The Third Taiwan-Japan Workshop on Mechanical and Aerospace Engineering Hualian, TAIWAN, R.O.C. Nov. 28-29, 2005 s A Finite Element Approach for Estimation of Young’ Modulus of Single-walled Carbon Nanotubes Cheng-Wen Fan 1, Jhih-Hua Huang 2, Chyanbin Hwu 2 and Yu-Yang Liu2 1 Department of Mechanical Engineering, Yung-Ta Institute of Technology & Commerce, Ping-Tung, Taiwan, R.O.C. 2 Institute of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan, R.O.C. s ABSTRACT: In this paper, the Young’ modulus of single-walled carbon nanotubes is estimated by a finite element approach. Individual carbon nanotube is simulated as a frame-like structure and the primary bonds between two nearest-neighboring atoms are treated as beam members. The beam properties for input into a finite element model are calculated via the concept of energy equivalence between molecular mechanics and structural mechanics. s To verify this approach, the computed result of Young’ modulus of graphite sheets is examined, which shows good agreement with available literatures. Moreover, finite element s models of both armchair and zigzag carbon nanotubes are established and the Young’moduli s of these tubes are then effectively predicted. The relations of the obtained Young’moduli to the diameters of the nanotubes are also discussed. All the modeling and computing work in this paper are performed by the finite element commercial code ANSYS. s KEYWORDS: carbon nanotubes, Young’modulus, finite element model INTRODUCTION s Since Iijima’ discovery in 1991 (Iijima, 1991), carbon nanotubes (CNTs) have stimulated extensive research activities devoted to nanomechanics and their applications in nanoengineering. Due to their exceptional mechanical and electrical properties: small size, low density, high stiffness, high strength etc., CNTs represent a very promising material in many areas of science and industry. The elastic properties of multi- and singled-walled nanotubes (MWNTs and SWNTs) have been the subject of numerous research works in experimental, molecular dynamics, and elastic continuum modeling approaches. For examples, experimental investigations conducted s by Treacy et al. (1996), who used TEM to measure the Young’modulus of MWNTs, reported a mean value of 1.8 Tpa with a variation from 0.40 to 4.15 Tpa; Krishnam et al. (1998) also s used TEM to observe the vibration of a SWNT and obtained the Young’ modulus ranged from 0.90 to 1.70 Tpa; Wang et al. (1997) conducted bending tests on cantilevered tubes using s atomic force microscopy and estimated a Young’modulus of 1.28 Tpa; Poncharal et al. (1999) observed the static and dynamic mechanical deflections of cantilevered MWNTs, which is induced by an electric field, and reported a modulus about 1.0 Tpa for small-diameter tubes. Besides the experimental works on CNTs, many researchers try to analyze the elastic properties of CNTs by theoretical modeling techniques. These modeling approaches can be generally classified into two categories. One is the molecular dynamics (MD) method (Iijima et al., 1996; Gau et al., 1998; Zhang et al., 1998; Zhou et al., 2000; Belytschko et al., 2002), which is based on the force field and total potential energy related to the interatomic potentials for CNTs in a macroscopic sense. In this method, the bonding and nonbonding potentials are represented in terms of the force constants and the deformation among the atomic bonds, and then elastic moduli are determined by applying different small-strain deformation modes. However, the computational expense of MD simulations limits the size of CNTs that can be studied by this technique. The other approach is the continuum/finite element method (Zhang et al., 2002a, 2002b; Jin and Yuan, 2003; Li and Chou, 2003). Since a nanotube can be well described as a continuum solid beam or shell subject to tension, bending, or torsional forces, it is reasonable to model the nanotube as a frame- or shell-like structure, then the elastic properties of such a structure can be obtained by classical continuum mechanics or finite element method. However, due to the uncertainty of the CNT wall thickness and modulus for both of the above modeling techniques, the obtained elastic properties of SWNTs or MWNTs have scattered values, for example, the results of axial s Young’ modulus ranged from about 1.0 Tpa to 5.5 Tpa can be found in the existing literatures. s The purpose of this paper is to estimate the Young’ moduli of SWNTs by a finite element (FE) approach. Since the carbon nanotube can be treated as a frame-like structure, the primary bonds