A STABILIZED MIXED FINITE ELEMENT FORMULATION FOR FINITE STRAIN DEFORMATION. Roxana Cisloiu. BS, Technical University Gh.Asachi, Iasi, Romania, 1991

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1 A STABILIZED MIXED FIITE ELEMET FORMULATIO FOR FIITE STRAI DEFORMATIO by Roana Cslo BS, Tchncal Unrsy Gh.Asach, Ias, Romana, 99 MS, Ws rgna Unrsy, Sbmd o h Grada Facly of h School of Engnrng n paral flfllmn of h rqrmns for h dgr of Docor of Phlosophy Unrsy of Psbrgh 6

2 UIERSITY OF PITTSBURGH SCHOOL OF EGIEERIG Ths dssraon was prsnd by Roana Cslo I was dfndd on Fbrary 7, 6 and approd by Dr. Roy D. Marangon, Assoca Profssor, Dp. of Mchancal Engnrng Dr. Lara Schafr, Asssan Profssor, Dp. of Mchancal Engnrng Dr. Wllam S. Slaghr, Assoca Profssor, Dp. of Mchancal Engnrng Dssraon Adsor: Dr. Mchal R. Loll, Assoca Profssor, Dp. of Indsral Engnrng

3 ABSTRACT A STABILIZED MIXED FIITE ELEMET FORMULATIO FOR FIITE STRAI DEFORMATIO Roana Cslo, PhD Unrsy of Psbrgh, 6 Whn mprong h crrn sa of chnology n h fn lmn mhod, lmn formlaon s a ry mporan ara of nsgaon. Th obc of hs dssraon s o dlop a robs low-ordr rahdral lmn ha s capabl of mshng complcad gomrs whch canno b mshd wh sandard brc lmns. Ths lmn wll b applcabl o a larg class of nonlnar marals ha ncld narly ncomprssbl and ncomprssbl marals and capabl of analyzng small and larg dformaon as wll as larg roaons. Dlopmn of sch an lmn wll parclarly bnf larg sran mal-formng applcaons. Lnar rahdral lmns ar ry praccal for sral rasons ncldng hr smplcy and ffcncy. Dsp hr adanags, hs lmns ha nown shorcomngs n hr prformanc whn appld o ncomprssbl or narly ncomprssbl marals bcas of hr ndncy o loc. To orcom hs problm a sablzd md formlaon s proposd for rahdral lmns ha can b lzd n sold mchancs and larg dformaon problms.

4 An ancd sran drd from a bbbl fncon s addd o h lmn o prod h ncssary sablzaon. Th nqnss of h proposd formlaon ls whn h fac ha dos no rqr any gomrc or maral dpndn paramrs and no spcfc maral modl so ha h formlaon s complly gnral. Th lmn was mplmnd hrogh a sr-programmabl lmn no h commrcal fn lmn sofwar, ASYS. Usng h ASYS plaform, h prformanc of h lmn was alad by dffrn nmrcal nsgaons ncompassng boh small and larg dformaon, lnar and nonlnar marals as wll as nar and flly ncomprssbl condons. Th lmn formlaon was sd wh sral sandard mal formng problms sch as mal rson and pnch forgng ha ar nown o prnc dffcls drng larg dformaons. Th rsls wr compard wh analycal rsls or ohr aalabl fn lmn rsls n h lrar. Fnally, conclsons ar drawn and possbl fr nsgaons ar dscssd sch as h applcaon of h nw lmn n D rzonng, dynamc problms and ansoropc marals.

5 TABLE OF COTETS ABSTRACT... LIST OF TABLES... LIST OF FIGURES... ACOWLEDGEMETS.... ITRODUCTIO.... SIGIFICACE.... MOTIATIO AD OBJECTIES.... OLIEAR FIITE ELEMET METHODS LITERATURE REIEW MIXED FORMULATIO METHODS Gnral Aspcs Sably of Md Mhods: Th Pach Ts Two-Fld Md Formlaon Thr-Fld Md Formlaon.... REIEW OF STABILIZED MIXED METHODS Bbbl Sablzaon Sablzaon by Addng Msh-Dpndn Trms Md Enhancd Sran Sablzaon Orhogonal Sb-Grd Scal Mhod...

6 ..5 Fn Incrmn Calcls Mhod Eqalnc bwn bbbl mhods and prssr sablzd mhods THEORETICAL DEELOPMET.... OERALL APPROACH.... DEELOPMET OF THE THREE-FIELD MIXED EHACED FORMULATIO.... Formlaon of h Prncpl of ral Wor..... Slcon of Appropra Srss and Sran Masrs Lnarzaon of h Prncpl of ral Wor REDUCTIO OF THE THREE-FIELD TO A TWO-FIELD FORMULATIO UPDATED LAGRAGIA JAUMA FORMULATIO EHACED DEFORMATIO GRADIET FIITE ELEMET APPROXIMATIO. MATRIX FORMULATIO Shap Fncons Mar Formlaon ELEMET IMPLEMETATIO ASPECTS ITEGRATIO RULES OLIEAR ITERATIE ALGORITHM Sac Condnsaon Procdr won Raphson Procdr STRESS AD STRAI UPDATE ALGORITHM UMERICAL IESTIGATIOS OF LIEAR ICOMPRESSIBLE MATERIALS HOMOGEEOUS DEFORMATIO TESTS EXPASIO OF A THIC-WALL CYLIDER UDER PRESSURE... 68

7 5. COO S PROBLEM TEST OF BEDIG CAPABILITY UMERICAL IESTIGATIOS OF OLIEAR MATERIALS I LARGE DEFORMATIOS OLIEAR HOMOGEEOUS DEFORMATIO TESTS UPSETTIG OF A BILLET METAL EXTRUSIO HYPERELASTIC CATILEER BEAM COCLUSIOS SUMMARY SUGGESTIOS FOR FUTURE WOR... 8 APPEDIX A... APPEDIX B... BIBLIOGRAPHY... 5

8 LIST OF TABLES Tabl. mrcal Ingraon for Trahdral Elmn Tabl. Radal dsplacmns for h hc wall cylndr, Tabl. Radal dsplacmns for h hc wall cylndr, Tabl 4. Radal dsplacmns for h hc wall cylndr,... 7 Tabl 5. Coo s problm, rcal dsplacmn of op cornr nod for dffrn msh szs Tabl 6. Coo s problm, aal nsl srss for dffrn msh szs Tabl 7. Erson: Comparson bwn rahdral and hahdral msh dsplacmns... 97

9 LIST OF FIGURES Fgr. Lnar dsplacmn/lnar prssr rangl wh bbbl fncon... 6 Fgr. olm Coordnas Fgr. Ingraon Pon Locaon for Trahdral Elmn [] Fgr 4. won-raphson Procdr[]... 6 Fgr 5. Srss and sran n h drcon of loadng for h naal comprsson s Fgr 6. Fn lmn modl of n cb formd by 5-rahdra Fgr 7. Thc walld cylndr srsss for... 7 Fgr 8. Coo s problm gomry... 7 Fgr 9. Coo s Problm: ormal Srss n drcon Fgr. Coo s Problm: ormal Srss n y drcon Fgr. Coo s Problm: Shar Srsss Fgr. Fn lmn modl of h pr bndng s... 8 Fgr. Pr Bndng Ts: Aal Srss for ν... 8 Fgr 4. Pr Bndng Ts: Shar Srss for ν Fgr 5. Pr Bndng Ts: Aal Srss for ν Fgr 6. Pr Bndng Ts: Shar Srss for ν Fgr 7. Srss-Sran cr of MISO maral Fgr 8. onlnar Unaal Comprsson Ts: Sran n loadng drcon

10 Fgr 9. onlnar Unaal Comprsson Ts: Hydrosac Prssr Fgr. Ra-Dpndn Unaal Comprsson Ts: Eqaln Srss Fgr. Homognos Dformaon Ts wh lnar dsplacmn: Eqaln Srss Fgr. Fn lmn modl of h psng problm... 9 Fgr. Upsng of a bll: Eqaln Srss... 9 Fgr 4. Upsng of a bll: Prncpal srss n scond drcon Fgr 5. Fn Elmn modl of h mal rson Fgr 6. Erson procss: Eqaln Srss Fgr 7. Erson procss: Eqaln Sran Fgr 8. Hyprlasc Canlr Bam: Fn Elmn modl... Fgr 9. Hyprlasc Canlr Bam: Eqaln Srss... Fgr. Hyprlasc Canlr Bam: Prncpal Srss n scond drcon... 4 Fgr. Hyprlasc Canlr Bam: Hydrosac Prssr... 5

11 ACOWLEDGEMETS I ha larn a gra dal from hos who ha agh m and word wh m or hs yars and I wold l o graflly acnowldg all of hm. Frs of all, I wold l o han my adsor, Dr. Mchal Loll, for hs fah n m, for always ncoragng m, for hs nhsasc and pr gdanc. Who hs sppor hs wor cold no ha bn nhr sard nor compld. My gra apprcaon also gos o wondrfl popl of ASYS, Inc. and spcally o Jn Wang who hlpd m wh gra panc a crcal and opporn ms. I am grafl o hm for hs hoghfl and cra commns and mor gnrally for plorng wh m h bondars of profssonal frndshp. Many hans o h comm mmbrs: Dr. Roy Marangon, Dr. Lara Schafr and Dr. Wllam Slaghr. I prss my grad o h saff of h Mchancal Engnrng Dparmn and spcally o Glnda Hary who has bn for m, no only h grada admnsraor b always a ry dar frnd. I am dply ndbd o my long sandng frnds Ma Palamara, Srgy Sdoro, Raffalla d a, Bran Enns, Zhaochn Yang, Jm Cordl and Jon Chambrs and las, b no las, I shold han o my hsband and son for hr panc and forbaranc whls I ha spn so mch of or m worng on hs dgr.

12 . ITRODUCTIO. SIGIFICACE Trahdral lmns ar ry arac bcas h mos powrfl msh gnraors sd oday prodc hs lmns. Thy ar smpl and lss sns o dsoron and hr mplmnaon lads o lowr mmory rqrmns and compaonal coss. Bcas of adancs n hardwar and paralll compaons chnqs, mor complcad gomrs ar now bng modld. Mshng of hs gomrs bcoms rmly dffcl sng qadrlaral and brc lmns. Th srngn ncssy of a fn modl ha can asly nrfac wh CAD modls, oghr wh h fac ha ranglar and rahdral mshs ar ry robs and fas, has brogh abo h nd for dlopng hgh qaly rahdral lmns. As nod n h lrar, non of hs lmns dlopd o da prform wll n all saons. Hrn ls h man moaon of h proposd rsarch. Sandard dsplacmn basd fn lmns loc n wo dffrn saons: bndng (shar locng) and ncomprssbl or nar ncomprssbl marals (olmrc locng). Consan sran rahdral lmns show sr olmrc locng and sff bhaor n bndng. Locng can gnrally b dfnd as h ndncy for h fn lmn solon o approach zro bcas of rsrcons n h mdm bng modld (shar sran or ncomprssbly consrans). Whn prssd n a dscr form, locng s a condon of an or consrand sysm. In hs saons h nrpolaon fncons ar ncapabl of rprsnng h dformaons ha dlop.

13 Th nrpolaon fncons shold nsr ha any ancpad consrans ar handld who or rsrcng h sysm. Falr o do so cass solons o loc whch lmaly lads o rronos rsls. Dsplacmns ar ndr prdcd by facors of 5 o, mang hs lmns complly slss [6]. Ths, n bndng domnad problms, consan sran lmns don ha h capacy o rprsn h crar bcas of hr lac of dformaon mods rslng n a ry sff answr. Ths ffc can b allad f h msh s rfnd or f hghr ordr lmns ar sd; boh of hs solons ar drmnal o h compaonal m. Th mos dffcl saon ha canno b sold by rfnng h msh s h olmrc locng. Undr narly ncomprssbl (Posson s rao clos o.5 or bl modls approachs nfny) and ncomprssbl condons, h dsplacmns ar no accraly prdcd. Th olmrc sran, whch s drmnd from h dras of h dsplacmns, wll hrfor no b accraly prdcd. Any small rror n h olmrc sran wll ransform no a largr rror n h srsss and hydrosac prssrs, whch n rn wll ha a drmnal ffc on h dsplacmns [4]. Ths s d o h fac ha rnal loads ar a any momn balancd by srsss a h prncpl of ral wor. To lmna olmrc locng, wo dffrn chnqs ha old [6]. Th frs ss ml-fld lmns n whch h prssrs or h srsss and sran flds ar consdrd as ndpndn arabls. Th scond ss h rdcd ngraon procdrs n whch cran rms of h nrnal forcs ar ndr ngrad. Boh of hs chnqs ha hr own shorcomngs. Whn ml-fld mhods ar sd, h rslng lmns possss nsabls n h addonal flds. Th sam shorcomng mrgs from h rdcd ngraon chnqs as wll. Or h las sral yars dffrn srags ha bn dlopd for rdcng and aodng h olmrc locng and prssr oscllaons n fn lmn solons.

