EVALUATION OF THERMOELASTIC PROPERTIES OF CARBON NANOTUBE-BASED COMPOSITES USING FINITE ELEMENT METHOD

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1 Proceedings of the Iernational Conference on Mechanical ngineering 009 (ICM009) 6-8 Deceber 009, Dhaka, Bangladesh ICM09-AM- VALUATION OF THRMOLASTIC PROPRTIS OF CARBON NANOTUB-BASD COMPOSITS USING FINIT LMNT MTHOD Sushen Kirtania and Debabrata Chakraborty Departe of Mechanical ngineering, Tezpur UniversityAssa, India. Departe of Mechanical ngineering, Indian Institute of Technology Guwahati,Assa, India ABSTRACT This article deals with the deterination of the theroelastic properties of single-walled carbon nanotube (SWNT)-reinforced coposites using finite elee (F) analysis. A full three-diensional (-D) F analysis has been perfored using general purpose F software ANSYS. SOLID45 elees ebodied in ANSYS have been used for odeling a square represeative volue elee (RV). Both carbon nanotubes (CNTs) and atrix aterials are assued to be isotropic. ffect of differe iporta paraeters on the properties of the CNT-based coposites has been studied. Prese work concludes that by adding ~% of CNT in epoxy, the effective axial Young s odulus as well as axial coefficie of theral expansion (CT) of coposites could be increased and decreased by 776.6% and 9% copared to the Young s odulus and CT of the epoxy, respectively. It is observed that axial CT of the SWNT/POXY coposites could be reduced to zero corresponding to a volue fraction of ~%. Keywords: C-Based Coposites, Finite lee Analysis, Coefficie of Theral xpansion.. INTRODUCTION Aong any poteial applications of nanotechnology, nanocoposites have been one of the rece research areas. Carbon nanotubes due to their inhere advaages like high strength, stiffness, resilience along with superior thero-electro-echanical properties are believed to be ideal reinforcing aterials for high perforance structural coposites. There have been good nubers of works reported in the broad area of nano-coposites in rece ties. One of the ajor difficulties in the study of CNT-based coposites is experieal characterization of such aterials due to their sall size. On the other hand, odeling and siulation of nanocoposites can be easily analyzed by a coputer. Deterination of theroelastic properties is one of the iporta tasks where siulation can be used advaageously. Therefore, coputational approach played a significa role in the develope of the CNT-based coposites by providing siulation results to help in understanding, analyzing and designing of such nanocoposites Carbon nanotubes first discovered by Suio Iijia in 99 [] and subsequely there were any papers published to deterine the Young s odulus [-5], Poisson s ratio [6-7] and CT [4, 8-9] of CNTs. Fro the above literatures, it was found that the Young s odulus, Poisson s ratio, and CT of CNTs are in the order of TPa, 0.8, and K, respectively. Due to superior echanical, theral and electrical properties of CNTs, they provide the ultiate reinforcing aterials for the develope of a new class of nanocoposites [0-]. The echanical load carrying capacities of carbon nanotubes in nanocoposites have been investigated in soe experieal works [-]. Qian and Dickey [] conducted experies and concluded that with only % (by weight) addition of CNTs in polystyrene (PS), elastic odulus and breaking stress have been observed to increase by 6%-4% and ~5%, respectively. Jia et al. [4] explained the reasons and possibility of strong (C-C) bond between the CNTs and atrix, as well as the iportance of ierface in CNT-based coposites. Lusti and Gusev [5] perfored F analysis of Young s odulus and CT of CNT/epoxy coposites for differe orieation of CNTs in atrix, and concluded that CNTs could be ore efficie as reinforcee copared to conveional glass or carbon fibers. Guo et al. [6] copared the CT and Young s odulus of PAN/SWNT coposites by perforing experies. The effective echanical properties of CNT-based coposites are evaluated using a -D nanoscale represeative volue elee based on the -D elasticity theory and solved by the FM [7-8] and observed that with the addition of.6% volue fraction of the CNTs in a atrix, axial Young s odulus of the ICM009 AM-

