DESIGNING WITH ANISOTROPY.

Transcription

1 DESIGNING WITH ANISOTOPY. PAT : QUASI-HOMOGENEOUS ANISOTOPIC LAMINATES. P. Vannucci, X. J. Gong & G. Vrchry. ISAT - Institut Suériur d l Automobil t ds Transorts, LMA - Laboratoir d chrch n Mécaniqu t Acoustiqu, 49, u Madmoisll Bourgois - BP, 587 Nvrs Cdx, Franc. Poalo.Vannucci@u-bourgogn.fr SUMMAY : In this ar, w show th xistnc of a articular class of solutions to th roblm of quasi-homognous laminats, that is laminats that hav th sam bhaviour in strtching and in bnding, with no couling. W call quasi-trivial this ty of solutions, to mhasis th fact that it is not ncssary to solv dirctly th quations govrning th roblm of quasi-homognity to find thm. A vry scial fatur of this class of solutions is th fact that no rdtrmind orintation is fixd for th lis. A surrising rsult is th xistnc of a grat numbr of solutions. KEYWODS : laminats, uncouling, quasi-homognity, olar rrsntation, lan tnsors, comosits dsign, invrs roblms. INTODUCTION. It is wll nown by th dsignrs of comosit structurs that a laminat mad of anisotroic, but idntical, lis has not, gnrally saing, th sam bhaviour in bnding and in strtching, and morovr thr is a couling btwn ths two rsonss. At rsnt, no gnral rul is nown to dsign laminats that bhav li homognous matrials. Th roblm of uncouling is rsntd in Part, whil in this Part w considr th roblm of finding an uncould laminat that has also th sam quivalnt lastic rortis in mmbran and bnding bhaviour. This asct has bn rathr nglctd for a long riod. It sms that th xistnc of such laminats has vn not bn susctd until th wor of Vrchry and his co-authors. In 988, Kandil & Vrchry [7] gav som sufficint conditions to obtain th sam in-lan and flxural bhaviour, with limitd xamls. So w wr lad to dfin th conct of quasi-homognous laminats, as laminatd lats with th sam lastic bhaviour as classical homognous lats, i.. without couling and

2 with th sam lastic bhaviour in bnding and in strtching. Such matrials hav a much simlr bhaviour than ordinary laminats, whil rtaining th advantags of comosits. Still starting by th olar rrsntation thory of in-lan lasticity tnsors, w considr hrin th gnral roblm of quasi-homognity, and formulat th quations that govrn th roblm. It is raidly rcognisd that ths quations ar non-linar, li oftn in invrs roblms, vn in linar lasticity. Though th non linarity of th systm of govrning quations b low, as it is comosd of linar and quadratic quations, it aars immdiatly that a comlt solution of such a systm is in gnral a vry hard tas. Fortunatly, thans to th vry scial structur of th quations, w hav bn abl to find a articular class of solutions, rathr gnral, asy to b dtctd and surrisingly oulatd by a grat numbr of solutions. To rcognis th fact that ths solutions ar found with no nd to solv dirctly th govrning quations, w hav calld thm quasi-trivial. Th quasi-trivial solutions hav som imortant rortis. Th most imortant on, scially for alications, is th fact that thy dtrmin only a stacing squnc, th lis forming grous with no dfind orintation. Thrmolastic rortis ar also invstigatd in th ar, and w show how th two classs of quasi-trivial solutions for lastic and thrmolastic rortis coincid rfctly, which is a vry imortant fatur for alications. Th ar nds with rsults of xrimntal tsts rformd, which rfctly confirm and illustrat th rdictions. LAMINATES BEHAVIOU BY POLA TENSOS EPESENTATION. Whn a lat is comosd surosing various lis, th bhaviour of th laminat is givn by th so calld Classical Laminatd Plat Thory CLPT, basd uon th wll-nown hyothss of Kirchhoff-Lov s for instanc [], [], [], [4]. Th mchanical bhaviour of th lat is so condnsd in th quation : N A M = B B ε D χ whr N is th vctor of mmbran forcs and M that of th bnding momnts, whil ε and χ ar rsctivly th vctors of middl lan strains and curvaturs. Th tnsors A and D dscrib th mmbran and th flxural bhaviour, whil B tas into account th couling btwn mmbran and bnding. For a gnral stch li in Fig., th comonnts of A, B and D ar N= + if odd and N= if vn : z+ z z+ z z + z A = Q B = Q D= Q. = = Hr, Q is th classical tnsor rlating strsss to strains in two dimnsional lasticity. It is wll nown that xrss th abov tnsors in rotatd frams is rathr comlicat in Cartsian co-ordinats. To this uros, and in ordr to obtain simlifications, w will us th so-calld olar rrsntation of fourth-ordr lan tnsors dvlod and usd by Vrchry and his coworrs [5] - [], and also Parts and, in ths sam rocdings. This olar mthod incororats in a mor systmatic and owrfull way nown rsults and mthods such as thos introducd by Tsai and Pagano [] and []. Th comonnts of a gnric fourth-ordr lan tnsor L, having th symmtris of lasticity, can b xrssd as : =

