Nonlocal Theory Research Articles

Investigating nonlinear forced vibration of functionally graded nanobeam based on the nonlocal strain gradient theory considering mechanical impact, electromagnetic actuator, thickness effect and nonlinear foundation

The present study proposes an investigation to nonlinear vibration response of a functionally graded (FG) nanobeam based on nonlocal strain gradient theory (NSGT) taking into consideration the longitudinal magnetic field and the thickness effect. The nonlinear terms from the governing equation are associated with the curvature of the nanobeam, the nonlinear foundation, the mean axial extension of the nanobeam and electromagnetic actuator for pinned-pinned nanobeam and the effect of discontinuities marked by the presence of the Dirac function (mechanical impact).The governing differential equation of motion and the boundary conditions are modeled within the framework of a simple supported Euler-Bernoulli nanobeam which accounts for the nonlinear von-Kármán strain and using so-called Hamilton's principle. To truncate the continuous system having an infinite degree of freedom, the Galerkin-Bubnov procedure is applied. The resulting nonlinear differential equation is studied by Optimal Auxiliary Functions Method. An explicit analytical solution is proposed for a complex problem near the primary resonance. Our proper procedure is a powerful tool to solve a nonlinear problem without presence of any small parameters. The main characteristic of our technique consists in the existence of some auxiliary functions derived from the expressions of the solution formed for initial linear equation and the form of nonlinear terms calculated for the solution of the linear equation. Also, the presence of some convergence-control parameters assures the rapid convergence of the solutions after only a single iteration. These parameters can be evaluated by means of some rigorous mathematical procedures. We have a large freedom to select the auxiliary functions and the number of convergence-control parameters. The proposed approach is proved to be very accurate, simple, and easy to implement for complicated nonlinear problems. The potential energy is analyzed in detail for different values of the parameters. The local stability of motion near the primary resonance is established using homotopy perturbation method.

Nonlocal Near-Field Radiative Heat Transfer by Transdimensional Plasmonics.

Using transdimensional plasmonic materials (TDPM) within the framework of fluctuational electrodynamics, we demonstrate nonlocality in dielectric response alters near-field heat transfer at gap sizes on the order of hundreds of nanometers. Our theoretical study reveals that, opposite to the local model prediction, propagating waves can transport energy through the TDPM. However, energy transport by polaritons at shorter separations is reduced due to the metallic response of TDPM stronger than that predicted by the local model. Our experiments conducted for a configuration with a silica sphere and a doped silicon plate coated with an ultrathin layer of platinum as the TDPM show good agreement with the nonlocal near-field radiation theory. Our experimental work in conjunction with the nonlocal theory has important implications in thermophotovoltaic energy conversion, thermal management applications with metal coatings, and quantum-optical structures.

Electro-mechanical responses of transversely isotropic piezoelectric nano-plate based on the nonlocal strain gradient theory with flexoelectric effect

Electro-mechanical responses of transversely isotropic piezoelectric nano-plate based on the nonlocal strain gradient theory with flexoelectric effect

Propagation of plane waves in nonlocal semiconductor nanostructure thermoelastic solid with fractional derivative due to ramp-type heat source

In this paper, propagation of thermoelastic waves in semi-conducting nanostructure medium is studied. First, the governing equations are formulated with the help of coupled nonlocal elasticity theory, thermoelastic three-phase-lag model, and equation of motion. Heat equation with fractional order is incorporated to study thermal effects on the propagation of wave. The problem is solved using Fourier series and Laplace transform to obtain the exact expressions of physical fields. The distributions of the nondimensional carrier density and stresses are obtained and illustrated graphically. The effects of nonlocal parameter, fractional order parameter, and time parameter on these wave characteristics are concerned and discussed in detail. Some semiconductor materials as Silicon (Si) and Germanium (Ge) are used to make the numerical simulation and some comparisons in different thermal memories are made.

