",isbn:"978-1-83969-221-5",printIsbn:"978-1-83969-220-8",pdfIsbn:"978-1-83969-222-2",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!0,hash:"ec438b5e4be44dc63870c1ace6a56ed2",bookSignature:"Dr. Marcos Roberto Tovani Palone",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10710.jpg",keywords:"Orofacial Cleft, Cleft Lip, Surgery, Cleft Palate, Oral Surgical Procedures, Orthodontics, Dental Treatment, Comprehensive Dental Care, Speech Therapy, Speech-Language Pathology, Pediatric Treatment, Therapy",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"February 3rd 2021",dateEndSecondStepPublish:"March 3rd 2021",dateEndThirdStepPublish:"May 2nd 2021",dateEndFourthStepPublish:"July 21st 2021",dateEndFifthStepPublish:"September 19th 2021",remainingDaysToSecondStep:"2 days",secondStepPassed:!0,currentStepOfPublishingProcess:3,editedByType:null,kuFlag:!1,biosketch:"Dr. Marcos Roberto Tovani Palone received his Ph.D. from Ribeirão Preto Medical School, University of São Paulo, Brazil. 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\n\t\t\t
1. Introduction
\n\t\t\t
The geometric structure of carbon nanotubes (CNTs) can be considered to be the curling of graphene (graphite sheet) (Gao & Li, 2003\n\t\t\t\t; Shen, 2004). The physical parameters of carbon in graphite are widely adopted in the molecule dynamics (MD) simulations or other theoretical studies of CNTs, such as the bond energy and length of C-C, the bond angle of C-C-C, etc (Belytschko et al., 2002; Mayo et al., 1990; Xin et al., 2007, 2008). Previous researches have shown that the elastic modulus of small CNTs changes a lot when the radius varies. However, the elastic modulus of CNTs fairly approaches that of graphene, while the radius of CNTs is large enough. Therefore, it is necessary to have a clear understanding of the mechanical properties of graphene for the further realization of the properties of CNTs.
\n\t\t\t
Numerous researchers carried out experiments to measure the effective elastic modulus of CNTs (Krishnan et al.,1998; Poncharal et al., 1999). They reported the effective Young’s modulus of CNTs ranging from 0.1 to 1.7 TPa, decreasing as the diameter increased, and the average was about 1.0~1.2 TPa. MD simulations have provided abundant results for the understanding of the buckling behavior of CNTs. The Young’s modulus of the CNTs was predicted about 1.0~1.2 TPa through various MD methods (Jin & Yuan, 2003; Li & Chou, 2003; Lu, 1997). Hu et al. (Hu et al., 2007) proposed an improved molecular structural mechanics method for the buckling analysis of CNTs, based on Li and Chou’s model (Hwang et al., 2010) and Tersoff-Brenner potential (Brenner, 1990). Due to the different methods employed on various CNTs in these researches, the reported data scattered around an average of 1.0 TPa.
\n\t\t\t
The elastic properties were also discussed in the theoretical analysis by Govinjee and Sackman (Govinjee and Sackman, 1999) based on Euler beam theory, which showed the size dependency of the elastic properties at the nanoscale, which does not occur at continuum scale. Harik (Harik, 2001) further proposed three non-dimensional parameters to validate the beam assumption, and the results showed that the beam model is only proper for CNTs with small radius. Liu et al (Liu et al., 2001) reported the decrease of the elastic modulus of CNTs with increase in the tube diameter. Shell model was also used in some researches (Wang et al., 2003), to study axially compressed buckling of multi-walled CNTs. And studies by Sudak (Sudak, 2003) reported that the scale effect of CNTs should not be ignored. Wang et al (Wang et al., 2006) investigated the buckling of CNTs and the results showed that the critical buckling load drawn with the classical continuum theory is higher than that with considering the scale effect. There are some other researches also reporting clear scale effect on the vibration of CNTs (Wang & Varadan, 2007; Zhang et al., 2005). Li and Chou (Li & Chou, 2003) put forward a truss model for CNTs and their studies showed the radius-dependence of elastic modulus of SWCNTs. Finite element analysis (FEA) is also employed in researches on the mechanical properties of CNTs (Yao & Han, 2007, \n\t\t\t\t\t2008\n\t\t\t\t; \n\t\t\t\t\tYao et al., 2008\n\t\t\t\t).
\n\t\t\t
An equivalent model is established in this chapter, based on the basic principles of the anisotropic elasticity and composite mechanics, for the analysis of the elastic properties of graphite sheet at the nanometer scale. With this equivalent model, the relationship between the nanotube structure and the graphite sheet is built up, and the radial scale effect of the elastic properties of CNTs is investigated.
\n\t\t
\n\t\t
\n\t\t\t
2. Constitutive equations of orthotropic system
\n\t\t\t
The essential difference between the basic equations of anisotropic materials and those of isotropic ones is in their constitutive equations, which means the usage of the anisotropy Hooke law for the anisotropic constitutive equation and the isotropy one for the other. The phenomenon reflected from the anisotropy equations is more accurate than from the isotropy ones, though this distinction also makes the calculation with the anisotropy equations much more complicated.
\n\t\t\t
One of the anisotropic systems, with three mutually perpendicular principal axes of elasticity, is called an orthotropic system (Fig. 1). If the three principal axes of elasticity are defined as\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the constitutive equation of orthotropic system can be obtained as follow,
\n\t\t\t
Figure 1.
The principal elastic axes of orthotropic system: a) right-hand coordinate system; b) left-hand coordinate system
\n\t\t\t\tEquation (3) is also correct if we make exchanges of C for S and S for C.
\n\t\t\t
Figure 2.
Sketch of macroscopic homogeneous orthotropic material
\n\t\t\t
The unidirectional fiber composed composite materials can be treated as orthotropic systems, for which the three axes in the right-handed coordinate system in Fig. 2 are the principal material axes and the axis 1 is along the fiber length. Thus, the components of equation (1) can be given minutely in detail,
3. Macro-mechanics fundamental principle of composite material
\n\t\t\t
\n\t\t\t\t
3.1. Constitutive equations of monolayer under plane stress
\n\t\t\t\t
The monolayer composite with unidirectional fiber can be considered as a homogeneous orthotropic material in the macro analysis. Fig. 3 displays the three principal material axes and the axis 3 is perpendicular to the mid-plane of the monolayer. Suppose the monolayer is in a plane stress state, there are the in-plane stresses,
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the axis flexibility matrix. The equation (10) can also be given as,
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the converted axis stiffness matrix at the plane stress state. \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tin equation (12) and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t in equation (2) have the following relations,
The values of the flexibility and stiffness in equation (11) and equation (13) can be obtained through micromechanics calculations or experiments.
\n\t\t\t
\n\t\t\t
\n\t\t\t\t
3.2. Off-axis flexibility and stiffness of monolayer under plane stress
\n\t\t\t\t
The off-axis flexibility and stiffness are often used in the mechanical analysis on the unidirectional fiber monolayer. As displayed in Fig. 4, axes 1 and 2 are along the principal axes of material, x and y are off-axis, and the anticlockwise angle from x-axis to the 1-axis is positive. Thus, we obtain,
The equation (22) and (23) are the constitutive equations in the Oxy coordinate system, where the off-axis stiffness\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and the off-axis flexibility\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are given as follows,
3.3. Constitutive equations in classical laminated plate theory
\n\t\t\t\t
The so-called classical laminated plate theory or the classical laminate theory, refers to the use of the straight normal hypothesis in elastic shell theory, neglects a number of secondary factors, and has been an acknowledged laminated plate theory. In the classical laminated plate theory, the transverse shear strain\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t23\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t31\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, and the normal direction strain \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tare supposed to be zero.
\n\t\t\t\t
Figure 5.
Sketch of laminated thin plate and the Cartesian coordinates
\n\t\t\t\t
As a result of straight normal assumption, the deformation of laminated plate can be described with the mid-plane deformation. If the mid-plane strain is \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand thickness of each layer in the laminated plate is the same, the mid-plane stress of an n-layers laminated plate is
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the mid-plane stress of the laminated plate, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the stress of the kth layer, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the converted stiffness of the kth layer.
\n\t\t\t
\n\t\t
\n\t\t
\n\t\t\t
4. Equivalent model of graphite sheet
\n\t\t\t
\n\t\t\t\t
4.1. The basic idea of the equivalent model
\n\t\t\t\t
All the C atoms in the graphite sheet are connected with the σ bonds and the bonds form a hexagonal structure (Fig.6). Fig. 7 is a schematic diagram of the graphite sheet, thick solid lines in which represent the C-C bond in graphite. If each C-C bond is longer (shown in thin lines), that will form a network structure, as shown in Fig. 8.
\n\t\t\t\t
Figure 6.
The bonding relationship in C-C covalent bonds
\n\t\t\t\t
As can be seen from Fig. 8, the network structure is formed by the three groups of parallel fibers and they are into 60 degree angles with each other. If we consider each group of fiber as a composite monolayer, the mechanical properties of the entire network structure can be obtained with the laminated plate theory. Comparing Fig. 7 and Fig. 8, we find that the network structure in Fig. 8 can also be formed if the three graphite sheets in Fig. 7 are staggered and stacked one on top of the other.
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In summary, here we put forward a new original equivalent model used to study the mechanical properties of graphite sheet. The analysis steps are, treat the network structure shown in Fig. 8 as a laminated composite plate with three layers orthotropic monolayer of unidirectional fiber (each fiber in the monolayer is just the covalent C-C bond in series), the mechanical properties of fiber can be deduced from the physical parameters of graphite, and the 1/3 of the converted stiffness of the network structure can be considered as the converted stiffness of the graphite sheet at the plane stress state.
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Figure 7.
The atomic structure of a graphite sheet
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Figure 8.
Effective network structure of laminated graphite sheets
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\n\t\t
\n\t\t
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4.2 Mechanical properties of graphite sheet at plane stress state
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The hexagonal plane composed of the σ bonds is defined as the σ-plane, and the energy of interactions between any C atoms in the σ-plane are considered to be functions of the position of the C atoms. With all the weak interactions (e.g. the electronic potential, the van der Waals interactions) neglected, the total potential energy of the graphite sheet can be expressed as.
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the axial stretching energy of C-C bond, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the C-C-C bond angle potential.
\n\t\t\t
Establish a local coordinate system in the σ-plane with the C-C bond direction as the x′ axis (Fig. 9), we can obtain the whole potential,
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the axial stretching energy of C-C bond in the local coordinate system and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t the angle bending energy. According to the basic principles of molecular mechanics, the two bond energy can be written as:.
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are the displacement, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are the MD force field constants.
\n\t\t
\n\t\t
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4.3 Elastic constants of monolayer in the equivalent model
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If we consider the C-C bond as a single elastic fiber in the equivalent model, and the 1 axis for the fiber is defined as the axial direction of the C-C bond, we have,
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the elastic stiffness factor of the fiber at the 1 direction, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are the C-C bond length and the cross-sectional area of the equivalent fiber, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the elastic modulus of the fiber at the 1 direction.
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The 2 axis for the fiber is defined as the vertical direction of the C-C bond in the σ-plane. The fiber interactions at the 2 direction are mainly reflected through the C-C-C angle bending potential (Fig. 10).
Set \n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t to be the elastic stiffness factor of the fiber at the 2 direction, we obtain,
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are the C-C-C angle change and the displacement at the 2 direction when the model is loaded at the 2 direction, t is the effective thickness of the equivalent fiber layer, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the elastic modulus of the fiber at the 2 direction.
\n\t\t\t
Figure 11.
Shearing deformation of the fiber in the equivalent model of graphite
\n\t\t\t
When the fiber layer in the equivalent model is subjected to a shearing force P (Fig. 11), there will be a horizontal displacement \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\tτ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t for the equivalent fiber element and an angle deflection\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. As to the displacement of the bottom atom in of the figure, it can be obtained,
The strain energy induced by the angle deflection \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t of a fiber is,
4.4. Test for the mechanical constants of the monolayer in the equivalent model
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From equation (34), (37) and (44), we obtain the mechanical constants of the monolayer in the equivalent model\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, of which \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are independent quantities, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis a function of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Both \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are related to MD parameters of the C-C bond energy
\n\t\t\t\t
There are several empirical potentials and relevant parameters for C-C bond energy. In this section, the following potential energy function and parameters are used to verify the equivalent model provided in this chapter. The Morse potential is employed for the bond stretching action and harmonic potential for the angle bending. The short-range potential caused by the deformation of C-C bond is described as bellow,
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are the potentials of bond stretching and angle bending, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are the corresponding force constants. \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\trepresents the distance between any couple of bonded atoms, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\trepresents all the possible angles of bending, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are the corresponding reference geometry parameters of grapheme. \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\n\t\t\t\t\tdefines the steepness of the Morse well. The values of all these parameters are listed in Table 1.
There are other scholars working on the C-C stretching force constants through experiments or theoretical calculations, who reported the values along the C-C bond like\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t729\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tnN\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\tnm\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t880\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tnN\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\tnm\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t708\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tnN\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\tnm\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand so on. They also obtained the constants at the direction perpendicular to the C-C bond, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t432\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tnN\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\tnm\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t398\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tnN\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\tnm\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t (\n\t\t\t\t\t\tYang & Zeng, 2006\n\t\t\t\t\t). Noting equation (47) and (48), we can see that the data obtained here based on the current equivalent model are in good agreement with the results from other researchers.
\n\t\t\t
\n\t\t
\n\t\t
\n\t\t\t
5. Elastic properties of graphite sheet
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\n\t\t\t\t
5.1. Flexibility of monolayer in the equivalent model
\n\t\t\t\t
Graphite sheet can be considered as the network structure formed by the three groups of parallel fibers which are into 60 degree angles with each other. Based on the result of the last section, we can get the axial flexibility of the fiber monolayer,
where it has\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tν\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tν\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t21\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t can be calculated according to the equation (34), (37) and (44). If we consider \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tν\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t21\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t to be 0.3 with reference to the parameters of general materials, assume the fiber thickness of 0.34nm, and apply the data in Table 1, the follows can be calculated,
where the unit of the values is\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t10\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tnm\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t/nN\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
\n\t\t\t\t
Substituting \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand the values in equation (51) into equation (28), we get the off-axis flexibility of the \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t60\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\to\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t60\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\to\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t monolayers,
The unit of the values is also \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t10\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tnm\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t/nN\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t in the last two equations.
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\n\t\t\t
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5.2. Elastic properties of the equivalent model of graphite
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The graphite sheet is a whole layer structure with no delamination and the strain did not change in the thickness. Thus we can use formula (29) to calculate the stiffness,
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the converted stiffness of the graphite sheet, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the stiffness of the kth layer in the equivalent model. The flexibility of the graphite sheet is,
where the unit of the values is\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t10\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tnm\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t/nN\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
\n\t\t\t\t
A series of well acknowledged experiment values (\n\t\t\t\t\t\tYang & Zeng, 2006\n\t\t\t\t\t) of elastic constants of perfect graphite are listed in Table 2. Data obtained in current work are in good agreement with those results, which justifies the present equivalent model. It should be noticed that there is an obvious error of the\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. However, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\thas little effect on the mechanical properties of the graphite sheet, the error on it will not influence the reasonable application of the current equivalent model in the mechanics analysis of graphite.
Data obtained in current work and elastic constants of perfect graphite crystals from other experiment
\n\t\t\t\t
Both the result in equation (56) and the data listed in Table 2 indicate that the graphite is in-plane isotropic at plane stress state. And according to the classical theory of composite mechanics, it is known that the \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t laminated plates with \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are in-plane isotropic. We can see that the graphite sheets being in-plane isotropic is mainly due to their special structure of C-C-C angles.
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According to the points made above, we try to explain why the elastic properties of CNTs are anisotropic to some extent. One of the possible reasons is that the curling at different curvature from graphene to CNTs makes the C-C-C angles change (e.g. the angles in an armchair CNT with 1nm diameter are not fixed 120o, but about 118 o), for which the quasi-isotropy of graphite sheet is disrupted and the orthotropy introduced. Having an general realization of the CNTs, one should find that the change of the C-C-C angle is obviously related to the change of CNT radius, especially when it is a small tube in diameter. Thus, the orthotropy of CNTs is diameter related, which is in good agreement with the other research results referring to the mechanical properties of CNTs changing due to their various radial size. The changes of C-C-C angles are also somehow related to the chirality of CNTs. However, the extent of the changes of chiral angles is about 0º ~30º, and the change of C-C-C induced by the difference of chiral angles is little when to curl the graphite sheet at the same curvature. It agrees with that many studies reporting that the difference of elastic properties among various chiral CNTs decreases with the increase in the diameter of CNTs.
