Numerical modeling of metal cutting processes using the Particle Finite Element Method

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1 Numrical modlig of mal cuig rocsss usig h Paricl Fii Elm Mhod by Jua Maul Rodriguz Prio Advisors Jua Carlos Ca Trá Xavir Olivr Olivlla Barcloa, Ocobr 3

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3 Absrac Mal cuig or machiig is a rocss i which a hi layr or mal, h chi, is rmovd by a wdg-shad ool from a larg body. Mal cuig rocsss ar rs i big idusris (auomoiv, arosac, hom aliac, c.) ha maufacur big roducs, bu also high ch idusris whr small ic bu high rcisio is dd. Th imorac of machiig is such ha, i is h mos commo maufacurig rocsss for roducig ars ad obaiig scifid gomrical dimsios ad surfac fiish, is cos rrs 5% of h valu of all maufacurd roducs i all idusrializd couris. Cuig is a comlx hysical homa i which fricio, adiabaic shar bads, xcssiv haig, larg srais ad high ra srais ar rs. Tool gomry, rak agl ad cuig sd lay a imora rol i chi morhology, cuig forcs, rgy cosumio ad ool war. Th sudy of mal cuig is difficul from a xrimal oi of viw, bcaus of h high sd a which i aks lac udr idusrial machiig codiios (xrims ar difficul o carry ou), h small scal of h homa which ar o b obsrvd, h coiuous dvlom of ool ad workic marials ad h coiuous dvlom of ool gomris, amog ohrs rasos. Simulaio of machiig rocsss i which h workic marial is highly dformd o mal cuig is a major challg of h fii lm mhod (FEM). Th ricial roblm i usig a covioal FE modl wih lagragia msh ar msh disorio i h high dformaio. Tradiioal Lagragia aroachs such as FEM cao rsolv h larg dformaios vry wll. Elm disorio has b always mar of cocr which limid h aalysis o icii chi formaio i som sudis. Isad, FEM wih a Eulria formulaio rquir h kowldg of h chi gomry i advac, which, udoubdly, rsrics h rag of cuig codiios caabl of big aalyzd. Furhrmor srrad ad discoiuous chi formaio cao b simulad. Th mai objciv of his work is rcisly o coribu o solv som of h roblms dscribd abov hrough h xsio of h Paricl Fii Elm Mhod (PFEM) o hrmo-mchaical roblms i solid mchaics which ivolv larg srais ad roaios, mulil coacs ad graio of w surfacs, wih h mai focus i h umrical simulaio of mal cuig rocss. I his work, w xloi h aricl ad lagragia aur of PFEM ad h advaags of fii lm discrizaio o simula h diffr chi shas (coiuous ad srrad) ha aar wh cuig marials lik sl ad iaium a diffr cuig sds. Th w igrdis of PFEM ar focusd o h isrio ad rmoio of aricls, h us of cosraid Dlauay riagulaio ad a ovl rasfr oraor of h iral variabls. 3

4 Th rmoio ad isrio of aricls circumvs h difficulis associad o lm disorio, allowig h saraio of chi ad workic wihou usig a hysical or gomrical cririo. Th cosraid Dlauay imrovs mass cosrvaio ad h chi sha hrough h simulaio, ad h rasfr allows us o miimiz h rror du o umrical diffusio. Th hrmo-mchaical roblm, formulad i h framwork of coiuum mchaics, is igrad usig a isohrmal sli i cojucio wih imlici, smi-xlici ad IMPLEX schms. Th ool has b discrizd usig a sadard hr-od riagl fii lm. Th workic has b discrizd usig a mixd dislacm-rssur fii lm o dal wih h icomrssibiliy cosrai imosd by lasiciy. Th mixd fii lm has b sabilizd usig h Polyomial Prssur Projcio (PPP), iiially alid i h liraur o h Soks quaio i h fild of fluid mchaics. Th bhavior of h ool is dscribd usig a No-Hooka Hyrlasic cosiuiv modl. Th bhavior of h workic is dscribd usig a ra dd, isoroic, fii srai j lasolasiciy wih hr diffr yilds fucios usd o dscrib h srai hardig, h srai ra hardig ad h hrmal sofig (Simo, Johso Cook, Bakr) of diffr marials udr a wid variy of cuig codiios. Th fricio a h ool chi irfac is modld usig h Noro-Hoff fricio law. Th ha rasfr a h ool chi irfac icluds ha rasfr du o coducio ad fricio. To valida h roosd mixd dislacm-rssur formulaio, w rs hr bchmark roblms which valida h aroach, amly, lai srai Cook s mmbra, h Taylor imac s ad a hrmo-mchaical racio s. Th isohrmal-implex sli rsd i his work has b validad usig a hrmo-mchaical racio s. Bsids, i ordr o xlor h ossibiliis of h umrical modl as a ool for assisig i h dsig ad aalysis of mal cuig rocsss a s of rrsaiv umrical simulaios ar rsd i his work, amog hm: cuig usig a ra idd yild fucio, cuig usig diffr rak agls, cuig wih a dformabl ool ad a friciolss aroach, cuig wih a dformabl ool icludig fricio ad ha rasfr, h rasiio from coiuous o srrad chi formaio icrasig h cuig sd. W hav assmbld svral umrical chiqus which abl h simulaio of orhogoal cuig rocsss. Our simulaios dmosra h abiliy of h PFEM o rdic chi morhologis cosis wih xrimal obsrvaios. Also, our rsuls show ha h suiabl slcio of h global im igraio schm may ivolv savigs i comuaio im u o 9 ims. Furhrmor, his work rs a ssibiliy aalysis o cuig codiios by mas of a Dsig of Exrims (DoE). Th Dsig of Exrims carrid ou wih PFEM has b comard wih DoE carrid ou wih AdvaaEdg, Dform, Abaqus ad Exrims. Th rsuls obaid wih PFEM ad ohr umrical simulaios ar vry similar, whil, a comariso of umrical simulaios ad xrims show som diffrcs i h ouu variabls ha 4

5 dd o h fricio homa. Th rsuls suggs ha is cssary o imrov h modlizaio of h fricio a h ool-chi irfac. 5

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7 Ackowldgms Muchas rsoas ha coribuido a qu és rabajo udis llvars a cabo. Mis uors d sis docoral, los rofsors Jua Carlos Ca Trá, y Xavir Olivr Olivlla, or darm la ooruidad d dsarrollar mi sis docoral su gruo d ivsigació. E scial, a Jua Carlos l agradzco l imo qu m ddicó y los coocimios qu m rasmiió. Al rofsor Pdro Arrazola, d la Uivrsidad d Modragó, or rmiirm ralizar la sacia su cro d ivsigació dura los rimros mss d. Al rofsor Pär José d la Uivrsidad Tcológica d Lula (LTU) or darm la ooruidad d dsarrollar ua sacia osdocoral su gruo d ivsigació. A mi amigo y comañro d docorado Maul Caicdo, u scial agradcimio or los momos qu comarimos, ochs qu rasochamos rsolvido roblmas qu aarcía usras sis, uos cuaos viajs y uas cuaas crvzas. A los rofsors Rafal Wylr y Joaquí Hrádz or sus cosjos y sugrcias rsoals y ciíficas qu m hiciro dura mis sudios d docorado. Al rofsor Orlado Porras, l agradzco or l imo ddicado y l coocimio qu m rasmiió dura l dsarrollo d mi sis d masría y los cursos d rmodiámica, rasfrcia d calor y mcáica dl mdio coiuo, cursos qu furo fudamals ara l dsarrollo d la sis d docoral. Tambié l agradzco la ooruidad qu m bridó d rsar los rsulados d mi sis la Uivrsidad d los Ads. Por úlimo quiro agradcrl or orm coaco co mis acuals dircors d sis, los rofsors Olivr y Ca. Al rofsor Doria Liro u scial agradcimio or rmiirm rsar los avacs d mi sis docoral la Uivrsidad Nacioal. A Rosa Maria Ola or oda la colaboració rsada odos los rámis admiisraivos qu rquir l docorado. Al rofsor Orla Cácrs l agradzco las alabras d alio qu simr llgaba l momo ooruo. Al rofsor Ré Mzia, l agradzco or iiciarm l mudo d los méodos uméricos la clas d Aálisis Numérico, camo d ivsigació fudamal l dsarrollo d mi disració docoral. U agradcimio scial a Ré or sr ua d las rsoas qu m abrió las uras d CIMNE. Al rsoal admiisraivo d CIMNE or simr sar disusos a colaborar lo rlacioado co la bca, l corao, los cogrsos, r oras cosas. A mis amigos Marco Javir Sáchz, Adrés Gozálz, Juliá Eriqu Rodríguz, Adrés Hrádz, Carlos Galvis or sr quis so. 7

8 A mi familia, mi Paá (QEPD), mi Mamá, mis hrmaos (Frado y Srgio) o go alabras ara agradcrl or odo lo qu ha hcho or mí dura odos sos años. A Mauricio Garay Ruiz or simr sar disuso a colaborar si srar rcibir ada a cambio. Al Miisrio d Cicia Iovació d Esaña or la ayuda Prdocoral d Formació d Prsoal Ivsigador d rfrcia BES asociada al royco d ivsigació BIA8-4, lidrado or l Prof. Xavir Olivr qu m rmiió fiaciar mis sudios docorals la Uivrsidad Poliécica d Caaluña. Al cro d Ivsigació CIMNE or bridarm soor coómico dura los rimros 6 mss dl docorado. 8

9 Tabl of cos Numrical modlig of mal cuig rocsss usig h Paricl Fii Elm Mhod... Iroducio Moivaio Objciv ad Sco of h rs rsarch Mal cuig mchaics Chi shas Procsss zos Forcs i mal cuig Mchaical ad hrmal coac a h ool-chi irfac Numrical Simulaio of Mal Machiig. (Fii Elm): Sa of h Ar Problm Formulaio Numrical ram of h icomrssibiliy cosrai du o lasiciy Tim igraio schms (imlici, xlici, smi-xlici) Coac algorihms Adaiv rmshig, Error simaors, Trasfr oraors Workic-chi saraio criria Chi sgmaio ad brakag Numrical simulaio of Mal Machiig (Mshlss Mhods): Sa of h Ar Smooh Paricls Hydrodyamics (SPH) Fii Poi S Mhod (FPM) Cosraid Naural Elm Mhod (CNEM) Discr Elm Mhod (DEM) Imrovd Eulria Formulaio: Sa of h Ar Muli marial Eulria Mhod (MMEM) Volum of Solid (VOS) Marial Poi Mhod (MPM) or Poi i Cll (PIC) Paricl Fii Elm Mhod (PFEM) Thsis Ouli Th Paricl Fii Elm Mhod i h umrical simulaio of mal cuig rocsss Problm sam Basic oaio Th could Thrmo-mchaical IBVP wih fricioal Coac Cosrais Balac quaios

10 .3. Boudary codiios Iiial codiios Boudary codiios a h ool-chi irfac Global oraor sli for fii dformaio lasiciy Isohrmal lasolasic s Thrmolasic s a fixd cofiguraio Cosiuiv modls Tool cosiuiv modl Workic Cosiuiv Modl Fricioal coac cosrais Normal bhavior Tagial bhavior Ha rasfr a h ool chi irfac Variaioal Formulaio. Wak Form of h IBVP Icludig Fricioal Coac Cosrais Numrical Igraio Algorihm Th icrmal boudary valu roblm. Fii lm discrizaio Th icrmal boudary valu roblm. Tim discrizaio Workic cosiuiv law: im discrizaio Discrizaio of h fricioal coac modl....9 Mshig i h Paricl Fii Elm Mhod (PFEM) Mshig i h Paricl Fii Elm Mhod: umrical simulaio of mal cuig rocss Numrical modlig of mal cuig rocsss usig PFEM Pla srai Cook s Mmbra roblm Taylor imac s Thrmo-mchaical racio s Machiig sl usig a ra idd yild fucio Machiig a AISI 434 usig diffr rak agls Imlici, IMPLEX or xlici im igraio schms i h umrical simulaio of mal cuig rocsss? Machiig a iaium alloy (Ti6Al4V) a diffr cuig sds. Th ffc o cuig forcs ad chi shas Orhogoal cuig of AISI 434 sl usig a dformabl ool. A friciolss aroach Orhogoal cuig of AISI 434 sl usig a dformabl ool: Ha rasfr ad fricio bw h ool ad h workic Orhogoal cuig of 4CD4 sl: A xrimal comariso A Ssibiliy Aalysis o Gomric ad Cuig Codiios usig h Paricl Fii Elm Mhod (PFEM) Dsig of Exrims (DoE) Orhogoal cuig simulaio of 4CD4 sl usig h Paricl Fii Elm (PFEM)... 7

11 4.3 Numrical ad Exrimal validaio of h PFEM sragy A Dsig of Exrims wih PFEM ad is comariso wih a DoE wih h commrcial sofwar (Abaqus, AdvaEdg ad Dform) Coclusios Cocludig rmarks O h gral faurs of h roosd soluio schm O h mixd dislacm-rssur formulaio for hrmolaso-lasic roblm O h im igraio schm of h could hrmomchaical roblm O h mshig schm usig h aricl fii lm mhod (PFEM) O h simulaio chology O lis of rsarch... 85

