Effects Of Nonlocal Parameter Research Articles

A Modified Couple Stress-Based Model for the Nonlinear Vibrational Analysis of Nano-Disks Using Multiple Scales Method

In this article, the nonlinear vibrational behavior of a nano-disk was analyzed using the multiple scales method (MSM). The modified couple stress theory was used to consider the small-scale effect via the application of nonlocal parameter. Employing Hamilton's principle, two coupled nonlinear differential equations were derived based on the nonlinear von-Karman strain-displacement relation and the classical plate theory. The Galerkin-based procedure was utilized to obtain a Duffing-type nonlinear ordinary differential equation with a cubic nonlinear term and solved by the application of MSM. The effects of nonlocal parameter, aspect ratio, different boundary conditions, and the nonlinear shift frequencies, were obtained on the overall behavior of the nano-disk. Results indicate that increasing the central dimensionless amplitude of the nano-disk, the nonlinear frequency, and the shift index exhibit an increasing behavior, while the increase in the non-dimensional nonlocal parameter, causes a decrease in the nonlinear frequency ratios and the shift index. Additionally, the increase in h/r increases the effect of dimensionless central amplitude on the nonlinear frequencies ratios. Additionally, comparison of the current results with those previously published in the literature shows good agreements. This indicates that the MSM can ease up the solution, and hence, can be applied to the solution of nonlinear nano-disks with high accuracy.

Transmission phenomenon at the interface between isotropic and semiconductor nanostructure based on nonlocal theory

In this article, we analyzed the response of elastic waves after striking the interface between two elastic mediums. The upper-medium is isotropic while the lower medium is semiconductor nanostructure in nature. Both the mediums are set on rotation with some fix angular frequency . A fractional ordered heat conduction model of three-phase lag along with the nonlocal theory of elasticity is adopted to formulate the problem. The incident wave after striking the interface results in five coupled quasireflected waves and three elastic transmitted waves. The ratios of the amplitude of reflection and transmitted waves with incident wave are calculated. Variations in obtained amplitude ratios with respect to incident angle have been shown graphically for silicon (Si) semiconductor. The effects of nonlocal parameter, rotational frequency and fractional order parameter on the reflection and transmitted coefficients in the presence of voids parameters are studied. The results obtained through the current analysis may be applicable in the field of seismology and semiconductors nanostructure appliances.

Eringen’s nonlocal and modified couple stress theories applied to vibrating rotating nanobeams with temperature effects

This study develops a comprehensive vibrational analysis of rotating nanobeams on visco-elastic foundations with thermal effects based on the modified couple stress and Eringen’s nonlocal elasticity theories. This approach accurately simulates the nonlocal stress and size effects. Higher-order shear deformation beam theory and the generalized differential quadrature method are used to obtain the numerical results. The effects of nonlocal parameters, length scale, Winkler–Pasternak coefficients, thermal gradient, slenderness ratios, rotating velocity, and viscoelastic coefficient are demonstrated and discussed in detail. Mode switching and the importance of the correct choice of theory and associated size effect parameters are highlighted.

Plane waves in nonlocal viscothermoelastic medium with temperature-dependent properties under three-phase-lag theory

The present investigation deals with the study of propagation of plane waves in a homogeneous, isotropic, nonlocal viscothermoelastic half space with temperature-dependent properties in the context of three-phase-lag (TPL) model. The Kelvin–Voigt model of linear viscoelasticity is employed to describe the viscoelastic nature of isotropic material. A set of coupled longitudinal waves and one independent transverse wave are found to travel with distinct speeds in the medium. Reflection coefficients and energy ratios of various reflected waves are presented when (1) a set of coupled longitudinal waves is made incident and (2) a transverse wave is made incident. The results for partition of the energy for various values of the angle of incidence are computed numerically and presented graphically for copper material. It has been verified that there is no dissipation of energy at the boundary surface during reflection. The effects of nonlocal parameter, viscosity and temperature dependent properties on reflection coefficients are depicted graphically. A comparison between the TPL model and GNIII model is also presented by different characteristics of wave propagation like phase velocity and reflection coefficients.

