The discovery of carbon nanotubes (CNTs) has stimulated considerable experimental
and theoretical studies. Various promising applications have been proposed based
on their unique geometrical and mechanical properties Li
and Chou (2004). The potential use of CNTs as reinforcing materials in nano-composites
has originated the need to explore their mechanical properties.
Due to these difficulties in experimental investigation of CNTs, FE modeling
techniques have been also developed to estimate physical properties of nanotubes.
Carbon nanotubes were simulated extensively using molecular dynamics and continuum
mechanics. The atomistic approaches include classical molecular dynamics, tight-bonding
molecular dynamics and density functional theory Tserpesa
and Papanikos (2005). Despite the fact that these approaches can be used
for any problem associated with molecular or atomic motions, their huge computational
tasks restricted their application to small number of molecules or atoms Shokrieh
and Rafiee (2010a).
The continuum mechanics approaches, on the other hand, mainly involve classical
continuum mechanics and continuum sell modeling Tserpesa
and Papanikos (2005) Since a nanotube can be well described as a continuum
solid beam or shell subjected to tension, bending or torsional forces, it is
reasonable to model nanotube as a frame-or shell-like structure, then the mechanical
properties of such structure can be obtained by classical continuum mechanics
for finite element method.
However, due to the uncertainty of the nanotubes characteristics for both of
the above modeling techniques, the obtained mechanical properties of nanotubes
are widely scattered. The predicted Youngs modulus of CNTs yields a widespread
range of about 1.0 to 5.5 TPa (Selmi et al., 2007).
Experimentally determined Youngs modulus of SWCNTs also scattered in a
relatively large interval of 2.8-3.6 TPa (Selmi et al.,
2007; Thostensona et al., 2001).
The main objective of this article is to develop finite element model of single-walled CNTs (SWCNTs) and investigate the Youngs modulus of SWCNTs based on nanoscale continuum modeling. By employing frame elements to simulate carbon-to-carbon bonds, a finite element model is presented for simulating of elastic modulus of SWCNTs. Furthermore, the effect of nanotube diameter and structure is also studied in this work.
Carbon nanotube simulation: There are several ways to view a SWCNT.
The most widely used is by reference to rolling up grapheme sheet to form a
hollow cylinder with end caps. The cylinder is composed of hexagonal carbon
rings, while the end caps of pentagonal rings. The atomic structure of nanotubes
depends on tube chirality, which is defined by the chiral vector and the chiral
angle (Thostensona et al., 2001). The hexagonal
pattern is repeated periodically leading to binding of each carbon atom to three
neighboring atoms with covalent bonds. This covalent bond is a very strong chemical
bond and plays significant role to the impressive mechanical properties of graphitic
and as a consequence, of all carbon-related nano-structures (Thostensona
et al., 2001; Lau et al., 2006).
These bonds have a characteristic bond length aCC and bond
angle in the 3D space. The displacement of individual atoms under an external
force is constrained by the bonds. Therefore, the total deformation of the nanotube
is the result of the interactions between the bonds. By considering the bonds
as connecting load-carrying elements and the atoms as joints of the connecting
elements, CNTs may be simulated as space-frame structures Tserpesa
and Papanikos (2005). In Fig. 1, a typical nanotube in
the form of a 3D frame structure illustrated.
As mentioned above, by treating CNTs as space-frame structures, their mechanical behavior can be analyzed using classical structural mechanics methods. In this work, a 3D FE model able to assess the mechanical properties of SWCNTs is proposed. The 3D FE model is developed using ANSYS commercial FE code. For the modeling of the C-C bonds, 3D elastic BEAM4 element is used Fig. 2. The specific element is a uni-axial element with tension, compression, torsion and bending capabilities. It has six degrees of freedom at each node: Translations in the nodal x, y and z directions and rotations about the nodal x, y and z-axes. The element is defined by two or three nodes as well as its cross-sectional area, two moments of inertia, two dimensions and the material properties.
To calculate the elastic modulus of beam elements, a linkage between molecular
and continuum mechanics is used. In its general formula, the potential energy
is described as Shokrieh and Rafiee (2010a),
where, Vr, Vè, Vö, Vù,
VvdW, Vel are bond stretching, bond angle bending, dihedral
angle torsion, inversion terms, van der Walls interaction and electrostatic
interaction, respectively. Various functional forms may be used for these potential
energy terms depending on the particular material and loading conditions considered.
In some papers, the effects of Vö, Vù, VvdW,
are neglected under the uniaxial loading and small strain (Shokrieh
and Rafiee, 2010a; Tserpesa and Papanikos, 2005;
Fan et al., 2009). In addition, in most cases
where continuum methods have been used to analyze carbon nanotubes embedded
in an elastic Figure medium, a linear behavior of the reinforcements has been
assumed such as reported in Fan et al. (2009)
and Shokrieh and Rafiee (2010b) works.
|| The hexagonal structure of a typical CNT
||Finite element modeling concept of the hexagonal structure
of a CNT
This assumption leads to accurate predictions only in cases where very small
nanotube deformations take place. Consequently, all these methods cannot be
used for modeling the mechanical behavior of the composites. Therefore, in the
present work, the tensile behavior of the isolated carbon nanotubes is simulated
using the progressive fracture model developed by Tserpes
et al. (2008).