14 Unfornaly, ry fw conrbons n h ara of lmn chnology wr sogh o mpro h prformanc of h lnar rahdral lmns. Th focs has always bn on sablzng h low-ordr qadrlaral and hahdral lmns and sms ha only n h las hr-for yars som anon has bn drcd owards h smpl lmns. Thrfor hr ar ry fw pblcaons rlad o h sablzaon of ranglar and rahdral lmns and mos of hm ar drcd owards fld lmns [, 7, 9-, 7, 5, 9, 5-6, 4-4]. Among h fw wors of sablzng hs lmns, som formlaons addrss h problm of small dformaons [4-6,-] and ohr apply only o hyprlasc marals [-, 7, 44] or J plascy [5,, 8]. A prsn hr s no formlaon applcabl o boh gnral fn sran dformaon and o a larg class of nonlnar marals.. MOTIATIO AD OBJECTIES Th ncrasng s of aomad msh gnraors and rmshrs has rggrd h nd for accra and ffcn ranglar and rahdral lmns. Ths s spcally r n h modlng of mal formng procsss. A h prsn m, hr s no a mshng program aalabl ha can dscrz compl gomrcal shaps of formd pars who sng ranglar and rahdral lmns. Fn lmn modls sd n analyss of mal formng procsss ms b abl o rprsn h narly ncomprssbl nar of laso-plasc dformaon of marals drng formng. For hs o b possbl, h ss of olmrc locng has o b addrssd snc can cas sr arfcal sffnss ha lms h flow of h maral. Bcas lnar rangls and rahdral lmns dlopd wh md formlaons sll sffr from olmrc locng and prssr nsabls, smlaon of mal formng s somwha lmd [9-4].

15 As dscssd n., spcal sablzng chnqs ms b dlopd o aod locng n hs lmns. In hs sdy a md ancd sran formlaon s proposd o spcfcally addrss h abo ss. Hnc, h followng obcs wll b aand n h prsn rsarch proc:. Dlop a sablzaon chnq for h for-nod rahdral lmn ha wll allow larg dformaon analyss o b prformd wh a larg ary of nonlnar marals and n narly ncomprssbl and ncomprssbl condons. Th sablzng procdr wll b basd on h ancd sran approach of R.Taylor [47] drd from a bbbl fncon.. Implmn h nw formlaon no a sr programmabl lmn ha nrfacs wh h commrcal fn lmn sofwar, ASYS.. Prform nmrcal nsgaons o assss h conrgnc and accracy of h nw lmn. 4. Us h dlopd lmns o smla mal formng procsss.. OLIEAR FIITE ELEMET METHODS For mor han a dcad, nonlnar fn lmn chnqs ha bcom poplar n h analyss of mal formng, fld-sold nracon and fld flow problms. In rcn yars, h aras of bomchancs and lcromagncs ha sard o s nonlnar fn lmns. Dsp hs ffors, hr ar sll nmros nracabl nonlnar problms for whch solons ha no bn oband. A larg sgmn of hs problms can b cagorzd as larg dformaon problms ha ar appld o ry complcad gomrs and hghly nonlnar marals. 4

16 A problm s dfnd as nonlnar f h forc-dsplacmn rlaonshp dpnds on h crrn dformaon sa. onlnars can ars from hr dffrn sorcs: maral, gomry and nonlnar bondary condons. Maral nonlnary rsls from h nonlnar rlaonshp bwn srsss and srans. onlnars casd by bondary condons or loads can b fond n conac and frcon problms sch as mal formng, gars, crash, and h nrfrnc of mchancal componns. Ths problms ar nonlnar bcas nsananos changs n sffnss occr or m. Gomrc nonlnary rsls from h nonlnar rlaonshp bwn srans and dsplacmns on on hand, and h nonlnar rlaonshps bwn srsss and forcs on h ohr hand. Ths yp of nonlnary s mahmacally wll dfnd b q dffcl o sol nmrcally and nclds problms sch as larg sran manfacrng, crash and mpac phnomnon. As sad n scon., h prsn rsarch wor wll focs on a formlaon for gnral fn sran dformaon. Ths yp of nonlnary can b sold by hr approachs: Lagrangan Formlaon, Elran Formlaon and Arbrary Lagrangan-Elran Formlaon (ALE) [6]. In h Lagrangan mhod h fn lmn msh s aachd o h maral and mos hrogh spac along wh. I sally dscrbs h dformaon of srcral lmns. A shorcomng of hs mhod s ha h msh dsoron s as sr as h dformaon of h obc. Rcn adancs n adap mshng and rzonng ha mprod hs problm. Th Lagrangan approach can b classfd no wo cagors: h Toal Lagrangan mhod (TL) and h Updad Lagrangan mhod (UL). In TL h qlbrm s prssd wh rspc o h orgnal ndformd sa, whch s h rfrnc confgraon. In h UL h crrn confgraon acs as h rfrnc sa. Snc h formlaon bng proposd n h prsn sdy appls o larg dformaons nolng lmn dsorons whn ASYS, h Updad Lagrangan formlaon wll b sd. 5

17 . LITERATURE REIEW. MIXED FORMULATIO METHODS.. Gnral Aspcs I was mphaszd n scon. ha h man dsadanag of h low-ordr ranglar and rahdral lmns s h ndncy o loc. On way o orcom olmrc locng s by mployng ml-flds lmns n whch h prssrs or srss and sran flds ar consdrd as ndpndn arabls, and hs hy ar nrpolad ndpndnly of h dsplacmns. Ml-fld lmns ar formlad basd on ml-fld wa forms or araonal prncpls, also nown as md araonal prncpls. Ths lmns ar dsgnd only whn spcfc consrans s, sch as ncomprssbly, and hy ar nffc n h absnc of sch consrans. In mos cass ncldng hs sdy, h hydrosac prssr s sd as an addonal ndpndn fld. Ths yp of formlaon s also nown as a md /p formlaon. In h md /p formlaon, h prssr s oband a global ll nsad of bng calclad from olmrc sran. In sch an approach h solon accracy s ndpndn of Posson s rao or bl modls. 6

18 Th man fars of a md formlaon accordng o [49] ar: ) Th conny rqrmns on h shap fncons ar dffrn. Th addonal arabl can b dsconnos n or bwn lmns as no dras of hs ar prsn. ) If h focs s mor on h addonal arabl hn an mprod appromaon can rsl n a hghr accracy han was oband for h pr dsplacmn formlaon. ) Th qaons rslng from md formlaons ofn ha zro dagonal rms. Ths conss a sgnfcan problm snc h Gassan lmnaon procss sd n lmn solon bcoms nsabl. 4) Th addonal nmbr of arabls nlargs h sz of h algbrac problm b hs dsadanag can b dal wh by sabl ra mhods. Th gras concrn of hs md mhods ha was no mnond abo s hr sably. Ths wll b dscssd n h n scon... Sably of Md Mhods: Th Pach Ts. Th man problm of h md mhods s choosng h nrpolaon fncon for h addonal arabl, whch n or cas s h hydrosac prssr. I was shown mahmacally [4] ha for cran chocs of h shap fncons, h md formlaons do no yld manngfl rsls. Ths mahmacal crron, whch prsss h rqrmn rlad o h shap fncons n md formlaons, s ofn calld h Ladyzhnsaya-Babša-Brzz condon (LBB condon) [4]. To sablsh f hs condon s sasfd for an lmn s a ry dffcl as bcas h formlaon of hs condon has a ry mahmacal characr. Ths, prs n hs ara rd o rplac hs condon wh a mor smpl procdr for drmnng whhr h condon s sasfd. On sch condon, calld h consran con condon, has pron o b 7

19 ry ffc n drmnng f an lmn prforms wll n ncomprssbl and narly ncomprssbl saons [6]. I s no a prcs mahmacal condon b rahr a qc and smpl way of rfyng an lmn. Accordng o [49], f w consdr h dsplacmn arabl as h prmary arabl and h addonal arabl as h consran arabl hn h sably of an lmn can b oband f hs condon s sasfd for any solad pach on h bondars of whch w consran h mamm nmbr of prmary arabls and h mnmm nmbr of consran arabls. If n rprsn h oal nmbr of dsplacmn qaons afr mposng h bondary condons and n p rprsns h oal nmbr of ncomprssbly consrans hn h consrand rao s dfnd as r n n p. Ths rao shold mmc h bhaor of h rao bwn h nmbr of qlbrm qaons and h nmbr of ncomprssbly condons for h gornng sysm of paral dffrnal qaons. In wo dmnsons h dal al wold b r /, and for hr dmnsons r /. A al of r lss han h dal al ndcas h ndncy o loc. If r hr ar mor consrans of h prssr han hr ar dsplacmns dgrs of frdom aalabl and sr locng s ancpad. A al mch largr han h dal al ndcas ha no nogh ncomprssbly consrans ar prsn so hs condon wll b poorly appromad. Md dsplacmn-prssr formlaons wh qal ordr of nrpolaon for boh and p do no pass h Babša-Brzz condons nlss spcal sablzaon chnqs ar sd. Th man goal n h lmn chnology s o prodc mor ffcn cods and hs s possbl only f h nrpolaon spacs of dsplacmn and prssr concd. 8

20 Ths moad an ns rsarch ffor o fnd formlaons whch wold ma possbl o crcmn h LBB condon and s qal ordr nrpolaon fncons. Ths cagory s calld sablzd md mhods and s rad nsly n.. For shornss rasons, only h lnar lasc md /p and /p/ε formlaons of Znwcz and Taylor [49] wll b prsnd n h n scons as hy sr as a bass of h md ancd sran formlaon dlopd n hs sdy... Two-Fld Md Formlaon Th man problm n h applcaon of a pr dsplacmn formlaon o ncomprssbl and narly ncomprssbl problm ls n h drmnaon of h hydrosac prssr, whch s rlad o h olmrc par of h sran (for soropc marals). Md formlaons ar basd on dcomposng h srss nsor no s daorc and hydrosac componns. s pi s p (.) whr s s h daorc srss and p r( ). (.) Th cons rlaon lnng s and h sran nsor s spplmnd by a consran qaon rlang h prssr and h olmrc sranε ε ε. (.) In h cas of lasc marals, ε p (.4) whr s h bl modls of h maral rlad wh Posson s rao by E (.5) ( ν ) 9

21 In h ncomprssbl lm, ν.5 and and h qaon (.4) bcoms ε (.6) Th daorc sran s prssd as ε d ε ε (.7) Eqlbrm and ral Wor Many ngnrng problms can b sold by fndng an approma (fn lmn) solon for h dsplacmns, dformaons, srsss, forcs, and ohr sa arabls n a sold body ha s sbcd o a loadng hsory. Th ac solon of sch a problm rqrs ha boh forc and momn qlbrm b manand a all ms or any arbrary olm of h body. L b h olm occpd by a par of h body n h crrn confgraon and S h srfac bondng hs olm. L h srfac racon a any pon on S b h forc pr n of crrn ara and l h body forc a any pon n h olm b b pr n crrn olm. Forc qlbrm for h olm s hn: S ds bd (.8) Th r or Cachy srss mar s dfnd by n (.9) whr n s h n oward normal o S a h pon. Usng (.9) and applyng h Grn s horm w can rwr (.8) as b (.) Th momn qlbrm qaon lads o h rsls ha h r Cachy srss mar ms b T symmrc:. Th bass for h dlopmn of any fn lmn formlaon s h nrodcon of som locally basd spaal appromaons o pars of h solon.

22 To dlop sch an appromaon w rplac h qlbrm qaon (.) by an qaln wa form - a sngl scalar qaon or h nr body whch s oband by mlplyng h pon ws dffrnal qaon by an arbrary, cor-ald s fncon and ngra. Th s fncon can b magnd as an arbrary ral dsplacmn fld,, whch s complly arbrary cp ha ms oby any prscrbd nmac consrans and ha sffcn conny. Th do prodc of hs s fncon wh h qlbrm qaon rsls n a sngl scalar qaon ha s ngrad or h nr body o rprsn h prncpl of ral wor. Ths samn has a ry smpl physcal nrpraon: h wor don by h rnal forcs sbcd o any ral dsplacmn fld s qal o h wor don by h qlbrang srsss on h dformaon of h sam dsplacmn fld. T ε d T bd S T ds (.) wh Cε as h cons rlaon and h daorc par as s D l ε d l ow h qlbrm qaon s rwrn sng (.) and rang p as an ndpndn arabl ε sd ε pd b d ds,,, (.) S and n addon w shall mpos a wa form of (.4),.. h prssr cons qaon: p p( ε ) d (.) Mar Formlaon Sbsng now h ndpndn appromaons of, p,, p as, and p p,p p (.4) p p

23 whr A B T [,... ] ar h nrpolaon fncons for dsplacmns and prssrs and A B T A B T A B T A B T [,... ], [,... ] p [ p p,...p ],p [ p p,... ], ar h dsplacmns, ral dsplacmns, prssr, and ral prssrs a ach nod. If w nsr h abo rlaons n h wa form (.) and (.) and ma s of h Lmma of calcls of araons w g h lmn ws lnar qaons sysm: p p pp p R whr (.5) B T DBd (.6) s h lmn sffnss mar ha conncs h dsplacmns hrogh h dffrnal opraor of h shap fncons, B, and h daorc par of h maral cons mar D, p T ( p ) ( ) pd (.7) ar h md rms dpndng on dsplacmns as wll as on h prssrs, wh T [,, ] (.8)..., D pp T p p d (.9) s h rm whch dpnds only on h prssr. R T T bd ds (.) S If ncomprssbly ss, h prssr rm bcoms zro. Th fndamnal problm now rlas o fndng ffc nrpolaon fncons for boh dsplacmns and prssrs so ha accra fn lmn solons ar oband.

