Prediction of the critical buckling load of multi-walled carbon nanotubes under axial compression

In this paper, we propose a new explicit analytical formula of the critical buckling load of double-walled carbon nanotubes (DWCNT) under axial compression. This formula takes into account van der Waals interactions between adjacent tubes and the effect of terms involving tube radii differences generally neglected in the derived expressions of the critical buckling load published in the literature. The elastic multiple Donnell shells continuum approach is employed for modelling the multi-walled carbon nanotubes. The validation of the proposed formula is made by comparison with a numerical solution. The influence of the neglected terms is also studied.     

INVESTIGATION OF THE DYNAMIC BUCKLING OF DOUBLEWALLED CARBON NANOTUBE SUBJECTED TO AXIAL PERIODIC DISTURBING FORCES

IntroductionThediscoveryofthefirstcarbonnanotubes[1]hasattractedwideattentionandstimulatedextensivestudies[2 - 5 ].Thestudiesshowedthatthecarbonnanotubesexhibitsuperiormechanical,electronicandchemicalproperties.Onthemechanicalbehavior,thecarbonnanotubespossessexceptionallyhighstrength ,stiffnessandelasticmodulus.Theestimatemodulusofthecarbonnanotubemayreachashighas 1TPa.Itisthelargestofallknownmaterials.Thestrengthorstiffnessishigherthananyknownfiber[3].Thecarbonnanotubeareusedascompositemat…     

Rippling effect on the structural response of electrostatically actuated single-walled carbon nanotube based NEMS actuators

Several nonlinear phenomena have shown to have significant effect on the electromechanical performance of single-walled carbon nanotube (SWCNT) based nanoelectromechanical (NEMS) devices. To name few: the van der Waals forces, the Casimir forces, the tip charge concentration and the rippling phenomenon. Some of these effects have been take care of in previous investigation; however, some have been disregarded in the mechanical models suggested for simulation of the SWCNT based structures. In this paper, the influence of rippling deformation on the vibration characteristics of SWCNT based actuators is investigated using a nonlinear Euler-Bernoulli beam theory that incorporates the effect of rippling deformation using an improved function including some correcting terms for the SWCNT curvature (rippling deformation). The influence of the Casimir and the van der Waals attraction forces are considered in the proposed model as well as the size-dependent behavior assuming the so-called Eringen nonlocal elasticity theory. The dynamic response of CNT is investigated based on time history and phase portrait plots of the CNT based nano-actuator. It is shown that the rippling deformation can significantly decrease the static as well as the dynamic pull-in voltage of the SWCNT based actuator. The rippling deformation of SWCNT decreases the dynamic pull-in time as well. Effect of various factors such as the DC actuation load and the Casimir attractive forces on the dynamic stability and the pull-in characteristics of the nano-actuator are examined. Results of the present study are beneficial to accurate design and fabrication of electromechanical CNT based actuators. Comparison between the obtained results and those reported in the literature by experiments and molecular dynamics, verifies the integrity of the present numerical analysis.     

多场作用下输流单层碳纳米管的动力学特性

以非局部弹性理论为基础,采用欧拉-伯努利梁模型,考虑管型区域内滑移边界条件以及碳纳米管的小尺度效应,应用哈密顿原理获得了温度场与轴向磁场共同作用下的输流单层固支碳纳米管（SWCNT）的振动控制方程以及边界条件,依靠微分变换法（DTM法）对此高阶偏微分方程进行求解,通过数值计算研究了多场中单层固支输流碳纳米管的振动与失稳问题。结果表明：温度场、轴向磁场强度、Knudsen数及小尺度参数都会对系统振动频率以及失稳临界流速产生影响。     

Nonlinear vibration of embedded single-walled carbon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory

Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube (SWCNT) subjected to a harmonic load is investigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.     

The thermal effect on axially compressed buckling of a double-walled carbon nanotube

The thermal effect on axially compressed buckling of a double-walled carbon nanotube is studied in this paper. The effects of temperature change, 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 with thermal effect is presented for axially compressed buckling of a double-walled carbon nanotube embedded in an elastic matrix under thermal environment. Based on the model, an explicit formula for the critical axial stress is derived in terms of the buckling modes of the shell and the parameters that indicate the effects of temperature change, surrounding elastic medium and the van der Waals forces. Based on that, some simplified analysis is carried out to estimate the critical axial stress for axially compressed buckling of the double-walled carbon nanotube. Numerical results for the general case are obtained for the thermal effect on axially compressed buckling of a double-walled carbon nanotube. It is shown that the axial buckling load of double-walled carbon nanotube under thermal loads is dependent on the wave number of axially buckling modes. And a conclusion is drawn that at low and room temperature the critical axial stress for infinitesimal buckling of a double-walled carbon nanotube increase as the value of temperature change increases, while at high temperature the critical axial stress for infinitesimal buckling of a double-walled carbon nanotube decrease as the value of temperature change increases.     

