Nanoscale Beams Research Articles

A comparative study of 1D nonlocal integral Timoshenko beam and 2D nonlocal integral elasticity theories for bending of nanoscale beams

In this paper, the bending behavior of nanoscale beams is studied using the 1D nonlocal integral Timoshenko beam theory (NITBT) and the 2D nonlocal integral elasticity theory (2D-NIET) using two types of nonlocal kernels, i.e., the two-phase kernel and a modified kernel which compensates the boundary effects. The governing equations are solved using the finite element method and the COMSOL code. Mesh sensitivity study and numerical verifications are presented. The main differences and similarities in both theories at the nanoscale are revealed. For both theories and both kernels, a softening behavior is found by increasing the nonlocal parameter and decreasing the phase parameter, for different boundary and load conditions. In contrast to the differential theory, no paradoxical behavior for the cantilever conditions is found for both theories. The sensitivity of the 2D-NIET to the nonlocal parameter is found higher than that of the NITBT. The normalized transverse deflection for the 2D-NIET is found independent of the boundary and load conditions. Also, the normalized transverse deflection varies linearly versus the normalized nonlocal parameter for both theories with the two-phase kernel and any condition except for the simply supported beam under distributed load condition in the NITBT. The boundary effect, resulting in a different softening near the boundaries, reduces by increasing the phase parameter. The modified kernel in the 2D-NIET is found sensitive to the pinned not to the free and fixed boundaries. It is in detail shown that for the 2D-NIET especially with the modified kernel, by increasing the nonlocal parameter, the deflection increases with almost the same ratio through the entire length of the beam and for all the boundary conditions. The obtained results can be used for modeling of various beam problems with nonlocal effects at the nanoscale.

Memory and dynamic response of a thermoelastic functionally graded nanobeams due to a periodic heat flux

This study highlights the introduction of a new mathematical model for functionally graded thermoelastic nanobeams (FGNB) which contains a free choice of the kernel function and time delay. The basic equations that govern the proposed model have been constructed on the basis of Hamilton’s principle, Euler-Bernoulli's assumptions, Eringen’s theory and three-phase-lag memory dependent heat conduction. The FGNB is due to a heat flux depending on time. The nanoscale beams can be regarded as non-homogeneous composite structures and typically vary from the ceramic at the bottom of the beam to the metal on the top with a continuous structural transmission along with the thickness of the beam. Utilizing the Laplace integral transform, the problem is solved analytically. The variations in moment distribution, temperature, displacement and deflection are illustrated graphically for the various kernel functions, time delays, the influence of the nonlocal parameter, and periodic pulse. Compared with the current thermoelastic models, the implications of the present model are discussed.

Study of frequency shift and thermoelastic damping in transversely isotropic nano-beam with GN III theory and two temperature

In this study, a nanoscale beam of transversely isotropic thermoelastic (TIT) medium with two temperature and with Green–Naghdi (GN) III theory of thermoelasticity for free vibrations with simply supported boundaries have been examined. Euler–Bernoulli (EB) beam theory is used to formulate a mathematical model of the nanoscale beam in a closed form. The lateral deflection, frequency shift, thermal moment and thermoelastic damping have been solved. A program in MATLAB software is developed to find the numerical values for different physical quantities. The lateral deflection, frequency shift, thermal moment and thermoelastic damping for the varying two temperature, different modes, frequency of time harmonic sources, thickness of nano-beam, and material medium of nano-beam has been represented graphically and discussed. The results for some particular cases have also been compared to some earlier work done.

Scientific Visualization of the Results of a Numerical Experiment of the Nonlinear Dynamics of a Nanoscale Beam Structure

The paper presents the results of scientific visualization of the nonlinear dynamics of contact interaction of a nanoscale beam structure under the action of an external harmonic load. The beam structure consists of two beams obeying the kinematic hypotheses of Euler-Bernoulli and S.P. Timoshenko. The constructed mathematical model takes into account geometric and constructive nonlinearities. The size-dependent behavior of the structure is implemented on the basis of the modified moment theory of elasticity. The resulting system of partial differential equations is reduced to a system of ordinary differential equations by the second order finite difference method. The Cauchy problem is solved by the fourth order Runge-Kutta method. In this work, using the methods of scientific visualization of the results of applying the methods of nonlinear dynamics, the influence of the size-dependent parameter and external load on the vibrations of the beam structure is investigated. As methods for studying nonlinear dynamics, the work uses wavelet spectra based on the mother Morlet, Fourier power spectra, signals. The use of scientific visualization methods makes it possible to develop specific recommendations for the operating conditions of the beam structure. This, in turn, makes it possible to avoid unwanted vibration modes of beam nanostructures, which are widely used as sensitive elements of sensors of micro and nano electromechanical systems.

