An adaptive Strategy for the Multi-scale Analysis of Plate and Shell Structures with Elasto-plastic Material Behaviour

Transcription

1 TECHNISCHE MECHANIK, 36, 1-2, (2016), submittd: Sptmbr 7, 2015 An adaptiv Stratgy for th Multi-scal Analysis of Plat and Shll Structurs with Elasto-plastic Matrial Bhaviour W Wagnr, F Gruttmann Modlling of structurs on diffrnt scals has bn a popular subjct in th past Within such a stratgy th structural bhaviour is modld on a macro-lvl, dscribing th structur itslf, whras th matrial bhaviour is modld on a micro-lvl Hr typically RVEs ar usd Th propr choic of boundary conditions for th RVE is a difficult task in cas of shll structurs It should b mntiond that th corrct calculation of matrial paramtrs on th macro lvl is crucial for any associatd nonlinar analysis Hr, rsults hav bn prsntd for homognous and layrd structurs for composit matrials in Gruttmann and Wagnr (2013) In th prsnt papr w will discuss th influnc of matrial nonlinar bhaviour, hr th lasto-plastic bhaviour, within th abov dscribd stting Typically ths calculations ar vry tim consuming, vn if th FE-modl is paralllizd Thus w will discuss possibilitis to chang matrial modls on th structural modl during th loading procss, starting with lastic matrial modls without a scond scal and switching to a two-scal approach, whr ncssary 1 Introduction Finit shll lmnts which ar basd on th first ordr shar dformation thory ar in gnral abl to dscrib th global dformation bhaviour of thin plat and shll structurs Howvr for som strss componnts only an avrag shap through th thicknss can b obtaind Various mthods hav bn dvlopd to obtain th complicatd local dformation bhaviour in inhomognous thin structurs So calld multi dirctor shll formulations with an appropriat numbr of global dgrs of frdom at th nods yild approximat solutions of th thr dimnsional boundary valu problm, g Rddy (2004) Th application of brick lmnts or solid shll lmnts provids likwis a computationally xpnsiv approach, g Klinkl t al (1999, 2006) For laminats ach layr must b discrtizd with svral lmnts in thicknss dirction to obtain satisfactory rsults Th numrical ffort for such a full-scal solution lads for practical problms to an unrasonabl numbr of unknowns Th nhancmnt of th displacmnt fild by layr-wis linar (zig-zag) functions through th thicknss, s g Carrra (2003), could b anothr option Applications for thin shll structurs with a complicatd 3D-strss stat could b photovoltaic laminats, OLED, PLED dvics, picoltric dvics, thin CFRP-parts of lightwight structurs in arospac or automotiv industry, among many othrs In th prsnt papr th shlls ar tratd as a homognous continuum with ffctiv proprtis obtaind through a homognization procdur to avoid larg-scal computations A larg numbr of paprs xists on computational homognization mthods for gnral htrognous matrials, s g Zohdi and Wriggrs (2005); D Borst and Ramm (2011) for a survy and nw dvlopmnts Computational homognization procdurs for thin structurd shts hav bn proposd in Grs t al (2007); Conn t al (2010) Th thory in Grs t al (2007) is basd on a Rissnr Mindlin kinmatic, whras in Conn t al (2010) a Kirchhoff-Lov kinmatic is adoptd Rprsntativ volum lmnts (RVE) xtnding through th full thicknss of th structur ar introducd At th top and bottom surfacs of th RVE strss boundary conditions ar applid, whras priodicity constraints ar applid at th latral surfacs Basd on ths prliminaris th ssntial faturs and nw aspcts of th prsnt formulation ar summarizd as follows: (i) Th undrlying shll formulation basd on th Rissnr Mindlin thory with inxtnsibl dirctor fild is summarizd A two-scal modl is introducd and a variational formulation and associatd linarization for th coupld global local boundary valu problm is prsntd 142

