Eringen's Nonlocal Elasticity Theory Research Articles

Stability analysis of sandwich double nanobeam-system with varying cross-section interconnected by Kerr-type three-parameter elastic layer

In this study, the endurable buckling load of the elastically connected parallel sandwich nano-beams with varying cross-sections is assessed. In this regard, a layered beam system consisting of two parallel axially loaded tapered sandwich nanobeams that are interconnected via a Kerr-type three-parameter elastic foundation is considered. The geometric properties are assumed to be changed exponentially along the length of the beam element. The governing equilibrium equations of the system are described by a set of three coupled homogeneous differential equations, which originates in the context of Eringen's non-local elasticity theory and Euler beam model. Then, the numerical differential quadrature technique is used to estimate the endurable axial critical loads. Eventually, a comprehensive parameterization research is performed to investigate the sensitivity of linear buckling resistance to tapering ratio, nonlocal parameter, stiffness of elastic connections, volume fraction exponent, and thickness ratio. The research work of the present study is novel, and the attained numerical outcomes can be used as benchmarks for future researches in this field.

Bending analysis of FGSP nanoplate resting on elastic foundation by using nonlocal quasi-3D theory

In this paper, the bending response of a functionally graded saturated porous (FGSP) nanoplate resting on the Pasternak elastic foundation is analyzed within the framework of quasi-3D higher-order shear deformation theory (quasi-3D HSDT) for the first time. The material properties are presumed to change gradually along the thickness direction following three patterns of porosity distribution: uniform, non-uniform symmetric, and asymmetric. The theory of poroelasticity developed by Biot is utilized in modeling the stress-strain relationships for the saturated condition. Moreover, the nanoscale effects of the structures are considered by Eringen's nonlocal elasticity theory. The governing equations are derived by using the principle of minimum potential energy according to quasi-3D HSDT, which ensures transverse shear stress-free on the upper and lower surfaces of the nanoplate. Based on the obtained closed-form solution, the impacts of the porosity distribution patterns, porosity coefficient, Skempton coefficient, geometrical parameters, elastic foundation, and nonlocal parameters on the bending behavior have been explored. According to the findings, when the pores are saturated by the fluid, the plate stiffness increases. Additionally, increasing the values of the nonlocal parameter for FGSP nanoplates leads to an increase in deflection and stresses. Finally, the present study quantitatively reveals the size-dependent effects of a saturated porous medium.

Dynamic stability of the sandwich nano-beam system

This article explores the stochastic stability of a three nanobeam system connected by Kerr type elastic layer, subjected to the magnetic effects and compressive axial forces. While considering nano beams rotary inertia is taken into account and differential equations of nanobeams are given according to Eringen's nonlocal elasticity theory. The impact of a longitudinal magnetic field on the behavior of the nanobeam system is described using Lorentz magnetic forces. The objective of this research paper is to investigate the moment and almost sure stability of the system. It analyzes the system's stability under white noise excitation by employing the contact transformation method and perturbation approach. Notably, we consider the case where the system exhibits low damping. The study focuses on examining the system's dynamic behavior and stability characteristics under these conditions. The stochastic stability of this system is investigated through moment Lyapunov exponents and Lyapunov exponents for the first time and interesting conclusions regarding the stability region of this system were obtained by varying various system parameters. These conclusions are presented in a visual manner and discussed.

Investigation of bio-thermo-mechanical responses based on nonlocal elasticity theory and fractional Pennes equation

A nonlocal fractional bio-thermo-mechanical model is developed in the context of modified fractional Pennes equation and nonlocal elasticity theory of Eringen. In the example analysis, a one-dimensional homogeneous and isotropic biological tissue model is assumed and the solutions of coupling fractional bio-thermo-mechanical equations are obtained by the method of Laplace transform. The heat conduction process and deformation in biological tissue under the action of a pulsing heat flux are investigated. The influences of fractional order parameter, pulsing heat flux characteristic time and nonlocal elastic parameter on the heat transfer, heat-induced deformation and thermal stress are analyzed.

