Eringen's Nonlocal Elasticity Theory Research Articles

A nonlocal quasi-3D trigonometric plate model for free vibration behaviour of micro/nanoscale plates

In this work, a nonlocal quasi-3D trigonometric plate theory for micro/nanoscale plates is proposed. In order to introduce the size influences, the Eringen's nonlocal elasticity theory is utilized. In addition, the theory considers both shear deformation and thickness stretching effects by a trigonometric variation of all displacements within the thickness, and respects the stress-free boundary conditions on the top and bottom surfaces of the plate without considering the shear correction factor. The advantage of this theory is that, in addition to considering the small scale and thickness stretching effects (epsilon(z)not equal 0), the displacement field is modelled with only 5 unknowns as the first order shear deformation theory (FSDT). Analytical solutions for vibration of simply supported micro/nanoscale plates are illustrated, and the computed results are compared with the available solutions in the literature and finite element model using ABAQUS software package. The influences of the nonlocal parameter, shear deformation and thickness stretching on the vibration behaviors of the micro/ nanoscale plates are examined

Analytical Solution for Thermomechanical Vibration of Double-Viscoelastic Nanoplate-Systems Made of Functionally Graded Materials

In this article, based on the nonlocal elasticity theory of Eringen, dynamic characteristics of a double-FGM viscoelastic nanoplates-system subjected to temperature change with considering surface effects (surface elasticity, tension and density) is studied. Two Kirchhoff nanoplates are coupled by an internal Kelvin–Voigt viscoelastic medium and also are limited to the external Pasternak elastic foundation. The material properties of the simply supported functionally graded nanoplates are assumed to follow power law distribution in the thickness direction. The governing equations of motion for three cases (out-of-phase vibration, in-phase vibration and one nanoplate fixed) are derived from Hamilton's principle. The analytical approach is employed to determine explicit closed-form expression for complex natural frequencies of the system. Numerical results are presented to show variations of the frequency of double-FGM viscoelastic nanoplates corresponding to various values of the nonlocal parameter, temperature change, power law index, aspect ratio and transverse and shear stiffness coefficients of the Pasternak elastic foundation. Moreover, influence of higher order modes, viscoelastic structural damping and damping coefficient of the viscoelastic medium on vibration characteristics are investigated. Numerical results show that natural frequency is greatly influenced by surface elastic modulus and residual surface stress.

Thermomechanical Vibration Behavior of FG Nanobeams Subjected to Linear and Non-Linear Temperature Distributions

In this study, thermomechanical vibration analysis of functionally graded (FG) nanobeams subjected to in-plane thermal loads are carried out by presenting a Navier-type solution and employing a semi-analytical differential transform method (DTM) for the first time. Two types of thermal loading, namely, linear and non-linear temperature rises through the thickness direction are considered. Thermomechanical properties of FG nanobeam are supposed to vary smoothly and continuously throughout the thickness based on power-law model and material properties are assumed to be temperature-dependent. Eringen non-local elasticity theory is exploited to describe the size dependency of FG nanobeam. Using Hamilton's principle, the non-local equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FG nanobeams including size effect and they are solved applying DTM. According to numerical results, it was revealed that the proposed modeling and semi-analytical approach can provide accurate frequency results of the FG nanobeams as compared to analytical results and also some cases in the literature. A parametric study is included to examine the effects of several parameters, such as temperature rise, gradient index, small-scale parameter and boundary conditions on the normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behaviour of a FG nanobeams is significantly influenced by these effects. The new results can be used as benchmark solutions for analyses of FG nanobeams.

Analytical modeling for the determination of nonlocal resonance frequencies of perforated nanobeams subjected to temperature-induced loads

This paper is concerned with the investigation of thermal loads and small scale effects on free dynamics vibration of slender simply-supported nanobeams perforated with periodic square holes network and subjected to temperature-induced loads. The Euler–Bernoulli beam model (EBM) and shear beam model (SBM) developed for the determination of resonance frequency are derived by modifying the standard Timoshenko beam equations. The small scale effect is included by using the Eringen's nonlocal elasticity theory while the thermal loads effect is included by considering the additional axial thermal force in the standard differential equations. Numerical results are shown that the resonance frequency change, the thermal loads and the small scale effects are depended on size and number of holes. Thus, numerical results are discussed in detail for a properly investigation of the dynamic behavior of perforated nanobeams which are of interest in the development of resonant devices integrated in micro/nanoelectromichanical systems (M(N)EMS).