2 coposites increased by % for the case of long CNT fibers. The tensile strength, odulus and electrical conductivity of a pitch coposite fiber with 5 wt % of purified SWNTs are enhanced by ~90%, ~50%, and 40% respectively, as copared to the corresponding values in unodified isotropic pitch fibers [9]. Han and lliott [0] perfored olecular dynaics siulation and deterined axial and transverse elastic oduli using consta-strain energy iniization technique and reported that ierfacial bonding effect is iporta. Literature review reveals that even though nuber of work have been reported in the direction of characterization of CNT-based coposites, especially using F ethod, not any work in thero-elastic characterization in CNT-based coposites have been reported. Therefore the prese work ais at F based estiation of thero-elastic properties of CNT-based coposites and studying the effect of differe iporta paraeters on such properties which will be useful in design of such coposites.. FORMULATION OF THRMOLASTIC PROPRTIS OF CARBON NANOTUB-BASD COMPOSITS In this prese work, it has been assued that the CNTs and atrix in a RV are linear elastic, isotropic and hoogeneous aterials, with given Young s odulus and Poisson s ratios. It has also been assued that the CNTs and atrix are perfectively bonded with no slip at the ierface in the RV to be studied. The RV has been odeled by taking differe aterials like epoxy to steel and differe volue fraction of SWNT in the atrix ranging fro 0.5% to ~5%. The F esh of a cross section of the CNT-coposites of the square RV is shown in fig.. length of the CNTs has taken as 00 n keeping the aspect ratio of the SWNT as 06. ven though it was reported in the literature [5] that the elastic odulus of CNTs does not change beyond an aspect ratio of 00, but is was observed in the prese work that after an aspect ratio of 06, change in elastic odulus is insignifica.. ffective Young s Modulus of the Nanocoposites Based on Strength of Materials Approaches Figure shows a siple strength of aterials odel for calculating the effective axial Young s odulus of a long CNT reinforced inside the atrix of a square RV. All the nodes at one end are fully restrained and the nodes at other ends are subjected to unifor tensile load (F). The axial Young s odulus have been evaluated using F / Ac () Δ La / La where, F (i.e. F) stands for the total axial force acting at one end, A c is the cross sectional area, L a (i.e. L) is the initial axial length and ΔL a is elongation of the nanocoposites in axial direction. In calculating the cross sectional area of the nanocoposites, the thickness t of the CNT is taken as 0.4 n [, 6] which is the ierlayer spacing of graphite. The volue fraction of the CNT in atrix of the square RV is defined by π( r0 ri ) V () 4a πri where, r 0 is the outer radius of the CNT, r i is the inner radius of the CNT and a is the thickness (or width) of the nanocoposites odel. Fig. A siple strength of aterials odel for calculating effective axial Young s odulus of a long CNT reinforced inside the atrix of a square RV Total nuber of nodes 9,79, Total nuber of elees 8,400, Total nuber of CNT layer, Total nuber of atrix layers 6, Thickness of CNT layer 0.4 n, Thickness of each atrix layer 0.5 n, Volue fraction of the CNT in the atrix i.e..056% Fig. F esh of a cross section of the CNT-coposites (square RV) The square RV is used for calculation of the effective Young s odulus as well as CT of the CNT-based coposites. The diaeter of CNTs has been chosen as.88 n which is equal to the diaeter of zigzag (4, 0) CNTs. The thickness of CNT layer t 0.4 n but thickness of atrix layers are differe for differe volue fraction of CNT in coposites. The For a fiber coposite under uniaxial loading, the dependence of the effective Young s odulus in ters of the odulus and the volue fraction of each constitue can be estiated by the rule of ixture (ROM) []. The sae equations of the ROM are used to predict the effective Young s odulus of the CNT-based coposites. The longitudinal elastic odulus or effective axial odulus, of the nanocoposites with long CNT is + V V () where and are the elastic odulus of the CNT and atrix, respectively, and V and V are the volue fractions of the CNT and atrix, respectively. Where V + V (4) These ROM forulae are applied to verify the coputational results of the effective Young s odulus ICM009 AM-

