Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies

A modified continuum model of electrically actuated nanobeams is presented by incorporating surface elasticity in this paper. The classical beam theory is adopted to model the bulk, while the bulk stresses along the surfaces of the bulk substrate are required to satisfy the surface balance equations of the continuum surface elasticity. On the basis of this modified beam theory the governing equation of an electrically actuated nanobeam is derived and a powerful technology, analog equation method (AEM) is applied to solve this complex problem. Beams made from two materials: aluminum and silicon are chosen as examples. The numerical results show that the pull-in phenomena in electrically actuated nanobeams are size-dependent. The effects of the surface energies on the static and dynamic responses, pull-in voltage and pull-in time are discussed.     

Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects

In this paper, the linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams are studied based on the Gurtin–Murdoch surface stress theory. Firstly, the constitutive equations of fractional viscoelasticity theory are considered, and based on the Gurtin–Murdoch model, stress components on the surface of the nanobeam are incorporated into the axial stress tensor. Afterward, using Hamilton's principle, equations governing the two-dimensional vibrations of fractional viscoelastic nanobeams are derived. Finally, two solution procedures are utilized to describe the time responses of nanobeams. In the first method, which is fully numerical, the generalized differential quadrature and finite difference methods are used to discretize the linear part of the governing equations in spatial and time domains. In the second method, which is semi-analytical, the Galerkin approach is first used to discretize nonlinear partial differential governing equations in the spatial domain, and the obtained set of fractional-order ordinary differential equations are then solved by the predictor–corrector method. The accuracy of the results for the linear and nonlinear vibrations of fractional viscoelastic nanobeams with different boundary conditions is shown. Also, by comparing obtained results for different values of some parameters such as viscoelasticity coefficient, order of fractional derivative and parameters of surface stress model, their effects on the frequency and damping of vibrations of the fractional viscoelastic nanobeams are investigated.     

Non-linear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects

Analytical solution for the steady-state response of an Euler–Bernoulli nanobeam subjected to moving concentrated load and resting on a viscoelastic foundation with surface effects consideration in a thermal environment is investigated in this article. At first, based on the Eringen's nonlocal theory, the governing equations of motion are derived using the Hamilton's principle. Then, in order to solve the equation, Galerkin method is applied to discretize the governing nonlinear partial differential equation to a nonlinear ordinary differential equation; solution is obtained employing the perturbation technique (multiple scales method). Results indicate that by increasing of various parameters such as foundation damping, linear stiffness, residual surface stress and the temperature change, the jump phenomenon is postponed and with increasing the amplitude of the moving force and the nonlocal parameter, the jump phenomenon occurs earlier and its frequency and the peak value of amplitude of vibration increases. In addition, it is seen that the non-linear stiffness and the critical velocity of the moving load are important factors in studying nanobeams subjected to moving concentrated load. Presence of the non-linear stiffness of Winkler foundation resulting nanobeam tends to instability and so, the jump phenomenon occurs. But, presence of the linear stiffness will lead to stability of the nanobeam. In the next sections of the paper, frequency responses of the nanobeam made of temperature-dependent material properties under multi-frequency excitations are investigated.     

Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory

The aim of this paper is to study the free vibration of nanobeams with multiple cracks. The analysis procedure is based on nonlocal elasticity theory. This theory states that stress at a point is a function of strains at all points in the continuum. The nonlocal elasticity theory becomes significant for small length scale in micro and nanostructures. The effects of nonlocality, crack location and crack parameter are investigated on the natural frequencies of the cracked nanobeam. In this study, analytical solutions are given for cracked Euler–Bernoulli nanobeams of different boundary conditions.     

