Nonlocal Elasticity Theory Research Articles

A layer-wise theory for the dynamic analysis of functionally graded composite nanobeams under thermal environment using nonlocal boundary conditions and Chebyshev collocation technique

In this study, a layer-wise theory satisfying displacement continuity at each layer’s interface and the Chebyshev collocation technique are adopted to analyze the vibration behavior of functionally graded composite nanobeams under non-uniform temperature variation in the thickness direction. The temperature-dependent mechanical properties vary according to the rule of mixture. Boundary conditions are modified by including the effect of size parameter and named as nonlocal boundary conditions. These nonlocal boundary conditions along with Eringen’s nonlocal elasticity theory are developed to capture the effect of size-dependency for such a nanobeam based on the first-order shear deformation theory (FSDT) and physical neutral plane. The energy principle and the variational approach are used to obtain force-moment equilibrium equations, which lead to governing differential equations and local boundary conditions. Two types of sandwich constructions are considered, namely, Type 1 (functionally graded-homogeneous-functionally graded) and Type 2 (homogeneous-functionally graded-homogeneous), by maintaining the material continuity at the interfaces. Natural frequencies are analyzed in the fundamental mode for various values of volume fraction index, thickness of the layers, size scale parameter, and thermal environment. The convergence rate of the Chebyshev collocation technique is also shown over the differential quadrature method.

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.

Torsional Dynamics of Axially Graded Viscoelastic Carbon Nanotubes

Torsional vibration analysis of the axially functionally graded carbon nanotubes has been carried out. Nonlocal stress gradient elasticity theory has been used in continuum mechanics model of the carbon nanotube. Variation of the material properties of the axially graded nanostructure has been assumed in exponential form. Differently from the majority of literature works, viscous damping and nonlocal parameters have been assumed in grading form. Energy functional for the carbon nanotube has been achieved with minimum potential energy principle and weak form solution has been obtained with the Ritz Method. Effects of material grading, nonlocality and viscoelasticity to the torsional dynamics of axially graded carbon nanotube have been investigated. Results of the present work could be useful in modeling and production of axially functionally graded nanostructures.

Exact closed-form solutions for the free vibration analysis of multiple cracked FGM nanobeams

Exact closed-form solutions for the free vibration analysis of multiple cracked FGM nanobeams with arbitrary boundary conditions based on the Nonlocal Elasticity Theory (NET) and the Timoshenko beam theory are presented. The NET considering the size effect of nanostructures is applied. FGM properties vary nonlinearly along the height of the beam. A crack model using two springs with stiffness depending on the crack depth is applied. The proposed solutions are provided explicitly as functions of integration constants determined from the standard boundary conditions. Frequency equations, that established in the form of the determinant of the matrix 3 × 3 order for arbitrary boundary conditions, are applied to analyse free vibration. Expressions for the mode shapes are given explicitly. The effects of geometry, material, nonlocal and crack parameters on the free vibration of the multiple cracked nanobeam are then analysed in detail.

Dynamic Analysis of a Timoshenko–Ehrenfest Single-Walled Carbon Nanotube in the Presence of Surface Effects: The Truncated Theory

The main objective of this paper is to study the free vibration of a Timoshenko–Ehrenfest single-walled carbon nanotube based on the nonlocal theory and taking surface effects into account. To model these effects on frequency response of nanotubes, we use Eringen’s nonlocal elastic theory and surface elastic theory proposed by Gurtin and Murdoch to modify the governing equation. A modified version of Timoshenko nonlocal elasticity theory—known as the nonlocal truncated Timoshenko beam theory—is put forth to investigate the free vibration behavior of single-walled carbon nanotubes (SWCNTs). Using Hamilton’s principle, the governing equations and the corresponding boundary conditions are derived. Finally, to check the accuracy and validity of the proposed method, some numerical examples are carried out. The impacts of the nonlocal coefficient, surface effects, and nanotube length on the free vibration of single-walled carbon nanotubes (SWCNTs) are evaluated, and the results are compared with those found in the literature. The findings indicate that the length of the nanotube, the nonlocal parameter, and the surface effect all play important roles and should not be disregarded in the vibrational analysis of nanotubes. Finally, the results show how effective and successful the current formulation is at explaining the behavior of nanobeams.

