Using a C 1 triangular finite element based on a refined model for analyzing buckling and post-buckling of multilayered plate/shell structures

Transcription

1 Using a C 1 triangular finit lmnt basd on a rfind modl for analyzing buckling and post-buckling of multilayrd plat/sll structurs F. Dau O. Polit M.Touratir LAMEFIP/ENSAM, Esplanad ds arts t métirs, Talnc-Franc LMpX, 1 Cmin Dsvallièrs, Vill d Avray-Franc LMSP-UMR CNRS-ENSAM/ESEM, 151 Bd d l Hopital, Paris-Franc 1 Introduction Multilayrd bam, plat and sll modls and finit lmnts ar ndd in structural mcanics for analyzing, dimnsionning and dsigning tis kind of structurs. In t fild of multilayrd slls wr transvrs sar strss ffcts ar of grat importanc, many ig ordr sll toris xist but vry fw numrical tools av bn dvloppd, s for xampl t rviw papr 4]. T aim of tis work is to prsnt a nw finit lmnt, simpl to us, fr from classical numrical problms and vry fficint for computing bot displacmnts and strsss for multilayrd sll applications. Tis nw C 1 sll finit lmnt is basd on t rfind kinmatic modl givn in 7] wic incorporats : a cosin distribution for t transvrs sar strains avoiding t us of sar corrction factors, t continuity conditions btwn layrs of t laminat for bot displacmnts and transvrs sar strsss, t satisfaction of t boundary conditions at t top and bottom surfacs of t sll, t us of only fiv indpndnt gnralizd displacmnts (tr translations and two rotations). A conforming finit lmnt mtod and ig-ordr finit lmnt approximations, Argyris intrpolation for t transvrs displacmnt and Ganv intrpolation for mmbran displacmnts and transvrs sar rotations, ar bot rtaind in tis work. Som unavoidabl gomtric sll considrations ar firstly prsntd to introduc ncssary tools for sll dscription. In t scond part of tis papr, t sll modl basd on a rfind kinmatic approac is dvloppd. T nxt part is ddicatd to t finit lmnt approximations and t prsnt triangular finit lmnt. Finally, numrical rsults, compard wit xprimntal ons, ar prsntd. Good rsults on classical tsts for 1

2 multilayrd plats and slls for bot linar static and dynamic analysis av alrady bn obtaind using tis nw lmnt. In tis papr, critical buckling and post buckling problms ar spcially studid to sow t fficincy of tis nw finit lmnt for multilayrd structurs. 2 Gomtric considrations A sll C wit a middl surfac S and a constant ticknss is dfind by, s 3] : C = { M R 3 : OM(ξ, ξ 3 ) = Φ(ξ) + ξ 3 a 3 ; ξ Ω; 1 2 (ξ) ξ3 1 2 (ξ) } wr t middl surfac is dscribd by a map Φ from a paramtric bidimnsional domain Ω as : Φ : Ω R 2 S R 3 ξ = (ξ 1, ξ 2 ) Φ(ξ) (1) For a point on t sll middl surfac, covariant bas vctors ar usually obtaind as follows : a α = Φ(ξ 1, ξ 2 ),α ; a 3 = a 1 a 2 a 1 a 2 = t 3 (2) In Eq. (2) and furtr on, latin indics i, j,... tak tir valus in t st {1, 2, 3} wil grk indics α, β,... tak tir valus in t st {1, 2}. T summation convntion on rpatd indics and t classic notation ( ),α = ( ) ar usd. ξ α For any point of t sll, covariant bas vctors ar now dducd as : g α = r(ξ 1, ξ 2, ξ 3 ),α = (δ β α ξ 3 b β α) a β = µ β α a β and g 3 = a 3 (3) T mixd tnsor m β α must also b introducd and is dfind by t rlation : m β α = (µ 1 ) β α = 1 µ {δβ α + ξ3 (b β α 2Hδβ α )} (4) wr µ = dt(µ β α) = 1 2Hξ 3 + (ξ 3 ) 2 K ; H = 1 2 tr(bβ α) ; K = dt(b β α). Trfor, t covariant mtric tnsor a αβ, covariant b αβ and mixt b β α curvatur tnsors can b dfind. Ts tnsors and som rlations btwn tm ar rcalld raftr : a αβ = a βα = a α. a β a α = a αβ a β a = dt(a b αβ = b βα = a α. a 3,β b β α = a βγ αβ ) (5) b γα Finally, t surfac lmnt ds and t volum lmnt dv ar classically givn by : ds = adξ 1 dξ 2 dv = µdsdξ 3 (6) All ts rlations ar classic and mor dtails in ordr to obtain t Cristoffl symbols and otr diffrntial gomtric ntitis could b found in Brnadou 3]. 2

