Eringen's Nonlocal Elasticity Research Articles

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.

Effect of non‐local modified couple stress theory on the responses of axially moving thermoelastic nano‐beams

AbstractThe present study examines the lateral thermal vibrations of a nanobeam subjected to an axial motion while being influenced by a sinusoidal thermal load. The partial differential equation describing the system was obtained using the extended Hamiltonian principle. Also considered were the Euler‐Bernoulli (EB) beam, the nonlocal couple stress theory, and Eringen's nonlocal elasticity model. This type of axially moving beam has multiple applications in design, including industrial, civil, structural, chemical, and computer engineering. The Laplace transform technique is utilized to translate partial differential equations into a thermoelastic differential equation of the sixth order. This study investigates the impact of nanobeam size and velocity on thermo‐mechanical characteristics. To explore the impacts of axial velocity, pulse width, nonlocal index, material length scale coefficient, and phase lag coefficients on the examined studied fields, such as lateral vibration and temperature change for the moving nanobeam are calculated. The specified factors were discovered to impact the flexibility and dynamic response of the nanobeam substantially.

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.

Nonlocal boundaries and paradoxes in thermoelastic vibrations of functionally graded Non-Uniform cantilever nanobeams and annular nanoplates

The paradoxical behaviour of free vibrations of cantilever nanostructures using Eringen's nonlocal elasticity model is always a debatable issue for researchers in the past few years. According to this, the frequency of cantilever nanobeams increases with the increase in the value of the nonlocal parameter while it decreases for other boundary conditions. In thermoelastic vibrations, the frequency of nanobeams increases and decreases with the increase in temperature at the surfaces for clamped-free and other boundary conditions, respectively. Therefore, investigation of these structures is of critical importance. In view of this, the authors have proposed the modified boundary conditions for the well-established differential model of Eringen’s nonlocal elasticity theory to deal with the free edges of the structural components. Using these conditions, the vibration analysis of nonuniform functionally graded nanobeams and axisymmetric annular nanoplates is presented for clamped-free boundary conditions on the basis of first-order shear deformation theory and physical neutral plane. The thickness of these structural elements (nanobeams and annular nanoplates) is assumed to be varying linearly and parabolically in the axial direction. The temperature-dependent mechanical properties of the material are obtained by the power law. The nonlocal governing equations and modified boundary conditions for the structural elements are discretized using the generalized differential quadrature method and solved by MATLAB to obtain the fundamental frequency. For the present modifications, the behaviour of frequency parameter for different values of nonlocal parameter, volume fraction index, temperature difference, thickness parameter, and taper parameter is discussed.

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.

Vibration of initially stressed nonlocal irregular nanoplate using wave propagation approach

The vibration analysis of pre-stressed nonlocal irregular nanoplate is discussed in this study. To consider the nonlocal effects, Eringen's nonlocal elasticity model are adapted. Biot`s theory is used to generate the new governing equation as well as its closed-form solutions including the initial stress parameter. Governing equations considering small length scale, initial stress, and surface irregularity were solved by the wave propagation approach. A novel frequency equation is explored in relation to a small length scale, initial stress, and surface irregularity. To understand the reactions of the considered nanoplate in terms of natural frequency, the numerical calculations are done and the results are shown graphically. The findings provide a better declaration for vibration analysis of the nonlocal initially stressed irregular nanoplates. It has been observed that, the increase of the small length scale parameter decreasing the natural frequencies and the increase of initial stress and surface irregularity parameters increasing the natural frequencies throughout the computation frequencies of vibrating nanoplate. In addition, the existence of nonlocal, initial stress, and irregularity parameters effect the natural frequency of vibrating nanoplates significantly. Furthermore, the findings obtained could be relevant in designing of nanodevices in which nanoplate are the most common nano element.

A fractional nonlocal elastic model for lattice wave analysis

This paper investigates the link between lattice elasticity and nonlocal continuum mechanics from one-dimensional and multi-dimensional wave propagation problems. Eringen (1983) closed the bridge between one-dimensional linear lattice elasticity (called Lagrange lattice) and nonlocal elastic continuum. The Born-Kármán wave dispersive properties of the one-dimensional lattice are accurately fitted by Eringen's nonlocal differential elastic law using asymptotic or phenomenological arguments. A fractional version of Eringen's nonlocal model based on the introduction of a fractional Laplacian for the nonlocal differential operators can improve the accuracy in matching the wave dispersive curve of lattice dynamics. A multi-dimensional generalization of Lagrange lattice for cubic crystals is the lattice of Gazis et al. composed of central and angular interactions. We show in this paper that the differential Eringen's nonlocal elasticity differs from the lattice-based nonlocal elasticity continuum. In particular, Eringen's nonlocal elasticity preserves the isotropic property, whereas the cubic lattice is only isotropic in the low frequency regimes. The fractional generalization of Eringen's model accurately fitted the wave dispersive properties of the cubic crystal in the first Brillouin zone. The fractional generalization of Eringen's model highlights an isotropic response in the low frequency regimes and a cubic symmetry in the high frequency regime.

