Buckling Of Single-walled Carbon Nanotubes Research Articles

Axial buckling behavior of single-walled carbon nanotubes: Atomistic structural instability analysis

To reveal the atomistic scale mechanism of buckling of single-walled carbon nanotubes (SWCNTs) under axial compression, we carried out molecular dynamics (MD) simulation and atomistic structural instability (ASI) analysis. The ASI analysis enables us to reveal the development of ‘latent’ instability modes until buckling of structure, which cannot be clarified with MD alone. Euler-type buckling was found in relatively thin and long nanotube models, while buckling triggered by change of cross-sectional shape (i.e., radial buckling) was found in thick and short models. With growing radius, a crossover between the Euler-type and radial buckling modes was clearly found in the ASI analysis. Even when the deformation behavior was apparently the same, the buckling can be triggered by different instability modes. We also analyzed the structural instability using the Flügge theory based on linear continuum mechanics (LCM) to compare the results with the ASI analysis. Initial eigenvalues and development of instability modes were different between the ASI and LCM results, leading to a significant deviation of buckling strain and instability crossover points. We investigated the effect of chirality of CNT which is difficult to explain by LCM.

Effect of Material Nonlinearity on Spatial Buckling of Nanorods and Nanotubes

We show the importance of incorporating material nonlinearity for accurate determination of spatial buckling of nanorods and nanotubes. Both the nanorods and nanotubes are modeled as a special Cosserat rod whose nonlinear material laws are obtained using the recently proposed helical Cauchy-Born rule. We first present Euler buckling of solid diamond nanorods whose normalized buckling load, obtained from fully atomistic calculations, exhibits an interesting trend. The buckling load starts from unity at large aspect ratio of the nanorod, then as the aspect ratio is decreased, the buckling load increases slowly and finally decreases rapidly. We attribute this trend to material nonlinearity of the nanorod’s core at large compressive strain. We also discuss how surface stress affects buckling in nanorods. We then present the effect of compression and twist on buckling of single-walled carbon nanotubes. Interestingly, for highly twisted nanotubes, fully atomistic calculations show the first buckled mode to be different from a typical Euler buckling mode. Both the observations about nanorods and nanotubes are accurately replicated in the finite element special Cosserat rod simulation when the material nonlinearity is also incorporated. However, the simulation results exhibit completely different trend when only linear material laws are incorporated.

Flexural vibration and buckling analysis of single-walled carbon nanotubes using different gradient elasticity theories based on Reddy and Huu-Tai formulations

The aim of the present work is to analyze free flexural vibration and buckling of single-walled carbon nanotubes (SWCNT) under compressive axial loading based on different constitutive equations and beam theories. The models contain a material length scale parameter that can capture the size effect, unlike the classical Euler-Bernoulli or Reddy beam theory. The equations of motion of the Reddy and the Huu-Tai beam theories are reformulated using different gradient elasticity theories, including stress, strain and combined strain/inertia. The equations of motion are derived from Hamilton's principle in terms of the generalized displacements. Analytical solutions of free vibration and buckling are presented to bring out the effect of the nonlocal behavior on natural frequencies and buckling loads. The presented theoretical analysis is illustrated by a numerical example, and the results are qualitatively compared by another results.

Tuning the Shear Modulus of Single-Walled Carbon Nanotube by Feeding Van Der Waals Molecules

Torsional buckling of single-walled carbon nanotubes filled with light weight molecular via molecular dynamics is reported. The model accounts for the deformation of CNTs, and interactions among gas molecules; between gas and carbon atoms. The effect of particle loading is predicted to significantly change CNT’s critical torsional moment and stiffness. This is therefore an approach by which the torsional mechanical properties and oscillation frequencies of carbon nanotubes may be tuned. Importantly, the predicted changes in torsional siffness are unique relative to conventional linear elastic materials and are indicative of nonlinear oscillations due to nonlinear mechanical effects. CNTs subjects to large deformations reversibly switch into different morphological patterns. Each shape change corresponds to an abrupt release of energy and a singularity in the stress-strain curve. At higher torsional angle, van der Waals (VDW: He, Ar, H2) molecules reveal a stability effect on carbon nanotubes.

