Non-linear vibration of nanobeams with various boundary condition based on nonlocal elasticity theory

In this study, nonlinear vibrations of Euler-Bernoulli nanobeams with various supports condition is investigated. The non-linear equations of motion including stretching of the neutral axis are derived. Forcing and damping effects are included in the analysis. Exact solutions for the mode shapes and frequencies are obtained for the linear part of the problem. For the non-linear problem approximate solutions using perturbation technique is applied to the equations of motion. The different of support cases are investigated and the cases analyzed in detail. The method of multiple time scale that is a perturbation technique is applied to the equations of motion. Natural frequencies and mode shapes for the linear problem are found for the nanobeam. Nonlinear frequencies are calculated; amplitude and phase modulation figures are presented for different cases. Frequency-response curves are drawn.     

Torsion of functionally graded nonlocal viscoelastic circular nanobeams

The elastostatic problem of functionally graded circular nanobeams under torsion, with nonlocal elastic behavior proposed by Eringen, is preliminarily formulated. Exact solutions are detected for nanobeams with arbitrary axial gradations of elastic properties and radially quadratic distributions of shear moduli. Extension of the treatment to nonlocal viscoelastic composite circular nanobeams is then performed. An effective solution procedure based on Laplace transform is developed, providing a new correspondence principle in nonlocal viscoelasticity for functionally graded materials. Displacements, shear strains and stresses are established for nonlocal viscoelastic nanobeams made of periodic fiber-reinforced materials, with polymeric matrix described by a Maxwell model connected in series with a Voigt model.     

Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium

Based on nonlocal theory, this article discusses vibration of CDVGS¹ systems. The properties of each single layer graphene sheet (SLGS) are assumed to be orthotropic and viscoelastic. The two SLGSs are simply supported and coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak layer. This model is aimed at representing dynamic interactions in nanocomposite materials with dissipation effect. By considering the Kirchhoff plate theory and Kelvin–Voigt model, the governing equation is derived using Hamilton's principle. The equation is solved analytically to obtain the complex natural frequency. The parametric study is thoroughly performed, concentrating on the series effects of viscoelastic damping structure, aspect ratio, visco-Pasternak medium, and mode number. In this system, in-phase (IPV) and out-of-phase (OPV) vibrations are investigated. The numerical results of this article show a perfect correspondence with those of the previous researches.     

Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions

In this paper, the thermal effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying DTM. According to the numerical results, it is revealed that the proposed modeling and semi analytical approach can provide accurate frequency results of the FG nanobeams as compared to analytical results and also some cases in the literature. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as thermal effect, material distribution profile, small scale effects, mode number and boundary conditions on the normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behavior of an FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.     

An analytical study on the nonlinear free vibration of functionally graded nanobeams incorporating surface effects

Nonlinear free vibration of simply supported FG nanoscale beams with considering surface effects (surface elasticity, tension and density) and balance condition between the FG nanobeam bulk and its surfaces is investigated in this paper. The non-classical beam model is developed within the framework of Euler–Bernoulli beam theory including the von Kármán geometric nonlinearity. The component of the bulk stress, σ_zz, is assumed to vary cubically through the nanobeam thickness and satisfies the balance conditions between the FG nanobeam bulk and its surfaces. Accordingly, surface density is introduced into the governing equation of the nonlinear free vibration of FG nanobeams. The multiple scales method is employed as an analytical solution for the nonlinear governing equation to obtain the nonlinear natural frequencies of FG nanbeams. Several comparison studies are carried out to demonstrate the effect of considering the balance conditions on free nonlinear vibration of FG nanobeams. Lastly, the influences of the FG nanobeam length, volume fraction index, amplitude ratio, mode number and thickness ratio on the normalized nonlinear natural frequencies of the FG nanobeams are discussed in detail.     

Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams

The surface and nonlocal effects on the nonlinear flexural free vibrations of elastically supported non-uniform cross section nanobeams are studied simultaneously. The formulations are derived based on both Euler–Bernoulli beam theory (EBT) and Timoshenko beam theory (TBT) independently using Hamilton’s principle in conjunction with Eringen’s nonlocal elasticity theory. Green’s strain tensor together with von Kármán assumptions are employed to model the geometrical nonlinearity. The differential quadrature method (DQM) as an efficient and accurate numerical tool in conjunction with a direct iterative method is adopted to obtain the nonlinear vibration frequencies of nanobeams subjected to different boundary conditions. After demonstrating the fast rate of convergence of the method, it is shown that the results are in excellent agreement with the previous studies in the limit cases. The influences of surface free energy, nonlocal parameter, length of non-uniform nanobeams, variation of nanobeam width and elastic medium parameters on the nonlinear free vibrations are investigated.     

