Nanoscale finite element models for vibrations of single-walled carbon nanotubes: atomistic versus continuum

By the atomistic and continuum finite element models, the free vibration behavior of single-walled carbon nanotubes （SWCNTs） is studied. In the atomistic finite element model, the bonds and atoms are modeled by the beam and point mass elements, respectively. The molecular mechanics is linked to structural mechanics to determine the elastic properties of the mentioned beam elements. In the continuum finite element approach, by neglecting the discrete nature of the atomic structure of the nanotubes, they are modeled with shell elements. By both models, the natural frequencies of SWCNTs are computed, and the effects of the geometrical parameters, the atomic structure, and the boundary conditions are investigated. The accuracy of the utilized methods is verified in comparison with molecular dynamic simulations. The molecular structural model leads to more reliable results, especially for lower aspect ratios. The present analysis provides valuable information about application of continuum models in the investigation of the mechanical behaviors of nanotubes.     

Variational principles for buckling and vibration of MWCNTs modeled by strain gradient theory

Variational principles for the buckling and vibration of multi-walled carbon nanotubes （MWCNTs） are established with the aid of the semi-inverse method. They are used to derive the natural and geometric boundary conditions coupled by small scale parameters. Hamilton＇s principle and Rayleigh＇s quotient for the buckling and vibration of the MWCNTs are given. The Rayleigh-Ritz method is used to study the buckling and vibration of the single-walled carbon nanotubes （SWCNTs） and double-walled carbon nanotubes （DWCNTs） with three typical boundary conditions. The numerical results reveal that the small scale parameter, aspect ratio, and boundary conditions have a profound effect on the buckling and vibration of the SWCNTs and DWCNTs.     

GUIDED CIRCUMFERENTIAL WAVES IN DOUBLE-WALLED CARBON NANOTUBES

A model of guided circumferential waves propagating in double-walled carbon nan- otubes is built by the theory of wave propagation in continuum mechanics,while the van der Waals force between the inner and outer nanotube has been taken into account in the model.The dispersion curves of the guided circumferential wave propagation are studied,and some dispersion characteristics are illustrated by comparing with those of single-walled carbon nanotubes.It is found that in double-walled carbon nanotubes,the guided circumferential waves will propagate in more dispersive ways.More interactions between neighboring wave modes may take place.In particular,it has been found that a couple of wave modes may disappear at a certain frequency and that,while a couple of wave modes disappear,another new couple of wave modes are excited at the same wave number.     

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.     

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.     

Deformation of metallic single-walled carbon nanotubes in electric field based on elastic theory

The electromechanical properties of metallic single-walled carbon nanotubes (SWCNTs) in the electric field are demonstrated with a column shell model in this paper.A hemisphere model is introduced to determine the charge distribution and the local electric field in SWCNTs.By treating the SWCNT as an elastic column shell,the analytical solutions of the charged SWCNT’s axial strain and the radial strain are obtained.SWCNTs with a larger aspect ratio show greater deformation.The greatest radial deformation appears at the end of the tube.The significant axial strain can be induced in CNTs with a large length (around 100 nm) even though the applied electric field is not strong enough.When SWCNTs are fixed at both ends,the radius of SWCNTs becomes small along the axial position.     

POSTBUCKLING OF CARBON NANOTUBES SUBJECTED TO CYCLIC LOAD

This study focuses on the postbuckling and the nonlinear behavior of single-walled carbon nanotubes subject to a cyclic axial compressive load through the use of molecular dynamic simulation based on the Tersoff-Brenner potential. It reveals the bifurcation behavior in the buckling process simulated with very fine time steps. In the whole cycle of nonlinear deformation, the carbon nanotubes exhibit the profound hysteretic behavior and the energy absorption ability. The molecular dynamics simulation indicates that the carbon nanotube behaves approximately as an ideal plastic spring when the cyclic strain is applied within the same postbuckling mode. In comparison, the theory of continuum mechanics gives a good prediction about the critical buckling strength, but only provides a rough estimation for the post-buckling behaviors.     

On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method

This paper deals with the capabilities of linear and nonlinear beam theories in predicting the dynamic response of an elastically supported thin beam traversed by a moving mass. To this end, the discrete equations of motion are developed based on Lagrange’s equations via reproducing kernel particle method (RKPM). For a particular case of a simply supported beam, Galerkin method is also employed to verify the results obtained by RKPM, and a reasonably good agreement is achieved. Variations of the maximum dynamic deflection and bending moment associated with the linear and nonlinear beam theories are investigated in terms of moving mass weight and velocity for various beam boundary conditions. It is demonstrated that for majority of the moving mass velocities, the differences between the results of linear and nonlinear analyses become remarkable as the moving mass weight increases, particularly for high levels of moving mass velocity. Except for the cantilever beam, the nonlinear beam theory predicts higher possibility of moving mass separation from the base beam compared to the linear one. Furthermore, the accuracy levels of the linear beam theory are determined for thin beams under large deflections and small rotations as a function of moving mass weight and velocity in various boundary conditions.     