Nonlinear nonlocal-surface energy-based vibrations of a bidirectionally excited nanobeam at its supports

This study concerns with nonlinear axial and lateral vibrations of nanoscaled beams under action of bidirectional support excitations. Using the Euler-Bernoulli beam model through considering the von-Karman terms, small-scale effect, and surface energy, the nonlinear equations of motion of an excited nanobeam are derived in terms of purely dynamical deformations. By utilizing the Galerkin approach on the basis of the appropriate modes, the resulting nonlinear equations are reduced to the nonlinear ordinary differential equations, and these are solved for the unknown elastic deformations due to the most general form of support excitations. The roles of the excitation amplitude, excitation frequency, geometry of the nanostructure, surface energy, and nonlocality on the nonlinear dynamic response are investigated constructively. The importance of nonlinear analysis in capturing the more realistic dynamic response of the bi-directionally excited nanobeam at its ends is also revealed and explained methodically.

A semi-analytical study on the nonlinear pull-in instability of FGM nanoactuators

In this paper, a new semi-analytical solution for estimating the pull-in parameters of electrically actuated functionally graded (FG) nanobeams is proposed. All the bulk and surface material properties of the FG nanoactuator vary continuously in thickness direction according to power law distribution. Here, the modified couple stress theory (MCST) and Gurtin-Murdoch surface elasticity theory (SET) are jointly employed to capture the size effects of the nanoscale beam in the context of Euler-Bernoulli beam theory. According to the MCST and SET and accounting for the mid-plane stretching, axial residual stress, electrostatic actuation, fringing field, and dispersion (Casimir or/and van der Waals) forces, the nonlinear nonclassical equation of motion and boundary conditions are obtained derived using Hamilton principle. The proposed semi-analytical solution is derived by employing Galerkin method in conjunction with the Particle Swarm Optimization (PSO) method. The proposed solution approach is validated with the available literature. The freestanding behavior of nanoactuators is also investigated. A parametric study is conducted to illustrate the effects of different material and geometrical parameters on the pull-in response of cantilever and doubly-clamped FG nanoactuators. This model and proposed solution are helpful especially in mechanical design of micro/nanoactuators made of FGMs.

Tunable beam splitter using bilayer geometric metasurfaces in the visible spectrum.

Metasurfaces have been widely investigated for their capabilities of manipulating wavefront versatilely and miniaturizing traditional optical elements into ultrathin devices. In this study, a nanoscale tunable beam splitter utilizing a bilayer of geometric metasurfaces in the visible spectrum is proposed and numerically examined. Inspired by the diffractive Alvarez lens and multilayer geometric metasurfaces, opposite quadratic phase distributions are imparted on both layers, and a varying linear phase gradient will arise through relatively lateral displacement between two layers, generating tunable angles of deflection. In addition, such geometric metasurfaces offer opposite directions of phase gradients for orthogonal circularly polarized incidences, leading to effective polarization beam splitting. Results prove that the splitting angles can be tuned precisely, and the energy split ratio can be effectively changed according to the ellipticity of the polarized incidence. This design could find significant applications in optical communication, measurement, display, and so on.

Pull-in and freestanding instability of actuated functionally graded nanobeams including surface and stiffening effects