Flexomagnetic response of buckled piezomagnetic composite nanoplates

In this paper, the equation governing the buckling of a magnetic composite nanoplate under the influence of an in-plane one-dimensional magnetic field, assuming the concept of flexomagnetic and considering the resulting flexural force and moment, is investigated for the first time by different analytical boundary conditions. To determine the equation governing the stability of the nanoplate, the nonlocal strain gradient theory has been used by taking into account the classical plate theory. The axial magnetic force, which is originated from the magnetic field, is investigated. After extracting the governing differential equation, the critical buckling load is obtained for different support conditions. The effect of nonlocal parameter, sheet aspect ratio and the effect of one-dimensional magnetic field on critical load are discussed. It was earned that if the nanoplate is rectangular so that the value of aspect ratio is less than one, the flexomagnetic response will be more noticeable.

A novel size-dependent nonlocal strain gradient isogeometric model for functionally graded carbon nanotube-reinforced composite nanoplates

The paper presents a novel nonlocal strain gradient isogeometric model for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates. To observe the length scale and size-dependency effects of nanostructures, the nonlocal strain gradient theory (NSGT) is considered. The present model is efficient to capture both nonlocal effects and strain gradient effects in nanoplate structures. In addition, the material properties of the FG-CNTRC are assumed to be graded in the plate thickness direction. Based on the higher order shear deformation theory (HSDT), the weak form of the governing equations of motion of the nanoplates is presented using the principle of virtual work. Afterward, the natural frequency and deflection of the nanoplates are made out of isogeometric analysis (IGA). Thanks to higher order derivatives and continuity of NURBS basic function, IGA is suitable for the weak form of NSGT which requires at least the third-order derivatives in approximate formulations. Effects of nonlocal parameter, strain gradient parameter, carbon nanotube (CNT) volume fraction, distributions of CNTs and length-to-thickness ratios on deflection and natural frequency of the nanoplates are examined and discussed in detail. Numerical results are developed to show the phenomenon of stiffness-softening and stiffness-hardening mechanisms of the present model.

Lateral buckling analysis of nanotubes and nanorings under uniform external pressure: a closed-form nonlocal solution

This study presents a modified cylindrical shell theory to predict the critical lateral buckling of both nanotubes and nanorings under uniform external pressure. To this end, nonlocal elasticity theory is incorporated into Donnell’s cylindrical shell model and closed-form critical lateral buckling solutions for different nanotubes and nanorings are derived from the coupled displacement equations. In this regard, a novel method is presented for decoupling the displacement equations to facilitate the derivation of the equations and results. Also, the effects of nonlocal parameter, aspect ratio, and mode numbers are investigated for lateral buckling of short and long nanotubes as well as nanorings. Results show that increasing the nonlocal effects, aspect ratio, and mode numbers of the nanotubes can reduce the value of the critical lateral pressure. Finally, the numerical results are compared with existing literature and show good agreement. The results of the new model are more accurate to capture the size effects for nanotubes and nanorings. These findings are useful for designing nanoscale devices.

Transverse vibration of tapered nanobeam with elastic supports

A tapered nanobeam is modelled by Euler–Bernoulli beam theory and nonlocal theory of elasticity. A solution technique based on homotopy perturbation method is proposed to analyse the dynamic behavior of the tapered nanobeam. Beam becomes tapered along the width exponentially. The effects of flexible and rigid supports on the vibration of tapered nanobeam are analysed. Influences of tapered ratio and nonlocal parameter on the shape mode of deflection are also scrutinized. Homotopy perturbation method shows its effectiveness in this analysis. The results reveal that the effects of nonlocal parameter and tapered ratio are significant for the dynamic behavior of tapered nanobeam.

Stoneley wave propagation in nonlocal isotropic magneto-thermoelastic solid with multi-dual-phase lag heat transfer

In the present paper we have investigated the Stoneley wave propagation at the interface of two dissimilar homogeneous nonlocal magneto-thermoelastic media under the effect of hall current applied to multi-dual-phase lag heat transfer. The secular equations of Stoneley waves have been derived by using appropriate boundary conditions. The wave characteristics such as attenuation coefficients, temperature distribution and phase velocity are computed and have been depicted graphically. Effect of nonlocal parameter and hall effect are studied on the attenuation coefficient, phase velocity, temperature distribution change, stress component and displacement component. Also, some particular cases have been discussed from the present study.

