Prediction of Thermal Conductivity of Aligned Short Fibre Composites with Different Fibre Aspect Ratios

Transcription

1 Predctn f Theral Cnductvty f Algned Shrt Fbre Cpstes wth Dfferent Fbre Aspect Rats Predctn f Theral Cnductvty f Algned Shrt Fbre Cpstes wth Dfferent Fbre Aspect Rats Jae-Kn Lee Schl f Mechancal and Auttve Engneerng, Cathlc Unversty f Daegu, Gyengsan, Gyengbuk, Krea Receved: 8 July 006 Accepted: 4 Octber 006 SUMMARY A clsed-fr slutn usng the dstrbutn f fbre aspect rats s prpsed fr predctng the theral cnductvty f algned shrt fbre cpstes. The present del s the extended versn f Hatta and Taya s del, where Eshelby s equvalent nclusn ethd dfed fr fnte vlue fractn f fbres s eplyed. The valdty f usng the average fbre aspect rat fr predctng the theral cnductvty s exaned n ters f the dstrbutn f the aspect rats such as ts average and standard devatn, fbre vlue fractn, and cnsttuents theral cnductvty rat. The gudelne fr the selectn f ether the actual dstrbutn r the average fbre aspect rat s presented n ters f the theral cnductvty rat and the ceffcent f the varatn f the aspect rat fr the better predctn f the theral cnductvty f the cpstes.. ITRODUCTIO The use f shrt fbre cpstes s ncreasng due t ther lw cst and relatve ease f a fabrcatn. Dstrbutns f fbre aspect rats and fbre rentatns n these cpstes are nevtable durng fabrcatn prcesses. In rder t desgn and fabrcate the cpstes, ther therechancal prpertes ust be predcted accurately. Thus, uch attentn has been pad t the understandng f the effect f crstructure n the ther-echancal prpertes f the cpstes. Many researches have attepted t predct the ther-echancal prpertes f the shrt fbre cpstes -. It s well knwn that the st ppular apprach fr predctng the echancal prpertes f the cpstes wth the dstrbutn f the fbre aspect rats s t replace the dstrbutn wth an average aspect rat 3,7,8,. A new del-based Rapra Technlgy, 007 averagng ethd has been prpsed fr analysng the effect f the aspect rat dstrbutn n the vscsty f se-cncentrated plyer-fbre suspensns nstead f cnventnal averagng technques because f ther pr representatn f the aspect rat dstrbutn. Algned shrt fbre cpstes wth a sngle aspect rat have been exaned t study ther theral cnductvtes,4. It was shwn that the theral cnductvty s strngly dependent n the aspect rat. In real shrt fbre cpstes, the dstrbutns f the aspect rats and fbre rentatns are unavdable, ang whch the effect f the latter n the theral cnductvty has been extensvely nvestgated,6,0. By cnsderng the cpstes wth a cnstant aspect rat f fbres, the theral cnductvty has been shwn t be sgnfcantly affected by the fbre rentatn. Hwever, studes f the effect f the aspect rat dstrbutn n the cnductvty are very lted. The theral cnductvty f cpstes havng an aspect rat dstrbutn has been predcted usng the lanate analgy apprach 0. Fu and Ma 0 have pnted ut that Hatta and Taya s del can accurately predct the theral cnductvty f shrt fbre cpstes havng a sngle aspect rat. But ths del s nt sutable fr the predctn f the theral cnductvty f a cpste havng an aspect rat dstrbutn. In ths paper, a clsed-fr slutn fr predctng the theral cnductvty f an algned shrt fbre cpste wth a dstrbutn f aspect rats s prpsed nstead f usng the average aspect rat. The present del s the extended versn f Hatta and Taya s del fr a three-phase cpste, where Eshelby s equvalent nclusn ethd dfed fr a fnte fbre vlue fractn s selected as the fraewrk f an analytcal study 3,4. The valdty f usng the average aspect rat fr predctng the theral cnductvty s exaned n ters f the dstrbutn f the aspect rats such as ts average and standard devatn, fbre vlue fractn, and cnsttuents theral cnductvty rat. Gudelnes Plyers & Plyer Cpstes, Vl. 5,. 4,