Imperfect and multilayered magneto-electro-elastic nanoplates bending response analysis based on the nonlocal state-space approach

The present article is about the 3D static responses of the multilayered magneto-electro-elastic (MEE) nanoplates with imperfect interfaces and nonlocal effect analytically by the state-space method. To model the imperfect interfaces, which have been assumed to be electro-magnetically weakly conducting, the spring type layer model has been used. The nonlocal effect is taking into account in the modeling through to the Eringen nonlocal elasticity theory. The propagator matrix technique is utilized to derive the solution at any position in the transverse direction of the MEE nanoplates. The nonlocal length effect on the whole amount state variables has been highlighted in numerical examples. It has been carried out that, this parameter influences weakly the stresses, the electric and the magnetic displacements, respectively; but it has a strong influence on the elastic displacements, the electric and the magnetic potentials. Furthermore, the effects of the staking sequence and the interface parameters have been clearly illustrated.

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.

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.

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.

Exploring the benefits of functionally graded carbon nanotubes (FG-CNTs) as a platform for targeted drug delivery systems

Background and ObjectiveModern therapeutic systems have benefited from the use of functionally graded carbon nanotubes (FG-CNTs) to enhance their efficiency. Various studies have shown that the study of dynamic response and stability of fluid-conveying FG-nanotubes can be improved by considering a Multiphysics framework for the modeling of such a complex biological environment. However, despite noticing important aspects in modeling, the previous studies have drawbacks such as underrepresenting the effect of varying composition of the nanotube on magnetic drug release in drug delivery systems. The present work has the novelty of studying the combined effects of fluid flow, magnetic field, small-scale parameters, and functionally graded material on the performance of FG-CNTs for drug delivery applications. Additionally, the lack of an inclusive parametric study is resolved in the present study by evaluating the significance of different geometrical and physical parameters. As such, the achievements support the development of an efficient drug delivery treatment. MethodsThe Euler–Bernoulli beam theory is implemented to model the nanotube and Hamilton's principle based on Eringen's nonlocal elasticity theory is used to derive the constitutive equations of motion. To add the effect of slip velocity on the CNT's wall, a correction factor is applied to velocity based on the Beskok–Karniadakis model. Resultsdemonstrate that the dimensionless critical flow velocity increases by 227% as the magnetic field intensity increases from 0 to 20 T, and improves the system stability. On the contrary, drug loading on the CNT has the opposite effect, as the critical velocity decreases from 10.1 to 8.38 using a linear function for drug loading, and it decreases to 7.95 using an exponential function. By employing a hybrid load distribution, an optimum material distribution can be achieved. ConclusionsTo benefit from the potential of CNTs in drug delivery systems while minimizing the instability problems, a suitable design for the drug loading is required prior to the clinical implementation of the nanotube.

Nonlinear dynamic analysis and structural optimization design of Ni/Mg modified single wall carbon nanotube for hydrogen storage

The combustion calorific value of hydrogen is three times of gasoline and the burning product of hydrogen is water, which makes hydrogen the representative of clean energy. The hollow part of single-walled carbon nanotube (SWNT) is an excellent container space, which has great potential as hydrogen storage materials. This paper considers the micro-scale effect of SWNT based on Eringen's nonlocal elastic theory, and constructs a nanobeam model and analyzes the nonlinear dynamic characteristics. Based on these, the effects of different modification components on hydrogen storage properties of SWNT is discussed. The analysis exhibits that different modification positions also have a great influence on the hydrogen storage efficiency. The results of theoretical analysis and numerical simulation show that the modification components' optimization plays a certain role in improving the hydrogen storage efficiency of SWNT. The maximum increase hydrogen storage efficiency can be 12.7% by selecting appropriate decorative elements and appropriate positions.