3 3 T sll modl 3.1 T displacmnt fild Lt us dnot by u (k) i (ξ 1, ξ 2, ξ 3 = z, t), i {1, 2, 3} t curvilinar componnts of t displacmnt fild associatd wit t contravariant bas vctors a i. T rfind displacmnt fild is basd on continuity rquirmnts from 2] and follows classical plat/sll assumption σ 33 = 0. In a layr (k), it is xprssd as 5] : u (k) u (k) u (k) 1 (ξ 1, ξ 2, z, t) = µ α 1 v α (ξ 1, ξ 2, t) z v 3,1 (ξ 1, ξ 2, t) + F1 α(k) (z) γα(ξ 0 1, ξ 2, t) 2 (ξ 1, ξ 2, z, t) = µ α 2 v α(ξ 1, ξ 2, t) z v 3,2 (ξ 1, ξ 2, t) + F2 α(k) (z) γα 0(ξ1, ξ 2, t) 3 (ξ 1, ξ 2, z, t) = v 3 (ξ 1, ξ 2, t) (7) wr t is t tim and t classical summation on rpatd indics is usd. In Eq. (7), v i ar displacmnts of a point on t middl surfac and γα 0 is t transvrs sar strain at z = 0, wil Fβ α(k) ar functions of t normal transvrs co-ordinat z dfining t distribution of t transvrs sar strsss troug t ticknss. Ty ar dfind by : F1 1(k) (z) = f 1 (z) + g (k) 1 (z) F1 2(k) (z) = g (k) 2 (z) F2 1(k) (z) = g (k) 3 (z) F2 2(k) (z) = f 2 (z) + g (k) 4 (z) (8) In Eq. (8), t ticknss functions f 1, f 2, g (k) 1,...,g (k) 4 dpnd on cofficints a (k) i, d (k) i, b 44, b 55 and trigonomtric functions as follows : f 1 (z) = f(z) π b 55f (z) f 2 (z) = f(z) π b 44f (z) g i (k) (z) = a i (k) z + d i (k) i = 1, 2, 3, 4. and k = 1, 2, 3,..., N. (9) wit f(z) = πz sin π and f (z) stands for f(z) drivativ wit rspct to z co-ordinat. N rprsnts t numbr of layrs. Ts cofficints ar dtrmind from t boundary conditions on t top and bottom surfacs of t sll, and from t continuity rquirmnts at t layr intrfacs for displacmnts and strsss, s Béakou 2]. From Eq. (7), classical sll modls can b dducd : t classical sll tory (Koïtr tory), calld CST modl wit : f 1 (z) = f 2 (z) = 0 t g k i (z) = 0 t first ordr sar dformation tory (Nagdi tory), calld FSDT modl wit : f 1 (z) = f 2 (z) = z t g k i (z) = 0 Hraftr, t suprscript (k) for u α (k) componnts is omittd in ordr to simplify t finit lmnt dscription. 3