2 Jae-Kn Lee fr predctng the theral cnductvty f cpstes wth the average fbre aspect rat are prpsed, based n the theral cnductvty rat and the ceffcent f varatn f the fbre aspect rats. Fgure. Hstgra shwng the dstrbutn f fbre aspect rats easured fr cr-structural bservatn f the cpste, where the fbre vlue fractn crrespndng t the class ark α s f. AALYTICAL MODELLIG. Dstrbutn f Fbre Aspect Rats The dstrbutn f the aspect rats can be experentally easured fr the cr-structural bservatn f cpstes, and graphcally represented as ether a hstgra r a plygn. As shwn n Fgure, the aspect rats f the fbres are dstrbuted between α n and α ax, and the nterval f the aspect rat fr the graphcal representatn s usually subdvded nt suffcently sall class ntervals. If the daeters f the shrt fbres are assued t be cnstant, the frequency hstgra can be easly cnverted nt the relatve frequency f the fbre vlue fractn f fr each class ark α as shwn n Fgure. The subscrpt represents the class nterval, rangng fr t the ttal nuber f the class ntervals. The theral cnductvty f the cpstes s cputed by the present del depcted n Sectn. fr the actual dstrbutn f the aspect rats. In rder t nvestgate analytcally the effect f certan paraeters relatng t the aspect rat dstrbutn, such as ts average and standard devatn, n the theral cnductvty, the relatn between fbre vlue fractn and aspect rat s selected as the nral dstrbutn. The prbablty densty functn f the fbre vlue fractn s a functn f the aspect rat, whch s represented as ρ α α ρα ( )= exp σ π σ () where α, α, σ, and ρ dente the aspect rat, average aspect rat, the standard devatn f the aspect rat, and a scalng factr, respectvely. α n s assued t be and α ax s autatcally deterned by the syetry f the nral dstrbutn. Integratng ρ(α) fr α n t α ax, the cuulatve prbablty f the prbablty densty functn shwn n Eq. () s always less than because f the prbablty relatng t the utsde f the ntegral nterval nt beng sued. S, the scalng factr ρ s ntrduced nt Eq. (), whch s cputed as ρ = erf α α σ ax () where erf eans the errr functn. T cpute the theral cnductvty, the cntnuus densty functn n Eq. () s subdvded nt several class ntervals. The fbre vlue fractn crrespndng t the class ark α s then cputed by ultplyng the densty, the length f the class nterval, and ttal fbre vlue fractn f, whch s dented as f. Whenever the dstrbutn f the aspect rats s gven, the theral cnductvty can be btaned wth the help f Sectn... Theral Cnductvty f Cpstes Let s cnsder the nfnte dan f a cpste (D) n whch nuerus shrt fbres (Ω ) are dstrbuted, as shwn n Fgure (a). Fbres wth the sae daeter and dfferent lengths are assued t be algned n the drectn f x 3, and the shapes f the fbres are sulated as an ellpsdal spherd wth dfferent aspect rats. Fgure (a) s cnverted nt Fgure (b) usng Eshelby s equvalent nclusn ethd 3. The atrx and fbre are assued t be strpc, whse theral cnductvtes are dented by k and k f, respectvely. Subscrpts and f represent the atrx and fbre, respectvely and bld-faced letters dente a vectr r atrx. The cpste s subjected t a cnstant heat flux (q ) alng the x 3 drectn. When ne fbre s ebedded n the cpste, t s cnsdered t be the representatve fbre f thse fbres n the -th class. By usng Eshelby s equvalent nclusn ethd 3 wth Mr-Tanaka s ean feld apprach 4, heat fluxes n the atrx and -th class fbre dans, q and q f, are expressed as qf = Kf ( + + )= * K + + ( ) q = K + (3) ( ) (4) where K s the theral cnductvty atrx, s the average f a dsturbed teperature gradent vectr n the 74 Plyers & Plyer Cpstes, Vl. 5,. 4, 007

