Application of Halpin-Tsai Method in Modelling and Sizedependent Vibration Analysis of CNTs/fiber/polymer Composite Microplates

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1 JCAMECH Vol. 47 No. June 0 pp 45-5 DOI: 0.059/jcamech Applcaton of Halpn-Tsa Method n Modellng and Szedependent Vbraton Analyss of CNTs/fber/polymer Composte Mcroplates Ghorbanpour Aran H. Baba Akbar Zare E. Haghparast Faculty of Mechancal Engneerng Unversty of Kashan Kashan Iran Insttute of Nanoscence& Nanotechnology Unversty of Kashan Kashan Iran Abstract Receved: 6 Jan 0 Accepted: 4 Apr. 0 In the present study modellng and vbraton analyss of Carbon nanotubes/ fber/ polymer composte mcroplates are nvestgated. The governng equatons of the Carbon nanotubes/ fber/ polymer composte mcroplates are derved based on frst order shear deformaton plate theory rather than other plate theores due to accuracy and smplcty of polynomal functons. The modfed couple stress theory s employed because of ts capablty to nterpret the sze effect. Halpn-Tsa model s utlzed to evaluate the materal propertes of two-phase composte conssted of unformly dstrbuted and randomly orented Carbon nanotubes through the epoy resn matr. Afterwards the structural propertes of carbon nanotubes renforced polymer matr whch s assumed as a new matr and then renforced wth E-Glass fber they are calculated by fber mcromechancs approach. Employng Hamlton s prncple the equatons of moton are obtaned and solved by Hybrd analytcal numercal method. The nfluences of varous parameters such as the weght percentage of sngle-walled carbon nanotube aspect rato and sze effect on the vbraton characterstcs of mcroplate are dscussed n detals. Results ndcate that the stablty of Carbon nanotubes/fber/polymer composte mcroplates can be mproved by addng approprate values of Carbon nanotubes. In addton ncrease n the frequences s more pronounced n the case of mcroplates renforced wth SWCNT compared wth MWCNT. These fndngs can be used n desgn and manufacturng of marne vessels and arcrafts. Keywords: Halpn-Tsa model modfed couple stress theory multscale composte plate vbraton analyss. Introducton Carbon nanotubes (CNTs) are molecular structures of Tel.: ; Fa: E-mal address: aghorban@kashanu.ac.r graphtc carbons wth ecellent and remarkable propertes. The eceptonal mechancal propertes of 45

2 Ghorbanpour Aran et al. the CNTs combned wth ther low densty offer scope for the development of nanotube renforced composte materals. Due to ther etraordnary specfc stffness and strength the study of mcro- and nano-compostes renforced wth CNTs represent tremendous opportunty for researchers and ndustralsts. Therefore the presence of the CNTs can mprove the strength and stffness of structures as well as electrcal and thermal conductvtes to composte systems. In mcro- and nano-scale studes t s mportant to consder the sze effect. Due to lack of materal length scale parameters conventonal plate models based on classcal contnuum theores do not consder the sze effects whle the sze effects have been observed [-3] epermentally. Therefore sze-dependent plate models based on sze-dependent contnuum theores that contan addtonal materal length scale parameters have been developed. Several szedependent contnuum theores have been developed to account for the sze effects such as the classcal couple stress theory (CCST) [4] and the nonlocal elastcty theory [5] wth two materal length scale parameters and the stran gradent theory (SGT) [6] wth three materal length scale