A Finite Element Approach for Estimation of Young s Modulus of Single-walled Carbon Nanotubes

Transcription

1 The Third Taiwan-Japan Workshop on Mechanical and Aerospace Enineerin Hualian, TAIWAN, R.O.C. Nov. 8-9, 005 A Finite Element Approach for Estimation of Youn s Modulus of Sinle-walled Carbon Nanotubes Chen-Wen Fan 1, Jhih-Hua Huan, Chyanbin Hwu and Yu-Yan iu 1 Department of Mechanical Enineerin, Yun-Ta Institute of Technoloy & Commerce, Pin-Tun, Taiwan, R.O.C. Institute of Aeronautics and Astronautics, National Chen Kun University, Tainan, Taiwan, R.O.C. ABSTRACT: In this paper, the Youn s modulus of sinle-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-neihborin atoms are treated as beam members. The beam properties for input into a finite element model are calculated via the concept of enery equivalence between molecular mechanics and structural mechanics. To verify this approach, the computed result of Youn s modulus of raphite sheets is examined, which shows ood areement with available literatures. Moreover, finite element models of both armchair and ziza carbon nanotubes are established and the Youn s moduli of these tubes are then effectively predicted. The relations of the obtained Youn s moduli to the diameters of the nanotubes are also discussed. All the modelin and computin work in this paper are performed by the finite element commercial code ANSYS. KEYWORDS: carbon nanotubes, Youn s modulus, finite element model INTRODUCTION Since Iijima s discovery in 1991 (Iijima, 1991), carbon nanotubes (CNTs) have stimulated extensive research activities devoted to nanomechanics and their applications in nanoenineerin. Due to their exceptional mechanical and electrical properties: small size, low density, hih stiffness, hih strenth etc., CNTs represent a very promisin material in many areas of science and industry. The elastic properties of multi- and sinled-walled nanotubes (MWNTs and SWNTs) have been the subject of numerous research works in experimental, molecular dynamics, and elastic continuum modelin approaches. For examples, experimental investiations conducted by Treacy et al. (1996), who used TEM to measure the Youn s 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 used TEM to observe the vibration of a SWNT and obtained the Youn s modulus raned

2 from 0.90 to 1.70 Tpa; Wan et al. (1997) conducted bendin tests on cantilevered tubes usin atomic force microscopy and estimated a Youn s modulus of 1.8 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 modelin techniques. These modelin approaches can be enerally classified into two cateories. One is the molecular dynamics (MD) method (Iijima et al., 1996; Gau et al., 1998; Zhan et al., 1998; Zhou et al., 000; Belytschko et al., 00), which is based on the force field and total potential enery related to the interatomic potentials for CNTs in a macroscopic sense. In this method, the bondin and nonbondin potentials are represented in terms of the force constants and the deformation amon the atomic bonds, and then elastic moduli are determined by applyin 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 (Zhan et al., 00a, 00b; Jin and Yuan, 003; i and Chou, 003). Since a nanotube can be well described as a continuum solid beam or shell subject to tension, bendin, 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 modelin techniques, the obtained elastic properties of SWNTs or MWNTs have scattered values, for example, the results of axial Youn s modulus raned from about 1.0 Tpa to 5.5 Tpa can be found in the existin literatures. The purpose of this paper is to estimate the Youn s 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-neihborin 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 enery equivalence between molecular mechanics and structural mechanics. The Youn s moduli of both armchair and ziza 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 savin the effort of proram codin and makin this approach more popular, all the modelin and computin work are performed by the finite element commercial code ANSYS. FINITE EEMENT MODEING As mentioned above, CNTs can be treated as a frame-like structure with their bonds as beam members and carbon atoms as joints. In (i and Chou, 003), the stiffness method was used to simulate the mechanical behavior of nanotubes, they established the linkae between

3 the force constants in molecular mechanics and the element stiffness in structural mechanics throuh the enery equivalence concept. The key point of this concept is that the simplest harmonic forms of the various steric potential enery of nanotube bonds are adopted under the assumption of small deformation, i.e. where 1 1 U k r r k r r r ( 0 ) r ( ), (1) 1 1 U k ( 0 ) k ( ), () 1 U U U k ( ), (3) Ur stands for bonded stretchin enery, Uθ for bonded anle bendin enery, U is the combination of the dihedral anle torsion enery U and the improper (out-of-plane) torsion eneryu ; k r, k and k are the bond stretchin force constant, bond anle bendin force constant and torsional resistance respectively, and the symbols r, and represent the bond stretchin increment, the bond anle chane and the anle of bond twistin, respectively. By comparin these enery forms to their counterparts in structural mechanics, which are U U A M N N EA d ( ), (4) EA EA M EI EI d ( ), (5) EI T T GJ d ( ), (6) UT GJ GJ are the strain enery of a uniform beam of lenth, cross-section where U A, U M and UT A and moment of inertia I under axial force, pure bendin and pure torsion respectively; E and G are Youn s modulus and shear modulus; denotes the total relative rotation anle, is the total axial stretchin deformation, α is the relative torsion anle and J the polar moment of inertia. The concept of enery equivalence between the two systems implies that both U A represent the stretchin enery, both U and UM the bendin enery, both U and U the torsional enery. Then it is reasonable to assume that is equivalent to r, and T α equivalent to, and equivalent to. Therefore, by comparin Eqs.(1)-(3) with Eqs.(4)-(6), a direct relationship between the element stiffness the force constants in molecular mechanics EA, EI Ur GJ and k r, k and k can be obtained as follows and and EA k r, EI k, GJ k. (7) Eq. (7) constructs the base of the stiffness method employed by (i and Chou, 003), and they