Analytical solutions for vibration of Bi-directional functionally graded nonlocal nanobeams

This article provides analytical solutions for the vibration of bidirectional functionally graded (BDFG) nanobeams. The material characteristics of BDFG nanobeams vary along the axial and the thickness directions. The nonlocal elasticity theory of Eringen is followed to model the small-scale effects. The nanobeam's vibrational behavior is formulated by Euler-Bernoulli and Timoshenko beam theories. Hamilton's principle is used to derive the governing equations of motion. Using the Laplace transform, analytical solutions to the governing equations are established, and sets of closed-form frequency equations for four boundary conditions of nonlocal nanobeams are presented. The effects of material constants in both the axial and the thickness directions are investigated. The accuracy of the presented frequency equations is demonstrated by comparisons with a few numerical results available in the literature. It is discovered that when the material parameter along the axis increases, clamped nanobeams have a higher natural frequency, but simply supported, cantilevered, and propped cantilevered nanobeams have a lower. Nonetheless, a rising material parameter along the thickness direction exhibited identical decreasing patterns in natural frequency for all boundary conditions.

Size dependent free vibration analysis of 2D-functionally graded curved nanobeam by meshless method

The free vibration of two-directional functionally graded (2D-FG) thick curved nanobeam with concentrated mass is investigated for various boundary conditions. Hamilton’s principle is employed to develop the governing equations of the 2D-FG curved nanobeam using the first-order shear deformation theory (FSDT). The small scale effect is captured by the nonlocal elasticity theory of Eringen. The nanobeam is functionally graded in the transverse and circumferential directions of the curved nanobeam. The governing equations of the system are obtained as three coupled partial differential equations with non-constant coefficients. An appropriate meshless formulation is developed to discretize the governing equations based on local weak formulation and radial basis function. The presented meshless method is employed to study the free vibration of the one-directional and two-directional functional graded curved nanobeams. Simple-simple, free-free, clamped-clamped, and clamped-free edge conditions are investigated. Moreover, an analytical solution is developed for simple-simple transversely 1D-FG curved nanobeam. In the numerical results, the natural frequencies and mode shapes of shadow and deep curved nanobeam are studied and the influence of effective parameters such as nonlocal parameter, FG power indexes, opening angle, edge conditions, and concentrated mass are investigated. It is seen that shear locking effect is eliminated in present meshless formulation.

An analytical solution for vibration response of CNT/GPL/fibre/polymer hybrid composite micro/nanoplates

In the present article, a closed-form solution is carried out for the vibration response of CNT/GPL/fiber/polymer hybrid composite macro/micro/nanoplates resting on elastic support. Multi-layered fiber/polymer hybrid plates with either functionally graded carbon nanotube reinforced composite (FG-CNTRC) material or functionally graded graphene platelets reinforced composite (FG-GPLRC) material are two different types of plate considered here. The distribution of reinforcing nanocomposites can be uniform or functionally graded through the plate layers. The governing vibration equations are developed in the framework of the nonlocal elasticity theory of Eringen and using the first-order shear deformation plate theory. In the analytical solution procedure, the partial differential equations of motion are simplified to a series of homogeneous ordinary differential equations using a displacement field for Levy-type of the boundary conditions. The general solution of the obtained equations is expressed in terms of exponential functions. Using that the displacements should satisfy the remaining boundary conditions, the vibrational frequencies are derived. Finally, the influences of different parameters including geometry and material properties such as nonlocal parameter, number and angle of layers, volume fraction, and type of distribution of reinforcing nanocomposites are investigated and discussed. Natural frequencies for various boundary conditions are illustrated. The numerical results illustrate that the effects of the nonlocal parameter on the frequency decrease when one of the boundaries is free, and these effects increase when one of the boundaries is clamped for both FG-CNTRC and FG-GPLRC plates.

