Nonlocal Small-scale Parameter Research Articles

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.

How does porosity affect the free vibration of single-layered graphene sheets?

This paper aims to investigate the influence of porosity and length size on the free vibration of single-layered graphene sheets (SLGSs). Frequency analysis is performed using a finite element based molecular structural mechanics (MSM) approach mimicking the SLGSs as frame-like structures constructed out of the beam elements. Defining a porous unit cell, 320 SLGSs with different arrangements and values of porosities and various length sizes ranging from 4 to 32 nm are considered. Results reveal that increasing porosity as well as length size both decrease the natural frequencies of SLGSs, significantly. To improve the applicability of the results, a nonlocal small scale parameter introduced by the analytical solutions for vibration of nanoplates in the literature is calibrated in such a way that the obtained frequencies by MSM match the analytical solutions based on the nonlocal theory of elasticity. Both neural network and genetic programming processes are successfully implemented for the calibration. The proposed calibrated parameter can be easily applied to evaluate the natural frequencies of SLGSs for certain values of porosities and length sizes.

Nonlocal Analysis of Natural Vibrations of Carbon Nanotubes

When the atomic level is considered, the dynamic properties have a major influence on nanostructures behavior because of their ultrahigh or very high natural frequencies. Vibrations are essential in the analysis of resonators, oscillators, and sensors for carbon nanotube-based devices. The present paper is dedicated to the eigenfrequencies analysis of single-walled carbon nanotubes. In the analysis, the small-scale coefficient is introduced, and the nonlocal elasticity theory is applied in the modal analysis of the carbon nanotubes. The correlation between nonlocal small-scale parameter and the vibrational behavior of the carbon nanotubes is studied. The determination of nonlocal parameter is based on the structural finite element model of the nanotube. A detailed parametric study is realized to investigate the effects of the length-to-diameter ratio L/D, nonlocal coefficient, the influence of boundary conditions on eigenfrequencies of carbon nanotubes. The main contribution of this work is the evaluation of the nonlocal parameter for single-walled carbon nanotubes based on the finite element approximation in modeling the dynamic behavior instead of commonly used MD simulations. It was clearly displayed that the application of the nonlocal beam models involves analysis of the small-scale parameter, whose value determines the final eigenfrequencies. The highest values of eigenfrequencies were observed for the local beam model. It was presented that the eigenfrequencies for carbon nanotubes decrease when the nanotube length increases for all analyzed modes. On the other hand, when the diameter of carbon nanotubes increases, the eigenfrequency values rise. The numerical validation of the nonlocal parameter revealed the high dependence on the boundary conditions and the mode number. However, the influence of the mode number and the boundary conditions is vanishing for long single-walled carbon nanotubes.

Does vibration amplitude influence the evaluation of nonlocal small scale parameter of single layered graphene sheets?

ABSTRACTThis study aims to evaluate the nonlocal small scale parameter for large amplitude vibration of single layered graphene sheets (SLGSs) comparing nonlinear resonant frequencies obtained via nonlocal continuum and molecular dynamics (MD) simulations. Nonlinear governing equations of motion are numerically solved employing the pseudo-spectral method to obtain the frequency response. Results reveal that the calibrated small scale parameter decreases when the vibration amplitude increases. Also, from MD simulations it is seen that for all length sizes after an ultimate vibration amplitude around 31% length size, the graphene sheets start to fracture.

Thermo-mechanical vibration of rotating axially functionally graded nonlocal Timoshenko beam

The free vibration analysis of rotating axially functionally graded nanobeams under an in-plane nonlinear thermal loading is provided for the first time in this paper. The formulations are based on Timoshenko beam theory through Hamilton’s principle. The small-scale effect has been considered using the nonlocal Eringen’s elasticity theory. Then, the governing equations are solved by generalized differential quadrature method. It is supposed that the thermal distribution is considered as nonlinear, material properties are temperature dependent, and the power-law form is the basis of the variation of the material properties through the axial of beam. Free vibration frequencies obtained are cantilever type of boundary conditions. Presented numerical results are validated by comparing the obtained results with the published results in the literature. The influences of the nonlocal small-scale parameter, angular velocity, hub radius, FG index and also thermal effects on the frequencies of the FG nanobeams are investigated in detail.

