Nonlocal material properties of single-walled carbon nanotubes (original) (raw)
Related papers
2012
This paper investigates the size effects in the dynamic torsional response of single-walled carbon nanotubes (SWCNTs) by developing a modified nonlocal continuum shell model. The purpose is to facilitate the design of devices based on CNT torsion by providing a simple, accurate, and efficient continuum model that can predict the frequency of torsional vibrations and the propagation speed of torsional waves. To this end, dispersion relations of torsional waves are obtained from the proposed nonlocal model and compared to classical models. It is seen that the classical and nonlocal models predict nondispersive and dispersive behavior, respectively. Molecular dynamics simulations of torsional vibrations of (6,6) and (10,10) SWCNTs are also performed, the results of which are compared with the classical and nonlocal models and used to extract consistent values of the nonlocal elasticity constant. The superiority and accuracy of the nonlocal elasticity model in predicting the size-dependent dynamic torsional response of SWC-NTs are demonstrated.
Bending Vibrations of Carbon Nanotubes by Using Nonlocal Theory
Procedia Engineering, 2014
The topic of this article is the investigation of the free bending vibration of single-walled carbon nanotubes (SWCNTs). The underlying theory of frequency computation is based on nonlocal theory of beam bending. The solutions of frequency equation are given for four types of boundary conditions: clamped-free (C-F), simply-simply supported (S-S), clamped-simply supported (C-S) and clamped-clamped (C-C). The graphical outputs of computations are given only for limited number of cases that are able to characterize influences of parameters in question.
Influence of the nonlocal parameter on the transverse vibration of double-walled carbon nanotubes
Journal of the Mechanical Behavior of Materials, 2015
A high-order nonlocal continuum beam model is proposed, which can be applied to study the transverse vibrations of double-walled carbon nanotubes (DWCNTs), including those that could have initial deformations due to defects or external actions. A beam element is developed adopting Hermite cubic polynomials as shape functions, and mass and elastic stiffness matrix are presented. The influence of the nonlocal parameter on the vibrational properties of DWCNTs is studied. Using the proposed model, it was found that the nonlocal parameter has a strong influence on the natural frequencies.
Mechanical Sciences, 2018
In this work, a recently proposed nonlocal theory of bending is used in the analysis of eigenfrequencies of single-walled carbon nanotubes (SWCNTs). The nanotube vibration is analyzed in the form of a homogenized continuum. Classical treatment where a nanotube is approximated by standard beam theory, is replaced by the more sophisticated nonlocal method of material interactions where a nonlocal parameter is used. The eigenfrequencies are computed by the combination of analytical as well as numerical methods for four different carbon nanotube (CNT) supports. Various types of supports are considered for the analysis: fixed-simply supported, fixed-free, simply-simply supported and fixed-fixed. Due to the huge amount of computed data, only outcomes of eigenfrequency computations for the nanobeams of armchair type with fixed and simply supported ends, and different nonlocal parameters are represented in the form of graphs at the end of the article. The study shows how the nanotube eigenfrequencies depend on nonlocal parameters as well as on the length and diameter of CNTs. The obtained results are in good agreement with the results published in papers which were gained by different procedures.
Int. J. Eng. Appl. Sciences, 2009
Static analysis of carbon nanotubes (CNT) is presented using the nonlocal Bernoulli-Euler beam theory. Differential quadrature (DQ) method is used for bending analysis of numerical solution of carbon nanotubes. Numerical results are presented and compared with that available in the literature. Deflection and bending moment are presented for different boundary conditions. It is shown that reasonable accurate results are obtained.
Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory
Nonlinear free vibration of single-walled carbon nanotubes (SWCNTs) is studied in this paper based on von Ká rmá n geometric nonlinearity and Eringen's nonlocal elasticity theory. The SWCNTs are modeled as nanobeams where the effects of transverse shear deformation and rotary inertia are considered within the framework of Timoshenko beam theory. The governing equations and boundary conditions are derived by using the Hamilton's principle. The differential quadrature (DQ) method is employed to discretize the nonlinear governing equations which are then solved by a direct iterative method to obtain the nonlinear vibration frequencies of SWCNTs with different boundary conditions. Zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs are considered in numerical calculations and the elastic modulus is obtained through molecular mechanics (MM) simulation. A detailed parametric study is conducted to study the influences of nonlocal parameter, length and radius of the SWCNTs and end supports on the nonlinear free vibration characteristics of SWCNTs.
