Eringen's Elasticity Theory Research Articles

Nonlinear nonlocal damping effects under magnetic loads of a ferromagnetic-viscoelastic nanotube exposed to a nonlinear elastic medium with nonlocal viscosity

This study presents innovative results on a nonlinear dynamic model of a ferromagnetic-viscoelastic nanotube (FVN) whose dissipation effects are captured by material behavior with a linear dissipation law operating in a nonlinear geometric regime. It investigates a possible system derived from a Kelvin–Voigt type dissipation model that describes internal viscoelastic damping in the form of a parallel spring and a dashpot. In addition, a class of losses by linear viscous media in local and nonlocal damping is presented in the modeling. Magnetic load effects are studied using axial rod theory regardless of rotational inertia and shear effects. The governing equation is derived by Hamilton's principle for the FVN subjected to a transverse magnetic field resting on a nonlinear elastic foundation with viscose effects. Formulas have been developed to obtain the nonlinear damping characteristics of a single-walled nanotube. Then, multiple scale method (MSM)-based solutions and Eringen elasticity theory are applied to demonstrate the aforementioned effects on a narrow clamped-clamped (CC) nanotube in detail. The approximate analytical solution is verified by numerical integration methods and a good agreement between the approximate analytical and numerical results is observed. The effect of all important parameters such as local and nonlocal viscosity damping, nonlinear damping coefficient, linear and nonlinear stiffness of the elastic medium due to changes in amplitude vibration of the magnetic field (AVMF) and displacement responses (DR) have been investigated. The original research proposed in this paper may enable the systematic study of nonlinear damping in nanomechanical oscillators, which may help to reveal the underlying physical mechanism.

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.

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.

Investigating the free vibration of viscoelastic FGM Timoshenko nanobeams resting on viscoelastic foundations with the shear correction factor using finite element method

This article intends to investigate the free vibration damping characteristics of functionally graded viscoelastic nanobeams embedded in viscoelastic foundations using the infinite Kelvin–Voigt model of the Winkler–Pasternak type. This model is based on the nonlocal strain-driven gradient elasticity theory of Eringen. A power law model is then adopted to describe the continuous variation in the material properties of functionally graded material nanobeams. The study was carried by means of the first-order shear deformation theory with the shear correction factor using the finite element method. For the purpose of validating the developed model, it was decided to compare the results obtained with those reported about elastic nanobeams in the literature. Moreover, the influence of the power law exponent, structural damping coefficient, foundation coefficient, nonlocal parameter, and boundary conditions on the free frequency response of viscoelastic nanobeams was also investigated.

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.

Bending and hygro-thermo-mechanical vibration analysis of a functionally graded porous sandwich nanoshell resting on elastic foundation

This article presents the finite element method (FEM) used four-unknown shear deformation theory for the static bending and hygro-thermo-mechanical vibration analysis of sandwich functionally graded porous (SFGP) doubly curved nanoshells resting on the elastic foundation (EF). The configurations of SFGP nanoshells include a homogenous core made of full ceramic while the top and bottom layers vary through the thickness with the law of uneven porosity distribution. The governing equations are obtained by using Hamilton’s principle and the nonlocal elasticity theory of Eringen (nonlocal theory). For the first time, a four-node quadrilateral element with ten degrees of freedom (DOFs) for each node using Lagrangian and Hermitian interpolation functions to approximate the membrane and bending displacement fields are proposed to analyze the SFGP nanoshells. The numerical results are compared with other exact solutions to evaluate the performance of the proposed method. Furthermore, influences of geometrical parameters and material properties such as the power-law index the porosity coefficient the nonlocal coefficient and EF-stiffness ) on the static bending, free vibration of SFGP nanoshells are comprehensive studied.

Thermomagnetic modeling of a nonlocal viscoelastic half-space exposed to an internal heat source through a two-phase delay model

In this article, a thermomagnetic viscoelastic nonlocal model has been introduced that incorporating the nonlocal elasticity theory of Eringen and the Kelvin–Voigt type. In the heat conductivity equation, the dual-phase-lag thermoelastic model (DPL) has been used to account for microstructural effects caused by high-rate heat transfer processes. The proposed DPL thermoelastic model illustrates the effects of material deficiencies and the thermomechanical relationship due to the extremely rapid heating. Within the framework of the nonlocal DPL model, a one-dimensional magneto-thermoelastic problem of half-space whose surface is exposed to an instantaneous heat source and magnetic field has been studied. With the eigenvalue approach technique and Laplace transform method, analytical solutions for the studied physical fields are provided. The transformed expressions to the field quantities are reversed into the physical domain by applying Zakian's algorithm. The effects of the nonlocal parameter, the viscosity and the applied heat source have been shown graphically on the differences in the fields within the medium.

