Nonlocal Beam Theory Research Articles

Nonlinear vibration and parametric excitation of magneto‐thermo elastic embedded nanobeam using homotopy perturbation technique

AbstractThe present study focuses on the modeling and analyzing the nonlinear vibration patterns and parametric excitation of embedded Euler–Bernoulli nanobeams subjected to thermo‐magneto‐mechanical loads. The Euler–Bernoulli nanobeam is developed with external parametric excitation. The governing equation of motion is derived by utilizing nonlocal continuum theory and nonlinear von Karman beam theory. Subsequently, the homotopy perturbation technique is employed to determine the vibration frequencies. Finally, the modulation equation of Euler–Bernoulli nanobeams is derived for simply supported boundary condition. In order to validate our findings, we conduct a comparative analysis against existing literature, thereby underscoring the effectiveness and robustness of our proposed methodology. The influence of stress, magnetic potential, temperature, damping coefficient, Winkler coefficient, and nonlocal parameters are tested numerically on nonlinear frequency‐amplitude and parametric excitation–amplitude responses. The numerical examples indicate the significant impact of physical variables on the nonlinear frequency and parametric excitation. The primary objective of this study is to investigate the effects of external physical variables on the dynamic behavior of nanobeams, particularly in nonlinear regimes. This study provides insights into the design and control of nanostructures under complex load conditions, contributing to developing advanced materials and nanosystems.

Modelling architected beam using a nonlocal derivative-free shear deformable beam theory

Modelling architected beam using a nonlocal derivative-free shear deformable beam theory

Transverse vibration analysis of double-walled carbon nanotubes in an elastic medium under temperature gradients and electrical fields based on nonlocal Reddy beam theory

This work studies the mechanical behavior of double-walled carbon nanotubes (DWCNTs) under temperature gradients and electrical fields based on the nonlocal reddy beam model, focusing on the free vibration of DWCNTs. The DWCNTs include the interaction of van der Waals (vdW) forces between the outer and inner tubes, and they are supposed to be embedded in a Winkler-Pasternak foundation. Eringen’s nonlocal elasticity theory is taken into account to consider the small-scale effects. The governing equations are achieved according to Hamilton's principle and then discretized and solved using the generalized differential quadrature method. The impact of various parameters, such as nonlocal parameters, length and ratio of thickness-to-radius, Winkler, Pasternak coefficients and piezoelectric effects and radius to length ratio, has been studied. The outcomes show that with increasing outer radius of the nanotube, the first dimensionless frequency decreases. In addition, the results illustrate that when the length of the nanotube and the Winkler spring constant increase, the initial frequencies are increased. Moreover, the desirable value for ratios of L/r2 of DWCNTs in order to obtain the minimum and maximum dimensionless natural frequencies for Winkler spring model Kw is approximately 0.7 and 6.3, respectively, whereas this value for the shear constant of Pasternak Gp is about 1.25 and 4.75.

On the Static Flexural Response of Shear Deformable Isotropic Rectangular Nanobeams under the Parabolic Loading

