• Structural Engineering and Mechanics
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• v.64 no.4
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• pp.391-402
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• 2017

In this paper, a new nonlocal trigonometric shear deformation theory is proposed for thermal buckling response of nanosize functionally graded (FG) nano-plates resting on two-parameter elastic foundation under various types of thermal environments. This theory uses for the first time, undetermined integral variables and it contains only four unknowns, that is even less than the first shear deformation theory (FSDT). It is considered that the FG nano-plate is exposed to uniform, linear and sinusoidal temperature rises. Mori-Tanaka model is utilized to define the gradually variation of material properties along the plate thickness. Nonlocal elasticity theory of Eringen is employed to capture the size influences. Through the stationary potential energy the governing equations are derived for a refined nonlocal four-variable shear deformation plate theory and then solved analytically. A variety of examples is proposed to demonstrate the importance of elastic foundation parameters, various temperature fields, nonlocality, material composition, aspect and side-to-thickness ratios on critical stability temperatures of FG nano-plate.

• Advances in nano research
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• v.7 no.3
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• pp.191-208
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• 2019

In the present work the dynamic analysis of the functionally graded rectangular nanoplates is studied. The theory of nonlocal elasticity based on the quasi 3D high shear deformation theory (quasi 3D HSDT) has been employed to determine the natural frequencies of the nanosize FG plate. In HSDT a cubic function is employed in terms of thickness coordinate to introduce the influence of transverse shear deformation and stretching thickness. The theory of nonlocal elasticity is utilized to examine the impact of the small scale on the natural frequency of the FG rectangular nanoplate. The equations of motion are deduced by implementing Hamilton's principle. To demonstrate the accuracy of the proposed method, the calculated results in specific cases are compared and examined with available results in the literature and a good agreement is observed. Finally, the influence of the various parameters such as the nonlocal coefficient, the material indexes, the aspect ratio, and the thickness to length ratio on the dynamic properties of the FG nanoplates is illustrated and discussed in detail.

• Structural Engineering and Mechanics
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• v.70 no.1
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• pp.55-66
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• 2019

This work deals with the size-dependent wave propagation analysis of functionally graded (FG) anisotropic nanoplates based on a nonlocal strain gradient refined plate model. The present model incorporates two scale coefficients to examine wave dispersion relations more accurately. Material properties of FG anisotropic nanoplates are exponentially varying in the z-direction. In order to solve the governing equations for bulk waves, an analytical method is performed and wave frequencies and phase velocities are obtained as a function of wave number. The influences of several important parameters such as material graduation exponent, geometry, Winkler-Pasternak foundation parameters and wave number on the wave propagation of FG anisotropic nanoplates resting on the elastic foundation are investigated and discussed in detail. It is concluded that these parameters play significant roles on the wave propagation behavior of the nanoplates. From the best knowledge of authors, it is the first time that FG nanoplate made of anisotropic materials is investigated, so, presented numerical results can serve as benchmarks for future analysis of such structures.

• Steel and Composite Structures
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• v.49 no.4
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• pp.361-380
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• 2023

A size dependent bending behavior of piezoelectrical flexoelectric layered perforated functionally graded (FG) composite nanobeam rested on an elastic foundation is investigated analytically. The composite beam is composed of regularly cutout FG core and two piezoelectric face sheets. The material characteristics is graded through the core thickness by power law function. Regular squared cutout perforation pattern is considered and closed forms of the equivalent stiffness parameters are derived. The modified nonlocal strain gradient elasticity theory is employed to incorporate the microstructure as well as nonlocality effects into governing equations. The Winkler as well as the Pasternak elastic foundation models are employed to simulate the substrate medium. The Hamiltonian approach is adopted to derive the governing equilibrium equation including piezoelectric and flexoelectric effects. Analytical solution methodology is developed to derive closed forms for the size dependent electromechanical as well as mechanical bending profiles. The model is verified by comparing the obtained results with the available corresponding results in the literature. To demonstrate the applicability of the developed procedure, parametric studies are performed to explore influences of gradation index, elastic medium parameters, flexoelectric and piezoelectric parameters, geometrical and peroration parameters, and material parameters on the size dependent bending behavior of piezoelectrically layered PFG nanobeams. Results obtained revealed the significant effects both the flexoelectric and piezoelectric parameters on the bending behavior of the piezoelectric composite nanobeams. These parameters could be controlled to improve the size dependent electromechanical as well as mechanical behaviors. The obtained results and the developed procedure are helpful for design and manufacturing of MEMS and NEMS.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.30 no.5
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• pp.405-413
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• 2017

In this paper, we study the biaxial buckling analysis of nonlocal MEE(magneto-electro-elastic) nano plates based on the first-order shear deformation theory. The in-plane electric and magnetic fields can be ignored for MEE(magneto-electro-elastic) nano plates. According to magneto-electric boundary condition and Maxwell equation, the variation of magnetic and electric potentials along the thickness direction of the MME plate is determined. In order to reformulate the elastic theory of MEE(magneto-electro-elastic) nano-plate, the nonlocal differential constitutive relations of Eringen is used. Using the variational principle, the governing equations of the nonlocal theory are discussed. The relations between nonlocal and local theories are investigated by computational results. Also, the effects of nonlocal parameters, in-plane load directions, and aspect ratio on structural responses are studied. Computational results show the effects of the electric and magnetic potentials. These computational results can be useful in the design and analysis of advanced structures constructed from MEE(magneto-electro-elastic) materials and may be the benchmark test for the future study.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.15 no.2
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• pp.1109-1117
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• 2014