4 3.2 T strain fild Aftr som algbraic calculations, t covariant strain tnsor componnts ar obtaind in t local contravariant basis a i as follows : ɛ = ɛ ij (a i a j ) wit 2ɛ αβ = 1 ( ɛ 0 µ αβ + ɛ 0 βα + Fα(z) ν ɛ 1 νβ + Fβ ν (z) ɛ 1 να + G ν α(z) ɛ 2 νβ + G ν β(z) ɛ 2 να +z { (b λ β 2Hδλ β ) (ɛ0 αλ + F α ν(z) ɛ1 νλ + Gν α (z) ɛ2 νλ )+ (b λ α 2Hδλ α ) ( ɛ 0 βλ + F β ν(z) ɛ1 νλ + Gν β (z) )}) ɛ2 νλ (10) 2ɛ α3 = F ν α (z) γ0 ν wit G ν α(z) = F ν α(z) δ ν α z and F ν α (z) stands for F ν α(z) drivativ wit rspct to z coordinat. By convninc, t following notations av bn introducd in Eq. (10) to caractriz t mcanical ffcts : mmbran strain : ɛ 0 αβ = v α β b αβ v 3 bnding strain 1 : bnding strain 2 : ɛ 1 αβ = β α β ɛ 2 αβ = bλ α v λ β + b λ α βv λ + v 3 αβ transvrs sar strain : γ 0 α = β α + b β α v β + v 3,α wr t notation β stands for t covariant drivation wit rspct to t ξ β curvilinar co-ordinat. 4 T finit lmnt approximation 4.1 T discrt wak form of t boundary valu problm T discrt formulation of t sll boundary valu problm in linar lasticity is dducd from t following functional : a( u, u ) Ω = f( u ) Ω + F( u ) C, u (11) wr Ω is t triangulation of t multilayrd structur and C is t dg of t msd structur. In addition, u is t finit lmnt approximation of t displacmnt fild u givn by Eq. (7) and u is t finit lmnt approximation of t corrsponding virtual vlocity fild u. Linar functions f and F rprsnt t body (including inrtia trms) and surfac loads, actually surfac and lin loads rspctivly, du to t intgration prformd trougout t ticknss in Eq. (11). T suprscript introducd in Eq. (11) indicats t finit lmnt approximation. It is also usd for finit lmnt approximation of t gnralizd displacmnts in Eq. (7), dnotd by v i and θ α wit i = 1, 2, 3 and α = 1, 2. T gomtry of t sll is approximatd by t classical tr nod triangular finit lmnt. 4

5 4.2 T gnralizd displacmnt approximations In a conforming finit lmnt approac, t displacmnt fild, givn by Eq. (7) indicats tat v 3 must b approximatd by a C 1 -continuous function. T otr gnralizd displacmnts v α and θ α av to b dfind in t Sobolv spac H 1 (Ω ). Tos functions must b at last C 0 -continuous. Trfor, Argyris intrpolation for t dflxion and t Ganv intrpolation for t otr gnralizd displacmnts ar usd. Not tat t Argyris intrpolation is xactly of continuity C 1 and t Ganv intrpolation involvs a smi-c 1 continuity wic is not ndd r. Du to vry long xprssions for ts intrpolations, t radr is rfrrd to itr t original paprs 1] and 6] or a rcnt book 3]. T dgrs of frdom associatd wit on finit lmnt in t local curvilinar basis ar givn as : for a cornr nod : v 1 v 1,1 v 1,2 v 2 v 2,1 v 2,2 v 3 v 3,1 v 3,2 v 3,11 v 3,22 v 3,12 (12) θ 1 θ 1,1 θ 1,2 θ 2 θ 2,1 θ 2,2 wil, for a mid-sid nod : v 1 v 1,n v 2 v 2,n v 3,n (13) θ 1 θ 1,n θ 2 θ 2,n wr p,n is t drivativ wit rspct to t normal dirction of t dg lmnt. Tn, aving drivativs in t prvious st of dgrs of frdom (dof), t following mtodology is usd to prscrib kinmatic boundary conditions. For a givn p function wit t condition p(ξ 1 = 0, ξ 2 ) = 0 to satisfy ξ 2, tn t first ordr drivativs bcom : and so on for t scond ordr drivativs. p,1 (0, ξ 2 p(, ξ 2 ) p(0, ξ 2 ) ) = lim 0 0 p,2 (0, ξ 2 p(0, ξ 2 + ) p(0, ξ 2 ) ) = lim 0 = 0 (14) 4.3 T lmntary matrics T stiffnss matrix T lmntary stiffnss matrix K ] is obtaind by computing t bilinar form givn in Eq. (11) at t lmntary lvl as : a( u, u ) Ω = = = /2 Ω /2 ] ( E T /2 Ω Ω ɛ ] T C(k) ] ɛ ] µdz adω /2 E ] T A ] E ] adω = Q ]T K ] Q ] B ] T C(k) ] B ] µdz) E ] adω (15) 5

