Constitutive Modeling of Progressive Damage in Composite Laminates

Transcription

1 Consttutv Modlng of Progrssv amag n Compost Lamnats Chatanya A. nadayalu *, Adt Chattopadhyay and Xu Zhou Arzona Stat Unvrsty, Tmp, AZ A procdur has bn dvlopd for smulatng progrssv damag n compost lamnats usng a combnaton of an accurat strss analyss tchnqu and a quascontnuum approxmaton of th damagd matral. An mprovd lamnat thory ncorporatng a layrws dformaton fld has bn usd to modl th compost lamnat. Stat varabl consttutv quatons orgnally dvlopd for polymrc matrals hav bn modfd n ordr to accuratly account for th ffcts of damag on th nonlnar stran rat dpndnt dformaton of polymr matrx composts. Th prsnt rsarch ffort taks a phnomnologcal (known as contnuum damag mchancs or CM) approach to modl damag at th mcromchancal lvl (n th matrx). Th ffctv stffnss or complanc of th damagd sold s dfnd n trms of th actual damag stat rprsntd by a proprly slctd damag varabl. Th fnt lmnt modl for nonlnar analyss of lamnatd shlls s coupld wth th mcromchancs-basd mchanstc modl for modllng th ntaton and voluton of damag and for prdctng th ffctv compost proprts. Nomnclatur u,v,w mchancal dsplacmnts along x, y, z drctons z locaton rlatv to lamnat mdplan k U n-plan dsplacmnt at th kth layr of th lamnat k U 3 transvrs dflcton φ rotatons of th normal to th rfrnc plan about th x and y axs k k θ, ψ layrws structural unknowns dfnd at ach lamna j u, j w possbl jumps n dsplacmnt fld du to dlamnaton allowng slppng and sparaton btwn layrs H ( z - z j ) havsd unt stp functon g ( z), h( z) charactrstcs of n-plan zgzag dformaton N numbr of layrs n th lamnat N numbr of dlamnatons n th lamnat ( ) ( ε, ~ ) ε, stran and strss tnsors for th actual htrognous sold ~ ffctv stran and ffctv strss tnsors for th dalzd (homognzd) sold nomnal strss tnsor dvatorc strss tnsor I frst nvarant of th nomnal strss tnsor * Graduat Rsarch Assstant, partmnt of Mchancal and Arospac Engnrng Profssor, partmnt of Mchancal and Arospac Engnrng Assstant Profssor Rsarch, partmnt of Mchancal and Arospac Engnrng

2 kk hydrostatc strss J scond nvarant of th dvatorc strss tnsor ( ) dvatorc componnts ( ~ ) ffctv stat varabls f plastc dsspaton scalar damag varabl ( ~, ) F ( Y ) damag voluton potntal F fully coupld plastc potntal Y lastc nrgy rlas du to damag voluton E 4 th ordr matral lastcty tnsor kl ε kl lastc stran φ lastc stran nrgy dnsty E Young s modulus ν Posson s rato p cumulatv plastc stran rat p W plastc work rat p ε plastc stran rat tnsor λ scalar rat varabl p ε dvatorc nlastc stran rat 0 maxmum plastc stran rat n controls th rat dpndnc of th matral Z rsstanc to molcular flow (ntrnal strss) α hydrostatc ffct stat varabl q matral constant rprsntng th hardnng rat Z matral constant rprsntng th maxmum valu of Z α matral constant rprsntng th maxmum valu of α Z 0 matral constant rprsntng th ntal valu of Z α 0 matral constant rprsntng th ntal valu of α F ( ) fbr componnts M ( ) matrx componnts I. Introducton POSSIBLY th most mportant consdraton n prdctng th prformanc of nw compost structural dsgn concpts s th prdcton of th ntaton and th voluton of damag. Compost structurs contan nhrnt strss concntratons causd by matral htrognty. Ths strss concntratons may b rlvd by crp and strss rlaxaton n th matrx or by th propagaton of mcromchancal damag. Ths may lad to a svr rducton n load carryng capablty and can oftn rsult n structural falur. Hnc progrssv damag analyss s an mportant lmnt n th dsgn and analyss of lamnatd compost structurs.