基于非局部弹性应力场理论的纳米尺度效应研究:纳米梁的平衡条件、控制方程以及静态挠度

该文成功地解答了3个关于非局部应力理论用于纳米梁的问题:(ⅰ) 在绝大多数研究中,非局部效应增加导致纳米结构体刚度下降,其现象表现为弯曲挠度增加,固有频率减少,屈曲载荷下降,但为什么Eringen 的非局部弹性理论给出了完全相反的结论;(ⅱ) 为什么在某些研究结果中,非局部效应消失或是对研究结果无影响,比如纳米悬臂梁在集中载荷作用下的弯曲挠度; (ⅲ) 在高阶控制方程中,为什么高阶边界条件不存在．通过应用非局部弹性理论和精确变分原理分析纳米梁的弯曲问题,推导出全新的平衡条件、控制方程、边界条件和静态响应．这些方程和条件包含了与之前的相关研究结果符号相反的高阶微分项,这一差别导致了纳米效应对结构体的影响结果完全相反． 还证明之前为大家所公认的纳米梁静态或动态平衡条件实际上没有达到平衡,只有用等效弯矩代替非局部弯矩时,才可达到平衡．这些结论通常是可以被其它方法,比如应变梯度理论、耦合应力模型以及相关实验所证明．     

Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen's differential model

This study analyzes the nonlinear free vibration and post-buckling of nanobeams with flexoelectric effect based on Eringen's differential model. The nanobeam is modeled based on Timoshenko beam's theory. The von-Kármán strain–displacement relation together with the electrical Gibbs free energy and Hamilton's principle are employed to derive equations of motion. The nonlinear free vibration frequencies are obtained for pinned–pinned (P–P) and clamped–clamped (C–C) boundary conditions. Multiple scales method is employed to obtain the closed-form solution for the nonlinear governing equations. By employing this methodology, the natural frequencies of nanobeams are obtained and their post-buckling behavior is examined. The influence of nonlocal parameter, amplitude ratio, and input voltage on the top surface and flexoelectricity constant on nonlinear free vibration and post-buckling characteristics of nanobeam is investigated. In this paper, it is concluded that the flexoelectricity has a significant effect on free vibration of the beams in nano-scale and its effect has to be considered in designing nano-electro-mechanical systems (NEMS) such as nano- generators and nano-sensors.     

Exact modes for post-buckling characteristics of nonlocal nanobeams in a longitudinal magnetic field

An exact mode solution that investigates the prebuckling and postbuckling characteristics of nonlocal nanobeams with fixed–fixed, hinged–hinged, and fixed–hinged boundary conditions in a longitudinal magnetic field is determined. The geometric nonlinearity arising from mid-plane stretching is considered to obtain the nonlinear governing equation of motion by virtue of Hamilton's principle. The influences of the nonlocal and magnetic parameters on the prebuckling and postbuckling dynamics of nanobeams with various boundary conditions are evaluated, indicating that the critical buckling force can be decreased with the increase of the nonlocal parameter while can be increased with increasing the magnetic parameter. It is demonstrated that the first natural frequency of the nanobeam with fixed–fixed and fixed–hinged conditions in postbuckling configuration is increased from zero to a constant value for higher values of the nonlocal parameter with increasing the axial force. The second natural frequency of the buckled nanobeam is always decreased with an increase of the nonlocal parameter. The results show that the internal resonance between the first and second modes of the postbuckling nanobeams can be quickly and easily activated by increasing the nonlocal parameters, especially for fixed–fixed and hinged–hinged boundary conditions. In addition, the results obtained by exact mode solution are compared those obtained by classical mode solution. It is found that the classical mode is valid only for nonlocal nanobeams with the hinged–hinged boundary conditions.     

Nonlinear secondary resonance of nanobeams under subharmonic and superharmonic excitations including surface free energy effects

The main objective of this study is to predict both the subharmonic and superharmonic resonances of the nonlinear oscillation of nanobeams in the presence of surface free energy effects. To this purpose, Gurtin–Murdoch elasticity theory is adopted to the classical beam theory in order to consider the surface Lame constants, surface mass density, and residual surface stress within the differential equations of motion. The Galerkin method together with the method of multiple scales is utilized to investigate the size-dependent response of nanobeams under hard excitations corresponding to various boundary conditions. A parametric analysis is carried out to indicate the influence of the surface elastic parameters on the frequency-response as well as amplitude-response of the nonlinear secondary resonance including multiple vibration modes and interactions between them. It is seen that for the superharmonic excitation, except for the clamped–free boundary condition, the jump phenomenon is along the hardening direction, while in the clamped–free end supports, it is along the softening direction. Moreover, it is revealed that for the subharmonic excitation, within a specific range of the excitation amplitude, the nanobeam is excited, and this range shifts to lower external force by incorporating the surface free energy effects. It is found that in the case of superharmonic excitation, the value of the excitation frequency associated with the bifurcation point at the peak of the frequency-response curve increases by taking the surface free energy effect into consideration.     