Nonlocal thermoelasticity and its application in thermoelastic problem with temperature-dependent thermal conductivity

Thermoelastic analysis at the nanoscale is becoming important due to the miniaturization of the device and wide application of ultrashort lasers, and the classical thermoelastic theory is no longer applicable under extreme environments, i.e. extremely high temperature gradient or heat flux, extremely short action time, and extremely small structure size. The nonlocal thermoelastic model is developed to predict the thermoelastic behavior of nanostructures under extreme environments in this paper. The governing equations with temperature-dependent thermal conductivity are solved by Kirchhoff and Laplace transformation. As a numerical example, the transient thermoelastic responses of a slab with temperature-dependent thermal conductivity are investigated. From numerical results, the effects of nonlocal parameters and the temperature-dependent thermal conductivity are discussed, systematically. The results show that the above parameters have significant effects on the transient thermoelastic responses, which is crucial to predict the thermoelastic response accurately for the design and processing of the nanostructures.

A nonlocal Layerwise theory for free vibration analysis of nanobeams with various boundary conditions on Winkler-Pasternak foundation

In this study, a nonlocal Layerwise theory is presented for free vibration analysis of nanobeams resting on an elastic foundation. Eringen's nonlocal elasticity theory is used to consider the small-scale effect on behavior of nanobeam. The governing equations are obtained by employing Hamilton's principle and Layerwise theory of beams and Eringen's nonlocal constitutive equation. The presented theory takes into account the in-plane and transverse normal and shear strain in the modeling of the nanobeam and can predict more accurate results. The governing equations of the beam are solved by Navier's method for Simple-Simple boundary conditions and semi-analytical methods to obtain the natural frequency for various boundary conditions including Clamped-Simple (C-S), Clamped-Clamped (C-C) and Free-Free (F-F) boundary conditions. Predictions of the present theory are compared with benchmark results in the literature. Effects of nonlocal parameter, Pasternak shear coefficient, Winkler spring coefficient, boundary conditions, and the aspect ratio on the free vibration of nanobeams are studied. The flexural mode and thickness mode natural frequencies of the nanobeam are predicted. It is shown that the predictions of present method are more accurate than the equivalent single layer theories. The theoretical developments and formulation presented herein should also be served to analyze the mechanical behavior of various nanostructures with various loading and boundary conditions.

Nonlinear vibration and stability analysis of a size-dependent viscoelastic cantilever nanobeam with axial excitation

In the present paper, nonlinear forced vibrations of an axial moving nanobeam which is vertically influenced by an external harmonic excitation and gravity is analyzed by considering the effects of linear damping. Considering certain assumptions, a nonlinear Euler-Bernoulli beam theory is developed. With the implementation of the nonlocal elasticity theory, the governing integro-partial-differential equation is obtained by using the Hamilton principle. The multiple scale method is employed to obtain a steady-state response for the size-dependent viscoelastic nanobeam with fixed-free boundary conditions. Subsequently, the trivial and non-trivial steady-state response and the bifurcation point types are examined. Finally, the effects of damping coefficient and nonlocal parameter on stability and bifurcation of trivial and non-trivial solutions are studied. It is found that the effect of nonlocal parameter on the steady-state response and the bifurcation point types is quite important.

Three-dimensional nonlocal thermoelasticity in orthotropic medium based on Eringen’s nonlocal elasticity

ABSTRACT The present article deals with thermoelastic boundary value problem in three-dimensional orthotropic medium based on Eringen’s nonlocal elasticity. A time-dependent thermal shock is applied on the traction-free surface of homogeneous orthotropic solid and the problem is treated in the context of Green-Naghdi model type III. The governing equations of three-dimensional problem of generalized thermoelasticity are obtained as a vector matrix differential equation form by employing normal mode analysis to the considered equations which are then solved by eigenvalue approach. The effect of nonlocal parameter on the displacements and temperature are presented graphically.

