Effects Of Nonlocal Parameter Research Articles

Propagation of in-plane wave in viscoelastic monolayer graphene via nonlocal strain gradient theory

The behaviors of monolayer graphene sheet have attracted increasing attention of many scientists and researchers. In this study, the propagation behaviors of in-plane wave in viscoelastic monolayer graphene are investigated. The constitutive equation and governing equation for in-plane wave propagation is developed by employing Hamilton’s principle and nonlocal strain gradient theory. By solving the governing equation of motion, the closed-form dispersion relation between phase velocity and wave number is derived and an asymptotic phase velocity can be acquired. The effects of wave number, material length scale parameter, nonlocal parameter and damping coefficient on in-plane wave propagation behaviors are discussed in the numerical studies. It is found that, when exciting wavelengths or structural dimensions become comparable to the material length scale parameters and nonlocal parameters, the scaling effects on wave propagation behaviors are significant. For nanoscaled graphene sheet, the effects of nonlocal parameter, material length scale parameter and damping coefficient on phase velocity are tiny at low wave numbers while significant at high wave numbers. The phase velocity would increase with the increase of material length scale parameter or the decrease of nonlocal parameter and damping coefficient. Furthermore, results indicate that the asymptotic phase velocity can be increase by increasing material length scale parameter or decreasing nonlocal parameter.

Surface layer and nonlocal parameter effects on the in-phase and out-of-phase natural frequencies of a double-layer piezoelectric nanoplate under thermo-electro-mechanical loadings

The in-phase and out-of-phase natural frequencies of double-layer piezoelectric nanoplates (DPNP) are studied under thermo-electro-mechanical loadings using the surface layer and nonlocal elasticity theories. For this purpose, two nanoplates are coupled via an elastic medium that induces van der Waals forces while the DPNP is surrounded by the Winkler and shear moduli of elastic foundations. Both in-phase and out-of-phase fundamental frequencies are investigated and compared under different boundary conditions. The Hamilton’s principle is employed to derive the governing equations and the finite difference method is subsequently used to solve them. The finite difference solution thus obtained is validated against the results obtained from the Navier’s method. A different aspect of the study involves the investigation of the effects of nonlocal parameters, thermo-electro-mechanical forces, various boundary conditions, and elastic foundations on the surface layer vibration of a DPNP. It is observed that the effects of surface energy layers on the in-phase and out-of-phase vibrations increase with increasing values of the nonlocal parameter and external electric voltage but that they reduce with rising mechanical and thermal loadings, elastic foundation, and stiffness for the boundary conditions defined. The results of the present work are expected to be used in designing NEMS/MEMS components using smart composite nanostructures.

Vibration analysis of quadrilateral graphene sheets subjected to an in-plane magnetic field based on nonlocal elasticity theory

Graphene sheets have wide application, in which the integration of multiple disparate fields for the realization of expanded functionalities is of great significance. This study investigates the vibration behavior of quadrilateral single-layered graphene sheets (SLGSs) in a magnetic field using classic plate theory and incorporating nonlocal elasticity theory, concerning the small scale effect. The element-free kp-Ritz method is employed to perform the numerical simulation. The efficiency of the proposed numerical tool is verified by published results. The effect of nonlocal parameters, skew angles, magnetic parameters and boundary conditions on the vibration behavior of parallelogram SLGSs is studied. The results show that skew angles and the magnetic field help increase the fundamental frequencies of SLGSs, which indicates the potential application of SLGSs as highly sensitive mass detectors. Moreover, 14 different quadrilateral SLGSs, with different nonlocal parameters and magnetic parameters for different boundary conditions, are simulated.

A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams

In this paper, size dependent bending and free flexural vibration behaviors of functionally graded (FG) nanobeams are investigated using a nonlocal quasi-3D theory in which both shear deformation and thickness stretching effects are introduced. The nonlocal elastic behavior is described by the differential constitutive model of Eringen, which enables the present model to become effective in the analysis and design of nanostructures. The present theory incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect, and furthermore accounts for both shear deformation and thickness stretching effects by virtue of a hyperbolic variation of all displacements through the thickness without using shear correction factor. The material properties of FG nanobeams are assumed to vary through the thickness according to a power law. The neutral surface position for such FG nanobeams is determined and the present theory based on exact neutral surface position is employed here. The governing equations are derived using the principal of minimum total potential energy. The effects of nonlocal parameter, aspect ratio and various material compositions on the static and dynamic responses of the FG nanobeam are discussed in detail. A detailed numerical study is carried out to examine the effect of material gradient index, the nonlocal parameter, the beam aspect ratio on the global response of the FG nanobeam. These findings are important in mechanical design considerations of devices that use carbon nanotubes.