A nonlocal higher-order model including thickness stretching effect for bending and buckling of curved nanobeams

An analytical approach for static bending and buckling analyses of curved nanobeams using the differential constitutive law, consequent to Eringen’s strain-driven integral model coupled with a higher-order shear deformation accounting for through thickness stretching is presented. The formulation is general in the sense that it can be deduced to examine the influence of different structural theories, for static and dynamic analyses of curved nanobeams. The governing equations derived using Hamiltons principle are solved in conjunction with Naviers solutions. The formulation is validated considering problems for which solutions are available. A comparative study is made here by various theories obtained through the formulation. The effects various structural parameters such as thickness ratio, beam length, rise of the curved beam, and nonlocal scale parameter are brought out on bending and stability characteristics of curved nanobeams.     

非局部因子和表面效应对微纳米材料振动特性的影响

基于非局部理论和表面效应模型,导出表面吸附物对微纳米材料的动力学方程,研究非局部因子和表面能对微纳米传感器振动特性的影响．结果显示,非局部因子、表面能、吸附物种类、附加刚度和基底种类对微纳米结构的振动特性有重要影响．     

Size-dependent dynamics of a FG Nanobeam near nonlinear resonances induced by heat

This study explores heat-induced nonlinear vibration of a functionally graded (FG) capacitive nanobeam within the framework of nonlocal strain gradient theory (NLSGT). The elastic FG beam, which is firstly deflected by a DC voltage, is driven to vibrate about its deflected position by a periodic heat load. The nano-structure, which consists of a clamped-clamped nanobeam, is modeled assuming Euler–Bernoulli beam assumption which accounts for the nonlinear von-Karman strain and the electrostatic and intermolecular forcing. To simulate the static and dynamic responses, a model reduction procedure is carried out by employing the Galerkin method. The method of Averaging as a regular semi-analytic perturbation method is applied to obtain governing equations of the steady-state responses. With the purpose of establishing the validity of the solution, a Shooting technique in conjunction with the Floquet theory is used to capture the periodic motions and then examine their stability. The nonlinear resonance frequency of the FG nanobeam near its fundamental natural frequency (primary resonance) and near principal parametric resonance is investigated while the emphasis is placed on studying the effect of various parameters including DC voltage, amplitude of the periodic heat source, material index, damping ratio, and small scale parameters. The main objective of this study is to model a miniature structure which can be used as either a sensitive remote temperature sensor or a high-efficiency thermal energy harvester.     

Thermal effects on buckling of shear deformable nanocolumns with von Kármán nonlinearity based on nonlocal stress theory

Thermal buckling of nanocolumns considering nonlocal effect and shear deformation is investigated based on the nonlocal elasticity theory and the Timoshenko beam theory. By expressing the nonlocal stress as nonlinear strain gradients and based on the variational principle and von Kármán nonlinearity, new higher-order differential governing equations with corresponding higher-order nonlocal boundary conditions both in transverse and axial directions for instability of nanocolumns are derived. New analytical solutions for some practical examples on instability of nanocolumns are presented and analyzed in detail. The paper concluded that the critical buckling load is significantly increased in the presence of nonlocal stress and the results confirm that nanocolumn stiffness is enhanced by nanoscale size effect and reduced by shear deformation. The critical temperature change is increased with larger diameter to length ratio and higher nonlocal nanoscale. It is also concluded that at low and room temperatures the buckling load of nanocolumns increases with increasing temperature change, while at high temperature the buckling load decreases with increasing temperature change.     

