also subjected to the uniform temperature rise. The non-classical theory presented for curved nanotubes contains a nonlocal parameter and a material length scale parameter which can capture the size effect. A power law distribution function is used to describe the graded properties through the thickness direction of curved nanotubes. The even dispersion pattern is used to model the porosities distribution. The high-order shear deformation theory and the von Kármán type of geometric non-linearity are utilized to obtain the nonlinear governing equations of the structure. The size-dependent equations of motion for the large amplitude vibrations of curved nanotubes are obtained by employing Hamilton’s principle. The analytical solutions are extracted for the curved nanotube with immovable hinged-hinged boundary conditions. Size-dependent frequencies of the curved nanotube exposed to thermal field are obtained using the two-step perturbation technique and Galerkin procedure. The effects of important parameters such as nonlocal and length scale parameters, temperature field, elastic foundation, porosity, power law index and geometrical parameters are studied in detail.

     

Nonlinear thermo-mechanical response of temperature-dependent FG sandwich nanobeams with geometric imperfection

In this paper, the nonlinear dynamic response of functionally graded (FG) sandwich nanobeam associated with temperature-dependent material properties by considering the initial geometric imperfection is investigated. The size-dependent behavior of the FG sandwich nanobeam is simulated based on the nonlocal strain gradient theory, and Von Karman nonlinear hypothesis is used to model the geometrical nonlinearity. Moreover, the geometric imperfection is considered as a slight curvature satisfying the first mode shape, and four different FG sandwich patterns including two asymmetric configurations and two symmetric configurations are taken into account. The governing equation of the FG sandwich nanobeam subjected to thermal and harmonic external excitation loadings is derived on the basis of Hamilton’s principle. The numerical results are obtained by employing the multiple-scale method, which are also validated by comparison with two previous studies. Furthermore, comprehensive investigations into the influences of size-dependent parameters, external temperature variation, geometric imperfection amplitude, gradient index and sandwich configuration on the nonlinear characteristics of imperfect FG sandwich nanobeams are conducted through numerical results.

     

Axial magnetic field effect on wave propagation in bi-layer FG graphene platelet-reinforced nanobeams

This paper aims to investigate the size-dependent wave propagation in functionally graded (FG) graphene platelet (GPL)-reinforced composite bi-layer nanobeams embedded in Pasternak elastic foundation and exposed to in-plane compressive mechanical load and in-plane magnetic field. The small-scale effects are taken into account by employing the nonlocal strain gradient theory that contains two different length scale parameters. The present two nanobeams are made of multi-composite layers. Each layer is composed of a polymer matrix reinforced by uniformly distributed and randomly oriented GPLs. The GPLs weight fraction is graded from layer to other according to a new piece-wise rule and then four distribution types will be established. Our technique depends on applying the four-variable shear and normal deformations theory to model the wave propagation problem. The equations of motion are obtained using Hamilton principle. These equations are then analytically solved to obtain the wave frequencies and phase velocities of the waves. The calculated results are compared with those published in the literature. The impacts of the length scale parameters, foundation stiffness, in-plane magnetic field, weight fraction of graphene, graphene platelets distribution type and beam geometry on the propagating waves in the FG GPLs nanobeams are discussed in details. It is found that the strength of the composite beams may be enhanced with increasing in the GPLs weight fraction and magnetic field leading to an increment in the phase velocity and wave frequency of the present system.

     

Stability analysis of restrained nanotubes placed in electromagnetic field

In this present work, buckling analysis of restrained nanotubes placed in electromagnetic field is studied on the basis of Euler–Bernoulli beam theory in conjunction with Eringen’s nonlocal elasticity theory. The modal displacement function is assumed for the stability analysis in order to discretize the derived governing equation. A Fourier sine series with Stoke’s transformation is utilized to investigate the buckling response. The essential advantage of this transformation is its ability of dealing with various boundary conditions to determine the buckling loads. For demonstrate the effects of various parameters such as Hartmann parameter, spring parameter and mode number on the stability response and critical buckling load of electromagnetic nanobeam a detailed study is presented. Variations of buckling loads, critical buckling loads and buckling load ratios of the nanobeam are exhibited with a number of tables and plotted figures. The results obtained from the analysis are discussed on the tables and figures.

