Effects Of Nonlocal Elasticity Research Articles

Effect of Nonlocal Elasticity and Phase Lags on the Magneto Thermoelastic Waves in a Composite Cylinder with Hall Current

In this study, the effect of nonlocal scale value and two phases lags on the free vibration of generalized magneto thermoelastic multilayered LEMV (Linear Elastic Material with Voids)/CFRP (Carbon Fiber Reinforced Polymer) composite cylinder is studied using nonlocal form of linear theory of elasticity. The governing equation of motion is established in longitudinal axis and variable separation model is used to transform the governing equations into a system of differential equations. To investigate vibration analysis from frequency equations, the stress free boundary conditions are adopted at the inner, outer and interface boundaries. The graphical representation of the numerically calculated results for frequency shift, natural frequency, and thermoelastic damping is presented. A special care has been taken to inspect the effect of nonlocal parameter on the aforementioned quantities. The results suggest that the nonlocal scale and the phase lag parameters alter the vibration characteristics of composite cylinders significantly.

Plane waves in Moore–Gibson–Thompson thermoelasticity considering nonlocal elasticity effect

The Moore–Gibson–Thompson (MGT) generalized thermoelasticity theory together with the Eringen’s nonlocal elasticity model is used to study the propagation of harmonic plane waves in an infinite elastic medium that conducts heat. Two sets of coupled longitudinal thermoelastic waves have been initiated. These waves are dispersive in nature and experience attenuation. In conjunction with the coupled waves, there also exists one uncoupled vertically shear–type (SV-type) wave. The SV-type wave is dispersive in nature but without any attenuation. The elastic nonlocality parameter influences all these waves. Furthermore, the SV-type wave faces critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. We also explore the problem of reflection of thermoelastic waves at a stress-free insulated and isothermal plane boundary of the medium considered here. The reflection coefficients and their respective energy ratios of the reflected waves to that of incident wave are determined analytically and numerically by considering an appropriate material. The numerical results are presented graphically to highlight the effects of various parameters of interest. Some important observations about the prediction of the MGT model in comparison to the other existing models (Lord–Shulman (LS) and the Green–Nagdhi theory of type III (GN-III) models) are highlighted.

Nonlinear Vibration Analysis of Fractional Viscoelastic Nanobeam

PurposeConsidering the size-dependent influence ignored by classical continuum mechanics, a new non-classical Euler–Bernoulli beam model is proposed in this paper. The new fractional viscoelastic nanobeam model is set up using the fractional Kelvin–Voigt viscoelastic model and Hamilton’s principle. And the new model studies the total effects of nonlocal elasticity, modified couple stress, and surface energy.MethodsThe model represented the fractional integral-partial differential governing equation is solved by Galerkin’s and predictor–corrector methods. First, the effects of nonlocal elasticity, modified couple stress, surface energy, and their coupling impact on the nonlinear time response of free vibration of fractional viscoelastic nanobeam are analyzed. Then, in the frame of nonlocal couple-stress elasticity and surface energy theory, the effects of different parameters on the nonlinear time responses of free and forced vibration of fractional viscoelastic nanobeam are discussed.Results and ConclusionThe numerical results show that the fractional order must be considered in the modeling of viscoelastic nanobeam, and the system’s damping enhances by the increase of fractional order. Moreover, owing to the correlation between fractional order and excitation frequency, the nonlinear time responses of fractional order to free and forced vibration are different.

Size-Dependent Vibration of Porous Bishop Nanorod with Arbitrary Boundary Conditions and Nonlocal Elasticity Effects

Size-Dependent Vibration of Porous Bishop Nanorod with Arbitrary Boundary Conditions and Nonlocal Elasticity Effects

Size-dependent vibration of single-crystalline rectangular nanoplates with cubic anisotropy considering surface stress and nonlocal elasticity effects