Wave Properties of nonlocal Euler beam model in Carbon nanotubes
2014
— In this paper Eringen’s nonlocal theory of elasticity is applied to study the wave properties of single walled carbon nanotubes. The effects of wave number and nonlocal scaling parameter (e0a) on the frequency, phase velocity and group velocity are discussed for a particular material of carbon nanotube.
Applied Mechanics
The main objective of this paper is to study the free vibration of a Timoshenko–Ehrenfest single-walled carbon nanotube based on the nonlocal theory and taking surface effects into account. To model these effects on frequency response of nanotubes, we use Eringen’s nonlocal elastic theory and surface elastic theory proposed by Gurtin and Murdoch to modify the governing equation. A modified version of Timoshenko nonlocal elasticity theory—known as the nonlocal truncated Timoshenko beam theory—is put forth to investigate the free vibration behavior of single-walled carbon nanotubes (SWCNTs). Using Hamilton’s principle, the governing equations and the corresponding boundary conditions are derived. Finally, to check the accuracy and validity of the proposed method, some numerical examples are carried out. The impacts of the nonlocal coefficient, surface effects, and nanotube length on the free vibration of single-walled carbon nanotubes (SWCNTs) are evaluated, and the results are compar...
Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes
Journal of the Mechanics and Physics of Solids, 2008
This paper investigates the transverse and torsional wave in single-and double-walled carbon nanotubes (SWCNTs and DWCNTs), focusing on the effect of carbon nanotube microstructure on wave dispersion. The SWCNTs and DWCNTs are modeled as nonlocal single and double elastic cylindrical shells. Molecular dynamics (MD) simulations indicate that the wave dispersion predicted by the nonlocal elastic cylindrical shell theory shows good agreement with that of the MD simulations in a wide frequency range up to the terahertz region. The nonlocal elastic shell theory provides a better prediction of the dispersion relationships than the classical shell theory when the wavenumber is large enough for the carbon nanotube microstructure to have a significant influence on the wave dispersion. The nonlocal shell models are required when the wavelengths are approximately less than 2.36 Â 10 À9 and 0.95 Â 10 À9 m for transverse wave in armchair (15,15) SWCNT and torsional wave in armchair (10,10) SWCNT, respectively. Moreover, an MD-based estimation of the scale coefficient e 0 for the nonlocal elastic cylindrical shell model is suggested. Due to the small-scale effects of SWCNTs and the interlayer van der Waals interaction of DWCNTs, the phase difference of the transverse wave in the inner and outer tube can be observed in MD simulations in wave propagation at high frequency. However, the van der Waals interaction has little effect on the phase difference of transverse wave.
International Journal of Advanced Structural Engineering, 2012
The transverse vibration of a single-walled carbon nanotube (SWCNT) with light waviness along its axis is modeled by the nonlocal Euler-Bernoulli and Timoshenko beam theory. Unlike the Euler-Bernoulli beam model (EBM), the effects of transverse shear deformation and rotary inertia are considered within the framework of the Timoshenko beam model (TBM). The surrounding elastic medium is described as both Winkler-type and Pasternak-type foundation models. The governing equations are derived using Hamilton's principle, and the Galerkin method is applied to solve these equations. According to this study, the results indicate that the frequency calculated by TBM is lower than that obtained by EBM. Detailed results show that the importance of transverse shear deformation and rotary inertia become more significant for stocky SWCNTs with clamped-clamped boundary conditions. Moreover, the influences of the amplitude of waviness, nonlocal parameter, medium constants, boundary conditions and aspect ratio are analyzed and discussed. It is shown that waviness in the curved SWCNT causes an obvious increase in the natural frequency in comparison with the straight SWCNT, especially for a compliant medium, pinned-pinned boundary condition, short SWCNT and large nonlocal coefficient.