Reza Nazemnezhad

Free axial vibration of axially functionally graded (AFG) nanorods is studied by focusing on the inertia of lateral motions and shear stiffness effects. To this end, Bishop's theory considering the inertia of the lateral motions and shear stiffness effects and the nonlocal theory considering the small scale effect are used. The material properties are assumed to change continuously through the length of the AFG nanorod according to a power-law distribution. Then, nonlocal governing equation of motion and boundary conditions are derived by implementing the Hamilton's principle. The governing equation is solved using the harmonic differential quadrature method (HDQM), After that, the first five axial natural frequencies of the AFG nanorod with clamped-clamped end condition are obtained. In the next step, effects of various parameters like the length of the AFG nanorod, the diameter of the AFG nanorod, material properties, and the nonlocal parameter value on natural frequencies are investigated. Results of the present study can be useful in more accurate design of nano-electro-mechanical systems in which nanotubes are used. © 2018 Techno-Press, Ltd.

In this paper, nonlocal free longitudinal vibration of thick nanorods is investigated by focusing on the inertia of lateral motions and shear stiffness effects. To this end, Bishop and nonlocal theories are used. Then, by implementing the Hamilton's principle nonlocal governing equation of motion and boundary conditions are derived. The governing equation is solved analytically for fixed-fixed and fixed-free end conditions and the first five longitudinal natural frequencies of nanorod are obtained. In the next step, effects of various parameters like the length of nanorod, the diameter of nanorod and the nonlocal parameter value on natural frequencies are investigated. This study can be a useful reference for modeling of the multi-walled carbon nanotubes in which the interlayer shear plays a significant role in their various mechanical behaviors. © 2018 IOP Publishing Ltd.

The essence of nonlocal elasticity for vibration analysis of multi-layer graphene sheets (MLGSs) is investigated in the following study. The van der Waals interactions of every two adjacent layers are considered in analysis which results in interlayer shear effect. The proposed formulation is according to sandwich model (SM). Molecular Dynamic (MD) simulation is implemented to verify our model. We present an investigation to obtain the consistent values for SM parameters to comply with MD results. Afterward, the SM is integrated with nonlocal elasticity. The new calibrated values for nonlocal parameter lead to the best conformity of nonlocal SM with MD. © 2018 Elsevier Masson SAS

In this paper, surface energy and elastic medium effects on torsional vibrational behavior of nanorods are studied. The surface elasticity theory is used to consider the surface energy effects and the elastic medium is modeled as torsional springs attached to the nanorod. At the next step, Hamilton’s principle is utilized to derive governing equations and boundary conditions. Then, with the aid of an analytical method, natural frequencies are obtained and effects of various parameters on torsional frequencies are studied in details. It is concluded from the present study that the surface energy can make nanorods unstable depending on the nanorod dimension and frequency number. Results of the present study can be useful in design of nanoelectromechanical systems like drive shafts. © 2018 Materials and Energy Research Center. All rights reserved.

The following study addresses the free vibration analysis of a bilayer graphene sheet (BLGS) embedded in an elastic medium in the presence of shear and tensile-compressive effects of van der Waals (vdWs) interactions. To ascertain the contribution of each force, the effects are considered separately and simultaneously. To model the geometry of the BLGS, the sandwich plate theory and the Hamilton's principle are considered to derive the governing equations of motion. The Harmonic differential quadrature method is applied to solve the coupled equations and obtain the natural frequencies and related mode shapes. The results reveal that the contribution of tensile-compressive modulus of elastic medium is the most in changing the frequency of BLGSs. © 2018 IOP Publishing Ltd.

The shear and tensile-compressive effects of van der Waals (vdWs) bindings on the nonlocal vibrational behavior of bilayer graphene nanoribbons (BLGNRs) are simultaneously investigated in the present study. To idealize the structure of BLGNRs incorporating interlayer shear and tensile-compressive influences, a nonlocal sandwich beam (NSB) theory is employed to model the nanoribbon layers as faces and vdWs interactions as core of the NSB. The effects of interlayer moduli are investigated on the first four nonlocal natural frequencies of BLGNRs for various nonlocal parameters, considering that the small-scale effect causes mode shapes involving tensile-compressive effect of vdWs interactions be excited later. © 2018 Taylor & Francis Group, LLC.

