The present paper talks about the free vibration analysis of simply supported Single-and Double-Walled Carbon Nanotubes (SWCNTs and DWCNTs). Refined 2D Generalized Differential Quadrature (GDQ) shell methods and an exact 3D shell model are compared. A continuum approach (based on an elastic three-dimensional shell model) is used for natural frequency investigation of SWCNTs and DWCNTs. SWCNTs are defined as isotropic cylinders with an equivalent thickness and Young modulus. DWCNTs are defined

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... two concentric isotropic cylinders (with an equivalent thickness and Young modulus) which can be linked by means of the interlaminar continuity conditions or by means of van der Waals interactions. Layer wise approaches are mandatory for the analysis of van der Waals forces in DWCNTs. The effect of van der Waals interaction between the two cylinders is shown for different DWCNT lengths, diameters and vibration modes. The accuracy of beam models and classical 2D shell models in the free vibration analysis of SWCNTs and DWCNTs is also investigated. Technologies 2015, 3 260 Keywords: single-walled carbon nanotube; double-walled carbon nanotube; free vibrations; van der Waals interaction; three-dimensional shell model; exact solution; two-dimensional shell models; generalized differential quadrature method Introduction Research about Carbon NanoTubes (CNTs) has demonstrated their exceptional mechanical properties [1] . In view of these exceptional mechanical properties (the elastic modulus has been shown to be greater than 1 TPa and the tensile strength exceeds that of steel by over one order of magnitude), CNTs are considered to be ideal reinforcements in composite structures [2] . CNTs are closed graphene sheets with a cylindrical shape, they were discovered in Japan by Iijma [3] in 1991. When a continuum elastic model is applied to CNT analysis, it is of central importance to accurately quantify the elastic properties of Single-Walled CNTs (SWCNTs) and Double-Walled CNTs (DWCNTs) [4] . The behavior of CNTs can be simulated by means of three different basic methods [5]: Molecular Dynamic (MD) simulations, atomistic-based modelling approaches and continuum approaches. In the MD approaches, the simulations are based on the definition of an appropriate potential energy function (e.g., Tersoff-Brenner or Lennard-Jones functions) [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] . In the atomistic-based modelling approaches, CNTs are investigated by means of an atomistic finite element model with beam elements and concentrated masses. The beams simulate the interatomic covalent forces and the masses (which are located at the ends of the beams) represent the carbon positions [16] [17] [18] [19] [20] [21] . The continuum approaches are based on the assumption that carbon nanotubes (which have a discrete molecular structure) are continuum isotropic elastic cylinders which can be analyzed via beam or shell models. When a continuum elastic model is applied to CNT analysis, it is of central importance to accurately quantify the elastic properties of SWCNTs and DWCNTs [22, 23] . The high computational cost of the MD simulations and the atomistic-based modelling approaches in the case of complex CNT networks does not allow fast analyses. Continuum approaches are preferred to MD and atomistic-based models because of their better computational cost. In order to apply a continuum model, it is necessary to correctly define effective CNT wall thickness, Young modulus and Poisson ratio even if a carbon nanotube has a discrete molecular structure. Extensive studies have been conducted to analyze this feature [24] [25] [26] [27] . A final conclusion has not yet been reached, as demonstrated by the different thickness and Young modulus values shown in the papers analyzed in this contribution. The equivalent properties discussed in the papers proposed in the following review are not always the same for a given elastic stiffness. Continuum approaches to analyze free vibrations of single-walled and multi-walled carbon nanotubes can use beam or shell models. The most important investigations about beam models for Single-Walled Carbon Nanotubes (SWCNTs) can be found in [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] . Further works about Double-Walled Carbon Nanotube (DWCNT) simulations via beam models can be found in [42] [43] [44] [45] [46] [47] [48] [49] [50] . Shell models are used for the analysis of SWCNTs and DWCNTs in [51] [52] [53] [54] [55] [56] [57] [58] and [58] [59] [60] [61] [62] [63] [64] [65] , respectively. The use of shell models for the vibration analysis of CNTs is usually more complicated than the use of beam models, but shell models allow the analysis of CNTs with small length/diameter ratios.