Nonlocal second-order strain gradient elasticity model and its application in wave propagating in carbon nanotubes

Recent experimental studies indicate that Young’s modulus of carbon nanotubes increases steeply with tube diameter decreasing. The consideration of this effect is of great importance for the fabrication and exploitation of nano-electromechanical devices. Nevertheless, the rapid stiffness enhancement effect noticed from experimental observation maybe unable to be predicted by using size-dependent elasticity models available in literatures. It is strongly necessary to further shed light on the size-dependent mechanical mechanism and characterize the rapid strengthening effect of stiffness for nano-sized materials. To achieve this goal, the nonlocal second-order strain gradient elasticity model is established by introducing the second-order strain gradient field with nonlocal effect into the stored energy function of nonlocal first-order strain gradient elasticity theory. With the aids of the laws of thermodynamics, the constitutive relations are obtained. The Hamilton principle is used to derive the governing equations of equilibrium and boundary conditions. The proposed model is applied to investigate the problem of wave propagating in carbon nanotubes. The new dispersion relations derived are presented for evaluating the influences of size-dependent parameters on the characteristics of wave propagation. The results show that present model can predict the rapid increasing effect of carbon nanotubes with the decrease of tube size.

     

Application of the Green function method to flow-thermoelastic forced vibration analysis of viscoelastic carbon nanotubes

In this paper, the Green function method (GFM) is implemented for forced vibration analysis of carbon nanotubes (CNTs) conveying fluid in thermal environment. The Eringen’s nonlocal elasticity theory is used to take into account the size effect of CNT with modeling the CNT wall–fluid flow interaction by means of slip boundary condition and Knudsen number (Kn). The derived governing differential equations are solved by GFM which demonstrated to have high precision and computational efficiency in the vibration analysis of CNTs. The validity of the present analytical solution is confirmed by comparing the results with those reported in other literature, and good agreement is observed. The analytical examinations are accomplished, while the emphasis is placed on considering the influences of nonlocal parameter, boundary conditions, temperature change, structural damping of the CNT, Knudsen number, fluid velocity and visco-Pasternak foundation on the dynamic deflection response of the fluid-conveying CNTs in detail.     

Size-dependent effects on critical flow velocity of a SWCNT conveying viscous fluid based on nonlocal strain gradient cylindrical shell model

This article investigates vibration and instability analysis of a single-walled carbon nanotube (SWCNT) conveying viscous fluid flow. For this purpose, the first-order shear deformation shell model is developed in the framework of nonlocal strain gradient theory (NSGT) for the first time. The proposed model is a conveying viscous fluid in which the external force of fluid flow is applied by the modified Navier–Stokes relation and considering slip boundary condition and Knudsen number. The NSGT can be reduced to the nonlocal elasticity theory, strain gradient theory or the classical elasticity theory by inserting their specific nonlocal parameters and material length scale parameters into the governing equations. Comparison of above-mentioned theories suggests that the NSGT predicts the greatest critical fluid flow velocity and stability region. The governing equations of motion and corresponding boundary conditions are discretized using the generalized differential quadrature method. Furthermore, the effects of the material length scale, nonlocal parameter, Winkler elastic foundation and Pasternak elastic foundation on vibration behavior and instability of a SWCNT conveying viscous fluid flow with simply supported and clamped–clamped boundary conditions are investigated.     

Nonlocal frequency analysis of nanoscale biosensors

As a first endeavor, we propose nonlocal elasticity theory for carbon nanotube based cantilever biosensors. By using the frequency-shift of the fundamental vibration mode, we develop new nonlocal frequency sensor equations utilizing energy principles. Two physically realistic configurations of the added mass, namely, point mass and distributed mass are considered. Exact closed-form expressions relating the frequency-shift and the added mass have been derived for both the cases. The proposed nonlocal sensor-equations are general in nature and depend on three non-dimensional calibration constants namely, the stiffness calibration constant, the mass calibration constant and the nonlocal calibration constant. Explicit analytical expressions of these calibration constants are derived. An example of a single wall carbon nanotube with attached multiple strands of deoxythimidine is considered to illustrate the analytical results. Molecular mechanics simulation is used to validate the new nonlocal sensor equations. The optimal values of nonlocal parameter are obtained from the molecular mechanics simulation results. The nonlocal approach generally predicts the frequency shift accurately compared to the local approach. Numerical results show the importance of considering the distributed nature of the added mass while using the nonlocal theory.     

