Development of solid-shell elements for large deformation simulation and springback prediction

Transcription

1 Faculté ds Scincs Appliqués Anné Académiqu 8-9 Dvlopmnt of solid-shll lmnts for larg dformation simulation and springback prdiction ravail présnté par Nhu Huynh NGUYEN Ingéniur mécanicin pour l obtntion du grad d Doctur n Scincs d l Ingéniur Sptmbr 9

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3 o my mothr for hr sacrifics to sons o my wif for hr undrstanding and ncouragmnt o my sincr frinds for thir prcious hlps

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5 Acknowldgmnts All acknowldgmnts to th hlp I hav rcivd whil complting this thsis ar worthlss. First of all I would lik to thank th financial support from Scholarship Foundation of Vitnams Govrnmnt; without that financial support for yar in Vitnam and.5 yars in Blgium I would not hav a chanc of doing this thsis. I would lik to xprss my dp motion to my family in Vitnam and to my Vitnams frinds in Blgium. hanks to thir cars I always flt th familial air narby. Spcial thanks ar dvotd to Dr. Quoc Vit BUI. I dply thank Prof. M. HOGGE and Prof. J.P. PONHO for thir prcious supports and thir patinc. I also thank all th mmbrs of th Rsarch am in M& Dpartmnt Univrsity of Lièg for thir dirct or indirct supports. Finally I would lik to thank th jury mmbrs of my thsis for thir participation in rading th thsis.

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7 NOAIONS AND SYMBOLS Oprator D X x [ X ] X { } Maning Diffrntial oprator Scalar Vctor with componnts x i Matrix olumn vctor dx / X im drivativ of X ( dt X ranspos of X ( x Sgn + or sign of scalar x x x = x if x > ; x = if x < X : Y Doubl contractd product of X with Y X I First invariant of X : X I = r( X X II Scond invariant of X : X II = [ r ( X r ( X ] X hird invariant of X : X = dt( X III X' Dviator of X : X' = X - X I I δ Kronckr dlta δ ij = if i=j δ ij = if i j X ~ A quantity with tilt bar is assumd in natural (isoparamtric coordinat systm or physical coordinat systm III Scalar Maning A u Boundary along which displacmnts ar spcifid A σ Boundary along which surfac tractions ar spcifid L ontour lin π Functional µ λ Lamé cofficints ρ Mass dnsity ν Poisson ratio G Shar modulus γ ij Shar strain A Surfac of a body t im V Volum of a body W Work W Stord nrgy function of a hyprlastic matrial S(E i

8 st ordr Maning nsor i Basic unit vctor b Body forc pr unit mass u urrnt displacmnt vctor F xt Extrnal forc vctor F int Intrnal forc vctor X ( X X X Matrial coordinat vctor Natural (isoparamtric coordinat vctor ( h h U Nodal displacmnt vctor U = [ u v w... un vn wn ] n Outward normal vctor to th surfac A x ( x x x Spatial coordinat vctor t * Spcific tractions along A σ u * t Spcific displacmnts along A u raction vctor strss vctor nd and highr Maning ordr nsor ε auchy s strain tnsor σ auchy s strss tnsor F Dformation gradint tnsor Grad U Matrial displacmnt gradint tnsor Grad U = U grad u Spatial displacmnt gradint tnsor grad u = u P First Piola-Kirchhoff strss tnsor E Grn-Lagrang tnsor E Grn-Lagrang vctor (Voigt notation: E = [ E E E E E E] J Jacobian matrix b Lft auchy-grn tnsor (Fingr dformation tnsor v - Lft strtch dformation tnsor v = FR 4 onstitutiv moduli tnsor 4 onstitutiv moduli matrix dimnsion (6x6 Right auchy-grn tnsor = F F U - Right strtch tnsor U = R F R - Rotation tnsor R = FU S Scond Piola-Kirchhoff strss (PK tnsor S Scond Piola-Kirchhoff strss vctor (Voigt notation: S = [ S S S S S S] K angnt stiffnss matrix I Unity scond ordr tnsor b = FF ii

9 FIGURE LIS.. onnction of solid-shll lmnt (whit color with standard solid lmnt (gry color..... Hxahdral solid lmnt Pur bnding of a rctangl..... Errors in strsss in pur bnding Dgnration of a D solid lmnt into a dgnratd shll lmnt ANS mthod illustration spcial cas: X-Y aligns Mid-surfac of lmnt in isoparamtric coordinats Mmbran patch tst Eignvalu analyss of lmnt stiffnss matrics in-plan bnding mod ircular cantilvr urvd cantilvr bam: displacmnts Scordlis-Lo roof Scordlis-Lo roof: convrgnc of finit lmnt solution Rgular block... 5 A ubic patch tst... 5 A Strss in patch lmnts Four nod lmnt urvd bam lmnt Initial configuration Ω currnt configuration Ω and isoparamtric t configuration Ω onfiguration of low-ordr solid-shll lmnt Dgnration from D to D of ANS-solid lmnt Solid lmnt spcial cas (XYZ ( Pur bnding in X dirction Sampling points for ANSn lmnt Sampling points for D.ANSn lmnt Distortd lmnt - sampling points for strains Solid-shll algorithm Static condnsation algorithm [SIM9] Rgular and distortd cubs wo dformation mods Squar plat wo bnding mods wo warping mods of EASv6s & Q Squar plat gomtry Displacmnt vrsus applid forc for fully clampd plat Displacmnt vrsus applid forc for simply supportd plat antilvr undr pur bnding wistd bam wistd bam cas of load along Z wistd bam cas of load along Y Squar plat undr uniformd prssur A quartr of th plat - distortd msh Pinchd cylindr with two rigid nd diaphragms iii

10 4..8 onvrgnc invstigation for th pinchd cylindr Morly sphrical shll onvrgnc of finit lmnt solution hick-walld cylindr antilvr bam Displacmnt of tst point Load-dflction curvs for displacmnts at points A(u and B(v Dformd hmisphr at F = 5 (without any magnification Slit annular plat - initial configuration h dformd configuration at maximum load (without any magnification Load-dflction curvs for displacmnts at points A and B Schmatic diagram of multiplicativ dcomposition Finit strain thory Matrial strss updat algorithm antilvr at larg lasto-plastic dformation EAS9 (MEAFOR rsults with various lmnts along thicknss EAS (MEAFOR rsults with various lmnts along thicknss ANS rsults with various lmnts along thicknss SS7n rsults with various lmnts along thicknss ANS lmnt - σ xz (msh ANSn lmnt - σ xz (msh SS7 lmnt - σ xz strss (msh SS7n lmnt - σ xz strss (msh EAS9 lmnt - σ xz strss (MEAFOR msh Pinchd cylindr EAS lmnt (MEAFOR Pinchd cylindr: Forc-Displacmnt SS7 lmnt - σ xz (msh SS7n lmnt - σ xz (msh EAS lmnt - σ xz (EASMEAFOR 6 6 (max. valu has bn imposd to EAS lmnt - σ yz (EASMEAFOR 6 6 (min. valu has bn imposd to SS7 lmnt - σ yz (msh SS7n lmnt - σ yz (msh a Rfrnc - σ xz (EASMEAFOR b Rfrnc - σ xz (EASMEAFOR 4 4 (maximum valu has bn imposd to a Rfrnc - σ yz (EASMEAFOR b Rfrnc - σ yz (EASMEAFOR 4 4 (minimal valu has bn imposd to SS7 lmnt - von Miss strss (msh SS7n lmnt - von Miss strss (msh Pinchd cylindr: solid-shll lmnts Pinchd cylindr: solid-shll and EAS Initial position (lft and final position (right for stamping Dfinition of angls (point E and point F ar th farthst contact points iv

11 from th cntrlin Discrtization of th modl : dfinition of th zons Punch forc vs. punch displacmnt EAS lmnt (quasi-static; abl 5../cass to 5; pnalty = Punch forc vs. punch displacmnt EAS lmnt (abl 5../cass 6 to Rfrnc solution of SRI lmnt (abl 5..4 /cas 7 quivalnt plastic strain simulation with MEAFOR Punch forc vs. punch displacmnt SRI lmnt (Pnalty=. ; abl5..4/cass to Punch forc vs. punch displacmnt SRI lmnt (Pnalty=. ; abl5..4/cass 5 to hosn solution of D.ANSn lmnt (abl 5..5 /cas 6 von Miss strss simulation with FEAP Punch forc vs. punch displacmnt ANSn lmnt (abl5..5/cass to Punch forc vs. punch displacmnt ANSn lmnt (abl5..5/cass 5 to Rfrnc solution of D.SS4n lmnt (abl 5..6/cas 6 von Miss strss simulation with FEAP Punch forc vs. punch displacmnt D.SS4n lmnt (abl5..6/cass to Punch forc vs. punch displacmnt D.SS4n lmnt (abl5..6/cass 4 and Punch forc vs. punch displacmnt [NUM] (BS = Bnchmark simulation rsult Punch forc vs. punch displacmnt [NUM] (BE = Bnchmark Exprimnt rsult Dfinition of th angl btwn contact points which ar th farthst from th cntrlin Angl btwn contact points which ar th farthst from th cntrlin [NUM] Opn angl btwn th lins AB and D bfor spring back (abl 5../as 6 - quivalnt plastic strain Opn angl btwn th lins AB and D aftr spring back (abl 5..6/as - quivalnt strss - FEAP Angl bfor and aftr springback at th final punch strok of 8.5 mm. 45 v

12 ABLE LIS.. Faturs of low-ordr solid lmnts Dformation mods of bilinar lmnt D.Q Indx transformation EAS lmnts Equivalnt bilinar EAS - HR lmnts Intrior nodal coordinats Displacmnts of th intrior nods Displacmnts at intrior nods of lmnts Vrtical displacmnt at th block s cntr... 5 A Location of innr nods... 5 A Rsults of EAS Dominant faturs of dgnratd shll and solid-shll Eignvalus of rgular cub Distortd cub - location of nods Eignvalus of distortd cub Eignvalus of th squar plat Eignvalus of th squar plat Summary Summary of mployd lmnts Numrical rsults Non-dimnsional dflction α Normalization of dimnsionlss dflction α - lampd plat (υ = Normalization of dimnsionlss dflction α - Simply supportd plat (υ = Normalization of dimnsionlss dflction α - lampd plat (υ = Normalization of dimnsionlss dflction α - Simply supportd plat (υ = ntr dflction Rgular msh - Normalization of displacmnt Irrgular msh - Normalization of displacmnt Morly sphrical shll - Normalizd displacmnts at tst point Normalizd radial displacmnt at R i Displacmnts du to pinchd forc F = onvrgnc of σ xz (MPa ransvrs shar strsss alculation with D.EAS7 lmnt alculation with D.SRI lmnt alculation with D.ANSn lmnt alculation with solid-shll lmnt Springback angls - calculation with D.EAS7 lmnt Springback angls - calculation with D.SRI lmnt Springback angls - calculation with D.ANSn lmnt Springback angls - calculation with D.SS4n lmnt... 4 vi

13 haptr. INRODUION INRODUION Sinc th sixtis of th th cntury th dvlopmnt of numrical mthods has bn th bas for dvloping various advancd nginring simulation tools. Still nowadays at th bginning of th st cntury applications of numrical mthods in simulation and prdiction of industrial problms and/or tchnological procsss bcom mor and mor important h most popular numrical mthods is th finit lmnt mthod with applications for simulation of biomchanic problms [HOL96] lastoplasticity problms [SIM88a] thrmalmchanic problms [HOG76] and contact problms [PON99] tc. hin shll structurs whos numrical analysis is th targt of this thsis appar in many products such as th outr-body of a car th fuslag and wings of an airplan tc. Modling ths parts with standard solid lmnts would rquir a hug numbr of lmnts and lads to prohibitiv computational costs. For instanc to prvnt locking ffcts i.. artificial stiffnss in th modl modling a bam with hxahdral solid lmnts rquirs a minimum of about - 5 lmnts through th thicknss. In such cass a low-ordr shll lmnt can rplac - 5 or mor solid lmnts which improvs computational fficincy immnsly. Furthrmor modling thin structurs with standard solid lmnts oftn lads to lmnts with high aspct ratios which dgrads th conditioning of th quations and th accuracy of th solution. Howvr for crtain problms in structural analysis displacmnt dgrs of frdom (DOF at th nods of th lmnt ar mor advantagous for analysis than displacmnt and rotational DOF s. For xampl considr complx structurs which consist of both thin and thick walls. For th sak of ffctivnss shll lmnts should b usd for thin-walld parts and solid lmnts should b usd for thick-walld parts. If both solid and shll lmnts hav th sam DOF s (.g. only displacmnts at nods th analysis procss xhibits on typ of DOF only no rquirmnt on transition lmnts xhibiting displacmnt and rotational DOF.g. Figur... Standard solid lmnt Solid-shll lmnt Figur..: onnction of solid-shll lmnt (whit color with standard solid lmnt (gry color

14 h dformation procsss also with contact and friction of shll lmnts such as happning in mtal forming ar asir to considr if shll lmnts hav thos configuration displacmnt DOF. Motivatd by ths argumnts th thsis concntrats on dvloping an lmnt that has simpl kinmatics (only displacmnt DOF s at nods as solid lmnts but is as ffctiv in computation as shll lmnts. A class of thos finit lmnts calld solid-shll has bn rcntly invstigatd by many rsarchrs ([HAU98] [HAU] [QUOab] [AN5] [JE8] bcaus that lmnt is not only capabl of modling complicatd structurs but it can also b usd to simulat mtal forming problms. Litratur shows indd that solid-shll lmnt is th most suitabl choic for th abov mntiond tasks.. OBJEIVES OF HE HESIS h us of low-ordr lmnts in finit lmnt computations rmains a popular fatur in solid mchanics bcaus of th following rasons. First thy rquir a simplr manipulation for mshing spcially for a distortd configuration. Scond ths lmnts facilitat mor convnint manipulations in th adaptiv h-typ of msh rfinmnt. Finally using low-ordr lmnts will rmarkably sav computation tim spcially for simulation with larg numbr of DOF s such as in nonlinar problms. In this thsis high-ordr lmnts ar thrfor not considrd. hy would in addition xhibit mor difficultis to dal with at contact surfac intrfacs. From now on for th sak of brifnss lt s call th low-ordr standard solid lmnt in thr-dimnsion (D is th ight-nod hxahdral lmnt and th low-ordr standard solid lmnt in twodimnsion (D is th four-nod quadrilatral lmnt. Although th structur of th low-ordr standard displacmnt lmnts is straightforward thy should not b usd dirctly in th following situations: h lmnts tnd to b too stiff in bnding.g. for slndr bams or thin plats undr bnding. h lmnts ar too stiff in narly incomprssibl or incomprssibl bhavior. In othr words four-nod quadrilatral lmnts and ight-nod hxahdral lmnt in two and thr dimnsions rspctivly hav a major drawback sinc thy lad to locking in th incomprssibl limit. It mans thy do not possss th proprty of bing uniformly convrgnt. In addition vn in comprssibl problms th us of ths standard lmnts lads to poor accuracy particularly in bnding-dominatd problms whn coars mshs ar usd. h linar and nonlinar finit lmnt analysis of plats and shlls has attractd much attntion in rcnt dcads. It is ncssary to captur th bnding-strtching coupling of thin shll bhavior. Hnc on of th motivations for dsigning nw lmnts is thir potntial ability to captur th mmbran-bnding coupling corrctly. So far thr ar two ways in which this could b don. On is to us lmnts basd on spcific shll thoris (.g. th Vlasov [VLA49] Flugg [FLU7] thoris tc.. hr ar considrabl controvrsis rgarding th rlativ mrits and drawbacks of ths thoris. Each thory has bn obtaind by carrying out approximations to diffrnt dgrs whn th D fild quations ar rducd to th particular class of shll quations.

15 h scond approach is calld dgnratd shll approach D solid lmnts can b rducd (dgnratd into shll lmnts having only mid-surfac nodal variabls - ths ar no longr dpndnt on th various forms of shll thoris proposd and should b simpl to us. hy ar in fact quivalnt to a Mindlin typ curvd shll lmnt (quivalnt to a shar dformabl thory s [PRA] [SO95]. With th standard procdur a plat or shll thory is usd as a basis for th finit lmnt formulation. It bgins with th fild quations of th D thory and maks various assumptions which lad to th plat or shll thory. Manwhil in th rduction from th thr to two dimnsions an analytical intgration ovr th thicknss was includd. h mid-surfac gomtry (in th cas of shlls and th fild variabls ar approximatd using discrtizd nodal valus and suitabl intrpolation functions. Intgration of various lmnt stiffnss and forc trms is carrid out ovr th rfrnc surfac. Exampls of such an approach includ th simpl fact lmnt and many lmnts drivd from th classical thin plat thory: th Mindlin-Rissnr plat thory shallow shll thory or vn high-ordr shll thoris. urrntly w can introduc th solid-shll thory as th third approach for capturing th bnding-strtching coupling of thin shll bhavior. h low-ordr solidshll lmnt has two nods along vrtical sids. Naturally th low-ordr solid-shll lmnt obys th straight normal assumption of Mindlin-Rissnr thoris. Without any assumptions bcaus of possssing solid configuration intgration of th solid-shll lmnt stiffnss is carrid out ovr th lmnt s physical volum. h solid-shll lmnts ar combinations of solid lmnts with shll lmnts. h lmnts ar gnrally usd for nonlinar problms (finit strain contact tc. so thy hav to satisfy som rquirmnts.g. fr from all locking typs simpl kinmatics tc. h following faturs of th solid lmnts and shll lmnts ar found in th solid-shll lmnts: Faturs from th solid lmnts: - Sam DOF s as solid; - Intgrating dirctly D matrial modl (vrsus plan strss assumption; - In contrast to th dgnratd shll concpt th complt D strain tnsor and strss tnsor ar usd (strains and strsss in thicknss dirction ar also includd. Faturs from th shll lmnts: - Us of a shll lmnt mthod to rmov transvrs shar locking. h solid-shll lmnts hnc ar applicabl to thin-walld structurs. - h normal to th mid-surfac rmains straight. hrough invstigating th litraturs thr ar two assumd strain mthods that hav bn xploitd so far to dvlop solid-shll lmnts. hy ar th Assumd Natural Strain (ANS of Dvorkin and Bath [DVO84] and th Enhancd Assumd Strain (EAS proposd by Simo and Rifai [SIM9]. h strain fild of th EAS lmnt is additionally modifid to b a complt polynomial fild. h strain fild of th ANS lmnt is rplacd by an incompatibl strain fild that satisfis th pur bnding rquirmnt. Othrwis thr is a sound variational mthod: Mixd Enhancd Strain (MES mthod proposd by Kaspr and aylor [KASb] whr strsss ar indpndnt from th strain fild. h MES mthod rquirs mor variabls (apart from th displacmnt fild and strain fild as rquird by th EAS and ANS mthods th MES furthr considrs th strss fild as variabls than th EAS and th ANS thus is not an attractiv approach.

16 Objctiv of th thsis is to dvlop a finit lmnt that is ffctiv for simulation of thin-walld bhavior in mtal forming procsss. It mans th lmnt givs prcis rsults whil proposing a low computational cost. Application of that lmnt is mainly xploitd in springback simulation. Springback rlats to th chang in shap btwn th fully loadd and subsqunt unloadd configurations. h springback ffct is ncountrd during a stamping opration. his can rsult in th formd componnt bing out of tolranc and thus crats major problms in assmbly or installation. Accurat dscription of th contact is on of th main factors which rndrs mtal forming simulation prdictiv or not. Lt s considr th solid-shll lmnt. ontact algorithms ar mor asily applid for solid lik lmnts thanks to th gomtrical dscription of th lowr and uppr surfacs than for shll lmnts whr nods li in th mid-plan. Howvr du to limitation of a thsis only prformanc of th solid-shll for springback simulation is mainly xploitd in this thsis. abl..: Faturs of low-ordr solid lmnts Elmnt Faturs Application domain Standard EAS ANS Solid-shll h standard lmnt xhibits dficincis as: Volumtric locking; Shar locking; Poisson locking. h standard lmnt which is nhancd by EAS paramtrs is: locking fr; high in computational cost. h standard lmnt which adopts th ANS mthod is: only shar locking fr; chap in computational cost. h standard lmnt which adopts both th ANS and th EAS mthods is: locking fr; chapr in computational cost than th EAS lmnt. Bing applicabl to all of problms but rquir vry fin msh to convrg Mtal forming Incomprssibl matrial tc hin-walld structurs ollaps of shlls tc hin-walld structurs Mtal forming Incomprssibl matrial tc.. APPLIAION DOMAIN Nowadays aims of th nw D solid lmnts ar:. No locking for incomprssibl matrials;. Good bnding bhavior;. No locking in th limit of vry thin lmnts; 4

17 4. Distortion insnsitivity; 5. Good coars msh accuracy; 6. Simpl implmntation of nonlinar constitutiv laws. h first two aims ar ssntial rquirmnts of solid structurd analyss. h third aim is usually rquird for structural lmnts such as plat shll and bam lmnts. h fourth aim is important bcaus in discrtizing an arbitrary gomtry th xistnc of distortd lmnts is invitabl. In addition lmnts can gt highly distortd during nonlinar simulations including finit dformations. h fifth aim rsults from th fact that many nginring problms hav to b modld as D problms. Du to computr limitations quit coars mshs hav to b usd oftn to solv ths problms. hus an lmnt which provids good coars msh accuracy is valuabl in ths situations. h sixth aim is associatd with th fact that mor and mor nonlinar computations involving nonlinar constitutiv modls hav to b prformd to dsign nginring structurs. hus an lmnt formulation which allows a straightforward implmntation of such constitutiv quations is dsirabl. h third aim bcoms incrasingly important sinc it nabls th solid lmnts to simulat shll problms. his maks th simulation work bcom mor ffctiv spcially for simulation of complicatd structurs. his spars th nd for introducing finit rotations as variabls in thin shll lmnts rsults in simplr contact dtction on uppr and lowr surfacs and provids th possibility to apply D constitutiv quations straight away. h EAS lmnts can satisfy all rquirmnts xcpt th third on. h solidshll lmnts ar th ons who could satisfy all of that 6 six rquirmnts. Du to thir dominant prformanc th solid-shll lmnts ar applicabl to various applications in structural analyss. h solid-shll lmnts could b applid for both solid and structural nginring problms and in both linar and nonlinar applications (finit strain contact tc.. oncrtly th solid-shll lmnts ar suitabl choic for mtal forming simulations civil-nginring structurs impact/crash analysis tc. SUMMARY OF HE HESIS Nowadays in computational mchanics thr is a trnd to trat plats and shlls as a D continuum using solid finit lmnts or D-lik plat and shll lmnts taking into account thicknss changs throughout dformation and using D matrial laws. h EAS lmnts ar vry suitabl to that trnd; thy ar applicabl to almost any nginring problms. Howvr th EAS lmnts ar snsitiv to distortd msh (.g. bnding patch tst and thy xhibit poor prformancs in bnding for vry thin-walld structurs. Most important is th fact that th EAS lmnts ar vry tim consuming for calculation. In contrary Rducd Intgration (RI lmnts ar computational tim saving lmnts. hy could b fr from volumtric locking and shar locking. Howvr in som situations thy ar not stabl (du to hourglass mods s [HAN98]. h solid-shll lmnts ar attractiv ons. urrntly thy attract much considration of rsarchrs. hy possss prformanc of th EAS lmnts whil thy ar insnsitiv to distortd msh. Furthrmor thy ar stabl and tim saving lmnts. 5

18 Obviously bcaus of having th solid s configuration th solid-shll lmnts ar suitabl for handling contact in mtal forming simulation particularly for simulation of sht mtal products whos ratio btwn lngth and thicknss is larg. hy hav only translational DOF s of solid lmnts so thy can b asily combind with th standard solid lmnts in problms daling with complx structurs manwhil thy can also work as shll lmnts. For transvrs shar locking rmoval th lmnts mploy th ANS tchniqus bcaus th ANS mthod is chapr (highr prformanc but chapr computational cost than th EAS mthod in rmoving transvrs shar locking. o gt rid of volumtric locking and mmbran locking th lmnts adopt tchniqus of th EAS mthod. Du to th us of ANS tchniqus that wr originally applid for plat and shll lmnts th solid-shll lmnts ar abl to simulat thin and modratly thick-walld structurs. h thsis includs six chaptrs and is structurd as follows. h first chaptr prsnts th objcts for th rsarch. h scond chaptr introducs background mthods which will b incorporatd in th solid-shll lmnts. h third chaptr dvlops an altrnativ ANS tchniqu and applis it to th solid-shll lmnts. As a rsult in that chaptr a nw solid-shll lmnt basd on th altrnativ ANS tchniqu is proposd. Elastic applications of th just dvlopd solid-shll lmnt ar illustratd in haptr 4. In haptr 5 plasticity thory and numrical problms in plasticity dformation ar prsntd. h thsis spcially concntrats on trating a currnt industrial problm: spring back prdiction. Rsults in haptr 4 and haptr 5 dmonstrat th capabilitis of th proposd solid-shll lmnt. haptr 6 withdraws conclusions and thn maks som rmarkabl futur dvlopmnts. 6

19 haptr. BAKGROUND ON HE DEVELOPMEN OF SOLID-SHELL ELEMENS INRODUION h solid-shll is a solid lmnt that has incorporatd shll faturs.g. ANS tchniqu for shar locking and curvatur thicknss locking rmovals and EAS or RI tchniqus for othr locking ffcts. In ordr to bttr undrstand th solid-shll concpt lt s start considring som faturs of th low-ordr standard solid lmnt and th obstacls that th solid lmnt is facing. h difficultis whn using th wll-known dgnratd shll lmnts ar also invstigatd. Latr in th chaptr all th mthods that concrn th solid-shll lmnts: th EAS (formulatd in Grn-Lagrang strain [KLI97] and th classical ANS mthods [DVO84] (applid for finit dformation solid lmnts ar introducd. hs mthods hav bn implmntd in a MALAB cod. In this chaptr sparatd prformancs of th ANS and EAS lmnts ar prsntd. Dtails for thir combination and co-opratd prformancs ar prsntd latr in haptr whr w also prsnt how th solid-shll lmnt rmdis all th obstacls that th low-ordr standard solid lmnts hav to ovrcom.. HREE DIMENSIONAL SANDARD ELEMEN In this sction w invstigat th low-ordr standard solid lmnt. Lt s considr th following trilinar displacmnt fild which is convntionally mployd for th ight-nod standard lmnt Figur... o facilitat undrstanding th analysis is rstrictd to a rctangular prismatic gomtry lmnt so that th physical systm (XYZ and th isoparamtric (natural systm ( can b usd intrchangably. h displacmnt filds u v and w ar linarly intrpolatd with th hlp of cofficints a i b i and c i (i =... 7 stmming from a trilinar assumption: u = a + a X + a Y + a Z + a XY + a YZ + a XZ + a XYZ v = b + b X + b Y + b Z + b XY + b YZ + b XZ + b XYZ w = c + c X + cy + cz + c4 XY + c5yz + c6 XZ + c7 XYZ or undr th form: u = ( α + α X ( β + β Y( γ + γ Z v = ( α + α X ( β + β Y( γ + γ Z * * * * * * (..a (..b ** ** ** ** ** ** w = ( α + α X ( β + β Y( γ + γ Z ** whr α α... γ ar constants. h Grn-Lagrang strain componnts usd in larg dformation thoris ar givn by: 7

20 E com ( u E X.5( X X v X v X X X XX u u u + + w w E v Y.5( u Yu Y + v Y v Y + w Y w Y YY E w Z.5( u Zu Z + v Z v Z + w Z w Z ZZ = = + EXY u Y + v X ( u Xu Y + v X v Y + w X w Y E ZX u Z + w X ( u Xu Z + v X v Z + w X w Z EYZ v Z + w Y ( u Yu Z + v Y v Z + w Y w Z ε nonlinar trms com ( u (.. whr: - h suprscript com and th subscript (u man that th strain fild is compatibl with th displacmnt fild. u - ui A ; ui = u v w ; A = X Y Z ; A com - h infinitsimal strains ar ε ( = { ε ε ε γ γ γ }. u X Y Z XY XZ YZ For th sak of simplicity locking ffcts ar simply considrd with infinitsimal strains. It mans w considr infinitsimal strains ε instad of Grn-Lagrang strain E. All typs of locking and thir rmdis for low-ordr solid lmnts will b mntiond in dtails in th following sction Z 7 5 Y X 4 Gauss intgration points nodal points Figur..: Hxahdral solid lmnt.. Difficultis with low-ordr standard lmnts Low-ordr lmnts ar prfrrd in nonlinar structural mchanics bcaus of thir low computational cost and simplicity in daling with th gomtry. Howvr in many cass spcially in pur bnding problms th low-ordr solid lmnt xhibits a low-prcision rsult du to stiffning ffcts known as locking. Locking trminology A numbr of diffrnt concpts to dfin xplain and quantify th locking ffcts hav bn discussd in th past. In th squl it is trid to classify som of thm in two diffrnt groups. 8

21 Mathmatical point of viw: In th mathmatical litratur th trm locking is not as popular as in nginring litratur. From a mathmatical point of viw it is rathr an ill-conditioning of th undrlying mchanical problm or th systm of partial diffrntial quations to b mor prcis. h crucial proprty is th prsnc of a crtain small scal paramtr within th quations. his paramtr lads to a high ratio of th cofficints in th discrtizd systm of quations (.g. th stiffnss. hus th parasitic trms volving from unbalancd shap functions ar ovrly nlargd. h lmnt locks if thr is no uniform convrgnc with rspct to this paramtr (i.. th rat of convrgnc in th rang of coars mshs dpnds on this paramtr. According to Wilson t al. [WIL7] ffct of th ill-conditioning will b minimizd by th us of a computr with high (doubl for xampl prcision or by rstricting th application of th lmnt to thickwalld structurs. Mchanical point of viw: h simplst way to xplain locking is to associat th ffct with th prsnc of parasitic (or spurious strains or strsss. With parasitic w man such strains (strsss that do not show up in th xact solution of a crtain problm. hs ar for instanc transvrs shar strains in th cas of pur bnding of a plat lmnt (transvrs shar locking or mmbran strains in th cas of inxtnsional bnding of shlls (mmbran locking or volumtric strain in th cas of incomprssibl bhavior (volumtric locking. In fact th wll-known locking phnomnon of displacmnt basd finit lmnts for thin-walld bams plats shlls and solids is causd by an unbalanc of th trial functions. It should also b mntiond that a natural stratgy to rmdy locking ffcts is th dsign of highr ordr finit lmnts. Howvr thy ar not attractiv for nonlinar structural analysis bcaus of xpnsiv computation and complicatd configuration. Locking of crtain low-ordr standard displacmnt basd finit lmnts coms along in diffrnt ways namly as volumtric locking if incomprssibl or narly incomprssibl matrials ar usd or as mmbran shar and curvatur thicknss locking if th strss and strain spac is not compatibl du to th spatial discrtization. h phnomna prsntd hraftr ar th svr locking ffcts that happn with th low-ordr standard solid lmnt. Othr svr locking ffcts which happn to th solid-shll lmnts ar mntiond in th nxt chaptr.... Volumtric locking If narly-incomprssibl or incomprssibl matrial bhavior is concrnd th low-ordr standard solid lmnts suffr from volumtric locking. In daling with this locking th following condition on th volumtric strain ε v is rquird during th dformation procss: u v w ε = tr( ε = + + (.. v X Y Z With th abov tri-linar displacmnt fild (.. th volumtric strain corrspondingly calculatd as: com ε v is 9

