• Advances in nano research
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• v.5 no.2
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• pp.141-169
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• 2017

In this paper, a nanobeam connected to a rotating molecular hub is considered. The vibration behavior of rotating functionally graded nanobeam based on Eringen's nonlocal theory and Euler-Bernoulli beam model is investigated. Furthermore, axial preload and porosity effect is studied. It is supposed that the material attributes of the functionally graded porous nanobeam, varies continuously in the thickness direction according to the power law model considering the even distribution of porosities. Porosity at the nanoscopic length scale can affect on the rotating functionally graded nanobeams dynamics. The equations of motion and the associated boundary conditions are derived through the Hamilton's principle and generalized differential quadrature method (GDQM) is utilized to solve the equations. In this paper, the influences of some parameters such as functionally graded power (FG-index), porosity parameter, axial preload, nonlocal parameter and angular velocity on natural frequencies of rotating nanobeams with pure ceramic, pure metal and functionally graded materials are examined and some comparisons about the influence of various parameters on the natural frequencies corresponding to the simply-simply, simplyclamped, clamped-clamped boundary conditions are carried out.

• Steel and Composite Structures
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• v.36 no.3
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• pp.293-305
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• 2020

This article presented a nanoscale modified continuum model to investigate the free vibration of functionally graded (FG) porous nanobeam by using finite element method. The main novelty of this manuscript is presenting effects of four different porosity models on vibration behaviors of nonlocal nanobeam structure including size effect, that not be discussed before The proposed porosity models are, uniform porosity distribution, symmetric with mid-plane, bottom surface distribution and top surface distribution. The nano-scale effect is included in modified model by using the differential nonlocal continuum theory of Eringen that adding the length scale into the constitutive equations as a material parameter constant. The graded material is distributed through the beam thickness by a generalized power law function. The beam is simply supported, and it is assumed to be thin. Therefore, the kinematic assumptions of Euler-Bernoulli beam theory are held. The mathematical model is solved numerically using the finite element method. Results demonstrate effects of porosity type, material gradation, and nanoscale parameters on the free vibration of nanobeam. The proposed model is effective in vibration analysis of NEMS structure manufactured by porous functionally graded materials.

• Steel and Composite Structures
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• v.29 no.3
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• pp.349-362
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• 2018

Thermal buckling behavior of porous functionally graded nanobeam integrated with piezoelectric sensor and actuator based on the nonlocal higher-order shear deformation beam theory is investigated for the first time. Its material properties are assumed to be temperature-dependent and varying along the thickness direction according to the modified power-law rule. Note that the porosity with even type is considered herein. The equations of motion are obtained through Hamilton's principle. The influences of several parameters (such as type of temperature distribution, external electric voltage, material composition, porosity, small-scale effect, Ker foundation parameters, and beam thickness) on the thermal buckling of FG nanobeam are investigated 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.10 no.4
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• pp.313-325
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• 2021

This paper is concerned with the buckling behavior of 2D and quasi-3D problem of functionally graded nanobeam founded on high order shear deformation beams theory and made by two different types of porous distribution materials in Nano- and micro-scales. The used Quasi-3D formulation takes into account the transverse shear effect and uses only three variables. Both formulations do not include the correction factor that is required in the first shear deformation theory proposed by Timoshenko. Governing equations are derived using the principle of virtual work. Analytical resolutions for buckling of FG nanobeam are introduced under tow different boundary conditions, and the results obtained are compared to those proposed in literatures.

• 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).

• Structural Engineering and Mechanics
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• v.71 no.1
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• pp.89-98
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• 2019

In this study, the mechanical bending behaviors of functionally graded porous nanobeams are investigated. Four types of porosity which are, the classical power porosity function, the symmetric with mid-plane cosine function, bottom surface distribution and top surface distribution are proposed in analysis of nanobeam for the first time. A comparison between four types of porosity are illustrated. The effect of nano-scale is described by the differential nonlocal continuum theory of Eringen by adding the length scale into the constitutive equations as a material parameter comprising information about nanoscopic forces and its interactions. The graded material is designated by a power function through the thickness of nanobeam. The beam is simply-supported and is assumed to be thin, and hence, the kinematic assumptions of Euler-Bernoulli beam theory are held. The mathematical model is solved numerically using the finite element method. Numerical results show effects of porosity type, material graduation, and nanoscale parameters on the static deflection of nanobeam.

• Structural Engineering and Mechanics
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• v.54 no.4
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• pp.693-710
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• 2015

This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler-Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton's principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

• Steel and Composite Structures
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• v.18 no.4
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• pp.909-924
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• 2015

This paper investigates the vibration phenomenon of a nanobeam subjected to a time-dependent heat flux. Material properties of the nanobeam are assumed to be graded in the thickness direction according to a novel exponential distribution law in terms of the volume fractions of the metal and ceramic constituents. The upper surface of the functionally graded (FG) nanobeam is pure ceramic whereas the lower surface is pure metal. A nonlocal generalized thermoelasticity theory with dual-phase-lag (DPL) model is used to solve this problem. The theories of coupled thermoelasticity, generalized thermoelasticity with one relaxation time, and without energy dissipation can extracted as limited and special cases of the present model. An analytical technique based on Laplace transform is used to calculate the variation of deflection and temperature. The inverse of Laplace transforms are computed numerically using Fourier expansion techniques. The effects of the phase-lags (PLs), nonlocal parameter and the angular frequency of oscillation of the heat flux on the lateral vibration, the temperature, and the axial displacement of the nanobeam are studied.

• Steel and Composite Structures
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• v.20 no.5
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• pp.963-981
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• 2016

A nonlocal trigonometric shear deformation beam theory based on neutral surface position is developed for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The present model is capable of capturing both small scale effect and transverse shear deformation effects of FG nanobeams, and does not require shear correction factors. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived by employing Hamilton's principle, and the physical neutral surface concept. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.