3 Wth rspct to thr scal, progrssv damag modls may b rfrrd to th mcro-scal (mcromchancs), n whch th matral structur s pcws dscontnuous and htrognous and th lamna strsss prdctd by a global lamnat analyss ar appld to th mcro-modl to comput th mcromchancal strsss n ach consttunt of th compost at vry pont of th structur, and th macro-scal -5 (contnuum mchancs), n whch th actual htrognous matral s approxmatd an quvalnt homognzd contnuum by th us of th coupls of ffctv stat varabls,.g., ( ~ ε, ~ ), n th stat and dsspaton potntals n lu of th classcal stat varabls, ( ) ε,, rprsntng stran and strss tnsors for th dalzd (homognzd) and th actual htrognous sold, rspctvly. In th prsnt work, progrssv damag analyss of lamnatd composts s prformd usng a mcromchancal modl. A stat varabl consttutv formulaton 6 for charactrzng th nonlnar, stran rat dpndnt dformaton of polymrc matrals has bn modfd n ordr to accuratly account for th ffcts of damag on thr mchancal bhavour. Th rvsd polymr consttutv quatons hav bn ncorporatd nto a classcal sold mchancs basd mcromchancs modl 3 to prdct damag and th consstnt rducton of ffctv compost proprts durng th varous phass of ts voluton n tm. A fnt lmnt program basd on a rfnd layrws thory 7,8 s usd n th lamnat analyss. A macro-mcro approach s usd to rlat th lamna strsss prdctd by th global lamnat analyss to th mcromchancal strsss n ach consttunt of th compost. Th lamnat thory, th progrssv damag algorthm, and th mcromchancal modl usd n th prsnt study ar dscrbd n th followng sctons. Numrcal xampls ar prsntd n th fnal scton. II. Mthodology A. Equatons of Moton Th constructon of a modl for compost lamnats starts wth th formulaton of th consttutv rlatons. A rfnd layrws lamnat thory dvlopd by Km t. al. 7,8 s usd to modl th mchancal dsplacmnt fld. Orthotropc compost plats wth varous thcknsss and ply stackng squncs ar consdrd. Th coordnat systm for th plat s takn to b wth th x-y plan paralll to th plan of th plat and th z coordnat normal to th plan of th plat masurd from th cntr of thcknss (Fg. ). Th thory starts wth a gnral frst ordr dsplacmnt fld suprposd by layrws functons. To account for th prsnc of dlamnatons, th assumd dsplacmnt fld s supplmntd wth Havsd unt stp functons, whch allow th ntroducton of ndpndnt dsplacmnt flds abov and blow th layrs. Th rfnd dsplacmnt fld taks th followng form U U k k 3 ( x, y, z, t) u ( x, y, t) + φ ( x, y, t) z k k + θ ( x, y, t) g( z) + ψ ( x, y, t) h( z) + N j j ( x, y, z, t) w( x, y, t) + u ( x, y, t) H ( z z ) j N j j w ( x, y, t) H ( z z ) j () Not that prfctly bondd ntrfacs can b asly smulatd by sttng and w to b zro at ths ntrfacs. Th through-lamnat-thcknss functons, g( z) and h ( z), ar usd to addrss th zgzag natur of th n-plan zgzag dformaton and tak th followng form j u j g( z) snh( z / h) h( z) cosh( z / h) ()

