Journal of Nanomaterials

Volume 2016 (2016), Article ID 8641954, 13 pages

http://dx.doi.org/10.1155/2016/8641954

## An Atomistic-Based Continuum Modeling for Evaluation of Effective Elastic Properties of Single-Walled Carbon Nanotubes

Department of Mechanical and Industrial Engineering, College of Engineering, Sultan Qaboos University, P.O. Box 33, 123 Al-Khod, Oman

Received 22 September 2015; Revised 31 January 2016; Accepted 18 February 2016

Academic Editor: Ilaria Armentano

Copyright © 2016 M. S. M. Al-Kharusi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The mechanical behavior of SWCNTs is characterized using an atomistic-based continuum method. At nanoscale, interatomic energy among carbon atoms and the corresponding force constants are defined. Subsequently, we used an atomistic finite element analysis to calculate the energy stored in the SWCNT model, which forms a basis for calculating effective elastic moduli. In the finite element model, the force interaction among carbon atoms in a SWCNT is modeled using load-carrying structural beams. At macroscale, the SWCNT is taken as cylindrical continuum solid with transversely isotropic mechanical properties. Equivalence of energies of both models establishes a framework to calculate effective elastic moduli of armchair and zigzag nanotubes. This is achieved by solving five boundary value problems under distinct essential-controlled boundary conditions, which generates a prescribed uniform strain field in both models. Elastic constants are extracted from the calculated elastic moduli. While results of Young’s modulus obtained in this study generally concur with the published theoretical and numerical predictions, values of Poisson’s ratio are on the high side.

#### 1. Introduction

Extensive research work done by researchers from science and engineering background in composite materials opens new prospects for future short and long term technologies, which will reshape the practical application of modern composites. Currently, the research themes on nanocomposites and/or composites with nanoreinforcements face the challenges of characterization, fabrication, and application. Significant amount of experimental and numerical research work is done to characterize the nanoreinforcement. But further research is needed to bring these to the level of practical application. These nanocomposites are becoming favorable candidates for materials with a bright future in a wide variety of industries such as transport, defense, electronics, and biomedicine, to name a few. Hence, it is important that the mechanical properties of these composite constituents, particularly the carbon nanotubes (CNTs), be predicted accurately. Further, the potential use of carbon nanotubes (CNTs) as a reinforcing material in nanocomposites and light weight composite structures has triggered a need to explore their mechanical properties and assess their deformation under mechanical loading. The unique structure and geometric configuration of CNTs along with their high stiffness, low density, and large aspect ratio have propelled an increasing demand in furthering the research to quantify their elastic properties as well as to explore possible applications in different fields.

Various experimental and theoretical approaches have been developed or used to characterize the elastic behavior of SWCNTs. Several investigators [1, 2] have conducted experimental studies to investigate the mechanical properties of carbon nanotubes. These experiments were mainly based on atomic force microscopy (AFM) and transmission electron microscopy (TEM) and were able to confirm that CNTs possess superior mechanical properties. However, the experimental error bars are too large to state exact characteristics of CNTs of different configurations, sizes, and structures. The wide scatter in the experimentally reported values of the elastic constants of the CNT can be attributed to the lack of proper direct measuring techniques at nanometer scale, difficulties in test specimen preparation, and the dissimilarities in the method of nanotubes manufacture [3]. Such high complexity in the experimental characterization has prompted many researches to pursue a variety of theoretical studies on determining the effective mechanical properties of nanotubes.

The theoretical approaches found in literature can be divided into three main categories: the atomistic methods, the continuum mechanics modeling, and the equivalent continuum modeling using finite element method. The atomistic approaches include classical molecular dynamics (MD) [4], tight-binding molecular dynamics, and density functional theory (DFT) [5]. Phenomenological interatomic potential energy functions are used in these approaches to model the nanoscale systems in order to determine the force applied by the carbon atoms. Therefore, the more realistic and accurate these potentials are, the more closely the results match the experimental data and the better they reflect the actual properties of real system. Although these atomistic methods can simulate any problem associated with molecular or atomic motions, their huge computational tasks bound their application to problems with small number of molecules or atoms.