2 (ii) For th solution of th two-scal problm a FE 2 mthod for small strains is dscribd Quadrilatral lmnts ar usd for th discrtization of th shll structur, whras solid shll lmnts ar applid on th RVE, which xtnds through th total thicknss of th shll A rlation btwn in plan displacmnts and shll strains is dvlopd on th latral surfacs (iii) Th nonlinar coupld local and global boundary valu problms ar simultanously solvd in a Nwton itration schm (iv) An indicator, basd on idas in Zinkiwicz and Zhu (1987), is dvlopd which allows th switch btwn diffrnt matrial modls Thus th FE 2 mthod is only usd, whr it should b ncssary Th approach of changing matrial modls is fully rvrsibl Thus, unloading is possibl 2 Two-scal Shll Modl Thory Ω h σ D ε h - h + F h B B Figur 1 Computational homognization of a layrd shll Lt B b th thr dimnsional Euclidan spac occupid by a shll with thicknss h in th rfrnc configuration With ξ i w dnot a convctd coordinat systm of th body Th thicknss coordinat ξ 3 = z is dfind in th rang h z h +, whr h and h + ar th z coordinats of th outr surfacs Thus, an arbitrary rfrnc surfac Ω with boundary Γ is introducd Th shll is loadd statically by loads ˉp in Ω and by boundary forcs ˉt on Γ σ In th following Grk indics rang from 1 to 2 and commas dnot partial diffrntiation with rspct to ξ α Position vctors of th initial rfrnc surfac and currnt surfac ar dnotd by X(ξ α ) and x(ξ α ), rspctivly Furthrmor, a dirctor ˉD(ξ α ) with ˉD(ξ α ) = 1 is introducd as a vctor fild prpndicular to Ω Th unit dirctor fild ˉd(ξ α ) of th currnt configuration is obtaind by orthogonal transformations and is a function of th rotational paramtrs ˉω Within th Rissnr Mindlin thory transvrs shar strains ar accountd for, thus ˉd x, α 0 Hnc, th displacmnt fild follows from th diffrnc of th position vctors in shll spac ū = ū 0 + z (ˉd ˉD) ū 0 = x X (1) Th shll strains ar drivd from th Grn Lagrangian strain tnsor using kinmatic assumption (1) and ar arrangd in a vctor as ε(ū 0, ˉω) = ε 11, ε 22, 2 ε 12, κ 11, κ 22, 2 κ 12, γ 1, γ 2 T, (2) with th mmbran strains ε αβ, curvaturs κ αβ and transvrs shar strains γ α, rspctivly According to Fig 1 a rprsntativ volum lmnt (RVE) at an intgration point i of a typical finit shll lmnt is introducd Th domain B i xtnds through th total thicknss h of th shll and has th siz l x l y h 143

3 Th displacmnt fild is split in an avragd part ū and a fluctuation part ũ u = ū + ũ (3) Th avragd displacmnts ū according to (1) ar a linar function of th thicknss coordinat, whras ũ dscribs warping and thicknss chang Hnc, th dformation gradint F = 1 + Grad u is dfind in a standard way and th Grn Lagrangian strain tnsor follows from E = 1 2 (FT F 1) Th wak form of quilibrium can now b writtn with v = ū 0, ˉω, u T and admissibl variations δv = δū 0, δ ˉω, δu T g(v, δv) = Ω (σ δε ˉp δū 0 ) da n2 ˉt δū 0 ds + Γ =1 σ NGP i=1 1 A i Ω i h + h S : δe ˉμ dz da = 0 (4) Hr, σ dnots th vctor of strss rsultants σ = n 11, n 22, n 12, m 11, m 22, m 12, q 1, q 2 T (5) with mmbran forcs n αβ = n βα, bnding momnts m αβ = m βα and shar forcs q α On th RVE s S dnots th Scond Piola-Kirchhoff strss tnsor with P = F S and th virtual Grn-Lagrangian strain tnsor is introducd via δe = 1 2 (δft F + F T δf) Furthrmor n1 and n2 dnot th numbr of shll lmnts without or with a two-scal modl introducd NGP is th numbr of Gauss points for ach lmnt and A i = l x l y is th rfrnc ara of th RVE It holds for th total numbr of shll lmnts: n = n1 + n2 For th finit lmnt formulation of th nxt sction w nd to driv th linarization of q (4) With consrvativ loads ˉp and ˉt on obtains L g(v, δv), Δv := g(v, δv) + Dg Δv (6) whr g(v, δv) is givn in (4) and Dg Δv = Ω n2 (Δσ δε + σ Δδε) da + =1 NGP i=1 1 A i Ω i h + h (ΔS : δe + S : ΔδE) ˉμ dz da (7) with Δσ = D Δε, ΔS = C ΔE and ΔδE = 1 2 (δft ΔF + ΔF T δf) Th matrial matrix C is a standard output of a library of constitutiv laws in a matrial dscription Th linarizd virtual shll strains Δδε ar drivd for finit rotations in Wagnr and Gruttmann (2005) Th strss rsultant vctor σ and th matrix of linarizd strss rsultants D ar spcifid in th nxt sction 3 Two-scal Shll Modl Finit Elmnt Formulation W dscrib a finit lmnt formulation basd on a standard displacmnt mthod applying th isoparamtric concpt Th rfrnc surfac of th shll is discrtizd with n quadrilatral isoparamtric shll lmnts n1 n2 Ω h = Ω + Ω, =1 =1 (8) whr th subscript h rfrs to th finit lmnt approximation Initial gomtry, displacmnts and rotations ar intrpolatd with bilinar functions N I (ξ, η) which ar arrangd in th matrix N Th nodal dgrs of frdom ar thr displacmnts and two or thr rotations 144