parameters. Consderng the dffcultes n determnng materal length scale parameters the modfed couple stress theory (MCST) [7] takes an advantage over the aforementoned szedependent contnuum theores contanng only one materal length scale parameter. Also MCST ncludes asymmetrc couple stress tensor respect to CCST. Tsatas s nvestgaton [] can be noted as one of the frst works that employed modfed couple stress theory (MCST) for plate. He developed a Krchhoff plate model based on a MCST for statc analyss of mcroplates. Then Yn et al. [9] and Jomehzadeh et al. [0] utlzed ths model to study the vbraton of mcroplates. Asghar [] studed geometrcally nonlnear mcro-plate formulaton wth Krchhoff plate theory of sotropc plates. He presents the general form of boundary condtons consderng the sharp corners. Also Sahman and Ansar [] presented SGT for the free vbraton of functonally graded (FG) mcro-plates based on non-classcal hgher-order shear deformable plate model and Mor-Tanaka homogenzaton technque. They obtaned that n the hgher length scale parameters the materal property gradent nde has more effects on the vbraton behavor of FG mcro plates for MCST and stran gradent theory (SGT) wth respect to classcal plate theory (CPT). Mohammadmehr et al. [3] presented the szedependent effect on the bucklng and vbraton analyss of double-bonded nanocomposte pezoelectrc plate renforced by boron ntrde nanotube. Usng MCST they studed the effects of materal length scale parameter elastc foundaton coeffcents and transverse and longtudnal wave numbers etc. on the dmensonless natural frequency. Motvated by the aforementoned deas the vbraton analyss of CNTs/ fber/ polymer multscale composte mcroplate s nvestgated n ths study for the frst tme. Halpn-Tsa model and fber mcromechancs approach are used to determne the materal propertes of the multscale composte plate. Also frst order shear deformaton plate theory (FSDT) s selected for the dsplacement feld. To consder the sze effects MCST are utlzed to obtan stran energy. Fnally the equatons of moton for mcroplate are derved based on Hamlton s prncple and solved by means of hybrd analytcal numercal method for smply supported boundary condtons. Effects of varous parameters such as the weght percentage of sngle-walled carbon nanotube aspect rato and sze effect on the dmensonless frequences of mcroplates are nvestgated. The natural frequences of plates are compared wth the prevous studes n lterature to verfy the presented research. The result of ths work can be useful to control and mprove the performance of composte devces whch are employed n mltary equpment's.. Mcromechancs modellng Consder a multscale composte mcroplate composed of matr ( m ) and fber ( f ) phases. Note that matr phase conssts of CNTs and resn where CNTs were assumed to be unformly dstrbuted and randomly orented through the matr. To predct the materal propertes of multscale composte a combnaton of Halpn Tsa model and mcromechancs approach were used n herarchy as shown n Fgure. Fg.. A herarchcal confguraton for modellng of three phase multscale composte.. Halpn Tsa model The elastc propertes of matr phase can be calculated by a sem-emprcal method. Accordng to Halpn-Ts model the tensle modulus of matr phase 46

