Delivered by Ingenta to: ETH-Bibliothek Zurich IP : 109.162.214.36 Tue, 31 Jan 2012 16:00:14 Copyright © 2011 American Scientific Publishers All rights reserved Printed in the United States of America Journal of Computational and Theoretical Nanoscience Vol. 8, 2214–2219, 2011 Buckling Analysis of Multi-Walled Carbon Nanotubes with Consideration of Small Scale Effects A. Tourki Samaei ∗ and M. M. Mirsayar School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846, Iran The effect of small scale and van der Waals forces between adjacent tubes on the column buckling of multi-walled carbon nanotubes are investigated. The governing equations for the aforementioned problems are derived using the nonlocal elasticity theory and Timoshenko beam theory. It is found that both small scale effects and van der Waals interaction depend on the length, radius, and buckling modes. In addition, it is shown that considering both of small length scale and van der Waals interaction into the analysis decrease MWCNTs buckling strain and increase the critical strain when the buckling mode is increased. Keywords: Multi-Walled Carbon Nanotube, Van der Waals Interaction, Size Effect, Critical Strain. 1. INTRODUCTION Micro and nanoscale structures, materials and devices have been increasingly utilized among the scientific communi- ties. These nanoscale structures have a vast area of article applications including aerospace, micro electromechanical systems, superfast microelectronics, etc. In 1991, Iijima 1 discovered carbon nanotube, and, subsequently, compre- hensive researches on CNTs have been conducted on their superior mechanical, electrical, physical, chemical, ther- mal conductivity, and electronic properties. Two methods have been used to fundamentally under- stand the behavior of CNTs: continuum mechanics and atomistic molecular dynamic simulations. 2 3 Yakobson et al. 4 studied the axially compressed buckling modes of single-walled CNTs. Using MD simulations and compared their results with a simple continuum shell model. They found that all changes of buckling patterns in the atomic MD simulations can be predicted using a continuum shell model. In addition, Guduru et al. 5 studied the uniaxial and shell buckling experiments on MWCNT using nanoinden- tation technique. They showed that theoretically computed buckling loads are approximately 40% to 50% smaller than experimentally measured buckling loads. However as con- trolled experiments in nanoscale are difficult and atomistic simulations are difficult to formulate exactly, theoreti- cal modelings of micro/nanoscale structures become an ∗ Author to whom correspondence should be addressed. important issue concerning approximate analysis of struc- tures. Hence, researchers have tried to expand the classical continuum mechanics to the atomic-scale approach. 6 The theory of nonlocal elasticity, introduced by Eringen 7 in 1972, implements the nonlocal elasticity con- cept for screw dislocations and surface waves in solids. Peddieson et al. 8 used the nonlocal elasticity version of Euler-Bernoulli beam theory to study the size-effect in micro and nano-scale structures. Many researchers studied the bending, buckling and vibration of micro and nano- scale beams, rods, tubes and plates using the concept of nonlocal elasticity. 9–11 Also, a nonlocal Euler-Bernoulli beam model has been used to investigate the small-scale effect and van der Waals interaction on the column buck- ling behavior of MWNTs with hinge ends. 9 The results demonstrate that the nonlocal continuum model can effec- tively play a more important role in nanoscale structures. Nowadays, it is still challenging to study the buckling behavior of CNTs by means of experimental tests due to the scale difficulties. Reddy, 12 reformulated various avail- able beam theories using the nonlocal elasticity theory for beam bending, vibration and buckling modeling. The first- order shear deformation beam theory of Timoshenko was developed to include the effect of transverse shear defor- mation which had been neglected in the Euler-Bernoulli beam theory. Wang et al. 13 used stress gradient theory on a Timoshenko beam to complete a nonlocal continuum model and presented the influence of shear forces using the nonlocal constitutive differential equations. They studied the small scale effect on the elastic buckling of micro and nano-scale rods and tubes based on nonlocal Timoshenko 2214 J. Comput. Theor. Nanosci. 2011, Vol. 8, No. 11 1546-1955/2011/8/2214/006 doi:10.1166/jctn.2011.1946