24 ..4 Thr-Fld Md Formlaon For many cons modls sch as hyprlascy ha can ha mlpl dformaon sas for h sam srss ll s mor connn o s a hr-fld araonal form [47, 49]. Ths formlaon s mor gnral and mor sabl for ansoropc, nlasc marals and fn dformaon problms. I mploys appromaons of dsplacmn, prssr and sran. Th s of hs yp of appromaon has ld o sccssfl lowr-ordr qadrlaral or hahdral lmns ha can b sd n narly ncomprssbl cass for a larg class of marals. Assmng an ndpndn appromaon for h hydrosac prssr, p, and h olmrc sran, ε, h sam problm as n.. can b formlad by nrodcng wo consran qaons wh wo Lagrang mlplrs n h prncpl of ral wor. Ths, f h wo consrans ar: T T ε ε ε ε ε whr [,,,,, ] (.) y z ε p (.) h prncpl of ral wor can b prssd by (.) and h wo addonal wa samns of qaons (.) and (.) wrn as: p( ε ε ) d ε ( ε p) d T (.) Thn, sng fn lmn appromaons for and p flds from (.4) and ε ε (.4) a md appromaon s oband n h followng mar formlaon:

25 T p p T p R p p (.5) ε whr, p and R ar h sam as n (.6), (.7) and (.9) and, p T T p d d (.6) Usally s dncal wh p so ha p s symmrc pos dfn. In h cas whn p s connos and ε s dsconnos h olmrc sran can b lmnad from h hrd qaon ladng o a sysm of qaons n only wo nnowns, and p.. REIEW OF STABILIZED MIXED METHODS Sablzd fn lmn mhods wr nally dlopd for applcaon n h Galrn fn lmn mhod o sol problms n ngnrng and mahmacs ha prodc nmrcal appromaons ha ddn ha h sably proprs of h connos problm. Sablzd mhods amp o mpro h sably bhaor who compromsng accracy. Incomprssbl fld dynamcs ha always bn h fron ln of rsarch n hs ara. Sral approachs ha achd sccss and ha bn ndd o h sold mchancs as wll. Som of hm ar dmonsrad o b rlad ndr spcfc condons [] b hy can b gnrally classfd no h followng cagors: 4

26 ) Sablzaon by mployng fn lmn appromaons nrchd wh so-calld bbbl fncons. ) Sablzaon by addng msh-dpndn prrbaon rms, whch dpnd on h rsdals of h gornng qaon. ) Md-ancd sran sablzaon 4) Orhogonal sb-grd scal mhod (OSGS) 5) Fn ncrmn calcls (FIC) All h mhods, wh h cpon of ) and ), lz a wghng paramr ha s appld o h addonal rms. Ohr approachs ar arag nodal prssrs [-], arag nodal dformaon gradn chnqs [5], or compos lmns [48] b hy ar no of gra nrs o h proposd rsarch snc hy wll b hard o mplmn no commrcal fn lmn sofwar. Each of hs mhods wll b dscssd n h n scons... Bbbl Sablzaon Th frs ffor o sablz h low ordr lmns was h MII lmn nrodcd by Arnold, Brzz and Forn n []. Ths s an arac lnar rangl wh a dsplacmn nrpolaon ancd wh a cbc bbbl fncon. Ths lmn has hr rnal dgrs of frdom pr r (wo dsplacmns and on prssr) and an qaln nrnal nod d o h bbbl fncon wh wo dgrs of frdom (dsplacmns) (s Fgr ). Th bbbl fncon s a hghr ordr polynomal ha anshs on h bondary of h lmn and s qal o ny n h cnr of h rangl. Sch a bbbl fncon can b wrn n rms of ara coordnas as: ( ξ ) 7ξξ ξ (.7) 5

27 Fgr. Lnar dsplacmn/lnar prssr rangl wh bbbl fncon Th dsplacmn fld wh h bbbl fncon can b appromad as: (.8) whr ar h nodal dsplacmns and ar nrnal paramrs. Smlarly, h prssrs ar nrpolad as: p p (.9) For h lnar rangl and rahdral lmn h shap fncons ar h ara, rspcly h olm coordnas ξ (.) Bcas h nrnal paramrs ar dfnd sparaly for ach lmn, a paral solon can b prformd a h lmn ll o oban a s of qaons n rms of rnal dgrs of frdom as dscssd n h md /p formlaon. Adanags of sng bbbl fncon: - Sasfs LBB condon and h md pach s - Fw dgrs of frdom - Easy o mplmn n larg sran dformaon bcas dos no rqr h nrodcon of ohr paramrs 6

28 Dsadanags: - Only margnally sabl - Ths lmn mgh no b robs nogh snc n many ss h prssr solon sll had small ampld oscllaons. - In ransn problms, h acclraons wll also nol h bbbl mod and wll affc h nral rms... Sablzaon by Addng Msh-Dpndn Trms Th da of hs mhod s o modfy h dscr qaons nsad of h nrpolaon fncons. Th smpls form of hs yp of sablzaon s o add a non-zro dagonal rm hrogh a pnaly-l rm n h prssr cons qaon. Ths was frs nrodcd by Brzz and Parana n [] for sablzng h fn lmn appromaon of Sos problm. mros ohr alrnas of hs mhod ha bn dlopd [-5, 9-, 4-44]. Rcallng from scon.., h splng of h Cachy srss no h daorc and olmrc pars as: s pi s p (.) and nong ha h daorc srss s rlad o h daorc sran hrogh h rlaon: s d Gε G (.) Th qlbrm qaons n h absnc of nral forcs ar: s p b (.) Sbsng h cons qaon for h daorc par (.) no (.) 7

29 p G b (.4) or n nsor form: G ( ) p b whr s h Laplacan opraor and s h gradn opraor. Th prssr cons qaon n rms of dsplacmn s: p ε (.5) Tang h drgnc of h qlbrm qaon (.4) and sng (.5) w oban: 4G p b (.6) Ths qaon was sd o consrc h addonal rm for h wa form, whch wold ohrws b zro. Brzz and Parana [] addd a wghd form and s h body forc o zro for smplcy. Thn, hy ngrad by pars and gnord h bondary rms o oban a mor arac form of h prssr cons qaon: p T ε p d β p p d (.7) From dmnsonal consdraons wh h frs rm, h paramr β shold ha a al proporonal o L 4 /F whr L s lngh and F s forc. In h wor of Hghs al.[9], β s gn n rms of h characrsc lmn lngh h and a non nga, non dmnsonal sably paramr, α as: αh β (.8) G 8

30 Ths mhod and alrnas o ha bn appld o ncomprssbl lnar lascy and o h Sos flow [9-,, -5, 9-, 4]. Th abo formlaon was ndd o fn dformaon by O. laas al [] and appld o a lnar dsplacmn, lnar prssr rahdral lmn. Thy prod a formlaon for fn lascy n boh rfrnc and crrn confgraon, whch s hn lnarzd o allow an mplmnaon n a won-raphson schm. Thr formlaon can b sd for h hyprlasc marals. Adanags - o oscllaons n h srss fld - Srss concnraons ar wll appromad Dsadanags - Addon of h non-zro dagonal rms dos no ha a srong horcal fondaon - Rqrs h choc of a paramr - Dpnds on h lmn lngh (mamm dg lngh) whch also changs ndr h larg dformaon assmpon - Dpnds also on h maral hrogh h shar modls.. Md Enhancd Sran Sablzaon Ths mhod s prsnd n Znwcz and Taylor [49] and appld by R. Taylor [47] o a lowordr rahdral lmn n boh small dformaon and fn dformaon. I ss a hr-fld appromaon nolng connos, p and dsconnos olm chang ε oghr wh an ancd sran formlaon. Th ancd srans ar addd o h rglar srans o prod h ncssary sablzaon for h narly ncomprssbl cas. Hs formlaon sars from h fnconal and as no consdraon a hyprlasc maral for whch a sran nrgy fncon 9

31 ss. Ths approach hogh s no longr ald for marals for whch w canno dfn a sran nrgy fncon. Small dformaon cas Splng h sran no hr daorc and olmrc componns, h md sran can b wrn as: ε ε ε d (.9) Th fnconal, s araon and hn lnarzaon ar gn as: [ ] ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),, ( Π Π Π Π Π Π T T T d d dp d d p d d d d p p Saonary d p W p ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε D ε ε ε ε T T d d T T d Mar Formlaon Th dsplacmn, prssr, and olm chang n ach lmn ar wrn sng h nrpolaon fncons n rms of olm coordnas,ξ, and corrspondng nodal als as: (.4) ε p ˆ ˆ ) ( ˆ ˆ ) ( ˆ ˆ ) ( α α α α α α ε ξ ξ ε ξ ξ ξ ξ p p Srans ar compd sng h sran-dsplacmn mar as: (.4) α ξ (ξ B ε α ˆ ) ) ( An ancd sran formlaon s consdrd as (ξ B ε ˆ ) ) ( ξ (.4) whr ar h ancd sran paramrs and s oband from h dras of a bbbl mod û ) B (ξ 4 ) ( ξ ξ ξ ξ ξ (.4)

32 Ths, h md sran s: d ε I (B ˆ B ˆ ) ε (.44) Rplacng all hs appromaons n h lnarzd araon of h fnconal w oban a lnar sysm of qaons wh for nnown ncrmns of dsplacmns, ancd sran paramrs, prssrs and olmrc changs. Th angn nsor s: p T p T T (.45) p p p T T T p p p B B B T T T I d d I DI d d B Dd T d T T Dd 9 d (.46) Th rms nolng h ancd mods ar smlar o, p, b hy ha B nsad of B. Fn Elascy Cas Π Π [ W ( C) p( J ε )] d Π Saonary W : C p( J ε ) ( J ε ) pd Π C (.47)

33 ( ) ) ( ) ( ) ( : : : ) ( Π Π d pd J d J p d J p d W W ε ε C C C C C C Th angn rms rslng from h frs wo ngrals can b spl no a cons par and a gomrc par. Th rms nolng h ancd srans wll b rplacd wh h appropra rms and wll ha h sam form as hos nolng rglar srans. ( ) d d d Jd d p J d p J II p J p p c ε ε ε ε ε D D I B B I : B DI I B T T T T d d T T T g T T T T d T d d d T 9 9 ) ( 9 (.48) If h nrpolaon for and p s connos n h whol doman and h olmrc chang s pc ws connos, h solon can b prformd by sng sac condnsaon a h lmn ll. Afr lmnang and ε, a sysm of qaons n ncrmnal nodal dsplacmns and prssrs has o b sold. Adanags - Rsls ar fr of prssr oscllaons - Enhancd rms ar maral and msh ndpndn - Appls o ncomprssbl lasc, hyprlasc marals and o boh small and larg dformaons

34 Dsadanags - Incrasd compaonal cos d o h nrodcon of h addonal arabls b compnsad by h possbly of prformng sac condnsaon - Bhas poorr n bndng han md hahdral lmns wh dsconnos prssr and olm chang - Ehbs som locng ndncy n bndng..4 Orhogonal Sb-Grd Scal Mhod Th sb-grd scal approach was proposd frs by Hghs []. Mor rcnly h mhod of orhogonal sb-grd scal was nrodcd by Codna [] and has bn appld o ncomprssbl fld dynamcs. Chmn al. ndd hs mhod o sold mchancs n h con of ncomprssbl lascy [9] and J plascy [7-8]. An qal ordr nrpolaon s sd for boh h dsplacmn and prssr. Th basc da of h mhod s o dcompos h connos flds no a coars componn and a fn componn, corrspondng o dffrn scals. Snc h solon of h problm conans componns from boh scals s ncssary ha h fn lmn appromaon ncld h ffc of boh scals. Th coars scal can b sold by a sandard fn lmn nrpolaon, whch canno sol h fn scal. For h problm o b sabl h ffc of h fn scal has o b an no consdraon. Ths, h dsplacmn fld can b appromad as; ~ (.49) h and h daorc srsss can b spl no wo corrspondng conrbons as: s s ~ s (.5) h Ths rsls n h followng md formlaon n h wa form as:

35 ~ ( ~ )( ( s s ) d ( ~ ) pd ( ~ ) p h ( ( ~ ) d h h h h bd (.5) whch ngrad by pars ylds hr qaons of whch on s rlad o h fn scal: s s ( )( ) ( )( ~ h sh d h s ) d ( h )( p) d ( h ) bd ( h ) ~ ( )( s ) d ( ) ( s) d ( ) p ( ) d p( ) d h h ~ ~ ~ ~ bd d (.5) Th frs qaon sols h balanc of momnm and nclds a sablzaon rm whch dpnds on h sb-grd srsss. For lnar lmns hs rm s zro. Th hrd qaon nforcs h ncomprssbly condon and has a sablzaon rm ha dpnds on h sb-grd scal. Th scond qaon s complly dfnd n h sb-grd scal and canno b sold by h fn lmn msh. To fnd h sb-grd dsplacmns ha ar nrodcd n h scond sablzaon rm, Codna proposd o loo n h spac orhogonal o h fn lmn spac. Sch a mhod has alrady bn appld wh sccss n fld mchancs. From h scond qaon, shold b nod ha hs dsplacmns ar drn by h rsdal of h coars scal and hs hy can b prssd as: ( p P ( )) ~ τ p (.5) h whr π ( p) s h procon of h gradn of prssr ono h fn lmn spac h P h and τ s h sam as β from scon... Ths h fnal formlaon bcoms: 4