Nonlinear stress-driven nonlocal formulation of Timoshenko beams made of FGMs

Motivated by the paradoxical results obtained from the differential nonlocal elasticity theory in some cases (e.g., bending and vibration problems of cantilevers), several attempts have been recently made to develop nonlocal beam models based on the integral (original) formulation of Eringen’s nonlocal theory. These models can be classified into two main groups including strain- and stress-driven ones which have the capability of capturing the softening and hardening behaviors of material caused by nanoscale (nonlocal) effects, respectively. In the present paper, a novel stress-driven nonlocal formulation is developed for the nonlinear analysis of Timoshenko beams made of functionally graded materials. To this end, the governing equations are first derived in the context of integral form of stress-driven nonlocal model. The proposed model can be used for arbitrary kernel functions, and the paradox related to cantilever is resolved by it. The governing equations of stress-driven model in differential form together with corresponding constitutive boundary conditions are also derived. The Timoshenko beam under various end conditions is considered as the problem under study whose nonlinear static bending is analyzed. Furthermore, the generalized differential quadrature method is employed in the solution procedure. The effects of nonlocal parameter, FG index, length-to-thickness ratio and nonlinearity on the deflection of fully clamped, fully simply supported, clamped–simply supported and clamped–free beams are investigated. The presented formulation and results may be helpful in understanding nonlocal phenomena in nano-electro-mechanical systems.

Vibration analysis of porous nanocomposite viscoelastic plate reinforced by FG-SWCNTs based on a nonlocal strain gradient theory

This paper investigates the size dependent effect on the vibration analysis of a porous nanocomposite viscoelastic plate reinforced by functionally graded-single walled carbon nanotubes (FG-SWCNTs) by considering nonlocal strain gradient theory. Therefore, using energy method and Hamilton\'s principle, the equations of motion are derived. In this article, the effects of nonlocal parameter, aspect ratio, strain gradient parameter, volume fraction of carbon nanotubes (CNTs), damping coefficient, porosity coefficient, and temperature change on the natural frequency are perused. The innovation of this paper is to compare the effectiveness of each mentioned parameters individually on the free vibrations of this plate and to represent the appropriate value for each parameter to achieve an ideal nanocomposite plate that minimizes vibration. The results are verified with those referenced in the paper. The results illustrate that the effect of damping coefficient on the increase of natural frequency is significantly higher than the other parameters effect, and the effects of the strain gradient parameter and nonlocal parameter on the natural frequency increase are less than damping coefficient effect, respectively. Furthermore, the results indicate that the natural frequency decreases with a rise in the nonlocal parameter, aspect ratio and temperature change. Also, the natural frequency increases with a rise in the strain gradient parameter and CNTs volume fraction. This study can be used for optimizing the industrial and medical designs, such as automotive industry, aerospace engineering and water purification system, by considering ideal properties for the nanocomposite plate.

Nonlocal theory of thermoelastic diffusive materials and its application in structural dynamic thermo-elasto-diffusive responses analysis

With the rapid development of ultrafast heating technologies and its application in the micro-machining of micro/nano-electromechanical devices, where the nonlocal effects on elastic deformation and heat/mass transfer will increase and classical/generalized theory of thermoelastic diffusion coupling does not hold any more. In this work, a nonlocal theory of thermoelastic diffusive materials is developed, and the spatial nonlocal effects of heat and mass transport are both considered for the first time. With the aid of thermodynamic principles, the constitutive relations involving size-dependent characteristic lengths are derived. In the aspect of the application, the proposed theoretical model is further used to investigate the dynamic thermo-elasto-diffusive responses of a one-dimensional semi-infinite medium with one end subjected to time-dependent shock loads, and the semi-analytical techniques based on Laplace transform methods are applied to obtain the transient solutions. The effects of nonlocal parameters on structural dynamic thermo-elasto-diffusive responses are also evaluated and discussed to be expected to provide new insights on the thermoelastic diffusion coupling at the micro/nanoscale.