温度场作用下输流悬臂碳纳米管的颤振失稳分析

以非局部弹性理论为基础,采用欧拉-伯努利梁模型,考虑碳纳米管的小尺度效应,应用哈密顿原理获得了温度场作用下的输流悬臂单层碳纳米管（SWCNT）的振动控制方程以及边界条件,依靠微分变换法（DTM法）对此高阶偏微分方程进行求解,通过数值计算研究了温度场中悬臂单层输流碳纳米管的振动与颤振失稳问题。结果表明：管内流体流速、温度场中温度变化情况与小尺度参数都会对系统振动频率以及颤振失稳临界流速产生影响。其中,小尺度效应将会降低悬臂输流系统的稳定性,使系统更为柔软;而高温场与低温场对系统动态失稳的影响不同,低温场中随温度变化值的增加,系统的稳定性提高;高温场这一作用效果恰好与之相反。     

基于非局部效应和表面效应的输流碳纳米管稳定性分析简

应用非局部黏弹性夹层梁模型分析双参数弹性介质中输送脉动流碳纳米管的稳定性. 新模型中同时考虑了由管道内、外壁上的薄表面层引起的表面弹性效应和表面残余应力，经典的欧拉梁模型因此通过引入非局部参数和表面参数得到了改进. 用平均法对其控制方程进行求解，得到了管道稳定性区域. 数值算例揭示了纳米材料的非局部效应、表面效应及两个弹性介质参数对管道固有频率、临界流速和动态稳定性的复杂影响，结论可为纳米流体机械的结构设计和振动分析提供理论基础.     

DYNAMIC STABILITY ANALYSIS OF EMBEDDED MULTI-WALLED CARBON NANOTUBES IN THERMAL ENVIRONMENT

In the present paper, the dynamic stability of multi-walled carbon nanotubes(MWCNTs) embedded in an elastic medium is investigated including thermal environment effects. To this end, a nonlocal Timoshenko beam model is developed which captures small scale effects.Dynamic governing equations of the carbon nanotubes are formulated based on the Timoshenko beam theory including the effects of axial compressive force. Then a parametric study is conducted to investigate the influences of static load factor, temperature change, nonlocal parameter, slenderness ratio and spring constant of the elastic medium on the dynamic stability characteristics of MWCNTs with simply-supported end supports.     

Nonlinear vibrations of a single-walled carbon nanotube for delivering of nanoparticles

The capability of carbon nanotubes (CNTs) in efficient transporting of drug molecules into the biological cells has been the focus of attention of various scientific disciplines during the past decade. From applied mechanics points of view, translocation of a nanoparticle inside the pore of a CNT would result in vibrations. The true understanding of the interactive forces between the moving nanoparticle and the inner surface of the CNT is a vital step in factual realization of such vibrations. Herein, by employing the nonlocal Rayleigh beam theory, nonlinear vibrations of single-walled carbon nanotubes (SWCNTs) as nanoparticle delivery nanodevices are studied. The existing van der Waals interactional forces between the constitutive atoms of the nanoparticle and those of the SWCNT, frictional force, and both longitudinal and transverse inertial effects of the moving nanoparticle are taken into account in the proposed model. The nonlinear-nonlocal governing equations are explicitly obtained and then numerically solved using Galerkin method and a finite difference scheme in the space and time domains, respectively. The roles of the velocity and mass weight of the nanoparticle, small-scale effect, slenderness ratio, and vdW force on the maximum longitudinal and transverse displacements as well as the maximum nonlocal axial force and bending moment within the SWCNT are examined. In general, the obtained results reveal that the nonlinear analysis should be performed when the nanotube structure is traversed by a moving nanoparticle with high levels of the mass weight and velocity.     

A new method for characterizing the interphase regions of carbon nanotube composites

The elastic properties of a carbon nanotube (CNT) reinforced composite are affected by many factors such as the CNT–matrix interphase. As such, mechanical analysis without sufficient consideration of these factors can give rise to incorrect predictions. Using single-walled carbon nanotube (SWCNT) reinforced Polyvinylchloride (PVC) as an example, this paper presents a new technique to characterize interphase regions. The representative volume element (RVE) of the SWCNT–PVC system is modeled as an assemblage of three phases, the equivalent solid fiber (ESF) mimicking the SWCNT under the van der Waals (vdW) forces, the dense interphase PVC of appropriate thickness and density, and the bulk PVC matrix. Two methods are proposed to extract the elastic properties of the ESF from the atomistic RVE and the CNT-cluster. Using atomistic simulations, the thickness and the average density of interphase matrix are determined and the elastic properties of amorphous interphase matrix are characterized as a function of density. The method is examined in a continuum-based three-phase model developed with the aid of molecular mechanics (MM) and the finite element (FE) method. The predictions of the continuum-based model show a good agreement with the atomistic results verifies that the interphase properties of amorphous matrix in CNT-composites could be approximated as a function of density. The results show that ignoring either the vdW interaction region or the interphase matrix layer can bring about misleading results, and that the effect of internal walls of multi-walled carbon nanotubes (MWCNTs) on the density and thickness of the dense interphase is negligible.     