Nonlocal Vibration Analysis for Micro/Nano Beam on Winkler Foundation via DTM

In the present study, vibration of micro/nano beams on Winkler foundation is studied using Eringen's nonlocal elasticity theoy. Hamilton’s principle is employed to derive the governing equations. Differential transform method is used to obtain result. Simply supported and clamped–clamped boundary conditions are used to study natural frequencies. The effect of nonlocal parameter and Winkler elastic foundation modulus on the natural frequencies of the nonlocal Euler-Bernoulli beam is investigated and tabulated. The differential transform method is applicable for micro/nano beams and gives high accuracy results.

Numerical study on the free vibration of carbon nanocones resting on elastic foundation using nonlocal shell model

Employing the variational differential quadrature method, the free vibration of carbon nanocones (CNCs) embedded in an elastic foundation, is studied based on nonlocal elasticity theory. On the basis of the first-order shear deformation theory, the energy functional of the CNC is presented and then discretized by employing the generalized differential quadrature method in the axial direction and periodic differential operators in the circumferential direction. According to Hamilton’s principle and using matrix relations, the reduced forms of mass and stiffness matrices are readily obtained. The results of present study are compared to those obtained by molecular mechanics to verify the proposed approach. In addition, the effects of nonlocal parameter, boundary conditions, semi-apex angle and both Winkler and Pasternak coefficients of elastic foundation are examined on the vibrational behavior of CNCs. The results indicate that the increase in nonlocal parameter and elastic foundation coefficients decreases and increases the fundamental frequency of CNCs, respectively.

Nonlocal transient electrothermomechanical vibration and bending analysis of a functionally graded piezoelectric single-layered nanosheet rest on visco-Pasternak foundation

ABSTRACTThis study develops the transient thermoelectromechanical vibration and bending analysis of a functionally graded piezoelectric nanosheet rest on visco-Pasternak’s foundation. Nonlocal elasticity theory as well as classical plate theory is used to implement basic equations of the nanosheet. The plate is resting on visco-Pasternak’s foundation and subject to mechanical, thermal, and electrical loadings. Hamilton’s principle is used for derivation equations of motion in terms of displacement components. As a first case study and for validation of the responses of the system, nonlocal vibration analysis of nanosheet is studied. The effects of nonlocal parameter and nonhomogeneous index of nanosheet are studied on the fundamental frequencies of the system. As the main objective of this study, the electrothermal bending results of the nanosheet are studied. The effects of some important parameters such as nonlocal parameter, nonhomogeneous index, thickness, distribution of electric potential, and damping are calculated on the maximum deflection of the sheet under various thermal and electrical loadings.

A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams

This paper investigates buckling characteristics of a curved functionally graded (FG) nanobeam based on nonlocal strain gradient elasticity theory accounting the stress for not only the nonlocal stress field but also the strain gradients stress field. The modeling of nanobeam is carried out via a higher order refined beam theory which captures shear deformation influences needless of any shear correction factor. Power-law model is adopted to describe continuous variation of material properties of curved FG nanobeam. The governing equations of nonlocal strain gradient curved FG nanobeam in the framework of refined hyperbolic beam model are obtained using Hamilton’s principle and solved implementing an analytical solution for simply-supported and clamped boundary conditions. To validate the present model, the results are compared with those of straight FG nanobeams by extending the radius of nanobeam to infinity. The effects of nonlocal parameter, length scale parameter, power-law exponent, boundary conditions and slenderness ratio on the buckling response of curved FG nanobeams are investigated.

