8 TH INTERNTIONL CONFERENCE ON COMPOSITE MTERILS ELSTIC-VISCOPLSTIC HOMOGENIZTION NLYSIS OF PLIN-WOVEN GFRP LMINTES WITH MISLIGNED PLIN FBRICS S. Knmru, T. Mtsud * Deprtment of Engneerng Mechncs nd Energy, Unversty of Tsuku, Tsuku, Jpn * Correspondng uthor (mtsud@kz.tsuku.c.p) Keywords: Pn-Woven Lmnte, Homogenzton, Msgnment, Vscopstcty Introducton Pn-woven mntes mde of pn frcs nd poymer mters hve hgh specfc strength, hgh specfc stffness nd good formty. They re therefore used n mor ndustr sectors, such s erospce, uto nd mrne ndustres. Ths cn mke pn-woven mntes encounter severe condtons ncudng hgh stress nd hgh temperture. Thus, t s requred to nyze not ony estc ut so nestc ehvor of pn-woven mntes. In these knds of nyses, the msgnment of pn frcs n mntes, whch s ced "mnte msgnment" herefter, ecomes n mportnt ssue ecuse such msgnment cn ffect the mechnc propertes of pn-woven mntes [-]. Therefore, t s of gret mportnce to nvestgte the effects of mnte msgnment on the mechnc propertes of pn-woven mntes. When nyzng mechnc propertes of pnwoven mntes, the mthemtc homogenzton theory sed on unt ce nyss [] ecomes one of the most usefu theores. Therefore, ths theory hs een redy pped to the estc proems [] nd mcroscopc dmge proems [6, 7]. In most studes, pn frcs n pn-woven mntes were ssumed to hve no msgnment ced "n-phse" mnte confgurton, nd unt ces were defned s shown n Fg. (), resutng n no consderton for mnte msgnment. However, Tkno et. [] conducted mcroscopc dmge nyss of pnwoven gss fer-renforced pstc (GFRP) mntes wth not ony n-phse ut so "out-ofphse" mnte confgurtons. Here, the out-ofphse mens tht pn-woven frcs possess hf unt ce msgnment s shown n Fg. (). In ths cse, the unt ce ecomes two tmes rger thn tht of n-phse s shown n the fgure. Zemn nd Šenoh [2] further dopted qus-qurter of ce Unt ce () () Fg. Two types of mnte confgurtons of pn frcs; () n-phse () out-of-phse.
msgnment n ddton to the two ovementoned mnte confgurtons n sttstc nvestgton of the estc propertes of pn-woven cron fer-renforced pstc (CFRP) mntes. Resuts of these nyses reveed sgnfcnt nfuence of msgnment on the mechnc propertes of pn-woven mntes. In the nyses, however, the msgnment ws restrcted to hf or qus-qurter of unt ces. In ddton, the nyses for the msgned cses requred twce the voume of unt ces defned for gned cses. The uthors [,8-], on the other hnd, hve performed estc-vscopstc nd creep nyses of fer-renforced compostes usng homogenzton theory for non-ner tme-dependent mters [], whch s referred to s "tme-dependent homogenzton theory" herefter. In one of these studes [], the uthors proposed method whch ws e to de wth oth the n-phse nd out-ofphse mnte confgurtons n pn-woven mntes usng the sme unt ce defned for the nphse cse, whch ws sed on pont-symmetrc oundry condton for unt ce nyss [2]. Ths method, however, ws so mted to the hf ce msgnment, nd ws not e to de wth rtrry msgnment. In ths study, estc-vscopstc ehvor of pnwoven GFRP mntes wth rtrry mnte msgnment s nyzed usng the tme-dependent homogenzton theory. In the nyss, the uthors empoy sc ce, whch s qurter of n ordnry unt ce, s n nyss domn usng nove oundry condton for unt ce nyss. Ths enes one sc ce to de wth rtrry mnte msgnment, vodng not ony geometry nd mesh generton of sc ce for every msgnment, ut so the nfuence of mesh dependence. From the nyss resuts, t s shown tht the msgnment of pn frcs ffects vscopstc propertes of the pn-woven mntes. 2 Tme-dependent homogenzton theory for pn-woven mntes wth msgned pn frcs Consder pn-woven mnte wth msgned pn frcs s ustrted n Fg. 2. For the mnte, sc ce nd Crtesn coordntes y (, 2, ) re defned s shown n the fgure. ccordng to the conventon homogenzton theory [], the mcroscopc veocty fed u ( y, t) n s expressed s u ( y, t) F ( t) y u ( y, t), () where ( ) ndctes dfferentton wth respect to t, F () t denotes the mcroscopc deformton grdent, nd u stnds for the pertured veocty from the mcroscopc one F () t y. Then the mcroscopc strn rte hs the foowng expresson: ( y, t) E ( t) ( y, t), (2) where E nd ndcte respectvey the mcroscopc strn rte nd the pertured strn rte descred s E ( F F ) 2 nd ( u, u, ) 2, n whch ( ), represents the dfferentton wth Bsc ce Fg. 2 Pn-woven mnte wth msgned pn frcs. In the fgure, nd ndcte the mount of msgnment n the y - nd y2 -drectons, respectvey.