磁场对不同温度场中输流悬臂碳纳米管动态特性的影响

本文在采用经典欧拉-伯努利梁模型的基础上,引入考虑小尺度效应的非局部弹性理论,着重研究不同温度场中输流悬臂单层碳纳米管系统（SWCNT）在外加纵向磁场作用下的颤振失稳问题。基于哈密顿原理获得了该流固耦合系统的振动控制方程及相应的边界条件,应用微分变换法（DTM法）求解此高阶偏微分方程,通过数值计算研究了不同温度场中施加纵向磁场对系统动力学特性的影响。结果表明：施加纵向磁场在不同温度场中都将增强输流悬臂碳纳米管的动态稳定性。然而,这种增强程度却与温度场的变化量有关,在不同温度变化量下,磁场对系统稳定性的增强程度有一个峰值,这意味着,实际应用中,为了提高这类流固耦合系统的动态稳定性,一味提高纵向磁场强度并不可取。     

Dynamic and buckling analysis of polymer hybrid composite beam with variable thickness

This work deals with a study of the dynamic and buckling analysis of polymer hybrid composite(PHC) beam. The beam has variable thickness and is reinforced by carbon nanotubes(CNTs) and nanoclay(NC) simultaneously. The governing equations are derived based on the first shear deformation theory(FSDT). A three-phase HalpinTsai approach is used to predict the mechanical properties of the PHC. We focus our attention on the effect of the simultaneous addition of NC and CNT on the vibration and buckling analysis of the PHC beam with variable thickness. Also a comparison study is done on the sensation of three impressive parameters including CNT, NC weight fractions, and the shape factor of fillers on the mechanical properties of PHC beams,as well as fundamental frequencies of free vibrations and critical buckling load. The results show that the increase of shape factor value, NC, and CNT weight fractions leads to considerable reinforcement in mechanical properties as well as increase of the dimensionless fundamental frequency and buckling load. The variation of CNT weight fraction on elastic modulus is more sensitive rather than shear modulus but the effect of NC weight fraction on elastic and shear moduli is fairly the same. The shape factor values more than the medium level do not affect the mechanical properties.     

原位微拉曼测试技术在碳纳米管纤维和薄膜材料力学性能研究中的应用

利用宏观应力联合原位微拉曼测试技术对双壁碳纳米管(Carbon Nanotube,CNT)纤维和薄膜材料的力学性能进行了实验分析,探讨了拉伸加载期间纤维和薄膜内CNT的载荷响应及其与宏观力学性能的关联,揭示了两种材料力学性能差异性的微观机理。实验分析表明,CNT纤维和薄膜的拉伸变形呈现弹性、强化和损伤断裂三个阶段,但其内的CNT只发生弹性变形,没有塑性形变,且没有明显的损伤或键的断裂,纤维和薄膜呈现阶段性拉伸变形的原因可归结为滑移。纤维的弹性模量显著高于薄膜,是薄膜的4.7倍,原因是弹性阶段纤维中CNT的轴向伸长对宏观应变的贡献较大。纤维和薄膜的拉伸强度相差较小,原因是强化阶段薄膜内不断有大量CNT进入承载队伍,这也使得薄膜具有比纤维更高的韧性。     

DYNAMIC BUCKLING OF DOUBLE-WALLED CARBON NANOTUBES UNDER STEP AXIAL LOAD

An approximate method is presented in this paper for studying the dynamic buckling of double-walled carbon nanotubes （DWNTs） under step axial load. The analysis is based on the continuum mechanics model, which takes into account the van der Waals interaction between the outer and inner nanotubes. A buckling condition is derived, from which the critical buckling load and associated buckling mode can be determined. As examples, numerical results are worked out for DWNTs under fixed boundary conditions. It is shown that, due to the effect of van der Waals forces, the critical buckling load of a DWNT is enhanced when inserting an inner tube into a single-walled one. The paper indicates that the critical buckling load of DWNTs for dynamic buckling is higher than that for static buckling. The effect of the radii is also examined. In addition, some of the results are compared with the previous ones.     

