Length Of Nanobeam Research Articles

Surface energy effects on thermoelastic vibration of nanomechanical systems under Moore–Gibson–Thompson thermoelasticity and Eringen’s nonlocal elasticity theories

Surface and size effects are frequently important in the design of high-performance nano-sized devices. However, due to the influence of the small-scale impact, the classical theory fails to predict the size-dependent behavior of nanoscale devices. In this regard, the current work attempts to investigate the dynamic and mechanical behaviors of nanobeams using nonlocal elasticity theory and the recently proposed Moore–Gibson–Thompson (MGT) thermoelasticity theory while taking into account the surface effects. In the context of nonlocal elasticity and surface elasticity theories, the governing equation of motion for an Euler–Bernoulli nanobeam that is simply supported at both ends and has a uniform load on the top surface is first derived. Thereafter, the solution for deflection and temperature distribution along the thickness direction of the nanobeam is obtained based on MGT theory using the finite Fourier sine transform and Laplace transform techniques. The influences of nonlocal parameter, phase-lag time, residual surface tension, surface elastic modulus, thickness and length of nanobeam on deflection and temperature over time are systematically analyzed in depth. Some important points are highlighted regarding size dependency and surface effects on thermoelastic vibration of nanobeams in the present context. When surface and small-scale effects are included, this approach may be valuable in understanding the dynamic and mechanical behaviors of nanomechanical systems.

Core-shell heterostructure-enabled stress engineering in vanadium dioxide nanobeams

In strongly correlated materials (SCMs), especially for vanadium dioxide (VO2), manipulating physical properties through stress engineering is an important issue for the use of ultrafast metal-insulator transition (MIT) in device applications. Recent research efforts have mainly focused on modulation and related phenomena of physical properties by epitaxial and mechanical stresses in VO2 films or anisotropic nanocrystals. However, inhomogeneous stress in such planar and nanocrystal systems leads to complications induced by phase competitions or the creation of intermediate phases. Here, we demonstrate the core-shell heterostructures-enabled stress engineering on MIT, which provides accommodation of uniform axial stress and control of phase transition pathways in VO2 nanobeams. Specifically, the VO2 nanobeams with an amorphous Al2O3 shell undergo a simple and direct MIT at lower temperatures without intermediate phases, distinctly different from pristine nanobeams with coexisting phases. For the core-shell nanobeams, the VO2 core sustains a uniform compressive stress state along the nanobeam length caused by shell formation, which can be attributed to the different thermal behaviors coupled to the elastic modulus between the VO2 and shell. Our results can lead to better engineering of phase transitions in SCMs, providing the beneficial effect of suppressing inhomogeneities during the MIT process.

Stability analysis of nanobeams placed in electromagnetic field using a finite element method

In this work, the buckling analysis of the nanobeams placed in electromagnetic field is presented via a nonlocal finite element method based on Eringen’s nonlocal elasticity theory. The governing differential equation is derived by implementing minimum total potential energy principle. A finite element method is proposed for solution of nonlocal buckling of nanobeam placed in electromagnetic field. The contribution of this article is the use of interpolation functions and the nonlocal elasticity theory to form the stiffness matrices and geometric stiffness matrices of the electromagnetic nanobeam for buckling analysis. A detailed study is performed to indicate the influences of some parameters such as Hartmann parameter (Ha), mode number (n), nonlocal parameter (e0a), length of nanobeam (L), and boundary conditions on the buckling loads of Euler-Bernoulli nanobeams. The values of buckling load of electromagnetic nanobeams are obtained via a finite element solution, examined through several numerical examples, and demonstrated by tables and graphs.

Monolithic Silicon-Based Nanobeam Cavities for Integrated Nonlinear and Quantum Photonics

Photonic resonators allowing to confine the electromagnetic field in ultra-small volumes and with long decay times are crucial to a number of applications requiring enhanced nonlinear effects. For applications to integrated photonic devices on chip, compactness and optimized in-plane transmission become relevant figures of merit as well. Here we optimize an encapsulated Si/SiO$_2$ photonic crystal nanobeam cavity at telecom wavelengths by means of a global optimization procedure, where only the first few holes surrounding the cavity are varied to decrease its radiative losses. This strategy allows to achieve close to 10 million intrinsic quality factor, sub-diffraction limited mode volumes, and in-plane transmission above 65\%, in a structure with a record small footprint of around $8~\mu$m$^2$. We address and quantitatively assess the dependence of the main figures of merit on the nanobeam length and fabrication disorder. Finally, we theoretically give a realistic estimate of the single-photon nonlinearity in such a device, which holds promise for prospective experiments in low-power nonlinear and quantum photonics.

