Size-dependent Nanobeams Research Articles

Free vibration of the bi-dimensional functionally grated (2D-FG) nanobeams

In this paper, the free vibration of the bi-dimensional functionally grated (2D-FG) size dependent nanobeams was investigated by finite element method. The material properties of 2D-FG nanobeam are assumed to vary in both axial and thickness directions by power gradation laws. The size-dependent 2D-FG nanobeam is studied on the basic of nonlocal continuum model. Based on the Euler-Bernoulli beam theory, a finite nanobeam element is derived and employed in vibration analysis of the beam. A parametric study has been carried out to show the effect of material distribution, nonlocal effect on the natural frequencies of the beams.

Nonlinear vibration and stability analysis of a size-dependent viscoelastic cantilever nanobeam with axial excitation

In the present paper, nonlinear forced vibrations of an axial moving nanobeam which is vertically influenced by an external harmonic excitation and gravity is analyzed by considering the effects of linear damping. Considering certain assumptions, a nonlinear Euler-Bernoulli beam theory is developed. With the implementation of the nonlocal elasticity theory, the governing integro-partial-differential equation is obtained by using the Hamilton principle. The multiple scale method is employed to obtain a steady-state response for the size-dependent viscoelastic nanobeam with fixed-free boundary conditions. Subsequently, the trivial and non-trivial steady-state response and the bifurcation point types are examined. Finally, the effects of damping coefficient and nonlocal parameter on stability and bifurcation of trivial and non-trivial solutions are studied. It is found that the effect of nonlocal parameter on the steady-state response and the bifurcation point types is quite important.

Buckling and Nonlinear Vibration of Size-Dependent Nanobeam based on the Non-Local Strain Gradient Theory

Based on the nonlocal strain gradient theory, a Euler-Bernoulli nanobeam model subjected to the compressive axial force and resting on the Winkler-Pasternak layer is developed to study buckling and free nonlinear vibration problems. Critical buckling force and nonlinear frequency of simply supported nanobeam are analytically derived. Comparison of obtained analytical solutions with published and numerical ones shows accuracy of the present solutions. Effects of the scale factor, the aspect ratio, the Winkler parameter and the Pasternak parameter on the critical buckling force ratio and the vibration response of the nanobeam are studied in this work.

Thermal buckling and free vibration of Euler–Bernoulli FG nanobeams based on the higher-order nonlocal strain gradient theory

A size-dependent Euler–Bernoulli beam model is derived within the framework of the higher-order nonlocal strain gradient theory. Nonlocal equations of motion are derived by applying Hamilton’s principle and solved with an analytical solution. The solution is obtained using the Navier solution procedure. In the case of simply supported boundary conditions, the analytical solutions of natural frequencies and critical buckling temperature for free vibration problems are obtained. The paper investigates the thermal effects on buckling and free vibrational characteristics of functionally graded size-dependent nanobeams subjected to various types of thermal loading. The influence of higher-order and lower-order nonlocal parameters and strain gradient scale on buckling and vibration are investigated for various thermal conditions. The obtained results are compared with previous research.

Small scale and spin effects on free transverse vibration of size-dependent nano-scale beams

This study aims to investigate the transverse vibration of size-dependent nano-beam which spins about its longitudinal axis. To this end, nano-beams are modeled based on the Euler–Bernoulli and Timoshenko theories. To consider the size dependency of the nano-beams, the nonlocal elasticity theory is used. By using an exact analytical solution, the governing equations of motions are solved. Comprehensive results are presented to show the effects of various boundary conditions, nonlocal scale parameter, rotational speed, and geometrical parameters on natural frequencies and critical speeds of spinning nano-beam. As spinning nano-beams are the main part of any rotary nano-machines, conclusions of present analysis promote researcher’s information about the vibrational behavior of spinning nano-beams.

Stability and Nonlinear Vibration Analysis of an Axially Loaded Nanobeam Based on Nonlocal Strain Gradient Theory

The size-dependent stability mechanism and nonlinear dynamics of a nano-scale beam under an axial time-dependent load are theoretically studied based on nonlocal strain gradient theory. First, with the consideration of geometric nonlinearity of the nanobeam, the equation of motion is derived with the aid of Hamilton’s principle. Then, by using both Floquet technology and Bolotin’s method, a linear analysis is employed to investigate the size-dependent stability mechanism of the nanobeam for pinned-pinned boundary conditions. It is demonstrated that the results obtained based on the Floquet technology and Bolotin’s method agree well with each other in most cases. Furthermore, the analytical expressions for stability boundaries of the nanobeam system are derived. It is found that when the size-dependent nanobeam system is reduced to a classical one, the obtained expressions for stability boundaries are in good agreement with previous experimental results. Finally, the nonlinear responses of the nanobeam are presented in the form of time traces, bifurcation diagrams and phase portraits. It is shown that the axially excited nanobeam always undergoes a limit cycle oscillation. For all results regarding stability boundaries and nonlinear responses, the size-dependent effects are found to be significant.