3 Vol. 47 No. June 0 can be epressed as [4 5]: m 3 l [ ( NTLVNT E ) LVNT d NT 5 DVNT ] Eepoy DVNT E NT dnt E NT lnt L E epoy 4t E epoy t E NT dnt E NT d NT D E epoy 4t E epoy t (a) (b)) (c) w V NT NT w NT / / NT epoy NT epoy (d) where E V w and represent Young s modulus volume fracton mass fracton and densty respectvely. Also d NT l N T and t are dameter length and thckness of CNT respectvely. Hence the subscrpts NT and epoy are related to CNT and the resn epoy respectvely. Therefore the shear modulus of matr phase can be calculated by: m m E G m ( ) m epoy n whch s Posson rato. (a) (b).. Mcromechancs approach The elastc propertes of the mcroplate can be determned by Mcromechancs approach as follows []: E E f V f E m V m (3a) f m V V f m V V f m f m f m f m f m ( ) E E ( ) E E f f m m E E E E E where E and f m V V G f m G G (3b) (3c) c f f m m V V f V f m V m (3d) (3e) E are longtudnal and transverse Young s modulus respectvely. Also G s n plane shear modulus. 3. Fundamental relatons As shown n Fgure consder rectangular mcroplate wth length a wdth b and thckness h. The global coordnate system ( y and z as) s located n mdplane so that the orgn s at one corner of mcroplate. Accordng to FSDT dsplacement feld of mcroplate can be epressed as follows [7]: w ( y t) U( y z t) u( y t) z z ( y t) w ( y t) V ( y z t) v( y t) z zy ( y t) y W ( y z t) w ( y t) (4) n whch u v and w represent dsplacement through the y and z aes respectvely. Also and y are the rotatonal about and y aes respectvely. Fg.. Schematc fgure of CNTs/fber/polymer multscale composte mcroplate The lnear stran relatons can be descrbed as: u w v w y z yy z k y s y y yz y y u v w y y y y (5) n whch and are normal strans and shear strans respectvely and k s s shear correcton factor. It should be noted that zz 0. Based on the Hook s law the consttutve equatons can be epressed as: Q Q yy Q Q yy y 0 0 Q y yz Q44 0 yz Q z 55z where (6) 47

4 Ghorbanpour Aran et al. E E Q Q Q E Q Q44 G3 Q66 G Q55 G3 pq ( p q y z ) are stresses and Q ( rs and rs ) are the materal constants. Mcroplate s made of sngle orthotropc layer whle renforced by fbers. 6 4 and 5 denote the local coordnate aes where as s along fbers drecton as s normal to fbers drecton and 6 as s shown shear n-plan drecton. Also 4 and 5 represent shear out of plan drectons. Therefore the consttutve equatons must be transformed to the global coordnates. So the consttutve equatons of multscale composte can be rewrtten as follows []. Q Q Q 0 0 yy Q Q Q6 0 0 yy y Q Q6 Q y yz Q44 Q 45 yz z Q45 Q 55 z where Q ( j 6 and 45) are gven as: j 4 4 Q m Q n Q m n Q 4 m n Q Q n Q m Q m n Q 4 m n Q Q m n Q m n Q ( m n ) Q 4 m n Q Q m nq Q ( m n) Q ( m n) Q Q6 Q m nq ( m n ) Q ( m n ) Q66 Q66 m n Qm n Q m n Q ( m n ) Q66 Q44 m Q44 n Q55 Q45 ( Q55 Q44) Q55 n Q44 m Q55 E (7) () (9) n whch m cos and n sn ( beng the angle between global as and fber aes). 4. Hamlton's prncple Hamlton s prncple s employed to derve the moton equatons as follows []: n whch t t ( Us K ) dt 0 (0) U s and K represent stran energy and knetc energy respectvely. MCST s used to derve equatons of stran energy as [7]: yyyy zz zz Us dv V yy zz yz yz () m myyyy mzz zz dv V myy mzz myz yz pq s symmetrc curvature tensor and m pq s devatory part of the couple stress tensor. Also m pq s the stress whch s palpable n the small scale and cannot be gnored and s defned as: m pq M pq M l0 G (a) (b) jq jp (c) pq epj eqj where l 0 and e pj represent the materal length scale parameter and alternate tensor respectvely. Also G s equal to Q 66 n ths paper. So the symmetrc curvature tensor can be rewrtten by substtutng Eq. (5) nto Eq. (c) as follows: w y y w yy y y y zz y y w w y 4 y y (3a) (3b) (3c) (3d) v u y (3e) z z 4 y 4 y v u y (3f) yz z. 4y 4 y y y Substtutng Eq. (5) nto Eq. () and usng Eqs. ()-(3) stran energy can be obtaned. Also the knetc energy of mcroplate can be determned by: c U V W K dv. V t t t (4) Substtutng Eq. (4) nto Eq. (4) knetc energy can be calculated. Now usng Hamlton s prncple and puttng the coeffcents of u v w and y equal to zero the moton equatons are obtaned. For the sake of brevty these equatons are not 4

