Stress Distribution on a Single-Walled Carbon Nanohorn Embedded in an Epoxy Matrix Nanocomposite Under Axial Force

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1 Cpyght 00 Amecan Scentc Publshes All ghts eseved Pnted n the Unted States Ameca Junal Cmputatnal and Theetcal Nanscence Vl.7, 7, 00 Stess Dstbutn n a Sngle-Walled Cabn Nanhn Embedded n an Epxy Matx Nancmpste Unde Axal Fce K. Mmen and R. S. Yassa Depatment Mechancal Engneeng-Engneeng Mechancs, Mchgan Technlgcal Unvesty, Hughtn, MI 4993, US Cabn Nantubes CNTs) have been a subject nteest mst the eseaches ate the dscvey, due t the supe mechancal ppetes and ablty t be used as the encement phase n nancmpstes. The the cabn stuctue whch was dscveed a ew yeas ate the dscvey CNTs was Cabn Nanhns CNHs). The stuctue CNH s cne-shaped cmpaed t the cylndcal shape CNT s stuctue. Ths cne-shaped stuctue causes dcultes n ndng the stess dstbutn n CNHs when embedded n the matx phase nancmpstes. In ths pape the gvenng deental equatn the stess dstbutn n the CNHs placed n a nancmpste matx s deved usng sme smplyng assumptns. It has been shwn hee that whle the stess dstbutn s symmetc the specal case CNT, t s nn-symmetc the geneal CNHs and the maxmum stess shts twad the tp CNH. Keywds: Analytcal Mdelng, Nan-Stuctues, Stess Tanse, Cabn Nanhn.. INTRODUCTION Few yeas ate the dscvey CNTs by Ijma et al. n 99, CNHs wee ntduced by Has et al.n 994. CNTs shw unque mechancal ppetes, such as lage elastc mdulus whch can each up t 5 TPa 3 8 cmpaed wth elastc mdulus cabn bes whch s m 0. t 0.8 TP 9 and stength up t 63 GPa. 0 Dscvey CNTs made a geat deal nteest amng the scentsts t exple new devces and applcatns.on the the hand, cabn nanhns have a hgh stength sp cabn-cabn bnd, and they can be pduced wth puty hghe than 90%.Ths s because CNHs dn t need the use catalytc patcles synthess n cntast t CNTs. In addtn the suace aea the cabn nanhns s 300 t 400 m /g cmpaed t 78 m /g the mult-walled CNTs and 85 m /g sngle-walled CNTs.Futheme, suace aea CNHs can be mpved by a pst teatment pcess up t me than 000 m /g. 3 7 Due t the abve advantages, t s expected that CNH enced nancmpstes have a hghe mechancal ppetes than the CNT enced nancmpstes.in ths pape the deental equatns gvenng the dstbutn stess n a cabn nanhn whch s placed n an epxy matx unde axal ladng wll be nvestgated usng Cx mdel 8 and t wll be cmpaed wth the stess dstbutn mdel the CNTs. 9. PROBLEMDEFINITION The epesentatve vlume element RVE) ths pblem s shwn n Fgue and the sde vews the CNH n the RVE ae shwn n Fgue.We assumed that L s the lngtudnal length the RVE.In addtn L epesents the lngtudnal length the embedded CNH.We have cnsdeed L and L nstead smply cnsdeng L and L because n the case ze cne angle,.e., a cylndcal tube lke CNT, the stess s symmetc abut the mddle plane. The ute adus the CNH s = a z tan ; whee a epesents the CNH s ute damete at z =0. Cnsdeng the CNH s thckness as t then the elatn between the ute and nne adus the CNH wuld be = t. The equlbum equatns n absence bdy ces, axsymmetc pblem n tems cylndcal cdnate,, z) ae: 0 Auth t whm cespndence shuld be addessed. z = 0 a) J. Cmput. The. Nansc. 00, Vl. 7, N /00/7/00/007 d:0.66/jctn