Vibration analysis of a rotating cantilever double-tapered AFGM nanobeam

Nano-structures such as carbon nanotube, nanobeams, nanorods, nanoplates, nanowires, and nanorings are tremendously used in various small-scale devices and investigating their dynamical behavior has been a hot research topic and can be beneficial in manufacturing and designing new devices. Therefore, free vibration analysis of a rotating cantilever double-tapered axially functionally graded material nanobeam is presented in this work. Due to the nano-scale dimension, classical beam theories are incompetent at describing the behavior of nanobeams. Thus, the nonlocal Eringen elasticity theory is adopted which considers the nonlocal scale effect for the small dimension effect of the structure of a nano-scale mechanism. The proposed nanobeam structure is assumed to taper linearly in two different axes, and its material is changing nonlinearly along its length. The equation of motion of the proposed system is found utilizing the nonlocal Eringen theory, and it is solved using a semi-analytical technique, differential transform method. Mode shapes and natural frequencies are extracted as the solution of the equation of motion of the system. Furthermore, the effects of several parameters such as nonlocal scale effect, rotational speed, hub radius, and taper ratios on the natural frequencies are investigated. Finally, a comparison between the presented work and other reported results show an excellent agreement.

Multiquadric radial basis function approximation method and asymptotic numerical method for nonlinear and linear static analysis of single-walled carbon nanotubes

In this paper, a numerical algorithm combining the meshless collocation technique based on the globally supported Multiquadric Radial Basis Function (MQRBF) approximation method and the Asymptotic Numerical Method (ANM) is developed to investigate the nonlinear and linear static behaviors of Single-Walled Carbone Nanotubes (SWCNTs) based on a Nonlocal Continuum Beam Model (NCBM). The NCBM which accounts for the effects of small scale, shear deformation and geometric nonlinearity of von-Kármán type is constructed by implementing the Nonlocal Elasticity Theory of Eringen (NET) into the First-Order Shear Deformation Elastic Beam Theory (FOSDEBT) and used to model the SWCNTs as a Continuous Elastic Nanobeam (CEN). Based on the Total Lagrangian Formulation (TLF) the nonlocal nonlinear governing equations of SWCNT are elaborated in a quadratic matrix strong form. These equations are then solved numerically by applying the proposed algorithm. This algorithm permits to transform the quadratic nonlocal nonlinear governing equations into a sequence of linear ones shared the same tangent stiffness matrix which are then solved numerically by the MQRBF approximation method, to get the whole branch of solution a continuation procedure is required. In order to highlight the reliability and efficiency of the present numerical algorithm, comparative studies are conducted.

A size-dependent moving Kriging meshfree model for deformation and free vibration analysis of functionally graded carbon nanotube-reinforced composite nanoplates

In this paper, a size-dependent meshfree model using the higher-order shear deformation plate theory in conjunction with the nonlocal Eringen elasticity theory for bending and free vibration analyses of functionally graded carbon nanotube-reinforced composite (FG CNTRC) nanoplates is presented. Configurations of carbon nanotubes (CNTs) are carried out for the uniform and functionally graded distributions via the plate thickness. Effective material properties are computed by the extended rule of mixture. The differential equation form of nonlocal elasticity theory is utilized to take account of size-dependent effects. Based on the principle of virtual work, discretized governing equations for nanoplates are obtained. Thereafter, the displacement and natural frequency of the FG CNTRC nanoplates are determined by a moving Kriging meshfree method. Essential boundary conditions are directly enforced at nodes the same with the finite element method because the moving Kriging shape function satisfies the Kronecker delta function property. Numerical results prove that the present model is simple, stable and well accurate prediction for nanostructures. Moreover, the stiffness-softening mechanisms are found when using the nonlocal elasticity theory leading to a rise of deflection and a decrease of natural frequency of FG CNTRC nanoplates.