36 s ( )( s ) d ( )( p) d ( ) bd ( ) p π lm ( h ) d τ ( p [ p π h ]) h h h ( p π ) h n d h h h d (.54) I s mporan o no ha no sablzaon rm appars n h frs qaon for low-ordr lmns and h only sablzaon rm appars n h ncomprssbly qaon. Th angn sffnss mar s: p T τ τ p p pp τ τ T p ππ (.55) Adanags: - Crcmns h src LBB condon - Rsls ar fr of olmrc locng and prssr oscllaons and comparabl qalaly o h md qadrlaral and hahdron. - Corrc falr mchansms wh localzd parns of plasc dformaons ar oband whch show ha h mhod s no nflncd by h msh drconal bas. Dsadanags: - Th formlaon s applcabl only for small dformaon. - Rqrs a paramr dpndng on lmn lngh and shar modls whch mas h formlaon hard o nd o fn dformaon cas. - Inrodcs a nw fld arabl, Π h whch s connos and hrfor h sac condnsaon procdr canno b prformd a lmn ll. 5

37 ..5 Fn Incrmn Calcls Mhod Th FIC mhod s a complly dffrn approach n sablzng h md /p lmns, b ylds h sam formlaon as h pros dscssd mhod (OSGS). Th bass of h FIC formlaon s h sasfacon of h sandard qaons of qlbrm n a doman of fn sz by prssng h dffrn rms of h dffrnal qaon sng a Taylor panson srs and ranng h hghr ordr rms. Th rslng qaons ar ancd wh som addonal rms, whch nrodc h ncssary sably o orcom h olmrc locng. Ths mhod was frs dlopd by Ona for adc-dffs and fld flow problms [] and was lar appld o ncomprssbl solds n [4-6]. Ths mhod allows h s of lnar rangl and rahdral n qas-ncomprssbl and flly ncomprssbl sold problms. For any problm n mchancs, h qaons prssng balanc of momnm, mass, ha, c. can b wrn n a doman of fn sz wh h as h characrsc lngh. By pandng h balanc qaon n Taylor panson and ranng h lowr ordr rm, w g: h r r (.56) whr r s h sandard form of h h dffrnal qaon for h nfnsmal problm and h s h characrsc lngh of h doman. Ths mhod s no sfl for obanng an analycal solon b pros o b ry sfl for fndng an approma solon, whch conrgs o h analycal on whn h grd sz nds o zro. If w consdr h md /p formlaon as n (.) and (.) and rplacng r by h qlbrm qaon frs and hn by h ncomprssbly qaon, w oban h followng: 6

38 ) ( n r ds p d p r d p p r d h ds b d pd d s S n S d τ τ ε ε ε (.57) whr τ ar coffcns rfrrd as nrnsc m paramrs wh h followng al: G h 8 τ, whch s ry smlar wh h hrscally al of G h τ chosn n h pros wors. Snc h las rms form h frs qaon and h scond qaon ar no rlan for sold mchancs problms hy ar omd n h fnal formlaon. Also nrodcng h noaon b s p r π π whr (.58) π s h par of h dffrnal qaon no conanng h prssr gradn Ths rm, onc nrodcd, has o b waly nforcd by mans of a Lagrang mlplr (π ) yldng h fnal s of gornng qaons as:, ) ( d p d p p d p p ds b d pd d s n S d π π τ π τ ε ε ε (.59) Mar formlaon s smlar wh h on oband n h OSGS mhods. 7

39 Adanags - FIC mhod s basd on a ry srong horcal fondaon bcas mrgs narally from h gornng qaons - Rsls ar accra and fr of prssr oscllaons - Can b sd n a smplr form by nglcng h ffc of h procd prssr gradn rms. - I s applcabl o non-lnar dynamcs. Dsadanags - Sablzaon paramrs ar a fncon of h maral proprs and characrsc lngh - Enson o fn dformaon s mpdd by h characrsc lngh paramrs whos conssn dfnon rmans sll an opn qson...6 Eqalnc bwn bbbl mhods and prssr sablzd mhods I can b concldd from h pros scons ha h OSGS mhod prodcs h sam addonal rms n h prssr cons qaon as FIC mhod and boh rsls ar ry smlar wh h rms prodcd by h prssr sablzd mhods wh h only dffrnc n h rm π, h procd gradn of prssr, whch s no so sgnfcan for h cas of sac analyss. Hang sablshd ha hs hr mhods ar smlar, h normal qson o as s whhr h bbbl mhods ar qaln or no o hs mhods. Answrng hs qson wll lad s mmdaly o h mos ffcn and sabl mhod o adop for or rahdral lmn. Sral ahors prod h qalnc bwn h bbbls and sablzd mhods. Hghs [] sablshs a rlaonshp bwn bbbl fncon mhods and sablzd 8

40 mhods and hn of boh mhods o sbgrd scal mhods. H also dnfs h orgn of h τ paramrs ha cold ha nr bn pland bfor as bng drd from h lmn Grn s fncon. Anohr qalnc proof s offrd by R. Prr n [8] who has shown ha by lmnang h cbc bbbl sng sac condnsaon w rcor h sablzd mhods. If w wr h dsplacmn fld as n h OSGS mhods: ~ ( φ ( ) ~ (.6) ( ) h ( ) ) h ( ) whr h s an appromaon of ~ ar h nrnal d.o.f. wh corrspondng bbbl fncon ϕ. Th n smplfyng condons wr sd for sac condnsaon: Ω h ( φ ~ ) : ( φ ~ ) d A ( φ ~ : ( φ ~ ) d ) : ( φ ~ ) d ~ (.6) By rplacng (.6) n h dscr gornng qaons (qlbrm and consran) and sng (.6) w oban a sysm of hr qaons as n h OSGS mhod o of whch h scond on rfrs only o h nrnal dgrs of frdom or bbbl paramrs. Solng hs qaon for h bbbl paramrs and sbsng hm n h prssr qaon by sng a ry wll nown formla 9 φ d h (.6) mas( ) w g h qaln form of h prssr qaon of h bbbl mhod as: 9

41 ε pd h ε pd p p ( ) φ d C h p p ( ) d (.6) I s asy o obsr now h by h procss of sac condnsaon and ndr som condons h mhod of nrchng h dsplacmn fld wh a cbc bbbl fncon prodcs h sam addonal rms n h prssr qaon as h prssr sablzd mhods. Th only dffrnc s ha h bbbl mhods do no prodc h procon of prssr gradn ha appars n h OSGS and FIC mhods. Thrfor, f all mhods ar qaln ndr cran condons, sms naral o choos h mhod ha wold b h mos ffcn and ass o mplmn. W chos o mplmn a md /p ancd sran approach wh h ancd sran drd from a bbbl fncon for h followng rasons: Rasonably sabl Conssn n nonlnar (larg dformaon) analyss o maral or msh dpndn paramrs Compaonally ffcn (lss dgrs of frdom pr lmn).

42 . THEORETICAL DEELOPMET In h prsn wor, a md ancd sran formlaon s nrodcd for a lowr-ordr rahdral sold lmn ha s applcabl o small and larg dformaons and larg roaons. Th nqnss of h formlaon ls whn h fac ha no spcfc gomrc or maral modl paramrs ar rqrd and no spcfc maral modl s chosn whch mas h formlaon as gnral as possbl. Th horcal formlaon s dlopd from h prncpl of ral wor and has a hr-fld form. Th proposd lmn has a nod n ach cornr wh dsplacmn and prssr as rnal dgrs of frdom and a cnr nod wh h olmrc sran and dsplacmn as nrnal dgrs of frdom. Th sablzaon rm coms from an ancd sran drd from a bbbl fncon. Two bbbl fncons, conformng and nonconformng, ar sdd for obanng opmal rsls. Two formlaons, a gnral hr-fld formlaon and s rdcon o a wo-fld form ar prsnd n h n scons. For ffcncy rasons, h rdcd wo-fld form was mplmnd hrogh a sr-programmabl lmn no h commrcal fn lmn sofwar, ASYS.

43 . OERALL APPROACH Snc h proposd formlaon s applcabl o gnral fn sran dformaon, has o a no consdraon h followng facors [6]: Gomry changs drng dformaon; h crrn confgraon s dffrn from h rfrnc confgraon and dffrn from h confgraons a any ohr m. A larg sran dfnon has o b sd. Cachy srss canno b pdad by smply addng s ncrmn d o sranng of h maral. Cachy srsss a m ha o a no accon h rgd body roaons. Implmnaon of h non-lnar bhaor shold b basd on an ncrmnal approach. Th qlbrm of h body ms b sablshd n h crrn confgraon. Th basc da of h nonlnar fn lmn formlaon s o lnarz h wa form of h gornng qaons of h problm and o sol hs qaons for h fn lmns dscrzd doman. Ths lads o an ncrmnal approach, accordng o whch h solon a ach sp s oband from h solon a h pros sp. A sp s consdrd a load ncrmn n a sac analyss and a m sp n ransn analyss. Th proposd hr-fld formlaon follows h md ancd sran formlaon of Taylor [47] wh h dffrnc ha sars from h wa form nsad of an nrgy fnconal ( h wa form s mor gnral- appls o problms ha don ha a araonal prncpl) and ss h Jamann ra of Cachy srss wh s conga dformaon ra nsad of scond Pola- rchhoff srss congad wh Grn-Lagrang sran. In hs formlaons for boh small and larg dformaon problms s assmd ha hr ss a sord nrgy fncon for h maral, prssd n rms of h rgh Grn dformaon nsor, whch s no always h cas.

44 Th prsn formlaon dos no assm h snc of sch a fncon b mploys a ra form of h cons qaon whch s sabl for boh ra-dpndn and ra-ndpndn maral cons laws. Ths s d o h fac ha h Jamann ra of Cachy srss s mployd as an obc srss ra n h cons law. An Updad Lagrangan Jamann (ULJ) procdr s mployd o sol for gomrc nonlnars bcas of s ably o handl larg dsplacmns, larg roaons, and larg sran analyss. For h fn lmn mplmnaon, a smlar approach o Bah [6] wll b sd.. DEELOPMET OF THE THREE-FIELD MIXED EHACED FORMULATIO.. Formlaon of h Prncpl of ral Wor Formlang h prncpl of ral wor conss h bass of h horcal dlopmn snc obanng h s of lnarzd qaons s h man goal of an lmn formlaon. Thy ar oband by dffrnang h prncpl of ral wor and ranng only h lnar rms (all hghr ordr rms ar gnord). Th prncpl of ral wor, whch s h wa form of h qaons of qlbrm, s sd as h basc qlbrm samn for h proposd formlaon [6]. Ths wa form prsss h qlbrm sa n h form of an ngral or h nr olm of h body and prods a srongr mahmacal bass for sdyng h appromaon han drcly dscrzng h dffrnal qaons of qlbrm and rqrng hm o b sasfd pon ws. Snc h frs formlaon has hr ndpndn arabls, dsplacmn, prssr and olmrc sran, h Cachy srsss ha o b modfd wh h corrspondng ndpndn

45 prssr fld and h srans ha o b modfd wh h corrspondng ndpndn olmrc sran as []: d P P P ( P P) (.) ε ε, d ε ε ( ε ε ) (.) whr s h Cachy srss from cons law, d s h daorc par of h Cachy srss, P s h prssr drd from Cachy srss, P s h nrpolad prssr, ε s h modfd ancd sran wh h sablzng rm from h bbbl fncon d ε s h daorc par of h sran, ε s h nrpolad olmrc sran and s h roncr dla. Th nrnal ral wor can b prssd now as: W n ( ε ) ε d (.) Th wo consrans mposd by h wo addonal arabls ar: ε ε (.4) P P (.5) Th nrodcon of h wo addonal fld arabls has o b waly nforcd n h prncpl of ral wor by mans of wo Lagrang mlplrs as: Wn ( ε ) ε d ( ε ε ) Pd ( P( ε ) P) ε d (.6) 4

46 Th agmnd prncpl of ral wor s dffrnad agan o oban h ncrmnal prncpl of ral wor, whch ylds, by sbsng h arabls wh hr fn lmn appromaons, h mar formlaon or h lnarzd sysm of qaons ha has o b sold. Drng h dffrnaon procss w wll oban h ras of h Cachy srss and of srans and hrfor w nd o appropraly dfn hm. Ths s don n h n scon... Slcon of Appropra Srss and Sran Masrs Whn a small sran appromaon s no longr ald w ha o s appropra masrs of srss and sran. Th approach ha w wll follow s o s srss and sran masrs ha ar conga so ha h prncpl of ral wor can b prssd as n (.6). Many of h marals w wsh o modl ar hsory dpndn and hrfor s common for h cons qaons o appar n ra form. W hrfor nd o dfn h ra of h Cachy srss for s n h maral cons law whch rlas h ncrmns of srss wh h ncrmns of sran. On of h obc srsss ha can b appld s h Jamann ra of Cachy srss prssd by McMng and Rc n []. Th ss ha arss whn sng h scond Pola - rchhoff srss, whch s a marallybasd srss, s ha rmans consan drng h roaon bcas s componns ar assocad wh a maral bass. Th problm s ha h componns of h Cachy srss,, ar assocad wh h crrn drcons n spac and, hrfor, h Cachy srss ra, D, wll b nonzro f hr s pr rgd body roaon n hogh from a cons pon of w h maral s nchangd. Ths, h ncrmn of Cachy srss, D, ms b ddd no wo dffrn pars: on d o h rgd body moon only and on assocad o h ra form of h srss-sran law. 5

