Axisymmetric compressive buckling of multi-walled carbon nanotubes under different boundary conditions

The paper studies the axisymmetric compressive buckling behavior of multi-walled carbon nanotubes (MWNTs) under different boundary conditions based on continuum mechanics model. A buckling condition is derived for determining the critical buckling load and associated buckling mode of MWNTs, and numerical results are worked out for MWNTs with different aspect ratios under fixed and simply supported boundary conditions. It is shown that the critical buckling load of MWNTs is insensitive to boundary conditions, except for nanotubes with smaller radii and very small aspect ratio. The associated buckling modes for different layers of MWNTs are in-phase, and the buckling displacement ratios for different layers are independent of the boundary conditions and the length of MWNTs. Moreover, for simply supported boundary conditions, the critical buckling load is compared with the corresponding one for axial compressive buckling, which indicates that the critical buckling load for axial compressive buckling can be well approximated by the corresponding one for axisymmetric compressive buckling. In particular, for axial compressive buckling of double-walled carbon nanotubes, an analytical expression is given for approximating the critical buckling load. The present investigation may be of some help in further understanding the mechanical properties of MWNTs.     

A glance on the effects of temperature on axisymmetric dynamic behavior of multiwall carbon nanotubes

In this paper the effects of temperature on theradial breathing modes(RBMs) and radial wave propagation in multiwall carbon nanotubes(MWCNTs) are investigated using a continuum model of multiple elastic isotropicshells.The van der Waals forces between tubes are simulatedas a nonlinear function of interlayer spacing of MWCNTs.The governing equations are solved using a finite elementmethod.A wide range of innermost radius-to-thickness ratioof MWCNTs is considered to enhance the investigation.Thepresented solution is verified by comparing the results withthose reported in the literature.The effects of temperature onthe van der Waals interaction coefficient between layers ofMWCNTs are examined.It is found that the variation of thevan der Waals interaction coefficient at high temperature issensible.Subsequently,variations of RBM frequencies andradial wave propagation in MWCNTs with temperatures upto 1 600 K are illustrated.It is shown that the thick MWCNTs are more sensible to temperature than the thin ones.     

On the van der Waals interaction of carbon nanotubes as electromechanical nanothermometers

Electromechanical carbon nanothermometers are devices that work based on the interactions and relative motions of double-walled carbon nanotubes (DWCNTs). In this paper, the mechanics of carbon nanotubes (CNTs) constituting two well-known configurations for nanothermometer, namely shuttle configuration and telescope configuration are fully investigated. Lennard-Jones (LJ) potential function along with the continuum approximation is employed to investigate van der Waals (vdW) interactions between the interacting entities. Accordingly, semi-analytical expressions in terms of single integrals are obtained for vdW interactions. Acceptance condition and suction energy are studied for the shuttle configuration. In addition, a universal potential energy is presented for the shuttle configuration consisting of two finite CNTs. Also, for the telescope configuration, extensive studies are performed on the distributions of potential energy and interaction force for various radii and lengths of CNTs. It is found that these geometrical parameters have a considerable effect on the potential energy.     

Influence of interparticle interactions on the kinetics of self-assembly and mechanical strength of nanoparticulate aggregates

Computer simulations based on Discrete Element Method have been performed in order to investigate the influence of interparticle interactions on the kinetics of self-assembly and the mechanical strength of nanoparticle aggregates.Three different systems have been considered.In the first system the interaction between particles has been simulated using the JKR (Johnson,Kendall and Roberts) contact theory,while in the second and third systems the interaction between particles has been simulated using van der Waals and electrostatic forces respectively.In order to compare the mechanical behaviour of the three systems,the magnitude of the maximum attractive force between particles has been kept the same in all cases.However,the relationship between force and separation distance differs from case to case and thus,the range of the interparticle force.The results clearly indicate that as the range of the interparticle force increases,the self-assembly process is faster and the work required to produce the mechanical failure of the assemblies increases by more than one order of magnitude.     

DYNAMIC BUCKLING BEHAVIOR OF MULTI-WALLED CARBON NANOTUBES SUBJECTED TO STEP AXIAL LOADING

This paper studies the dynamic buckling behavior of multi-walled carbon nanotubes (MWNTs) subjected to step axial loading.A buckling condition is derived,and numerical results are presented for MWNTs u...     

多尺度法二义性的一种解释

多尺度法是为解决含小参数系统发展起来的应用最广泛的摄动法之一. 在求解高阶近 似方程时,多尺度法一般只求特解. 用多尺度法求解van der Pol 方程的三阶解时 将出现矛盾. 以van der Pol方程为例,证明了忽略一阶修正量中的一阶谐波 项使得混合偏导数不能交换顺序,从而导致了多尺度法的二义性和另一个数学矛盾. 在求解一阶修正量时采用含有一阶谐波项的全解,消除了二义性和该矛盾. 该 方法所求得的近似解与数值解进行了比较,结果非常吻合,验证了其合理性.     

