• Advances in materials Research
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• v.5 no.4
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• pp.245-261
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• 2016

Thermo-mechanical buckling problem of functionally graded (FG) nanoplates supported by Pasternak elastic foundation subjected to linearly/non-linearly varying loadings is analyzed via the nonlocal elasticity theory. Two opposite edges of the nanoplate are subjected to the linear and nonlinear varying normal stresses. Elastic properties of nanoplate change in spatial coordinate based on a power-law form. Eringen's nonlocal elasticity theory is exploited to describe the size dependency of nanoplate. The equations of motion for an embedded FG nanoplate are derived by using Hamilton principle and Eringen's nonlocal elasticity theory. Navier's method is presented to explore the influences of elastic foundation parameters, various thermal environments, small scale parameter, material composition and the plate geometrical parameters on buckling characteristics of the FG nanoplate. According to the numerical results, it is revealed that the proposed modeling can provide accurate results of the FG nanoplates as compared some cases in the literature. Numerical examples show that the buckling characteristics of the FG nanoplate are related to the material composition, temperature distribution, elastic foundation parameters, nonlocality effects and the different loading conditions.

• Structural Engineering and Mechanics
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• v.69 no.2
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• pp.131-143
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• 2019

In this study, the nonlinear vibration analysis of the composite nanoplate is studied. The composite nanoplate is fabricated by the functional graded (FG) core and lipid face sheets. The material properties in the FG core vary in three directions. The Kelvin-Voigt model is used to study the viscoelastic effect of the lipid layers. By using the Von-Karman assumptions, the nonlinear differential equation of the vibration analysis of the composite nanoplate is obtained. The foundation of the system is modeled by the nonlinear Pasternak foundation. The Bubnov-Galerkin method and the multiple scale method are used to solve the nonlinear differential equation of the composite nanoplate. The free and force vibration analysis of the composite nanoplate are studied. A comparison between the presented results and the reported results is done and good achievement is obtained. The reported results are verified by the results which are obtained by the Runge-Kutta method. The effects of different parameters on the nonlinear vibration frequencies, the primary, the super harmonic and subharmonic resonance cases are investigated. This work will be useful to design the nanosensors with high biocompatibility.

• Structural Engineering and Mechanics
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• v.73 no.3
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• pp.225-238
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• 2020

This work treats the axisymmetric buckling of functionally graded (FG) porous annular/circular nanoplates based on modified couple stress theory (MCST). The nanoplate is located at the elastic medium which is simulated by Kerr foundation with two spring and one shear layer. The material properties of the porous FG nanostructure are assumed to vary through the nanoplate thickness based on power-law rule. Based on two variables refined plate theory, the governing equations are derived by utilizing Hamilton's principle. Applying generalized differential quadrature method (GDQM), the buckling load of the annular/circular nanoplates is obtained for different boundary conditions. The influences of different involved parameters such as boundary conditions, Kerr medium, material length scale parameter, geometrical parameters of the nanoplate, FG power index and porosity are demonstrated on the nonlinear buckling load of the annular/circular nanoplates. The results indicate that with increasing the porosity of the nanoplate, the nonlinear buckling load is decreased. In addition, with increasing the material length scale parameter to thickness ratio, the effect of spring constant of Kerr foundation on the buckling load becomes more prominent. The present results are compared with those available in the literature to validate the accuracy and reliability. A good agreement is observed between the two sets of the results.

• Steel and Composite Structures
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• v.33 no.2
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• pp.307-318
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• 2019

This is the first attempt to consider the nonlinear bending analysis of porous functionally graded (FG) thick annular and circular nanoplates resting on Kerr foundation. The size effects are captured based on modified couple stress theory (MCST). The material properties of the porous FG nanostructure are assumed to vary smoothly through the thickness according to a power law distribution of the volume fraction of the constituent materials. The elastic medium is modeled by Kerr elastic foundation which consists of two spring layers and one shear layer. The governing equations are extracted based on Hamilton's principle and two variables refined plate theory. Utilizing generalized differential quadrature method (GDQM), the nonlinear static behavior of the nanostructure is obtained under different boundary conditions. The effects of various parameters such as material length scale parameter, boundary conditions, and geometrical parameters of the nanoplate, elastic medium constants, porosity and FG index are shown on the nonlinear deflection of the annular and circular nanoplates. The results indicate that with increasing the material length scale parameter, the nonlinear deflection is decreased. In addition, the dimensionless nonlinear deflection of the porous annular nanoplate is diminished with the increase of porosity parameter. It is hoped that the present work may provide a benchmark in the study of nonlinear static behavior of porous nanoplates.

• Steel and Composite Structures
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• v.32 no.2
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• pp.281-292
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• 2019

Nonlocal elasticity and Reddy plant theory are used to study the vibration response of functionally graded (FG) nanoplates resting on two parameters elastic medium called Pasternak foundation. Nonlocal higher order theory accounts for the effects of both scale and the effect of transverse shear deformation, which becomes significant where stocky and short nanoplates are concerned. It is assumed that the properties of FG nanoplate follow a power law through the thickness. In addition, Poisson's ratio is assumed to be constant in this model. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction of nanoplate with surrounding elastic medium. Using Hamilton's principle, size-dependent governing differential equations of motion and corresponding boundary conditions are derived. A differential quadrature approach is being utilized to discretize the model and obtain numerical solutions for various boundary conditions. The model is validated by comparing the results with other published results.