3 h/ h/ y z z + x h/ h/ y z z + x a b Figur : Gnral stch a odd numbr of lis b vn numbr of lis. L = T + T + cos 4Φ + 4 cos Φ L = T + T cos 4Φ L = T + T + cos 4Φ 4 cos Φ L = T cos 4Φ L = sin 4Φ + sin Φ L = sin 4Φ + sin Φ. W can obtain in this way : i. olar comonnts of tnsor A for a laminat : T = T z z + = T = T z z + = 4iΦ! Φ + δ = z z iφ! = + Φ + δ = z z = 4i i + 4 ii. olar comonnts of tnsor B for a laminat :

4 +! T = T z z = +! T = T z z =! 4iΦ! 4i = z z = Φ + δ +! iφ! i = z z = Φ + δ + 5 iii. olar comonnts of tnsor D for a laminat : + ~ T = T z z = + ~ T = T z z = ~ ~ 4iΦ i = z z = Φ + δ + ~ ~ iφ i = z z = 4 Φ + δ + 6 Hr, th angls Φ and Φ ar th angls dfind for th ly by its own rfrnc fram, and δ is th angl that such fram forms with th global rfrnc fram of th laminat. By Eqs., th comonnts of tnsors A, B and D for th laminats may b obtaind as functions of th abov final comonnts T, tc. QUASI-HOMOGENEOUS LAMINATES Mathmatically saing, a quasi-homognous laminat, that is a laminat which bhavs li an homognous matrial in th linar lastic rang, is charactrisd by th following two rortis: A h D = B = h. 7 In th rcding quations, h is th total thicnss of th lat. As alrady said, w will dal in th following with laminats comosd of idntical lis, that is of lis with th sam hysical rortis and thicnss. In such a cas, Eqs. 4, 5 and 6 ar considrably simlifid, and it can b asily sn that : and : ~ ~ T = T = T T = T = T T! = T! =. 8

5 4iΦ 4iΦ 4iδ = = iφ iφ iδ = = +! 4iΦ! 4iΦ 4iδ = z z =!! iφ iφ i δ = z z = + ~ ~ 4iΦ 4iΦ 4iδ = z z = + ~ ~ iφ i i Φ δ = z z. = + 9 Finally, th quasi-homognity conditions 7, by virtu of quations 8 and 9, bcom simly : h iφ h ~ ~ iφ iδ 4 iδ = = = = N h h iφ h ~ ~ iφ iδ 4 iδ = = = = N h z + z + z z! 4iΦ! = 4i z z = = δ + iδ! = z z =. iφ! = + Introducing th tnsor : C = A D h h, th quasi-homognity conditions, quations, bcom simly : C C B B 4iδ 4iδ c = * 4iδ iδ iδ c = * iδ 4iδ i 4 δ b = * 4iδ iδ i δ b = * iδ = + = = + = = = = =. Hr * = if N is odd, and * = if it is vn. Th cofficints c and b ar givn by th following ruls : c = +, b = for N odd c = +, b = for N vn.