非线性动力系统线性模型数值计算的Taylor变换法

将非线性动力系统化为连续变化的线性系统,并导出任意自治或非自治非线性动力系统的瞬时线性化方程,该线性方程的连续变化描述了系统的全部复杂动力行为.进一步采用Taylor变换法求解系统的线性化方程,得到一种非线性动力系统数值计算的新方法,避免了指数矩阵展开的乘积运算.计算实例表明该方法在不增加计算机时的前提下,精度高于传统的Houbolt,Wilson-θ及Newmark-β等方法.计算了Duffing方程和van　Pol方程的混沌及周期特性.     

两自由度耦合van der Pol振子的拟主振动解

本文运用非线性系统的模态方法研究了两自由度耦合van der Pol振子。从退化系统稳定的主振动解出发,得到了原系统的拟主振动解,并给出了系统周期运动的条件,讨论了系统周期解、概周期解的分叉。     

Bifurcation and dynamic response analysis of rotating blade excited by upstream vortices

A reduced model is proposed and analyzed for the simulation of vortex-induced vibrations (VIVs) for turbine blades. A rotating blade is modelled as a uniform cantilever beam, while a van der Pol oscillator is used to represent the time-varying characteristics of the vortex shedding, which interacts with the equations of motion for the blade to simulate the fluid-structure interaction. The action for the structural motion on the fluid is considered as a linear inertia coupling. The nonlinear characteristics for the dynamic responses are investigated with the multiple scale method, and the modulation equations are derived. The transition set consisting of the bifurcation set and the hysteresis set is constructed by the singularity theory and the effects of the system parameters, such as the van der Pol damping. The coupling parameter on the equilibrium solutions is analyzed. The frequency-response curves are obtained, and the stabilities are determined by the Routh-Hurwitz criterion. The phenomena including the saddle-node and Hopf bifurcations are found to occur under certain parameter values. A direct numerical method is used to analyze the dynamic characteristics for the original system and verify the validity of the multiple scale method. The results indicate that the new coupled model is useful in explaining the rich dynamic response characteristics such as possible bifurcation phenomena in the VIVs.     

Pull-in instability analyses for NEMS actuators with quartic shape approximation

The pull-in instability of a cantilever nano-actuator model incorporating the effects of the surface, the fringing field, and the Casimir attraction force is investigated. A new quartic polynomial is proposed as the shape function of the beam during the deflection, satisfying all of the four boundary values. The Gaussian quadrature rule is used to treat the involved integrations, and the design parameters are preserved in the evaluated formulas. The analytic expressions are derived for the tip deflection and pull-in parameters of the cantilever beam. The micro-electromechanical system (MEMS) cantilever actuators and freestanding nanoactuators are considered as two special cases. It is proved that the proposed method is convenient for the analyses of the effects of the surface, the Casimir force, and the fringing field on the pull-in parameters.     

Analysis of humidity-dependent adhesion between a probe tip and a surface

Adhesive forces commonly exhibit a monotonic increase or a maximum with increasing relative humidity. However, anomalous behavior has been reported. Here, a numerical model of adhesive forces, comprised mainly of capillary and van der Waals forces, between a tip and a surface is established. It is described by a power law that considers the geometry, the liquid bridge wetting radius, the contact angle, and the separation distance. Capillary forces (sum of surface tension and Laplace pressure) and van der Waals forces are calculated. The latter cannot be neglected in the adhesion even at high humidity. Decrease in adhesion with increasing relative humidity can be attributed to a blunt tip shape, which is validated by experimental data. Specifically, the decrease in adhesion is attributed primarily to a transition from a rounded to a blunt tip shape. Structuring objects at the micro- or nanoscale can either increase or decrease adhesion as a function of relative humidity. This has a wide range of applications in robotic manipulation and can provide a better understanding of adhesion mechanisms in atomic force microscopy in ambient air.     

相对湿度对材料表面粘附力影响的研究

利用自制微摩擦及粘附力测试装置考察了在微载荷条件下,相对湿度对Si(100)材料表面粘附力的影响,分析了在大气环境中水分子的毛细作用力和范德华力对粘附力的贡献,并以BET吸附模型为基础推导出考虑湿度影响的粘附力计算公式.结果表明:在微载荷条件下,相对湿度对材料表面的粘附力影响十分显著,随着相对湿度升高粘附力增加,特别是相对湿度RH在40%～80%之间时,粘附力变化最为显著;当相对湿度RH小于20%时,范德华力大于水的毛细作用力且占主导地位;当相对湿度RH大于20%后,水分子的毛细作用力不断增加,同时范德华力因水膜的存在而降低,水的毛细作用力占主导地位.     

范德华力对磁头/硬盘系统气体润滑特性的影响

通过数值方法分析范德华力对磁头承载力的影响以及随飞行姿态的变化规律.结果表明,范德华力主要作用在磁头滑块尾部的凸台上,当飞行高度低于10 nm后,范德华力使得滑块的承载能力明显降低,从而导致飞行高度下降.范德华力主要与飞行高度和俯仰角有关,随着飞行高度和俯仰角减小,范德华力使承载能力和俯仰转矩明显减小.范德华力对侧翻角的影响不敏感,随着侧翻角的增加,范德华力使滑块的承载能力和侧翻转矩略微减小.在磁头设计时,通过增加俯仰角和减小尾部凸台面积可以减小范德华力的影响.     