Nonlocal free vibration in the pre- and post-buckled states of magneto-electro-thermo elastic rectangular nanoplates with various edge conditions

Considering the small scale effect together with the influences of transverse shear deformation, rotary inertia and the magneto-electro-thermo-mechanical coupling, the linear free vibration of magneto-electro-thermo-elastic (METE) rectangular nanoplates with various edge supports in pre- and post-buckled states is investigated herein. It is assumed that the METE nanoplate is subjected to the external in-plane compressive loads in combination with magnetic, electric and thermal loads. The Mindlin plate theory, von Kármán hypothesis and the nonlocal theory are utilized to develop a size-dependent geometrically nonlinear plate model for describing the size-dependent linear and nonlinear mechanical characteristics of moderately thick METE rectangular nanoplates. The nonlinear governing equations and the corresponding boundary conditions are derived using Hamilton’s principle which are then discretized via the generalized differential quadrature method. The pseudo-arc length continuation approach is used to obtain the equilibrium postbuckling path of METE nanoplates. By the obtained postbuckling response, and taking a time-dependent small disturbance around the buckled configuration, and inserting them into the nonlinear governing equations, an eigenvalue problem is achieved from which the frequencies of pre- and post-buckled METE nanoplates can be calculated. The effects of nonlocal parameter, electric, magnetic and thermal loadings, length-to-thickness ratio and different boundary conditions on the free vibration response of METE rectangular nanoplates in the pre- and post-buckled states are highlighted.

Size-dependent nonlinear vibration analysis of carbon nanotubes conveying multiphase flow

An understanding of the dynamic behavior of the carbon nanotubes (CNTs) conveying fluid is very important for exploring the applications in nanoscale systems. Due to the molecular network disruption, the passing flow has multiphase nature. In this regard, the nonlinear vibration behavior of the CNT conveying multiphase flow is investigated by considering the small scale effects based on the nonlocal theory. The effect of the multiphase flow on the CNT's vibration behavior is modeled by the resultant random uncertainty in the external excitation along with considering the slip flow velocity profile. After extraction of the governing equation by implementing Hamilton's principle and discretizing it by the Galerkin method, the resulting equations are solved numerically. Due to the stochastic nature of the differential equations, the statistical parameters of the response have been obtained by Monte–Carlo simulation. By studying the deflection of the midpoint of the CNT and also considering corresponding upper and lower limit band (confidence interval), extended results of uncertainty effects have been obtained. Moreover the effect of nonlocal parameter, flow velocity and Knudsen number on the statistical dynamic behavior of the system have been investigated. The results show that as the molecular behavior of the flow increases the uncertainty in the system and the confidence interval increase.

Buckling analysis of biaxially compressed double-layered graphene sheets with various boundary conditions based on nonlocal elasticity theory

This paper investigates the buckling behavior of biaxially compressed orthotropic double-layered graphene sheet (DLGS) embedded in an elastic medium based on nonlocal elasticity theory. The DLGS is modeled as a nonlocal double-layered plate which contains small scale effect and van der Waals interaction forces. The van der Waals interactions between the graphene layers are simulated as a set of linear springs using the Lennard-Jones potential model. Using the principle of virtual work, equilibrium equations are derived based on first order shear deformation theory and the nonlocal differential constitutive relations of Eringen. Differential quadrature method is employed to solve the governing equations for various combinations of free, simply supported or clamped boundary conditions. The present formulation and method of solution are validated by comparing the results, in the limit cases, with those available in the open literature. Finally, the effects of nonlocal parameter, geometric properties, elastic medium, compression ratio, mode numbers and boundary conditions are investigated on the buckling load of DLGS in detail. It is observed that in DLGSs with width of 20 nm and lengthes larger than 30 nm, the buckling load is independent of aspect ratios and boundary conditions.