4 Insrting th intrpolation functions for th displacmnts and virtual displacmnts into th linarizd wak form (6) considring (4) and (7) yilds n1 L g(v h, δv h ), Δv h = δv G k G Δv G + f G =1 δv G T k G δv 1 n2 + 0 K L =1 δv i K L i δv NGP K L NGP Δv G ΔV 1 ΔV i ΔV NGP + f G (σ i ) F L 1 F L i F L NGP (9) Th indics G and L rfr to th global and local boundary valu problms, rspctivly Th matrics of th first row in (9) follow from th global part of th linarizd wak form Th lmnt rsidual vctor and th tangntial lmnt stiffnss matrix rad f G (σ i ) = (B T σ N T ˉp) da N T ˉt ds k G (D i ) = (B T D B + G) da (10) (Ω ) (Γ σ) (Ω ) whr th matrics B and G ar drivd in Wagnr and Gruttmann (2005) Th vctor of strss rsultants σ i and linarizd strss rsultants D i ar spcifid blow Th matrics of th scond to th last row in (9) ar associatd with th local boundary valu problms at Gauss points 1 i NGP of shll lmnt and occur only, if a two-scal modl is usd A local boundary valu problm can b dfind at Gauss point i δv T i (K L i ΔV i + F L i ) = 1 N δv T (k L Δv + f L ) (11) A i =1 Hr, th total numbr of lmnts usd for th discrtization of th RVE is dnotd by N Th lmnt rsidual vctor f L and th tangntial lmnt stiffnss matrix k L rad f L = B T S dv k L = ( B T C B + G) dv (12) (V ) (V ) whr B and G ar th virtual strain displacmnt matrix and th gomtrical matrix of 8 nodd solid shll lmnts, rspctivly Th lmnt displacmnt vctor v is now split in a part v Ω which contains th intrnal displacmnts and a part v Γ which contains th boundary displacmnts of th RVE v = vω v Γ = va v b = a V i A ε i In Eq (13) a is th standard assmbly matrix and A (x, y, z) will b spcifid in th following (13) Assuming small strains th rlation of th boundary displacmnts to th avragd strains Ē is writtn as ūx ū y ū z = Ē 11 Ē 12 Ē 13 Ē 21 Ē 22 Ē 23 Ē 31 Ē 32 Ē 33 x y z (14) 145

