Wave dispersion in viscoelastic single walled carbon nanotubes based on the nonlocal strain gradient Timoshenko beam model

Based on the nonlocal strain gradient theory and Timoshenko beam model, the properties of wave propagation in a viscoelastic single-walled carbon nanotube (SWCNT) are investigated. The characteristic equations for flexural and shear waves in visco-SWCNTs are established. The influence of the tube size on the wave dispersion is clarified. For a low damping coefficient, threshold diameter for shear wave (SW) is observed, below which the phase velocity of SW is equal to zero, whilst flexural wave (FW) always exists. For a high damping coefficient, SW is absolutely constrained, and blocking diameter for FW is observed, above which the wave propagation is blocked. The effects of the wave number, nonlocal and strain gradient length scale parameters on the threshold and blocking diameters are discussed in detail.     

Wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes with surface and nonlocal effects

In this paper, the transverse wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes is investigated based on nonlocal elasticity theory with consideration of surface effect. The governing equation is formulated utilizing nonlocal Euler-Bernoulli beam theory and Kelvin-Voigt model. Explicit wave dispersion relation is developed and wave phase velocities and frequencies are obtained. The effect of the fluid flow velocity, structural damping, surface effect, small scale effects and tube diameter on the wave propagation properties are discussed with different wave numbers. The wave frequency increases with the increase of fluid flow velocity, but decreases with the increases of tube diameter and wave number. The effect of surface elasticity and residual surface tension is more significant for small wave number and tube diameter. For larger values of wave number and nonlocal parameters, the real part of frequency ratio raises.     

Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory

The governing equation of wave motion of viscoelastic SWCNTs (single-walled carbon nanotubes) with surface effect under magnetic field is formulated on the basis of the nonlocal strain gradient theory. Based on the formulated equation of wave motion, the closed-form dispersion relation between the wave frequency (or phase velocity) and the wave number is derived. It is found that the size-dependent effects on the phase velocity may be ignored at low wave numbers, however, is significant at high wave numbers. Phase velocity can increase by decreasing damping or increasing the intensity of magnetic field. The damping ratio considering surface effect is larger than that without considering surface effect. Damping ratio can increase by increasing damping, increasing wave number, or decreasing the intensity of magnetic field.     

Dispersion of dilatation wave propagation in single-wall carbon nanotubes using nonlocal scale effects

Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with micro- or nano-structures. This paper investigates a model of wave propagation in single-wall carbon nanotubes (SWCNTs) with small scale effects are studied. The equation of motion of the dilatation wave is obtained using the nonlocal elastic theory. We show that a dispersive wave equation is obtained from a nonlocal elastic constitutive law, based on a mixture of a local and a nonlocal strain. The SWCNTs structures are treated within the multilayer thin shell approximation with the elastic properties taken to be those of the graphene sheet. The SWCNT was the (40,0) zigzag tube with an effective diameter of 3.13 nm. Nonlinear frequency equations of wave propagation in SWCNTs are described through the effect of small scale. The phase velocity and the group velocity are derived, respectively. The nonlinear dispersion relation is analyzed with different wave numbers versus scale coefficient. It can be observed from the results that the dispersion properties of the dilatation wave are induced by the small scale effects, which will disappear in local continuous models. The dispersion degree can be strengthened by increasing the scale coefficient and the wave number. Furthermore, the characteristics for the group velocity of the dilatation wave in carbon nanotubes can also be tuned by these factors.     

Transverse Vibration of Tapered Single-Walled Carbon Nanotubes Embedded in Viscoelastic Medium

Based on the nonlocal theory, Euler-Bernoulli beam theory and Kelvin viscoelastic foundation model, free transverse vibration is studied for a tapered viscoelastic single-walled carbon nanotube (visco-SWCNT) embedded in a viscoelastic medium. Firstly, the governing equations for vibration analysis are established. And then, we derive the natural frequencies in closed form for SWCNTs with arbitrary boundary conditions by applying transfer function method and perturbation method. Numerical results are also presented to discuss the effects of nonlocal parameter, relaxation time and taper parameter of SWCNTs, and material property parameters of the medium. This study demonstrates that the proposed model is available for vibration analysis of the tapered SWCNTs-viscoelastic medium coupling system.     

