A nonlocal meshless solution for flexural vibrations of double-walled carbon nanotubes

Keywords: Free transverse vibration Vibration mode shapes Double-walled carbon nanotubes (DWCNTs) Nonlocal Rayleigh beam theory Reproducing kernel particle method (RKPM) a b s t r a c t The true understanding of free vibration of double-walled carbon nanotubes (DWCNTs) plays a vital role in optimal design and dynamic control of such nanostructures. This paper is aimed to examine free flexural vibration of lengthy DWCNTs with arbitrary boundary conditions in the framework of nonlocal elasticity theory. The DWCNTs are embedded in an elastic medium and are subjected to initially axial forces. Equivalent continuum structures associated with the innermost and outermost tubes of the DWCNT are considered. The transverse and rotational interactions of the DWCNT with the surrounding elastic medium are also taken into account. The generalized equations of motion of lengthy DWCNTs are established based on the nonlocal Rayleigh beam theory. Seeking an analytical solution to the developed equations, particularly in their general form, is a very problematic task. As an alternative solution, an efficient numerical scheme is proposed. The effects of slenderness ratio, small-scale parameter, lateral and rotational stiffness of the surrounding matrix, and initially axial force on the first five natural frequencies of DWCNTs under different boundary conditions are comprehensively scrutinized. The discovery of carbon nanotubes (CNTs) has opened up a new world in the field of nanotechnology. Since the past three decades, four major forms of carbon materials have come to the real world; those are C 60 by Kroto et al. [1] in 1985, multi-walled carbon nanotube (MWCNT) by Iijima [2] in 1991, single-walled carbon nanotube (SWCNT) by Bethune et al. [3] in 1993, and carbon nanofiber. Subsequent studies revealed that CNTs integrate astonishing rigid and toughness properties, such as exceptionally high elastic properties, huge elastic strain, and fracture strain sustaining capability [4–8]. Beyond any exaggeration, CNTs are the strongest fibers recognized to date. The Young's modulus of a SWCNT is around 1 TPa, which is five times greater than steel, while its density is only 1200–1400 kg/m 3 [9]. It implies that the macro-scale structures made of CNTs will be extremely lighter and stronger than steel frames. Further studies reported that the frequency of flexural vibration of CNTs could reach the level of GHz [10,11] or even THz [12]. These evidences indicate that CNTs could be regarded as the most capable reinforcement materials for the next generation of high frequency engineered structures. Thereby, they have attracted …