A nonlocal Timoshenko beam theory for vibration analysis of thick nanobeams using differential transform method

This article presents the solution for free vibration of nanobeams based on Eringen nonlocal elasticity theory and Timoshenko beam theory. The small scale effect is considered in the first theory, and the transverse shear deformation effects as well as rotary inertia are taken into account in the latter one. Through variational formulation and the Hamilton principle, the governing differential equations of free vibration of the nonlocal Timoshenko beam and the boundary conditions are derived. The obtained equations are solved by the differential transformation method (DTM) for various frequency modes of the beams with different end conditions. In addition, the effects of slenderness and on vibration behavior are presented. It is revealed that the slenderness affects the vibration characteristics slightly whilst the small scale plays a significant role in the vibration behavior of the nanobeam.

Thermo-mechanical vibration analysis of a single-walled carbon nanotube embedded in an elastic medium based on higher-order shear deformation beam theory

In this study, the thermal effect on the free vibration characteristics of embedded Single-walled carbon nanotubes (SWCNTs) based on the size-dependent Reddy higher order shear deformation beam theory subjected to in-plane thermal loading is investigated by presenting a Navier-type solution and employing a semi-analytical Differential transform method (DTM) for the first time. In addition, the exact nonlocal Reddy beam theory solution presented here should be useful to engineers designing nanoelectromechanical devices. The smallscale effect is considered based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton’s principle, and they are solved by applying DTM. Numerical results reveal that the proposed modeling and semi-analytical approach can provide more accurate frequency results of the SWCNTs compared to analytical results and some cases in the literature. The detailed mathematical derivations are presented, and numerical investigations are performed, whereas emphasis is placed on investigating the effect of several parameters such as small-scale effects, boundary conditions, mode number, thickness ratio, temperature change, and Winkler spring modulus on the natural frequencies of the SWCNTs in detail. The vibration behavior of SWCNTs is significantly influenced by these effects. Results indicate that the inclusion of size effect results in a decrease in nanobeam stiffness and leads to a decrease in natural frequency. Numerical results are presented to serve as benchmarks for future analyses of SWCNTs.

Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method

This study investigates the small scale effect on the nonlinear bending vibration of a rotating cantilever and propped cantilever nanobeam. The nanobeam is modeled as an Euler---Bernoulli beam theory with von Karman geometric nonlinearity. The axial forces are also included in the model as the true spatial variation due to the rotation. Hamilton's principle is used to derive the governing equation and boundary conditions for the Euler---Bernoulli beam based on Eringen's nonlocal elasticity theory. The differential quadrature method as an efficient and accurate numerical tool in conjunction with a direct iterative method is adopted to obtain the nonlinear vibration frequencies of nanobeam. The effect of nonlocal small---scale, angular speed, hub radius and nonlinear amplitude of rotary nanobeam is discussed.

Closed-form elasticity solution for three-dimensional deformation of functionally graded micro/nano plates on elastic foundation

This paper addresses the static deformation of simply supported rectangular micro/nano plates made of functionally graded (FG) materials based on the three-dimensional nonlocal elasticity theory of Eringen. The plates are assumed to be simply supported and rested on a Winkler-Pasternak elastic foundation. Elasticity modulus is assumed to obey an exponential law along the thickness direction of the micro/nano plate. Using the Fourier series, a displacement field is defined that satisfies simply supported boundary condition and reduces three elasticity equations to two independent equations. The closed-form bending response is achieved by exerting boundary conditions of the lateral surfaces. Numerical results are presented to investigate the influences of the gradient index of the material properties, nonlocal parameter and stiffness of elastic foundation on the mechanical behavior of the plates.

Differential quadrature based nonlocal flapwise bending vibration analysis of rotating nanobeam using the eringen nonlocal elasticity theory under axial load

The present paper deals with the dynamic behavior of rotating nanobeam subjected to follower force using the nonlocal elasticity theory. The nonlocal elasticity theory is used to analyze the mechanical behavior of nanoscale materials. The used method of solution is the Differential Quadrature Method (DQM). It is shown that the frequencies are affected when the size effect is taken into consideration. The effects of the nonlocal small scale parameter and the hub radii on the natural frequencies of the rotating nanobeam with respect to the rotating angular velocity are investigated in the present work. This study could be useful in design of the next generation of nanodevices such as blades of a nanoturbine, nanogears, nanoscale molecular bearings, that make use of the vibration analysis of rotating nanobeam. POLYM. COMPOS., 37:3175–3180, 2016. © 2015 Society of Plastics Engineers