Exact closed-form solution for non-local vibration and biaxial buckling of bonded multi-nanoplate system

In recent years, nonlocal elasticity theory is widely used for the analytical and computational modeling of nanostructures. This theory, developed by Eringen, has shown to be practical for the vibration and buckling analysis of nanoscale structures and reliable for predesign procedures of nano-devices. This paper considers buckling and dynamic analysis of multi-nanoplate systems. This type of system can be relevant to composite structures embedded with graphene sheets. Exact solutions for the natural frequencies and buckling loads of multi-nanoplate systems have been proposed by considering that the multi-nanoplate system is embedded within an elastic medium. Nonlocal elasticity theory is utilized for the mathematical establishment of the system. The solutions of the homogenous system of differential equations are obtained using the Navier’s method and trigonometric method. An asymptotic analysis is proposed to show the influence of increasing number of nanoplates in the system. Analytical expressions are validated with existing results in the literature for some special cases. Numerical results based on the analytical expressions is shown to quantify the effects of the change in nonlocal parameter, stiffness coefficients of the elastic mediums and the number of layers on the natural frequencies and buckling load.     

Modified continuum model for stability analysis of asymmetric FGM double-sided NEMS: Corrections due to finite conductivity,surface energy and nonlocal effect

Finite conductivity, surface energy and nonlocal effect can influence the electromechanical performance of micro/nano-electromechanical systems (MEMS/NEMS). However, these factors are yet ignored on stability analysis of MEMS/NEMS fabricated from functionally graded materials (FGM). In this paper, dynamic stability of double-sided NEMS fabricated from non-symmetric FGM is investigated incorporating finite conductivity, surface energy and nonlocal effect. The Gurtin–Murdoch model and Eringen's elasticity are employed to consider the surface energy and nonlocal effect, respectively. Effect of finite conductivity of FGM on electrostatic and Casimir attractions is incorporated via relative permittivity and plasma frequency of the material. The stability analysis of the nanostructure is conducted by plotting time history and phase portraits. Moreover, bifurcation analysis is conducted to investigate the stability of the fixed points of the nano-structure. The validity of the proposed model is examined by comparing the results of the present study with those reported in the literature. The impact of various parameters i.e. finite conductivity, nonlocal parameter, surface stresses and material characteristics on the dynamic instability of the NEMS are addressed.     

Stochastic FEM on nonlinear vibration of fluid-loaded double-walled carbon nanotubes subjected to a moving load based on nonlocal elasticity theory

This paper adopts stochastic FEM to study the statistical dynamic behaviors of nonlinear vibration of the fluid-conveying double-walled carbon nanotubes (DWCNTs) under a moving load by considering the effects of the geometric nonlinearity and the nonlinearity of van der Waals (vdW) force. The Young’s modulus of elasticity of the DWCNTs is considered as stochastic with respect to the position to actually characterize the random material properties of the DWCNTs. Besides, the small scale effects of the nonlinear vibration of the DWCNTs are studied by using the theory of nonlocal elasticity. Based on the Hamilton’s principle, the nonlinear governing equations of the fluid-conveying double-walled carbon nanotubes under a moving load are formulated. The stochastic finite element method along with the perturbation technique is adopted to study the statistical dynamic response of the DWCNTs. Some statistical dynamic response of the DWCNTs such as the mean values and standard deviations of the non-dimensional dynamic deflections are computed and checked by the Monte Carlo Simulation, meanwhile the effects of the nonlocal parameter, aspect ratio and the flow velocity on the statistical dynamic response of the DWCNTs are investigated. It can be concluded that the nonlocal solutions of the dynamic deflections get larger with the increase of the nonlocal parameters due to the small scale effect, and as the flow velocity increases, the maxima non-dimensional dynamic deflections of the DWCNTs get larger.     

The influence of the intermolecular surface forces on the static deflection and pull-in instability of the micro/nano cantilever gyroscopes

In this paper, the effects of van der Waals and Casimir forces on the static deflection and pull-in instability of a micro/nano cantilever gyroscope with proof mass at its end are investigated. The micro/nano gyroscope is subjected to coupled bending motions which are related by base rotation and nonlinearities due to the geometry and the inertial terms. It is actuated and detected by capacitance plates which are placed on the proof mass. The extended Hamilton principle is used to find the equations governing the static behavior of the clamp-free micro/nano gyroscopes under electrostatic, Casimir and van der Waals forces. The equations of static motion are discritized by Galerkin’s decomposition method. The nonlinear equilibrium equations are solved analytically using homotopy perturbation method (HPM). The static response of the micro/nano gyroscopes to variations in the DC voltage across the drive and sense electrodes is obtained and the effects of different parameters on pull-in instability are investigated. The presented results can be used for accurate estimations of the instability and performance of the micro/nano gyroscopes.     

