Comparison Between Different Finite Element Methods for Foreseeing the Elastic Properties of Carbon nanotube Reinforced Epoxy Resin Composite

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1 Comparison Between Dierent Finite lement Methods or Foreseeing the lastic Properties o Carbon nanotube Reinorced poy Resin Composite Abdolhosein.Fereidoon, smaeel.saeedi, and Babak.Ahmadimoghadam Abstract In this paper the mechanical properties o carbon nanotube C.N.T reinorced epoy resin composite are studied with several methods in order to determine the best inite element modeling method. Composite eective module in dierent aspect ratios and volume ractions have been calculated. ective ratio which achieved by dierent orientations o C.N.Ts in matri has been investigated too. Two methods have been applied. In the irst method by embedding a C.N.T into representative volume element with dierent alignments the equivalent module were calculated.in the second method, the nanotube was modeled as a beam element in a representative surace element. In both methods by substituting the obtained equivalent module in surace element, composite eective module has been calculated. Finally, the achieved results are compared with Halpin-Tsai method and also eperimental results. Inde Terms Carbon nanotube, inite elemen, resin. I. INTRODUCTION Carbon nanotube (C.N.T) has peculiar properties such as elastic modulus and high tensile strength. Both theoretical [1 4] and eperimental [ 5 7] methods were applied in order to determine the Young s modulus and Poisson s ratio o C.N.Ts. Results were shown that it has quite broad variation o the Young s modulus ranging rom 200 to 4 TPa, tensile strength rom 10 to 200 GPa, and bending strength o about 20 GPa [8 10]. Thereore it is one o the most promising materials with potential as an ultimate reinorcing material in nanocomposite [11-12]. Addition o C.N.T to matri increases the toughness and stiness and strength between composite layers. Addition o only 1% (by weight) C.N.T to matri, can increase the stiness o the matri about 36%-46%. [13]. In order to calculate elastic properties o carbon nanotube composite some analytical methods such as role o mitures and also Halpin-Tsai [15], Mori-Tanaka [14] relations has been applied. In the case o numerical methods, [16-19] has been used volume elements including C.N.T. Selmi, Friebel [20] has modeled twisted ibers o C.N.T by Abdolhosein.ereidoon is Assistant proessor Department o mechanical engineering semnan university.iran ( ab.ereidoon@gmail.com ). Ssmaeel.saeedi is Master o Science student Department o mechanical engineering semnan university.iran ( es_saeedi@yahoo.com ). Babak.ahmadimoghadam is Master o Science student Department o mechanical engineering semnan university.iran ( numerical methods. As the aspect ratio o C.N.T is very high and positions o them in matri is randomly in many cases, so inding a good inite element modeling method which can mystery the nanotube condition in matri and also has the most accordance with laboratorial conditions is very important. In this study, the eective elastic modulus o the C.N.T illed nanocomposite is eamined by using two inite element methods. ects o aspect ratio and volume raction and alignment o C.N.Ts on the elastic modulus are investigated. Representative volume element by solid beam model and Representative volume element is adopted in this study to predict these eects. In order to compare the numerical results with analytic ones, the analytic model proposed by Halpin and Tsai [20] is employed. The numerically predicted aial tensile modulus is also veriied by comparing it with eperimental results. II. MTHOD A. Representative volume element In model (a) similar to [16-20] has been used rom a representative volume element that has a C.N.T inside o itsel. The C.N.T dimensions are based on [17-18]. Dimensions o representative volume element were deined as ollow: 3 V L (L)lement length 120 (nm), (V) element volume The capability o C.N.T revolution in matri was considered in dimensions. Classic equivalent representative volume element module calculation method [21] was deined below. Relation between stresses and strain or one transversely isotropic material can be written as ollow. ε ε y ε 1 y y 1 1 σ σ y σ (2) ISBN: WC 2008