Buckling analysis of graphene sheets embedded in an elastic medium based on the kp-Ritz method and non-local elasticity theory

Graphene finds application in nanoscale electronic devices, nano sensors and nanocomposites in which graphene sheets usually rest on an elastic medium. Buckling of graphene sheets can occur in such structures, which significantly influences its performance. Therefore, it is essential to investigate the buckling behavior of graphene for better engineering design and manufacture. The present work provides an element-free kp-Ritz framework to analyze the buckling behavior of graphene. The element-free kp-Ritz method is first implemented to investigate the buckling behavior of single-layered graphene sheets (SLGSs) embedded in elastic foundations, based on the classical plate theory (CLP), incorporating the non-local elasticity theory, which takes small effects into account when dealing with nanostructures. A Winkler-type model is adopted to simulate the functioning of the elastic medium. Numerical solutions for critical buckling loads of SLGSs are obtained by solving the governing differential equations derived from the principle of minimum potential energy using the element-free kp-Ritz method. The concepts of non-local parameter effects and non-local effects are distinguished. The influence of non-local parameters for both side length and aspect ratio on critical buckling loads of SLGSs is analyzed. Additionally, the effects of non-local parameters and the Winkler modulus parameter on the buckling patterns are discussed.

Vibration analysis of orthotropic circular and elliptical nano-plates embedded in elastic medium based on nonlocal Mindlin plate theory and using Galerkin method

In the present study a continuum model based on the nonlocal elasticity theory is developed for free vibration analysis of embedded orthotropic thick circular and elliptical nano-plates rested on an elastic foundation. The elastic foundation is considered to behave like a Pasternak type of foundations. Governing equations for vibrating nano-plate are derived according to the Mindlin plate theory in which the effects of shear deformations of nano-plate are also included. The Galerkin method is then employed to obtain the size dependent natural frequencies of nano-plate. The solution procedure considers the entire nano-plate as a single super-continuum element. Effect of nonlocal parameter, lengths of nano-plate, aspect ratio, mode number, material properties, thickness and foundation on circular frequencies are investigated. It is seen that the nonlocal frequencies of the nano-plate are smaller in comparison to those from the classical theory and this is more pronounced for small lengths and higher vibration modes. It is also found that as the aspect ratio increases or the nanoplate becomes more elliptical, the small scale effect on natural frequencies increases. Further, it is observed that the elastic foundation decreases the influence of nonlocal parameter on the results. Since the effect of shear deformations plays an important role in vibration analysis and design of nano-plates, by predicting smaller values for fundamental frequencies, the study of these nano-structures using thick plate theories such as Mindlin plate theory is essential.

The effect of vacant defect on bending analysis of graphene sheets based on the Mindlin nonlocal elasticity theory

Abstract In the paper, effects of eccentric vacant defect on bending analysis of circular graphene sheets have been studied based on nonlocal elasticity theory considering the Mindlin theory of the plates. The governing equations have been derived for circular graphene sheet including eccentric vacant defect. Because of existence of an eccentric vacant defect on the plate, the anti-symmetric problem has been obtained. So, the constitutive equations are partial differential equations system. Considering this fact that there is not any analytical solution available for solving the nonlinear constitutive equations based on the first order shear deformation theory, a new semi-analytical polynomial method is applied which has been presented by the authors recently. By applying the mentioned method, the governing equations have been solved. Regarding this fact that no study has been done heretofore in case of the effects of eccentric vacant defect on bending analysis of graphene sheets, the obtained results of local analysis have been compared with the results of Abaqus software. At the following, the effects of nonlocal parameters, size and location of vacant defect on the results have been surveyed.

Coupled twist–bending static and dynamic behavior of a curved single-walled carbon nanotube based on nonlocal theory

The static and dynamic behavior of a curved single-walled carbon nanotube which is under twist–bending couple based on nonlocal theory is analyzed. The nonlocal theory is used to model the mechanical behavior of structure in small scale. The obtained differential equations are solved using a simply supported boundary condition and Navier analytical method. Moreover the twisted vibration and bending of curved nanotube is analyzed and also the armchair model is assumed in this study. The following parameters were studied in this paper: the effect of nonlocal parameter, the curved nanotube’s opening angel, the Young’s modulus and the mode number is studied. The results were verified with the previous literature which showed an excellent agreement. The results of this paper can be used as a benchmark for future investigations.