Because of fasttechnological development, electrostatic nanoactuator devices like nanosensors, nanoswitches, and nanoresonators are highly considered by scientific community. Thus, this article presents a new solution technique in solving highly nonlinear integro-differential equation governing electrically actuated nanobeams made of functionally graded material. The modified couple stress theory and Gurtin–Murdoch surface elasticity theory are coupled together to capture the size effects of the nanoscale thin beam in the context of Euler–Bernoulli beam theory. For accurate modelling, all the material properties of the bulk and surface continuums of the FG nanoactuator are varied continuously in thickness direction according to power law. The nonlinearity arising from the electrostatic actuation, fringing field, mid-plane stretching effect, axial residual stress, Casimir dispersion, and van der Waals forces are considered in mathematical formulation. The nonlinear nonclassical equilibrium equation of FG nanobeam-based actuators and associated boundary conditions are exactly derived using Hamilton principle. The new solution methodology is combined from three phases. The first phase applies Galerkin method to get an integro-algebraic equation. The second one employs particle swarm optimization method to approximate the integral terms (i.e. electrostatic force, fringing field, and intermolecular forces) to non-integral cubic algebraic equation. Then, solved the system easily in last phase. The resulting algebraic model provides means for obtaining critical deflection, pull-in voltage, detachment length, minimum gap, and freestanding effects. A reasonable agreement is found between the results obtained from the present method and those in the available literature. A parametric study is performed to investigate the effects of the gradient index, material length scale parameter, surface energy, intermolecular forces, initial gap, and beam length on the pull-in response and freestanding phenomena of fully clamped and cantilever FG nanoactuators.

Nonlinear Primary Frequency Response Analysis of Self-Sustaining Nanobeam Considering Surface Elasticity

This paper is focused to investigate the effects of nonlinear sources, including viscoelastic foundation and geometrical nonlinearity along with the surface elasticity and residual surface stress effects on the primary frequency response of a harmonically excited nanoscale Bernoulli-Euler beam. Due to large surface-area-to-volume ratio, the theory of surface elasticity as well as residual surface stress effects are taken into account within the beam models. The Galerkin approach accompanied by trigonometric shape functions is utilized to reduce the governing PDEs of the system to ODEs. The multiple scales perturbation method theory is applied to compute the nonlinear frequency response of nanobeam. The effects of linear and nonlinear viscoelastic damping coefficient of the medium, crystallographic directions of [100] and [111] of anodic alumina, geometrical nonlinear term and geometrical property on the nonlinear primary frequency response of nanoscale beam are investigated. The results show that theses parameters have a significant effect on the nonlinear frequency response of nanobeams in the case of primary resonance.

Effects of nonlocal thermoelasticity on nanoscale beams based on couple stress theory

A modified nonlocal Bernoulli–Euler model was established in this paper to analyze thermoelastic interactions in isotropic nanobeams subjected to ramp‐type heating, depending on the modified couple stress and generalized thermoelasticity theories. To solve this problem, the thermoelasticity theory with one thermal relaxation time (Lord–Shulman [LS]) is proposed. Due to the modified couple stress, the analytical expressions for deflection, temperature, flexure moment, and displacement in the nanobeam were obtained by using the Laplace transform technique. In addition, comparisons were shown in graphs to estimate the effects of the various parameters such as the couple stress, and the ramping time parameter on all the fields studied under consideration. Also, the numerical results are provided to demonstrate the small‐scale impact on the resonator of the nanobeams.

Effect of memory dependent derivative on forced transverse vibrations in transversely isotropic thermoelastic cantilever nano-Beam with two temperature

The present research deals with the study of forced vibrations in transversely isotropic thermoelastic (TIT) nanoscale beam with two temperature (2T). Memory dependent derivative theory of thermoelasticity for clamped-free/cantilever nano-beam has been considered. The mathematical model is prepared for the nanoscale beam in a closed form with the application of Euler Bernoulli beam theory. Laplace transform method is employed to solve the problem. Forced vibrations due to exponential decaying time varying load acting vertically downward along the thickness direction of the nano-beam, Uniform load, Time harmonic load have been considered. Dynamic analysis for these forced vibrations and Static analysis has been carried out in this research. The dimensionless expressions for lateral deflection, thermal moment, temperature change, and axial stress are solved for these three forced vibrations. Response ratio has also been calculated. The analytical results have been numerically analysed using programming in MATLAB. The effect of kernel function has been depicted graphically on the lateral deflection, thermal moment, temperature change, axial stress and response ratio for all the three types of forced vibrations. Some particular cases have also been discussed.