2 Fig. 1. Modeling the eective module o composite by representative volume element 4-independent constants (Young's modules and y, Poison's ratios ν y and ν y) correspond to stresses and vertical strains. For calculating these 4 constants, we need 4 equations that will be obtained rom using a simple load case along width and length directions o representative volume element.[17-18]. Fith independent material constant- the shear modules G y can be obtained rom using a simple (torsion) load case. Finite lement or speciying the authority o this method and calculating eective elastic module has been used. For matri modeling has been used rom 8-node tetrahedral element and 20-node heagonal or C.N.T modeling. Interace layer, elastic speciications o interace layer between C.N.T and matri relevant to [22] has been equal to matri speciication. periments show that in most cases, the C.N.T directions in matri are completely random. Thereore, with isotropic assumption or resultant composite, instead o using equation (1), the eect o C.N.T dierent direction positioning in matri was surveyed in 4 models. ο In model a-1, C.N.T turned about90 in a representative volume element with ied dimensions and it's equivalent module was calculated. Fig. 1 In model a-2 with iing the representative volume element that due to volume raction and iing the C.N.T end distance rom the representative volume element edges, ο C.N.T has turned about 90 in matri, also the dimensions o representative volume element was changed proportionally with the swiveling angle. Finally in 45 degree that has the largest area, has minimum thickness. quivalent elastic module in dierent angles was calculated. Fig. 2. B. Calculating the eective elastic module In models a-1 and a-2 or calculating the composite eective elastic module in C.N.T random distribution status in matri, has been used a surace element. This element was divided to smaller elements and the elastic speciications o each surace were selected randomly rom the results that achieved in models a-1 and a-2. In other words, surace element contains some parts that have C.N.Ts with random positions. Fig. 2. ect o C.N.T rotation in representative volume element. In (a-1) model volume raction is equal to 0.4% and in (a-2) model, volume raction is equal to 2.4 % ventually, load applied on surace element in plane stress status and the changing o element surace length was concluded and based on the (3), composite eective elastic module was obtained. (Table 1). PL eq (3) δ A Where (P) orce on aial direction, (L) representative volume element length, ( eq ) equivalent module, (A) cross sectional area o representative volume element Table. 1. results comparison in model a-1, a-2 a-1 model a-2 model e m The main problem in these two modeling methods is to obtain the composite eective elastic module in dierent volume ractions. By this reason in b-1 and b-2 models respectively the sensitive rate o equivalent module o representative volume element in normal loading (ero angles) proportion to changing the element length and/or to changing in width o it has been calculated. Length and diameter o C.N.T was considered constant.fig. 3. Fig. 3. ect o change in with and height in equivalent module o C.N.T in aial tensile load ISBN: WC 2008

3 Fig. 4. Stress concentration in matri around the two ends o C.N.T As you can see, eective elastic module o representative volume element proportion to length changing is more sensitive than width changing, by this reason stress concentration is in matri around the two ends o C.N.T, so by approaching the boundaries o representative volume element to the 2 ends o C.N.T up to 10(nm), the equivalent module value will increase approimately 4 times. Fig. 4. C. Representative surace element In this method or calculating the eective module, representative surace element and C.N.T as beam element has been used. Applied elements or matri are 8-node quadratic and or C.N.T modeling in matri has been used beam element with hollow cross section with 10nm diameter and 0.4(nm) thickness [17-18]. Finally or modeling o C.N.T connection with matri has been used beam elements It is better note that the connection beams are used only or orce transmission and displacement o C.N.Ts in matri, so the connection beams are assumed completely rigid. One o the major problems in nano scale