Axial vibration analysis of a Rayleigh nanorod with deformable boundaries

Microsystem Technologies - In this study, the free axial vibration of Rayleigh nanorods with axial restraints is studied via Eringens’ nonlocal elasticity theory. This higher order elasticity...     

Dynamic modeling of embedded nanoplate systems incorporating flexoelectricity and surface effects

Microsystem Technologies - In this research, vibration characteristics of a flexoelectric nanoplate in contact with Winkler-Pasternak foundation are investigated based on nonlocal elasticity theory...     

Nonlocal vibration and instability analysis of carbon nanotubes conveying fluid considering the influences of nanoflow and non-uniform velocity profile

In this article, the influences of non-uniform velocity profile attributable to slip boundary condition and viscosity of fluid on the dynamic instability of carbon nanotubes (CNTs) conveying fluid are investigated. The nonlocal elasticity theory and the Euler–Bernoulli beam theory are employed to derive partial differential equation of nanotubes conveying fluid. Furthermore, a dimensionless momentum correction factor (MCF) is obtained as a function of Knudsen number (Kn) so as to insert the effects of non-uniform velocity profile into the equation of motion. In continuation, complex eigen-frequencies of the system are attained with respect to different boundary conditions, the momentum correction factor, slip boundary condition and nonlocal parameter. The results delineate that considering the effects of non-uniform velocity profile could diminish predicted critical velocity of flow. Therefore, the divergence instability occurs in the lower values of flow velocity. In addition, the MCF decreases through enhancement of Kn; hence, the effects of non-uniform velocity profile are more noticeable for liquid fluid than gas fluid.     

Chaotic dynamics and forced harmonic vibration analysis of magneto-electro-viscoelastic multiscale composite nanobeam

In this article, the damping forced harmonic vibration characteristics of magneto-electro-viscoelastic (MEV) nanobeam embedded in viscoelastic foundation is evaluated based on nonlocal strain gradient elasticity theory. The viscoelastic foundation consists of Winkler–Pasternak layer. The governing equations of nonlocal strain gradient viscoelastic nanobeam in the framework of refined shear deformable beam theory are obtained using Hamilton’s principle and solved implementing an analytical solution. In addition, a parametric study is presented to examine the effect of the nonlocal strain gradient parameter, magneto-electro-mechanical loadings, and aspect ratio on the vibration characteristics of nanobeam. From the numerical evaluation, it is revealed that the effect of electric and magnetic loading on the natural frequency has a predominant influence.

     

Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method

This study investigates the small scale effect on the nonlinear bending vibration of a rotating cantilever and propped cantilever nanobeam. The nanobeam is modeled as an Euler–Bernoulli beam theory with von Kármán geometric nonlinearity. The axial forces are also included in the model as the true spatial variation due to the rotation. Hamilton’s principle is used to derive the governing equation and boundary conditions for the Euler–Bernoulli beam based on Eringen’s nonlocal elasticity theory. The differential quadrature method as an efficient and accurate numerical tool in conjunction with a direct iterative method is adopted to obtain the nonlinear vibration frequencies of nanobeam. The effect of nonlocal small–scale, angular speed, hub radius and nonlinear amplitude of rotary nanobeam is discussed.     

Stability analysis of restrained nanotubes placed in electromagnetic field

In this present work, buckling analysis of restrained nanotubes placed in electromagnetic field is studied on the basis of Euler–Bernoulli beam theory in conjunction with Eringen’s nonlocal elasticity theory. The modal displacement function is assumed for the stability analysis in order to discretize the derived governing equation. A Fourier sine series with Stoke’s transformation is utilized to investigate the buckling response. The essential advantage of this transformation is its ability of dealing with various boundary conditions to determine the buckling loads. For demonstrate the effects of various parameters such as Hartmann parameter, spring parameter and mode number on the stability response and critical buckling load of electromagnetic nanobeam a detailed study is presented. Variations of buckling loads, critical buckling loads and buckling load ratios of the nanobeam are exhibited with a number of tables and plotted figures. The results obtained from the analysis are discussed on the tables and figures.

