THE HOMOGENIZED VISCOELASTIC AND RATE DEPENDENT PLASTIC MODEL FOR PLAIN WEAVE FABRIC REINFORCED POLYMER COMPOSITES

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1 18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS THE HOMOENIZED ISCOELASTIC AND RATE DEPENDENT PLASTIC MODEL FOR PLAIN WEAE FABRIC REINFORCED POLYMER COMPOSITES 1 Introducton S. L 1, C. Cho 1 *, N. Wang 1, K. K. Cho 2 1 Dpartmnt of Mchancal Engnrng, Inha Unvrsty, Inchon, Kora, 2 Dpt. of Elctrcal and Computr Engnrng, Illnos Insttut of Tchnology, IL, U.S. * Corrspondng author(cdcho@nha.ac.kr) Kywords: Plan wav fabrc, Stran rat dpndnt bhavor, scolastcty, Prony srs, Composts rnforcd by wovn fabrc ar of ncrasng ntrst n many dvrs and unqu applcatons such as arospac, automotv, marn, and mltary. Th wovn fabrc composts hav bn found mor ffctv than th undrctonal composts du to ts rnforcmnt n all drctons wthn a sngl layr. In addton, thy also hav othr advantags ncludng bttr mpact rsstanc, bttr toughnss, and good conformablty to complx shap. Ths proprts mak th wovn fabrc composts attractv n many structural applcatons. Hnc, th analyss of wovn fabrc composts to charactrz thr bhavor s an ara of actv ntrst to rsarchrs. In gnral, th bhavor of composts strongly dpnds on th gomtrc and mchancal proprts of ts consttunts,.., rnforcng fbrs and matrx, and th fabrc archtctur. Many rsarchrs hav nvstgatd ths homognzaton tchnqu for analyzng bhavors of whol compost. Ishkawa and Chou [1] dvlopd thr analytcal ondmnsonal modls, known as mosac, crmp, and brdgng modls. Ths modls consdr only th undulaton of th yarns n th loadng drcton by usng classcal lamnat thory. Nak and Shmbkar [2] and Nak and ansh [3] xtndd th abov modl to two dmnsonal lastc modls. Thy usd paralll-srs assumptons for thr mcromchancal modls basd on classcal lamnat thory. Nak [4] dvlopd thr-dmnsonal lastc modls of wovn and bradd fabrc rnforcd composts. Ths modls wr ncorporatd n computr cod for falur analyss of fabrc composts. Th yarns ar dvdd n many slcs and ths slcs ar homognzd assumng snusodal undulaton of yarns. Th homognzaton of th matral proprts s basd on so-stran assumpton. Karayaka and Kurath [5] proposd mcro-mchancal modl basd on mxd boundary condtons, whch ar so-stran and so-strss homognzaton tchnqu. Tab and Jang [6] dvlopd a mcro-mchancal modl wth nonlnar consttunt. In ths modl, th rprsntatv volum cll (RC) was dvdd nto many sub-clls and th quvalnt proprts wr obtand by an avragng tchnqu. Charactrzaton of stran rat dpndnt bhavor of composts s ssntal to prdct th dynamc bhavor such as mpact or crp phnomna assocatd wth thm. Hgh stran rat xprmnts hav bn prformd on poxy n undrctonal comprsson only to dtrmn a rhologcal modl for th poxy [7]. A sprng n paralll wth two Maxwll lmnts s usd to gt th tm-dpndnt consttutv for th vsco-lastc matral. Wang [8] showd that two Maxwll lmnts ar nough to gt th hgh stran rat rspons of poxy, though th poxy was dfnd as nonlnar modl. Karm [9] usd a two-trm lnar sprng dashpot systm to gt th hgh stran rat of th poxy. In gnral, th constants of rat dpndnt bhavor ar dtrmnd from xprmntal rsults nvolvng strss and stran curv, stran rat, and th tm of load applcaton. A common form for ths consttunt mploys a Prony srs. In ths papr, rat dpndnt mchancal bhavor of plan wovn carbon fabrc rnforcd plastc compost s nvstgatd. Th polymr matrx s assumd sotropc and vsco-lastc n lastc rgon. Th vsco-lastc bhavor s dfnd as tm doman Prony srs. Rat-dpndnt bhavor n plastc rgon s nvstgatd by Splt-Hopknson Prssur Bar (SHPB) tst. Th coffcnts of Prony srs ar obtand nductvly from th SHPB tst rsult.