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13 Iroducio. Moivaio Char Th sam gi is cosidrd o b drivig forc ad h mos imora chology of h Idusrial Rvoluio.Wa sam gi (76), wih is larg mal cylidrs ad ohr ars of urcdd dimsioal accuracy, i ld o h firs major dvloms i mal cuig. Raid failurs of h ools could b avoidd oly by cuig vry slowly, as a cosquc 7 workig days o bor ad fac o of Wa s larg cylidrs wr rquird. From 76 o 86 h kowldg of how o machi diffr shas i marials lik cas iro, wrough iro ad fw cor allows usig high carbo sl ools was acquird. Durig his cury, h rsarch focus was basd o h imrovm of h qualiy ad cosiscy of ool sls usig a rial ad rror aroach. Wih h ivio of h Bssmr (855) ad Sims-Mari (865) sl makig rocsss, sl raidly rlac wrough iro as h workhors of cosrucio marials, bcaus hs rocsss allow h raid roducio of larg quaiis of basic sl durig a shor riod of im. Du o h high srgh ad siffss of sl, i is much mor difficul ha wrough iro o machi, ad cuig sds has o b lowrd v furhr o maiai rasoabl ool lif. Durig h las cury, h iciv o rduc h cos by acclraig ad auomaig h cuig rocss has b h major drivig forc bhid h chological dvloms i mal cuig. Maily, roduciviy has b icrasd wih h highr cuig sds achivabl usig high sd sl ad cmd carbid ool, boh rrsig a advaag ovr radiioal carbo sl chology. Also, dsigrs ad maufacurs hav oimizd h sha of h ools o lgh ool lif a high cuig sds, whil lubricas maufacurs hav dvlod w coolas ad lubricas o imrov surfac fiish ad rmi icrasd ras of rmoval. Th sudy of mal cuig is difficul from a xrimal oi of viw, bcaus of h high sd a which i aks lac udr idusrial machiig codiios (xrims ar difficul o carry ou), h small scal of h homa which ar o b obsrvd, h coiuous dvlom of ool ad workic marials ad h coiuous dvlom of ool gomris, amog ohrs rasos. I h las yars hr hav b may ams o dvlod mahmaical modls which will rdic quaiaivly h bhaviour of work marial durig cuig from kowldg of hir roris. From h mchaisic oi of viw, Taylor (97), Ers ad Mrcha (945), Oxly (959) ad Row ad Sick 3

14 (967), ad D Vor al. (98). Mor comlx mahmaical modls wr dvlod usig Fii Elm Aalysis. Th firs umrical simulaio of mal cuig was carrid ou by Tay l al. (974). As rord i [], a mahmaical modl of mal cuig should b abl o giv h followig iformaio:. rdicio of ool lif. rdicio of h accuracy of h comos big machid 3. rdicio of h surfac fiish o h como big machid 4. rdicio of chi corol 5. rdicio of h loads o h ool, h workic ad h fixurs Th aalyical ad mirical modls hav coribud a gra dal o h fild rovidig iformaio abou chi formaio rocsss i h avrag ss. Isad, umrical simulaio of mal cuig rocsss has bcom icrasigly mor oular du o is abiliy o rovid daild isigh h rocss. Such simulaios hav h rosc o rlac h difficul ad cosly xrims usd for ool ad rocss dsig. Mos of h ims, h aalyical or mirical modls ar h rfrrd oios a idusry du o is simliciy. For wo rasos, h mirical, mchaisic ad FEA modls ar ro o b criicizd. Firs, if h chos ius o h modl do o mach h codiios i racic, h forcas will b a ucraiy. Scod, if h irals of h modl do o mbody h corrc marial cosiuiv quaio, or hav o way of accouig for h fricio homa hrough h scodary shar zo, h agai h forcas will b ucrai. Furhrmor, FEA is also criicizd du o is larg comuaioal cos dd o carry ou a umrical simulaio. Mor ha 3 yars of sudy of mal cuig rflc h comlxiy of h rocss. Th log ah sard wih h ool ad rocss dsig i 76 usig a rial a rror aroach ad coius, oday, wih h ool ad rocss dsig usig mchaisic ad umrical modls of mal cuig. Nvrhlss, is cssary o imrov h xrimal dvics ad h mchaisic ad umrical modls o icras our udrsadig abou mal cuig. Th abov cosidraios cosiu solid ad comllig rasos o ursu a li or rsarch i h fild of umrical simulaios of mal cuig rocsss.. Objciv ad Sco of h rs rsarch Th cral goal of his work is o xd h Paricl Fii Elm Mhod (PFEM) o hrmo-mchaical roblms i solid mchaics which ivolv larg srais ad roaios, mulil coacs ad graio of w surfacs, wih h mai focus i h umrical simulaio of mal cuig rocss. I his work, w xloi h aricl ad lagragia aur of PFEM ad h advaags of fii lm discrizaio o simula h diffr chi shas (coiuous ad srrad) ha aar wh cuig marials lik sl ad iaium a diffr cuig sds. Also, i his work, w dvlod a mixd (dislacm/rssur) sabilizd liar riagl fii lm basd i h 4

15 work of Dohrma ad Bochv [, 3], abl o dal wih h icomrssibiliy cosrai du o lasic homa..3 Mal cuig mchaics Mal cuig or machiig is a rocss i which a hi layr or mal, h chi, is rmovd by a wdg-shad ool from a larg body. Mal cuig rocsss ar rs i big idusris (auomoiv, arosac, hom aliac, c.) ha maufacur big roducs, bu also high ch idusris whr small ic bu high rcisio is dd. Th imorac of machiig is such ha, i is h mos commo maufacurig rocsss for roducig ars ad obaiig scifid gomrical dimsios ad surfac fiish, is cos rrs 5% of h valu of all maufacurd roducs i all idusrializd couris. Cuig is a comlx hysical homa i which fricio, adiabaic shar bads, xcssiv haig, larg srais ad high ra srais ar rs. Tool gomry, rak agl ad cuig sd lay a imora rol i chi morhology, cuig forcs, rgy cosumio ad ool war. Du o h comlx hysical rocss ha aks lac i cuig, machiig rocsss is o of h mos irsig idusrial roblms o b aalysd from h umrical oi of viw. Figur. Four ys of chi dformaio i mal cuig.[] 5

16 Figur. Localizaio of h rimary ad h scodary shar zos.[7] This scio iroducs h mai cocs rlad o cuig mchaics. Amog hm, h ys of chis ha ca b formd ddig o marial ad cuig codiios, h wo rgios whr lasic flow aks lac ad as a cosquc a icras of mraur du o lasic work, h cuig forcs acig o h ool, ad h mai vocabulary associad o agls ad dimsios i which cuig forcs dds o, ad fially, h mchaical ad hrmal coac i h chi-ool irfac..3. Chi shas Figur shows h wid rag of chi flows ha ar fr o b formd, ddig o h marial ad cuig codiios. Th formaio of all ys of chis ivolvs a sharig of h work marial i h rgio of a la xdig from h ool dg o h osiio whr h ur surfac of h chi lavs h work surfac. A vry larg amou of srai aks lac i his rgio i a vry shor irval of im, ad o all mals ad alloys ca wihsad his srai wihou fracur. Th vas majoriy of chi availabl, ca b classifid as follows: discoiuous, coiuous, coiuous wih build-u-dg ad srrad chi. Discoiuous chi (Figur (a)): This y of chi is yically obaid wh machiig bril marials a low cuig sds, somims h chi is formd i sara sgms or somims i sgms wih vry hi marial jucios. Wih his y of chi is yical o obai high surfac roughss. Ohr facors ha ifluc h formaio of discoiuous chis ar a high fricio bw h work ic ad h chi, high fd ra ad cuig dhs. Coiuous chi (Figur (b)): wh machiig ducil marials a high cuig sds ad low fd ras ad dhs of cu. A good surfac fiish is obaid 6

17 wh his y of chi is formd. A low fricio coffici bw h ool ad h chi courag h formaio of his kid chi. Coiuous chi wih build-u-dg (Figur (c)): wh machiig ducil marials a low o mdium cuig sds, h fricio bw h chi ad h rak fac is high ad h chi may wld oo h ool fac. This formaio is calld a build-u-dg. Srrad chi (Figur (d)): Ths chis ar smi-coiuous i h ss ha hy ossss a saw-ooh aarac ha is roducd by a cyclical chi formaio of alraig high shar srai followd by low shar srai. This y of chi is rlad wih marials lik iaium alloys, ickl alloys ad sur ausiic sailss sls..3. Procsss zos Durig machiig rocsss, h major dformaios ar cocrad i wo (a) (b) Figur 3. Forcs i mal cuig: (a) forcs acig o h chi i orhogoal cuig ad (b) forcs acig o h ool ha ca b masurd. rgios clos o h cuig ool dg. Ths rgios ar usually calld rimary ad scodary dformaio zos. (Figur ) Th rimary dformaio zo xds from h i of h cuig ool o h jucio bw h udformd work marial ad h dformd chi. Th workic is subjcd o larg dformaio a a high srai ra i his rgio. Th haig is du o rgy dissiaio from lasic dformaio. Mal cuig xrims hav idicad ha h hickss of h rimary shar zo is oly a fw housadhs of a cimr. Th rimary shar zo is iclid a agl (shar agl) wih h la of work. Scodary shar zo rsuls from h fricio bw h ool ad h chi as h chi slids alog h rak fac of h ool. A h scodary shar zo, ha is grad du o lasic dformaio ad fricio bw h ool ad h chi. This rgio is usually dividd i wo rgios, h sickig ad h slidig. 7

18 Figur 4. Orhogoal cuig gomrical aramrs..3.3 Forcs i mal cuig Th forcs alid agais h chi by h ool ca b sarad io wo muually rdicular comos (Figur 3 (a)): fricio forc ad h ormal forc o fricio. Th fricio forc F is h fricioal forc rsisig h flow of h chi alog h rak fac. Th ormal forc N o fricio is rdicular o h fricio forc. I addiio o h ool forcs acig o h chi, hr ar wo forc comos alid by h workic o h chi: shar forc ad ormal forc o shar. Th shar forc F S is h forc ha causs shar dformaio o occur i h shar la, ad h ormal forc o shar F N is h rdicular o h shar forc. No of hs forcs F, N, FS, F N ca b masurd i h machiig oraios, bcaus h dircios i which hy ac vary wih diffr ool gomris ad cuig codiios. Howvr, i is ossibl o masur h cuig forc ad h hrus forc. Th cuig forc F C is i h dircio of cuig, ad h hrus forc F T is rdicular o h cuig forc (Figur 3(b)). Mahmaical quaios o rla h four comos ha cao b masurd o h wo forcs ha ca b masurd ar rs by Groovr i []. Th ool i orhogoal cuig has oly wo lms of gomry (Figur 4): () rak agl ad () clarac agl. Th rak agl drmis h dircio ha h chis flows as i is formd from h workic; ad h clarac agl rovids a small clarac bw h ool flak ad h wly grad machid surfac. 8

19 .3.4 Mchaical ad hrmal coac a h ool-chi irfac To imrov h machiabiliy ad h rformac of cuig ools is madaory o udrsad h movm of h chi ad of h work marial across h facs ad aroud h dg of h ool. I mos sudis his has b rad as a classical fricio siuaio, i which fricioal forcs d o rsrai movm across h ool surfac, ad h forcs hav b cosidrd i rms of a coffici of fricio bw h ool ad work marials (Coulomb fricio law). Howvr, daild sudis of h ool/work irfac hav show ha his aroach is iaroria o mos mal cuig codiios [4-7]. A his sag, i is cssary, o xlai why Figur 5. Normal ad fricioal srss disribuios o h ool rak fac.[] classical fricio cocs do o aly ad o suggs a mor suiabl modl for aalyzig his siuaio. Coulomb basd i may commo xamls of h slidig of o solid surfac ovr aohr, rooss ha h forc F rquird o iiia or coiu slidig is roorioal o h forc N ormal o h irfac a which slidig is akig lac F N (.) Th fricio coffici μ is dd oly o hs forcs ad is idd of h slidig ara of h wo surfacs. Bowd ad Tabor [8], Archard [9] dmosrad ha his roorioaliy rsuls from h fac ha ral solid 9

20 surfacs ar vr comlly fla o a molcular scal, ad hrfor mak coac oly a h os of h hills, whil h vallys ar sarad by a ga. Wh h ormal forc is icrasd o such a x ha h ral ara of coac is a larg roorio of h aar coac ara, a yical cas i machiig, i is o logr ossibl for h ral coac ara o icras roorioaly o h load. I h xrm cas, whr h wo surfacs ar comlly i coac, h ral ara of coac bcoms idd of h ormal forc, ad h fricioal forc bcoms ha rquird o shar h marial across h whol irfac. Wh wo marials of diffr srghs ar i coac, as i mal cuig, h forc rquird o mov o body ovr h ohr bcoms ha rquird o shar h wakr of h wo marials across h whol ara. This forc is almos idd of h ormal forc, bu is dircly roorioal o aar ara of coac - a rlaioshi dircly oosd o ha of classical fricio cocs. Du o h iadquacy of Coulomb fricio law o sudy h comlx homa ha aks lac a h ool chi irfac, svral cosiuiv modls hav b roosd which ry o xlai h comlx homa which aks lac a h ool chi irfac. A summary abou som of h w dvlod fricio modls is rsd i h followig lis: Arrazola al. ()[6] rs a fricio law which cosidr h sick ad h sli homa a h chi-ool irfac (Figur 5). I h formr zo h shar srsss ar dfis as follows: ( x) k wh mk ad <x l (.) Whil i h lar h shar is giv by: ( x) (x) wh mk ad l <x lc (.3) Whr k is h shar flow srss of h workic marial, is h fricio coffici, is h ormal comrssiv srss acig a h ool fac, l is h siz of h sizur rgio, l c is h coac lgh bw h ool ad h workic ad m is a cosa ha aks a valu of m. Usui ad Shirakashi drivd a mirical quaio as a fricio modl, which rlas h fricioal srss o h ormal srss k k (.4) Whr k is h shar flow srss of h workic marial ad is a fricio coffici xrimally obaid for diffr workic ool marial combiaios. Childs al. [] modifid his modl by mulilyig k wih a fricio facorm, whr m :

21 mk k Dirikolu al. [] mad a furhr modificaios o his modl by mulilyig k wih a friciom, whr m ad iroducig a xo : k mk (.6) Mor iformaio abou h fricio modl a h ool-chi irfac is giv i [4-7, -].4 Numrical Simulaio of Mal Machiig. (Fii Elm): Sa of h Ar Th s of umrical ools usd i h umrical simulaio of mal cuig should b abl o rrs h comlx hrmo-mchaical homa akig lac isid h ool ad h workic. A h sam im, h s of umrical igrdis should b as siml as ossibl i ordr o dcras h comuig im o g accura rsuls. As rsuls, h followig qusio ariss: Wha ar h s of umrical ools ha allow simulaig mal cuig rocsss wih high accuracy, bu wih a low comuaioal cos? I ordr o aswr his qusio, firs of all, h followig lis rs a summary abou h sa of h ar i h umrical simulaio of mal cuig rocsss. This summary icluds msh ad mshlss sragis, diffr formulaios (lagragia, ulria, ALE), sragis o dal wih h icomrssibiliy cosrai imosd du o h lasic bhavior of h workic, im igraio schms, coac algorihms o rrs h coac bw h ool ad h workic, ha hav b usd or ca b usd i h umrical simulaios of mal cuig rocsss. This summary icluds advaags ad drawbacks accordig o h oi of viw of h auhor of his work, abou h diffr umrical sragis caabl o rrs h comlx homa akig lac i machiig..4. Problm Formulaio A Fii Elm Lagragia calculaio mbds a comuaioal msh i h marial domai ad solvs for h osiio of h msh a discr ois i im. As a cosquc h Fii Elm Lagragia formulaio is rlad o h roblm of msh disorio. Th calculaio rocss ca v b imossibl o coiu wh h Jacobia drmias bcom gaiv a som igraios ois. A ims, Fii Elm Lagragia formulaios us a cririo o sara h chi from h workic. Thos criria icludd

