Eringen's Nonlocal Theory Research Articles

New static and dynamic analyses of macro and nano FGM plates using exact three-dimensional elasticity in thermal environment

In this article, the static and dynamic behaviors of macro and nano inhomogeneous plates made of functionally graded materials (FGMs) have been investigated based on three-dimensional elasticity theory in combination with the nonlocal theory of Eringen. Variation of deflection along the thickness has been considered as the result of three-dimensional analysis. The governing equations have been derived and developed considering the nonlocal theory for nano-scale plates. Conveniently, the results of macro-sized plates were concluded by neglecting the small scale effects. The results were compared with the some related papers available in the literature. Since the three-dimensional elasticity has been applied, the obtained results are more accurate than other plate theories. In addition, if the plate is subjected to thermal loading, only the results of three-dimensional analysis will be correct. Since the nonlocal analysis has been assumed, the small-scale effects and related parameters have been studied for the nano-scale plates. Finally, the effects of some other important parameters have been surveyed on the results, such as the size of plate, loading, FGM parameters, and boundary conditions.

Stochastic stability of a magnetically affected single-layer graphene sheet resting on a viscoelastic foundation

Abstract In this paper, we analyzed the stochastic stability of a single-layer graphene sheet resting on a viscoelastic foundation and influenced by the in-plane magnetic field. The mechanical model of a graphene sheet is given as an orthotropic and viscoelastic nanoplate while the viscoelastic foundation is of the Kelvin-Voigt type. We assume that the graphene sheet is influenced by the in-plane random forces variable with time and exerted in-plane magnetic field. Based on the Eringen's nonlocal continuum theory and Kirchhoff – Love plate theory, the governing equation of motion is derived by considering the influence of the Lorentz forces obtained from the classical Maxwell relations. In order to investigate the stochastic stability of such system, the maximal and moment Lyapunov exponents are considered by using the perturbation method. The predicted approximated analytical results for the p -th moment Lyapunov exponents are validated by the Monte Carlo method . Moreover, the boundaries of almost-sure and moment stability of the viscoelastic nanoplate are determined as functions of different system parameters. The influences of the nonlocal and magnetic field parameters, stiffness and damping coefficients and spectral density on the moment Lyapunov exponents are investigated through several numerical examples. Presented results reveal that the applied in-plane magnetic field could be successfully used to improve stability performances of nano-electromechanical systems based on graphene sheets.

Bending analysis of embedded nanoplates based on the integral formulation of Eringen's nonlocal theory using the finite element method

Due to the capability of Eringen's nonlocal elasticity theory to capture the small length scale effect, it is widely used to study the mechanical behaviors of nanostructures. Previous studies have indicated that in some cases, the differential form of this theory cannot correctly predict the behavior of structure, and the integral form should be employed to avoid obtaining inconsistent results. The present study deals with the bending analysis of nanoplates resting on elastic foundation based on the integral formulation of Eringen's nonlocal theory. Since the formulation is presented in a general form, arbitrary kernel functions can be used. The first order shear deformation plate theory is considered to model the nanoplates, and the governing equations for both integral and differential forms are presented. Finally, the finite element method is applied to solve the problem. Selected results are given to investigate the effects of elastic foundation and to compare the predictions of integral nonlocal model with those of its differential nonlocal and local counterparts. It is found that by the use of proposed integral formulation of Eringen's nonlocal model, the paradox observed for the cantilever nanoplate is resolved.

Poisson's ratio effects on the mechanics of auxetic nanobeams

Poisson ratio is an important mechanical property that reveals the deformation patterns of materials. A positive Poisson ratio is a feature of the majority of materials. Some materials, however, display auxetic behaviors (i.e. possess negative Poisson ratios). Indeed, auxetic and non-auxetic materials display different deformation mechanisms. Revealing these differences and their effects on the mechanics of these materials is of a significant importance. In this study, effects of Poisson ratio on the mechanics of auxetic and non-auxetic nanobeams are revealed. A parametric study is provided on effects of Poisson ratio on the static bending and free vibration behaviors of auxetic nanobeams. The general nonlocal theory is employed to model the nonlocal effects. Unlike Eringen nonlocal theory, the general nonlocal theory uses different attenuation functions for the longitudinal and lateral strains. This theory can reveal the Poisson ratio-nonlocal coupling effects on the mechanics of nanomaterials. The obtained results showed that Poisson ratio is an essential parameter for determining mechanical behaviors of nanobeams. It is revealed that auxetic and non-auxetic nanobeams may reflect softening or hardening behaviors depending on the ratio of the nonlocal fields of the longitudinal and lateral strains of the beam.