3 Predctn f Theral Cnductvty f Algned Shrt Fbre Cpstes wth Dfferent Fbre Aspect Rats atrx, s the dsturbed teperature gradent vectr n the -th class fbres, * s the egen-teperature gradent vectr f the equvalent nclusn prble. Snce s a cnstant teperature gradent vectr n the atrx wthut any fbre generated by the heat flux q, ther relatnshp s gven by q = K (5) Snce the ntegratn f the dsturbed heat flux ver the entre cpste dan s reduced t zer, the fllwng relatnshp s derved fr Eqs. (3)~(5): ( ) + = f q fq 0 f = (6) where f and f represents the ttal fbre vlue fractn and the fbre vlue fractn wthn the -th class ark, respectvely. They are related by the expressn f = = f (7) Insertng Eqs. (3) and (4) nt Eq. (6), s gven by * + f ( )= 0 = (8) The teperature gradent vectr n the fbre s related t * thrugh Eshelby s tensr S as where I dentes the 3 3 dentty atrx. The relatn s further splfed t: * * * *( ) k k k3 k ( ) 3 A + A + A + + A + A k * = B where A kk, A k, and B k are defned by the fllwng equatns: {( ) ( ) + } k Akk = Kf K fk S f k I + K A ( k )= f K K S I k f ( ) = B = K K D q k f k () () ( )( ) (3) k (4) By extendng Eq. () t all the class arks, all the egen-teperature gradent vectrs can be expressed n the parttned atrx fr as A A A ( ) A A A A( ) A A( ) A( ) A( )( ) A ( A A A ( ) A ) * * * * ( ) whch s further reduced nt a splfed fr as fllws: A * = Dq D D = q D ( ) D (5) (6) The egen-teperature gradent vectr fr each class ark s deterned by the equatn * = A Dq = Fq (7) The ttal teperature gradent vectrs n the atrx and fbre f the -th class ark, and, are defned as f = S * (9) = + (8) Fr Eqs. (3)~(5), (8), and (9), the relatnshp between egenteperature gradent vectrs fr the k-th class ark s derved: ( )( ) + ( ) k K + fk Kf K S fk Kf K ( f) f = k + K K * ( ) = ( f) S I K K (0) * k = + + f (9) Ther vlue average n the entre cpste s equal t the teperature gradent vectr n the cpste, whch s gven by = f f ffq c f = = ( ) + = + (0) The relatnshp between the heat flux, teperature gradent vectr, and theral cnductvty atrx f the cpste s expressed as Plyers & Plyer Cpstes, Vl. 5,. 4,

4 Jae-Kn Lee q = K = q c c c () Fnally, the theral cnductvty f the cpste K c s deterned by Eqs. (5), (0), and (), whch s reduced t Fgure. Analytcal del fr cputng heat fluxes and teperature gradent vectr n bth fbre and atrx; (a) rgnal prble, whch s cnverted t (b) Eshelby s equvalent nclusn prble Kc = K + ff = () 3. RESULTS AD DISCUSSIO 3. The Effect f Fller Shape The effect f the fller shape n the theral cnductvty f the cpste s exaned by cparng the predcted results wth the experental nes fr the lterature 5. The del cpste s a plyer atrx renfrced wth slca, aluna, and alunu ntrde, where the theral cnductvtes f the atrx and fllers are 0.95,.5, 36, 0W/K, respectvely. The present del s used t predct the theral cnductvty, and the fller shape s assued t be a perfect sphere. The theral cnductvtes by the predctn and the experent are