Nonlocal Parameter Research Articles

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.

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.

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.

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.

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.

Vibration analysis of piezoelectric semiconductor beams with size-dependent damping characteristic

In this paper, the vibration analysis of the piezoelectric semiconductor (PS) beam with size-dependent damping characteristic is conducted based on the nonlocal PS model. The nonlocal constitutive relations including the stress, electric displacement as well as current density of electrons are achieved. The one-dimensional vibration governing equations are obtained by integrating the three-dimensional governing equations along the cross section. Then, the natural frequency and damping characteristic are given by solving the governing equations of the PS beam with simply-supported boundary conditions. Through the numerical results, the effect of the nonlocal parameter, initial electron concentration, length to radius ratio and different scales on the vibration behavior of the system is demonstrated. The main innovation of the manuscript is that the peak value of the damping shifts to the smaller initial electron concentration because of the nonlocal effect, and the damping characteristics of the PS beam at nanoscale, micrometer scale and millimeter scale are obviously different.

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.

A unified local-nonlocal integral formulation for dynamic stability of FG porous viscoelastic Timoshenko beams resting on nonlocal Winkler-Pasternak foundation

We present a study on the nonlocal dynamic stability of functionally graded (FG) porous viscoelastic Timoshenko beams resting on a Winkler-Pasternak foundation. The porosity distributions are assumed to be symmetric or antisymmetric about the mid-plane of the beams, and the Kelvin-Voigt viscoelastic model is considered to depict the beams’ viscoelastic properties. Unlike most studies on this topic, we consider both the beam’s bending deformation and the foundation reaction as nonlocal, simultaneously, by uniting the equivalently differential forms of two well-posed local-nonlocal integral models (i.e., strain-driven (ε-D) and stress-driven (σ-D) strategies) which are strictly equipped with a set of constitutive constraint conditions, through which both the softening and hardening effects on the stiffness of structures due to size reduction can be addressed. The generalized differential quadrature method (GDQM) is utilized for discretizing variables presented in the differential problem formulation, from which the numerical solution of the dynamic instability region (DIR) of different bounded beams can be obtained. After conducting comparative studies to validate the proposed model, the effects of nonlocal parameters, static force factor, porosity distribution and viscoelastic property on the DIR are investigated. Moreover, the influence of incorporating the nonlocality of the foundation is also presented.

Mathematical modelling of micropolar thermoelastic problem with nonlocal and hyperbolic two-temperature based on Moore–Gibson–Thompson heat equation

The aim of this article is to study a problem of thermomechanical deformation in a homogeneous, isotropic, micropolar thermoelastic half-space based on the Moore–Gibson–Thompson heat equation under the influence of nonlocal and hyperbolic two-temperature (HTT) parameters. The problem is formulated for the considered model by reducing the governing equations into 2D and then converting to dimensionless form. Laplace transform and Fourier transform techniques are employed to obtain the system of differential equations. In the transformed domain, the physical quantities like displacement components, stresses, thermodynamic temperature, and conductive temperature are calculated under the specific types of normal force and thermal source at the boundary surface. A numerical inversion technique is used to recuperate the equations in the physical domain to exhibit the influence of nonlocal and HTT in the form of graphs. Particular cases of interest are also discussed in the present problem. The present study finds applications in a wide range of problems in engineering and sciences, control theory, vibration mechanics, and continuum mechanics.

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.

Investigation of Free Vibrations of Stepped Nanobeam Embedded In Elastic Foundation

The free vibration of stepped nanobeams embedded in the elastic foundation was investigated using Eringen's nonlocal elasticity theory. It is fixed at the system ends with a simple-simple support. The stepped nanbeam’s equations of motion are obtained by using Hamilton's principle. Multi-time scale, which is the perturbation methods, was used for the analytical solution of the equations. To observe the effects of nano size effect, elastic basis coefficient and step location, natural frequencies of the first three modes of the system were obtained for different non-local parameter values, elastic foundation coefficients, step rates and step positions. In the results, it was seen that the non-local parameter had a negative effect on the natural frequency. The elastic foundation coefficient has been shown to reduce vibration amplitudes.