On the hybrid atomistic-continuum model for vibrational analysis of α-, β-, and γ-graphyne circular nano-plates

The natural frequency of graphyne structures for all phases is calculated by molecular dynamics simulation. Tersoff potential function was used in molecular dynamics (MD) simulations. In this paper, the vibration behavior of α-, β-, and γ-graphyne circular nano-plates was investigated using a hybrid atomistic-continuum Model. The MD frequencies are higher than those obtained from the classical local plate model, so the use of the Eringen-nonlocal elasticity model is practically unusable. The obtained equations based on the stress-driven elasticity, modified couple stress theory, and strain gradient theory were solved by generalized differential quadrature rule (GDQR), which has a faster convergence than generalized differential quadrature method (GDQ). The value of the size parameter for each diameter was calculated using the molecular dynamics results. By changing the phase from alpha to beta and beta to gamma, value of stiffness is increased. Therefore, for a smaller size parameter, continuum mechanics results match the molecular dynamics results. The results show that the size effect parameter is not a fixed value and has different values in different conditions. For all three phases and all diameters, the calculated frequencies are greater than those from classical theory. This result shows that the graphyne structure is fantastic and peculiar.

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.

On the vibration dynamics of heterogeneous panels under arbitrary boundary conditions

This work is promoted to analyze the time-dependent behavior of composite panels. Furthermore, different curves and boundary conditions are also covered. The material properties for the composite panel are assumed to be vary continuously within z-direction. The effective material properties are estimated via Voigt’s micro-mechanical model by a modified-power law rule for arbitrary power index. The governing motion equations for the dynamics of imperfect panels are evaluated by rewriting the classical Hooke’s law through Eringen nonlocal elasticity for a higher-order shear deformable theory. Then, a numerical solution for imperfect functionally graded doubly-curved panels with arbitrary boundary conditions is carried out. The panel considered in the current study has pure ceramic/metal-rich surfaces within top and bottom of the panel in which the synthesize of imperfections in the composite material are described by a cosine model of porosity distribution. Ultimately, the sensitivity of both time-dependent transverse displacement and stresses are studied for material composition, porosity level, geometrical parameters, nonlocality and excitation frequency that supported by simply-supported, clamped, free and the combination of them as well as different curves (i.e., spherical, elliptical, hyperbolic and cylindrical).

Dynamics of imperfect inhomogeneous nanoplate with exponentially-varying properties resting on viscoelastic foundation

This article tries to investigate the dynamic deflection response of exponentially functionally graded material (E-FGM) nanoplate considering the role of porosities when embedded in a visco-elastic foundation and subjected to moving load, for the first time. The effective material properties are found using an exponential model of the rule of mixture. Next, the governing equations for the nanoplates while resting on a visco-Winkler foundation are found based on the third-order shear deformation theory in conjunction with Eringen nonlocal elasticity by developing Hamilton's principle. To solve the time-dependent governing motion equations, a state-space method is developed to find the response of the structure including simply-supported edges under moving load. Through some examples, the validation of the approach is provided before investigating the roles of nonlocality, volume fraction index (exponential parameter), porosity index, visco-elastic foundation coefficients, and velocity and time span of moving load on the forced vibration of embedded E-FGM nano-size plate under moving load.

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.

A Quasi-3D Higher-Order Theory for Bending of FG Nanoplates Embedded in an Elastic Medium in a Thermal Environment

This paper presents the effects of temperature and the nonlocal coefficient on the bending response of functionally graded (FG) nanoplates embedded in an elastic foundation in a thermal environment. The effects of transverse normal strain, as well as transverse shear strains, are considered where the variation of the material properties of the FG nanoplate are considered only in thickness direction. Unlike other shear and deformations theories in which the number of unknown functions is five and more, the present work uses shear and deformations theory with only four unknown functions. The four-unknown normal and shear deformations model, associated with Eringen nonlocal elasticity theory, is used to derive the equations of equilibrium utilizing the principle of virtual displacements. The effects due to nonlocal coefficient, side-to-thickness ratio, aspect ratio, normal and shear deformations, thermal load and elastic foundation parameters, as well as the gradation in FG nanoplate bending, are investigated. In addition, for validation, the results obtained from the present work are compared to ones available in the literature.