2 2 Rat-dpndnt consttunt 2.1 Homognzaton Rao, Mahajan and Mttal [1] proposd a homognzaton tchnqu basd on th prncpls of avrag strss-avrag stran as follows. 1 ( xydv, ) (1) () 1 X whr m () ( xydv, ) C Cmn yn H C ( ) () s unform stran, (2) s unform strss, X m s a prodc functon rprsntng charactrstc mods of th unt cll and C H s th quvalnt homognzd stffnss coffcnts. In th prsnt study, w usd ths tchnqu to nvstgat th unt modl of plan wovn consstng 3 layrs and of ths 2 hav.25 mm thcknss, fabrc layrs and a sngl rsn layr. 2.2 Quas-statc comprsson tst Fg. 2 Instron kN Unvrsal tstng machn Th spcmns wr manufacturd as 48 pls and by vacuum bag moldng mthod. Th stackng squnc was [45/-45//9] 6s. Thy wr curd at 35 F and 41 ps for 4 hours. Th spcmns wr plan wav carbon fabrc/poxy lamnats of sz mm. For th quas-statc tst, an unvrsal matral tst machn, Instron 6567, of capacty 5kN s usd, as shown n fg. Th comprsson tst s dsplacmnt control. And th stran rat s approxmatly.2 /s. Fg. 1 Rprsntatv Unt cll and two fabrc layrs To valdat ths modl, a xampl analyss s prformd. Th compost s mad of Hrculs AS4 grapht fbrs mprgnatd wth Hrculs poxy matrx. Th fbr damtr s.7 mm and fbr volum fracton s 5%. Th dal yarn fbr packng dnsty of.78 s usd n ths analyss. Tabl 1. Yarn and rsn proprts Matral E 11 E (Pa) (Pa) (Pa) ν 12 ν 23 Yarn Rsn Rsn vsco-lastcty Polymrs ar gnrally vsco-lastc but som xhbt vsco-plastcty,.. thy undrgo prmannt dformaton upon loadng. Most xprmntal rsults on fbr-rnforcd polymrs ndcat that rgardlss to th mod of loadng, th modulus ncrass wth ncrasng stran rat, and th strngth and fractur stran may ncras dpndng on th rang of stran rat. Snc th sam rat-dpndnt bhavor had bn obsrvd for unrnforcd polymrc matrals, t s assumd that th ratdpndnt charactrstcs of th compost can b dscrbd usng tm-dpndnt or vscolastc proprty of th polymr. Th rat-dpndnt consttutv quaton for carbon/poxy compost matral s xptrssd n trms of a tm-dpndnt rlaxaton modulus. Th tm-dpndnt rlaxaton modulus s found by consdrng how th matral rlaxs undr constant stran. Th consttutv quaton s thn xprssd by lnar hrdtary law. Snc all th hgh stran rat xprmntal rsults avalabl to us ar carrd out at constant stran rat, th hrdtary law basd on constant stran rat s

3 PAPER TITLE usd to fnd matral proprts approprat for a hgh stran rat consttutv law. Thus, th homognzd consttunt can b xprssd as: 1 t () ( x, ytdv, ) g ( ) ( ) R s tsds 1 ( x, ytdv, ) X t m () mn R y n (3) E E g ()( s ts) ds H E ( ) (4) W can assum that th vscolastc matral s dfnd by a Prony srs xpanson of th dmnsonlss rlaxaton modulus; N () t E (5) 1 P g t s/ whr, ( t s) ds 2.4 Rat-dpndnt plastc bhavor For hgh stran rat tstng, a modfd SHPB apparatus was usd. Th SHPB apparatus conssts of 16 mm damtr Inconl 718 strkr, ncdnt and transmttr bars whch ar 3, 16 and 16 mm n lngth, rspctvly. Tabl 2 Matral homognzaton comparson among 3 layr unt-cll modl, 1 layr