6 Equations ar th systm to solv in th sarch for quasi-homognous laminats comosd of idntical lis. As it can b noticd, such a rorty dos not dnd uon th mchanical rortis of th singl ly, as it was to b xctd. In ordr to fix a solution, th orintation of on of th lis must b fixd a riori. In th following, w will considr that th first ly from th bottom has δ=. It is an asy tas to vrify that th trivial solution, that is th solution which givs th sam orintation for ach ly in a laminat, is ossibl for ach N. THE CLASS OF QUASI-TIVIAL SOLUTIONS. No gnral solution of systm is rsntly availabl. Nvrthlss, a articular class of solutions is rathr asy to b found, onc and for all, for ach laminat : w will call such class th st of quasi-trivial solutions of th roblm of quasi-homognity. Th xistnc of a articular class of solutions to systm is suggstd by som culiar rortis of cofficints c and b, in Eqs.. In fact, it can b noticd asily that cofficints c ar symmtric about th middl lan, whil cofficints b ar anti-symmtric which suggsts immdiatly th wll-nown rul of symmtrical stacing squncs for uncould laminats, and thy ar linarly incrasing from th lowr to th ur ly. Morovr, th sum of thm is zro, in both cass. So, if w assum that a grou is comosd by a crtain numbr of lis having th sam orintation, w will hav a solution ach tim that th sums of th cofficints c and b of such a grou ar zro. W will call it a quasi-trivial solution and such a grou a saturatd grou. It is immdiatly rcognisd that a grou is saturatd indndntly of th orintation of its lis, which mans that quasi-trivial solutions dtrmin in rality only a stacing squnc, but not th orintations of ach layr. Consquntly, mchanically saing, ach quasi-trivial solution corrsonds to g- infinitis of diffrnt solutions, whr g is th numbr of saturatd grous, that is of ossibl diffrnt orintations. All thm ar quasi-homognous, but ar distinguishd by th final mchanical rortis of th laminat. Evidntly, for th sam laminat, it is in gnral ossibl to find diffrnt quasi-trivial solutions, with diffrnt numbrs of orintations. Bfor going on to show th mthod to find quasi-trivial solutions, it is worth listing som rortis of this class of solutions. A saturatd grou must hav lis both in th ur and in th lowr art of th middl lan in fact, lt us suos that a grou of lis, all bing in th sam art with rsct to middl lan, is saturatd with rsct to cofficints c thn, th sum of its b cofficints is diffrnt from zro, du to th fact that ths last ar anti-symmtric and linarly variabl along th thicnss. Such a stacing squnc can hav th sam mmbran and bnding bhaviour, but should not b uncould. Th rcding rorty imlis that th lowr thortical numbr of lis in a saturatd grou is two. Consquntly, th maximum numbr of diffrnt saturatd grou, that is of diffrnt orintations, in a N-lis laminat is strictly lss than N/. In fact, xctd th cas of cofficints c null s blow, in a saturatd grou thr must b at last lis, in ordr to hav a null sum both of cofficints c and b, and this automatically limits th numbr of diffrnt orintations to lss than N/. In addition, as th cofficints c hav null sum also on on half of th laminat, symmtric solutions ar ossibl, and th numbr of diffrnt orintations, for what said abov, is strictly limitd to N/4. Morovr, a symmtric solution is ossibl only if a saturatd grou of c cofficints xists in only on half of th stacing squnc. Anothr imortant rorty of such class of solutions, is that a solution with g diffrnt orintations dscnds from anothr with g- grous. As a consqunc of this rorty, if a

7 laminat has not quasi-trivial solutions with g diffrnt orintations, it will not hav quasitrivial solutions also for mor than g orintations. This circumstanc givs a critrion for nding th rocdur of th sarch for quasi-trivial solutions. It must b strssd out also th xistnc of articular cass in quasi-trivial solutions. In fact, in som cass, it occurs that a cofficint c is zro, and consquntly also its symmtric, c -. Thn, by virtu of th anti-symmtry of cofficints b, a quasi-trivial solution is immdiatly idntifid, corrsonding to a saturatd grou formd by ths two symmtric lis. It is worth noticing that, du to th fact that cofficints c ar quadratically varying and that thy ar symmtric, not mor than two lis in a laminat can hav null cofficint c. To find laminats which hav this rorty, it is sufficint to os qual to zro th xrssions giving cofficints c in Eqs.. Th first fiv laminats that can njoy this rorty ar th laminats with, 7, 6, 97 and 6 lis, whr th lis with null cofficint c ar rsctivly th numbr,, 8, 8 and 5 from th bottom, and thir symmtric ons. In th following, w will giv a labl, namly a numbr, to ach diffrnt saturatd grou in a laminat, bginning from zro. To asily dtct qual solutions, ach squnc is ordrd, in th sns that, rocding from th bottom of th stacing squnc, th first orintation ncountrd is always lablld by, thn th scond by, th third by, and so on. QUASI-HOMOGENEOUS LAMINATES FO THEMO-ELASTIC EFFECTS It is ossibl to rform for thrmolastic ffcts an analysis similar to th rcding on. Classically, w will considr a linarly through thicnss varying thrmal fild, which is th ral situation of a laminat constitutd by lis of th sam matrial, subjctd to a diffrnc of tmratur btwn th ur and th lowr surfac, onc th thrmal flux is stationary. It must b strssd out that now th assumtion of all th lis with th sam hysical rortis has now a hysical rlvanc. Morovr, th assumd conditions ar thos usd in th manufacturing rocss of laminats, so thy hav a fundamntal imortanc in ractic. W assum th following distribution of tmratur across th thicnss : τ z = τo + τ h z, 4 whr τ o is th valu corrsonding to middl lan, z =, and τ is th diffrnc of tmratur btwn th two facs. Th gnral bhaviour in th rsnc of such a thrmal fild is givn by N A M = B o B ε D τ χ o S S τ, 5 h V whr, S and V ar thr scond ordr tnsors dscribing thrmal strsss for th laminat in strtching, couling and bnding rsctivly. Liwis in th cas of lastic constants, on can writ down th olar comonnts of, S and V : for a gnric scond ordr lan tnsor L w hav its Cartsian comonnts xrssd as : L = T+ cos Φ L = T cos Φ L = sin Φ. 6 Th conditions of quasi-homognity for thrmal ffcts ar :

8 which giv, in th cas of idntical lis, and V = S = h h, 7 ~ T = T= T, T! =, 8 h iφ h ~ ~ iφ iδ 4 iδ = = z z = = + N h iφ! i! δ = z z =. = + It is aarnt that Eqs. 9 corrsond xactly to Eqs. 4. Togthr with th fact that in Eqs. only two sts of cofficint, b and c, xists for four quations, this imlis that thr is a rfct coincidnc btwn lastic and thrmolastic quasi-trivial solutions. 9 ESULTS AND DISCUSSION. Th abov mntiond rortis of quasi-trivial solutions hav bn widly usd in th formulation of an automatic algorithm abl to find all quasi-trivial solutions. By mans of this, w hav analysd a crtain numbr of laminats, and found for ach on th total numbr of quasi-trivial solutions for th roblm of quasi-homognity. A qustion ariss from considration of ths rsults : ar all th solutions ffctivly indndnt solutions? Ta for instanc a solution of th ind [ ] and anothr of th ind [ ]. It is aarnt that th scond on can b considrd as a articular cas of th first on, whn for th third and fourth layr it is assumd th sam orintation of th saturatd grou lablld as. So, w can considr th scond on as a drivd solution, and th first on as a ur, or indndnt, solution. W hav dcidd to considr only th indndnt solutions as ral distinct ons. In ffct, undr a mathmatical oint of viw, two dndnt solutions ar th sam on : thy diffrntiat thmslvs only for having chosn th sam orintation for two or mor distinct grous. Tabl : Numbr and ty of quasi-trivial indndnt solutions. N. of Total solutions orintations orintations 4 orintations 5 orintations 6 orintations lis gnric. sym. gnric. sym. gnric. sym. gnric. sym. gnric. sym. gnric. sym