5 Vol. 47 No. June 0 presented here. 5. Soluton Procedure In the case of smply supported mcroplate the dsplacement components can be defned accordng to the Naver s procedure whch automatcally satsfy the boundary condtons at the mcroplate edges. Based on ths procedure the dsplacement components can be epressed n the followng forms: y sn e ( ) cos sn e t uyt U y ( ) cos sn e t v y t V y w( y t) W sn sn y e ( y t) P cos sn e ( y t) Q cos y y m n t t t (5) m n n whch and. m and n a b show the wave or mode numbers along the and y drectons respectvely. Also U V W P and Q are unknown coeffcents of each mode numbers and denotes the natural frequency. Insertng the dsplacement components from Eq. (5) nto the moton equatons fnal relaton s obtaned as a matr form: K K U 0 K K V K33 K34 K35 W K43 K44 K 45 P K53 K54 K 55 Q 0 Matr K () for whch the components of matr K have been mentoned n Append A. 6. Numercal results In ths secton effects of varous parameters such as weght percentage of sngle walled carbon nanotube (SWCNT) sze effect fbers orentaton aspect rato and thckness on the vbraton characterstcs of mcroplates are dscussed n detals. The values of the dfferent parameters n ths study were obtaned accordng to Rafee et al. [9]. Fgure 3 shows the weght percentage effect of SWCNT on dmensonless frequences versus aspect rato (a/b) of multscale composte mcroplates. As can be seen ncreasng aspect rato of mcroplate leads to ncrease dmensonless frequences of mcroplate. Also ncreasng weght percentage of SWCNT leads to ncrease stffness of mcroplate and consequently leads to mproved dmensonless frequences. Dmensonless frequency ( ) a/b Fg. 3. The weght percentage effect of SWCNT on dmensonless frequences versus aspect rato of mcroplate Fgure 4 llustrates the dmensonless natural frequency versus aspect rato of mcroplate n varous aspect ratos (length-to-dameter) of CNTs. It can be found that ncreasng length-to-dameter rato of CNTs causes the ncrease of dmensonless frequences and t s because of the fact that nternal structure of mcroplate are further strengthened. Dmensonless natural frequences versus length-to-thckness rato of mcroplate n varous weght percentages of CNTs are studed and depcted n Fgure 5. Accordng to ths fgure dmensonless frequences of mcroplate are reduced by ncreasng length-to-thckness rato (a/h) of mcroplate. It s obvous that a hgher length-tothckness rato whch s related to thnner mcroplate has a lower stffness. Moreover SWCNT renforced composte mcroplates ndcate better resstance to reduced natural frequences compared to mult-walled carbon nanotube (MWCNT) renforced composte mcroplates. Dmensonless frequency ( ) % SWCNT 3% SWCNT % SWCNT % SWCNT 0% L/d=00 L/d=300 L/d=500 L/d= a/b Fg. 4. The nfluence of SWCNTs aspect rato on the dmensonless frequences versus aspect rato of mcroplate 49