Stability of perforated nanobeams incorporating surface energy effects

This paper aims to present an analytical methodology to investigate influences of nanoscale and surface energy on buckling stability behavior of perforated nanobeam structural element, for the first time. The surface energy effect is exploited to consider the free energy on the surface of nanobeam by using Gurtin-Murdoch surface elasticity theory. Thin and thick beams are considered by using both classical beam of Euler and first order shear deformation of Timoshenko theories, respectively. Equivalent geometrical constant of regularly squared perforated beam are presented in simplified form. Problem formulation of nanostructure beam including surface energies is derived in detail. Explicit analytical solution for nanoscale beams are developed for both beam theories to evaluate the surface stress effects and size-dependent nanoscale on the critical buckling loads. The closed form solution is confirmed and proven by comparing the obtained results with previous works. Parametric studies are achieved to demonstrate impacts of beam filling ratio, the number of hole rows, surface material characteristics, beam slenderness ratio, boundary conditions as well as loading conditions on the non-classical buckling of perforated nanobeams in incidence of surface effects. It is found that, the surface residual stress has more significant effect on the critical buckling loads with the corresponding effect of the surface elasticity. The proposed model can be used as benchmarks in designing, analysis and manufacturing of perforated nanobeams.

A coupled nonlinear nonlocal strain gradient theory for functionally graded Timoshenko nanobeams

A coupled nonlinear nonlocal strain gradient theory (NSGT) is developed for functionally graded Timoshenko nanoscale (FGTN) beams. This highly nonlinear-scale-dependent continuum model is solved numerically for dynamical responses for the first time. Hamilton’s energy minimisation technique is used for rotation-equation of the nanobeam in addition to axial-displacement and transverse-displacement equations; these equations are coupled both nonlinearly and linearly, where decoupling is not possible. The functional nature of the material of the nanobeam is incorporated via the Mori–Tanaka mixture scheme. Small-scale influences are taken into account via the NSGT which is able to incorporate both softening/hardening effects. The coupled/nonlinear model is reduced to a high-dimensional system using Galerkin’s method. The dynamics of the FGTN beam is analysed and the influence of functionally graded nature as well as the softening/hardening effects due to the NSGT is highlighted.

Optimum Design of Nano-Scaled Beam Using the Social Spider Optimization (SSO) Algorithm

In this research study, the optimum cross-sectional dimensions of nano-scale beam elements are investigated under different load conditions. Euler-Bernoulli beam model based on nonlocal elasticity theory is utilized for the analysis of the beam. Two types of nano-scaled beams are modeled; carbon nanotubes (CNTs) and Boron nitride nanotubes (BNNTs). The novel meta-heuristic based optimization algorithm called Social Spider Optimization (SSO) algorithm is employed to find the beam designs with the objective of minimizing the cross-sectional area. Furthermore, the obtained optimum cross-sectional dimensions, critical stress and displacement values of the beams are compared according to the material type, beam length, and load conditions.

Small scale and spin effects on free transverse vibration of size-dependent nano-scale beams

This study aims to investigate the transverse vibration of size-dependent nano-beam which spins about its longitudinal axis. To this end, nano-beams are modeled based on the Euler–Bernoulli and Timoshenko theories. To consider the size dependency of the nano-beams, the nonlocal elasticity theory is used. By using an exact analytical solution, the governing equations of motions are solved. Comprehensive results are presented to show the effects of various boundary conditions, nonlocal scale parameter, rotational speed, and geometrical parameters on natural frequencies and critical speeds of spinning nano-beam. As spinning nano-beams are the main part of any rotary nano-machines, conclusions of present analysis promote researcher’s information about the vibrational behavior of spinning nano-beams.

Nonlinear Analysis for Bending, Buckling and Post-buckling of Nano-Beams with Nonlocal and Surface Energy Effects

The modeling and analysis for mechanical response of nano-scale beams undergoing large displacements and rotations are presented. The beam element is modeled as a composite consisting of the bulk material and the surface material layer. Both Eringen nonlocal elasticity theory and Gurtin–Murdoch surface elasticity theory are adopted to formulate the moment–curvature relationship of the beam. In the formulation, the pre-existing residual stress within the bulk material, induced by the residual surface tension in the material layer, is also taken into account. The resulting moment-curvature relationship is then utilized together with Euler–Bernoulli beam theory and the elliptic integral technique to establish a set of exact algebraic equations governing the displacements and rotations at the ends of the beam. The linearized version of those equations is also established and used in the derivation of a closed-form solution of the buckling load of nano-beams under various end conditions. A discretization-free solution procedure based mainly upon Newton iterative scheme and a selected numerical quadrature is developed to solve a system of fully coupled nonlinear equations. It is demonstrated that the proposed technique yields highly accurate results comparable to the benchmark analytical solutions. In addition, the nonlocal and surface energy effects play a significant role on the predicted buckling load, post-buckling and bending responses of the nano-beam. In particular, the presence of those effects remarkably alters the overall stiffness of the beam and predicted solutions exhibit strong size-dependence when the characteristic length of the beam is comparable to the intrinsic length scale of the material surface.