4 selected the values of the force constants raphite sheets as k r k r, k and k accordin to the experience with = 938 kcal mol A = N/nm, k = 16 kcal mol /rad = N nm/rad 1-10, and k is adopted as 40 kcal mol /rad = N nm/rad 1-10, which is numerically proven to have little influence on CNTs Youn s modulus in their paper. By self-developed proram, the computed elastic properties of the SWNTs are obtained and their dependence on the tube diameter is also discussed. Note that the Poisson s ratio, which is one of the primary property of materials, does not appear explicitly in the formulation of i and Chou briefly reviewed above. However, by the selected values of the force constants, the restriction ofthe Poison s ratiothat should be a positive number smaller than 0.5 is violated when calculatin its value from Eq. (7) and the well known relationship of E and G, i.e. G E / (1 ), for an isotropic and uniform beam. In fact, all the other available choices of the values of the force constants found in literatures can t 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 advantae of the FE approach proposed here is that no extra effort is needed for proram codin and consequently make the present approach more feasible for mechanical analysis of nanotubes. In our finite element modelin work, the BEAM4 element in ANSYS is selected to simulate the carbon bonds while the atoms are nodes. Fi.1 (modified from Tserpes and Papanikos, 005) depicts how the hexaonal lattice of the CNT is simulated as structure elements of a space frame, where stands for the initial lenth of C-C bond. ac c

5 Fi. 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 Youn s modulus E, the Poisson s ratio and the diameter of the circular 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 By Eq. (8) and the values of k r k d 4, k r k E r. (8) 4 k and k iven above, the numerical values of E and d are calculated as 5.49 Tpa and nm respectively. The element lenth is set to be equal to the initial C-C bond lenth a of 0.14 nm. As to the Poisson s ratio, several values of 0.05, c c 0.1, 0. 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 diits overflow/underflow error durin the computation performed by ANSYS. Thus, we adjust the dimensions as follows in , Fin 10 F, Ein E, (9) where the oriinal dimensions of lenth, force F and modulus E are m, N and N/m respectively, and the subscript in denotes the real input values. After such adjustment, the numerical parts of the input data prepared for the BEAM4 element can be listed as follows

6 1 5 in in E.49 10, A 1.69, 0.3, in in d in where A / 4 is the area of the cross section, d in = 1.47 is the diameter of the circular section. YOUNG S MODUUS OF A GRAPHITE SHEET A VERIFICATION CASE To verify the feasibility of the model described above, we first calculate the Youn s modulus of a raphite 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 treatin the raphite 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 Fi.. Then the Youn s modulus can be determined from classical elasticity theory, i.e. Y F / A FH H / H HWt, (10) where Y is the Youn s modulus of the sheet, F is the total force applied on the nodes at one end, A Wt is the cross-sectional area of the sheet with width W and thickness t, H and H are the initial lenth and the induced elonation respectively. The thickness t is taken as the interlayer spacin of raphite, 0.34 nm. Fi.. FE model of the raphite sheet An enery form in elasticity theory to calculate the Youn s modulus is also used as an alternative approach, i.e. Y 1 1 V, (11) U N U N Wt H where V WtH is the total volume of the raphite sheet, and U N is the total strain enery. In this approach, prescribed displacements instead of the tensile forces are applied at the end nodes. Table 1 lists the computed Youn s moduli of the raphite sheets by the ANSYS code for

7 different model sizes. From the results shown in Table 1, it can be seen that the obtained Youn s moduli of the raphite sheets are fairly closed to the commonly accepted value 1.05 Tpa (Kelly, 1981) and those presented by i and Chou (003). In addition, we can also observe that the calculatin result is weakly affected by model size. Table 1. The computed Youn s moduli of the raphite sheets for different model sizes Width (nm) Heiht (nm) Youn s modulus (TPa) Eq. (10) Eq. (11) i and Chou (003) YOUNG S MODUUS OF A SWNT A successful verification of the present FE model is achieved in last section. In the followin, we will apply this method and employ the same parameters of the BEAM4 element to estimate the Youn s moduli of SWNTs. Two main types of SWNTs, ziza and armchair, are considered. Fi.3 depicts the typical FE model for calculatin the Youn s modulus of a sinle SWNT. As shown in Fi.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 to calculate theyoun s modulus Y of a SWNT are slihtly modified as follows and where At dt F / At Ft Y /, (1) dt t t t 1 1 Y, (13) V t U N U N dt t is the cross-sectional area, t and d are respectively the initial wall thickness and the diameter of the SWNT, t and t are the initial tube lenth and the elonation in the axial direction, Vt dtt denotes the total volume of the tube, U N is the total strain enery. Note that different values of the wall thickness of the SWNTs, ranin from to 0.69 nm, have been used in many researchers studies. Moreover, they will result in widely scattered values of the Youn s moduli of SWNTs. In majority of the studies, it has been assumed that the wall thickness is equal to the interlayer spacin of raphite, i.e nm. Thus, in order to make a reasonable comparison with existin literatures in which t = 0.34 nm is specified, we also select this value to be the wall thickness in our FE model.