Dynamic response of a size-dependent nanobeam to low velocity impact by a nanoparticle with considering atomic interaction forces

In the present paper, low velocity impact response of a size-dependent nanobeam in a thermal field with uniform temperature distribution has been investigated. The van-der Waals interaction force based on description of Lennard–Jonses is considered as the impact force between nanoparticle and nanobeam. According to third-order shear deformation beam theory, the governing equations are obtained using Hamilton's principle based on nonlocal strain-gradient theory. The Galerkin's method was adopted to solve the differential equations of nanobeam with simply supported and clamped boundary conditions. Afterward, the system of time-dependent equations by applying the fourth-order Runge–Kutta method is solved. The parametric study is presented to examine the effect of particle radius, initial velocity, temperature environment, the nonlocal parameter, and the length-scale parameter on the impact response of nanobeam.

Bifurcation and Chaos in Size-Dependent NEMS Considering Surface Energy Effect and Intermolecular Interactions

The impetus of this study is to investigate the chaotic behavior of a size-dependent nano-beam with double-sided electrostatic actuation, incorporating surface energy effect and intermolecular interactions. The geometrically nonlinear beam model is based on Euler-Bernoulli beam assumption. The influence of the small-scale and the surface energy effect are modeled by implementing the consistent couple stress theory proposed by Hadjesfandiari and Dargush together with Gurtin-Murdoch elasticity theory. The governing differential equation of motion is derived using Hamilton’s principle and discretized to a set of nonlinear ODE through Galerkin’s method. Nonlinearities stemmed from different sources such as mid-plane stretching, electrostatic and interatomic forces lead to an intensive nonlinear dynamics in nano-electro-mechanical devices so that the systems exhibit rich dynamic behavior such as periodic and chaotic motions. Poincare portrait is utilized in order to present the system dynamic response in discrete state-space. Bifurcation analysis has been performed with a change in the magnitude of AC voltage corresponding to the various values of DC voltage and excitation frequency. Then, we compare some ranges of AC voltage amplitude, in which the system response becomes stable for these cases. Fast Fourier transformation is also carried out to analyze the frequency content of the system response.

Static bending and buckling of perforated nonlocal size-dependent nanobeams

Due to their superior electronic, mechanical and thermal properties of nanomaterials, they have been widely used in nano-electromechanical systems (NEMS). Nanobeams are common elements in these systems. In some situations perforation is necessary in nanostructures for manufacturing process and technological reasons. Therefore, studying of static bending and buckling behavior of perforated nanobeams is essential for accurate and reliable design of nanostructures. Because of the importance of size effects on the mechanical performance of these structures the size-scale effect should be considered in their design. In this paper an analytical model capable of predicting bending response and critical buckling load of perforated nanobeams is presented for the first time. The model is formulated based on both Timoshenko and Euler–Bernoulli beam theories with nonlocal differential form of Eringen model. The present model is validated by comparing the obtained predictions with that of the available published results in literature for fully beam. Effects of size-scale and perforation parameters (perforation size and number of cutouts) on the static and buckling behavior of the perforated nanobeams are analyzed.

Vibration analysis of parabolic shear-deformable piezoelectrically actuated nanoscale beams incorporating thermal effects

ABSTRACTThermoelectric-mechanical vibration behavior of functionally graded piezoelectric (FGP) nanobeams is first investigated in this article, based on the nonlocal theory and third-order parabolic beam theory by presenting a Navier-type solution. Electro-thermo-mechanical properties of a nanobeam are supposed to change continuously throughout the thickness based on the power-law model. To capture the small-size effects, Eringen's nonlocal elasticity theory is adopted. Using Hamilton's principle, the nonlocal governing equations for the third-order, shear deformable, piezoelectric, FG nanobeams are obtained and they are solved applying an analytical solution. By presenting some numerical results, it is demonstrated that the suggested model presents accurate frequency results of FGP nanobeams. The influences of several parameters, including external electric voltage, power-law exponent, nonlocal parameter, and mode number on the natural frequencies of the size-dependent FGP nanobeams are discussed in detail. The results should be relevant to the design and application of the piezoelectric nanodevices.

Scale-dependent effects on wave propagation in magnetically affected single/double-layered compositionally graded nanosize beams

This article deals with the wave propagation analysis of single/double layered functionally graded (FG) size-dependent nanobeams in elastic medium and subjected to a longitudinal magnetic field employing nonlocal elasticity theory. Material properties of nanobeam change gradually according to the sigmoid function. Applying an analytical solution, the acoustical and optical dispersion relations are explored for various wave number, nonlocality parameter, material composition, elastic foundation constants, and magnetic field intensity. It is found that frequency and phase velocity of waves propagating in S-FGM nanobeam are significantly affected by these parameters. Also, presence of cut-off and escape frequencies in wave propagation analysis of embedded S-FGM nanobeams is investigated.