6 Using t displacmnt fild u in Eq. (7) and t strain componnts in Eq. (10), t matrix B ] can asily b dducd. It is computd by t following rlation : B ] = Ep]Go] (16) In t abov xprssion of matrix B ], Ep] is a matrix containing ticknss functions of t co-ordinat z wras Go] is a matrix including only diffrntial gomtry ntitis suc as mtric tnsor, curvatur tnsor and Cristoffl symbols (s Brnadou 3]). T matrix E ] (idntically for E ] adding t astrisk suprscript), wic may b sn as a gnralizd displacmnts matrix, is givn by : E ] T = v 1 v 1,1 v 1,2. v 2 v 2,1 v 2,2. v 3 v 3,1 v 3,2 v 3,11 v 3,12 v 3,22. θ 1 θ 1,1 θ 1,2. θ 2 θ 2,1 θ 2,2 ] T finit lmnt approximations, dfind at t abov sction 4.2, ar dirctly usd to xprss t matrix E ] as a function of t lmntary dgrs of frdom vctor Q ] (s Eq. (12) and Eq. (13)). Finally, A ] contains t matrial baviour matrix for a multilayrd finit lmnt, rsulting from t intgration wit rspct to t ticknss co-ordinat, and diffrntial gomtry ntitis suc as mtric, curvatur, Cristoffl symbols, Extnsion to t gomtric non linarity T multilayrd structur is now considrd in a Total Lagrangian configuration, so tat its ms is dnotd Ω (0) wit dg C (0) wr (0) indicats t initial (fixd) configuration usd. Lagrangian co-ordinats will raftr b writtn ξ i as prviously. T discrt boundary valu problm statd abov for linar analysis Eq. (11) is trfor formulatd by t following Total Lagrangian functional availabl for non-linar analysis : J( u, u ) Ω(0) = a( u, u ) Ω(0) f( u ) Ω(0) F( u ) C(0) = 0, u (18) Absnc of followd forcs is considrd r. Tus, t main fatur of t non-linar formulation is incorporatd into t virtual intrnal powr and t following xprssion is obtaind : /2 a( u, u ) Ω(0) = ] ɛ T ] C(k) ɛ ] µdz adω (0) (19) Ω(0) /2 All t quantitis in Eq. (19) rfr to t initial (fixd) configuration. Virtual strain rats and strains can b split into tir linar (indx L) and non-linar (indx NL) parts as follows : ɛ ] = ɛ L ] + ɛ NL ] (17) ɛ ] = ɛ L ] + ɛ NL ] (20) T gomtrically non-linar formulation now considrd is basd on Von-Karmann assumptions wr dflxion is modratly larg, wil rotations and strains rmain small. Non-linar virtual strain rat matrix in Eq. (20) is tn givn by : ] T 1 ɛnl = 2 (v 3,1 + b α 1 v α )(v3,1 + b α 1 v α 1 ) 2 (v 3,2 + b α 2 v α )(v3,2 + b α 2 v α ) (v 3,1 + b α 1v α )(v3,2 + b λ 2vλ ) + (v 3,2 + b α 2v α )(v3,1 + b λ 1vλ ) ] (21)