ELSTIC-VISCOPLSTIC HOMOGENIZTION NLYSIS OF PLIN- WOVEN GFRP LMINTES WITH MISLIGNED PLIN FBRICS respect to y. The equrum of s expressed n rte form s. (), Let v ( y, t) e n rtrry vrton of the pertured veocty fed u defned n t t. Then, the ntegrton y prts nd the dvergence theorem ow Eq. () to e trnsformed to wek form: v d n vd, (), where sgnfes the oundry of, nd n ndctes the unt vector outwrd norm to. Now, s dvded nto (, 2,,2) s shown n Fg.. Then, the ntegr term n the ove equton cn e expressed s n vd n vd. () Frst, on, the dstrutons of u,.e. v, nd re pont-symmetrc wth respect to ts center, ecuse the ntern structure stsfes pontsymmetry regrdng the pont. Therefore, the oundry ntegr term for vnshes,.e.: n v d. (6) The sme stuton exsts on,, 7 nd 9-2. Thus, we hve n v d, (7) () nv d, (8) nv 7 2 d, (9) 2 9 7 (c) 8 () 6 7 2 9 n v d. () By contrst, v nd on 2 nd those on 8 respectvey dstrute perodcy, whe n tkes the opposte drectons on these oundres. In consequence, the foowng reton s otned: n vd n vd. () 2 8 2 8 The sme stuton on nd 6 yeds n vd n vd, (2) 6 6 Susttuton of Eqs. (6)-(2) nto Eq. () eds to n v d, () whch ows Eq. () to e wrtten s v d. (), Ths resutng equton hs the sme form s tht otned n the orgn work y Ohno et. [], whch enes the uthors to reud the tmedependent homogenzton theory s foows. Consttuents of the mter re ssumed to exht ner estcty nd non-ner vscopstcty s chrcterzed y c ( ), () where c nd represent the estc stffness nd vscopstc strn rte of the consttuents, stsfyng c c ck c nd k. Susttuton of the ove equton nd Eq. (2) nto Eq. () resuts n c u v d E c v d c v d. pq p, q,,, (6) Let nd e functons whch re determned y sovng the foowng oundry vue proems for, respectvey []: c v d c v d, (7) pq p, q,, pq p, q,, c v d c v d. (8) Then Eq. (6) hs the foowng souton for the pertured veocty fed: u ( y, t) ( y) E ( t) ( y, t). (9) By susttutng Eq. (2) nd the ove equton nto Eq. (), the evouton equton of mcroscopc stress nd the reton etween mcroscopc stress rte nd strn rte E re derved []: Fg. Boundres (, 2,,2) of the sc ce; () upper surfce, () ottom surfce nd (c) sde surfces.
,,,, c E c, (2) pq pk q p q k c E c, pq pk q p q k (2) where ndctes Kronecker s det, stnds for the voume verge n defned s d, n whch sgnfes the voume of. It s emphszed tht the present theory enes one sc ce to e suffcent to de wth rtrry msgnment y chngng the re rtos of - 8. nyss condtons In the present nyss, the mnte msgnment n the y - nd y2 -drectons were consdered s ustrted n Fg.. Denotng the mount of msgnment n the y - nd y2 -drectons s nd, respectvey, eght cses of ther comntons were seected,.e. =, /, / 2, /,, / /, / 2 / 2 nd, where ndctes the wdth nd depth of the sc ce. The sc ce ws defned nd dvded nto eght-node soprmetrc eements (28 eements, 299 nodes) s shown n Fg.. The fer undes were regrded s trnsversey sotropc estc mters, whe the epoxy mtrx s n sotropc estc-vscopstc mter. Mter propertes used re shown n Te. []. The pn-woven mntes were suected to mcroscopc unx - od t constnt strn rte of s t room temperture. Three cses of odng drectons,.e. the y -, y2 - nd -drectons, were consdered. 