The presented paper investigates size-dependent vibration of single-crystalline rectangular nanoplates with cubic anisotropy based on combined surface stress and nonlocal elasticity effects. For this, the fundamental equations are derived with a discussion about the anisotropy formulations. The generalized differential quadrature method (GDQM) is used to solve the differential equation for applicable boundary conditions. The convergence and stability of the results are verified to obtain a suitable number of grid points. At first, the effective Young’s modulus versus the thickness is verified by the existing experiments. Then, the problem is simulated for a complete set of single crystal orientation and a wide range of size scale parameters and material types. The results indicate that size scale parameters have different vibration effects at different material orientations and boundary conditions. The size-scale effects decrease the natural frequency of CCFF (two edges clamped) and CCCC (fully clamped) nanoplates rotates from [100] to [110] axis for FCC (Face Centered Cubic ) materials in contrast with cantilevered nanoplates (CFFF).

L–S theory for the propagation of the photo-thermal waves in a semiconducting nonlocal elastic medium

The model of the equations of generalized thermoelasticity based on the Lord–Shulman's theory with nonlocal elasticity effect is used to study the photothermal waves in a semiconducting medium. The exact expressions for the displacement components, temperature, carrier density, and stress components are obtained by using normal mode analysis. Numerically simulated results are obtained and presented graphically for silicon. The comparisons are shown graphically to explore the effects of the time and the elastic nonlocal parameter on the different physical quantities.

Postbuckling Analysis of a Nonlocal Nanorod Under Self-Weight

This paper presents the postbuckled configurations of simply supported and clamped-pinned nanorods under self-weight based on elastica theory. Numerical solution is considered in this work since closed-form solution of postbuckling analysis under self-weight cannot be obtained. The set of nonlinear differential equations of a nanorod including the effect of nonlocal elasticity are investigated. The constraint equation at boundary condition technique is introduced for the solution of postbuckling analysis. In order to solve the set of nonlinear differential equations, the shooting method is utilized, where the set of these equations along with boundary conditions are integrated by the fourth-order Runge-Kutta algorithm. Numerical results are obtained and the highlighting influences of the nonlocal elasticity on postbuckling behavior of nanorods are discussed. The obtained results indicate that the rotation angle and the postbuckled configurations of nanorods are varied by changing the nonlocal elasticity parameter. The effect of nonlocal elasticity shows the softening behavior in comparison with the Euler beam. The present formulation together with constraint boundary condition technique is an effective solution for postbuckling analysis of a nanorod under self-weight including the effect of nonlocal elasticity.

Nonlocal elasticity effect on Linear Vibration of Nano-circular plate using Adomian decomposition method

In this study, the small scale effect on the linear free-field vibration of a nano-circular plate has been investigated using nonlocal elasticity theory. The formulation is based on the classical theory and the linear strain in cylindrical coordinates. To take into account the small scale and the linear geometric effects, the governing differential equation based on the nonlocal elasticity theory was extracted from Hamilton principle while the inertial effect, as well as the shear stresses effect was ignored. Effect of nonlocal parameter is investigated by solving the governing equation using Adomian decomposition method (ADM) for the clamped and simply supported boundary conditions. By using this method, the first five axisymmetric natural frequencies and displacements of nano-circular plate are obtained one at a time and some numerical results are given to illustrate the influence of nonlocal parameters on the natural frequencies and displacements of the nano-circular plate. For the purpose of comparison, the linear equations were solved by the analytical method. Excellent agreements were observed between the two methods. This indicates that the latter method can be applied to seek the linear solution of nano-circular plates with high accuracy while simplifying the problem.

Confinement-Induced Control of Similarity Solutions in Premelting Dynamics and Other Thin Film Problems

We study the combined effects of nonlocal elasticity and confinement induced ordering on the dynamics of thermomolecular pressure gradient driven premelted films bound by an elastic membrane. The confinement induced ordering is modeled using a film thickness dependent viscosity. When there is no confinement induced ordering, we recover the similarity solution for the evolution of the elastic membrane, which exhibits an infinite sequence of oscillations. However, when the confinement induced viscosity is comparable to the bulk viscosity, the numerical solutions of the full system reveal the conditions under which the oscillations and similarity solutions vanish. Implications of our results for general thermomechanical dynamics, frost heave observations and cryogenic cell preservation are discussed. Finally, through its influence on the viscosity, the confinement effect implicitly introduces a new universal length scale into the volume flux. Thus, there are a host of thin film problems, from droplet breakup to wetting/dewetting dynamics, whose properties (similarity solutions, regularization, and compact support) will change under the action of the confinement effect. Therefore, our study suggests revisiting the mathematical structure and experimental implications of a wide range of problems within the framework of the confinement effect.