22 lm dlig basd o a gomrical disac of h ool i o closs workic lm, lasic srai ad srai rgy dsiy. Ohr ims, Fii Elm Lagragia formulaios ar usd wih msh adaiviy ad auomaic rmshig, ad as a cosquc his sragy dos o rquir a chi saraio cririo. Usig rmshig mas ha h filds of sa variabls hav o b mad from o old msh o h w o. This maig is o a sraigh forward ask ad iroducs som umrical diffusio o h sa variabls. This chiqu has b succssfully alid i simulaios of coiuous ad srrad chi formaio. A lagragia dscriio of moio ad adaiv rmshig was usd o simula orhogoal cuig i [3, 4] ad [5]. Isad, a Lagragia formulaio lus a od saraio as a cririo o rdic chi ic saraio was usd i [6], [7] ad [8]. Fii Elm Eulria formulaios hav b usd by may auhors o simula coiuous chi formaio a sady sa. Fii Elm Eulria formulaio avoid h roblm of msh disorio bu ds a rdfid chi sha o dvlo h umrical simulaio, whil Fii Elm lagragia formulaio is abl o rdic chi formaio from icii o sady sa. A ror assumio of h chi sha is vry difficul o obai sic i dds o may facors. I h Fii Elm Eulria formulaio h marial flows hrough h fixd msh. Th mai disadvaag of Eulria formulaios is ha is o asily adaabl for modlig h ucosraid flow of h marial as h chi volvs durig h rocss. As a cosquc Fii Elm Eulria formulaios cao simula srrad ad discoiuous chi formaio. A xaml of a Fii Elm Eulria formulaio alid o h umrical simulaio of mal cuig is rsd i [9]. I ordr o avoid h disadvaags of Fii Elm Lagragia ad Eulria formulaios, ohr comuaioal chiqus hav b ivsigad. O of hm is h Fii Elm Arbirary Lagragia Eulria (ALE) formulaio i cojucio wih adaiv msh chiqus [3-3]. Th ALE formulaio combis h bs faurs of ur Lagragia aalysis (i which h msh follows h marial) ad Eulria aalysis (i which h msh is fixd ad h marial flows hrough h msh). I ALE framwork msh moio is idd of marial moio, for ha raso high qualiy fii lm mshs ar rsrvd durig h umrical simulaio of machiig rocss. ALE formulaio dos o d a cririo o sara h chi ad h workic. Grally, ALE formulaio is comuaioally char ha a Lagragia formulaio, bu ds a rformd chi, which us som rsricio o a ALE formulaio o rdic diffr chi shas (srrad, discoiuous). ALE formulaio ca b usd o simula chi formaio from icii o sady sa, bu h roblm is o dfi a msh moio schm i ordr o rsrv a high qualiy fii lm msh durig h simulaio. Numrical simulaios i 3D usig ALE ar difficul o carry ou, bcaus h msh moio i ordr o rsrv a high qualiy msh is mor difficul i 3D

23 ha i D. A daild iformaio o us of ALE formulaios i modlig mal machiig ar rsd by [3, 33], ad [3]..4. Numrical ram of h icomrssibiliy cosrai du o lasiciy Th mos commo fii lms usd i h umrical simulaios of mal cuig ar h followig: a la srai quadrilaral isoaramric fii lm usd i [3, 6, 34], a 6 odd isoaramric riagular lms usd i [3] ad [4], a hacd four od quadrilaral wih -oi quadraur usd i [5] ad a 3 odd liar riagl lus h Avrag Nodal Prssur formulaio o dal wih h icomrssibiliy cosrai usd i [35]. Th 6 odd isoaramric riagular lm rsd i [4], is ow usd i h commrcial sofwar AdvaEdg ad h 4 odd quadrilaral wih rducd igraio is usd i Dform. Thos sofwar ar h mos commo umrical ools usd i idusry i h umrical simulaio of mal cuig rocsss. Also, a umbr of diffr fii lms hav b dvlod o imrov h oor rformac of liar riagls ad rahdral udr icomrssibl ad arly icomrssibiliy codiios. Ths fii lms ca b classifid i four grous maily: ) Mixd Ehacd Elm, ) Prssur sabilizd fii lms, 3) Comosi rssur filds ad 4) Avrag Nodal Prssur/Dformaio Gradi. Th followig lis rs a summary abou h advaags ad h disadvaags of ach of h imrovd liar riagl ad rahdral, ad i cas i is availabl a rfrc which aly h imrovd riagl i h umrical simulaios of mal cuig rocsss..4.. Mixd hacd lms Ehacd Srai Tchiqu, ssially cosiss i augmig h sac of discr srais wih local fucios, which may o driv from admissibl dislacms. A suiabl choic of hs addiioal mods ca imrov h umrical rformac of low-ordr lms ad, mor imoraly, i ca graly allvia h wll-kow volumric lockig homo i h arlyicomrssibl rgim. Mos of h schms akig advaag of h Ehacd Srai chiqu hav b dsigd i cocio wih quadrilaral lms, bcaus liar dislacm fii lms richd wih a hacd srai locks. Isad, h mixd liar dislacm liar rssur fii lm richd wih hacd srais is abl o dal wih h icomrssibiliy cosrai. Liraur rviw [36], rmarks ha h sraighforward xsio of h Ehacd Srai aroach o larg dformaio roblms grally lads o usabl mhods, rrsig a disadvaag of Ehacd Srai Tchiqu. As h auhor kowldg, Mixd hacd fii lms hav o b alid i h umrical simulaios of mal cuig. 3

24 .4.. Prssur sabilizaio Th rssur fild i mixd liar dislacm liar rssur fii lms wh usd i h umrical simulaios of icomrssibl or arly icomrssibl marials rss uhysical oscillaios. Mahmaically, i mas ha qual ordr irolaio for dislacm ad rssur dos o saisfy Babuska-Brzzy codiio. I ordr o rmov hs udsirabl oscillaios, a liraur ovrviw shows four diffr sragis maily: characrisic basd sli (CBS)[37], Fii Calculus (FIC) [38], Orhogoal Subgrid Scals (OSS) [39-4] ad h Polyomial Prssur Projcio (PPP) [, 3] Th characrisic basd sli (CBS) was origially dvlod i h fild of fluid mchaics [37]. This mhod is basd o h iroducio of a arificial comrssibiliy io h mass cosrvaio quaio, i such a way ha h fial rsuls do o dd o h arificial comrssibiliy. Th ohr mai igrdi of CBS is h fracioal s mhod usd i h im igraio of momum balac. This fracioal s rooss a sli of momum quaio i wo quaios such ha is sum is qual o h balac of momum quaio. Th quaio sli is quival o sli h vlociy uda i a im s io dviaoric ad hydrosaic comos. I summary, CBS algorihm uss four mai ss: ) Comu h vlociy uda usig a xlici im igraio schm of h quaio of balac of momum; i his im igraio schms hydrosaic forcs ar o ak io accou; ) Usig h balac o mass, calculas h odal rssur ad 3) Usig h vlociy obaid i () ad h gradi of h rssur fild obaid a () uda h vlociy fild. 4) Giv h udad vlociy usig a xlici igraio schm g h valu of odal dislacms ad uda odal osiios. Afr his four ss, h x im s sar. O advaag of CBS algorihm is h ossibiliy o valua h rssur i a coml xlici way, bu a h sam im CBS allows o solv h rssur usig a imlici schm, i cas i is dd. As h auhor kowldg, hr is o a coml CBS imlici schm for vlociy ad rssur, so dos o mar if h rssur is igrad imlicily CBS algorihm is codiioally sabl, ad as a cosquc hr is a rsricio i h maximum allowd im ss. Th Fii Calculus (FIC) [38] mhod is h basd o h saisfacio of h balac of momum ad mass cosrvaio i a domai of fii siz ad raiig highr ordr rms i h Taylor xasios usd o xrss h diffr rms of h diffrial quaios ovr h balac domai. Th modifid diffrial quaios coai addiioal rms which mai fucio is o surss h uhysical oscillaios of h rssur fild. Th mixd liar dislacm/liar rssur rahdral could b usd wih imlici, xlici ad smi-imlici im igraio schms. This rrss a advaag i comariso wih ohr modifid rahdral fii lms. Also, FIC mhod ds 5 dgrs of frdom r od ( dislacm, rssur, rojcd rssur gradis) i cas of liar riagl ad 7 (3 dislacm, rssur, 3 4

25 rojcd rssur gradis) i cas of rahdral. Th umbr of dgrs of frdom r od rrss a disadvaag of FIC, i rms of comuig im, rquird mmory ad a xra odal variabl o rasfr bw rmshigs. Aohr disadvaag of FIC is ha, h rm addd o h mass cosrvaio quaio dds o h shar modulus, h msh siz ad a cosa ha is roblm dd, Orhogoal Subgrid scals (OSS) was alid i h fild of solid mchaics i [39-4]. Orhogoal Subgrid scals aroach is basd o wo mai igrdis: ) a mixd qual ordr irolaio of h rssur ad dislacm filds ad ) a dcomosiio of h ukows io rsolvabl ad sub grid scals orhogoal o h fii lm sac. Th ida bhid Orhogoal Subgrid scals is o aroxima h ffc of h coiuous soluio which cao b caurd by h fii lm soluio which is h caus of volumric lockig. Th mai uros of Orhogoal Subgrid Scals is o dfi a sragy o ovrcom h rquirms of Babuska-Brssi codiios ad i cosquc mak ossibl h us of qual ordr coiuous irolaio for dislacm ad rssur. As a cosquc of addig a subgrid scal dislacm, xra dgrs of frdom ar addd o a od. Furhrmor, a rm ha dds o msh siz, shar modulus, ad a cosa ha is roblm dd is addd o h mass cosrvaio quaio. Th, subgrid scals ds 5 dgrs of frdom r od ( dislacm, rssur, rojcd rssur gradis) i cas of liar riagl ad 7 (3 dislacm, rssur, 3 rojcd rssur gradis) i cas of rahdral.. I h framwork of imlici dyamics, a saggrd schm i which h dislacm ad rssur ar solvd imlicily, whil h rssur gradi is solvd xlicily is usually usd as roosd i [4]. As a cosquc h rssur gradis dgrs of frdom addd rrs a mior cos i rms of comuig im. I is imora o rmark ha h rms addd o h balac quaios usig FIC ad subgrid scals o ovrcom Babuska-Brzzi codiios ar xacly h sam. Big h diffrc bw FIC ad subgrid scals h ida o g h rms ha sabiliz h fii lm soluio. Polyomial Prssur Projcio (PPP) was iiially formulad ad alid o sabiliz soks quaios i [, 3]. PPP is basd o wo igrdis. Firs, us a mixd qual ordr h rssur ad vlociy filds ad scod, ad L rssur rojcio. FIC ad OSS iroduc h rojcio of h rssur gradi oo h dislacm sac as a w dd variabl, ad us h diffrc bw hs wo filds o rlax h coiuiy quaio, whil PPP uss a rojcio o a discoiuous sac ad as a cosquc ca b imlmd i a lmary lvl. FIC ad OSS us msh dd ad roblms dd aramrs whil PPP dos o d. PPP has o b alid i h umrical simulaios of mal cuig rocsss. Mor dail abou h 5

26 xsio of PPP o solid mchaics roblms will b rsd i h x char Comosi rssur filds Ths fii lms forc a cosa rssur fild ovr a grou of riagls or rahdrals o rduc rssur cosrais. Th mos rrsaiv fii lms ar F-Bar[43, 44] ad Comosi Triagls [45, 46]. Th F-bar is aohr sragy o dal wih h icomrssibiliy cosrais usig liar riagls. F-bar formulaio was roosd i [44] i h framwork of imlici dyamics ad i [43] i h framwork of xlici dyamics. I rlis ssially o h rlaxaio of h xcssiv volumric cosrai yical of low ordr lms hrough h forcm of h icomrssibiliy cosrai ovr a ach of simlx lms. A imora asc of h rs mhod is ha i rsrvs h dislacm-basd forma of h corrsodig fii lm quaios. A h sam im, his mhod rss a ucovioal siffss forma ha sm from h fac ha h iral forc vcor of a aricular lm dds o h odal dislacms of all lms of is ach, brakig h yical lmary assmbly of h iral forc vcor ad h siffss marix. A riagular lm i which a six-od riagl is cosrucd from four 3- od riagls wih liar dislacm filds i ach subriagl ad a coiuous liar srai fild ovr h assmblag is rsd i [45]. Thy hav calld his fii lm a comosi riagls (CT). This lm rss som advaags i rms of coac sarch ad imosiio i comariso wih 6-od-riagl ad furhrmor his lm is lockig fr i comariso wih hr od riagl. Furhrmor, his lm dos o saisfy BB codiio. A imrovd CT riagl i ordr o saisfy BB codiios usig a cosa volumric srai ad a liar dviaoric srai ovr h six-od fii lm is rsd i [46] Avrag Nodal srais / Avrag Nodal srsss Comus h avrag volumric srai/ volumric srss or srais/srsss a ods basd o surroudig riagls or rahdrals. Th, h lmary volumric srai/ volumric srss or srais/srsss ar qual o h avrag of h odal valus ha blogs o h lm. Avrag Nodal Prssur (ANP) was rsd i [47] ad [48] i h framwork of xlici dyamics ad by [49] i h framwork of imlici dyamics. ANP is a siml liar rahdro lm ha ca b usd i alicaios ivolvig arly icomrssibl marials or icomrssibl marials modlld usig a 6