6 Ghorbanpour Aran et al. Effect of CNT content on the dmensonless natural frequency under varous fbers orentaton s presented n Fgure 6. Increasng volume fracton of CNT dmensonless natural frequency decreases from 0 to /4 radan and t ncreases from /4 to / radan. Therefore the results of ths fgure can help n selectng an approprate combnaton of volume fracton of CNT and fbers orentaton n order to optmze the desgn of multscale composte mcroplates. Dmensonless frequency ( ) =0 =/6 =/4 =/3 =/ V CNT Fg. 6. Effect of CNT content on the dmensonless natural frequency under varous fbers orentaton.4 Fg. 5. Dmensonless natural frequency versus length-tothckness rato of mcroplate n varous weght percentages of CNTs Fgure 7 shows the sze effect on dmensonless natural frequences n varous aspect ratos of mcroplate. Accordng to ths the results of MCST are as the same of CPT when the materal lengths scale parameter s equal to zero. Also MCST gves the hgher natural frequences respect to CPT. In addton selectng a hgher materal lengths scale parameter causes ncreased dmensonless frequences appromately. In order to verfy the relablty of the results the comparson between the frequences of sotropc mcroplate n the present study and the work whch s done by Ke et al. [0] s performed. As can be seen n Fgure there s a good agreement between the results of present study and the work done by Ke et al. [0]. 7. Concluson Ths paper develops the vbraton analyss of CNTs/fber/polymer multscale composte mcroplates usng MCST for the frst tme. To determne the materal propertes of the multscale composte plate Halpn-Tsa model and fber mcromechancs approach are employed. The moton equatons for mcroplate are derved usng FSDT and solved by means of Naver soluton for smply supported Dmensonless frequency ( ) l Fg. 7. The sze effect on dmensonless natural frequences n varous aspect rato of mcroplate Dmensonless frequency ( ) Present work Ke et al. [0] a=4b a=3b a=b a=b a/b Fg.. Comparson between the frequences of sotropc mcroplate n the present study and the work whch s done by Ke et al. [0] the weght percentage of SWCNT sze effect fbers orentaton aspect rato and thckness on the vbraton characterstcs of multscale composte mcroplate. It s found that: Dmensonless frequences of mcroplates are ncreased by ncreasng weght percentage of 50

7 Vol. 47 No. June 0 SWCNT aspect rato of mcroplate and length-todameter rato of CNTs. Dmensonless frequences of mcroplate are reduced by ncreasng length-to-thckness rato of mcroplates. Increase n the frequences s more pronounced n the case of mcroplates renforced wth SWCNT compared wth MWCNT. The varaton of fbers orentaton up to /4radan leads to decrease n dmensonless natural frequency. The varaton of fbers orentaton more than /4 radan leads to ncrease n dmensonless natural frequency. MCST desres CPT whle the materal length scale parameter s equal to zero. Also MCST gves the hgher natural frequences respect to CPT.. Append The components of matr K can be obtaned as follows: 3 3 K I0Q K I0Q66K I0M K 4 c I0MK I0 K (A) 4 3 K I0MK I0Q66K I0Q6 KI 3 (A) IM 0 K IQ 0 KIQ 0 K 3 K I0M K3 I0Q66 K3 I0Q6K 4 3 (A3) IM 0 K3 IQ 0 K3 IQ 0 K 3 IQ 0 6 K IQ 0 K3 IQ 0 66 K3 4 3 c IM 0 K3 IM 0 K3 I0 K 3 K c c c K33 I K4 I K4 I0 K4 3 I0M K4 QI K4 4Q66I K Q6 IK 4Q I K QI K I0M K4 I0M K4 QI K4 4 4 (A4) (A5) 3 c 3 K34 I0M K4 I K4 3QI K 4 4 QI K4 I0M K4 Q I K4 3 Q66 I K4 Q6 IK 3 3 c K35 I0M K4 QI K I K4 4 4 /I0M K4 Q I K4 Q I K4 3 3Q6 IK Q66 I K4 3 K43 c IK 3QI K I0MK 3 3 I0M KQ6I K Q66IK 3 QI K Q I K K44 I0ksQ55 K Q66 IK Q I K 3 3 MI K /I0MK I0M K 4 c 3 MIK IK Q I K (A6) (A7) (A) (A9) K45 Q I K Q I K Q66 IK Q6 I K MI K MIK (A0) 5 I0MK 3 c K53 I0M K3 I K3 Q66I K3 3 3 Q I K Q I K3 3Q6 IK 3 I0M K3 Q I K3 5 K54 Q66 I K3 I0M K3 Q6 IK 3 4 Q I K Q I K3 MI K3 3 M I K3 K55 I0ksQ44 K3 I0M K3 Q6IK c 3 IK3 Q66 I K3 Q I K MI K3 MI K3 I0M K3 (A) (A) (A3) where 5