4 k In Eq. (), varabls u, φ, w,, ( N - ) k ψ θ, j u j and w, rprsnt th unknowns, rsultng n to a total of 5 + 4N + 3 structural unknowns for an N -layrd lamnat. Ths mpls that th total numbr of structural unknowns s a functon of th numbr of layrs and dlamnatons. Ths can b computatonally ntnsv f multlayrd lamnats ar usd. To ovrcom ths ssu, strss fr boundary condtons on all fr surfacs, ncludng dbondng ntrfacs and contnuty of th transvrs shar strsss and th n-plan dsplacmnt at ply ntrfacs ar mposd. Ths procdur lads to a total of 7 + 5N. It must b notd that th numbr of unknowns s ndpndnt of th numbr of pls, but dpnds on th numbr of dlamnatons. Th fnt lmnt mplmntaton of th rfnd layrws thory s dscrbd n dtal n rfrnc 4. Natural frquncs and mod shaps computd for compost plats wth and wthout dlamnaton hav shown xcllnt corrlaton wth xprmntal rsults. B. Polymr Consttutv Rlatons Th prsnt unfd formulaton, drvd from th prncpls of thrmodynamcs, assums th sam potntal to dscrb both th plastc dsspaton and th damag dsspaton. In othr words, n ths modl, t s supposd that damag cannot ntat wthout plastc dformaton. A ruckr-pragr-typ yld functon govrns th plastc dsspaton and may b wrttn as ( ) α I J f + (3) whr, I kk dnots th frst nvarant of th nomnal strss tnsor, and J j, th scond j nvarant of th dvatorc strss tnsor. Symbols dnotd by ( ) rprsnt dvatorc componnts, whras th tld ( ~ ) rfrs to th ffctv stat varabls. A consstnt coupld yld functon s obtand as th xtnson of th ruckr-pragr uncoupld yld functon of th vrgn nonlnar matral, whn th classcal stat varabl n Eq. (3) s rplacd by th ffctv varabl ~ usng th hypothss of stran quvalnc. In othr words, th ruckr-pragr-typ yld functon govrns th plastc dsspaton coupld wth damag n spac of ffctv varabls ( ~ ) α~ J ( ~ ) f kk + whr, th classcal ffctv strss concpt has bn usd, ~ ( ). Th scalar damag varabl rprsnts th actual contnuum damag stat, th voluton of whch s govrnd by th varaton of th lastc stran nrgy. Th fully coupld plastc potntal may b wrttn as whr, th damag voluton potntal F ( Y ) F ( ~, ) f + F ( Y ), s supposd to b a lnar functon of th lastc nrgy rlas du to damag voluton Y. In ths smplfd approach th varabl Y assocatd wth th sotropc damag ntrnal varabl contans th contrbuton of th lastc (rvrsbl) nrgy only, whras th nlastc (rrvrsbl) stord nrgy s not rlasd by th onst and growth of damag. (4) (5) F Y Y ( ) (6)

5 For lastc stran th ansotropc lastcty law coupld wth sotropc damag s assumd E ε kl kl ( ) (7) Hnc, applyng Eq. (7) th lastc stran nrgy dnsty s furnshd: φ dε E kl ε kl ( ) dε E ε ε ( ) kl kl (8) Whn th constant strss condton s usd, d [( ) dε d] 0, E kl kl ε kl (9) a varaton of th lastc stran nrgy du to th contnuous damag growth s rprsntd by dφ dε ε d ( ) (0) Evntually, applyng Eq. (7) and Eq. (8), th varabl as follows Y assocatd wth th ntrnal (scalar) varabl s dfnd Y dφ d const ε E kl ε ε kl φ () In gnral, a modl basd on ansotropc lastcty coupld wth sotropc damag s not consstnt. So, confnng th formulaton to th sotropc lastcty law ε + ν ν kk δ E E () and ntroducng th dcomposton to dvatorc and hydrostatc trms + δ, ε + δ, kk kk (3) th shap and volum chang law coupld wth damag s stablshd: + ν, E kk ν kk, E (4)

6 as wll as th corrspondng shar and hydrostatc nrgy portons: E 3 E kk H ) ( ) S ( ν ν φ φ φ (5) or, quvalntly, + + kk J E 3 E J ν ν φ (6) Substtuton of Eq. (6) for φ n Eq. () ylds th followng formula for th lastc stran nrgy rlas rat (thrmodynamc forc Y assocatd wth th sotropc damag rprsntd by a scalar varabl : ( ) + + kk J E 3 E J Y ν ν (7) To dtrmn th cumulatv plastc stran rat, p, th prncpl of th quvalnc of th plastc work rat s mployd. Th plastc work rat, p W, can b xprssd as p W p ~ ε (8) Whn th gnralzd normalty rul (assocatv thory), Y F, F p λ λ ε (9) s appld togthr wth th dsspatv potntal (quaton (5)) and th cumulatv plastc stran rat (Eq. (8)) and followng a smlar format to th Bodnr-Partom modl, th followng stat quatons ar furnshd:

