• Advances in nano research
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• v.7 no.3
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• pp.145-155
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• 2019

In this article, the influence of small scale effects on the free vibration response of curved magneto-electro-elastic functionally graded (MEE-FG) nanobeams has been investigated considering nonlocal elasticity theory. Power-law is used to judge the through thickness material property distribution of MEE nanobeams. The Euler-Bernoulli beam model has been adopted and through Hamilton's principle the Nonlocal governing equations of curved MEE-FG nanobeam are obtained. The analytical solutions are obtained and validated with the results reported in the literature. Several parametric studies are performed to assess the influence of nonlocal parameter, magnetic potential, electric voltage, opening angle, material composition and slenderness ratio on the dynamic behaviour of MEE curved nanobeams. It is believed that the results presented in this article may serve as benchmark results in accurate analysis and design of smart nanostructures.

• Advances in nano research
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• v.6 no.4
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• pp.339-356
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• 2018

In this article, the critical buckling of a single-walled carbon nanotube (SWCNT) embedded in Kerr's medium is studied. Based on the nonlocal continuum theory and the Euler-Bernoulli beam model. The governing equilibrium equations are acquired and solved for CNTs subjected to mechanical loads and embedded in Kerr's medium. Kerr-type model is employed to simulate the interaction of the (SWNT) with a surrounding elastic medium. A first time, a comparison with the available results is made, and another comparison between various models Winkler-type, Pasternak-type and Kerr-type is studied. Effects of nonlocal parameter and aspect ratio of length to diameter of nanobeam, as well as the foundation parameters on buckling of CNT are investigated. These results are important in the mechanical design considerations of nanocomposites based on carbon nanotubes.

• Steel and Composite Structures
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• v.43 no.2
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• pp.185-200
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• 2022

A Closed-form solution for dynamic response of a functionally graded (FG) nonlocal nanobeam due to action of moving harmonic load is presented in this paper. Due to analyzing in small scale, a nonlocal elasticity theory is utilized. The governing equation and boundary conditions are derived based on the Euler-Bernoulli beam theory and Hamilton's principle. The material properties vary through the thickness direction. The harmonic moving load is modeled by Delta function and the FG nanobeam is simply supported. Using the Laplace transform the dynamic response is obtained. The effect of important parameters such as excitation frequency, the velocity of the moving load, the power index law of FG material and the nonlocal parameter is analyzed. To validate, the results were compared with previous literature, which showed an excellent agreement.

• Computers and Concrete
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• v.31 no.2
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• pp.139-149
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• 2023

This article investigates the statically analysis regarding the thermal buckling behavior of a nonuniform small-scale nanobeam made of functionally graded material based on classic beam theories along with the nonlocal Eringen elasticity. The material distribution of functionally graded structures is composed of temperature-dependent ceramic and metal phases in axial and thickness directions, called two-dimensional functionally graded (2D-FG). The partial differential (PD) formulations and end conditions are extracted by using to the conservation energy method. The porosity voids are assumed in the nonuniform functionally graded (FG) structure. The thermal loads are in the axial direction of the beam. The extracted nonlocal PD equations are also solved by employing generalized differential quadrature method (GDQM). Last but not least, the information acquired is used to produce miniature sensors, providing a unique perspective on the growth of nanoelectromechanical systems (NEMS).

• Advances in nano research
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• v.16 no.5
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• pp.489-500
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• 2024

This paper studies the free vibration analysis of the piezo-magneto-thermo-elastic FG nanobeam submerged in a fluid environment. The problem governed by the partial differential equations is determined by refined higher-order State Space Strain Gradient Theory (SSSGT). Hamilton's principle is applied to discretize the differential equation and transform it into a coupled Euler-Lagrange equation. Furthermore, the equations are solved analytically using Navier's solution technique to form stiffness, damping, and mass matrices. Also, the effects of nonlocal ceramic and metal parts over various parameters such as temperature, Magnetic potential and electric voltage on the free vibration are interpreted graphically. A comparison with existing published findings is performed to showcase the precision of the results.

• Smart Structures and Systems
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• v.19 no.2
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• pp.115-126
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• 2017

In this paper, size dependent bending and free flexural vibration behaviors of functionally graded (FG) nanobeams are investigated using a nonlocal quasi-3D theory in which both shear deformation and thickness stretching effects are introduced. The nonlocal elastic behavior is described by the differential constitutive model of Eringen, which enables the present model to become effective in the analysis and design of nanostructures. The present theory incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect, and furthermore accounts for both shear deformation and thickness stretching effects by virtue of a hyperbolic variation of all displacements through the thickness without using shear correction factor. The material properties of FG nanobeams are assumed to vary through the thickness according to a power law. The neutral surface position for such FG nanobeams is determined and the present theory based on exact neutral surface position is employed here. The governing equations are derived using the principal of minimum total potential energy. The effects of nonlocal parameter, aspect ratio and various material compositions on the static and dynamic responses of the FG nanobeam are discussed in detail. A detailed numerical study is carried out to examine the effect of material gradient index, the nonlocal parameter, the beam aspect ratio on the global response of the FG nanobeam. These findings are important in mechanical design considerations of devices that use carbon nanotubes.