Vibration analysis of higher-order nonlocal strain gradient plate via meshfree moving Kriging interpolation method

In this paper, a higher-order nonlocal strain gradient plate model combining nonlocal elasticity theory with strain gradient theory in the form of adding signs is constructed. The governing equation for the higher-order nonlocal strain gradient plate model is derived by Hamilton’s principle. Two higher-order parameters with nonlocal stress and strain gradient are introduced here to explain the size effect and dispersive behaviour. The meshfree moving Kriging interpolation method is utilized to address the frequency of higher-order nonlocal strain gradient plate under complicated boundary conditions. Nonlocal elasticity theory, strain gradient theory in the form of add signs and minus signs on the effect of frequency are compared by degenerating the higher-order nonlocal strain gradient plate model. The other goal of this paper is to compare the present model with higher-order nonlocal strain gradient theory in the form of minus signs on the effect of frequency. It is shown that these two items can describe the size effect well. The present method can also produce lower natural frequencies corresponding to ultrahigh-order mode shapes and applied in a discrete medium.

Refined quasi-3D shear deformation theory for buckling analysis of functionally graded curved nanobeam rested on Winkler/Pasternak/Kerr foundation

In the present work, a novel refined three-variable quasi-3D shear deformation theory incorporates a correction factor is developed to analyze the buckling behavior of multi-directional functionally graded (FG) curved beams. The proposed displacement field is formulated in accordance with the Euler-Bernoulli beam theory. The research investigated two types of coated Functionally Graded (FG) nanobeams: Hardcore (HC) FG curved nanobeams and Softcore (SC) FG curved nanobeams. Three different material distributions are taken into consideration: a bidirectional material distribution referred to as “2D-FG,” a unidirectional transverse material distribution known as “T-FG,” and a unidirectional axial material distribution called “A-FG.” Eringen’s nonlocal elasticity theory is employed to capture small-scale effects. The total potential energy principle is utilized to derive the equilibrium equations of curved nanobeams. A novel solution, utilizing Galerkin’s method, has been developed to effectively address a range of boundary conditions. The curved FG beam is supported by an elastic foundation following the Winkler/Pasternak/Kerr model. A comprehensive analysis has been conducted to examine the impacts of various FG schemes, curved beam geometry, nonlocal parameter, elastic foundations, and various boundary conditions on the dimensionless critical buckling loads. This analysis aims to provide a comprehensive understanding of how each of these factors influences the critical buckling loads of the nanobeams.

Vibrations of Nonlocal Polymer-GPL Plates at Nanoscale: Application of a Quasi-3D Plate Model

An analysis is performed in this research to obtain the natural frequencies of a graphene-platelet-reinforced composite plate at nanoscale. To this end, the nonlocal elasticity theory is applied. A composite laminated plate is considered where each layer is reinforced with GPLs. The amount of GPLs may be different between the layers, which results in functionally graded media. To establish the governing equations of the plate, a quasi-3D plate model is used, which takes the non-uniform shear strains as well as normal strain through the thickness into account. With the aid of the Hamilton principle, the governing equations of the plate are established. For the case of a plate that is simply supported all around, natural frequencies are obtained using the well-known Navier solution method. The results of this study are compared with the available data in the open literature, and, after that, novel numerical results are provided to explore the effects of different parameters. It is depicted that, with the introduction of GPLs in the matrix of the composite media, the natural frequencies of the plate enhance. Also, a proper graded pattern in GPL-reinforced composite plates, i.e., an FG-X pattern, results in the maximum frequencies of the plate. In addition, the introduced quasi-3D plate theory is accurate in the estimation of the natural frequencies of thick nanocomposite plates at nanoscale.