This study focuses only on the interlayer effects of van der Waals (VdWs) interactions (including simultaneous effects of shear and tensile-compressive effects) on the free transverse vibrational behavior of bilayer graphene sheets by implementing the classical continuum mechanics theory. To this end, the classical sandwich plate theory and the Hamilton’s principle are involved to obtain the governing equations and the harmonic differential quadrature method is employed to calculate the natural frequencies and related mode shapes. The results show the shear effect of VdWs interactions has significant influences on primary natural frequencies and mode shapes. Therefore it is a main determinant and can safely assume the pure shear effect while designing sensors, actuators, accelerometers and resonators. Finally, the potential depth parameter is introduced to consider the simultaneous effects of shear and tensile-compressive forces. © 2018, The Brazilian Society of Mechanical Sciences and Engineering.

The elastic moduli of the bulk and surface as well as the equivalent mass density are applied to the continuum model of a nanobeam to predict the free longitudinal vibrations of aluminum (Al) nanobeams, especially in cases in which applying the classical theories results in significant errors- i.e. the cross section area of the nanobeams is lower than 4 × 4 nm2. To this end, the bulk elastic modulus and the surface elastic modulus of the Al nanobeam are extracted using surface elasticity theory, a bulk-surface model and molecular dynamics (MD) simulation. These elastic moduli are extracted by several strain values and the results show that the surface elastic modulus is very sensitive to the strain value, so that when the strain value increases from 0.001 to 0.02, the surface elastic modulus decreases significantly while the bulk elastic modulus shows fewer changes. Moreover, the size dependency of the mass density in the continuum nanobeam models is investigated in detail. Also, the natural frequencies of the Al nanobeams are obtained by MD simulation and they are compared with those obtained by the continuum model. Comparisons of the results show that the extracted bulk and surface elastic moduli are acceptable, and that the modified continuum model can predict the dynamic behavior of the Al nanobeam. © 2017 IOP Publishing Ltd.

Van der Waals (vdWs) bindings acting as interlayer shear force between graphene layers of multi-layer graphene sheets (MLGSs) are considered in vibration analysis in the present study. To idealize the structure of MLGS incorporating interlayer shear interactions, a sandwich model (SM) is represented which laminates the graphene layers. The layers stick together with vdWs bonds. The bonds are modeled as core layers between every two adjacent layers. Molecular dynamic (MD) simulation is carried out to validate the results obtained by SM. Afterward, the values for bending rigidity and layers thicknesses are obtained so as to match SM frequencies with MD results. It is observed the SM can predict the vibration behavior of MLGSs well for different values of aspect ratio. The present paper deals with a new method of incorporating shear effect and a novel investigation of integrating SM with MD. © 2017 Elsevier Inc.

This paper investigates surface energy effects, including the surface shear modulus, the surface stress, and the surface density, on the free torsional vibration of nanobeams with a circumferential crack and various boundary conditions. To formulate the problem, the surface elasticity theory is used. The cracked nanobeam is modeled by dividing it into two parts connected by a torsional linear spring in which its stiffness is related to the crack severity. Governing equations and corresponding boundary conditions are derived with the aid of Hamilton’s principle. Then, natural frequencies are obtained analytically, and the influence of the crack severity and position, the surface energy, the boundary conditions, the mode number, and the dimensions of nanobeam on the free torsional vibration of nanobeams is studied in detail. Results of the present study reveal that the surface energy has completely different effects on the free torsional vibration of cracked nanobeams compared with its effects on the free transverse vibration of cracked nanobeams. © 2017, Shanghai University and Springer-Verlag Berlin Heidelberg.