cpared as a functn f the fller vlue fractn n Fgure 3, where the cnductvty f the cpste s nralsed by that f the atrx. The theral cnductvtes f the cpste wth the slca predcted by the present del shw excellent agreeents wth the experental results ver all fller vlue fractns. Hwever, the results fr the aluna and the alunu ntrde shw large dfferences. Wng and Bllapally 5 ade the cr-structural studes f the cpstes, and bserved that the slca fllers were perfectly sphercal but aluna nes were nt, althugh they were re nearly sphercal than alunu ntrde, whch was the st rregularly shaped fller. The re the fller shape departs fr the sphere, the greater the agntude f the errr beces. It can be cncluded fr the results that the theral cnductvty f the cpste s strngly dependent n the fller shape. Fgure 3. Theral cnductvty f plyer cpstes as a functn f fller vlue fractn, where the fller shape s assued t be a perfect sphere 3. The Effect f Aspect Rat and Theral Cnductvty Rat The effects f bth the aspect rat and the rat f the theral cnductvty f the fbre t that f the atrx n the theral cnductvty f the cpste are nvestgated by usng cpstes wth algned shrt fbres f the sae aspect rat. The results are shwn as functns f the aspect rat and cnductvty rat n Fgure 4, where the fbre vlue fractn s 0. and the predcted theral cnductvty s nralsed by the theral cnductvty f the atrx. As the aspect rat ncreases, the theral cnductvty fr all cnductvty rats ncreases rapdly at the lwer aspect rats, but at hgher aspect rats t ncreases slwly and s then saturated. The predcted theral 76 Plyers & Plyer Cpstes, Vl. 5,. 4, 007

5 Predctn f Theral Cnductvty f Algned Shrt Fbre Cpstes wth Dfferent Fbre Aspect Rats Fgure 4. ralsed theral cnductvty f the cpste as functns f the average aspect rat and theral cnductvty rat, where the fbre vlue fractn s fxed at 0. rats f 0 and 0, the axu errrs ccur at the theral cnductvty rats f 30 and 75, respectvely. Fr hgher average aspect rats such as 30 and 40, t ccurs at the theral cnductvty rats f abut 00. It s knwn fr the present study that f f s arund 0., cnsderable attentn has t be pad t predctng the theral cnductvty f the cpste wth ether lwer cnductvty rats and average aspect rats r hgher cnductvty rats and average aspect rats. cnductvty s shwn t be a nnlnear functn f the aspect rat. The aspect rat at whch the slpe f the predcted theral cnductvty changes rapdly ncreases as the cnductvty rat ncreases. It s bvus that bth the aspect rat dstrbutn and the theral cnductvty rat nfluence the theral cnductvty f the cpste. Therefre, these paraeters ust be cnsdered tgether t predct t. 3.3 The Effect f Aspect Rat Dstrbutn 3.3. Unfr Dstrbutn When predctng the elastc cnstants f cpstes, the use f the average aspect rat s justfed nly f the aspect rat dstrbutn s nt s scattered. S, the unfr dstrbutn f the aspect rats s cnsdered as the extree case and s eplyed t nvestgate the theral cnductvty f the cpste, whse average aspect rat s vared fr 0 t 40. The predcted results wth the aspect rat dstrbutn are nralsed by thse wth the average aspect rat, and shwn n Fgure 5. The devatn fr the unty s defned as the errr, whch s shwn t be always less than fr all cndtns. Ths eans that the theral cnductvty s cnsstently verestated by usng the average aspect rat. Fr a gven fbre vlue fractn and theral cnductvty rat, the axu errr ncreases as the average aspect rat ncreases. Increasng