A. Mioduchowski

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.

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.

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.

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.

Free vibration of multiwall carbon nanotubes

A multiple-elastic shell model is applied to systematically study free vibration of multiwall carbon nanotubes (MWNTs). Using Flugge [Stresses in Shells (Springer, Berlin, 1960)] equations of elastic shells, vibrational frequencies and associated modes are calculated for MWNTs of innermost radii 5 and 0.65 nm, respectively. The emphasis is placed on the effect of interlayer van der Waals (vdW) interaction on free vibration of MWNTs. Our results show that the interlayer vdW interaction has a crucial effect on radial (R) modes of large-radius MWNTs (e.g., of the innermost radius 5 nm), but is less pronounced for R modes of small-radius MWNTs (e.g., of the innermost radius 0.65 nm), and usually negligible for torsional (T) and longitudinal (L) modes of MWNTs. This is attributed to the fact that the interlayer vdW interaction, characterized by a radius-independent vdW interaction coefficient, depends on radial deflections only, and is dominant only for large-radius MWNTs of lower radial rigidity but less pron...

A circular inclusion with inhomogeneously imperfect interface in plane elasticity

A general method is presented for the rigorous solution of a circular inclusion embedded within an infinite matrix in plane elastostatics. The bonding at the inclusion-matrix interface is considered to be imperfect with the assumption that the interface imperfections are circumferentially inhomogeneous. Using analytic continuation, the basic boundary value problem for four analytic functions is reduced to two coupled first order differential equations for two analytic functions. The resulting closed-form solutions include a finite number of unknown constants determined by analyticity and certain other auxiliary conditions. The method is illustrated using a particular class of inhomogeneous interface. The results from these calculations are compared to the corresponding results when the imperfections in the interface are circumferentially homogeneous. These comparisons illustrate, for the first time, how the circumferential variation of the parameter describing the imperfection has a pronounced effect on the average stresses induced within the inclusion.

Uniformity of Stresses Within a Three-Phase Elliptic Inclusion in Anti-Plane Shear

In this paper, we show that a three-phase elliptic inclusion under uniform remote stress and eigenstrain in anti-plane shear admits an internal uniform stress field provided that the interfaces are two confocal ellipses. The exact closed-form solution is used to quantify the effect of the interphase layer on the residual stresses within the inclusion and the dependency of this effect on the aspect ratio of the elliptic inclusion.

An elliptic inclusion with imperfect interface in anti-plane shear

Abstract A semi-analytic solution is developed for the problem associated with an elliptic inclusion embedded within an infinite matrix in anti-plane shear. The bonding at the inclusion-matrix interface is assumed to be homogeneously imperfect. The interface is modeled as a spring (interphase) layer with vanishing thickness. The behaviour of this interphase layer is based on the assumption that tractions are continuous but displacements are discontinuous across the interface. Complex variable techniques are used to obtain infinite series representations of the stresses induced within the inclusion. The results obtained demonstrate how the (non-uniform) stress field and the average stresses inside the inclusion vary with the aspect ratio of the inclusion and the parameter describing the imperfect interface. In addition, it is shown that, in some cases (depending on the aspect ratio of the ellipse), it is possible to identify specific values of the interface parameter which correspond to maximum peak stress along the interface.