It is well known that size-dependent effects, which are insignificant for macroscopic structures, become significant for small-scale structures like nanobeams. In addition, effects of the beam transverse shear deformation, which are insignificant for slender beams, become significant for shear deformable beams. Hence the need arises for a beam theory which is simple-to-use as well as which accounts for just-mentioned features of shear deformable nanobeams. Recently, the authors have developed a single variable new first-order shear deformation nonlocal beam theory (NFSDNBT, doi: 10.1007/s40430-019-2128-6). The NFSDNBT is applicable for the flexure of linear isotropic nanobeams undergoing small deformations. Displacement functions of the NFSDNBT give rise to the constant transverse shear strain through the beam thickness. Hence similar to the nonlocal Timoshenko beam theory (NTBT), the NFSDNBT also requires a shear correction factor. In the NFSDNBT, nonlocal differential stress-strain constitutive relations of Eringen are utilized, through which size-dependent effects have been taken into account. These relations relate not only the beam axial stress with the beam axial strain but also the beam transverse shear stress with the beam transverse shear strain. The governing equation of the NFSDNBT is obtained by utilizing beam gross equilibrium equations and it has a strong resemblance with the governing equation of the nonlocal Bernoulli-Euler beam theory (NBEBT). In this paper, the NFSDNBT is utilized for finding the static flexural response of shear deformable isotropic rectangular nanobeams under the action of parabolically distributed transverse loading. For the simply-supported, clamped-clamped and cantilever nanobeams, effects of variations in values of the nonlocal parameter of Eringen and beam thickness-to-length ratio on the maximum non-dimensional beam transverse displacement are presented. Obtained results are compared with corresponding results obtained by utilizing the NTBT and NBEBT so as to demonstrate the efficacy of the NFSDNBT. Along-the-length profiles of the non-dimensional beam transverse displacement for the just-mentioned cases of nanobeams, for various values of the nonlocal parameter and beam thickness-to-length ratio are also presented.

Free in-plane bending vibration of flexible L-shaped nanostructures based on the nonlocal beam theory

Free in-plane bending vibration of flexible L-shaped nanostructures based on the nonlocal beam theory

Size-dependent investigation of the 2D-FG CNTRC nanobeams using nonlocal strain gradient beam theory subjected to hygro-thermo conditions

In the present study, the vibration frequency and wave propagation investigation of the multidirectionally FG nanobeams reinforced with carbon nanotubes is studied comprehensively for the first time. A nonlocal model including both softening and stiffness enhancement impacts of the nanosize beam according to nonlocal strain gradient beam theory is presented to consider the size-dependent effect. The transverse response is separated into bending and shear parts to consider the shear impacts exactly. The effects of the moisture and thermal fields are considered nonlinearly through the thickness according to the comprehensive relations. Hamilton’s principle is utilized to achieve the dynamic equations of motion, whereas an analytical solution method is utilized to find the vibration characteristics of the nanobeam for several boundary conditions. The simulation results indicate that by reducing/raising the FG indexes, the impacts of the strain gradient coefficient on the frequency of the nanobeam will increase/decrease.

Mechanical Behavior and Physical Properties of Protein Microtubules in Living Cells Using the Nonlocal Beam Theory

A biomechanical model for vibrational analysis with characteristics of protein microtubules based on the nanobeam shape inside the cellular structure is presented. Young’s modulus of protein microtubules and unexplained length-dependent flexural rigidity are studied using a higher-order nonlocal shear deformation theory. The governing equations are provided by employing the principle of virtual work. The protein microtubules are considered simply supported for all numerical studies. The obtained results are critically discussed together with the theories as well as demonstrated in each case graphically.

Fractal nonlocal thermoelasticity of thin elastic nanobeam with apparent negative thermal conductivity

This study treats a new model of nonlocal fractal thermoelasticity beam theory for nanomaterials characterized by an apparent negative thermal conductivity which occurs in shaped graded materials. The model is based on the new concept of "product-like fractal measure" introduced by Li and Ostoja-Starzewski in their formulation of fractal continuum media with fractal dimension The basic setups of the theory are formulated, yet, we have restricted our analysis for the static case. The effects of fractal dimension on field variables, in particular the lateral deflection and the temperature increment of the resonator were analyzed analytically and graphically. It was observed that the physics of the thin elastic nanobeam is affected by the numerical value of the fractal dimension and the characteristic lengths of the theory. Comparable scenarios observed in complex material structures are obtained and discussed.