7 p Z ε 0 xp F λ Y ( Y ) n f + F f + F f + F 3 f + F p ( ) ( ) αδ + ( ) J + ν 3( ν ) kk, ( ) E E + J J 3( ν ) kkδ kk ( ) E J J (0) p p p whr p dnots th cumulatv plastc stran rat, p ( 3) ε ε, whr ε s th dvatorc nlastc stran rat. 0 rprsnts th maxmum plastc stran rat, and n controls th rat dpndnc of th matral. Z s an sotropc stat varabl whch rprsnts th rsstanc to molcular flow (ntrnal strss). Th rat of voluton of th ntrnal strss stat varabl Z and th hydrostatc strss ffct stat varabl α ar furnshd by: ( Z Z ) Z q p () ( α ) α q α p () whr q s a matral constant rprsntng th hardnng rat, and Z, and α ar matral constants rprsntng th maxmum valus of Z and α, rspctvly. Th ntal valus of Z and α ar dfnd by th matral constants Z 0 and α 0. C. Mcromchancs Modl Thr s a larg dffrnc n th mchancal proprts of th consttunts of a fbr-rnforcd compost matral. Both ths and thr gomtrc arrangmnt rsult n a vry complx local stat of strss n a fbr-rnforcd compost matral. Th mor accurat numrcal lamnat modls basd on mcromchancs can oftn b computatonally xpnsv. And gvn th dvrgnc btwn rspons tm rqurmnts and computatonal tm rqurmnts, thr s a nd to rduc th tm rqurd to dtrmn th strsss n ach phas of th matral wthout any loss of accuracy. Ths crtra ar mt by th modl dvlopd by Goldbrg 9, n whch a unt cll, th smallst matral unt for whch th rspons can b consdrd as rprsntatv of th ntr lamna, s dvdd up nto an arbtrary numbr of horzontal rctangular slcs. And thn a smplfd classcal sold mchancs approach s appld wthn ach slc to obtan th ffctv lastc constants and mcrostrsss. Th soluton of Goldbrg, hncforth calld th mcromchancal modl, s accurat, provdd crtan assumptons ar satsfd. Ths assumptons ar (s Fg. and Fg. 3): () Th fbrs hav a crcular cross-scton that s constant along th lngth of th fbr. () Th fbrs ar arrangd n a prodc squar array. () Adjacnt consttunts (phass) ar prfctly bondd. (v) All matrals ar homognous. Th matrx s sotropc, wth a rat dpndnt, nonlnar dformaton rspons computd usng th consttutv quatons dscrbd n th prvous scton and th fbrs ar transvrsly sotropc and lnarly lastc. (v) Hygrothrmal ffcts ar gnord wthn ach phas of th compost and th stran-dsplacmnt rlatons assum small strans. (v) Th volum avrag of th

