C. Q. Ru

Vibration of an embedded multiwall carbon nanotube

This paper studies resonant frequencies and the associated vibrational modes of an individual multiwall carbon nanotube embedded in an elastic medium. The analysis is based on a multiple-elastic beam model [Phys. Rev. B. 62 (2000) 16962], which considers intertube radial displacements and the related internal degrees of freedom. New intertube resonant frequencies and the associated non-coaxial vibrational modes are calculated. The results show that non-coaxial intertube resonance will be excited at the higher resonant frequencies of multiwall carbon nanotubes, and thus the latter does not keep the otherwise concentric structure at ultrahigh frequencies. In particular, because non-coaxial intertube vibration will distort the concentric geometry of the multiwall carbon nanotubes, it could significantly affect some of their important physical (such as electronic and optical) properties.

Effect of van der Waals forces on axial buckling of a double-walled carbon nanotube

An elastic model is presented to study infinitesimal buckling of a double-walled carbon nanotube under axial compression. A simple formula is derived for the critical axial strain, which clearly indicates the role of the van der Waals forces between the inner and outer tubes characterized by two parameters. In particular, the analysis shows that inserting an inner tube into a single-walled nanotube does not increase the critical axial strain as compared to the single-walled nanotube under otherwise identical conditions, despite the fact that the total critical axial force of the double-walled nanotube could be increased due to an increase in the cross-sectional area.

Axially compressed buckling of pressured multiwall carbon nanotubes

Abstract 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.

Degraded axial buckling strain of multiwalled carbon nanotubes due to interlayer slips

A multiple-shell model is presented for infinitesimal axially compressed buckling of a multiwalled carbon nanotube embedded within an elastic matrix. In contrast to an existing single-shell model which treats the entire multiwalled nanotube as a singlelayer elastic shell, the present model assumes that each of the nested concentric tubes is an individual elastic shell and the deflections of all shells are coupled through the van der Waals interaction between adjacent nanotubes. By examining a doublewalled carbon nanotube, it is found that the change in interlayer spacing has a negligible effect on the axial buckling strain provided that the innermost radius is at least a few nanometers. Under this condition, a single equation is derived which determines the deflection of the multiwalled carbon nanotube, and it is shown that infinitesimal axial buckling of a N-walled carbon nanotubes is equivalent to that of a single layer elastic shell whose bending stiffness is approximately N times the effective bending...

Sound wave propagation in multiwall carbon nanotubes

This article studies transverse sound wave propagation in individual multiwall carbon nanotubes. The present model predicts that there exist (N-1) critical frequencies (within terahertz range) for an N-wall carbon nanotube. When the frequency is below all critical frequencies, vibrational mode is almost coaxial and the associated sound speed can be predicted satisfactorily by the existing single-elastic beam model. However, when the frequency is higher than at least one of the critical frequencies, non-coaxial vibrational modes emerge which propagate at various speeds significantly higher or lower than the speed predicted by the single-elastic beam model. Hence, terahertz sound waves in multiwall carbon nanotubes exhibit complex phenomena and are essentially noncoaxial. In particular, terahertz sound waves in multiwall carbon nanotubes propagate at various speeds, depending not only on the frequency but also on the noncoaxial vibrational modes.

Applicability and Limitations of Simplified Elastic Shell Equations for Carbon Nanotubes

This paper examines applicability and limitations of simplified models of elastic cylindrical shells for carbon nanotubes. The simplified models examined here include Donnell equations and simplified Flugge equations characterized by an uncoupled single equation for radial deflection. These simplified elastic shell equations are used to study static buckling and free vibration of carbon nanotubes, with detailed comparison to exact Flugge equations of cylindrical shells. It is shown that all three elastic shell models are in excellent agreement (with relative errors less than 5%) with recent molecular dynamics simulations for radial breathing vibration modes of carbon nanotubes, while reasonable agreements for various buckling problems have been reported previously for Donnell equations. For general cases of buckling and vibration, the results show that the simplified Flugge model, which retains mathematical simplicity of Donnell model, is consistently in better agreement with exact Flugge equations than Donnell model, and has a significantly enlarged range of applicability for carbon nanotubes. In particular, the simplified Flugge model is applicable for carbon nanotubes (with relative errors around 10% or less) in almost all cases of physical interest, including some important cases in which Donnell model results in much larger errors. These results are significant for further application of elastic shell models to carbon nanotubes because simplified shell models, characterized by a single uncoupled equation for radial deflection, are particularly useful for multiwall carbon nanotubes of large number of layers.@DOI: 10.1115/1.1778415#