Natural Frequency and Stability Tuning of Cantilevered CNTs Conveying Fluid in Magnetic Field

This paper investigates the dynamics of cantilevered CNTs conveying fluid in longitudinal magnetic field and presents the possibility of controlling/tuning the stability of the CNT system with the aid of magnetic field. The slender CNT is treated as an Euler-Bernoulli beam. Based on nonlocal elasticity theory, the equation of motion with consideration of magnetic field effect is developed. This partial differential equation is then discretized using the differential quadrature method (DQM). Numerical results show that the nonlocal small-scale parameter makes the fluid-conveying CNT more flexible and can shift the unstable mode in which flutter instability occurs first at sufficiently high flow velocity from one to another. More importantly, the addition of a longitudinal magnetic field leads to much richer dynamical behaviors of the CNT system. Indeed, the presence of longitudinal magnetic field can significantly affect the evolution of natural frequency of the dynamical system when the flow velocity is successively increased. With increasing magnetic field parameter, it is shown that the CNT system behaves stiffer and hence the critical flow velocity becomes higher. It is of particular interest that when the magnetic field parameter is equal to or larger than the flow velocity, the cantilevered CNT conveying fluid becomes unconditionally stable, indicating that the dynamic stability of the system can be controlled due to the presence of a longitudinal magnetic field.

Imperfection Sensitivity of Nonlinear Vibration of Curved Single-Walled Carbon Nanotubes Based on Nonlocal Timoshenko Beam Theory.

Imperfection sensitivity of large amplitude vibration of curved single-walled carbon nanotubes (SWCNTs) is considered in this study. The SWCNT is modeled as a Timoshenko nano-beam and its curved shape is included as an initial geometric imperfection term in the displacement field. Geometric nonlinearities of von Kármán type and nonlocal elasticity theory of Eringen are employed to derive governing equations of motion. Spatial discretization of governing equations and associated boundary conditions is performed using differential quadrature (DQ) method and the corresponding nonlinear eigenvalue problem is iteratively solved. Effects of amplitude and location of the geometric imperfection, and the nonlocal small-scale parameter on the nonlinear frequency for various boundary conditions are investigated. The results show that the geometric imperfection and non-locality play a significant role in the nonlinear vibration characteristics of curved SWCNTs.

Nonlinear vibration behavior of a rotating nanobeam under thermal stress using Eringen’s nonlocal elasticity and DQM

This study is concerned with the small-scale effect on the nonlinear flapwise bending vibration of rotating cantilever and propped cantilever nanobeams. Euler–Bernoulli beam theory is used to model the nanobeam with nonlinearity. Nonlinear strain–displacement relations are employed to account for geometric nonlinearity of the system. The axial forces are modeled as the true spatial and thermal variations due to the rotation. Hamilton’s principle is used to derive the nonlinear governing equation and nonlocal nonlinear boundary conditions based on Eringen’s nonlocal elasticity theory. Finally, the differential quadrature method is used in conjunction with the direct iterative method to derive the nonlinear vibration frequencies of the nanobeam. The effects of the angular velocity, nonlocal small-scale parameter, temperature change and nonlinear amplitude on nonlinear vibration of the rotary nanobeam are discussed. The results of this work can be used in nanosensors, nanomotors, nanoturbines and NEMS applications.

Coupling pseudo-spectral method and integral quadrature rule for nonlinear vibration analysis of graphene sheets carrying concentrated nano objects

In present study the potential application of graphene with attached particles as nano resonant sensors is investigated using nonlocal elasticity. An elastic connection between the nano plate and the attached nanoparticle, is considered as a spring and a combination of pseudo-spectral and integral quadrature methods is implemented to determine the frequency shift. Results reveal that for soft connections the predicted frequency shift is greater than rigid connections that means considering a rigid connection instead of elastic one will underestimate the frequency shift of nano resonant sensors. Also, it is seen that nonlocal small scale parameter has a decreasing effect on the frequency shift of nano resonant sensors.

