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.     

Non-local continuum modeling of carbon nanotubes: Physical interpretation of non-local kernels using atomistic simulations

The longitudinal, transverse and torsional wave dispersion curves in single walled carbon nanotubes (SWCNT) are used to estimate the non-local kernel for use in continuum elasticity models of nanotubes. The dispersion data for an armchair (10,10) SWCNT was obtained using lattice dynamics of SWNTs while accounting for the helical symmetry of the tubes. In our approach, the Fourier transformed kernel of non-local linear elastic theory is directly estimated by matching the atomistic data to the dispersion curves predicted from non-local 1D rod theory and axisymmetric shell theory. We found that gradient models incur significant errors in both the phase and group velocity when compared to the atomistic model. Complementing these studies, we have also performed detailed tests on the effect of length of the nanotube on the axial and shear moduli to gain a better physical insight on the nature of the true non-local kernel. We note that unlike the kernel from gradient theory, the numerically fitted kernel becomes negative at larger distances from the reference point. We postulate and confirm that the fitted kernel changes sign close to the inflection point of the interatomic potential. The numerically computed kernels obtained from this study will aid in the development of improved and efficient continuum models for predicting the mechanical response of CNTs.     

单壁碳纳米管的扭转力学特性研究

利用基于高阶Cauchy-Born准则所建立的单壁碳纳米管本构模型,针对不同手性的单壁碳纳米管的扭转力学特性进行了研究.研究发现结构呈现对称性的锯齿型和扶手型单壁碳纳米管具有完全对称的扭转特性,而结构不对称的手性型单壁碳纳米管具有正反相异的扭转特性.同时,针对一系列手性不同的单壁碳纳米管的扭转力学特性展开了详细的研究.研究的部分结果与采用其他方法得到的结果进行了对比,证实了所提出方法以及预测结果的有效性和可行性.     

双层碳纳米管在扭矩作用下的屈曲

考虑双层碳纳米管层间范德华力的作用,利用连续介质力学的壳体理论,建立了扭矩作用下碳纳米管屈曲问题的双层弹性壳体模型,给出了相应的临界屈曲扭矩,分析了双层碳纳米管层间范德华力对临界屈曲扭矩的影响。     

Small scale effect on flow-induced instability of double-walled carbon nanotubes

Small scale effect on flow-induced instability of double-walled carbon nanotubes (DWCNTs) is investigated using an elastic shell model based on Donnell’s shell theory. The dynamic governing equations of DWCNTs are formulated on the basis of nonlocal elasticity theory, in addition, the van der Waals (vdW) interaction between the inner and outer walls is taken into account in the nonlocal shell modeling. The instability of DWCNTs that is induced by a pressure-driven steady flow is investigated. The numerical computations indicate that as the flow velocity increases, DWCNTs have a destabilizing way to get through multi-bifurcations of the first and second bifurcations in turn. It is concluded that the natural frequency of DWCNTs and the critical flow velocity of the flow-induced instability are strictly related to the ratio of the length to the outer radius of DWCNTs, the pressure of the fluid and the small scale effects. Furthermore, it is interesting to observe that as the small scale effects are considered, the natural frequencies and the critical flow velocities of DWCNTs decrease as compared to the results with the classical (local) continuum mechanics, therefore, the small scale effects play an important role on performing the instability analysis in the fluid-conveying DWCNTs.     

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.     

Buckling behavior of perfect and defective DWCNTs under axial, bending and torsional loadings via a structural mechanics approach

The buckling behavior of perfect and defective double-walled carbon nanotubes (DWCNTs) under axial compressive, torsional and bending loadings is investigated using a structural mechanics model. The effects of van der Waals (vdW) forces are further modeled using a nonlinear spring element. Critical buckling loads, critical buckling moments and the effects of vacancy defects were studied for armchair nanotubes with various aspect ratios. The results show that vacancy defects greatly reduce the critical buckling load of DWCNTs. The density of defects plays an important role in buckling of DWCNTs. The results of this numerical model are in good agreement with their comparable existing works.     

Can a single-wall carbon nanotube be modeled as a thin shell?