modeling is applying the boundary conditions on the representative element. For surveying the eect o dierent boundary conditions on the results o composite equivalent module, 3 types o boundary conditions or top and bottom suraces o representative surace element was deined. (a) Top and bottom lines o representative surace element were constrained along y-ais. (b) Bottom line o representative surace element has symmetry conditions. (c) Top and bottom lines o representative surace element don't have any boundary conditions. Results have been surveyed in dierent angle positioning o C.N.T. Fig. 5. quivalent module in irst status o boundary conditions has so much deviation in angles o positioning between 15 and 75. Because the swiveling the representative surace element that happens by angular position o C.N.Ts in matri. The most suitable type o boundary condition is iing the top and bottom edges along Y direction that due to increase the equivalent module about 6%. Thereore, or modiying the results, they have to be subtracted 3% rom equivalent module in irst status. And ater that the results can be used or calculating the element eective module. Fig. 5. Boundary condition eects on equivalent module o representative surace element Because the lengths o the representative surace elements was selected equally to the C.N.T lengths, or volume raction and aspect ratio we can decrease the quantity o C.N.Ts in the matri by reducing in representative volume surace. For minimum value, aspect ratio25 and volume raction0.5, the minimum thickness or a C.N.T that can completely positioned inside the representative surace element is (nm). In this model has been used rom 100 nm thickness. Also or low volume raction and low matri elastic module in proportion to C.N.T elastic module, we neglect rom the beam interaction and matri. In this method has been assumed that C.N.Ts is in very small elements that aligned along the length o one C.N.T.Fig. 6. A cod has developed in commercial FM sotware ANSYS and The angle o C.N.Ts position in surace element has changed rom ero to 90 degree and with applying loads the equivalent module has been calculated. These operations or each angle have been repeating 5 times and equivalent elastic module in each status was obtained rom the average o these 5 iterations. The rate o equivalent elastic module or volume raction in 1% and dierent angles or positioning can be seen in Fig. 7. Figure 6. Beam C.N.Ts in matri by align orientation and random position and stress distribution in matri ater loading. ISBN: WC 2008

4 3 1+ 2aη Lφ η Tφ C + 8 1η Lφ 8 1ηTφ 1 M, 1 η L M η T + 2a NT + 2 M M M (4) Figure 7. Rate o equivalent elastic module in dierent angles or positioning Where a L / d ) is the aspect ratio with L ( being the length and d the diameter o the nanotubes, φnt is the C.N.T volume raction and,, M and are the modulus o C.N.Ts, matri, and composite, C respectively In align case Hlpin-Tsai relation written as: 1 + ζηv (5) c m 1ηV ( / m ) 1 η (6) / ( ) ζ Where m V is volume raction o C.N.T, and and m are Young s module o iber and matri respectively. The parameter ζ is dependent on the geometry and the boundary conditions o the reinorcing phase. For the case o aligned short-iber, ζ can be epressed by: Fig. 8. ect o changing the aspect ratio on eective module l ζ V (7) D For a low volume raction ( V ) the second term is negligible compared to the irst term. ective elastic module in representative surace element module and Halpin-Tsai equation and eperimental results in 1% volume raction an`d dierent aspect ratios was compared with each other ig. 9, 10. (a) Figure 9. ect o changing the volume raction on eective module For calculating the eective composite module ollowed the section 2-2.Result was plotted. Fig. 8, 9. As seen in the results, increase in aspect ratio greater than 250 in not eective to increase module. D. Theoretical methods The most common models used is (4) Halpin-Tsai equation [23 26]. The Halpin-Tsai equation or reinorced with randomly oriented C.N.Ts is given by [27]: ISBN: WC 2008