Free vibration analysis of graded porous circular micro/nanoplates with various boundary conditions based on the nonlocal elasticity theory

AbstractIn this article, the free vibrations of graded porous circular nanoplates subjected to various boundary conditions have been investigated. It is assumed that the performance of graded porous materials varies continuously in the whole thickness, and two cosine forms of non‐uniform porosity distribution along its thickness are considered. In order to capture the size effect, the Eringen's nonlocal elastic theory is applied. Then, Mindlin plate theory, combined with Hamilton's principle, is utilized to derive the governing equations of motion. Various types of boundary conditions are assumed for the plate with a combination of clamped, simply supported and free edges. Finally, the governing equations of motion are solved numerically using the shooting technique. The effects of various factors such as porosity coefficient, porosity distribution pattern, thickness‐diameter ratio, and the nonlocal scale effect and boundary conditions on the natural frequencies of grade porous circular nanoplates are discussed in detail.

Nonlocal chaotic analysis of double layered viscoelastic nanoplates embedded in Winkler–Pasternak elastic foundation and under in‐plane biaxial loads

AbstractIn this paper, based on the extended Melnikov method the homoclinic analysis for chaotic vibration of a double layered viscoelastic nanoplates (DLNPs) embedded in Winkler–Pasternak elastic foundation and under in‐plane excited biaxial loads using the Eringen's nonlocal elasticity theory has been investigated. The governing nonlinear coupled partial differential equations (PDEs) of motion for DLNP considering viscoelastic Kelvin–Voigt model in the Kirchhoff's thin plate theory in conjunction with the von‐Karman geometrical nonlinear strain–displacement relations have been derived. Then, applying the Galerkin method the governing nonlocal coupled PDEs of motion of DLNP have been converted into the nonlinear coupled ordinary differential equations (ODEs) using mode summation technique and discretized in terms of time. The criteria for the homoclinic transverse intersection for synchronous and asynchronous types of buckling are studied to predict the homoclinic orbits and chaotic motions applying the generalized Melnikov method. Besides, the critical buckling load has been determined for asynchronous and synchronous types of buckling. The obtained results are validated with comparing them with the results presented in the literature for a special case. Then, the effect of changes of different parameters including nonlocal small‐scale parameter, foundation coefficients, biaxial load ratio, geometric aspect ratio on the critical buckling load and ratio of nonlocal to local natural frequency have been studied. The critical threshold surface for the synchronous and asynchronous manifolds to predict transversely condition for the chaotic motion are discussed on the parametric space.

Reflection of temperature rate–dependent coupled thermoelastic waves on a non-local elastic half-space

Based upon the Green–Lindsay (GL) theory of thermoelasticity and the Eringen non-local elasticity theory, we describe the new constitutive relations and the field equations for generalized non-local thermoelastic model. The propagation of harmonically time-dependent thermoelastic plane waves of assigned frequency is studied in a homogeneous, isotropic, non-local infinite elastic solid by employing these equations. Reflection phenomenon of thermoelastic waves at a stress-free thermally insulated or isothermal flat boundary of a non-local thermoelastic solid half-space is also studied in case of incident coupled longitudinal wave. The amplitude 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.

Free vibration of multiple-nanobeam system with nonlocal Timoshenko beam theory for various boundary conditions

The free vibration of multiple-nanobeam system is studied for various edge boundary conditions and number of coupled nanobeams. The size effect is captured using the Eringen's nonlocal elasticity theory. The governing equations of the beams which are coupled in the multiple-nanobeam system are obtained by Timoshenko beam theory. The vibration of multiple-nanobeam system with the number of m coupled nanobeams is described by 2 m coupled partial differential equations. A meshless formulation is presented to discretize the governing equations to a set of ordinary differential equations in time domain. The number of nanobeams of multiple-nanobeam system and the boundary conditions of each nanobeam can be arbitrary. The accuracy of results is examined by comparison of the predictions with available analytical results in the literature for especial cases, and good agreement is seen. In the numerical results the free vibration frequencies and mode shapes of multiple-nanobeam system for various edge boundary conditions are presented and the effects of parameters such as coupling stiffness, nonlocal parameters and number of nanobeams of multiple-nanobeam system are investigate. The presented method can be very useful for analysis of multiple-nanobeam system with arbitrary number of nanobeams, arbitrary boundary conditions, coupling stiffness and length to thickness ratio.