Complex dynamics, released energy, and multistability in a single-layered graphene sheet under periodic loads

Graphene was found in 2004. It’s a carbon allotrope built around a single layer of atoms organized in a two-dimensional (2D) honeycomb lattice nanostructure. In the contribution, the complex behavior of a graphene sheet under periodic loads is investigated using the usual nonlinear analysis tools. From the dynamic states and the material’s symmetry investigations, it is found that the graphene model considered in this work can exhibit the simultaneous existence of up to six different patterns by varying initial conditions for fixed parameters. The Helmholtz theorem is also used to calculate the Hamilton energy of excited graphene. As a result, when the damping coefficient and the nonlocal parameter were close to zero, the system’s energy was null. This supports the fact that when graphene is in its resting state, vibrations are null inside the material. In contrast, when the damping coefficient and the nonlocal parameter were simultaneously increased, the energy of the graphene sheet increased until it reached a peak value, justifying the oscillations observed in the material. The dependency of the material on the initial condition is also used to support the coexistence of the energy found.

Investigation on the thermoelastic response of a porous microplate in a modified fractional-order heat conduction model incorporating the nonlocal effect

In recent years, the application of porous materials in practical engineering has become increasingly common. Under extreme thermal conditions, how to accurately describe the heat conduction and structural response of porous materials is a key issue in the structural safety and functional design of porous materials. Among them, it is particularly important to consider the size-dependent and memory-dependent effects for small-sized structures or microdevices of porous materials. To address these issues of porous structures and to guide their applications as well as optimization, this paper first investigates the thermoelastic transient response of porous microplates subjected to thermal and stress shocks at the left boundary based on the Atangana-Baleanu (AB) fractional-order generalized thermoelastic theory by considering nonlocal effects and fractional-order strain, which is combined with the thermoelasticity theory of porous materials and the dual-phase-lag (DPL) heat conduction model. The governing equations are then established and solved using the Laplace transform and its numerical inversion. Finally, the effects of newly defined fractional order parameters, nonlocal parameters and the initial magnitude of the mechanical shock on the dimensionless temperature, volume fraction field, displacement and stress distribution are discussed and displayed graphically.

Vibration properties of an elastic gold nanosphere submerged in viscoelastic fluid

In this paper, we propose a novel, simple and accurate analytical study based on nonlocal elasticity theory to forecast small-scale effects on the radial vibration of anisotropic gold nanospheres submerged in viscoelastic fluid (VEF). Eringen’s model is used to determine the motion equation for anisotropic nanospheres, with the fluid assumed to be viscoelastic and compressible. The frequency equation is derived by imposing the fluid-nanosphere interface continuity conditions. A comparison with the literature results is conducted to demonstrate the validity and correctness of this analysis, which indicates a very good agreement. The importance of small-scale effects in the radial vibration, which need to be included in the nonlocal elasticity model of submerged nanospheres, is eventually revealed by numerical examples. It is discovered that the nanosphere size, nonlocal parameter, and glycerol–water mixture have a significant impact on the vibration behaviors. Our results show that the small scale is crucial for the radial vibration of gold nanoparticles when the gold nanosphere is smaller than [Formula: see text]. Thus, the resulting frequency equation is very useful to interpret experimental measurements of the vibration characteristics of submerged gold nanospheres in VEF.

Influence of a magnetic field on a nonlocal thermoelastic porous solid with memory-dependent derivative

AbstractA novel model of a nonlocal magneto-thermoelastic porous solid in the context of the three-phase-lag model with a memory-dependent derivative is introduced. The effect of a magnetic field on a nonlocal thermoelastic porous medium in the context of a three-phase-lag model with memory-dependent derivatives was studied. The normal mode analysis is used to solve the problem of an isothermal boundary to obtain the exact expressions of physical fields. The numerical results are represented to estimate the effects of the magnetic field, time delay, and the nonlocal parameter on the behavior of all of the field variables such as temperature, displacement, and stresses. Comparisons are given for the results in the absence and presence of the magnetic field as well as the locality. Comparisons are also given for the results for different values of time delay. To the best of the author’s knowledge, this model is reported for the first time. Some particular cases are also deduced from the present investigation.

