EXTENDED MULTISCALE FINITE ELEMENT METHOD FOR GEOMETRICALLY NONLINEAR ANALYSIS OF THIN COMPOSITE PLATES ON BENDING PROBLEMS

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1 21 st Intrnatonal Confrnc on Compost Matrals X an, th August 2017 XTNDD MULTISCAL FINIT LMNT MTHOD FOR GOMTRICALLY NONLINAR ANALYSIS OF THIN COMPOSIT PLATS ON BNDING PROBLMS Mngfa Rn 1, J Cong 1 1 Faculty of Vhcl ngnrng and Mchancs, Dalan Unvrsty of Tchnology No.2 Lnggong Road, Ganjngz Dstrct, Dalan Cty, Laonng Provnc, Chna mal: rnmf@dlut.du.cn ywords: xtndd multscal fnt lmnt mthod, Thn compost plats, Nonlnar, Larg dflcton, Couplng ffcts ABSTRACT An ffcnt multscal fnt lmnt mthod s dvlopd for gomtrcally nonlnar analyss of thn compost plats on larg dflcton bndng problm. On th bass of our xstng work about small dflcton problms, th couplng ffcts of compost lamnats ar also consdrd by ntroducng th addtonal couplng trms among translatons and rotatons nto th multscal bas functons. Thn, th dcouplng dsplacmnt boundary condtons for dflctons and rotatons ar mployd to construct th multscal bas functons numrcally at th mcro scal. Aftr that, th quvalnt tangnt stffnss matrx and quvalnt scant stffnss matrx scant stffnss of macroscopc lmnts can b obtand basd on th Total-Lagrang ncrmntal approachs. Furthrmor, th qulbrum macro scal fnt lmnt tratons for ach load stp can b prformd drctly and th macroscopc rspons flds wll b obtand ffcntly. Fnally, th mcroscopc rspons flds can b obtand through downscal computaton asly. Numrcal xampls dmonstrat that th dvlopd mthod posssss hgh computng accuracy and ffcncy compard wth th convntonal fnt lmnt mthod. 1 INTRODUCTION Compost structurs hav bn wdly usd n th fld of arospac du to ts hgh mchancal proprts and outstandng dsgnablty charactrstc. For thn compost structurs subjctd to bndng load, t s usually a larg-dflcton problm du to small bndng stffnss. In ordr to obtan hghr stffnss-wght proprts, ths structurs oftn hav complcatd mcrostructurs such as hrarchcal stffnrs, mcroscopc hols [1,2] as shown n Fg. 1. Du to ths mcrostructurs, analyss of th thn compost lamnats conductd by th convntonal fnt lmnt mthod (FM) s oftn tm consumng. Thrfor, t s ncssary to sk for a numrcal computaton mthod wth hgh computng ffcncy and prcson. (a) (b) Fg. 1 Thn compost structurs wth dffrnt mcrostructurs. Th multscal fnt lmnt mthod (MsFM) s namd offcally by Hou t al [3, ]. Th basc da of MsFM s to captur th proprts of mcrostructur ffctvly by constructng th multscal bas functons numrcally. On th bass of th multscal bas functons, th problms can b solvd at th macro scal drctly wth hgh computng ffcncy and accuracy. It s worth mntonng that,

2 Mngfa Rn, J Cong th MsFM s not rstrctd by th prodcty assumpton and th scal sparaton assumpton. Thus, Th MsFM has bn wdly usd [5, 6]. Th xtndd multscal fnt lmnt mthod (MsFM) s dvlopd by Zhang Hongwu t al [7] basd on th MsFM. In ordr to consdr th couplng ffcts among dffrnt drctons for th mult-dmnsonal vctor fld problms, th addtonal couplng trms of th multscal bas functons ar ntroducd. It has bn succssfully appld n th rsarch of htrognous porous mda and compost matrals [8-10]. In ths papr, an ffcnt multscal fnt lmnt mthod s dvlopd for gomtrcally nonlnar analyss of thn compost lamnats wth mcroscopc hols on larg dflcton bndng problms. Frst, on th bass of our xstng work for small-dflcton analyss of thn compost plats, th couplng ffcts of compost lamnats ar consdrd by ntroducng th addtonal couplng trms among translatons and rotatons nto th multscal bas functons and th dcouplng dsplacmnt boundary condtons for dflctons and rotatons ar rconstructd to construct th multscal bas functons accordng to thr coupld dsplacmnt mods of thn plats. Thn, th quvalnt tangnt stffnss matrx and quvalnt scant stffnss matrx scant stffnss of macroscopc lmnts can b obtand basd on th Total-Lagrang ncrmntal approachs. Fnally, th qulbrum macro scal fnt lmnt tratons for ach load stp can b prformd drctly and th macroscopc and mcroscopc rspons flds wll b obtand ffcntly through th macro scal and downscal computaton rspctvly on th bass of multscal bas functons Svral numrcal xampls dmonstrat that th dvlopd mthod posssss hgh computng accuracy and ffcncy compard wth th convntonal fnt lmnt mthod. 2 MTHODOLOGY 2.1 Incrmntal macro and mcro scal fnt lmnt formulaton On th bass of ncrmntal thory and Nwton-Raphson traton mthod, th multscal fnt lmnt computaton can b prformd for larg dflcton problms of thn compost lamnats. Th ncrmntal macro and mcro scal fnt lmnt formulatons from to t t ar drvd hr accordng to Total-Lagrang formulaton. At th mcro scal, th wak form of qulbrum quatons and boundary condtons at t t can b drvd as q. (1) basd on th prncpl of vrtual dsplacmnt. T t T t t T T t t T T t t G 0B 0F dxdy G N 0 fdxdy G N 0T ds (1) Th smplfd form can b xprssd as q. (2): t t t tt t t L0 0 L1 0 0 T 0 0 C 0 u u P u (2) whr 0 t T and 0 t C ar th tangnt and scant stffnss matrcs of a macroscopc lmnt, 0 th quvalnt mcroscopc nodal forc wthn a macroscopc lmnt. Thy can b xprssd as: T A B G B B B B dxdyg t T t t L0 0 L1 L0 0 L1 L0 0 L1 T t 0 T t G HN 0 F HN dxdyg T A B t T t t 0C G BL0 0BL1 BL0 0BL1 dxdyg tt T T tt T T tt P G N f dxdy G N T ds t tt P q. (2) s th ncrmntal mcro scal fnt lmnt formulaton. It s worth notng that th numbr of s

3 21 st Intrnatonal Confrnc on Compost Matrals X an, th August 2017 ncrmnt stp and traton stp should b st to on whn q. (2) s appld to construct th multscal bas functons. At th macro scal, a compost lamnat s dscrtzd nto th macroscopc fnt lmnts. It s assumd that th ncrmntal macroscopc nodal dsplacmnt wthn a macroscopc lmnt satsfy N U u N U (3) whr s th multscal bas functon of a macroscopc lmnt. Th wak form of qulbrum quatons and boundary condtons at t t can b drvd as follows T T t T t t T T T t t T T T t t G N B F dxdy G N N f dxdy G N N T ds () q. () can b furthr smplfd as follows t t t tt t t L0 0 L1 0 0 T 0 0 C 0 U U P U (5) whr 0 t T and 0 t C ar th tangnt and scant stffnss matrcs of whol structur, quvalnt macroscopc nodal forc of whol structur. Thy can b xprssd as: T A B t T T t t L0 0L1 G N BL0 0BL1 BL0 0BL1 dxdyn G G N HN F HN dxdyn G t T T T t 0 0 T A B G N B B B B dxdyn G t T T t t 0 C L0 0 L1 L0 0 L1 P G N N f dxdy G N N T ds tt T T T tt T T T tt t 0 t P s th q. (5) s th ncrmntal macro scal fnt lmnt formulaton whch s appld to calculat th ncrmntal macroscopc nodal dsplacmnt. Thn, th macroscopc nodal dsplacmnt can b obtand by q. (6). tt t 0 0 U U U (6) 2.2 Multscal bas functons Th multscal bas functons ar th dsplacmnt functons of th macroscopc lmnts whch ar mployd to stablsh th rlatonshp btwn th macro scal and th mcro scal stffnss matrcs. Bsds, th multscal bas functons ar also appld to obtan th mcroscopc stran and strss flds n th downscal computaton. Thus, t can b sn that th constructon of multscal bas functons s a ky stp n th MsFM. Consdr that th couplng ffcts ar obvous as a rsult of layup confguratons and coupld dsplacmnt mods btwn dflctons and rotatons for th thn compost plats. Th dsplacmnts of a mcroscopc nod wthn a macroscopc lmnt can b xprssd as

