Postbuckling of double-walled carbon nanotubes with temperature dependent properties and initial defects under combined axial and radial mechanical loads

Buckling and postbuckling analysis is presented for a double-walled carbon nanotube subjected to combined axial and radial loads in thermal environments. The analysis is based on a continuum mechanics model in which each tube of a double-walled carbon nanotube is described as an individual orthotropic shell with presence of van der Waals interaction forces and the interlayer friction is negligible between the inner and outer tubes. The governing equations are based on higher order shear deformation shell theory with a von Kármán-Donnell-type of kinematic nonlinearity and include thermal effects. Temperature-dependent material properties, which come from molecular dynamics simulations, and initial point defect, which is simulated as a dimple on the tube wall, are both taken into account. A singular perturbation technique is employed to determine the interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, double-walled carbon nanotubes subjected to combined axial and radial mechanical loads under different sets of thermal environments. The results reveal that temperature change only has a small effect on the postbuckling behavior of the double-walled carbon nanotube. The axially-loaded double-walled carbon nanotube subjected to radial pressure has an unstable postbuckling path, and the structure is imperfection–sensitive. In contrast, the pressure-loaded double-walled carbon nanotube subjected to axial compression has a very weak “snap-through” postbuckling path, and the structure is virtually imperfection–insensitive.     

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.     

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.     

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 finite element analysis for vibrations of double-walled carbon nanotubes

The large-amplitude free vibration analysis of double-walled carbon nanotubes embedded in an elastic medium is investigated by means of a finite element formulation. A double-beam model is utilized in which the governing equations of layers are coupled with each other via the van der Waals interlayer forces. Von-Karman type nonlinear strain-displacement relationships are employed where the ends of the nanotube are constrained to move axially. The amplitude-frequency response curves for large-amplitude free vibrations of single-walled and double-walled carbon nanotubes with arbitrary boundary conditions are graphically illustrated. The effects of material constant of the surrounding elastic medium and the geometric parameters on the vibration characteristics are investigated. For a double-walled carbon nanotube with different boundary conditions between inner and outer tubes, the nonlinear frequencies are obtained apparently for the first time. Comparison of the results with those from the open literature is made for the amplitude-frequency curves where possible. This comparison illustrates that the present scheme yields very accurate results in predicting the nonlinear frequencies.     

Buckling and Postbuckling of Laminated Thin Cylindrical Shells under Hygrothermal Environments

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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.     

冲击扭矩作用下碳纳米管动力屈曲的研究

为了研究碳纳米管在冲击扭矩作用下的动力屈曲,采用了连续模型将碳纳米管模拟成半无限长的弹性连续圆柱壳。将冲击扭矩作用下碳纳米管的动力屈曲问题归结为由于扭转应力波传播导致的分叉问题,此分叉问题被化为一个非线性方程组的求解。最后进行了数值分析,讨论了碳纳米管的不同参数对动力屈曲的影响,发现碳纳米管有极强的抗冲击性,临界屈曲剪应力可高达几百吉帕。     

The effects of chirality and boundary conditions on the mechanical properties of single-walled carbon nanotubes

The effects of chirality and boundary conditions on the elastic properties and buckling behavior of single-walled carbon nanotubes are investigated using atomistic simulations. The influences of the tube length and diameter are also included. It is found that the elastic properties of carbon nanotubes at small deformations are insensitive to the tube chirality and boundary conditions during compression. However, for large deformations occurred upon both compression and bending, the tube buckling behavior is shown to be very sensitive to both tube chirality and boundary conditions. Therefore, while the popular continuum thin shell model can be successfully applied to describe nanotube elastic properties at small deformation such as the Young’s modulus, it cannot correctly account for the buckling behavior. These results may allow better evaluation of nanotube mechanical properties via appropriate atomistic simulations.     

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。     

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.     

Postbuckling analysis of axially loaded functionally graded cylindrical panels in thermal environments

A postbuckling analysis is presented for a functionally graded cylindrical panel of finite length subjected to axial compression in thermal environments. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded cylindrical panel are based on Reddy’s higher order shear deformation shell theory with a von Kármán–Donnell-type of kinematic nonlinearity and including thermal effects. Two cases of the in-plane boundary conditions are considered. The nonlinear prebuckling deformations and initial geometric imperfections of the panel are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical panels under axial compression. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of axially loaded, perfect and imperfect, functional graded cylindrical panels with two constituent materials and under different sets of thermal environments. The influences played by temperature rise, volume fraction distributions, the character of in-plane boundary conditions, transverse shear deformation, panel geometric parameters, as well as initial geometric imperfections are studied.     