Investigation of the vibration behavior of nano piezoelectric rod using surface effects and non-local elasticity theory

Due to the importance of the analysis of piezoelectric nanorods and the different conditions they are exposed to in engineering problems, in this research, for the first time, both nonlocal elasticity theory and surface energy theory have been used simultaneously to model a piezoelectric nanorod. The natural frequency values of the system have been extracted using the Galerkin weighted residual method. By applying an external excitation voltage, the forced vibrations of nonlocal nanorods with surface effects are presented. The results are reported for different vibrational modes, and the impact of different parameters such as dimensions, nonlocal parameters, and surface effects are investigated. The results of this study prove that the nonlocal elastic theory is more effective than the surface energy theory on the vibration behavior of piezoelectric nanorods. Also, the effect of piezoelectric properties is strengthened due to the nanoscale of the rods.

Buckling Analysis of Beam on Winkler Foundation by Using Mdqm And Nonlocal Theory

In the present work buckling analysis of beam using Eringen nonlocal elasticity theory is being carried out. The associated governing differential equation is solved by the modified differential quadrature method (MDQM). The present MDQM employs Chebyshev polynomial for the determination of weighting coefficient matrices. The results obtained from present analysis are being validated with those reported in literature. Effect of number of interpolation points on the accuracy of the results is also investigated. It is found that seven numbers of interpolations are required to achieve reasonable accurate results for various nonlocal parameter values. It is also observed that the effect of nonlocal parameter on critical buckling load for the higher modes is higher and more nonlinear than the lower modes. Further the effect of (i) nonlocal parameter, (ii) Winkler elastic foundation moduli and (ii) boundary conditions on the critical buckling loads are being investigated and discussed.

Bending Vibration and Buckling Analysis Of Nonhomogeneous Nanotubes Using Nonlocal Elasticity Theory and Gdq Method

In this paper structural analysis of nonhomogeneous nanotubes has been carried out using nonlocal elasticity theory. Governing differential equations of nonhomogeneous nanotubes are derived. Nonlocal theory of elasticity has been employed to include the scale effect of the nanotubes. Nonlocal parameter, elastic modulus, density and diameter of the cross sections are assumed to be functions of spatial coordinates. General Differential Quadrature (GDQ) method has been employed to solve the governing differential equations of the nanotubes. Various boundary conditions have been applied to the nanotubes. Present results considering nonlocal theory are in good agreement with the results available in the literature. Effect of various geometrical and material parameters on the structural response of the nonhomogeneous nanotubes has been investigated. Present results of the nonhomogeneous nanotubes are useful in the design of the nanotubes.

Vibration and Small Scale Effects of Skew Graphene Sheets Using Nonlocal Elasticity Theory

Nonlocal elasticity theory is a popularly growing technique for the realistic analysis of nano structures. In the present work nonlocal elasticity plate theory has been employed and vibration analyses of skew graphene sheets are carried out. Relevant governing differential equations are reformulated using the nonlocal differential constitutive relations suggested by Eringen. The equations of motion including the nonlocal theory are derived. All edges of the skew graphene sheets are assumed to be simply supported. Naviers approach has been employed to solve the governing differential equations. Bauers skew plate analysis has been extended to include the nonlocal elasticity plate theory. Vibration response of the skew graphene sheets is studied. Effects of the (i) size of the graphene sheets (ii) modes of vibration (iii) nonlocal parameter and (iv) skew angle of graphene sheet on nonlocal vibration frequencies are investigated. It has been observed that the vibration response of the skew graphene sheets are influenced significantly by the nonlocal parameter.