In this paper, nonlinear free vibration of nanobeams with various end conditions is studied using the nonlocal elasticity within the frame work of Euler-Bernoulli theory with von Kármán nonlinearity. The equation of motion is obtained and the exact solution is established using elliptic integrals. Two comparison studies are carried out to demonstrate accuracy and applicability of the elliptic integrals method for nonlocal nonlinear free vibration analysis of nanobeams. It is observed that the phase plane diagrams of nanobeams in the presence of the small scale effect are symmetric ellipses, and consideration the small scale effect decreases the area of the diagram.

This study comprehensively investigates tensile-compressive and shear effects of van der Waals (vdWs) interactions on free vibration of cantilever bilayer graphene nanoribbons (CBGNRs) to be answered to this question that how and how much the effects of each of these factors are in comparison with the other one when they are considered simultaneously. To this end, the CBGNRs are modeled based on sandwich beam theory in which each nanoribbon plays role of sandwich layer and vdWs interactions are equivalent to the sandwich core. At the first step a geometrical–analytical method is presented to calculate the equivalent tensile-compressive and shear moduli of vdWs interactions. After that, a set of coupled governing equations of motion and boundary conditions are derived and solved numerically by the harmonic differential quadrature method. This study shows that for designing multi-layer GNR based applications, such as resonators, the shear effect of vdWs interactions must be considered. © 2016, Springer Science+Business Media Dordrecht.

A new study is conducted with the aid of molecular dynamics (MD) simulation to investigate the effect of shear modulus value of the interlayer van der Waals (vdWs) interactions on free vibration of cantilever multi-layer graphene nanoribbons (MLGNRs). The corresponding calibrated nonlocal parameters of the nonlocal model are obtained accordingly. The vdWs interactions are treated as the cores between every two adjacent graphene layers and their equivalent shear modulus is calculated using MD simulation. The obtained resonant frequencies via the nonlocal sandwich model are compared to the MD simulation results to calibrate the nonlocal parameter. Results reveal a strong conclusion that the calibrated nonlocal parameter is dependent on the values of interlayer shear modulus. © 2016 Elsevier Ltd

In this paper, the buckling analysis of FG circular/annular nanoplates under uniform in-plane radial compressive load with a concentric internal ring support and elastically restrained edges is studied using an exact analytical approach within the framework of nonlocal Mindlin plate theory. The material properties vary according to a power-law distribution of the volume fraction of the constituents whereas Poison's ratio is set to be constant. In solving this problem, the circular/annular FG nanoplate is first divided into an annular segment and a core circular/annular segment at the location of the internal ring support; accordingly solutions for two segments brought together by using the interfacial conditions. It is observed that an internal ring support can increase the buckling capacity, accordingly this capacity is maximized when the internal ring support is located at an optimal position. Furthermore, the effects of small scales on the maximum buckling load corresponding to the optimal radius of the internal ring support are investigated for various parameters such as radius and thickness of the FG nanoplate, boundary conditions, mode numbers and material properties. © 2015 Elsevier Inc.

The aim of this study is to investigate the interlayer shear effect on nonlocal free vibration of multilayer graphene nanoribbons (MLGNRs) based on Reddy beam theory. The major novelty of the study is using the higher order shear deformation theory of Reddy to take the van der Waals' interaction of the layers into account and calibrating the nonlocal parameter with the aids of molecular dynamics (MD) simulations to match the best results. Furthermore, the shear correction factor of nonlocal Timoshenko theory is calculated for modeling the MLGNRs. MLGNRs have broad applications in mechanical devices, such as resonators, sensors and actuators. Accordingly the present results can be used as a reference for future works in which the interlayer shear effects has substantial influences on the mechanical behavior of MLGNRs. The present study outstands for its modeling simplicity and low computational time and cost as well as excellent accuracy. © 2015 Elsevier Masson SAS. All rights reserved.

