{'fraction_non_alphanumeric': 0.0487062404870624, 'fraction_numerical': 0.007229832572298325, 'mean_word_length': 4.868303571428571, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Vibrations analysis of multi-walled carbon nanotubes by refined shell theories

Much research has been carried out on carbon nanotubes (CNTs) since their discovery in Japan by Ijima [1] in 1991. Carbon nanotubes exhibit superior mechanical properties and hold substantial promise by providing strong, light, and high toughness characteristics. CNTs are theoretically assumed to be made by rolling a graphene sheet about an opportune vector − → T . A chiral vector −→ C h, perpendicular to the vector −→ T , can be defined. The chiral vector is expressed in terms of base vectors − →a 1 and −→a 2 as: − → C h = n− →a 1 + m−→a 2 (1) where n and m are lattice translational indices. If the chiral vector is described by (n, 0) or (m,m) indices, the structure is zigzag or armchair, respectively [2]. There are two basic methods in simulating mechanical behavior of the nanostructures: atomistic-based modelling approaches and continuum ones. In the former, the vibrational behavior of CNTs is investigated by using an atomistic finite element model with beam elements and concentrated masses [2]. The beams simulate the interatomic covalent forces and the masses are located at the ends of the beams representing the carbon position: the computational effort for these methods does not permit simulation of real size multi-walled CNTs. For these reasons, continuum approaches are preferred to atomistic-based models. Among the various problem related to the modelling of CNTs, the present work considers the free vibrations response. Wang [3] and Aydogdu [4] studied vibrations problem in simple supported multi-walled carbon nanotubes (MWCNTs) via beam models (Timoshenko and Parabolic Shear Deformation Theory (PSDT)). He et alii used an elastic multiple shell model for the vibration and buckling analysis of MWCNTs [5], [6]; the effect of van der Waals (vdW) interaction on the vibration characteristics is considered by introducing a continuum model accounting for the dependence of vdW interaction coefficients on the change of interlayer spacing and the radii of tubes. In order to investigate the free vibrations of MWCNTs, Carrera’s Unified Formulation (CUF) [7] is extended in this work. MWCNTs are assumed as a multilayer cylindrical shell. The displacement components u, as well as their virtual variations δu, are modelled in the thickness direction z as: δu(α, β, z) = Fτ (z)δuτ (α, β) , u(α, β, z) = Fs(z)us(α, β) (2) where Fτ and Fs are the thickness functions, (α, β, z) is the orthogonal curvilinear reference system for a shell geometry, u = (u, v, w) is the vector containing the three displacements components.