     

Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model

This paper aims to investigate the size scale effect on the buckling and post-buckling of single-walled carbon nanotube (SWCNT) rested on nonlinear elastic foundations using energy-equivalent model (EEM). CNTs are modelled as a beam with higher order shear deformation to consider a shear effect and eliminate the shear correction factor, which appeared in Timoshenko and missed in Euler–Bernoulli beam theories. Energy-equivalent model is proposed to bridge the chemical energy between atoms with mechanical strain energy of beam structure. Therefore, Young’s and shear moduli and Poisson’s ratio for zigzag (n, 0), and armchair (n, n) carbon nanotubes (CNTs) are presented as functions of orientation and force constants. Conservation energy principle is exploited to derive governing equations of motion in terms of primary displacement variable. The differential–integral quadrature method (DIQM) is exploited to discretize the problem in spatial domain and transformed the integro-differential equilibrium equations to algebraic equations. The static problem is solved for critical buckling loads and the post-buckling deformation as a function of applied axial load, CNT length, orientations and elastic foundation parameters. Numerical results show that effects of chirality angle, boundary conditions, tube length and elastic foundation constants on buckling and post-buckling behaviors of armchair and zigzag CNTs are significant. This model is helpful especially in mechanical design of NEMS manufactured from CNTs.

     

Meshless local boundary integral equation method for simply supported and clamped plates resting on elastic foundation

Simply supported and clamped thin elastic plates resting on a two-parameter foundation are analyzed in the paper. The governing partial differential equation of fourth order for a plate is decomposed into two coupled partial differential equations of second order. One of them is Poisson’s equation whereas the other one is Helmholtz’s equation. The local boundary integral equation method is used with meshless approximation for both the Poisson and the Helmholtz equation. The moving least square method is employed as the meshless approximation. Independent of the boundary conditions fictitious nodal unknowns used for the approximation of bending moments and deflections are always coupled in the resulting system of algebraic equations. The Winkler foundation model follows from the Pasternak model if the second parameter is equal to zero. Numerical results for a square plate with simply and/or clamped edges are presented to prove the efficiency of the proposed formulation.     

Nonlinear resonance responses of size-dependent functionally graded cylindrical microshells with thermal effect and elastic medium

The nonlinear resonance responses of functionally graded (FG) cylindrical microshells with the elastic medium is investigated by considering thermal and scale effects. First, using the modified couple stress theory, the nonlinear dynamics model for FG microshell are established. Then the reduced nonlinear differential equations are derived by Galerkin’s method and static condensation. Finally, subharmonic, superharmonic and primary resonances of FG cylindrical microshells are analyzed by a perturbation method. In addition, the bifurcation characteristics of the nonlinear dynamic responses are investigated by some numerical examples. The effects of key parameters (modal damping, excitation frequency, foundation medium, scale parameter and thermal effect) on the nonlinear resonance responses are also discussed by numerical simulation.

     

Magneto-electro-elastic analysis of piezoelectric–flexoelectric nanobeams rested on silica aerogel foundation

This article explores that the study on bending of magneto-electric-elastic nanobeams relies on nonlocal elasticity theory. The Vlasov’s model foundation utilizes the silica aerogel foundation. The guiding expressions of nonlocal nanobeams in the considered framework are used extensively and where parabolic third-order beam theory is achieved after using Hamilton’s principle. Parametric work is introduced to scrutinize the influence of the magneto-electro-mechanical loadings, nonlocal parameter, and aspect ratio on the deflection characteristics of nanobeams. It is noticed that the boundary conditions, nonlocal parameter, and beam geometrical parameters have significant effects on dimensionless deflection of nanoscale beams.

     

Size-dependent effects on critical flow velocity of fluid-conveying microtubes via nonlocal strain gradient theory

Size-dependent Timoshenko and Euler–Bernoulli models are derived for fluid-conveying microtubes in the framework of the nonlocal strain gradient theory. The equations of motion and boundary conditions are deduced by employing the Hamilton principle. A flow-profile-modification factor, which is related to the flow velocity profile, is introduced to consider the size-dependent effects of flow. The analytical solutions of predicting the critical flow velocity of the microtubes with simply supported ends are derived. By choosing different values of the nonlocal parameter and the material length scale parameter, the critical flow velocity of the nonlocal strain gradient theory can be reduced to that of the nonlocal elasticity theory, the strain gradient theory, or the classical elasticity theory. It is shown that the critical flow velocity can be increased by increasing the flexural rigidity, decreasing the length of tube, decreasing the mass density of internal flow, or increasing the shear rigidity. The critical flow velocity can generally increase with the increasing material length scale parameter or the decreasing nonlocal parameter. The flow-profile-modification factor can decrease the critical flow velocity. The critical flow velocity predicted by classical elasticity theory is generally larger than that of nonlocal strain gradient theory when considering the size-dependent effect of flow.     

