Refined 2D and Exact 3D Shell Models for the Free Vibration Analysis of Single- and Double-Walled Carbon Nanotubes

Abstract

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 as 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.

Boundary Conditions in 2D Numerical and 3D Exact Models for Cylindrical Bending Analysis of Functionally Graded Structures

“…This number of frequencies depends on the order used in each exponential matrix A * * and the number of mathematical layers employed for the shell curvature and/or the FGM law approximations. The results shown in [32][33][34][35][36][37][38][39][40] confirm that = 3 for the exponential matrix and = 100 for the mathematical layers are appropriate to always calculate the correct free frequencies for each structure, lamination sequence, number of layers, material, FGM law, and thickness value.…”

“…The three-dimensional exact shell model employed for the comparisons proposed in Section 4 has been elaborated in [32][33][34][35][36][37][38][39][40] for the free frequency analyses of single-layered and multilayered isotropic, orthotropic, composite, FGM, and sandwich plates and shells with constant radii of curvature and for the free vibrations of single-and double-walled carbon nanotubes including van der Waals interactions. The equilibrium equations proposed in the general orthogonal curvilinear coordinate system ( , , ) are valid for both plates and shells with constant radii of curvature.…”

“…U 1 (0) is the vector U defined at the bottom of the whole multilayered shell or plate (first layer 1 with 1 = 0). All the steps and details, here omitted for the sake of brevity, can be found in [32][33][34][35][36][37][38][39][40] where the proposed exact 3D shell model has been elaborated and tested for the first time for several applications. Several validations for the 3D exact shell model have been shown in [32][33][34][35][36][37][38][39][40].…”

Nonlinear free and forced vibrations of curved single walled carbon nanotube on a Pasternak elastic foundation

“…The governing di erential equation of curved singlewalled carbon nanotube on a Pasternak elastic foundation under harmonic excitation (see Figure 3) is: d 2 T (t) dt 2 + 1 T (t) + 2 T (t) 2 + 3 T (t) 3 = F cos( t):…”

“…Carbon nanotubes (CNTs) are perfect materials for a wide range of applications [3][4][5]. Carbon nanotubes possess excellent mechanical properties such as extremely high strength, sti ness, and resilience.…”