27 aly formulaio. Th lm rvs volumric lockig by dfiig odal volums ad valuaig avrag odal rssurs i rms of hs volums. Avrag Nodal Prssur (ANP) dfis h odal volum as follows: a () V V (.7) dim a () Whr V is h lm volum, dimis h dimsio of h roblm o a solv ad V is h odal volum. This dfiiio of odal volum, allows o dfi a odal volum chag raio ad as a rsul a odal rssur. Th avrag of odal rssurs is usd as h modifid lmary rssur. This dfiiio of odal volum rducs h volumric lockig dcy of liar riagls ad rahdral ad allows a accura rdicio of dformd shas ad forcs. Bu, ANP formulaio is foud o roduc cosidrabl chckrboard-y hydrosaic rssur flucuaios, which limis h rag of alicabiliy of ANP rssur formulaios. I is imora o rmark ha ANP work wll for volumric lockig bu rs lockig du o bdig.[5]. ANP has b alid i h umrical simulaio of high sd cuig i [5]. Th criical im s of ANP formulaios imosd by sabiliy is mor or lss 7 ims grar i comariso wih CBS formulaios [48]. I rms of comuig im i rrss a gra advaag of ANP formulaios. Nod Basd Uiform Srai Elms (NBUSE) [5] ad Avrag Nodal Dformaio Gradi (ANDG) [53] rs a formulaio usig liar riagl ad rahdral ha is fr of volumric ad shar lockig. Ths formulaios ar basd o h dfiiio of h odal dislacm gradi (NBUSE) ad Avrag odal dformaio gradi (ANDG). Ths avrag gradis ar dfid as follows: V ( ) ( ) a a u (.8) () a V V u ( ) ( ) a a F (.9) () a () () Whr V is h lm volum, u is h lm dislacm gradi, a () u is h odal dislacm gradi, F lm dformaio gradi ad a a F is h odal dformaio gradi. Th usig u / F a ad h cosiuiv quaio, odal srsss ar calculad ad fially, modifid lmary srsss ar calculad as h avrag of odal srsss. As rord i [5] ad [53] hs formulaio ca lad o h rsc of o-hysical low-rgy mods. A imrovd odal dformaio gradi basd o Sramli uwid Prov- V F 7

28 Galrki (SUPG) which rmov h o-hysical rgy mods was roosd i [53]. A sabilizaio sragy o rmov h uhysical rgy mods i h framworks of imlici dyamics was roosd i [5], v so, h rssur fild i som xamls rss uhysical oscillaios. Th gra advaag of avrag odal formulaios is ha ar lockig fr wihou icrasig h umbr of dgrs of frdom r od. NBUSE has b alid i h umrical simulaio of mal cuig rocsss i [54]. A Mixd Discrizaio Tchiqu (MD) [55] which is basd o h followig igrdis. ) Msh h solid body i quadrilaral or hxahdral zos; h divid ach quadrilaral or hdrahdral io riagls or rahdros. ) Th dviaoric bhavior is dfid o a lmary basis (riagl/rahdral), whil h volumric is avragd ovr a zo (quadrilaral/hxahdral). Th, a imrovm of (MD) is rsd i [56], h auhors of ha work call hir formulaio Nodal Mixd discrizaio (NMD). Th mai advaag of (NMD) is ha h avrag of h volumric bhavior is carrid ou i a odal basis rahr ha i a zo basis (quadrilaral/hxahdral), i imlis ha oly a msh of riagls or rahdral is dd. NMD uss a odal volumric srai ras, dfid as wighd avrag of h lmary surroudig valus. Th, a modifid lmary volumric srai ra is dfid as h avrag of h odal valus. Th diffrc bw NMD ad ANP is ha i NMD h cosiuiv modl is calld o a lm basis, whil i ANP h cosiuiv modl is calld i a odal basis for h volumric bhavior. Du o h similariis bw ANP ad NMD, is xcd ha NMD rss chckrboard-y hydrosaic rssur flucuaios, big a disadvaag of boh formulaios. I cas h hydrosaic ar of h srss sor dds liarly o h drmia of h dformaio gradi, ANP ad NMD ar xacly h sam..4.3 Tim igraio schms (imlici, xlici, smi-xlici) Th Fii Elm Mhod allows diffr im discrizaio schms. Th mos commo ar h imlici ad xlici im igraio schms. Each of hm has is advaags or disadvaags. Th imlici schm is ucodiioally sabl; i mas ha hr is o rsricio o h im s usd i h umrical simulaio. I imlici formulaios, mchaical roblm ca b solvd i a saic o dyamic way. Furhrmor, imlici formulaios ca b usd wih sadard ad mixd (dislacm/rssur) fii lms. Howvr, imlici schms ds h soluio of a liar sysm crai umbr of ims ach im s. Usually, h soluio of h liar sysm rrs mos of h comuig im. Imlmaio of a w cosiuiv quaio is a log ask, du o h rquirm o imlm h algorihmic cosiuiv sor. Morovr, i som cass a imlici schm dos o covrg, du o h high oliariis ivolvd i h roblm. 8

29 Fially, coac codiios dcras h siz of h im s usd i h umrical simulaio ad as a cosquc icras h comuig im of a imlici schm. Exlici formulaio solvs h mchaical roblm i a dyamical way. Th soluio of ach im s i a xlici schm is siml ad comuaioally ffici, rovidd us a lumd mass marix i h simulaio. Exlici schm dos o d h soluio of a liar sysm; his oic is a advaag if h umrical soluio is do usig aralll comuig. Imlmaio of a w cosiuiv quaio is a asy ask; i allows imlm siml or comlx cosiuiv quaio wihou a big ffor. Exlici schm ar codiioally sabl, i mas ha h im s usd i h simulaios should b lss or qual ha a giv criical im s, h criical im s corrsod o h im ha ak o a wav o ravl hrough h small fii lm of h msh. I cas of a lasic marial, h criical im s dds o h msh siz, lasic modulus, Poisso raio, dsiy of h marial ad a cosa ha dds o h fii lm usd. x x c (.) 3 G Th rsricio imosd o h im s by h xlici schms, allows cocludig ha for umrical simulaio which ivolvs log riod of comuig im or low sds, imlici schms ar mor favorabl i comariso wih xlici schms. O h corary, wh vlociis ar high ad h coac codiios ar comlx, is cssary o dcras h im s usd i imlici formulaios, so i his cas xlici formulaios aar as a ffciv ad a ffici ool, wih a rally irsig comuig im. O xaml i cuig mchaics i which xlici schm ar mor ffici ha imlici schm is high sd cuig. Now, h qusio is: A wha cuig sd, xlici schms ar mor comuaioally ffici ha imlici schms? I is imora o mio ha i h liraur hr is o a comariso bw xlici ad imlici im igraio schms, which shows udr wha codiio o schm is br ha h ohr. I h liraur, imlici schms hav b usd i [3, 6, 8, 57] ad xlici schms i [4, 5] ad [5]. Also, hr ar som mixd schms i which h hydrosaic ar of h balac of momum is igrad imlicily ad h dviaoric ar is igrad xlicily. Som xamls of mixd im igraio schms ar: Characrisic Basd Sli [58] ad Fii Calculus [38]. Ths sragis hav o b alid i h umrical simulaios of mal cuig. Th commrcial sofwar AdvaEdg uss a xlici im igraio schms, whil h sofwar Dform uss a imlici im igraio schm. 9

30 .4.4 Coac algorihms Modlig h comlx hrmo-mchaical homa ha aks lac a h ool chi irfac is of aramou imorac, bcaus umrical rsuls lik fd forc ad coac dds srogly o a accura modlig of h hrmomchaical coac a h ool-chi irfac. For ha raso, i h umrical simulaio of mal cuig rocsss is cssary o us accura, robus ad comuaioal ffici coac algorihms. Coac roblms usig Fii Elms imlis wo basic roblms: Firs, h way i which h coac cosrai is imosd ad scod, h coac dcio sragy. Coac cosrais ar imosd usig h aly aroach [59-65] ad h Lagragia mulilirs [6, 64], A mixd aly-lagragia formulaio which uss h advaags of aly ad lagragia formulaios is rsd i [66]. Svral coac formulaios dvlod u o ow, forc h coac cosrais a scific collocaio ois (coac dcio sragy). Th mos commo sragy is h od-o-sgm aroach dvlod by Hallquis al. [6] Is mai ida is ha a od locad o h slav surfac mus o ra h oosig masr sid sgm. This aroach ca b alid i a sigl ad wo ass algorihm. I a sigl ass algorihm oly ods o h slav sid ar chckd agais raio io h masr sgm, ad h ods o h masr sid ar fr o ra h slav sgms, whil i h wo ass algorihm, h ods o h slav surfac ar chckd agais raio io h masr sgm ad h ods o h masr surfac ar chckd agais raio io h slav sgm. Boh sarchig sragis hav disadvaags bcaus o ass algorihm allows raio of masr ods io slav sgms ad do o ass h ach s ad h wo ass is ro o lock du o ovrcosraiig of h dislacms o h coac surfac, bu i ass h ach s. Th od o sgm aroach has b xdd o hrmomchaical coac by Wriggrs, Zavaris, ad Mih i [67-69] Rcly, a coac algorihm sragy basd o followig igrdis [7, 7]: Th coiuiy cosrais imosiio alog h ir coulig boudary i a wak igral ss ad h us of sgm-o-sgm discrizaio sragis basd o h so-calld morar mhod was rsd. I coras o classical od-o-sgm formulaios, i h sgm o sgm discrizaio h coac cosrais ar o imosd oiwis a a fii umbr of slav ods bu ar dfid alog h ir coac boudary ad hrfor a mor coml coulig bw h dgrs of frdom of h coac surfacs is obaid. Th xsio of h morar mhod o hrmo-mchaical dyamic coac roblms icludig fricioal haig ad hrmal sofig ffcs a h coac irfac was rs by Hübr ad Wohlmuh i [7] Olivr al. [73, 74] roos h Coac Domai Mhod (CDM). I his mhod, coac cosrais ar forcd usig a sabilizd Lagrag mulilir formulaio basd o a irior aly mhod (his allows h codsaio of h iroducd Lagrag mulilirs ad lads o a urly dislacm driv roblm) ad coac dcio sragy is do wih a ficiv irmdia 3

31 rgio cocig h oial coac surfacs of h dformabl bodis (his ficiv irmdia rgio is buil usig Dlauay riagulaio). Olivr idifis h followig advaags of CDM i comariso wih ohr coac algorihms: ) Th soluio dos o dd o h choic of slav ad masr sids, as h coac airig is uiquly dfid via a cosrai Dlauay riagulaio ) h rformac of h coac domai mhod (CDM) is surior o classical od-o-sgm formulaios ad comarabl o morar basd coac algorihms. U o ow, h Coac Domai Mhod (CDM) has o b alid i h simulaios of hrmo-mchaicals roblms. Th summary rs abov abou coac imosiio ad coac sarch is focusd o imlici dyamics. I cas of xlici dyamics scial rocdurs hav b dvlod [6, 75], such as h momum rlad chiqus i which modificaios ar mad o acclraio, vlociis ad dislacms. Th mai goal of hs modificaios is o avoid h alizig ffc of h im s iroducd by h high siffss, associad wih aly aroachs. Prcisly, his sragy has b usd i h umrical simulaio of mal cuig rocsss by Marusich ad Oriz [4]. Brucho al. [76], roos h us h mric roris of h disac (or ga) fucio bw wo bodis i h formulaio of coac roblms. I his formulaio, h vcors ormal o h coac surfacs ar dfid hrough h gradi of his disac fucio. This coac sragy ca b alid wih i xlici ad imlici framworks. Th coac sragy rsd by Brucho has b alid i h umrical simulaio of high sd mal cuig rocsss by D Michli ad Mocli i [5]. Skho ad Cho [3] rs a vry siml coac sragy alid o h umrical simulaio of mal cuig rocsss usig a flow formulaio, such ha if a boudary od is foud li isid h ool h lgh of im s is dcrasd i such a way ha h od would li o h ool fac. Th, if h od a h sar of h im s is i a comrssiv srss sa, h h od is rsricd o mov alog h ool fac. A ovrviw abou h coac algorihms availabl i h liraur, shows ha coac modlig usig fii lms is o a siml a ask, a mor rsarch is d i ohr o dvlo a robus, ffici ad gral coac algorihm. Howvr, h bs algorihm abl o rdic h coac homa a h ool chi irfac ca b chos basd o h followig aramrs: ) comuaioally ffici, ) xac saisfacio of coac cosrais, 3) dcras marix ill-codiioig ad 4) o xra dgrs of frdom du o coac cosrais. Th s of aramrs rsd abov, will allows us i h x char o choos h mos aroria coac algorihm wih is rsciv imrovms o modl h coac a h ool chi irfac..4.5 Adaiv rmshig, Error simaors, Trasfr oraors I h umrical simulaios of mal cuig rocsss larg dformaio, marial ad gomrical oliariis ar rs. Du o his raso, msh 3

32 dgraio hrough h umrical simulaio is rs, h firs aroach o ackl his roblm was usig rdisord fii lm mshs [8, 34, 77], som of hs rfrcs mio h limiaios of rdisord mshs i h umrical simulaio of mal cuig rocsss. Aohr aroach o dal wih his roblm is adaiv rmshig. I his kid of umrical simulaios h magiud ad disribuio of rror simaors volv durig h icrmal soluio, showig ha is cssary o rfi h msh whr high gradis ar akig lac or d-rfi h msh whr rrors simaors ar small i ordr o rsrv a boudd comuaioal cos. Furhrmor, rmshig is usd o rsrv a adqua lm sha ad o rdic crackig i h umrical simulaio of srrad ad discoiuous chi formaio. Morovr, h boudary lms i coac wih h ool ar ro o disor ad irfr wih h ool, such a irfrc ca lad o losss of volum of h workic ad h udsirabl ffc of largig h ool radius, for ha raso coac boudary codiios is o imora rmsh idicaor. Grally, hr ss ar dd o rmsh h workic: ) Calcula rror simaors ad disorio mrics. If hs valus ar grar ha a giv olrac, go o. ) Cra a w fii lm msh 3) Daa rasfr from h old o w msh Rcly, svral rror simaors hav b roosd for laso-lasic roblms basd o mahmaical foudaios ad hysical cosidraios. Zikiwicz-Zhu [77, 78] rror simaor, calculas for ach od a imrovd srss ad dfis h rror as h diffrc bw his srss ad h o calculad by h sadard fii lm rocdur. Oriz ad Quigly [79] roos a adaaio sragy basd o qui-disribuig h variaio of h vlociy fild ovr h lms of h msh. Marusich ad Oriz [4] usd a adaaio cririo basd o h qui-disribuio of lasic owr, h lasic work criria has b alid i [4] o h umrical simulaio of mal cuig rocsss. L ad Bah [8] roos a oi wis idicaor for rror i srsss ad lasic srai icrms, basd o h diffrc bw smoohd (srss/lasic srai) ad h (srss/lasic srai) a gauss ois. Pric al. [8] rs a rror simaor basd o h rojcio ad smoohig of lasic owr ra ad h ra of fracur, as a cosquc, i is o oly abl o caur h rogrssio of lasic dformaio, bu also rovid rfid mshs a rgios of ossibl marial failur. This rror simaor has b alid i h umrical simulaio of high sd machiig i [5, 8]. Michli ad Mocli [5] rsd a adaiv rmshr could wih msh boxs usd o msh vry rcisly h ara whr adiabaic shar bads. Afr rror simaio ad disorio mrics ar calculad, h x s is msh graio. Srucurd ad usrucurd mshs ca b usd i h discrizaio of h domai. Also, i h liraur hr diffr rfim aroachs hav b roosd ad usd: h-vrsio, i which h dsiy of h fii lms is icrasd usig h sam irolaio ordr i h lms, 3