Nonlocal Free Vibrations of Metallic FGM Beams

This work aims to analyse the free-vibration response of functionally graded, simply supported beams with different gradient directions, taking into account nonlocal effects. To this purpose, the first-order shear deformation theory and the nonlocal elasticity theory of Eringen are used, in order to assess the influence of size dependency effects on the free-vibration responses of those beams. The influence of other factors such as the aspect ratio of the beams and the evolution of the constituents’ mixture through the beam thickness and along its length is also considered. In this last case, a mixture distribution is proposed, accounting for the boundary conditions’ characteristics. The finite element model is first verified against existing alternative solutions, to assess and illustrate its performance. Based on the conclusions achieved, a set of parametric studies is then developed. The results are discussed considering the material distribution profiles, and conclusions are drawn with respect to their relative performance under the analysed conditions.

Thermal stability analysis of functionally graded non-uniform asymmetric circular and annular nano discs: Size-dependent regularity and boundary conditions

In this article, the size-dependent thermal buckling analysis of nonuniform functionally graded asymmetric circular and annular nanodiscs is presented on the basis of Kirchhoff's plate theory, Eringen's nonlocal elasticity theory, and physical neutral plane. For the first time, the nonlocal regularity conditions and boundary conditions are obtained for the asymmetric discs. The thickness of the nanodiscs is assumed to be varying linearly and parabolically in the radial direction. The Power-law model is adopted to compute the temperature-independent effective mechanical properties of the functionally graded materials (FGMs). The size-dependent stability equation is obtained from Euler-Lagrange's equation which is derived from Hamilton's principle. This equation and corresponding boundary conditions are discretized by the differential quadrature method (DQM) and provide an eigenvalue problem. The numerical value of the lowest eigenvalue is reported as a critical temperature difference on the surfaces of the nanodiscs. The effects of various parameters such as nonlocal parameter, volume fraction index, nodal lines, and taper parameters for thickness variations are studied.

Applicability and efficacy of Galerkin-based approximation for solving the buckling and dynamics of nanobeams with higher-order boundary conditions

In this research effort, the general nonlocal elasticity theory is applied to an axially-loaded Euler-Bernoulli nanobeam with varying boundary conditions. In applying the general nonlocal elasticity theory, a modified version of Eringen's nonlocal elasticity theory, a sixth-order partial differential equation is obtained for the transverse governing equation of motion. Obtaining the sixth-order equation created the need for additional non-classical beam boundary conditions. These boundary conditions are determined using the weighted residual approach and are physically interpreted as higher-order force and moment terms. This study investigates the buckling and dynamic responses of nanobeams using a Galerkin-based approximation with different trial mode shapes, as it relates to the exact solution. As such, comparisons are made between using three distinct types guess functions in the approximation, namely, the exact unloaded classical mode shapes, exact unloaded Eringen nonlocal mode shapes, and exact unloaded general nonlocal mode shapes. Unloaded mode shapes refer to the mode shapes in which the applied axial load is neglected or set to zero. Additional studies are conducted focusing on direct comparison between Eringen nonlocal and general nonlocal beams for both equal and unequal nonlocal parameters. It is shown that selection of an appropriate guess function that satisfies all boundary conditions is necessary for the approximate solution. Additionally, the response of Eringen and general nonlocal nanobeams with equal nonlocal parameters and hinged-hinged boundary conditions is identical using both the exact and approximate solutions. Finally, it is demonstrated that the general nonlocal dynamic response follows a different trajectory than the Eringen nonlocal dynamic response as a function of the applied axial load. The findings of this investigation may be used by other researchers to simplify analysis procedures and to better understand the fundamental relationships existing in the size-dependent governing equations and boundary conditions for nanobeam-based systems.