5 Insrting th rlation of th avragd strains to th shll strains Ē 11 = ε 11 + z κ 11 Ē 22 = ε 22 + z κ 22 Ē 33 = 0 Ē 12 = Ē21 = ε 12 + z κ 12 (15) 2 Ē13 = 2 Ē31 = 2 ε 13 = γ 1 2 Ē23 = 2 Ē32 = 2 ε 23 = γ 2 into (14) yilds with furthr considrations, s Gruttmann and Wagnr (2013), ūx = ū y ε11 + zκ 11 ε 12 + zκ 12 2 ε 13 ε 12 + zκ 12 ε 22 + zκ 22 2 ε 23 x y z (16) Eq (16) is now rwrittn with th vctor of shll strains (2) as ūx ū y = x y xz y z z y 2 x 0 yz 1 2 x z 0 z ū I = A I (x, y, z) ε, ε 11 ε 22 2ε 12 κ 11 κ 22 2κ 12 γ 1 γ 2 (17) whr th indx rfrs to nod I of th considrd lmnt Th matrics A I ar submatrics of A introducd in (13) A = δ 1 A 1 δ I A I δ nl A nl (2 nl 8) with nl = 8 for 8 nodd solid shll lmnts { 1 if nod I has fixd dofs δ I = 0 ls (18) Introducing k αβ and f α with α, β = a, b as submatrics of k L and f L in (11) lads to δv T i (KL i ΔV i + F L i ) = 1 N δvi A i δε i =1 T { a T k aa a a T k ab A A T k ba a A T k bb A ΔVi Δε i a T f a + A T f b } (19) = 1 A i δvi δε i T { K L L T M ΔVi Δε i + Fa F b } With δv i 0 th intrnal dgrs of frdom ΔV i can b liminatd from th st of quations using K ΔV i + L Δε i + F a = 0 (20) which yilds ΔV i = K 1 (F a + L Δε i ) (21) 146

6 With (20) and (21) q (19) rducs to δvi T (KL i ΔV i + F L i ) = δεt i (D i Δε i + σ i ) (22) whr σ i = 1 (F b L T K 1 F a ) D i = 1 (M L T K 1 L) (23) A i A i ar th strss rsultants and linarizd strss rsultants of Gauss point i Finally (22) is insrtd into th linarizd coupld global-local boundary valu problm (9) n1 L g(v h, δv h ), Δv h = δv G k G Δv G + f G =1 δv G T k G (D i ) δε 1 n2 + 0 D =1 δε i D i δε NGP D NGP Δv G Δε 1 Δε i Δε NGP + f G (σ i ) σ 1 σ i σ NGP (24) Th coupld nonlinar systm of quations is simultanously solvd within a Nwton itration schm Th itration is trminatd for th actual load stp whn local quilibrium in all Gauss points is attaind along with th global quilibrium of th shll which is formulatd through th first row of (9) or (24) For furthr dtails, s Gruttmann and Wagnr (2013) 4 Adaptiv Stratgy for th us of diffrnt matrial modls As alrady said th calculation of structurs with a two-scal modl is vry tim consuming Thus an intrsting stratgy could b to us th sam 2D nonlinar shll lmnt togthr with diffrnt matrial laws at diffrnt load lvls Within this papr w hav in mind Non layrd shll structur with linar lastic matrial law-mat1 (EL) Layrd shll structur with lasto-plastic matrial law-mat4 (ELPL) ( only for tst rasons) Non layrd shll structur with a two-scal FE 2 -modl on lmnts 0 n2 with RVE on ach Gauss point-mat8 (RVE + ELPL) Thus a calculation starts fully lastic with MAT1 (EL) on a non layrd structur During th loading procss a switch to a layrd structur with an lasto-plastic matrial-mat4 (ELPL) or to a RVE with lasto-plastic matrial- MAT8 (RVE + ELPL) is introducd This switching procdur rquirs an indicator to dcid if a matrial chang is ncssary In arlir tims th indicator of Zinkiwicz and Zhu (1987) has bn succssfully usd in diffrnt applications for an adaptiv msh rfinmnt A typical formulation is ΔS T ΔS da α E σ, (25) whr E σ is an avragd systm nrgy basd on FE-strsss S GP calculatd at Gauss points E σ = 1 n n =1 S T GP S GP da (26) Basd on a standard FE-smoothing procdur nodal strsss S N could b providd, which can b usd to calculat strss diffrncs ΔS = S N S GP on ach lmnt If th first trm in Eq (25) is largr thn th scond on a 147

7 msh rfinmnt will b ncssary Typical α = 005 can b usd, which mans that an rror of 5% is admissibl in on lmnt In th sam sns on could introduc similar xprssions within a nonlinar lasto-plastic modl A first attmpt is th simpl modl S 2 V da α E σ and E σ = 1 n n =1 Y 2 0 da (27) with a yild strss Y 0 and th quivalnt strss S V = S11 2 S 11 S 22 + S S2 12 Now, if th first trm xcds th scond on a chang of th matrial modl will b ncssary Typical α = 09 is a possibl choic, which mans th lastic domain is admissibl in on lmnt up to 90% of th yild strss Th matrial modl is changd for th whol lmnt, if th abov dscribd condition is violatd onc during th loops on layrs and Gauss points In cas of th lasto-plastic matrial, strsss ar calculatd naturally at ach layr Using th two-scal modl th output of th RVE ar strss rsultants, s Eqs (5, 23) Thus th strss valus at top and bottom of th shll hav to b calculatd in a standard way via S ii+ = n ii h + m ii I i z +, S ii = n ii h + m ii I i z (28) Using th abov stratgy in a paralllizd FE-cod nds a dynamic paralllization of th lmnt loop Thus a balancing of th workload of ach procssor is ncssary du to th fact that lmnts with lastic matrial bhaviour as wll as lmnts with anothr lasto-plastic FE-modl on ach Gauss-point occur LOOP, Load LOOP, Itration(Nwton) LOOP, Elmnt (Stiffnss matrix and rsidual) P-01 P-02 P-03 N-Procssors in paralll du to work load NEXT, Elmnt SOLVE K T ΔV = R with Paralll-Solvr NEXT, Itration NEXT, Load Furthr informations on th paralllization of an lmnt loop in a FE-cod may b found g in Jarzbski t al (2015) 5 Unloading Spcial attntion has to b st on possibl unloading bhaviour Typically th procss of changing th matrial modl should b rvrsibl Thus a switch back to a fully lastic matrial modl must b possibl Having in mind that lasto-plastic strains hav bn rachd arlir a gnral matrial modl σ p = D(ε ε p ) for th strss rsultants is ncssary Hnc th qustion is how to calculat lasto-plastic shll strains ε p and lasto-plastic strss rsultants σ p For th lasto-plastic matrial- MAT4 (ELPL) on can calculat th strss rsultants σ p in a loop ovr all layrs 148

8 at ach Gauss point from th 3D-strsss S p σ p = 0, D p = 0 i = 1, nlay E i = ε m + z i κ S i p = C i p E i σ p = σ p + S i p dz i z D p = D p + C i p dz i nd i z stor for i : E i p, E i v (29) In cas of th two-scal modl-mat8 all shll valus ar calculatd dirctly on th RVE at ach Gauss point ε = ε m, κ, γ σ p = N p, M p, Q p stor on RVE: E p, E v D p (30) Thus, in cas of unloading, th lasto-plastic strains ε p = ε D 1 p σ p (31) ar stord in a history array and th switch to th fully lastic matrial-mat1 can b don via σ = D(ε ε p ) (32) 6 Exampls Th dvlopd algorithms ar implmntd in an xtndd vrsion of th gnral finit lmnt program FEAP Taylor (2011) With th first xampl w compar th finit lmnt solutions for a Two-span girdr subjctd to concntratd loads with analytical xprssions Th sam is don within th scond xampl for a squar plat 61 Two-span girdr with singl loads For th first xampl w choos a simpl Two-span girdr with singl loads Th lngth is givn with L = 100 cm and th cross sction data ar B = 10 cm, H = 2 cm Th xtrnal loading is givn by two 2 singl loads F = f B = 1 kn For th matrial law w introduc a classical lasto-plastic matrial bhaviour including linar hardning W choos th lasticity modulus E = kn/cm 2, Poisson s ratio ν = 03, initial yild strngth Y 0 = 30 kn/cm 2 and th hardning modulus C p = E/10000 kn/cm 2 An analytical solution (without hardning) including unloading is availabl on th basis of a plastic hing thory Th distribution of th bnding momnts in th ultimat load cas is prsntd in Figur 2 with 3 plastic hings Figur 2 Bnding momnts of 2-span girdr with singl loads Th associatd finit lmnt discrtization is prsntd in Figur 3, taking into account that symmtry conditions could b usd Th msh is chosn with 40 4-nod shll lmnts in lngth dirction and 4 lmnts in transvrs dirction, rspctivly Th lmnt formulation is basd on a finit rotation thory including an intrfac to diffrnt 2D- and 3D-matrial formulations, for dtails s Wagnr and Gruttmann (2005) 149