On vibrations of nanobeam systems

Differential form of Eringen's nonlocal elastic theory has some inaccuracies in analyzing small-scale structures with different boundary conditions. For this reason, Eringen's two-phase local/nonlocal integral model as a reliable well-posed form of the theory attracted a great deal of attention in modeling size-dependent effects. Accordingly, in this work a comprehensive study on vibration behavior of double-layered nanobeam systems (DNBS) is presented within the framework of Eringen's two-phase local/nonlocal integral model. DNBS's vibrational behavior is formulated for three different cases by having in-phase vibration, out-phase vibration and by fixing the underneath layer and analyzing beams on a medium. Governing equations are presented for different boundary conditions especially cantilever DNBS model where the differential form of Eringen's nonlocal theory is unable to model. In order to show the influence of nonlocal terms in Eringen's two-phase model and the elastic coupling term on dynamic behavior of such structures, a comprehensive parametric study is carried out. It is shown that these parameters can have a significant effect on the natural frequency terms which could lead to more accurate and reliable dynamic models with different boundary conditions.

Nonlocal Dynamic Stability of DWCNTs Containing Pulsating Viscous Fluid Including Surface Stress and Thermal Effects Based on Sinusoidal Higher Order Shear Deformation Shell Theory

AbstractIn this study, the dynamic stability of double-walled carbon nanotubes (DWCNTs) conveying pulsating viscous fluid located on visco-Pasternak medium is investigated. The effects of small scale are considered using Eringen's non-local theory. The surrounding medium of DWCNT is anticipated as a visco-Pasternak foundation including the normal, shear, and damping forces. The van der Waals (vdWs) force is considered among the inner and outer layers of DWCNT. Three types of temperature changes, i.e. uniform, linear and sinusoidal temperature rise, through the thickness are studied. The equations of motion are obtained using higher order sinusoidal shear deformation shell theory (SSDT), in which the surface effects are included. Dynamical equations of DWCNT are extracted using the energy method and Hamilton's principle. Due to orthogonal conditions, the governing equations are solved based on Bolotin method. The effects of different parameters such as surface stress, non-local parameter, fluid velocity, Knudsen's number, fluid density, dimensions ratio and various temperature loadings on the dynamic stability regions of DWCNTs are discussed.

Analytical solution for static and dynamic analysis of magnetically affected viscoelastic orthotropic double-layered graphene sheets resting on viscoelastic foundation

By considering the small scale effect based on the nonlocal Eringen's theory, the static and dynamic analysis of viscoelastic orthotropic double-layered graphene sheets subjected to longitudinal magnetic field and mechanical load is investigated analytically. For this objective, first order shear deformation theory (FSDT) is proposed. The surrounding medium is simulated by visco-Pasternak foundation model in which damping, normal and transverse shear loads are taken into account. The governing equations of motion are obtained via energy method and Hamilton's principle which are then solved analytically by means of Navier's approach and Laplace inversion technique in the space and time domains, respectively. Through various parametric studies, the influences of the nonlocal parameter, structural damping, van der Waals (vdW) interaction, stiffness and damping coefficient of the foundation, magnetic parameter, aspect ratio and length to thickness ratio on the static and dynamic response of the nanoplates are examined. The results depict that when the vdW interaction is considered to be zero, the upper layer deflection reaches a maximum point whereas the lower layer deflection becomes zero. In addition, it is observed that with growing the vdW interaction, the effect of magnetic field on the deflection of the lower layer increases while this effect reduces for the upper layer deflection.