Effect of nonlocal elasticity on vibration analysis of multi-layer graphene sheets using sandwich model

Abstract The essence of nonlocal elasticity for vibration analysis of multi-layer graphene sheets (MLGSs) is investigated in the following study. The van der Waals interactions of every two adjacent layers are considered in analysis which results in interlayer shear effect. The proposed formulation is according to sandwich model (SM). Molecular Dynamic (MD) simulation is implemented to verify our model. We present an investigation to obtain the consistent values for SM parameters to comply with MD results. Afterward, the SM is integrated with nonlocal elasticity. The new calibrated values for nonlocal parameter lead to the best conformity of nonlocal SM with MD.

Bending and vibrational behaviors of piezoelectric nonlocal nanobeam including surface elasticity

ABSTRACTThis manuscript illustrates coupled effects of nonlocal elasticity and surface properties on static and vibration characteristics of piezoelectric nanobeams using thin beam theory. The mechanical and piezoelectric surface nanoscale properties are governed by Gurtin–Murdoch model. The length scale effect is imposed to the problem by including nonlocal elasticity theory to describe the long–range atoms interactions. The governing equations are derived by Hamilton’s principle and solved numerically using the finite-element method. The proposed model is verified and validated with previous published works. Numerical results illustrate the effects of nonlocal parameter, surface elasticity, and boundary conditions on the bending and dynamic characteristics of the nanobeam. It is found that, the nonlocal effect and the surface piezoelectricity effect play a significant role on the static deflection and the natural frequencies. The obtained results are in good agreement with previous published works. This study should have useful insights on the design, fabrication and applications of piezoelectric-beam-based nanodevices.

An uncoupled higher-order beam theory and its finite element implementation

A beam problem is though classical but not theoretically settled down at present. Different from the previous work, the current study starts with defining the generalized displacements. Together with the assumptions and the shear stress free condition, the axial displacement is first mathematically expanded in two terms and then decomposed into an orthogonal form in terms of the generalized displacements. The generalized stresses are accordingly defined, and the uncoupled constitutive relations are derived for beam problems after the generalized strains are properly measured. The principle of virtual work is proposed and the variationally consistent higher-order beam theory is eventually established, which can reduce to the variationally asymptotic lower-order beam theory. With these fundamentals, the finite element method is finally formulated as easy as for a three-dimensional elastic problem, and applied to typical problems. The results show that the higher-order beam element can capture the effect of the clamped end and the load jump via smoothly modeling the warping of cross section by using a locally refined mesh while accurately modeling the deflection. With the current framework, modern beam structures accounting for the effect of nonlocal elasticity, small scales and material heterogeneities can be readily solved.

A variational approach for large deflection of ends supported nanorod under a uniformly distributed load, using intrinsic coordinate finite elements

• Very large deflection is presented by using intrinsic coordinate finite elements. • The Lagrange multiplier technique is applied to impose the boundary condition. • The finite element method is used for the numerical solution of the nonlinear system. • The coupled effects of nonlocal elasticity and surface stress are included. This paper presents a variational formulation that can be used for large deflection analysis of ends supported nanorod including the coupled effects of nonlocal elasticity and surface stress under a uniformly distributed load. The variational formulation involving the strain energy due to bending of nonlocal elasticity including the surface stress effect and virtual work done by a uniformly distributed load, is expressed in terms of the intrinsic coordinates. The Lagrange multiplier technique is applied to impose the boundary conditions which accomplished in the formulation. The validity of the variational approach is ensured by Euler's equation, which identical to the one derived by the force equilibrium consideration of an infinitesimal nanorod segment. The finite element method and Newton–Raphson iterative procedure based on the variational formulation are used to solve a system of nonlinear equations. Moreover, the very large deflection configurations of ends supported nanorod are highlighted in this study.