8 Ghorbanpour Aran et al. 3 4 ba 00 ba 00 ba 00 ba 00 K = cos( )sn( y) cos( )sn( y) ddy K = sn( )cos( y) cos( )sn( y) ddy (A4) K = sn( )cos( y) sn( )cos( y) ddy K = sn( )sn( y) sn( )sn( y) ddy. References: [] Fleck N.A. Muller G.M. Ashby M.F. and Hutchnson J.W. (994). Stran gradent plastcty: theory and eperment Acta Metallurgca et Materala 4(): [] Stolken J.S. and Evans A.G. (99). A mcrobend test method for measurng the plastcty length scale Acta Materala 46(4): [3] Chong A.C.M. Yang F. Lam D.C.C. and Tong P. (00). Torson and bendng of mcron-scaled structures Journal of Materals Research (4): [4] Koter W.T. (964). Couple stresses n the theory of elastcty I and II Proceedngs of the Konnkljke Nederlandse Akademe van Wetenschappen Seres B 67: [5] Erngen A.C. (97). Nonlocal polar elastc contnua Internatonal Journal of Engneerng Scence 0():. [6] Lam D.C.C. Yang F. Chong A.C.M. Wang J. and Tong P. (003). Eperments and theory n stran gradent elastcty Journal of the Mechancs and Physcs of Solds 5(): [7] Yang F. Chong A.C.M. Lam D.C.C. and Tong P. (00). Couple stress based stran gradent theory for elastcty Internatonal Journal of Solds and Structures 39(0): [] Tsatas G.C. (009). A new Krchhoff plate model based on a modfed couple stress theory Internatonal Journal of Solds and Structures 46(3): [9] Yn L. Qan Q. Wang L. and Xa W. (00). Vbraton analyss of mcroscale plates based on modfed couple stress theory Acta Mechanca Sold a Snca 3(5): [0] Jomehzadeh E. Noor H.R. and Sad A.R. (0). The sze-dependent vbraton analyss of mcro plates based on a modfed couple stress theory Physca E: Low-dmensonal Systems and Nanostructures 43(4) [] Asghar M. (0). Geometrcally nonlnear mcro-plate formulaton based on the modfed couple stress theory Internatonal Journal of Engneerng Scence 5: [] Sahman S. and Ansar R. (03). On the free vbraton response of functonally graded hgherorder shear deformable mcroplates based on the stran gradent elastcty theory Composte Structures 95: [3] Mohammadmehr M. Mohandes M. and Morad M. (04). Sze dependent effect on the bucklng and vbraton analyss of double bonded nanocomposte pezoelectrc plate ren- forced by BNNT based on modfed couple stress theory Journal of Vbraton and Control do: 0.77/ [4] Km M. Park Y.B. Okol O.I. and Zhang C. (009). Processng characterzaton and modelng of carbon nanotube-renforced multscale compostes Compostes Scence and Technology 69(3-4): [5] Bhardwaj G. Upadhyay A.K. Pandey R. and Shukla K.K. (03). Non-lnear fleural and dynamc response of CNT renforced lamnated composte plates Compostes Part B: Engneerng 45(): [] Reddy J.N. (004). Mechancs of Lamnated Composte Plates and Shells: Theory and Analyss CRC Press Boca Raton FL Second Edton. [7] GhorbanpourAran A. and Haghparast E. (05). Sze-dependent vbraton of aally movng vscoelastc mcroplates based on snusodal shear deformaton theory Internatonal Journal of Appled Mechancs. [] Ghorbanpour Aran A. Khoddam Maragh Z. and Khan Aran H. (05). Orthotropc patterns of Pasternak foundaton n smart vbraton analyss of magnetostrctve nanoplate Journal of Mechancal Engneerng Scence do: 0.77/ [9] Rafee M. He X.Q. Maresh S. and Lew K.M. (04). Modelng and stress analyss of smart CNTs/fber/polymer multscale composte plates Internatonal Journal of Appled Mechancs 6(3): [0] Ke LL. Wang Y.S. Yang J. and Ktporncha S. (0). Free vbraton of sze-dependent Mndln mcroplates based on the modfed couple stress theory Journal of Sound and Vbraton 33: