Volume 32 Issue 4 - June 1, 2020 PDF
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A Review of Mechanical Analyses of Rectangular Nanobeams and Single-, Double-, and Multi-Walled Carbon Nanotubes Using Eringen’s Nonlocal Elasticity Theory
Chih-Ping Wu*, Jung-Jen Yu
Department of Civil Engineering, National Cheng Kung University
 
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This article was intended to present an overview regarding the papers examining assorted structural behavior of rectangular nanobeams and single-, double-, and multi-walled carbon nanotubes (CNTs) with various boundary conditions embedded in an elastic medium. The review article contains 228 references, of which 13 papers are contributed from the research group of the first author, Professor Chih-Ping Wu.

    This review article introduced the Erigen nonlocal constitutive relations, the Ru and the He et al formulae for the van der Waals interactive force models, the Winkler- and the Pasternak-type foundation models, nonlocal Timoshenko beam theories, and nonlocal classical shell theories (CSTs). For illustrative and comparison purposes, the strong- and weak-form formulations of a nonlocal mixed Timoshenko beam theory (TBT) based on the Reissner mixed variational theorem (RMVT) and the principle of virtual displacements (PVD) were presented. Based on the Hamilton principle, the Euler-Lagrange equations for various nonlocal CSTs were also presented.
Comparisons of the vibration frequencies of single-walled CNTs among the results obtained using the molecular dynamic (MD) simulation, assorted nonlocal CSTs, including the nonlocal Flugge and the nonlocal Sanders theories, and Wu’s nonlocal TBTs were carried out. The results showed the small length scale effect softens the gross stiffness of CNTs, such that the frequencies obtained using the nonlocal beam/shell theoris are always less than those using the local beam/shell theories. On the basis of the MD solutions, it is shown that the nonlocal TBT is suitable for the analysis of thin and long CNTs only. The nonlocal CST solutions are more accurate than the nonlocal TBT solutions for CNTs with the length-to-diameter ratio greater than 10, in which the relative errors are below about 5% for the nonlocal CST solutions and those are below about 10% for the nonlocal TBT as compared with the MD solutions.


Fig. 1. The configuration, geometric parameters, and coordinates of the outmost wall of an embedded multi-walled carbon nanotube.


(a)
(b)

Fig. 2 The variations in the lowest frequency ratios of an simply supported, embedded triple-walled carbon nanotube with (a) the stiffness of its surrounding medium kw, (b) the shear modulus of its surrounding medium.
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