Effects of dimensional parameters and various boundary conditions on axisymmetric vibrations of multi-walled carbon nanotubes using a continuum model

Axisymmetric vibrations of multi-walled carbon nanotubes (MWCNTs) with finite length are investigated in this paper. A multi-walled carbon nanotube is modeled with multiple elastic isotropic shells. Based on a continuum model and considering van der Waals forces between tubes, a two-dimensional finite element model is developed to obtain axisymmetric natural frequencies and mode shapes of MWCNTs. First, the axisymmetric vibrational characteristics of single-walled carbon nanotubes are investigated, and then, they are compared with those of MWCNTs. The effects of van der Waals forces on radial vibration of MWCNTs are also explained. Moreover, influences of end conditions on radial and axial natural frequencies of carbon nanotubes (CNTs) with wide range of length-to-thickness ratio are studied by considering the free-free (F-F) and simply supported (S-S) end conditions. Besides, it is focused on dependence of axisymmetric mode frequencies on dimensional parameters such as length, diameter, as well as number of layers of MWCNTs. Through this, explicit expressions are found for calculating the radial breathing mode and longitudinal mode frequencies of MWCNTs.     

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.     

Thermal vibration and apparent thermal contraction of single-walled carbon nanotubes

It is of fundamental value to understand the thermo-mechanical properties of carbon nanotubes. In this paper, by using molecular dynamics simulation, a systematic numerical investigation is carried out to explore the natural thermal vibration behaviors of single-walled carbon nanotubes and their quantitative contributions to the apparent thermal contraction behaviors. It is found that the thermo-mechanical behavior of single-walled carbon nanotubes is exhibited through the competition between quasi-static thermal expansion and dynamic thermal vibration, while the vibration effect is more prominent and induces apparent contraction in both radial and axial directions. With increasing temperature, the anharmonic interatomic potential helps to increase the bond length, which leads to thermally induced expansion. On the other hand, the higher structural entropy and vibrational entropy of the system cause the carbon nanotube to vibrate, and the apparent length of nanotube decreases due to various vibration modes. Parallel analytical and finite element analyses are used to validate the vibration frequencies and provide helpful insights. The unified multi-scale study has successfully decoupled and systematically analyzed both thermal expansion and contraction behaviors of single-walled carbon nanotube from 100 to 800 K, and obtained detailed information on various vibration modes as well as their quantitative contributions to the coefficient of thermal expansion in axial and radial directions. The results of this paper may provide useful information on the thermo-mechanical integrity of single-walled carbon nanotubes, and become important in practical applications involving finite temperature.     

Elongational and shear rheology of carbon nanotube suspensions

Rheological behavior of concentrated suspensions of chemical vapor deposition carbon nanotubes in uniaxial elongation and simple shear is studied experimentally and theoretically. Nanotubes are suspended in viscous host liquids—castor oil or its blends with n-decane. The elongational measurements are performed by analyzing self-thinning (due to surface tension effect) liquid threads of nanotube suspensions. A quasi-one-dimensional model is used to describe the self-thinning process, whereas corrections accounting for thread nonuniformity and necking are introduced a posteriori. The effects of nanotube concentration and aspect ratio, viscosity of the suspending liquid, and initial diameter of the self-thinning thread in uniaxial elongation are elucidated. The results for uniaxial elongation are compared with those for simple shear. The correspondence in the results of the shear and elongational measurements is addressed and interpreted. The results conform to the Herschel–Bulkley rheological constitutive equation (i.e., power law fluids with yield stress). However, the yield stress in elongation is about 40% higher than in simple shear flow, which suggests that the original Herschel–Bulkley model need modification with the yield stress being a function of the second invariant of the deviatoric stress tensor. The present effort is the first to study capillary self-thinning of Herschel–Bulkley liquids, which are exemplified here by suspensions of carbon nanotubes.     

A structural mechanics approach for predicting the mechanical properties of carbon nanotubes

Based on molecular mechanics, a structural mechanics model of carbon nanotubes (CNTs) was developed with special consideration given to the bending stiffness of the graphite layer. The potentials associated with the atomic interactions within a CNT were evaluated by the strain energies of beam elements which serve as structural substitutions of covalent bonds in a CNT. In contrast to the original model developed by Li and Chou (Int. J. Solids Struct. 40(10):2487–2499, 2003), in the current model the out-of-plane deformation (inversion) of the bond was distinguished from the in-plane deformation by considering a rectangular cross-section for the beam element. Consequently, the model is able to study problems where the effect of local bending of the graphite layer in a carbon nanotube is significant. A closed-form solution of the sectional properties of the beam element was derived analytically. The model was verified through the analysis of rolling a graphite sheet into a carbon nanotube. Using the present model, the buckling behavior of nanotubes under bending is simulated. The predicted critical bending angle agrees well with molecular dynamics simulations.     