22 com ε v = ( a + b + c + ( b4 + c6 X + ( a4 + c5 Y + ( a6 + b5 Z + ( c7 XY + a7yz + b7zx (..4 h volumtric strain (..4 can b constraind to b zro by imposing that th cofficints of ach trm in (..4 vanish as: a + b + c = (..5a b4 + c6 = (..5b a4 + c5 = (..5c a6 + b5 = (..5d a = ; b = ; c = ( com But th volumtric strain ε v is gnrally non-zro sinc th prsnc of th trms a 7 b 7 and c 7 in th sns that thy com from isolatd trms of u or v or w s (... hs isolatd trms ar diffrnt from zro in ordr to assur th compltnss com of intrpolation functions. In othr words th trms a 7 b 7 and c 7 mak ε v usually diffrnt from zro. Forcing th incomprssibility condition (.g. for ν =.5 in lasticity or for incomprssibl plasticity such as J von Miss plasticity will impos a7 = b7 = c7 = and thus an xcssiv stiffnss is gnratd by this condition hnc th nam locking. onsquntly th constraint of an incomprssibl matrial (.. cannot gnrally b fulfilld by th normal strains of th pur displacmnt lmnt. h ffct of this dficincy on th dformation bhavior of an lmnt can b xplaind by using th intrnal nrgy. h intrnal nrgy π int of an lmnt consists of a dviatoric trm π d and a volumtric trm π v. It is dfind by π u int = ( K u and th rlativ contribution of dviatoric and volumtric dformation to th lmnt stiffnss can b shown to b: π π π = + = G ε : ε dv + κ ε : ε dv = G ε : ε dv + κ tr( ε dv int d v d d v v d d V V V V ( + ν with: - bulk modulus κ = G ; ( ν - volumtric strains ε v = ε I = tr( ε ; i = ; ii - dviatoric strains ε = ε ε. d v (..6 In contrast to all othr known locking ffcts which ar primarily kinmatic or gomtric ffcts volumtric locking dpnds on a matrial paramtr Poisson s ratio ν. hrfor also th trm Poisson locking is somtims usd in th litratur. It is straightforward to considr th bulk modulus as th critical paramtr. For ν = thr is no spurious volumtric locking at all; th ffct bcoms mor and mor pronouncd as ν.5 bcaus limκ (ν.5 =. In solid mchanics this ffct can occur.g. for rubbr matrials but also for mtals in th rang of plastic dformations (yilding. h bulk modulus κ bcoms vry larg for ν. If tr (ε is not vanishing th stiffnss of on lmnt or a group of lmnts will thus b much largr than th stiffnss of th ral incomprssibl continuum for which th trm π v is vanishing. For th standard solid lmnt sinc th incomprssibl condition (.. cannot b accomplishd consquntly an undsirabl stiffnss is addd to th rigidity of th

23 lmnt and it maks th lmnt stiffr than th ral continuum. In othr words th nonzro volumtric strain ε lads to volumtric locking. com v... Shar locking ransvrs shar locking can occur in shar dformabl bam plat and shll lmnts. In principl it is also prsnt in solid lmnts if ths ar applid to th analysis of thin-walld structurs. Howvr with solid lmnts it is simply calld shar locking bcaus thr is no distinct transvrs dirction in a solid lmnt. On th contrary th solid-shll lmnts tak th shll bhavior in thicknss dirction hnc th transvrs dirction is distinct. Shar locking in th solid lmnts is on of th most svr locking ffcts bcaus it dos not only slows down th convrgnc but also can ssntially prclud an analysis with a rasonabl amount of numrical ffort in practical applications. From th classical thory of lasticity whn th lmnt is subjctd to a pur bnding situation such as bnding in X dirction around Z axis (Figur.. th shar strain γ XY in this cas must vanish. Howvr with th abov trilinar fild (.. com th shar strain γ XY in (.. is calculatd as: com u v γ XY = + = ( a + b + ( a5 + b6 Z + a4 X + b4y + a7 XZ + b7yz. (..7 Y X hat shar strain is gnrally non-zro. It is qual to zro only whn th cofficints of ach trm vanish as: a + b = ; a5 + b6 = ; (..8a a4 = ; b4 = ; a7 = ; b7 = ; (..8b Equations (..8a contain cofficints from both th contributing intrpolation functions (u and v which ar rlvant to th dscription of th shar strain fild γ. com Hnc ths cofficints can corrctly rprsnt a tru zro condition on shar strain γ XY whn a = b and a5 = b6. Each quation (..8b contains only an isolatd trm from u or v ( a 4 or a 7 and ( b 4 or b 7. In gnral ths isolatd trms ar diffrnt from zro in ordr to assur th compltnss of intrpolation functions s (... As a consqunc ths cofficints mak γ also diffrnt from zro. In othr words th com XY prsnc of cofficints a 4 b 4 a 7 and 7 com XY com γ XY caus shar locking hnc thy ar b in calld inconsistnt trms [HA89]. h sam argumnts ar applid for th othr shar strain componnts com γ YZ com γ XZ : com w v γ YZ = + = ( b + c + ( b6 + c4 X + b5y + c5z + b7 XY + c7 XZ y z com u w γ XZ = + = ( a + c + ( a5 + c4 Y + a6 X + c6z + a7 XY + c7yz Z X (..9a (..9b com h shar strain γ YZ is qual to zro only whn th cofficints of ach trms in (..9a vanish as: b + c = ; b6 + c4 = ; (..a b = ; c = ; b = ; c = ; (..b 5 Similarly th shar strain trms in (..9b vanish as: com γ XZ is qual to zro only whn th cofficints of ach

24 a a 6 + c = ; = ; c 6 a 5 = ; + c 4 = ; a 7 = ; c 7 = ; (..a (..b onsquntly non-physical ffcts with compatibl shar strains caus th socalld shar locking for th standard lmnts by introducing artificial flxural stiffnss. his phnomnon is ssntial with th vanishing of th thicknss of standard lmnts in th modling of bnding dominatd problms. abl..: Dformation mods of bilinar lmnt (D.Q Mod c c c c 4 c 5 c 6 c 7 c 8 u X Z XZ w X Z XZ ε X c c 7 Y ε Z c 6 c 8 X γ XZ c 4 c 5 c 7 X c 8 Z F a F M = Z X = M Z X b F F a ontinuum mchanics γ = XZ For th sak of mor clarity lt s invstigat dformation mods of a bam in bnding Figur... Dformation of th bam is assumd to b indpndnt of Y th width dirction. In that cas th displacmnt fild (.. bcoms: u = c + c X + c Z + c XZ ( X Z 5 7 w = c + c X + c Z + c XZ ( X Z Figur..: Pur bnding of a rctangl b FEM rprsntativ γ in gnral (.. It mans th trilinar ight-nod hxahdral lmnt rducs to th bilinar fournod quadrilatral lmnt. Lt s considr dformation mods of th displacmnt fild (... All of thos dformation mods ar tabulatd in abl... In abl.. c and c ar rigid body mods. c to c 6 ar constant strain mods and c 7 and c 8 ar linar strain mods. Whn an in-plan bnding dformation happns it mans bnding in X-dirction around Z-axis th mod No. 7 is activ only th cofficint c 7 is non-zro thus lading to a parasitic linar shar strain γ XZ in X-dirction (s abl... In othr words it is impossibl to find a linar combination of mods that lads to a linar variation of ε X in Z-dirction without bing accompanid by shar XZ

25 strain γ XZ. his phnomnon is th manifstation of (transvrs shar locking. h sam is in turn tru for th bnding in Z-dirction (s abl.. - mod 8. Lt s invstigat analytical solution of a bam of rctangular cross sction which is bnt by two qual and opposit coupls M (Figur... Strss componnts of th bam ar: MZ σ Z = I σ = τ = X XZ (.. whr I is th ara momnt of inrtia of th bam cross sction. From (.. and strss-strain rlations w attain displacmnt componnts by intgrating strain-displacmnt rlations s [DUR58]. Finally th analytical formulations for displacmnts ar: M M u( X Z = XZ + Z + EI M w = ( X + ν Z X + EI M ( X Z (..4 whr and ar constants of intgration. h suprscript M mans that th displacmnt fild is calculatd by th continuum mchanics. If w impos th boundary conditions as: symmtric plan of th bam is th plan which gos through X=; vrtical displacmnt at th 4 cornrs is qual to zro. hn th analytical solution of th problm in Figur.. is: M M u( X Z = XZ = αxz EI M M M w( X Z = ( a X ν ( b Z = α( a X α( b Z EI EI (..5 whr α M = EI and α M = ν EI ar constants. h constant α is a function dpnds on matrial proprtis for Poisson s ratio qual to zro that constant is α =. h analytical strains calculatd from (..5 ar: ε ε γ M X M Z M XZ = α Z if ν = = α Z if ν u w = + = Z X (..6 Obviously th solution (..5 satisfis th pur bnding condition zro shar strain constraint. For finit lmnt solution whn pur bnding occurs only th mod 7 (abl.. is activ it mans: u = c XZ ( X Z 7 w ( X Z = (..7 Hnc compar with th analytical solution th form of rror in th numrical solution is:

26 Error( w = β ( a X + β ( b Z (..8 ( X Z y z x y z x a σ XZ b σ Z (υ hrfor th rrors in strains ar followd by th rrors in displacmnt as: if ν = Error( ε Z = βz if ν Error( γ = c X XZ 7 Figur..: Errors in strsss in pur bnding 4 (..9 hs rrors caus parasitic strsss as shown in Figur... In fact th Error( γ XZ causs shar locking th Error( ε Z causs Poisson thicknss locking (will b discussd in th nxt sction. In conclusion shar locking happns bcaus normal strains of linar lmnts ar coupld by shar strains. Elmnts do not hav pur bnding mods to bhav corrctly for pur bnding load cass. h consqunc is thr ar parasitic shar strains appar simultanously with normal strains which ar physical in pur bnding cass. hs shar strains ar important compard to th normal strains. onsidr th pur bnding problm undr th point of viw of th continuum mchanics thory. In pur bnding th applid momnt is constant and thn shar strss must vanish sinc th shar strss is th drivativ of th bnding momnt with rspct to th axial coordinat. Whn solving th pur bnding problm by th finit lmnt mthod sinc th shar stiffnss is oftn significantly gratr than th bnding stiffnss th spurious shar absorbs a larg part of th nrgy inducd by th xtrnal forcs and th prdictd dflctions and strains ar much too small. In othr words th additional shar strss in th lmnt (which dos not occur in th actual bam causs th lmnt to rach quilibrium with smallr displacmnts i.. it maks th lmnt appar to b stiffr than it actually is and givs bnding displacmnts smallr than thy should b. Long slndr structurs in bnding hav gratr curvatur than do short dp ons and whn modld with low-ordr lmnts will b affctd mor by shar locking. Incrasing th numbr of lmnts will allow a mor accurat modling of th curvatur and rduc th ffcts of shar locking. Shar locking is prvntd through th us of high ordr lmnts.

27 ... Poisson thicknss locking Also in bnding problms.g. bnding in X dirction around Z axis (Figur.. a linar distribution of strain ε ovr th thicknss in th Z dirction s M Z M (..6b ε Z = α Z is thortically xpctd. Howvr th standard lmnt (.. givs only a constant strain (with rspct to Z as: com w ε Z = = c 6 + c 8 X (.. z com his constant (with rspct to Z approximation of th strain ε Z in th thicknss M dirction is contrary to th linar variation along Z of ε Z. In th ral structur du to MZ Poisson ffct th normal strain ( ε M X = which linarly vary in th Z dirction EI would caus a transvrs normal strain also linarly varis along Z ( ε M = νε M. com Howvr it is not th cas with ε Z calculatd by th low-ordr intrpolation function s (... onsquntly th transvrs normal strss in Z dirction which is calculatd as com com com σ = E Z [( ν ε Z νε X ] ( + ν ( ν + Z X (.. is not qual to th analytical solution along th thicknss whn bnding occurs. It lads to an undsird locking which is known as Poisson thicknss locking phnomnon. In gnral Poisson thicknss locking is du to th rsulting incorrct-linar distribution of th normal strss in thicknss dirction. hat locking ffct dos not diminish with msh rfinmnt in all dirctions xcpt th thicknss dirction (layrs... Solution for a locking fr lmnt As discussd abov th low-ordr standard solid lmnt suffrs from locking ffcts. It has bn pointd out that th stiffning in th cas of incomprssibility is causd by rdundant trms in th normal strains. Using th SRI mthod will b discussd at th nd of this chaptr is on of th bst way for liminating volumtric locking. Rstriction of th SRI mthod is that th applid matrial laws must allow a dcoupling strss and also strain into volumtric parts and dviatoric parts. Volumtric locking could also b vanishd by insrting suitabl nhancing componnts in ordr that th incomprssibl condition (.. b satisfid. h us of th EAS mthod with nin volumtric mods (s Sction.4 as proposd by Andlfingr and Ramm [AND9] assurs that all thr normal strains ε ii consist of th sam polynomial filds thn no spurious constraint is producd. Disadvantag of this way is highly computational cost for ach lmnt bcaus th nhancing strains making th lmnt systm of quations largr. In th thsis only thr EAS intrnal paramtrs ar usd to rmov volumtric locking for solid-shll lmnt s haptr. Shar locking for solid lmnts can b rmovd by th ANS mthod. Dtaild discussions ar prsntd in Sction.. o ovrcom Poisson thicknss locking additional trms with linar distribution in thicknss dirction must b introducd for th transvrs normal strains. his also assurs 5

28 th strss fild σ z corrctly varis linarly along th thicknss in bnding situations. Using th EAS mthod th trms ( of EAS7 lmnt (s Sction.4 blow can b usd to circumvnt Poisson thicknss locking. In fact th constant normal strain in thicknss dirction is nhancd with a linar xtnsion ovr th thicknss and linar in in-plan dirction according to th EAS mthod. In gnral it is quit possibl to us standard solid lmnts for th analysis of shll-typ structurs if on can ovrcom th following problms as pointd out by Wilson t al. [WIL7]:. Most solid lmnts hav not had ability to rprsnt accuratly bnding momnts.. Du to th full intgration i.. Gauss intgration (s Figur.. th lmnt will bhav badly for isochoric matrial bhavior i.. for high valus of Poisson's ratio or plastic bhavior (du to volumtric locking s SRI mthod at th nd of this chaptr for a rfrnc solution.. Errors in th transvrs shar caus th lmnt to b vry stiff (transvrs shar locking. 4. For simulation of thin shlls bcaus th thicknss of th lmnt is rlativly small compard to th in-plan dimnsions thr ar rlativly larg stiffnss cofficints in th thicknss dirction of numrical problms that ar introducd. his ffct maks th simulation problm illconditiond. h dsign of th solid-shll lmnts aims to ovrcom ths disadvantags of th standard solid lmnt. Indd th first two problms can b solvd by th us of th EAS mthod. h third and fourth problms can b minimizd by applying th ANS mthod originally dvlopd for thin shll lmnts. Practically th solid-shll lmnts adopt th EAS tchniqu for th in-plan shar strains and in-plan normal strains and adopt th ANS tchniqu for th transvrs shar strains and transvrs normal strain. As a rsult th solid-shll lmnts may ovrcom all of th abov difficultis. hanks to intrinsic prformanc of th ANS mthod th solid-shll lmnt can b applid not only for thin but also for modratly thick shll structurs.. INRODUION O ONVENIONAL SHELLS In this sction th convntional shll lmnts ar brifly introducd. Som basic concpts mntiond hr ar usful for building th solid-shll lmnts. W will brifly discuss about disadvantags of th convntional shll lmnts compard with th solidshll lmnts such as modifid matrial modls variation of thicknss strains contradictions of assumptions for shll lmnts tc. Basically th convntional shll finit lmnts ar dvlopd from on of th two following approachs:. Dvlop th formulation for shll lmnts by using classical strain displacmnt and momntum (or quilibrium quations for shlls to dvlop a wak form of th momntum (or quilibrium quations. 6

29 . Dvlop th lmnt dirctly from a continuum lmnt by imposing th structural assumptions on th wak form or on th discrt quations; this is calld th continuum basd approach. For xampl th kinmatic assumptions will b imposd on th discrt quations i.. th continuum finit lmnt will b modifid so that it bhavs lik a shll [AHM7] [DVO84]. h first approach also calld classical shll thoris is difficult particularly for nonlinar shlls sinc th govrning quations for nonlinar shlls ar vry complx and awkward to dal with. hy ar usually formulatd in trms of curvilinar componnts of tnsors and faturs such as variations in thicknss junctions and stiffnrs ar gnrally difficult to incorporat. hr is still a disagrmnt as to what ar th bst nonlinar classical shll quations [SO95]. h continuum-basd approach on th othr hand is straight forward yilds xcllnt rsults is applicabl to arbitrarily larg dformations and is widly usd in commrcial softwars and rsarchs. h popular continuum-basd mthod usd in structural analysis is calld th dgnratd continuum approach s.g. Ahmad t al. [AHM7]... lassical shll thoris Earlir a shll was considrd as a curvd form of a plat and its structural action is a combination of strtching and bnding [ZIEb]. It is possibl to prform a finit lmnt analysis of a shll by using what is calld a fact rprsntation - i.. th shll surfac is rplacd by a flat triangular and/or quadrilatral plat lmnts in which a mmbran stiffnss (mmbran lmnt is suprposd on a bnding stiffnss (plat bnding lmnt. Such a modl is undrstandably inaccurat in th sns that with vry coars mshs thy do not captur th bnding-strtching coupling of thin shll bhavior. Hnc th motivation for dsigning lmnts is twofold: mid-surfac curvatur has to b takn into account and th lmnt has to captur th mmbran-bnding coupling corrctly. hr ar two typs of kinmatic assumptions thos that admit transvrs shar strains and thos that don't. h thory which admit transvrs shar strains ar calld Mindlin-Rissnr thoris whras th thory which dos not admit transvrs shar strains is calld Kirchhoff-Lov thory. h ssntial kinmatic assumptions in ths shll thoris ar:. h normal to th mid-surfac rmains straight and normal (Kirchhoff-Lov thory.. h normal to th mid-surfac rmains straight (Mindlin-Rissnr thory. Shll thoris s [ZIEb] provd that th Kirchhoff-Lov assumptions ar th most accurat in prdicting th bhavior of thin shlls. For thick shlls th Mindlin- Rissnr assumptions ar mor accurat bcaus transvrs shar ffcts bcom important. ransvrs shar ffcts ar particularly important in composits. Mindlin- Rissnr thory can also b usd for thin shlls. In that cas th normal will rmain approximatly normal and th transvrs shar strains will almost vanish. h assumptions for Kirchhoff-Lov shll thory ar: h shll is thin compard to th radius of curvatur i.. t/r <<. h linar and angular dformations of th shll ar small. 7

30 h transvrs normal strss is ngligibl: σ ZZ =. 4 h normals to th rfrnc plan bfor dformation rmain normal straight and inxtnsibl aftr dformation. h first assumption is th most basic sinc it implis th othr thr. It mans th assumption cannot b violatd without violating assumptions -4. Assumption in ffct mans that th stat of th dformd shll can b rlatd dirctly to th stat of th non-dformd shll. Assumption is rasonabl for thin shlls (xcpt for plasticity and has no furthr implications bsids simplifying th drivation of th govrning quations. Assumption 4 has two implications; first th inxtnsibility assumption implis zro normal strain ( ε ZZ = ; scond th normal rmains normal this assumption liminats th possibility of transvrs angular distortions and consquntly lad to nglct th transvrs sharing ( γ = γ = s th coordinat systm in Figur... XZ YZ Mindlin-Rissnr thory is applid for thin to thick shlls. h kinmatic assumptions ar:. h normal to th mid-surfac rmains straight throughout dformation.. h lngth of th normal rmains unchangd throughout dformation.. ransvrs normal strsss ar ngligibl σ =. Whn Poisson s ratio is not qual to zro th lattr two assumptions ar contradictory bcaus th normal must strtch whn σ ZZ =. Rissnr s and Mindlin s thoris diffr from ach othr in th way thy solv this problm. Whil Rissnr assums a cubic variation in thicknss dirction of th transvrs normal strsss Mindlin manipulats th matrial law in ordr to comply with his assumptions. Although concptually diffrnt both thoris practically lad to th sam rsults for transvrs displacmnts shar forcs and bnding momnts in actual structural analysis. h mthod of Mindlin is simplr so it is popularly applid for numrical plat and shll modls. In ordr to construct th finit lmnt formulas th Mindlin mthod rquirs th variational principl with only a displacmnt fild but th Rissnr mthod nds two filds (displacmnt and strss variational principl. Although ths lmnts hav th advantag of bing abl to account for th transvrs shar that occurs for thick shll low-ordr forms of ths lmnts ar subjct to shar locking. Employing th first and scond assumptions maks th transvrs shar strains ar constant ovr thicknss dirction. Manwhil mploying th third assumption rquirs modifications of th D-matrial law. his work is not simpl in such an approach spcially for complicatd matrial laws which ar dscribd only for D-continuum. In ordr to us D-matrial laws dirctly th normal strss in th thicknss dirction must b takn into account. his lads to an intrpolation of th xtnsibl dirctor vctor (dfind in Figur.. s [BE96] [BIS97]. ZZ.. Dgnratd shll lmnts hr wr various shll lmnts whos formulations ar drivd from th dgnration concpt introducd by Ahmad t al. [AHM7]. h cor of this concpt is 8

31 th discrtization of a D mathmatical modl with D lmnts and thir subsqunt rduction into D lmnts. As classifid abov th dgnratd shll lmnts ar built from a so-calld continuum basd approach (M. In comparison th M to shll thory it is not ncssary to dvlop th complt formulation i.. dvloping a wak form discrtizing th problm by using finit lmnt intrpolations tc. h dgnration of this D shll lmnt is don by liminating th nods with th sam ( coordinats into a singl nod locatd at th mid-surfac of th lmnt as shown in Figur... h procdur whn crating a shll lmnt using th dgnratd solid approach is to liminat nods by nforcing diffrnt constraints on th bhavior of th lmnt. First nods on th mid-surfac ar rmovd (nods 7 to Figur.. corrsponding to assuming constant transvrs strains. hn opposit nods (&9 & 5& 7&5; & 4& 6&4 8&6 ar linkd by assuming qual displacmnts (u v and w and assigning two rotational DOF s ( θ x and θ y to ach pair of nods. Finally th motion of ach straight lin is dscribd by fiv DOF s in on nod lying on th rfrnc surfac Figur..b. Assumptions: For th shar dformabl shlls th following assumptions ar mad:. h fibrs (lin conncts bottom nod with top nod rmain straight. h unit vctor along ach fibr is calld a dirctor vctor;. h lmnt is in a plan strss stat so σ ZZ = ;. h longation of th fibrs is govrnd by consrvation of mass and/or th constitutiv quation. h first assumption will b calld th modifid Mindlin-Rissnr assumption. It diffrs from what w call th classical Mindlin-Rissnr assumption which rquirs th normal to rmain straight; th fibrs ar not initially normal to th midlin. For th M shll lmnt to satisfy th classical Mindlin-Rissnr assumptions it is ncssary for th fibrs to b alignd as closly as possibl with th normal to th midlin. his can b accomplishd by placing th slav nods (nods of th original solid lmnt so that th fibrs ar as clos to normal to th midlin as possibl in th initial configuration. Othrwis th bhavior of th dgnratd shll lmnt may dviat substantially from classical Mindlin-Rissnr thory and may not agr with th physical bhavior. Obviously it is impossibl to align th fibrs with th normal xactly along th ntir lngth of th lmnt whn th motion of th continuum lmnt is. ontrarily if th fibrs ar inclind too much with rspct to th normal whil th transvrs normal strain is takn into account thr is an ffct call curvatur thicknss locking occurs s haptr for mor dtails. Instad of th third assumption many authors assum that th fibrs ar inxtnsibl. Inxtnsibility contradicts th plan strss assumption: th fibrs ar usually clos to th z dirction and so if σ ZZ = th strain in th z dirction gnrally cannot vanish. h assumption of constant fibr lngth is inconsistnt with th consrvation of mass: if th shll lmnt is horizontally strtchd it must bcom thinnr to consrv mass. hrfor if th thicknss strain is calculatd through th constitutiv quation via th plan strss rquirmnt consrvation of mass is nforcd. h important fatur of th third assumption is that th xtnsion of th fibrs is not govrnd by th quations of motion or quilibrium. From th third assumption it follows automatically that th 9

32 quations of motion or quilibrium associatd with th thicknss mods ar liminatd from th systm. Solid lmnt Slav nod Z Y X 8 Dlt midsurfac nods 5 7 Z Y X 9 4 Dgnratd shll lmnt onnct opposit nods 7;5 Z 6;4 Y 8;6 5; ;9 4; ; 5 DOF s/nod ; 7 Z 6 Y Dirctor Mastr nod X X Figur..: Dgnration of a D solid lmnt into a dgnratd shll lmnt h third assumption can b rplacd by an inxtnsibility assumption if th chang in thicknss is small. In that cas ffct of th thicknss strain on th position of th slav nods is nglctd so that th nodal intrnal forcs do not rflct changs in th thicknss. h thory is thn applicabl only to problms with modrat strains (on th ordr of. for instanc. h dgnratd shll lmnts ar in gnral chap in computational cost du to th rducd numbr of DOF and using coars msh. h major shortcoming of th lmnts is th problms of locking for thin shlls. Howvr th dgnratd shll lmnts ar attractiv sinc thy propos a simpl mthod without discrtization of th govrning shll quation as in th cas of th dirct formulations - th classical shll thoris. W hav prsntd abov a brif discussion about shll lmnts. In th nxt

33 chaptr w will discovr prformancs of th on calld solid-shll in comparison with th dgnratd and classical shll lmnts in simulating thin-walld structurs.. INOMPAIBLE DISPLAEMEN ELEMEN Incompatibl mthod also calld assumd displacmnt mthod is drivd from th potntial nrgy variational principl. In th thsis th incompatibl mthod is prsntd as a rfrnc to motivat for th EAS mthod. Hnc th incompatibl mthod is brifly introducd and only considrd in linar lastic thory. h standard solid lmnts pos th following difficultis: locking phnomna for bnding and incomprssibl problms. By adding incompatibl displacmnts to -D isoparamtric nods th mntiond difficultis ar cancld... Finit lmnt formulation onsidr a continuum body occupying a volum V in a spac of boundary surfac A. Assum that th body forc and th tnsion forc ar consrvativ and th objct is in static stat. Undr th thory of linar lasticity th principl of minimum potntial nrgy can b statd as: π = ( ε ε u t u b ρ u ( com 4 com * * ε( u ε( u dv da dv V Aσ V (.. Whn th solid continuum is discrtizd into a finit numbr of lmnts th abov variational is rwrittn in th form as: π Nl ( 4 ( * * = Du Du u b dv da = u t u (.. V Aσ ( whr Nl : numbr of lmnts. In th finit lmnt formulation th lmnt displacmnts u ar intrpolatd in trms of nodal displacmnts that may b at both boundary nods (Srndipity lmnts and intrnal nods (Lagrang lmnts. Elmnts can also b formulatd by adding to th original lmnt displacmnts u which ar in trms of nodal displacmnts U highr-ordr displacmnts λ which ar not xprssd in trms of nodal displacmnts of th boundary nods. For xampl th displacmnts u and v for th standard fournod quadrilatral lmnt ar basd on bilinar intrpolation functions. hy ar incomplt in quadratic trms s (... Improvmnt of th prformanc of a fournod lmnt can b mad by adding trms such that th displacmnts ar complt in quadratic trms. Wilson t al. [WIL7] suggstd th addition of incompatibl displacmnts that vanish at all cornr nods. In ths cass th lmnt strain can b xprssd as ε = Du

34 whr Du = [ B B ] u U λ (.. Λ B u B λ ar compatibl strain matrix and incompatibl strain matrix rspctivly. U is nodal displacmnt vctor and U Λ = [ U = [ Λ V Λ W... U Λ V inmod ] W... Λ is incompatibl mod vctor of lmnt: U Nnod V Nnod W Nnod ] (..4 with Nnod is th numbr of nods pr lmnt; inmod is th numbr of incompatibl mods pr lmnt. Introducing (.. into (.. w hav: π Nl = ( U kuu U + Λ kλu U + ( Λ kλλ Λ u = ( ( U fxt ( Λ f λ (..5 whr: - h standard stiffnss matrix of th lmnt is: with k uu = ( B V 4 u BudV - h incompatibl-standard stiffnss matrix of th lmnt is dfind as: k λ u 4 = ( Bλ V BudV - h incompatibl stiffnss matrix of th lmnt is dfind as: k λλ = ( B V 4 λ B λ dv - h standard nodal forc vctor is: (..6 (..7 (..8 f xt = ( N u b* dv + ( N u t * da (..9 V - h incompatibl nodal forc vctor is dfind as: λ σ A f λ = ( N b* λ dv + ( N λ t * da (.. V N ar th incompatibl shap functions. σ A h incompatibl vctor Λ consists of intrnal variabls hnc it can b condnsd out of th variational formulation by stting π / Λ = in ordr to gt Λ in function of U as:

35 Λ = [ kλλ ] ( kλu U f λ (.. hn th total potntial nrgy π ( u is rwrittn: π Nl ( ( = U k U U f u = (.. ( whr th quivalnt lmnt stiffnss matrix is k = k ( k ( k k (.. uu λu λλ λu and th quivalnt nodal forc vctor is f = f ( k ( k f (..4 xt λu λλ λ Assmbling k into th global stiffnss matrix K and F finally th total potntial nrgy π ( u is: f into global nodal vctor π ( u = ( U K U ( U F (..5 whr: - U is th global nodal displacmnt vctor; - - = Nl K k ; = = Nl F f. = Lt s tak th first variation of ( u π π with rspct to global displacmnt vctor U and impos it qual to zro ( / U = th quation for displacmnt solution is u K U = F (..6 Aftr th solution procss solving systm (..6 all th nodal displacmnts ar known. Othr variabls strains and strsss ar obtaind as in th standard mannr. In th nxt sction th EAS mthod basd on th thr fild Hu-Washizu variational principl is prsntd. h EAS mthod is considrd as a gnralizd approach of th incompatibl mthod as pointd by Simo and Rifai [SIM9]..4 EAS ELEMEN Du to thir fficincy and simpl gomtry low-ordr solid lmnts ar oftn prfrrd in structural mchanics. As mntiond abov th low-ordr standard displacmnt lmnts xhibit in many cass svr stiffning ffcts known as locking. Shar locking occurs whn simulating thin-walld structurs by th low-ordr standard displacmnt lmnts whr pur bnding mods ar spoild by parasitic shar strains. Mmbran locking is ncountrd in high aspct ratio lmnts whn bnding mods

36 cannot b sparatd from mmbran strains and thus not allowing th vrification of pur inxtnsional mods. For incomprssibl or narly incomprssibl conditions volumtric locking may also occur; in this cas dviatoric mods always com along undsirably with volumtric strains. h class of EAS lmnts prsntd blow allows th systmatic dvlopmnt of low-ordr lmnts with nhancd accuracy for coars mshs. h EAS lmnts hav bn applid to simulat gomtrically and matrially nonlinar problms du to th fact that thy prform wll in svr situations as th narly incomprssibl limit and pur bnding situations. ompard with almost all finit lmnts th EAS lmnts show vry good coars msh accuracy. In gnral a lowordr fr-locking lmnt can b dvlopd basd on EAS tchniqu..4. Variational formulation Initially proposd by Simo and Rifai [SIM9] for small strains th EAS mthod which involvs th thr fild variational principl of Hu-Washizu was latly xtndd to th finit strain thory by Simo and Armro [SIM9] and opn to th thrmomchanically coupld bhavior [ADA5]. Whn incorporatd with th ANS tchniqu that assums dirctly on strain componnts a formulation in trms of th Grn-Lagrang strains is howvr mor favourabl than th on basd on th displacmnt gradint. In th light of this considration th following form of th Hu- Washizu principl is takn as th variational principl for a hyprlastic matrial: π ( π + π int xt S ( E V V com ( u = = W dv + S : [ E E] dv + u E S * * * + ( u u tda u t da u b ρdv Au Aσ V whr: - π int and π xt : intrnal and xtrnal nrgis rspctivly; - h script * dnots prscribd valus; - u : admissibl displacmnt fild; - S : admissibl scond Piola-Kirchhoff strss fild; - b t : body forc and surfac traction vctors rspctivly; - W S (E : stord nrgy function; A - (.4. com E (u : admissibl Grn-Lagrang strain fild. In th thsis th lttr com in uppr position dsignats for compatibl quantitis. In (.4. th body undr considration occupy a volum V and has th boundary = A σ A u whr σ A dnots th prscribd traction ( t * parts whil * A u dnots th prscribd displacmnt ( u ons. As th thsis concntrats on quasi-static problms hnc th inrtial forc in (.4. is not considrd. Furthrmor th body forc and surfac traction ar assumd to b consrvativ. In (.4. thr indpndnt filds ar th strain fild E th strss fild S and th displacmnt fild u. According to th EAS mthod th indpndnt strain fild is proposd as: 4

37 strain tnsor fild mod com nh ( u E E = E + E (.4. h strain tnsor nh E in (.4. is namd th nhancd assumd strain fild. h mod E is namd th modifid strain fild. Introducing th modifid strain mod E into (.4. w hav th Hu-Washizu principl as th variational basis for th EAS mthod: π π + π mod = mod : [ ( ] ( int xt = W dv + S E dv + u E S S ( E u E V V Onc th modifid strain tnsor com * * * + ( u u tda u t da u b ρdv Au Aσ V mod (.4. mod E is obtaind th gradint dformation tnsor mod F can b consistntly drivd through th us of a polar dcomposition s Sction mod.5.. In (.4. thr filds u E and S ar indpndnt whil th two filds t and S rlat togthr through th auchy s strss thorm. In ordr to pass th patch tst s dtails in Sction.4.4 th approximation of nh E and S ar chosn to satisfy th following orthogonality condition as proposd in [SIM9]: nh S : E dv = V (.4.4 Applying (.4.4 in combination with (.4. th thr fild variational (.4. rducs to a two fild variational principl as blow: π whr: δ * * nh ( = W com nh dv + ( u u t da da ρdv ( + ( S u t u b u E S E E (.4.5 V h first variation δπ nh ( ue immdiatly follows: π π π A u δ nh = δ + δ ( ue int xt (.4.6 π - int : th variation of th intrnal trm δ π int com nh = ( δ E + δ E : S V and th scond Piola-Kirchhoff strss S mod W S mod mod 5 dv Aσ mod S is givn by * V (.4.7 = (.4.8 E - δ π xt : th variation of th xtrnal trm δπ = xt Aσ * * δ u t da δ u b ρ V dv (.4.9 h orthogonality (consistncy condition (.4.4 can b intrprtd such that th variationally consistnt strss fild of an EAS lmnt is complmntary to th nhancd strain intrpolation.

38 Following th finit lmnt mthod th approximation of th currnt gomtry vctor x and displacmnt vctor u at th lmnt lvl ar rad as: h ( x ( = N ( x and u( = N ( U (.4. whr: - suprior indx rfrs to quantitis at th lmnt lvl; - for ight-nod hxahdral lmnt N ( = [ N N N... N 8] is matrix of shap functions in which: - h (.4. N I N I = N I (.4. N I with th standard trilinar shap functions as N h ( x and ( = ( + I ( + I ( + I I ; I=-8 is nodal numbr. (.4. 8 h U ar th vctors of nodal coordinats and nodal displacmnts rspctivly. hy ar dfind as follows x U h h = [ x = [ U y V z W x U y V z... W... x 8 y U 8 8 z V 8 8 ] W 8 ] ( h suprscript h rfrs to nodal valus. h displacmnt variation and incrmnt ar rquird by th linarization of th variational. h displacmnt variation δu ( ar intrpolatd as: δu( = N ( δu (.4.5 h ( h displacmnt incrmnt u( ar intrpolatd in th sam mannr: h u( = N ( U (.4.6 ( In th following sctions th suprscript in u will b omittd. h variation δ E and incrmnt E of Grn-Lagrang strain at th lmnt lvl ar intrpolatd as: δe com h = B h ( δu and com h E = B h U ( ( U U ( (.4.7 whr th compatibl strain-displacmnt matrix B is a function of displacmnts u s [ZIEb] such as: with com com com com B = B B... B ] (.4.8 [ 8 6

39 B com i u X u N v N w N X i X X i X X i X v u N v N w N Y Y i Y Y i Y Y i Y w u N v N w N Z Z i Z Z i Z Z i Z = + v u u N + u N v N + v N w N + w N Y X X i Y Y i X X i Y Y i X X i Y Y i X w u u Z Ni + X u X Ni Z v Z Ni + X v X Ni Z X Z w Z Ni X + w X N i Z w Z v Y u Y Ni Z + u Z Ni Y v Y Ni Z + v Z Ni Y w Y Ni Z + w Z Ni Y (.4.9 In ordr to simplify th notation w ar going to introduc Voigt notation. hus com componnts of Grn-Lagrang strain vctor E and PK vctor S ar transformd from th corrsponding matrics with indics as listd in th tabl blow. abl.4.: Indx transformation Matrix indx ( ( ( Vctor indx nh h nhancd strains E which will b prsntd in dtails blow as functions of nhancd strain intrpolation matrix Γ s (.4.45 and intrnal strain paramtrs α nh nh variation δ E and incrmnt E of th nhancd strains ar: E nh nh nh = Γα ; δ E = Γ δα ; E = Γ α (.4. his variation and incrmnt ar rquird by th linarization of th wak form s ( Linarization of discrt wak form Employing th Voigt notation (abl.4. w rwrit th intrnal trm (.4.7 at th lmnt lvl: δ δ π (ue nh π nh int(u E com mod + nh mod δ ( E dv 6 S 6 δ ( E 6 S 6 = V V h first variation of th xtrnal trm (.4.9 is π * * δ = δu b ρdv δu t da. xt W us V δ π int and A σ π xt 7 dv (.4. (.4. δ to formulat th linarization of th wak form by mploying th truncatd aylor sris about th k th itration: π ( π π π δ δ δ δ ( δπ ( u E ( u E nh ( δπ nh ( u E = + = + nh nh nh nh (u k+ E k+ int xt k + (u k E k nh (u k Ek δ π = + D nh (u E (u E k k k k whr D( is th Gataux drivativ oprator s Simo and Hughs [SIM98]. (.4.

40 o allviat th notation th right subscript k dsignating th itrativ indx is omittd. In ordr to calculat ( u E nh w lt th right hand sid of (.4. qual to zro th rsult is: nh ( δπ int + δπ xt = D( δπ nh ( u E (u E ( δπ int + δπ xt h ( δπ int ( ( ( ( h = U α = U α h h U α U α - (.4.4 Introducing (.4.5 (.4.7 and (.4. into (.4. w gt th intrnal virtual work as: with: whr: δπ = ( Bδ U S dv + ( Γδα S U α h mod mod h int( V V = f B S dv ; mod int V = δu f + δα f (h ( int nh = f Γ S mod nh V Introducing (.4.5 into (.4. of th xtrnal virtual work w gt: δ π dv dv (.4.5 (.4.6 (h xt = δu fxt (.4.7 f xt * = N b ρdv + V A σ N * t ρda Obsrv that th lmnt xtrnal forc vctor xprssion of th standard displacmnt lmnt. Substituting (.4.5 and (.4.7 into (.4.4 w hav: D ( δπ ( ( δπ int ( δπ ( int ( u E = U + α U α nh h int h δ ( fint ( f ( nh δ h h ( (h h ( h = U U + α U + U U ( fint ( f ( nh δ ( (.4.8 f xt (.4.8 has th usual (h ( + δu α + α α α α (h ( h (h ( = δu kuu + δα k αu U + δu ku α + δα k αα α (.4.9 h constitutiv tnsor in th physical spac is xprssd through th following strss-strain rlationship: 4 ijkl [ ] S = E (ijmod = mod kl 6 6 In (.4.9 th stiffnss matrics ar stablishd and listd hraftr. (.4. 8

41 h standard stiffnss matrix of th lmnt includs th matrial part gomtrical part k go that is k mat and f int kuu = = kmat + kgo = ( dv + h h B B B S U U 4 mod V V dv (.4. whr for a gomtrical nonlinar thory th strain-displacmnt matrix B s (.4.9 is a function of th displacmnts u. Furthrmor th strain-displacmnt matrix contains th drivativs of th shap functions with rspct to th global co-ordinats X in th rfrnc configuration. omponnts of th gomtrical part (.69 of [ZIEb]: k go ( 4 4 G G G G G G G G G 8 8 = go k is dfind as s (.4. whr G IJ ( I J = -8 is dfind for a nod combination I and J as GIJ = GIJ I ; with I is th unit matrix of dimnsion ( and: GIJ = N I K SKLN J LdV ; K L = -. (.4. V Enhancd-compatibl stiffnss matrix of th lmnt is f int 4 k u ( ku = Γ B dv α α = α = (.4.4 whr Γ is dfind in th nxt sction xprssion ( Enhancd stiffnss matrix of th lmnt is k f E 4 = = = dv Γ Γ α E α mod nh nh αα mod f V V (.4.5 ombination of (.4. (.4.5 (.4.7 (.4.9 with (.4. and (.4.4 and (.4.5 w gt th discrt linarizd systm of quations to solv for th incrmnt U h and α for mor dtails s Klinkl and Wagnr [KLI97]: k k uu αu h k U fxt f uα int = (.4.6 k α f αα nh h algorithm for solving th systm (.4.6 is listd in haptr - Figur.4.. Sinc E nh is not rquird to nhanc intr-lmnt continuity w could liminat α at th lmnt lvl bfor assmbling th lmnt matrics to th global matrics. From (.4.6 w withdraw th formula for α : h [ k ] ( + k U α = αα f nh αu (.4.7 9

42 matrix Introducing (.4.7 into (.4.6 w finally gt th condnsd lmnt stiffnss ( k and lmnt rsidual forc vctor r : k uu [ k ] [ k ] k = k (.4.8 αu αα αu r f f [ k ] [ k ] f = xt int + u αα nh α (.4.9 Assmbling lmnt matrics th global systm has th form: K U h = R (.4.4 Aftr condnsing th global systm (.4.4 has th similar form as th global systm of th standard displacmnt FEM..4. EAS paramtrs mod com nh In this sction w invstigat th modifid strain E = E( u + E undr th framwork of th EAS approach. h nhancing strain fild for an lmnt in th nh artsian coordinat systm E is usually assumd s [KLI97] by E J nh nh = E J whr: - J is th Jacobian matrix (.4.4 X Y Z J = X Y Z (.4.4 X Y Z - J is th Jacobian matrix at th cntr of th lmnt ( = ; - is th transformation matrix that maps quantitis in th physical spac to th natural spac: J J J JJ JJ J J J J J JJ JJ J J J J J JJ JJ JJ = (.4.4 JJ JJ JJ ( JJ + JJ ( JJ + JJ ( JJ + JJ J ( + ( + ( + J JJ JJ JJ JJ JJ JJ JJ JJ JJ JJ JJ ( JJ + JJ ( JJ + JJ ( JJ + JJ with J J ( i j. ij - is natural-physical transformation matrix at th lmnt cntr ( =. For nhancing strain intrpolation in th natural coordinat systm th following formulation is usd: E nh = M ( α (.4.44

43 whr α is th vctor of intrnal paramtrs. h dimnsion of α is various it dpnds on th typ of th fr-locking EAS lmnts in volumtric mmbran and shar rsponss. It is qual to th numbr of th additionally nhancd mods dfind in abl.4. blow. h natural - physical mapping is ralizd at lmnt s cntr to obtain uniqu valus for th paramtrs α i. Othrwis valus of th α i paramtrs will vary according to intgration points. hs intrnal strains fill in th availabl compatibl strain fild to allviat parasitic trms. h additional factor J / J in (.4.4 is introducd to b abl to nforc th orthogonality condition (.4.4. h justification of its us as suggstd by aylor t al. [AY76] is basd on th sam considrations which ld to th approximation of th local-global transformation at th lmnt cntr ( = i.. using rathr than th transformation. In fact th factor J / J rlats th transformation of an infinitsimal volum lmnt to th lmnt cntr as wll thus facilitating th nforcmnt of th orthogonality condition for th constant strss and strain stats i.. th patch tst s th nxt sction for a mor dtaild analyss. nh Aftr combination of (.4.4 and (.4.44 th nhancing strain fild E in th physical coordinat systm is writtn as: nh J E = M( J α = Γα J with Γ = M ( (.4.45 J h nhancing matrix M according to Andlfingr and Ramm [AND9] is dfind: η ς ης η η ης ης ς ς ης ης M= η ς ης η ης ς η ης ς ης η ς η ς ης ης (.4.46 whr ; η and ς. In (.4.46 non-zro trms in th thr uppr-rows ar applid for nhancing th additional mods of th normal strains; non-zro trms in th thr lowr-rows ar applid for nhancing th additional mods of th shar strains. For th sak of comprhnsion lt s considr th compatibl strain ε( in th natural spac for th ight-nod solid lmnt: u = a + a + a + a + a + a + a + a ( u ε com ( ( = = a + a4 + a6 + a7 (.4.47 Whn additionally nhancd by th non-zro trms in th first row of matrix M in (.4.46 th xprssion ε( will bcoms a complt tri-linar polynomial as: E = ε + E mod com nh ( ( ( = ( a + a + a + a + ( d + d + d + d (.4.48

44 whr di ( i = 4 ar componnts of vctor α. h modifid strains as (.4.48 will hlp th EAS lmnt to satisfy th incomprssibl condition and pur bnding condition bcaus thr is no inconsistnt trm (s Sction.. in th nhancd strains. EAS lmnt EASv EAS6s EAS9 EAS EAS5 EAS EAS4 EAS abl.4.: EAS lmnts Additional mods Dtaild mods v + s (only mods 5 7 in (.4.46 ar adoptd to nhancd normal strains v + 6s (mods 8 in (.4.46 ar adoptd to nhancd shar strains v + 6s (5 7 + (8 6v + 6s (5 7; (8 v + s (5 7 + (8 9 9v + s (5 7; (8 9 9v + 5s (5 7; (8 9;46 48 v + 8s (5 7;4 45; (8 9;46 48;5 54 apability Fr volumtric locking Fr shar locking Fr volumtric & shar locking Improvd incomprssibility bhavior with rspct to EAS9 Improvd bnding bhavior with rspct to EAS9 otally fr volumtric locking & fr shar locking otally fr shar & volumtric locking otally fr shar & volumtric locking applid for distortd msh h nhancing matrix M in (.4.46 whn introducd in th xprssion (.4.45 nh nh to calculat E will mak E satisfy th orthogonality condition (.4.4. In othr words this matrix M is dsignd in such a way that th EAS lmnts ar locking fr whil pass th patch tst s th nxt sction for a dtaild xprssion. onsquntly as chosn in (.4.46 th matrix M xpands th compatibl strain fild up to th complt tri-linar fild (EAS. It mans th nhancd lmnt has additional mods and 54 mods in total (4 compatibl mods and nhancd mods. Howvr th numbr of additional mods should b suitabl to ach problm so as to limit th calculation tim. h list in abl.4. givs som suggstions for rducing additional mod lmnts. Abov a formulation of th EAS lmnts in th Grn-Lagrang strains has bn prsntd. his formulation was implmntd in a MALAB cod. h numrical rsults at th nd of this chaptr show prformancs of th EAS lmnts and also assur quality of th implmntation..4.4 Patch tst h argumnts in this sction ar valid for both th incompatibl mthod (Sction. and th EAS mthod. For lmnts such as th ight-nod hxahdral th

45 mthod for driving th incompatibl shap functions N λ (Sction. for th incompatibl lmnts (or nhancing matrix M (.4.46 for th EAS lmnts is so dtail. Not only filling highr ordr trms in th standard displacmnt fild but th incompatibl shap functions also hav to satisfy th patch tst rquirmnt. hat is th ncssary condition for an lmnt to convrg to th corrct solution. nh Dsignat th spac of th nhancd strain fild as ε and th spac of com admissibl strain fild dfind in th standard fashion as ε. h nhancd strain intrpolation and th compatibl strain intrpolation ar indpndnt in a sns as: ε nh ε com = {} (.4.49 onsidr th nhancd stiffnss matrix (.4.5 and us Γ in (.4.45 w hav: 4 J 4 dv ( ( k ( ( αα = Γ Γ = M M dv J V V (.4.5 In ordr to considr positiv dfinit of positiv valus ( J k αα th constant valus ( J and can b liminatd from th xprssion. In addition rows of M s (.4.46 hav bn assumd to b linarly indpndnt. Sinc th constitutiv matrix 4 is positiv dfinit th assumption of linarly indpndnt rows of M assur th nhancd stiffnss matrix positiv dfinit. 4 k αα = Γ ΓdV (in fact th intgral V 4 ( ( V M M dv also Lt s considr rigid body motions or constant strain conditions. In nonlinar problms w considr th incrmnt quantitis. Hnc th constant strain condition is h E = const. Dnot as th st of nodal displacmnts which corrsponds to on of U th rigid body cass or a constant strain stat. hn w also hav h E = B U = const. h Dnot α as th valus of intrnal variabls in th cas of th motion U. In ordr to h pass th patch tst th EAS lmnt rquirs α to b zro whnvr U corrsponds to rigid body motions or constant strain conditions. Sinc th matrix positiv dfinit th condition for rducs to: h k ] U + f = [ α u nh 4 h mod Γ B U dv + Γ S dv = V V k is always αα α = from th scond st of quations in (.4.6 ( h mod Pay attntion that th trm ( B U is in fact som constant strss stat S that is corrspond to rigid body motions or constant strain conditions. onsquntly th rquirmnt for th patch tst to b satisfid rducs to: Γ dv = (.4.5 V alculation of th lft hand sid xprssion will b prformd in th natural spac whr th prsnc of th Jacobian (.4.4 is vidnt as: ( ΓdV = Γ J d d d (.4.5 V

46 his will gnrally lad to non-zro valu of V Γ dv and hnc th lmnt will not pass th patch tst xcpt whn th lmnt is a paralllpipd. In this lattr cas th Jacobian will consist of constants and Γ dv will b qual to zro and th patch tst will V b passd. As a rmdy it was proposd (aylor t al. [AY76] to rplac J by th constant valus computd at th origin ( = of natural coordinats as: V J ΓdV = Γ J d d d = M J d d d ( J J M( d d d J M( d d d = = = (.4.54 In fact th orthogonality condition (.4.4 did imply xprssion (.4.5 bcaus: nh : S dv = α ( ΓS dv = E V V (.4.55 W s that xprssion (.4.55 is th strong form of th patch tst condition. h condition in (.4.55 is valid for an arbitrary strss fild whil th patch tst condition is only valid for a constant strss fild S. mod h us of J to approximat J is quivalnt to th introduction of a gomtric approximation by rplacing th original hxahdral (for D: quadrilatral lmnt into a paralllpipd of th sam volum (for D: paralllogram of th sam ara..4.5 Equivalnc btwn EAS and Hllingr-Rissnr lmnts A so-calld hybrid strss mthod can b drivd from th Hllingr-Rissnr (HR principl which consists of th strss fild and th displacmnt fild. Bfor comparing with th EAS lmnts th formulation of HR lmnt is brifly introducd. From th Hu-Washizu principl in (.4. on obtains th convntional strss-displacmnt HR S 4 functional by liminating E = E = ( S. h rsult is: = + = [ S( S + SE ] dv + u S u π ( π π 4 com int xt ( V * * * ( u u tda u t da u b ρdv Au Aσ V (.4.56 h HR lmnts with compatibl displacmnt and assumd strss filds that ar built as follow: whr: h u U ( S ( = N ( H = P( h h β = P β (.4.58 S H is assumd strss at lmnt lvl; 4

47 - P ( is th matrix that nhancs th strss fild; - P = P( with is dfind in (.4.4; h - β is vctor of intrnal strss variabls. h displacmnt trial functions N ( should b compatibl ( continuous across intr-lmnt boundaris bcaus thr ar first drivativs of th displacmnt fild in th HR-functional. But strss trial functions P ar not subjct to drivation so could b chosn to b incompatibl ( continuous. his hlp for liminating th h intrnal strss variabls β at th lmnt lvl mor asily. For th sak of simplicity th following abbrviations ar dfind: 4 H = ( P ( P dv (.4.59 V G = ( P B dv (.4.6 V h lmnt stiffnss matrix of th HR lmnts is drivd from HR principl has th form as: k = ( G ( H G (.4.6 h h (.4.6 is achivd aftr liminating strss paramtrs β from th systm of quations. hn th lmnt stiffnss matrix (.4.6 can b usd for th standard displacmnt formulation. Aftr th solution procss th strsss can b obtaind at th lmnt lvl as: SH = P ( H G d (.4.6 Simo and Rifai [SIM9] gnralizd th incompatibl displacmnt mthod to th EAS mthod. Both of ths mthods ar th dual ons of th hybrid strss mthod. Hnc th EAS and th incompatibl displacmnt mthods with rspct to th hybrid strss mthod ar corrspondnt in som spcial cass. In this sction som rlationships btwn ths mthods ar introducd. hs rlationships provid a hlpful thortical basis for dvlopmnt and xploitation of th incompatibl and EAS mthods and also th hybrid mthod. It was provn by inspction [AND9] that th stiffnss matrix of th EAS lmnts is quivalnt to th stiffnss matrix of HR lmnts if th polynomials in M ( (.4.46 and P ( (.4.58 ar complmntary. It mans a polynomial trm usd for a strain componnt in M ( is not considrd for th corrsponding strss componnt in P (. onsquntly th hybrid strss and nhancd assumd strain filds ar orthogonal to ach othr as: ( P( M ( d d d = - nh or ( SH E dv = V (.4.6 5

48 6 In abl.4. a rctangl low-ordr lmnt with intgration will provid an quivalnt stiffnss matrix whthr assumd by a strain fild (EAS mthod or by a strss fild (hybrid strss mthod. abl.4.: Equivalnt bilinar EAS - HR lmnts EAS lmnt HR lmnt EAS7 : ( = M PS : ( = P EAS4: ( = M HR8: ( = P EAS: ( [ ] = M HR: ( = P.5 ANS ELEMEN Application of th EAS mthod that is dscribd in th prcding sctions to shar dformabl lmnts dos not work satisfactorily in all situations. Particularly in th cas of svrly distortd mshs or too thin structurs ths lmnts do not prform wll. h mthod that is most widly usd in ths situations is th ANS mthod. In this contxt w focus on th problm of transvrs shar locking in solid lmnts although th sam hav bn don in a similar way for shll lmnts. hr ar also a numbr of publications on application of th ANS concpt to ovrcom volumtric locking shar locking and mmbran locking. Howvr th ANS mthod is only ffctiv in transvrs shar locking rmoval [BIS97]. h ANS abbrviation mans Assumd Natural Strain. Hr strain componnts ar assumd in th natural (isoparamtric spac. h principal ida of th ANS mthod is to choos a crtain intrpolation for th transvrs shar strains instad of driving th strains dirctly from th intrpolation of th displacmnts. Hnc th mthod also namd mixd intrpolation it mans both th intrpolation for displacmnt fild and intrpolation for strain fild ar rquird by th mthod. h bilinar ANS plat lmnt is th most widly usd lmnt in both scintific and commrcial finit lmnt packags (.g. ADINA ANSYS. It is also known as MI4 lmnt (MI = Mixd Intrpolation of nsorial omponnts or Dvorkin- Bath lmnt [BA96].

49 h natural strains at an intrior point of th MI4 lmnt ar obtaind by linar intrpolations of th strains on boundary lins. Finally th physical strains which ar rquird at numrical intgration points for valuating th lmnt stiffnss and intrnal forc arrays ar obtaind by tnsorial transformation of th natural strain componnts instad of th standard isoparamtric drivativ transformations. h procdur of th natural strains and transforming tnsorially to physical coordinats has bn found to play a ky rol in improving lmnt prformanc whn th msh is distortd or curvd. h ANS mthod can also b combind with th EAS mthod to improv th inplan bnding bhavior. For an xplicit dfinition of an ANS lmnt on has to spcify two things namly: - h numbr of nods and th corrsponding shap functions for th displacmnts (which ar th standard shap functions of a displacmnt lmnt and - h numbr and location of th sampling points and th corrsponding shap functions. h ANS lmnts do not contain any spurious zro nrgy mod and show good convrgnc bhavior [PAR86]. All th ANS lmnts can b usd in linar analysis and in larg displacmnt and larg strain analyss.g. in th simulations of structural problms and collaps of shlls. h ANS was concivd as on of svral compting mthods with which to solv shar locking problms. Its most notworthy fatur is that unlik many forms of rducd intgration lmnts it producs no rank dficincy. Furthrmor it is asily xtndibl to gomtrically nonlinar problms..5. Kinmatics in natural coordinat systm h ANS mthod rquirs th intrpolation of all assumd strains in natural coordinat systm. hrfor it is ncssary to dfin a convctd dscription which naturally prsrvs th objctivity (in th convctd dscription th matrial bas vctors rflct th gomtrical and kinmatic aspcts hnc th corrsponding componnts ar indiffrnt with rspct to thir matrial bas vctors. o this nd lt s dnot th position vctors of th rfrnc configuration Ω and th currnt configuration Ω t in th local coordinat systm by X ( and x ( rspctivly. h convctd basis vctor G i and its componnts G ij in th initial basis systm ar dfind by: i G = X / ; G G = ; i = (.5. i whil th contravariant vctor i j G ij j G and its componnts ji G ar dfind following: j j j ji - G G = δ ; G = G G = G G ; ij = (.5. i i Similarly th convctd basis vctor basis systm ar dfind: g i i ji i g i and its componnts g ij in th currnt i i = x / = G + u / ; g g = ; i = (.5. i i j g ij 7

50 and its contravariant vctor xprssions: j g and its componnts ij g ar dfind through th following j j i ij - g g = δ ; g = g g = g g ; ij = (.5.4 i i j ij j h dformation gradint in th form of th convctd vctors is calculatd as: x( i F( = = g i G (.5.5 X ( whil th Grn-Lagrang strain tnsor taks th following form E = ( F F ( ( ( or altrnativly: E u j I = u i ( g ij G ij u i i G G u j j = i EijG G i j ( = Gi + G j + G G ij = j (.5.6 (.5.7 In th contxt of larg dformation th ANS mthod modifis shar componnts of Grn-Lagrang strain tnsor E. Hnc th variational quation should b writtn in matrial configuration (or total Lagrang formulation in trms of th Grn-Lagrang strain tnsor E and its nrgy conjugatd quantity S th scond Piola-Kirchhoff strss tnsor..5. lassical ANS formulation In this sction w prsnt th classical ANS tchniqu for th rmoval of transvrs shar locking in an ight-nod hxahdral lmnt. For th sak of simplicity lt s considr th cas with a rctangular prismatic gomtric configuration. It mans th physical spac (XYZ is chosn to b idntical to th natural spac ( s (Figur.5.. Instad of th standard computation which lads to shar locking th transvrs shar strains E and E ar assumd according to Dvorkin and Bath [DVO84] to b intrpolatd through th us of crtain sampling points as follows ~ E ~ E XZ YZ ~ E ~ E = ( E = ( E ( A ( D + ( + E + ( + E ( ( B (.5.8 whr E ( A E( E( B and E ( D ar th natural shar strains at points A B and D situatd on th mid-surfac of th solid lmnt (Figur.5. rspctivly. Valus of ths sampling strains can b dirctly drivd through th us of th covariant componnts in th contravariant bas vctors as by E ij in (.5.6: 8