Analogies for simply supported nonlocal Kirchhoff plates of polygonal planform

In the present paper, we consider nonlocal linear elastic Kirchhoff plates, where we restrict to thin, isotropic, and homogeneous plates under the action of static transverse forces, utilizing the differential equation form of the Eringen nonlocal continuum theory. For the case of simply supported plates of polygonal planform, we derive analogies between the solutions of the nonlocal Kirchhoff theory and its local counterpart. First, we extend the Marcus decomposition method for local Kirchhoff plates, where we show that, analogous to the local case considered by Marcus, the moment sum and the nonlocal deflection are both governed by Poisson boundary value problems, which correspond to auxiliary (local) membrane problems. In the present context, it eventually follows that there is a nonvanishing term responsible for a correction of the deflection due to the nonlocal effect, while the moments and shear forces of the local and the nonlocal plate coincide. The nonlocal correction of the deflection turns out to be governed by a membrane-type boundary value problem again. From this fact, it follows immediately that the correction for the deflection due to the nonlocal effect can be derived alternatively from both the local deflection and the local moment sum, which represents a substantial simplification of the nonlocal computations. For the sake of demonstration, we consider examples with closed form solutions. We first consider (as limiting case) infinite plates under the action of single forces, where the singular behavior of Green’s function of the nonlocal Kirchhoff theory can be clarified. Then we discuss the bending of nonlocal plate strips for comparison sake, as well as equilateral triangular plates. To our best knowledge, no results for bending of nonlocal triangular plates have been presented in the literature so far. The present paper has been strongly influenced by a former contribution on analogies between simply supported polygonal Kirchhoff plates rigid in shear and shear-deformable plate solutions. This paper was co-authored together with our teacher, the late Professor Franz Ziegler, to whom we dedicate the present paper. In light of this latter contribution, and extending an idea promoted by Professor Franz Ziegler and his co-workers in the framework of various other fields, we finally show in the Appendix that the deflections of simply supported nonlocal Kirchhoff plates can be considered as deflections of a local “background” Kirchhoff plate under the action of additional fictitious eigenstrain loadings, such as thermal moments.

Thermoelastic Damping in FGM Nano-Electromechanical System in Axial Vibration Based on Eringen Nonlocal Theory

The thermo-elastic damping is a dominant source of internal damping in micro-electromechanical systems (MEMS) and nano-electromechanical systems (NEMS). The internal damping cannot neither be controlled nor minimized unless either mechanical or geometrical properties are changed. Therefore, a novel FGMNEM system with a controllable thermo-elastic damping of axial vibration based on Eringen nonlocal theory is considered. The effects of different parameter like the gradient index, nonlocal parameter, length of nanobeam and ambient temperature on the thermo-elastic damping quality factor are presented. It is shown that the thermo-elastic damping can be controlled by changing different parameter.

Flapwise bending vibration analysis of rotary tapered functionally graded nanobeam in thermal environment

ABSTRACTIn this article, transverse vibration of rotary functionally graded size-dependent tapered Bernoulli–Euler nanobeam in thermal environment at low temperature has been investigated based on Eringen's nonlocal theory for cantilever and propped cantilever boundary conditions. Material properties of FG nanobeam are supposed to be temperature dependant and vary continuously along the thickness according to the power-law form. The axial force is also included in the model as the true spatial variation due to the rotation. The nonlocal equations of motion are derived through Hamilton's principle and they are solved by the differential quadrature method. Validations are done by comparing available literatures and obtained results, which reveal the accuracy of the applied method. 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 angular velocity, material distribution profile, different boundary conditions, small-scale parameter and rate of cross-section change on the first three nondimensional natural frequencies of the rotary FG nanobeam in detail. Numerical results are presented to serve as benchmarks for the application and the design of nanoelectronic, nanodrive devices and nanomotor, in which nanobeams act as basic elements. They can also be useful as valuable sources for validating other approaches and approximate methods.The results of this article are suitable in designation of micromachines, such as micromotors and micro-rotors.

Application of modified Adomian decomposition method to pull-in instability of nano-switches using nonlocal Timoshenko beam theory

In this paper, the pull-in instability of doubly clamped nano-switches subjected to electrostatic and intermolecular forces are investigated. To this end, using Eringen's nonlocal elsticity theory, the static governing equation of nonlocal Timoshenko beam model is derived. The obtained equation which is a high-order nonlinear ordinary differential equation owing to the electrostatic and intermolecular forces is solved through the use of modified Adomian decomposition method (MADM). This method presents an analytical approximate solution by which the small scale effect on the static pull-in instability of nanobeam is examined. The results obtained from the present method are found to be in reasonable agreement with ones generated by the differential quadrature method (DQM) and experiment. A parametric study is carried out to study the effect of nonlocal parameter on the mid-point deflection, external forces, bending moment and shear force of doubly clamped nanobeams. Moreover, the results of nonlocal Timoshenko beam model are compared with those of nonlocal Euler–Bernoulli and classical beam models.

Linear and non-linear free vibration of nano beams based on a new fractional non-local theory

PurposeRecently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems.Design/methodology/approachIn this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition.FindingsThe theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio.Originality/valueA new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.

