Finite Element Modeling and Free Vibration Analysis of Functionally Graded Nanocomposite Beams Reinforced by Randomly Oriented Carbon Nanotubes

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1 Finite Eleent Modeling and Fee ibation Analysis of Functionally Gaded Nanocoposite Beas Reinfoced by Randoly Oiented Cabon Nanotubes Benedict Thoas 1, Pasad Inada 2, Taapada Roy 3 & B. K. Nanda 4 1,2&3 Depatent of Mechanical Engineeing, National Institute of Technology, Roukela, India 4 ice Chancello, ee Suenda Sai Univesity of Technology, Bula, Sabalpu, India E-ail : benedict_thoas@ediffail.co 1, pasad220088@gail.co 2, taapada@nitkl.ac.in 3 Abstact This aticle deals with the finite eleent odeling and fee vibation analysis of functionally gaded nanocoposite beas einfoced by andoly oiented staight single-walled cabon nanotubes (SWCNTs). Nanostuctual ateials can be used to alte echanical, theal and electical popeties of polye-based coposite ateials, because of thei supeio popeties and pefect ato aangeent. Tioshenko bea theoy is used to evaluate dynaic chaacteistics of the bea. The Eshelby Moi Tanaka appoach based on an equivalent fibe is used to investigate the ateial popeties of the bea. The equations of otion ae deived by using Hailton s pinciple. The finite eleent ethod is eployed to discetize the odel and obtain a nueical appoxiation of the otion equation. Diffeent SWCNTs distibutions in the thickness diection ae intoduced to ipove fundaental natual fequency and dynaic behavio of unifo functionally gaded nanocoposite bea. Results ae pesented in tabula and gaphical fos to show the effects of vaious ateial distibutions, cabon nanotube oientations, shea defoation, slendeness atios and bounday conditions on the dynaic behavio of the bea. The fist five noalized ode shapes fo functionally gaded cabon nanotube einfoced coposite (FG-CNTRC) beas with diffeent bounday conditions and diffeent cabon nanotubes (CNTs) oientation ae pesented. The esults show that the above entioned effects play vey ipotant ole on the dynaic behavio of the bea. Keywods Cabon nanotubes, Moi Tanaka appoach, FG bea, Finite eleent analysis, ibation. I. INTRODUCTION Functionally Gaded Mateial (FGM) belongs to a class of advanced ateial chaacteized by vaiation in popeties as the diension vaies. The oveall popeties of FMG ae unique and diffeent fo any of the individual ateial that fos it.fgm concept was oiginated in Japan in 1984 duing the space plane poject. Cabon nanotube (CNT) is a new fo of cabon, configuationally equivalent to two diensional gaphene sheet olled into a tube. It is gown now by seveal techniques in the laboatoy and is just a few nanoetes in diaete and seveal icons long. CNT exhibits extaodinay echanical popeties: the Young's odulus is ove 1 Tea Pascal. It is stiff as diaond. The estiated tensile stength is 200 Giga Pascal. These popeties ae ideal fo einfoced coposites, nanoelectoechanical systes (NEMS).Cabon Nanotubes (CNTs) ae also known fo thei good echanical and electical popeties. The use of CNT with Functionally Gaded Mateial (FGM) povides ipoved echanical, electical as well as theal popeties. Main advantage of using CNT based FGM is that we can obtain these popeties as pe ou equieent just by vaying the distibution and coposition of CNT. The CNT based functionally gaded ateials ae used in wind tubines, tissue engineeing, thin fils of shape eoy alloys, nanoelectoechanical systes ico sensos, ico actuatos, telecounications and tanspot industy. Most studies on cabon nanotube-einfoced coposites (CNTRCs) have focused on thei ateial popeties [2 7]. Seveal investigations have shown that the addition of sall aounts of cabon nanotube can consideably ipove the echanical, electical and theal popeties of polyeic coposites [4 7]. Wuite and Adali [5] exained the deflection and stess of nanocoposite einfoced beas using a ulti-scale 97