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\n\t\t\t\t
5.3. Elastic properties of the equivalent model with various C-C-C bond angles
\n\t\t\t\t
The C-C-C bond angles will change from the constant 120º to smaller values when the graphite sheet curling to CNTs. In the same way, the angles between the fibers in the equivalent model will change to less than \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand the effect of the change on the elastic properties of the model will be investigated in this section.
\n\t\t\t\t
Substituting equation (51) and the angle \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t between the fibers (responding to the changed C-C-C angles) into equation (28), we can obtain the off-axis flexibility of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t monolayers, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Substitute those two into equation (55) to get the approximate converted flexibility of the equivalent model (responding to the changed C-C-C angles):
The elastic modulus of the equivalent model responding to the graphite with changed C-C-C angles, at the direction of 0o fiber or at the vertical direction, can be obtained from equation (57). The variation of the modulus with the changes of the C-C-C angles are displayed in Fig. 12 and Fig. 13, and also displayed the variation of\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tν\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tν\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t21\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t in Fig.14~Fig.16.
\n\t\t\t\t
Figure 12.
Elastic modulus of the equivalent model of graphene along the 0º fiber
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Figure 13.
Elastic modulus of the equivalent model of graphene at perpendicular direction to the 0º fiber
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Figure 14.
G12 of the equivalent model of graphene
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Figure 15.
Formula: Eqn154.wmf>of the equivalent model of graphene
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Figure 16.
Formula: Eqn155.wmf>of the equivalent model of graphene
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We can see from Fig. 12 and Fig. 13 that the elastic modulus along the 0o fiber of the equivalent model of graphene almost increases linearly, and the elastic modulus at the vertical direction decreases, with the decrease in the C-C-C angle. And Fig. 14 ~ Fig. 16 show that the G12, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tν\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tν\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t21\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t of the equivalent model also changes a lot with the changes of the C-C-C angle.
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6. Scale effect of elastic properties of CNTs
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6.1. Equivalent model of single-walled carbon nanotubes (SWCNTs)
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CNTs can be considered as curling graphite sheet. The C-C-C bond angles will change from the constant 120º to smaller values when the graphite sheet curling to CNTs and the change of the C-C-C angle is related to the radius of the formed CNTs. According to that, we consider CNTs same as graphene with changed C-C-C angles and the elastic properties of CNTs are consistent with those of graphene with changed C-C-C angles.
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\n\t\t\t\t\t
6.1.1. Zigzag SWCNTs
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Taking the zigzag SWCNTs as an example, we study the effect of the changing in diameter of SWCNTs on the value of C-C-C angle. The SWCNT structure and the geometric diagram are shown in Fig. 17, with which we can obtain,
With\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tν\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t21\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, the last equation becomes,
where the \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t does not change a lot when the graphene curls to the SWCNT so that we make\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\t\t∠\n\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\t\t∠\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\t\t∠\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t.
Thus, with the diameter of the zigzag SWCNT provided, setting the AB direction in Fig. 17 as the direction of the 0o fiber, we can calculate the C-C-C angles in zigzag SWCNTs as,
Different C-C-C angles in zigzag SWCNTs with different diameters
\n\t\t\t\t\t
The curve drawn from equation (63) is displayed in Fig. 18, which shows that the smaller the tube radius is the more sensitive the changing in C-C-C angle is. For the SWCNTs with diameter 0.4nm, 1.0nm, 2.0nm and 4.0nm, the C-C-C angles in the tubes decrease 7.3%, 1.2%, 0.3% and 0.08% from 120º in the graphene.
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\n\t\t\t
\n\t\t\t
\n\t\t\t\t
6.1.2 Armchair SWCNTs
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Taking the armchair SWCNTs as another example, we study the effect of the changing in diameter of SWCNTs on the value of C-C-C angle. The SWCNT structure and the geometric diagram are shown in Fig. 19, with which we can obtain,
With\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tArc\n\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.0783\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the last equation becomes,
where the \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t does not change a lot when the graphene curls to the armchair SWCNT so that we make\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ttan\n\t\t\t\t\t\t\t\t∠\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\ttan\n\t\t\t\t\t\t\t\t∠\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t∠\n\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
Thus, with the diameter of the armchair SWCNT provided, setting the AB direction in Fig. 19 as the direction of the 0o fiber, we can calculate the C-C-C angles in armchair SWCNTs as,
The curve drawn from equation (69) is displayed in Fig. 20, which shows, as for the zigzag SWCNTs, that the smaller the tube radius is the more sensitive the changing in C-C-C angle is. For the armchair SWCNTs with diameter 0.4nm, 1.0nm, 2.0nm and 4.0nm, the C-C-C angles in the tubes decrease 5.9%, 0.94%, 0.23% and 0.06% from 120º in the graphene. Comparing the armchair SWCNTs with the zigzag ones with same diameter, the changing in C-C-C angle in the armchair SWCNTs is smaller.
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6.2. Scale effect of elastic properties of SWCNTs
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With the usual continuum model of nanotubes, many researchers use the same isotropic material constants for CNTs with different radius so that the changes of the material properties due to changes in the radial size of CNTs can not be taken into account. In this
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Figure 20.
Different C-C-C angles in armchair SWCNTs with different diameters
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section, axial deformation of the SWCNTs with different diameter is analyzed with the finite element method to study the effect of the changes of the material properties for CNTs with different radius on the mechanical properties.
\n\t\t\t\t
According to the theoretical model described above in this chapter, changes in the elastic properties of CNTs are mainly attributed to the changes of the C-C-C angles when to make CNTs by curling the graphene. The C-C-C angles in CNTs are related to the tube radius, while the corresponding C-C-C angles in the graphene induce varying degrees of anisotropy. Thus, we should choose different anisotropic material parameters for different radial size CNTs based on the parameters of the graphene with different C-C-C angles. Some elastic constants of graphite sheet and SWCNTs are listed in Table 3, the values for SWCNTs calculated with equation (57), (63) and (69). The subscript 1 and 2 in the table identify the along-axis direction of SWCNTs and the circumferential direction.
\n\t\t\t\t
In the finite element simulation, the same axial strain is applied to each SWCNT, with one end of the tube fixed and at the other end imposed the axial deformation. The length of the tube is 6nm and the axial compression strain is 5%. Two series of the elasticity parameters, the isotropic ones (from equation (56)) and the anisotropic ones (from equation (57)), are both tried to obtain the axial forces in SWCNTs for comparison. With the results of the FEA, we use the following equation to define the scale effect of SWCNTs,
where R\n\t\t\t\t\t\n\t\t\t\t\t\tANISO\n\t\t\t\t\t and R\n\t\t\t\t\t\n\t\t\t\t\t\tISO\n\t\t\t\t\t are the axial forces obtained with the anisotropic parameters and the isotropic ones. The scale effect of SWCNTs with different radial size is shown in Fig. 21.
\n\t\t\t\t
FEA results show that the anisotropy of the SWCNTs gradually increased with the decreases in the diameter, leading to more and more obvious scale effect. It can be seen from Fig. 21 that compared with the armchair SWCNTs the zigzag ones show more apparent scale effect and the scale effect of SWCNTs with diameter greater than 2nm is negligible (<0.05%), for tubes with any chirality. However, for zigzag SWCNTs with very small diameter, the scale effect could be very obvious up to 4.4%. That is in good agreement with the other researcher’s results. Thus, the scale effect should be considered in the accurate calculation about the mechanical behaviour of small SWCNTs.
With the data in Table 3 and the results from Fig. 21 we can see that, for SWCNTs with diameter greater than 2nm, the change of C-C-C angle due to the change in tube diameter is negligible so that the tubes could be treated as isotropic materials with the elastic parameters of graphene. For SWCNTs with diameter smaller than 1nm, the change of C-C-C angle due to the change in tube diameter should not be neglected and it is better for the tubes to be treated as anisotropic materials with the elastic parameters calculated from equation (57), (63) and (69).
\n\t\t\t
\n\t\t
\n\t\t
\n\t\t\t
7. Conclusion
\n\t\t\t
A new equivalent continuum model is presented to theoretically investigate the elastic properties of the graphite sheet. By comparison, the equivalent model can properly reflect the actual elasticity status of graphite sheet. Further more this equivalent model is employed to study the radial scale effect of SWCNTs.
\n\t\t\t
The C-C-C bond angles will change from the constant 120o to smaller values when the graphite sheet curling to CNTs. The change of the C-C-C angle is obviously related to the change in CNT radius. Then the relationship between the anisotropy and the changes of the C-C-C angles of CNTs is deduced. The present theory not only clarify some puzzlement in the basic mechanical research of CNTs, but also lay the foundations for the application of continuum mechanics in the theoretical analysis of CNTs.