On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory

This paper aims to present nonlinear forced vibration characteristics of nanobeams including surface stress effect. By considering the local geometrical nonlinearity based on von Karman relation, a new formulation of the Timoshenko beam model is developed through the Gurtin–Murdoch elasticity theory in which the effect of surface stress is incorporated. By using a variational approach on the basis of Hamilton’s principle, the size-dependent equations of motion and associated boundary conditions are obtained. The generalized differential quadrature (GDQ) method is employed to discretize the non-classical governing differential equations over the spatial domain by using the shifted Chebyshev–Gauss–Lobatto grid points. Subsequently, a Galerkin-based numerical approach is put to use in order to reduce the set of nonlinear equations into a time-varying set of ordinary differential equations of Duffing-type. In the next step, the time domain is discretized via spectral differentiation matrix operators which are defined based on the derivatives of a periodic base function. Finally, the pseudo arc-length method is employed to solve the resulting nonlinear parameterized algebraic equations. The frequency–response curves for forced vibration behavior of nanobeams including the effect of surface stress are predicted corresponding to various values of beam thickness, length to thickness ratio and surface elastic constants. It is revealed that by incorporating the surface stress effect, the maximum amplitude occurs at lower excitation frequencies and the wide of region of the response tends to decrease.     

Bending stiffening of graphene and other 2D materials via controlled rippling

In this paper, the effects of rippling on the bending stiffness of a monolayer graphene are studied. The initial rippling of the surface is modeled by cosine functions with a hierarchical topology. Considering both large displacement and small scale effect, the governing equilibrium equations are determined and solved. Then an equivalent bending stiffness is calculated for a rippled graphene and the effects of rippling, material discreteness, and structural dimension on its stiffness are discussed in details. The results quantify how the rippling strongly increases the effective bending stiffness of graphene and interacts with the discrete nature of the material not only because of increase in the moment of inertia. This approach can be applied to ripples design of 2D materials in order to achieve stiffening in bending as required in specific applications.     

Size and surface effects on the mechanical behavior of nanotubes in first gradient elasticity

First gradient elasticity theory considering surface energy is employed to investigate the axisymmetric deformation of circular nanotubes, in which the microstructural and surface effects are taken into account. The governing equilibrium equation is derived and the corresponding analytical solutions are obtained. It is demonstrated that the total stress decreases with the increase of the intrinsic bulk size, while it increases with the increase of the directional surface energy length parameter. These parameters have strong influences on the mechanical behavior of the embedded nanotubes and thus should be considered in the practical analysis of nanostructures.     

Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method

In this paper nonlocal Euler–Bernoulli beam theory is employed for vibration analysis of functionally graded (FG) size-dependent nanobeams by using Navier-based analytical method and a semi analytical differential transform method. Two kinds of mathematical models, namely, power law and Mori-Tanaka models are considered. The nonlocal Eringen theory takes into account the effect of small size, which enables the present model to become effective in the analysis and design of nanosensors and nanoactuators. Governing equations are derived through Hamilton's principle and they are solved applying semi analytical differential transform method (DTM). It is demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of FG nanobeams. The good agreement between the results of this article and those available in literature validated the presented approach. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as small scale effects, different material compositions, mode number and thickness ratio on the normalized natural frequencies of the FG nanobeams in detail. It is explicitly shown that the vibration of a FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.     

Dynamic behavior of thin and thick cracked nanobeams incorporating surface effects

Free transverse vibration of cracked nanobeams is investigated in the presence of the surface effects. Two nanobeam types, thin and thick, are studied using two beam theories, Euler–Bernoulli and Timoshenko. The influences of crack severity and position, surface density, rotary inertia and shear deformation, nanobeam dimension, mode number, satisfying balance condition between the surface layers and the bulk, boundary conditions and satisfying compatibility and boundary conditions with appropriate resultant moment and shear force are studied in details. It is found out that satisfying compatibility and boundary conditions with the resultant moment and shear force in presence of the surface effects and considering surface density neglected in previous work have significant effects on the natural frequencies of cracked nanobeams. In addition, rotary inertia and shear deformation cause a reduction in the crack and surface effects on the natural frequencies.     