Shock-induced thermal wave propagation and response analysis of a viscoelastic thin plate under transient heating loads

This paper is devoted to the thermal shock analysis for viscoelastic materials under transient heating loads. The governing coupled equations with time-delay parameter and nonlocal scale parameter are derived based on the generalized thermo-viscoelasticity theory. The problem of a thin plate composed of viscoelastic material, subjected to a sudden temperature rise at the boundary plane, is solved by employing Laplace transformation techniques. The transient responses, i.e. temperature, displacement, stresses, heat flux as well as strain, are obtained and discussed. The effects of time-delay and nonlocal scale parameter on the transient responses are analyzed and discussed. It can be observed that: the propagation of thermal wave is dynamically smoothed and changed with the variation of time-delay; while the displacement, strain, and stress can be rapidly reduced by nonlocal scale parameter, which can be viewed as an important indicator for predicting the stiffness softening behavior for viscoelastic materials.     

Effects of strain and higher order inertia gradients on wave propagation in single-walled carbon nanotubes

Dispersion relation of single-walled carbon nanotubes (SWCNTs) is investigated. The governing equations of motion of SWCNTs are derived on the basis of the gradient shell model, which involves one strain gradient and one higher order inertia parameters in addition to two Lamé constants. The present shell model can predict wave dispersion in good agreement with those of molecular dynamic (MD) simulations available in the literature. The effects of two small scale parameters on the angular frequency and phase velocity in the longitudinal, torsional and radial directions are studied in detail. The numerical results show that the angular frequency and phase velocity increase with the increase of strain gradient parameter, whereas decrease with inertia gradient parameter increases. In addition, analytical expressions of the cut-off frequencies and asymptotic phase velocities are given. It is found that the number of cut-off frequencies is dependent on the circumferential wave number, and two asymptotic phase velocities exist for nonzero value of strain gradient parameter, while only one exists when the strain gradient parameter is excluded.     

Wave characteristics of single-walled fluid-conveying carbon nanotubes subjected to multi-physical fields

Wave propagation in single-walled carbon nanotubes (SWCNTs) conveying fluids and placed in multi-physical fields (including magnetic and temperature fields) is studied in this paper. The nanotubes are modelled as Timoshenko beams. Based on the nonlocal beam theory, the governing equations of motion are derived using Hamilton's principle, and then solved by Galerkin approach, leading to two second-order ordinary differential equations (ODEs). Numerical simulations are carried out to verify the analytical model proposed in the present study, and determine the influences of the nonlocal parameter, the fluid velocity and flow density, the temperature and magnetic field flux change, and the surrounding elastic medium on the wave behaviour of SWCNTs. The results show that the nonlocal parameter has a considerable influence on dynamic behaviour of the nanotube and the fluid flow inside it. The results also show that the magnetic and temperature fields play an important role on the wave propagation characteristics of SWCNTs.     

Early transient responses of multi-span stepped single-walled carbon nanotubes using the method of reverberation ray matrix

The early transient responses of multi-span stepped single walled carbon nanotubes (SWCNTs) under impact loadings are studied by the method of reverberation ray matrix (MRRM). The dynamics model of the carbon nanotubes is established in the Fourier phase space on the basis of the nonlocal Timoshenko beam model. The wave solutions of SWCNTs with arbitrary boundary conditions are obtained by the wave method. The reverberation ray matrix of the multi-span stepped SWCNTs is the product of scattering, phase and permutation matrices, which can be determined by the impact loadings, continuous conditions and boundary conditions. The early transient responses can be calculated by the inverse Fourier transform of the sum of initial ray groups. It can be found that the early transient displacement response in the very short time subjected to the impact loading is very small, while the transient transverse shear strain becomes large in the very short time. The influences of nanotubes span number, nanotubes type and boundary conditions on the early transient responses of multi-span stepped SWCNTs are investigated.     

Effect of Longitudinal Magnetic Field on Vibration Characteristics of Single-Walled Carbon Nanotubes in a Viscoelastic Medium

The effect of longitudinal magnetic field on vibration response of a sing-walled carbon nanotube (SWCNT) embedded in viscoelastic medium is investigated. Based on nonlocal Euler-Bernoulli beam theory, Maxwell’s relations, and Kelvin viscoelastic foundation model, the governing equations of motion for vibration analysis are established. The complex natural frequencies and corresponding mode shapes in closed form for the embedded SWCNT with arbitrary boundary conditions are obtained using transfer function method (TFM). The new analytical expressions for the complex natural frequencies are also derived for certain typical boundary conditions and Kelvin-Voigt model. Numerical results from the model are presented to show the effects of nonlocal parameter, viscoelastic parameter, boundary conditions, aspect ratio, and strength of the magnetic field on vibration characteristics for the embedded SWCNT in longitudinal magnetic field. The results demonstrate the efficiency of the proposed methods for vibration analysis of embedded SWCNTs under magnetic field.     