7 and non-linar strain matrix is : ɛnl ] T = 1 2 (v 3,1 + b α 1 v α ) (v 3,2 + b α 2 v α ) 2 (v 3,1 + b α 1v α )(v 3,2 + b λ 2v λ ) 0 0 ] (22) Consistnt linarization procdur From ts last quations, it is vidnt tat Eq. (19) is non-linar wit rspct to displacmnts, and a Nwton algoritm as to b usd to find a numrical solution of Eq. (18). For a Nwton-typ mtod, knowldg of t tangnt stiffnss is rquird and can b drivd using standard linarization procdurs. Applying it to Eq. (18), w obtain : J( u, u ) Ω(0) = J( ū, u ) Ω(0) + D u J( ū, u ). u (23) wit u = ū + u, wr ū rfrs to a known stat. According to Eq. (23), t linarizd form of t functional Eq. (18) is dducd and w now av to find t solution of t following quation : D u a( ū, u ) Ω(0). u = a( ū, u ) Ω(0) + f( u ) Ω(0) + F( u ) C(0), u (24) T lft mmbr of Eq. (24) as to b computd and givs t tangnt oprator, wil otr quantitis in t rigt mmbr ar known vctors as ty dpnd only on t known stat ū T tangnt stiffnss matrix T tangnt stiffnss matrix is now drivd from t lft mmbr of Eq. (24). For an arbitrary finit lmnt Ω (0) of t ms Ω (0), t tangnt oprator is found as : D u a( ū, u ) Ω(0). u = Ω (0) Ω (0) Ω (0) E ] T A ] E ] adω (0)+ E ] T A ( ū ) ] E ] adω (0)+ E ] T A ( σ ) ] E ] adω (0) In Eq. (25), t matrix A ] as bn givn in (s Eq. (15)), as wll as vctors E ] and E ] = E ] from Eq. (17). T matrix A ( ū )] dpnds on matrial proprtis and on bot linar and quadratic known stat ū. Finally, t matrix A ( σ )] is linkd to t in-plan strsss. Following t procdur givn in to driv t linar stiffnss matrix K ], it is asy to comput t tangnt stiffnss matrix, dnotd K T ], so tat : (25) D u a( ū, u ) Ω(0). u = Q ]T K T ] Q ] (26) wr K T ] = K ] + K ( ū ) ] + K ( σ ) ] (27) In Eq. (27), to comput itr matrix K ] or K ( ū ) ] or K ( σ ) ], w us Eq. (15) rspctivly basd itr on A ] or A ( ū )] or A ( σ )]. 7

8 5 Numrical rsults on non linar tst Only buckling and post-buckling analysis ar prsntd r. For bot analysis, numrical simulations ar compard wit rsults issud from xprimnts. Exprimnt and modlisation ar firstly dscribd and rsults ar tn discussd. Exprimnt : Exprimntal conditions ar now dscribd. tst configuration : A rctangular sandwic plat is st in a prssing-macin wic prmits to rac 100kN of comprssion forc. Plat dimnsions ar : 390mm 200mm. T sandwic is mad of 12.7mm nida for cor and rspctivly 0.55mm of laminatd glass fibrs/poxy matrix for skins. A mcanical kncap joining is ralizd in ordr to nsur simply supportd boundary conditions, s Fig. 1. A constant vlocity of 0.9mm/mn for t forc application is satisfid during t tst. Figur 1: Exprimntal buckling and post-buckling. instrumntations and masurs : Critical load, displacmnts and strains of t plat during a tst ar obtaind by masurmnts and will stand for rfrnc valus in numrical simulations. Trfor, snsors and gags av bn arrangd on t plat and on t prssingmacin, s Fig. 1 for displacmnt snsors position. obsrvations : Exprimntal tsts av bn rpatd for a good rprsntation of masurmnts. Disprsions av bn obsrvd during xprimnts du to variations of initial conditions (plat prloadd or not, comprssion paralll or prpndicular to sts dirctions). A rprsntativ tst, rfrncd by tst12, as bn rtaind for latr comparison wit numrical simulations. Simulations : Boundary conditions, ms and mcanical proprtis usd for simulations ar dscribd blow : gomtry: T rctangular plat dscribd abov is modlizd. boundaris conditions and loading: T sandwic panl can rotat (s Fig. 1) from its smallst dgs and is fr of constraints on t otrs. A constant prssur is firstly imposd all ovr t plat to initiat bnding dformd sap. So, incrmntal in plan comprssion load is applid on t two opposit dgs of t panl to simulat post buckling baviour. 8