8 out-of-plan strsss (3-drcton) s zro. Ths assumpton mpls that th rgon of ntrst s n a stat of gnralzd plan strss. Thus, only a rprsntatv volum lmnt (RVE) comprsng a sngl fbr and ts surroundng phas, nd b analyzd. Th RVE and th coordnat systms usd n th analytcal modl ar shown n Fg. 3 and Fg. 4. Th soluton for th rspctv phass (consttunts) s of th form: F M F 33 M 33 Q Q Q Q A B Q ε + Q ε A 3 B 3 A B ε + Q ε ε + Q ε A 3 B 3 ε + Q ε ε + Q ε + ε + ε + ε AI BI I + ε + ε AI 3 BI 3 (3) whr Q ar th ffctv stffnss matrx trms and th varous ε I ar collctons of th nlastc stran trms 9.Complt dtals of th soluton ar gvn lswhr (s Goldbrg 00). Th rsultng mcromchancal modl provds pont-ws strsss n all phass of th compost undr gnral mchancal loadng and prdcts compost lastc proprts from knowldg of th ndvdual consttunts. In ordr to rlat th stat of strss at a pont n a lamna to th stat of strss n ach consttunt of th matral, th rgorous condtons undr whch th mcromchancal modl s xact must b rlaxd, and a nw st of condtons must b assumd to xst undr whch th mcromchancal modl prdcts th stat of strss n a typcal fbr and ts surroundng matrx. Ths rlaxd st of condtons s as follows: th total strans appld to th unt cll and th nlastc strans n ach slc ar assumd to b known. All slcs ar assumd to b n a plan strss stat,.., only n-plan strans and strsss ar consdrd. Each slc thn s sparatd nto two sub-slcs, on composd of fbr matral and th othr composd of matrx matral. Along th fbr drcton (drcton ), th strans ar assumd to b qual n ach sub-slc of th slc, and th strsss ar combnd usng volum avragng. Th nplan transvrs normal strsss (drcton ) and n-plan shar strsss (drcton ) ar assumd to b qual n ach sub-slc of th slc, and th strans ar combnd usng volum avragng. Lkws, all th avrag strans n ach slc ar assumd to b qual, and th strsss ar combnd usng volum avragng. Th out-of-plan normal strans (drcton 3) ar assumd to b unform n ach slc, and th volum avrag of th out-of-plan strsss n ach sub-slc s assumd to b zro. Th ffctv proprts, strsss and strans for ach slc ar computd by applyng ths assumptons along wth th consttutv quatons for th fbr and matrx and solvng th rsultng srs of quatons. Th quvalnt strsss, strans and proprts for th unt cll can thn b computd by applyng lamnat thory to ach slc. Th advantag of ths typ of modllng approach ovr othr mcromchancs mthods s that th bhavour of ach slc s dcoupld, and th rspons of ach slc can b dtrmnd ndpndntly. Ths procss rsults n a srs of small matrx quatons, nstad of on larg systm of quatons, whch rducs th complxty of th analyss sgnfcantly and ncrass th computatonal ffcncy. For lastc matrals, th proprty of transvrs sotropy of th matral could b mantand by usng a combnaton of horzontal and vrtcal slcs and th concpt of suprposton 0. Howvr, for th nonlnar matrals usd n ths study, suprposton cannot b appld, and du to th fact that only horzontal slcng s usd, th nhrnt transvrs sotropy of th compost unt cll s lost. Smlarly, th assumpton of crcular cross-sctond fbrs s rplacd by th assumpton that t s composd of a st of rctangular cross-sctons. Each rctangular cross-scton, though, has th sam ara as th quvalnt curvlnar cross-scton. III. Rsults and scusson Numrcal rsults ar prsntd for progrssv damag of a smply-supportd and cantlvr cross-ply ( +0/ +90/ +0) and angl-ply, ( +30/ -60/ +30), ( +45/ -45/ +45), plats subjctd to a unform load. Th lamnat gomtry and th fnt lmnt msh usd ar shown n Fg. 5 (a 0.5m, b 0.5m). Th thcknss of ach ply s m, gvng a total lamnat thcknss of h 0.00m and a lngth to thcknss raton of a/h 5. Th matral usd n th currnt study s IM7/977- grapht/poxy. Th consttunt proprts of IM7 carbon fbrs and 977- poxy ar gvn n Tabl. Usng ths valus for consttunt proprts and a fbr volum fracton of 40 prcnt, ffctv compost proprts wr calculatd usng th mcromchancal modl. Fgurs 6 through 9 llustrat th dsplacmnt rspons of th structural modl for varous stackng squncs and boundary condtons usng gomtrcally nonlnar analyss. splacmnt prdctons usng th unfd damag

