• Advances in nano research
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• v.7 no.1
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• pp.1-11
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• 2019

In this article, wave propagation characteristics in magneto-electro-elastic (MEE) nanotube considering shell model is studied in the framework nonlocal theory. To account for the small-scale effects, the Eringen's nonlocal elasticity theory of is applied. Nonlocal governing equations of MEE nanotube have been derived utilizing Hamilton's principle. The results of this investigation have been accredited by comparing them of previous studies. An analytical solution of governing equations is used to obtain phase velocities and wave frequencies. The influences of different parameters, such as different mode, nonlocal parameter, length parameter, geometry, magnetic field and electric field on wave propagation responses of MEE nanotube are expressed in detail.

• Advances in materials Research
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• v.6 no.2
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• pp.93-128
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• 2017

In this paper, thermal vibration behavior of nanoscale beams made of functionally graded (FG) materials subjected to various types of thermal loading are investigated. A Reddy shear deformation beam theory which captures both the microstructural and shear deformation effects without the need for any shear correction factors is employed. Material properties of FG nanobeam are assumed to be temperature-dependent and vary gradually along the thickness according to the power-law form. The influence of small scale is captured based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying analytical solution. The comparison of the obtained results is conducted with those of nonlocal Euler-Bernoulli beam theory and it is demonstrated that the proposed modeling predict correctly the vibration responses of FG nanobeams. The effects of nonlocal parameter, material graduation, mode number, slenderness ratio and thermal loading on vibration behavior of the nanobeams are studied in detail.

• Smart Structures and Systems
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• v.17 no.5
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• pp.837-857
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• 2016

In the present study, for first time the size dependent vibration behavior of a rotating functionally graded (FG) Timoshenko nanobeam based on Eringen's nonlocal theory is investigated. It is assumed that the physical and mechanical properties of the FG nanobeam are varying along the thickness based on a power law equation. The governing equations are determined using Hamilton's principle and the generalized differential quadrature method (GDQM) is used to obtain the results for cantilever boundary conditions. The accuracy and validity of the results are shown through several numerical examples. In order to display the influence of size effect on first three natural frequencies due to change of some important nanobeam parameters such as material length scale, angular velocity and gradient index of FG material, several diagrams and tables are presented. The results of this article can be used in designing and optimizing elastic and rotary type nano-electro-mechanical systems (NEMS) like nano-motors and nano-robots including rotating parts.

• Advances in nano research
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• v.12 no.1
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• pp.37-47
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• 2022

In this paper, the non-linear static analysis of Timoshenko nanobeams consisting of bi-directional functionally graded material (BFGM) with immovable ends is investigated. The scratching in the FG nanobeam mid-plane, is the source of nonlinearity of the bending problems. The nonlocal theory is used to investigate the non-linear static deflection of nanobeam. In order to simplify the formulation, the problem formulas is derived according to the physical middle surface. The Hamilton principle is employed to determine governing partial differential equations as well as boundary conditions. Moreover, the differential quadrature method (DQM) and direct iterative method are applied to solve governing equations. Present results for non-linear static deflection were compared with previously published results in order to validate the present formulation. The impacts of the nonlocal factors, beam length and material property gradient on the non-linear static deflection of BFG nanobeams are investigated. It is observed that these parameters are vital in the value of the non-linear static deflection of the BFG nanobeam.

• Advances in nano research
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• v.4 no.4
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• pp.265-279
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• 2016

Present study interests with the longitudinal forced vibration of nanorods. The nonlocal elasticity theory of Eringen is used in modeling of nanorods. Uniform, linear and sinusoidal axial loads are considered. Dynamic displacements are obtained for nanorods with different geometrical properties, boundary conditions and nonlocal parameters. The nonlocal effect increases dynamic displacement and frequency when compared with local elasticity theory. Present results can be useful for modeling of the axial nanomotors and nanoelectromechanical systems.