Effect of the Stone–Wales defect on the structure and mechanical properties of single-wall carbon nanotubes in axial stretch and twist

The effect of the Stone–Wales defect due to the rotation of a pair of neighboring atoms on the equilibrium structure and mechanical properties of single-wall carbon nanotubes in axial stretch and twist is considered. The position of carbon atoms in a test section consisting of a number of repeated units hosting a solitary Stone–Wales defect is computed by minimizing the Tersoff–Brenner potential. The energy invested in the defect is found to decrease as the radius of the nanotube becomes smaller. Numerical computations for nanotubes with zigzag and armchair chiralities show that inclined, axial, and circumferential defect orientations have a strong influence on the mechanical response in axial stretch and twist. Stretching may cause the defect energy to become negative, revealing the possibility of spontaneous defect formation leading to failure. In some cases, stretching may eliminate the defect and purify the nanotube. When the tube is twisted around its axis, a neck develops at the location of the defect, signaling possible disintegration.     

Natural vibrations in a system of nanotubes

Natural vibrations in a system of parallel micro-and nanotubes attached horizontally to an elastic substrate are analyzed. It is shown that several first eigenfrequencies corresponding to flexural vibrations of a single nanotube can be identified with the use of the linear shell theory within the frequency spectrum of an “integrated system” consisting of a substrate and nanotubes. This allows the flexural rigidity of a single nanotube to be evaluated. The resultant conclusion is supported by finite-element modeling based on the three-dimensional theory of electroelasticity. Results of a modal analysis of gallium arsenide nanotubes are presented. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 160–171, March–April, 2008.     

Study of dynamic stability of the double-walled carbon nanotube under axial loading embedded in an elastic medium by the energy method

In this paper, the dynamic stability of single- and double-walled carbon nanotubes (SWCNT and DWCNT) under dynamic axial loading is investigated using the continuum mechanics model and the minimum total energy method. The natural frequencies of the SWCNT and the critical dynamic axial load of the SWCNT and DWCNT are obtained using the Rayleigh-Ritz method. The effects of the elastic medium and the van der Waals forces between the two layers in the DWCNT are taken into account using the Winkler model and Lennard-Jones theory, respectively. The effect of the small length scale is also considered using the Eringen Model. The critical dynamic axial load is increased by inserting an inner carbon nanotube (CNT) into an isolated CNT embedded in an elastic medium.     

Free vibration of carbon nanotube reinforced composite plate on point Supports using Lagrangian multipliers

Chebyshev polynomial functions are used in the Lagrangian multipliers method to study the free vibration characteristics of rectangular moderately thick composite plates reinforced with carbon nanotubes (CNTs). Plate is resting on point supports. Distribution of CNTs across the plate thickness is considered to be either uniform or functionally graded. Properties of the plate are obtained using a refined rule of mixtures approach which includes the efficiency parameters to capture the size dependent characteristics of the composite plate. Using a Ritz solution method, an eigenvalue problem is established which results in natural frequencies and mode shapes of the plate. Based on the developed solution method, number and position of point supports are arbitrary and also various boundary conditions may be assumed for the four edges of the plate. After performing comparison studies for isotropic homogeneous plates on point supports, parametric studies are provided to explore the vibration characteristics of the carbon nanotube reinforced composite plates on point supports. It is shown that, frequencies of the plate increase as the volume fraction of CNTs increases.     

Effect of small size on dispersion characteristics of wave in carbon nanotubes

Effect of small size on dispersion characteristics of waves in multi-walled carbon nanotubes is investigated using an elastic shell model. Dynamic governing equations of the carbon nanotube are formulated on the basis of nonlocal elastic theory. The relationship between wavenumber and frequency of wave propagation is obtained from the solution of the eigenvalue equations. The numerical results show that the dispersion characteristics of wave in the multi-walled carbon nanotube are affected by the small size. Effect of small size is not obvious for the smaller wavenumber, and it will arise and increase gradually with the increase of the wavenumber. Effect of the small size will decrease as the inner radius of carbon nanotubes increases. In addition, the explicit expressions of the cut-off frequencies are derived. The results show that the cut-off frequencies cannot be influenced by the small size of carbon nanotubes.     

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.     

Asymptotic frequencies of various damped nonlocal beams and plates

A striking difference between the conventional local and nonlocal dynamical systems is that the later possess finite asymptotic frequencies. The asymptotic frequencies of four kinds of nonlocal viscoelastic damped structures are derived, including an Euler–Bernoulli beam with rotary inertia, a Timoshenko beam, a Kirchhoff plate with rotary inertia and a Mindlin plate. For these undamped and damped nonlocal beam and plate models, the analytical expressions for the asymptotic frequencies, also called the maximum or escape frequencies, are obtained. For the damped nonlocal beams or plates, the asymptotic critical damping factors are also obtained. These quantities are independent of the boundary conditions and hence simply supported boundary conditions are used. Taking a carbon nanotube as a numerical example and using the Euler–Bernoulli beam model, the natural frequencies of the carbon nanotubes with typical boundary conditions are computed and the asymptotic characteristics of natural frequencies are shown.     