We study free vibration analysis of sigmoid functionally graded materials(S-FGM) nano-scale plates, using a nonlocal elasticity theory of Eringen in this paper. This theory has ability to capture the both small scale effects and sigmoid function in terms of the volume fraction of the constituents for material properties through the plate thickness. Numerical solutions of S-FGM nano-scale plate are presented using this theory to illustrate the effect of nonlocal theory on natural frequency of the S-FGM nano-scale plates. The relations between nonlocal and local theories are discussed by numerical results. Further, effects of (i) power law index (ii) nonlocal parameters, (iii) elastic modulus ratio and (iv) thickness and aspect ratios on nondimensional frequencies are investigated. In order to validate the present solutions, the reference solutions are compared and discussed. The results of S-FGM nano-scale plates using the nonlocal theory may be the benchmark test for the free vibration analysis.

• Advances in nano research
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• v.6 no.2
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• pp.163-182
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• 2018

Based on the Reissner mixed variational theorem (RMVT), the authors present a nonlocal Timoshenko beam theory (TBT) for the nonlinear free vibration analysis of multi-walled carbon nanotubes (MWCNT) embedded in an elastic medium. In this formulation, four different edge conditions of the embedded MWCNT are considered, two different models with regard to the van der Waals interaction between each pair of walls constituting the MWCNT are considered, and the interaction between the MWCNT and its surrounding medium is simulated using the Pasternak-type foundation. The motion equations of an individual wall and the associated boundary conditions are derived using Hamilton's principle, in which the von $K{\acute{a}}rm{\acute{a}}n$ geometrical nonlinearity is considered. Eringen's nonlocal elasticity theory is used to account for the effects of the small length scale. Variations of the lowest frequency parameters with the maximum modal deflection of the embedded MWCNT are obtained using the differential quadrature method in conjunction with a direct iterative approach.

• Advances in nano research
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• v.9 no.1
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• pp.33-45
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• 2020

Geometrically nonlinear buckling of functionally graded magneto-electro-elastic (FG-MEE) nanoshells with the use of classical shell theory and nonlocal strain gradient theory (NSGT) has been analyzed in present research. Mathematical formulation based on NSGT gives two scale coefficients for simultaneous description of structural stiffness reduction and increment. Functional gradation of material properties is described based on power-law formulation. The nanoshell is under a multi-physical field related to applied voltage, magnetic potential, and mechanical load. Exerting a strong electric voltage, magnetic potential or mechanical load may lead to buckling of nanoshell. Taking into account geometric nonlinearity effects after buckling, the behavior of nanoshell in post-buckling regime can be analyzed. Nonlinear governing equations are reduced to ordinary equations utilizing Galerkin's approach and post-buckling curves are obtained based on an analytical procedure. It will be shown that post-buckling curves are dependent on nonlocal/strain gradient parameters, electric voltage magnitude and sign, magnetic potential magnitude and sign and material gradation exponent.

• Wind and Structures
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• v.26 no.4
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• pp.205-214
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• 2018

In this paper, the thermo-mechanical buckling characteristics of functionally graded (FG) size-dependent Timoshenko nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution for the first time. 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 governing equations are derived based on Timoshenko beam theory through Hamilton's principle and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate critical buckling temperature results of the FG nanobeams as compared to 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 material distribution profile, small scale effects and aspect ratio on the critical buckling temperature of the FG nanobeams in detail. It is explicitly shown that the thermal buckling of a FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

• Smart Structures and Systems
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• v.22 no.1
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• pp.105-120
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• 2018

In this paper, the nonlinear free and forced vibration responses of sandwich nano-beams with three various functionally graded (FG) patterns of reinforced carbon nanotubes (CNTs) face-sheets are investigated. The sandwich nano-beam is resting on nonlinear Visco-elastic foundation and is subjected to thermal and electrical loads. The nonlinear governing equations of motion are derived for an Euler-Bernoulli beam based on Hamilton principle and von Karman nonlinear relation. To analyze nonlinear vibration, Galerkin's decomposition technique is employed to convert the governing partial differential equation (PDE) to a nonlinear ordinary differential equation (ODE). Furthermore, the Multiple Times Scale (MTS) method is employed to find approximate solution for the nonlinear time, frequency and forced responses of the sandwich nano-beam. Comparison between results of this paper and previous published paper shows that our numerical results are in good agreement with literature. In addition, the nonlinear frequency, force response and nonlinear damping time response is carefully studied. The influences of important parameters such as nonlocal parameter, volume fraction of the CNTs, different patterns of CNTs, length scale parameter, Visco-Pasternak foundation parameter, applied voltage, longitudinal magnetic field and temperature change are investigated on the various responses. One can conclude that frequency of FG-AV pattern is greater than other used patterns.