between two nearest-neighboring atoms can be modeled as beam elements in view of the concept of finite element method. The beam properties for input into a finite element model are calculated via the concept of energy equivalence between molecular s mechanics and structural mechanics. The Young’ moduli of both armchair and zigzag nanotubes with different diameters will be calculated by this approach, and the relations of the obtained moduli to the diameters of the nanotubes will be also discussed. For saving the effort of program coding and making this approach more popular, all the modeling and computing work are performed by the finite element commercial code ANSYS. FINITE ELEMENT MODELING As mentioned above, CNTs can be treated as a frame-like structure with their bonds as beam members and carbon atoms as joints. In (Li and Chou, 2003), the stiffness method was used to simulate the mechanical behavior of nanotubes, they established the linkage between the force constants in molecular mechanics and the element stiffness in structural mechanics through the energy equivalence concept. The key point of this concept is that the simplest harmonic forms of the various steric potential energy of nanotube bonds are adopted under the assumption of small deformation, i.e. 1 1 U r k r ( r 0 ) 2 kr ( ) 2 , r r (1) 2 2 1 1 U k( 0 ) 2 k( )2 , (2) 2 2 1 U k( ) 2 , U U (3) 2 where U r stands for bonded stretching energy, U θ for bonded angle bending energy, U is the combination of the dihedral angle torsion energy U and the improper (out-of-plane) torsion energy U ; kr , k and k are the bond stretching force constant, bond angle bending force constant and torsional resistance respectively, and the symbols , and r represent the bond stretching increment, the bond angle change and the angle of bond twisting, respectively. By comparing these energy forms to their counterparts in structural mechanics, which are 1 L N2 1 N 2 L 1 EA U A dL ( ) 2 , L (4) 2 0 EA 2 EA 2 L 1 LM2 2 EI 2 1 EI U M dL (2 2 , ) (5) 2 0 EI L 2 L 1 L T2 1 T 2 L 1 GJ UT dL ( )2 , (6) 2 0 GJ 2 GJ 2 L where U A , U M and U T are the strain energy of a uniform beam of length L, cross-section A and moment of inertia I under axial force, pure bending and pure torsion respectively; E and G are Young’modulus and shear modulus; is the total axial stretching deformation, 2α s L denotes the total relative rotation angle, is the relative torsion angle and J the polar moment of inertia. The concept of energy equivalence between the two systems implies that both U r and U A represent the stretching energy, both U and U M the bending energy, both U and UT the torsional energy. Then it is reasonable to assume that is equivalent to , and L r 2α equivalent to , and equivalent to . Therefore, by comparing Eqs.(1)-(3) EA EI GJ with Eqs.(4)-(6), a direct relationship between the element stiffness , and and L L L the force constants in molecular mechanics kr , k and k can be obtained as follows EA EI GJ kr , k, k. (7) L L L Eq. (7) constructs the base of the stiffness method employed by (Li and Chou, 2003), and they selected the values of the force constants kr , k and k according to the experience with graphite sheets as -2 mol =6.53 -7 N/nm , kr = 938 kcal 1 A 10 k= 126 kcal /rad 2 =8.79 -10 N mol 1 10 nm/rad 2 , and k is adopted as 40 kcal /rad 2 =2.79 -10 N mol 1 10 nm/rad 2 , which is numerically proven to have little influence on CNTs’ s Young’modulus in their paper. By self-developed program, the computed elastic properties of the SWNTs are obtained and their dependence on the tube diameter is also discussed. Note that the Poisson’ratio which is one of the primary property of materials, does s , not appear explicitly in the formulation of Li and Chou briefly reviewed above. However, by h o sn ao e s s t the selected values of the force constants, the restriction of t P i o’r i that should be a positive number smaller than 0.5 is violated when calculating its value from Eq. (7) and the well known relationship of E and G, i.e. G E / 2(1 ) , for an isotropic and uniform beam. In fact, all the other available choices of the values of the force constants found in literatures t can’satisfy this restriction either. This is the main defect necessary to be solved or avoided for theoretical completeness of this approach. In the present study, a finite element method implemented by the commercial code ANSYS, instead of the stiffness method, is developed under the same theoretical base described above. The major advantage of the FE approach proposed here is that no extra effort is needed for