Eringen's Theory Research Articles

Thermo-mechanical vibration analysis of non-local refined trigonometric shear deformable FG beams

In this article, thermo vibration analysis of functionally graded (FG) nanobeams subjected to linear and uniform temperature distribution is studied. The structure is modelled by new efficient shear beam theory considering the effect of shear deformation without shear correction factor. The mentioned theory satisfies the zero boundary traction conditions on the beam surfaces and can be used for transverse shear strains trigonometric distribution. Material properties of the nanostructure are assumed temperature-dependent and FG based on the distribution of power law. The influence of size scale is captured utilising Eringen theory of non-local elasticity. Based on the present theory, the motion equations are obtained using principle of Hamilton. It is found that the applied theory is very simple and leads to accurate results for thermo vibration analysis of FG nanobeams.

On the electro-thermo-mechanical vibration characteristics of elastically restrained functionally graded nanobeams using differential transformation method

In this investigation, free vibration of piezoelectric functionally graded nanobeams in the presence of thermal field effects for various elastic boundary conditions is studied. Formulations are established in the framework of nonlocal elasticity theory of Eringen accompanied by Euler–Bernoulli beam theory. Further, the material characteristics are supposed to change through the thickness based on the power law. Discretization of the equations of motion and the relative boundary conditions are carried out by utilizing the differential transformation method. The presented model is validated through obtaining numerical results for conventional boundary conditions in comparison with the corresponding benchmark results. Finally, numerical results are given in details to demonstrate the impact of material gradient, nonlocal effect, piezoelectric voltage, along with temperature changes on vibration characteristic of piezoelectric functionally graded nanobeams, with various elastic boundary conditions.

Free Vibration of Single Walled Carbon Nanotube Resting on Exponentially Varying Elastic Foundation

Abstract In this article, free vibration of SingleWalled Carbon Nanotube (SWCNT) resting on exponentially varying Winkler elastic foundation is investigated by using Differential Quadrature Method (DQM). Euler-Bernoulli beam theory is considered in conjunction with the nonlocal elasticity theory of Eringen. Step by step procedure is included and MATLAB code has been developed to obtain the numerical results for different scaling parameters as well as for four types of edge conditions. Obtained results are validated with known results in special cases showing good agreement. Further, numerical as well as graphical results are illustrated to show the effects of nonuniform parameter, nonlocal parameter, aspect ratio,Winkler modulus parameter and edge conditions on the frequency parameters.

On wave dispersion characteristics of magneto-electro-elastic nanotubes considering the shell model based on the nonlocal strain gradient elasticity theory

In the present study, wave propagation analysis of magneto-electro-elastic (MEE) nanotubes considering the shell model is explored in the framework of the nonlocal strain gradient elasticity theory. To take the small-scale effects into account, the nonlocal elasticity theory of Eringen is applied. Nonlocal governing equations of MEE nanotube have been derived utilizing Hamilton’s principle. The outcomes of this paper have been validated by comparing them with previous investigations. An analytical solution of governing equations is used to obtain phase velocities and wave frequencies. The influence of different parameters, such as different mode, nonlocal parameter, length parameter, geometry, magnetic field and electric field on wave propagation characteristics of MEE nanotube are reported in detail.

Free Vibration Analysis of Variable Cross-Section Single-Layered Graphene Nano-Ribbons (SLGNRs) Using Differential Quadrature Method

In this article, free vibration of variable cross-section (non-uniform) single layered graphene nano-ribbons (SLGNRs) is investigated by using Differential Quadrature Method (DQM). Here characteristic width of the cross section is assumed to vary exponentially along the length of the nano-ribbon while the thickness of the cross section is kept constant. Euler–Bernoulli beam theory is considered in conjunction with the nonlocal elasticity theory of Eringen. Step by step procedure is included and MATLAB code has been developed to obtain the numerical results for different scaling parameters as well as for four types of edge conditions. Convergence study is carried out to illustrate the efficiency of the method and obtained results are validated with known results in special cases showing good agreement. Further, numerical as well as graphical results are depicted to show the effects of nonuniform parameter, nonlocal parameter, aspect ratio and edge conditions on the frequency parameters.