Axially compressed buckling of pressured multiwall carbon nanotubes

This paper studies axially compressed buckling of an individual multiwall carbon nanotube subjected to an internal or external radial pressure. The emphasis is placed on new physical phenomena due to combined axial stress and radial pressure. According to the radius-to-thickness ratio, multiwall carbon nanotubes discussed here are classified into three types: thin, thick, and (almost) solid. The critical axial stress and the buckling mode are calculated for various radial pressures, with detailed comparison to the classic results of singlelayer elastic shells under combined loadings. It is shown that the buckling mode associated with the minimum axial stress is determined uniquely for multiwall carbon nanotubes under combined axial stress and radial pressure, while it is not unique under pure axial stress. In particular, a thin N-wall nanotube (defined by the radius-to-thickness ratio larger than 5) is shown to be approximately equivalent to a single layer elastic shell whose effective bending stiffness and thickness are N times the effective bending stiffness and thickness of singlewall carbon nanotubes. Based on this result, an approximate method is suggested to substitute a multiwall nanotube of many layers by a multilayer elastic shell of fewer layers with acceptable relative errors. Especially, the present results show that the predicted increase of the critical axial stress due to an internal radial pressure appears to be in qualitative agreement with some known results for filled singlewall carbon nanotubes obtained by molecular dynamics simulations.     

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.     

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。     

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.     

Axial buckling of multi-walled carbon nanotubes and nanopeapods

In this paper, we investigate both pre- and post-buckling behaviors of multi-walled carbon nanotubes and multi-walled carbon nanopeapods by incorporating into the applied forces of a prescribed beam equation both van der Waals interactions between the adjacent walls of the nanotubes and the interactions between the fullerenes and the inner wall of the nanotube. Two beam theories are employed. First, we utilize Donnell’s equilibrium equation to derive an axial stability condition for the multi-walled carbon nanotubes and multi-walled carbon nanopeapods. We then determine analytically the critical forces for single-walled and double-walled nanotubes and nanopeapods. Given the outer nanotube of a fixed radius, we observe that the critical force and strain derived from the axial buckling stability criterion decrease as a result of the molecular interactions between the adjacent layers of the nanotubes and the molecular interactions between the embedded fullerenes and the inner carbon nanotube, which is in agreement with existing literature. Next, we utilize an Euler–Bernoulli beam equation incorporating the curvature effect to obtain the post-buckled axial bending displacement for the multi-walled nanotubes and nanopeapods. We find that the interactions between molecules generate an inward force, which tends to resist any applied forces. While the inward force induced by the fullerenes to the inner wall of the nanotube vanishes as we increase the applied force, the inward force induced by the layers increases as the applied force increases. The main contribution of this paper is the incorporation of both van der Waals interactions and the curvature effect into prescribed beam theories to accurately measure the critical forces and the buckled displacements of multi-walled nanotubes and nanopeapods subject to a small external force. Our analysis is relevant to future nano devices, such as biological sensors and measuring devices for small forces arising from electrical charges or Casimir forces.     

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.     

Thermomechanical postbuckling of composite laminated cylindrical shells with local geometric imperfections

The effect of local geometric imperfections on the buckling and postbuckling of composite laminated cylindrical shells subjected to combined axial compression and uniform temperature loading was investigated. The two cases of compressive postbuckling of initially heated shells and of thermal postbuckling of initially compressed shells are considered. The formulations are based on a boundary layer theory of shell buckling, which includes the effects of the nonlinear prebuckling deformation, the nonlinear large deflection in the postbuckling range and the initial geometric imperfection of the shell. The analysis uses a singular perturbation technique to determine buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of cross-ply laminated cylindrical shells with or without initial local imperfections, from which results for isotropic cylindrical shells follow as a limiting case. Typical results are presented in dimensionless graphical form for different parameters and loading conditions.