Delivered by Ingenta to: Chinese University of Hong Kong IP: 46.161.57.235 On: Sat, 02 Jul 2016 00:10:45 Copyright: American Scientific Publishers Copyright © 2010 American Scientific Publishers All rights reserved Printed in the United States of America Journal of Computational and Theoretical Nanoscience Vol. 7, 583–593, 2010 A Precise Model to Predict the Structural and Elastic Properties of Single-Walled Carbon Nanotubes Yuzhou Sun 1 2 and K. M. Liew 1 ∗ 1 Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong 2 Department of Civil Engineering and Architecture, Zhongyuan University of Technology, Zhengzhou 450007, China This paper presents a continuum analysis for the structural and elastic properties of single-walled carbon nanotubes (SWCNTs). An SWCNT is viewed as a rolled-up graphite sheet, and the trans- formation is appropriately written into a set of equations with three geometrical parameters. The microscale bond deformation in a representative cell is calculated exactly, and the atomic energy is evaluated with the Brenner potential. The structural properties of SWCNTs are determined by mini- mizing the atomic energy, and the elastic constants are calculated from the second-order derivatives of the strain energy density with respect to the set geometrical parameters. The dependence of the elastic constants on the chirality and tube radius is discussed. The paper also investigates the precision of the Cauchy-Born rule and the higher-order Cauchy-Born rule, which have previously been used to derive the continuum constitutive model of SWCNTs. Keywords: Carbon Nanotubes, Continuum Model, Elastic Constants, Cauchy-Born Rule. 1. INTRODUCTION The unique nanostructure of Carbon nanotubes (CNTs) gives them remarkable physical, electrical, and mechanical properties, and attractive prospects for application. In addition to a large amount of experimental work, theo- retical modeling and analysis play an important role in understanding their subtle and complex behavior. Mod- eling approaches can be classified into two categories: atomic simulations and continuum simulations. Atomic simulations such as molecular dynamics 1–4 can capture the microscale mechanism of nanostructures and yield results that are, in many cases, explicit in nature. However, they consume a large amount of computational resources, and thus the computation is limited to a very small size. 5 6 Con- tinuum simulations are much faster than atomic simulations in the analysis of systems of engineering interest, which makes them attractive. Continuum simulations can also dis- play certain properties of CNTs that are difficult to capture using atomic simulations. Moreover, phenomenological continuum-based material parameters such as the Young’s modulus can be well defined and measured in contin- uum simulations. Several efficient continuum-based meth- ods have been developed and applied in the study of CNTs. Govindjee and Sackman adopted the Euler beam theory to ∗ Author to whom correspondence should be addressed. study the elastic properties of CNTs and showed the depen- dency of the elastic properties at nanoscale dimensions. 7 Ru treated an SWCNT as a single-layer elastic shell with an effective bending stiffness. 8 9 Li and Chou developed a molecular structural mechanics approach to study the elastic properties of SWCNTs. 10 11 He et al. 12 and Liew et al. 13 14 developed a continuum model to account for the van der Waals interaction between the different walls of CNTs. Wang et al. 15 and Hu et al. 16 applied non-local shell theory to study flexural and longitudinal wave propaga- tions in SWCNTs. Knowledge of the elastic property of a material is the first step toward its use as a structural element in various applications. However, the current inves- tigations have not achieved good consistency for the elas- tic constants of CNTs. This research presents a systemic study of the structural and elastic properties of SWCNTs with a precise continuum model. An SWCNT is consid- ered to have been formed by the rolling up of a graphite sheet into a cylindrical shape. However, this rolling is not a grid transformation. With three geometrical parameters, the rolling process is appropriately written as a set of equa- tions. The microscale bond deformation in a representative cell is calculated exactly, and the atomic energy is evaluated with the Brenner potential. 17 The structural properties of SWCNTs are determined by minimizing the atomic energy. Regarding the physical meaning of the set parameters, the elastic constants (Young’s modulus, shear modulus, and Poisson’s ratio) are calculated from the second-order J. Comput. Theor. Nanosci. 2010, Vol. 7, No. 3 1546-1955/2010/7/583/011 doi:10.1166/jctn.2010.1398 583