• Steel and Composite Structures
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• v.36 no.2
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• pp.143-161
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• 2020

In nanosized structures as the surface area to the bulk volume ratio increases the classical continuum mechanics approaches fails to investigate the mechanical behavior of such structures. In perforated nanobeam structures, more decrease in the bulk volume is obtained due to perforation process thus nonclassical continuum approaches should be employed for reliable investigation of the mechanical behavior these structures. This article introduces an analytical methodology to investigate the size dependent, surface energy, and perforation impacts on the nonclassical bending behavior of regularly squared cutout nanobeam structures for the first time. To do this, geometrical model for both bulk and surface characteristics is developed for regularly squared perforated nanobeams. Based on the proposed geometrical model, the nonclassical Gurtin-Murdoch surface elasticity model is adopted and modified to incorporate the surface energy effects in perforated nanobeams. To investigate the effect of shear deformation associated with cutout process, both Euler-Bernoulli and Timoshenko beams theories are developed. Mathematical model for perforated nanobeam structure including surface energy effects are derived in comprehensive procedure and nonclassical boundary conditions are presented. Closed forms for the nonclassical bending and rotational displacements are derived for both theories considering all classical and nonclassical kinematics and kinetics boundary conditions. Additionally, both uniformly distributed and concentrated loads are considered. The developed methodology is verified and compared with the available results and an excellent agreement is noticed. Both classical and nonclassical bending profiles for both thin and thick perforated nanobeams are investigated. Numerical results are obtained to illustrate effects of beam filling ratio, the number of hole rows through the cross section, surface material characteristics, beam slenderness ratio as well as the boundary and loading conditions on the non-classical bending behavior of perforated nanobeams in the presence of surface effects. It is found that, the surface residual stress has more significant effect on the bending deflection compared with the corresponding effect of the surface elasticity, Es. The obtained results are supportive for the design, analysis and manufacturing of perforated nanobeams.

• Steel and Composite Structures
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• v.18 no.2
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• pp.425-442
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• 2015

This paper addresses theoretically the bending and buckling behaviors of size-dependent nanobeams made of functionally graded materials (FGMs) including the thickness stretching effect. The size-dependent FGM nanobeam is investigated on the basis of the nonlocal continuum model. The nonlocal elastic behavior is described by the differential constitutive model of Eringen, which enables the present model to become effective in the analysis and design of nanostructures. The present model incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect, and furthermore accounts for both shear deformation and thickness stretching effects by virtue of a sinusoidal variation of all displacements through the thickness without using shear correction factor. The material properties of FGM nanobeams are assumed to vary through the thickness according to a power law. The governing equations and the related boundary conditions are derived using the principal of minimum total potential energy. A Navier-type solution is developed for simply-supported boundary conditions, and exact expressions are proposed for the deflections and the buckling load. The effects of nonlocal parameter, aspect ratio and various material compositions on the static and stability responses of the FGM nanobeam are discussed in detail. The study is relevant to nanotechnology deployment in for example aircraft structures.

• Smart Structures and Systems
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• v.19 no.1
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• pp.105-113
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• 2017

A microstructure-dependent dynamic model for silicon nanobeams with axial motion is developed by considering the effects of nonlocal elasticity and surface energy. The nanobeam is considered to subject to both transverse and longitudinal loads arising from nanostructural surface effect and all positive directions of physical quantities are defined clearly prior to modeling so as to clarify the confusions of sign in governing equations of previous work. The nonlocal and surface effects are taken into consideration in the dynamic behaviors of silicon nanobeams with axial motion including circular natural frequency, vibration mode, transverse displacement and critical speed. Various supporting conditions are presented to investigate the circular frequencies by a numerical method and the effects of many variables such as nonlocal nanoscale, axial velocity and external loads on non-dimensional circular frequencies are addressed. It is found that both nonlocal and surface effects play remarkable roles on the dynamics of nanobeams with axial motion and cause the frequencies and critical speed to decrease compared with the classical continuum results. The comparisons of the non-dimensional calculation values by present and previous studies validate the correctness of the present work. Additionally, numerical examples for silicon nanobeams with axial motion are addressed to show the nonlocal and surface effects on circular frequencies intuitively. Results obtained in this paper are helpful for the design and optimization of nanobeam-like microstructures based sensors and oscillators at nanoscale with desired dynamic mechanical properties.

• Advances in aircraft and spacecraft science
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• v.10 no.1
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• pp.51-65
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• 2023

In this paper, the effect of deepness on in-plane free vibration behavior of a curved functionally graded (FG) nanobeam based on nonlocal elasticity theory has been investigated. Differential equations and boundary conditions have been developed based on Hamilton's principle. In order to figure out the size effect, nonlocal theory has been adopted. Properties of material vary in radial direction. By using Navier solution technique, the amount of natural frequencies has been obtained. Also, to take into account the deepness effect on vibrations, thickness to radius ratio has been considered. Differences percentage between results of cases in which deepness effect is included and excluded are obtained and influences of power-law exponent, nonlocal parameter and arc angle on these differences percentage are studied. Results show that arc angle and power law exponent parameters have the most influences on the amount of the differences percentage due to deepness effect. It has been observed that the inclusion of geometrical deep term and material distribution results in an increase in sensitivity of dimensionless natural frequency about variation of aforementioned parameters and a change in variation range of natural frequency. Finally, several numerical results of deep and shallow curved functionally graded nanobeams with different geometry dimensions are presented, which may serve as benchmark solutions for the future research in this field.