Effects of van der Waals force on nonlinear vibration of electromechanical integrated electrostatic harmonic actuator

An electromechanical integrated electrostatic harmonic actuator is promising for the miniaturization of electromechanical devices. As the dimensions of the actuator decrease, the effects of the van der Waals force become obvious. In this study, by considering the nonlinearity of electrostatic and van der Waals forces, nonlinear vibration equations of the flexible ring of an electrostatic harmonic actuator are deduced. Using these equations, the nonlinear free vibration and nonlinear forced response of the actuator are investigated. The effects of the van der Waals force on the nonlinear vibration of the flexible ring are analyzed. Results show that these effects of the van der Waals force are relatively obvious under some conditions and should be considered.     

范德华力对磁头/硬盘薄膜气体润滑动态特性的影响

利用摄动法分析了纳米间隙下范德华力对磁头动态飞行特性的影响规律.结果表明,在飞行高度低于5 nm后,范德华力降低了磁头的承载能力和飞行高度,使飞行高度和俯仰气膜刚度降低,对侧翻气膜刚度没有明显影响.范德华力对磁头滑块飞高振幅的影响主要在于特征振动频率区间,因此,通过磁头设计提高磁头特征振动频率,可以减弱范德华力对飞高振幅的影响.     

A cohesive law for carbon nanotube/polymer interfaces based on the van der Waals force

We have established the cohesive law for interfaces between a carbon nanotube (CNT) and polymer that are not well bonded and are characterized by the van der Waals force. The tensile cohesive strength and cohesive energy are given in terms of the area density of carbon nanotube and volume density of polymer, as well as the parameters in the van der Waals force. For a CNT in an infinite polymer, the shear cohesive stress vanishes, and the tensile cohesive stress depends only on the opening displacement. For a CNT in a finite polymer matrix, the tensile cohesive stress remains the same, but the shear cohesive stress depends on both opening and sliding displacements, i.e., the tension/shear coupling. The simple, analytical expressions of the cohesive law are useful to study the interaction between CNT and polymer, such as in CNT-reinforced composites. The effect of polymer surface roughness on the cohesive law is also studied.     

基于定量内聚力模型的金属陶瓷界面强度的研究

建立了一个基于分子之间范德华作用力的内聚力模型,并将其应用于小尺寸范围.通过该模型研究了纳米金属陶瓷交界面的结合强度,交界面的内聚力强度与内聚能和纳米金属陶瓷的界面原子密度以及范德华力的相关参数有关.模型结果显示,界面距离h和张开位移v对法向内聚应力σcohesive有很大影响,而晶粒尺寸d对其影响不大;切向内聚应力τcohesive与滑移位移u和晶粒尺寸d都有关系.该模型还可以用于其它以范德华力为主要机制的未完全粘合界面的情况.     

Torsional buckling of a double-walled carbon nanotube embedded in an elastic medium

The torsional buckling of a double-walled carbon nanotube embedded in an elastic medium is studied in this paper. The effects of surrounding elastic medium and van der Waals forces between the inner and outer nanotubes are taken into account. Using continuum mechanics, an elastic double-shell model is presented for the torsional buckling of a double-walled carbon nanotube. Based on the model, a condition is derived in terms of the buckling modes of the shell and the parameters describing the effect of van der Waals interaction and surrounding elastic medium. A simplified analysis is also carried out estimate the critical torque for torsional buckling of the double-walled carbon nanotube.     

Influence of interparticle interactions on the kinetics of self-assembly and mechanical strength of nanoparticulate aggregates

Computer simulations based on Discrete Element Method have been performed in order to investigate the influence of interparticle interactions on the kinetics of self-assembly and the mechanical strength of nanoparticle aggregates. Three different systems have been considered. In the first system the interaction between particles has been simulated using the JKR （Johnson, Kendall and Roberts） contact theory, while in the second and third systems the interaction between particles has been simulated using van der Waals and electrostatic forces respectively. In order to compare the mechanical behaviour of the three systems, the magnitude of the maximum attractive force between particles has been kept the same in all cases. However, the relationship between force and separation distance differs from case to case and thus, the range of the interparticle force. The results clearly indicate that as the range of the interparticle force increases, the self-assembly process is faster and the work required to produce the mechanical failure of the assemblies increases by more than one order of magnitude.     

考虑范德华力曲率效应的双壁碳纳米管外压屈曲

针对双壁碳纳米管外压屈曲问题,研究了层间范德华力的曲率效应对临界外压的影响。应用弹性双层圆柱壳模型,考虑层间范德华力不仅与层间距有关而且与挠度曲率的变化有关,导出了外压屈曲临界压力解析公式。计算得出在不同半径、不同长细比下,外压屈曲临界压力的数值结果,并与经典壳的结果和忽略范德华力曲率效应的结果做了比较。结果显示,对于小半径的双壁碳纳米管曲率效应对外压屈曲有效明显的影响。