     

Size-dependent effects on critical flow velocity of fluid-conveying microtubes via nonlocal strain gradient theory

Size-dependent Timoshenko and Euler–Bernoulli models are derived for fluid-conveying microtubes in the framework of the nonlocal strain gradient theory. The equations of motion and boundary conditions are deduced by employing the Hamilton principle. A flow-profile-modification factor, which is related to the flow velocity profile, is introduced to consider the size-dependent effects of flow. The analytical solutions of predicting the critical flow velocity of the microtubes with simply supported ends are derived. By choosing different values of the nonlocal parameter and the material length scale parameter, the critical flow velocity of the nonlocal strain gradient theory can be reduced to that of the nonlocal elasticity theory, the strain gradient theory, or the classical elasticity theory. It is shown that the critical flow velocity can be increased by increasing the flexural rigidity, decreasing the length of tube, decreasing the mass density of internal flow, or increasing the shear rigidity. The critical flow velocity can generally increase with the increasing material length scale parameter or the decreasing nonlocal parameter. The flow-profile-modification factor can decrease the critical flow velocity. The critical flow velocity predicted by classical elasticity theory is generally larger than that of nonlocal strain gradient theory when considering the size-dependent effect of flow.     

Nonlocal free dynamic analysis of periodic arrays of single-walled carbon nanotubes in the presence of longitudinal thermal and magnetic fields

This paper deals with transverse vibrations of three-dimensional vertically aligned periodic arrays of single-walled carbon nanotubes acted upon by both longitudinal magnetic and thermal fields. For this purpose, a nonlocal higher-order beam theory is employed for modeling of the nanosystem. Accounting for the intertube van der Waals forces, a discrete-based model as well as a continuous-based model is established. For various geometries of the nanosystem and under various applied thermal and magnetic fields, the results of the continuous model are successfully checked with those of the discrete model. Through a comprehensive parametric study, the effects of the nanotube’s radius, slenderness ratio, nonlocality, ensemble’s population, temperature change, and strength of magnetic field on fundamental frequencies of the nanosystem are addressed and displayed.     

Characterizing the nonlinear behaviour of double walled carbon nanotube based nano mass sensor

This work deals with the evaluation of nonlinear behaviour of a curved double walled carbon nanotube (DWCNT) when used for mass sensing applications. The DWCNT is considered to be doubly clamped at a source and a drain. To judge the nonlinear behaviour, equations of motion have been derived using Euler beam theory and Hamilton principle, considering nonlinear van der Waals interaction nonlinear oscillations of a double walled carbon nanotube excited harmonically near its primary resonance are considered. The nonlinear free vibration of double-walled carbon nanotubes based on the elasticity theory is studied in this paper. Modelling of the weak van der Waals force of attraction between the inner and outer tubes is represented using a spring element. The equation of motion involves four nonlinear terms due to the curved geometry and the stretching of the central plane. The dynamic response of the double walled carbon nanotube based mass sensor is analyzed in the context of the time response, Poincaré maps, and fast Fourier transformation diagrams. The results show the appearance of instability and chaos in the dynamic response as the mass on carbon nanotube is changed. The appearance of regions of periodic, subharmonic, and chaotic behavior is observed to be strongly dependent on mass, inner and outer tubes and the geometric imperfections of double walled carbon nanotube.

     

Nonlinear vibration of nanobeam with attached mass at the free end via nonlocal elasticity theory

In the present study, nonlinear free and forced vibration of Euler–Bernoulli nanobeam with attached nanoparticle at the free end is investigated based on nonlocal elasticity theory. The effects of the different nonlocal parameters (γ) and mass ratios (α) as well as effects of fixed-free boundary conditions on the vibrations are determined. To obtain the equation of motion of the system, the Hamilton’s principle is employed. The stretching of neutral axis which introduces cubic nonlinearity is included into the equation for deriving nonlinear equation. And also effects of damping and forcing are included into the equations. The approximate solutions of the equations are derived by using the multiple scale method. Fundamental frequencies, frequency shift and mode shapes for the linear problem are estimated for a nonlocal Euler–Bernoulli nanobeam with an attached nanoparticle and graphically represented the frequency shift and mode shapes. Nonlinear frequencies are derived depending on amplitude and phase modulation. Frequency–response curves are drawn for different nonlocal parameters and different modes.

