Vertical deformation of member OA (x10^{4}).
1. Introduction
The intended applications of carbon nanotubes have steadily increased since their discovery by Ijima [4]. They range from the nanoscale, as in the tip of an atomic electron microscope, to the macroscale, as in the preliminary design of the space elevator cable.
In modeling CNTs, molecular as well as quantum mechanics have been the primary tools for analysis. Also, closed form expressions were developed to study the response of CNTs in different environments [3,8,12].
On the other hand, some attempted to use structural mechanics, and built corresponding finite element models, to study the behavior of CNTs, as evident in several publications [58]. Some of these publications simplified the property relations between molecular mechanics and structural mechanics. They assumed the structural bending stiffness
However, in [1], with a simple proof, we showed that the main assumption used by various authors to equate the element bending stiffness to the bond bending stiffness (
In [1], we derived an expression for the axial deformation of zigzag CNTs that accounts for the axial and bending structural stiffnesses under simple tension. While molecular mechanics uses the bond angle between two bonds to describe bond bending deformations, structural mechanics uses the bending within one 3D frame element for this definition. Comparing the deformation equation in structural mechanics to the equivalent equation derived for molecular mechanics, leads us to a “consistent” frame bending stiffness for an infinitely long zigzag CNT. For large diameter tubes, the frame bending stiffness tends to half the bond bending stiffness. This later case is representative of a graphene sheet. For small diameters,
In a recent paper (Chen et al., [9]), the radial elastic modulus of the original Molecular Structure Mechanics model (MSM) was compared to the one from the Molecular Dynamics (MD) simulation. In that paper, it was pointed to the fact that a modification to the original MSM model was suggested in our previous paper (Kasti, [1]).
In this chapter, we extend our previous work to armchair carbon nanotubes under simple tension. In addition, we summarize the equivalent results for zigzag CNTs under simple tension and torsion.
We start with a brief review of molecular and structural mechanics and we refer to the work of Chang and Gao [3]. Then, the relation between the structural bending stiffness
2. Bond Energies and the Finite Element Method
This section deals with the molecular and structural mechanics formulations of bond energies. Also, a short review of the finite element method is presented as it applies to the modeling of carbon nanotubes subjected to mechanical loading.
2.1. Characterization of the Atomic Structure and Molecular/Structural Mechanics of Carbon Nanotubes (CNTs)
The geometry of a CNT could be described with the pair (
For zigzag CNTs (Fig. 1a), the value of
The bond energies between carbon atoms include the stretching
where
The subscript “_{o}” refers to the initial equilibrium configuration.
As far as structural mechanics, the linear elastic deformation is assumed to be the combination of axial, bending and torsional deformations. Their corresponding strain energies are expressed as:
where
2.2. Review of the Finite Element Method for 3D Space Frames
For linear elastic behavior of 3D space frames, the axial, bending and torsional strain energies can be expressed as in equation (3).
The bonding between two carbon atoms is modeled by placing a 3D space frame element between them, Fig. 2. When this procedure is repeated throughout the tube, a finite element mesh is obtained with the carbon atoms becoming the nodes in the mesh.
Each node is assumed to have six degrees of freedom, three translational and three rotational.
The stiffness matrix
3. Work of Chang and Gao [3]
Chang and Gao derived closed form expressions for carbon nanotubes subjected to simple tensile loading using molecular mechanics.
Representing units of zigzag and armchair carbon nanotubes are shown in Fig. 3 with α and β being the internal angles. Equivalent equations to the ones of Chang and Gao will be derived in the next section for the armchair CNT.
4. Axial and Bending Stiffnesses of Armchair CNTs: Molecular versus Structural Mechanics
We start by expressing the results of Chang and Gao [3] in a more suitable form using the principle of minimum total potential energy. We will split the approach into bond stretching and bond bending deformations.
4.1. Bond Stretching – Molecular/Structural Mechanics
Due to bond stretching, it is easy to verify that
We start with the molecular energy expression
where
Expression (5) takes the following form in structural mechanics:
Since both
To determine the tube deformation, we let
4.2. Bond Bending – Molecular Mechanics
Due to bond bending, the total potential energy is written as:
Let
Minimizing the total potential energy with respect to α gives
Since cosβ = cos(π/2n)cos(α/2) and cos
The vertical deformation of the tube can be expressed as:
where
Solving for
Substituting
In the next Section, we will derive an equivalent expression in terms of the material properties of structural mechanics. This will allow us to deduce a relation between
4.3. Bond Bending  Structural Mechanics: Armchair Carbon Nanotube under Simple Tension
Let
Also, let
where
For an infinite cylinder and due to symmetry, the radial and tangential rotations at
In addition, due to multiple symmetries (for
and
where (
Following a similar procedure to Kasti [1], one can show that
And, the only force in the inclined member OA in Fig. 3 is the vertical force F and the moment is (1/2)F.a.sin(α/2).
Due to multiple symmetries (for
and the corresponding elastic modulus Y_{s}
The corresponding axial deformation at the end of a zigzag CNT under simple tension is given by [1]:
The tangential deformation at the end of a zigzag CNT unit under simple torsion is given by following expression for
5. Discussion and Validation of the Results obtained in Section 4 – Armchair CNTs
Comparing the molecular mechanics expression Eq. (12) with the structural mechanics Eq. (16), we obtain
Thus, in general, the bending stiffness to be used in the structural mechanics varies with the bond bending stiffness
For long CNT tubes with large diameters,
To validate the closed form solution Eq. (16), we compared the axial deformation of member OA (Fig. 3) and the change in radius to the results from a finite element model in ABAQUS [14]. The results are shown in Tables 1 and 2 below for