8 Fi. 3. FE model for calculatin the Youn s modulus of a sinle SWNT Table lists the computed Youn s moduli of both ziza and armchair SWNTs with different tube diameters by Eqs. (1) and (13), and these results are also displayed in Fi.4. From Fi.4, it can be seen that the trend is similar for both ziza and armchair SWNT, and the effect of the tube chirality is not sinificant. It is also observed that the effect of the diameter on the Youn s moduli of both ziza and armchair SWNTs is evident, especially for diameters smaller than 0.8 nm. With increasin tube diameter, the Youn s moduli of ziza SWNTs increase slihtly faster than those of armchair SWNTs, and this increasin tendency diminished radually for both types as the diameters larer than 1.1 nm. As pointed out in i and Chou (003), this increase for both ziza and armchair SWNTs is due to the effect of nanotube curvature. Hiher curvature, which results in more sinificant distortion of C-C bonds, leads to a smaller Youn s modulus. In addition, the converent values of the Youn s moduli estimated in this FE model are about 1.04 Tpa for ziza SWNTs and Tpa for armchair SWNTs, which is in reasonable areement with that obtained by i and Chou (003), i.e Tpa for both types.

9 Table. The computed Youn s moduli of both ziza and armchair SWNTs with different tube diameters Tube type Diameter (nm) Youn s modulus (TPa) Eq. (1) Eq. (13) ziza(5,0) ziza (10,0) ziza (15,0) ziza (0,0) ziza (5,0) armchair (3,3) armchair (6,6) armchair (9,9) armchair (1,1) armchair (15,15) Youn's modulus (TPa) Diameter (nm) Ziza [1] Ziza [13] Arm-chair [1] Arm-chair [13] Fi. 4. Youn s moduli of SWNTs versus tube diameters CONCUSIONS A FE approach for estimatin the Youn s moduli of the ziza 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 linkae between the force constants in molecular mechanics and the elastic properties of the beam member in structural

10 mechanics is established throuh the enery equivalence concept. We obtain the Youn s moduli of about 1.04 Tpa for ziza and Tpa for armchair SWNTs at lare tube diameters. These results are comparable to those found in the existin literatures. Furthermore, dependence of the Youn s moduli to tube diameters has been also investiated. The investiation shows that the Youn s moduli increase with increasin diameter, and then radually approaches to a value close to that of raphite sheet when the tube diameter becomes lare. The main advantae of the proposed method is the sinificant savin of the proram codin time and the fair accuracy in estimatin the Youn s moduli of the SWNTs when compared to the other numerical methodoloies. It can be concluded that this approach is a valuable tool for studyin the mechanical behavior of carbon nanotubes. ACKNOWEDGEMENT The support from National Science Council (NSC), R.O.C., throuh rant NSC93-1-E is appreciated by the authors. REFERENCES Belytschko, T., Xiao, S., Schatz, G. and Ruoff, R., Atomistic Simulation of Nanotube Fracture, Phys. Rev. B, 65, pp , 00. Gao, G., Cain, T. and Goddard III, W., Eneretics, Structure, Mechanical and vibrational Properties of Sinle-walled Carbon Nanotubes, Nanotechnoloy, 9, pp , Iijima, S., Helical Microtubes of Graphitic Carbon, Nature, 354, pp.56-58, Iijima, S., Brabec, C., Maiti, A. and Bernholc, J., Structural Flexibility of Carbon Nanotubes, J. Chem. Phys., 104, pp , Jin, Y. and Yuan, F.G., Simulation of elastic Properties of Sinle-walled Carbon Nanotubes, Composite Science and Technoloy, 63, pp , 003. Kelly, B., Physics of Graphite, Applied Science Publishers, ondon, Krishmen, A., Dujardin, E., Ebbesen, T.W., Yianilos, P.N. and Treacy, M.M.J., Youn s Modulus of Sinle-walled Nanotubes, Phys. Rev. ett., 58, pp , i, C. and Chou, T.W., A Structural Mechanics Approach for the Analysis of Carbon Nanotubes, Int. J. Solids Struc., 40, pp , 003. u, J.P., Elastic Properties of Carbon Nanotubes and Nanoropes, Phys. Rev. ett., 79, pp , Poncharal, A., Parks, D.M. and Boyce, M.C., Electrostatic Deflections and Electromechanical Resonance of Carbon Nanotubes, Science, 83, pp , Treacy, M.M.J., Ebbesen, T.W. and Gibson, J.M., Exceptionally Hih Youn s Modulus Observed for Individual Carbon Nanotubes, Nature, 381, pp , 1996.

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