51 E.5( g G E.5( g G E.5( g G E = = (.5.9 E ( g G E ( g G E ( g G Onc th transvrs shar strains ar assumd all assumd strain-displacmnt matrics can b immdiatly formulatd as: ~ ~ ~ ~ ~ ~ ~ ~ ~ B = [ B B B B B B B ] B8 ~ B I = ( ( N ( ( N I I g g + N + N N I I I g g g N N N I I I + N ( A ( D g g g I g + ( + ( N + ( + ( N I I g g + N + N I I g g ( ( B ; I=-8 (.5. hn th stiffnss matrix is formulatd as in th standard mannr. Exprssions (.5.8- ar only valid whn th physical spac is idntical to th natural spac i.. (XYZ (. For th gnral cas whr th mid-surfac quadrilatral is not a rctangl and th X-Y fram is not alignd to th fram th natural shar strain componnts must b firstly intrpolatd in th covariant spac as dfind in (.5.. his allows taking into account th lmnt distortion. ~ E E ( ~ E E (D 5 D 8 H 4 E (A Z O Y X 6 B 7 G E (B (- D H ( G B E A Figur.5.: ANS mthod illustration spcial cas: X-Y aligns - F E A (-- (- Figur.5.: Mid-surfac of lmnt in isoparamtric coordinats F 9

52 From (.5.8 and (.5.9 th assumd shar strains in th gnral cas can b computd by: ~ E s ~ E = ~ E ( ( g = ( ( g ( E = ( E G G ( A ( D ( A ( D + ( + E + ( + E + ( + ( g + ( + ( g ( ( B G G ( ( B (.5. whr E ( A E( E( D and E ( B ar natural shar strains valuatd by displacmnt intrpolations at points A D and B rspctivly s Figur.5.. ~ larly E is constant with rspct to and discontinuous at = ± (btwn ~ lmnts whil E is constant with and discontinuous at = ±. h strain tnsor can b quivalntly xprssd in both th natural spac and th physical (artsian spac. h transformation of th strain componnts btwn th natural spac and th physical spac is don by using th transforming matrix ( as dfind in (.4.4. In ordr to allviat th shar locking th natural shar strains E ~ ~ and E ar rplacd by th assumd natural shar strains E and E bfor doing th transformation from th natural spac to th physical spac. Finally th physical assumd strain vctor is: E.5( XX E g G E.5( YY E g G ~.5( E ZZ E g G ANS E = = = ( g G (.5. E XY E ~ ~ E ( ( ( + ( + ( ( XZ E g G A g G ~ ~ E ( ( ( ( ( g G D + + g G YZ E ( B hs assumd strains can b implmntd in th standard solid lmnt in a straightforward mannr. hs assumptions allow th lmnt to rprsnt pur bnding mods without any spurious shar ffct. h modifid shar strains lad to th nw oprator matrix ɶ B (.5. in th natural spac. Howvr th formulation can b slightly modifid so that no xplicit strain valuation at th sampling points is ncssary in th numrical calculation. h physical assumd strain-displacmnt matrix at nod I of th ight-nod solid lmnt is: 4

53 ~ B ANS I = ( ( N ( ( N I I g g + N + N N I I I g g g N N N I I I + N ( A ( D g g g I g + ( + ( N I + ( + ( N I g g + N + N I I g g ( ( B ( Som variational basis for th ANS mthod h ANS lmnt is locking-fr rank sufficint and distortion insnsitiv vn with coars mshs as it has bn pointd out by Park and Stanly [PAR86]. h ANS mthod has originally bn drivd from nginring intuition without a convincing variational background. Firstly a rstrictd form of th mthod was proposd in 969 for four-nod plan strss lmnt by assuming a constant shar strain that is indpndnt to th dirct strains [MIL9a]. In 98 Hughs and zduyar [HUG8] usd th mthod to avoid shar locking for plats; latr in 984 it was applid succssfully by Dvorkin and Bath to four-nod shll lmnt for gomtric and matrial nonlinar analysis [DVO84]. In fact th mathmatical justifications basd on th Hu-Washizu and mixd functionals hav bn providd a coupl of yars latr in sparat publications.g. Militllo and Flippa [MIL9ab]. In ANS mthod thr ar two strain filds: th drivd-displacmnt strain fild and th assumd natural strain fild. If w considr ths filds as indpndnt filds th ANS formulation can b intrprtd by a Rissnr typ functional: th functional that uss th strains and displacmnts as indpndnt filds. Dpartur from th thr filds gnral Hu-Washizu functional (.4. th displacmnts u strsss S and strains E ar indpndntly varid lt s r-writ th functional (.4.: π ( u E S = π + π int xt = + V Au W ( u S ( E * dv + V u tda S [ E Aσ 4 com ( u u t * E] dv + da V * u b ρdv (.5.4 From π ( u E S on obtains th convntional strss-displacmnt Hillingr-Rissnr functional by liminating E by: E = E S 4 = ( S (.5.5 Anothr Rissnr typ strain-displacmnt functional is obtaind by liminating S through: S = S E = which yilds: 4 E (.5.6

54 π 4 4 com * * ( u E = [ E E + E E( u ] dv + ( u u tda u t da u b dv V Au Aσ * V ρ (.5.7 Stting π π com E = E (u and π * u = u on u A rducs π ( ue to th potntial nrgy functional com 4 com * ( u = int + xt == ( E( u E( u dv u t da u b dv V Aσ * V ρ (.5.8 Partial Strain Assumption It is common practic to assum only a part of th strains to b indpndnt filds. For instanc with MI4 lmnt indpndnt assumptions ar only mad for th transvrs shar strains [BA96] whras th bnding strains ar ntirly drivd from displacmnts: E = [ E ] a E b (.5.9 com whr E a stands for th assumd strain fild and E b = E b stands for th drivddisplacmnt (bnding strain fild. h π ( ue functional (.5.7 rquirs obvious modification in th volum trm: π ( u E a = V [ E a 4 Eb ] 4 + aa ba Au ( u * 4 4 ab bb E u tda.5e a dv +.5Eb com a Aσ u t * da V * u b ρdv (.5. h rsulting principls tak a particularly simpl form if th constitutiv coupling trms ab and ba vanish in that cas: π ( u E = π a( u E + π b( u + π a a xt (.5. whr π a( ue a is a mixd strain-displacmnt nrgy involving E a ; π b( u is a potntial com nrgy involving th E b ; π xt is th xtrnal nrgy. Up to now a compatibl displacmnt fild and a discontinuous strain fild ar involvd. Hnc us π ( ue a is a suitabl functional for th ANS mthod. h lmnt displacmnt fild is intrpolatd as: u = N U (.5. c h c whr N c is th compatibl shap functions as dfind in (.4. displacmnt vctor as dfind in (.4.4. h strain filds drivd from th displacmnts ar: - bnding strains: - shar strains: h U c is th nodal E com = B c U h (.5. b b c E com = B c U h (.5.4 s s c 4

55 c c whr B b and B s ar parts of strain-displacmnt matrix that rlat to bnding strains and shar strains rspctivly (th lttr c stands for compatibl valus. h indpndnt strains in π ( ue a ar: - bnding strains as th drivd-displacmnt bnding strains (.5.; - shar strains (th suprscript a stands for ANS valus: E = B a (.5.5 a a s a with B s is natural assumd strain-displacmnt matrix and a is strain cofficint vctor. Introduc (.5. - (.5.5 into (.5. and carrying out th intrpolations at th lmnt lvl w will hav: whr h cc h aa ca h c π h = ( ( U + ( ( c c K U U a c a K a Uh K a - Uc f (.5.6 K K cc ca = = c 4 c aa a ( Bb b Bb dv ; K = ( Bs V c 4 a c ( Bs s Bs dv ; f = ( N c b* dv + ( N c t* da V On prforming th variations w obtain th matrix quation: K ( K cc b ca K - K ca aa U a h c f = c 4 V V From th scond quation of (.5.8 w obtaind th shar strain cofficints: a = ( K aa ( K ca U h c = Q U c h c Introduc (.5.9 into (.5.8 givs th statically condnsd systm: 4 s B a s dv σ A (.5.7 (.5.8 (.5.9 cc aa h c ( Kb + Qc K Qc Uc = f (.5. In (.5. cc b potntial nrgy principl. Whil K is th bnding stiffnss matrix which is also obtainabl from th aa Q c K Qc stands for th nw shar stiffnss matrix. h systm of quations (.5. now contains only nodal displacmnt vctor standard displacmnt mthod. h U c as in th A variational justification of th ANS formulation as prsntd abov has bn don by Militllo and Flippa [MIL9ab]. his study is basd on two hybrid xtnsions of Rissnr-typ functional that uss strains and displacmnts as indpndnt filds. Howvr th work of Militllo and Flippa is not applicabl to all typs of matrial modls. h matrial is firstly rquird to b dcoupld as in (.5.. Furthrmor th proposal of Militllo and Flippa currntly is only valid to transvrs shar locking rmoval. Manwhil th ANS mthod could apply for anothr locking ffct such as curvatur thicknss locking (haptr. By ths argumnts w s that variational bas of th ANS mthod is still an opn problm. Howvr as it has bn pointd out [BA96] that a variational basis of an lmnt might not xist but whthr th lmnt is

56 usful and ffctiv can of cours b dtrmind only by a dpr analysis of th formulation. Advantags of th ANS mthod ar simplicity whil rmaining ffctiv. h ANS lmnts could b applid for both structural (plat shll and continuum (solid lmnts. Numrical rsults in litratur show that ANS lmnts ar locking-fr rank sufficint and distortion insnsitiv vn with coars mshs. Furthrmor th mthod is asy to b implmntd in any cod. Bcaus of its attractions th ANS mthod has bn bing dvlopd by many authors as Dvorkin and Bath [DVO84] Park and Stanly [PAR86] Btsch and Stin [BE95] Bath t al. [BA] tc..6 NUMERIAL RESULS his sction invstigats prformancs of th EAS and ANS lmnts. In th following tsts th low-ordr standard solid lmnt is dsignatd as Q. Whil th standard solid lmnt Q which mploys th classical ANS tchniqu [DVO84] for allviating transvrs shar locking and curvatur locking is dsignatd as ANS. h EAS lmnts ar dsignatd as EASx whr x is th numbr of intrnal paramtrs. h additional lttr D. stands for lmnts in D without this additional lttr mans lmnts ar in D. h ANS EAS lmnts usd in th following tsts ar implmntd in a MALAB cod according to th thoris prsntd in this chaptr..6. Mmbran patch tst Lt s considr a patch tst as suggstd by McNal and Hardr [MA85] and originally aimd to chck th mmbran bhavior of plat and shll lmnts. In ordr to adapt to D lmnts th numbr of nods has bn doubld (Figur.6. as Vu-Quoc and an [QUOa]. An imposd displacmnt fild at th boundary nods is chosn to caus a constant strss fild in th plat. abl.6.: Intrior nodal coordinats X Y Z -h/ -h/ -h/ -h/ h/ h/ h/ h/ Practically considr a rctangl plat of dimnsions L W h =.4... h matrial paramtrs ar takn as E = 6 and ν =.5. In th original problm dsignd for plan strss problms McNal and Hardr [MA85] mployd th following boundary conditions: u = ( X + Y / ; v = ( Y X / + (.6. which lad to th corrsponding rfrnc solutions of constant strains and strsss: 44

57 ε ε = γ = ; X = Y XY σ X = σ Y = ; τ XY = 4. (.6. W h 4 Y Z X L Figur.6.: Mmbran patch tst Motivatd by this rsult th abov boundary conditions (.6. ar also adoptd for th modifid doubld-surfac mmbran patch tst. Additionally th condition w = has also b imposd to a bottom nod.g. nod to prvnt rigid body motions. Bcaus th plat is too thin L/h=4 it is possibl to assur a constant strss stat in th plat whn apply ths conditions. Numrical rsults show that th ANS lmnt passs th modifid patch tst; i.. th computd strsss ar constant all ovr th plat and consistnt with (.6.. h computd displacmnts of intrior nods also fully agr with (.6.. W also not that th EAS lmnts also pass this patch tst as prviously rportd in [KLI6] [QUOa]..6. Out-of-plan bnding patch tst Lt s r-considr th abov plat Figur.6. but in bnding situation. Again a patch tst for D lmnts (plats and shlls is xtndd for D lmnts. In ordr to crat a constant strss stat th original boundary conditions s [MA85] for th displacmnt w and rotations θ at th rfrnc surfac of th plat ar: ( w = X + XY + Y ( / ; ( x θ y w w θ X = = + θ = = + Y X ( Y X / ; Y ( X Y /. (.6. h boundary conditions (.6. ar not dirctly applicabl to th solid lmnts bcaus thy contain th rotations. If th cross sctions of th plat ar assumd plan (it is plausibl bcaus th plat is thin th boundary conditions can b modifid as in [QUOa] in such a way that: w = w = ( X + XY + Y / ; (

58 h h u = ± Y Y θ = ( X + / ; h h v = = ( Y + X / θ X ; i.. diffrnt displacmnts ar imposd to th xtrior nods on th top (uppr sign and bottom (lowr sign surfacs of th plat rspctivly. h thortical strsss s [MA85] at th top and bottom surfacs of th plat ar: σ σ = ±.667 ; τ = ±. (.6.5 X = Y XY abl.6.: Displacmnts of th intrior nods Nod u v w abl.6.: Normalizd displacmnts at intrior nods Nod Displacmnt ANS EASv6s EASv8s 5 u v w u v w 7 u v w 8 u v w As also alrady rportd by Vu-Quoc and an [QUOa] all EAS lmnts ar unabl to convrg to th analytic solution listd in abl.6.. h displacmnts of th EASv8s in abl.6. ar idntical to thos rportd by Vu-Quoc and an [QUOa]. his hlps to valuat th quality of our EAS implmntation in th MALAB cod. In contrast th classical ANS nabls th xact solution. his prsnts th suprior prformanc of th ANS tchniqus ovr th EAS ons in th rmoval of shar locking and in working with distortd msh s abl Eignvalus analyss of a rctangl his is on of th basic tsts. h ignvalus of stiffnss matrics of rgular mshs ar calculatd. W xamin a singl lmnt of rctangl shap with lngth is qual to unit matrial proprtis ar Young s modulus E = and Poisson s ratio ν =. In ordr to chck whthr th lmnt is fr from shar locking ignvalus of pur bnding mod of th stiffnss matrix ar analysd. As rfrncs lt s tak fr- 46

59 shar locking lmnts as D.EAS4 (s abl.4. basd on th EAS mthod proposd by Simo and Rifai [SIM9] and DSG lmnt proposd by Bischoff t al. [BIS] to compar with th D.ANS lmnt..e+5.e+4 Eignvalu.E+.E+.E+.E+.E- D.ANS/D.EAS4/D.DSG D.Q Lx Ly Aspct Ratio Lx/Ly Figur.6.: Eignvalu analyss of lmnt stiffnss matrics in-plan bnding mod h Poisson s ratio is st to zro ν = in ordr to prvnting th lmnt from volumtric locking and Poisson thicknss locking. h rsults show that with various aspct ratios (lngth is fixd thicknss is changd th D.ANS lmnt displayd rigid body mods and no spurious zro nrgy mods. Figur.6. shows that bhavior of th prsnt D.ANS solid is compltly coincidnt with rsults givn by D.EAS4 and DSG. It mans all of thm ar fr from shar locking at high aspct ratio. Manwhil th standard lmnt D.Q shows stiffr (locking bhavior whn aspct ratio incrasing..6.4 ircular cantilvr bam at larg displacmnts h following xampl shows th applicability for thin D-bams. Lt s considr a circular cantilvr of dimnsion R = α = 45 and cross sction s Figur Matrial paramtrs ar lastic modulus E = and Poisson ratio ν =. h cantilvr is clampd at on nd and loadd by a forc P at th othr xtrmity. Z Y R P X Figur.6.: ircular cantilvr 47

60 h problm has bn modld by Slavkovic t al. [SLA94] who usd 8 intrnal strain paramtrs to nhanc th solid lmnt. h problm has also bn modld by Klinkl t al. [KLI97] who also usd EAS solid lmnt. Both th nhancd lmnts of authors in [SLA94] and [KLI97] ar EAS lmnts but nhancd by diffrnt mods. Howvr ths lmnts ar shar and volumtric locking fr. h larg dformation rspons will b calculatd for diffrnt vrtical tip loads. In this tst bhavior of th classical ANS mthod is vry attractiv. Figur.6.4 shows that displacmnts of th cantilvr discrtizd by only ANS lmnts ar quit comparabl to th rsults of Slavkovic and Klinkl with 6 lmnts. h ANS lmnt is softr in displacmnt u (curv 5 but stiffr in displacmnt v (curv 6 whil giv a wll approximatd displacmnt w (curv 7 to th rfrncs [SLA94] and [KLI97]. his is th rsult of shar locking rmoval in thicknss dirction (z. Applid forc P (N u (EAS v w w (Q Klinkl; Slavkovic (6 lmnts D.ANS-u D.ANS-v D.ANS-w ( lmnts Figur.6.4: urvd cantilvr bam: displacmnts ntr displacmnt w (mm.6.5 Scordlis-Lo roof with rigid nd diaphragms onsidr a shll of radius R = 5 thicknss t =.5 lngth L = 5 and opn angl α = 4 undr a gravity load p = 9 (pr unit ara distributd on th shll surfac (Figur.6.5. Both nds of th shll ar constraind with only a fr movmnt in th axial 8 dirction. h matrial paramtrs ar: E = 4. and ν =.. h vrtical dflction of th mid-sid fr dg v =.4 is takn as th rfrnc solution (McNal and Hardr [MA85]. Du to th symmtry of th structur only a singl quartr of th shll is modld. Diffrnt typs of discrtization ar considrd togthr with various lmnts (Figur.6.6. h ANS lmnt dlivrs a good solution with a rathr coars msh (4 4. In contrast th EAS9 lmnt which has 9 incompatibl mods rquirs a finr msh (6 6 to rach th corrct solution. 48

61 Z X Y Fr dg p L Rigid diaphragm R Figur.6.5: Scordlis-Lo roof.6 Normalizd dflction ANS EAS9 EAS5 Elmnt pr sid Q Figur.6.6: Scordlis-Lo roof: convrgnc of finit lmnt solution With an incras of nhancd mods th EAS5 lmnt appars to giv a bttr rsult mor thortically xpctd than th EAS9 lmnt. Howvr instad of starting from a low valu of displacmnt at coars mshs and thn progrssivly incrasing this valu with th rfinmnt of msh as sn for th EAS9 lmnt th EAS5 lmnt givs first a highr valu of dflction at coars mshs and thn lowr valus du to a msh rfinmnt. his can b xplaind by th ffcts of th high-ordr incompatibl mods (mods 4-9 s (.4.46 thy mak th EAS5 soft by fairly allviating th shar locking. As rgards th Q standard lmnt a rathr slow convrgnc is found. Obviously th shar locking contributs to this bhavior. 49

62 .6.6 Rgular block with narly incomprssibl matrial In ordr to invstigat th prformanc of th ANS and EAS lmnts in volumtric locking conditions a rgular block of dimnsions 5 clampd at bottom and loadd by a uniform prssur of q = 5/unit ara acting on a top ara of at th cntr is considrd [AND9] s Figur.6.7. h matrial has an lastic 5 modulus E =. and Poisson ratio ν =.4999 i.. narly incomprssibl matrial. Du to symmtry only a quadrant of th block is modld by a msh of lmnts. q 5 z O y x Figur.6.7: Rgular block In abl.6.4 th vrtical displacmnt simulatd by diffrnt lmnts is listd. ak th rsult of th fr-volumtric locking SRI lmnt as th rfrnc. Obviously th volumtric locking rspons is obsrvd with th standard lmnt. Sinc volumtric locking cannot b rmovd by th ANS tchniqus th ANS lmnt is narly as stiff as th standard lmnt Q in this problm. abl.6.4: Vrtical displacmnt at th block s cntr Elmnt Q ANS EASv6s EAS9v EAS EAS SRI w w/w SRI ompar to th EAS w s that EASv6s with volumtric mods is rathr stiff. h EAS lmnt with 9 volumtric mods givs as good rsult as th EAS. Not that th shar mods can assist th volumtric mods in volumtric locking rmoval. It xplains why th locking rspons can b mor rmovd with th additional introduction of shar nhancd mods. For xampl th EAS with shar mods bsids 9 volumtric mods givs a quit bttr rsult than th EAS9v with uniqu 9 volumtric mods. h SRI tchniqu shows a bttr prformanc in rmoving th incomprssibl locking in comparison with th EAS tchniqu. 5

63 ONLUSION hrough th numrical tsts w s that th transvrs shar locking tratmnt is idally suitd by th ANS mthod. In th cas th Poisson s ratio is diffrnt from zro th ANS mthod givs lss accuracy but always rmains too much bttr than th standard displacmnt-basd mthod (s Sction.6.5 for xampl. h EAS mthod is also usful to shar locking rmoval but computational cost of th EAS mthod is mor xpnsiv than th ANS bcaus th EAS mthod rquirs calculation of intrnal variabls. h volumtric locking is ffctivly rmovd by th EAS mthod. h SRI is compltly suitabl for volumtric locking rmoval but this mthod cannot pass th patch tst (s Appndix in th nxt pag so it is not takn into considr in th thsis. Howvr th SRI lmnts ar still usd by many authors bcaus of its simplicity. hr was a combination btwn mthods of ANS with EAS proposd by Andlfingr and Ramm [AND9] for four-nod dgnratd-shll lmnts. In ordr to improv th lmnt prformanc Andlfingr and Ramm us th EAS mthod for th mmbran and bnding componnts whil th transvrs shar componnt is formulatd according to th ANS mthod. o allviat th shar locking th shar strains ar rfrrd to natural coordinats. Nxt th combination of Andlfingr and Ramm was continually xtndd to gomtrically nonlinar Rissnr-Mindlin shll by Bischoff and Ramm [BIS97]. Authors in [BIS97] did apply th ANS mthod for avoiding curvatur thicknss locking by us of intrpolation functions for transvrs normal strains at nodal points instad of an valuation at th intgration quadratur points. Othr xampl of EAS-ANS combination rsults in a simpl shll modl built dirctly from basis of ANS shll modl of Dvorkin and Bath whr th EAS tchniqus ar intgratd for mmbran locking was introducd by Slavkovic t al. [SLA94] tc. Othr succssful combination of th EAS concpts and ANS mthod is th gnrating of th solid-shll lmnts. h solid-shll concpt was built on modifying assumptions of th standard shll thory (Rissnr-Mindlin shll modl. In fact th solid-shll lmnts form a class of finit lmnt modls which ar intrmdiat btwn thin shll and convntional solid lmnts. Dtail for th solid-shll lmnts ar prsntd in th chaptr following. 5

64 APPENDIX of haptr Fr-volumtric locking lmnt - SRI lmnt If narly incomprssibl matrial bhavior is us th lmnts suffr from volumtric locking. Using th SRI mthod is on of th bst ways for liminating volumtric locking [PON95]. By th way us th SRI mthod with rducd intgration for volumtric part and full intgration for dviatoric part assurs that all thr normal strains satisfy th volumtric constraint ε =. ii onsidr th SRI mthod starting from th potntial nrgy functional (.5.8: π ( u W com dv + π S( E xt (A. = V h first variation of th functional is: V δe : SdV + δ π xt = (A. W split th strain E and strss S of (A. additivly into volumtric (dilatational parts and dviatoric parts. h volumtric parts ar: Ev Sv = E I i = - ii = S I i = - ii And th dviatoric parts ar: E = E E d S = S S d v v (A. (A.4 (A.5 (A.6 Rwriting th xprssion (A. with tnsors dcoupld into volumtric and dviatoric parts w hav: δ Ed : SddV + δ Ev : SvdV + δ Ed : SvdV + δ Ev : S ddv + δπ xt = (A.7 V V V V Assum that: δ E : S dv = δ E : S dv = d v v d V V So th wak form (A.7 bcoms: δ E : S dv + δ E : S dv + δπ = d d v v V V xt (A.8 (A.9 h scond intgrand of th xprssion (A.9 is calculatd by th rducd intgration but this is only applid for in-plan componnts intgration instad of whil th intgration in thicknss dirction rmains unchangd. Rstriction of th SRI mthod is that th applid matrial laws must allow a dcoupling strss fild and strain fild into volumtric parts and dviatoric parts. It mans th tangnt moduli tnsor 4 could b also split into a volumtric part and a dviatoric part as: 5

65 4 = 4 d + 4 v (A. As mntiond abov th SRI is th bst mthod to rmov volumtric locking. Its svr problm is cannot pass th patch tst. In ordr to undrstand th problm lt s considr a cubic patch tst. h patch tst for convrgnc is a fascinating ara in th dvlopmnt of nonconforming finit lmnt mthods. It has bn intuitionally proposd by Irons sinc th mid-96s. By th arly 97s th tst had bcam a powrful and practical tool for valuating and chcking nonconforming lmnts. W considr th following xampl: a unit cub modld by svn lmnts - distortd msh [MA85]. Matrial paramtrs ar: lastic modulus E = 6 and Poisson ratio ν =.5. Y X Z Figur A. ubic patch tst abl A: Location of innr nods Nods oordinats X Y Z h outr nods subjct to following conditions: u = - (X+Y+Z/ v = - (X+Y+Z/ w = - (X+Y+Z/ (A. 5

66 hs conditions assur a uniform strain in th cubic. Rfrnc solution of th problm is analytically drivd as: ε X = ε Y = ε Z = γ XY = γ YZ = γ ZX = - σ X = σ Y = σ Z = (A. τ XY = τ YZ = τ ZX = 4 h D lmnt patch tst is usd to vrify whthr volum lmnts can xactly rproduc a constant strain stat for any configuration. If this is th cas thn th lmnt will convrg to th analytically xact solution (assuming that th matrials ar lastic and dformations ar small as th msh is rfind. Dpnding on th lmnt typ and problm howvr convrgnc may b too slow for practical purposs. h valus of strss at th Gauss points obtaind with th EAS9 ar prsntd in abl A. Elmnt σ X = σ X vol. + σ X dv. abl A: Rsults of EAS9 τ XY τ XZ σ X vol. σ X dv E E E E E E E Rf. 4 4 A quit similar trnd is obsrvd with diffrnt schms of nhancd mods (EAS5 EAS tc. In contrast with th SRI lmnts whr th hydrostatic strss σ vol. is spurious and hnc ths lmnts fail to pass th patch tst th EAS lmnts rally pass th patch tst s Figur A σ z is spurious with SRI Figur A. Strss in patch lmnts (MEAFOR [ME8] 54 σ z is smooth with EAS As statd by McNal and Hardr [MA85] if an lmnt producs corrct rsults for th patch tst th rsults of any problm solvd with that lmnt will convrg

67 toward th corrct solution as th lmnts ar subdividd. Many authors supposd that an lmnt that dos not pass th patch tst should not b trustd. On th othr hand passing th patch tst dos not guarant satisfaction sinc th rat of convrgnc may b too slow for practical us. Howvr in th thsis th SRI lmnt is not furthrmor considrd bcaus mploymnt of th SRI tchniqu may b th caus maks th solidshll lmnts cannot pass th bnding patch tst s [AR7] and [REE7]. 55

68 haptr. SOLID-SHELL ELEMENS FOR FINIE DEFORMAION INRODUION For th analysis of nonlinar mchanical bhavior of structurs low-ordr lmnts ar widly applid bcaus of thir fficincy and simpl gomtry. Howvr th standard pur displacmnt lmnts usually xhibit svr stiffning ffcts known as locking. oncrtly in thin-walld structurs pur bnding mods ar spoild by th parasitic shar strains shar locking occurs. For narly incomprssibl matrials and incomprssibl conditions volumtric locking occurs whn dviatoric mods com along undsirably with volumtric strains. In this work w attmpt to dvlop a lowordr solid-shll lmnt which is fr from all kinds of locking ffcts. Indd transvrs shar locking and curvatur thicknss locking can b circumvntd by using th ANS mthod whil mmbran locking and volumtric locking can b rmovd by th EAS mthod. h solid-shll lmnts ar wll applicabl for gomtrically nonlinar problms ([HAU98] [QUOab] or for both gomtrically and matrially (lastoplastic nonlinar problms ([HAU] [AN5] [JE8]. In comparison with othr shll lmnts th solid-shll lmnts nabl an asy connction with othr continuum lmnts du to thir solid topology (Figur... h solid-shll lmnts hr rfr to th finit lmnt modls which ar applicabl to shll analyss and possss no rotational DOF s. hy ar diffrnt from th dgnratd shll lmnts in th sns that th lattr lmnts ar quippd with both translational and rotational DOF s. h solid-shll lmnts hav th charactristics of solid lmnts whr strains can b xtndd up to complt trilinar filds by intrnal paramtrs [AND9] or can b naturally assumd. hr ar svral advantags of th solid-shll lmnts compard to th dgnratd shll lmnts. First th solid-shll lmnts ar simplr in thir kinmatic and gomtric dscriptions. Scond no spcial ffort is rquird for matching th translational and rotational DOF s whn a structur consists of both solid and thin-walld rgions. h laborious task of dfining algbraic constraints or introducing solid-to-shll transition lmnts can b xmptd. hird th complication on handling finit rotational incrmnts can b avoidd. Nvrthlss formulating th robust solid-shll lmnts is indd mor dmanding than formulating th dgnratd shll lmnts. Howvr th lattr lmnts ar only plagud by shar and mmbran locking ffcts whil th formr lmnts ar also bothrd by Poisson thicknss and trapzoidal (curvatur locking ffcts s Sction.. Starting from th principl of shar locking rmoval by th availabl ANS mthod [DVO84] an ANS tchniqu with an altrnativ schm of sampling points which can b mployd for th solid-shll lmnts is invstigatd. In fact svral ways can b mployd for th intrpolation of natural strains such as linar intrpolations [DVO84] quadratic intrpolations [HAU] [PAR86]. In Sction. an altrnativ bilinar intrpolation is introducd. 56

69 abl..: Dominant faturs of dgnratd shll and solid-shll Dgnratd shll h kinmatic DOF ar th componnts of th displacmnt vctor and of th xtnsibl dirctor vctor of th rfrnc surfac. Locking ffcts (Poisson thicknss locking which occur if a D matrial law is usd along with constant normal thicknss strains can b avoidd. Solid-shll h kinmatic dgrs of frdom ar th componnts of th displacmnt vctor. h strsss ar valuatd from a D matrial law. his fatur is spcially usful for complicatd nonlinar constitutiv quations.. LOKING PHENOMENA WIH SOLID-SHELL h wll-known locking phnomnon of displacmnt basd finit lmnts for thin-walld bams plats and shlls is causd by unbalancs of th trial functions. his unbalanc dscribd in innumrabl paprs can b curd by ithr rduction or nhancmnt of th DOF to strngthn th intrpolations of variabl filds. With solid lmnts thr ar ssntial locking ffcts as: Shar locking; Poisson thicknss locking; urvatur thicknss locking (Sction..; Volumtric locking. Manwhil for shll lmnts thr ar also som shll-typical locking phnomna to fac with: ransvrs shar locking; Mmbran locking. h solid-shll lmnts ar solid lmnts whr th shll lik-bhaviors ar intgratd so th lmnts can b copd with thin-walld problms. With th assumption of straight normal to th lmnt mid-surfac th solid-shll formulation can tak into account transvrs shar ffcts. If th solid-shll lmnts ar applid for simulation of thick structurs th locking ffcts thy mt ar similar to locking ffcts happn with solid lmnts. On th contrary bing applid for thin and curv structur th solid-shll lmnts bhav similarly to shll lmnts so th anti-locking tchniqus for shll lmnts would b usful. In gnral mthods to rmdy locking ffcts may b classifid as follows: Incompatibl displacmnt modls by Wilson t al. [WIL7] and aylor t al. [AY76] dsignd by th xtnsion of th trial functions through additional incompatibl mods. ANS lmnts by Hughs & zduyar [HUG8] Dvorkin and Bath [DVO84]. 57