Physically consistent nonlocal kernels for predicting vibrational characteristics of single walled carbon nanotubes

In the present work, at first, the drawbacks of the continuum mechanics (CM) based method in predicting the vibrational and wave dispersion characteristics of carbon nanotubes (CNTs) are illustrated and discussed. For this purpose, the response of the Euler-Bernoulli and Timoshenko beam theories together with the differential form of the Eringen’s nonlocal elasticity theory under different boundary conditions are examined. The same problems are simulated via molecular dynamics (MD) approach. By considering the results of MD simulation as the benchmark solutions, it is shown that the optimum value of the length scale parameter of the nonlocal elasticity theory of Eringen depends on the nanotube boundary conditions, aspect ratio and vibrational mode number of the CNTs. To overcome these drawbacks, a kernel with unique length scale variable is introduced by combing different Gaussian kernels and specifying two related new scale variables. These newly defined scale parameters resolve former problem in shifting between different scales. Accordingly, an improved nonlocal constitutive relation is suggested. It is shown that the proposed model can accurately predict the vibrational and flexural vibrational dispersion relation (FVDR) characteristics of CNTs even when using Euler-Bernoulli theory (EBT).

Nonlinear transient isogeometric analysis of FG-CNTRC nanoplates in thermal environments

This paper presents size-dependency effects on nonlinear transient dynamic response of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates under a transverse uniform load in thermal environments. To consider the length scale and size-dependency effect of nanostructures, a nonlocal continuum theory of Eringen is adopted. The nonlocal governing equations for nanoplate theory are derived from the Hamilton’s principle and approximated by using isogeometric analysis associated with the higher-order shear deformation theory. A numerical model based on the von Kármán strains and Newmark time integration scheme is employed to solve geometrically nonlinear transient problems. The material properties of the FG-CNTRC nanoplate are assumed to be graded and temperature-dependent in the thickness direction, which are expressed through a micromechanical model. Effects of nonlocal parameter, carbon nanotube volume fraction, length-to-thickness ratio, distributions of carbon nanotubes and temperatures through thickness are investigated in detail. Several numerical results show the reliability of the present method.

Nonlocal vibrations and potential instability of monolayers from double-walled carbon nanotubes subjected to temperature gradients

Using nonlocal elasticity theory of Eringen, free transverse thermo-elastic vibrations of vertically aligned double-walled carbon nanotubes (DWCNTs) with a membrane configuration is going to be explored methodically. Accounting for nonlocal heat conduction along the side walls of DWCNTs with allowance of heat dissipation from the outer surfaces, the nonlocal temperature fields within the constitutive tubes are assessed for a steady-state regime. The van der Waals interactional forces between atoms of each pair of DWCNTs as well as the innermost and outermost tubes are displayed by laterally continuous springs whose constants are appropriately evaluated. By establishing suitable discrete and continuous models on the basis of the Rayleigh and higher-order beam theories, nonlocal in-plane and out-of-plane vibrations of thermally affected nanosystems of monolayers of DWCNTs are examined and discussed. The critical values of the temperature change are stated explicitly and the role of influential factors on this crucial factor as well as the free vibration behavior are investigated in some detail. Through conducting a fairly comprehensive numerical study, the influences of the slenderness ratio, temperature change in the low and high temperatures, number of nanotubes, small-scale parameter, intertube distance, and stiffness of the surrounding environment on the nonlocal-fundamental frequency are explained. The obtained results from this work could provide crucial guidelines for the design of membranes or even jungles of vertically aligned DWCNTs as thermal interface nanostructures.