VIBRATION OF FLUID-FILLED MULTI-WALLED CARBON NANOTUBES SEEN VIA NONLOCAL ELASTICITY THEORY

Vibration characteristics of fluid-filled multi-walled carbon nanotubes axe studied by using nonlocal elastic Fliigge shell model. Vibration governing equations of an N-layer carbon nanotube are formulated by considering the scale effect. In the numerical simulations, the effects of different theories, small-scale, variation of wavenumber, the innermost radius and length of double- walled and triple-walled carbon nanotubes are considered. Vibrational frequencies decrease with an increase of scale coefficient, the innermost radius, length of nanotube and effects of wall number are negligible. The results show that the cut-off frequencies can be influenced by the wall number of nanotubes.     

A CONTINUUM MODEL FOR AXIAL-STRAIN-INDUCED TORSION IN SINGLE-WALL CARBON NANOTUBES

Using the atomistic-based finite-deformation shell theory, we analytically investigate the coupling between the axial deformation and the torsion in single-wall carbon nanotubes. We find that the axial-strain-induced torsion（ASIT） response is limited only to chiral nanotubes. This response is affected by chiralities and radii of carbon nanotubes. Our results are similar to that of molecular dynamic simulations reported in the literatures.     

金沙江虎跳峡河段岸坡变形破坏的相关动力因子研究

石墨层和单臂碳纳米管都是以C---C共价键结合的. 在小变形条件下C---C键的势能可用谐和函 数来描述，这与梁单元的变形能具有相同的形式，因此可以用梁单元等效C---C键的作用. 提出了一种C---C键的等效梁单元有限元模型，该模型能够完备地替代谐和势描述C---C键的 伸长、面内键角变化、离面键角变化和扭转. 通过分析石墨层的典型受载情况得到了等效梁 单元的参数，以及等效梁单元参数与谐和势参数的关系，并用该模型计算了单臂碳纳米管的 杨氏模量和泊松比，计算结果为相关文献所验证.     

A structural mechanics approach for the analysis of carbon nanotubes

This paper presents a structural mechanics approach to modeling the deformation of carbon nanotubes. Fundamental to the proposed concept is the notion that a carbon nanotube is a geometrical frame-like structure and the primary bonds between two nearest-neighboring atoms act like load-bearing beam members, whereas an individual atom acts as the joint of the related load-bearing beam members. By establishing a linkage between structural mechanics and molecular mechanics, the sectional property parameters of these beam members are obtained. The accuracy and stability of the present method is verified by its application to graphite. Computations of the elastic deformation of single-walled carbon nanotubes reveal that the Young’s moduli of carbon nanotubes vary with the tube diameter and are affected by their helicity. With increasing tube diameter, the Young’s moduli of both armchair and zigzag carbon nanotubes increase monotonically and approach the Young’s modulus of graphite. These findings are in good agreement with the existing theoretical and experimental results.     

基于粗粒化方法的类超级碳纳米管自由振动研究

超级碳纳米管是在碳纳米管结构基础上，将每一根碳-碳键替换为碳纳米管而形成的新型结构。类超级碳纳米管是超级碳纳米管对应的宏观结构，在保持外观结构的基础上将尺度放大到宏观尺度。本文建立了类超级碳纳米管的粗粒化模型。基于粗粒化方法，研究了类超级碳纳米管的自由振动。分析了内外管半径以及长度对类超级碳纳米管振动行为的影响。与原结构有限元进行对比，结果表明粗粒化方法能有效的计算类超级碳纳米管的振动行为。     

基于粗粒化方法的类超级碳纳米管自由振动研究

超级碳纳米管是在碳纳米管结构基础上，将每一根碳-碳键替换为碳纳米管而形成的新型结构。类超级碳纳米管是超级碳纳米管对应的宏观结构，在保持外观结构的基础上将尺度放大到宏观尺度。本文建立了类超级碳纳米管的粗粒化模型。基于粗粒化方法，研究了类超级碳纳米管的自由振动。分析了内外管半径以及长度对类超级碳纳米管振动行为的影响。与原结构有限元进行对比，结果表明粗粒化方法能有效的计算类超级碳纳米管的振动行为。     