2 Intenational Jounal on Theoetical and Applied Reseach in Mechanical Engineeing (IJTARME) analysis. They found that a sall pecentage of nanotube einfoceent leads to significant ipoveents in bea stiffness. Studies showed that the addition of a sall aount of cabon nanotube can consideably ipove the echanical, electical and theal popeties of polyeic coposites. Thei esults ae vey useful and can be applied to the analysis of the global esponse of CNTRC in an actual stuctual eleent. Liao-Liang Ke et al. [10] investigated the nonlinea fee vibation of functionally gaded nanocoposite beas einfoced by aligned, staight single-walled cabon nanotubes (SWCNTs) based on Tioshenko bea theoy. The ateial popeties of functionally gaded cabon nanotube-einfoced coposites (FG-CNTRCs) wee assued to be gaded in the thickness diection and estiated though the ule of ixtue. They intoduced the CNT efficiency paaete to account fo load tansfe between the nanotube and polyeic phases. Howeve, the ule of ixtue is not applicable when CNTs ae oiented andoly in the atix. Thus, in this pape the Moi Tanaka appoach which is applicable to nanopaticle is eployed to pedict ateial popeties of coposites einfoced with andoly oiented, staight CNTs. The objective of the pesent wok is to study the fee vibations of functionally gaded nanocoposite beas einfoced by andoly oiented staight singlewalled cabon nanotubes within the faewok of Tioshenko bea theoy using finite eleent ethod. The ateial popeties of the FG-CNTRC ae assued to be gaded in the thickness diection and estiated though the Moi Tanaka ethod [11] because of its siplicity and accuacy even at a high volue faction of inclusions. Finally, the effects of CNTs oientation, Effect of vaiation of volue faction of CNT, slendeness atios and bounday conditions on the dynaic chaacteistics of the bea ae investigated. Using finite eleent analysis eigenvalues ae evaluated fo diffeent bounday conditions. Mode shapes ae plotted to visualize vibation esponse. II. MATERIAL PROPERTIES OF FG-CNTRC 2.1 Popeties of equivalent fibe: Using the esults obtained fo ulti-scale FEM, the investigated CNT and its inte-phase can be conveted into an equivalent fibe. Thus an ebedded cabon nanotube in a polye atix is eplaced with an equivalent long fibe fo pedicting the echanical popeties of the cabon nanotube/polye coposite. The equivalent fibe fo SWCNT with chial index of (10,10) is a solid cylinde with diaete of n. The ule of ixtue is used invesely fo calculating ateial popeties of equivalent fibe [13]: E L E LC M M E 1 1 M E E E T TC M 1 1 M G G G C M C M M whee E L, E T, G, υ, E LC, E TC, G C, υ C, E M, G M, υ M, and M ae longitudinal odulus of equivalent fibe, tansvese odulus of equivalent fibe, shea odulus of equivalent fibe, Poisson s atio of equivalent fibe, longitudinal odulus of coposites, tansvese odulus of coposites, shea odulus of coposites, Poisson s atio of coposites, odulus of atix, shea odulus of atix, Poisson s atio of atix, volue faction of the equivalent fibe and volue faction of the atix. Table 1: Mateial popeties of equivalent fibe: Mechanical popety Longitudinal Young s odulus (E L ) Tansvese Young s odulus (E T ) Longitudinal shea odulus (G ) Equivalent fibe [9] (GPa) (GPa) 5.13 (GPa) Poisson s atio (υ ) The odeling of bea stats with the calculation of ateial popeties. As CNT volue is assued to vay along the thickness only, ateial popeties fo each laye ae calculated fist and finally effective values fo entie bea ae calculated. Hee, linea vaiation of volue faction of CNT ( ) is consideed. It is calculated by following foula: 4 z h Hee, depends on ass faction and density of CNT and density of atix. Fo unidiectional CNT distibution =. 98