Thermal stress effects on size-dependent nonlinear axial vibrations of nanorods exposed to magnetic fields surrounded by nonlinear elastic medium

Taking von Kármán geometric nonlinearity into account, this research proposes a nonlinear model of nanorod, based on the nonlocal elasticity and axial beam theories. Imperfect nanorod is surrounded by nonlinear elastic medium and is subjected to the axial compression or various fields in terms of thermal and transvers magnetic loads. Pinned–pinned boundary conditions are immovable and free for the nanorod under axial compression considering thermal and magnetic loads. The governing equilibrium equations of the nanorod are derived by means of Hamilton's principle. The coupled nonlinear dimensionless differential equations are solved employing He's variational method. To evaluate the accuracy of the results, the results of this method are compared with the values obtained from the finite element method. Numerical results are provided to explore the influences of the low and high temperatures, nonlocal parameter, magnetic force of transverse field, amplitude of vibrations, and linear and nonlinear elastic materials for nanorods.

Exact analytical solutions for bending of bi-directional functionally graded nanobeams by the nonlocal beam theory using the Laplace transform

This paper presents a comprehensive analytical method to solve the fourth-order differential equation relevant to bending analysis of bi-directional functionally graded (BDFG) nanobeams obtained by the Eringen's nonlocal elasticity theory. Unlike usual numerical solutions (e.g., by following the Differential Quadrature method), analytical solutions following the Laplace transform are adopted. The material properties of the nanobeam vary along both the thickness and the axial directions by following the power law and the exponential law, respectively. Explicit equations are presented for bending displacements of BDFG nanobeams under various distributions of applied loads in combination with a few common boundary conditions. The effects of material inhomogeneity parameters along both the axial and the thickness directions, as well as that of the nonlocal parameter, are investigated. Comparisons with a few available results in the literature demonstrate the accuracy and unravel the versatility of the presented analytical method.

Effect of boundary conditions and constitutive relations on the free vibration of nonlocal beams

The free vibrations of nonlocal Euler and Timoshenko beams have been studied extensively, but there still remain some problems concerning boundary conditions and constitutive relations. For cantilever beams, a counterintuitive stiffening phenomenon is widely observed. And to the best of the authors’ knowledge, there is no work concerning about the Timoshenko beam with nonlocal shear force and local bending moment. This work reconsiders and explains these problems based on nonlocal differential theory. After obtaining the governing equation for nonlocal Euler beam, it is observed that the governing equation has the same form as that for the vibration of local beam with axial compressive force. Thus by comparing the nonlocal effects to the effects of end axial force, the stiffening phenomenon of cantilever beams is found to be caused by a meaningless term in boundary bending moment. This explanation is also suitable for nonlocal Timoshenko cantilever beams. Besides, for Timoshenko beams, the nonlocal effects on tensile and shear stress constitutive relations are considered separately. Three kinds of nonlocal Timoshenko beam models are solved. Numerical comparison shows that the effects of nonlocal tensile are bigger than that of nonlocal shear stress. Especially, for Timoshenko beams with nonlocal shear force only, the nonlocal effects lower the natural frequency for all boundary conditions. This finding can verify our explanation for the stiffening phenomenon of cantilever beams. In addition, it is also observed that, for both nonlocal beam theories, if all nonlocal internal forces are included in the governing equations, the eigenvalue equations and natural frequencies corresponding to local and nonlocal boundary conditions are the same.

A nonlocal continuum mechanics-based asymptotic theory for the buckling analysis of SWCNTs embedded in an elastic medium subjected to combined hydrostatic pressure and axial compression

Based on nonlocal continuum mechanics, the authors develop an asymptotic theory by which to examine the buckling behavior of simply supported, single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium under combined hydrostatic pressure and axial compression using the perturbation method. In the formulation, the Eringen nonlocal constitutive relations are used to account for the small length scale effect; the interactions between the SWCNT and its surrounding medium are modelled as a Winkler foundation model, and the half-thickness- to- the mid-surface radius ratio is selected as the perturbation parameter. After performing some mathematical processes, including non-dimensionalization, asymptotic expansion, and successive integration, among others, the authors separate the three-dimensional (3D) nonlocal elasticity equations into recursive sets of governing equations (GEs) based on the Donnell-type nonlocal classical shell theory (CST) for various order problems. The Donnell-type nonlocal CST is derived as a first-order approximation of the 3D nonlocal elasticity problem, where the GEs for higher-order problems retain the same differential operators as those of the Donnell-type nonlocal CST although with different nonhomogeneous terms. The current asymptotic solutions for the critical load parameters of the embedded SWCNT are obtained in order to assess the accuracy of various nonlocal CSTs and nonlocal beam theories available in the literature.