Analytical and numerical techniques to predict carbon nanotubes properties

In this paper, two different approaches for modeling the behaviour of carbon nanotubes are presented. The first method models carbon nanotubes as an inhomogeneous cylindrical network shell using the asymptotic homogenization method. Explicit formulae are derived representing Young’s and shear moduli of single-walled nanotubes in terms of pertinent material and geometric parameters. As an example, assuming certain values for these parameters, the Young’s modulus was found to be 1.71 TPa, while the shear modulus was 0.32 TPa. The second method is based on finite element models. The inter-atomic interactions due to covalent and non-covalent bonds are replaced by beam and spring elements, respectively, in the structural model. Correlations between classical molecular mechanics and structural mechanics are used to effectively model the physics governing the nanotubes. Finite element models are developed for single-, double- and multi-walled carbon nanotubes. The deformations from the finite element simulations are subsequently used to predict the elastic and shear moduli of the nanotubes. The variation of mechanical properties with tube diameter is investigated for both zig-zag and armchair configurations. Furthermore, the dependence of mechanical properties on the number of nanotubules in multi-walled structures is also examined. Based on the finite element model, the value for the elastic modulus varied from 0.9 to 1.05 TPa for single and 1.32 to 1.58 TPa for double/multi-walled nanotubes. The shear modulus was found to vary from 0.14 to 0.47 TPa for single-walled nanotubes and 0.37 to 0.62 for double/multi-walled nanotubes.     

单壁碳纳米管屈曲的原子/连续介质混合模型

用数学和力学研究所，上海 200072)//力学学报.--2004,36(6).--744～748 提供了一种运用原子/连续介质混合(hybrid atomic/continuum,HAC)方法解决纳米力学问题的思路. 通过在连续介质力学模型中引入利用分子力学方法获得物性参数，建立了预测单壁碳纳米管临界屈曲参数的HAC模型. 结果表明, HAC模型具有与连续介质力学模型可比拟的简洁性, 同时可表征纳米管微观结构特征对屈曲参数的影响. 计算结果表明，Zigzag纳米管的抗屈曲性能优于Armchair纳米管. 基于Tersoff-Brenner作用势的分子动力学结果证实了这一结论.     

Prediction of buckling characteristics of carbon nanotubes

In this paper, to investigate the buckling characteristics of carbon nanotubes, an equivalent beam model is first constructed. The molecular mechanics potentials in a C–C covalent bond are transformed into the form of equivalent strain energy stored in a three dimensional (3D) virtual beam element connecting two carbon atoms. Then, the equivalent stiffness parameters of the beam element can be estimated from the force field constants of the molecular mechanics theory. To evaluate the buckling loads of multi-walled carbon nanotubes, the effects of van-der Waals forces are further modeled using a newly proposed rod element. Then, the buckling characteristics of nanotubes can be easily obtained using a 3D beam and rod model of the traditional finite element method (FEM). The results of this numerical model are in good agreement with some previous results, such as those obtained from molecular dynamics computations. This method, designated as molecular structural mechanics approach, is thus proved to be an efficient means to predict the buckling characteristics of carbon nanotubes. Moreover, in the case of nanotubes with large length/diameter, the validity of Euler’s beam buckling theory and a shell model with the proper material properties defined from the results of present 3D FEM beam model is investigated to reduce the computational cost. The results of these simple theoretical models are found to agree well with the existing experimental results.     

A correlation for yield stress fluid flow through packed beds

Relatively few correlations are available for non-Newtonian fluid flows through packed beds, even though such fluids are frequently used in industry. In this paper, a correlation is presented for yield stress fluid flow through packed beds. The correlation is developed by introducing the yield stress model in place of the Newtonian model used in deriving Erguns equation. The resulting model has three parameters that are functions of the geometry and roughness of the particle surfaces. Two of the parameters can be deduced in the limit as the yield stress becomes negligible and the model reduces to Erguns equation for Newtonian fluids. The third model parameter is determined from experimental data. The correlation relates a defined friction factor to the dimensionless Reynolds and Hedstrom numbers and can be used to predict pressure drop for flow of a yield stress fluid through a packed bed of spherical particles. Conditions for flow or no-flow are also determined in the correlation. Comparison of model calculations, between a Newtonian and a yield stress fluid for flow penetration into a packed bed of spheres, shows the yield stress fluid initially performs similar to the Newtonian fluid, at large Reynolds numbers. At lower Reynolds numbers the yield stress effect becomes important and the flow rate significantly decreases when compared to the Newtonian fluid.