Terahertz Vibration of Short Carbon Nanotubes Modeled as Timoshenko Beams

Short carbon nanotubes of smaller aspect ratio (say, between 10 and 50) are finding significant application in nanotechnology. This paper studies vibration of such short carbon nanotubes whose higher-order resonant frequencies fall within terahertz range. Because rotary inertia and shear deformation are significant for higher-order modes of shorter elastic beams, the carbon nanotubes studied here are modeled as Timoshenko beams instead of classical Euler beams. Detailed results are demonstrated for double-wall carbon nanotubes of aspect ratio 10, 20, or 50 based on the Timoshenko-beam model and the Euler-beam model, respectively. Comparisons between different single-beam or double-beam models indicate that rotary inertia and shear deformation, accounted for by the Timoshenko-beam model, have a substantial effect on higher-order resonant frequencies and modes of double-wall carbon nanotubes of small aspect ratio (between 10 and 20). In particular, Timoshenoko-beam effects are significant for both large-diameter and small-diameter double-wall carbon nanotubes, while double-beam effects characterized by noncoaxial deflections of the inner and outer tubes are more significant for small-diameter than large-diameter double-wall carbon nanotubes. This suggests that the Timoshenko-beam model, rather than the Euler-beam model, is relevant for terahertz vibration of short carbon nanotubes.

Vibration of a double-walled carbon nanotube aroused by nonlinear intertube van der Waals forces

Vibration of a double-walled carbon nanotube aroused by nonlinear interlayer van der Waals (vdW) forces is studied. The interlayer vdW forces as a nonlinear function are described by the interlayer spacing. The inner and outer carbon nanotubes are modeled as two individual elastic beams. Detailed results are demonstrated for double-walled carbon nanotubes (DWCNTs) with an aspect ratios of 10 and 20, based on the simply supported, fixed, or free end conditions, respectively. Harmonic balance method is used to analyze the relation between the amplitudes of deflection and the frequencies of coaxial and noncoaxial free vibrations. Our results indicate that the nonlinear factors of vdW forces have little effect on the coaxial free vibration, and that the deflection amplitudes increase rapidly with the increasing frequency, which are almost the same with those of the linear free vibration. On the other hand, the nonlinear factors of vdW forces have a great effect on noncoaxial free vibration. The relation betwe...

The Effects of Surface Elasticity on an Elastic Solid With Mode-III Crack: Complete Solution

We examined the effects of surface elasticity in a classical mode-III crack problem arising in the antiplane shear deformations of a linearly elastic solid. The surface mechanics are incorporated using the continuum based surface/interface model of Gurtin and Murdoch. Complex variable methods are used to obtain an exact solution valid everywhere in the domain of interest (including at the crack tip) by reducing the problem to a Cauchy singular integro-differential equation of the first order. Finally, we adapt classical collocation methods to obtain numerical solutions, which demonstrate several interesting phenomena in the case when the solid incorporates a traction-free crack face and is subjected to uniform remote loading. In particular, we note that, in contrast to the classical result from linear elastic fracture mechanics, the stresses at the (sharp) crack tip remain finite.

New phenomena concerning the effect of imperfect bonding on radial matrix cracking in fiber composites

Abstract In this paper we study the effects of imperfect bonding on stress intensity factors (SIFs) calculated at a radial matrix crack in a fiber (inclusion) composite subjected to various cases of mechanical loading. We use analytic continuation to adapt and extend the existing series methods to obtain series representations of deformation and stress fields in both the inclusion and the surrounding matrix in the presence of the crack. The interaction between the crack and the inclusion is demonstrated numerically for different elastic materials, geometries and varying degrees of bonding (represented by imperfect interface parameters) at the interface. Some qualitatively new phenomena are predicted for radial matrix cracking, specifically the influence of imperfect bonding at the inclusion–matrix interface on the direction of crack growth. For example, in the case of an inclusion perfectly bonded to the surrounding matrix, the SIF at the nearby crack tip is greater than that at the distant crack tip only when the inclusion is more compliant than the matrix. In contrast, the effects of imperfect bonding at the inclusion–matrix interface allow for the SIF at the nearby crack tip to be greater than that at the distant crack tip even when the inclusion is stiffer than the matrix . In fact, for any given case when the inclusion is stiffer than the matrix, we show that there is a corresponding critical value of the imperfect interface parameter below which a radial matrix crack grows towards the interface leading eventually to complete debonding. In particular, this critical value of the imperfect interface parameter tends to a non-zero finite value when the stiffness of the inclusion approaches infinity. To our knowledge, these results provide, for the first time, a clear quantitative description of the relationship between interface imperfections and the direction of propagation of radial matrix cracks.