Differential quadrature based nonlocal flapwise bending vibration analysis of rotating nanobeam using the eringen nonlocal elasticity theory under axial load

The present paper deals with the dynamic behavior of rotating nanobeam subjected to follower force using the nonlocal elasticity theory. The nonlocal elasticity theory is used to analyze the mechanical behavior of nanoscale materials. The used method of solution is the Differential Quadrature Method (DQM). It is shown that the frequencies are affected when the size effect is taken into consideration. The effects of the nonlocal small scale parameter and the hub radii on the natural frequencies of the rotating nanobeam with respect to the rotating angular velocity are investigated in the present work. This study could be useful in design of the next generation of nanodevices such as blades of a nanoturbine, nanogears, nanoscale molecular bearings, that make use of the vibration analysis of rotating nanobeam. POLYM. COMPOS., 37:3175–3180, 2016. © 2015 Society of Plastics Engineers

Graphene-Based Resonant Sensors for Detection of Ultra-Fine Nanoparticles: Molecular Dynamics and Nonlocal Elasticity Investigations

Application of single layered graphene sheets (SLGSs) as resonant sensors in detection of ultra-fine nanoparticles (NPs) is investigated via molecular dynamics (MD) and nonlocal elasticity approaches. To take into consideration the effect of geometric nonlinearity, nonlocality and atomic interactions between SLGSs and NPs, a nonlinear nonlocal plate model carrying an attached mass-spring system is introduced and a combination of pseudo-spectral (PS) and integral quadrature (IQ) methods is proposed to numerically determine the frequency shifts caused by the attached metal NPs. In MD simulations, interactions between carbon–carbon, metal–metal and metal–carbon atoms are described by adaptive intermolecular reactive empirical bond order (AIREBO) potential, embedded atom method (EAM), and Lennard–Jones (L–J) potential, respectively. Nonlocal small-scale parameter is calibrated by matching frequency shifts obtained by nonlocal and MD simulation approaches with same vibration amplitude. The influence of nonlinearity, nonlocality and distribution of attached NPs on frequency shifts and sensitivity of the SLGS sensors are discussed in detail.

Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models

Recently, there have been an increasing number of studies on nonlocal theoretical models, of which different kinds of nonlocal elasticity approaches including softening and hardening models have been investigated extensively. In the present work, the longitudinal dynamic behaviors of some common one-dimensional nanostructures (e.g. nanorods/nanotubes) were examined using the hardening nonlocal approach. The hardening nonlocal longitudinal stress was employed and an eigenvalue analysis method was utilized in deriving some dynamic characteristics of the minute structures at nanoscale. The effects of a dimensionless nonlocal small scale parameter at molecular level unavailable in classical rods/tubes were investigated. Subsequently, nonlocal longitudinal vibration responses under various boundary conditions including soft and hard constraints were determined through a numerical method. The correlations between natural frequencies and the nonlocal nanoscale parameter were obtained and discussed in detail. Within the framework of hardening nonlocal stress theory, it concludes for the first time that the longitudinal free vibration frequencies of nanorods/nanotubes are higher than those based on the classical continuum mechanics but they are quite different from the softening nonlocal model. The strengthening effects on nonlocal stiffness of nanorods/nanotubes are observed and they are consistent with some other theoretical approaches or atomistic simulations. In particular, a case study on nonlocal natural frequencies of carbon nanotubes (CNTs) was presented and the results were calculated by atomistic simulations, classical continuum theory, softening and hardening nonlocal models, respectively. The validity of both existing softening and hardening nonlocal models was confirmed in comparison with the atomistic simulations. In addition, a comparative calculation for dimensional natural frequencies with respect to length of CNTs by different methodologies was provided to explain why the softening and hardening nonlocal models are both correct in nonlocal elasticity theory.

Large Amplitude Vibration Analysis of Graphene Sheets as Resonant Mass Sensors Using Mixed Pseudo-Spectral and Integral Quadrature Methods

The present paper investigates the potential application of graphene sheets with attached nanoparticles as resonant sensors by introducing a nonlocal shear deformation plate model. To take into account an elastic connection between the nanoplate and the attached nanoparticle, the nanoparticle is considered as a mass-spring system. Then, a combination of pseudo-spectral and integral quadrature methods is implemented to numerically determine the frequency shift caused by the attached mass-spring system for both clamped and simply supported boundary conditions. The obtained results are in a good agreement with those available in the literature, which reveals that the proposed combined method provides accurate results for structural problems related to concentrated objects. The results show that for soft connections with small spring constant values, the predicted frequency shift is greater than for rigid connections. This means that considering a rigid connection instead of elastic one will underestimate the frequency shift of nano resonant sensors. Additionally, it is shown that neglecting nonlocal small scale parameter results in overestimating the frequency shift of nano resonant sensors. The presented results can be useful as a guideline for designing plane shape nano resonant sensors like graphene-based mass sensors.