9 L = 100 f = 01 L = 100 B=10 Figur 3 FE-discrtization of th symmtric part 2-span girdr with singl loads In Figur 4 load dflction curvs ar prsntd Ths ar dpictd for th singl load vrsus th vrtical displacmnt undr th singl load Curvs ar shown basd on an analytical solution from th plastic hing bam thory with F pht = Y 0 (BH 2 /6L)(4 + 1) = 9 kn First rsults ar from a FE-modl shll with lasto-plastic matrial bhaviour (MAT4) and 11 layrs in thicknss dirction Scond rsults ar from a shll formulation with a multiscal FE 2 approach (MAT8), s Gruttmann and Wagnr (2013), dscribing th macroscopic bhaviour on th shll modl and th lasto-plastic matrial bhaviour (MAT4) on a rprsntativ volum modl (RVE), introducd at ach Gauss point on th shll lvl Th RVE has a siz of l x = l y = h and a discrtization of solid shll lmnts, s Klinkl t al (2006) Finally, for comparison, w prsnt rsults introducing a 3D FE-modl, again with th solid shll lmnt mntiond bfor Hr, w choos for th 3D-msh 11 lmnts in thicknss dirction All rsults ar vry clos togthr in th lastic rang Du to diffrnt gomtrical discrtization modls slightly diffrnt rsults occur, spcially in th plastic rang, which lads to th sam unloading bhaviour starting from diffrnt points Singl Load kn A n a l y t c i a l 2D - S h l l M A T 4( E L 2D - S h l l M A T 8( F E 3D - S h l l M A T 4( E L Displacmnt undr load cm Figur 4 Load dflction curvs for diffrnt modls without changing th matrial modl Figur 5 dpicts two load dflction curvs whn using th chang of matrial modls basd on th abov introducd rror indicator Ths curvs ar on on hand for th switch btwn lastic(mat1) and lasto-plastic matrial bhaviour (MAT4) and on th othr hand for th switch btwn lastic (MAT1) and lasto-plastic matrial bhaviour (MAT4) on a rprsntativ volum modl (RVE) Both curvs ar in prfct agrmnt with load dflction curvs without changing th matrial modls Prsnting th unloading bhaviour dmonstrats, that th matrial switching procdur holds in both dirctions Th distribution of matrial modls on th FE-msh is visualizd in Figur 6 Hr, w can s finit lmnts with lastic(rd) and lasto-plastic matrial(blu) on th 150

10 RVE at load lvls for F=6/7/8/9/925/95/925/90 kn Singl Load kn D - S h l l M A T 1-4( E 2D - S h l l M A T 4( E L 2D - S h l l M A T 1-8( F 2D - S h l l M A T 8( F E Displacmnt undr load cm Figur 5 Load dflction curvs for diffrnt modls with chang of th matrial modl Matrial distributio Matrial 1 Matrial 2 Matrial distribution Matrial distribution Matrial distribution Matrial distributio Matrial 1 Matrial 2 Matrial 1 Matrial 2 Matrial 1 Matrial 2 Matrial 1 Matrial 2 Figur 6 Elmnts with lastic (rd) and lasto-plastic matrial (blu) at diffrnt load lvls for F=6/7/8/9/925/95/925/90 kn (top to bottom, lft to right) 62 Squar plat with uniform load With th scond xampl w discuss th lasto-plastic bhaviour of a simply supportd squar plat with uniform load, s Figur 7 Th lngth is givn with L x = L y = 200 cm and th thicknss is h = 4 cm Th xtrnal uniform load is givn by q = 1 N/cm 2 For th matrial law w introduc again a classical lasto-plastic matrial 151