Three-dimensional asymptotic nonlocal elasticity theory for the free vibration analysis of embedded single-walled carbon nanotubes

Within the framework of three-dimensional (3D) nonlocal elasticity theory, the authors develop an asymptotic theory to investigate the free vibration characteristics of simply supported, single-walled carbon nanotubes (SWCNTs) non-embedded or embedded in an elastic medium using the multiple time scale method. Eringen’s nonlocal constitutive relations are adopted to account for the small length scale effect in the formulation. The interactions between the SWCNT and its surrounding medium are modeled as a two-parameter Pasternak foundation model. After performing a series of mathematical processes, including nondimensionalization, asymptotic expansion, and successive integration, etc., the authors obtain recurrent sets of motion equations for various order problems. The nonlocal classical shell theory (CST) is derived as a first-order approximation of the current 3D nonlocal elasticity problem, and the equations of motion for higher-order problems retain the same differential operators as those of the nonlocal CST, although with different nonhomogeneous terms. The current asymptotic solutions for the natural frequency parameters of non-embedded or embedded SWCNTs and their corresponding through-thickness modal stress and displacement component distributions are obtained to assess the accuracy of various nonlocal shell and beam theories available in the literature.

Bifurcation Analysis of an Electro-Statically Actuated Nano-beam Based on the Nonlocal Theory considering Centrifugal Forces

Abstract A nonlocal elasticity theory is a popular growing technique for mechanical analysis of the micro- and nanoscale structures which captures the small-size effects. In this paper, a comprehensive study was carried out to investigate the influence of the nonlocal parameter on the bifurcation behavior of a capacitive clamped-clamped nano-beam in the presence of the electrostatic and centrifugal forces. By using Eringen’s nonlocal elasticity theory, the nonlocal equation of the dynamic motion for a nano-beam has been derived using Euler–Bernoulli beam assumptions. The governing static equation of motion has been linearized using step by step linearization method; then, a Galerkin based reduced order model have been used to solve the linearized equation. In order to study the bifurcation behavior of the nano-beam, the static non-linear equation is changed to a one degree of freedom model using a one term Galerkin weighted residual method. So, by using a direct method, the equilibrium points of the system, including stable center points, unstable saddle points and singular points have been obtained. The stability of the fixed points has been investigated drawing motion trajectories in phase portraits and basins of attraction set and repulsion have been illustrated. The obtained results have been verified using the results of the prior studies for some cases and a good agreement has been observed. Moreover, the effects of the different values of the nonlocal parameter, angular velocity and van der Waals force on the fixed points have been studied using the phase portraits of the system for different initial conditions. Also, the influence of the nonlocal beam theory and centrifugal forces on the dynamic pull-in behavior have been investigated using time histories and phase portraits for different values of the nonlocal parameter.

Effect of residual surface stress on parametrically excited nonlinear dynamics and instability of double-walled nanobeams: an analytical study

A class of nonlinear Mathieu–Hill equation is established to determine the bifurcations and the regions of nonlinear dynamic instability of a short double-walled nanobeam, while the emphasis is placed on investigating the effect of residual surface stress on instability. To achieve this goal, first, a short double-walled nanobeam is modeled and embedded on a viscoelastic foundation and subjected to an axial parametric force. Second, based on the nonlocal elasticity and nonlinear von Karman beam theories, the nonlinear governing equation of motion is derived. Finally, Galerkin technique and multiple time scales method are used to solve the equation. Numerical examples are treated which show various discontinuous bifurcations. Also, infinitely stable and unstable solutions are addressed.