2 Stess Dstbutn n a Sngle-Walled CNH Embedded n an Epxy Matx Nancmpste Mmen and Yassa z = u w d) Assumng CNH and matx as elastc mateals, the cnsttutve equatns ae: = E = E = E 3a) 3b) 3c) Fg.. RVE a CNH wth length L embedded n a matx wth length L.The cmpste s then beng subjected t the veall stess alng the cylndcal axs. z z = 0 b) Assumng the u and w as dsplacement alng and z dectn, the gemetcal equatns CNH can be wtten as: = u a) = u b) = w c) z = z 3d) G The the stan cmpnents ae vanshed due t the pblem cnguatn and gemety appled ces,.e., axal ladng. In the next step we wll dene the bunday equatns the abve mentned gvenng equatns.the bunday cndtns 4) ae descbng the ce balance at the bundaes the RVE.The Eq.5) ae due t the ce balance at the CNH-matx nteace.accdng t the type ladng, thee s n tactn alng adal dectn n suace the RVE,.e., Eq. 4a). The stess s appled n the extemes the CNH alng ts axs symmety, whch leads t Eq.4b).In Eq.4b) ê z s the unt vect alng the z-axs. { T m =R = 0 4a) T m z=±l =± ê z 4b) The ce balance between the CNH and matx alng the adal dectn and at the exteme ends, lead t Eqs.5a, b). { T L z L = = T m L z L = 5a) T z=±l = T m z=±l 5b) a) b) Fg.. a) Sde vew and b) css-sectnal vew, A-A, the RVE shwn n Fgue.The nne and ute aduses, and 0, CNH change alng the z-axs as a unctn cne angle,, whle ts thckness, t, emans cnstant. 3. SOLUTION In geneal, the ntegatn Eq.b) wth espect t m t a encng CNH desgnated by supescpt : z d z d = 0 6) whee the st pat 6) can be dened as an aveage axal nmal stess ve the css sectn the CNH as belw: = z d 7) J. Cmput. The. Nansc. 7, 7, 00

3 Mmen and Yassa Stess Dstbutn n a Sngle-Walled CNH Embedded n an Epxy Matx Nancmpste Deentatn 7) wth espect t z and usng 6) leads t: d dz = tg z d tg tg 8) By assumng that: Nw ntegatng b) wth espect t m t R and usng 5b) leads t, R z m d R m z d = 0 7) Cnsdeng, m = R m z d 8) then by assumng, z and usng b) we have: = 9) z = 0 0) Ths s a st de lnea deental equatn n tems z whch ts slutn leads t: z = c ) Assumng z = = 0 whch means n matx penetatn nt the hllw pat the CNH we have: z = ) Usng ) and ) we have: c = 3) F smplcty we wll epesent z = by ;Swe btan: = = 4a) z = 4b) The bunday cndtn 4a) mples that thee s n adal nmal stess and shea stess n the ccumeence the matx.s we may ewte Eq.4a) as: m =R = 0 m z =R = 0 In a smla manne ewtng Eq.5a) leads t: L <z<l = = m L <z<l = z L <z<l = = m z L <z<l = 5a) 5b) 6a) 6b) m = g 9) wee gz) s an unknwn unctn that must be detemned. Insetng 9) n 7) and ntegatng wth espect t m t R leads t: g d R R m z d = 0 g = R m z m z = g R 0) Usng 0) and 5b), g = R ) Substtutng ) n 0), m z = ) R R ) and assumng that u/ w/,.e., the adal dsplacement alng the ad s much smalle than the axal dsplacement alng the ad, and d) and 3d), w z = G m wm z = Gm Usng 3b) and ) we have, 3a) 3b) = R Gm R wm 4) Reaangng 4) and ntegatng m t R leads t, ) R d = G m R dw m = R w Gm R m 5) wm R ln R/ / R J. Cmput. The. Nansc. 7, 7, 00 3