Single-wall carbon nanotubes (SWCNT) have been frequently modeled as thin shells, but the shell thickness and Young's modulus reported in literatures display large scattering. The order of error to approximate SWCNTs as thin shells is studied in this paper via an atomistic-based finite-deformation shell theory, which avoids the shell thickness and Young's modulus, but links the tension and bending rigidities directly to the interatomic potential. The ratio of atomic spacing (Δ≈0.14 nm) to the radius of SWCNT, Δ/R, which ranges from zero (for graphene) to 40% [for a small (5,5) armchair SWCNT (R=0.35 nm)], is used to estimate the order of error. For the order of error O[(Δ/R)³], SWCNTs cannot be represented by a conventional thin shell because their constitutive relation involves the coupling between tension and curvature and between bending and strain. For the order of error O[(Δ/R)²], the tension and bending (shear and torsion) rigidities of SWCNTs can be represented by an elastic orthotropic thin shell, but the thickness and elastic modulus cannot. Only for the order of error O(Δ/R), a universal constant shell thickness can be defined and SWCNTs can be modeled as an elastic isotropic thin shell.     

Atomistic-continuum modeling for mechanical properties of single-walled carbon nanotubes

The study attempts to explore the influences of the surface effect resulting in an initial relaxed unstrained deformation and the in-layer non-bonded van der Waals (vdW) atomistic interactions on the mechanical properties of single-walled carbon nanotubes (SWCNTs) using a proposed atomistic-continuum modeling (ACM) approach. The modeling approach incorporates atomistic modeling, by virtue of molecular dynamics (MD) simulation, for simulating the initial unstrained equilibrium state, and equivalent-continuum modeling (ECM), by way of finite element approximations (FEA), for modeling the subsequent static/dynamic behaviors.SWCNTs with various radius and two different chiralities, including zigzag and armchair type, are presented. To validate the proposed technique, the present results are compared with the literature data, including numerical and experimental values. Results show that the derived elastic moduli (1.2–1.4 TPa) when considering these two nanoeffects tend to be more consistent with the published experimental data. In specific, they can increase up to 17–23% Young’s modulus, 5–15% shear modulus, 6–11% natural frequencies and 10–30% critical buckling load of the SWCNTs, implying that without considering these two effects, the material behaviors of SWCNTs would be potentially underestimated.     

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.     

Nonlinear analysis of a SWCNT over a bundle of nanotubes

The deformation of a single wall carbon nanotube (SWCNT) interacting with a curved bundle of nanotubes is analyzed. The SWCNT is modeled as a straight elastic inextensible beam based on small deformation. The bundle of nanotubes is assumed rigid and the interaction is due to the van der Waals forces. An analytical solution is obtained using a bilinear approximation to the van der Waals forces. The analytical results are in good agreement with the results of two numerical methods. The results indicate that the SWCNT remains near the curved bundle provided that its curvature is below a critical value. For curvatures above this critical value the SWCNT breaks contact with the curved bundle and nearly returns to its straight position. A parameter study shows that the critical curvature depends on the stiffness of the SWCNT and the absolute minimum energy associated with the van der Waals forces but it is independent of the SWCNT's length in general. An analytical estimate of the critical curvature is developed. The results of this study may be applicable to composites of nanotubes where separation phenomena are suspected to occur.     

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.     

Study of dynamic stability of the double-walled carbon nanotube under axial loading embedded in an elastic medium by the energy method

In this paper, the dynamic stability of single- and double-walled carbon nanotubes (SWCNT and DWCNT) under dynamic axial loading is investigated using the continuum mechanics model and the minimum total energy method. The natural frequencies of the SWCNT and the critical dynamic axial load of the SWCNT and DWCNT are obtained using the Rayleigh-Ritz method. The effects of the elastic medium and the van der Waals forces between the two layers in the DWCNT are taken into account using the Winkler model and Lennard-Jones theory, respectively. The effect of the small length scale is also considered using the Eringen Model. The critical dynamic axial load is increased by inserting an inner carbon nanotube (CNT) into an isolated CNT embedded in an elastic medium.     