3 of the CNT-based coposite aterials.. The ffective Coefficie of Theral xpansion of the CNT-Reinforced Coposites The theral expansion of a solid can be anisotropic if the coefficies of theral expansion are direction depende. This situation occurs in coposite or nanocoposites aterials with a directional reinforcee. In the prese study, the axial and transverse linear CT of the nanocoposites have been evaluated using finite elee ethod (FM). The unifor teperature is applied on each node by fixing the nodes at one end (zero displacee). The axial CT of the coposites in the axial direction is given by ΔLa (5) La ΔT where, Δ T is the change in teperature. Siilarly, the coefficie of theral expansion of the nanocoposites in the transverse direction is given by ΔLt (6) Lt ΔT To verify the coputed CT of the CNT-based coposite aterials, following are the expressions developed for the two theral expansion coefficies using the theroelastic extreu principle []. V +V (7) V + V ( + ) V + ( +) V (8) where, and are the linear CT in axial and transverse direction, and are the CT for the CNT and atrix, and and are the Poisson s ratio for the CNT and atrix, respectively. The effective axial Poisson s ratio, is V + V (9) which is approxiated by the ROM expression [], as in axial effective Young s odulus of the nanocoposites.. RSULTS AND DISCUSSION The effective Young s odulus as well as the CT of the CNT-reinforced coposite has been evaluated considering a square RV using FM. The coputed Young odulus and CT are copared to the Young s odulus and CT of the atrix, respectively. To get a clear idea on the variation of the Young s odulus and CT of the differe types of nanocoposites, four types of atrix aterials have been chosen. The atrix aterials are epoxy, lead, titaniu and steel i.e. fro low strength to high strength aterials. ffect of volue fraction on the variation of the Young s odulus as well as the CT of the nanocoposites has also been studied. SOLID45 elees ebodied in ANSYS0 have been used for odeling the RV. Properties of epoxy have been chosen fro literature [4] and densities of epoxy and SWNT are chosen fro books [, 5], respectively. Properties of the atrices aterials and the SWNT [-7] are as follows SWNT: 000 GPa, 0.8, ρ 00 kg / poxy: Lead: Titaniu: Steel:.89 GPa, 6 GPa, 0.7, ρ , ρ 40 6 GPa, 0., ρ GPa, 0.9, ρ 7800 kg / kg/ kg / kg /. ffective Axial Young s Modulus of CNT-Reinforced Coposites Figure shows the -D view along with the applied boundary conditions of a F odel for the square RV with a long CNT. In the prese odel, the x-y plane is the transverse plane and the z-axis is the axial direction of the CNT which is shown in Fig.. All the nodes at z 0 are fully restrained and the nodes at z L a are subjected to unifor tensile load. The average displacee along z-direction is calculated for all the nodes in the cross section at z L a / and effective axial Young s odulus is calculated using q. (). Fig. A -D odel of F eshes for the square RV ( V ~8%) with a long CNT along with the applied boundary conditions Table : Coputed axial effective Young s odulus of the CNT-based coposites of RV by using volue fraction.056% Types nanocoposites % of increased (RO M) of / % of increased (Cop uted) SWNT/POXY SWNT/LAD SWNT/TITANIUM SWNT/STL Based on the forulation described in section, effective axial Young s odulus of the CNT-based coposites have been deterined fro the prese FA of the RV. Table shows the axial Young s odulus of the CNT-based coposites copared with that of the atrix for a consta volue fraction.056%. For coparison the strength of aterials solution based on ROM is calculated using q. () and also listed in Table. Results in Table show that by adding.056% CNT ICM009 AM-

in a atrix, the axial Young s odulus ( ) of CNT-based coposites could be increased by.75% copared to the Young s odulus of the atrix, when the ratio of the Young s odulus of CNT and atrix i.e. / 4.76.

$4 shows that perceage increase of effective axial Young s odulus of differe nanocoposites at a consta volue fraction of.056%. Titaniu. Fig 5.$

4 in a atrix, the axial Young s odulus ( ) of CNT-based coposites could be increased by.75% copared to the Young s odulus of the atrix, when the ratio of the Young s odulus of CNT and atrix i.e. / In the case of / 57, the Young s odulus of the coposites in the axial direction ( ) has been observed to have increased by about nine ties copared to that of the atrix. Fig. 4 shows that perceage increase of effective axial Young s odulus of differe nanocoposites at a consta volue fraction of.056%. Titaniu. Fig 5. The variation of the perceage of increase of effective axial Young s odulus of CNT-Titaniu coposites in differe volue fraction (0.5% to 0.%) Fig 4. The variation of the perceage of increase of effective axial Young s odulus of differe nanocoposites at a consta volue fraction (.056%) In practice the weight fraction of CNT in CNT-based coposites is liited to 0% [5] and hence in the prese analysis the range of the volue fraction is taken between 0.5%-0.%. Table shows the variation of with increasing volue fraction in a CNT/Titaniu coposite. Results in table show that there is a very high perceage increase of as the volue fraction is increase fro 0.5% to0.%. Sae trend has also been observed for CNT/poxy coposite. Table : Coputed effective axial Young s oduli of the CNT-Titaniu nanocoposites taking volue fraction 0.5% to 0.% V % of increased w.r (coputed) 0.5% % % % % % % of increased (ROM) Fro the results shown in Table and Table it could be observed that the effective axial Young s odulus of coposites calculated fro the prese study appear very close to those obtained fro ROM. Fig. 5. shows that by taking 0.5% to 0.% volue fraction of CNT in Titaniu atrix, the perceage of increase in effective axial Young s odulus of nanocoposites vary fro 4.% to 78.54% with respect to the Young s odulus of. ffective Coefficie of Theral xpansion of CNT-Reinforced Coposites The effective CT of CNT-based coposites have been evaluated by using FM. The effective axial CT of the coposites have been calculated by taking a consta volue fraction as well as by varying the volue fraction of the CNT in atrix. The sae -D F odel for the square RV with a long CNT is used which was used for the calculation of the effective Young s odulus of coposites. The CT s of SWNT, epoxy, lead, titaniu, and steel 6 are K, 58 0 K, 9 0 K, K, and 0 K, respectively... ffective CT of CNT-Reinforced Coposites at a Consta Volue Fraction The effective axial as well as transverse CT of the coposite are calculated by taking a consta volue fraction.056%. All the nodes at z 0 are fully 0 restrained and a unifor teperature, Δ T 0 C is applied on all nodes. The effective axial CT are calculated using equation (5). The calculated effective axial CT of the CNT-based coposites by using FM are listed in Table and copared with the theoretical values using equation (7). Table : Coputed axial CT of the CNT-based coposites of RV by using a consta volue fraction.056% Types nanocoposites of % reduction of (Coputa % reduction of (Theoreti tional) cal) SWNT/POXY SWNT/LAD SWNT/TITANIUM SWNT/STL For all the four coposite aterials the axial CT have ICM009 4 AM-

