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1、Mechanics Of Carbon NanotubesAbstract：Carbon nanotubes(CNTs) have been a research hotspot all over the world since their first discovery by Iijima(1991) The small dimensions, strength and the remarkable physical properties of these structures make them a very unique material with a whole range of pr
2、omising applications.1 Carbon nanotubes have a perfect mechanical properties such as incredible high strength and high hardness but low density due to their peculiar structure2. Clearly to determine the mechanical properties of carbon nanotubes is important to carbon nanotubes composites design In t
3、his dissertation, an overview of structure and mechanical properties of carbon nanotubes is proposed.1. IntroductionCarbon nanotubes, discovered in 1991 by Sumio Iijima, are members of the fullerene family. Their morphology is considered equivalent to a graphene sheet rolled into a seamless tube cap
4、ped on both ends. Single-walled carbon nanotubes (SWNTs) have diameters on the order of single-digit nanometers, and their lengths can range from tens of nanometers to several centimeters. SWNTs also exhibit extraordinary mechanical properties ideal for applications in reinforced composite materials
5、 and nano-electro-mechanical systems (NEMS): Youngs modulus is over 1 TPa and the tensile strength is an estimated 200 GPa. Additionally, SWNTs have very interesting band structures. Depending on the atomic arrangement of the carbon atoms making up the nanotube (chirality), the electronic properties
6、 can be metallic or semiconducting in nature, making it possible to create nanoelectronic devices, circuits, and computers using SWNTs.2. The structure of carbon nanotubesA single-walled carbon nanotube (SWNT) can be viewed as a graphene sheet that has been rolled into a tube3, the tube body is cons
7、ists of hexagonal carbon atoms subgrids with a hemispherical fullerene at both end, as shown in fig1.14. Fig 1.2 Schematic diagram of a monolayer graphene layer rolls into a single-walled carbon nanotubeFig 1.1 Different chiral single-walled carbon nanotubes(a)Armchair type; (b)Zigzag type; (c)Chira
8、l typeA multi-walled carbon nanotube (MWNT) is composed of concentric graphitic cylinders with closed caps at both ends and the graphitic layer spacing is about 0.34 nm3. Ignore the fullerene at the end, the hollow cylinder structure can be seen as a model curled from monolayer graphene layer accord
9、ing to a certain victor Ch as is shown in fig 1.2. Here we define the Ch as the chiral vector as the following formula： Ch=na1+ma2 （1.1）Where n and m are all integer and they conform the relationship 0mn，a1 and a2 are two basis vectors of graphite and the angle between them is 60, the average length
10、 of the vector is a1=a2=3ac-c （1.2）Where ac-c=0.142nm is the length of C-C covalent bond5 The structure of carbon nanotubes can be determined absolutely by (n,m)， (n,m) is called the chriality of carbon nanotubes. Single-walled carbon nanotubes can be divided into armchair type (n,n)(nm), zigzag typ
11、e (n,0)(m0) and chrial type (n,m)(nm) according to the different value of n and m. The diameter of the single-walled carbon nanotube defined in fig 1.2 can be determined as the following formuladt= Ch=ac-c3(n2+m2+nm) (1.3)Single-walled carbon nanotubes are periodic along the axial. The periodic stru
12、cture curled from the rectangular surrounded by the chrial vector Ch and the translation vector T which is perpendicular to Ch. The translation vector T is determined by the following formula T=t1a1+t2 (1.4)Where t1=(2m+n)/dR, t2=-(2n+m)/dR,dR is for the greatest common divisor of (2m+n) and (2n+m).
13、 So the length of the translation T is by the formula 1.5T=| T |=3 ChdR=3ac-cn2+m2+nmdR (1.5)Chiral angle (the angle between chiral vector and a1) is also an important parameter to single-walled carbon nanotubes. Obviously is a function of n and mq=sin-1(3m)2n2+m2+nm (1.6)The chiral angle is used to
14、 separate carbon nanotubes into three classesdifferentiated by their electronic properties: armchair (n = m, = 30), zig-zag(m = 0, n 0, = 0), and chiral (0 |m| n, 0 iN1|ri-rj|+vri (3.5)Where the first part is the kinetic energy, the second part is the Coulomb interaction between electronics, the thi
15、rd part is the Coulomb potential of the nucleus. It is still too hard to solve the equations of multi-electronics system so a further approximation is necessary. There exists two common approximation at present: ：Hartree-Fock approximation and density functional theory (DFT). 3.2.3 Molecular Mechani
16、calMolecular Mechanical is also called Force Filed Method too. The energy function is expressed as followsUtotle=Ur+Uq+Uf+U+UvdW+Uel (3.6)Where Ur is for a bond stretch interaction, Uq for a bond angle bending, Uf for a dihedral angle torsion, U for an improper (out-of-plane) torsion, UvdW for a non
17、bonded van der Waals interaction, Uel for a electrostatic interactions. We could apply simplified energy expression according to different requirements and in different situations Li and Chou(2003) proposed a structural mechanics approach to modeling the deformation of carbon nanotubes 3. For sake o
18、f simplicity and convenience, they adopt the simplest harmonic forms and merge the dihedral angle torsion and the improper torsion into a single equivalent term, i.e.Ur=12kr(r-r0)2=12krr2 (3.7)Uq=12kq(q-q0)2=12kqq2 (3.8)Ut=Vf+V=12ktf2 (3.9)where kr , kq and kt are the bond stretching force constant,
19、 bond angle bending force constant and torsional resistance respectively, and the symbols r, q and f represent the bond stretching increment, the bond angle change and the angle change of bond twisting, respectively. According to the theory of classical structural mechanics, the strain energy of a u
20、niform beam of length L subjected to pure axial force is UA=120LN2EAdL=12N2LEA=12EAL(L)2 (3.10)where L is the axial stretching deformation. The strain energy of a uniform beam under pure bendingmoment M isUM=120LM2EIdL=12EIL2=12EAL(2)2 (3.11)where denotes the rotational angle at the ends of the beam
21、. The strain energy of a uniform beam under pure torsion T isUT=120LT2GJdL=12T2LGJ=12GJL()2 (3.12)It can be seen that in Eqs. (3.7)(3.12) both Ur and UA represent the stretching energy, both Uq and UM represent the bending energy, and both Ut and UT represent the torsional energy. It is reasonable t
22、o assume that the rotation angle 2a is equivalent to the total change q of the bond angle, L is equivalent to r, and is equivalent to f. Thus by comparing Eqs. (3.7)(3.9) and Eqs. (3.10)(3.12), a direct relationship between the structural mechanics parameters EA, EI and GJ and the molecular mechanic
23、s parameters kr , kq and kt is deduced as following:EAL=kr, EIL=kq, GJL=kt (3.13)Eq. (3.13) establishes the foundation of applying the theory of structural mechanics to the modeling of carbon nanotubes or other similar fullerene structures. As long as the force constants kr , kq and kt are known, the sectional stiness parameters EA, EI and GJ can be readily obtained. And then by following t