Assessment of Timoshenko Beam Models for Vibrational Behavior of Single-Walled Carbon Nanotubes Using Molecular Dynamics

In this paper, we study the flexural vibration behavior of single-walled carbon nanotubes (SWCNTs) for the assessment of Timoshenko beam models. Extensive molecular dynamics (MD) simulations based on second-generation reactive empirical bond-order (REBO) potential and Timoshenko beam modeling are performed to determine the vibration frequencies for SWCNTs with various length-to-diameter ratios, boundary conditions, chiral angles and initial strain. The effectiveness of the local and nonlocal Timoshenko beam models in the vibration analysis is assessed using the vibration frequencies of MD simulations as the benchmark. It is shown herein that the Timoshenko beam models with properly chosen parameters are applicable for the vibration analysis of SWCNTs. The simulation results show that the fundamental frequencies are independent of the chiral angles, but the chirality has an appreciable effect on higher vibration frequencies. The SWCNTs is very sensitive to the initial strain even if the strain is extremely small.     

Nonlocal shear stress for homogeneous fluids

It has been suggested that for fluids in which the rate of strain varies appreciably over length scales of the order of the intermolecular interaction range, the viscosity must be treated as a nonlocal property of the fluid. The shear stress can then be postulated to be a convolution of this nonlocal viscosity kernel with the strain rate over all space. In this Letter, we confirm that this postulate is correct by a combination of analytical and numerical methods for an atomic fluid out of equilibrium. Furthermore, we show that a gradient expansion of the nonlocal constitutive equation gives a reasonable approximation to the shear stress in the small wave vector limit.     

Rayleigh-Ritz axial buckling analysis of single-walled carbon nanotubes with different boundary conditions

Eringen's nonlocality is incorporated into the shell theory to include the small-scale effects on the axial buckling of single-walled carbon nanotubes (SWCNTs) with arbitrary boundary conditions. To this end, the Rayleigh-Ritz solution technique is implemented in conjunction with the set of beam functions as modal displacement functions. Then, molecular dynamics simulations are employed to obtain the critical buckling loads of armchair and zigzag SWCNTs, the results of which are matched with those of nonlocal shell model to extract the appropriate values of nonlocal parameter. It is found that in contrast to the chirality, boundary conditions have a considerable influence on the proper values of nonlocal parameter.     

钢板-空气背衬上含空腔粘弹性材料层的声反射特性

本文研究声波从水中入射到粘弹性材料层-钢板-空气上的反射特性，运用分层介质中的波动理论研究粘弹性层均匀情况的声特性，运用有限元方法研究粘弹性层含空气腔结构情况的声特性。结果表明，改变粘弹性材料消声特性需要综合调整材料的弹性参数；粘弹性层内的空腔结构对声特性产生很大影响，改变其谐振特性和反射系数的收敛特性。文中以含圆锥腔、圆台等复合腔的结构粘弹性层为例，计算了其反射系数的频响特性、谐振模态，并分析了结构的影响。     

Three-dimensional absolute and convective instabilities in mixed convection of a viscoelastic fluid through a porous medium

By using the mathematical formalism of absolute and convective instabilities we study the nature of unstable three-dimensional disturbances of viscoelastic flow convection in a porous medium with horizontal through-flow and vertical temperature gradient. Temporal stability analysis reveals that among three-dimensional (3D) modes the pure down-stream transverse rolls are favored for the onset of convection. In addition, by considering a spatiotemporal stability approach we found that all unstable 3D modes are convectively unstable except the transverse rolls which may experience a transition to absolute instability. The combined influence of through-flow and elastic parameters on the absolute instability threshold, wave number and frequency is then determined, and results are compared to those of a Newtonian fluid.     