Hygrothermal deformation of orthotropic nanoplates based on the state-space concept

This paper deals with the investigation of the effect of hygrothermal conditions on the bending of nanoplates using Levy type solution model employing the state-space concept. The nanoplates are assumed to be subjected to a hygrothermal environment. The two-unknown function plate theory is used to derive the governing differential equations on the basis of Eringen's nonlocal elasticity theory. The governing equations contain the small scale effect as well as hygrothermal and mechanical effects. These equations are converted into a set of first-order linear ordinary differential equations with constant coefficients. Analytical solution of bending response for nanoplates under combinations of simply supported, clamped and free boundary conditions is obtained. Comparison of the results with those being in the open literature is made. The influences played by small scale parameter, temperature rise, the degree of moisture concentration, boundary conditions, plate aspect ratio and side-to-thickness ratio are studied.

Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment

In this paper, the thermal effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams subjected to various types of thermal loading is investigated by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Two kinds of thermal loading, namely, linear temperature rise and nonlinear temperature rise are studied. 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 equations of motion are derived through Hamilton׳s principle and they are solved applying DTM. According to the numerical results, it is revealed that the proposed modeling and semi analytical approach can provide accurate frequency results of the FG nanobeams as compared to analytical results and also 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 thermal effect, material distribution profile, small scale effects, mode number and boundary conditions on the normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behaviour of a FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions

In this paper, the thermal effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying DTM. According to the numerical results, it is revealed that the proposed modeling and semi analytical approach can provide accurate frequency results of the FG nanobeams as compared to analytical results and also 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 thermal effect, material distribution profile, small scale effects, mode number and boundary conditions on the normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behavior of an FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

A theoretical analysis of the free axial vibration of non-local rods with fractional continuum mechanics

In this paper we present a theoretical analysis of free axial vibrations of rods described in terms of the fractional continuum mechanics. This formulation is able to capture the size effect arising in different problems of solid mechanics. The natural frequencies and modal shapes for clamped rods are obtained and the effects of the derivative order, as well as the corresponding length-scale parameters in the fractional model are discussed. The results are compared with those derived from the Eringen non-local elasticity theory, which is an approach widely used to model structures showing size effects.

Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments

In this paper, the thermal effect on buckling and free vibration 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 be temperature-dependent and vary continuously along the thickness according to the power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion 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 frequency 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 thermal effect, material distribution profile, small scale effects, beam thickness and mode number on the critical buckling temperature and normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the thermal buckling and vibration behavior of a FG nanobeams is significantly influenced by these effects.

A Semi-analytical Method for Vibrational and BucklingAnalysis of Functionally Graded Nanobeams Consideringthe Physical Neutral Axis Position

In this paper, a semi-analytical method is presented for free vibration and buckling analysis of functionally graded (FG) size-dependent nanobeams based on the physical neutral axis position. It is the first time that a semi-analytical dif- ferential transform method (DTM) solution is developed for the FG nanobeams vibration and buckling analysis. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form. The physical neutral axis position for mentioned FG nanobeams is determined. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton's princi- ple and they are solved applying DTM. It is demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of FG nanobeams. The good agreement between the results of this article and those available in liter- ature validated the presented approach. 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 neutral axis position, small scale effects, the material distribution profile, mode number, thickness ratio and boundary conditions on the normalized natural frequencies and dimensionless buckling load of the FG nanobeams in detail. It is explicitly shown that the vibra- tion and buckling behaviour of a FG nanobeams is significantly influenced by these effects.

Active vibration control of nanotube structures under a moving nanoparticle based on the nonlocal continuum theories

The active vibration control of nanotubes under action of a moving nanoscale particle is carried out using nonlocal elasticity theory of Eringen. The nanotube is simulated by an equivalent continuum structure under simply supported boundary conditions within the framework of nonlocal Rayleigh as well as Timoshenko beam theory. A Dirac-delta function is applied to model the location of the moving mass along the nanotube as well as its inertial effects. In the following a linear classical optimal control algorithm with a time varying gain matrix and displacement-velocity feedback is applied to vibration suppression in nanotube structure. The effects of the moving nanoparticle velocity, slenderness ratio of nanotube and small scale effect parameter on the dynamic deflection are studied in some detail for both the Rayleigh and Timoshenko theories. Finally the proficiency of the control algorithm in suppressing the response of the nanostructure under the effect of moving nanoparticle with different number of controlled modes and control forces is studied. The results of the present work can be used as a benchmark in future studies of nanomachines and nanosystems.