Nonlinear Buckling of Circular Nano Plates on Elastic Foundation

The following article investigates nonlinear symmetric buckling of moderately thick circular Nano plates with an orthotropic property under uniform radial compressive in-plane mechanical load. Taking into account Eringen nonlocal elasticity theory, principle of virtual work, first order shear deformation plate theory (FSDT) and nonlinear Von-Karman strains, the governing equations are obtained based on displacements. The differential quadrature method (DQM) as a numerical procedure is applied for solving the equations. In this analysis, for solving the stability equations, adjacent equilibrium methodis employed. In nonlinear buckling analyses and for obtaining the buckling load, generally the available nonlinear terms of the stability equation are neglected. However, in this study, for getting the most accurate data, nonlinear terms are considered and the non-dimensional buckling load is compared with the condition of considering or neglecting that of terms and the effect of that of terms are also studied. The accuracy of the present results is validated by comparing the solutions with available studies. The effects of nonlocal parameter, thickness, radiusand elastic foundation are investigated on non-dimensional buckling loads. The results of analyses based on local and non-local theories are compared. From the results, it can be seen that the effect of nonlocal parameter on simply support condition is less than clamped condition. It can be observed that with increasing the radius of the plate, the difference between local and non-local analyses,increases.

BENDING VIBRATION AND STABILITY OF A MULTIPLE-NANOBEAM SYSTEM INFLUENCED BY TEMPERATURE CHANGE

In this study, we analyzed the bending vibration and stability of a multiple-nanobeam system (MNBS) coupled in elastic medium and influenced by temperature change and compressive axial load. The MNBS is modeled as the system consisting of a set of m identical and simply supported nanobeams mutually connected by Winkler’s type elastic layers. According to the Euler - Bernoulli beam and nonlocal thermo-elasticity theory, the system of m coupled partial differential equations is derived and solved by means of the method of separation of variables as well as the trigonometric one. Analytical solutions for natural frequencies and critical buckling loads of elastic MNBS are obtained. The effects of nonlocal parameter, temperature change and the number of nanobeams on the natural frequencies and the buckling loads are investigated through numerical examples. Thus, this work can represent a starting point to examine dynamical behavior and design of complex nanobeam structures, nanocomposites and nanodevices under the influence of various physical fields.

Small scale effect on the pull-in instability and vibration of graphene sheets

In this paper, the small scale effect on the pull-in instability and frequency of graphene sheets subjected to electrostatic and van der Waals forces is studied. Eringen's nonlocal elasticity theory is used to account for the small scale effect, and the graphene sheet is modeled as a nonlocal plate. Reduce-order model incorporating nonlocal effect and van der Waals force is derived. Results show that nonlocal elasticity reduces the pull-in voltages and fundamental frequencies. The effect of nonlocal parameter on pull-in voltages and fundamental frequencies is more obvious if the size of nanoplate is smaller. In addition, an empirical formula for pull-in voltage is explored. The form of the empirical formula is relatively simple, therefore is convenient for engineers to indicate an experimental method to determine the nonlocal parameters of the materials and to design nano-switches.

Differential quadrature method for nonlocal nonlinear vibration analysis of a boron nitride nanotube using sinusoidal shear deformation theory

ABSTRACTThis article presents a nonlocal sinusoidal shear deformation beam theory (SDBT) for the nonlinear vibration of single-walled boron nitride nanotubes (SWBNNTs). The surrounding elastic medium is simulated based on nonlinear Pasternak foundation. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the SWBNNTs are derived using Hamilton's principle. Differential quadrature method (DQM) for the nonlinear frequency is presented, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory (TBT). The effects of nonlocal parameter, vibrational modes, length, and elastic medium on the nonlinear frequency of SWBNNTs are considered.

Nonlinear vibration and adhesion instability of Casimir-induced nonlocal nanowires with the consideration of surface energy

The following research work deals with the size-dependent dynamic instability of suspended nanowires in the presence of Casimir force and surface effects. Specifically, the Casimir-induced instability of nanostructures with circular cross-section and cylinder-plate geometry is studied. Following the Gurtin–Murdoch model and nonlocal elasticity, the governing equation of motion for nanowires is derived. To express the Casimir attraction of cylinder-plate geometry, two approaches, e.g. proximity force approximation (PFA) for small separations and Dirichlet asymptotic approximation for large separations are studied. To overcome the difficulties for solving a nonlinear problem, a step-by-step numerical method is utilized. The effects of nonlocal parameter, surface energy and vacuum fluctuations on the dynamic instability characteristic and adhesion time of nanowires are studied. It is observed that the phase portrait of Casimir-induced nanowires exhibit periodic and homoclinic orbits.