Size-dependent free vibration and buckling analysis of sigmoid and power law functionally graded sandwich nanobeams with microstructural defects

The investigation conducted in this paper aims to study free vibration and buckling behaviors of size-dependent functionally graded sandwich nanobeams. In order to take into account the small size effects, nonlocal elasticity theory of Eringen's is incorporated. Material properties of the functionally graded sandwich beams are supposed to change continuously through the thickness direction according to two forms of the volume fraction of constituents by power law functionally graded material and sigmoid law functionally graded material. These rules are modified to consider the effect of porosity, which covers four kinds of porosity distributions. Two types of sandwich nanobeams were provided: (a) homogeneous core and functionally graded skins and (b) functionally graded core and homogeneous skins. Third-order shear deformation theory without any shear correction factor in conjunction with Hamilton's principle is used to extract the governing equations of motions of porous functionally graded sandwich nanobeams and then solved analytically for two hinged ends. The effects of nonlocal parameter, length to thickness ratios, material graduation index, amount of porosity, porosity distribution shape, on the nondimensional frequency and critical buckling load of the functionally graded sandwich nanobeams made of porous materials are exhibited by a parametric study.

Mechanical analysis of cutout piezoelectric nonlocal nanobeam including surface energy effects

This manuscript tends to investigate influences of nanoscale and surface energy on a static bending and free vibration of piezoelectric perforated nanobeam structural element, for the first time. Nonlocal differential elasticity theory of Eringen is manipulated to depict the long–range atoms interactions, by imposing length scale parameter. Surface energy dominated in nanoscale structure, is included in the proposed model by using Gurtin–Murdoch model. The coupling effect between nonlocal elasticity and surface energy is included in the proposed model. Constitutive and governing equations of nonlocal-surface perforated Euler–Bernoulli nanobeam are derived by Hamilton’s principle. The distribution of electric potential for the piezoelectric nanobeam model is assumed to vary as a combination of a cosine and linear variation, which satisfies the Maxwell’s equation. The proposed model is solved numerically by using the finite-element method (FEM). The present model is validated by comparing the obtained results with previously published works. The detailed parametric study is presented to examine effects of the number of holes, perforation size, nonlocal parameter, surface energy, boundary conditions, and external electric voltage on the electro-mechanical behaviors of piezoelectric perforated nanobeams. It is found that the effect of surface stresses becomes more significant as the thickness decreases in the range of nanometers. The effect of number of holes becomes significant in the region 0.2≤ɑ ≤0.8. The current model can be used in design of perforated nano-electro-mechanical systems (PNEMS).

A single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams

In this paper, a displacement-based single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams is presented. In the present theory, the beam axial displacement consists of a bending component, whereas the beam transverse displacement consists of a bending component and a shearing component. Bending components do not take part in the cross-sectional shearing force, and the shearing component does not take part in the cross-sectional bending moment. The present theory utilizes linear strain–displacement relations. The displacement functions of the present theory give rise to the constant transverse shear strain through the beam thickness. Hence as is the case with the nonlocal Timoshenko beam theory, the present theory also requires a shear correction factor. In the present theory, nonlocal differential stress–strain constitutive relations of Eringen are utilized in order to take into account the size-dependent behaviour of nanobeams. In the present theory, beam gross equilibrium equations are utilized to derive the governing differential equation. As against other first-order shear deformation nonlocal beam theories, the present theory involves only one governing differential equation of fourth order and has only one unknown function. There exists a strong resemblance between expressions of the cross-sectional shearing force, cross-sectional bending moment and governing differential equation of the present theory and corresponding expressions of the nonlocal Bernoulli–Euler beam theory based on the nonlocal elasticity theory of Eringen. For the present theory, boundary conditions are described. The present theory also describes two distinct and physically meaningful clamped boundary conditions. Illustrative examples pertaining to the flexure of shear deformable isotropic nanobeams demonstrate the efficacy of the present theory.