33 h -vrsio, i which h fii lm is fixd ad h irolaio ordr of h lms is icrasd; ad h h-vrsio, which is a hybrid of h wo aroachs. Marusich ad Oriz [4] rs a h rfim schm lus coiuous Dlauay riagulaio a h ods i hir saial osiio. Skho ad Cho [3] cra a comlly w fii lm msh basd o a Dlauay-Vorooi y algorihm ach im o or mor lms of h msh hav go ovrly disord. Pric al. [8] rs a rmshig schm usig quadrilarals (cosrucd usig Dlauay riagulaio). Th rmshig schms rsd abov wr alid o h umrical simulaio of mal cuig i [3-5]. Afr craig a w msh, h rasfr of dislacm, mraurs, rssur as odal variabls ad hisory-dd variabls as Gauss oi variabls from h old msh o a w msh is rquird. Th mai goal of a rasfr oraor is h miimizaio of h umrical diffusio of h sa variabls. Oriz ad Quigly [79], roosd a rasfr oraor basd o h wak form of h quilibrium quaios i cojucio wih h irolaio of odal variabls ad aly i i h cox of srai localizaio roblms. Tha rasfr oraor has b alid i h umrical simulaio of mal cuig by Marusich ad Oriz [4]. L ad Bah [8], Pric al. [8, 8] rs a adaiv msh sragy for larg dformaio roblms, which uss a diffr rasfr oraor for odal variabls ad gauss oi variabls. Th odal variabls us h ivrs isoaramric maig. Th gauss oi variabls ar smoohd o h ods of h old msh followd by rasfr o h ods of h w msh ad, fially irolad o h Gauss ois. This rror simaor has b alid i h umrical simulaio of high sd machiig i [5, 8]..4.6 Workic-chi saraio criria Marial saraio is a comlx homo ivolvig may hysical rocsss occurrig a h micromchaical lvl. Fracur bgis a h micromchaical scal ad vually macroscoic fracur is obsrvd. As a cosquc, h chi saraio is o of h sumblig blocks i h umrical simulaio of mal cuig rocss. Th mhodologis usd i h liraur o modl chi saraio ca b classifid as follows: coiuous chi saraio alog a rdfid cuig la or chi saraio usig lm dlig or killig lms basd o som lm dlig idicaor, ad h las oio is basd o larg lasic dformaios ad coiuous rmshig. Thr ar basically wo ys of idicaors: hos basd o gomrical ad hysical cosidraios. Som xamls of h mos commo idicaors usd i h umrical simulaios wih FEM ar lisd blow: (a) a chi saraio cririo basd o h disac bw h ool i ad h ars od alog a rdfid cuig dircio, (b) cosa quival srai cririo, (c) maximum shar srss cririo, (d) Johso Cook fracur modl ad (d) Cockrof Laham cririo. 33

34 Th quival srai cririo has b a oular failur cririo for mal cuig simulaios [6]. I his aroach fracur is assumd o occur wh lasic srai calculad a h ars od o h cuig dg, rachs a criical valu. Th drawback of his mhod is, if ucorolld, od saraio roagas fasr ha h cuig sd, as a rsul formig a larg crack ahad h ool i. Similarly a criical srss cririo has also b suggsd whr od saraio is acivad oc h marial rachs a criical srss valu [83]. Th Johso Cook failur cririo is basd o h osulaio ha h criical quival fracur srai is a fucio of srss riaxialiy, srai ra ad mraur. Th Johso Cook fracur modl is smi-mirical i aur ad cssias h drmiaio of cosas from sil ss wih high riaxialiy, shar ss ad Hokiso bar orsio ss a varyig mraurs ad srai ras. Jcohso-Cook fracur modl has b usd o modl machiig of iaium alloy (Ti-6Al-4V) i [84]. Aohr fracur modl imlmd i machiig simulaio is h Cockcrof Laham fracur cririo [85, 86]. Adaiv rmshig has b usd i [3-5, 5] as a sragy o sara h chi form h workic..4.7 Chi sgmaio ad brakag Th rocss of sgmd ad discoiuous chi formaio ivolvs h roagaio of fracurs hrough h dformig chi. Chi sgmaio by shar localizaio is a imora characrisic which ca b obsrvd i a crai rag of cuig sds wh machiig marials lik iaium. Exrimal sudis hav show ha shar localizaio i machiig iaium alloys is du o h occurrc of hrmo-lasic isabiliy ad ducil fracur. Isad, srucural sls ca fracur i a ducil or a bril mar ddig o cuig codiios. Marusich ad Oriz [4] rs a bril fracur criria formulad i rms of h oughss, K IC, usd i cojucio wih a muli-fracurig algorihm ad a ducil fracur xrssd i rms of h fracur srai, drivd from Ric ad Tracy`s void growh cririo. Th bril or ducil fracur cririos ar usd ddig o h machiig codiios. Ow ad Vaz [5] rs a fracur criria basd o h quival lasic srai ad h ucould igraio of Lmair`s damag modl. Ths fracur criria ar abl o modl h marial failur (hrmal sofig/failur sofig) i roblms ivolvig adiabaic srais localizaio, whr high sd cuig is a rrsaiv xaml. Boroushaki al. [87] ad Umbrllo [85, 88] also icludd damag mchaics i h simulaio of crack roagaio basd o Lmair s modl. Umbrllo al. [89] is mloyd Brozzo`s cririo o rdic h ffc of hydrosaic srss o h chi sgmaio durig orhogoal cuig. 34

35 Cri al. [86] combid h Cockrof ad Laham cririo wih a cririo basd o h ffciv srss i ordr o oimiz h marial fracur i cuig oraios. Ch al. [84] us h Johso Cook fracur modl o simula srrad chi formaio i iaium alloy (Ti-6Al-4V) high sd machiig. 35

36 .5 Numrical simulaio of Mal Machiig (Mshlss Mhods): Sa of h Ar.5. Smooh Paricls Hydrodyamics (SPH) SPH simulaio of cuig Alumiium 66-T6 a diffr rak agls ad fds [3] SPH simulaio of cuig Alumiium 66-T6 [] Figur 6. Numrical simulaio of mal cuig rocsss usig SPH Th firs lm fr Lagragia chiqu alid o cuig rocss is SPH (Smoohd Paricl Hydrodyamics).Firs, Hisi al. [3] alid h SPH mhod for orhogoal cuig rocss simulaios. Th, Limido al. [] aly SPH o high sd umrical cuig of a Al66 Alumium alloy usig h commrcial sofwar Ls-Dya. Auhors rors ha SPH rsuls ar i good 36

37 agrm wih h xrimal obsrvaios ad umrical simulaios carrid ou wih AdvaEdg. Th mai advaag of SPH is ha i dos o d a fii lm msh o calcula drivaivs. Marial roris ad sa variabls ar availabl a a s of ois, calld aricls. This avoids svr roblms associad wih msh aglig ad disorio which usually occur i Fii Elm Lagragia formulaios ivolvig larg dformaio ad srai ras. I SPH h valu of a fucio, or is drivaiv ca b simad a ay aricl i basd i h s of aricls ha ar wihi a giv disac h from i aricl. O of h advaags of SPH is h aural workic-ool saraio; h workic mar flows aurally aroud h i ool. A addiioal advaag of SPH is coac hadlig, bcaus coac is modld as a aricl iracio ad fricio aramrs (lik Coulomb fricio aramr) do o hav o b dfid. Th mai disadvaag of SPH i comariso wih FEM is h ighbours sarch, bcaus udaig h daa bas of ighbor aricls aks usually a log im i comariso wih ohr calculaios dd durig ach im s. O xaml of h umrical simulaio of mal cuig rocsss usig SPH is show i Figur Fii Poi S Mhod (FPM) Uhlma al. [9, 9] aly h Fii Poi S Mhod (FPM) o modl cuig of Icol 78. FPM is a imlici schm which is basd o h diffrial form of h cosrvaio laws of mass, momum ad rgy. I dail, FPM is a gralizd Fii Diffrc schm basd o Movig Las Squar Mhod (MLS). Th mai advaags of FPM ar: ) Rmshig is avoidd by h msh fr aroach, ) Numrical losss du o rmshig dos o occur, 3) Bcaus Figur 7. Numrical simulaio of mal cuig rocsss usig Fii Poi S Mhod.[9] 37

38 FPM is a lagragia formulaio allows for a asy way o rrs fr ad dyamics boudaris. Som disadvaags of h FPM ar: ) Rlaivly high comuaioal cos i comariso o FEM, du o h high umbr of ighbours ha ach od has i FPM comard o FEM. I h rfrcd work, h auhors cosidr h ool as rigid, ad o ha rasfr bw h ool ad h ic. Morovr, auhors say ha FPM ds furhr dvlom o simula a mor ralisic chi formaio rocss. Thr ar som similariis bw SPH ad FPM, bcaus boh us a shr of ifluc o sudy aricls iracio, bu SPH uss for fild fucio aroximaio h krl aroximaio whil FPM uss Movig Las Squars. O xaml of h umrical simulaio of mal cuig rocsss usig Fii Poi S Mhod (FPM) is show i Figur Cosraid Naural Elm Mhod (CNEM) Figur 8. Numrical simulaio of mal cuig rocsss usig CNEM.[6] Illoul al. [6] alid CNEM o 3D umrical simulaio of orhogoal ad obliqu cuig of a Ti-6A-4V alloy. Th CNEM s sha fucios ar buil usig h cosraid Vorooi diagram. CNEM ivolvs hr mai ss. Firs, h cosraid Vorooi diagram is buil. Thus, for ach od, a Vorooi cll is gomrical dfid. Th, CNEM sha fucios ar comud. Fially, usig a variaioal formulaio, iral ad xral forcs, acclraio, vlociis ad dislacms ar calculad. I CNEM, sa variabls ar availabl o aricls, so hr is o umrical diffusio du o a uda of h Vorooi diagram. Furhrmor, CNEM dos o d chi-workic saraio criria. Th mai disadvaag of CNEM is h comur im d o calcula sha fucios. Aohr disadvaag is h cssiy o rmsh surfacs, du o surfacs folds, xcssiv logaios or whr ods bcom oo clos. Th umrical rsuls rsd by Illoul ad Lorog hav o b validad agais 38

39 xrims. Figur 8 shows a xaml of a umrical simulaio of mal cuig usig CNEM..5.4 Discr Elm Mhod (DEM) Flissr al. [4] alid DEM mhod o orhogoal cuig rocss simulaios. Th auhors rrsd h workic as a bulk of idical shrs arragd i a fac crd cubic laic. Paricls irac by a visco-lasolasic law glcig hrmal ffcs. Th umrical modl of mal cuig rsd i ha work was o validad agais xrimal rsuls. I coras o ohr mshlss mhods, which ar maily dsigd o solv arial diffrial quaios ha dscrib h hysical homa, DEM accous for h simulaios of aricl iracios. DEM dvlors rcommd his chology o roblms which ivolvs brakag, ruur ad larg dformaios, ad oghr wih coac of mulil bodis. Prcisly, h umrical igrdis dd for simulaio of orhogoal cuig. Howvr, a h sam im, DEM dvlors rcogiz ha FEM is surior o DEM for roblms whr small lasic srai ar of irs or for h ivsigaio o mod sha of srucural oscillaios. DEM ds o b furhr dvlod o rdic accura chi formaio. O xaml of h umrical simulaio of mal cuig rocsss usig Discr Elm Mhod (DEM) is show i Figur 9. Figur 9. Numrical simulaio of mal cuig rocsss usig DEM.[4] 39

40 Ebrhard ad Gaugl [9] show h alicabiliy of h DEM for modlig of a orhogoal cuig rocss of sl ad alumium. I ha work, aricls irac by a visco-laso-lasic law icludig hrmal ffcs. Th umrical rsuls ar xamid by comarig h simulad forcs acig o h ool wih xrimally obaid forcs. Rsuls rsd i [9] show ha usig a workic wih a rgular laic h cuig forc como ca b rroducd vry icly whras h assiv forc shows cosidrabl dviaio. Isad, modlig a cuig rocss wih a radom laic workic fails o rroduc basic qualiaiv characrisics of mal cuig. Furhrmor, Ebrhard ad Gaugl [9] idifis ha h mai challg of DEM is o fid aroria forc laws ad aramrs i ordr o syhsiz h solid wih corrc hysical roris whras h difficuly wih FEM is foud wih rgard o saraio of marial ad rmshig. Ebrhard ad Gaugl [9] suggs ha som igrdis of SPH ca b addd o DEM o ovrcom is drawbacks. Figur. Numrical simulaio of mal cuig rocsss usig Muli Marial Eulria Formulaio.[] 4