47 Accordng o [], f a m w aach o a maral pon a s of bas cors,,,, hy canno srch b hy can spn wh h sam spn as h maral, Ω T (.7) Thrfor, hr moon can b dscrbd as: & Ω (.8) Ths, f w consdr h Cachy srss nsor n h crrn confgraon as (.9) Tang h m dra w oban: J T & & (Ω Ω ) (.) whr J & s h ra of Cachy srss assocad wh h cons rspons, also calld h Jamann ra of Cachy srss. Th Jamann srss ra s an obc srss ra nsor ha s rlad o h ra of sranng as:. C Dε C d (.) J l l l l wh C l as h componns of maral cons nsor, Dε l d l l l as h componns of h ra of dformaon nsor, and s h locy. Snc h Jamann srss ra s dfnd n rms of boh ra of dformaon and pas hsory hs qaon prods a ln bwn h maral modl and h orall chang n Cachy srss. Wrn n ndcal noaon [8], & J & & ω & ω whr (.) 6

48 ω& s h spn nsor dfnd n (.7) wrn n ndcal noaon and & s h m ra of Cachy srss. Thn h Cachy srss ra bcoms: D C Dε Dω Dω (.) l l whr ω ar h componns of h roaon nsor. Ths mans ha an ngraon procss s always rqrd o ala h Cachy srsss. Ths s sabl for an analyss of pah-dpndn marals []. Undr hs crcmsancs h dffrnad or lnarzd prncpl of ral wor wh rspc o h crrn confgraon can b sd o formla h Updad Lagrangan Jamann fn lmn mhod. Th compl draon of h dffrnaon procss s prsnd n h n scon and Appnd A... Lnarzaon of h Prncpl of ral Wor Usng h agmnd nrnal ral wor from (.6) w can formla h prncpl of ral wor as: (.4) W n ( ) ( ) W W ε ε d ε ε Pd ( P( ε ) P) ε d W Tang h dra wh rspc o m of h nrnal ral wor w g: DW n Dε D ε d Dε ( P P ) d ( DP DP ) ε d d ( Dε Dε ) Pd ( ε ε ) D Pd (.5) 7

49 whr h dffrnaon of D D( d ) d Dε d. Ths rm s sally nsgnfcan n mos dformaon cass and s gnord. Also hs rm wll yld an nsymmrcal sffnss mar []. Th sam obsraons can b mad for h rms nolng D P, Dε whch ar ry small and can b gnord. Afr mang hs smplfyng assmpons, h lnarzd agmnd prncpl of ral wor bcoms: A B D W a D ε d Dε d ( C l D ε l ) ε d Dε Pd Dε Pd ε DP d (.6) Endng h frs wo ngrals, whch ar gropd as rm A, and sng h dfnon of Jamann srss ra from (.) w oban: A {[ C ε Dε C Dε ε ε D ε l C C l l l l Dε d Dε d Dω l d Dε l ε l Dε d ( Dε Dε ) d ε ( Dω Dω ) DP C ε ε C d Dω ( DP DP)] ε l l l Dε C Dε l D[ ε d d l Dε [ ε ( Dω Dω ) Dε ] l ( ε ε )] d } d d ε DPd d 8

50 A ε Cl ldε d [ ε ( Dω Dω ) Dε ] ε ε C C l l Dε l d ldε d ε ε C l ldε C l d d ε ldε d ε C l Dε DPd l d (.7) Th frs for ngrals n (.7) prodc h so-calld cons sffnss rm,, bcas coms from h maral cons law drcly or from h sranng; h ffh ngral whch wll b ndd n h n scon prodcs h so-calld srss sffnss rm or gomrc c sffnss, gom (d o gomrc nonlnars). B ( C l ε C ε C Dε l l l D[ ε Dε l l ) ε d l ( ε d ε )] d ε C l (.8) l Dε d ε C l l Dε d Rwrng (.6) sng h oband formlas for rms A and B w g h lnarzd prncpl of ral wor as: DW ε Cl l Dε d [ ε ( Dω Dω ) Dε ] a ε ε C Dε C ε l C l Dε d l Dε Pd l Dε l l d d Dε Pd ε ε C C ε C ε DPd l l l Dε l d l Dε d l Dε d d ε C l Dε l d ε DPd ε Cl l Dε d (.9) 9

51 Th fnal prsson for h lnarzd nrnal ral wor s: DW ε a ε Dε C DPd Pd ε DPd ( ε l C Dε ldε l ( ε l Dε Pd ε C ε C ) d l ε [ ε ( Dω Dω ) Dε ] ) C l ε C ldε d ldε l ldε d l ε ( Dε l C l ldε Dε l ) d d (.) Th draon of h gomrc sffnss rm (scond ngral) can b fond n Appnd A. Th fnal rsl for hs rm s: D Dω ε Dωε Dε ε Dε (.) Rrnng now o h lnarzd prncpl of ral wor and rplacng h nonlnar gomrc rm wh h abo formla wrn n rms of ancd srans, w oban h fnal formlaon: DW a ε Dε ( ε DPd Pd ε C, d l C ( Dε l ε Dε, d C Dε Pd, d l, d l l ) d D d ) ldε d ε DPd ε C ε l Dε ldε d d (.) From hr can b sn ha ach of h ngrals prodcs h spcfc sffnss rm assocad wh h corrspondng arabl yldng a symmrc sysm of ncrmnal qaons. 4

52 . REDUCTIO OF THE THREE-FIELD TO A TWO-FIELD FORMULATIO For ffcncy rasons and asnss of mplmnaon, h abo formlaon was rdcd o a md dsplacmn prssr form. For h wo-fld form h prncpl of ral wor has o b modfd accordngly by ang o h consran ε ε ha corrsponds o h olmrc sran arabl as: Wn ( ε ) ε d ( P( ε ) P) ε d (.) and by prssng h nrpolad olmrc sran from h scond rm as a fncon of h nrpolad prssr as: P C C l ε, d l l ε C l l, d h εl lε, d lε Clεl ε (.4) P, d P ε Clε l Clε l ε, (.5) whr C l l s h nsananos bl modls. 9 Undr hs crcmsancs h lnarzd prncpl of ral wor for h md /p formlaon bcoms: 4

53 DW C DPd ( DP DP ) ε ε n ( ε C DP C D ε l l Dε P Dε l l l ε d d ε d Dε d ε ( DP DP ) d [ ε ( Dω Dω ) Dε ] DPd C d DP C l ) d P DP d d P DP d [ ε ( Dω Dω ) Dε ]d l ε l l ε ε l d (.5) B, DP Cl Dε l (.6) If w sbs (.6) n (.5) w g: DW n D ( C ε l Dε C l l Dε ε l Pd ε 9 Dε C ) d DP C T mn l ε mn C DP Pd l op d opl Dε l d (.7) Th abo formla rprsns h fnal prsson for h ra of nrnal ral wor from whch h ncrmnal sysm of qaons can b ddcd by sbsng h arabls wh hr fn lmn nrpolaons. For hs o b possbl, (.7) has o b conrd n a mar formlaon whch can b asly mplmnd n a FORTRA program ha nrfacs wh ASYS solr capabls. 4

54 .4 UPDATED LAGRAGIA JAUMA FORMULATIO In ASYS an Updad Lagrangan Jamann (ULJ) procdr s mployd o sol gomrc nonlnars and hrfor h sam procdr wll b sd n h proposd formlaon and mplmnaon. Accordng o [6], hs s a ypcal formlaon for larg dsplacmns, larg roaons, and larg sran analyss. Th ULJ formlaon s applcabl o gnral lasc-plasc analyss and s ry ffc n larg sran analyss, bcas h srss and sran masrs sd ar hos o dscrb h maral rspons n a naral way. Accordng o hs procdr h arabls a m ar consdrd sold and nown and hy ar sd o calcla h solon a m by solng a s of lnarzd smlanos qaons. ASYS ss h won- Raphson mhod as an ra nmrcal algorhm for solng h nonlnar qlbrm qaons. Dals rgardng h nonlnar ra algorhm wll b gn n chapr 4. Inrodcon of hs nonlnar procdr a hs pon s sfl only for sng h crrn confgraon ( ) as h m whn all h sffnss marcs oghr wh nrnal forcs ar alad. Th basc qaons mployd n h U.L.J. formlaon ar h Jamann srss ra qaon dfnd accordng o (.) and h ncrmnal ral wor qaon (.7) from whch w can oban h followng wo qaons: Eqlbrm qaon d ( ) ( ) ( ) mn ( ) : D ( ) DP ) ( ε C 9 C ( ) l C ( ) mn l mn ε d op ε D C ( ) ( ) opl ε d D F ε d l ( ) F ( ) ( ) (.8) 4

55 Consran Eqaon ( ) l l ( ) ( ) ( ) C D ε Pd d DP Pd ( ) F ( ) p (.9) whr C ( ) l s h ncrmnal maral propry nsor a m, rfrrd o h confgraon a m ; s h modfd Cachy srss a m ; and ( ) Dε and DP ar h ncrmnal srans and prssrs whch ar rfrrd o h confgraon a m. All h qans wh (-) sprscrp man ha hy ar alad a sng h al of h dsplacmns oband from h pros raon. Snc w ar nrsd n fndng h solon n crrn confgraon n rms of h solon of pros confgraon h y rm ha wll prod hs s h dformaon gradn nsor..5 EHACED DEFORMATIO GRADIET In mos srcral problms w ar nrsd n fndng h dformaon of h srcr hrogho h loadng hsory n rms of a rfrnc confgraon ha s gn. Sppos a parcl s locad a poson X nally and mos o poson. Th locaon of h parcl n crrn confgraon s rlad o h locaon n h rfrnc confgraon by h dsplacmn and s gn by: X (.) Th dformaon gradn mar s dfnd as: ( X ) F I (.) X X X 44

56 Snc h proposd formlaon s basd on ancng h dformaon mod sch ha wll prn olmrc locng s ncssary o rla hs dformaon mod o a modfcaon of h dformaon gradn. Ths approach has bn appld sccssflly by Taylor [47] o rahdral lmns. Th ncrmnal dformaon gradn can prssd as: F F (.) or n ndcal noaon F ( D ) D (.) Th ancd dformaon gradn s ancd wh a rm, F, oband from h dras of h bbbl fncon as: F F F (.4) Th oal ancd dformaon gradn from h nal confgraon o h crrn confgraon s compd n rms of dformaon gradn a pros m sp and ncrmnal dformaon gradn as: F F F (.5) Th olm has o b modfd conssnly as: d F d F d (.6) 45

57 .6 FIITE ELEMET APPROXIMATIO. MATRIX FORMULATIO Th fn lmn appromaon for h dlopd formlaon s basd on assmng a lnar nrpolaon for boh h dsplacmn and prssr flds. Gnrally, h nrpolaon can b wrn as: a α a α (.7) whr a s a cor-ald fncon a any pon; α s a s of nrpolaon fncons also calld shap fncons and a α s a s of nodal arabls. If w s an soparamrc concp for or for-nod rahdral lmn hn h coordnas of any pon n h rahdron can b prssd as: 4 α α α (ξ) (.8) and h sam shap fncons ar sd o dfn h dsplacmn and prssr arabls as: ( ξ ) P( ξ ) 4 α 4 α α α ( ξ ) α ( ξ ) P α P (.9).6. Shap Fncons Th 4-nod rahdral lmn s dfnd by a s of olm coordnas, ξ, ξ, ξ, ξ 4 whch rprsn h rao of h shadd olm o h oal olm (s Fgr ) 46

58 4 P Fgr. olm Coordnas Usng h olm coordnas, h poson n ach rahdron s gn by: ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ z z z z z y y y y y (.4) and h shap fncons for h lnar rahdron ar smply h olm coordnas. Each of hm s ny for on nod and zro a h ohrs and ars lnarly rywhr. 4 4 ξ ξ ξ ξ ξ ξ ξ (.4) I s wll nown ha d o h Babša-Brzz condon (or nf-sp condon)[4] h choc of dncal shap fncons for boh dsplacmn and prssr arabls s no allowd nlss w add sablzaon rms o h araonal formlaon or nrch h spac of dsplacmns wh a bbbl fncon as n h MII lmn []. Taylor [47] showd ha s also possbl o consdr h rm addd o h srans and dld h bbbl fncon from h dsplacmn rms. As a solon o hs problm an ancd sran was consrcd from h dra of a bbbl fncon and was addd o prod h 47

59 ncssary sablzaon. Or lmn s basd on h sam approach and hrfor wll ha a nod n ach cornr and an nrnal nod locad n h cnr. Two bbbl fncons, conformng and nonconformng, ar sdd for an opmal slcon. Conformng Bbbl Fncons Ths fncons ar a famly of hgh ordr polynomals ha ar zro on h bondary of h lmn. A sandard conformng bbbl fncon s h cbc bbbl fncon. Th cbc bbbl fncon was sd n h dlopmn of MII lmn and also n h aalabl lrar rlad o ranglar and rahdral lmns [5, 9-,, -5, 9-]. If, for any rahdral lmn w dno h olmrc coordnas by ξ,,,,4, h conformng bbbls ar accordng o [9]: c 64ξξ ξ ( ξ ξ ξ ) (.4) Ths bbbl fncon was sd by Taylor [47] n h formlaon of hs md-ancd rahdral lmn. Taylor s rsls ndca ha h conformng bbbl fncon n (.5) dos no lmna all oscllaons for problms whr srong prssr gradns occr. Accordngly, n hs rsarch, w wll also consdr ohr choc for bbbl fncons. on-conformng Bbbl Fncons R. Prr showd n 995 [9], ha h conformng bbbls sasfy h LBB condon as long as h msh s rglar. H dmonsrad ha hs happns only f h rangl s qlaral. Hnc, h proposd an opmal bbbl fncon for whch h local sablzng ffc s h gras. Ths opmal bbbl fncon s a qadrac non-conformng on whch mans ha dos no ansh on h prmr b s ny n h cnr nod. Ths fncon s dfnd for any rahdron as: 48