70 Assumd strss lmnts by Pian [PIA86] basd on th Hllingr-Rissnr functional to nrich th strss spac. Rducd or slctd rducd intgration tchniqus.g. [DOL] to clar th parasitic strsss by a modifid numrical intgration. EAS lmnts by Simo and Rifai [SIM9] Simo and Armro [SIM9] basd on th Hu-Washizu functional and th xtnsion of th strain tnsor or th matrial dformation gradint by additional trms. h solid-shll lmnt would suffr transvrs shar locking and mmbran locking as th dgnratd shll if w do not apply any on of th rmdial mthods as: ANS EAS RI or assumd strss mthod. On th othr hand th solid-shll lmnts would show volumtric locking as solid lmnt if plasticity in ithr small or larg displacmnt occurs. o ovrcom thos th solid-shll formulation has to adopt th abov mthods. In ordr to improv th lmnt prformanc it mans rmov locking ffcts th EAS mthod could b usd for th mmbran and bnding strains whil th transvrs shar strain componnts ar formulatd according to th ANS mthod. In haptr w did discuss about svr locking ffcts (volumtric locking Poisson thicknss locking and shar locking that happn to th low-ordr solid and of cours also happn to th solid-shll lmnts. Hraftr undr mchanical point of viw (Sction.. othr locking phnomna happn to th solid-shll lmnts ar continuous discussd... urvatur (trapzoidal locking A furthr locking ffct obsrvd for solid lmnts is th phnomnon of a so calld curvatur locking or somtims trapzoidal locking. h phnomnon is only found in structurs whr th out-of-plan lmnt dgs ar not prpndicular to th mid-layr which is th cas for originally curvd as wll as for havily dformd structurs. Howvr curvatur locking only occurs whn th lmnts includ thicknss strains. It mans continuum lmnts and som xtnsibl-dirctor dgnratd shll lmnts suffr this locking. Othr dgnratd shll lmnts whos thicknss strains ar qual to zro do not suffr this locking. In brif curvatur locking happns whn th two following factors ar prsnt in th sam tim: Whn lmnt modls includ normal strains in thicknss dirction and h out-of-plan lmnt dgs ar not prpndicular to th mid-layr it will activat incompatibl normal strains in thicknss dirction. rapzoidal locking is th last nvisiond dficincy in th solid-shll lmnt dvlopmnt. his ignoranc is probably du to th fact that curvatur locking dos not occur in dgnratd shll lmnts and flat plat gomtry. urvatur locking was first put forward by McNal [MA87]. Lt s considr a simpl cas of four nod lmnt trapzoidal shap as Figur... In th artsian coordinats th lmnt has th hight H = ; th avrag width W = H Λ whr Λ is th lmnt aspct ratio. h trapzoidal lmnt rlats to th isoparamtric spac by: X = Λ ( - α ; Z = ; Λα = tan(δ (.. 58

71 Λ(-α δ H = Z X δ Λ(+α a In artsian spac b In isoparamtric spac Figur..: Four nod lmnt onsidr th analytical displacmnts for in-plan bnding s (..5. For th sak of simplicity w assum M/EI = ; ν = and w do not considr th constant trm in vrtical displacmnt. Finally th solutions ar: u M = XZ w M (.. = - X²/ or in th form of isoparamtric coordinats u M = Λ[ - α ( ] w M = - Λ² [( - α( + (α (.. ]/ h displacmnt within th lmnt is calculatd s [MA94] as: u a = Λ( - α w a = - Λ²( - α (..4 + α²/ h compatibl strains in th lmnt ar: a ( α ε X = ( α ε a Z = Λ α a α( α γ XZ = Λ + ( α Whil th analytical strains ar: M ε = ε X M Z = (..5 (..6 M γ XZ = In th cas of rctangular α = th strain componnts ε X and ε Z ar corrct. In cas α th rror in ε X is small for α small. h rror in shar trm γ XZ (causs transvrs shar locking can b liminatd by th ANS mthod whn shar strain is intrpolatd by strains at sampling points in ordr to assur shar strains ar qual to zro in pur bnding s haptr. Hr w can s that apart from th inconsistnt trms (s Sction... that caus transvrs shar locking th distortd msh also causs a similar ffct. his locking ffct bcoms svr du to parasitic strain ε whn th curvatur of th structur is high with rspct to th thicknss i.. whn tan(δ = Λα >> a Z 59

72 . urvatur locking is a consqunc of gomtric irrgular (distortion on analysis accuracy. urvatur locking shows up if D or D solid lmnts ar usd to modl curvd thin-walld structurs (th nam trapzoidal locking rflcts th fact that in ths cass th individual lmnts hav a trapzoidal shap. Whn shll lmnts ar built on xtnsibl dirctor kinmatics and ar incorporatd unmodifid D constitutiv modls ths lmnts will also show svr locking bhavior in th cas of curvd-thin shll structurs. W mak conclusions that whn multipl trapzoidal lmnts ar usd to modl bnding problms th transvrs bnding strss/strain mod which should physically vanish is most dtrimntal to th lmnt accuracy and lads to a dficincy. In othr words th lmnt accuracy drops substantially if trapzoidal mshs ar usd. h obliqu dgs activat parasitic normal strains in thicknss dirction and may lad to locking. h ffct happns only for curvd structurs and is svr for thin solid and solid-shll lmnts. On mthod to rsolv this problm is using a naturally assumd strain intrpolation of th normal strain in thicknss dirction as proposd by Bischoff and Ramm [BIS97] Btsch and Stin [BE95]. h dtail formulation is prsntd in Sction.4 blow... Mmbran locking In ordr to undrstand mmbran locking it is ncssary to distinguish xtnsional bnding and inxtnsional bnding. h trm "inxtnsional bnding" rfrs to a class of plat and shll problms in which th potntial nrgy is dominatd by flxural strains as opposd to xtnsional strains. It mans th in-plan strains (ε X ε Y and γ XY bcom vanishingly small whn compard to th bnding strains (ε Z. In contrast th trm xtnsional bnding is rfrrd if th mid-surfac xprincs significant strtching or contraction; also calld combind bnding-strtching or coupld mmbranbnding. Mmbran locking also known as inxtnsional locking dos only occur in curvd bam and shll lmnts whn th curvatur is larg. It is somtims confusd with shar locking and volumtric locking bcaus ths affct th mmbran part of shll lmnts. Howvr thy ar compltly diffrnt phnomna. l = s/l s R w u Figur..: urvd bam lmnt For th sak of simplicity in ordr to undrstand mmbran locking asily lt s considr a curvd bam lmnt of lngth l and radius of curvatur R basd on classical thin bam thory s Figur... h displacmnt dgrs of frdom rquird ar th circumfrntial displacmnt u and th radial displacmnt w. h coordinat s follows 6

73 th middl lin of th curvd bam. h mmbran strain ε and th bnding strain χ ar dscribd by th strain-displacmnt rlations [PRA]: ε = u s + w/r (..7 χ = u s /R - w ss Bas on drivativs of (..7 obviously a dscription for u and a dscription for w ar rquird. Kinmatically admissibl displacmnt intrpolations for u and w ar: u = a + a w = b + b + b + b (..8 whr a to b cofficints ar th gnralizd DOF s which can b rlatd to th nodal dgrs of frdom U W and W s at th two nods. h strain fild intrpolations can b drivd as ε = (a /l + b /R + (b /R + (b /R (b /R (..9 χ = (a /Rl - b /l (6b /l Whn th abov curvd lmnt is applid for simulating an inxtnsional bnding problm th physical rspons rquirs that th mmbran strain (ε tnds to vanish. It mans: a /l + b /R = b /R = b /R = b /R = (..a (..b (..c (..d W can obsrv that constraint (..a has trms participating from both th u and w filds. It can thrfor rprsnt th condition (ε in a physical way. Howvr th thr rmaining constraint (..b to (..d hav no participation from th u fild. Lt s xamin what ths thr constraints imply for th physical problm. From th thr constraints w hav th conditions (b (b (b. Each of ths th conditions in turn implis following conditions (w s (w ss and (w sss. hs ar th spurious constraints du to b b and b must b diffrnt from zro s (..8. onsquntly th xist of b b and b causs mmbran locking. Apart from th EAS mthod thr ar two ways ar popularly applid for mmbran locking rmoval. Us high-ordr approximations for in-plan displacmnts or us RI tchniqu for inplan strains [SO8]. h first mthod rquirs a dramatical rduction of tim stp for xplicit tim intgration whil th scond mthod may caus spurious mods. Mmbran locking dos not occur in flat lmnts.g. th four-nod quadrilatral shll only manifsts mmbran locking in wrappd configurations. With th solid-shll lmnts du to th coupling of th transvrs normal strains with in-plan strains whn th msh is distortd mmbran locking may occur. Particularly it is svr with larg aspct ratios of th lmnts. For th solid-shll lmnt this locking is ovrcomd ffctivly by using th EAS mthod s [HAU] with an nhancing matrix M as: 6

74 M = (.. whr only in-plan strains which ar quivalnt to rows and 4 of th nhancing matrix M in (.4.46 ar nhancd. In conclusion mmbran locking occurs if xist two conditions. First th transvrs strains and th in-plan strains intrlock. Scond th intrpolations ar unabl to modl th inxtnsional bhavior in inxtnsional bnding problms. onsquntly a stiffning ffct occurs whn pur bnding dformations ar accompanid by parasitic mmbran strsss. In th following sctions w will formulat a solid-shll lmnt which incorporats an altrnativ ANS tchniqu that is diffrnt from th classical ANS tchniqu prsntd in Sction.5. his lmnt is fr from all locking ffcts that hav bn mntiond.. KINEMAIS OF SOLID-SHELL For th us of th ANS and th EAS mthods to dsign th fr-locking solid-shll lmnts th wak form is writtn in a local coordinat systm whr th two axs X and Y ar alignd with th lmnt mid-surfac and th third axis Z is alignd with th thicknss dirction. o as for prsntation som kinmatic dfinitions in Sction.5. will b rpatd in this sction. For th dvlopmnt of th low-ordr solid-shll lmnt w naturally adopt th assumption of Naghdi for shlls: th normal to th lmnt mid-surfac rmains straight but not ncssarily normal during th dformation. Also adoptd by imoshnko bam thory and th Mindlin-Rissnr plat thory this assumption is fulfilld by a linar approximation of th in-plan displacmnts ovr th shll thicknss [HAU98]. In this contxt th intrpolation formulations for gomtrical vctors can b rad as ( [( h ( ( h X = + X X u l( ] (.. ( [( h ( ( h x = + x x u l( ] (.. whr: = { } =[-] [-] [-] tri-unit cub in R. In th abov formulas X ( and x ( ar th gomtrical dscriptions of th solidshll lmnt in th initial and currnt coordinat systms rspctivly. h vctors h h X and X ar th position vctors of nods in uppr and lowr surfacs u( l( rspctivly of th lmnt in th initial coordinat systm. Similarly th position 6

75 vctors of nods in th currnt coordinat systm ar dnotd by x and h u ( x. h l ( Hr th subscript l is th indx for trms in lowr surfac and th subscript u is th indx for trms in uppr surfac. g g g G G Ω t Ω G E Ω E E ( Ε Ε Ε : unit vctors of isoparamtric systm : unit vctors of global (physical systm ( Figur..: Initial configuration Ω currnt configuration and isoparamtric configuration Ω h displacmnt vctor u is calculatd as u = x X (.. ( ( ( and th dformation gradint tnsor F is dfind by F ( x( = (..4 X ( h Grn-Lagrang strain tnsor is dfind as blow x ( x ( E ( = ( F( F( I = [( I] X X ( ( 6 Ω t (..5 Originally proposd by Dvorkin and Bath for small strains [DVO84] th ANS mthod rquirs a modification of th shar componnts of th Grn-Lagrang strain tnsor E whn daling with larg dformation. onsquntly it is ncssary to writ th variational quation in th matrial configuration (or th total Lagrang formulation in trms of th Grn-Lagrang strain tnsor E and its conjugatd scond Piola-Kirchhoff strss tnsor S. Morovr th ANS mthod rquirs th intrpolation of all assumd trms in natural coordinat systm. hrfor it is ncssary to dfin a convctd dscription. Lt s considr th position vctors X ( of th rfrnc configuration Ω and th

76 position vctors x ( of th currnt configuration Ω t in th local coordinat systm. As illustratd in Figur.. th convctd basis vctor G i and its componnts in th initial basis systm ar dfind by: i G = X / ; G = G G ; i = (..6 i whil th contravariant vctor ij i j j G and its componnts ji G ar dfind following: j j j ji - G G = δ ; G = G G = G G ; ij = (..7 i i i ji i Uppr surfac G 8 4 Z Y G 7 5 G X x 6 Lowr surfac Figur..: onfiguration of low-ordr solid-shll lmnt Similarly th convctd basis vctor componnts g ij ar dfind as: g i in th currnt basis systm and its g g i i i = x / = Gi + u / ; = ij = g g i j i (..8 and its contravariant vctor xprssions j g and its componnts ij g ar dfind through th following j j i ij - g g = δ ; g = g g = g g ; ij = (..9 i i j ij j Anticipating th transition to a shll lmnt on of th solid-lmnt dimnsion is idntifid as th shll thicknss dirction. W call th lmnt fac (4 as th lowr fac th lmnt fac (5678 as th uppr fac s Figur... All th dgs and 4-8 th ons that connct th lowr nod with th uppr nods ar thicknss dgs. h dformation gradint in th form of th convctd vctors bcoms x( i F( = = g i G (.. X ( whil th Grn-Lagrang strain tnsor taks th following form 64

77 E = ( F F ( ( ( or altrnativly: E G u j I = u i ( G g ij G ij u i i G G u G j j = E ij i G G i j ( = i + j + ij = G j (.. (... AN ALERNAIVE ANS EHNIQUE FOR RANSVERSE SHEAR LOKING REMOVAL h classical ANS tchniqu prsntd in haptr has bn widly adoptd and thus applid for shll [DVO84] and solid-shll lmnts ([HAU98] [HAU] [HAU] [QUOa] i.. D modling. Its D countrpart can b obtaind through a dgnration of th D vrsion and is applicabl to a D solid-shll lmnt (Figur... h D.ANS lmnt has only a singl sampling point (A for th assumd shar strain E ~ XZ (D cas. Such a configuration can b also obtaind through th application of a slctiv-rducd intgration to th shar part of th lmnt. onsquntly th assumd shar strain E ~ XZ bcoms in th D vrsion constant ovr th lmnt whil it is not th cas for th D vrsion. W altrnativly invstigat anothr ANS tchniqu whr th D and D vrsions always fatur a linarly assumd shar strain (along th thicknss dirction. o this nd lt s first start from th standard solid lmnt to dvlop a nw D ANS-solid lmnt (dsignatd by ANSn. Its D countrpart (dsignatd by D.ANSn can b immdiatly followd through a simpl dgnration s [NGU8]. E Z Y X D H A F B Z G Dgnration D D X E H A F G Figur..: Dgnration from D to D of ANS-solid lmnt.. ubic hxahdral ANS lmnt (ANSn onsidr a singl solid lmnt of tri-unit gomtry (siz s Figur... For th sak of simplicity th isoparamtric spac ( and th physical spac (XYZ ar takn idntical. Hnc it is possibl to invstigat th problm dirctly in th physical spac. Rcall th Grn-Lagrang strain componnts in (..: 65

78 = = v v ( v v ( v v ( v v.5( v v.5( v v.5( v v v ( Z Y Z Y Z Y Z X Z X Z X Y X Y X Y X Z Z Z Z Z Z Y Y Y Y Y Y X X X X X X Y Z X Z X Y Z Y X YZ ZX XY ZZ YY XX com u w w u u w w u u w w u u w w u u w w u u w w u u w w u u w u E E E E E E E (.. h displacmnt drivativs of displacmnt in X dirction ar dfind as follows I I I I I I X I I Y I I I I Z I I I I I I N N U U X u N N u U U Y u N N U U Z = = = = = = = = (.. If th lmnt is subjctd to a pur bnding in X dirction Figur.. and assuming that th Poisson ratio is qual to zro th following rlations ar hold: 8 I ; = = = = = = = = = = = V I W I U U U U U U U U U U (.. and thn v v v = = = = = = Z Y X Z Y X w w w (..4 Lt s invstigat th compatibl shar strain XZ E which can b dirctly drivd from (.. - (..4: Z X Z X Z XZ u u u u u E ( + = + = (..5 Figur..: Solid lmnt spcial cas (XYZ ( Y X Z Nodal coordinats: (--- 5(-- (-- 6(- (- 7( 4(-- 8(

79 whr th scond trm rprsnts th nonlinar gomtrical quantitis. Notic that: N N u = ( U + U (..6 Z u X 8 8 I I U I = U = I I = Z I = 5 = [ U + U6 ( U U6 ] (..7 F U 8 U 7 F 8 7 U 5 U 6 5 Z Y 6 F F F 4 U 4 X F U F U U F hus th standard lmnt clarly prsnts a strain E XZ which is a bilinar function in both and whil this strain componnt must b physically zro in th pur bnding cas. In othr words E XZ contributs to th so-calld parasitic transvrs shar strain in pur bnding problms. In ordr to rmov this shar locking w obsrv that this parasitic shar strain is only qual to zro at = s (..5 and (..6 or lin u/ Z in Figur..4. his motivats th us of sampling points which ar mployd latr for th intrpolation of assumd strains on th vrtical plan including th point = and th axis O. Indd th shar strain E XZ is qual to zro at th points A l Au l and u (Figur..4 undr th pur bnding condition. As a rsult th physical strain E ~ XZ is assumd to tak th following form: ~ ~ A ~ E XZ = ( E XZ + ( + EXZ (..8 A whr E ~ XZ and E ~ XZ ar drivd through th strains at sampling points on th fac A A on Figur..4: ( l u u l Figur..: Pur bnding in X dirction ~ E ~ E A XZ XZ = ( E = ( E Al XZ l XZ + ( + E + ( + E Au XZ u XZ (..9 67

80 u/ Z 8 u (in pur bnding 7 Z Y 4 l D X 5 O 6 E D u H A u O l D l A A l O u F B u B B l G Strain sampling points Sampling points: A l (--; A u (- B l ( - ; B u ( l ( - ; u ( D l (- - ; D u (- Figur..4: Sampling points for ANSn lmnt Similarly th assumd shar strain E ~ YZ is intrpolatd as ~ ~ D ~ B EYZ = ( EYZ + ( + EYZ (.. D whr E ~ B YZ and E ~ YZ ar assumd by using sampling points on th fac ( D l Du Bu Bl Figur..4 as ~ E ~ E D YZ B YZ = ( E = ( E Dl YZ Bl YZ + ( + E + ( + E Du YZ Bu YZ (.. Intrpolation of th transvrs shar strains as (..8 and (.. mak sur of prsrving th bilinar variation of th assumd strains as th drivd-displacmnt shar strains. h D vrsion (Figur..5 of th ANSn lmnt can b dducd in a straightforward mannr. Indd undr pur bnding in X dirction th bnding mod is activatd so that w/ X = w/ Z =. Hnc th compatibl transvrs shar strain bcoms: u u u E = EXZ = + = ( U + U 4 [ + ( U U( + ] Z Z X (.. Sinc this shar strain is qual to zro at = w can mploy sampling points on th vrtical lin through th point =. For xampl with th us of two sampling points A ( = = and B ( = = and th corrsponding assumd shap functions ar: 68

81 N A = ( ; N B = ( + (.. h assumd shar strain and B: A B XZ A XZ B XZ E ɶ xz can b linarly intrpolatd through two points A Eɶ = N E + N E (..4 4 B Assumd shar strain E ɶ XZ (arbitrary cas Z X Intgration sampling points Strain sampling points A u/ Z (pur bnding cas Figur..5: Sampling points for D.ANSn lmnt In th cas of D lmnts bing applid for th pur bnding in X dirction th abov ANS tchniqu for th shar locking rmoval can b also mployd for th pur bnding in Z dirction. W howvr not that th bnding in Z dirction is not considrd in th convntional shll lmnts. In th gnral cas whr th lmnt gomtry is not rgular th shar trm nds to b valuatd in th natural spac. h assumd strains ar thn transformd to th physical spac for th computation of th stiffnss matrics. All of ths problms ar addrssd blow... Distortd hxahdral ANSn lmnt As a rquirmnt th ordr of th assumd strain fild should b qual to th on of th drivation of th displacmnt fild so that th assumd strains may consistntly captur th strain fild rsulting from th drivation of th displacmnt in non-pur bnding mods. Notic that only th transvrs shar strains ( E and E ar considrd for shar locking rmoval by th ANS tchniqu. onsidr th Grn-Lagrang strain valus which can b drivd from (.. and (.. as: E ij = g ij G ij u u u u = G i + G j i j + i j (..5 69

82 7 In ordr to dtrmin th ordr of th function ij E with rspct to th natural variabls i w invstigat th constitunts of th strains in (..5. h vctor i G is a bilinar function of th natural coordinats as follows: { } k k k k i K i i i i Z Y X N Z Y X 8 = = = G (..6 whr } { k k k Z Y X ar rfrnc nodal coordinat vctor. h drivativs of th displacmnt vctor with rspct to th natural coordinat i is { } k k k k i k i i i i W V U N w v u 8 = = = u (..7 Basd on (..5 - (..7 w can xamin th ordr of th natural constitunts in th shar strain function E as + + = u u G u u G E (..8 Vctors G and u ar drivativs of th shap functions with rspct to hnc G and u ar functions of ( ; similarly vctors G and u ar functions of (. onsquntly E is a function of th natural coordinats of th typ: ] ( ( ( ( [ f E = (..9 larly (..9 shows that E is a linar function of. In ordr to rmov shar locking th naturally transvrs assumd shar strain E is intrpolatd on th fac that gos through = and contains th axis O s plan (P in Figur..6. On that natural fac th shar strain is quadratic with rspct to but linar with rspct to Figur..6: Distortd lmnt - sampling points for strains Z A B D E F G H X Y L U A A B B D D L L L U U U P O L O U

83 ~ as E = f [ ( ( ]. Hnc th naturally assumd shar strain E can b assumd to b a function whos ordr of natural variabls is qual to or lowr than th ordr of natural variabls in th drivd-displacmnt strain (..9. It can b: onstant in thicknss dirction O and linar in horizontal dirction O. his is th classical ANS mthod [DVO84]. Hauptmann and Schwizrhof [HAU98] Klinkl t al. [KLI6] Vu-Quoc and an [QUOa] us this intrpolation in thir solid-shll lmnts. his intrpolation rquirs only two sampling points A and s (.5.8 and Figur..6. Linar in both thicknss dirction O and horizontal dirction O. his intrpolation rquirs two sampling points along both O and O ~ ( A l Au and l u Figur..6. Similarly to (..8 th natural assumd strain E is intrpolatd as ~ ~ A ~ E = ( E + ( + E (.. ~ A ~ whr E and E ar valuatd by using sampling points on th fac that gos through A A : ( l u l u ~ E ~ E A = ( E = ( E Al l + ( + E + ( + E Au u (.. Linar in thicknss dirction O and quadratic in O dirction. his intrpolation rquirs two sampling points along O A A and and thr ( l u l u sampling points along O ( A l Ol l and Au Ou u Figur..6. his assumd intrpolation is th closst to th consistnt displacmnt-drivd strain E. Our numrical tsts show that this quadratic intrpolation for assumd strains in O axis significantly incrass computational tim with rspct to th corrsponding linar intrpolation whil th improvmnt is not rmarkabl. In th thsis th scond option is chosn to b invstigatd. Similar to E (..9 th shar strain E is a function of th natural coordinats as shown blow E = f [ ( ( ( ( ] (.. In ordr to ovrcom that shar locking occurs with th pur bnding in Y dirction w considr E at =. At that coordinat th shar strain E is quadratic with rspct to as E = f [ ( ( ] but E is linar with rspct to. ~ Similar to (.. th natural assumd strain E if linarly intrpolatd in both thicknss dirction O and horizontal dirction O is xprssd as ~ ~ ~ = (.. D B E ( E + ( + E 7

84 whr ~ ~ and B ar linar functions of with sampling points on facs paralll axis D E E O and go through a pair of points ( l Du ~ E ~ E D B = ( E = ( E Dl Bl D and B B rspctivly: + ( + E + ( + E D u Bu ( l u (..4 In ordr to allviat shar locking th natural shar strains E and E ar ~ ~ rplacd by th assumd natural shar strains E and E bfor doing th transformation from th natural spac to th physical spac. Finally th physical assumd strain vctor by th altrnativ ANS tchniqu is:.5( g G.5( g G E.5( g G E ( g G ~ E ANS E = = {( [( ( g G ( A + ( + ( g G ( ] + (..5 L A E U ~ E ( + [( ( g G ( + ( + ( ( ]} g G L U ~ E {( [( ( g G ( D + ( + ( g G ( ] + L DU ( + [( ( g G ( B + ( + ( g G ( ]} L BU hs assumd strains can b implmntd in th standard solid lmnt in a straightforward mannr. h physical assumd strain-displacmnt matrix at nod I (I = -8 of th solid-shll lmnt is similarly assumd: ~ B ANS I = N I g N I g N I g N I g + N I g {( [( ( N I g + N I g ( A + ( + ( N + ] + L I g N I g (..6 ( AU + ( + [( ( N I g + N I g ( + ( + ( N + ( ]} L I g N I g U {( [( ( N + + ( + ( + ] + } I g N I g ( DL N I g N I g ( DU ( + [( ( N I g + N I g ( B + ( + ( N + ( ] L I g N I g BU W can s that th ANS tchniqu with altrnativ sampling points (th bilinar assumd strains (..5 and th classical ANS tchniqu (.5. hav som common and uncommon faturs: 7

85 For both tchniqus th strain sampling points ar takn at coordinat = for E and = for E. For th altrnativ ANS tchniqu th assumd strains ar linar in th thicknss dirction. Manwhil for th classical ANS tchniqu th assumd strains ar constant in th thicknss dirction. Whn strain is not too larg th two tchniqus ar similar bcaus th strain at th mid-surfac (th classical ANS will b th avrag valu of th strains at th lowr and th uppr surfacs (th altrnativ ANS prsntd hr. As bing mployd in th dgnratd shll lmnts [AHM7] th classical ANS tchniqu uss sampling points on th mid-surfac which is usually considrd to b th rfrnc plan. ontrarily th altrnativ ANS tchniqu mploys sampling points locatd on th physical uppr and lowr surfacs but not on th rfrnc plan of th solid lmnt. For distortd D.ANSn lmnt th shar strain is similarly assumd as (..4 Figur..5 ~ = + (..7 A B E ( E + ( E hs assumd strains ar thn transformd to th physical spac by th transform matrix in (.4.4. h D ight-nod solid lmnt which dirctly mploys assumd strains as (..5 is hrinaftr rfrrd as ANSn. h lmnt can b applid for distortd msh modl. It is a D solid lmnt which is fr from shar locking for thin and modrat thick-wall structurs s Sction.6 blow for numrical illustration..4 OMBINED ANS-EAS SOLID-SHELL ELEMEN h solid-shll lmnt prsntd in this chaptr posssss th prformancs of th ANS lmnts in allviating transvrsal shar locking and curvatur locking. Furthr mor to b fr from volumtric locking Poisson thicknss locking (s [HAU98] [HAU] and mmbran locking th EAS mthod nds to b adoptd. h solid-shll lmnts which only mploy th ANS tchniqus [FEL] should not b applid for incomprssibl dformation problms bcaus thy xhibit poor prformanc for volumtric locking rmoval s [BIS97]. In th abov paragraphs curvatur locking was mntiond. In ordr to circumvnt curvatur locking svral authors.g. Bischoff and Ramm [BIS97] and Btsch and Stin [BE95] suggstd mploying an assumd strain approximation for th strain componnt E. ornr nods on th mid-surfac play th rol of th sampling points (points E F G and H Figur..6. h -continuous strain fild is thus givn by: ~ = E N E I ( ( I I ; I = E F G H (.4. 7

86 whr N ar th shap functions in mid-surfac I ( ( + ( = as: N = I I + I (.4. ( 4 Hauptmann t al. [HAU] did show that this kind of locking is minor compard to othr typs a bilinar intrpolation of th transvrs normal strain as (.4. is nough for subduing this locking. On simpl way to ovrcom curvatur locking is using fin msh on thicknss dirction. ~ Introducing th assumd natural strain E (.4. into th assumd strain vctor ANS E ~ (.5. or (..5 w rciv th strain fild of an lmnt that is fr from both transvrsal shar locking and curvatur locking. h classical ANS lmnt that is fr from transvrs shar locking and curvatur locking is givn by:.5( g G E.5( g G E 4 ~ ~ E.5( N g ( G ( ANS I I E = = I (.4. E ( g G ~ E ( ( g G ( A + ( + ( g G ( ~ E ( ( g G ( D + ( + ( g G ( B whr th transvrs normal strain (.4. which is assumd in ordr to allviat curvatur thicknss locking [BIS97] wr rwrittn as ~ E = 4 N.5( g ( G I ( I I (.4.4 Onc th transvrs shar strains ar assumd all strain-displacmnt matrics can b similarly formulatd. h classical ANS strain-displacmnt matrics ar givn by: ~ B ANS I = ( ( N ( ( N I I 4 J + N + N I I J ( I I I + N ( A ( D with I = -8 (nod numbr and J = E F G H. g g N N g g g N N g g ( N I I g g + ( + ( N I + ( + ( N I J g g + N + N I I g g ( ( B (.4.5 h altrnativ assumd natural strains and th associat strain-displacmnt matrics with curvatur locking rmoval ar formulatd similarly to th classical ANS ons (.4. and (