program coding and consequently make the present approach more feasible for mechanical analysis of nanotubes. In our finite element modeling work, the BEAM4 element in ANSYS is selected to simulate the carbon bonds while the atoms are nodes. Fig.1 (modified from Tserpes and Papanikos, 2005) depicts how the hexagonal lattice of the CNT is simulated as structure elements of a space frame, where a c stands for the initial length of C-C bond. c Fig. 1. Schematic finite element simulation of a CNT To determine the elastic properties of the BEAM4 element for input into the ANSYS code, we assume that the cross sections of the beam elements are identical and circular, and note that only the Young’modulus E, the Poisson’ratio and the diameter of the circular s s cross section d are needed to be prepared for the properties of the BEAM4 element. From the first two equations of Eq. (7), we have k Lk 2 d 4 , E r . (8) kr 4 k By Eq. (8) and the values of kr and k given above, the numerical values of E and d are calculated as 5.49 Tpa and 0.147 nm respectively. The element length is set to be equal to the initial C-C bond length ac of 0.142 nm. As to the Poisson’ratio several values of 0.05, c s , 0.1, 0.2 and 0.3 have been examined for the present FE model, and the tests prove that has little effect on the final result in our computation. Therefore, we set = 0.3 as a representative value in the FE model. Before we key in the input data of the BEAM4 element properties, the dimensions of the parameters stated above should be further adjusted to avoid digits overflow/underflow error during the computation performed by ANSYS. Thus, we adjust the dimensions as follows Lin 10 L , Fin 20 F , Ein E , 10 10 (9) where the original dimensions of length L, force F and modulus E are m, N and N/m2 respectively, and the subscript “ denotes the real input values. After such adjustment, the in” numerical parts of the input data prepared for the BEAM4 element can be listed as follows E in .49 12 , Ain .69 , 0.3 , 5 10 1 in where Ain in / 4 is the area of the cross section, din = 1.47 is the diameter of the circular d2 section. O N ’ D L S F R P IE H E A Y U G SMO U U O AG A H T S E T– VERIFICATION CASE To verify the feasibility of the model described above, we first calculate the Young’ s modulus of a graphite sheet, which can be rolled into a carbon nanotube. It is also expected to provide useful information about the selection of those force constants from the calculation result. By treating the graphite sheet as an elastic plane structure, a uniform tensile load is applied at one end of the sheet and the other end is set to be fixed, as depicted in Fig.2. Then s the Young’modulus can be determined from classical elasticity theory, i.e. F / Ag FH Yg , (10) / H tg H HW s where Yg is the Young’modulus of the sheet, F is the total force applied on the nodes at one end, Ag t g is the cross-sectional area of the sheet with width W and thickness t g , W H and are the initial length and the induced elongation respectively. The thickness t g is H taken as the interlayer spacing of graphite, 0.34 nm. Fig. 2. FE model of the graphite sheet s An energy form in elasticity theory to calculate the Young’modulus is also used as an alternative approach, i.e. 1 U N 2 1 U N 2 Yg , (11) Vg 2 Wtg H 2 where Vg g H is the total volume of the graphite sheet, and U N is the total strain Wt energy. In this approach, prescribed displacements instead of the tensile forces are applied at the end nodes. s Table 1 lists the computed Young’moduli of the graphite sheets by the ANSYS code for different model sizes. From the results shown in Table 1, it can be seen that the obtained s Young’moduli of the graphite sheets are fairly closed to the commonly accepted value 1.025 Tpa (Kelly, 1981) and those presented by Li and Chou (2003). In addition, we can also observe that the calculating result is weakly affected by model size. s Table 1. The computed Young’moduli of the graphite sheets for different model sizes Width (nm) Height (nm) s Young’modulus (TPa) Eq. (10) Eq. (11) Li and Chou (2003) 0.738 2.842 1.002 1.003 0.995 0.985 3.126 1.011 1.070 1.002 1.969 4.831 0.996 1.001 1.021 1.969 9.847 1.048 1.051 1.024 S YOUNG’ MODULUS OF A SWNT A successful verification of the present FE model is achieved in last section. In the following, we will apply this method and employ the same parameters of the BEAM4 element s to estimate the Young’moduli of SWNTs. Two main types of SWNTs, zigzag and armchair, s are considered. Fig.3 depicts the typical FE model for calculating the Young’modulus