Uncertain vibration characteristics of Bi-directional functionally graded sandwich nanoplate subjected to dynamic load

In this paper, we present a novel approach that combines isogeometric analysis (IGA) with the higher-order shear deformation theory (HSDT) and Monte Carlo simulation (MCS). Our objective is to investigate uncertain vibration characteristics of bi-functionally graded sandwich (BFGSW) nanoplates under dynamic loading. The small-scale effect observed in nanostructures is achieved by employing the nonlocal elasticity theory. This theory allows for the consideration of the mechanical behavior that arises at the nanoscale level. Within the stochastic design approach, the state function for design conditions is commonly formulated by incorporating input random variables, assumed distribution functions, and random responses obtained from computational models. This work primarily focuses on investigating vibrational characteristics of BFGSW nanoplates when the model parameters are considered as random quantities. The obtained results show that distribution characteristics of vibration of the BFGSW nanoplate depend significantly on the standard deviation of input parameters.

Thermoelastic dissipation of circular-cross-sectional ring including nonlocal and dual-phase-lagging effects

In the high-frequency region, thermoelastic damping (TED) is essential to ensure the accuracy of micro- or nano-precision sensors or actuators. The size-effect of molecules should be considered in the design and analysis since the atomic volume is not negligible in the nanoscale model. Moreover, a toroidal ring is easy to manufacture, so there is a high possibility that it will be developed in the near future. And thus, the 3-dimensional (3D) TED model of a circular cross-sectional ring is investigated by adopting dual-phase-lagging (DPL) and modified couple stress (MCS) theories based on nonlocal elasticity. Additionally, the size-dependent influences are specified by length-scale parameters in the elastic and thermal fields. Then the TED results are predicted as a quality factor (Q) with the peaks of the spectra and are compared separately by the causes of physical quantity. Moreover, the phenomenon is analyzed in terms of how the properties and length-scale parameters correlate with each other. Finally, it is confirmed that nonlocal elasticity should be considered when the size of the resonator decreases to the scale of the energy-carrying mean-free path. In this regard, this work provides an idea for the analysis of DPL TED of the toroidal ring, including nanoscale effects based on nonlocal elasticity and MCS theories.

Reflection, transmission, and dissipation of plane waves in sandwiched functionally graded thermo- electro-elastic nanoplates via nonlocal integral elasticity theory

On account of the Lord-Shulman thermo-electric-elastic (TEE) theory and the nonlocal integral elasticity effect, the propagation of the plane wave in sandwiched functionally graded material (FGM) nanoplates, especially its energy dissipation, is investigated. A mathematical model is established by using the extended Legendre polynomial series approach. After imposing the mechanical, electrical and thermal boundary conditions by using the Legendre series, the reflection and transmission as well as the dissipation energy ratios are obtained. The influences of the nonlocal effect, relaxation time, gradient index, and electric filed are discussed. It is found that the nonlocal effect on the reflection of P waves is consistent with their transmission, but opposite to reflection of SV waves. Meanwhile, the nonlocal effect increases the dissipation. Besides, the dissipation variation in the normal incidence mainly comes from the transmission waves. The study is beneficial for the design of wave characteristics in TEE FGM nanoplates.

Nonlocal elasticity of Klein‐Gordon type with internal length and time scales: Constitutive modelling and dispersion relations

AbstractIn this work, a nonlocal elasticity theory with nonlocality in space and time is presented by considering nonlocal constitutive equations with a dynamical scalar nonlocal kernel. Based on the proposed theory, we consider the isotropic nonlocal elasticity of Klein‐Gordon type including a characteristic internal time scale parameter in addition to the characteristic internal length scale parameter. The dispersion relations for homogeneous isotropic media in the framework of nonlocal elasticity of Klein‐Gordon type are analytically determined. The obtained results reveal the ability of the presented nonlocal elasticity model to predict, for the first time in the framework of nonlocal elasticity, in addition to the acoustic modes (low‐frequency modes), optic modes (high‐frequency modes) as well as frequency band‐gaps. The phase and group velocities for all four modes (acoustic and optic branches of longitudinal and transverse waves) are determined showing that all four modes exhibit normal dispersion with positive group velocity. The presented model allows for physically realistic dispersive wave propagation.

