Finite element modeling of single-walled carbon nanotubes

Abstract

A three-dimensional finite element (FE) model for armchair, zigzag and chiral single-walled carbon nanotubes (SWCNTs) is proposed. The model development is based on the assumption that carbon nanotubes, when subjected to loading, behave like space-frame structures. The bonds between carbon atoms are considered as connecting load-carrying members, while the carbon atoms as joints of the members. To create the FE models, nodes are placed at the locations of carbon atoms and the bonds between them are modeled using three-dimensional elastic beam elements. The elastic moduli of beam elements are determined by using a linkage between molecular and continuum mechanics.

In order to evaluate the FE model and demonstrate its performance, the influence of tube wall thickness, diameter and chirality on the elastic moduli (Young's modulus and shear modulus) of SWCNTs is investigated. The investigation includes armchair, zigzag and chiral SWCNTs. It is found that the choice of wall thickness significantly affects the calculation of Young's modulus. For the values of wall thickness used in the literature, the obtained values of Young's modulus agree very well with the corresponding theoretical results and many experimental measurements. Dependence of elastic moduli to diameter and chirality of the nanotubes is also obtained. With increased tube diameter, the elastic moduli of the SWCNTs increase. The Young's modulus of chiral SWCNTs is found to be larger than that of armchair and zigzag SWCNTs. The presented results demonstrate that the proposed FE model may provide a valuable tool for studying the mechanical behavior of carbon nanotubes and their integration in nano-composites.

Introduction

In the past decade, significant progress has been achieved in the area of nano-engineering. One of the specific fields, in which systematic research has been conducted, is the development of nano-composites. Since their discovery in 1991 by Iijima [1], carbon nanotubes (CNTs) have received great attention in terms of measurement of fundamental properties and potential applications. This is mainly due to their extraordinary physical (mechanical, thermal and electrical) properties as revealed from both theoretical and experimental studies. CNTs exhibit exceptionally high stiffness, strength and resilience. As a consequence, they may provide the ideal reinforcing material for a new class of nano-composites [2], [3]. It has been demonstrated [4] that with only 1% (by weight of matrix) of CNTs added in matrix material, the stiffness of the resulting composite film can increase by 36–42%, while the tensile strength by 25%. Recent investigations [3], [4] have shown that CNTs when aligned perpendiculars to cracks are able to slow down the crack growth by bridging up the crack faces. Although CNTs can be used as conventional carbon fibers to reinforce polymer matrix in order to form advanced nano-composites, they may be also used to improve the out-of-plane and interlaminar properties of current advanced composite structures [3].

The potential use of CNTs as reinforcing materials in nano-composites has originated the need to explore their exact mechanical properties, assess their deformation under mechanical loading and, as a subsequent step, identify the possible failure mechanisms that may appear. The characterization of CNTs is more complex than that of conventional materials due to the dependence of their mechanical properties on size and nano-structure. Noticeable progress has been accomplished in this area in the last few years. Two very useful critical reviews describing this progress can be found in Refs. [5], [6]. Both experimental and theoretical approaches have been adopted. Since direct experimental measurements are impractical due to the very small size of CNTs, the theoretical approaches provide good alternative.

The theoretical approaches followed can be divided into two main categories: the atomistic approaches and the continuum mechanics approaches. The atomistic approaches include classical molecular dynamics (representative works: Refs. [7], [8]), tight-binding molecular dynamics (representative work: Ref. [9]) and density functional theory (representative work: Ref. [10]). Although these approaches can be used to deal with any problem associated with molecular or atomic motions, their huge computational tasks bound their application to problems with small number of molecules or atoms. Therefore, their application to CNTs is practical only for single-walled carbon nanotubes (SWCNTs) with small number of atoms. The continuum mechanics approaches mainly involve classical continuum mechanics (representative works: Refs. [11], [12]) and continuum shell modeling (representative works: Refs. [13], [14], [15]). The advantage of continuum shell modeling is that can be used to calculate both static and dynamic properties of CNTs. However, as it simulates CNTs with shells, it neglects their atomic characteristics, eliminating in that way any possible effect that may have on the mechanical behavior of CNTs. Another disadvantage of continuum shell modeling is that it does not consider the interatomic forces. It is therefore obvious that there is a need for the development of new modeling techniques able to accurately calculate the mechanical behavior of both SWCNTs and multi-walled carbon nanotubes (MWCNTs) (generally, of atomic systems with many atoms) by taking into account their atomic characteristics, in affordable computational times. It is also necessary for the techniques to be able not only to simulate the behavior of CNTs as entities but their effectiveness as load-bearing members as well. Given the potentialities of continuum mechanics approaches, it is a challenge to investigate if and under which assumptions they can be used to satisfy this need.

In a recent paper, Li and Chou [12] have presented a structural mechanics approach for modeling the deformation of CNTs. Fundamental to their approach was the notion that CNTs are geometrical space-frame structures and therefore, can be analyzed by classical structural mechanics. For the simulation of CNTs as space-frame structures the authors used the stiffness matrix method. The approach of Li and Chou [12] has been applied in Refs. [12], [16] in order to calculate the elastic moduli of armchair and zigzag SWCNTs and MWCNTs, respectively. The calculation did not include chiral nanotubes.

In this paper, based on the concept of Li and Chou [12], a three-dimensional (3D) FE model for armchair, zigzag and chiral SWCNTs is proposed. Contrary to the stiffness matrix approach, the FE model requires specific values of elastic moduli and moments of inertia of the connecting beams. The paper is organized as follows: the atomic structure of SWCNTs is shortly described in Section 2. The development of the FE model of SWCNTs is described in Section 3. Section 4 comprises the description of the method of calculation of the elastic moduli (Young's modulus and shear modulus) of SWCNTs, through the use of the FE model, as well as the numerical results of the parametric studies conducted on the effect of tube wall thickness, diameter and chirality on the elastic moduli of SWCNTs. Finally, Section 5 concludes the paper.