Effect of nonlocal elasticity on the performance of a flexoelectric layer as a distributed actuator of nanobeams

The paper is concerned with the development of finite element model for the static analysis of smart nanobeams integrated with a flexoelectric layer on its top surface, using nonlocal elastic theory. The flexoelectric layer acts as a distributed actuator of the nanobeam. A layerwise displacement theory has been used to derive the element stiffness matrices from variational principles incorporating nonlocal effects. The finite element model for nonlocal response of the beams has been validated with the exact solution for the case of a simply supported standalone flexoelectric layer. Also, the finite element model of the simply supported smart beam has been validated with exact solutions and numerical models for the local elastic case. The performance of the flexoelectric actuator has been compared for different values of nonlocal parameters and different combinations of nonlocal and local elastic substrate and flexoelectric layer. Further, the model developed has been utlized for investigating the performance of the active flexoelectric layer in case of cantilever beam, for which the exact solutions are not available.

Dynamics of silicon nanobeams with axial motion subjected to transverse and longitudinal loads considering nonlocal and surface effects

A microstructure-dependent dynamic model for silicon nanobeams with axial motion is developed by considering the effects of nonlocal elasticity and surface energy. The nanobeam is considered to subject to both transverse and longitudinal loads arising from nanostructural surface effect and all positive directions of physical quantities are defined clearly prior to modeling so as to clarify the confusions of sign in governing equations of previous work. The nonlocal and surface effects are taken into consideration in the dynamic behaviors of silicon nanobeams with axial motion including circular natural frequency, vibration mode, transverse displacement and critical speed. Various supporting conditions are presented to investigate the circular frequencies by a numerical method and the effects of many variables such as nonlocal nanoscale, axial velocity and external loads on non-dimensional circular frequencies are addressed. It is found that both nonlocal and surface effects play remarkable roles on the dynamics of nanobeams with axial motion and cause the frequencies and critical speed to decrease compared with the classical continuum results. The comparisons of the non-dimensional calculation values by present and previous studies validate the correctness of the present work. Additionally, numerical examples for silicon nanobeams with axial motion are addressed to show the nonlocal and surface effects on circular frequencies intuitively. Results obtained in this paper are helpful for the design and optimization of nanobeam-like microstructures based sensors and oscillators at nanoscale with desired dynamic mechanical properties.

A Semi-Analytical Evaluation of Surface and Nonlocal Effects on Buckling and Vibrational Characteristics of Nanotubes with Various Boundary Conditions

In the present paper, the vibrational and buckling characteristics of nanotubes with various boundary conditions are investigated considering the coupled effects of nonlocal elasticity and surface effects, including surface elasticity and surface tension. The nonlocal Eringen theory is adopted to consider the effect of small scale size, and the Gurtin–Murdoch model the surface effect. Hamilton’s principle is employed to formulate the governing equation and differential transformation method (DTM) is utilized to obtain the natural frequency and critical buckling load of nanotubes with various boundary conditions. The results obtained match the available ones in the literature. Detailed mathematical derivations are presented and numerical investigations are performed. The emphasis is placed on the effects of several parameters, such as the nonlocal parameter, surface effect, aspect ratio, mode number and beam size, on the normalized natural frequencies and critical buckling loads of the nanotube. It is explicitly shown that the vibration and buckling of a nanotube is significantly influenced by these effects. Numerical results are presented which may serve as benchmarks for future analysis of nanotubes.