4  5.9844  5.9849 
8  5.9477  5.9470 
12  5.9358  5.9358 
16  5.9310  5.9306 



4  1.3538  1.3539 
8  2.6402  2.6401 
12  3.9383  3.9383 
16  5.2403  5.2402 
For zigzag CNT under simple tension, the equivalent stiffness is given by [1]:
Thus, in general, the bending stiffness to be used in the structural mechanics varies with the bond bending stiffness
For long CNT tubes with large diameters,
For a zigzag CNTs under simple torsion, the corresponding stiffness is given by [2]:
Thus, in general, the bending stiffness to be used in structural mechanics varies with the bond bending stiffness
For long CNT tubes with large diameters,
6. Deformation of Armchair CNTs due to Axial, Bending and Torsional Structural Stiffnesses
In Sections 4 and 5, a closed form expression was developed for the deformation of infinitely long armchair CNT under simple tension. It included the axial and bending stiffnesses of 3D frame elements. In this Section, we study the effect of the torsional stiffness of 3D space frames.
6.1. Bond Bending and Torsion  Structural Mechanics: Armchair Nanotube under Simple Tension
Similar work to Kasti [1] will show that the torsional stiffness does not enter the expression for the deformation of an infinitely long armchair carbon nanotube under simple tension.
When the axial, bending and torsional deformations are combined, we obtain the following formula:
To validate the closed form expression of Eq. (24), we compared the vertical deformation of member OA (Fig. 3) and the change in radius to the results from a finite element model in ABAQUS. The results are shown in Tables 3 and 4 below for



4  1.7686  1.7686 
8  1.7500  1.7500 
12  1.7461  1.7461 
16  1.7446  1.7446 



4  0.53131  0.53136 
8  0.96001  0.96001 
12  1.4088  1.4088 
16  1.8634  1.8634 
The contribution of each of the bond stiffnesses (axial, bending and torsion) to the total vertical deformation of an armchair carbon nanotube is shown in the following example.
Two long carbon nanotubes (40 armchair carbon units) with lattice translational indices “n” equal to 4 and 16, respectively, are modeled using MSC/Nastran [15]. The tubes are supported at the bottom and subjected to tensile loading at the top.
The resulting vertical deformations are compared to the closed form solution of Eq. (24), as shown in Fig. 4. In spite of the difference in boundary conditions between the closed form solution and the finite element modeling, the errors in the results are less than 3%.
Going through the same manipulations as for an armchair CNT, the vertical deformation at the end of a zigzag CNT under simple tension that accounts for bending and torsional deformations can be expressed as [1]:
where
When the torsional stiffness of 3D frame elements is included, in addition to the axial and bending deformations, the tangential deformation
where
For example, for n=4,
One point worth mentioning is that when the torsional stiffness is neglected, i.e.
Thus, in this case of negligible torsional stiffness, the axial and bending deformations are decoupled.
6.2. Elastic Modulus of an Armchair CNT
Similar to the previous work by Kasti [1], an elastic modulus
where
For a finite length cylinder, the elastic modulus obtained from the closed form expressions of Eq. (28) and ABAQUS are compared in Table 5 for a lattice translational index of 4 and variable tube length. The following values of stiffnesses were used:
The closed form results compare very well with the values from ABAQUS.
4  358.12  358.284  358.307 
Similar to the definition of an elastic modulus
where
7. Conclusions
Relations between the structural bending stiffness
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