Chaotic vibrations of double-layer graphene sheet system

Nonlinear vibrations of double-layer nanoplate system are studied by numerical simulation. The governing equations are derived on a base of the nonlocal elasticity theory, von Kármán nonlinear theory, and Kirchhoff’s hypotheses. Both layers are assumed to be with the same material properties, while interaction between them is taken by the Winkler model. Discretization of the system of governing differential equations is performed by the Bubnov-Galerkin method with the two-mode approximation model, and the obtained system of ordinary differential equations is analysed by the Runge–Kutta method. The study of a linearized system allows the construction of regions of instability. The application of nonlinear dynamics methods made it possible to determine the nature of oscillations when stability is lost, and when small-scale effects are manifested.

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.

Airy stress based nonlinear forced vibrations and internal resonances of nonlocal strain gradient nanoplates

This paper presents a novel investigation of nonlinear forced vibrations and internal resonance in nonlocal strain gradient nanoplates, taking into account in-plane motions. The study comprehensively models the nanoplate structure, employing Kirchhoff's plate theory, nonlocal elasticity theory, and strain gradient elasticity theory. The incorporation of the Airy stress function allows for the consideration of in-plane displacements. The large amplitude deformations in the nanoplate are described using the von Kármán geometric nonlinearity model, and the equations of motion are derived using the force-moment dynamic balance method. The coupled equations of motion are solved using the Galerkin scheme and a dynamic equilibrium technique. Detailed discussions are provided on the influence of nonlocal and strain gradient parameters on the nonlinear vibration response of the Airy stress nanostructure. Moreover, it is demonstrated that specific combinations of nonlocal and strain gradient parameters lead to various types of internal resonance (including two-to-one and three-to-one internal resonances), significantly affecting the nonlinear frequency responses of the nanoplate, resulting in rich nonlinear behaviours. This study contributes to the understanding of the complex dynamics of nanoplates subjected to external time-dependant loads and offers valuable insights for the design of diverse nanoplate systems, including graphene sheets, which are relevant to nanoelectromechanical systems (NEMS) applications.

New closed-form solutions for flexural vibration and eigen-buckling of nanoplates based on the nonlocal theory of elasticity

New closed-form solutions for flexural vibration and eigen-buckling of nanoplates based on the nonlocal theory of elasticity

Frequency stability analysis of a cantilever viscoelastic CNT conveying fluid on a viscoelastic Pasternak foundation and under axial load based on nonlocal elasticity theory

AbstractIn this paper, the frequency stability analysis of a fluid conveying cantilever viscoelastic carbon nanotube (CNT) in the Kelvin–Voigt material model with slip boundary condition (BC) regime on a viscoelastic Pasternak foundation and under axial load has been performed based on nonlocal Euler–Bernoulli thin beam theory. The governing partial differential equation of motion (EOM) and its associated BCs have been derived using the Hamilton principle. The governing EOM has been converted into an ordinary differential equation using mode summation technique and solved by applying the extended Galerkin method. Then, the frequency stability analysis has been carried out for the vibrational response of CNT in the state‐space form. The obtained results have been validated with the literature works. The effect of changes of various parameters like elastic stiffness, shear stiffness and damping coefficients of foundation, nonlocal scale‐effect parameter of the nanotube, fluid flow Knudsen number, mass ratio, structural damping of the nanotube and applied axial force have been investigated in terms of frequency to predict the occurrence of different instability modes for the system. It is concluded that flutter and divergence instabilities in the vibration of the viscoelastic CNT are significantly affected by changes of different above‐mentioned parameters.

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.