\n\t\t\t
Based on above theory the scale effect of CNTs is studied. It is showed that the scale effect of the zigzag CNTs is more significant than the armchair ones. For SWCNTs with diameter greater than 2nm, the change of C-C-C angle due to the change in tube diameter is negligible so that the tubes could be treated as isotropic materials with the elastic parameters of graphene, and the scale effect could also be neglected no mater what chirality they are. However, for SWCNTs with diameter smaller than 1nm, the change of C-C-C angle due to the change in diameter should not be neglected (the scale effect neither) and it is better for the tubes to be treated as anisotropic materials with the elastic parameters calculated from corresponding equations.
\n\t\t\t
It is theoretically demonstrated that the graphite sheet is in-plane isotropic under plane stress, which is mainly due to their special structure of C-C-C angles. Any deformation of the graphite molecule making changes in the C-C-C angles, e.g. curling, will introduce anisotropic elastic properties. That provides a direction for applying the composite mechanics to the research in the mechanical properties of CNTs, and also has laid an important foundation.
\n\t\t
\n\t
Acknowledgments
\n\t\t\t
The authors wish to acknowledge the supports from the Natural Science Foundation of Guangdong Province (8151064101000002, 10151064101000062).
\n\t\t
\n',keywords:null,chapterPDFUrl:"https://cdn.intechopen.com/pdfs/16989.pdf",chapterXML:"https://mts.intechopen.com/source/xml/16989.xml",downloadPdfUrl:"/chapter/pdf-download/16989",previewPdfUrl:"/chapter/pdf-preview/16989",totalDownloads:2324,totalViews:931,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"October 26th 2010",dateReviewed:"March 17th 2011",datePrePublished:null,datePublished:"August 17th 2011",dateFinished:null,readingETA:"0",abstract:null,reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/16989",risUrl:"/chapter/ris/16989",book:{slug:"carbon-nanotubes-polymer-nanocomposites"},signatures:"Qiang Han and Hao Xin",authors:[{id:"30529",title:"Prof.",name:"Qiang",middleName:null,surname:"Han",fullName:"Qiang Han",slug:"qiang-han",email:"emqhan@scut.edu.cn",position:null,institution:null},{id:"33728",title:"Dr.",name:"Hao",middleName:null,surname:"Xin",fullName:"Hao Xin",slug:"hao-xin",email:"jackyxh@163.com",position:null,institution:{name:"South China University of Technology",institutionURL:null,country:{name:"China"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Constitutive equations of orthotropic system",level:"1"},{id:"sec_3",title:"3. Macro-mechanics fundamental principle of composite material",level:"1"},{id:"sec_3_2",title:"3.1. Constitutive equations of monolayer under plane stress",level:"2"},{id:"sec_4_2",title:"3.2. Off-axis flexibility and stiffness of monolayer under plane stress",level:"2"},{id:"sec_5_2",title:"3.3. Constitutive equations in classical laminated plate theory",level:"2"},{id:"sec_7",title:"4. Equivalent model of graphite sheet",level:"1"},{id:"sec_7_2",title:"4.1. The basic idea of the equivalent model",level:"2"},{id:"sec_9",title:"4.2 Mechanical properties of graphite sheet at plane stress state",level:"1"},{id:"sec_10",title:"4.3 Elastic constants of monolayer in the equivalent model",level:"1"},{id:"sec_10_2",title:"4.4. Test for the mechanical constants of the monolayer in the equivalent model",level:"2"},{id:"sec_12",title:"5. Elastic properties of graphite sheet",level:"1"},{id:"sec_12_2",title:"5.1. Flexibility of monolayer in the equivalent model",level:"2"},{id:"sec_13_2",title:"5.2. Elastic properties of the equivalent model of graphite",level:"2"},{id:"sec_14_2",title:"5.3. Elastic properties of the equivalent model with various C-C-C bond angles",level:"2"},{id:"sec_16",title:"6. Scale effect of elastic properties of CNTs",level:"1"},{id:"sec_16_2",title:"6.1. Equivalent model of single-walled carbon nanotubes (SWCNTs)",level:"2"},{id:"sec_16_3",title:"6.1.1. Zigzag SWCNTs",level:"3"},{id:"sec_18_2",title:"6.1.2 Armchair SWCNTs",level:"2"},{id:"sec_19_2",title:"6.2. Scale effect of elastic properties of SWCNTs",level:"2"},{id:"sec_21",title:"7. Conclusion",level:"1"},{id:"sec_22",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBrenner\n\t\t\t\t\t\t\tD. 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School of Civil Engineering and TransportationSouth China University of TechnologyGuangzhou, People’s Republic of China
School of Civil Engineering and TransportationSouth China University of TechnologyGuangzhou, People’s Republic of China
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Coleman and Werner J. Blau",authors:[{id:"25377",title:"Prof.",name:"Jun",middleName:null,surname:"Wang",fullName:"Jun Wang",slug:"jun-wang"},{id:"28023",title:"Dr.",name:"Werner J.",middleName:null,surname:"Blau",fullName:"Werner J. Blau",slug:"werner-j.-blau"},{id:"36950",title:"Prof.",name:"Yu",middleName:null,surname:"Chen",fullName:"Yu Chen",slug:"yu-chen"},{id:"122500",title:"Dr.",name:"Rihong",middleName:null,surname:"Li",fullName:"Rihong Li",slug:"rihong-li"},{id:"122501",title:"Dr.",name:"Hongxing",middleName:null,surname:"Dong",fullName:"Hongxing Dong",slug:"hongxing-dong"},{id:"122502",title:"Prof.",name:"Long",middleName:null,surname:"Zhang",fullName:"Long Zhang",slug:"long-zhang"},{id:"122503",title:"Dr.",name:"Mustafa",middleName:null,surname:"Lotya",fullName:"Mustafa Lotya",slug:"mustafa-lotya"},{id:"122504",title:"Prof.",name:"Jonathan N.",middleName:null,surname:"Coleman",fullName:"Jonathan N. Coleman",slug:"jonathan-n.-coleman"}]},{id:"16814",title:"Design and Demonstration of Carbon Nanotubes(CNTs)-Based Field Emission Device",slug:"design-and-demonstration-of-carbon-nanotubes-cnts-based-field-emission-device",signatures:"Tian Jin-shou, Li Ji, Xu Xiang-yan and Wang Jun-feng",authors:[{id:"26918",title:"Dr.",name:null,middleName:null,surname:"Tian",fullName:"Tian",slug:"tian"}]},{id:"16815",title:"Reinforced Thermoplastic Natural Rubber (TPNR) Composites with Different Types of Carbon Nanotubes (MWNTS)",slug:"reinforced-thermoplastic-natural-rubber-tpnr-composites-with-different-types-of-carbon-nanotubes-mwn",signatures:"Sahrim Hj. Ahmad, Mou'ad.A.Tarawneh, S.Y.Yahya and Rozaidi Rasid",authors:[{id:"25298",title:"Dr.",name:"Mou'ad",middleName:null,surname:"Al-Tarawneh",fullName:"Mou'ad Al-Tarawneh",slug:"mou'ad-al-tarawneh"},{id:"26863",title:"Prof.",name:"Sahrim",middleName:null,surname:"Hj. Ahmad",fullName:"Sahrim Hj. Ahmad",slug:"sahrim-hj.-ahmad"}]},{id:"16816",title:"Carbon Nanotubes and Semiconducting Polymer Nanocomposites",slug:"carbon-nanotubes-and-semiconducting-polymer-nanocomposites",signatures:"Duong Huyen",authors:[{id:"43170",title:"Dr.",name:"Duong",middleName:"Ngoc",surname:"Huyen",fullName:"Duong Huyen",slug:"duong-huyen"}]},{id:"16817",title:"Carbon Nanotube-Based Thin Films: Synthesis and Properties",slug:"carbon-nanotube-based-thin-films-synthesis-and-properties",signatures:"Qiguan Wang and Hiroshi Moriyama",authors:[{id:"45902",title:"Prof.",name:"Hiroshi",middleName:null,surname:"Moriyama",fullName:"Hiroshi Moriyama",slug:"hiroshi-moriyama"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"64837",title:"Processing of Beta Titanium Alloys for Aerospace and Biomedical Applications",doi:"10.5772/intechopen.81899",slug:"processing-of-beta-titanium-alloys-for-aerospace-and-biomedical-applications",body:'\n
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1. Introduction
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Titanium was discovered in 1791, but it came into effective application only in the 1950s. After 115 years, i.e., in the year 1906, M. A Hunter at General Electric Company prepared pure titanium for the first time [1]. Since 1950s, titanium holds a prime position in aerospace, biomedical, automotive, and chemical processing industries due to unique features listed below:
Low density (60% of steel or super alloy’s density),
Higher tensile strength (Higher than ferritic stainless steel and comparable to martensitic stainless steel and Fe- base superalloys)
Higher operating temperature (Up to 595°C for commercially available alloys and >595°C for titanium aluminides)
Excellent corrosion resistance (Higher than stainless steel and biocompatible)
Forgeability
Castability (Mostly by investment casting)
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Despite being the fourth-most abundant structural metal available in the earth crust, its commercial exploitation has been low compared to steel and aluminium due to high cost of production.