Vibration analysis of two-phase local/nonlocal viscoelastic nanobeams with surface effects

According to defects of the differential form of pure nonlocal elasticity, two-phase nonlocal elasticity has recently attracted the attention of researchers for studying the size-dependent mechanical behaviors of nanostructures. Given this, in present work, damping vibration behavior of two-phase local/nonlocal viscoelastic nanobeams in conjunction with surface effects is studied. The governing equation of Euler–Bernoulli nanobeam and corresponding constitutive boundary conditions are presented. Using analytical solution, the influences of viscoelastic parameter, surface effects, local phase fraction and nonlocal parameter on the size-dependent damping vibration behavior of nanobeams are investigated. From the results, it can be observed in all boundary conditions that not only the real part but also the imaginary part of the complex frequencies of nanobeams decreases, while the impact of nonlocal elasticity is raised by increasing the nonlocal parameter ratio or decreasing the local phase fraction. Also, the variation of frequency ratios of nanobeams with different dimensions reveals that the influence of surface elasticity on the both parts of complex frequencies depends on the size of cross section and length of nanobeam.

Dynamic instability of Euler–Bernoulli nanobeams subject to parametric excitation

Little research on the dynamic instability of nanobeams caused by parametric resonance has been reported in the literature. This paper presents an accurate and analytical method for investigating the dynamic instability of nanobeams based on the nonlocal continuum mechanics. The governing equation of transverse vibration of nanobeams subject to axial dynamic load is derived using the Hamilton's principle and the nonlocal theory to establish the Mathieu-Hill equation of dynamic stability, based on which the equations of critical excitation frequencies are derived by the Bolotin's theory to determine the regions of dynamic instability. Especially, the matrix singularity problem encountered in solving the equations of critical frequencies is overcome by matrix transformation. For verifying the accuracy of obtained regions of dynamic stability, the dynamic responses of nanobeams are computed by the fourth-order Runge-Kutta approach. By comprehensively exploring the size dependence of dynamic stability of nanobeams, it is found that the size scale parameter influences the regions of dynamic instability mainly through the nonlocal natural frequency and the nonlocal Euler buckling load. As the size scale parameter increases, the nonlocal natural frequency and the nonlocal buckling load decrease, which leads to the reduction of the value and bandwidth of critical excitation frequencies. Moreover, the size effects are found to decrease with an increase of length of nanobeam.

Thermoelastic Damping in FGM Nano-Electromechanical System in Axial Vibration Based on Eringen Nonlocal Theory

The thermo-elastic damping is a dominant source of internal damping in micro-electromechanical systems (MEMS) and nano-electromechanical systems (NEMS). The internal damping cannot neither be controlled nor minimized unless either mechanical or geometrical properties are changed. Therefore, a novel FGMNEM system with a controllable thermo-elastic damping of axial vibration based on Eringen nonlocal theory is considered. The effects of different parameter like the gradient index, nonlocal parameter, length of nanobeam and ambient temperature on the thermo-elastic damping quality factor are presented. It is shown that the thermo-elastic damping can be controlled by changing different parameter.

Buckling of Eccentrically Loaded Carbon Nanotubes

In the present study, buckling of eccentrically loaded nanobeams in which the load is not applied at the centroid of cross section, has been studied. Eringen’s Nonlocal Elasticity Theory has been used in the formulation of governing equation of motion of the nanobeam. Simply supported and free boundary conditions for nanobeam have been taken consideration. The effect of nonlocal parameter, eccentricity of the load, nanobeam length on the buckling deflection and critical buckling load on nanobeam have been investigated. Present results can be useful in the design of nano-structures.