Comment on “Transverse vibration analysis of single-layered graphene sheet under magneto-thermal environment based on non-local plate theory [J. Appl. Phys. 116(16), 164303 (2014)]” and “Nonlocal elasticity based magnetic field affected vibration response of double single-walled carbon nanotube systems [J. Appl. Phys. 111(11), 113511 (2012)

The impact of a time-varying external magnetic field on carbon nanotubes has been studied using Maxwell’s equations in the mentioned articles. However, the Maxwell’s equations are erroneously presented in the papers. The incorrect Maxwell’s equations, their derivatives, and the Lorentz force equations have all been examined in this work. In this article, we have endeavored to ensure that the equations are formatted correctly and have made an effort to do so. We have made an effort to intuitively correct the equations’ errors. We did not comment on the other portions of the mentioned articles; instead, we restricted ourselves to the examination of the impact of Maxwell’s equations and Lorentz force equations.

Vibration and Buckling Analysis of Nano-Scale Beams via Nonlocal Elasticity and Timoshenko Beam Theory : A Differential Quadrature Approach

A single-elastic beam model is being developed based on Eringens nonlocal elasticity and Timoshenko beam theory. Nonlocal parameter takes into account the small scale effects in the analysis of nano-size structures. Derived herein is a decoupled sixth-order nonlocal governing differential equations for the frequency and stability analysis of nano-scale short beams. The effect of shear deformation is thereby included in the small scale analysis. A Differential Quadrature (DQ) approach is being applied and its elaborate method of solution is illustrated. The higher order DQ approach is found to be a good numerical technique for the convenient and rapid solution of the aforementioned nonlocal problems. Frequency and buckling results for various scale-based nonlocal parameters are shown. Effect of number of interpolation points on the accuracy of the results is also investigated. It is seen that there is a significant effect of aspect ratio and nonlocal parameter on the nonlocal frequency and buckling loads of nano-scale beams. Also, the present study could open up a new approach of solution technique for the analysis of nano-scale structures based on nonlocal Timoshenko theory.

Pullback exponential attractors in nonlocal Mindlin's strain gradient porous elasticity

ABSTRACT In this paper, we study the long-time dynamics of a pullback exponential attractors for non-autonomous one dimensional system in nonlocal Mindlin's strain gradient porous elastic theory recently developed by Aouadi [Well-posedness, lack of analyticity and exponential stability in nonlocal Mindlin's strain gradient porous elasticity. Z Angew Math Phys. 2022;73:111]. In this theory, the second gradient of deformation and the second gradient of volume fraction field are added to the set of independent constitutive variables by taking into account the nonlocal length scale parameters effect. By virtue of Galerkin method combined with the priori estimates, we prove the existence and uniqueness of global solution. Then we establish a Lipschitz stability result. We also prove the existence of pullback exponential attractors which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. Finally, we prove the upper-semicontinuity of minimal pullback attractors with respect to the perturbations parameter in natural energy space.

Non-local Analyses of Tapered Beams

In the present article bending, vibration and buckling analyses of a tapered beam using Eringen non-local elasticity theory is being carried out. The associated governing differential equations are solved employing Rayleigh-Ritz method. Both Euler-Bernoulli and Timoshenko beam theories are considered in the analyses. Present results are in good agreement with those reported in literature. Non-local analyses for tapered beam with simply supported - simply supported (SS) , clamped - simply supported (CS) and clamped - free (CF) boundary conditions are conducted and discussed. It is observed that the maximum deflection increases with increase in non-local parameter value for SS and CS boundary conditions. Further, vibration frequency and critical buckling load decrease with increase in non-local parameter value for SS and CS boundary conditions. Non-local parameter effect on deflection, frequency and buckling load for CF supports is found to be opposite in nature to that of SS and CS supports. In case of thick beams non-local structural response is observed to be sensitive to length to thickness ratio.

Small Scale Effects and Vibration of Graphene Sheets with Various Boundry Conditions

Elastic theory of graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived. Levy’s approach has been employed to solve the governing differential equations for various boundary conditions. Nonlocal theories are employed to bring out the small scale effect of the nonlocal parameter on the natural frequencies of the graphene sheets. Present vibration results associated with various boundary conditions are in good agreement with those available in literature. Further, effects of (i) nonlocal parameter, (ii) size of the graphene sheets and (iii) boundary conditions on nondimensional vibration frequencies are investigated. The theoretical development as well as numerical solutions presented here in should serve as reference for nonlocal theories of nanoplates and nanoshells.