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Pure Titanium has an hcp crystal structure. Due to the allotropic nature of titanium, the room temperature hcp crystal structure (alpha phase) will be transformed to bcc (beta phase) structure on heating to a particular temperature called beta transus temperature (882.5°C). Alloying elements of titanium are classified on the basis of their influence on the transus temperature. For example, if the transus temperature is increased on the addition of the certain elements, then they are called as alpha stabilisers (Al, O, N, and C); similarly there are some other elements which bring the transus temperature down and they are termed as beta stabilisers (V, Mo, Ta and Nb). The elements Sn and Zr have little or no effect on transus temperature and are termed as neutral elements.
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Beta alloys form the metastable beta phase upon quenching rather than undergoing martensitic transformation. A schematic representation of the beta isomorphous phase diagram is shown in the Figure 1. Beta alloys can also be classified as those which have alloy which has enough beta stabilisers to avoid the martensitic start (Ms) pass through upon quenching. Beta alloys are further classified into metastable and stable beta alloys based on the content of beta stabilisers. Commercially available beta alloys are metastable beta alloys and stable beta alloys are not commercially available [2]. The metastable beta phase can precipitate the fine alpha phase upon ageing/thermal treatment. Hence, beta alloys are hardenable and can attain a higher strength level than alpha + beta alloys and higher specific strength compared to many other alloys [3].
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Figure 1.
Beta isomorphous phase diagram.
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Corrosion resistance of beta alloys is also found to be better than that of alpha + beta alloys. Higher hydrogen tolerance makes beta alloys to perform better in the Hydrogen-rich environments [2]. Increased fracture toughness for a given strength level and amenability to room temperature forming and shaping are superior attributes compared to alpha + beta alloys [1]. Ti-13V-11Cr-3Al (B120VCA) was the first beta alloy produced/developed and used in the SR-71 (Surveillance aircraft) as a sheet product.
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Beta alloys’ inherent characteristics such as pronounced ductility owing to the crystal structure (bcc), heat treatability, and superior cold rollability make them an effective alternative to alpha + beta alloys [4]. Furthermore, beta alloys have lower beta transus temperature than the alpha + beta alloys [5]. Hence, beta alloys are considered to be the economical choice in perspective of processing compared to the alpha + beta alloys. For example, despite the higher formulation cost, Ti-15V-3Al-3Cr-3Sn alloy’s thinner gauges (<2 mm thick) cost one-tenth of those of Ti-6Al-4V [3].
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2. Processing of beta alloy
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2.1 Melting
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The initial step is the fabrication of ingot from sponge for conversion to mill products. The melting practices to produce beta titanium alloy ingots can be broadly categorised into Vaccum Arc Remelting (VAR) and Cold Hearth Melting.
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The conventional method used for the melting of beta titanium alloys is the Vacuum Arc Remelting (VAR) in a consumable arc furnace. In VAR, the furnace is initially evacuated for required vacuum and a dc arc is struck between the two electrodes. Here a consumable electrode (material to be melted) is employed as the cathode and starting materials such as titanium-based metal chips or machine turnings act as the anode. The consumable electrode can be fabricated from either of the two strategies.
From the compacted sponge and/or scrap
From plasma/electron beam hearth melting
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Among these methods, the first method of predensification by compacting using a hydraulic press is widely used to fabricate electrodes. Compacted electrodes with nominal alloy composition are made by the pressing of blended clean and uniform-sized titanium sponge and alloying elements devoid of any harmful inclusions. These compacts (called as briquettes) are then assembled with bulk scrap to form the first melt electrode (called as a stick) by appropriate welding methods.
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Finally, these fabricated electrodes are placed inside a vacuum furnace. When the electric arc is established, associated heat generation will result in the dripping of molten metal down to the water-cooled copper crucible to form the ingot. Initially, a layer of solid titanium or skull will be formed on the surface of cooled copper crucible which will hold the subsequently falling molten metal. In order to ensure chemical homogeneity, the ingots will be inverted and remelting will be performed. Ingots produced during first stage melting are again used as consumable electrodes during double or triple remelting. In addition to this, electrical coils are provided in most of the VAR furnaces to generate an electromagnetic field capable of stirring the molten metal thereby further enhancing the homogeneity. Cold hearth melting is another developing technique which uses either plasma arc (PAM) or electron beam (EBM) melting furnace.
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Proper monitoring should be ensured to control the solidification of beta titanium based ingots. Specifically, beta eutectoid compositions containing Fe, Mn, Cr, Ni and Cu are associated with depressed freezing temperatures [2]. This allows for solidification over a significant temperature range, consequently leading to solute segregation during solidification of the ingot. Such type of segregation results in regions with lower beta transus and results in a microstructure distinctive from the surrounding material. These solute segregated regions are clearly visible in beta titanium alloys subjected to heat treatment below/near to beta transus and are termed as beta flecks. Beta flecks, which range from a scale of few hundred micrometres to a few millimetres, can act as crack nucleation sites leading to fatigue failure. Beta flecks are mostly developed in large diameter ingots. However, beta isomorphous alloys containing Nb, Mo and V are not associated with these depressed solidification temperatures and are less prone to solute segregation.
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Lower values of tensile ductility and low cycle fatigue life of near-β Ti alloy Ti–10V–2Fe–3Al was found to be due to the presence of beta flecks [6]. Under tensile loading, crack nucleation occurred at beta fleck grain boundaries leading to intergranular and quasi-cleavage fracture. In the case of fatigue loading, the inhomogeneous strains developed due to the presence of beta flecks accelerated the crack nucleation and early crack propagation.
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2.2 Casting
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For an expensive material such as titanium, casting is the perfect choice in attaining a (near) net shape in the fabrication of components with complex geometry without incurring much wastage. A significant weight (35%) saving can be achieved by employing the titanium casting instead of stainless steel casting in B-777 aircraft [7]. In general, rammed graphite mould and investment casting were utilised in titanium casting. Investment casting is preferred to obtain thin sections and better surface finish [8]. Ti-5Al-5V-5Mo-3Cr castings followed by HIP (Hot Isostatic Pressing) possess a superior strength compared to hipped Ti-6Al-4V castings with almost same ductility [9]. To extend brake life of fighter aircraft (F-18 EF)Ti-15V-3Al-3Cr-3Sn castings were used instead of Ti-6Al-4V castings due to the higher specific strength of the former [10].
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2.3 Forging and rolling
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2.3.1 Ingot breakdown forging
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To exploit the ductile nature of the beta phase (bcc crystal structure), even for alpha and alpha + beta alloys, ingot break down forging is done above the beta transus temperature. In general, to avoid thermal stress cracking, titanium alloys are subjected to preheating before high-temperature forging.
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Forging is performed to produce billets and bars of titanium with the optimum combination of strength and ductility [11]. Forging is performed using hydraulic presses. Both straight-forging and upset forging are performed in case of Ti alloys. For greater deformation and larger size, upset-forging is preferred [1]. Higher reactivity of the titanium demands the inert / vacuum processing to prevent surface contamination during high-temperature processing [1]. Drawing operation of titanium is prone to galling and seizing. Hence, proper lubricants have to be employed to avoid those effects [1]. Compared to all other Ti alloys, beta alloys can withstand high pressure before cracking. Ti- 13V-11Cr-3Al can withstand up to 690 MPa without cracking. In contrast, Ti-6Al-4V can withstand 585 MPa before cracking [1].
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The microstructure of the ingots of beta alloys varies from small equiaxed grains (at the surface) to elongated columnar grains and large equiaxed grains at the bulk/centre of the ingot [4]. Beta Ti alloys are more suitable for low temperature working without being vulnerable to rupture or cracking compared to other Ti alloys [1] and this effect is attributed to the availability of enough slip systems to accommodate the deformations.