Time discretization effect on the nonlinear vibration of embedded SWBNNT conveying viscous fluid

In this study, the effect of time discretization on the nonlinear transverse vibration and instability of single-walled boron nitride nanotube (SWBNNT) conveying viscous fluid is investigated based on the nonlocal piezoelasticity theory. SWBNNT is considered as an Euler–Bernoulli beam and is subjected to combined mechanical loading, thermal changes and electrical field. The elastic medium is simulated as Winkler and Pasternak foundation. The interaction between the inner viscous fluid and SWBNNT is obtained using Navier–Stokes equation. The axial inertia is neglected and a new approach is introduced to decouple the mechanical and electrical fields considering charge equation. Motion equations are derived by Hamilton’s principle using the Von-Kármán nonlinearity theory. In the first approach, time and space domains are discretized using the method of multiple scale (MMS) and Galerkin procedure respectively, and in the second one differential quadrature method (DQM) is utilized to space discretization. Good agreement is shown between the results of first and second approach. Numerical results indicate the significant effects of aspect ratio, elastic medium and nonlocality on the frequency and instability of the SWBNNT.     

Biaxial buckling analysis of double-orthotropic microplate-systems including in-plane magnetic field based on strain gradient theory

Background/purposeThis paper deals with analysis of biaxial buckling behavior of double-orthotropic microplate system including in-plane magnetic field, using strain gradient theory.MethodsTwo Kirchhoff microplates are coupled by an internal elastic medium and also are limited to the external Pasternak elastic foundation. Utilizing the principle of total potential energy, the equilibrium equations of motion for three cases (out-of-phase buckling, in-phase buckling and buckling with a plate) are acquired. In this study, we assumed boundary conditions of all the edges are simply supported. In order to get exact solution for buckling load of system, Navier approach which satisfies the simply supported boundary conditions is applied.ResultsVariations of the buckling load of double-microplate system subjected to biaxial compression corresponding to various values of the thickness, length scale parameter, magnetic field, stiffness of internal and external elastic medium, aspect ratio, shear stiffness of the Pasternak foundation and biaxial compression ratio are investigated. Furthermore, influence of higher modes on buckling load is shown. By comparing the numerical results, it is found that dimensionless buckling load ratio for in-phase mode is more than those of out of phase and one microplate fixed. Also it is shown that the value of buckling load ratio reduces, when non-dimensional length scale parameter increases.ConclusionHowever, we found when properties of plate are orthotropic the buckling load ratio is more than isotropic state. Also, by considering the effect of magnetic field, non-dimensional buckling load ratio reduces.     

Large-amplitude vibration of non-homogeneous orthotropic composite truncated conical shell

In this study, the large-amplitude vibration of non-homogenous orthotropic composite truncated conical shell is investigated. It is assumed that the Young’s moduli and density of orthotropic materials vary exponentially through the thickness direction. The basic equations of non-homogenous orthotropic truncated conical shell are derived using the finite deflection theory with von Karman–Donnell-type of kinematic non-linearity. Then, foregoing equations are solved using the Superposition principle, Galerkin and Semi-inverse methods and the frequency- amplitude relationship is found. Finally, carrying out some computations, the effects of non-homogeneity, orthotropy and conical shell characteristics on the nonlinear vibration characteristics have been studied.     

Free vibration analysis of FGM cylindrical shell partially resting on Pasternak elastic foundation with an oblique edge

The free vibration characteristics of FGM cylindrical shells partially resting on elastic foundation with an oblique edge are investigated by an analytical method. The cylindrical shell is partially surrounded by an elastic foundation which is represented by the Pasternak model. An edge of an elastic foundation lies in a plane that is oblique at an angle with the shell axis. The motion of shell is represented based on the first order shear deformation theory (FSDT) to account for rotary inertia and transverse shear strains. The functionally graded cylindrical shell is composed of stainless steel and silicon nitride. Material properties vary continuously through the thickness according to a four-parameter power law distribution in terms of volume fraction of the constituents. The equation of motion for eigenvalue problem is obtained using Rayleigh–Ritz method and variational approach. To validate the present method, the numerical example is presented and compared with the available existing results.     

On a problem of the vibration of functionally graded conical shells with mixed boundary conditions

This paper presents a theoretical approach to solve vibration problems of functionally graded (FG) truncated conical shells under mixed boundary conditions. The material properties of FG shell are assumed to vary continuously through the thickness of the conical shell. The fundamental relations, motion and strain compatibility equations of FG truncated conical shells are derived by means of the Airy stress function method. Two cases of mixed boundary conditions are investigated. The basic equations are solved by using Galerkin method and fundamental cyclic frequencies of FG truncated conical shells are obtained. The results are compared and validated with the results available in the literature. The detailed parametric studies are carried out to investigate the influences of radius-to-thickness ratio, lengths-to-radius ratio, material composition and mixed boundary conditions on the fundamental cyclic frequencies of truncated conical shells.