$5 Axial CT [0-6 K - ] 0 5 0 5 Fig 6. The variation of the perceage of axial CT for differe type s nanocoposites by taking a consta volue fraction ~%.$

5 been observed to have decreased and the perceage of reduction of axial CT of nanocoposites is axiu for SWNT/POXY (~9%) and iniu for SWNT/STL (~5%) is shown in Fig Axial CT [0-6 K - ] Fig 6. The variation of the perceage of axial CT for differe type s nanocoposites by taking a consta volue fraction ~%.. ffect on Volue Fraction on the ffective CT of CNT-Reinforced Coposites ffective axial and transverse CT are calculated using q. (5) and (6), respectively. For coparison of the calculated values with the theoretical values, equations (7) and (8) have been used. The calculated effective axial and transverse CT of the CNT/POXY coposites by using FM for differe volue fractions are listed in Table 4. Table 4: Coputed effective axial and transverse CT of the CNT/POXY nanocoposites taking volue fraction fro 0.5% to 5.77% V (%) 6 0 K (Copu ted) 6 0 K (Theoreti cal) 6 0 K (Coput ed) 6 0 K (Theoreti cal) In this study, the axial as well as transverse CT of the SWNT/POXY coposites are calculated by varying volue fraction fro 0.5% to 5.77%. The variation of the axial CT of SWNT/POXY with volue fraction of CNT is plotted in Fig. 7. It can be seen that the axial CT 6 of the nanocoposites are 5. 0 K and K corresponding to it s volue fractions of 0.5% and 5.77%, respectively. Another iporta observation fro Fig. 7 is that the axial CT of the coposites is zero at a volue fraction ~%. The sae reduction trend of axial CT with CNT volue fraction is also observed for CNT/Titaniu coposites. The coputed effective CT using F results are very close to the coputational [5] and experieal [6] results Volue Fraction [%] Fig 7. The variation of the axial CT with respect to the volue fraction of the CNT in epoxy atrix 4. CONCLUSIONS In general, the load carrying capacity is enhanced and CT is reduced due to addition of CNTs in atrices. By adding ~% of CNT in a particular atrix, the axial Young s odulus of CNT-based coposites could be increased between. ties to 8.76 ties that of the atrix, depending upon the ratio of the Young s oduli of CNT and atrix. The increase in effective CT in transverse direction is very less copared to the increase in effective CT in axial direction. By adding % of CNT in epoxy atrix, the axial CT of CNT-based coposites could be reduced by 9% copared to the CT of the atrix. Increase in volue fractions of CNT in atrix lead to decrease in CT and the axial CT of the SWNT/POXY could be reduced to zero corresponding to a volue fraction of ~%. 5. RFRNCS. Iijia S., Helical icrotubules of graphitic carbon Nature (London), 54, pp , (99). Li C., Chou T-W, A structural echanics approach for the analysis of carbon nanotubes I. J. Solid Struct., 40, pp , (00). K.I. Tserpes, P. Papanikos, Finite elee odeling of single-walled carbon nanotubes Coposites: Part B, 6, pp , (005) 4. S. Kirtania and D. Chakraborty, Finite elee based characterization of carbon nanotubes J. Reinforced Plastics and Coposites, 40(5), pp , (007) 5. M. J. Treacy, T. W.bbesen and J. M. Gibson, xceptionally high Young s odulus observed for individual carbon nanotubes Nature, 8, pp , (996) 6. Lu J. P., lastic properties of carbon nanotubes and nanoropes Phys. Rev. Lett., 79(7), pp , (997) 7. T. Belytschko, S. P. Xiao, G. C. Schatz, and R. S. ICM009 5 AM-