Forced Vibration Study of Carbon Nanotubes Using Non-Local Elasticity and Timoshenko Beam Theories

Forced vibrations of the single wall carbon nanotubes (SWCNTs) using the non-local elasticity theory and Timoshenko beam theory is studied. Governing equations for the vibration response of the carbon nanotubes are derived employing Eringen’s nonlocal elasticity theory. Analysis is carried out for both EulerBernoulli beam theory and Timoshenko beam theory. Present forced vibration results for Euler-Bernoulli are in good agreement with those reported in literature. Effects of (i) nonlocal parameter (ii) length of the carbon nanotube and (iii) modes of vibration on the vibration response of the carbon nanotubes are investigated. The effect of nonlocal parameter on the vibration response are significant for small size carbon nanotube and higher modes of vibration.

Buckling Analysis of Orthotropic Graphene Sheets Supported on Pasternak Elastic Medium Using Nonlocal Elasticity Theory and Differential Quadrature Method

In this paper, the small scale effect on the buckling analysis of bi-axially compressed orthotropic Single-Layered Graphene Sheets (SLGS) supported on elastic medium is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction between the graphene sheet and supporting elastic medium. Using the principle of virtual work the governing differential equations are derived for rectangular orthotropic graphene sheets supported on elastic medium. Solutions for buckling loads for various boundary conditions are computed using Differential Quadrature Method (DQM). Parametric study has been performed to investigate the dependence of small scale effect on various graphene sheet parameters. It is observed that the nonlocal effect is significant in graphene sheets supported on elastic medium and has a decreasing effect on the buckling loads.

Coronary artery lipid accumulation prevention through vibrating piezo electric nano plates embedded in smart stent.

Considering the lipid concentration and side effects regarding the stents used by surgeons, a new heart stent model is proposed. In the new stent, a few piezo plates are designed and attached to the stents by which release of the lipids can take place due to the applied alternative voltages. Due to the vibrations of small-scale piezoelectric plates, the deposition of low-density-lipoproteins (LDL) floating in the blood flow in the coronary arteries is prevented. Small-scale effects are considered using nonlocal elasticity theory, and the interaction between fluid and solid is modeled using the Navier-Stokes equation. The effect of fluid parameters as well as applied voltage and geometry structure is reported. Developing of smart stents maybe the key for prevention of short time conventional stents failure.

Принцип Сен-Венана в задачах нелокальной теории упругости

Simulating the modern structural materials requires introduction of a model that takes into account structural features at the micro-level. The Eringen’s nonlocal elasticity theory model could be referred to such models. This model introduction was considered in comparison with the elasticity classical model. Main feature of the nonlocal model is that it takes into consideration the long-range interactions between the continuous medium particles; classical formulation is its special case. In this case, equations are having the integral differential form, which significantly complicates obtaining the analytical solutions. In this regard, the finite elements method was applied to find a solution using the isoparametric finite elements. Here, the main balance relations are satisfied, as in the classical elasticity theory model. However, the solutions obtained are differing to a larger extent from the classical solutions, since such solutions exhibit the edge effect in vicinity of the domain free boundaries. This effect, as well as preservation of the balance of forces, are demonstrated on the example of the Saint-Venant principle feasibility at stretching the rectangular plate. Solutions obtained in the nonlocal formulation have a significant decrease in the tensile stress decrease near the free boundaries and shear stresses in the cross section

Analytical solution for nonlocal forced vibration of elliptical nanorod under linear and nonlinear external torque