Non-linear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects

• Non-linear vibration of nanobeam considering thermal and surface effects is investigated. • The nanobeam resting on a viscoelastic foundation is subjected to moving concentrated load. • The Eringen's nonlocal and the Euler–Bernoulli theories are applied. • The Galerkin and the multiple scales methods are employed to solve the problem. • The influences of the various parameters on the jump phenomenon are studied. Analytical solution for the steady-state response of an Euler–Bernoulli nanobeam subjected to moving concentrated load and resting on a viscoelastic foundation with surface effects consideration in a thermal environment is investigated in this article. At first, based on the Eringen's nonlocal theory, the governing equations of motion are derived using the Hamilton's principle. Then, in order to solve the equation, Galerkin method is applied to discretize the governing nonlinear partial differential equation to a nonlinear ordinary differential equation; solution is obtained employing the perturbation technique (multiple scales method). Results indicate that by increasing of various parameters such as foundation damping, linear stiffness, residual surface stress and the temperature change, the jump phenomenon is postponed and with increasing the amplitude of the moving force and the nonlocal parameter, the jump phenomenon occurs earlier and its frequency and the peak value of amplitude of vibration increases. In addition, it is seen that the non-linear stiffness and the critical velocity of the moving load are important factors in studying nanobeams subjected to moving concentrated load. Presence of the non-linear stiffness of Winkler foundation resulting nanobeam tends to instability and so, the jump phenomenon occurs. But, presence of the linear stiffness will lead to stability of the nanobeam. In the next sections of the paper, frequency responses of the nanobeam made of temperature-dependent material properties under multi-frequency excitations are investigated.

Size-dependent dynamic analysis of rectangular nanoplates in the presence of electrostatic, Casimir and thermal forces

• The size dependent dynamic instability of fully clamped nanoplates is obtained. • The effects of Casimir forces as well as temperature change are examined. • Eringen's nonlocal theory is applied along with CLT to obtain the equation. • The dynamic governing equation is solved numerically by Galerkin method. In the present study by considering the small-scale effects, the dynamic instability of fully clamped and simply supported nanoplates is studied in the attendance of electrostatic, Casimir as well as thermal forces. To this end, by applying the nonlocal elasticity theory of Eringen along with the classical plate theory, the dynamic equilibrium equation of nanoplates is obtained by incorporating the in-plane thermal and transverse intermolecular distributed loads. The solution of the obtained nonlinear governing equation is done using the Galerkin method and the dynamic pull-in instability voltage of the nanoplates is compared with the available experimental results. Finally, the simultaneous effects of thermal force as well as nonlocal parameter on the dynamic response of nanoplates are examined in the presence of Casimir force in detail.

A new modified higher-order shear deformation theory for nonlinear analysis of macro- and nano-annular sector plates using the extended Kantorovich method in conjunction with SAPM

In this research, the nonlinear local and nonlocal analysis of an annular sector plate is studied and solved based on a new modified higher-order shear deformation theory. Due to the shortcomings of HSDT in the two-dimensional nonlinear analysis, it is modified by eliminating the defects, and a comprehensive theory is presented for analyzing the mechanical behavior of an annular sector sheet in general form. The strain field is developed by considering the von Karman assumptions and also the nonlocal theory of Eringen from which the classical local analysis can be deduced conveniently by neglecting the small-scale effects. Whereas the annular sector plate is assumed, the sector, annular/circular, rectangular and solid circular plates can be simulated. Afterward, the nonlocal constitutive equations are derived and solved by using the two-dimensional SAPM (Dastjerdi et al. in Compos B Eng 98, 78–87, 2016). Moreover, a combination of the extended Kantorovich method and one-dimensional SAPM is applied. Since the presented theory is relatively new and similar studying was not available in order to compare the results, a comparison is done with the results of lower-order theories. Finally, the effect of various parameters, such as boundary conditions, different theories, nonlocal and local analyses, loading and the size of the plate, on the mechanical behavior of an annular sector plate are investigated.

Influence of porosity and axial preload on vibration behavior of rotating FG nanobeam

In this paper, a nanobeam connected to a rotating molecular hub is considered. The vibration behavior of rotating functionally graded nanobeam based on Eringen's nonlocal theory and Euler–Bernoulli beam model is investigated. Furthermore, axial preload and porosity effect is studied. It is supposed that the material attributes of the functionally graded porous nanobeam, varies continuously in the thickness direction according to the power law model considering the even distribution of porosities. Porosity at the nanoscopic length scale can affect on the rotating functionally graded nanobeams dynamics. The equations of motion and the associated boundary conditions are derived through the Hamilton's principle and generalized differential quadrature method (GDQM) is utilized to solve the equations. In this paper, the influences of some parameters such as functionally graded power (FG-index), porosity parameter, axial preload, nonlocal parameter and angular velocity on natural frequencies of rotating nanobeams with pure ceramic, pure metal and functionally graded materials are examined and some comparisons about the influence of various parameters on the natural frequencies corresponding to the simply-simply, simply-clamped, clamped-clamped boundary conditions are carried out.