Torsion of cracked nanorods using a nonlocal elasticity model

This paper presents a nonlocal cracked-rod model from which we have analysed the torsional vibrations of a carbon nanotube with a circumferential crack. Several types of boundary conditions, including the consideration of a buckyball at the end of the nanotube, have been studied. The nonlocal Eringen elasticity theory is used to formulate the problem. The cracked rod is modelled by dividing the cracked element into two segments connected by a torsional linear spring whose stiffness is related to the crack severity. The effect of the nonlocal small-scale parameter, crack severity, cracked section position, different boundary conditions and attached mass are examined in this work.

Differential quadrature based nonlocal flapwise bending vibration analysis of rotating nanotube with consideration of transverse shear deformation and rotary inertia

This paper is concerned with the flapwise bending free vibration problem for rotating nanotubes. The small scale effect is taken into consideration based on nonlocal elasticity theory. In the nanotube model the effects of transverse shear deformation and rotary inertia are accounted. The nonlocal governing equations and the nonlocal boundary conditions are derived. These equations are solved analytically for the vibration frequencies of nanotubes via differential quadrature method. The vibration solutions obtained provide a better representation of the vibration behavior of short, stubby, nanotubes where the effects of small scale, transverse shear deformation and rotary inertia are significant. The effects of the nonlocal small scale parameter and the hub radii on the natural frequencies of the rotating carbon nanotube with respect to the rotating angular velocity are investigated in the present work. The results can provide useful guidance for the study and design of the next generation of nanodevices such as blades of a nanoturbine, nanogears, and nanoscale bearings, that make use of the vibration properties of rotating single-walled carbon nanotubes.

Axial wave propagation in coupled nanorod system with nonlocal small scale effects

This article deals with the axial wave propagation properties of a coupled nanorod system with consideration of small scale effects. The nonlocal elasticity theory has been incorporated into classical rod/bar model to capture unique features of the coupled nanorods under the umbrella of continuum mechanics theory. Nonlocal rod model is developed for coupled nanorods. The strong effect of the nonlocal scale has been obtained which leads to substantially different wave behavior of nanorods from those of macroscopic rods. Explicit expressions are derived for wavenumber, cut-off frequency and escape frequency of nanorods. The analysis shows that the wave characteristics of nanorods are highly over estimated by the classical rod model, which ignores the effect of small-length scale. The studies also shows that the nonlocal scale parameter introduces certain band gap region in axial or longitudinal wave mode, where no wave propagation occurs. This is manifested in the spectrum cures as the region, where the wavenumber tends to infinite or wave speed tends to zero. The effect of the coupled spring stiffness is also capture in the present analysis. It has been also shown that the cut-off frequency increases as the stiffness of the coupled spring increases and also the coupled spring stiffness has no effect on escape frequency of the axial wave mode in the nanorod. This cut-off frequency is also independent of the nonlocal small scale parameter. The present study may bring in helpful insights while investigating multiple-nanorod-system-models for future nano-optomechanical systems applications. The results can also provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave propagation properties of coupled single-walled carbon nanotubes or coupled nanorods.

Critical buckling temperature of single-walled carbon nanotubes embedded in a one-parameter elastic medium based on nonlocal continuum mechanics

In this paper, the critical buckling temperature of single-walled carbon nanotubes (SWCNTs), which are embedded in one-parameter elastic medium (Winkler foundation) is estimated under the umbrella of continuum mechanics theory. Nonlocal continuum theory is incorporated into Timoshenko beam model and the governing differential equations of motion are derived. An explicit expression for the non-dimensional critical buckling temperature is also derived in this work. The effect of the nonlocal small scale coefficient, the Winkler foundation parameter and the ratio of the length to the diameter on the critical buckling temperature is investigated in detail. It can be observed that the effects of nonlocal small scale parameter and the Winkler foundation parameter are significant and should be considered for thermal analysis of SWCNTs. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the thermal buckling properties of embedded single-walled carbon nanotubes.