The torsional vibration analysis of small-scaled rods is studied in the framework of Eringen's nonlocal theory. The equations of motion and related boundary conditions are deduced by employing the Hamilton principle. The nonlocal elasticity theory based on Eringen's model is used to cover the size-dependent behavior of the nanorods. Galerkin's analytical method is used to solve the equation of motion. In the present work, time-dependent torsional vibration in an elliptical nanorod under the linear and harmonic distributed external torques along with determined amplitude is considered with clamped-clamped end supports. When discussing free vibration analysis, the small-scale effect shows the sufficiency of natural frequencies in different modes. These results are best verified by comparison with results available in the literature. The influences of effective parameters such as nonlocal parameters, the value of aspect ratio, and excitation frequency variations in time-dependent angular displacement and non-dimensional angular displacement are discussed in detail. It is found that an increase in the nonlocal parameter leads to a rise in the value of non-dimensional angular displacement. Also, it is concluded that the aspect ratio and frequency of excitation are inversely related to angular displacement.

Ordinary state-based peri‑ultrasound modeling for monitoring crack propagation in plate structures using sideband peak count-index technique

This work presents a nonlocal theory based on ordinary state-based (OSB) peridynamics for numerical modeling of crack propagation under loadings and peri‑ultrasound modeling for elastic wave propagation that interacts with generated cracks in 3-D plate structures. Propagating cracks in plate structures for two types of loading – tensile and shear are considered. For both loading cases, at six different times after the start of the loading the analysis is carried out to investigate how the results vary for these six different stages of crack propagation. An ultrasonic excitation is applied at a point on the surface of the plate to generate guided waves propagating through the plate structure and interacting with generated cracks at different stages of crack formation. The out-of-plane velocity fields are recorded at the receiving material point for further analysis. A relatively new and promising nonlinear ultrasonic (NLU) technique called sideband peak count-index (or SPC-I) is adopted to extract the nonlinear response due to the generated cracks. Results show that when the crack propagation is at its early stage the SPC-I value first increases, and then decreases with time at the advanced stages of crack propagation. This is because for both types of loading (tensile or shear) the newly generated cracks are small cracks (or micro-cracks) in the beginning and then they evolve into macro-cracks under increasing loadings. The observation for SPC-I trends is consistent with other numerical modelings and experimental observations reported in the literature, and such qualitative matching gives confidence in the proposed method. The proposed method combining the peri‑ultrasound modeling and SPC-I technique presented in this paper can provide a new insight for monitoring crack propagation and give some guidance for experimental structural health monitoring (SHM) applications.

Nonlocal fiber-reinforced double porous material structure under fractional-order heat and mass transfer

PurposeThe purpose of this study is to analyze the thermo-diffusion process in a semi-infinite nonlocal fiber-reinforced double porous thermoelastic diffusive material with voids (FRDPTDMWV) in light of the fractional-order Lord–Shulman thermo-elasto-diffusion (LSTED) model. By virtue of Eringen’s nonlocal elasticity theory, the governing equations for the considered material are developed. The free surface of the substrate is governed by the inclined mechanical load and thermal and chemical shocks.Design/methodology/approachWith the aid of the normal mode technique, the solutions of the nondimensional coupled governing equations have been obtained.FindingsThe expressions of field variables are obtained analytically. By using MATHEMATICA software, various graphical implementations are presented to describe the impacts of angle of inclination, fractional-order and nonlocality parameters. The present model is also validated on the basis of some comparative studies with some preestablished cases.Originality/valueAs observed from the literature survey, many different studies have been carried out by taking into account the deformation analysis in nonlocal double porous thermoelastic material structures and thermo-mechanical interaction in fiber-reinforced medium under fractional-order thermoelasticity theories. However, to the best of the authors’ knowledge, no research emphasizing the thermo-elasto-diffusive interactions in a nonlocal FRDPTDMWV has been carried out. Moreover, the effect of fractional-order LSTED theory on fiber-reinforced thermoelastic diffusive half-space with double porosity has not been illuminated till now, which significantly defines the novelty of the conducted research.