60 4 4 q ξ (.4) Ths fncon was also sd n or formlaon d o h followng adanags: I has h hghs sablzng facor, spcally n hr dmnsons.. Snc s a qadrac fncon wll yld lnar srans whch ar flly conssn wh h lnar nrpolaons of and p. Impls a lowr compaonal cos d o h fac ha w s a qadrac fncon nsad of a cbc (D) or a qarc on (D). mrcal ngraon s also lss cosly. In ry ohr aspc h nonconformng bbbl s as smpl o s as h conformng on..6. Mar Formlaon Usng h sam nrpolaon fncons for all arabls, h dsplacmns and prssrs can b rprsnd as: α α α α α α α α α α α α ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ DP DP D D P P P P ) (, ) ( ) (, ) ( ) (, ) ( (.44) whr h nodal dsplacmns and prssrs oghr wh hr araons and ncrmns ar: [ ] [ ] [ ] [ ] [ ] [ ] T T T P P P ,,, DP DP DP DP D D D D D D P P P P P P P P T T T α α α α α α (.45) 49

61 and h dscrzd srans ar prssd wh h hlp of an ancd sran-dsplacmn mar B as : α α α α α α B ε B ε B ε D D (.46) Th nlargd sran-dsplacmn mar has hr addonal colmns corrspondng o h hr addonal nrnal dsplacmns and has o b alad wh rspc o h crrn confgraon. I has h form: (.47),4 B α Ealaon of h Carsan dras of h shap fncons a m can b don sng h dras a m and ncrmnal dformaon gradn dfnd n (.): T F F α α α α α ) ( ) ( (.48) Usng h abo fn lmn appromaons and by placng h araons n h lf hand sd and h ncrmns n h rgh hand sd, ach of h ngrals from (.7) ylds h spcfc sffnss mar assocad wh h corrspondng arabl as: 5

62 . Cons sffnss mar:, c T B (C C 9 T I I C)B d (.49). Dsplacmn prssr sffnss marcs: p T ( p ) B T C T I d (.5). Prssr sffnss mar: pp T d (.5) whr h og mar noaon was sd (ach scond ran nsor s wrn as a cor and ach forh ran nsor as a mar). Accordng o hs noaon w ha: T [ yy zz y yz z ] (.5) T [ ε ε ε ε ε ε ] ε (.5) yy zz T [ ] y yz z I (.54) Th lasc modl ar wrn also n rms of a mar as: C C C C4 C5 C6 C C C C4 C5 C6 C C C C4 C5 C6 C (.55) C4 C4 C4 C44 C45 C46 C 5 C5 C5 C54 C55 C56 C6 C6 C6 C64 C65 C66 Whn larg dformaons ar pcd h gomrc sffnss mar s addd o h cons sffnss mar o form h fnal sffnss mar whch s a sqar symmrc mar hang h dmnson of h oal nmbr of dgrs of frdom. 5

63 Conrng h formlaon of hs rm no a mar hang h sam sz as h cons sffnss mar s a rahr complcad as snc h og noaon canno b sd n hs cas d o h dscrpancs n h szs of h marcs. Th procdr sd n hs wor s olnd n h n paragraph. 4. Gomrc sffnss mar: [ ][ ][ ] [ ][ ][ ]d d g B B T,,,, (.56) whr [ ], s h ancd mar formd by h dras of h shap fncons oghr wh h dras of h bbbl fncon, oband by wrng h gradn of h ral dsplacmn mar as a cor of nn componns as follows: { T, α α } (.57) whr { } { } 4 4 4, α s h ancd cor of dgrs of frdom formd by h wl rnal dsplacmn dgrs of frdom and h hr nrnal dsplacmn dgr of frdom. Usng h abo arrangmn of cors h mar of h dras of shap fncons can b wrn as a 95 mar as: 5

64 ,4 α α (.58) Th Cachy srss mar ha appars n boh rms s wrn as a 99 dagonal mar as: [ ] 9) (9 (.59) Th sam procdr s sd for h scond rm of h gomrc sffnss mar. Th lnar sysm of qaons n mar forma wll b: p pp p p F F F DP D (.6) whr s a sqar mar of sz 55, (.6) g c,, D s h cor of rnal and nrnal dgrs of frdom of sz 5, 5

65 F s h nrnal forc rslng from h qlbrm qaon and, h F p s h nrnal forc rslng from h consran qaon. 5. Inrnal Forcs From h frs rm of h prncpl of ral wor h nrnal forc d o h dsplacmn dgrs of frdom can b compd as: F T ( B ) d (.6) whr I( P P) s h modfd Cachy srss a m oband from h sran ncrmn sng h Jamann srss ra and ngrad nmrcally or h sbsp. From h scond rm of h prncpl of ral wor, (ε ε ) Pd, w can comp h fcos nrnal forc d o h md /p formlaon as: F p DP Dε Cl Dε l Dε Pd DP Cl Dε l Pd ( DP DP) P d ( DP DP) d (.6) DP P P ) (.64) whr ( ) ( and h Cachy srsss a m ar sord from h pros sbsp. 54

66 4. ELEMET IMPLEMETATIO ASPECTS Th fn lmn mhod s accompand by a larg nmbr of nmrcal procdrs sch as nmrcal ngraon, nmrcal algorhms for solng nonlnar problms, mhods for solng a lnar sysm of qaons and ohr mhods no rlad o h scop of or rsarch opc. Snc h commrcal fn lmn cod ASYS wll b sd for mplmnaon, all h nmrcal mhods rlad o hs cod wll b sd. 4. ITEGRATIO RULES In h cas of dsord lmns and non-lnar bhaor nmrcal ngraon s ndd. For h proposd rahdral lmn a for pon Gass nmrcal ngraon rl was sd. f ( ξ ) d n w f ( ξ ) d (4.) 6 l Th facor /6 was rqrd bcas h wghng coffcns always add p ny whras h olm of h rahdron n olm coordnas qals /6. Th wghng facors and h locaon of h Gass ngraon pons ar lsd n Tabl and shown n Fgr. Fgr. Ingraon Pon Locaon for Trahdral Elmn [] 55

67 Tabl. mrcal Ingraon for Trahdral Elmn [] Typ Ingraon Pon Locaon Wghng Facor 4 cornr pons ξ ξ ξ ξ Prm ξ, ξ, ξ, ξ 4 for ohr locaons.5 Snc sffnss marcs and nrnal forcs ar all prssd as ngrals or h olm of h lmn w nd an prsson for h olm of h rahdral lmn. Ths s gn by h drmnan of h Jacoban ransformaon mar bwn olm and Carsan coordnas as: 6 ξ ξ ξ ξ ξ ξ ξ ξ ξ Jac d (4.) whch can b drd from: ξ ξ ξ ξ (4.) whr ar h Carsan coordnas of nod α. α Ths olm s a sgnd qany whch mans ha s pos f h cornr nods ar nmbrd n a spcal way. To aod problms wh nga olms, a spcal sbron was nsrd a h bgnnng of h sr lmn sbron n whch h olms for ach lmn wr calclad and chcd for nga als. 56

68 If h olm was fond o b nga h sgn was changd and sord so wold b accssbl a ach raon. Insd h Gass ngraon loop h lmn olm was always mlpld wh h sgn sad. 4. OLIEAR ITERATIE ALGORITHM I was shown n chapr ha by sng an Updad Lagrangan Formlaon h lmn marcs and load cors wr drd. Thy can b arrangd as a lnar sysm of qaons of h form prssd n (.6) or n an pandd form as: p () () (4) p () () (4) p p pp (4) (4) (44) (9,9) D() D () Dp(4) (9) F F F p F () () (4) (9) (4.4) whr h scond row and scond colmn n h lmn sffnss mar ar acally h las hr rows and las hr colmns of h ancd sffnss mar. In ohr words,, p marcs ar s pars of h ancd (5,5) mar. Th szs of h parond marcs and cors ar shown n h parnhss. Snc h proposd nrpolaons for and p ar connos n h whol doman, and h ancd sran paramrs ar nrodcd only nrnally n h lmn, h solon can b prformd n wo sps. In h frs sp h nrnal dsplacmns ar lmnad a h lmn ll by sac condnsaon. In h scond sp h rdcd sffnss mar oband from h frs sp, oghr wh h rsdals, ar assmbld no h global qaons o b sold sng h won- Raphson ra algorhm. 57

69 4.. Sac Condnsaon Procdr In ordr o sol for h nrnal dsplacmns, w frs nd o swap h rows and colmns corrspondng o h nrnal paramrs wh h rows and colmns corrspondng o h prssr arabls. Ths s don so h fnal lmn arrays wll ha only h dsplacmn and prssr dgrs of frdom. Th fnal sz of h sffnss mar, load cor and h cor of h rnal dgrs of frdom has o b 46. Th followng sps wr prformd n h sac condnsaon procdr:. Inrodc on row and on colmn of zros n h 6 h plac of h ancd sffnss mar whch wll b now (, ).. Swap h rows and colmns corrspondng o and p sch ha h nrnal paramrs wll b arrangd h las and h prssrs n h mddl. Th las row and h las colmn of h mar wll b all zros and wll b lmnad oghr wh h rsdals. Th sysm wll now ha h form: p () (4) () p pp p (4) (44) (4) p () D() (4) Dp(4) () D () F F F p F (4.5). Paron h marcs as: (66) (6) (6) D () D fnal F F (4.6) () p (4) p (4) pp (44) (4.7) () p (4) (4.8) 58

70 [ () (4)] p (4.9) D D fnal Dp (4.) F F F Fp (4.) 4. Sor ( ) *,( ) * F and rr hm a h bgnnng of h n raon o calcla h ra al of h nrnal dsplacmn dgrs of frdom. D r r [ ] (F D ) (4.) Th ncrmn of h nrnal dsplacmns s oband as an accmlaon of ach raon al and s sord f h raon conrgd. sbsp r D D (4.) 5. Sbs n h frs qaon o g h fnal sffnss mar of sz (6,6): fnal fnal D ( * * ) D F fnal fnal (4.4) 6. Calcla rdcd nrnal forc as: F F ( ) * F (4.5) fnal 7. From hr h assmbly of fnal lmn mar and rsdal no global qaons procds o fnd o h ncrmnal als of nodal dsplacmns and prssrs. 59

71 4.. won Raphson Procdr Th basc approach n an ncrmnal sp-by-sp solon s o assm ha h solon a m s nown and o sol for h solon a m sng h balanc bwn h rnally appld forcs F and h rsorng forcs ha corrspond o lmn srsss [6]: n F F (4.6) whr n F F F (4.7) and F s h ncrmn n nodal pon forcs ha can b appromad sng a angn sffnss mar whch corrsponds o h gomrc and maral condons a m. F D (4.8) whr D s h cor of ncrmnal nodal pon dsplacmns. Ths w oban, D F F (4.9) Solng for D w can calcla an appromaon o h dsplacmns a m as: D (4.) From h abo prsson, h srans, srsss and nodal pon forcs can b calclad a m, and h procdr can mo o h n m ncrmn. Snc hs procdr may b ry nsabl h wdly sd fll won-raphson raon chnq (sffnss mar s pdad a ach raon) s lzd n ASYS o sol h global sysm. 6

72 Th qaons sd n h won-raphson raon ar: () () D () () F D () F () (4.) wh h nal condons: F () () () F (4.) Th gnral algorhm for a sbsp a raon () s llsrad n Fgr 4 and can b dscrbd as follows[]:. Assm whch s sally h conrgd solon from pros m sp and for h frs raon of h frs sbsp s s zro.. Comp h pdad lmn sffnss mar, srans and srsss, whch n nlasc analyss ar oband by an ngraon procss, and sng (.6) comp h rsorng load.. Chc qlbrm and f h rsorng load s no qal (or a las o whn som olranc) o h appld forc hn comp D () sng frs qaon of (4.). 4. Upda h dsplacmn o oban h n appromaon sng scond qaon of (4.). 5. Rpa sps o 4 nl conrgnc s achd. 6

73 Fgr 4. won-raphson Procdr[] Conrgnc n or cas s assmd whn () R < ε R R rf and ( ) εrf () D < and ( P ) ε p Prf D < (4.) whr RF F n s h rsdal cor and ε, ε, ε ar olrancs and h corrspondng Rrf, rf, P rf ar rfrnc als and h norm sd s h Ecldan norm ha s h scalar masr of h magnd of h cor as: R (4.4) R Th olranc sd for h o-of-balanc conrgnc for dsplacmn dgr of frdom s.5 and.5 for h prssr dgr of frdom. Th dfal o-of-balanc rfrnc R p al sd was F, for dsplacmn and for prssr a rfrnc al of was sd. 4. STRESS AD STRAI UPDATE ALGORITHM Whn a won-raphson ra algorhm s appld o a hsory-dpndn problm, as n or cas, h nonconrgd solon oband from h raon procss s sally no on h acal 6