9 Th rsults ar summarisd in Tabl, whr thy ar rangd by total numbr of ur solutions, and by ur solutions with,, 4, 5, or 6 diffrnt orintations. In addition, for ach cas, th numbr of symmtric ur solutions is dtaild. At a first glanc, it is surrising to notic that, contrarily to what is commonly thought, symmtric solutions ar only a littl art of th whol. Th numbr of solutions is raidly incrasing, but not monotonically. Finally, w hav considrd th xistnc of so-calld anti-symmtric solutions, i.. solutions with two orintations charactrisd by having a stacing squnc of th ind [ ]. W hav found that only laminats whos numbr of lis is a multil of 4 can hav quasi-trivial anti-symmtric solutions for th quasi-homognity roblm. In Tabl, w show th numbr of anti-symmtric solutions as a function of th numbr of lis. Tabl : Numbr of anti-symmtric solutions. Numbr of layrs Anti-symmtric ur solutions EXPEIMENTAL ESULTS. W hav manufacturd and tstd som laminats dsignd according to th rvious analysis. W hav chosn a 9-layr lat. Th singl layr is a carbon-oxy ly, with stimatd lastic constants E =.9 Ga, E = 9.5 Ga, G = 5.7 Ga and ν =.6. Th thicnss of a singl ly is.5 mm, for a total thicnss of th lat of.75 mm. For a 9-layr laminat, w hav at our disosal 85 diffrnt quasi-trivial solutions, fiv of which ar symmtric. W hav slctd th following non symmtric stacing squnc : [ 5 5 ]. Th layrs lablld by ar orintd at, whil thos lablld by ar orintd at 9. Hnc th laminat, comosd of orthotroic lis, should b orthotroic itslf. Th xrimntal rsults, obtaind from tnsil tsts, ar shown in Tabl, along with rdictions from th classical laminatd lat thory. All th tsts hav confirmd that th comonnts of couling tnsor B wr null. Th rsults ar rathr satisfying, and thortical rdictions can b considrd as confirmd by xrimntal rsults. Tabl : Exrimntal rsults for th 9-layr lat. E x [GPa] E y [GPa] G xy [GPa] ν xy Mmbran Bnding CLPT stimat

10 CONCLUSIONS. Using th olar rrsntation of lan lasticity tnsors, w hav discovrd a articular class of solutions to th roblm of quasi-homognity, that w hav namd quasi-trivial solutions. In othr words, w hav found a gnral mthod to dsign th stacing squnc for a quasihomognous laminat, without distinction btwn symmtric and non symmtric solutions. Th quasi-trivial solutions hav som imortant rortis, from which w hav formulatd an algorithm abl to find all thm. Th larg numbr of solutions that w hav found can b considrd abl to solv most of th roblms found in ractical alications. EFEENCES.. Jons. M. : Mchanics of Comosit Matrials. Taylor & Francis, USA, Christnsn. M. : Mchanics of Comosit Matrials. Wily, Nw Yor, Tsai S. W., Han H. T.: Introduction to Comosit Matrials. Tchnomic, Tsai, S. W.: Comosit dsign guid. Dayton, USA, Vrchry G. : Ls invariants ds tnsurs d ordr quatr du ty d l élasticité Invariants of fourth ran tnsors with th symmtry of th lasticity tnsors. Procdings of th Euromch Colloquium 5,. 9-4.Villard-d-Lans, 979, Ed. J.-P. Bohlr, Editions du CNS, Paris, 98 in Frnch. 6. Kandil N., Vrchry G.: Nw mthods of dsign for stacing squncs of laminats. Procdings of Comutr Aidd Dsign in Comosit Matrials 88, Southamton, Kandil N., Vrchry G.: Nouvlls méthods d conction ds milmnts ds stratifiés. Comts rndus d JNC 6,. 89-9, 988 in Frnch. 8. Kandil N., Vrchry G.: Som nw dvlomnts in th dsign of stacing squncs of laminats. Procdings of ICCM 7, Canton, China, Vrchry G.: Dsigning with anisotroy. Txtil Comosits in Building Construction, art,. 9-4, Pluralis, Paris, 99.. Kandil N., Vrchry G.: Dsign of stacing squncs of laminatd lats for thrmolastic ffcts. CADCOMP 9, Comosit Matrials : Dsign and Analysis. Procdings of th nd Int. Confrnc On Comutr Aidd Dsign in Comosit Matrials Tchnology, Brussls 5-7 Aril 99. Eds. W. P. D Wild & W.. Blain. Comutational Mchanics Publications, Southamton, Tsai S. W., Pagano, N. J.: Comosit Matrials Worsho. Eds. Tsai S. W. t al., Tchnomic, 968.