The continuum mechanics based approaches employ theories of shells, trusses, and beams [6] to model CNTs. One advantage of the continuum shell modeling is that it can efficiently calculate both static and dynamics properties of CNTs. However, traditional continuum models cannot accurately describe the mechanical properties of CNT structures because they lack the atomistic representation and appropriate constitutive relations that govern material behavior at nanoscale. Hence, there is a need for the development of new modeling techniques to accurately capture the mechanical behavior of CNTs. Thus, nanomechanics continuum theories that integrate continuum mechanics theories with interatomic potentials of atomic and molecular structure were developed. Equivalent continuum modeling (ECM) approach is one of the major developments of continuum method. It has been regarded as a very efficient method, especially with nanostructures modeled at large scale. Molecular mechanics combined with finite element method (FEM) involving shell, beam, spring, rod, and combination of these models form the framework of ECM approach. Over the past years, many ECM models were presented in literature. Atomistic-based continuum multiscale modeling techniques were used to predict the mechanical behavior of CNTs considering the interatomic interactions at nanoscale [7]. Li and Guo [8] proposed an equivalent continuum beam model that is capable of modeling interatomic forces between carbon atoms to compute effective elastic constant of CNTs. The elastic constants of beam elements in a finite element model were determined using a linkage between molecular and structural mechanics. An improved beam element, which includes the bond inversion energy, is proposed by Lu and Hu [9] to evaluate mechanical properties of graphene and SWCNTs based on molecular mechanics. A finite element approach based on molecular mechanics was proposed by Sun and Zhao [10]. The chemical bond was simulated with a two-node elastic rod element and an elastic joint at each end. A tensile modulus of 0.4 TPa was found. Shokrieh and Rafiee [11] investigated Young’s modulus of CNTs based on a nanoscale continuum modeling by employing frame elements to simulate C-C bonds. Another continuum model that allows calculation of Young’s and shears moduli based on structural mechanics combined with FEA was developed by Muc [12]. Young’s moduli are derived from the natural frequencies of CNT structures. These models assume that the material is transversally isotropic.

Moreover, different kinds of atomistic finite elements including rods, trusses, beams, and springs have been used to model carbon-carbon (C-C) link in CNTs [6]. Giannopoulos et al. [13] constructed a computational FE model to simulate the SWCNT using linear interatomic potentials for C-C bonds. Meo and Rossi [14] developed a finite model including both nonlinear elastic and linear torsional spring elements to represent the modified Morse potential when simulating SWCNTs. A full nonlinear finite element model was developed using spring elements accounting for both C-C bond stretching and C-C-C bond angle variations to investigate the effect of chirality and the diameter on Young’s modulus of SWCNT [15]. The elastic constants, Young’s and shear modulus, of SWCNTs were numerically computed via finite element method incorporating Poisson’s effect in the estimation of Young’s and shear modulus of SWCNT [16]. The numerically calculated values of Young’s and shear modulus were approximately 1.046 TPa and 0.424 TPa, respectively, for a SWCNT having a thickness of 0.34 nm.

In spite of the variety of theoretical studies, there still remain differences in approaches and calculated values regarding the effect of geometric structure of CNTs on elastic constants, as evident by the wide scatter among elastic constants reported in the literature. The objective of this paper is to study the elastic behavior of single-walled carbon nanotubes (SWCNTs) using a multiscale modeling approach. At nanoscale, interatomic interactions among carbon atoms are modeled by a structural beam in atomistic finite element approach in conjunction with molecular structural mechanics. At macroscale, an equivalent continuum modeling method is proposed to compute directly the effective elastic moduli of SWCNTs having different chirality and configuration. The elastic constants, that is, Young’s modulus and Poisson’s ratio, are calculated from the elastic moduli.

#### 2. Atomic Structure of SWCNTs

Several methods to construct CNTs are reported in literature. In a common procedure, CNT is built by rolling a graphene sheet in two directions: a specific rolling direction and the circumference of the tube cross section, as shown in Figure 1. In general, two chiral indexes () are used to define the configuration of carbon nanotubes, where () normally is greater than (). The nanotubes with () are typically labeled as armchair, while the structure with () is usually labeled as zigzag. The translation vector, , is parallel to the tube axis and perpendicular to the tube chiral vector . The unit vectors of graphene sheet lie along two sequent “zigzag” lines and are represented by and . The vector has a different magnitude than and when added together, they equal the chiral vector . A complete graphical description of the variables and the relations governing the geometry of SWCNT can be found in [17]. The following formulae are used to define the diameter, , and the coordinate of 3-D SWCNT:where is the bond length between carbon atoms, , , are the generalized coordinates of an arbitrary point in nanotube, is the radius of nanotube (i.e., ), and are the coordinates of graphene sheet. The properties of CNTs are dependent on its structural configurations, the number of the concentric layers, and their structural defect and impurities. The present paper focuses on characterizing the effect of geometric configuration on mechanical properties of SWCNTs.