41 .6 Imrovd Eulria Formulaio: Sa of h Ar.6. Muli marial Eulria Mhod (MMEM) Eulria formulaios ca b imrovd usig a Muli-marial Eulria (MMEM) mhod [, 9] basd o fii lms. Th Muli-marial Eulria sragy is abl o dal wih larg dformaios ad fr surfac graio, which usually ak lac i h umrical simulaio of machiig rocsss. MMEM ovrcoms h mai disadvaag of sadard Eulria Formulaio. Tyically, h coac i MMEM is basd o mixurs hory. Th mai disadvaag of Muli marial Eulria formulaio is h coac sragy bw h chi ad h work ic, bcaus mixurs hory dos o rdic Figur. Numrical simulaio of mal cuig rocsss usig a Eulria Formulaio ad a Volum of Solid Aroach. [8] wll h fricio homa a h ool-chi irfac. Rcly, Viali al. [93] imrov MMEM usig X-FEM a h irfac whr wo marials com i coac, gig as a rsul a discoiuiy i h vlociy fild a irfacs ad as a cosquc rdicig wll h fricio homa. MMEM has b alid o h umrical simulaio of AISI 434 sl udr orhogoal cuig codiios i [], gig rsuls ha agr wll wih xrimal rsuls. Th umrical simulaio rsd i ha work usd a rigid ool; fricio is glcd ad dos o ak io accou ha rasfr bw h ool ad h workic. O xaml of h umrical simulaio of mal cuig rocsss usig h Muli Marial Elm Mhod is show i Figur. 4

42 .6. Volum of Solid (VOS) Al-Ahl al. [8] roos a w sragy o simula orhogoal cuig basd o a modifid Fii Elm Eulria formulaio ad a Volum of Solid (VOS) Aroach. Fii Elm Eulria VOS formulaio uss h advaags of a Eulria formulaio lus a umrical schm (VOS) ha ca rack h fr surfac of h marial ad modl h ucosraid flow of h chi wihou big limid o oly sady sa scarios wih a assumd sha for h dformd chi. Th rsuls from umrical simulaios showd good agrm i valus ad bhavior wih h os obaid from ALE ad xrims. Auhors rmarks ha Eulria VOS formulaio is oly abl o Figur. Numrical modlig of mal cuig rocsss usig Marial Poi Mhod (MPM).[5] rdic coiuous chi formaio ad cao hadl sgm chi. Th mai advaag of Eulria VOS formulaio is ha dos o rquir ay msh moio schm or rmshig sragy..6.3 Marial Poi Mhod (MPM) or Poi i Cll (PIC) Th marial oi mhod (MPM), iroducd iiially i fluid dyamics by [94] ad alid o orhogoal cuig simulaios by [95] ad [5]. I h marial oi mhod, sa variabls ar racd a a s of ois (marials ois) dfid iddly of a Eulria msh o which h quaios of moios ar formulad ad solvd. Mor i dail, h wak form is solvd o a backgroud msh a ach im s ad h comud acclraio is usd o 4

43 uda h aricl osiios. Lar h udad aricl daa is usd o risa h osiio ad coordias of h backgroud msh ods. As h msh is dfid i a arbirary way, h roblm of msh disorio, which lads o difficulis i Lagragia FEM, is avoidd. Aohr advaag of MPM is h asy way o solv fr surfac roblms. Th mai drawback of h marial oi mhod is rlad o h codiio of sabiliy for h rocdur of im igraio of dyamics quaios. Bcaus for a giv ir aricl disac quival o a crai msh siz, h criical im s usd i MPM is may ims smallr ha i cas of sadard FEM. Furhrmor, srai localizaio i MPM is vry ssiiv o h dsiy of marials ois usd i h calculaio. Also, MPM ds mor mmory i comariso wih sadard FEM, bcaus i is cssary o sav iformaio of h Eulria msh ad Lagragia aricls. Ambai ad co-workrs [5] comard MPM ad FEM i rms of h lasic srai fild ad mraur fild, fidig a good agrm bw h umrical simulaios. Also, h comariso shows ha MPM rovids a smoohr chi formaio ad lss srai localizaio ha FEM..7 Paricl Fii Elm Mhod (PFEM) Th iiial dvloms of h Paricl Fii Elm Mhod (PFEM) ook lac i h fild of fluid mchaics [96], bcaus PFEM facilias rackig ad modlig of fr surfacs. Lar o, h Paricl Fii Elm (PFEM) was alid i a variy of simulaio roblms: fluid srucur iracio wih rigid bodis, rosio rocsss, mixig rocsss, could hrmo-viscous rocsss ad hrmal diffusio roblms [97, 98]. Firs alicaios of PFEM o solid mchaics wr do i roblms ivolvig larg srais ad roaios, muli body coacs ad craio of w surfacs (rivig, owdr fillig ad machiig) [99]. I his work w xdd h Paricl Fii Elm Mhod o h umrical simulaio of mal cuig rocsss. I h PFEM, h coiuum mdium, cosidrd as a ifii ack of aricls ach of hm of ifiisimal siz, is rrsd by a fii s (or a cloud) of ifiisimal-sizd aricls. Th aricls of h cloud dscrib ad coai h roris of h coiuum mdium (dislacm, rssur, mraurs, srais, srsss, iral variabls, c.) a hir isaaous locaios, ad, wh cssary, h roris of h ohr aricls of h coiuum ar obaid by irolaio from hos i h cloud. Th PFEM is characrizd by wo basic igrdis: ) A Dlauay riagulaio is grad a ach im s cocig h aricls of h cloud o h basis of hir udad osiios (s []): his riagulaio is usd as h fii lm msh durig h im s, ad, wh cssary for comuaioal uross, h roris of h aricls (.g. a h ods of h msh) ar irolad o h Gauss ois. Du o h oimal roris of Dlauay riagulaios for miimizig agl disorios, his coiuous r-mshig miimizs h lm disorios makig h mhod suiabl ad advaagous, i rms of rliabiliy ad robusss, i fro of 43

44 mor classical fii lm mhods. Scifically h mhod bcoms suiabl for simulaio of hos idusrial roblms dislayig marial flow (cuig rocsss, graular marial flow, mal formig rocsss, c.). Figur 3. Skchd Paricl Fii Elm Mhod (PFEM) a im-s, ) Boudary rcogiio chiqus ca b aurally usd i h PFEM (.g. alha-sha chiqus). This facilias modlig hos comlx mchaical rocsss i which w boudaris, diffr from h iiial os, aar. 44

45 Figur 4 shows a umrical simulaio usig PFEM of coiuous chi formaio, bu also h mor comlx srrad chi formaio. Dails abou marial roris, ool vlociy, ad ool sha usd i h umrical simulaios will a) Coiuous chi formaio (mraur disribuio) (ha rasfr bw h ool ad h workic is glcd) b) Srrad chi formaio (mraur disribuio) c) Coiuous chi formaio (Vo Miss srss disribuio) d) Srrad chi formaio (Vo Miss srss disribuio) 4 cuig forc(n/mm) 5 5 cuig forc(n/mm) ool dislacm(mm) ) Coiuous chi formaio: rdicd cuig forc (sady) ool dislacm(mm) f) Srrad chi formaio: rdicd cuig forc (oscillaory) Figur 4. Numrical simulaio of coiuous ad srrad chi formaio usig PFEM chiqus b giv i h followig chars. 45

46 .8 Thsis Ouli Th rmaidr of his hsis is orgaizd as follows. Char xlais h umrical schm dvlod o modl mal cuig rocss usig PFEM. Dails ar giv abou h fii lm discrizaio of h govrig quaios usig mixd fii lms, h rmshig rocdur, h coac modlig a h ool chi irfac, h im igraio quaio of balac quaios ad igraio of h hrmo-lasolasic cosiuiv quaios. Som xamls ar rsd which validad h s of umrical ools dvlod i his hsis. I char 3 som xamls ar rsd which validad h s of umrical ools dvlod i his hsis, as h a s of umrical simulaios of mal cuig rocsss usig PFEM is rsd. This char rss a comariso of xlici, IMPLEX ad imlici im igraios schms i rms of comuig im a a giv cuig sd, a sudy which show a ddcy of chi sha o cuig sd for a iaium alloy (Ti6Al4V), a s of xamls which show h ifluc of h rak agl o chi sha, som xamls ar rsd which show h ifluc of ool siffss o cuig forcs ad chi sha ad fially, ad som xamls sudy h ifluc of fricio a h ool chi irfac. Char 4 rs a ssibiliy aalysis o gomric ad cuig codiios lik h ool vlociy, h ool radius, h rak agl, ad h udformd chi hickss usig PFEM by mas of a Dsig of Exrims (DoE) mhodology. Th, w comar h ssibiliy of rocss variabls lik cuig forcs, fd forcs, dformd chi hickss, ad coac lgh, c., o cuig codiios giv by PFEM wih h ssibiliy rdicd hrough (Abaqus, AdvaEdg, Dform) ad xrims. Also char 4 idifis h advaags ad drawbacks of PFEM ovr FEM ad mshlss sragis. Th las char summarizs h mai coclusios ad achivms of h hsis ad dscribs o lis of rsarch. 46

47 Rfrcs [] M. P. Groovr, Fudamals of Modr Maufacurig: Marials, Procsss, ad Sysms, 6. [] E. Tr ad P. Wrigh, Mal cuig, Fourh Ediio d.,. [3] M. Hisi ad D. Sgalma, "Simulaio of Orhogoal Cuig wih Smooh Paricls Hydrodyamics," Sadia Naioal Laboraoris997. [4] F. Flissr, T. Gaugl, ad P. Ebrhard, "Alicaios of h discr lm mhod i mchaical girig," Mulibody sysm dyamics vol. 8,. 8-94, 7. [5] R. Ambai, X.Pa, H. Yua, ad X. Zhag, "Alicaio of marial oi mhods for cuig rocss simulaios," Comuaioal Marials Scic, vol. 57,. -,. [6] L. Illoul ad P. Lorog, "O som ascs of h CNEM imlmaio i 3D i ordr o simula high sd machiig or sharig," Comur ad Srucurs, vol. 89, ,. [7] M. Vaz, D. R. J. Ow, V. Kalhori, M. Ludblad, ad L.-E. Lidgr, "Modllig ad Simulaio of Machiig Procsss," ARCHIVES OF COMPUTATIONAL METHODS IN ENGINEERING, vol. 4,. 73-4, 7. [8] K. S. Al-Ahl ad M. S. Gadala, "Th Us of Volum of Solid (VOS) i Simulaig Mal Cuig wih Chamfrd ad Blu Tools," Iraioal Joural of Mchaical Scics, vol. Vol. 53,. 3-3,. [9] E. Uhlma, R. Grsbrgr, M. Graf vo dr Schulburg, J. Kurr, ad A. Mas, "Th Fii Pois Mhod for h Mshfr Numrical Simulaio of Chi Formaio," rsd a h Cir Cofrc o Modllig of Machiig Oraios, Sa Sbasia, Sai, 9. [] D. J. Bso ad S. Okazawa, "Coac i a muli-marial Eulria fii lm formulaio," Comur Mhods i Alid Mchaics ad Egirig, vol. 93, , 4. [] J. Limido, C. Esiosa, M. Salaü, ad J. L. Lacom, "SPH mhod alid o high sd cuig modllig," Iraioal Joural of Mchaical Scics, vol. 49, , 7. [] C. R. Dohrma ad P. B. Bochv, "A sabilizd fii lm mhod for h Soks roblm basd o olyomial rssur rojcios," Iraioal Joural for Numrical Mhods i Fluids, vol. 46,. 83, 4. [3] P. B. Bochv, C. R. Dohrma, ad M. D. Guzburgr, "Sabilizaio of Low-Ordr Mixd Fii Elms for h Soks Equaios," SIAM Joural o Numrical Aalysis, vol. 44,. 8-, 8. 47

48 [4] L. Filic, F. Micari, S. Rizzui, ad D. Umbrllo, "A criical aalysis o h fricio modllig i orhogoal machiig," Iraioal Joural of Machi Tools ad Maufacur, vol. 47, , 7. [5] P. J. Arrazola, D. Ugar, ad X. Domíguz, "A w aroach for h fricio idificaio durig machiig hrough h us of fii lm modlig," Iraioal Joural of Machi Tools & Maufacur vol. 48, , 8. [6] P. J. Arrazola ad T. r. Özl, "Ivsigaios o h ffcs of fricio modlig i fii lm simulaio of machiig," Iraioal Joural of Mchaical Scics, vol. 5,. 3 4,. [7] A. J. Haglud, H. A. Kishawy, ad R. J. Rogrs, "A xloraio of fricio modls for h chi ool irfac usig a Arbirary Lagragia Eulria fii lm modl," War, vol. 65, , 8. [8] F. P. Bowd ad D. Tabor, Th Fricio ad Lubricaio of Solids, 954. [9] J. F. Archard, "Elasic Dformaio ad h Laws of Fricio," Procdigs of h Royal Sociy of Lodo. Sris A. Mahmaical ad Physical Scics, vol. 43,. 9-5, Dcmbr 4, [] T.H. Childs, K. Makawa, T. Obikawa, ad Y. Yama, Mal Machiig: Thory ad Alicaios. Amsrdam,. [] M. H. Dirikolu, T. H. C. Childs, ad K. Makawa, "Fii lm simulaio of chi flow i mal machiig," Iraioal Joural of Mchaical Scics, vol. 43, ,. [] T. H. C. Childs, M. I. Mahdi, ad G. Barrow, "O h Srss Disribuio Bw h Chi ad Tool Durig Mal Turig," CIRP Aals - Maufacurig Tchology, vol. 38, , 989. [3] G. S. Skho ad J. L. Cho, "Numrical simulaio of coiuous chi formaio durig o-sady orhogoal cuig simulaio," Egirig Comuaios, vol., 993. [4] T. D. Marusich ad M. Oriz, "Modllig ad simulaio of high-sd machiig," Iraioal Joural for Numrical Mhods i Egirig, vol. 38, , 995. [5] D. R. J. Ow ad M. Vaz Jr, "Comuaioal chiqus alid o high-sd machiig udr adiabaic srai localizaio codiios," Comur Mhods i Alid Mchaics ad Egirig, vol. 7, , 999. [6] J. S. Srkowski ad J. T. Carroll, "A Fii Elm Modl of Orhogoal Mal Cuig," Joural of Egirig for IdusryTrasacios of h Asm vol. 7, , 985. [7] K. Komvooulos ad S. A. Erbck, " Fii Elm Modlig of Orhogoal Mal Cuig," Joural of Egirig for Idusry, ASME Tras, vol. 3, ,