Magnetic field and thermal environment effects on sound transmission loss of double-walled cracked piezoelectric cylindrical nanoshells

Based on a non-local strain gradient theory (NSGT), this article analytically studies the sound transmission loss (STL) performance of a special type of double-walled partially cracked piezoelectric nanoshell structures of cylindrical shape undergoing initial electric potential and longitudinal magnetic field. The structure, which includes a specific number of axial and circumferential surface cracks, is struck by a plane wave in a thermal setting. Such cracks have minimal size compared to the nano-shell structure as a whole, minimizing the effects of curvature for this problem. The first-order shear deformation theory (FSDT) is utilized to explain the displacement components, and Hamilton’s concept is applied to obtain the coupled vibroacoustic equations that are used to generate the governing equations of the system for the two crack types, giving us an idea of the resulting structural stiffness in the presence of cracks using coefficients of crack compliance that are derived from the line spring model (LSM). The mechanical and acoustic properties are then checked to see their effect on the system response followed by a series of validation case studies. This evaluation also allows one to optimize a similar structure by selecting appropriate external parameters such as electric voltage, magnetic field intensity, different crack types, nonlocal and strain gradient parameters, variations of temperature, and incident angle of sound waves.

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.

Nonlocal theory of propagation and reflection of plane waves in higher order thermo diffusive semiconducting medium

The present article is related to propagation and reflection of elastic wave through a nonlocal generalized thermo-diffusive semiconducting elastic solid. The non-local theory is employed to study the wave behavior. Three phase lag model with higher order fractional order derivative is incorporated to discuss heat propagation through the medium, in addition with two phase lags diffusion equation. The Helmholz vector rule is applied to decompose the system into longitudinal and transverse components. The frequency dispersion relation indicates the presence of four coupled longitudinal and one un-coupled shear vertical wave. The speed of the waves is plotted against angular frequency for local and nonlocal medium. The cutoff frequency of the waves is also depicted graphically. The longitudinal P-wave is taken to be an incident wave at the free surface of the solid to compute the reflection coefficients. The influences of fractional order and nonlocal parameters on amplitude ratios are also studied. The effect of these parameters is found to be significant. The results are proved in the context of energy conservation. The results obtained from the current investigation are very useful for scientists working on problems of geophysics and various fields of mechanics.

A size-dependent meshfree approach for magneto-electro-elastic functionally graded nanoplates based on nonlocal strain gradient theory

In this paper, we introduce a novel size-dependent meshfree approach for analyzing free vibrations of magneto-electro-elastic (MEE) functionally graded (FG) nanoplates. This approach combines the nonlocal strain gradient theory (NSGT), the higher-order shear deformation theory (HSDT), and meshfree method for the first time. MEE materials are essentially mixed by the barium titanate and cobalt ferrite which are definitely coupled by both piezoelectric and piezomagnetic effects. Our approach captures both the nonlocal and strain gradient effects in nanoplates, which result in the reduced stiffness and increased stiffness, respectively, by adjusting two parameters. The kind of effective material properties of MEE-FG nanoplates are expressed using a power-law scheme. The coupled governing equations, based on the principle of extended virtual displacement, are solved using the moving Kriging (MK) meshfree method. Numerical examples are given to investigate the effect of geometrical parameters, initial electric voltage, power index, initial magnetic potential, strain gradient parameter, and nonlocal parameter on the natural frequency of MEE-FG nanoplates.

Nonlocal suppression of Biermann battery magnetic-field generation for arbitrary atomic numbers and magnetization

The Biermann battery term of magnetohydrodynamics (MHD) generates a magnetic field where electron density gradients and electron temperature gradients are perpendicular to one another. Kinetic simulations and experiments have shown that the rate of magnetic-field generation is lower than Biermann when the electron mean free path becomes comparable to or greater than the temperature gradient scale length, known as the nonlocal regime. We investigate the nonlocal suppression of the Biermann term using simplified Fokker–Planck simulations covering a wide range of parameters. We provide the first fit for nonlocal Biermann suppression that has physically accurate behavior for small and large values of a suitable nonlocality parameter, valid for an arbitrary atomic number, and that includes the effect of magnetization on nonlocality. The fit is intended to provide an approximate method to account for reduced magnetic-field generation in MHD codes and theory.