Reissner’s mixed variational theorem-based nonlocal Timoshenko beam theory for a single-walled carbon nanotube embedded in an elastic medium and with various boundary conditions

The nonlocal Timoshenko beam theories (TBTs), based on the Reissner mixed variational theorem (RMVT) and principle of virtual displacement (PVD), are derived for the analysis of a single-walled carbon nanotube (SWCNT) embedded in an elastic medium and with various boundary conditions, in which the Eringen nonlocal elasticity theory is used. The strong formulations of RMVT- and PVD-based nonlocal TBTs and their associated possible boundary conditions are presented. The interaction between the SWCNT and its surrounding elastic medium is simulated using the Pasternak foundation model. The generalized displacement and force resultant components induced in the loaded SWCNT are obtained using the meshless collocation method, in which the shape functions of the primary variables are constructed using the differential reproducing kernel (DRK) method. The results show that RMVT-based nonlocal TBT is superior to its PVD-based counterpart in that the convergent rate of the RMVT-based nonlocal TBT is faster than that of the PVD-based one, that the predictions of generalized force resultants obtained using the RMVT-based nonlocal TBT are more accurate than those obtained using the PVD-based nonlocal one, and that the highest order of the base functions used in the RMVT-based nonlocal TBT is lower than that used in the PVD-based one.

Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory

The nonlinear flexural vibration properties of double layered viscoelastic nanoplates are investigated based on nonlocal continuum theory. The von Kámán strain–displacement relation is employed to model the geometrical nonlinearity. Based on the classical plate theory, the formulations are derived by the Hamilton's principle in conjunction with Eringen's nonlocal elasticity theory, and are further discretized by the Galerkin's method. The coordinate transformation is adopted to obtain the nonlinear governing equations of motion in the modal coordinate system. On the basis of these equations, the frequency responses of double layered nanoplates with simply supported and clamped boundary conditions are derived by the method of multiple scales. The influences of small scale and other structural parameters (e.g. the aspect ratio of the plate, van der Walls (vdW) interaction and the viscidity of the plate) on the nonlinear vibration characteristics are discussed. From the result, the vdW interaction has obvious effects on the nonlinear frequency corresponding to the second nonlinear normal mode (NNM). The nonexistence of the internal resonance is also induced from the vdW forces between the plates. The influence of the elastic matrix is also discussed. The hardening nonlinearity is observed for the primary resonance. Additionally, some interesting phenomena different from the linear vibration are observed.

Electromechanical coupling characteristics of carbon nanotube reinforced cantilever nano-actuator

The electromechanical coupling characteristics of carbon nanotubes (CNTs) reinforced cantilever nano-actuator are investigated by considering surface effect, nonlocal scale effect containing the long-range forces among atoms, van der Waals force as molecular interaction, and Casimir force as macro effect of quantum field fluctuation. The extremely nonlinear governing equation is derived by utilizing energy methods based on Eringen's nonlocal elasticity theory and Young–Laplace's surface effect model. Some useful finding and results show that the pull-in voltage and deflection of nano-actuator increase with the increase of CNTs volume ratio, the increase of the nonlocal scale parameter enhances the pull-in voltage, but declines the pull-in deflection, and the surface effect becomes gradually significant as the thickness of CNTs reinforced nano-actuator decreases. In addition, it is found that van der Waals force and Casimir force could lead to the collapse of nano-actuator without applied voltage, where the influence of Casimir force is more significant than that of van der Waals force on the electromechanical coupling behavior of CNTs reinforced nano-actuator.

Exact analytical solution for free vibration of functionally graded micro/nanoplates via three-dimensional nonlocal elasticity

Using three-dimensional (3-D) nonlocal elasticity theory of Eringen, this paper presents closed-form solutions for in-plane and out-of-plane free vibration of simply supported functionally graded (FG) rectangular micro/nanoplates. Elasticity modulus and mass density of FG material are assumed to vary exponentially through the thickness of micro/nanoplate, whereas Poisson's ratio is considered to be constant. By employing appropriate displacement fields for the in-plane and out-of-plane modes that satisfy boundary conditions of the plate, ordinary differential equations of free vibration are obtained. Boundary conditions on the lateral surfaces are imposed on the analytical solutions of the equations to yield the natural frequencies of FG micro/nanoplate. The natural frequencies of FG micro/nanoplate are obtained for different values of nonlocal parameter and gradient index of material properties. The results of this investigation can be used as a benchmark for the future numerical, semi-analytical and analytical studies on the free vibration of FG micro/nanoplates.