modl and tst rsults Matral 1 layr (mmtxlam) E xx,e yy (Pa) xy (Pa) ν xy layr Tst Comparson btwn xprmntal and computatonal rsult for quas-statc comprsson tst Th comprsson tst was conductd on [45/- 45//9] 6S spcmn. Th lastc comprsson modulus of th fabrc s calculatd as Pa from 3-layr FEM analyss. Th proprts of rsn ar sam as n tabl 1. Fg. 4 shows th rsults of xprmnt and fnt lmnt analyss quas-statc comprsson of [45/-45//9] 6S CFRP spcmn. Th statc yld strss s obsrvd as 2 MPa at stran.8. In th xprmntal rsult, th strss valu ncrass aftr th drop at 2 MPa, whch ths can b rgardd as yld lmt and falur. Th strssstran bhavor shows vry good conformty btwn xprmntal and FEM rsults. Fg 3. SHPB apparatus 3 Rsults and Dscusson 3.1 aldaton of unt-cll modl Th lastc bhavor of 3 layr unt modl s compard wth valus obtand from mmtexlam and xprmnts [11]. Th rsults ar compard n tabl 2. Th 3-layr unt modl shows smlar rspons to tst rsult. Fg 4. Comparson btwn Quas-statc comprsson tst and FEM rsult 3.3 Rat dpndnt bhavor Th plastc bhavors undr stran rats of 4/s, 12/s, and 15/s ar plottd n Fg. 5, obtand from th SHPB tst. Th plastc proprts lk flow strss, ultmat strngth, and stran hardnng can b 3

4 obtand sparatly from ths graphs. In numrcal analyss, ach can b xprssd as a gnral mprcal plastc consttunt calld Johnson-Cook plastcty wthout tmpratur ffct. Th yld strss for Johnson-Cook stran-rat dpndnc s xprssd as blow. pl n pl A B 1 Cln (6) Th lft trm s for stran hardnng and th rght prsnts stran rat hardnng bhavor. A, B, n and C ar matral constants ar prsntd n Tabl 3. practcal complanc rang xpctd from SHPB tst ar shown n fg. 6. To fnd Prony constants, xpctd modulus of stran rats ar convrt to complancs and ach complanc s xprssd as ratos about statc complanc as shown n tabl 4. Tabl 4 Expctd lastc modulus at 8, 12, and 15 /s stran rats Stran rat (/s).2 (Qus-statc) Complanc (/pa) Complanc Rato (%) ~ ~ ~.1 2. ~ ~ ~ 18.5 Fg. 5 Strss-Stran curv rsult from SHPB tst Tabl 3 Th matral proprts of Johnson-Cook plastcty for th CFRP [45/-45//9] 6S composts. A B Matral n (MPa) (MPa) C CFRP [45/-45//9] 6S Though th xprmntal rsult shown n Fg.5 do not mply lastc bhavor, n composts th stmaton of xact flow strss s svrly dffcult bcaus of short plastc rang; and thrfor, th Johnson-Cook plastcty s usful to fnd rat dpndnt lastcty of compost matrals. 3.4 Prdcton of Prony coffcnts Th flow strss for ach stran rat calculatd from SHPB tst r can b usd to dfn th vsco-lastc coffcnts of Prony srs, P g and. Th Fg. 6 Expctd lastc modulus rang Th complanc rducton rato can b dtrmnd from th xpctd lastc modulus. And Prony constants can b xtractd by complanc rducton rato analyss lk tabl 5. Tabl 5 scolastc prony constants for th CFRP [45/- 45//9] 6S composts. Matral CFRP [45/-45//9] 6S P g K (s) Fg. 7 and Fg. 8 show th maxmum and th spcfc tm-normalzd complanc graphs calculatd by Prony constants lstd n tabl 5.

