A structural mechanics approach for the analysis of carbon nanotubes

Vibration and Elastic Buckling Analyses of Singlewalled Carbon Nanocones

“…Odegard et al [18] developed a model that links the molecular mechanics and solid mechanics, which is established by equating the molecular potential energy terms with the mechanical strain energy of a representative volume element of a continuum model. Li and Chou in the works [19][20][21][22] developed a similar approach to model CNTs and GSs similar to space-frame structures and investigated the elastic, vibrational and buckling characteristics of CNTs and/or GSs. Cantilevered and bridged single walled carbon nanotubes (SWCNTs) are taken into account to predict fundamental frequencies of SWCNTs [21] that are found to be in the range of 10 GHz -1.5 THz.…”

“…Li and Chou [22] reported that the buckling load in axial compression is higher than bending load. Tserpes and Papanikos [23] introduced an atomistic FE method based on the approach of Li and Chou [19] to model CNTs by using commercial FE codes; they identified the C-C bond thickness d, Young's modulus E and shear modulus G by using the AMBER force model [19]. By using the methods developed by Li and Chou [19] and Tserpes and Papanikos [23]; Hashemnia et al [24] and Sakhaee-Pour et al [25] examined natural frequencies and mode shapes of single-layered graphene sheets (SLGSs) and Sakhaee-Pour et al [25] predicted fundamental frequencies of SLGSs with equivalent lengths that are found to be in the range of 2.4 GHz -3.5 THz; Sakhaee-Pour et al [26] studied natural frequencies and mode shapes SWCNTs; Sakhaee-Pour [27] analyzed elastic buckling of SLGSs; Lee and Lee [28] studied vibrational behaviors of SWCNTs and SWCNCs, and predicted fundamental frequencies of SWCNCs below 20 GHz with a cone having the height of 20 Ǻ; Mir et al [29] studied natural frequencies and mode shapes of SWCNTs; Cheng et al [30] and Fan et al [31] examined mechanical properties of CNTs such as Young's modulus, shear modulus, natural frequency and buckling load; Avila et al [32] analyzed elastic and vibrational properties of GSs and CNTs; we studied twoand three-dimensional modal and transient analyses of SLGSs in [33].…”

“…It is noteworthy that transient dynamics analyses can be performed by using Newmark method due to the advantages of consistent mass matrix which does not yield singularity in numerical integrations. On the other hand, Scarpa and Adhikari [34,35] proposed a beam model considering the shear deformation effects and they found the C-C bond thickness d , Poisson's ratio ν, Young's modulus E and shear modulus G by using the AMBER force model constants [19]. Both of the models in Li and Chou [19] and Scarpa and Adhikari [34] yield the same deformation results as the structural mechanics stiffness constants in the AMBER force model are equal [33] if the corresponding element properties are used given in Table 1.…”

“…On the other hand, Scarpa and Adhikari [34,35] proposed a beam model considering the shear deformation effects and they found the C-C bond thickness d , Poisson's ratio ν, Young's modulus E and shear modulus G by using the AMBER force model constants [19]. Both of the models in Li and Chou [19] and Scarpa and Adhikari [34] yield the same deformation results as the structural mechanics stiffness constants in the AMBER force model are equal [33] if the corresponding element properties are used given in Table 1. Lee and Lee [28] used Timoshenko beam element formulations which include shear deformation effects but they employed Euler-Bernoulli beam element constants (i.e., see Table 1) [23]; this assumption affects the natural frequencies of SWCNC that are found to be lower than those of Euler-Bernoulli beam elements.…”

“…In addition, Lee and Lee [28] studied of vibrations of SWCNCs; however, they did not examine the effects of cone height and used Timoshenko beam formulations with Euler-Bernoulli beam parameters which may lead to lower modal frequencies than actual values. Motivated by these facts, vibration and elastic buckling analysis of SWCNCs are completed by using the MM based FE approach [19,23] where Euler-Bernoulli beam elements with consistent mass matrix are used. In vibration analysis of SWCNCs, free-free, free-clamped and clamped-clamped boundary conditions are considered.…”