t further causes the axu errr t bece saturated. Fr exaple, the axu errrs fr the average aspect rats f 0, 0, 30, and 40 shw 6, 9, 0, and 0%, respectvely. Accrdng t Fgure 5, the fbre vlue fractn causng the axu errr s abut 0.5 fr the case f the average aspect rat f 0 and gradually ncreases t saturatn at arund 0. wth ncreasng the average aspect rat. It s recended that the actual dstrbutn f the aspect rats be cnsdered fr the predctn f the theral cnductvty f the cpste wth f f abut 0.. The theral cnductvty rat generatng the axu errr gradually ncreases t be saturated as the average aspect rat ncreases. Fr the average aspect 3.3. ral Dstrbutn The nral dstrbutn s used fr exanng the effect f aspect rat dstrbutn n the accuracy f the predctn f the theral cnductvty, where the average aspect rat, standard devatn f the aspect rats, and cnductvty rat are cnsdered as the paraeters. Snce the axu errr s bserved t take place at f f abut 0. fr the extree dstrbutn f the aspect rats, the fbre vlue fractn s fxed as 0.. The theral cnductvty predcted wth the aspect rat dstrbutn s nralsed by that btaned wth the average aspect rat, and the results fr theral cnductvty rats f 0, 30, 50, 75, and 00 are shwn n Fgure 6. The axu errr fr the nvestgated cndtns ncreases wth ncreasng the theral cnductvty rat up t 75, beynd whch t s saturated. It s cnfred n Fgure 6 that ts values fr the theral cnductvty rats f 0, 30, 50, 75, and 00 are 4, 7, 8, 9, and 9%, respectvely. It s als bserved that bth the average aspect rat and standard devatn nducng the axu errr ncrease wth ncreasng the cnductvty rat. Fr the hgher theral cnductvty rats f 75 and 00, the axu errr ccurs under alst the sae cndtns,.e. the average aspect rat s abut 5 and the standard devatn s 5 r hgher. The sae results are bserved n Fgure 5(b) and (c), s the axu errr cncdes wth the aspect rat beng unfrly Plyers & Plyer Cpstes, Vl. 5,. 4,

6 Jae-Kn Lee Fgure 5. ralsed theral cnductvty f the cpste as functns f fbre vlue fractn and theral cnductvty rat, where the dstrbutn f the fbre aspect rats s unfr and the average aspect rat s taken as the paraeter; (a) average aspect rat (α )=0; (b) α -=0; (c) α =30; and (d) α =40 (a) (b) (c) (d) dstrbuted. The assuptn f a unfr dstrbutn as the extree case n Sectn 3.3. s justfed. Snce the theral cnductvty predcted wth the average aspect rat shws a neglgble errr n the regn f saller aspect rats and standard devatns, that regn can be excluded fr further analyss. The ceffcent f varatn (COV) s ntrduced t exane the effect f aspect rat dstrbutn n the qualtatve predctn f the theral cnductvty. COV s defned as the rat f the standard devatn f the aspect rat t the average aspect rat, and the lne fr COV= s drawn n Fgure 6. The regn lwer than the lne COV= has a hgher COV than, and COV ncreases wth ncreasng dstance fr the lne COV=. The dstrbutn f the aspect rats beces clser t the unfr dstrbutn wth ncreasng COV, whle the re fbres are dstrbuted near t the average aspect rat wth decreasng COV. The percent errrs n the predctn f the theral cnductvty wth the average aspect rat are deterned fr Fgure 6 as functns f the theral cnductvty rat and COV, wth the average aspect rat beng fxed at 30. The results are tabulated n Table. The errr decreases rapdly wth decreases n bth COV and the theral cnductvty rats. Whenever COV s saller than 0.5, the theral cnductvty fr all the theral cnductvty rats can be 78 Plyers & Plyer Cpstes, Vl. 5,. 4, 007