4 Mngfa Rn, J Cong u Nuuu Nuvv Nuww Nuxx Nuyy v Nvuu Nvvv Nvww Nvxx Nvyy wu wv ww wxx wyy x Nxuu Nxvv Nxww Nxx x Nxy y y N yuu N yvv N yww N yx x N yy y w N u N v N w N N whr N s th multscal bas functon of th macroscopc lmnt. Tak an addtonal couplng trms Nuv as an xampl. It mans th dsplacmnt fld u whn unt dsplacmnt v s appld on th macroscopc nod. 2.3 Boundary condton Th dcouplng dsplacmnt boundary condtons of macroscopc nods on th macroscopc lmnt boundary can b xprssd as follows: N N uw1 vw1 Nww1 1m N N xw1 yw1 0 0 (7) ( 1,2,3,) (8) 0 0 Th dcouplng dsplacmnt boundary condtons of mcroscopc nods on th macroscopc lmnt boundary can b xprssd as follows: N N N uw1 u uw1 1 vw1 v vw1 1 ww1 ww ww1 1 xw1 xw xw1 1 yw1 yw yw1 1 N N N N N N N N N N N N qs. (8) and (9) ar th dcouplng dsplacmnt boundary condtons corrspondng to th multscal bas functons multscal bas functons N 1 w. Undr ths dcouplng dsplacmnt boundary condtons, th N 1 w can b constructd. Th othr multscal bas functons can b (9)

5 21 st Intrnatonal Confrnc on Compost Matrals X an, th August 2017 constructd n a smlar way. 3 NUMRICAL XAMPLS A thn squar compost plat wth layup [0/90/90/0] s xamnd. Th plat s composd of 8 8 macroscopc lmnts and t s subjctd to th unform dstrbutd load wth all dgs clampd. Thr ar two knds of mcroscopc fnt lmnt modls, whr th numbrs of mcroscopc lmnts wth a macroscopc lmnt ar 1 1 and 3 3 (s Fg. 2). Th two knds of numrcal multscal modls ar dsgnatd as MsFM_1 1 and MsFM_3 3. Th lngth of plat s 12 nch. Th thcknss s nch. Th lamna constants ar gvn: 1= ps, 2= ps, G 12= G 13= G 23= ps, μ 12= Not that 1 nch s 2.5 cm and 1 ps s 6.9 N/m 2. Fg. 2 Multscal fnt lmnt modl of a thn squar compost plat: a macroscopc fnt lmnt modl; b mcroscopc fnt lmnt modl Th cntral dflctons obtand from th dvlopd mthod, togthr wth thos from rfrnc [11] ar prsntd n Tabl 1. P (ps) MsFM_1 1 MsFM_3 3 Rfrnc solutons[11] Tabl 1 Cntr dflctons of a smply supportd thn squar plat Th Load-dflcton curvs ar plottd n Fg. 3.

6 Mngfa Rn, J Cong Fg. 3 Load-dflcton curvs of a smply supportd thn squar plat subjctd to th unformly dstrbutd load It s obvous that th nonlnar rsults obtand by th dvlopd mthod ar n good agrmnt wth thos from rfrnc for thn compost plats. It mans that th dvlopd mthod has hgh computng accuracy. Bsds, th dvlopd mthod has a good convrgnc. CONCLUSIONS In ths papr, a novl multscal fnt lmnt mthod sutabl for small-dflcton analyss of thn compost plats wth aprodc macrostructur faturs s dvlopd on th bass of th thortcal framwork of th xtndd multscal fnt lmnt mthod. Th numrcal xampls llustrat that th dvlopd mthod has th charactrstc of hgh computng accuracy for thn compost plats n consdraton of layup confguratons and aprodc mcrostructur charactrstcs, spcally undr th rlaxd dcouplng dsplacmnt boundary condtons. Manwhl, th computr mmory s rducd and th computng ffcncy s mprovd sgnfcantly compard wth th convntonal fnt lmnt mthod. Th dvlopd mthod has grat potntal for strngth prdcton for compost structurs wth complcatd mcrostructurs. That wll b carrd out n our futur work. ACNOWLDGMNTS Ths work was supportd by th Natonal Natural Scnc Foundaton of Chna (No ), th Natonal Basc Rsarch Program of Chna (No. 201CB06506, No. 201CB09000). RFRNCS [1] Chn J F, Morozov V, Shankar. Progrssv falur analyss of prforatd alumnum/cfrp fbr mtal lamnats usng a combnd lastoplastc damag modl and ncludng dlamnaton ffcts. Compos Struct, 201, 11(1):6-79. [2] Wang B, Tan, Hao P, t al. Hybrd analyss and optmzaton of hrarchcal stffnd plats basd on asymptotc homognzaton mthod. Compos Struct, 2015, 132(11): [3] Hou T Y, Wu X H. A multscal fnt lmnt mthod for llptc problms n compost matrals and porous mda. J Comput Phys, 1997, 13(1): [] Hou T Y, Wu X H, Ca Z. Convrgnc of a multscal fnt lmnt mthod for llptc problms wth rapdly oscllatng coffcnts. Math Comput, 1999, 68(227): [5] fndv Y, Hou T Y, Gntng V. Multscal Fnt lmnt Mthods for Nonlnar Problms and Thr Applcatons. Commun Math Sc, 200, 2():

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