74 pah and hs, h sran and srss whch ar hsory-dpndn arabls ha o b ngrad or h ncrmn ach raon []. Thy canno b compd as h sm of h ngraons prformd n ach raon. W assm hogh ha h prmary arabls ar aryng lnarly or h ncrmn. In or formlaon h sran s dfnd as h ngral of h sran ra. Ths ngraon has o a no accon h fac ha h prncpal as of sran roa drng h dformaon. Thrfor h sran a h bgnnng of h ncrmn ms b road wh h amon of rgd body roaon ha occrs drng ha ncrmn. Ths s don sng h Hghs-Wng [7] algorhm. Accordng o Hghs-Wng, any nsor assocad wh a ra cons qaon s ngrad as: T ( a R a R a(dε) (4.5) whr a s h nsor, a ( s h ncrmn assocad wh h maral s cons bhaor whch dpnds on h sran ncrmn, Dε, dfnd by h cnral dffrnc formla as: D Dε sym( ) (4.6) whr / ( ) (4.7) / and R s h roaon mar compd from h polar dcomposon of h dformaon gradn a h mdpon confgraon: F (4.8) / R/ U/ 6

75 Th dsplacmn ncrmn a mdpon confgraon, D D / / (4.9) s compd from h ncrmn of dformaon gradn alad a h mdpon confgraon: / ( / ) ( D) I ( D ) (4.) and D B D (4.) yldng D D * (I / D ) (4.) ( ) ( ) If and h srans,, ar nown hn w procd o calcla h srsss U ε ( ) whch n nlasc analyss ar oband by h abo dscrbd ngraon procss as n (4.5). 64

76 5. UMERICAL IESTIGATIOS OF LIEAR ICOMPRESSIBLE MATERIALS Th md ancd sran formlaon proposd was mplmnd hrogh a srprogrammabl lmn no h commrcal fn lmn sofwar ASYS. Usng h ASYS plaform, h prformanc of h sablzd formlaon was sd consdrng sral D problms sng h nwly dlopd rahdral lmn. Th rsls of problms wr compard o hr analycal rsls n h lrar or wll nown solons wh md hahdral lmns wh B-bar formlaon []. Th sr-programmabl lmn was mplmnd wh for y opons sch ha for formlaons ar allowd. Ths was don o show h mpromn of h proposd formlaon as compard o h pr dsplacmn and md /p who sablzaon formlaons. Th for formlaons wr: md /p ancd sablzd wh cbc bbbl fncon, md ancd wh qadrac bbbl fncon, md /p who sablzaon and pr dsplacmn. For dffrn ss wr dlopd for lnar lasc marals, ach of hm bng carrd o a dffrn Posson s raos and dffrn msh szs: homognos dformaon ss, hcwalld cylndr ndr prssr, Coo s mmbran problm and wo canlr bam ss for sng h capably of h lmn n bndng. Two dffrn mshs wr sd n hs problms. Whnr h problm prmd (bndng ss), h msh was crad by gnrang, n hr drcons, blocs of s rahdral lmns sch ha h comparson wh h hahdral lmns wold b mor manngfl. In hs mannr, h nmbr of nods was h 65

77 sam n boh hahdral and rahdral mshs. I s mporan o no, howr, ha h nmbr of lmns n h rahdral mshs s s ms grar han h nmbr of hahdral lmns ha wr modld. Thrfor, som mnor dffrncs n lmn solons wr pcd snc h msh sz has an nflnc on h sran and srss als. 5. HOMOGEEOUS DEFORMATIO TESTS Two homognos dformaon ss wr carrd o o chc h capably of h lmn o ssan consan sran sas. Th pach ss wr also a ry sfl ool for h fn lmn dlopr o assss h corrcnss of hr formlaon and nal msas n programmng.. Unaal comprsson s Th fn lmn modl for hs s rprsns a cb wh sd lngh of mad p of s rahdral lmns. Thr of s adacn facs ar rsrand o mo only n hr own plans. Th cb s sbcd o 4 forcs (F F 7 -/6 and F 5 F 4 -/) ha prod a nform comprss srss of MPa. Th maral proprs sd ar E6 MPa and υ.. Th pcd rsls ar: srss n h drcon of loadng s -. MPa and sran n h sam drcon s Rsls of h homognos dformaon sng h ancd md /p wh boh bbbl fncons ar machng h pcd ons and ar dncal. Thrfor only rsls corrspondng o h qadrac bbbl fncons ar shown n Fgr 5. 66

78 (a) Srss n loadng drcon (b) Sran n loadng drcon Fgr 5. Srss and sran n h drcon of loadng for h naal comprsson s. Homognos dformaon of a n cb formd of 5-rahdra ndr lnar appld dsplacmn fld Th fn lmn modl consss of a bloc mshd wh 5 rahdral lmns and sbcd o a lnarly aryng dsplacmn fld appld o all nods as gn by h followng:..*.5* y.5* z.75.5*.* y.75* z z.5.5*.75* y.5* z Th maral proprs ar E.6MPa and h fn lmn modl s prsnd n Fgr 6. Fgr 6. Fn lmn modl of n cb formd by 5-rahdra 67

79 Th compd srans and srsss for hs dformaon fld shold b: ε ε y y., ε 4., ε yy yz., ε.5, ε 4., 6., yy yz zz z.5,. 6., 6., z zz. 8., Th md ancd rahdral lmn passd hs pach s snc h rsls oband ar acly as hos pcd. A par of h op fls wh srans and srsss for a fw slcd lmns oghr wh pcd srans and srsss s prsnd n Appnd B. 5. EXPASIO OF A THIC-WALL CYLIDER UDER PRESSURE Ths s was proposd by R. Taylor n [47] o assss h bhaor of a smlar sran ancd rahdral lmn n narly ncomprssbl marals. Dscrpon of h s An nfnly long hc walld cylndr wh nnr rads of ns and or rads of 9 ns s sbcd o an nrnal prssr of n. D o h asymmrc nar of h problm, h fn lmn modl can b rprsnd by a wdg of a 5 dgr angl scor and a n hcnss. Th maral proprs consdrd ar lnar lasc wh E.6 and Posson s raos wr ard from,.5,.,.49,.499 and.4999 o ala h robsnss of h formlaon. Th msh s nsrcrd and h nmbr of dsons pr radal drcon was ard from, o lmns. Rsls of ach formlaon ar compard wh h horcal rsls and rrors ar rpord n Tabls, and 4. 68

80 Tabl. Radal dsplacmns for h hc wall cylndr, Posson Rao Thory Md /p ancd Tra Error (%) Md /p Error(%) Pr Dsplac mn Error (%) Error (%) Md Enhancd Tra Md Tra Pr Dsplacmn Posson's Rao Tabl. Radal dsplacmns for h hc wall cylndr, Posson Rao Thory Md /p ancd Tra Error (%) Md /p Error(%) Pr Dsplac mn Error (%) Error(%) Posson's rao Md ancd Tra Md Tra Pr Dsplacmn 69

81 Tabl 4. Radal dsplacmns for h hc wall cylndr, Posson Rao Thory Md /p ancd Tra Error (%) Md /p Error(%) Pr Dsplac mn Error (%) Error(%) Md ancd Tra Md Tra Pr Dsplacmn Posson's rao Rsls and Dscssons Tabls, and 4 show rsls of radal dsplacmns a a nod locad on h nnr rads for all h Posson s raos consdrd and for all formlaons oghr wh h rrors wh rspc o h horcal rsls. Conor plos of radal and angnal srsss for h cas of for all hr formlaons ar shown n Fgr 7. Th choc of h bbbl fncon s nsgnfcan snc rsls show ha boh of hm lmna olmrc locng and sablz h lmn n h sam mannr. Ths can b assssd from h comparsons wh h ac solons for dsplacmns, radal and angnal srsss. Pr dsplacmn formlaon shows clar dnc of olmrc locng and h md /p formlaon mpros h bhaor n h ncomprssbl lm. 7

82 Th rror n h dsplacmns s graly rdcd b sll shows som olmrc locng bcas of hgh als of srsss as compard o h horcal srsss. As can b sn from abls, h ancd md /p rahdral lmn shows rrors lss han % n radal dsplacmn for boh comprssbl and almos ncomprssbl lnar lasc marals. Th mpromn wh rspc o h ohr wo formlaons, s mor obos for υ.4999 whr h pr dsplacmn formlaon gs rrors rangng bwn 8 and 76 % dpndng on h msh sz and rspcly h md /p formlaon gs rrors rangng bwn -7%. For comprssbl marals all formlaons bha smlarly whch s pcd. Plos of srss dsrbon n radal() and angnal(y) drcons prsnd n Fgr 7(a-f) show ha h md /p and pr dsplacmn formlaons prodc spros and ncorrc parns wh hgh srss als. In conras o hs formlaons, h srss dsrbons of h sablzd md /p rahdral ar fr of oscllaons and h parn shows clarly h araon of srsss as a fncon of rad. Th rfrnc als of srsss for hs parclar problm of an nfnly long cylndr wr calclad analycally and hy wr fond o b: rr rr r r r r,, θθ r r θθ r r.5.5 As can b sn form Fgr 7, h sablzd rahdral prdcd srsss ar n good agrmn wh h abo horcal ons; h radal srsss ar whn 4 % and h angnal srsss ar whn % whl h md /p and pr dsplacmn formlaons rrors ar ncrasng or % and 5% rspcly. Anohr rmar n faor of sablzd rahdral lmn s rlad o h fac ha h rrors ar no aryng oo mch from on Posson s rao o anohr whch pros ha h dlopd lmn s robs. 7

83 (a) Sablzd Md /p Trahdral, radal srss (d)sablzd Md /p Trahdral, angnal srss (b) Md /p Trahdral, radal srss () Md /p Trahdral, angnal srss (c) Pr Dsplacmn Trahdral, radal srss (f) Pr Dsplacmn Trahdral, angnal srss Fgr 7. Thc walld cylndr srsss for 7

84 5. COO S PROBLEM Ths problm has bn frqnly sd o assss fn lmns ndr combnd bndng and shar. Th problm rprsns a aprd panl clampd on on sd whl a sharng load acs on h oppos sd (s Fgr 8). Th hcnss was consdrd mm and a sa of plan sran was smlad by rsranng h pla o dform n h hcnss drcon. Maral proprs sd ar EMPa and Posson s rao was ard agan bwn. and Thr msh szs wr sdd hang, and lmns pr dg. 6 mm P F 44 mm 48 mm Fgr 8. Coo s problm gomry Rsls and Dscssons Th rsls ar compard for sablzd md /p, md /p and pr dsplacmn formlaons wh h rsls of h wll nown md hahdral lmns wh md ancd formlaon aalabl n ASYS. Ths lmns ar nown o bha corrcly n ncomprssbly and shar locng condons. 7

85 Tabls 5 and 6 rprsn h rcal dsplacmn of h op cornr nod P, h aal srsss for all formlaons and h rrors wh rspc o h hahdral lmns. Fgrs 9, and show conor plos of h normal srsss n and y drcon and shar srsss for all formlaons for h msh sz and Posson rao υ Th pr dsplacmn formlaon hbs sr locng bhaor as h rror n h dsplacmn of pon P rachs 4 % and h srsss ar hghly orsmad. Th md /p lmns show mprod accracy n dsplacmns (rror s s.7%) b h srsss show spros dsrbon and ar orsmad. Th sablzd rahdral lmns approach ry closly h hahdral lmn solon n boh dsplacmns and srss dsrbon. Thy yld accra rsls n on coars mshs wh s.6% rror n dsplacmns and whn 5% rror n srsss. Also hy yld a smooh srss dsrbon for all h saons consdrd. Th dffrnc ha can b nod n aal comprss srsss whn compard o h hahdral lmns s d o h nsrcrd msh ha was sd for h rahdral lmns whch prodcs hghly dsord lmns a h lf op cornr. As for h shar srsss, h md /p and sablzd md /p yld smlar rsls and dsrbon parns whch ar ry clos o h hahdral rsls. Ths s pland by h fac ha h shar componn of h srsss dos no conan h olmrc par and hrfor s no affcd by h ncrasng prssrs dlopd n cass of olmrc locng. 74

86 Tabl 5. Coo s problm, rcal dsplacmn of op cornr nod for dffrn msh szs Pr Msh Hahdral Sablzd Md /p Error (%) Error(%) Dsplacmn Error(%) Sz lmn Trahdral Trahdral Trahdral Msh Sz Hahdral lmn Tabl 6. Coo s problm, aal nsl srss for dffrn msh sz Sablzd Trahdral Error (%) Md /p Trahdral Error(%) Pr Dsplacmn Trahdral Error(%) Too larg Too larg Too larg 75

87 (a) Md /p ancd sran hahdral (b) Sablzd md /p rahdral (c) Md /p rahdral (d) Pr Dsplacmn Trahdral Fgr 9. Coo s Problm: ormal Srss n drcon 76

88 (a) Md /p ancd sran hahdral (b) Sablzd md /p rahdral (c) Md /p rahdral (d) Pr Dsplacmn Trahdral Fgr. Coo s Problm: ormal Srss n y drcon 77

89 (a) Md /p ancd sran hahdral (b) Sablzd md /p rahdral (c) Md /p rahdral (d) Pr Dsplacmn rahdral Fgr. Coo s Problm: Shar Srsss 78

90 5.4 TEST OF BEDIG CAPABILITY A pr bndng s of h nwly dlopd rahdral lmn was prformd o assss s bndng capabls n small dformaon lnar lasc problms. Snc w ar nrsd n assssng h prformanc n shar locng and olmrc locng, only rsls for Posson s raos of ν. and ν ar rpord. Rsls of mamm dflcons, shar and aal srsss ar compard wh h md /p hahdral lmn rsls wh md ancd sran. Ths lmn s wll nown for s sably and good prformanc n boh shar and olmrc locng. Dscrpon of h s Th s rprsns a canlr bam wh a rcanglar cross scon wh wdh of mm and hcnss of mm and lngh of mm clampd a h lf nd. A bndng momn s appld hrogh wo oppos dsplacmn loads along h rgh dgs (op and boom) sch ha an rm pr bndng cas s smlad. Th maral proprs ar an as E MPa and Posson s rao was consdrd. and Th msh sd was srcrd for drc comparson wh hahdral lmns. Gomry and fn lmn modl ar shown n Fgr. Conor plos of normal and shar srsss for ν., 6 ar shown n Fgrs and 4 and for ν ar shown n Fgr 5 and 6. 79