3 Intenational Jounal on Theoetical and Applied Reseach in Mechanical Engineeing (IJTARME) 2.2. Coposites einfoced with andoly oiented, staight CNTs: The effect of andoly oiented, staight CNTs is investigated in this section. The oientation of a staight CNT is chaacteized by two Eule angles α and β, as shown in Fig. 1. The base vectos e i and e i of the global (0 -x 1 x 2 x 3 ) and the local coodinate systes (0- x 1x 2x 3) ae elated via the tansfoation atix g: Whee g is given by: e i g e' cos cos sin sin sin g sin cos cos sin cos 0 sin cos The oientation distibution of CNTs in a coposite is chaacteized by a pobability density function p(α,β) satisfying the noalizing condition [3]. 2 /2 0 0 i j p(, )sindd 1 If CNTs ae copletely andoly oiented, the density function is: 1 p(, ) 2 j C 1 / E / E / E L 31 T 31 T / E 1 / E / E L T 31 T / E / E 1 / E L 31 T T / G / G / G Whee C is the stiffness tenso of the equivalent fibes and E T 31 E G 23 L ET 2(1 ) When CNTs ae copletely andoly oiented in the atix, the coposite is then isotopic, and its bulk odulus K and shea odulus G ae deived as [3] K K G G ( 3 K ) 3( ) ( 2 G ) 2( ) 1 3( K G ) k l 3( G k ) Fig.1.Repesentative volue eleent (RE) with andoly oiented, staight CNTs. [12] We found the Hill s elastic oduli of the einfocing phase fo the equality of two following atices [13]: n l l l k k l k k C p p 4G 2k l 4G 3( ) 1 G k G p 5 2 G (3 K G ) G (3K 7 G ) G (3 K G ) (3K 7 G ) 1 (2 k l )(3k 2 G l ) n 2l 3 G k 2 8G p ( ) n l 3 1 G p 5 8 G (3K 4 G ) 2( k l )(2 G l ) 3 K( G ) G (7 G ) 3( G k ) Whee k, l,, n, and p ae the Hill s elastic oduli fo the einfocing phase (CNTs). 99

4 Intenational Jounal on Theoetical and Applied Reseach in Mechanical Engineeing (IJTARME) III. PROBLEM FORMULATION Based on the fist-ode shea defoation (o the Tioshenko bea) theoy, the axial displaceent U and the tansvese displaceent of any point of the bea, W, ae given by [1] u( x, y, z, t) u ( x, t) z( x, t) w( x, y, z, t) w ( x, t) 0 0 Whee M M q Kq F denotes the ass atix, K denotes the stiffness atix. The eleents of stiffness and ass atix ae as given by [13]. I. RESULTS AND DISSCUSSION This theoy has been ipleented to a bea with h = 1, L/h = 20 to study the convegence of diensionless fundaental fequency fo is calculated by following foula: = It L A 2 2 EI Fig.2 : Bea Eleent [14] Stain displaceent and constitutive elations ae foed to calculate stain enegy and kinetic enegy of bea. Fo given case the final expessions fo these enegies ae, U 1 ( ) dadx 2 b xx xx xz xz v T b ( ) v u w dadx Hailton s pinciple is applied to get the diffeential equations in tes of two tanslational and one otational degee of feedo i.e. u, w and [13]. u u : I ( x) u I ( x) ( ) ( ) A x 11 B x 11 x x x x w w : I ( x) w A ( x) F ( x vt) x x u w : I ( x) I ( x) u B ( x) D ( x) k 55( ) s A x x x x x x 3.1. Finite Eleent Analysis To solve the above govening equation finite eleent analysis is ipleented. Finite eleent analysis ai is to find out the field vaiable (displaceent) at nodal points by appoxiate analysis. Hee eploying the appoxiate solution ethod the govening equations ae appoxiated by a syste of odinay diffeential equations. Finally, the equation of otion is deived as follows: Now, with the ateial popeties E = 10GPa; ν = 0.3 and ρ = 1150 Kg/ 3 esults of the fist five nondiensional fequencies of claped claped (C C), SFG-CNTR beas based on Tioshenko bea theoy with diffeent nube of eleent ae obtained and is shown in table 2. It is obseved that the convegence of the pesent esults is occus with a nube of eleent N = 100. To validate the esults bea slendeness atio L/h = 20 and = ae selected with claped claped condition of Tioshenko bea and ae veified with the esults given by [13].The effect of vaiation of eleents on diensionless fundaental fequency is shown in table 2. Mode No. Table 2.Effect of nube of eleents on nondiensional fundaental fequency: Table 3 shows effect of slendeness atio (L/h) on diensionless fundaental fequency. Fequencies ae calculated fo slendeness atio = 20, 40, 60, 80. It is found that as L/h atio inceases fequency also inceases. 100