Semi-analytical vibration analysis of functionally graded size-dependent nanobeams with various boundary conditions

In this paper, free vibration of functionally graded (FG) size-dependent nanobeams is studied within the framework of nonlocal Timoshenko beam model. It is assumed that material properties of the FG nanobeam, vary continuously through the thickness according to a power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The non-classical governing differential equations of motion are derived through Hamilton

Magnetic field effects on nonlocal wave dispersion characteristics of size-dependent nanobeams

In this paper, wave propagation analysis of functionally graded size-dependent nanobeams embedded in elastic foundation exposed to a longitudinal magnetic field is conducted based on nonlocal elasticity theory. Material properties of nanobeam change gradually according to the sigmoid function. Applying an analytical solution, the acoustical and optical dispersion relations are explored for various wave number, nonlocality parameter, material composition, elastic foundation constants and magnetic field intensity. It is found that frequency and phase velocity of waves propagating in S-FGM nanobeam are significantly affected by these parameters. Also the presence of cutoff and escape frequencies in wave propagation analysis of embedded S-FGM nanobeams is investigated.

Small-scale effects on hygro-thermo-mechanical vibration of temperature-dependent nonhomogeneous nanoscale beams

ABSTRACTIn this article, hygro-thermo-mechanical vibration analysis of functionally graded (FG) size-dependent nanobeams exposed to various hygro-thermal loadings is performed via a semi-analytical differential transform method (DTM). Three kinds of environmental conditions, namely, uniform, linear, and sinusoidal hygro-thermal loading, are investigated. Temperature-dependent material properties of a nonlocal FG beam change gradually according to the power-law distribution. A size-dependency description of the nanobeam is conducted using the nonlocal elasticity theory of Eringen. Applying DTM, the nonlocal coupled governing equations obtained from Hamilton's principle are solved. Finally, the impacts of moisture concentration, temperature rise, nonlocal parameters, material composition, and slenderness ratio on the vibrational characteristics of nanosize FG beams with arbitrary boundary conditions are explored. These findings can be used for the accurate design of FG nanostructures in various environmental conditions.

Nonlocal vibration analysis of FG nano beams with different boundary conditions

In this paper, the classical and non-classical boundary conditions effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams are investigated by presenting a semi analytical differential transform method (DTM) for the first time. Three kinds of mathematical models, namely; power law (P-FGM), sigmoid (S-FGM) and Mori-Tanaka (MT-FGM) distribution are considered to describe the material properties in the thickness direction. 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. 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, spring constant factors, various material compositions and mode number on the normalized natural frequencies of the FG nanobeams in detail. It is explicitly shown that the vibration of FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

Thermo-electro-mechanical vibration analysis of size-dependent nanobeam resting on elastic medium under axial preload in presence of surface effect

In the present manuscript, a nonclassical beam theory is developed to analyze free vibration of piezoelectric nanobeam by considering surface effects resting on Winkler–Pasternak elastic medium and thermal loading with axial preload. The nonclassical Eringen theory is utilized to incorporate the length-scale parameter to account for the small-scale effect, while the Gurtin–Murdoch model is employed to inject the surface effects including surface elasticity, surface stress and surface density. The governing equations are derived using Hamilton’s principle in the framework of Euler–Bernoulli beam theory. The governing partial differential equations of motions of system are reduced to a set of algebraic equations with the help of differential transformation method as a semi-analytical–numerical. The mathematical derivations and numerical results are presented in detail for various boundary conditions. Some numerical examples are illustrated in order to investigate the effect of several parameters such as the nonlocal parameter, piezoelectric voltage, surface effects, temperature change, axial preload and elastic medium parameters. Moreover, it is also indicated that the numerical results have good agreement with previous studies.

Effect of various thermal loadings on buckling and vibrational characteristics of nonlocal temperature-dependent functionally graded nanobeams

ABSTRACTIn this article, the thermal effects on buckling and free vibrational characteristics of functionally graded (FG) size-dependent nanobeams subjected to various types of thermal loading are investigated by presenting a Navier-type solution for the first time. Temperature-dependent material properties of FG nanobeams vary continuously along the thickness according to the power-law form. The small-scale effect is taken into consideration based on Eringen's nonlocal elasticity theory. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying an analytical solution. It is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams.

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.

Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment

In this paper, the thermal effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams subjected to various types of thermal loading is investigated by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Two kinds of thermal loading, namely, linear temperature rise and nonlinear temperature rise are studied. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form and the material properties are assumed to be temperature-dependent. 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 behaviour of a FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

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.