In this study, shear effect of van der Waals (vdWs) interactions on free vibration of cantilever multi-layer graphene nanoribbons (MLGNRs) is investigated by using nonlocal Timoshenko beam model and molecular dynamics (MD) simulations. To this end, it is assumed that GNR layers of MLGNRs are perfectly bonded that no delamination will occur in layers interfaces. To calibrate the small scale parameter of the nonlocal Timoshenko model, the first two frequencies of armchair type cantilever MLGNRs with various layers and lengths are extracted using MD simulations and matched with those of the nonlocal Timoshenko theory. Comparing frequencies obtained by the MD simulation and the Timoshenko model shows that if values of the bending rigidity and the interlayer shear modulus (or equivalent shear modulus of Timoshenko theory) are taken as 1.20. eV and 3.01. GPa, respectively, then it will be possible to use the Timoshenko model for the free vibration analysis of MLGNRs. Moreover, it is observed that the calibrated nonlocal parameter is dependent on the number of MLGNR layers, and its calibrated value increases by increasing the number of GNR layers. This study helps researchers to analyze the mechanical behavior of MLGNRs in which the interlayer shear has apparent impact. © 2015 Elsevier Ltd.

This paper provides the axisymmetric/asymmetric buckling analysis of moderately thick circular and annular functionally graded (FG) nanoplates under uniform compressive in-plane loads. The material properties change continuously through the thickness of the FG nanoplate, which can vary according to a power-law distribution of the volume fraction of the constituents whereas Poison’s ratio is set to be constant. To reflect the small scale effects on the critical buckling loads of the Mindlin model, the nonlocal elasticity of Eringen is employed. An exact closed-form solution is used to solve the nonlocal governing equations. The effect of small scales on buckling loads for different parameters such as diameter of the FG nanoplate, boundary conditions, material properties and mode numbers are investigated in details. It is found that the modes corresponding to the lowest critical buckling loads for circular and annular FG nanoplates may be axisymmetric or asymmetric. Therefore, the asymmetric modes as well as axisymmetric ones have been considered here. © 2015, Springer Science+Business Media Dordrecht.

In this paper, the natural frequencies of a functionally graded nanoplate are analyzed for different combinations of boundary conditions. Application of new materials and specially the functionally graded materials in the micro- and nano-scale devices and systems is increasingly spread. Therefore, the study of the natural frequencies of functionally graded materials for different boundary conditions seems to be necessary in the micro/nano-structures. The article presented here covers broad types of common boundary conditions for the free vibration of functionally graded rectangular nanoplates for the first time. The analytical solution method used here is new for this subject and solves the governing equations with no approximation. The size dependency is considered according to Eringen’s differential form of nonlocal elasticity theory. The elasticity modulus and mass density of the plate are varied along the thickness of the plate according to a power-law distribution of the constituents’ volume fractions. As the in-plane and out-of-plane displacement variables are coupled in the equations of motion, a new exact solution method is introduced to solve the displacement fields analytically. The method is capable of dealing with new combinations of boundary conditions which have not been studied before in the literature. The validity and accuracy of the present method is investigated by comparing some of the present results to their counterparts reported in the literature. The results presented here are new and discussed for the first time in this subject. As a novelty a detailed study is carried out to examine the effects of power-law distribution, the characteristic internal length, the plate aspect ratio and the mode number on the natural frequencies of functionally graded rectangular plates for different boundary conditions. It’s shown that the type of boundary condition affects considerably on the values of natural frequency but the behavior of the frequency variations is in the same manner for different combinations of boundary conditions. These results can be used as benchmark for future studies. © 2015, Springer Science+Business Media Dordrecht.