The influence of hole networks on the adsorption-induced frequency shift of a perforated nanobeam using non-local elasticity theory

Modelling the non-local frequency shift caused by the adsorption phenomenon of an adatom-perforated nanobeam system taking into account the influence of shear distortion as well as small-scale behavior has been proposed in this paper. The nanobeam structure is a one-dimensional system perforated with a network of periodic square holes. The small-scale behavior was modeled using the elasticity theory of Eringen, while the explicit shear force and bending moment were determined from the standard dynamic equations. Adatom-adatom energy and adatom-substrate energy were introduced using the van der Waals (vdW) interactions in the framework of Lennard-Jones (6–12) potential to determine the total shift in energy. Both the shear beam model (SBM) and Euler beam model (EBM) were derived in this work by modifying the system coupled equations. The frequency shift for the H/Au(100) system was calculated, which showed that its observed value depends on the size and number of holes as well as the mode number and adsorption density. These results have been interpreted in detail toward the appropriate design of mass detection devices.

Flapwise Bending Vibration Analysis of a Double Tapered Rotating Nonlocal Euler-Bernoulli Beam by the Differential Transform Method

Transverse (out-of-plane) vibrations of a rotating nanobeam tapered both in width and thickness are studied on the basis on the Euler-Bernoulli beam theory. The nonuniform nanobeam is modeled with allowance for a small-scale effect contained in the nonlocal Eringen elasticity theory. The axial force due to rotating movement (centrifugal stiffening) of the beam is included into the model. The governing partial differential equation of the structure is solved by implementing the differential transform method (DTM). The variations of the taper ratio, rotational velocity, hub radius, and nonlocal small-scale parameter are taken into consideration. Comparisons of the present results with available data are performed.

Wave dispersion analysis of magnetic-electrically affected fluid-conveying nanotubes in thermal environment

In this paper, the wave propagation analysis of fluid-conveying Magneto-Electro-Elastic (MEE) nanotube subjected to multi-physical fields is investigated via nonlocal strain gradient elasticity theory (NSGT). To take into account the small-scale effects, the nonlocal elasticity theory of Eringen is employed. Nonlocal governing equations of MEE nanotube have been derived utilizing Hamilton’s principle. The results of this study have been verified by checking them of antecedent investigations. An analytical solution of governing equations is used to acquire wave frequencies and phase velocities. The Knudsen number is considered to study the slip boundary wall of nanotube and flow. The effects of various parameters such as multi-physical fields, the Knudsen number, different mode, length parameter, nonlocal parameter, fluid velocity, fluid effect and the slip boundary condition on wave propagation characteristics of fluid-conveying MEE nanotube are investigated in detail.

Wave dispersion characteristics of fluid-conveying magneto-electro-elastic nanotubes

In this paper, the wave propagation analysis of fluid-conveying magneto-electro-elastic (MEE) nanotube incorporating fluid effect is investigated. To take into account the small-scale effects, the nonlocal elasticity theory of Eringen is employed. Nonlocal governing equations of MEE-FG nanotube have been derived utilizing Hamilton’s principle. The results of this study have been verified by checking them with antecedent investigations. An analytical solution of governing equations is used to acquire wave frequencies and phase velocities. The Knudsen number is applied to study the effect of slip boundary wall of nanotube and flow. Effect of Knudsen number, different modes, length parameter, nonlocal parameter, fluid velocity, fluid effect and slip boundary condition on wave propagation characteristics of fluid-conveying MEE nanotube is investigated, and the results are presented in detail.

A novel meshless particle method for nonlocal analysis of two-directional functionally graded nanobeams

Based on the Reddy–Bickford beam theory (RBBT), a comprehensive study on high-order bending, buckling and free vibration of two-directional functionally graded (FG) nanobeams is presented. The variation of the material properties is assumed to obey arbitrary power-law form in both axial and thickness directions. The equations of motion are derived using Hamilton’s principle, and small-scale effects are captured by nonlocal elasticity theory of Eringen. The RBBT formulation leads to complicated governing equations, and 2D FGM introduces additional stiffness terms. Accordingly, the governing equations cannot be solved by classical analytical schemes, and thus, symmetric smoothed particle hydrodynamics (SSPH) meshless method is adopted as an efficient numerical solution approach. The revised super-Gauss function is used as the kernel function. To validate the developed SSPH code, benchmark problems are studied and the results are compared to the analytical solutions. In this context, excellent agreement is observed. Several numerical examples are included to illustrate the effects of gradient indexes, boundary conditions, size scale parameters, aspect and elastic modulus ratios on static and dynamic responses of two-directional FG nanobeams.