3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory

This paper addresses a 3D elasticity analytical solution for static deformation of a simply-supported rectangular micro/nanoplate made of both homogeneous and functionally graded (FG) material within the framework of modified couple stress theory. The plate is assumed to be resting on a Winkler–Pasternak elastic foundation, and its modulus of elasticity is assumed to vary exponentially along thickness. By expanding displacement components in double Fourier series along in-plane coordinates and imposing relevant boundary conditions, the boundary value problem (BVP) of plate system, including its governing partial differential equations (PDEs) of equilibrium are reduced to BVP consisting only ordinary ones (ODEs). Parametric studies are conducted among displacement and stress components developed in the plate and FG material gradient index, length scale parameter, and foundation stiffnesses. From the numerical results, it is concluded that the out-of-plane shear stresses are not necessarily zero at the top and bottom surfaces of plate. The results of this investigation may serve as a benchmark to verify further bending analyses of either homogeneous or FG micro/nanoplates on elastic foundation.     

Static analysis of rectangular nano-plate using three-dimensional theory of elasticity

Analytical solution for bending of a simply supported rectangular graphene sheets based on three dimensional theory of elasticity, is studied employing non-local continuum mechanics. By applying the Fourier series solution to the both displacement and stress field along the in-plane rectangular coordinates direction, and to the governing equation and constitutive relations, the three-dimensional governing equations in term of displacement components can be obtained. Closed form solution for the bending behavior of nano-plate is obtained by exerting the surface tractions on the state equations. To validate the accuracy and convergence of the present approach, numerical results are presented and compared with the results available in the open literature. Effect of non-local parameter, aspect ratio, thickness to length ratio and half wave numbers in the bending behavior of plate are examined. Furthermore, these results may also serve as benchmark to further results into the two-dimensional plate theories.     

Instability of power-law fluid flows down an incline subjected to wind stress

In this paper we investigate the effect of a prescribed superficial shear stress on the generation and structure of roll waves developing from infinitesimal disturbances on the surface of a power-law fluid layer flowing down an incline. The unsteady equations of motion are depth integrated according to the von Kármán momentum integral method to obtain a non-homogeneous system of nonlinear hyperbolic conservation laws governing the average flow rate and the thickness of the fluid layer. By conducting a linear stability analysis we obtain an analytical formula for the critical conditions for the onset of instability of a uniform and steady flow in terms of the prescribed surface shear stress. A nonlinear analysis is performed by numerically calculating the nonlinear evolution of a perturbed flow. The calculation is carried out using a high-resolution finite volume scheme. The source term is handled by implementing the quasi-steady wave propagation algorithm. Conclusions are drawn regarding the effect of the applied surface shear stress parameter and flow conditions on the development and characteristics of the roll waves arising from the instability. For a Newtonian flow subjected to a prescribed superficial shear stress, using an analytical theory, we show that the nonlinear governing equations do not admit roll waves solutions under conditions when the uniform and steady flow is linearly stable. For the case of a general power-law fluid flow with zero shear stress applied at the surface, the analytical investigation leads to a procedure for calculating the characteristics of a roll waves flow. These results are compared with those yielded by the numerical procedure.     

Dual-phase-lag thermoelastic damping models for micro/nanobeam resonators

Thermoelastic damping (TED) affects the quality factors of vacuum-operated micro/nanobeam resonators significantly. In this work, by adopting the non-Fourier theory of dual-phase-lag (DPL) model, an analytical formula of TED in micro/nanobeam resonators with circular cross-section is first developed. Moreover, for micro/nanobeam resonators with rectangular cross-section, the series-form type of DPL-TED model is also proposed and compared with the modified existing model. The characteristics of TED spectra with the single-peak, dual-peak, and multiple-peak phenomena are explored. The simulation results reveal that the ratio of dual-phase-lag times and the characteristic dimension of beams such as the radius and thickness have significant influences on TED behaviors. In addition, temperature distributions in micro/nanobeams exhibit an apparent distinction under the DPL non-Fourier effect.     

Surface and nonlocal effects on response of linear and nonlinear NEMS devices

Nonlocal and surface effects become important for nanoscale devices. To model these effects on frequency response of linear and nonlinear nanobeam subjected to electrostatic excitation, we use Eringen’s nonlocal elastic theory and surface elastic theory proposed by Gurtin and Murdoch to modify the governing equation. Subsequently, we apply Galerkin’s method with exact mode shape including nonlocal and surface effects to get static and dynamic modal equations. After validating the procedure with the available results, we analyze the variation of pull-in voltage and frequency resonance by varying surface and nonlocal parameters. To do frequency analysis of nonlinear system, we solve nonlinear dynamic equation using the method of multiple scale. We found that the frequency response of nonlinear system reduces for fixed excitation as the surface and nonlocal effects increase. Also, we found that the nature of nonlinearity can be tuned from hardening to softening by increasing the nonlocal effects.     