5 PAPER TITLE 3.5 Comparson wth homognzd numrcal modl Fg. 9 compars th computatonal rsults for strssstran bhavor of CFRP [45/-45//9] 6S. Ths modl s homognzd by 3 layr unt cll modl wth Johnson-Cook plastcty. Bcaus of short prod dynamc analyss, thr ar strss rtardatons n lastc rang. Fg. 7 Complanc ncrmnt along tm Fg.7 ndcats a vry short vscolastc bhavor prod wth th statc complanc convrgng to 5. Th computatonal analyss s prformd by ABAQUS/xplct. Boundary vlocty and analyss tm for stran rats, 8 /s, 12 /s and 15 /s ar rprsntd n tabl 6. Tabl 6 Paramtrs for complanc analyss Stran rat (/s) Boundary vlocty (mm/s) Analyss tm prod(s) E E E-5 Fg. 9 Th rat dpndnt mchancal bhavor of CFRP [45/-45//9] 6S compost. SHPB rsults for dffrnt stran rat showd vry small dvaton n thr strss-stran curvs. Thus, th rat dpndnt bhavor was not sgnfcant. Ths can also b obsrvd n Fg.1, whch s magnfd vw of computatonal curvs of Fg.9 n rd box. Fg. 8 Complanc ncrmnt along tm n short tm rang. In Fg.8, for stran rat of 8 /s, analyss tm s 2.5E-5 and th normalzd complanc s 2 at th momnt. And, th complanc rducton rato s 25%. Lkws, at stran rat 12 /s and 15 /s complanc rducton ratos ar 2% and 18%, rspctvly. Fg. 1 Th rat dpndnt plastcty of CFRP [45/- 45//9] 6S compost. Fg. 1 shows Johnson-Cook plastc hardnng wth stran rat. Bcaus th rat dpndnt hardnng of Johnson-Cook plastcty s logarthm functon, th 5

6 flow strss dffrnc btwn 8 /s, 12 /s, 15 /s s not larg compard wth that of 1 /s, 1 /s, and 1 /s. Nvrthlss th numrcal rsults ft wll to th xprmntal rsult as shown n Fg. 11. composts", AIAA Journal, 1983, ol. 21, No. 12, pp [2] Nak N. K., Shmbrkar, P. S., "Elastc bhavor of wovn fabrc composts: I-lamna analyss", Journal of Compost Matral, 1992, ol. 26, No. 15, pp [3] Nak N. K., ansh,. K., "Falur bhavor of plan wav fabrc lamnats undr on-axs unaxal tnsl loadng: II analytcal prdctons", Journal of Compost Matral, 1996, ol. 3, No. 16, pp [4] Nak R. A., "Falur analyss of wovn and bradd fabrc rnforcd composts", Journal of Compost Matral, 1995, ol. 29, pp Fg. 11 Th suprposd total mchancal bhavor wth vscolastcty and rat dpndnt plastcty. Th matral modl nclud homognzd vscolastc and rat-dpndnt plastcty appld smultanously. In gnral, t s vry dffcult to masur or vn obsrv plastc bhavor n composts, partcularly th rat dpndnt bhavor n short tm lk mpact loadng. Thrfor, from practcal pont of vw th computatonal analyss and tchnqus of ths study can of sgnfcant hlp to composts and mpact ngnrng flds. In addton, t can b xtndd to nclud long trm vscolastcty. 4 Concluson Unt cll modl of plan wovn polymr compost s valdatd aftr comparson aganst commrcal softwar, mmtexlam. Th stran rat dpndnt matral consttunt s nclusvly xprssd n th lastc and plastc rgons. Ths consttunt bhavor s analyzd for wovn fabrc polymr compost undr dynamc bhavor lk mpact or crp. Acknowldgmnt Ths work was supportd by Inha unvrsty. Rfrncs [1] Ishkawa, T., and Chou T. W., "On-dmnsonal mcromchancal analyss of wovn fabrc [5] Karayaka, M. and Kurath, P., "Dformaton and falur bhavor of wovn compost lamnats", Journal of Engnrng Matral Tchnology, 1994, ol. 116, pp [6] Tab A, Jang Y, "Wovn fabrc compost matral modl wth matral nonlnarty for nonlnar fnt lmnt smulaton", Intrnatonal Journal of Solds Structur, 1999, ol. 36, No. 18, pp [7] Wks, C. A. and Sun, C. T., Modlng non-lnar rat-dpndnt bhavor n fbr-rnforcd composts, Compost Scnc and tchnology, 58, 1998, [8] Wang, L., Strss wav propagaton for nonlnar vscolastc polymrc matrals at hgh stran rats, Th Chns Journal of Mchancs- Srs A, ol.19, No. 1 [9] Mohammd R. K., "Consttutv modlng and falur crtra of carbon-fbr rnforcd polymr unr hgh stran rats", PhD dssrtaton, Unvrsty of Akron, 25 [1] M.. Rao, P. Mahajan and R. K. Mttal, Effct of archtctur on mchancal proprts of carbon/carbon compostss, Compost structur, ol 83, pp , 28 [11] K. N. Shvakumar, P. Challa and D. R. Rddy, "Mcromchancs and Lamnat Analyss of Txtl Fabrc Compost Usrs Manual", NASA lnn Rsarch Cntr, 2.