Magneto-electro-elastic analysis of piezoelectric–flexoelectric nanobeams rested on silica aerogel foundation

This article explores that the study on bending of magneto-electric-elastic nanobeams relies on nonlocal elasticity theory. The Vlasov’s model foundation utilizes the silica aerogel foundation. The guiding expressions of nonlocal nanobeams in the considered framework are used extensively and where parabolic third-order beam theory is achieved after using Hamilton’s principle. Parametric work is introduced to scrutinize the influence of the magneto-electro-mechanical loadings, nonlocal parameter, and aspect ratio on the deflection characteristics of nanobeams. It is noticed that the boundary conditions, nonlocal parameter, and beam geometrical parameters have significant effects on dimensionless deflection of nanoscale beams.

Nonlinear bending and postbuckling analysis of FG nanoscale beams using the two-phase fractional nonlocal continuum mechanics

The geometrically nonlinear bending and postbuckling of nanoscale beams are investigated herein according to the two-phase fractional nonlocal continuum model. It is considered that the beams have been made from functionally graded materials (FGMs), and the Bernoulli-Euler beam model is employed for their modeling. The variational form of the governing fractional equation is obtained first by means of an energy approach. Thereafter, a novel numerical solution method is proposed named as fractional variational differential quadrature method (FVDQM). In FVDQM, which is applied to the variational statement of the problem in a direct way, a combination of the differential quadrature method and matrix operators is utilized. The efficiency of the proposed fractional nonlocal model is evaluated by molecular dynamics (MD) simulations. Selected numerical results are given to explore the influences of fractional order, nonlocality, length-to-thickness ratio and FG index on the nonlinear bending and postbuckling responses of FG nanobeams with various types of boundary conditions.

Nonlocal coupled thermoelastic wave propagation band structures of nano-scale phononic crystal beams based on GN theory with energy dissipation: An analytical solution

This paper deals with the band structures of thermoelastic waves in nano-scale phononic crystal or metamaterial beams considering nano-size effects. The nonlocal coupled thermoelastic governing equations are derived using the Green–Naghdi theory of the generalized coupled thermoelasticity with energy dissipation and Eringen’s nonlocal theory. The derived governing equations are analytically solved and the field quantities including the temperature and the deflection are obtained in the closed forms. Using the proposed analytical solution, the transfer matrix between two unit-cells are obtained using the thermal and mechanical continuity conditions on the interfaces between the unit-cells and between the two sections of each unit-cell. The band structures of the phononic crystals are obtained using the Bloch–Floquet theorem. The detailed discussions are presented for the band structures of nonlocal thermoelastic waves in nano-scale aluminum/epoxy phononic crystal or metamaterial beams. Also, the effects of the small-scale parameter and the thickness of the epoxy layers on the band structures are studied and discussed by using the derived analytical solution.

Stability and Nonlinear Vibration Analysis of an Axially Loaded Nanobeam Based on Nonlocal Strain Gradient Theory

The size-dependent stability mechanism and nonlinear dynamics of a nano-scale beam under an axial time-dependent load are theoretically studied based on nonlocal strain gradient theory. First, with the consideration of geometric nonlinearity of the nanobeam, the equation of motion is derived with the aid of Hamilton’s principle. Then, by using both Floquet technology and Bolotin’s method, a linear analysis is employed to investigate the size-dependent stability mechanism of the nanobeam for pinned-pinned boundary conditions. It is demonstrated that the results obtained based on the Floquet technology and Bolotin’s method agree well with each other in most cases. Furthermore, the analytical expressions for stability boundaries of the nanobeam system are derived. It is found that when the size-dependent nanobeam system is reduced to a classical one, the obtained expressions for stability boundaries are in good agreement with previous experimental results. Finally, the nonlinear responses of the nanobeam are presented in the form of time traces, bifurcation diagrams and phase portraits. It is shown that the axially excited nanobeam always undergoes a limit cycle oscillation. For all results regarding stability boundaries and nonlinear responses, the size-dependent effects are found to be significant.