A nonlinear surface-stress-dependent model for vibration analysis of cylindrical nanoscale shells conveying fluid

A nonlinear surface-stress-dependent nanoscale shell model is developed on the base of the classical shell theory incorporating the surface stress elasticity. Nonlinear free vibrations of circular cylindrical nanoshells conveying fluid are studied in the framework of the proposed model. In order to describe the large-amplitude motion, the von Kármán nonlinear geometrical relations are taken into account. The governing equations are derived by using Hamilton's principle. Then, the method of multiple scales is adopted to perform an approximately analytical analysis on the present problem. Results show that the surface stress can influence the vibration characteristics of fluid-conveying thin-walled nanoshells. This influence becomes more and more considerable with the decrease of the wall thickness of the nanoshells. Furthermore, the fluid speed, the fluid mass density, the initial surface tension and the nanoshell geometry play important roles on the nonlinear vibration characteristics of fluid-conveying nanoshells.     

Flexoelectric and surface effects on the electromechanical behavior of graphene-based nanobeams

In this novel work, the electromechanical behavior of graphene-based nanocomposite (GNC) beams with flexoelectric and surface effects were investigated using size-dependent Euler-Bernoulli theory, linear piezoelectricity and Galerkin's weighted residual method along with modified strength of materials and finite element (FE) approaches. In addition, analytical and FE models were developed to study the static response of flexoelectric GNC nanobeams with various boundary conditions: cantilever, simply-supported and clamped-clamped. The developed models predict that the effective piezoelectric coefficients of GNC are responsible for the actuation capability of a graphene layer in the transverse direction due to the applied field in its axial direction and the predictions by both the models are found to be in good agreement. Results reveal that the flexoelectric and surface effects on the static response of GNC nanobeams are significant and should be taken into account. The electromechanical response of GNC nanobeams can be tailored to achieve the required coupled electromechanical characteristics of a vast range of NEMS using various boundary conditions and thickness of nanobeam as well as volume fraction of graphene. Our fundamental study sheds a light on the possibility of developing high-performance and lightweight graphene-based NEMS such as nanosensors, nanogenerators and nanoresonators using non-piezoelectric graphene.     

Unified theory of 2n+1 order size-dependent beams: Mathematical difficulty for functionally graded size-effect parameters solved

New insights on theoretical modeling of size-dependent functionally graded (FG) nanobeams are provided by establishing a unified theory of 2n+1 order shear deformable model with the aids of nonlocal strain gradient elasticity. The unified model covers Euler-type (n = 0), Reddy-type (n = 1), 5th (n = 2), 7th (n = 3) order beam and etc., and the limiting situation n → ∞ predicts nonlocal strain gradient Timoshenko model. The mathematical difficulty for FG nonlocal parameter is particularly emphasized, and an attempt is made for the first time to overcome the difficulty. Theoretically, the governing equations and boundary conditions of 2n+1 order nonlocal strain gradient beams, especially with FG nonlocal parameter and FG strain gradient parameter, are systematically formulated. The difficulty for FG nonlocal parameter is satisfactorily solved with by adopting the present 2n+1 order beam theory. Analytically, solutions to bending and buckling analyses within the unified model are obtained, from which the analytical solutions for Euler- and Timoshenko-type beam can be recovered. Numerically, bending deflection and buckling critical load for Euler beam, Reddy beam, 5th-11th order beam and Timoshenko beam are depicted, of which the benchmark solutions for the 5th to 11th order beam are given for the first time. Meanwhile, potential extensions of the unified model into fractional order is discussed, where benchmark solutions for n = 1.1, 0.88, 0.77, 0.4and0.2 are listed. The influences of FG nonlocal parameter, dimensionless height and Poisson's ratio (or the ratio E/G) on the bending deflection and buckling critical load are systematically studied. The present work mainly contributes to theoretical developments and greatly facilitates the mechanical analysis of beam-type structures.