4 Stess Dstbutn n a Sngle-Walled CNH Embedded n an Epxy Matx Nancmpste Mmen and Yassa Substtutn 5) nt ) esults n, m z = R w Gm R m wm R ln R/ / R 6) Puttng 6) nt 3b) leads t: w m = R wr m wm R ln R/ / R w m w m R w =wm m R ln/ / R ln R/ / R 7) R 4 ln R/ 4 ) R 3R ) R R ln R R 3 R3 3 ) R R R ln ) R R 8 4 ) R R4 ln 34 ) R4 3) Assumng bth matx and the CNH, due t the act that we cnsdeed the nancmpste unde axal ladng, and usng c), 3c) and ntng that = g, then we have: = E w m = Em wm m = m = R ln/ ) 8) R ln R/ ) R m =R m = w m R wm E m tg R ) ) R ln ) R R ln ) R ) R R ln R 9) Cnsdeng the ce balance the cmpste alng the z axs: R = d Usng 7), 9) and 30) we have: m m =R m = = R ln R/ ) R d 30) R 4 ln R/ 4 ) R 3R R m = R E m tg w m R wm Fm 8) and assumng n matx penetatn nsde the CNH, then: d dz = tg 3) Fm 6), 9b), 3), and 3) t llws that, d dz = tg tan = = G m R R E m R m = R R 4 ln ) 4 ) 3R R R w m R wm R ln ) ) R { Gm tan R ln R R R ) R ) ) R G m tan R R 4 ln ) 4 ) 3R R R R ln R ) R R R3 3 3 R R R R ln ) )) R R 4 8 R4 R ) ln 38 )} R4 33) 4 J. Cmput. The. Nansc. 7, 7, 00

5 Mmen and Yassa Stess Dstbutn n a Sngle-Walled CNH Embedded n an Epxy Matx Nancmpste Due t 8) and n matx penetatn nsde the CNH.e., = 0 and = the llwng equatn can be btaned: d dz = 34) Insetng 5) n 38) we have: wr m wm = d R ln R/ R G m R 35) dz Assumng that the bundng between the CNH and matx t be peect,.e., n sldng ccus at the CNH- Matx nteace, we have: m = = = m z = = Em E = 36) Assumng lw vlume actn the CNH, 37) and usng 33), 35), 36), and 37) we have: d dz d dz = 39) whee z), z), and z) have been dened n 40a c). n 40c) s the appled axal stess t the RVE, whch s cnstant.deental Eq.39) s a secnd de lnea ne whch ts cecents ae unctns ndependent vaable, z.theee t can be slved usng pwe sees methd. = G m R { Gm tan R R R ln ) ) R R G m tan R 4 ln ) 4 3R R R ) R R ln R R R3 3 3 R R R ln R/ / R R ) 4 8 R4 R ln 38 )} R4 40a) = R Em m E R ) R R 4 ln 4 3R R 40b) = R m R R 4 ln ) 4 3R 0 R 40c) Equatn 39) s cmpletely cnsstent wth the pevus wks n CNTs and CNHs. 4. RESULTS AND DISCUSSION Hee the deental Eq.39) has been slved numecally the case epxy matx and CNH whle sx deent cne angles have been cnsdeed.hee we have used the mechancal ppetes CNT, due t the lack data the mechancal ppetes CNH.It s justed because bth CNT and CNH have the same nte atmc bndng,.e., sp.theee, t s expected that the mechancal ppetes the CNH wuld nt be much deent m the mechancal ppetes CNT.The ppetes these mateals ae cnsdeed as llws; = 0, 0, 5,,, and 0, E m =.4 GPa, E = 000 GPa, m = 0 5, a = 0 8 nm, t = 0 34 nm, G m = E m / m, = GPa, L =00 nm and R>5 max = 50 nm. 3 4 F slvng ths pblem Maple has been used.ths pblem has been slved wth Runge-Kutta-Fehlbeg 6 methd whch esults n a th de accuate slutn.the CNH s mean stess dstbutn vesus the dstance m ts tp has been pltted alng ts lngtudnal axs symmety n Fgue 3.F slvng ths pblem we need tw bunday cndtns; Theee we used the value mean be stess at z = 0 and z = L the case pen-ended CNTs. As t has been shwn n Fgue 3, the stess dstbutn alng the CNH s a nn-lnea unctn the cne angle. Meve the maxmum mean nmal stess des nt ccu at the mddle the CNH as t des n the case penended CNTs and cnventnal d-shape bes. 7 Fm Fgue 3 we can cnclude that the lage the cne angle, the uthe the devatn lcatn maxmum mean stess m the mddle the be. In the case ze cne angle, the maxmum ccus at the mddle the be whch s plausble due t the act that n ths case the gvenng deental equatns wuld be exactly the same as the case hllw cylndcal be. In addtn t the devatn maxmum mean nmal stess, the ate change mean stess s nt symmetc n the case CNH.I we classy the stess dstbutn n the CNH nt tw sectns, ne m the CNH s tp t max value stess and the the m the max value stess t the ppste end CNH, t can be seen that the ate change mean stess s hghe at the st sectn cmpaed t the the sectn Fg.4).Futheme the J. Cmput. The. Nansc. 7, 7, 00 5