Effects of nonlocal elasticity and slip condition on vibration and stability analysis of viscoelastic cantilever carbon nanotubes conveying fluid

In this study, effects of nonlocal elasticity and slip condition on free vibration and flutter instability analysis of viscoelastic cantilever carbon nanotubes (CNTs) conveying fluid are investigated. Nonlocal Euler–Bernoulli beam theory is employed to establish the governing equations of motion for the vibrational behavior of CNT. The material property of the CNT is simulated by Kelvin–Voigt viscoelastic constitutive relation. The slip boundary conditions of CNT conveying fluid are considered based on Knudsen number (Kn). The equation of motion and associated boundary conditions are derived by using the extended form of Hamilton’s principle. The extended Galerkin method is then employed to discretize the equation of motion and boundary conditions which are then solved to calculate the eigenvalues and critical flutter speeds. Detailed results are demonstrated for the dependence of internal moving fluid, nonlocal parameter, structural damping coefficients and Knudsen number on the dimensionless eigenvalues and flutter boundaries of the cantilever CNT. It can be concluded that the structural damping coefficients and Knudsen number does not have a significant effect on the eigenvalues. However, the results show that the structural damping coefficients, Knudsen number and nonlocal parameter have significant effect on the flutter boundaries of the cantilever CNTs.

Nonlinear Vibration Analysis of Single-Walled Carbon Nanotube With Shell Model Based on the Nonlocal Elasticity Theory

In this paper, nonlinear vibration of a single-walled carbon nanotube (SWCNT) with simply supported ends is investigated based on von Karman's geometric nonlinearity and nonlocal shell theory. The SWCNT is designated as an individual shell, and the Donnell's formulations of a cylindrical shell are used to obtain the governing equations. The Galerkin's procedure is used to discretized partial differential equations (PDEs) into the ordinary differential equations (ODEs) of motion, and the method of averaging is applied to obtain an analytical solution of the nonlinear vibration of (10,0), (20,0), and (30,0) zigzag SWCNTs. The effects of the nonlocal parameters, nonlinear parameters, different aspect ratios, and different circumferential wave numbers are investigated. The results of the classical and the nonlocal models are compared with different nonlocal elasticity constants (e0a). It is shown that the nonlocal parameter predicts different resonant frequencies in comparison to the local models. The softening and/or hardening nonlinear behaviors of the CNTs may change against the nonlocal parameters. Hence, considering the geometrical nonlinearity and the nonlocal elasticity effects, the dynamical models of the SWCNTs predict their vibration behaviors accurately and should not be ignored during theoretical modeling.

Surface effects on static bending of nanowires based on non-local elasticity theory

The surface elasticity and non-local elasticity effects on the elastic behavior of statically bent nanowires are investigated in the present investigation. Explicit solutions are presented to evaluate the surface stress and non-local elasticity effects with various boundary conditions. Compared with the classical Euler beam, a nanowire with surface stress and/or non-local elasticity can be either stiffer or less stiff, depending on the boundary conditions. The concept of surface non-local elasticity was proposed and its physical interpretation discussed to explain the combined effect of surface elasticity and non-local elasticity. The effect of the nanowire size on its elastic bending behavior was investigated. The results obtained herein are helpful to characterize mechanical properties of nanowires and aid nanowire-based devices design.

Postbuckling of Unknown-Length Nanobeam Considering the Effects of Nonlocal Elasticity and Surface Stress

The objective of this paper is to study the postbuckling behaviors of an unknown-length nanobeam combined with small-scale effects. The concept of variable-arc-length elastica is firstly applied on the problem of nanobeams. The span length is not changed while the arc length is varied increasingly. The nanobeam is on a clamped support at one end, while the other end is an overhanging part through a frictionless slot subjected to axial compression. At this end, the nanobeam is movable only in a horizontal direction. The governing equation is developed by the moment–curvature relationship based on the classical Euler–Bernoulli beam theory, including the effects of nonlocal elasticity, residual surface stress, and both combined effects. The shooting–optimization technique with two-point boundary condition is employed to solve the differential equations in this problem. The results, including nonlocal elasticity, reveal that nanobeams have decreased structural stiffness; meanwhile, the residual surface tension and both combined effects have increased strength. The postbuckling loads decrease as the arc length of nanobeams is increased. The equilibrium configurations are close to an anti-loop for very large deflections. The friction force at the nanoslot is also considered.