In this study, influences of balance condition and surface density in addition to the surface elasticity and stress on the nonlinear free vibration of Timoshenko and Euler–Bernoulli nanobeams are investigated. In order to consider the balance condition between the nanobeam bulk and its surfaces, it is assumed that the normal stress varies linearly along the nanobeam thickness. Accordingly, the surface density in addition to the surface stress and elasticity is introduced into the governing equations. Moreover, besides using the bulk density, the surface density is also employed to obtain the kinetic energy of the nanobeams. The multiple scale method is used to obtain an analytical solution for the nonlinear governing equations. This results in the modal response frequencies (the amplitude dependence of the response frequencies due to the nonlinearity) of nanobeams. It is observed that considering the surface density effect and satisfying the balance condition cause a reduction in the frequency ratios, and this reduction is a little more for Timoshenko nanobeams than Euler–Bernoulli ones for all types of boundary conditions used. It is believed that this work is a comprehensive study for investigating effects of all components of the surface energy on the linear and nonlinear free vibrations of Timoshenko and Euler–Bernoulli nanobeams with different boundary conditions while the balance condition is also satisfied. © 2014, Springer Science+Business Media Dordrecht.

In this paper, nonlinear free vibration analysis of simply-supported nanoscale beams incorporating surface effects, i.e. surface elasticity, surface tension and surface density, is studied using the nonlocal elasticity within the frame work of Euler-Bernoulli beam theory with von kármán type nonlinearity. A linear variation for the component of the bulk stress, σzz, through the nanobeam thickness is used to satisfy the balance conditions between the nanobeam bulk and its surfaces. An exact analytical solution to the governing equation of motion is presented for natural frequencies of nanobeams using elliptic integrals. The effect of the nanobeam length, thickness to length ratio, mode number, amplitude of deflection to radius of gyration ratio and nonlocal parameter on the normalized natural frequencies of aluminum and silicon nanobeams with positive and negative surface elasticity, respectively, is investigated. It is observed that the surface effects increase natural frequencies of the aluminum nanobeam for all values of the amplitude ratio and the silicon nanobeam at low amplitude ratios while at higher amplitude ratios the surface effects decrease the natural frequencies of the silicon nanobeam. Also, for all values of amplitude ratios, the normalized fundamental natural frequencies of silicon and aluminum nanobeams vary linearly with respect to the nonlocal parameter while this is not the case at higher mode numbers. © 2015 Elsevier Masson SAS.

Free vibration of cantilever multi-layer graphene nanoribbons (MLGNRs) with interlayer shear effect is investigated using molecular dynamics simulations (MD) and nonlocal elasticity. Because of similarity of MLGNRs to sandwich structures, sandwich formulations are expressed in the nonlocal form. By comparing the first two frequencies of MLGNRs with various layers and lengths obtained using MD simulations with those of the nonlocal sandwich formulation; the nonlocal parameter is calibrated to match the results of two methods. The results reveal that the calibrated nonlocal parameter for predicting the second frequencies is dependent on the number of MLGNR layers, and it increases by increasing the number of layers. © 2014 Elsevier B.V. All rights reserved.

Free vibration of functionally graded material (FGM) nanobeams is investigated by considering surface effects including surface elasticity, surface stress, and surface density as well as the piezoelectric field using nonlocal elasticity theory. The balance conditions between the nanobeam bulk and its surfaces are satisfied assuming a cubic variation for the normal stress, σ z z, through the piezoelectric FG nanobeam thickness. Accordingly, the surface density is introduced into the governing equation of the free vibration of nanobeams. The results are obtained for various gradient indices, voltage values of the piezoelectric field, nanobeam lengths, and mode numbers. It is shown that making changes to voltage values and modifying mechanical properties of piezoelectric FGM nanobeams are two main approaches to achieve desired natural frequencies. © 2013 Springer-Verlag Wien.

Using an exact analytical approach, free vibration analysis of thick circular/annular FG Mindlin nanoplates is investigated in this paper. Eringen nonlocal elasticity theory is employed to consider small scale effects on natural frequencies. The edges of the nanoplate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. The material properties change continuously through the thickness of the FG nanoplate, which can vary according to a power-law distribution of the volume fraction of the constituents whereas Poison's ratio is set to be constant. In order to confirm the reliability of the method considered, the results are compared with those presented in literature. Also the effects of various parameters such as radius of the nanoplate, boundary conditions, material properties, mode number and nonlocal parameter on the natural frequencies are investigated. © 2013 Elsevier Ltd.