6 Ruoff, Atoistic siulation of nanotube fracture Phy. Rev. B, 65, pp , (00) 8. Jiang H., Liu B., Huang Y., Hwang K. C., Theral expansion of single wall carbon nanotubes J. ngg. Mat. Technol., 6, pp , (004) 9. Kwon Y-K., Berber S., and Toanek D., Theral coraction of carbon fullerenes and nanotubes Phys. Rev. Lett., 9(), pp , (004) 0. Thostenson. T., Ren Z., Chou T-W., Advance in the science and technology of carbon nanotubes and their coposites: a review Copos. Sci. Technol., 6, pp , (00). Lau K-T., Hui D., The revolutionary creation of new advanced aterials-carbon nanotube coposites Coposites: Part B ng.,, pp. 6-77, (00). D. Qian and. C. Dickey, Load transfer and deforation echaniss in carbon nanotube-polystyrene coposites Appl. Phy. Lett., 77(0), pp , (000). L. S. Schadler, S. C. Giannaris, and P. M. Ajayan, Load transfer in carbon nanotube epoxy coposites Appl. Phy. Lett., 7(6), pp.84-44, (998) 4. Z. Jia, Z. Wang, C. Xu, Ji Liang, B. Wei, D. Wu and S. Zhu Study on poly(ethyl ethacrylate)/carbon nanotube coposites Mat. Sc. and ngg. A7, pp , (999) 5. H. R. Lusti and A. A. Gusev, Finite elee predictions for the theroelastic properties of nanotube reinforced polyers Modelling siul. Mater. Sc. ngg.,, pp. S07-S9, (004) 6. H. Guo, T. V. Sreekuar, T. Liu, M. Minus, S. Kuar, Structure and properties of polyacrylonitrile/single wall carbon nanotube coposites fils Polyer, 46, pp. 00-5, (005) 7. Y. J. Liu and X. L. Chen, valuations of effective aterial properties of carbon nanotube-based coposites using a nanoscale represeative volue elee Mechanics of aterials, 5, pp. 69-8, (00) 8. Y. J. Liu and X. L. Chen, Square represeative volue elees for evaluating the effective aterial properties of carbon nanotube-based coposites Coputational aterials Sc., 9, pp. -, (004) 9. R. Andrews, D. Jacques, A. M. Rao, T. Raell, and F. Derbyshire, Nanotube coposite carbon fibers Appl. Phy. Lett., 75(9), pp.9-, (999) 0. Yue Han and Jaes lliott, Molecular dynaics siulations of the elastic properties of polyer/carbon nanotube coposites Coputational aterials Sc., 9, pp. 5-, (007). Dresselhaus M. S., Dresselhaus G., and Saito R., Physics of carbon nanotubes Carbon, (7), pp , (995). Carl T. Herakovich, Mechanics of Fibrous Coposites John Wiley & Sons, Inc., (998). R. A. Schepery, Theral expansion coefficie of coposite aterials based on energy principle J. of cop. aterials, (), pp , (968) 4. M. R. Nedele and M. R. Wisno, Three-diensional finite elee analysis of the stress conceration at a single fiber breaks Cop. Sc. and Tech., 5, pp. 57-4, (994) 5. M. Meyyappan, Carbon Nanotubes Science and Applications CR Press LLC, (005) 6. NOMNCLATUR Sybol Meaning Unit A Cross sectional area c n Young s odulus GPa Young s odulus of Matrix Young s odulus of CNT GPa L Initial axial length n a L Initial transverse length n t Δ Change in axial length n L a Δ L t Change in transverse length n r i Inner radius of CNT n r 0 Outer radius of CNT n T Teperature K ΔT Change in teperature K V Volue fraction of Matrix V V Volue fraction of CNT Axial CT of Nanocoposites K Transverse CT of K Nanocoposites CT of Matrix K CT of CNT Poisson s ratio of CNT 7. MAILING ADDRSS Poisson s ratio of Matrix Assista Prof. Sushen Kirtania Departe of Mechanical ngineering, Tezpur University P.O.:- Napaa, Tezpur-78408, Assa, India Phone : /8/9 (xtn 5857) Fax: /06 -ail: sushen.kirtania@gail.co K ICM009 6 AM-

Buckling Behavior of 3D Randomly Oriented CNT Reinforced Nanocomposite Plate

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