Application of the molecular mechanics method to simulation of buckling of single-walled carbon nanotubes

The molecular mechanics (MM) method based on the Modified Morse force field is used to determine the critical buckling parameters and post-critical deformation modes of single-walled carbon nanotubes (SWCNTs) twisted at the edges. Computer simulation of the buckling and post-critical deformation of nanotubes is performed using two versions of the MM method: the standard MM method and a mixed method combining the molecular mechanics and molecular structural mechanics approaches (MM/MSM method). Computer simulation shows that the MM/MSM method gives acceptable critical twisting angles, buckling modes, and post-critical deformed configurations of nanotubes, comparable to the critical twisting angles, modes, and configurations obtained using the standard MM method. More precisely, for the (10,10) armchair SCWNT 12.2919nm long, the solutions obtained by both methods are similar up to the beginning of fracture of the SCWNT (at a twisting angle up to 240° for each edge), and for the (10,0) zigzag SCWNT 16.8980nm long, the solutions obtained by both methods are similar up to the final twisting angle of 360° for each edge. The critical linear twisting angle for the (10,10) armchair SWCNT obtained in the present study is compared with the critical linear twisting angles obtained by other authors using the REBO, DREIDING, and MM3 force fields. It is concluded that the critical linear twisting angle obtained in the present study using the Modified Morse force field is unrealistically underestimated compared to the critical linear twisting angles obtained by other authors using the REBO, DREIDING, and MM3 force fields.

Continuum Shell Model for Buckling of Single-Walled Carbon Nanotubes with Different Chiral Angles

In this paper, an equivalent thick cylindrical shell model is proposed for the buckling analysis of short single-walled carbon nanotubes (SWCNTs) with allowance for different chiral angles. Extensive, molecular dynamics (MD) simulations are first performed using the adaptive intermolecular reactive bond order potential to determine the critical buckling loads/strains. The MD simulations buckling results are then used as reference solutions to calibrate the properties of the thick cylindrical shell model. Central to this development is the establishment of an empirical expression for the Young's modulus that is a function of both the diameter and the chiral angle of the SWCNT. For the shell model, we have assumed that the Poisson ratio ν = 0.19 and the shell thickness h = 0.066 nm . It will be shown that the proposed shell model furnishes good estimates of the critical buckling loads for SWCNTs with different chiral angles. The critical buckling strains are also evaluated from the critical buckling load with the aid of the stress–strain relation of SWCNTs.

Effect of higher-order deformation gradients on buckling of single-walled carbon nanotubes

A single-walled carbon nanotube (SWCNT) is treated as a tubular Bernoulli–Euler beam and the higher-order deformation gradients are involved to establish a higher-order multiscale beam model. The higher-order Cauchy–Born rule is used to calculate the deformation of bond vectors in the representative cell and the strain energy density is equivalent to the energy per unit surface area, calculated from the Brenner potential. On the basis of the classical Bernoulli–Euler beam theory, the second-order deformation gradients with respect to the axial direction are also considered. The total strain energy is expressed as an integral equation, in which all parameters are obtained by calculating the constitutive response around the circumference. The physical meaning of these parameters is discussed in detail. The global buckling of SWCNTs is studied, and the critical buckling force is obtained as the analytical formula for different boundary conditions. The critical buckling force is plotted against the tube chirality, tube radius and tube length, and it is discovered that the contribution of the higher-order terms rapidly becomes large when the tube radius or length is small enough.

Torsional Buckling of Single-Walled Carbon Nanotubes with Multi-Vacancy Defects

Torsional Buckling of Single-Walled Carbon Nanotubes with Multi-Vacancy Defects

Comparison of continuum-based and atomistic-based modeling of axial buckling of carbon nanotubes subject to hydrostatic pressure

Molecular dynamics simulation (MD) is used to investigate the effect of hydrostatic pressure on the axial buckling of single-walled carbon nanotube (SWCNT). Three different types of nanotube with small, medium and large aspect ratios (ARs) are considered. It is known that continuum-based models for axial buckling of nanotubes subjected to hydrostatic pressure are only applicable to those with small ARs, and that they are incapable of representing nanotubes with large ARs. In this paper, by employing MD simulations, the contribution of hydrostatic pressure to the axial buckling of nanotubes is studied for all ranges of ARs. In the case of small ARs, our MD results prove to be in agreement with those obtained from continuum-based models. However, concerning an SWCNT with a large AR, we found that the effect of hydrostatic pressure on the buckling properties is in total opposition to nanotubes with small ARs. This new trend is in contrast to the general conclusions, previously reported in continuum models, on how hydrostatic pressure affects the buckling properties. We conclude that the AR plays an essential role in determining the observed changes in the critical buckling loads when a nanotube is subjected to both an internal and an external hydrostatic pressure.