5 Intenational Jounal on Theoetical and Applied Reseach in Mechanical Engineeing (IJTARME) Table 3. Effect of slendeness atio on non-diensional fundaental fequency: Mode No. L/h = 20 L/h = 40 L/h = 60 L/h = Fig. 5 : Mode shape fo thid ode Table 4 copaes the effect of vaiation of volue faction of CNT on diensionless fundaental fequency. The fist five fequencies fo Claped Claped, SFG-CNTRC beas ae shown in the table: Table 4. Effect of vaiation of volue faction of CNT on diensionless fundaental fequency Mode No Coesponding Mode Shapes ae plotted fo fist 5 natual fequencies fo fist case of table 4. Figue 3-7 shows ode shape fo fist case with = Fig. 6 : Mode shape fo fouth ode Fig. 7 : Mode shape fo fifth ode Displaceent vesus tie diaga ae plotted to undestand the tie esponse with vaying CNT volue faction. Fig.8. shows the esponse of: i. Sinusoidal foce of agnitude 100 N which is acting at the cente of the bea. ii. With claped-claped bounday condition of FG bea and exciting fequency of 80 Hz. Fig. 3 : Mode shape fo fist ode s Fig. 4 : Mode shape fo second ode Fig. 8 : Plot of Displaceent s tie of CNT based FG bea unde sinusoidal loading 101

6 Intenational Jounal on Theoetical and Applied Reseach in Mechanical Engineeing (IJTARME). CONCLUSION CNT based FG Tioshenko bea has been odeled using finite eleent ethod. Fistly convegence of esults has been studied and then fee vibations as well as dynaic analyses have been caied out. Nondiensional fundaental fequencies ae calculated fo diffeent volue factions of CNT and slendeness atio. Mode shapes and displaceent histoy ae also pesented fo this bea. Results show that as volue faction of CNT inceases, fundaental fequency also inceases. The eason behind this is oe volue of CNT povides oe stiffness to the bea which esults in highe fequencies. Siila effect is obtained fo fequencies with incease in the slendeness atio. Non diensional fundaental fequency inceases as slendeness atio inceases. The esponse of displaceent vesus tie shows that fo inceasing values of volue faction displaceent goes on educing. Finally it can be concluded that, as volue faction inceases, stiffness of the bea inceases and it esults in inceased value of fundaental fequency and eduction in deflection. I. RERENCES [1] A. Chakaboty, S. Gopalakishnan, J.N. Reddy, A new bea finite eleent fo the analysis of functionally gaded ateials, Int. J. Mech. Sci. 45 (2003) [2] G.M. Odegad, T.S. Gates, K.E. Wise, C. Pak, E.J. Siochi, Constitutive odelling of nanotubeeinfoced polye coposites, Copos. Sci. Technol. 63 (2003) [3] Dong-Li Shi, Xi-Qiao Feng, Yonggang Y. Huang, Keh-Chih Hwang, Huajian Gao, The effect of nanotube waviness and aggloeation on the elastic popety of cabon nanotube einfoced coposites, J. Eng. Mate. Technol. 126 (2004) [4] J.D. Fidelus, E. Wiesel, F.H. Gojny, K. Schulte, H.D. Wagne, Theo-echanical popeties of andoly oiented cabon/epoxy nanocoposites, Copos. Pat A 36 (2005) [5] J. Wuite, S. Adali, Deflection and stess behaviou of nanocoposite einfoced beas using a ultiscale analysis, Copos. Stuct. 71 (2005) [6] Y. Han, J. Elliott, Molecula dynaics siulations of the elastic popeties of polye/cabon nanotube coposites, Coput. Mate. Sci. 39 (2007) [7] R. Zhu, E. Pan, A.K. Roy, Molecula dynaics study of the stess stain behavio of cabonnanotube einfoced Epon 862 coposites, Mate. Sci. Eng. A 447 (2007) [8] S.A. Sina, H.M. Navazi, H. Haddadpou, An analytical ethod fo fee vibation analysis of functionally gaded beas, Mate. Des. 30 (2009) [9] M. Shokieh, M. Roha Rafiee, On the tensile behavio of an ebedded cabon nanotube in polye atix with non-bonded intephase egion,copos. Stuct. 92 (2010) [10] L.L. Ke, J. Yang, S. Kitiponchai, Nonlinea fee vibation of functionally gaded cabon nanotubeeinfoced coposite beas, Copos. Stuct. 92 (2010) [11] T. Moi, K. Tanaka, Aveage stess in atix and aveage elastic enegy of ateials with Misfitting inclusions, Acta Metall. 21 (1973) [12] B. Sobhani Aagh, A.H. Nasollah Baati, Hedayati H.,Eshelby Moi Tanaka appoach fo vibational behavio of continuously gaded cabon nanotube-einfoced cylindical panels. Coposites: Pat B 43 (2012) [13] Heshati M., Yas M.H., Dynaic analysis of functionally gaded nanocoposite beas einfoced by andoly oiented cabon nanotube unde the action of oving load. Applied Matheatical Modelling 36 (2012) [14] Heshati M., Yas M.H., ibations of nonunifo functionally gaded MWCNTspolystyene nanocoposite beas unde action of oving load. Mateials and Design 46 (2013)