A nonlocal strain gradient theory for dynamic modeling of a rotary thermo piezo electrically actuated nano FG circular plate

Abstract In this paper, vibrational behavior of functionally graded piezoelectric material (FGPM) in a nano circular plate subjected to rotational and thermal loads is studied. Thermo-electro–elastic characteristics of FGPM nano circular plate are changed continuously along the thickness with the power-law model. This analysis is based on nonlocal strain gradient theory (NSGT), strain gradient theory (SGT) and nonlocal Eringen theory (NET) within the first order shear deformation theory (FSDT). Via Hamilton’s principle, the governing equations of FGPM nano circular plate obtained and solved using the differential quadrature method (DQM). It is seen that the vibrational behavior of FGPM nano circular plate is considerably affected by temperature rises, angular velocity, external voltage, FG power index, nonlocal parameter and material length scale for clamped and hinged boundary conditions.

Postbuckling analysis of functionally graded nanoplates based on nonlocal theory and isogeometric analysis

This study aims to investigate the postbuckling response of functionally graded (FG) nanoplates by using the nonlocal elasticity theory of Eringen to capture the size effect. In addition, Reddy’s third-order shear deformation theory is adopted to describe the kinematic relations, while von Kámán’s assumptions are used to account for the geometrical nonlinearity. In order to calculate the effective material properties, the Mori-Tanaka scheme is adopted. Governing equations are derived based on the principle of virtual work. Isogeometric analysis (IGA) is employed as a discretization tool, which is able to satisfy the C1-continuity demand efficiently. The Newton-Raphson iterative technique with imperfections is employed to trace the postbuckling paths. Various numerical studies are carried out to examine the influences of gradient index, nonlocal effect, ratio of compressive loads, boundary condition, thickness ratio and aspect ratio on the postbuckling behaviour of FG nanoplates.

Nonlocal magneto-thermo-vibro-elastic analysis of vertically aligned arrays of single-walled carbon nanotubes

Abstract This study concerns with transverse vibrations of magnetically-thermally affected vertically aligned arrays of single-walled carbon nanotubes (SWCNTs) using nonlocal elasticity theory of Eringen. Using nonlocal Rayleigh and Timoshenko beam theories, both discrete and continuous versions of equations of motion are presented. In contrast to the discrete models, the continuous models do not suffer from huge time and labor costs for nanosystems with high population. The capability and efficiency of the continuous models in capturing the frequencies of the discrete models are displayed, and a reasonably good agreement is obtained. Subsequently, the influences of radius of SWCNTs, slenderness ratio, population, small-scale parameter, strength of magnetic field, and variation of the temperature on the fundamental frequency are explained and discussed. Additionally, the role of shear deformation on the obtained results is explained and the limitations of the nonlocal Rayleigh beam model are revealed.

Analytical Buckling of FG Nanobeams on The Basis of A New One Variable First-Order Shear Deformation Beam Theory

In this work, buckling analysis of functionally graded (FG) nanobeams based on a new refined beam theory has been analyzed. The beam is modeled as an elastic beam subjected to unidirectional compressive loads. To achieve this aim, the new obtained beam theory has only one variable which lead to one equation similar to Euler beam theory and also is free of any shear correction factor. The equilibrium equation has been formulated by the nonlocal theory of Eringen to predict small-scale effects. The equation has been solved by Navier’s approach by which critical buckling loads have been obtained for simple boundaries. Finally, to approve the results of the new beam theory, various beam theories have been compared.

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.

Longitudinal vibrations of nano-rods by stress-driven integral elasticity

ABSTRACTIn elastic continuous structures defined on bounded domains, Eringen strain-driven integral model leads to ill-posed elastostatic problems since the constitutive stress field, got by convoluting the elastic strain field with an averaging kernel, conflicts with equilibrium requirements. The innovative stress-driven integral model, recently proposed in literature, is instead able to overcome inconsistencies of Eringen theory. In the present paper, size-dependent free vibrations of nano-rods are investigated by adopting the stress-driven nonlocal model equipped with Helmholtz kernel. Natural frequencies and mode shapes of nano-rods are established by an effective analytical solution strategy and are compared with those obtained by strain gradient theory.