87 An EAS lmnt that is fr from volumtric locking and mmbran locking [HAU] has an nhancing matrix M such as M = (.4.6 An EAS lmnt that is fr from volumtric locking mmbran locking and Poisson thicknss locking [QUOa] has an nhancing matrix M such as M = (.4.7 h EAS lmnt which is intrpolatd by a matrix M such as (.4.7 is nhancd by svn intrnal paramtrs hnc it is namd EAS7. Rcall that th nhancing matrix (.4.7 is xtractd from matrix ( hrfor th EAS7 always satisfis th patch tst. h trilinar trms in mods 49 5 and 5 of (.4.46 ar not takn into account bcaus thy only improv lmnt prformanc a littl whil making th computational cost incras significantly. Evntually a compltly original lmnt that combins both th advantags of th EAS and ANS formulations can b dsignd simply by combining th ANS strain fild (..5 and (.4. with th 7 EAS mods (.4.7 thus rsulting in th so-calld SS7n. It mans th solid-shll lmnt with 7 EAS mods and th altrnativ ANS tchniqu. In this lmnt th rsulting strain fild is mod ~ ANS nh E = E u + E α (.4.8 ( ( Similarly a solid-shll lmnt with 7 EAS mods (.4.7 and th classical ANS tchniqu (.4. rsults in th so-calld SS7. his lmnt has alrady bn prsntd in [QUOa]. In th thsis th EAS intrnal paramtrs ar usd to rmov volumtric locking for solid-shll lmnt. Instad of 9 mods (5-7 and 4-45 as proposd in [AND9] only thr mods 5-7 of (.4.46 ar adoptd for th solid-shll lmnt to limit computational cost. Dtail of th solid-shll implmntation is givn in Figur.4.. In gnral th implmntation is valid for th EAS lmnt. If th compatibl strains ar assumd as in (.4. or (..5 and thn introducd in stp of th algorithm (Figur.4. w thn hav th solid-shll lmnt formulation. 75

88 . Stp : Initial valus ALGORIHM FOR SOLID-SHELL IMPLEMENAION k=; U(k = ; ( k α = ;. Stp : Updat at lmnt lvl for itration (k +: Nodal displacmnt: U U (k from th last tim stp; tolranc ol = h h h ( k + U( k U( k 76 + EAS paramtrs: h α( k + = α( k [ kαα ( k ] ( kαu ( k U( k + f nh(k. Stp : at ach Gauss point for ach lmnt a. calculat: com ompatibl strains: E ( k+ as in (.. Enhancd strains (.4.45: nh J E + = Γ + α whr Γ = M J ( k ( k ( k+ ANS strains.g. (.4. for th classical ANS mthod: if (solid-shll lmnt mod ~ ANS nh ANS E ( k + = E( k+ + E( k+ ; ( k + = ɶ mod WS B B ( k + ; S ( k+ = mod E ls (EAS lmnt mod com nh mod W E ( k + = E( k+ + E( k+ ; S( k+ = E S mod ( k + nd if b. calculat tangnt matrics and intrnal forcs Sction.4. ( k+ ( + ( + ( + ( + U h ( + V V 4 mod kuu k = kmat k + kgo k = ( B B k dv + ( B S k 4 kα + ( Γ B + dv ; u k = ( ( V mod f ( B S dv ; f k = k k = k ( Γ Γ 4 αα ( k + ( k+ V mod f ( Γ S nh k = int( + ( + ( + ( + V V xt(k + = ( N c ( dv da k + b* + ( N c ( k + t * V Aσ k Updat at lmnt lvl [ ] [ ] k dv dv dv ( k+ = kuu( k+ kα u( k+ kαα ( k+ kα u( k+ [ k ] [ k ] = f f f ( k + xt( k + int ( k + + α u( k + αα ( k+ nh( k + r k α u ( k+ kαα ( k+ nh( k+ α( k+ K (k + R(k +. Sav EAS arrays: [ ] [ ] 4. Stp 4: Assmbl global matrics 5. Stp 5: Solv th incrmntal displacmnt and chck for global convrgnc U = K R ( k + ( k + ( k+ if R ( k + < ol or U( k + R ( k+ < ol goto nxt tim stp ls k=k+; rturn Stp nd if Figur.4.: Solid-shll algorithm f

89 .5 ENHANED QUANIIES.5. onsistnt dformation gradint h solid-shll lmnt prsntd in this chaptr has bn built on th modifid strain E mod (.4.8. h associatd modifid dformation gradint F mod is rquird if th lmnt is implmntd in a sourc cod bas on th updatd Lagrang formulation or whn a matrial algorithm for larg lastoplastic strains is ndd. h dformation gradint can b split into right-strtch tnsor U and rotation tnsor R as: F = RU (.5. Introduc U and R into th formulation for calculating Grn-Lagrang strain tnsor (..5 w hav: E = ( F F I = [ U ( R R U I] = ( U I (.5. larly Grn-Lagrang strain tnsor dpnds only on th right-strtch tnsor U. mod hus from th modifid strain E w can driv th associatd modifid right-strtch tnsor U mod. According to Hauptmann t al. [HAU] th modifid dformation gradint F mod is calculatd as mod mod F = RU (.5. W s that calculation of F mod rquirs twic of polar dcomposition. h first tim is calculation of rotation tnsor R in (.5.. h scond tim is calculation of modifid right-strtch tnsor U mod mod from E. hs calculations will incras computational cost of th algorithm whn th dformation gradint F mod is rquird..5. Local static condnsation h intrnal paramtrs of th EAS lmnt ar condnsd out at th lmnt lvl s (.4.7 bfor assmbling. As shown in th solid-shll algorithm (Figur.4. th intrnal paramtrs at itration (k+ is calculatd by: α + = α [ k ] ( k U + f (.5.4 h ( k ( k αα ( k αu( k (k nh(k his procdur rquirs for ach lmnt that apart from th intrnal paramtrs at itration (k - α must b stord for th calculation of α othr quantitis as k k and αu ( k nh(k (k 77 ( k + αα (k f also nd to b kpt. Whn solving a problm with a hug numbr of DOF s a larg mmory spac must b rsrvd for storing ths itms. Simo t al. [SIM9] did propos a local static condnsation algorithm. According to this local algorithm th intrnal paramtrs α ar not calculatd at Stp. Instad thy ar ( k + calculatd at th nd of Stp b (Figur.4. from k αα ( k + k and αu ( k+ f nh(k +. onsquntly it is not ncssary to sav EAS arrays that usually dmand a significant mmory allocation. Dtails of th local static condnsation algorithm ar prsntd in Figur.5..

90 LOAL SAI ONDENSAION ALGORIHM Lt { h h U n U n } b th corrct solutions at tim t n and { h h U n +( k U n +( k } b solutions at a givn itration (k within th intrval [t n t n+ ]. Fix this itration and comput α for ach lmnt by mans of th following sub-itration (at th n +( k lmnt lvl:. Stp : Initial valus k = ; α = α ; tolranc tol ; alculat n+ (. Stp : Updat at lmnt lvl for itration (k +: nh mod com nh omput: E ; E = E E ( k + n ( k + ( k+ + ( k+ Us constitutiv quations to comput: 4. Stp : omput incrmnt: S mod ( k+ α (k+ = [ k αα ( k + ] fnh(k + st for convrgnc: if ( α tol or (k ls nd if ( k + fnh( k+ < stop updat α = α ( k + ( k α k=k+; rturn Stp com E ( k + W = E S mod ( k+ Figur.5.: Static condnsation algorithm [SIM9] In th algorithm of Simo t al. [SIM9] Figur.5. th intrnal paramtrs ar calculatd by an approximation formulation: α = α [ kαα ] ( f (.5.5 ( k+ ( k ( k nh(k Hnc a limitd numbr of itrations ( hav to b ralizd to gt a corrct solution..6 NUMERIAL RESULS AND DISUSSION In th thsis th solid-shll which adopts 7 EAS paramtrs (.4.7 and th classical ANS tchniqus (.4. is dsignatd by SS7. Whil th solid-shll lmnt with th altrnativ ANS tchniqu that is prsntd in th abov sction xprssion (..5 is dsignatd by SS7n. h additional lttr n stands for th lmnts that mploy th nw altrnativ ANS tchniqu. 78

91 .6. Patch tsts h patch tst has bn originally proposd in th mid-sixtis as a simpl mans to proof convrgnc of an lmnt. Bsid a numrical vrification thr is also th possibility of a thortical analysis. h patch tst chcks whthr a constant distribution of any stat variabl within an arbitrary lmnt patch (i.. a distortd msh can b rprsntd xactly. It is spcially usful for finit lmnt formulations which violat th compatibility condition (and thus cannot b provn to b consistnt such as th ANS EAS tc. Natur of th patch tst is to vrify an lmnt s ability to rprsnt a constant strain/strss fild and thus nsur compltnss and an ability to convrg in th limit as th lmnt siz dcrass. h mmbran patch tst (for mmbran constant strss stat and th bnding patch tst (for bnding constant strss stat prsntd in haptr Sction.6 ar satisfid by th classical ANS SS7 lmnts and th proposd ANSn SS7n lmnts..6. Eignvalu analysis of an incomprssibl cub In ordr to stimat th bhavior of th ANSn and SS7n lmnts at th narly incomprssibl limit an ignvalu analysis of a unit cub is prformd as in [AND9]. h matrial paramtrs ar lastic modulus E =. and Poisson ratio ν = h cub is considrd in rgular configuration and furthrmor in distortd configuration to chck th snsitivity to distortd msh of lmnts s Figur.6.. abl.6. shows th ignvalus of 8 dformabl mods of th rgular cub (Figur.6.a th six zro ignvalus for th six rigid body mods ar not shown a Rgular shap b Distortd shap Figur.6.: Rgular and distortd cubs h six ignvalus of th constant strain-stats which ar idntical for all lmnts ar printd in italic lttrs. hir corrsponding ignmods can b idntifid as thr shar mods two tnsion mods and on dilatation (or incomprssibl mod (Figur.6.. h ignvalus of th EASv EAS6v and EAS9v in abl.6. ar totally idntical to thos valus of MEAFOR [BUI]. his assurs th quality of our EAS implmntation in th MALAB cod. 79

92 abl.6.: Eignvalus of rgular cub Mod Q EASv EAS6v EAS9v SS7 SS7n ANS ANSn a warping mod b dilatation mod Figur.6.: wo dformation mods For a volumtric-locking fr bhavior th lmnts should contain only on infinit ignvalu it is th ignvalu of th dilatation mod (mod numbr 8 ignvalu = 5 is considrd as infinit. In th cas of th standard lmnts Q six dviatoric mods (mods -7 ar always mixd up by volumtric strains which whn ν.5 lads to six unralistic infinit ignvalus. h sam consquncs ar found with ANS and ANSn it mans Q and both ANS and ANSn ar not volumtric locking fr. With th introduction of nhancd mods th most important part of volumtric locking is rmovd in th EASv lmnt which has now a bttr volumtric locking 8

93 rspons in comparison to th Q. Howvr highr ordr parts in th intrpolation functions still xist and this might caus volumtric locking in crtain cass. With th additional nhancd mods th EAS6v bcoms narly fr volumtric locking. Introduc mor nhancd mods for xampl EAS9v w obtains totally fr volumtric locking lmnt. As w might obsrv aftr rmoving volumtric locking th infinit ignvalu bcoms finit. h solid-shll lmnts SS7 and SS7n ar almost fr volumtric locking xcpt a warping mod (Figur.6. has a modrat ignvalu of 9.6 (mod 7. abl.6.: Distortd cub - location of nods x y z abl.6.: Eignvalus of distortd cub Mod Q EASv EAS6v EAS9v SS7 SS7n ANS ANSn For distortd cas (Figur.6.b th gnral trnd is quit similar as in th rgular cas. abl.6. shows th ignvalus of 8 dformabl mods of th distortd cub. h ANS and ANSn lmnts and th Q standard lmnt ar lockd bcaus of th prsnc of many infinit ignvalus (mods -7. With th introduction of nhancd mods in volum volumtric locking is rmovd. Howvr in this distortd configuration th EAS lmnts cannot rmov compltly th locking ffct. For xampl it xists two locking ignmods (mods 6 7 in th EAS6v. Fortunatly th ignvalu is rathr modrat and w xpct a vry mild locking. h solid-shll lmnts SS7 an SS7n also show two locking ignmods (mods 6 7 as th EAS6v. Anyway th ANSn and SS7n sm to b bttr than ANS and SS7 rspctivly in this distortd configuration s abl.6. (by comparing ignvalus of mod of th 8

94 ANS and ANSn; and by comparing ignvalus of mod 6 of th SS7 and SS7n. In th distortd configuration th SS7n and ANSn lmnts hav only on mor modrat ignvalu compar to thir ignvalus in th rgular configuration rspctivly. hs argumnts improv that th ANSn and SS7n lmnts ar lss snsitiv to distortd msh than th ANS and SS7 lmnts rspctivly. h distortd cas of th cub was also analysd by Jttur [JE8]. In [JE8] th incompatibl lmnt th SRI EAS4 and EAS9 lmnts wr invokd. Analysd rsults in [JE8] showd that th incompatibl lmnt bhavs similarly as th EASv lmnt. Volumtric locking was almost rmovd by th EAS9v lmnt. Hnc in this cas th EAS9v and EAS4 giv similar rsults. Only th SRI lmnt is compltly volumtric locking fr in this cas..6. Eignvalus of a squar plat Eignvalus of a squar plat with zro Poisson s ratio is invstigatd to chck prformanc of lmnts in bnding [HAU]. Dimnsions and matrial paramtrs of th plat ar givn in Figur.6.. E =.9 ν =. Z Y X h =. a a = 5 Figur.6.: Squar plat Eignvalus of lmnts ar shown in abl.6.4. Rsult of EASDEAS solidshll lmnt of Hauptmann t al. [HAU] is also gathrd to compar with th solidshll lmnts prsntd in this thsis. h EASDEAS lmnt incorporatd th classical ANS tchniqu for rmoval of transvrs shar locking and incorporatd th EAS mthod for mmbran and Poisson thicknss locking rmovals a Bnding in X b Bnding in Y Figur.6.4: wo bnding mods 8

95 All of th lmnts hav six rigid body mods. h standard lmnt Q suffrs transvrs shar locking as w s its ignvalus for bnding mods in X and Y axs (Figur.6.4 ar infinit. hs ignvalus of Q ar qual to 7.7 whil th corrspondnt valus of non-locking lmnts ar smallr than. (abl.6.4 mods and. Furthrmor th Q lmnt and also th EASv6s lmnt may suffr a mild locking causd by two warping mods (Figur.6.5 bcaus ths mods wr affctd by transvrs shar strains. h warping-mod ignvalus of th Q and th EASv6s ar qual to 75.9 whil th quivalnt valus of othr lmnts in abl.6.4 ar smallr than. (mods 9 and. abl.6.4: Eignvalus of th squar plat Eign mod Q EASv6s ANS-S SS7 ANSn SS7n E abl.6.5: Eignvalus of th squar plat - Summary Elmnt Rigid body Eignvalus (of dformabl mods mods <. <. Max. Q EASv6s EASDEAS (Hauptmann ANS ANSn SS SS7n

96 Figur.6.5: wo warping mods of EASv6s & Q.6.4 Squar plat at larg displacmnts and strains onsidr a squar plat of dimnsion a a h with a = and h =. Young modulus is E = 5 and Poisson s ratio is ν =.. h plat is considrd in two cass: fully clampd and simply supportd at all dgs. A concntratd forc P is applid at th cntr of th plat. P h a a Figur.6.6: Squar plat gomtry Du to symmtry only on quartr of th plat is analysd s Figur.6.6. By adopting th assumption on small displacmnts and strains analytical rsults for th displacmnt at th cntr of th plat can b obtaind thanks to imoshnko s plat thory. hs analytical rsults ar: for th clampd plat: w max =.56Pa / D (.6.a for th simply supportd plat: w max =.6Pa / D 84 (.6.b whr D = Eh /( ν is th flxural rigidity of th plat cross-sction. h analytical solution with th assumption of small displacmnts and strains offrs a rfrnc to xamin th corrctnss of th numrical rsults. h problm is modld by a coars msh of 5 5 lmnts. In both clampd and supportd cass th ANS and ANSn giv th sam rsults similarly th SS7 and SS7n giv th sam rsults. Hnc in this xampl w only xpos rsults of th ANSn and SS7n. onsidr first th cas whr th plat is clampd Figur.6.7. At a rathr modrat loading (P < displacmnts and strains in th plat rmain small. Hnc numrical rsults from larg strain vrsion should match th analytical solution. Rsults ar also compard to rsults from rducd intgration (RI lmnt of Li and scotto [LI97] with automatic hourglass control. Notic that if th RI rsults match quit wll

97 th EAS5 and SS7n this is at th xpns of layrs ovr th plat thicknss for th RI whil only on layr is rquird by othr lmnts. h lmnts drivd from th RI tchniqu undrgo Poisson thicknss locking. hrfor for RI lmnts mor than on layr should b usd ovr thicknss in ordr to obtain a good rsult of strss distribution along thicknss in bnding dominatd problms. ntr displacmnt w (mm ANSn 5 SS7n EAS5 4 RI 4 5 imoshnko Applid forc P (N 4 5 Figur.6.7: Displacmnt vrsus applid forc for fully clampd plat ntr displacmnt w (mm ANSn EAS5 SS7n 4 4 RI 5 imoshnko Applid forc P (N Figur.6.8: Displacmnt vrsus applid forc for simply supportd plat h EAS5 RI and SS7n lmnts ar vry satisfactory by approaching th analytical curv up to a loading lvl about P =. With incrasing loading gomtrically nonlinar ffcts bcom important and this du to mmbran ffcts maks th plat stiffr. It xplains why all numrical rsults ar lowr than th thortical ons. Also larg strains might happn and this will hav an influnc on th bhavior of th structur. h accuracy of th altrnativ ANS tchniqu is confirmd whn a vry good agrmnt is found in comparison of th SS7n lmnt with th EAS5 and RI lmnts bing dvlopd in th framwork of larg displacmnt and strains. onsidr now th cas whn th plat is simply supportd Figur.6.8. h rsults of th EAS5 SS7n and th RI ar rathr clos to ach othr. Howvr whil th EAS5 and SS7n rsults approach th analytical solution at a modrat loading (P < 85

98 as xpctd it is not th cas of th RI lmnt. h dviation of th RI rsults from th analytical solution at small strains. Whn th applid forc is ovr 5N th EAS5 and SS7n continu giving th sam rsult but thy ar a littl stiffr than th RI lmnt. h bhavior of th EAS5 in both clampd and simply supportd cass ar totally idntical to thos valus of MEAFOR [BUI]. his assurs th quality of our EAS implmntation in th MALAB cod. ONLUSION h ANSn and SS7n lmnts which bas on th altrnativ ANS tchniqu satisfy th patch tsts. By analysing ignvalus w s that thy ar fr from volumtric locking and transvrs shar locking. h solid-shll lmnts ar full-intgratd schm hnc th stiffnss matrix is stabl; it mans thr is not hourglass mods. It should b notd that th drivation of th solid-shll stiffnss matrix is carrid out on th flattnd (unwarpd solid gomtry thn globalizd to th actual gomtry. If th lmnt is too warpd or taprd crtain tsts ar only approximatly satisfid. Poor rsults can b xpctd if th lmnt is xcssivly warpd. Unlik othrs plan-strss shll formulations for mtal forming simulation th solid-shll lmnts provid a natural and fficint way for shll contact problm sinc doubl-sid surfacs of shll ar availabl and th transvrs normal strss is includd. Faturing an appropriat combination of th ANS and th EAS mthods ddicatd to allviat locking ffcts th solid-shll lmnts ar fr from locking du to parasitic shar strains distortd gomtris and incomprssibl matrials. hrough an invstigation of th ANS mthod in th rmoval of shar locking an altrnativ schm of sampling points which nabls a linar distribution instad of a constant valu of shar strains is dvlopd in this chaptr. As it was rvald by numrical rsults th solid-shll lmnts with this altrnativ schm offr a comparabl prformanc in comparison with that mployd th classical ANS schm. In th nxt chaptr prformancs of th ANS and ANSn th SS7 and SS7n solidshll lmnts for lastic applications ar continuously invstigatd with various linar and nonlinar tsts. 86

99 haptr 4. ELASI APPLIAIONS INRODUION In haptr a solid-shll lmnt which intgrats th altrnativ ANS tchniqu has bn dvlopd. In this chaptr various numrical tsts ar prsntd to dmonstrat th capabilitis of th proposd solid-shll lmnt. For th sak of clarity all dnotations for lmnts in haptr and haptr ar rcalld blow. h standard solid lmnt with full intgration is dsignatd by Q whil th solid lmnt which mploys th classical ANS tchniqu [DVO84] for allviating transvrs shar locking and curvatur locking is dsignatd by ANS. h solid-shll lmnt dscribd in haptr which adopts 7 EAS paramtrs and th classical ANS tchniqus (.4. is dsignatd by SS7 whil th solid-shll lmnt with th altrnativ ANS tchniqu prsntd in haptr (..5 is dsignatd by SS7n. h additional lttr n stands for th lmnts that mploy th altrnativ ANS tchniqu. h EAS lmnts ar dsignatd by EASx whr x is th numbr of intrnal paramtrs. For linar and nonlinar lasticity tsts in this chaptr all of ths lmnts ar implmntd in a MALAB cod. abl 4..: Summary of mployd lmnts Nam yp Dscription Q Standard solid lmnt Only compatibl strains ANS lassical ANS lmnt Linar transvrs shar strains (.4. ANSn Nw ANS lmnt Bi-linar transvrs shar strains (..5 EASx Enhancd assumd strain lmnt x nhancd mods (abl.4. SS7 Solid-shll lmnt 7 nhancd mods (.4.7 Linar transvrs shar strains (.4. SS7n Nw solid-shll lmnt 7 nhancd mods (.4.7 Bi-linar transvrs shar strains ( LINEAR APPLIAIONS In this sction w invstigat prformancs of th ANSn and SS7n lmnts. Various tsts which includ shar mmbran and volumtric locking ar takn into considrd. All of tsts in this sction ar linar problms. Nonlinar problms ar considrd latr in sction antilvr bam undr pur bnding onsidr a cantilvr of dimnsion clampd at lft nd and loadd by a constant momnt (inducd by forcs P =.5 at right nd Figur 4... his tst prsntd by handra and Prathap [HA89]. h lastic modulus is E = 6 and Poisson s ratio is ν =.. From th thory of th strngth of matrials th analytical solutions for vrtical displacmnt w and maximum normal strss ar: 87

100 ML 4 M bh w = = 6 ; σ max = = 6 with W = (4.. EI W 6 A msh of on lmnt is usd. Sinc th structur undrgos pur bnding an nhancmnt in volumtric locking is quit uslss. hrfor w will mploy hr shar-nhancd mods. With this rathr modrat nhancmnt th EASs is alrady abl to attain th xpctd rsults both in displacmnt and normal strss. h strss rsults in abl 4.. ar avrags of absolut valus at uppr and lowr surfacs of th cantilvr. Excpt th Q standard lmnt is too stiff du to shar locking all th othr lmnts ANS ANSn SS7 an SS7n ar shar locking fr. b= P P h= P L= P Figur 4..: antilvr undr pur bnding abl 4... Normalizd rsults Q EASs ANS ANSn SS7 SS7n w σ max wistd bam with warping ffcts In ordr to tst th warping ffct on lmnts McNal and Hardr [MA85] proposd th twistd cantilvr in bnding in-plan dirction (P v = and out-of-plan dirction (P h = Figur 4... h cantilvr lngth is L = th width is w =. and th thicknss is t =.. h cantilvr is twistd 9 from root to tip. Young s modulus is E = 9 6 and Poisson s ratio is ν =.. t Y w X Z P h P v Figur 4..: wistd bam 88

101 . Normalizd tip dflction ANS; ANSn SS7; SS7n EAS5 EAS NEXHEX ASQBI Q Numbr of lmnts (n 6 Figur 4..: wistd bam cas of load along Z Normalizd tip dflction ANSn SS7n EAS Numbr of lmnts (n 4 5 Q EAS 4 Figur 4..4: wistd bam cas of load along Y h cantilvr is fixd at lft nd and loadd by a unit forc at right nd. h rfrnc solutions ar: In-plan dirction (P v = : w rf = ; Out-of-plan dirction (P h = : v rf = h cantilvr is modld with a msh of n lmnts whr n is numbr of lmnts along th cantilvr lngth. 89

102 Numrical rsults for th cas of bnding in-plan dirction ar listd in Figur 4.. whr rsults of ASQBI lmnt of Blytschko and Bindman [BEL9] and NEWHEX lmnt of Frdriksson and Ottosn [FRE7] ar also takn into considrd. Both ASQBI and NEWHEX lmnts ar assumd strain lmnts whr lmnt undrintgration is adoptd. h SS7 and SS7n lmnts convrg with vry coars msh (n = whil th convrgnc with othr lmnts rquirs finr msh: n = for ASQBI and EAS n = for NEWHEX and n = 9 for th EAS5. h ANS and ANSn lmnts also convrg as quickly as SS7. h Q lmnt continus to show poor prformanc in this bnding cas as usual. Numrical rsults for th cas of bnding in out-of-plan dirction ar listd in Figur In this cas all of th lmnts (ANSn SS7n and EAS only convrg with rathr fin msh (n 5. Howvr with coars mshs th ANSn and SS7n lmnts giv bttr rsults than th EAS5. In contrary th EAS convrgs as fast as th SS7n. h Q lmnt suffr shar locking hnc show poor prformanc in this bnding cas. Solid-shll in Samcf is a volumtric quadratur shll lmnt which can prform th thicknss dformation [JE8]. hat lmnt adopts th ANS mthod for shar locking rmoval. Furthrmor EAS mods ar also adoptd to rmov othr locking ffcts. hat solid-shll lmnt passs th bnding and mmbran patch tsts [JE8]. For this twistd bam problm th solid-shll in Samcf convrgs with a msh of lmnts (for both bnding in-plan dirction and bnding out-of-plan dirction. 4.. lampd and simply supportd plats undr uniformd prssur A squar plat of dimnsion a a h with fixd thicknss h = and various valus of width a = ( is modld. All dgs of th plat ar clampd or simply supportd. A uniform prssur p = loads on th uppr fac of th plat. his tst is prsntd by handra and Prathap [HA89]. p h a a Figur 4..5: Squar plat undr uniformd prssur h symmtry of th structur allows to simulat a quartr of th plat s Figur h matrial proprtis ar Poisson s ratio υ =. and th lastic modulus E is artificially dpndnt on th lngth a s (4... h analytical rsult on th cntr dflction can b obtaind [IM59]: 9

103 4 pa w =.α with D Eh D = (4.. ( ν whr D is th flxural stiffnss of plat and dimnsionlss dflction α dpnds on boundary condition s abl 4... In ordr to rspct th condition of small displacmnts th cntr dflction has to b limitd to a small valu. Hr a typical valu w = h/ is rspctd. orrspondingly th Young s modulus taks th following valu from (4..: 4 ( ν D ( ν pa E = =.α (4.. h h w abl 4... Dimnsionlss dflction α (from [BLE] and [IM59] h/a.... Plat thick thin vry thin vry thin lampd α Simply supportd α For th comparison rason th numrical dimnsionlss cofficint α num which is dfind blow drivd from (4.. will b mployd: w num 4 α D (4..4 pa num = onsidr first th cas whr all th dgs of th plat ar clampd abl 4... Numrical analysis shows that th EAS5 SS7 and SS7n lmnts giv vry satisfactory rsults for all cass ranging from thick to thin plats. Whil th Q lmnt dlivrs a poor prdiction spcially whn shar locking and Poisson thicknss locking (du to Poisson ratio is diffrnt from zro bcoms important with a dcras of th plat thicknss. h EAS9 lmnt only givs good rsults for thick plat. Whn th aspct ratio (a/h is ovr th EAS9 is too much wors than th ANS and ANSn lmnts. Howvr th ANS and ANSn lmnts xhibit a stabl tndncy of convrgnc but cannot rach th dsird valu bcaus Poisson s ratio is not qual to zro. abl 4... Normalization of dimnsionlss dflction α - lampd plat (υ =. h/a.... Msh Q EAS EAS ANS ANSn SS SS7n

104 Lt s pass now to th cas whr th plat is simply supportd at all dgs s abl Locking rspons occurs in th cas of Q - standard lmnt. h EAS5 lmnt show again its prformanc with a quick convrgnc toward th xpctd rsults but th EAS9 lmnt continus to giv poor rsult at high aspct ratio. h SS7n and SS7 solid-shll lmnts show good prformanc as th EAS5 at coars msh for thick and thin plat. As xpctd th ANSn and ANS lmnt bhav bttr than th EAS9 as in th clampd cas whn th aspct ratio (a/h is ovr. abl Normalization of dimnsionlss dflction α - Simply supportd plat (υ =. h/a.... Msh Q EAS EAS ANS ANSn SS SS7n abl Normalization of dimnsionlss dflction α - lampd plat (υ =. h/a.... Msh Q EAS EAS ANS ANSn SS SS7n abl Normalization of dimnsionlss dflction α - Simply supportd plat (υ =. h/a.... Msh Q EAS EAS ANS ANSn SS SS7n

105 In ordr to chck prformanc of lmnts whn thr is no Poisson thicknss locking lt s impos υ =. s abls 4..5 and For both clampd and simply supportd cass th EAS9 lmnt continus to giv poor rsults whn a/h >. Hnc w can conclud that du to shar locking th EAS9 lmnt shows bad prformanc. h ANS and ANSn lmnts ar only shar locking fr. Whn thr is no Poisson thicknss locking thy bhav as wll as th SS7 and SS7n. Du to fr from Poisson thicknss locking prformanc of th othr lmnts (SS7 SS7n and EAS5 ar almost similar to th cas whr Poisson thicknss locking xists Squar clampd plats with concntratd loads A squar plat of dimnsion a a h = is clampd at all dgs and loadd by a concntratd load P = 6.67 at th cntr. Matrial proprtis ar E = 4 and υ =.. Du to symmtry on quadrant of th plat is modld with on layr of and 4 4 lmnts. h structur has bn modld with EAS lmnts by Andlfingr and Ramm [AND9]. By adopting th assumption on small displacmnts analytical rsults on th cntr displacmnt can obtaind thanks to th Kirchhoff s plat thory: Pa w =.56 with D whr D is th flxural stiffnss of th plat. Eh D = (4..5 ( ν abl ntr dflction Msh 4 4 Q EAS EAS ANS ANSn SS SS7n hory. Applying (4..5 th thortical displacmnt is imposd w = at th cntr of th plat. Without th prsnc of gomtrical (nonlinar stiffnss th EAS5 SS7n and SS7 lmnts giv vry good approximation s abl his thin plat bnding problm is modld by coars mshs hnc th Q standard lmnt is too stiff. Bttr rsults ar givn by th EAS9 lmnt but with both vry coars msh and finr msh 4 4 th EAS9 is always wors than th ANSn and ANS lmnts. h solid-shll lmnts who adopt th classical ANS tchniqu (ANS and SS7 or th altrnativ ANS tchniqu (ANSn and SS7n bhav idntically in this tst. 9