Buckling and post-buckling of magneto-electro-thermo-elastic functionally graded porous nanobeams

The problem of the nonlinear thermal buckling and post-buckling of magneto-electro-thermo-elastic functionally graded porous nanobeams is analyzed based on Eringen’s nonlocal elasticity theory and by using a refined beam model. The beams with immovable clamped ends are exposed to the external electric voltages, magnetic potentials, a uniform transverse load and uniform temperature change. For the first time, the four types of porosity distribution in the nanobeam are considered and compared in complex electric–magnetic fields. Besides, the new formula of the effective material properties is proposed in this paper to simultaneously estimate the material distribution and porosity distribution in the thickness direction. The generalized variation principle is used to induce the governing equations, then the approximate analytical solution of the METE-FG nanobeams based on physical neutral surface is obtained by using a two-step perturbation technique. Finally, detailed parametric analyses are performed to get an insight into the effects of different physical parameters, including the slenderness ratio, small-scale parameter, volume fraction index, external electric voltages, magnetic potentials, porosity coefficient and different porosity distributions, for providing an effective way to improve post-buckling strength of porous beams.

     

Dynamic response of double-FG porous beam system subjected to moving load

In this paper, the vibration response of the double-FG porous beam system (DFGPBS) acted by a moving load is investigated. The DFGPBS composed of two parallel FG porous beams with their material properties varying along both the axial and transverse directions, i.e., bi-directional FG material distribution, is taken into account. The porous imperfection is simulated by distributing the porosity along the beam thickness with even and uneven patterns. The governing equations of this bi-directional DFGPBS under a moving load are established with the aid of the Hamilton principle associated with the Timoshenko beam theory. The Ritz method is adopted to discrete the differential governing equations, which are solved by the Newmark-β approach. The validation of the present model is performed by comparing the numerical results with two previous works. Then, the parametric study is carried out to investigate the influences of bi-directional gradient indices, porosity volume fraction, boundary conditions, stiffness of elastic layer, and velocity of the moving load on the vibration response of bi-directional DFGPBSs excited by a moving load. It is demonstrated that the vibration response of the double-beam system subjected to moving loads can be governed by tailoring the distribution of the bi-directional FG materials. The present work can be used to guide the multi-functional design of a double-beam system under dynamic loadings.

     

Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method

This study investigates the small scale effect on the nonlinear bending vibration of a rotating cantilever and propped cantilever nanobeam. The nanobeam is modeled as an Euler–Bernoulli beam theory with von Kármán geometric nonlinearity. The axial forces are also included in the model as the true spatial variation due to the rotation. Hamilton’s principle is used to derive the governing equation and boundary conditions for the Euler–Bernoulli beam based on Eringen’s nonlocal elasticity theory. The differential quadrature method as an efficient and accurate numerical tool in conjunction with a direct iterative method is adopted to obtain the nonlinear vibration frequencies of nanobeam. The effect of nonlocal small–scale, angular speed, hub radius and nonlinear amplitude of rotary nanobeam is discussed.     

Nonlinear effects of thermo-sensitive nanobeams via a nonlocal thermoelasticity model with relaxation time

This paper presents a theoretical nonlocal model for a thermo-sensitive nanobeam based on the generalized thermoelasticity theory with thermal relaxation time. The present nanobeam is subjected to a sinusoidal pulse varying heat and its thermal conductivity is considered to be variable. This article deals with a nonlinear coupling partial differential equation since the thermal conductivity depends on temperature. The nonlocal theories of coupled thermoelasticity can be extracted as limited and special case of the present model. The effect of the variability thermal conductivity parameter, the nonlocal parameter, the relaxation time and the pulse width of the sinusoidal pulse on the distribution of lateral vibration, the temperature and the displacement of the nanobeam is investigated.     

粘弹性地基上损伤弹性Timoshenko梁动力学行为

建立了粘弹性地基上损伤弹性Timoshenko梁在有限变形情况下的运动微分方程,这是一组非线性偏微分方程.为了便于分析,首先利用Galerkin方法对该方程组进行简化,得到一组非线性常微分方程.然后利用Matlab软件进行数值模拟,考察了载荷参数、地基粘性参数和弹性参数、损伤对梁振动的影响.采用非线性动力学中的各种数值方法,如时程曲线、相平面图、Poincare截面和分叉图,发现增大地基的粘弹性参数,有利于增强结构运动的稳定性,而损伤会降低结构运动的稳定性.