2 Fg. Lmnte msgnment. () Fg. Bsc ce of pn-woven GFRP mntes nd ts fnte eement mesh; () fu vew nd () fer undes n the sc ce. Resuts of nyss p p () Fgures 6-8 show the mcroscopc stress-strn retons of the pn-woven mntes suected to the unx tenson n the y -, y2 - nd - drectons, respectvey. Frst, t s seen from Fg. 6 tht, wth odng n the y -drecton, the fow stress vres dependng on the mnte msgnment, nd the fow stress for s out % hgher thn tht for t E.2, showng sgnfcnt nfuence of mnte msgnment on the mcroscopc vscopstc ehvor of the mntes. Wth the y2 -drecton odng, the chnge of fow stress s so confrmed n Fg. 7. In ths cse, the fow stresses for nd for so ecome the mxmum nd mnmum vues, respectvey. When oded n the -drecton, on the other hnd, the mntes exhted much more vscopstc nonnerty (Fg. 8) thn tht oserved n the onxs odng mentoned ove. Ths ws ecuse the vscopstcty of the epoxy mtrx ecme domnnt n off-xs odng due to sher deformton of the mntes [], menng tht the deformton mechnsm ws qute dfferent from tht of the onxs odng. Nevertheess, the mnte msgnment so ffects the fow stress of the mntes s shown n Fg. 8. It shoud e noted however tht the fow stress decreses wth the ncrese of mnte msgnment, n contrst to the on-xs odng. Fer undes Mtrx Te. Mter constnts of gss fers nd epoxy Gss fer E f 8. f. Epoxy E g m.. p m. 2. 2. n 2 p
Mcroscopc Stress [MP] Mcroscopc Stress [MP] Mcroscopc Stress [MP] ELSTIC-VISCOPLSTIC HOMOGENIZTION NLYSIS OF PLIN- WOVEN GFRP LMINTES WITH MISLIGNED PLIN FBRICS 2 2 2 2..2 Mcroscopc Strn E Fg. 6 Mcroscopc stress-strn retons of the pn-woven GFRP mntes wth mnte msgnment suected to unx odng n the y -drecton...2 Mcroscopc Strn E Fg. 7 Mcroscopc stress-strn retons of the pn-woven GFRP mntes wth mnte msgnment suected to unx odng n the y -drecton. 2 6 2 2..2 Mcroscopc Strn E Fg. 8 Mcroscopc stress-strn retons of the pn-woven GFRP mntes wth mnte msgnment suected to unx odng n the -drecton. References [] N. Tkno, M. Zko nd S. Skt Threedmenson mcrostructur desgn of wovenfrc composte mters y the homogenzton method: st report, effect ofmsmtched y-up of woven frcs on the strength. Trnsctons of the Jpn Socety of Mechnc Engneers Seres, Vo. 6, pp. 8-, 99. [2] J. Zemn nd M. Šenoh Homogenzton of nced pn weve compostes wth mperfect mcrostructure: prt I - theoretc formuton. Internton Journ of Sods nd Structures, Vo., pp. 69-67, 2. [] T. Mtsud, Y. Nmy, N. Ohno nd M. Tokud Estc vscopstc ehvor of pn-woven GFRP mntes Homogenzton usng reduced domn of nyss. Composte Structures, Vo. 79, pp. 9-, 27. [] E. Snchez-Penc Non-Homogeneous Med nd
Vrton Theory. Lecture Notes n Physcs, Vo. 27, 98. [] J. M. Guedes nd N. Kkuch Preprocessng nd postprocessng for mters sed on the homogenzton method wth dptve fnte eement methods. Computer Methods n pped Mechncs nd Engneerng, Vo. 8, pp. 98, 99. [6] N. Tkno, Y. Ohnsh, M. Zko nd K. Nshyu The formuton of homogenzton method pped to rge deformton proem for composte mters. Internton Journ of Sods nd Structures, Vo. 7, pp. 67-6, 2. [7] V. Crve nd C. Pogg homogenzton procedure for the numerc nyss ofwoven frc compostes. Compostes Prt, Vo. 2, pp. 2-2, 2. [8] T. Mtsud, N. Ohno, H. Tnk nd T. Shmzu Homogenzed n-pneestc-vscopstc ehvor of ong fer-renforced mntes. JSME Internton Journ Seres, Vo., pp. 8-, 22. [9] T. Mtsud, N. Ohno, H. Tnk nd T. Shmzu Effects of fer dstruton onestc-vscopstc ehvor of ong fer-renforced mntes. Internton Journ of Mechnc Scences, Vo., pp. 8-98, 2. [] T. Mtsud nd Y. Fukut Mut-sce creep nyss of nge-py CFRP mntes sedon homogenzton theory. Journ of Sod Mechncs nd Mters Engneerng, Vo., pp. 66-672, 2. [] N. Ohno, X. Wu nd T. Mtsud Homogenzed propertes of estc-vscopstc compostes wth perodc ntern structures. Internton Journ of Mechnc Scences, Vo. 2, pp. 9-6, 2. [2] N. Ohno, T. Mtsud nd X. Wu homogenzton theory for estc-vscopstc compostes wth pont symmetry of ntern dstruton. Internton Journ of Sods nd Structures, Vo. 8, pp. 2867-2878, 2.