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2.3.2 Secondary forging
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Secondary forging refers to the forging process employed to obtain the final shape/components. The temperature required for this kind of forging is lower than that for ingot breakdown forging. Unlike alpha and alpha + beta alloys, beta alloys show a significant increase in strength at high strain rates [1]. Hence, higher pressures are to be applied for forging of beta alloys; the pressure required to induce crack during forging is higher for beta alloys compared to alpha and alpha + beta alloys [1]. Beta titanium alloys have a broader range of forging temperature compared to alpha/alpha + beta alloys.
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Due to the lower beta transus temperature, beta alloys have lower hot working temperature compared to alpha and alpha + beta alloys, For example, Ti–10V–2Fe–3Al has a secondary working temperature range between 700–870°C [12]. Types of forging and features are given in the Table 1.
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S.No.
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Forging type
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Features
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1
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Open-Die Forging
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Simple shapes can be made
The previous step to closed – die forging
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2
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Closed-Die Forging
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It is also called an impression die forging
More complex shapes can be obtained
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3
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Hot-die forging
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Die is maintained at a higher temperature compared to open and closed die forging
Less energy is required to produce the shape
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4
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Isothermal Forging
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Preheating the metal is required
Dies are at the same temperature as the metal
Capable of producing near net shape components with less energy required
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5
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Precision forging
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No machining required before assembling
Highly sophisticated and expensive method
Aero engine fan blades and even aerofoil shapes are precision forged
Unlike other alloys, rolling of titanium requires higher working pressure and extreme control in temperature. Cylindrical rollers are used to produce the strips, sheet and plate. In contrast, grooved rollers are employed in producing the rounds and other structural shapes. In sheet and plate rolling process, cross rolling is done to reduce the anisotropy in mechanical properties. Texture strengthening is less pronounced in the beta alloys compared to alpha alloys [1]. The lower rate of strain hardening of the beta alloy makes it more acquiescent to cold working.
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In Ti-3.5Al-5Mo-6V-3Cr-2Sn-0.5Fe alloy, rolling and ageing in the sub-transus (alpha + beta field) temperature yielded a better combination of the strength and ductility compared to working in the beta field [13]. Sheet beta Ti alloys are amenable to cold rolling. Cold rolling has a strong effect upon mechanical properties. For example, Rosenberg [14] reported the effect of cold rolling on tensile strength, yield strength and ductility of Ti-15-3 alloy:
Ductility (Rolled alloy) = EL (un-rolled) − 0.65 × Percentage of reduction (%)
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Two high roll mill and three high roll mill are commonly used for rolling titanium and its alloys.
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2.4 Thermomechanical processing
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Material processing performed with the aid of both mechanical force and thermal/ heat treatment can be termed as thermomechanical processing. The primary objective of this processing is to obtain a component in functional design with pre-determined microstructure and corresponding mechanical properties. Thermomechanical processing of beta Ti alloys can be done both above transus temperature (Super-transus processing) and below the transus temperature (Sub-transus processing). Super-transus processing with hot deformation is optimised to obtain fine recrystallised beta grains. Sub-transus processing is optimised to obtain fine beta grains with controlled alpha phase morphology [12]. Size, volume fraction, morphology, and the spatial distribution of the alpha precipitates formed during the thermomechanical processing have a vital influence over the mechanical properties of the end product.
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In Ti-15V-3Al-3Cr-3Sn alloy, Boyer et al. [15], showed the usefulness of thermomechanical treatment for attaining a wide range of tensile strength (from 1070 to 1610 MPa.)
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2.5 Heat treatment
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Heat treatment is the basic metallurgical process through which optimization of hardness, tensile strength, fatigue strength and fracture toughness can be achieved. All the metastable beta alloys are heat treatable to attain higher strength than alpha + beta alloys.
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Duplex ageing treatment yielded a superior combination of mechanical properties with no precipitation free zone and finer alpha precipitation compared to single ageing in Ti-15V-3Al-3Cr-3Sn-3Zr [16] and Ti-3Al-8V-6Cr-4Mo-4Zr [17]. The rate of heating to ageing temperature was found to have a substantial effect on the evolution of microstructure and mechanical properties [18]. Choice of solution treatment temperature is important. For example, for Ti-1Al-8V-5Fe (Ti185), solution treatment near beta transus temperature leads to a highest tensile and yield strength [19].
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Solution treatment followed by ageing in metastable beta alloys will lead to a microstructure consisting of soft alpha in the beta grain boundaries. Hence, this softer alpha phase may lead to the decline in the HCF behaviour [20] and tensile ductility by augmenting the intergranular fracture [17]. For example, Sauer and Luetjering [21] have also reported the adverse effect of alpha phase layers along the beta grain boundaries on the tensile and fatigue behaviour of Ti-5Al-2Sn-4Zr-4Mo-2Cr-1Fe (β CEZ).
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2.6 Surface processing for aerospace application
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Modifying the surface is an effective and economical way to enhance the tribological and fatigue properties of the material. Thermo-chemical and mechanical surface modification techniques are common in beta alloys.
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2.6.1 Thermo-chemical surface modification
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In order to enhance the surface hardness, wear resistance and near-surface strength, thermo-chemical surface processing techniques such as nitriding and carburising are employed. Among various thermo-chemical surface processing techniques, nitriding is extensively used. In this process, the nitrogen is fused into the titanium base alloy. Among the various technologies used for Nitriding, i.e., gas nitriding, laser nitriding, plasma nitriding, Ion nitriding and gas Nitriding are used widely [22]. Titanium nitrides will be formed on the surface as a result of the nitriding and these nitrides increase the surface hardness drastically and improve the tribological properties at the expense of the ductility of the material. Increased hardness due to TiN formation was made use in flap tracks of Military airplanes [23]. However, nitriding has a negative influence on the tensile strength and fatigue strength of the material.
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2.6.2 Mechanical surface modification
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Mechanical surface modifications such as shot peening, ball burnishing and laser peening are developed to enhance the fatigue behaviour of the target material by inducing the residual compressive stress and work hardening effect in near surface region. Both crack nucleation and crack propagation during fatigue loading were found to be affected by the surface modification treatment. However, surface roughness will be significantly increased at the end of the mechanical surface modification such as shot peening and this may lead to early crack initiation.
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Since 1970s, shot peening is being employed in enhancing the mechanical behaviour of Ti alloys in aerospace industries [24]. Schematic representation of shot peening is shown in the Figure 2. Shot peening of beta alloys, i.e. Ti-10V-2Fe-3Al and Ti-3Al-8V-6Cr-4Mo-4Zr yielded a marginal increase in the fatigue life compared to electro polished sample [25]. In LCB beta alloy, in order to compensate the residual compressive stress induced in the surface after peening, substantial tensile residual stress formed in the subsurface region and this deteriorated the fatigue behaviour compared to polished sample [26]. It is important to control the shot peening conditions to get the desired enhancement in fatigue life.
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Figure 2.
Schematic representation of shot peening.
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Unlike shot peening and laser peening, roller burnishing reduces the surface roughness by stressing the surface with a roller ball with optimised pressure. Schematic representation of the roller burnishing is shown in the Figure 3. Roller burnishing of Ti-10V-2Fe-3Al beta alloy induced deeper and higher magnitude residual stress compared to shot peening. In roller burnishing of LCB beta alloy, higher the rolling pressure, deeper was the site for fatigue crack nucleation [27]. In Beta C (Ti-3Al-8V-6Cr-4Mo-4Zr) alloy, deep rolling ended up with deeper residual stress distribution compared to shot peening, but the magnitude of the residual stress remained high for the shot peened sample. A marginal increase in fatigue life was achieved through deep rolling of Beta C alloy [28].
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Figure 3.
Schematic representation of the ball burnishing.
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Compared to shot peening, laser peening has unique features like the capability of inducing deeper and stable residual stress with extreme control in operation. Conventionally, laser peening is performed using Nd: Glass lasers after applying the coating, i.e. black paint on the target surface. To make this process simple, economical and more portable, LPwC (Laser peening without Coating) was developed in 1995 [29]. LPwC has proven to be an effective technique by inducing a relatively high compressive residual stress. For example, a residual stress of approx. −825 MPa was induced at a depth of ~75 μm from the surface in LCB (Ti-6.8Mo-4.5Fe-1.6Al) beta alloy [30].