106 4..5 Simply supportd squar plat with various thicknss and distortd msh his tst illustrats th distortd msh insnsitivity of th solid-shll lmnts and thir prformanc in shar locking and Poisson thicknss locking rmoval at high aspct ratio (a/h. onsidr a plat of dimnsions a a h (Figur 4..6 with a = and diffrnt valus of th thicknss h (abl h plat is simply supportd along th four dgs and loadd by a unit concntratd forc at th cntr. Young s modulus is E = 7 and th Poisson s ratio is ν =.5. h problm is symmtric hnc only on quartr of th plat is considrd in two cass: rgular msh and distortd msh of 4 4 lmnts (Figur P/4 6 (55h/ y z x O Figur 4..6: A quartr of th plat - distortd msh abl Rgular msh - Normalization of displacmnt h Q ASQBI NEW- HEX EAS9 ANSn ANS SS7n SS abl Irrgular msh - Normalization of displacmnt h Q ASQBI NEW- HEX EAS9 ANSn ANS SS7n SS

107 h analytical solution of displacmnt at th cntr of this problm is givn by (.6.b. Numrical solutions ar listd in abl 4..8 for th rgular msh and in abl 4..9 for th distortd msh. Rsults from th SS7n and ANSn lmnts ar compard with rsults of th EAS9 ASQBI of Blytschko and Bindman [BEL9] and with NEWHEX of Frdriksson and Ottosn [FRE7]. Whn th aspct ratio is lss than 4 (i.. h >.5 all th lmnts xcpt Q giv good rsults for both rgular and distortd mshs. Whn th aspct ratio is largr than (i.. h <. ASQBI and NEWHEX only giv good rsults for rgular msh whil rsults of th EAS9 dtriorat rapidly vn for rgular msh. On th contrary th bhavior of ANSn and SS7n is stabl with both distortion and high aspct ratio (a/h =. hs lmnts can thus b considrd as robust Pinchd cylindr with rigid nd diaphragms onsidr a cylindr of innr radius r = thicknss t = and lngth L = 6 s Figur Young s modulus is E = 6 and th Poisson s ratio is ν =.. h concntratd forcs F = apply at th mid-lngth of th cylindr. F r L/ L F Figur 4..7: Pinchd cylindr with two rigid nd diaphragms Rfrnc dflction w = is coincidnt with th loadd points. Du to symmtry only on-ighth of th cylindr is modld. h structur is dominatd by inxtnsional bnding rspons hnc mmbran locking may occur. Furthrmor th thin structur with highly curv gomtry also caus curvatur thicknss locking and shar locking. In this tst a solid-shll lmnt ANSDEAS of Hauptmann and Schwizrhof [HAU98] is usd for comparison. h ANSDEAS adopts th ANS mthod for transvrs shar locking rmoval and adopts th EAS mthod to nhanc th mmbran strains. h ANSDEAS lmnt is suprior in mmbran dominatd problms as statd in [HAU98]. Mindlin shll and th solid-shll lmnt of Samcf [JE8] s Sction 4.. ar also takn into comparison. Figur 4..8 shows that th vrtical displacmnts of th altrnativ ANS mthod ANSn lmnt (only with N < and SS7n lmnt ar bttr than ANSDEAS and 95

108 th EAS lmnts. h SS7n EAS5 and ANSDEAS lmnts convrg with vry fin msh (. h Samcf solid-shll lmnt bhavs as wll as th SS7n lmnt. h Mindlin (Samcf lmnt convrgs at vry coars msh; but dos not provid th corrct rsult whn th msh is finr. Normalizd displacmnt ANSn SS7n EAS9 4 EAS5 4 5 ANSDEAS [HAU98] 6 Mindlin (Samcf 7 Solid-shll (Samcf 5 Elmnts pr ara N Figur 4..8: onvrgnc invstigation for th pinchd cylindr 4..7 Morly sphrical shll A bnchmark tst for shll lmnts of McNal and Hardr [MA85] is considrd. h structur consists of a thin hmisphrical shll Figur h middl radius of th shll is R = th thicknss t =.4. Matrial proprtis ar lastic modulus E = and Poisson s ratio υ =.. oncntratd loads F of opposit signs position at vry 9 in th quatorial plan. abl 4... Morly sphrical shll - Normalizd displacmnts at tst point Msh Q EAS9 EAS5 ANSn ANS SS7n SS h thortical displacmnt of tst point is u =.94. Bcaus of th symmtry a on-fourth of th structur nds to b modld. On lmnt ovr thicknss will b fixd for all computation whil diffrnt kinds of msh in othr dirctions will b trid. 96

109 omputd rsults with th EAS lmnts as wll as th solid-shll lmnts ar compard. Z Fr 8 Sym R Sym st point X F= Fr F= - (on quadrant Y Figur 4..9: Morly sphrical shll Normalizd dflction ANSn.6 SS7n EAS9.4 4 EAS5 5 Q Elmnt pr sid Figur 4..: onvrgnc of finit lmnt solution As th sphrical shll is vry thin D modling may lad to an ill-conditiond problm sinc th distanc btwn two corrsponding nods in th thicknss dirction is too small in comparison to othr dirctions in th cas of a too coars msh. Hnc a rfinmnt of msh will b hlpful to handl this problm. Figur 4.. shows rsults of th solid-shll lmnts vrsus th EAS and standard lmnts with varity of mshs. h EAS9 convrgs to th xact solution for a vry fin msh ( whil th EAS5 convrgs for a coarsr msh (6 6. Howvr th solid-shll lmnts giv good 97

110 rsults vn at xtrmly coars msh (. h classical ANS lmnts ANS and SS7 lmnts and th altrnativ ons ANSn and SS7n lmnts ar in this cas totally quivalnt hick-walld cylindr Expansion of a thick cylindr with various Poisson s ratio (ν = as dscribd in [MA85] is considrd to invstigat th prformanc of th solid-shll lmnts in volumtric locking conditions. Elastic modulus is E =. h innr radius of th cylindr is R i =. th outr radius is R o = 9. and th thicknss is t =. s Figur 4... Plan strain conditions ar assumd in th thicknss dirction. h innr surfac of th cylindr is loadd by a prssur q = /unit ara. A part of th cylindr as dscribd in Figur 4.. is modld by a 5 msh. h analytical solution of th problm is givn by ( + ν pri u = [ R / r ( r] o + ν (4..6 E( R R o i Numrical solutions at th innr radius of lmnts ar tabulatd in abl 4... h rsults show that th ANS and ANSn lmnts dlivr idntical rsults. h sam rmark is also shown by th SS7 and SS7n lmnts. Obviously volumtric locking rspons can b obsrvd for th Q lmnt. Sinc volumtric locking cannot b rmovd by th ANS tchniqus th ANS lmnt s rspons is narly as stiff as th Q standard lmnt for this problm. i R = R =9 Sym Sym o Figur 4..: hick-walld cylindr his volumtric locking can b rmovd with th us of th EAS intrnal paramtrs or mor prcisly with th introduction of nhancd volumtric mods. Indd th ANS tchniqu combind with th EAS tchniqu to rsult in th solid-shll lmnts SS7 and SS7n. h solid-shll lmnts ar fr from not only shar locking but also volumtric locking s abl 4... Morovr numrical rsults show th 98

111 EAS9 lmnts with volumtric locking mods and th EAS with 9 volumtric locking mods giv almost xact rsults as th SS7n. abl 4... Normalizd radial displacmnt at R i ν Q ANSn ANS EAS9 EAS SS7n SS NONLINEAR APPLIAIONS In this sction w invstigat prformancs of th ANSn and SS7n lmnts in nonlinar problms. All locking ffcts (shar locking mmbran locking and volumtric locking ar prsntd in problms. 4.. antilvr in larg displacmnt onsidr a cantilvr undr transvrs lin load s Figur 4... Gomtry of th cantilvr ar L b h =.. h lastic modulus is E =. 5 and Poisson s ratio is υ =.. h cantilvr is clampd at on nd and suffr a lin load q = at th othr nd. Rfrnc solution is numrical rsults of ANSDEAS lmnt (s Sction 4..6 of Hauptmann t al. [HAU]. q h L tst point b Figur 4..: antilvr bam A discrtization with a msh of lmnts is usd. h load - displacmnt diagram in Figur 4.. is obtaind by using tn qual load stps. It is shown that whn th applid load is small or modrat q <.q th EAS5 lmnt prforms as wll as th SS7n and ANSDEAS (Hauptmann t al. [HAU98] lmnts whil th ANSn lmnt is a littl stiffr bcaus of Poisson ffct. Whn th applid load continus incrasing th EAS5 du to transvrs shar locking bcoms as stiff as th ANSn lmnt du to Poisson thicknss locking. Manwhil th othr lmnts SS7n and ANSDEAS giv idntically bttr rsults than ANSn and EAS5 do. 99

112 q q ANSn ANS SS7n SS7 EAS5 Q ANSDEAS [HAU] w A Figur 4..: Displacmnt of tst point 4.. Morly sphrical shll larg dformation cas h sam data ar givn as th numrical tst in Sction 4..7 xcpt th thicknss is thinnr t =. and th applid load is largr F=5.. h msh is composd of 6 6 lmnts. h total load is applid in 5 qual stps. h problm was considrd by Vu-Quoc and an [QUOa] and Klinkl t al. [KLI6] for invstigating bhavior of thir solid-shll lmnts. his tst is considrd as on of th most svr bnch-mark problms for nonlinar analysis of shll [QUOa]. Pinching forc F/ ANSn-u ANSn-v SS7n-u SS7n-v EAS5-u EAS5-v u-[quoa] v-[quoa] Displacmnt u/r and v/r Figur 4..: Load-dflction curvs for displacmnts at points A(u and B(v h inward and outward displacmnts at th point A and B ar plottd vrsus th pinching load s Figur 4... Both mmbran and bnding strains contribut to th displacmnts at th load points. h structur is a doubly-curvd shll with high aspct

113 ratio (R/t = hnc curvatur thicknss locking transvrs shar locking and mmbran locking may simultanously occur. From abl 4.. th data show that th prsntd solid-shll lmnt (SS7n is quit quivalnt to th solid-shll lmnt of Vu- Quoc and an. ompar with th EAS9 and EAS5 th ANSn lmnt which is fr from shar locking and curvatur thicknss locking dlivr a vry good rsult. Hnc w can conclud that transvrs shar locking and curvatur thicknss locking ar th rason for bad prformanc of th EAS9 and EAS5 lmnts in this problm. A B Figur 4..4: Dformd hmisphr at F = 5 (without any magnification abl 4..: Displacmnts du to pinchd forc F = 5 Elmnt u A u A /u rf v B v B /v rf EAS EAS ANSn SS7n Rfrnc [QUOa] Slit annular plat undr lin forc A circular annular plat has a slit cut (lin AB Figur 4..5 along th radial dirction. h plat is clampd at on nd of th slit and suffrs a lin forc p =.8 at th fr nd. h innr radius is R i = 6 th outr radius is R o = th plat thicknss is h =.. Young modulus is E =. 6 and Poisson s ratio is υ =.. h total load is applid in qual stps. A msh of 6 lmnts is usd to modl th plat. Rfrnc solution is numrical rsult of HS hybrid-strss solid-shll lmnt of Sz t al. [SZE]. h HS lmnt adopts th ANS mthod for transvrs shar locking and trapzoidal locking rmovals; and strss componnts ar assumd indpndntly from th ons obtaind from th displacmnt fild. Numrical rsults of th HS lmnt ar: vrtical displacmnt at point A: w A =.68; vrtical displacmnt at point B: w B = 7.57.

114 R o p=.8 R i A B Figur 4..5: Slit annular plat - initial configuration p=.8 B A Fixd Figur 4..6: h dformd configuration at maximum load (without any magnification h annular plat is a thin-walld structur (R/h > hnc transvrs shar locking may appar. From numrical rsults in Figur 4..7 w s that th SS7n lmnt prforms as wll as th HS lmnt of Sz t al. h ANSn lmnt is slightly stiffr than th SS7n and HS lmnts. Normally whn Poisson s ratio is qual to zro if only transvrs shar locking xists th ANSn and SS7n should giv similar rsults. In this tst mayb mmbran locking occurs hnc th ANSn is littl stiffr than th SS7n. Du to transvrs shar locking and high aspct ratio th EAS9 lmnt bhavs too stiff but still too much bttr than th Q standard lmnt.

115 Normalizd of load ANSn-w A ANSn-w B SS7n-w A SS7n-w B EAS9-w A 6 EAS9-w B. 7 Q-w A. 8 Q-w B. 9 HS-w A [SZE] HS-w B [SZE] Vrtical dflctions at points A and B Figur 4..7: Load-dflction curvs for displacmnts at points A and B ONLUSION In this chaptr various bnch-mark tsts (linar and nonlinar problms with various locking ffcts hav bn invokd to dmonstrat prformanc for th SS7n solid-shll lmnt. h SS7n shows good prformancs for incomprssibl bhavior and for bnding bhavior of thin and thick-walld structurs. Howvr all of th numrical tsts in this chaptr ar limitd to linar matrial modl. In comparison with th classical shll lmnts th solid-shll lmnts allow a straightforward intgration of D matrial modls sinc thy do not rsort to th plan strss assumption. his advantag spcially bcoms important for implmntation of nonlinar matrial modls. his argumnt is assurd in th nxt chaptr whr prformancs of th SS7n lmnt with nonlinar matrial modls ar considrd.

116 haptr 5. PLASI APPLIAIONS INRODUION In th last chaptr applications of th SS7n lmnt in linar lasticity and nonlinar lasticity wr prsntd. In ordr to xploit its prformancs in plasticity fild th SS7n lmnt has bn also implmntd in FEAP (Finit Elmnt Analysis Program [AY]. In this chaptr w invstigat plastic bhavior of th ANSn and SS7n lmnts. First of all availabl plastic thory in FEAP is brifly prsntd. hn numrical tsts ar invstigatd to look for diffrncs in plastic bhavior btwn ANS and ANSn tchniqus. o carry out that work strsss in largly plasticity-dformd structurs ar analysd. Latr apart from th just mntiond tsts a spcial car is takn for a springback simulation. Springback or lastic rcovry rlats to th chang in shap btwn th fully loadd and unloadd configurations that th matrial ncountrs during a stamping opration. his rsults in th stamping componnt bing out of tolranc and can crat major problms in th assmbly or installation. Springback prdiction of sht mtal aftr forming is an important issu in controlling th manufacturing procsss. o this nd a bnchmark tst for high strngth stl will b invstigatd with th ANSn and SS7n lmnts. 5. FINIE SRAIN HEORY 5.. Multiplicativ split onsidr a body Ω which contains a lin vctor dx bfor dformation (initial configuration. Aftr dformd to th currnt configuration th lin vctor dx is transformd to dx by a transformation mapping F (Figur 5... h lin vctor dx has undrgon both lastic and plastic dformation to b transformd to dx. h intrmdiat configuration is dfind as in which th lin vctor dx has bn unloadd to a strss fr stat charactrizd by lin vctor dp. In othr words th lin vctor dx in initial configuration has undrgon purly plastic dformation to bcom th lin vctor dp in th intrmdiat configuration. hat pur plastic transformation can b xprssd by: p dp = F dx (5.. And th pur lastic transformation from th intrmdiat configuration to th currnt configuration is ralizd by: dx = F dp (5.. hn w can writ: p dx = F dp = F F dx (5.. h finit strain plasticity formulation rlis on th local multiplicativ dcomposition of th dformation gradint F that is drivd from (5.. as: 4

117 F = F F p (5..4 p whr F is th dformation causd by th lastic strtching and rotation and F is th plastic dformation. his is th classical multiplicativ dcomposition of L s [SIM88a]. Z F Initial (rfrnc configuration dx X x dx urrnt configuration F p dp F X Y Du to th total Lagrang formulation of th variational quations for th solidshll lmnts th plasticity modl hraftr will b formulatd using th right auchy strain tnsor. As pointd out by Simo [SIM88a] th rturn-mapping algorithm of infinitsimal plasticity can b carrid ovr to th prsntd formulation without any modification. With th hyprlastic modls th lastic prdictor in th rturn-mapping algorithm is xactly calculatd by using th strain nrgy function. h Grn-Lagrang strain tnsor is dfind rlativly to th rfrnc configuration as: E = E p ( = Figur 5..: Schmatic diagram of multiplicativ dcomposition F ( F F p I = ( I F p I = ( p I with with = F F p = whr is th right auchy strain tnsor. onsidr a gnral form of strain nrgy function: F p F p (5..5 p W = W ( (5..6 Assuming hyprlastic rspons th scond Piola-Kirchhoff strss tnsor S is dfind as: S = W ( p Intrmdiat strss-fr configuration (5..7 5

118 5.. Yild condition onsidr a yild surfac dfind in strain spac with a gnral functional form givn by Φ( p Q (5..8 whr Q is a suitabl st of intrnal plastic variabl vctor. h volution of th intrnal plastic variabl vctor Q can b dtrmind by a rat quation in th form: Qɺ = p γɺ H( Q (5..9 p whr H( Q is a prscribd function of th gnralizd plastic hardning moduli; th initial condition is Q = at th rfrnc configuration. h trm γ is plastic consistncy paramtr. 5.. Flow rul As in th infinitsimal thory in this sction bginning with th principl of maximum plastic dissipation th volution of plastic flow is xprssd dirctly in trms of kinmatic variabls rlatd to th multiplicativ dcomposition. Without loss of gnrality th lastoplastic bhavior is assumd to b charactrizd by variabls { p Q}. Furthrmor assum that an lastoplastic potntial p function can b dcoupld into intrnal-indpndnt contribution W ( and intrnal-dpndnt contribution W Q (Q as: p Q W = W ( + W ( Q (5.. h plastic dissipation at th stat dfind by { p Q} is: D p p ( Q; ɺ p Qɺ W ( : ɺ p W = + : Qɺ p Q 6 (5.. In local form th maximum plastic dissipation formulatd in strain spac may b statd as follows. Giv a stat { p Q} among all admissibl right auchy strain tnsors satisfying th yild critrion th actual strain tnsor is th on for which plastic dissipation attains its maximum. Lt s considr th maximum plastic dissipation in point of viw of optimization thory th problm may b statd as: p p Maximiz { ( Q; ɺ p Qɺ W [ : ɺ p W D = + : Qɺ ]} p p Q 6 p subjct to K = { R Φ( Q } (5.. whr K is th spac of admissibl right auchy strain tnsors at fixd plastic variabls { p Q}; tnsor is symmtric hnc consists of 6 indpndnt componnts. As shown in (5.. and (5.. with a fixd st of { p W Q} : Qɺ constant Q th maximum of W ( p p p D only dpnds on th trm [ : ɺ ] p. Hnc th

119 7 maximum of p D is quivalnt to th minimum of ] : ( [ p p p W ɺ. hus th maximum plastic dissipation problm can b changd to: Minimiz { ] : ( [ p p p W ɺ } subjct to } ( { 6 Φ = Q p R K (5.. W can solv th problm (5.. by th mthod of Lagrang multiplirs to inquality constraints. h Lagrang functional for th problm (5.. is dfind: ( : ( Q p p p p p W L Φ + = γɺ ɺ (5..4 SRAIN-BASED ELASOPLASI ONSIUIVE MODEL. Stp : Multiplicativ dcomposition p p p p F F F F F F F = = =. Stp : Hyprlastic strss-strain rlations S p = ( W. Stp : constitutiv tnsors Elastic: 4 = W 4 Plastic p W M = 4 4. Stp 4: Flow rul Q S M S Φ = = ( : ( : p p p p p γɺ ɺ ɺ ɺ 5. Stp 5: Hardning law ( Q H Q p γɺ ɺ = 6. Stp 6: Loading/unloading conditions γɺ ; ( Φ Q p ( = Φ Q p γɺ Figur 5..: Finit strain thory

120 whr γɺ is positiv and blongs to th st of squar intgrabl L (Ω functions that p dfind by th positiv con K : K p = { ɺ γ L ( Ω ɺ γ } (5..5 Rsulting from (5..4 and (5..5 th Kuhn-uckr conditions ar: p p L p p p Φ( = ɺ = ( : ɺ Q S M = ɺ γ ɺ γ (5..6 p Φ( Q p ɺ γ Φ ( Q = p p W ( whr th constitutiv tnsor M( = 4. p h flow rul (5..6a and loading/unloading conditions (5..6b-d ar associativ with th multiplicativ dcomposition (5..4. h prsntd thory is summarizd in Figur Elastoplastic tangnt moduli h rquirmnt for th load point to rmain on th yild surfac during plastic dformation is calld th consistncy condition. It nabls to dtrmin th plastic multiplir γɺ. his condition stats that plastic loading ( γɺ rquirs Φ( p Q / t = as: or Φ( t p Q = Φɺ ( p Φ Φ Q = : ɺ + p : ɺ p Φ + : Qɺ = Q (5..7 Φ : ɺ Φ -[ : ɺ p Φ = + : Qɺ ] p (5..8 Q p alculat ɺ from (5..6a thn introduc th rsult into (5..8 w can driv th xprssion for th plastic consistncy paramtr: Φ : ɺ ɺ γ = Φ Φ Φ ( : M : : H p Q im diffrntiating of th lastic constitutiv quation (5..7 w hav: W W p Φ γ p S ɺ = 4 : ɺ + 4 : ɺ = A : ɺ - 4 ɺ W whr A = 4 and W : ɺ p Φ = -ɺ γ from (5..6a. p By insrting (5..9 into (5.. w hav th xprssion: (5.. Sɺ p = A : ɺ (5.. 8

121 with th lastoplastic tangnt moduli: A p = A - Φ 8 Φ Φ 8 Φ Φ : M : Q p : H 5. J MAERIAL MODEL 5.. Multiplicativ split and lastic rspons h matrial modl dscribd abov will b applid for J matrials. h J modls ar wll suitd to th matrials whos lastic volumtric rspons is uncoupld with lastoplastic dviatoric rspons this bhavior is obsrvd in mtal in plasticity for instanc. Dnot J = dt( F th volum-prsrving part of th dformation gradint part is dfind: / F = J F (5.. with dt( F = it mans F satisfis th incomprssibl condition. h right auchy strain tnsors which ar associatd with F and F ar also dfind: / = F F = J dt( = and similarly = F F = p p p / p J (5.. Account for uncoupld volumtric/dviatoric rspons th nrgy function (5.. is in th form: p vol dv p Q W ( Q = W ( J + W ( + W ( Q (5.. h uncoupld nrgy function in (5.. rsults in uncoupld volumtric dviatoric strss-strain rlationships. In mtal plasticity th plastic dformation is isochoric i.. fully incomprssibl. Whil lastic dformation is comprssibl and small in many applications. As an xampl an nrgy function from [SIM9b] is consistd as: W W W vol dv Q = K(ln J = G[( ε = H iso ε p / / + ( ε + ( σ = K[ln( λ λ λ ] y + ( ε y ] p / σ ε + δ ( σ y σ y xp( δε p (5..4 whr - G and K ar constant. hy ar shar modulus and bulk modulus rspctivly; - δ > is saturation xponnt; - ln( ε i = λi ; i = ; λ i ar principal lastic strtchs (from ignvalus of ; - J = dt( F ; σ and σ ar th first yild strss and th saturation yild strss; - y - p y ε is th quivalnt plastic strain; - H iso is th linar hardning modulus. 9

122 Following (5..7 th scond Piola-Kirchhoff for J matrial is dcomposd into th hydrostatic and dviatoric parts as: vol dv p W ( J W ( S = + dv p - / W ( vol = Jp + J DEV[ ] = S + S dv (5..5 vol whr p = W ( J J is th hydrostatic prssur. In (5..5 w did us th rlation (s [HOL] pag 4: J = J - and th drivativ (s [HOL] pag 9: (5..6 / - = J [ I ] (5..7 and th dnotation DEV[ - ] = ( [ : ( ] ( In (5..5 th hydrostatic strss is prsntd by th trm Jp and DEV [ ] givs th physically corrct dviator strsss in th rfrnc configuration. 5.. Flow rul and yild function h yild function for J matrial modl can b assumd to only dpnd on th dviatoric part of right auchy strain tnsor as: p Φ( Q (5..9 According to th argumnts in th last sction with th Kuhn-uckr conditions (5..6 lads to a flow rul as: ɺ p Φ Φ - / Φ S = ɺ γ = ɺ γ = ɺ γ J DEV[ ] ɺ γ (5.. p Φ( Q p ɺ γ Φ ( Q = p whr similar to (5..6a S ɺ is also st: dv p -/ W p Sɺ = J DEV[ : ] (5.. p h von Miss yild condition in form of spatial trms in strss spac is: φ( τ Q = dv[ τ α] + q σ y q = H iso κ whr: - σ y is th yild strss and (5..

123 - th Kirchhoff strss tnsor τ is th push-forward of th scond Piola-Kirchhoff strss S by: τ = FSF (5.. - th hardning variabls ar: Q = [ q α] (5..4 with α is back strss which dfins th location of th cntr of th yild surfac. his back strss is usd to account for Bauschingr s ffct (kinmatic hardning. h isotropic hardning is charactrizd by th intrnal variabl q with isotropic hardning modulus H iso >. H iso is a function of isotropic hardning variabl к for nonlinar hardning laws and for linar isotropic hardning laws H iso is a constant. h algorithm for th prsntd formulation is straightforward as follows. Intgrat th matrial (rfrnc dscription of th flow rul (5.. by an intgration schm such as backward Eulr diffrnc schm. Substitut th rsult into th hyprlastic strss-strain rlations (5..5. h dtaild algorithm is listd in Figur 5... h implmntation of th hyprlastic formulation of J-flow thory rducs to th classical radial rturn with th lastic prdictor computd by nrgy function valuation. W s that maximum of plastic dissipation (5.. lads to a rturn mapping algorithm s Figur 5... hat algorithm will look for th solutions on a path that maks th plastic dissipation stationary. oncrtly within a typical tim stp a trial lastic is calculatd first. hn th actual strss is dfind in th closst-point projction of th trial stat onto th lastic domain. For J flow thory th closst-point projction boils down to th classical radial rturn mthod.

124 REURN-MAPPING ALGORIHM FOR J-FLOW HEORY. Stp : Gomtry updat ; GRAD = = + = + = n n n n / - n n n n n J F F F F U F F U x x. Stp : Elastic prdictor (k= n n p n n dv n vol n p n p n W J J W J Q Q S = + = = ( ( / / ( ( ( (. Stp : hck for yilding ( ( ( ( ( Φ = Φ n p n k n k n Q IF ( k n+ Φ < OL HEN St ( ( ( k n n + + = and EXI (to th nxt tim stp ELSE continu Stp 4 and Stp 5 4. Stp 4: omput plastic consistncy paramtr (s (5..9 : : 8 ( / ( ( k n p k n k n J Φ Φ Φ Φ = H Q M γ 5. Stp 5: Updat stat variabls ( ( ( ( ( ( / ( ( ( ( / ( ( : k n k n k n k n k n k n k p n k p n k n k n k n k n J J = Φ = Φ = H Q Q M S S γ γ γ St k = k+ and REURN Stp Figur 5..: Matrial strss updat algorithm

125 5. PLASIIY APPLIAIONS In th sctions following w invstigat prformancs of th SS7n lmnt in fild of plasticity. 5.. antilvr at larg lastoplastic dformation onsidr a cantilvr bam of dimnsion L b h = 5.7 clampd at lft nd and loadd by a distributd forc (q = 7 at right nd Figur 5... his tst was prsntd by Huh and Kim [HUH]. h lastic modulus is E = 5 N/mm and Poisson s ratio is ν =.. Matrial law is lastoplastic with isotropic hardning as p σ = 4 + ε. q = 7 Z Y h X L=5 A b Figur 5..: antilvr at larg lasto-plastic dformation As proposd in th work of Huh and Kim [HUH] a vry fin msh of 4 lmnts (along L h and b rspctivly is first mployd. By adopting th assumption on a plain strain stat ths authors can mploy a D modling. Hr with D solid lmnts to simulat D modling th width of bam (Y dirction will b prsntd by only on finit lmnt and th plain strain stat will b obtaind by stting to zro displacmnt in th width dirction. For th comparison btwn diffrnt approachs th displacmnt at th tip point A will b considrd. h SRI EAS9 EAS (MEAFOR softwar [ME8] ANS ANSn SS7 and SS7n lmnts will b involvd in th computation. h SRI lmnt is on of th bst lmnts in rmoving volumtric locking. Howvr shar locking in this problm is important. onsquntly in Figur 5.. w s that th SRI lmnt with layrs (i.. 4 lmnts is stiffr than th RI lmnt [HUH] or th EAS9 with 4 or layrs. Rsults givn by th EAS9 with layrs ar convrgd to th rsults givn by th EAS with 4 or layrs s Figur 5... If only layr is adoptd both th EAS 9 and EAS lmnts show stiff bhaviour s Figurs 5.. and 5... h EAS always givs vry good rsults in both shar and volumtric locking rmovals [AND9]. Hnc with a vry fin msh of 4 lmnts rsult of th EAS (MEAFOR lmnt is considrd as a rfrnc. W s that this rfrnc is a littl softr than th rsult of th RI lmnt in [HUH].