Dynamic response of biaxially loaded double-layer viscoelastic orthotropic nanoplate system under a moving nanoparticle

In this paper, dynamic behavior of double layered nanoplate systems (DLNPS) with respect to a moving nanoparticle is investigated. Both layers of DLNPS are assumed to be orthotropic and each layer is bearing a biaxial load while internal damping effects are also taken into account. Furthermore, coupling between layers are modeled using Kelvin-Voigt viscoelastic theory and moving nanoparticles path are assumed to be linear and circular with constant velocities. Governing equations of motion are derived by using D'Alembert's principle, Kirchhoff-Love plate and Eringen's nonlocal theory. Galerkin's and Laplace transform methods is used to solve the governing equations and analytical solution is presented for linear moving nanoparticle while close-form solution is obtained for circular moving nanoparticle. In order to clarify the influence of different parameters such as small scale effect, stiffness and damping in coupling, biaxial compression and tension of layers, path of the moving mass, etc. on dynamic behavior of each layer, parametric study is presented. Accordingly, with the brand new discussions in moving atoms, molecules, nanocars, nanotrims, point loads on different nanosctructures using scanning tunneling microscopes (STM) and atomic force microscopes (AFM), this study could be a step forward in understanding such kind of behaviors.

Dynamic pull-in and pull-out analysis of viscoelastic nanoplates under electrostatic and Casimir forces via sinusoidal shear deformation theory

In the present research, dynamic pull-in and pull-out analysis of viscoelastic nanoplate switch under electrostatic and intermolecular Casimir forces are studied. The Viscoelastic properties of the nanoplate are considered using Kelvin-Voigt model. Sinusoidal shear deformation theory is utilized for mathematical modeling of the structure due to its accuracy of polynomial function than other plate theories. The surface effects are considered based on Gurtin-Murdoch theory to enhance the accuracy of results. Considering size effects based on Eringen's nonlocal theory, motion equations are derived using Hamilton's principle and solved by Galerkin's method. Considering two boundary conditions of clamped in all edges (CCCC) and clamped in two edges and free in tow another edges (CCFF), the influences of various parameters such as small scale effect, surface layer, viscoelastic damping coefficient and applied voltage on the pull-in voltage, pull-in time, pull-in deflection and pull-out voltage are discussed in details. Numerical results indicate that considering small scale effects, the pull-in time and voltage as well as pull-out voltage are increased and decreased, respectively, for CCFF and CCCC boundary conditions. In addition, as the effective gap distance increases, the pull-out voltage tends toward the pull-out voltage. The results of this investigation can be used in control and optimum design of smart nano-electrical devices in advanced applications as smart controller.

Vibration analysis of functionally graded piezoelectric nanoscale plates by nonlocal elasticity theory: An analytical solution

This paper investigated the free vibration analysis of functionally graded piezoelectric materials (FGPMs) nanoscale plates based on the Eringen's nonlocal Kirchhoff plate theory under simply supported edge conditions. The material properties vary continuously along the thickness direction based on the power-law distribution in terms of the volume fractions of the constituents. The material compositions are selected from the PZT family. Using Hamilton's principle, the governing differential equations are derived and the Navier's solution is used to attain the natural frequencies. The accuracy of the method is validated by comparing the results with the previous studies. Finally, the effects of the nonlocal parameter, various gradient indexes, mode numbers, aspect ratio and side-to-thickness ratio on natural frequencies are also studied.

On boundary conditions for buckling and vibration of nonlocal beams

In this paper, we argue that the boundary conditions for Eringen's nonlocal beam theory have to take on the discrete forms similar to those for the first order central finite difference beam model for beam buckling and vibration problems. The latter finite difference beam model has been found to be analogous to the physical Hencky bar-chain and therefore these two models can be regarded as one discrete beam model. Based on the phenomenological similarities between this discrete beam model and the Eringen's nonlocal beam theory, one may calibrate the Eringen's small length scale coefficient e0 which is expected to be a constant appropriate to each material. When using the classical continuous nonlocal boundary conditions, we find that e0 is not a constant but it depends on the cases of boundary conditions (such as pinned-pinned, clamped-clamped, clamped-pinned ends). However, this conundrum may be resolved if we use the discrete boundary conditions instead. By adopting the discrete boundary conditions, analytical solutions for e0 may be obtained and the e0 values finally converge to 0.289 for buckling problems and 0.408 for free vibration problems regardless of the cases of boundary conditions.