9 ms : Du to symmtry, only a quartr of t plat panl is modlizd using N = 1 (two lmnts), N = 2 (four lmnts) and N = 4 (sixtn lmnts) mss, s Fig. 2. Convrgnc av bn acivd wit t N = 1 coars ms for t critical buckling load and wit t N = 2 ms for t post buckling displacmnts and strains valus. Figur 2: Discrtisations N = 1, N = 2, N = 4. matrial proprtis : Elastic modulus, in N/mm 2, ar givn in Tab. 1. In tis E 1 E 2 E 3 G 12 G 13 G 23 ν 12 Cor (Nida) Skins (Glass fibr/epoxy matrix) Tabl 1: Matrial proprtis of t sandwic plat. tabl, valus ar issud from xprimntal caractrization xcptd for nida lastic modulus E 1, E 2 and for skins saring modulus G 13 and G 23. A snsibility analysis as provd ts last matrial cofficints ad no influnc. Rsults: First of all, numrical critical buckling load obtaind for t prsnt lmnt compard wit xprimntal valus is givn in Tab. 2. N DOF Critical load Numrical valus Exprimntal valu Tabl 2: Critical load : xprimntal and numrical valus. On t otr and, numrical and xprimntal rsults for t transvrs displacmnt v 3 and t strain componnts ɛ xx = ɛ 11, ɛ yy = ɛ 22 at t plat cntr ar compard in Fig. 3 and Fig. 4. Fig. 3 sows ow t transvrs displacmnt volution is influncd by bnding pr-loading of t plat. T slop of loading in t linar part can b adjustd by varying initial prssur bfor going on post-buckling part. T bst slop corrlation is consrvd for strains prdictions. Howvr bnding pr-loading valu may b, t limit valu is always in good agrmnt wit xprimntal rsults. Discrpancis can b obsrvd during t linar part of t loading, s Fig. 4. Exprimntal rsults sow non linar matrial pnomnon wic occur during tis loading stag. Trfor, linar matrial assumption mployd in t modl is not convnint. Howvr, good valuation of limit valu can b obtaind during t postbuckling non linar part. 9

10 Figur 3: Evolution of transvrs displacmnt v 3 at t plat cntr. Figur 4: Evolution of strains ɛ xx and ɛ yy. Finally, good valuations of global (v 3 transvrs displacmnt) and local valus (ɛ xx and ɛ yy ) can b obtaind for a low cost (only two lmnts ar ncssary in tis cas) using tis nw lmnt. 6 Conclusion In tis papr, a nw six nod multilayrd triangular finit lmnt as bn prsntd to analys t baviour of composit laminatd and sandwic slls. Tis nw lmnt : is basd on a rfind kinmatic wic prmits to satisfy intrlaminar continuity for bot displacmnts and transvrs sar strsss, and boundary conditions at t top and bottom surfacs of t sll or plat, uss ig ordr polynomia to intrpolat t gnralizd displacmnts : fild compatibility for transvrs sar strains is tn assurd and locking pnomnon is avoidd. Buckling and post-buckling analysis of a sandwic plat av bn xprimntd and simulatd using t prsnt lmnt. Comparisons btwn numrical and xprimntal valus av bn stablisd and good rsults av bn obtaind for bot critical load and strains volutions. Our futur work points towards invstigations on non linar constitutiv law in ordr to rcovr xprimntal rsults prsntd in prvious sction. Furtrmor, som non linar static, buckling and post-buckling tsts on multilayrd slls will b considrd. 10

11 Rfrncs 1] Argyris, J.H., Frid, I., Scarpf, D.W. T TUBA family of plat lmnts for t matrix displacmnt mtod. Aro. J. Royal Aronaut. Soc., 72, (1968). 2] Béakou, A., Touratir, M. A rctangular finit lmnt for analysing composit multilayrd sallow slls in statics, vibration and buckling. Int. Jour. Num. Mt. Eng., 36, (1993). 3] Brnadou, M. Finit Elmnt Mtods for Tin Sll Problms. Jon Wily and Sons (1996). 4] Carrra, E. Toris and Finit Elmnts for Multilayrd, Anisotropic, Composit Plats and Slls. 9(2), (2002). 5] Dau, F. A doubly curvd C 1 finit sll lmnt basd on a rfind modl for multilayrd/sandwic sll structurs. Tès d Doctorat, ENSAM Enginring Scool, Paris-Franc (2004). 6] Ganv, H.G., Dimitrov, Tc.T. Calculation of arc dams as a sll using an IBM-370 Computr and curvd finit lmnts. In Tory of slls, pp Nort- Holland, Amstrdam (1980). 7] Touratir, M. A rfind tory of laminatd sallow slls. Int. J. Solids Struc., 29(11), (1992). 11