     

Coupled twist–bending static and dynamic behavior of a curved single-walled carbon nanotube based on nonlocal theory

The static and dynamic behavior of a curved single-walled carbon nanotube which is under twist–bending couple based on nonlocal theory is analyzed. The nonlocal theory is used to model the mechanical behavior of structure in small scale. The obtained differential equations are solved using a simply supported boundary condition and Navier analytical method. Moreover the twisted vibration and bending of curved nanotube is analyzed and also the armchair model is assumed in this study. The following parameters were studied in this paper: the effect of nonlocal parameter, the curved nanotube’s opening angel, the Young’s modulus and the mode number is studied. The results were verified with the previous literature which showed an excellent agreement. The results of this paper can be used as a benchmark for future investigations.

     

Longitudinal vibration of carbon nanotubes with elastically restrained ends using doublet mechanics

Free axial vibration analysis of axially restrained carbon nanotubes (CNTs) is studied within the framework of doublet mechanics theory. Fourier sine series are utilized for describing the axial deflection of the carbon nanotube. An eigenvalue approximation is constructed for vibrational modes with the aid of Stokes’ transformation to deformable axial springs. This unclassified approximation bridges the gap between the deformable and rigid boundary conditions. The comparison studies are carried out to verify the efficiency and accuracy of the proposed analytical model by assigning proper values to elastic spring coefficients. The results indicate that the axial springs and small scale parameter of carbon nanotube have considerable effects on the axial vibration behavior of NTs. Similarly, the dependencies of the vibration frequencies on material scale parameter and axial restraints are significant. Similar higher order effects are predicted for other nano or micro structures, all of that confirmed the smaller is stiffer phenomenon.

     

Transverse vibration of pre-tensioned nonlocal nanobeams with precise internal axial loads

This paper investigates the transverse vibration of a simply supported nanobeam with an initial axial tension based on the nonlocal stress field theory with a nonlocal size parameter. Considering an axial elongation due to transverse vibration, the internal axial tension is not precisely equal to the external initial tension. A sixth-order nonlinear partial differential equation that governs the transverse vibration for such nonlocal nanobeam is derived. Using a perturbation method, the relation between nat...     

Free and forced vibration analysis of viscoelastic damped FG-CNT reinforced micro composite beams

In this paper, free and forced vibration analysis of viscoelastic microcomposite beam reinforced by functionally graded single-walled carbon nanotubes (FG-SWCNTs) is studied using the modified couple stress theory (MCST). The material properties of micro composite beam by generalized rule of mixtures carbon nanotubes are estimated. In addition, these properties are stated as uniform, and functionally graded (FG) distributions in the thickness direction. Energy method and Hamilton’s principle are employed to establish the governing equations of motion for the vibration of viscoelastic damped micro composite beam reinforced by SWCNTs based on the Kelvin–Voigt model. The influences of material length scale parameter, structural damping coefficient and different distributions of SWCNTs on non-dimensional complex natural frequency and amplitude vibration of the viscoelastic micro composite beam are investigated. The results reveal that the lowest vibration amplitude of FG microcomposite beam by the FG-X and the highest occurs by FG-◊. Moreover, in the presence of external periodic load and the absence of structural damping coefficient, the vibration amplitude increases and FG microcomposite beam becomes unstable, even though the amplitude of vibration decreases with increasing structural damping coefficient. It is shown that the natural frequency of SWCNT reinforced composite is more than the frequency of multi-walled carbon nanotubes because SWCNT have more stiffness. In addition, the results illustrate that the experimental data by Lei et al. agree well with those predicted by the MCST in the present work.

     

Molecular dynamics simulation of vibrational behavior of annular graphene sheet: Identification of nonlocal parameter

To obtain accurate results from the nonlocal plate theory, the nonlocal parameter should be properly determined. In this paper, practicing a molecular dynamics simulation, the nonlocal plate theory was presented for free vibration analysis of annular graphene sheets. Accuracy and stability of results are validated by published results. Calculations are performed for different boundary conditions and geometrical properties. Results reveal that inner and outer radius induce significant effects on the nonlocal parameter. This nonlocal parameter extracted based on Molecular Dynamics (MD) in the nonlocal theory can determine the natural frequencies of annular graphene sheets, conveniently, whereas the molecular dynamics simulation demands a lot of time.