9 modl ar prsntd and compard wth th prdctons of th macro-/mcro-modl wthout damag. In ach fgur, t may b sn that th mcromchancs-basd mchanstc modl undrprdcts th dsplacmnt compard to th damag modl. It can b sn that damag ffcts bcom domnant aftr a short tm n ach nstanc. Th dffrnc s ndscrnbl only n th cas of th cantlvr cross-ply ( +0/ +90/ +0) (Fg. 9). Ths may b attrbutd to th fact that strsss and strans ar localzd at th fxd nd and damag s rstrctd to that zon. Ths localzd damag (Fgs. 9 ) may coalsc nto an xtndd clustr and affct th dsplacmnt rspons ovr tm. Also, basd on th assumpton that a non-rgd moton can b factorzd nto a rgd moton and a blnd-shap (non-rgd) moton componnt, t can b stablshd that for th cantlvr boundary condton th rgd body moton componnt of th maxmum dsplacmnt s gratr than th dformaton componnt whch n turn lads to lss damag. Clos nspcton of th damag dstrbuton (n th matrx) contours n Fgs. 0 8 rvals a pronouncd dstncton n th damag growth rat and pattrns btwn th stff cross-ply lamnat and th rlatvly mor flxbl angl-ply lamnats, th dgs bng smply-supportd n both cass. Th plabl angl-ply lamnats wll dform wth lss damag compard to th stffr cross-ply lamnat ovr th sam prod of tm, all othr condtons bng qual. Ths ablty to dform radly rtards damag n th angl-ply lamnats. Addtonally, th damag dstrbuton s concntratd n th top ply n th cross-ply lamnat suggstng that whn t s strssd, a major porton of th load s born by th top 0 o ply. On th othr hand, n th angl-ply lamnats, th damag s dstrbutd mor vnly across th thcknss. IV. Concluson A mcromchancs-basd progrssv damag modl for lamnatd fbr-rnforcd compost structurs has bn dvlopd and mplmntd. Th modl combns a fnt lmnt modl for lamnat analyss and a unfd ruckr-pragr-typ plastcty soluton for dtrmnng mcromchancal strss dstrbutons, ffctv compost proprts, and an stmat of th xtnt of damag wthn th matrx phas of th compost. Ths mchanstc approach to damag analyss has many advantags ovr phnomnologcal modls, ncludng th dtrmnaton of a mans by whch consstnt rducton of ffctv compost proprts can b calculatd for damag analyss. Numrcal rsults wr prsntd for progrssv damag n 3-layr cross-ply and angl-ply cantlvr plats subjctd to a unform load. Acknowldgmnts Th support of ths rsarch by th Army Rsarch Offc through Grant AA s gratfully acknowldgd. Rfrncs Murakam, S. and Kamya, K., Consttutv and amag Evoluton Equatons of Elastc-Brttl Matrals Basd on Irrvrsbl Thrmodynamcs, Intrnatonal Journal of Mchancal Scncs, Vol. 39, No. 4, 997, pp Robbns,. H., Rddy, J. N. and Rostam-Abad, F., Towards Hrarchcal Modlng of amag n Compost Structurs, 45 th AIAA/ASME/ASCE/AHS/ASC Structurs, Structural ynamcs & Matrals Confrnc, AIAA, Palm Sprngs, CA, Hayakawa, K. and Murakam, S., Spac of amag Conjugat Forc and amag Potntal of Elastc-Plastc-amag Matrals, amag Mchancs n Engnrng Matrals, dtd by G. Z. Voyadjs, J.-W. W. Ju and J.-L. Chaboch, Elsvr Scnc, Amstrdam, 998, pp Hayakawa, K. and Murakam, S., Thrmodynamcal Modlng of Elastc-Plastc amag and Exprmntal Valdaton of amag Potntal, Intrnatonal Journal of amag Mchancs, Vol. 6, 997, pp Skrzypk, J. and Ganczarsk, A., Modlng of Matral amag and Falur of Structurs, Sprngr, Brln, Goldbrg R.K., Robrts,G.. and Glat, A., Implmntaton of an Assocatv Flow Rul Includng Hydrostatc Strss Effcts nto th Hgh Stran Rat formaton Analyss of Polymr Matrx Composts, NASA/TM , Natonal Aronautcs and Spac Admnstraton, Washngton,.C., Km, H. S., Zhou, X. and Chattopadhyay, A., Intrlamnar Strss Analyss of Shll Structurs wth Pzolctrc Patch Includng Thrmal Loadng, AIAA Journal, Vol. 40, No., 00, pp Km, H. S., Chattopadhyay, A. and Ghoshal, A., ynamc Analyss of Cross-ply Compost Lamnats wth Embddd Multpl lamnaton, AIAA/ASME/ASCE/AHS/ASC 44th Structurs, Structural ynamcs and Matrals Confrnc, AIAA, Norfolk, VA, 003.

10 9 Goldbrg R.K., Implmntaton of Fbr Substructurng Into Stran Rat pndnt Mcromchancs Analyss of Polymr Matrx Composts, NASA/TM-00-08, Natonal Aronautcs and Spac Admnstraton, Washngton,.C., Whtny J.M., A Lamnat Analogy for Mcromchancs, Procdngs of th Amrcan Socty for Composts, Eghth Tchncal Confrnc, dtd by G. M. Nwaz, Tchnomc Publshng Co., Lancastr, PA, 993, pp

11 Tabl. Matral proprts of th fbr and matrx IM7 fbrs E (GPa) E (GPa) G (GPa) G 3 (GPa) ν ν polymr matrx ε (/sc) E (Gpa) ν 0 (/sc) n Z 0 (MPa) Z (MPa) q 9E E α 0 α Fgur. Plat gomtry and coordnat systm. Fbr Fgur. Cross-scton of a fbr-rnforcd compost matral wth fbrs packd n a squar array. Unt Cll Slcd Unt Cll Crcular Fbr Slcd Unt Cll Rctangular Fbr Slcs Porton of Slcd Unt Cll Analyzd Slc of Unt Cll Fgur 4. Schmatc llustratng slc gomtry and coordnat systm usd n th analytcal modl. Fgur 3. Schmatc llustratng rlatonshp btwn squar RVE and slcs.