5 (b) Fig. 9. (a),(b). Comparison between representative surace element results by eperimental results and Halpin-Tsai ormulation in variable aspect ratio Figure10. Comparison between representative surace element results by eperimental results and Halpin-Tsai ormulation in variable volume ractions III. RSULT AND CONCLUSION As you can see in igure, representative surace element model shows good coverage with eperimental results. Composite eective module with aspect ratios more than 250 doesn't show large increment but C.N.Ts aligning has so aects on composite eective module increment. Results show that this model is useul or simulating the dierent types o short ibers in matri but in C.N.T case or long length (in short C.N.T this value is minimum 500 (nm) usage o this type o modeling is unsuitable. In addition, in this modeling type iber distance rom the element edges and the distribution procedure is so signiicant. In second model C.N.Ts positioned similar to beams in one representative surace element. C.N.Ts directions in element are aligned in dierent angle and the positioning places o them were random. Results showed that this model or aspect ratios more than 50 has proper answers. More than that, achieved result range shows that representative surace element model is usable or large ranges o aspect ratios and volume ractions and results have proper coverage with eperimental results. For using o representative surace element model in low aspect ratios, needs more solutions and gets averages.instead o using volumetric element and beam, using rom surace element and beam can give us more accurate results. In third model or connecting beam elements to matri have been used link and spring elements.but we should considering that insuicient DOF especially in 0 and 90 angles, achieved results have so many deviation. RFRNCS [1] Sanche D, Artacho and Soler J M 1999 Phys. Rev. B [2] Ruo R S and Lorents D C 1995 Carbon [3] Lu J P 1997 Phys. Rev. Lett [4] Hernande, Goe C, Bernier P and Rubio A 1998 Phys. Rev. Lett [5] Treacy M M J, bbesen T W and Gibson J M 1996 Nature [6] Wong W, Sheehan P and Lieber C M 1997 Science [7] Yu M F, Files B S, Arepalli S and Ruo R S 2000 Phys. Rev. Lett [8] Wagner HD, Lourie O, Feldman Y, Tenne R. Appl Phys Lett 1998;72(2): [9] Li F, Cheng HM, Bai S, Su G. Appl Phys Lett 2000;77(20): [10] Lau KT, Chipara M, Ling HY, Hui D. Composites, Part B 2004;35: [11] Thostenson T, Ren Z, Chou TW. Compos Sci Technol 2001;61: [12] Lau KT, Hui D. Composites, Part B 2002;33: [13] D. Qian,.C. Dickey, R. Andrews, T. Rantell, Load transer and deormation mechanisms in carbon nanotube polystyrene, Applied Physics Letters 76 (2000) [14] Mori T and Tanaka K 1973 Acta Metall [15] Halpin J C 1969 J. Comp. Mater [16] BY N. HU 1, H. FUKUNAGA 1, C. LU 2, M. KAMYAMA 1, B. YAN 1. Prediction o elastic properties o carbon nanotube reinorced [17] Y.J. Liu *, X.L. Chen. valuations o the eective material properties o carbon nanotube-based using a nanoscale representative volume element [18] X.L. Chen, Y.J. Liu. Square representative volume elements or evaluating the eective material properties o carbon nanotube-based [19] Young Seok Song, Jae Ryoun Youn. Modeling o eective elastic properties or polymer based carbon nanotube [20] A. Selmi a, C. Friebel b, I. Doghri, H. Hassis. Prediction o the elastic properties o single walled carbon nanotube reinorced polymers: A comparative study o several micromechanical models [21] M.W. Hyer, Stress Analysis o Fiber-Reinorced Composite Materials, irst ed., McGraw-Hill, Boston, [22] BY N. HU 1, H. FUKUNAGA 1, C. LU 2, M. KAMYAMA 1 AND B. YAN1 Prediction o elastic properties o carbon nanotube reinorced [23] Halpin, J. C.; Kardos, J. L. Polym. ng. Sci. 1976, 16, [24] Talreja, R.; Manson, J.-A.. In Comprehensive Composite Materials, Vol. 2: Polymer Matri Composites, Kelly, A.; Zweben, C. H., ds.; lsevier Science Ltd., New York; 2000, p [25] Cadek, M.; Coleman, J. N.; Barron, V.; et al. Appl. Phys. Lett. 2002, 81, [26] Halpin J. C, Tsai S. W, nvironmental actors in composite materials design ; US Air Force Technical Report AFML TR (1967). [27] Richard P. Wool Carbon Nanotube Composites with Soybean Oil Resin. chapter 14 Carbon Nanotube Composites with Soybean Oil Resins [28] Halpin, J. C.; Kardos, J. L. Polym. ng. Sci. 1976, 16, [29] Talreja, R.; Manson, J.-A.. In Comprehensive Composite Materials, Vol. 2: Polymer Matri Composites, Kelly, A.; Zweben, C. H., ds.; lsevier Science Ltd., New York; 2000, p [30] Cadek, M.; Coleman, J. N.; Barron, V.; et al. Appl. Phys. Lett. 2002, 81, ISBN: WC 2008