Effects of Rippling Deformation and Mid-Plane Stretching on Non-linear Vibration for Embedded Carbon Nanotube

Carbon nanotubes (CNTs) were discovered by Iijima (1991). At present, CNT is the chief research subject in the area of the fullerene, and it is one of the most promising research fields in mechanics, physics, chemistry, and material science. Because of their novel electronic, mechanical, and other physical and chemical properties, CNT holds substantial promise as building blocks for nanoelectronics, nanodevices, nanocomposites, and nanomechanical sensors (Jiao et al., 2010; Ansari et al., 2010; Ma et al., 2010; Shokrieh and Rafiee, 2010; Wan et al., 2005; Ho et al., 2010; Mehdipour et al., 2011). Accurate theoretical models for vibrational behavior of CNTs are important for several reasons. For instance, natural frequencies of CNTs play an important role on nanomechanical resonators using them. In addition, the effective Young’s modulus of a nanotube may be determined indirectly from its measured natural frequencies or mode shapes if a sufficiently precise theoretical model is used. The molecular dynamics (MD) method simulates CNTs accurately. However, MD simulation is limited to systems with a small number of atoms (say, less than 1016) and remains time-consuming and expensive (Yaghmaei and Rafii-Tabar, 2009; Zhang et al., 2009; Gibson et al., 2007). For large-scale systems, continuum mechanics approach has widely and successfully modeled mechanical and vibrational characteristics of CNTs (Gibson et al., 2007; Fu et al., 2006; Ranjbartoreh et al., 2007). The continuum modeling approach needs much less computational effort and is much cheaper than the MD simulations and experimental verification.