126 Nxt th dformation is calculatd by th ANS mthod with various layrs of lmnt along th thicknss dirction. onsidr th load-displacmnt curv th ANS and ANSn lmnts giv idntical rsults hnc only th rsults of th ANSn ar prsntd hr. W s that with layrs th ANSn lmnt is softr than th SRI whn P = q b 4.N. h rason is du to shar locking. At load lvl P > 4.N th cantilvr is in largr dformation ffct of volumtric locking bcom mor important hnc bhavior of two-layrd ANSn lmnt is stiffr than th SRI lmnt Figur With 4 layrs th ANSn lmnt is always softr than th SRI lmnt. Whn volumtric locking bcoms mor srious (bcaus of larg plasticity dformation th ANSn lmnt with 4 layrs approachs th bhavior of th SRI lmnt. Incrasing numbr of layrs to ANSn bhavior bcoms too soft. ompar to th rfrnc rsult (EAS bhaviour of th ANSn is too stiff. W s that for this problm th ANSn lmnt which is only fr from shar locking should not b usd EAS94 EAS94. EAS EAS94. 5 RI(4 [HUH] 6 SRI(4 MEAFOR Figur 5..: EAS9 (MEAFOR rsults with various lmnts along thicknss Now w invstigat th problm with th SS7n lmnt (bcaus th SS7 and SS7n lmnts giv idntical rsults. From Figur 5..5 w s that rsult of th SS7n lmnt with layrs along th thicknss is only clos to th rsult of th EAS whn shar locking is important (i.. P 4.N. Whn P > 4.N SS7n with layrs is a littl stiffr than th EAS and RI lmnts. Incrasing layrs along th thicknss to 4 or th SS7n vn bcoms stiffr. If a vry coars msh is adoptd i.. with only layr along th thicknss th SS7n is too stiff. Hnc in this problm layrs of SS7n ar th most suitabl choic. 4

127 EAS4 EAS4 EAS4 EAS94 EAS4 4 RI(4 [HUH] Figur 5..: EAS (MEAFOR rsults with various lmnts along thicknss Forc P(N ANSn(4 ANSn(4 4 ANSn( Q(4 5 EAS(4 MEAFOR 6 RI(4 [HUH] 7 SRI(4 MEAFOR Displacmnt v (mm Figur 5..4: ANS rsults with various lmnts along thicknss 5

128 EAS(4 MEAFOR RI(4 [HUH] 5 6 Forc P(N SS7n SS7 (4 4 SS7n SS7 (4 4 SS7n SS7 (4 SS7n SS7 (4 Displacmnt v (mm Figur 5..5: SS7n rsults with various lmnts along thicknss It was prsntd in haptr that th ANS and ANSn lmnts ar diffrnt in assumd straind componnts E xz and E yz. hs lmnts will giv idntical rsults if transvrs shar strains ar not important. As in this problm th ANS and ANSn lmnts giv idntical rsults bcaus th cantilvr is not too thin (aspct ratio hnc transvrs shar strsss givn by th ANS and ANSn ar similar s Figurs onsquntly σ xz strss givn by th SS7 and SS7n lmnts ar similar s Figurs abl 5... onvrgnc of σ xz (MPa min max min max min max min max min max EAS EAS SS SS7n Distribution of σ xz strss in a msh of 4 givn by th EAS9 lmnt (MEAFOR Figur 5.. and th solid-shll lmnts (SS7 and SS7n Figurs 5..8 and 5..9 ar similar. Howvr th ultimat valus givn by th EAS9 lmnt is largr in comparison with th solid-shll lmnts (-47MPa vs. -4MPa and 6MPa vs. 4MPa. h convrgnc valu of σ xz is found in abl 5.. with both solid-shll and EAS lmnts (bold lttrs with msh of 4 5 lmnts. Data in abl 5.. shows that th EAS and EAS9 lmnts go to th convrgnc from highr valus. In contrary th SS7 and SS7n lmnts go to th convrgnc from lowr valus. 6

129 Figur 5..6: ANS lmnt - σ xz (msh 4 Figur 5..7: ANSn lmnt - σ xz (msh 4 7

130 Figur 5..8: SS7 lmnt - σ xz strss (msh 4 (th rsult is shown in th structural coordinat systm Figur 5..9: SS7n lmnt - σ xz strss (msh 4 (th rsult is shown in th structural coordinat systm 8

131 Figur 5..: EAS9 lmnt - σ xz strss (MEAFOR msh 4 9

132 5.. Pinchd cylindr at larg lastoplastic dformations onsidr a cylindr of innr radius r = thicknss t = and lngth L = 6 Figur 5... Young s modulus is E = and Poisson s ratio is ν =.. A coupl of opposit concntratd forcs F applid at th mid-lngth of th cylindr. Both nds of th cylindr ar pinchd only a fr movmnt in th axial dirction Y is possibl. Matrial law is lastoplastic with isotropic hardning as σ = 4.+ ε p. Z r F Y X F L Figur 5..: Pinchd cylindr W invstigat dflction coincidnt with th point load against loads. h dflction rspons is strongly dominatd by inxtnsional circumfrntial bnding. Du to symmtry only on ighth of th cylindr is modld. A rsult with D lmnts and msh of 4 4 lmnts proposd by Wriggrs t al. [WRI96b] will b mployd as th rfrnc s Figur 5... Wriggrs did us D nhancd lmnt prsntd by Simo and Armro in [SIM9a]. hat lmnt is quivalnt to th EAS (6v + 6s lmnt s abl.4.. It mans th rfrnc lmnt is volumtric and shar locking fr. Lt s invstigat convrgnc of th EAS lmnt. W also s that th EAS lmnt of MEAFOR Figur 5.. convrgs with a msh of 4 4 lmnts. h EAS (4 4 lmnts of MEAFOR givs an idntical rsult as th rfrnc [WRI96b] Figur EAS (6 6 EAS (4 4 EAS ( 4 EAS ( Figur 5..: EAS lmnt (MEAFOR

133 Applid forc F 5 5 ANSn (6 6 ANS (6 6 D (Wriggrs SRI (MEAFOR EAS (MEAFOR EAS (MEAFOR Displacmnt w Figur 5..: Pinchd cylindr: Forc-Displacmnt his cylindr is a thin shll structur (R/t = >> and subjct to larg dformation. onsquntly transvrs shar strains ar important. In this problm rsults of th ANS and ANSn lmnts ar not totally coincidnt s Figur 5... Now w considr th rsultant strsss from th ANS and ANSn tchniqus with a coars msh: 6 6 lmnts. Lt s concntrat in ultimat transvrs shar strsss at th lft nd of th cylindr and at th ara whr th load is applid. All rsults ar considrd in th structural coordinat systm. Data in Figurs show th σ xz strsss calculatd by th SS7 and SS7n lmnts. Rfrnc rsult of th σ xz strss is in Figur 5..b. Pay attntion that th concntratd load is only applid at th cntr nod. As a consqunc th lmnt containing th cntr nod is singular du to th applid load is unphysical. Hnc in following analyss w will considr strsss at point B instad of strsss at point A s Figur Whn using th sam coars msh (6 6 th SS7n givs th rsults which ar closr to th rsults of th rfrnc - EAS (4 4 - than th SS7 dos s abl 5... In abl 5.. transvrs shar strsss of th EAS (6 6 ar xtractd from Figurs In that tabl transvrs shar strsss of th ANS and ANSn lmnts (6 6 ar also introducd. W s that transvrs shar strsss givn by th ANSn ar closr to th rfrnc than th rsult of ANS. ransvrs shar strsss prdictd by th EAS (6 6 ar wors than th valus prdictd by th SS7n (xcpt th minimal valu of σ yz abl σ xz σ yz abl 5... Normalizd transvrs shar strsss (rfrnc valus ar rsults of EAS lmnt ANS (6 6 ANSn (6 6 SS7 (6 6 SS7n (6 6 EAS (6 6 Rfrnc valus Max Min Max Min Data from Figurs show σ yz strss calculatd by th SS7 and SS7n lmnts. Rfrnc rsult of σ yz strss is in Figur 5..b. Whn a coars msh is

134 adoptd compar with th rfrnc strsss in aras I and II (Figurs and b w can s that th SS7n lmnt giv lss wors rsults than th SS7 lmnt. In dtail strss in ara I givn by th rfrnc is.4 whil valus givn by th SS7 and SS7n ar 7.6 and 9.6 rspctivly. Strss in ara II givn by th rfrnc is -5. whil valus givn by th SS7 and SS7n ar -.9 and -.7 rspctivly. h maximal and minimal valus of σ yz ar listd in abl 5... B A Figur 5..4: SS7 lmnt - σ xz (msh 6 6 Figur 5..5: SS7n lmnt - σ xz (msh 6 6

135 Figur 5..6: EAS lmnt - σ xz (EASMEAFOR 6 6 (du to singularity at load point max. valu has bn imposd to 6. Figur 5..7: EAS lmnt - σ yz (EASMEAFOR 6 6 (du to singularity at load point min. valu has bn imposd to -5.7

136 II I II Figur 5..8: SS7 lmnt - σ yz (msh 6 6 II I II Figur 5..9: SS7n lmnt - σ yz (msh 6 6 4

137 Figur 5..a: Rfrnc - σ xz (EASMEAFOR 4 4 Figur 5..b: Rfrnc - σ xz (EASMEAFOR 4 4 (du to singularity at load point max. valu for drawing has bn imposd to 4. 5

138 Figur 5..a: Rfrnc - σ yz (EASMEAFOR 4 4 II I II Figur 5..b: Rfrnc - σ yz (EASMEAFOR 4 4 (du to singularity at load point min. valu for drawing has bn imposd to

139 As analysing abov th diffrncs in transvrs shar strsss calculatd by th classical ANS tchniqu and th altrnativ ANS tchniqu lad to diffrnc in von- Miss strss and Load-Displacmnt curv calculatd by th SS7 and SS7n lmnts s Figurs Figur 5..: SS7 lmnt - von Miss strss (msh 6 6 Figur 5..: SS7n lmnt - von Miss strss (msh 6 6 7

140 Nxt w compar rsults of th SS7 and SS7n lmnts with rsults of solid lmnt (EAS and shll lmnt prsntd in [WRI96b]. h shll lmnt in [WRI96b] is basd on a quasi-kirchhoff-thory which mans that th assumption of th classical Kirchhoff-Lov kinmatics is rspctd via a pnalty constraint. A rducd intgration for th pnalty trm was applid in ordr to obtain locking-fr bhavior in bnding dominatd problms. In Figur 5..4 th load-displacmnts curvs for diffrnt lmnts ar plottd. In gnral th rsults of th SS7 and SS7n (6 6 lmnts ar a littl stiffr than th rsults of D calculation (EAS 4 4 lmnts. Howvr with th sam coars msh (6 6 th solid-shll lmnts SS7 and SS7n ar comparabl to th convntional shll lmnt (RI shll lmnts [WRI96b] and bttr than rsults of th EAS Figur h SS7n lmnt shows a load dcras (at w as RI shll lmnt dos (at w. his consqunc ariss as a rsult of th rlativly coars msh. Hnc it maks th plastic zon cannot dvlopd continuously. his consqunc can b cancld by using a finr msh as appar for th EAS rsults (6 6 vs In Figur 5..4 th curv of th SS7 lmnt is smoothr than th SS7n. 6 Applid forc F SS7n (6 6 5 SS7 (6 6 D (Wriggrs RI Shll (Wriggrs EAS (MEAFOR EAS (MEAFOR Snap-through Displacmnt w Figur 5..4: Pinchd cylindr solid-shll lmnts

141 For this problm bhavior of th EAS and EAS (with both coars msh and fin msh ar similar s Figurs Hnc it is rasonabl to us intrnal paramtrs for th EAS as Wriggrs t al. [WRI96b]. Applid forc F SS7n (6 6 SS7 (6 6 EAS (MEAFOR 4 4 EAS (MEAFOR 4 4 EAS (MEAFOR Displacmnt w Figur 5..4: Pinchd cylindr solid-shll lmnts By this numrical tst w s that th SS7n lmnt prdicts transvrs shar strsss bttr than th SS7 lmnt abl 5... For analyzing a thin-walld structur with a coars msh both solid-shll lmnts SS7 and SS7n can work as wll as th convntional shll lmnts.g. th shll lmnt in [WRI96b]. h othr solid-shll lmnts such as th solid-shll lmnt of Samcf ( lmnts [JE8] provids a rsult which approximats th rsult of D lmnt of Wriggrs (4 4 lmnts. Hnc using th solid-shll lmnts to simulat thin-walld structurs will provid a ral D modl with a chap computational cost. 9

142 5.. Springback of unconstraind cylindrical bnding h following bnchmark initially proposd at th NUMISHEE confrnc [NUM] has bn chosn as a rfrnc cas for comparing finit lmnt formulations with various tim intgration schms. his bnchmark is rcommndd to invstigat springback analysis and complx contact tratmnt. It consists of an initially flat blank bnt into a cylindrical shap and thn unloadd. Bcaus thr is no blankholdr th problm is calld unconstraind bnding. h initial gomtry and th loading procss ar dscribd in Figur 5..6 (lft sid. h loading procss is stoppd whn punch and di ar concntric (right sid of Figur R =.5 mm R =5 mm R = 4 mm Punch R Blank R R Di 9 Figur 5..6: Initial position (lft and final position (right for stamping Aftr th loading phas th unloading phas taks plac and som springback occurs. h amount of springback is quantifid in th following way s Figur Angl btwn lin AB and lin D (Figur 5..7a bfor and aftr springback at th final strok of 8.5 mm. Othr spcifications of th problm ar: - h tools ar assumd rigid. - Blank dimnsion ar: lngth:.mm; thicknss: mm; width: mm. - Plan strain is assumd during all simulations. - Friction cofficint: µ = h punch spd is kpt constant btwn: (-5 mm/sc. - otal punch strok: 8.5 mm. - Blank matrial: isotropic stl with mchanical proprtis:.577 σ = 645.4( ε +. MPa; E = 7.5 GPa ; ν =..

143 A Unit: mm B D Punch E F a btwn lin AB and lin D b btwn contactd points Figur 5..7: Dfinition of angls (point E and point F ar th farthst contact points from th cntrlin First of all a paramtr study ar undrtakn with all lmnts (EAS SRI ANS and solid-shll in ordr to dtrmin th snsitivity of th numrical solution with rspct to som numrical paramtrs. Indd w considrd th influnc on th rsults of th discrtization th pnalty paramtrs for contact tratmnt and th tim intgration schm with thir associatd numrical paramtrs. All computations with th EAS and SRI lmnts wr carrid out by using MEAFOR softwar [ME8]. All computations with th ANS and solid-shll lmnts wr carrid out by using FEAP program [AY]. his is a plan strain problm hnc instad of using th SS7n lmnt its D vrsion - D.SS4n lmnt will b adoptd. h D.SS4n is rsultd from th D.ANSn in (..7 and th EAS4 lmnt abl.4.. Simulation with D.EAS7 lmnts abl 5.. is a survy of som rprsntativ rsults. First of all th blank is dividd into thr zons (s Figur 5..8 whos lngths ar rspctivly givn by 8 and mm rspctivly. Insid any of th thr zons any lmnt has th sam lngth. hn ach of th zons has bn discrtizd by imposing n lmnts through th thicknss and rspctivly n qual lmnts in th first zon and n and n 4 in th two rmaining zons. h diffrnt numrical paramtrs ar thn systmatically xplord. First msh siz was changd to find out a good msh quality (abl 5../ass -4. Aftr having st th msh th pnalty paramtrs ar varid (abl 5../ass 5-6. hn intgration schm is changd from quasi-static to implicit hung-hulbrt (abl 5../ass 6-8 with paramtrs ar α M = -.97 and α F =.. his choic for α M and α F satisfis th unconditional stability and scond ordr accuracy of hung-hulbrt schm [PON99]. Numbr of lmnts through th thicknss dirction was also considrd (abl 5../ass 6-9. Each input paramtr was changd to gt a bttr and bttr rsult. h abl 5../as with vry fin msh is introducd to gt a rfrnc solution. h aras which ar snsibl to msh siz ar dformd aras (mshd by n n n lmnts. W only survy th changs of n n and n bcaus th ara with n 4 lmnts is nithr dformd nor contactd by th punch or di (s Figur 5..8.

144 Punch R Blank n lmnts n lmnts n lmnts n4 lmnts R R Di Figur 5..8: Discrtization of th modl: dfinition of th zons hn whn a fin nough discrtization has bn attaind for th msh th pnalty paramtrs is varid. For rasonabl variations it can b sn in Figur 5.. that thr is almost no diffrnc in th forc curvs. h conclusion is th sam whn th implicit hung-hulbrt schm has bn usd instad of th quasi-static algorithm. As far as on avoids using only on lmnt through th thicknss (as 9 in Figur 5..9 on can also s that th rsults ar quit stabl. All th punch forc - displacmnt curvs xcpt th cas of using lmnt through th thicknss narly coincid with th rfrnc curv - as (abl 5... So w can conclud that with lmnts through th thicknss th msh is rfind nough. abl 5... alculation with D.EAS7 lmnt n lmnts n lmnts n 4lmnts n lmnts Msh on Blank as Intgration Msh Siz Not Pnalty schm n ; n ; n ; n 4 QS. ; ; 4; QS. 6; ; 4; QS. ; ; 8; 4 QS. ; ; ; 5 QS 5. ; ; ; 6 -H 5. ; ; ; onvrgnc of D.EAS7 7 -H 5. ; ; ; 8 -H. ; ; ; 9 QS. ; ; ; QS. 5; 7; 5; Rfrnc * Bold numbrs or lttrs indicat what has changd from on cas to anothr -H: hung - Hulbrt schm QS: Quasi - static

145 Punch forc (N cas : EAS-xx4x cas : EAS-6xx4x cas : EAS-xx8x cas 4: EAS-xxx n lmnts n lmnts n 4lmnts n lmnts Msh on Blank cas 5: : EAS-xxx-QS; pnalty=5. Punch displacmnt (mm Figur 5..9: Punch forc vs. punch displacmnt EAS lmnt (quasi-static; abl 5../cass to 5; pnalty =. For msh discrtization paramtr n is th most snsitiv. Indd a rough discrtization in this zon lads to oscillations in th punch forc vrsus punch displacmnt curv as can b sn in Figur 5..9 (as. his can b asily xplaind sinc it is th lowr sid of lmnts locatd in this zon that do hav a sliding contact with th shouldr of th di (radius R - s Figur As far as th contact with punch is considrd th radius R is much largr than R so it is much lss snsitiv to discrtization. Actually if n is too small th numbr of nods in contact with th di can b rducd to on. As a consqunc this lads to oscillations in th curv. As can b sn in Figur 5..9 n = lad to almost no oscillation. Punch forc (N 5 5 as 6: EAS-xxx -H; pnalty=5. (convrgnc as 7: EAS-xxx -H; pnalty=5. as 8: EAS-xxx -H; pnalty=. as 9: EAS-xxx QS; pnalty=. as : EAS-5x7x5x QS; pnalty=. (Rfrnc n lmnts n lmnts n 4lmnts n lmnts Msh on Blank 5 Punch displacmnt (mm Figur 5..: Punch forc vs. punch displacmnt EAS lmnt (abl 5../cass 6 to

146 In Figur 5.. w s that th rsults got by using diffrnt pnalty paramtrs which ar tims of diffrnc (5. and ar gnrally similar. For this problm solutions got by quasi-static schm and dynamic schm ar almost similar. Solutions with or lmnts on th thicknss ar similar xcpt th cas of using lmnt on th thicknss - as 9 maks th bhavior stiffr. Simulation with D.SRI lmnts In this sction a similar study was carrid out but this tim using D.SRI lmnts. Diffrnt rsults ar tabulatd in abl First of all w startd th D.SRI computations with a rasonabl msh for th D.EAS7 simulation (rasonabl mans that th forc curv is quit clos to th rfrnc on in this cas.g. n = ; n = ; n = ; n 4 = (s abl 5../as 6. As can b sn in Figur 5.. th rsulting curv xhibits a lack of smoothnss. hanging th numrical paramtrs whil kping n = dos not affct too much th rsults (s again Figur 5.. whil incrasing n will lad to th rfrnc solution - s Figur for n qual to or largr than. As a first conclusion w can stat that vn if D.SRI lmnt ar much chapr than D.EAS7 and convrg to th rfrnc solution as th msh is rfind thy should b usd with som car as thy do not xhibit a coars msh accuracy as th D.EAS7 dos. h D.SRI lmnt convrgs with a msh of lmnt (abl 5..4/as 6. Mshs of cass 7 and 8 ar vry fin so thy ar th bst solutions of D.SRI lmnts Figur 5..: Rfrnc solution of SRI lmnt (abl 5..4 /cas 7 quivalnt plastic strain simulation with MEAFOR 4

147 abl alculation with D.SRI lmnt as Intgration Msh Pnalty schm n ; n ; n ; n 4 QS 5. ; ; ; QS 5. ; ; ; -H. ; ; ; 4 -H. ; ; ; 5 QS. ; ; ; 6 -H. ; ; ; 7 QS. 5; 5; 5; 8 -H. 85; 7; 85; n lmnts n lmnts n 4lmnts n lmnts Msh on Blank Not onvrgnc of D.SRI 5 as : SRI-xxx QS; (Pnalty 5. as : SRI-xxx QS; (Pnalty 5. as : SRI-xxx -H; (Pnalty. as 4: SRI-xxx -H; (Pnalty. onvrgnc of EAS-xxx -H; pnalty=5. Punch forc (N 5 5 Rfrnc (EAS-5x7x5x n lmnts n lmnts n 4lmnts n lmnts Msh on Blank Punch displacmnt (mm Figur 5..: Punch forc vs. punch displacmnt SRI lmnt (Pnalty=. ; abl5..4/cass to 4 Simulation with D.ANS lmnts Bfor invstigat springback with th solid-shll lmnts lt s considr springback bhavior of D.ANSn lmnts. For this springback tst bhavior of th ANS and ANSn is similar so only rsults of th D.ANSn ar prsntd in abl W invstigat th ANS computations with a msh with which th EAS lmnt convrgd i.. n = ; n = ; n = ; n 4 = (s abl 5../as 6. Du to volumtric locking happns with th D.ANSn lmnt w should start with a highr numbr of lmnt along th thicknss to rduc locking.g. n =. Howvr as sing in Figur 5..5 th rsulting curv still xhibits locking. 5

148 Punch forc (N 5 5 as 5: SRI-xxx QS as 6: SRI-xxx -H (convrgnc as 7: SRI-5x5x5x QS as 8: SRI-85x7x85x -H Rfrnc (EAS-5x7x5x n lmnts n lmnts n 4lmnts 5 n lmnts Msh on Blank Punch displacmnt (mm Figur 5..: Punch forc vs. punch displacmnt SRI lmnt (Pnalty=. ; abl5..4/cass 5 to 8 h punch forc - displacmnt curv with th D.ANSn lmnt is stiffr than th rfrnc curv s Figur Incrasing numbr of lmnt along th thicknss n = 4 (abl 5..5/as th rsulting curv is closr to th rfrnc on. o gt mor stringnt rsult lt's trying incrasing lmnt numbr in contact ara with punch n = (abl 5..5/as. Data in Figur 5..4 shows that incrasing n has only ffct of rducing oscillation of contact forc. Manwhil incrasing numbr of lmnt along th thicknss n = 5 (abl 5..5/as 4 w gt bttr rsult. h convrgd rsult of th ANSn lmnt is obtaind with pnalty paramtr qual to.+ abl 5..5/as 4. If w us a highr pnalty valu (5.+ th computational cost incrasing whil rsult is not improvd Figur h consqunc is th sam whn intgration schm is changd from quasi-static to Nwmark (bcaus th hung-hulbrt schm is not availabl in FEAP. n lmnts n lmnts n 4lmnts abl alculation with D.ANSn lmnt as Intgration Msh Pnalty schm n ; n ; n ; n 4 QS. ; ; ; QS. ; 4; ; QS. ; 4; ; 4 QS. ; 5; ; 5 QS. ; 5; ; 6 QS 5. ; 5; ; 7 Nwmark. ; 5; ; n lmnts Msh on Blank Not hosn rsult of D.ANSn 6

149 Figur 5..4: hosn solution of D.ANSn lmnt (abl 5..5 /cas 6 von Miss strss simulation with FEAP Rfrnc (abl5../as ANS-xxx (abl5..5/as ANS-x4xx (abl5..5/as ANS-x4xx (abl5..5/as ANS-x5xx (abl5..5/as 4 Punch forc (N n lmnts n lmnts n 4lmnts n lmnts Msh on Blank Punch displacmnt (mm Figur 5..5: Punch forc vs. punch displacmnt ANSn lmnt (abl5..5/cass to 4 7

150 Rfrnc (abl5../as ANS-x5xx (abl5..5/as 5 ANS-x5xx (abl5..5/as 6 ANS-x5xx (abl5..5/as 7 Punch forc (N n lmnts n lmnts n 4lmnts n lmnts Msh on Blank Punch displacmnt (mm Figur 5..6: Punch forc vs. punch displacmnt ANSn lmnt (abl5..5/cass 5 to 7 W s that ANS computation with 5 layrs of lmnt along th thicknss givs th sam maximal punch forc as th on of th EAS lmnt ( s Figurs Howvr bhaviors of th ANS and EAS lmnts ar diffrnt. h ANS lmnt is stiffr whn punch displacmnt is smallr than mm. Whn punch displacmnt is largr than mm ANS bhavior is softr. his consqunc is rasonabl bcaus th ANS lmnt is only shar-locking fr. Simulation with solid-shll lmnt Finally springback prdiction for this unconstraind bnding problm is invstigatd with th solid-shll lmnt: D.SS4n. With this tst bhavior of th solidshll lmnts using th classical tchniqu and th altrnativ ANS tchniqu is similar. Hnc only rsults of th latr ar prsntd in abl W bgin th computation with a coars msh for EAS simulation i.. n = ; n = ; n = 4; n 4 = (abl 5../as. hn th msh is mad finr to gt bttr rsult. Data from Figur 5..8 shows that th solid-shll lmnt convrgd with rathr coars msh n n n n 4 = 6 7. n lmnts n lmnts n 4lmnts abl alculation with solid-shll lmnt as Intgration Msh Pnalty schm n ; n ; n ; n 4 QS. ; ; 4; QS. 5; ; 6; QS. 6; ; 7; 4 QS 5. 6; ; 7; 5 Nwmark. 6; ; 7; n lmnts Msh on Blank Not onvrgnc of solid-shll 8

151 Figur 5..7: Rfrnc solution of D.SS4n lmnt (abl 5..6 /cas von Miss strss simulation with FEAP Aftr having st th msh th pnalty paramtrs ar varid (abl 5..6/ass 4. hn intgration schm is changd from quasi-static to Nwmark (abl 5..6/ass 5. It can b sn in Figurs that thr is almost no diffrnc in th forc curvs. h conclusion is th sam whn th Nwmark schm has bn usd instad of th quasi-static algorithm. W s that whn th punch forc is largr than N th solidshll lmnt is a littl softr than th rfrnc rsult. Rfrnc (abl5../as SS4n-xx4x (as SS4n-5xx6x (as SS4n-6xx7x (as. convrgnc Punch forc (N n lmnts n lmnts n 4lmnts n lmnts Msh on Blank Punch displacmnt (mm Figur 5..8: Punch forc vs. punch displacmnt D.SS4n lmnt (abl5..6/cass to 9

152 EAS-xxx (abl5../as 6 SS4n-x5xx (as 4 SS4n-x5xx (as 5 Punch forc (N n lmnts n lmnts n 4lmnts n lmnts Msh on Blank Punch displacmnt (mm Figur 5..9: Punch forc vs. punch displacmnt D.SS4n lmnt (abl5..6/cass 4 and 5 Validation of simulation rsults In ordr to validat rsults of th solid-shll lmnt th forc curv obtaind has bn compard to both xprimntal (BE - Figur 5..4 and numrical (BS - Figur 5..4 rsults publishd in th NUMISHEE procding [NUM]. As can b sn from thos figurs numrical rsults from th D.SS4n match quit wll th xprimntal rfrnc rsults (curv BE-. It should b notd that numrical rsults xhibit a quit larg disprsion which can b attributd to th varity of finit lmnt cods as wll as th varity of lmnts (shll continuum quads triangls and tim intgration algorithm (implicit quasi-static. Punch forc kn BE- BS- BS-4 BS-5 BS-6 BS-7 D.SS4n-6 7 (abl 5..6/as D.EAS7- (abl5../as 6 5 Punch displacmnt (mm Figur 5..4: Punch forc vs. punch displacmnt [NUM] (BS = Bnchmark simulation rsult 4

153 Punch forc N D.SS4n-6 7 (abl 5..6/as BE- BE- BE- BE-4 EAS- 5 (abl5../as 6 Punch displacmnt (mm Figur 5..4: Punch forc vs. punch displacmnt [NUM] (BE = Bnchmark Exprimnt rsult Whn punch forc is smallr than N rsults of th D.EAS7 and D.SS4n ar coincidnt but thy ar a littl diffrnt from th xprimnt rsults BE-. Whn punch forc is largr than N only th D.EAS7 givs th idntical rsults with rsults of th BE-. Manwhil th D.SS4n is a littl softr than th D.EAS7. As a mor local rsult of th bnchmark it was also askd to valuat th angl α - s Figur 5..7 and Figur 5..4 for dfinition and illustration for diffrnt punch strok i and 8.5 mm. Rsults dlivrd by th D.SS4n ar vry closd to th xprimntal valus (BE- s Figur h convrgnc rsults of th othr lmnt ar also prsntd in Figur S abl for angl α at diffrnt punch stroks in all cass of simulation with th D.EAS7 D.SRI D.ANSn D.SS4n lmnts. abl Springback angls - calculation with D.EAS7 lmnt as Angl ( btwn lin AB and lin D Angl ( btwn farthst points (from cntrlin at strok bfor SB aftr SB 7mm 4mm mm 8.5mm

154 contact ara ½ of opn angl Figur 5..4: Dfinition of th angl btwn contact points which ar th farthst from th cntrlin 6 Angl ( mm 8.5mm 7mm 4mm 8 BE 4 BE 8 BE 6 4 BE D.SS4n 4 D.ANSn Figur 5..4: Angl btwn contact points which ar th farthst from th cntrlin [NUM] D.SRI D.EAS7 BS-6 BS-5 BS-4 BS- BS- Participants BS- BS- BS-9 BS-8 BS-7 BS-6 BS-5 BS-4 BS- BS- BS- BE-4 BE- BE- BE-

155 abl 5..8.: Springback angls - calculation with D.SRI lmnt as Angl ( btwn lin AB and lin D Angl ( btwn farthst points (from cntrlin at strok bfor SB aftr SB 7mm 4mm mm 8.5mm abl Springback angls - calculation with D.ANSn lmnt as Angl ( btwn lin AB and lin D Angl ( btwn farthst points (from cntrlin at strok bfor SB aftr SB 7mm 4mm mm 8.5mm abl 5... Springback angls - calculation with D.SS4n lmnt as Angl ( btwn lin AB and lin D Angl ( btwn farthst points (from cntrlin at strok bfor SB aftr SB 7mm 4mm mm 8.5mm Springback simulation In ordr to valuat th springback th tools ar progrssivly rmovd and th rsulting opning angl as dfind in Figur 5..7a is masurd as shown in Figurs and

156 987 (½ opn angl Figur 5..44: Opn angl btwn th lins AB and D bfor spring back (abl 5../as 6 - quivalnt plastic strain - MEAFOR 6.75 (½ opn angl Figur 5..45: Opn angl btwn th lins AB and D aftr spring back (abl 5..6/as - quivalnt strss - FEAP 44