• Structural Engineering and Mechanics
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• v.88 no.4
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• pp.335-354
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• 2023

In this study, analysis of wave propagation characteristics for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates is performed using first-order shear deformation theory (FSDT) and nonlocal strain gradient theory. Uniform distribution (UD) and three types of functionally graded distributions of carbon nanotubes (CNTs) are assumed. The effective mechanical properties of the FG-CNTRC nanoplate are assumed to vary continuously in the thickness direction and are approximated based on the rule of mixture. Also, the governing equations of motion are derived via the extended Hamilton's principle. In numerical examples, the effects of nonlocal parameter, wavenumber, angle of wave propagation, volume fractions, and carbon nanotube distributions on the wave propagation characteristics of the FG-CNTRC nanoplate are studied. As represented in the results, it is clear that the internal length-scale parameter has a remarkable effect on the wave propagation characteristics resulting in significant changes in phase velocity and natural frequency. Furthermore, it is observed that the strain gradient theory yields a higher phase velocity and frequency compared to those obtained by the nonlocal strain gradient theory and classic theory.

• Structural Engineering and Mechanics
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• v.66 no.2
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• pp.249-262
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• 2018

In this paper, reaction of functionally graded (FG) thick nanoplates resting on a viscoelastic foundation to a moving nanoparticle/load is investigated. Nanoplate is assumed to be thick by using second order shear deformation theory and small-scale effects are taken into account in the framework of Eringen's nonlocal theory. Material properties are varied through the thickness using FG models by having power-law, sigmoid and exponential functions for material changes. FG nanoplate is assumed to be on a viscoelastic medium which is modeled using Kelvin-Voight viscoelastic model. Galerkin, state space and fourth-order Runge-Kutta methods are employed to solve the governing equations. A comprehensive parametric study is presetned to show the influence of different parameters on mechanical behavior of the system. It is shown that material variation in conjunction with nonlocal term have a significant effect on the dynamic deformation of nanoplate which could be used in comprehending and designing more efficient nanostructures. Moreover, it is shown that having a viscoelastic medium could play an important role in decreasing these dynamic deformations. With respect to the fresh studies on moving atoms, molecules, cells, nanocars, nanotrims and point loads on different nanosctructures using scanning tunneling microscopes (STM) and atomic force microscopes (AFM), this study could be a step forward in understanding, predicting and controlling such kind of behaviors by showing the influence of the moving path, velocity etc. on dynamic reaction of the plate.

• Structural Engineering and Mechanics
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• v.60 no.4
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• pp.707-727
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• 2016

In this paper, thermal vibration of a nonlocal functionally graded (FG) plates with arbitrary boundary conditions under linear and non-linear temperature fields is explored by developing a refined shear deformation plate theory with an inverse cotangential function in which shear deformation effect was involved without the need for shear correction factors. The material properties of FG nanoplate are considered to be temperature-dependent and graded in the thickness direction according to the Mori-Tanaka model. On the basis of non-classical higher order plate model and Eringen's nonlocal elasticity theory, the small size influence was captured. Numerical examples show the importance of non-uniform thermal loadings, boundary conditions, gradient index, nonlocal parameter and aspect and side-to-thickness ratio on vibrational responses of size-dependent FG nanoplates.

• Advances in nano research
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• v.7 no.5
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• pp.325-335
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• 2019

In the present study, nonlocal strain gradient theory (NSGT) is developed for wave propagation of functionally graded (FG) nanoscale plate in the thermal environment by considering the porosity effect. $Si_3N_4$ as ceramic phase and SUS304 as metal phase are regarded to be constitutive material of FG nanoplate. The porosity effect is taken into account on the basis of the newly extended method which considers coupling influence between Young's modulus and mass density. The motion relation is derived by applying Hamilton's principle. NSGT is implemented in order to account for small size effect. Wave frequency and phase velocity are obtained by solving the problem via an analytical method. The effects of different parameters such as porosity coefficient, gradient index, wave number, scale factor and temperature change on phase velocity and wave frequency of FG porous nanoplate have been examined and been presented in a group of illustrations.

• Structural Engineering and Mechanics
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• v.64 no.6
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• pp.683-693
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• 2017

According to a generalized nonlocal strain gradient theory (NSGT), dynamic modeling and free vibrational analysis of nanoporous inhomogeneous nanoplates is presented. The present model incorporates two scale coefficients to examine vibration behavior of nanoplates much accurately. Porosity-dependent material properties of the nanoplate are defined via a modified power-law function. The nanoplate is resting on a viscoelastic substrate and is subjected to hygro-thermal environment and in-plane linearly varying mechanical loads. The governing equations and related classical and non-classical boundary conditions are derived based on Hamilton's principle. These equations are solved for hinged nanoplates via Galerkin's method. Obtained results show the importance of hygro-thermal loading, viscoelastic medium, in-plane bending load, gradient index, nonlocal parameter, strain gradient parameter and porosities on vibrational characteristics of size-dependent FG nanoplates.