49 [8] A. J. Shih, "Fii lm aalysis of h rak agl ffcs i orhogoal mal cuig," Iraioal Joural of Mchaical Scics, vol. 38,. -7, 995. [9] A. Raczy, M. Elmadagli, W. J. Alhof, ad A. T. Alas, "A ulria fii-lm modl for drmiaio of dformaio sa of a cor subjcd o orhogoal cuig," METALLURGICAL AND MATERIALS TRANSACTIONS A, vol. 35, , 4. [3] M. S. Gadalaa, M. R. Movahhdya, ad J. Wagb, "O h msh moio for ALE modlig of mal formig rocsss," Fii Elms i Aalysis ad Dsig, vol. 38, ,. [3] M. S. Gadala, "Rc rds i ALE formulaio ad is alicaios i solid mchaics," Comur Mhods i Alid Mchaics ad Egirig, vol. 93, , 4. [3] L. Olovsso, L. Nilsso, ad K. Simosso, "A ALE formulaio for h soluio of wo-dimsioal mal cuig roblms," Comurs & Srucurs, vol. 7, , 999. [33] R. Rakoomalala, P. Joyo, ad M. Tourair, "Arbirary Lagragia- Eulria hrmomchaical fii-lm modl of marial cuig," Commuicaios i Numrical Mhods i Egirig, vol. 9, , 993. [34] H. T. Y. Yag, M. Hisi, ad J. M. Shih, "Adaiv D fii lm simulaio of mal formig rocsss," Iraioal Joural for Numrical Mhods i Egirig, vol. 8, , 989. [35] P. O. D. Michli ad K. Moclli, "Exlici F.E. formulaio wih modifid liar rahdral lms alid o high sd formig rocsss," INTERNATIONAL JOURNAL OF MATERIAL FORMING, vol.,. 4-44, 8. [36] F. Auricchio, L. Birão da Viga, C. Lovadia, ad A. Rali, "A aalysis of som mixd-hacd fii lm for la liar lasiciy," Comur Mhods i Alid Mchaics ad Egirig, vol. 94, , 5. [37] A. J. Chori, "A Numrical Mhod for Solvig Icomrssibl Viscous Flow Problms," Joural of Comuaioal Physics, vol. 35,. 8-5, 997. [38] E. Oña, J. Rojk, R. L. Taylor, ad O. C. Zikiwicz, "Fii calculus formulaio for icomrssibl solids usig liar riagls ad rahdra," Iraioal Joural for Numrical Mhods i Egirig, vol. 59, , 4. [39] C. A. d. Saracibar, M. Chiumi, Q. Valvrd, ad M. Crvra, "O h orhogoal subgrid scal rssur sabilizaio of fii dformaio J lasiciy," Comur Mhods i Alid Mchaics ad Egirig, vol. 95,. 4 5, 6. [4] M. Chiumi, Q. Valvrd, C. A. d. Saracibar, ad M. Crvra, "A sabilizd formulaio for icomrssibl lasiciy usig liar 49

50 dislacm ad rssur irolaios," Comur Mhods i Alid Mchaics ad Egirig, vol. 9, ,. [4] M. Chiumi, Q. Valvrd, C. A. d. Saracibara, ad M. Crvraa, "A sabilizd formulaio for icomrssibl lasiciy usig liar riagls ad rahdra," Iraioal Joural of Plasiciy, vol., , 3. [4] M. Crvra, M. Chiumia, Q. Valvrdb, ad C. A. d. Saracibara, "Mixd liar/liar simlicial lms for icomrssibl lasiciy ad lasiciy," Comur Mhods i Alid Mchaics ad Egirig, vol. 9, , 3. [43] F. M. A. Pirs, E. A. d. S. No, ad D. R. J. Ow, "O h fii lm rdicio of damag growh ad fracur iiiaio i fiily dformig ducil marials," Comur Mhods i Alid Mchaics ad Egirig, vol. 93, , 4. [44] E. A. d. S. No, F. M. A. Pirs, ad D. R. J. Ow, "F-bar-basd liar riagls ad rahdra for fii srai aalysis of arly icomrssibl solids. Par I: formulaio ad bchmarkig," Iraioal Joural for Numrical Mhods i Egirig, vol. 6, , 5. [45] G. T. Camacho ad M. Oriz, "Comuaioal modllig of imac damag i bril marials," Iraioal Joural of Solids ad Srucurs, vol. 33, , 996. [46] Y. Guo, M. Oriz, T. Blyschko, ad E. A. Ro, "Triagular comosi fii lms," Iraioal Joural for Numrical Mhods i Egirig, vol. 47, ,. [47] J. Bo ad A. J. Buro, "A siml avrag odal rssur rahdral lm for icomrssibl ad arly icomrssibl dyamic xlici alicaios," Commuicaios i Numrical Mhods i Egirig, vol. 4, , 998. [48] J. Bo, H. Marrio, ad O. Hassa, "Sabiliy ad comariso of diffr liar rahdral formulaios for arly icomrssibl xlici dyamic alicaios," Iraioal Joural for Numrical Mhods i Egirig, vol. 5,. 9-33,. [49] F. M. Adrad Pirs, E. A. d Souza No, ad J. L. d la Cusa Padilla, "A assssm of h avrag odal volum formulaio for h aalysis of arly icomrssibl solids udr fii srais," Commuicaios i Numrical Mhods i Egirig, vol., , 4. [5] M. A. Puso ad J. Solbrg, "A sabilizd odally igrad rahdral," Iraioal Joural for Numrical Mhods i Egirig, vol. 67, , 6. [5] P. O. D. Michli ad K. Moclli, "D high sd machiig simulaios usig a w xlici formulaio wih liar riagular lms," Iraioal Joural of Machiig ad Machiabiliy of Marials, vol. 9,. 66-8,. 5

51 [5] C. R. Dohrma, M. W. Hisi, J. Jug, S. W. Ky, ad W. R. Wikowski, "Nod-basd uiform srai lms for hr-od riagular ad four-od rahdral mshs," Iraioal Joural for Numrical Mhods i Egirig, vol. 47, ,. [53] J. Bo, H. Marrio, ad O. Hassa, "A avragd odal dformaio gradi liar rahdral lm for larg srai xlici dyamic alicaios," Commuicaios i Numrical Mhods i Egirig, vol. 7, ,. [54] F. Grco, D. Umbrllo, S. D. Rzo, L. Filic, I. Alfaro, ad E. Cuo, "Alicaio of h odal igrad fii lm mhod o cuig: a rlimiary comariso wih h "radiioal" FEM aroach," Advacd Marials Rsarch,. 7-8,. [55] J. Mari ad P. Cudall, "Mixd discrizaio rocdur for accura modllig of lasic collas," Iraioal Joural for Numrical ad Aalyical Mhods i Gomchaics, vol. 6,. 9-39, 98. [56] C. Douray ad E. Dzik, "Nodal Mixd Discrizaio for rahdral lms," rsd a h 4h Iraioal FLAC Symosium o Numrical Modlig i Gomchaics, Miaolis, 6. [57] M. Bäkr, J. Röslr, ad C. Simrs, "A fii lm modl of high sd mal cuig wih adiabaic sharig," Comurs & Srucurs, vol. 8, ,. [58] J. Rojk, E. Oña, ad R. L. Taylor, "CBS-basd sabilizaio i xlici solid dyamics," Iraioal Joural for Numrical Mhods i Egirig, vol. 66, , 6. [59] A. Curir ad P. Alar, "A GENERALIZED NEWTON METHOD FOR CONTACT PROBLEMS WITH FRICTION," Joural D Mcaiqu Thoriqu E Aliqu, vol. 7,. 67-8, 988. [6] J. O. Hallquis, G. L. Goudrau, ad D. J. Bso, "Slidig irfacs wih coac-imac i larg-scal Lagragia comuaios," Comur Mhods i Alid Mchaics ad Egirig, vol. 5,. 7-37, 985. [6] J. H. Hgaard ad A. Curir, "A augmd Lagragia mhod for discr larg-sli coac roblms," Iraioal Joural for Numrical Mhods i Egirig, vol. 36, , 993. [6] R. Michalowski ad Z. Mroz, "Associad ad o-associad slidig ruls i coac fricio roblms," Archiwum Mchaiki Sosowaj, vol. 3, , 978. [63] D. Prić ad D. R. J. Ow, "Comuaioal modl for 3-D coac roblms wih fricio basd o h aly mhod," Iraioal Joural for Numrical Mhods i Egirig, vol. 35, , 99. [64] P. Wriggrs ad J. C. Simo, "A o o ag siffss for fully oliar coac roblms," Commuicaios i Alid Numrical Mhods, vol.,. 99-3,

52 [65] P. Paadooulos ad R. L. Taylor, "A mixd formulaio for h fii lm soluio of coac roblms," Comur Mhods i Alid Mchaics ad Egirig, vol. 94, , 99. [66] J. C. Simo ad T. A. Laurs, "A augmd lagragia ram of coac roblms ivolvig fricio," Comurs &am; Srucurs, vol. 4,. 97-6, 99. [67] P. Wriggrs ad G. Zavaris, "Thrmomchaical coac a rigorous bu siml umrical aroach," Comurs &am; Srucurs, vol. 46, , 993. [68] P. Wriggrs ad C. Mih, "Coac cosrais wihi could hrmomchaical aalysis A fii lm modl," Comur Mhods i Alid Mchaics ad Egirig, vol. 3,. 3-39, 994. [69] G. Zavaris, P. Wriggrs, ad B. A. Schrflr, "O augmd Lagragia algorihms for hrmomchaical coac roblms wih fricio," Iraioal Joural for Numrical Mhods i Egirig, vol. 38, , 995. [7] K. A. Fischr ad P. Wriggrs, "Friciolss D Coac formulaios for fii dformaios basd o h morar mhod," COMPUTATIONAL MECHANICS, vol. 36,. 6-44, 5. [7] M. Tur, F. J. Fumayor, ad P. Wriggrs, "A morar-basd fricioal coac formulaio for larg dformaios usig Lagrag mulilirs," Comur Mhods i Alid Mchaics ad Egirig, vol. 98, , 9. [7] S. Hübr ad B. I. Wohlmuh, "Thrmo-mchaical coac roblms o o-machig mshs," Comur Mhods i Alid Mchaics ad Egirig, vol. 98, , 9. [73] J. Olivr, S. Harma, J. C. Ca, R. Wylr, ad J. A. Hrádz, "A coac domai mhod for larg dformaio fricioal coac roblms. Par : Thorical basis," Comur Mhods i Alid Mchaics ad Egirig, vol. 98, , 9. [74] S. Harma, J. Olivr, R. Wylr, J. C. Ca, ad J. A. Hrádz, "A coac domai mhod for larg dformaio fricioal coac roblms. Par : Numrical ascs," Comur Mhods i Alid Mchaics ad Egirig, vol. 98, , 9. [75] T. Blyschko ad M. O. Nal, "Coac-imac by h iball algorihm wih aly ad Lagragia mhods," Iraioal Joural for Numrical Mhods i Egirig, vol. 3, , 99. [76] J. Brucho, H. Digo, ad T. Couz, "Usig a sigd disac fucio for h simulaio of mal formig rocsss: Formulaio of h coac codiio ad msh adaaio. From a Lagragia aroach o a Eulria aroach," Iraioal Joural for Numrical Mhods i Egirig, vol. 78,. 98-8, 9. 5

53 [77] K. Komvooulos ad S. A. Erbck, "Fii Elm Modlig of Orhogoal Mal Cuig," Joural of Egirig for Idusry, vol. 3, , 99. [78] O. C. Zikiwicz ad J. Z. Zhu, "Th surcovrg ach rcovry ad a osriori rror simas. Par : Error simas ad adaiviy," Iraioal Joural for Numrical Mhods i Egirig, vol. 33, , 99. [79] M. Oriz ad J. J. Quigly Iv, "Adaiv msh rfim i srai localizaio roblms," Comur Mhods i Alid Mchaics ad Egirig, vol. 9, , 99. [8] N.-S. L ad K.-J. Bah, "Error idicaors ad adaiv rmshig i larg dformaio fii lm aalysis," Fii Elms i Aalysis ad Dsig, vol. 6, , 994. [8] D. Prić, M. Vaz Jr, ad D. R. J. Ow, "O adaiv sragis for larg dformaios of laso-lasic solids a fii srais: comuaioal issus ad idusrial alicaios," Comur Mhods i Alid Mchaics ad Egirig, vol. 76,. 79-3, 999. [8] D. Prić, C. Hochard, M. Duko, ad D. R. J. Ow, "Trasfr oraors for volvig mshs i small srai laso-lasiciy," Comur Mhods i Alid Mchaics ad Egirig, vol. 37, , 996. [83] C. Sh ad X. Dg, "Fii lm aalysis of h orhogoal mal cuig rocss," Joural of Marials Procssig Tchology, vol. 5,. 95-9,. [84] G. Ch, C. R, X. Yag, X. Ji, ad T. Guo, "Fii lm simulaio of high-sd machiig of iaium alloy (Ti 6Al 4V) basd o ducil failur modl," THE INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY, vol. 56,. 7-38,. [85] D. Umbrllo, "Fii lm simulaio of covioal ad high sd machiig of Ti6Al4V alloy," Joural of Marials Procssig Tchology, vol. 96, , 8. [86] E. Cri, M. Lucchi, ad T. Ala, "FEM simulaio of orhogoal cuig: srrad chi formaio," Joural of Marials Procssig Tchology, vol. 95,. 7-6, 999. [87] H. Borouchaki, P. Laug, A. Chroua, ad K. Saaoui, "Adaiv rmshig i larg lasic srai wih damag," Iraioal Joural for Numrical Mhods i Egirig, vol. 63,. -36, 5. [88] D. Umbrllo, S. Rizzui, J. C. Ouiro, R. Shivuri, ad R. M'Saoubi, "Hardss-basd flow srss for umrical simulaio of hard machiig AISI H3 ool sl," Joural of Marials Procssig Tchology, vol. 99, , 8. [89] D. Umbrllo, J. Hua, ad R. Shivuri, "Hardss-basd flow srss ad fracur modls for umrical simulaio of hard machiig AISI 5 53