To model the bond stretching, a simple analytical Morse function is used to
represent the experimentally determined bond energy curves of diatomic molecules,
which can be written as Shokrieh and Rafiee (2010a),
where, Deij represent the energy required to stretch
the bond rij from its equilibrium distance to infinity, is the bond
Δrij length variation and aij is equal to (ke/Deij)1/2
where, Ke is the force constant at the minimum of the well. For a
Nanotube system the modified potential energy is expressed as (Shokrieh
and Rafiee, 2010a; Tserpes et al., 2008):
where, Estretch is the bond energy due to bond stretching and Eangle
is the bond energy due to bond angle-bending, r is the current bond length and
θ is the current angle of the adjacent bond. The other parameters of the
potential are (Tserpes et al., 2008):
||r0= 1.421 1010 m
||De= 6.03105 1019 Nm
||β= 2.625 1010 m-1
||θ0= 2.094 rad
||Kθ= 0.9 10-18 Nm rad-2
||Ksexic= 0.754 rad4
For strains above 10%, as bond stretching dominates nanotube fracture and the effect of angle-bending potential is very small, only the bond stretching potential is considered. By differentiating the stretching energy term in (4), the stretching force of atomic bonds is obtained in the molecular forcefield as
The relationship between stress o and bond strain for the C-C bonds could be calculated using the elements cross-sectional area equal to 1.6910-20 m for C-C bond as shown in Fig. 3. The strain of the bond is defined by .As may be seen, the stress-strain relation is highly non-linear especially at large strains and the inflection point (peak force) occurs at about 19% strain.
The initial stiffness is set at 6.5 TPa, according to initial slope of the
C-C bond stress-strain curve Fig. 3. The nanotube is loaded
by an incremental force at one end while the other end being fully constrained.
Zero transverse displacement is applied to the loading end in order to prevent
nanotube buckling at high loads (Fig. 4).
||The stress-srain curve of the modified morse potential
It should be noted that before we feed in the input data of the BEAM4 element
properties, the dimensions of the parameters stated above were further adjusted
to avoid possible digits of overflow/underflow error during the computation
performed by ANSYS as suggested by Fan et al. (2009).
RESULTS AND DISCUSSION
In this section, we will use the finite element results to compute the axial Youngs modulus of carbon nanotubes of various types and sizes. In addition, comparison of these results to those found in the literature will be given. We also discuss the influence of tube size and type on the mechanical properties we obtained.
To compute the axial Youngs modulus from the numerical results, following equation was used:
where, E is the axial Youngs modulus, ó and ε are the axial
stress and strain respectively, F is the total force applied on one end of the
tube, A is the cross section area of the nanotube, which is defined as A=π
Dn tn (where Dn=nanotube diameter, thickness
tn=0.34 nm is the interlayer graphite distance) Shokrieh
and Rafiee (2010a). Nanotube radius was estimated by using Selmi
et al. (2007).
where, an= 0.142 nm is the C-C bond length and n, m are chirality index of nanotubes.
||Iso view of the FE meshes of the (a) (12,0), (b) (14,0) and
(c) (22,0) SWCNTs along with the applied boundary conditions
Six zigzag type single walled carbon nanotubes of different sizes are simulated and their axial Youngs modulus calculated by (7) are listed in Table 1 and depicted in Fig. 5.
Form the results it could be seen that the axial Youngs moduli whose values are about 1.2 Tpa for all cases increase slightly with increasing diameter. From Fig. 5, the effect of tube diameter on the Youngs modulus is also clearly observed. For smaller tubes, for example, diameter less than 1.0 nm, the Youngs modulus exhibits a strong dependence on the tube diameter.
However, for tube diameters larger than 1.0 nm, this dependence becomes very
weak. The general tendency is that the Youngs modulus increases with increasing
tube diameter. The lower Youngs modulus at smaller nanotube diameter could
be attributed to the higher curvature, which results in a more significant distortion
of C-C bonds. As the nanotube diameter increases, the effect of curvature diminishes
gradually Li and Chou (2003).
Our computational results are comparable to those obtained from experiments
and theoretical studies. The obtained results for CNT are in a good agreement
with experimental results which was reported by Wong et
al. (1997). He pinned MWCNTs at one end to molybdenum disulfide surfaces
and measured the bending force versus displacement along the unpinned lengths.
||The relation between the axial Young's modulus and diameter
of carbon nanotubes
|| Comparison between Youngs modulus of carbon nanotube
reported by developed methods and experimental observations in litreture
Numerical results which were reported by Jin and Yuan (2003)
using Molecular dynamics are also comparable to present work. Comparison between
reported data for nanotube Youngs modulus in literature and obtained results
in this study is presented in Table 2.
A finite element simulation technique for SWNTs has been developed which can be easily performed by commercial code ANSYS. The key modeling concept is that simulating molecular bonds are presented as beam elements. We propose and verified a simplifying method to model non-linear nature of covalent bond between to carbon atoms in the nanotube wall. This method can significantly save the modeling and computing effort when finite element analysis is performed. Numerical results for axial Youngs modulus are presented to illustrate the accuracy of the established finite element models. In addition, the relations between these mechanical properties and the nanotube size are also investigated to give a better understanding of the variation of mechanical properties of nanotubes. From the above results and the outstanding advantage that the present modeling concept can be easily extended to cases of MWNTs with higher number of layers, this method will be an effective and convenient tool in studying the mechanical behavior of MWNTs.
The authors would like to thank Teknology University of Petronas for the financial support.