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2.7 Surface processing for bio-medical application
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In the case of implant materials, the interaction between the biological environment and the implanted materials occurs on the biomaterial surface. Clinical success of implant materials is greatly dependent on various surface characteristics viz. chemical inertness, texture, corrosion resistance and surface energy [31] . In the case of orthopaedic implants, the surface should possess more bone forming ability and for blood contacting devices, it should not initiate any blood clot formation. Hence surface modification of biomedical grade beta titanium alloys is very significant. Oxide layer formation will occur spontaneously on the surface of titanium on exposure to air. This TiO2 film possesses a thickness of about 1.5 to 10 nm at room temperature. Chemical stability and structural characteristics of this oxide film greatly influence the biocompatibility of titanium implant materials. Some of the potential methods to enhance the properties of native TiO2 film are anodisation, sol–gel methods, acidic and alkaline treatments [32]. In addition to these, specific surface topographies and roughness induced by mechanical surface modifications (sandblasting, grit blasting, peening) have improved the clinical success of implant materials. An overview of the various surface modification techniques employed for biomedical beta titanium alloys is schematically shown in Figure 4.
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Figure 4.
Overview of surface modification of beta titanium alloys for biomedical application [33].
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In dental applications, Laser Nitriding has proved to be an effective process in enhancing the surface hardness, the coefficient of friction and corrosion resistance of the Ti-20Nb-13Zr and wear and corrosion resistance of Ti-13Nb-13Zr biomedical-beta alloys [34, 35]. Plasma nitrided beta 21S (Ti-15Mo-3Nb-3Al-0.2Si) alloy showed higher hardness but inferior corrosion resistance compared to the untreated alloy [36]. In line with the Nitriding, carburising of Ti-13Nb-13Zr (a biomedical beta alloy used for artificial joints) improved the surface hardness and wear resistance through the formation of the titanium carbide [37].
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2.8 Powder metallurgy
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As mentioned in the introduction (Section 1), a major limiting factor for the titanium application is its high production cost. In addition to the high raw material cost, the forging, machining contribute majorly to the production cost. This limitation instigated the industries to work towards processing methods through which the near net shape (NNS) could be obtained. Despite the higher cost involved, Powder metallurgy of titanium is capable of yielding almost same or better mechanical properties compared to wrought and cast components along with accurate net shape capability. This merit is mainly attributed to the absence of texture, segregation and nonuniformity in the grain size encountered in conventional processing.
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Even for the components made through powder metallurgy route, solution treatment followed by ageing (STA) leads to an enhancement in mechanical properties such as tensile strength and yield strength compared to the as-sintered condition [38]. Ti-10V-2Fe-3Al and Ti-11.5Mo-6Zr-4.5Sn alloys have been produced through powder metallurgy route. However, 90% of the powder metallurgy is focussed on the alpha + beta alloy Ti-6Al-4V.
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Guo et al. [39] reported a remarkable increase in the mechanical properties of Ti-10V-2Fe-3Al powder alloy compared to the wrought and cast products through isothermal forging of the sintered alloy. Jiao et al. [40] studied the model of alpha phase spatial distribution in laser additive manufactured Ti-10V-2Fe-3Al. The influence of nano-scale alpha precipitates on tensile properties of age hardened laser additive manufactured Ti-5Al-5Mo-5V-1Cr-1Fe (Ti-55,511) alloy was studied by He et al. [41] and the authors reported that precipitated nanoscale alpha precipitates have led to a decline in ductility.
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3. Applications
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3.1 Aerospace applications
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A recent forecast released by Airbus Industries [42], confirms the promising development of air transport requiring 37,400 aircraft at a value of 5.8 trillion US dollars business in the next 20 years. However, reducing the fuel consumption to control the emission of CO2 and NOx is the driving factor for the aerospace industries and this could be possible by reducing the overall weight [43]. Similarly, in space application weight of the payload is more crucial than civil/cargo aviation. Ti-6Al-4V is a workhorse for the aerospace industry for several decades and 65% of total titanium production in the United States belongs to Ti-6Al-4V alloy [3].Even though the alpha +beta alloys dominated the scene, beta alloys with their unique characteristics such as excellent hardenability, heat treatability to high strength levels and a high degree of sheet formability, are becoming increasingly important for the aerospace sector. Beta alloys and their aerospace application are listed in the Table 2.
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S. No.
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Alloy
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Application/components
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1
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Ti-15V-3Al-3Cr-3Sn
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Landing gear, springs, sheet, plate and airframe castings, environmental control system ducting
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2
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Ti-6V-6Mo-5.7Fe-2.7Al
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Fasteners
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3
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Ti-13V-11Cr-3Al
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Airframe, landing gear and springs
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4
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Ti-3Al-8V-6Cr-4Mo-4Zr (β-C)
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Springs and fasteners
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5
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Ti-11.5Mo-6Zr-4.5Sn
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Rivbolts—Boeing 747
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6
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Ti-5Al-5Mo-5V-3Cr
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Aircraft landing gear, Fuselage components and high lift devices
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7
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Ti-10V-2Fe-3Al
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Aircraft landing gear
High strength forging
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8
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Beta 21s
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Nozzle assembly parts in Boeing 777
Planned to use in Pratt & Whitney PW4168 engine components
Titanium is the ultimate choice for biomedical applications as they outperform conventionally used biomedical alloys such as 316L stainless steel and cobalt-chromium alloys [47]. The formation of a nanometre thick oxide layer on titanium when exposed to any environment imparts high corrosion resistance and superior biocompatibility [48]. All classes of titanium α, α + β, near β and β alloys are widely used for biomedical applications.
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Despite being initially developed for aerospace applications, CP titanium and Ti-6Al-4V are still the most widely used Ti grades being used for biomedical applications. However, CP Ti is associated with lower wear resistance and Ti-6Al-4V when implanted inside the body releases Al and V ions which can lead to severe neurological disorders and allergic reactions. Moreover, the elastic modulus values of these alloys (~110 GPa) are almost four times than that of human cortical bone (20–30 GPa) which can lead to stress shielding effect. This led to the development of β-Ti alloys composed of non-toxic elements and their inherent lower elastic modulus assists in reducing the stress shielding effect when used for orthopaedic applications [3]. Alloy systems based on Ti-Nb, Ti-Mo, Ti-Ta and Ti-Zr are potential materials for biomedical applications. Some of these β-Ti alloys initially developed are Ti-15Mo-5Zr-3Al, Ti-12Mo-6Zr-2Fe (TMZF), Ti-15Mo-3Nb-0.3O (21SRx) and Ti-13Nb-13Zr possessing modulus values in the range of 70–90 GPa.
\n
In the early 1990s, medical device industry focused on developing these low modulus β-Ti alloys for orthopaedic applications. Initially, two β-Ti alloys Ti-13Nb-13Zr specified by ASTM F1713 and Ti-12Mo-6Zr-2Fe (TMZF) specified by ASTM F1813 received Food and Drug Administration approval as implant materials. Among these, TMZF alloy possesses an elastic modulus of about 74–85 GPa, with a yield strength of 1000 MPa. During the early 2000s, this metastable β-Ti alloy was used for making hip stems, which rub against a modular neck made from a cobalt-chromium based alloy. However, in 2011, the US Food and Drug Administration recalled the use of this TMZF alloy due to the unacceptable level of wear debris formation. Another β-Ti alloy 21SRx is derived from the aerospace alloy 21S from which aluminium was eliminated over biocompatibility concerns. In addition, alloys such as Ti-29Nb-13Ta-4.6Zr and Ti-35Nb-7Zr-5Ta are receiving increasing attention due to their lower elastic moduli of about 65 and 55 GPa, respectively, lower than other β-Ti alloys [50].