Postbuckling scrutiny of highly deformable nanobeams: A novel exact nonlocal-surface energy-based model

This paper deals with analytical postbuckling analysis of elastic slender beam-like nanostructures by considering both surface energy and nonlocality effects. Using Euler-Bernoulli beam theory, the continuum-based governing equation is established for axially compressed nanobeams with large displacements. To capture the postbuckling behavior of the nanostructure more accurately, an exact solution is proposed on the basis of elastica approach. The explicit expressions of postbuckling load and maximum displacements are obtained for nanobeams with various ends. The predicted results are successfully checked with those of a Galerkin-modal-based approach. The postbuckled shapes of nanobeams are then presented based on the classical elasticity theory and the newly developed size-dependent models to show the roles of the nonlocality and surface energy. The effects of both nanobeam's length and diameter on the nonlinear trends of load-displacement are displayed. The present work can be regarded as a benchmark for further numerical/analytical studies on postbuckling of nonlocal-surface energetic ensembles of vertically aligned nanostructures.

Surface effects on vibration analysis of elastically restrained piezoelectric nanobeams subjected to magneto-thermo-electrical field embedded in elastic medium

In the present study, a generalized nonlocal beam theory is utilized to study the magneto–thermo–mechanical vibration characteristic of piezoelectric nanobeam by considering surface effects rested in elastic medium for various elastic boundary conditions. The nonlocal elasticity of Eringen as well as surface effects, including surface elasticity, surface stress and surface density are implemented to inject size-dependent effects into equations. Using the Hamilton’s principle and Euler–Bernoulli beam theory, the governing differential equations and associated boundary conditions will be obtained. The differential transformation method (DTM) is used to discretize resultant motion equations and related boundary conditions accordingly. The natural frequencies are obtained for the various elastic boundary conditions in detail to show the significance of nonlocal parameter, external voltage, temperature change, surface effects, elastic medium, magnetic field and length of nanobeam. Moreover, it should be noted that by changing the spring stiffness at each end, the conventional boundary conditions will be obtained which are validated by well-known literature.

Simulating the superheating of nanomaterials due to latent heat release in surface reconstruction

Surface reconstruction and phase transformation in nanomaterials can result in a significant rise in temperature due to the release of stored energy called “latent heat.” To simulate this behavior, we propose a hybrid continuum partial differential equation for non-Fourier heat transfer with a stochastic source term modeled using a kinetic Monte Carlo (KMC) algorithm for time-dependent rates to account for the latent heat release as the temperature is changing. The parameters required for the method are obtained through independent atomistic calculations. This includes energy barriers for KMC rates obtained using nudged elastic band calculations, and the non-Fourier thermal parameters obtained using a novel thermal parameter identification scheme described in this paper. As a demonstration of the approach, we study the superheating of silicon nanobeams observed in non-equilibrium molecular dynamics (NEMD) simulations. In these simulations, silicon (001) surfaces undergo a reconstruction from the ideal diamond crystalline surface to a reconstructed structure involving the formation of rows of dimers along a 〈110〉 direction. This reconstruction is accompanied by latent heat release that causes the nanobeam to dramatically superheat. The evolution of the nonuniform temperature profile along the nanobeam predicted by the continuum–KMC method is in excellent agreement with the NEMD results. A logarithmic dependence of the superheating temperature on nanobeam length is observed and theoretically discussed.

Free transverse vibration analysis of size dependent Timoshenko FG cracked nanobeams resting on elastic medium

In the present study, free transverse vibration of a cracked functionally graded (FG) size dependent Timoshenko nanobeam which is resting on polymer elastic foundation is investigated. It is supposed that the material properties of the FG nanobeam are varying continuously across the thickness according to the power-law distribution. To considering the small scale effect, the Eringen's nonlocal theory is used and for accounting the effect of polymer elastic foundation, the Winkler model is proposed. For this purpose, the equations of motion of the FG Timoshenko nanobeam and boundary conditions are obtained by using Hamilton's principle. To find the analytical solutions for equations of motion of FG nanobeam, the separation of variable method is employed. Two cases of boundary conditions i.e., simply supported-simply supported (SS) and clamped---clamped (CC) are investigated in the present work. Numerical results are demonstrating the good agreement between the results of present article and some cases available in the literature. The emphasis of the present study is based on investigating the effect of various parameters such as crack severity, crack position, gradient index, mode number, nonlocal parameter, elastic foundation parameter and nanobeam length. It is clearly revealed that the vibrational behavior of a FG nanobeam is significantly depending on these effects. Also, these numerical results can be serving as benchmarks for future studies of FG nanobeams.