Molecular mechanics method applied to problems of stability and natural vibrations of single-layer carbon nanotubes

The molecular mechanics (MM) method is used to determine the frequencies and natural vibration shapes and to determine the buckling critical parameters and the postcritical deformation shapes of single-walled carbon nanotubes with twisted ends. The following two variants of the MM method are used: the standard MM method and the mixed method of molecular mechanics/molecular structure mechanics method (MM/MSM). Computer simulation shows that the MM/MSM method allows one to obtain acceptable values of frequencies and natural vibration shapes as well as of critical angles of twist, appropriate buckling modes, and postcritical deformation configurations of nanotubes compared with the same characteristics of nanotube free vibrations and buckling obtained by the standard MM method.     

Numerical model for composite material with polymer matrix reinforced by carbon nanotubes

Due to the high stiffness and strength, as well as their ability to act as conductors, carbon nanotubes are under intense investigation as fillers in polymeric materials. The nature of the carbon nanotube/polymer bonding and the curvature of the carbon nanotubes within the polymer have arisen as particular factors in the efficacy of the carbon nanotubes to actually provide any enhanced stiffness or strength to the nanocomposite. Here the effects of carbon nanotube curvature and interface interaction with the matrix on the nanocomposite stiffness are investigated using nanomechanical analysis. In particular, the effects of poor bonding and thus poor shear lag load transfer to the carbon nanotubes are studied. In the case of poor bonding, carbon nanotubes waviness is shown to enhance the composite stiffness.     

A single degree of freedom ‘lollipop’ model for carbon nanotube bundle formation

Current carbon nanotube (CNT) synthesis methods include the production of ordered, free-standing vertically aligned arrays, the properties of which are partially governed by interactions between adjacent tubes. Using material parameters determined by atomistic methods, here we represent individual CNTs by a simple single degree of freedom ‘lollipop’ model to investigate the formation, mechanics, and self-organization of CNT bundles driven by weak van der Waals interactions. The computationally efficient simple single degree of freedom model enables us to study arrays consisting of hundreds of thousands of nanotubes. The effects of nanotube parameters such as aspect ratio, bending stiffness, and surface energy, on formation and bundle size, as well as the intentional manipulation of bundle pattern formation, are investigated. We report studies with both single wall carbon nanotubes (SWCNTs) and double wall carbon nanotubes (DWCNTs) with varying aspect ratios (that is, varying height). We calculate the local density distributions of the nanotube bundles and show that there exists a maximum attainable bundle density regardless of an increase in surface energy for nanotubes with given spacing and stiffness. In addition to applications to CNTs, our model can also be applied to other types of nanotube arrays (e.g. protein nanotubes, polymer nanofilaments).     

Nonaxisymmetric Electroelastic Vibrations of a Hollow Cylinder with Radial Axis of Physicomechanical Symmetry

An approach is proposed to calculate the natural frequencies and modes of vibrations of cylindrically anisotropic piezoelectric cylinders. The frequency equation is derived, and the frequency spectrum is analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 7, pp. 68–72, July 2005.     

Physical constants for one type of nonlinearly elastic fibrous micro-and nanocomposites with hard and soft nonlinearities

Sets of physical constants are tabulated for three structural models of fibrous composites with fibers of four types: Thornel-300 carbon microfibers, graphite whiskers, carbon zigzag nanotubes, and carbon chiral nanotubes. The matrix for all the types of composites is always éPON-828 epoxy rosin (in some cases with polystyrene or pyrex additive). The values of the physical constants are commented on and used to study the distinctions in the evolution of three types of waves (plane longitudinal, plane transverse, and cylindrical) propagating in materials with soft and hard nonlinearities __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 47–60, December 2005.     

On the problem of integrability of the axisymmetric boundary-value problem of dynamics for an inhomogeneous anisotropic finite cylinder

A closed solution is obtained for the axisymmetric boundary-value problem of dynamics for a finite cylinder with exponential elasticity and inertial inhomogeneity and a certain relationship between elastic constants on the basis of correlations of the linear theory of elasticity of an anisotropic inhomogeneous body. The boundary conditions are arbitrary on the curvilinear surface and are given in mixed form on the ends. The method of finite integral transforms is employed. Specific cases for cylinders of transverscly isotropic and isotropic homogeneous material are discussed. Institute of Architecture and Civil Engineering, Samara, Russia. Translated from Prikladnaya Mekhanika, Vol. 35, No. 4, pp. 19–29, April, 1999.     

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.     

Influence of orthotropy parameters on the stress state of hollow cylinders with elliptic cross-section

An approach based on the use of discrete Fourier series is employed to analyze the stress state of orthotropic and transversely isotropic elliptical hollow cylinders __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 82–90, December 2007.