A unified formulation for modeling of inhomogeneous nonlocal beams

In this article, buckling and free vibration of functionally graded (FG) nanobeams resting on elastic foundation are investigated by developing various higher order beam theories which capture shear deformation influences through the thickness of the beam without the need for shear correction factors. The elastic foundation is modeled as linear Winkler springs as well as Pasternak shear layer. The material properties of FG nanobeam are supposed to change gradually along the thickness through the Mori–Tanaka model. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. From Hamilton\'s principle, the nonlocal governing equations of motion are derived and then solved applying analytical solution. To verify the validity of the developed theories, the results of the present work are compared with those available in literature. The effects of shear deformation, elastic foundation, gradient index, nonlocal parameter and slenderness ratio on the buckling and free vibration behavior of FG nanobeams are studied.

Dynamic modeling of nonlocal compositionally graded temperature-dependent beams

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 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 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, aspect ratio 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 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.

Nonlinear dynamic analysis of SWNTs conveying fluid using nonlocal continuum theory

By employing the nonlocal continuum field theory of Eringen and Von Karman nonlinear strains, this paper presents an analytical model for linear and nonlinear dynamics analysis of single-walled carbon nanotubes (SWNTs) conveying fluid with different boundary conditions. In the linear analysis the natural frequencies and critical flow velocities of SWNTs are computed. However, in the nonlinear analysis the effect of nonlocal parameter on nonlinear dynamics of cantilevered SWNTs conveying fluid is investigated by using bifurcation diagram, phase plane and Poincare map. Numerical results confirm existence of chaos as well as a period-doubling transition to chaos.

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.

Closed-form solutions in stress-driven two-phase integral elasticity for bending of functionally graded nano-beams

Strain-driven and stress-driven integral elasticity models are formulated for the analysis of the structural behaviour of fuctionally graded nano-beams. An innovative stress-driven two-phases constitutive mixture defined by a convex combination of local and nonlocal phases is presented. The analysis reveals that the Eringen strain-driven fully nonlocal model cannot be used in Structural Mechanics since it is ill-posed and the local-nonlocal mixtures based on the Eringen integral model partially resolve the ill-posedeness of the model. In fact, a singular behaviour of continuous nano-structures appears if the local fraction tends to vanish so that the ill-posedness of the Eringen integral model is not eliminated. On the contrary, local-nonlocal mixtures based on the stress-driven theory are mathematically and mechanically appropriate for nanosystems. Exact solutions of inflected functionally graded nanobeams of technical interest are established by adopting the new local-nonlocal mixture stress-driven integral relation. Effectiveness of the new nonlocal approach is tested by comparing the contributed results with the ones corresponding to the mixture Eringen theory.

Static stability analysis of embedded flexoelectric nanoplates considering surface effects

In this paper, electromechanical buckling behavior of size-dependent flexoelectric nanoplates is investigated based on nonlocal and surface elasticity theories. Flexoelectricity represents the coupling between strain gradients and electrical polarizations. Flexoelectric nanoplates can tolerate higher buckling loads compared with conventional piezoelectric nanoplates, especially at lower thicknesses. The flexoelectric nanoplate is in contact with a two-parameter elastic foundation, which consists of infinite linear springs and a shear layer. Nonlocal elasticity theory of Eringen is applied in the analysis of flexoelectric nanoplates for the first time. The residual surface stresses which are usually neglected in the modeling of flexoelectric nanoplates are incorporated into nonlocal elasticity to provide better understanding of the physics of the problem. Applying an analytical solution which satisfies various boundary conditions, the governing equations obtained from Hamilton’s principle are solved. The reliability of the present approach is verified by comparing the obtained results with those provided in literature. Finally, the influences of nonlocal parameter, surface effect, plate geometrical parameters, elastic foundation and boundary conditions on the buckling characteristics of flexoelectric nanoplates are explored.