Prediction of compressive post-buckling behavior of single-walled carbon nanotubes in thermal environments

In the present investigation, the axial buckling and post-buckling configurations of single-walled carbon nanotubes (SWCNTs) are studied including the thermal environment effect. For this purpose, Eringen’s nonlocal elasticity continuum theory is implemented into the classical Euler–Bernoulli beam theory to represent the SWCNTs as a nonlocal elastic beam model. A closed-form analytical solution is carried out to analyze the static response of SWCNTs in their post-buckling state in which the axial buckling load is assumed to be beyond the critical axial buckling load. Common sets of boundary conditions, named simply supported–simply supported (SS–SS), clamped–clamped (C–C), and clamped–simply supported (C–SS), are considered in the investigation. Selected numerical results are given to represent the variation of the carbon nanotube’s mid-span deflection with the applied axial load corresponding to various nonlocal parameters, length-to-diameter aspect ratios, temperature changes, and end supports. Moreover, a comparison between the post-buckling behaviors of SWCNTs at low- and high-temperature environments is presented. It is found that the size effect leads to a decrease of the axial buckling load especially for SWCNTs with C–C boundary conditions. Also, it is revealed that the value of the temperature change plays different roles in the post-buckling response of SWCNTs at low- and high-temperature environments.     

Diameter-dependent thermal-oxidative stability of single-walled carbon nanotubes synthesized by a floating catalytic chemical vapor deposition method

In this paper, purified single-walled carbon naotubes (SWCNTs) with three different diameters were synthesized using a floating catalytic chemical vapor deposition method with ethanol as carbon feedstock, ferrocene as catalyst, and thiophene as growth promoter. The thermal-oxidative stability of different-diameter SWCNTs was studied by using thermal analysis (TG, DTA), transmission electron microscopy (TEM), and X-ray photoelectron spectroscopy (XPS) analysis. The results indicate that small diameter SWCNTs (∼1 nm) are less stable and burn at lower temperature (610 °C), however, the larger diameter SWCNTs (∼5 nm) survive after burning at higher temperature (685 °C), the oxidation rate varies inversely with the tube diameter of SWCNTs, which may be concluded that the higher oxidation-resistant temperature of larger diameter SWCNTs can be attributed to the lower curvature-induced strain by rolling the planar graphene sheet for the larger diameter, so small tubes will become thermodynamically unstable.     

Wave propagation in double walled carbon nanotubes by using doublet mechanics theory

Flexural and axial wave propagation in double walled carbon nanotubes embedded in an elastic medium and axial wave propagation in single walled carbon nanotubes are investigated. A length scale dependent theory which is called doublet mechanics is used in the analysis. Governing equations are obtained by using Hamilton principle. Doublet mechanics results are compared with classical elasticity and other size dependent continuum theories such as strain gradient theory, nonlocal theory and lattice dynamics. In addition, experimental wave frequencies of graphite are compared with the doublet mechanics theory. It is obtained that doublet mechanics gives accurate results for flexural and axial wave propagation in nanotubes. Thus, doublet mechanics can be used for the design of electro-mechanical nano-devices such as nanomotors, nanosensors and oscillators.     

Wave propagation analysis in nonlinear curved single-walled carbon nanotubes based on nonlocal elasticity theory

Theoretical predictions are presented for wave propagation in nonlinear curved single-walled carbon nanotubes (SWCNTs). Based on the nonlocal theory of elasticity, the computational model is established, combined with the effects of geometrical nonlinearity and imperfection. In order to use the wave analysis method on this topic, a linearization method is employed. Thus, the analytical expresses of the shear frequency and flexural frequency are obtained. The effects of the geometrical nonlinearity, the initial geometrical imperfection, temperature change and magnetic field on the flexural and shear wave frequencies are investigated. Numerical results indicate that the contribution of the higher-order small scale effect on the shear deformation and the rotary inertia can lead to a reduction in the frequencies compared with results reported in the published literature. The theoretical model derived in this study should be useful for characterizing the mechanical properties of carbon nanotubes and applications of nano-devices.     

具有不规则自由表面的粘弹性介质中声波模拟的有限差分算法

声波在具有不规则自由表面的粘弹性介质传播的数值模拟工作在地震勘探,地震预测中非常重要,特别是当模型有起伏的自由界面和较强的衰减特性时更是如此。基于镜像方法和直接方法,本文发展了一种二维有限差分算法,可以模拟不规则自由表面引起的声波散射问题。该方法将自由表面条件与速度应力方程结合求解粘弹性波动方程,在垂直和水平自由段及拐点处设置相应的边界条件。该方法假设自由表面穿过剪切应力和相应参数的网格点。为了提高计算精度,分别计算了水平和垂直方向上应力镜像值,而对质点速度,采用先水平方向后垂直方向分析进行镜像计算和更新。将粘弹性水平自由表面镜像方法和不同倾斜度平滑自由表面的改进方法的数值结果进行对比,并验证了算法的精度。