11 bhaviour including linar hardning with matrial paramtrs of th last xampl, but with an initial yild strngth of Y 0 = 36 kn/cm 2 q L y L x Figur 7 Simply supportd squar plat with uniform load For th associatd finit lmnt discrtization a msh with 40 4-nod shll lmnts, is chosn in both dirctions Possibl symmtry conditions ar not takn into account In Figur 8 load dflction curvs for th valu of th uniform load vrsus th vrtical cntr displacmnt ar prsntd An analytical solution (without hardning) is availabl on th basis of a yild lin thory with q ylt = 6Y 0 (h/l x ) 2 = 864 N/cm 2 First rsults ar from a FE-modl shll with lasto-plastic matrial bhaviour (MAT4) and 11 layrs in thicknss dirction Scond rsults ar from a FE-modl shll with a multi-scal FE 2 approach (MAT8) dscribing th macroscopic bhaviour on th shll modl and th lasto-plastic matrial bhaviour (MAT4) on a rprsntativ volum modl (RVE), introducd at ach Gauss point on th shll lvl Th RVE has a siz of l x = l y = h and a discrtization of solid shll lmnts Finally, for comparison, w prsnt 3D-rsults using th solid shll lmnt Hr, w choos for th 3D-msh 5 lmnts in thicknss dirction Associatd lmnt typs hav bn dscribd in th xampl bfor All rsults ar vry clos togthr in th lastic rang Du to diffrnt gomtrical discrtization modls slightly diffrnt rsults occur, spcially in th plastic rang, which lads to th sam unloading bhaviour starting from diffrnt points Th diffrnt us of matrial modls on th FE-msh is visualizd in Figur 9 Th distribution of finit lmnts with lastic (rd) and lasto-plastic matrial (blu) on th RVE is prsntd at load lvls q=40/50/60/80/825/875/825/80/775 N/cm 2 (top to bottom, lft to right) Th typical bhaviour with plastic ffcts along th diagonals ( yild lins ) can b sn clarly uniform load q N/cm D - S h l l M A T 1-4( E L P 2D - S h l l M A T 1-8( F E 2) 3D - S h l l M A T 4( E L P L ) A n a l y t i c a l cntr displacmnt cm Figur 8 Load dflction curvs for diffrnt discrtizations and matrial modls 152

12 7 Conclusions In this papr a multi-scal approach for shll structurs is prsntd With rspct to shlls th introducd RVE on th lmnt Gauss point has a natural thicknss h This rquirs th formulation of spcial boundary conditions Th total st of quations contain quilibrium formulations on th structural lvl as wll as on th lvl of Gauss points It is possibl to solv th whol st of quations simultanously Basically th formulation allows th mixtur of lmnt concpts W hav shown that it is possibl to mix lmnts with convntional matrial dscription with lmnts, whr RVE s ar introducd to dscrib th matrial bhaviour Calculations basd on a multi-scal approach ar vry tim consuming Thus an adaptiv stratgy has bn introducd to switch automatically btwn diffrnt discrtization and matrial modls For this an indicator in dpndnc on th indicator of Zinkiwicz and Zhu (1987) has bn proposd Th xampls show th fficint practical applicability of th proposd mthod, also in th unloading rgim Furthr rsarch may b focusd on th formulation of indicators to switch btwn diffrnt matrial modls Matrial distribution Matrial 1 Matrial 2 Matrial distribution Matrial 1 Matrial 2 Matrial distribution Matrial 1 Matrial 2 Matrial distribution Matrial 1 Matrial 2 Matrial distribution Matrial 1 Matrial 2 Matrial distribution Matrial 1 Matrial 2 Matrial distribution Matrial 1 Matrial 2 Matrial distribution Matrial 1 Matrial 2 Figur 9 Elmnts with lastic (rd) and lasto-plastic matrial (blu) at diffrnt load lvls for q=40/50/60/80/825/875/825/80/775 N/cm 2 (top to bottom, lft to right) 153

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