Bending, Buckling and Free Vibration Analysis of Size-Dependent Nanoscale FG Beams Using Refined Models and Eringen’s Nonlocal Theory

In this study, a theoretical unification of twenty-one nonlocal beam theories are presented by using a unified nonlocal beam theory. The small-scale effect is considered based on the nonlocal differential constitutive relations of Eringen. The present unified theory satisfies traction free boundary conditions at the top and bottom surface of the nanobeam and hence avoids the need of shearing correction factor. Hamilton’s principle is employed to derive the equations of motion. The present unified nonlocal formulation is applied for the bending, buckling and free vibration analysis of functionally graded (FG) nanobeams. The elastic properties of FG material vary continuously by gradually changing the volume fraction of the constituent materials in the thickness direction. Closed-form analytical solutions are obtained by using Navier’s solution technique. Non-dimensional displacements, stresses, natural frequencies and critical buckling loads for FG nanobeams are presented. The numerical results presented in this study can be served as a benchmark for future research.

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.

Nonlinear vibration and dynamic instability analysis nanobeams under thermo-magneto-mechanical loads: a parametric excitation study

The nonlinear vibration behavior and dynamic instability of Euler–Bernoulli nanobeams under thermo-magneto-mechanical loads is the main objective of the present paper. Firstly, a short Euler–Bernoulli nanobeam is modeled and exposed to an external parametric excitation. Based on the nonlocal continuum theory and nonlinear von Karman beam theory, the nonlinear governing differential equation of motion is derived. Secondly, to transport the partial differential equation to the ordinary differential equation, Galerkin method is applied. Then, multiple scales method, as an analytical approach, is used to solve the equation. At the end, modulation equation of Euler–Bernoulli nanobeams is obtained. Then, to evaluate the dynamic instability of the system, trivial and nontrivial steady-state solutions are discussed. Emphasizing the effect of parametric excitation, for considering the instability regions, bifurcation points are studied and investigated. As a result, it can be observed that the damping coefficient plays an effective role as well as parametric excitation in stability and frequency response of the system.

Vibration suppression of a boron nitride nanotube under a moving nanoparticle using a classical optimal control procedure

The current research investigates the vibration of single-walled boron nitride nanotube (SWBNNT) induced by a moving nanoparticle. In order to decrease the forced vibration of SWBNNT under a moving nanoparticle, a linear classical optimal control procedure is used to manipulate the movement of a nanoparticle as a drug material inside a nano-structure with consideration of various size-dependent procedures based on Rayleigh beam model. The Pasternak substrate is utilized to model the elastic medium. Hamilton, Galerkin and Newmark methods are also employed to solve the motion equations. Different size-dependent beam theories have been considered, such as the classical beam theory, nonlocal beam theory, strain gradient beam theory and nonlocal strain gradient beam theory, in order to reveal small-scale effects. The effects of the small length scale considered herein, the velocity of a moving nanoparticle, electrical potential field, the number of vibration modes and actuators and also resultant control force on the vibration behavior of the SWBNNT are investigated in detail.

A Jacobi polynomial based approximation for free vibration analysis of axially functionally graded material beams

The paper presents an effective approximation for free vibration analysis of axially functionally graded material (AFGM) beams based on the Jacobi polynomial theory. An arbitrary-order derivative of the Jacobi polynomial is expressed as the expression of low-order components, whereas its boundary values are fully defined by the polynomial parameters. The particular feature is used to derive the generalized eigenvalue equation for free vibration analysis of AFGM beams in conjunction with the Euler-Bernoulli, the Timoshenko and the nonlocal strain gradient beam theories. Several numerical examples in the literature are presented to demonstrate potential applications of the Jacobi polynomial approach. A fast convergence of the approximation error for natural frequency results has confirmed high accuracy of the proposed approach. The Legendre and the Chebyshev polynomials are special cases of the Jacobi basis function. This guarantees the flexibility of the presented method for free vibration analysis of AFGM beams with nonuniform geometries and axially varying material properties.