Studying the influence of surface effects on vibration behavior of size-dependent cracked FG Timoshenko nanobeam considering nonlocal elasticity and elastic foundation

Free transverse vibration of a size-dependent cracked functionally graded (FG) Timoshenko nanobeam resting on a polymer elastic foundation is investigated in the present study. Also, all of the surface effects: surface density, surface elasticity and residual surface tension are studied. Moreover, satisfying the balance condition between the nanobeam and its surfaces was discussed. According to the power-law distribution, it is supposed that the material properties of the FG nanobeam are varying continuously across the thickness. Considering the small-scale effect, the Eringen’s nonlocal theory is used; accounting the effect of polymer elastic foundation, the Winkler model is proposed. For this purpose, the equations of motion of the FG Timoshenko nanobeam and boundary conditions are obtained using Hamilton’s principle. To find the analytical solutions for equations of motion of the FG nanobeam, the separation of variables method is employed. Two cases of boundary conditions, i.e., simply supported–simply supported (SS) and clamped–clamped (CC) are investigated in the present work. Numerical results are demonstrating a good agreement between the results of the present study and some available cases in the literature. The emphasis of the present study is on investigating the effect of various parameters such as crack severity, crack position, gradient index, mode number, nonlocal parameter, elastic foundation parameter and nanobeam length. It is clearly revealed that the vibrational behavior of a FG nanobeam is depending significantly on these effects. Also, these numerical results can be serving as benchmarks for future studies of FG nanobeams.

Free vibration of fractional viscoelastic Timoshenko nanobeams using the nonlocal elasticity theory

In this article, the free vibration of a fractional viscoelastic Timoshenko nanobeam is studied through inserting fractional calculus as a viscoelastic material compatibility equations in nonlocal beam theory. The material properties of a single-walled carbon nanotube (SWCNT) are used and two solution procedures are proposed to solve the obtained equations in the time domain. The former is a semi-analytical approach in which the Galerkin scheme is employed to discretize the governing equations in the spatial domain and the obtained set of ordinary differential equations is solved using a direct numerical integration scheme. On the contrary, the latter is entirely numerical in which the governing equations of system on the spatial and time domains are first discretized using general differential quadrature (GDQ) technique and finite difference (FD) scheme, respectively and then the set of algebraic equations is solved to arrive at the time response of system under different boundary conditions. Considering the second solution procedure as the main approach, its validity and accuracy are verified by the semi-analytical approach which is more difficult to enter various boundary conditions. Numerical results are also presented to get an insight into the effects of fractional derivative order, nonlocal parameter, viscoelasticity coefficient and nanobeam length on the time response of fractional viscoelastic Timoshenko nanobeams under different boundary conditions.

Torsional Vibration of Cracked Nanobeam Based on Nonlocal Stress Theory with Various Boundary Conditions: An Analytical Study

In this paper, the torsional vibration of cracked nanobeam was studied based on a nonlocal elasticity theory. The location of the crack is simulated by a torsional spring which links segments of nanobeam together. Also, different boundary conditions, including clamped–free, clamped–clamped and clamped–torsional spring, were considered. Furthermore, a detailed parametric study was conducted to investigate the influence of crack location, nonlocal parameter, length of nanobeam, spring constant and end supports on the torsional vibration.

Nonlocal nonlinear free vibration of nanobeams with surface effects

Abstract In this paper, nonlinear free vibration analysis of simply-supported nanoscale beams incorporating surface effects, i.e. surface elasticity, surface tension and surface density, is studied using the nonlocal elasticity within the frame work of Euler–Bernoulli beam theory with von karman type nonlinearity. A linear variation for the component of the bulk stress, σzz, through the nanobeam thickness is used to satisfy the balance conditions between the nanobeam bulk and its surfaces. An exact analytical solution to the governing equation of motion is presented for natural frequencies of nanobeams using elliptic integrals. The effect of the nanobeam length, thickness to length ratio, mode number, amplitude of deflection to radius of gyration ratio and nonlocal parameter on the normalized natural frequencies of aluminum and silicon nanobeams with positive and negative surface elasticity, respectively, is investigated. It is observed that the surface effects increase natural frequencies of the aluminum nanobeam for all values of the amplitude ratio and the silicon nanobeam at low amplitude ratios while at higher amplitude ratios the surface effects decrease the natural frequencies of the silicon nanobeam. Also, for all values of amplitude ratios, the normalized fundamental natural frequencies of silicon and aluminum nanobeams vary linearly with respect to the nonlocal parameter while this is not the case at higher mode numbers.