Dynamic torsional buckling of multi-walled carbon nanotubes embedded in an elastic medium

In this paper the dynamic torsional buckling of multi-walled carbon nanotubes （MWNTs） embedded in an elastic medium is studied by using a continuum mechanics model. By introducing initial imperfections for MWNTs and applying the preferred mode analytical method, a buckling condition is derived for the buckling load and associated buckling mode. In particular, explicit expressions are obtained for embedded double-walled carbon nanotubes （DWNTs）. Numerical results show that, for both the DWNTs and embedded DWNTs, the buckling form shifts from the lower buckling mode to the higher buckling mode with increasing the buckling load, but the buckling mode is invari- able for a certain domain of the buckling load. It is also indicated that, the surrounding elastic medium generally has effect on the lower buckling mode of DWNTs only when compared with the corresponding one for individual DWNTs.     

The coupled effects of mechanical deformation and electronic properties in carbon nanotubes

Coupled effects of mechanical and electronic behavior in single walled carbon nanotu besare investigated by using quantum mechanics and quantum molecular dynamics. It is found that external applied electric fields can cause charge polarization and significant geometric deformation in metallic and semi-metallic carbon nanotubes. The electric induced axial tension ratio can be up to 10% in the armchair tube and 8.5% in the zigzag tube. Pure external applied load has little effect on charge distribution，but indeed influences the energy gap. Tensile load leads to a narrower energy gap and compressive load increases the gap. When the CNT is tensioned under an external electric field, the effect of mechanical load on the electronic structures of the CNT becomes significant, and the applied electric field may reduce the critical mechanical tension load remarkably. Size effects are also discussed.     

Microbuckling of a doublewalled carbon nanotube embedded in an elastic matrix

Microbuckling of a doublewalled carbon nanotube (DWCNT) in an elastic (polymer) matrix is studied. The investigations are made within the scope of the piecewise homogeneous body model by utilizing the three-dimensional linearized theory of stability of deformable bodies. Flexural and axisymmetric microbuckling modes are considered. The DWCNT is modeled as concentrically-nested two circular hollow cylinders between which there is free space. It is assumed that on the inner surface of the outer tube (cylinder) and on the outer surface of the inner tube (cylinder) of the DWCNT full slipping conditions occur. At the same time, it is assumed that the difference between the radial displacements of the adjacent surfaces of the tubes resists with the van der Waals forces. On the interface between the matrix and DWCNT complete contact conditions are satisfied. Numerical results on the influence of the problem parameters on the critical deformation are presented and discussed. Also, numerical results related to the cases where the interlayer space is ignored and where full contact between the tubes is assumed are presented and compared with the mentioned results. In particular, it is established that full slipping between the tubes causes the values of the critical deformation to decrease significantly with respect to those obtained in the case where complete contact conditions occur between the tubes. Moreover, it is established that an increase in the values of the van der Waals forces also causes a decrease in the values of the critical compressing strain and the magnitude of this decrease depends on the thicknesses of the tubes of the DWCNT.     

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

An elastic double-shell model based on continuum mechanics is presented to study the dynamic torsional buckling of an embedded double-walled carbon nanotube. Based on the presented model, a condition is derived to predict the buckling load of the embedded double-walled nanotube, and the effect of the van der Waals forces to the buckling load is discussed when an inner nanotube is inserted into an embedded outer one. In particular, the paper shows that the buckling load of the embedded double-walled nanotube is always between that of the isolated inner nanotube and that of the embedded outer nanotube for both dynamic and static torsional buckling, due to the effect of the van der Waals forces. This result is different from that obtained by the existing analysis neglecting the difference of the radii for the embedded double-walled nanotube, which indicates that disregarding the difference of the radii of multi-walled nanotubes cannot properly describe the effect of the van der Waals forces between interlayer spacing. In particular, for static torsional buckling of a double-walled nanotube, it is shown that the critical buckling load cannot only be enhanced, but also be reduced when inserting an inner nanotube into an isolated single-walled one. Additionally, it is shown that the elastic medium always increases the critical buckling load of double-walled nanotubes. The critical buckling load of embedded double-walled nanotubes for dynamic torsional buckling is proved to be no less than that for static torsional buckling.     

Buckling of embedded multi-walled carbon nanotubes under combined torsion and axial loading

This paper describes an investigation into elastic buckling of an embedded multi-walled carbon nanotube under combined torsion and axial loading, which takes account of the radial constraint from the surrounding elastic medium and van der Waals force between two adjacent tube walls. Depending on the ratio of radius to thickness, the multi-walled carbon nanotubes discussed here are classified as thin, thick, and nearly solid. Critical buckling load with the corresponding mode is obtained for multi-walled carbon nanotubes under combined torsion and axial loading, with various values of the radius to thickness ratio and surrounded with different elastic media. The study indicates that the buckling mode (m, n) of an embedded multi-walled carbon nanotube under combined torsion and axial loading is unique and it is different from that with axial compression only. New features for the buckling of an embedded multi-walled carbon nanotube under combined torsion and axial loading and the meaningful numerical results are useful in the design of nanodrive device, nanotorsional oscillator and rotational actuators, where multi-walled carbon nanotubes act as basic elements.