\n
Apart from orthopaedics, titanium is extensively used in the dental applications [49]. In the case of orthodontic wire material, it should possess three general characteristics viz. large spring back (ability to be deflected over longer distances without permanent deformation), lower stiffness and high formability [51]. The initially utilised materials for orthodontic wire application were gold based alloys containing copper, palladium, platinum or nickel. However, spring back values of these gold alloys were limited owing to their lower yield strength. In the 1960s gold was replaced by stainless steel and cobalt-chromium based alloy (elgiloy). These materials continue to be the standard orthodontic wire material for the past 70 years and possess higher springiness and strength with comparable corrosion resistance. During the early 1970s, nickel-titanium alloy Nitinol (Nickel Titanium Naval Ordinance Laboratory) was also used for orthodontic wires. Even though Nitinol orthodontic archwires are widely used owing to their superior superelastic properties, their use is hampered by reduced formability during the final stages of treatment. Moreover, there are serious concerns over the nickel ion release from these materials in the oral environment. It was later demonstrated that orthodontic wires made from β-Ti alloy Ti-11.3Mo-6.6Zr-4.3Sn (TMA alloy) possess enhanced spring back and formability, along with reduced stiffness. TMA alloys possess ideal elastic modulus values lower than that of stainless steels and higher than nitinol [51]. The higher surface roughness associated with these TMA wires can, however, lead to arch wire-bracket sliding friction due to the high coefficient of friction of TMA alloys. One of the most successful approaches to tackle this problem is the ion implantation process which renders the TMA wires with lower surface roughness and reduced friction coefficients. Another beta titanium alloy Ti-6Mo-4Sn was also investigated for orthodontic wire applications. By proper heat treatment procedures, this alloy exhibited an elastic modulus of 75 GPa and a tensile strength of 1650 MPa [52]. Ti-13V-11Cr-3Al, metastable Ti-3Al-8V-6Cr-4Mo-4Zr, metastable Ti-15V-3Cr-3Al-3Sn, near-beta Ti-10V-2Fe-3Al were also researched for dental archwire applications.
\n
Though beta titanium alloys possess superior haemocompatibility, which is beneficial for cardiovascular devices, they are not fully exploited for cardiovascular applications. Despite higher haemocompatibility, no β-Ti alloy based stents have been commercialised which can be attributed to their lower ductility and modulus as compared to 316L stainless steel and cobalt-chromium based stent materials. Recently, research based on the development of new β-Ti alloy compositions for coronary stent applications has been getting increased attention. Initial studies on Ti-12Mo (wt %) and ternary Ti-9Mo-6W (wt %) demonstrated a ductility of about 46% and 43% respectively [53]. Apart from this, initial investigations on Ti-50Ta, Ti-45Ta-5Ir and Ti-17Ir for stent applications were performed by Brien et al. [54]. Among the three alloys, Ti-17Ir exhibited a favourable elastic modulus of 128 GPa owing to the eutectoid Ti3Ir phase precipitation; iridium content will also assist in improving the fluoroscopic visibility of the stents during interventional procedures [54].
\n
\n
\n
\n
4. Conclusions
\n
Beta titanium alloys have shown much promise and extensive research and development work has been devoted to this group of alloys over the last four decades. For aerospace applications, their heat treatability, high hardenability, high strength to weight ratio and excellent hot and cold workability are major attractions. For orthopaedic applications, their corrosion resistance to biofluids, biocompatibility and low elastic modulus coming close to that of human bone are the important attractive features. Accordingly, development of cost-effective processing techniques has also assumed importance. Problems unique to beta titanium alloys such as high degree of proneness to segregation, high loads to be applied during hot working etc. have since been resolved. Powder processing and additive manufacturing of the alloys have recently received attention and hold promise. Surface modification has been an important part of the developmental efforts and has taken a prominent place, especially for biomedical applications. Coming years are bound to witness increased exploitation of this group of alloys, particularly in biomedical and aerospace applications.
\n
\n
Acknowledgments
\n
The authors would like to express their gratitude to the Management of Vellore Institute of Technology (VIT)—Vellore campus, Tamil Nadu, India, for allowing us to submit this manuscript.
\n
\n
Conflict of interest
\n
The authors declare that there is no conflict of interest regarding the publication of this article.
\n
\n',keywords:"beta titanium alloys, titanium melting, thermomechanical processing, surface modification, aerospace and biomedical applications",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/64837.pdf",chapterXML:"https://mts.intechopen.com/source/xml/64837.xml",downloadPdfUrl:"/chapter/pdf-download/64837",previewPdfUrl:"/chapter/pdf-preview/64837",totalDownloads:880,totalViews:0,totalCrossrefCites:0,dateSubmitted:"June 20th 2018",dateReviewed:"October 8th 2018",datePrePublished:"December 18th 2018",datePublished:null,dateFinished:null,readingETA:"0",abstract:"The unique combination of attributes—high strength to weight ratio, excellent heat treatability, a high degree of hardenability, and a remarkable hot and cold workability—has made beta titanium alloys an attractive group of materials for several aerospace applications. Titanium alloys, in general, possess a high degree of resistance to biofluid environments; beta titanium alloys with high molybdenum equivalent have low elastic modulus coming close to that of human bone, making them particularly attractive for biomedical applications. Bulk processing of the alloys for aerospace applications is carried out by double vacuum melting followed by hot working. There have been many studies with reference to super-solvus and sub-solvus forging of beta titanium alloys. For alloys with low to medium level of molybdenum equivalent, sub-solvus forging was demonstrated to result in a superior combination of mechanical properties. A number of studies have been carried out in the area of heat treatment of beta titanium alloys. Studies have also been devoted to surface modification of beta titanium alloys. The chapter attempts to review these studies, with emphasis on aerospace and biomedical applications.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/64837",risUrl:"/chapter/ris/64837",signatures:"Sudhagara Rajan Soundararajan, Jithin Vishnu, Geetha Manivasagam and Nageswara Rao Muktinutalapati",book:{id:"8408",title:"Titanium Alloys",subtitle:"Novel Aspects of Their Manufacturing and Processing",fullTitle:"Titanium Alloys - Novel Aspects of Their Manufacturing and Processing",slug:"titanium-alloys-novel-aspects-of-their-manufacturing-and-processing",publishedDate:"November 27th 2019",bookSignature:"Maciej Motyka, Waldemar Ziaja and Jan Sieniawsk",coverURL:"https://cdn.intechopen.com/books/images_new/8408.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"101690",title:"Associate Prof.",name:"Maciej",middleName:null,surname:"Motyka",slug:"maciej-motyka",fullName:"Maciej Motyka"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Processing of beta alloy",level:"1"},{id:"sec_2_2",title:"2.1 Melting",level:"2"},{id:"sec_3_2",title:"2.2 Casting",level:"2"},{id:"sec_4_2",title:"2.3 Forging and rolling",level:"2"},{id:"sec_4_3",title:"2.3.1 Ingot breakdown forging",level:"3"},{id:"sec_5_3",title:"Table 1.",level:"3"},{id:"sec_6_3",title:"2.3.3 Rolling",level:"3"},{id:"sec_8_2",title:"2.4 Thermomechanical processing",level:"2"},{id:"sec_9_2",title:"2.5 Heat treatment",level:"2"},{id:"sec_10_2",title:"2.6 Surface processing for aerospace application",level:"2"},{id:"sec_10_3",title:"2.6.1 Thermo-chemical surface modification",level:"3"},{id:"sec_11_3",title:"2.6.2 Mechanical surface modification",level:"3"},{id:"sec_13_2",title:"2.7 Surface processing for bio-medical application",level:"2"},{id:"sec_14_2",title:"2.8 Powder metallurgy",level:"2"},{id:"sec_16",title:"3. Applications",level:"1"},{id:"sec_16_2",title:"3.1 Aerospace applications",level:"2"},{id:"sec_17_2",title:"3.2 Biomedical applications",level:"2"},{id:"sec_19",title:"4. Conclusions",level:"1"},{id:"sec_20",title:"Acknowledgments",level:"1"},{id:"sec_20",title:"Conflict of interest",level:"1"}],chapterReferences:[{id:"B1",body:'Froes FH. Titanium: Physical Metallurgy Processing and Application. Materials Park, Ohio: ASM International; 2014'},{id:"B2",body:'Lütjering G, Williams JC. Titanium. 2nd ed. Berlin, Germany: Springer; 2007'},{id:"B3",body:'Bania PJ. Beta titanium alloys and their role in the titanium industry. Journal of Metals. 1994;46:16-19. DOI: 10.1007/BF03220742'},{id:"B4",body:'Froes FH, Bomberger HB. The Beta titanium alloys. Journal of Metals. 1985;37:28-37. DOI: 10.1007/BF03259693'},{id:"B5",body:'Kolli R, Devaraj A. A review of metastable Beta titanium alloys. Metals (Basel). 2018;8:506. DOI: 10.3390/met8070506'},{id:"B6",body:'Zeng WD, Zhou YG. 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DOI: 10.1016/j.actbio.2008.03.002'}],footnotes:[],contributors:[{corresp:null,contributorFullName:"Sudhagara Rajan Soundararajan",address:null,affiliation:'
School of Mechanical Engineering, Vellore Institute of Technology (VIT), India
School of Mechanical Engineering, Vellore Institute of Technology (VIT), India
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