Rayleigh–Ritz axial buckling analysis of single-walled carbon nanotubes with different boundary conditions

Eringen's nonlocality is incorporated into the shell theory to include the small-scale effects on the axial buckling of single-walled carbon nanotubes (SWCNTs) with arbitrary boundary conditions. To this end, the Rayleigh–Ritz solution technique is implemented in conjunction with the set of beam functions as modal displacement functions. Then, molecular dynamics simulations are employed to obtain the critical buckling loads of armchair and zigzag SWCNTs, the results of which are matched with those of nonlocal shell model to extract the appropriate values of nonlocal parameter. It is found that in contrast to the chirality, boundary conditions have a considerable influence on the proper values of nonlocal parameter.

Sanders shell model for buckling of single-walled carbon nanotubes with small aspect ratio

Abstract In this paper, the buckling behaviour of single-walled carbon nanotubes (CNTs) is revisited by resorting to Donnell and Sanders shell models, which are put in parallel and shown to lead to very distinct results for CNTs with small aspect ratio (length-to-diameter). This paper demonstrates inability of the widely used Donnell shell theory while it shows the validity and accuracy of the Sanders shell theory in reproducing buckling strains and mode shapes of axially compressed CNTs with small aspect ratios. The results obtained by the later shell theory are close to molecular dynamics simulation results.The Sanders shell theory could capture correctly the length-dependent buckling strains of CNTs which the Donnell shell theory fails to achieve. In view of this study, researchers should adopt the Sanders thin shell theory from hereon instead of the Donnell theory when analyzing CNTs with small aspect ratios.

Buckling Temperature of a Single-Walled Carbon Nanotube Using Nonlocal Timoshenko Beam Model

The thermal buckling of single-walled carbon nanotube (SWCNT) subjected to a uniform temperature rise is studied using Timoshenko beam model, including the effects of transverse shear deformation and rotary inertia. The governing buckling equations of SWCNT are established on the basis of nonlocal continuum theory. An analytical solution to the equations is derived for the determination of critical buckling temperature. The solution can be further reduced to obtain the results of Euler beam model. According to the analysis, for mode 4, the Euler beam model overpredicts the critical buckling temperature by 57%, 13% and 3.5% for L/d = 10, 30 and 60, respectively. The Timoshenko beam model is able to predict the buckling temperature of the SWCNT at higher-order modes with small length-to-diameter ratios. In addition, the critical buckling temperature decreases with increasing the nonlocal value.

Atomistic finite element model for axial buckling of single-walled carbon nanotubes

An atomistic finite element model is developed to study the buckling behavior of single-walled carbon nanotubes with different boundary conditions. By treating nanotubes as space-frame structures, in which the discrete nature of nanotubes is preserved, they are modeled using three-dimensional elastic beam elements for the bonds and point mass elements for the atoms. The elastic moduli of the beam elements are determined via a linkage between molecular mechanics and structural mechanics. Based on this model, the critical compressive forces of single-walled carbon nanotubes with different boundary conditions, geometries as well as chiralities are obtained and then compared. It is indicated that at low aspect ratios, the critical buckling load of nanotubes decreases considerably with increasing aspect ratios, whereas at higher aspect ratios, buckling load slightly decreases as the aspect ratio increases. It is also indicated that increasing aspect ratio at a given radius results in the convergence of buckling envelops associated with armchair and zigzag nanotubes.