• Advances in nano research
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• v.12 no.2
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• pp.139-150
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• 2022

This paper presents sets of explicit analytical equations that compute the static displacements of nanobeams by adopting the nonlocal elasticity theory of Eringen within the framework of Euler Bernoulli and Timoshenko beam theories. Castigliano's theorem is applied to an equivalent Virtual Local Beam (VLB) made up of linear elastic material to compute the displacements. The first derivative of the complementary energy of the VLB with respect to a virtual point load provides displacements. The displacements of the VLB are assumed equal to those of the nonlocal beam if nonlocal effects are superposed as additional stress resultants on the VLB. The illustrative equations of displacements are relevant to a few types of loadings combined with a few common boundary conditions. Several equations of displacements, thus derived, matched precisely in similar cases with the equations obtained by other analytical methods found in the literature. Furthermore, magnitudes of maximum displacements are also in excellent agreement with those computed by other numerical methods. These validated the superposition of nonlocal effects on the VLB and the accuracy of the derived equations.

• Advances in nano research
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• v.9 no.3
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• pp.157-172
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• 2020

The aim of this present research is the effect of the higher-order terms of the governing equation on the forced longitudinal vibration of a nanorod model and making comparisons of the results with classical nonlocal elasticity theory. For this purpose, the free axial vibration along with forced one under the two various linear and harmonic axial concentrated forces in zigzag Single-Walled Carbon Nanotube (SWCNT) are analyzed dynamically. Three various theories containing the classical theory, which is called Eringen's nonlocal elasticity, along with Rayleigh and Bishop theories (higher-order theories) are established to justify the nonlocal behavior of constitutive relations. The governing equation and the related boundary conditions are derived from Hamilton's principle. The assumed modes method is adopted to solve the equation of motion. For the free axial vibration, the natural frequencies are calculated for the various values of the nonlocal parameter only based on Eringen's theory. The effects of the nonlocal parameter, thickness, length, and ratio of the excitation frequency to the natural frequency over time in dimensional and non-dimensional axial displacements are investigated for the first time.

• Structural Engineering and Mechanics
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• v.54 no.6
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• pp.1061-1073
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• 2015

This paper presents a nonlocal sinusoidal shear deformation beam theory (SDBT) for the nonlinear vibration of single walled carbon nanotubes (CNTs). The present model is capable of capturing both small scale effect and transverse shear deformation effects of CNTs, and does not require shear correction factors. The surrounding elastic medium is simulated based on Pasternak foundation. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the CNTs are derived using Hamilton's principle. Differential quadrature method (DQM) for the natural frequency is presented for different boundary conditions, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory (TBT). The effects of nonlocal parameter, boundary condition, aspect ratio on the frequency of CNTs are considered. The comparison firmly establishes that the present beam theory can accurately predict the vibration responses of CNTs.

• Computers and Concrete
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• v.25 no.4
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• pp.343-354
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• 2020

In this paper, vibration characteristics of chiral double-walled carbon nanotubes entrenched on Kelvin's model. The Eringen's nonlocal elastic equations are being combined with Kelvin's theory to observe small scale response. A nonlocal model has been formulated to explore the frequency spectrum of chiral double-walled CNTs along with diversity of indices and nonlocal parameter. Wave propagation is proposed technique to establish field equations of model subjected to four distinct end supports. The significance of scale effect in relevance of length-to-diameter and thickness- to- radius ratios are discussed and displayed in detail.

• Structural Engineering and Mechanics
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• v.62 no.6
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• pp.695-702
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• 2017

In this work, a nonlocal zeroth-order shear deformation theory is developed for the nonlinear postbuckling behavior of nanoscale beams. The beauty of this formulation is that, in addition to including the nonlocal effect according to the nonlocal elasticity theory of Eringen, the shear deformation effect is considered in the axial displacement within the use of shear forces instead of rotational displacement like in existing shear deformation theories. The principle of virtual work together of the nonlocal differential constitutive relations of Eringen, are considered to obtain the equations of equilibrium. Closed-form solutions for the critical buckling load and the amplitude of the static nonlinear response in the postbuckling state for simply supported and clamped clamped nanoscale beams are determined.