Vibration Analysis of Nano-Beam with Consideration of Surface Effects and Damage Effects

AbstractOn the basis of Euler-Bernoulli beam theory, surface elastic theory, the strain equivalent assumption and modiffed couple stress theory, the nonlinear governing equations of the nano-beam are derived. In addition, the Galerkin method and the Harmonic Balance Method are adopted so as to give a solution to the equations. In the example, the effects of nano-beam length, nano-beam thickness, damage factor and surface efect to curves of amplitude-frequency response of the nano-beam are discussed. The results show that damage effects should be taken into consideration and the frequency can be controlled by load and structure size of nano-beam.

Nonlinear free vibration of piezoelectric nanobeams incorporating surface effects

In this study, the nonlinear free vibration of piezoelectric nanobeams incorporating surface effects (surface elasticity, surface tension, and surface density) is studied. The governing equation of the piezoelectric nanobeam is derived within the framework of Euler–Bernoulli beam theory with the von Kármán geometric nonlinearity. In order to satisfy the balance conditions between the nanobeam bulk and its surfaces, the component of the bulk stress, σzz, is assumed to vary linearly through the nanobeam thickness. An exact solution is obtained for the natural frequencies of a simply supported piezoelectric nanobeam in terms of the Jacobi elliptic functions using the free vibration mode shape of the corresponding linear problem. Then, the influences of the surface effects and the piezoelectric field on the nonlinear free vibration of nanobeams made of aluminum and silicon with positive and negative surface elasticity, respectively, have been studied for various properties of the piezoelectric field, various nanobeam sizes and amplitude ratios. It is observed that if the Young’s modulus of a nanobeam is lower, the effect of the piezoelectric field on the frequency ratios (FRs) of the nanobeam will be greater. In addition, it is seen that by increasing the nanobeam length so that the nanobeam cross section is set to be constant, the surface effects and the piezoelectric field with negative voltage values increases the FRs, whereas it is the other way around when the nanobeam cross section is assumed to be dependent on the length of the nanobeam.

Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity

In this paper, to consider all surface effects including surface elasticity, surface stress, and surface density, on the nonlinear free vibration analysis of simply-supported functionally graded Euler–Bernoulli nanobeams using nonlocal elasticity theory, the balance conditions between FG nanobeam bulk and its surfaces are considered to be satisfied assuming a cubic variation for the component of the normal stress through the FG nanobeam thickness. The nonlinear governing equation includes the von Kármán geometric nonlinearity and the material properties change continuously through the thickness of the FG nanobeam according to a power-law distribution of the volume fraction of the constituents. The multiple scale method is employed as an analytical solution for the nonlinear governing equation to obtain the nonlinear natural frequencies of FG nanobeams. The effect of the gradient index, the nanobeam length, thickness to length ratio, mode number, amplitude of deflection to radius of gyration ratio and nonlocal parameter on the frequency ratios of FG nanobeams is investigated.

Nonlocal nonlinear free vibration of functionally graded nanobeams

In this paper, nonlinear free vibration of functionally graded (FG) nanobeams with immovable ends, i.e. simply supported-simply supported (SS) and simply supported-clamped (SC), is studied using the nonlocal elasticity within the frame work of Euler–Bernoulli beam theory with von kármán type nonlinearity. The material properties are assumed to change continuously through the thickness of the FG nanobeam according to a power-law distribution. The analytical solution for the nonlinear natural frequency is established using the method of multiple scale. The small scale effects on the linear/nonlinear nonlocal frequency to the linear/nonlinear classical frequency ratios (the linear/nonlinear frequency ratios) are examined for various parameters such as the FG nanobeam length, the FG nanobeam thickness to length ratio (the thickness ratio), the vibration amplitude to the radius of gyration ratio (the amplitude ratio), and the boundary condition. As a main result, it is observed that while the linear frequency ratios are independent of the gradient index, the nonlinear frequency ratios vary with the gradient index.