Adhesion-driven buckling of single-walled carbon nanotube bundles

Buckling of a thin single-walled carbon nanotube (SWNT) bundle that is partially bound on another straight free-standing SWNT bundle is reported. The buckling of the SWNT bundle is purely due to the adhesion interaction between two SWNT bundles. The deformation curvature of the buckled SWNT bundle is experimentally measured by transmission electron microscopy, and is theoretically modeled by a continuum model based on nonlinear elastica theory. Our results reveal that the binding strength of the bundle interface and the bulk elastic modulus of the SWNT bundle can be associated by its buckling curvature. Our results show that the bulk elastic moduli of the tested SWNT bundles are significantly lower than the Young’s modulus of individual SWNTs. The reported adhesion-driven nanotube buckling provides a potential new approach to quantify the elastic modulus and the binding strength of bundled nanotubes.

WITHDRAWN: Nonlocal Shell Model for Axial and Torsional Buckling of Single-Walled Carbon Nanotubes

WITHDRAWN: Nonlocal Shell Model for Axial and Torsional Buckling of Single-Walled Carbon Nanotubes

Mechanical buckling of single-walled carbon nanotubes: Atomistic simulations

Various geometric sizes and helical types (i.e., armchair, zigzag, and chiral) of single-walled carbon nanotubes (CNTs) are considered in molecular dynamics simulations in order to systematically examine the length-to-radius ratio and chirality effects on the buckling mechanism. The buckling strain is getting smaller as the CNT becomes slender for most nanotubes, which implies that the slender nanotubes have lower buckling resistance regardless of the radius of the CNTs. The applicability of the continuum buckling theory, which has been well developed for thin tubes, on predicting the buckling strain of the CNT is also examined. In general, the corresponding buckling strain and buckling type predicted by the continuum buckling theory could agree reasonably well with simulation results except at the transition region due to the competition of two buckling mechanisms.

Thermal Buckling of Carbon Nanotubes

In this article, we analyze that the thermal buckling of single-walled carbon nanotubes (SWCNTs) and double-walled carbon nanotubes (DWCNTs) are subjected to axial compression due to temperature rise using Euler beam and Timoshenko beam theories. According to the analysis, the Euler beam model overpredicts the critical buckling temperature, especially at relatively small length-to-diameter ratios and higher-order modes. Therefore, the Timoshenko beam theory is able to predict the critical buckling temperature for SWCNTs and DWCNTs. In addition, the effects of rotary inertia and shear deformation on SWCNTs are larger than DWCNTs.

ON INSTABILITY OF SINGLE-WALLED CARBON NANOTUBES WITH A VACANCY DEFECT

This technical note is concerned with the buckling of single-walled carbon nanotubes with one atomic vacancy. An elastic beam theory is developed to predict the buckling strain of defective CNTs, and the strain prediction via the continuum mechanics model is verified from comparison studies by molecular dynamics simulations. The results demonstrate the effectiveness of the continuum mechanics theory for longer CNTs. In addition, a local kink is revealed in the morphology of the buckling of shorter defective CNTs via molecular dynamics.

The buckling of single-walled carbon nanotubes upon bending: The higher order gradient continuum and mesh-free method

The bending buckling of single-walled carbon nanotubes (SWCNTs) is studied in the theoretical scheme of the higher order gradient continuum. The deformation of the underlying lattice vectors is approximated with an extended Cauchy–Born rule in which the effect of the second order deformation gradient is considered, and the continuum constitutive responses are determined by minimizing the energy of the representative cell. A mesh-free method is developed to implement the numerical modeling of SWCNTs, and their bending buckling behavior is numerically simulated with the developed method. The results are compared with those obtained with a full atomistic simulation, and it is revealed that the developed mesh-free method can accurately exhibit the bending deformation of SWCNTs. Different types of carbon nanotubes (CNTs) are studied, and the buckling mechanism is investigated.

Molecular dynamics simulations of the torsional instability of carbon nanotubes filled with hydrogen or silicon atoms

Torsional buckling of single-walled carbon nanotubes filled with hydrogen or silicon atoms via molecular dynamics is reported. The formation of a concentric layered atomic cluster in the tubes with the half filled atoms is observed at a lower torsional angle. However, a spill-over phenomenon accompanied by torsional buckling is found at a higher torsional angle that reveals the effect that being fully filled with hydrogen and silicon atoms has on the stability of carbon nanotubes. The higher critical torsional angle of carbon nanotubes fully filled with foreign atoms is attributed to the extra energy necessary to break the atomic chain structure.