Flexural wave propagation in single-walled carbon nanotubes

339

146

In this paper, the small scaling parameter e0 of the nonlocal Timoshenko beam theory is calibrated for the free vibration problem of single-walled carbon nanotubes (SWCNTs). The calibration exercise is performed by using vibration frequencies generated from molecular dynamics simulations at room temperature. It was found that the calibrated values of e0 are rather different from published values of e0. Instead of a constant value, the calibrated e0 values vary with respect to length-to-diameter ratios, mode shapes, and boundary conditions of the SWCNTs. In addition, the physical meaning of the scaling parameter is explored. The results show that scaling parameter assists in converting the kinetic energy to the strain energy, thus enabling the kinetic energy to be equal to the strain energy. The calibrated e0 presented herein should be useful for researchers who are using the nonlocal beam theories for analysis of micro and nano beams/rods/tubes.

Section: MD Calibration Of E 0 and Discussioncontrasting

confidence: 45%

Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics

Duan¹,

Wang²,

Zhang³

2007

339

146

“…Since the first publication of nonlocal elasticity theory by Eringen and his associate, 1-3 many articles have been published on the application of this model in nanomechanics, particularly in the early 21st century, such as Peddieson et al, 4 Sudak, 5 Zhang et al, 6 Wang, 7,8 Lu et al, 9 Xu, 10 Wang and Hu, 11 etc. Almost all articles presented a simplified nonlocal beam model by assuming that the beam midplane is governed by a second-order ordinary differential equation, while in the transverse direction the classical Euler-Bernoulli beam model or the Timoshenko beam model applies.…”

Section: Introductionmentioning

confidence: 99%

Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams

Lim

Wang

2007

126

This article presents a complete and asymptotic representation of the one-dimensional nanobeam model with nonlocal stress via an exact variational principle approach. An asymptotic governing differential equation of infinite-order strain gradient model and the corresponding infinite number of boundary conditions are derived and discussed. For practical applications, it explores and presents a reduced higher-order solution to the asymptotic nonlocal model. It is also identified here and explained at length that most publications on this subject have inaccurately employed an excessively simplified lower-order model which furnishes intriguing solutions under certain loading and boundary conditions where the results become identical to the classical solution, i.e., without the small-scale effect at all. Various nanobeam examples are solved to demonstrate the difference between using the simplified lower-order nonlocal model and the asymptotic higher-order strain gradient nonlocal stress model. An important conclusion is the discovery of significant over-or underestimation of stress levels using the lower-order model, particularly at the vicinity of the clamped end of a cantilevered nanobeam under a tip point load. The consequence is that the design of a nanobeam based on the lower-order strain gradient model could be flawed in predicting the nonlocal stress at the clamped end where it could, depending on the magnitude of the small-scale parameter, significantly over-or underestimate the failure criteria of a nanobeam which are governed by the level of stress.

“…So far, no experi- 49 who adopted the second-order strain gradient constitutive relation, proposed e 0 = 0.288 for the flexural wave propagation study in a single walled carbon nanotube through the use of the nonlocal Timoshenko beam model and MD simulations. Eringen 50 himself proposed e 0 as 0.39 based on the matching of the dispersion curves via nonlocal theory for plane wave and Born-Karman model of lattice dynamics at the end of the Brillouin zone, ka = , where a is the distance between atoms and k is the wave number in the phonon analysis.…”

Section: Small Length Scale Effect and Vibrations Of Nonlocal CImentioning

confidence: 99%

Free vibration of nanorings/arches based on nonlocal elasticity

Wang

Duan

2008

This paper deals with the free vibration problem of nanorings/arches. The problem is formulated on the basis of Eringen's nonlocal theory of elasticity in order to allow for the small length scale effect. Exact vibration frequencies are derived for the nanorings/arches and the effects of small length scale, defects, and elastic boundary conditions are investigated. The small length scale effect lowers the vibration frequencies. The defects and the use of elastic boundary conditions ͑instead of fixed restraints͒ also significantly reduce the frequencies and alter the vibration mode shapes of circular rings/arches. The results presented should be useful to engineers who are designing nanorings/ arches for microelectromechanical and nanoelectromechanical devices.