Effect of small size on dispersion characteristics of wave in carbon nanotubes

Effect of small size on dispersion characteristics of waves in multi-walled carbon nanotubes is investigated using an elastic shell model. Dynamic governing equations of the carbon nanotube are formulated on the basis of nonlocal elastic theory. The relationship between wavenumber and frequency of wave propagation is obtained from the solution of the eigenvalue equations. The numerical results show that the dispersion characteristics of wave in the multi-walled carbon nanotube are affected by the small size. Effect of small size is not obvious for the smaller wavenumber, and it will arise and increase gradually with the increase of the wavenumber. Effect of the small size will decrease as the inner radius of carbon nanotubes increases. In addition, the explicit expressions of the cut-off frequencies are derived. The results show that the cut-off frequencies cannot be influenced by the small size of carbon nanotubes.     

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…     

Detailed investigation on single water molecule entering carbon nanotubes

The behavior of a water molecule entering carbon nanotubes (CNTs) is studied. The Lennard-Jones potential function together with the continuum approximation is used to obtain the van der Waals interaction between a single-walled CNT (SWCNT) and a single water molecule. Three orientations are chosen for the water molecule as the center of mass is on the axis of nanotube. Extensive studies on the variations of force, energy, and velocity distributions are performed by varying the nanotube radius and the orientations of the water molecule. The force and energy distributions are validated by those obtained from molecular dynamics (MD) simulations. The acceptance radius of the nanotube for sucking the water molecule inside is derived, in which the limit of the radius is specified so that the nanotube is favorable to absorb the water molecule. The velocities of a single water molecule entering CNTs are calculated and the maximum entrance and the interior velocity for different orientations are assigned and compared.     

A single degree of freedom ‘lollipop’ model for carbon nanotube bundle formation

Current carbon nanotube (CNT) synthesis methods include the production of ordered, free-standing vertically aligned arrays, the properties of which are partially governed by interactions between adjacent tubes. Using material parameters determined by atomistic methods, here we represent individual CNTs by a simple single degree of freedom ‘lollipop’ model to investigate the formation, mechanics, and self-organization of CNT bundles driven by weak van der Waals interactions. The computationally efficient simple single degree of freedom model enables us to study arrays consisting of hundreds of thousands of nanotubes. The effects of nanotube parameters such as aspect ratio, bending stiffness, and surface energy, on formation and bundle size, as well as the intentional manipulation of bundle pattern formation, are investigated. We report studies with both single wall carbon nanotubes (SWCNTs) and double wall carbon nanotubes (DWCNTs) with varying aspect ratios (that is, varying height). We calculate the local density distributions of the nanotube bundles and show that there exists a maximum attainable bundle density regardless of an increase in surface energy for nanotubes with given spacing and stiffness. In addition to applications to CNTs, our model can also be applied to other types of nanotube arrays (e.g. protein nanotubes, polymer nanofilaments).     

Wave propagation in carbon nanotubes embedded in an elastic matrix

This paper reports the results of an investigation into the characteristics of wave propagation in carbon nanotubes embedded in an elastic matrix, based on an exact shell model. Each of the concentric tubes of multi-walled carbon nanotubes is considered as an individual elastic shell and coupled together through the van der Waals forces between two adjacent tubes. The matrix surrounding carbon nanotubes is described as a spring element defined by the Winkler model. The effects of rotatory inertia and elastic matrix on the wave velocity, the critical frequency, and the amplitude ratio between two adjacent tubes are described and discussed through numerical examples. The results obtained show that wave propagation in carbon nanotubes may appear in a critical frequency at which the wave velocity changes suddenly; the elastic matrix surrounding carbon nanotubes debases the critical frequency and the wave velocity, and changes the vibration modes between two adjacent tubes; the rotatory inertia based on an exact shell model obviously influences the wave velocity at some wave modes. Finally, a comparison of dispersion solutions from different shell models is given. The present work may serve as a useful reference for the application and the design of nano-electronic and nano-drive devices, nano-oscillators, and nano-sensors, in which carbon nanotubes act as basic elements.     

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。