7 Predctn f Theral Cnductvty f Algned Shrt Fbre Cpstes wth Dfferent Fbre Aspect Rats Fgure 6. ralsed theral cnductvty f cpste as functns f the average aspect rat and standard devatn, where the cnductvty rat s taken as the paraeter and the fbre vlue fractn s fxed at 0.; (a) theral cnductvty rat (k f )=0; (b) k f =30; (c) k f =50; (d) k f =75; and (e) k f =00 (a) (b) (c) (d) (e) Table. The percentage errrs n the predctn f the theral cnductvty wth the average aspect rat are deterned fr Fgure 6 as functns f the theral cnductvty rat and COV. The average aspect rat s fxed at 30 k f = COV= Plyers & Plyer Cpstes, Vl. 5,. 4,

8 Jae-Kn Lee Table. The percentage errrs n the predctn f the theral cnductvty wth the average aspect rat are deterned fr Fgure 6 as functns f the theral cnductvty rat and COV. The average aspect rat s fxed at 0 k f = COV= predcted wthn an accuracy f abut 5% wth the average aspect rat. The results f the errr fr COV f and hgher are suarzed n Table, where the average aspect rat s fxed at 0. Althugh COV ncreases, the errrs d nt change fr all theral cnductvty rats. Ths eans that COV f s alst cnsstent wth a unfr dstrbutn f the aspect rats. As bserved n the abve nvestgatn, the cndtns generatng the errr are deterned by the cbned effect f the average aspect rat, standard devatn, and theral cnductvty rat. Therefre, the use f the aspect rat dstrbutn s recended t predct the theral cnductvty f cpste wth hgh cnductvty rats and COVs. 4. COCLUSIOS A clsed-fr slutn fr predctng the theral cnductvty f algned shrt fbre cpstes wth the aspect rat dstrbutn s prpsed, and appled t the unfr and nral dstrbutns f the aspect rats. The valdty f usng the average aspect rat fr predctng the theral cnductvty s exaned n ters f the dstrbutn f the aspect rats such as ts average and standard devatn, fbre vlue fractn, and cnsttuents theral cnductvty rat. Fbre vlue fractns f arund 0. are fund t be the st senstve t the predctn f the theral cnductvty wth the average aspect rat. The gudelne fr predctng the theral cnductvty f cpstes wth the average fbre aspect rat s prpsed by usng the theral cnductvty rat and COV. Whenever COV s saller than 0.5, the theral cnductvty f cpstes fr all theral cnductvty rats nvestgated can be predcted wthn an accuracy f abut 5% wth the average aspect rat. REFERECES. Hatta H. and Taya M., Jurnal f Appled Physcs, 58(7) (985) Hatta H. and Taya M., Internatnal Jurnal f Engneerng Scence, 4(7) (986) Taka Y. and Taya M., Jurnal f Cpste Materals, (987) Hatta H., Taya M., Kulack F.A. and Harder J.F., Jurnal f Cpste Materals, 6(5) (99) Chang J.S. and Cheng C.H., Cpstes Scence and Technlgy, 55 (995) Chen C.H. and Wang Y.C., Mechancs f Materals, 3 (996) Hne P.J., Lust H.R. and Gusev A.A., Cpstes Scence and Technlgy, 6 (00) Juhln E., Chen X. and Papathanasu T.D., Plyers and Plyer Cpstes, 0(3) (00) Lust H.R., Hne P.J. and Gusev A.A., Cpstes Scence and Technlgy, 6 (00) Fu S.Y. and Ma Y.W., Jurnal f Appled Plyer Scence, 88 (003) Hassan A., Yahya R., Yahaya A.H. and Tahr A.R.M., Jurnal f Renfrced Plastcs and Cpstes, 3(9) (004) Huq A.M.A. and Azaez J., Plyer Engneerng and Scence, 45(0) (005) Eshelby J.D., Prc. f the Ryal Scety f Lndn, A4 (957) Mr T. and Tanaka K., Acta Metallurgca, (973), Wng C.P. and Bllapally R.S., Jurnal f Appled Plyer Scence, 74 (999) Plyers & Plyer Cpstes, Vl. 5,. 4, 007