Free Vibration of Single-Walled Carbon Nanotubes Using Nonlocal Truncated Timoshenko-Ehrenfest Beam Theory

Abstract

Carbon nanotubes with their outstanding mechanical, physical and electrical properties have stimulated a significant amount of scientific and technological research due to their uniqueness compared to conventional materials. As a result, an extensive study on their mechanical properties has been conducted, and the static and dynamic behavior of single- walled and multi-walled carbon nanotubes has been examined using Euler-Bernoulli and Timoshenko beam models. The main objective of this paper is to study the free vibration behaviour of single-walled carbon nanotubes (SWCNT) using the nonlocal truncated Timoshenko beam theory. According to the Hamilton principle, the equation of motion of Timoshenko single-walled carbon nanotubes is calculated taking into account the truncated theory; and the general corresponding boundary conditions are derived. Finally, some numerical examples are performed to evaluate the effects of the nonlocal coefficient and the length of the nanotube. The obtained results are validated by comparing them with those found in the literature, and they show the accuracy and efficiency of the developed model. Particularly, the results demonstrate that the present formulation is highly efficient and capable of satisfactorily describing the behavior of nanobeams.

Dynamic Analysis of a Timoshenko–Ehrenfest Single-Walled Carbon Nanotube in the Presence of Surface Effects: The Truncated Theory

“…In the present study, the free vibration of a Timoshenko-Ehrenfest single-walled carbon nanotube, based on the nonlocal theory and taking surface effects into account, has been developed. A modified version of Timoshenko's nonlocal elasticity theory-known as the nonlocal truncated Timoshenko beam theory-is put forth to investigate the free vibration behavior of single-walled carbon nanotubes, as indicated in [10,32]. Starting from the Timoshenko truncated theory for the beam model and using the geometric and variational approach, the truncated Timoshenko-Ehrenfest single-walled carbon nanotube theory has been derived.…”

“…Consider the ashlar nanotube element in Figure 2; according to the paper [10], the equilibrium of the the applied loads at the abscissa z-axis is imposed: equilibrium to the vertical translation and rotation around the center of the right section of all the forces identified on the elementary ashlar, starting from its equilibrium. Separate contributions are provided for the translational f I1 and rotational m inertias, taking into account the following formulas:…”

“…Due to their exceptional mechanical, physical, and electrical properties, nanotubes have attracted great attention from researchers and have a wide range of applications. As a result, detailed studies on their mechanical properties have been conducted [1][2][3], and static and dynamic behaviors of single-walled and multi-walled carbon nanotubes have been proposed using Euler-Bernoulli and Timoshenko beam models [4][5][6][7][8][9][10]. Atomistic and continuous modeling approaches were used to analyze the mechanical properties of nanotubes, depending on the level of investigation.…”

“…However, classical theories, such as the Euler-Bernoulli or Timoshenko beam model [5][6][7][8], or even higher-order theories have proven to be inadequate because they do not capture the influence of the nonlocal effect. To overcome this drawback, models incorporating nonlocal effects have been proposed, based on the theory of nonlocal elasticity [9][10][11][12][13][14] developed by Eringen (see, in particular, [15,16]). Numerous theoretical and numerical methods have been developed to investigate the influence of nonlocal effects on the free vibrations of structures and nanostructures.…”

“…To model these effects on the frequency response of nanotubes, we use Eringen's nonlocal elastic theory and surface elastic theory to modify the governing equations. A modified version of Timoshenko nonlocal elasticity theory-known as the nonlocal truncated Timoshenko beam theory-is put forth to investigate the free vibration behavior of single-walled carbon nanotubes (SWCNTs), as indicated in [10,34]. Using Hamilton's principle, the governing equations and the corresponding boundary conditions are derived.…”