91 Fgr. Fn lmn modl of h pr bndng s Rsls and Dscssons From h wo ss of plos can b ddcd ha h rahdral lmn s prformanc n shar locng s no as good as s n olmrc locng for hs rm cas of pr bndng. En hogh h dsplacmns ar whn rrors of % and h normal srsss ar accpabl, h shar srsss ha largr als han hos oband sng md ancd sran hahdral lmns. Th shar srss dsrbon also shows dnc of locng. Whn boh shar and ncomprssbly consrans s, h sablzd rahdral brngs som mpromn as compard o h md /p and pr dsplacmn formlaon ha s complly locd as can b sn from Fgr 5. Ths rmar hlps s concld ha h sablzd rahdron s prformng poorr han h md /p ancd sran hahdral lmn n pr bndng problms. I s hogh ncssary o no ha n bndng domnan problms, as was sn for Coo s problm, hs lmaon dsappars and h prformanc s sgnfcanly mprod. 8

92 (a) Md /p ancd sran hahdral (b) Sablzd Md /p Trahdral (c) Md /p Trahdral (d) Pr Dsplacmn Trahdral Fgr. Pr Bndng Ts: Aal Srss for ν. 8

93 (a) Md /p ancd sran hahdral (b) Sablzd md /p rahdral (c) Md /p rahdral (d) Pr Dsplacmn rahdral Fgr 4. Pr Bndng Ts: Shar Srss for ν. 8

94 (a)md ancd sran Hahdral (b) Sablzd md Trahdral (c) Md /p Trahdral (d) Pr Dsplacmn Trahdral Fgr 5. Pr Bndng Ts: Aal Srss for ν

95 (a) Md ancd sran Hahdral (b) Sablzd Md /p Trahdral (c) Md /p Trahdral (d) Pr Dsplacmn Trahdral Fgr 6. Pr Bndng Ts: Shar Srss for ν

96 6. UMERICAL IESTIGATIOS OF OLIEAR MATERIALS I LARGE DEFORMATIOS Th bhaor of h proposd formlaon n larg dformaons and larg roaons sng nonlnar marals wll b llsrad n a nmbr of bnchmar problms wh mphass on smlaons of mal formng problms. Ths ar sally ry dffcl nonlnar problms d o boh css lmn dsorons and conac condons oghr wh h prsnc of h ncomprssbly consran. Prformanc of h sablzd md /p rahdral lmn was sd agans h prformanc of h md /p hahdral lmn wh B-bar formlaon for problms whr olmrc locng s praln or wh ancd sran n bndng domnad problms. As n h pros chapr h bhaor of h sablzd formlaon was agan compard o hos of md /p who sablzaon and pr dsplacmn for assssng h mprod prformanc ha h sablzaon brngs. I s mporan o no ha analyss wr carrd o sng boh h cbc and h qadrac bbbl fncons. Snc dncal rsls wr oband for ach bbbl fncon, only h cbc bbbl fncon rsls ar prsnd n hs scon. 6. OLIEAR HOMOGEEOUS DEFORMATIO TESTS Th sam wo ss dscrbd n chapr 5. for lnar sng wr sd o rfy h corrcnss of codng for h fn sran dformaon cas. Ths pach ss wr passd n h gomrcally nonlnar cas and brf rsls ar prsnd n h n scon. 85

97 . Unaal Comprsson Ts. Two marals wr sd for hs s. Th frs maral sd for h naal comprsson s s ml lnar soropc (MISO) dfnd by a for pon srss-sran cr wh h followng proprs: E4, ν.. Th srss/sran cr of h laso-plasc maral s shown n Fgr 7. Th appld load s a nform prssr n rcal drcon sch ha h fnal hgh wll b rdcd o 5%. Rsls ar compard o h md /p hahdral wh B-bar formlaon and agan wh h ohr wo formlaons, h md /p who sablzaon and h pr dsplacmn. Fgr 7. Srss-Sran cr of MISO maral Th prdcd sran rsls n h loadng drcon ar prsnd n Fgr 8. Th conor plos (s Fgr 8 and 9) show boh h ndformd gomry and h dformd gomry. Thy show a homognos dformaon as was pcd and h dsplacmns, srsss, srans and prssrs of h sablzd rahdral ar n clln agrmn wh h hahdral lmn. Th dsplacmns and srans of h md /p who sablzaon ar ry dffrn from h rfrnc modl and n from h pr dsplacmn formlaon whch prforms as br as h sablzd formlaon for hs spcal cas of dformaon. 86

98 (a) Md /p Hahdral Elmn (b) Sablzd Md Lnar Trahdral (c) Md /p Trahdral (d) Pr Dsplacmn Trahdral Fgr 8. onlnar Unaal Comprsson Ts: Sran n loadng drcon. 87

99 (a) Md /p hahdral (b) Sablzd Md Trahdral Fgr 9. onlnar Unaal Comprsson Ts: Hydrosac Prssr. Th scond maral s modld by a ra dpndn sco-plascy modl n whch h yld srngh s nrodcd by an orsrss powr law (PERZYA modl) and h sac yld srngh s dfnd by sng a blnar soropc modl. Th paramrs nrodcd n h modl ar: E.5, ν., 9, m (sran ra hardnng paramr)., γ (scosy paramr).4 accordng o h modl y pl & ε q γ m Th sam bondary condons as bfor ar manand and a nform prssr n h rcal drcon s appld n hr load sps as follows:. Apply a srss qal o h yld srss (9).. Apply a srss qal o 8.. p a consan srss a 8. 88

100 Fgr shows h qaln srss for h sablzd rahdral compard o h hahdral lmn wh md /p and B-bar formlaon. As can b obsrd h rsls ar dncal and hy show a consan dformaon. (a) Md /p Hahdral (B-bar) (b) Sablzd Md Trahdral Fgr. Ra-Dpndn Unaal Comprsson Ts: Eqaln Srss. Homognos dformaon of a n cb formd of 5-rahdra ndr lnar appld dsplacmn fld Th sam gomry as n h lnar cas (s scon 5.) was sd now wh a blnar soropc maral and compard o a consan prssr rahdral (SOLID 87 n ASYS). Th maral proprs ar an as follows: E 4, ν., y 8 and E T. A lnar dsplacmn fld s appld as: (..*.5* y.5* z) *.45 (.75.5*.* y.75* z ) *.45 z (.5.5*.75* y.5* z) *.45 89

101 Fgr shows h qaln srss for h dformd gomry a h las sp oghr wh h nal gomry. As can b sn h sablzd rahdral lmn passs h consan dformaon s and h rsls ar dncal wh h ons for consan prssr rahdral. (a)\consan prssr Trahdral (SOLID87) (b) Sablzd Md Trahdral Fgr. Homognos Dformaon Ts wh lnar dsplacmn: Eqaln Srss. 6. UPSETTIG OF A BILLET To nally assss h prformanc of h nw md ancd sran rahdral lmn ndr condons of laso-plasc fn dformaon, a bll psng procss was analyzd. In h psng problm, h bll was modld as n Fgr and was sbc o a dsrbd rcal dsplacmn load appld or on hrd of s op cross-sconal ara. Ths spcfc psng gomry and load was chosn bcas nown rsls ar aalabl n h lrar [6]. Th orall obc of h formng procss bng modld was o ach a 65% comprsson of h hgh of h bll. 9

102 In h smlaon, a sa of plan sran s consdrd and no conac pars wr dfnd and h maral proprs of h bll wr gn h followng als: Elasc modls (E): GPa Poson s rao (ν):.. Yld srss ( y ): 5MPa Tangn Modls (E T ):.GPa Th fn lmn modl sd n hs smlaon has a srcrd msh ha prms drc comparson wh hahdral lmn msh. Rsls ar compard wh a hahdral lmn wh B-bar formlaon. Fgr. Fn lmn modl of h psng problm Fgrs and 4 conan conor plos of h qaln srss of h dformd bll and prncpal srss n h scond drcon. As shown n h fgrs, ry good agrmn was oband bwn h prdcd dformaon and srss rsls of h hahdral lmns and or nw formlaon. Eamnng h conor plos of all hr formlaons w can no ha h sablzd formlaon s h only formlaon ha prdcs h corrc dformd shap and dsplacmn, as compard o h hahdral msh. 9

103 Th md /p formlaon who sablzaon prdcs clos als of qaln srss b whn loong a h dformd shap can b sn ha h dsplacmns ar hghly ndrsmad, ndcang sr olmrc locng. In fac h dformaon of h md /p formlaon prdcs an pward drcon of plasc flow whch s oally dffrn from h ral on. Th pr dsplacmn formlaon also shows clar dnc of locng by nspcon of h dformd shap and hgh als of prncpal srss. Th sablzaon ffc of h md ancd sran formlaon s ry obos from h plos of h scond prncpal srss whch show a smooh dsrbon n conras o h polld dsrbon of srsss oband wh h md formlaon who sablzaon. En hogh h rsls ar mprod a lo compard wh h ohr formlaons, hr ar sll som dffrncs whn compard o h hahdral ons. Th only dffrnc n h prdcd rsls rlas o h dformaon of h bll arond h pnch cornr whr h rahdral modl prdcs a smallr slop bwn h pnch fac and h fr srfac han h hahdral modl. In ha ara h msh s hghly dsord and h comprss srss corrspondng o ha lmn s ry hgh as compard o h hahdral srss. W ha o p n mnd hogh ha any sharp cornr rprsns a snglary for h srsss and hrfor s pcd ha h srsss wold ha hgh als whch consqnly chang h conor plo so ha h comparson o hahdral lmns sms drmnal o h rahdral lmns. Bcas of hs rasons, n Fgrs and 4, (c) h qaln srss and scond prncpal srss for hahdral modl was plod sng h sam conor rang as for h rahdral and as can b sn, or spposon ha h snglary a h sharp cornr s h only ara ha dffrs from h rfrnc plo was confrmd. Th sablzd rahdral modl shows a smooh dsrbon wh als smlar o h hahdral on rywhr cp a ha cornr. 9

104 (a)md /p Hahdral (aomac conor) (b) Md /p Hahdral ( sam conor rang) (c) Sablzd md /p Trahdral (d) Md /p Trahdral () Pr Dsplacmn Trahdral Fgr. Upsng of a bll: Eqaln Srss 9

105 (a) Md /p Hahdral (aomac conor) (b) Md /p Hahdral (sam conor rang) (c) Sablzd M d /ptrahdral (d)md /p Trahdral () Pr Dsplacmn Trahdral Fgr 4. Upsng of a bll: Prncpal srss n scond drcon 94

106 6. METAL EXTRUSIO Th scond formng procss sd o assss h qaly of h nw rahdral lmn was a larg sran rson procss ha was also sdd n h lrar [6]. Smlaon of mal rson s anohr complcad mal formng problm n whch larg srans ar pcd. Ths procss s sally ry hard o b carrd o o complon who sng rmshng procdrs. Ths s d o hr cass: h rm dsoron of h lmns spcally n h angld par of h d, h ncomprssbl nar of h dformaon and h snc of h bondary conac condons. Dscrpon of h modl In h fn lmn modl, h wor-pc was dfnd wh a rcanglar cross-scon wh dmnsons of 9.5 mm and an orall lngh of 45 mm. As shown n Fgr 5, h worpc was forcd o pass bwn wo rgd formng ools by mposng a laral dsplacmn of 45mm n h horzonal drcon. Th ppr ool was dfnd o b a rgd arg srfac and s shap was dsgnd o nsr ha h fnal hgh of h bll was rdcd by 5%. A frconlss conac par was dfnd bwn h ppr rgd ool and h bll. Th boom conac par was modld by mposng symmry bondary condons along h cnrln of h wor-pc. An orall sa of plan sran was assmd n h modl. For h wor-pc, a blnar soropc modl was dfnd for h maral bhaor wh h followng proprs: Elasc Modls (E):.GPa Yld Srss ( y ): 8MPa Posson s raon (ν):., Tangn Modls (E T ) : MPa. 95

107 In ordr o rdc h css msh dsoron whn h maral nrs h d and also o rdc h m an o ach sady-sa condons, a fll was crad on h rgh op cornr. Th msh ha was sd s an nsrcrd msh. Th rfrnc rsls ar agan consdrd from a fn lmn modl wh hahdral lmn wh B-bar md /p formlaon. Rsls of qaln srss ar plod n Fgr 6 and qaln sran n Fgr 7. Fgr 5. Fn Elmn modl of h mal rson Rsls and Dscsssons From h conors of qaln srss w can s ha h dformaon rachs a sady sa condon and msh s homognos n hs ara. As llsrad n h qaln plasc srss and sran rsls prsnd n Fgrs 6 and 7, h sablzd rahdral modl was fond o ha good agrmn wh h hahdral modl. From a qala prspc, h rahdral modl hbd a ry smlar dformaon parn and sran dsrbon o h hahdral modl hrogho h rson procss. Qanaly, h orall dsplacmns rsls bwn h wo formlaons wr narly dncal. 96