12 b a Fgur 5. Lamnat gomtry and fnt lmnt msh Fgur 6. splacmnt rspons of a (0/90/0) smplysupportd bam wth and wthout damag Fgur 7. splacmnt rspons of a (30/-60/30) smply-supportd bam wth and wthout damag Fgur 8. splacmnt rspons of a (45/-45/45) smply-supportd bam wth and wthout damag Fgur 9. splacmnt rspons of a (0/90/0) cantlvr bam wth and wthout damag

13 a) At t 0.50ms b) At t 0.500ms c) At t 0.750ms d) At t 0.880ms Fgur 0. amag dstrbuton (prcntag rducton n matrx stffnss) n th bottom layr of a (0/90/0) smply-supportd bam loadd by unform tractons of 0.75MPa

14 a) At t 0.50ms b) At t 0.500ms c) At t 0.750ms d) At t 0.880ms Fgur. amag dstrbuton (prcntag rducton n matrx stffnss) n th mddl layr of a (0/90/0) smply-supportd bam loadd by unform tractons of 0.75MPa

15 a) At t 0.50ms b) At t 0.500ms c) At t 0.750ms d) At t 0.880ms Fgur. amag dstrbuton (prcntag rducton n matrx stffnss) n th top layr of a (0/90/0) smplysupportd bam loadd by unform tractons of 0.75MPa

16 a) At t 5ms b) At t 50ms c) At t 75ms d) At t 00ms Fgur 3. amag dstrbuton (prcntag rducton n matrx stffnss) n th bottom layr of a (30/-60/30) smply-supportd bam loadd by unform tractons of 0.75MPa

17 a) At t 5ms b) At t 50ms c) At t 75ms d) At t 00ms Fgur 4. amag dstrbuton (prcntag rducton n matrx stffnss) n th mddl layr of a (30/-60/30) smply-supportd bam loadd by unform tractons of 0.75MPa

18 a) At t 5ms b) At t 50ms c) At t 75ms d) At t 00ms Fgur 5. amag dstrbuton (prcntag rducton n matrx stffnss) n th top layr of a (30/-60/30) smply-supportd bam loadd by unform tractons of 0.75MPa

19 a) At t 5ms b) At t 50ms c) At t 75ms d) At t 00ms Fgur 6. amag dstrbuton (prcntag rducton n matrx stffnss) n th bottom layr of a (45/-45/45) smply-supportd bam loadd by unform tractons of 0.75MPa

20 a) At t 5ms b) At t 50ms c) At t 75ms d) At t 00ms Fgur 7. amag dstrbuton (prcntag rducton n matrx stffnss) n th mddl layr of a (45/-45/45) smply-supportd bam loadd by unform tractons of 0.75MPa

21 a) At t 5ms b) At t 50ms c) At t 75ms d) At t 00ms Fgur 8. amag dstrbuton (prcntag rducton n matrx stffnss) n th top layr of a (45/-45/45) smply-supportd bam loadd by unform tractons of 0.75MPa

22 a) At t 0.50ms b) At t 0.500ms c) At t 0.75ms d) At t.0ms Fgur 9. amag dstrbuton (prcntag rducton n matrx stffnss) n th bottom layr of a (0/90/0) cantlvr bam loadd by unform tractons of 0.75MPa

23 a) At t 0.50ms b) At t 0.500ms c) At t 0.75ms d) At t.0ms Fgur 0. amag dstrbuton (prcntag rducton n matrx stffnss) n th mddl layr of a (0/90/0) cantlvr bam loadd by unform tractons of 0.75MPa

24 a) At t 0.50ms b) At t 0.500ms c) At t 0.75ms d) At t.0ms Fgur. amag dstrbuton (prcntag rducton n matrx stffnss) n th top layr of a (0/90/0) cantlvr bam loadd by unform tractons of 0.75MPa

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