of a single SWNT. As shown in Fig.3, the prescribed forces (or displacements) are applied at one end of the tube, while the other end is fixed. Similar to Eqs. (10) and (11), the equations used on’ ou s s u to calculate the Y ug m dl Y of a SWNT are slightly modified as follows F / At FLt Y , (12) t / Lt t L dt L and 1 U N 2 1 U N , 2 Y (13) Vt 2 t 2 dtL where At dt is the cross-sectional area, t and d are respectively the initial wall thickness and the diameter of the SWNT, Lt and t are the initial tube length and the elongation in L the axial direction, Vt dtLt denotes the total volume of the tube, U N is the total strain energy. Note that different values of the wall thickness of the SWNTs, ranging from 0.064 to ’ 0.69 nm, have been used in many researchersstudies. Moreover, they will result in widely s scattered values of the Young’ moduli of SWNTs. In majority of the studies, it has been assumed that the wall thickness is equal to the interlayer spacing of graphite, i.e. 0.34 nm. Thus, in order to make a reasonable comparison with existing literatures in which t = 0.34 nm is specified, we also select this value to be the wall thickness in our FE model. s Fig. 3. FE model for calculating the Young’modulus of a single SWNT s Table 2 lists the computed Young’ moduli of both zigzag and armchair SWNTs with different tube diameters by Eqs. (12) and (13), and these results are also displayed in Fig.4. From Fig.4, it can be seen that the trend is similar for both zigzag and armchair SWNT, and the effect of the tube chirality is not significant. It is also observed that the effect of the s diameter on the Young’moduli of both zigzag and armchair SWNTs is evident, especially for s diameters smaller than 0.8 nm. With increasing tube diameter, the Young’moduli of zigzag SWNTs increase slightly faster than those of armchair SWNTs, and this increasing tendency diminished gradually for both types as the diameters larger than 1.1 nm. As pointed out in Li and Chou (2003), this increase for both zigzag and armchair SWNTs is due to the effect of nanotube curvature. Higher curvature, which results in more significant distortion of C-C s bonds, leads to a smaller Young’modulus. In addition, the convergent values of the Young’ s moduli estimated in this FE model are about 1.042 Tpa for zigzag SWNTs and 1.038 Tpa for armchair SWNTs, which is in reasonable agreement with that obtained by Li and Chou (2003), i.e. 1.04 Tpa for both types. s Table 2. The computed Young’ moduli of both zigzag and armchair SWNTs with different tube diameters s Young’modulus (TPa) Tube type Diameter (nm) Eq. (12) Eq. (13) zigzag(5,0) 0.3917 1.0186 0.9903 zigzag (10,0) 0.7830 1.0389 1.0175 zigzag (15,0) 1.1750 1.0418 1.0381 zigzag (20,0) 1.5670 1.0427 1.0405 zigzag (25,0) 1.9590 1.0434 1.0422 armchair (3,3) 0.4070 1.0042 1.0313 armchair (6,6) 0.8140 1.0134 1.0346 armchair (9,9) 1.2210 1.0162 1.0363 armchair (12,12) 1.6280 1.0176 1.0368 armchair (15,15) 2.0350 1.0185 1.0378 1.07 Young's modulus (TPa) ) Zigzag  Zigzag  1.05 Arm-chair  Arm-chair  1.03 1.01 0.99 0.2 0.5 0.8 1.1 1.4 1.7 2.0 2.3 Diameter (nm) s Fig. 4. Young’moduli of SWNTs versus tube diameters CONCLUSIONS s A FE approach for estimating the Young’moduli of the zigzag and armchair SWNTs has been developed and implemented by a commercial tools ANSYS. Based on the fact that the nanotubes can be treated as a frame-like structure, a simple linkage between the force constants in molecular mechanics and the elastic properties of the beam member in structural mechanics is established through the energy equivalence concept. We obtain the Young’ s moduli of about 1.042 Tpa for zigzag and 1.038 Tpa for armchair SWNTs at large tube diameters. These results are comparable to those found in the existing literatures. Furthermore, s dependence of the Young’ moduli to tube diameters has been also investigated. The s investigation shows that the Young’ moduli increase with increasing diameter, and then gradually approaches to a value close to that of graphite sheet when the tube diameter becomes large. The main advantage of the proposed method is the significant saving of the s program coding time and the fair accuracy in estimating the Young’moduli of the SWNTs when compared to the other numerical methodologies. It can be concluded that this approach is a valuable tool for studying the mechanical behavior of carbon nanotubes. ACKNOWLEDGEMENT The support from National Science Council (NSC), R.O.C., through grant NSC93-2212-E-006-025 is appreciated by the authors. 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