6 Stess Dstbutn n a Sngle-Walled CNH Embedded n an Epxy Matx Nancmpste Mmen and Yassa means utue desgn new geneatn nancmpstes enced wth CNHs. 5. CONCLUSION The supe ppetes the CNHs cmpaed t CNTs, such as hghe puty and suace aeas, pmse an nceasng applcatn CNHs n utue nancmpstes. In ths wk a gus mathematcal amewk t mdel the stess dstbutn at the nteace a CNH and cmpste matx s deved.it has been shwn that the stess dstbutn alng the CNH s a nn-lnea unctn the cne angle.als we shwed that the lage the cne angle, the bgge s the devatn mean stess dstbutn m the symmetc case.althugh the mula btaned n ths pape ae the case CNHs but t s als applcable t the CNTs because CNTs ae specal case CNHs wth ze cne angle. Fg. 3. Mean stess,, vesus CNH dstance alng the axes CNH, L.Whle the maxmum stess ccus at the mddle the CNT, the case CNH, the pstn maxmum stess mves twad the tp CNH. ate change maxmum mean stess s lwe at the hghe angles but t wll change abuptly at the smalle angles.these eatues happen due t the act that the stess tanses t the CNH due t the shea stess between the matx and CNH.The amunt tanseed stess at the exteme ends the CNH s neglgble and t ses t a maxmum smewhee n between.on the the hand, the css-sectn aea the CNH changes m a mnmum value t a maxmum ne.whle the stess s pptnal t the amunt tanseed shea stess, t s nvesely pptnal t the css-sectn aea.it s expected that the deved mathematcal amewk pvdes a cmputatnal NOMENCLATURE z Axal cdnate the RVE a Radal cdnate the RVE R Radus the RVE Hal cne angle the CNH b Oute adus the CNH Inne adus the CNH a CNH s ute damete at z = 0 t Thckness CNH s wall Stan Cnstant appled axal stess t the RVE Aveage axal nmal stess Shea Stess Pssn s at G Shea Mdulus Shea stan m Paametes asscated t matx mateal Paametes asscated t CNH u Radal dsplacement w Axal dsplacement T Tactn L Length the RVE a Repesentatve Vlume Element. b Cabn Nan Hn. Acknwledgments: We appecate Ms.Sghat the help and the Mchgan Technlgcal Unvesty pvdng the nancal suppt. Reeences Fg. 4. Rate change mean stess alng the CNH,.e., /.. S.Ijma, Natue 56, ).. P.J.F.Has, M.L.H.Geen, and S.C.Tsang, J. Chem. Sc.- Faaday Tans. 89, ). 3. E.W.Wng, P.E.Sheehan, and C.M.Lebe, Scence 77, ). 6 J. Cmput. The. Nansc. 7, 7, 00

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