Two types of scale effects on the nonlinear forced vibration of axially moving nanobeams

Various non-classical continuum mechanics models appearing in previous studies cannot perfectly explain the mechanical properties of micro- and nanomaterials. Establishing a reasonable continuum mechanics model that comprehensively reflects the scale effect on material deformation is of great practical significance for objectively explaining the variation law of mechanical properties of micro- and nanomaterials under the combined action of different scale effects. Based on nonlocal strain gradient theory, a new scale-dependent model is proposed for axially moving nanobeams. In this study, an asymptotic expansion is performed using the multiscale time method to obtain the amplitude-frequency response curve of the equilibrium solutions for the forced vibration problem. Afterwards, the effects of various system parameters, especially the scale parameters, on the resonance curve are examined. Finally, the effects of nonlocal parameters and material characteristic length parameters on the amplitude-frequency response curves are investigated through typical numerical examples. The numerical results show that the nonlocal parameters promote the emergence of the main resonance, whereas the material characteristic length parameters suppress the emergence of the main resonance. Moreover, these parameters also affect the response amplitude and the skewness and jumping point of the amplitude-frequency characteristic curve.

Nonlinear Dynamics of Composite Microsheet with Graphene Skins in Non-uniform Thermal Field

The effects of different parameters on the nonlinear dynamic characteristics of macrofiber composite (MFC) microsheet with graphene (GP) skin under non-uniform thermal field are investigated. Firstly, the physical parameters of the MFC–GP structure are calculated by the mixing rule, and the constitutive equations of the structure are set up by employing the Eringen theory. The nonlinear dynamic equations of the microsheet are obtained by using Hamilton’s principle. Then, the heat conduction equations of the microsheet are considered, adopting Green and Naghdi’s generalized thermoelasticity theory. According to the Galerkin weighted residual method, the thermoelasticity coupling equations of the structure are obtained. Meanwhile, the influence of the positive piezoelectric effect of GP and MFC on the vibration response of the structure is also investigated. The nonlinear dynamic governing equations including displacement, coupled thermoelasticity, and electricity field are discretized by the Galerkin method. The effects of non-local parameter, volume fraction of GP, and thermal and electricity coupling coefficients on structural dynamic behavior are discussed in the numerical simulation.

Size-dependent free vibration and buckling analysis of sigmoid and power law functionally graded sandwich nanobeams with microstructural defects

The investigation conducted in this paper aims to study free vibration and buckling behaviors of size-dependent functionally graded sandwich nanobeams. In order to take into account the small size effects, nonlocal elasticity theory of Eringen's is incorporated. Material properties of the functionally graded sandwich beams are supposed to change continuously through the thickness direction according to two forms of the volume fraction of constituents by power law functionally graded material and sigmoid law functionally graded material. These rules are modified to consider the effect of porosity, which covers four kinds of porosity distributions. Two types of sandwich nanobeams were provided: (a) homogeneous core and functionally graded skins and (b) functionally graded core and homogeneous skins. Third-order shear deformation theory without any shear correction factor in conjunction with Hamilton's principle is used to extract the governing equations of motions of porous functionally graded sandwich nanobeams and then solved analytically for two hinged ends. The effects of nonlocal parameter, length to thickness ratios, material graduation index, amount of porosity, porosity distribution shape, on the nondimensional frequency and critical buckling load of the functionally graded sandwich nanobeams made of porous materials are exhibited by a parametric study.

Nonlocal Dispersion Analysis of a Fluid – Conveying Thermo Elastic Armchair Single Walled Carbon Nanotube Under Moving Harmonic Excitation

In this work, the nonlocal elastic waves in a fluid conveying armchair thermo elastic single walled carbon nanotube under moving harmonic load is studied using Eringen nonlocal elasticity theory via Euler Bernoulli beam equation. The governing equations that contains partial differential equations for single walled carbon nanotube is derived by considering thermal and Lorenz magnetic force. The small scale interactions induced by the nano tubes are simulated by the non-local effects. The time domain responses are obtained by using both modal super position method and Newmarks’s direct integration method. The effect of nonlocal parameter, thermal load, magnetic field of the moving harmonic load on the dynamic displacement of SWCNT is discussed. The results obtained show that the dynamic displacement of fluid conveying SWCNT ratio is significantly affected by the load velocity and the excitation frequency. This type of results presented here, will provide useful information for researchers in structural nano science to understand the small scale response of elastic waves coupled with thermo elasticity and some field forces.