54 barig sl," Marials Scic ad Egirig: A, vol. 374,. 9-, 4. [9] E. Uhlma, R. Grsbrgr, ad J. Kuhr, "Cuig Simulaio wih h Mshfr Fii Pois Mhod," Procdia CIRP, vol. 8, , 3. [9] P. Ebrhard ad T. Gaugl, "Simulaio of cuig rocsss usig msh-fr Lagragia aricl mhods," COMPUTATIONAL MECHANICS,. -8,. [9] D. J. Bso, "A mixur hory for coac i muli-marial Eulria formulaios," Comur Mhods i Alid Mchaics ad Egirig, vol. 4, , 997. [93] E. Viali ad D. J. Bso, "Coac wih fricio i muli-marial arbirary Lagragia Eulria formulaios usig X-FEM," I. J. Numr. Mh. Eg vol. 76, , 8. [94] F. H. Harlow, M. A. Elliso, ad J. H. Rid, "Th aricl-i-cll comuig mhod i fluid dyamics," Mhods Comu. Phys, vol. 3, , 964. [95] Z. Więckowski, "Th marial oi mhod i larg srai girig roblms," Comur Mhods i Alid Mchaics ad Egirig, vol. 93, , 4. [96] S. R. Idlsoh, E. Oña, ad F. D. Pi, "Th aricl fii lm mhod: a owrful ool o solv icomrssibl flows wih frsurfacs ad brakig wavs," Iraioal Joural for Numrical Mhods i Egirig, vol. 6, , 4. [97] E. Oña, S. R. Idlsoh, M. A. Cligua, ad R. Rossi, "Advacs i h aricl fii lm mhod for h aalysis of fluid mulibody iracio ad bd rosio i fr surfac flows," Comur Mhods i Alid Mchaics ad Egirig, vol. 97, , 8. [98] E. Oña, M. A. Cligua, ad S. R. Idlsoh, "Modlig bd rosio i fr surfac flows by h aricl fii lm mhod," ACTA GEOTECHNICA, vol , 6. [99] J. Olivr, J. C. Ca, R. Wylr, C. Gozálz, ad J. Hradz, "Paricl Fii Elm Mhods i Solid Mchaics Problms," Comuaioal Mhods i Alid Scics, vol. 7,. 87-3, 7. [] N. Calvo, S. R. Idlsoh, ad E. Oña, "Th xdd Dlauay ssllaio," Egirig Comuaios: I J for Comur-Aidd Egirig, vol., , 3. 54

55 Char Th Paricl Fii Elm Mhod i h umrical simulaio of mal cuig rocsss Numrical simulaios of mal cuig rocss has rovd o b aricularly comlx du o h divrsiy of h hysical homa ivolvd, icludig ha coducio, coac wih fricio, dyamics ffcs ad hrmo-mchaical coulig. This char rss h mahmaical ad umrical igrdis cssary o simula mal cuig, icludig h balac of momum ad is fii lm discrizaio, h balac of rgy ad is fii lm discrizaio. I addiio, h cosiuiv quaios for h ool ad h workic ad is im discrizaio, h adaaiv ad rmshig schm usig h Paricl Fii Elm mhod (PFEM) alid o h workic, h coac roblm bw h ool ad h workic, ad h global fii diffrcs im igraio schm ar rsd. I h followig scio, w rs h coiuum formulaio of h coul hrmo-mchaical muli-body fricioal coac roblm.. Problm sam Figur 5. Modl of usady chi formaio Th modl of usady chi formaio o b sudid is show i Figur 5. Th ool is cosidrd rigid or dformabl, havig a giv rak agl, a flak agl ad ool radius R. Th ool is movd a a cosa sd i h gaiv x 55

56 dircio. Th workic is cosidrd isoroic ad iiially usrssd, havig a high H ad a lgh L. Dislacm cosrais i h x ad y dircio ar imosd o h lf d ad o h boom of h workic. Th cu dh is d. Body acclraio, ha graio hrough lasic work dissiaio ad fricio, ad ha rasfr a h ool chi irfac ar icludd i h aalysis.. Basic oaio L dim b h sac dimsio ad I :, T h im irval of irs. L ad wih smooh boudaris ad, b h rfrc lacm of wo coiuum bodis (h workic) ad (h ool), wih marial aricls labld X ad X. ( i) ( i) ( i) () Do by x ( X, ) h dformaio ma of h body i, wih ( ) ( ) ( ) ( ) ( ) dislacm of a marial aricl u i ( X i, ) i ( X i, ) X i, wih marial ( i) ( i) ( i) ( i) ( i) ( i) vlociy V : ( X, ), wih acclraio A : V ( X, ), ( i) ( i) ( i) () dformaio gradi F D ( X, ) ad absolu mraur ( i) ( i) Crai codiios mus b imosd o h ma ( X, ) i ordr for i o rrs h dformaio of a marial body. I aricular, w assum (i) ( i) ( i) ( X, ) is o o o (wo or mor marials ois cao simulaously occuy h sam saial osiio), ad (ii) d( F ) (dformaio should ( i) ( i) rs h oriaio of h body). For ach im I, ( X, ) rrs a o-aramr family cofiguraios idxd by im I, which mas h rfrc lacm of h body oo is curr lacm S ( i) ( i) ( ) ( i) ( i) ( i) ( i) I h curr cofiguraio, h vlociy v ( x, ) ad acclraio a ( x, ) of a aricl which assums a oi x a im is giv by ( ) ( ) ( ) ( ) ( ) ( ) ( ) v i ( x i, ) v i ( i ( X i, ), ) V i ( X i, ) (.) ( i) ( i) ( i) ( i) ( i) ( i) ( i) a ( x, ) a ( ( X, ), ) A ( X, ) ( i) ( i) i. (.) v ( x, ) ( i) ( i) ( i) ( i) grad v ( x, ) v ( x, ) Th firs rm is kow as h local drivaiv ad h scod rm is h covciv ar of h im drivaiv. W will assum ha o coac forcs bw h ool ad h workic ar rs a h rfrc cofiguraio. Th movm of h ool caus h wo bodis com i coac ad roduc iraciv forcs. 56

57 ( i) ( i ) W will do as h coac surfac h ar of h boudary of h body such ha all marial ois whr coac occurs a ay im I ar icludd. Th curr lacm of h coac surfac is giv by ( i) ( i) ( i) (, )..3 Th could Thrmo-mchaical IBVP wih fricioal Coac Cosrais W dscrib blow h sysm of arial diffrial quaios govrig h voluio of h hrmo-mchaical iiial boudary valu roblm. To dscrib h workic bhavior, w will ado cosiuiv quaios ha icorora fii srai laso-lasiciy ad h mulilicaiv dcomosiio of h dformaio gradi. To dscrib h ool bhavior, w will us a No-Hooka marial modl. Fricioal coac cosrai will b iroducd usig a alizd chiqu ad h Noro Hoff Cosiuiv Law..3. Balac quaios Th could hrmo-mchaical iiial boudary valu roblm is govrd by h momum ad rgy balac quaios, rsricd by h scod law of hrmodyamics. Th marial form of h local govrig quaios for h body ca b wri as I h abov quaios ( i) ( i) ( i) ( i) ( i) V DIV( P ) B ( i) ( i) ( i) ( i) ( i) ( i) E DIV( Q ) Di R V (.3) is h rfrc dsiy, B is h rscribd forcs r ui of rfrc volum, is h rfrc divrgc oraor DIV, P ad S ar h firs ad h scod Piola-Kirchhoff srss sors, rscivly. V is h vlociy fild, E h iral rgy, Q h omial ha flux, R is h rscribd rfrc ha sourc ad i D is h iral dissiaio r ui rfrc volum. I addiio, h roy N ad firs Piola-Kirchhoff srss sor P ar formulad i rms of h fr rgy ad subjcd o h dissiaio iqualiy of rfrrd o as h Clausius Plak form of h scod law of hrmodyamics. ( i) i P ( i) F ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) D : N E (.4) P : F N I (.4), h fr rgy fucio is obaid from h iral rgy via h Lgdr rasformaio ( i) ( i) ( i) E N (.5) 57

58 Th omial ha flux Q is dfid by Fourir s Law, subjcd o h rsricio o h dissiaio by coducio D co D GRAD( ) Q (.6) ( i) ( i) co Th saial form of h local govrig quaios for h body ca b wri as ( i) ( i) ( i) ( x, ) ( i) ( i) ( i) ( i) v div( ) ( i) ( i) ( i) ( i) ( i) div( q ) Di r v b (.7) I hs quaios, h moio ad h absolu mraur ar rgardd as h rimary variabls i h roblm whil b h body forc r ui of volum ad r h ha sourc ar rscribd daa. I addiio, h ha flux q, h roy as wll as h Cauchy srss sor ar dfid via cosiuiv quaios. Ths cosiuiv quaios ar subjcd o h followig rsricios o h iral dissiaio ad h dissiaio arisig from ha coducio ( i) ( i) ( i) ( i) ( i) i d ( i) ( i) grad( ) co D J : q (.8) D whr w hav usd h Lgdr rasformaio ˆ (.9).3. Boudary codiios Th basic govrig quaios (.3) or (.7) ad h cosiuiv rsricios (.4) -(.6) or (.8) ar sulmd by h sadard boudary codiios for h mchaical fild ( i) ( i) ( i) o ( i) ( i) ( i) ( i) ( i) P N o (.) whr, ar rscribd dformaio ad omial racios. Toghr wih h aalogous ssial ad aural boudary codiios for h hrmal fild, amly, ( i) ( i) o ( i) ( i) ( i) Q N Q o Q (.) whr ad Q ar rscribd mraur ad ormal ha flux mas. As usual, i is assumd ha h followig codiios hold 58

59 .3.3 Iiial codiios ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) Q ( i) ( i) ( i) ( i) ( i) ( i) Q Q (.) Addiioally, w assum ha h followig iiial daa is scifid for h mchaical ad hrmal filds ( i) ( i) ( i) ( X, ) ( X ) ( i) ( i) ( i) ( i) (, ) ( ) i ( i) ( i) X X V X V X (, ) ( ).3.4 Boudary codiios a h ool-chi irfac (.3) () () For ach marial oi X a ay im I, w rquir ha h coac forc ad ormal ha coducio flux iducd o h body (h ool) a h marial oi () X b qual ad oosi o ha roducd o body (h () workic) a X. Mahmaically, hs quilibrium codiios ak h form hc () () () () () () ( X, ) d ( X, ) d () () () () () () hc X hc X Q (, ) d Q (, ) d ( i) ( i) hc fric (.4) Q Q R (.5) Whr Q is h ha coducio flux a h coac irfac ad R is h ha sourc du o fricio a h ool workic irfac. Th ha sourcs du o coducio ad fricioal dissiaio a h ool workic irfac ar rlad hrough h rlaioshis (.6), whr Q is h ha coducio flux ha dds uo h coac mraurs ad h coac rssur, ad D is h fricioal dissiaio. fric () () () () () () () Q ( X, ) d Q ( X, ) d Qd () () () () () () () () fric( X, ) fric( X, ) fric( X, ) R d R d D d fric (.6).4 Global oraor sli for fii dformaio lasiciy Th IBVP (.7) ca b wri i a siml way. Suos ha ( i) ( i) ( i) ( i) ( i) ( i) ( i) Z v ad Z v (.7) ( i) ( i) 59

60 Th quaios ca b wri i a gralizd form as ( i) ( i) ( i) Z A( Z ) f (.8) Whr A is a oliar lliic oraor ad f a rscribd fucio. Th Cauchy srss sor, h ha flux vcor q, h oal ad h ( i), lasic rois, ad h mchaical dissiaio Di will b rgardd as dd variabls i h roblm, dfid i rms of h rimary variabls Z ad a s of iral srai-lik variabls. Th s of iral variabls ar dfid i rms of a cosraid roblm of voluio driv by h rimary variabls, wih h fucioal form ( i) ( i) ( i) ( i) ( i) (, Z ) (.9) whr is a addiioal variabl drmid by mas of h Kuh-Tuckr codiios, as follows ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) (, Z ) ad (, Z ) (.) ( i) ( i) ( i) ad (, Z ) is h Miss yild fucio Grally, h oliar oraor A ca b dcomosd i wo simlr oraors A ad A, whr A A A []. Th us of h addiiv oraor sli alid o h could sysm of oliar ordiary diffrial quaios lads o h followig wo siml roblms.4. Isohrmal lasolasic s ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) Z v div( (,, (, ))) b (.) v ( x, ) alog wih a s of firs ordr diffrial quaios ha dscrib h voluio of h iral variabls ( i) ( i) ( i) ( i) ( i) (, Z ) (.) whr h cosiscy aramr is h Lagrag mulilir saisfyig h Kuh-Tuckr codiios ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) (, Z ) ad (, Z ) (.3).4. Thrmolasic s a fixd cofiguraio Z ( i) ( i) v ( i) ( i) ( ) ( ) ( ) ( ) ( ) ( i) ( i) ( i) i i i i i -div( q (,, (, ))) D r i (.4) alog wih a s of firs ordr diffrial quaios ha dscrib h voluio of h iral variabls 6

61 ( i) ( i) ( i) ( i) ( i) (, Z ) (.5) whr h cosiscy aramr..is h Lagrag mulilir saisfyig h Kuh-Tuckr codiios ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) (, Z ) ad (, Z ) (.6) Th s of firs ordr diffrial quaios ha dscrib h voluio of h iral variabls ar rsd i h x scio. Th addiiv oraor sli oghr wih a roduc formula algorihm will b usd o solv h could sysm of oliar ordiary diffrials quaios..5 Cosiuiv modls.5. Tool cosiuiv modl Tools mus b srog ough ad rigid ough o rsis fracur ad o giv a miimum dflcio udr load. I rms of marials roris, h lasic modulus ad h yild srss mus b highr i h ool ha i h workic (h ool dos o dform lasically). Morovr, du o h high mraurs rs i mal cuig, h volumric chag i rsos o h chag i mraur is rs i h ool, ad should b cosidrd i h modl. A h sam im, w cosidrd h ool marial as isoroic, commo assumio mad i mal lasiciy. A No-Hooka marial [3, 4] wih h followig fr rgy fucio is usd o rrs h homology miod abov. l( J) ˆ( C ) l ( J ) r ( C ) 3-3 ( ) (.7) J whr, ad ca b irrd as h shar modulus, h bulk modulus, ad h hrmal xasio coffici, rscivly. C,C,J ar h righ Cauchy-Gr sor, h volum rsrvig righ Cauchy-Gr sor, h Jacobia of h dformaio gradi ad h mraur, rscivly. Th fr rgy fucio ˆ( C ) saisfy h wo followig imora roris: ˆ( C ) is ivaria wh h curr cofiguraio udrgos a rigid body roaio, bcaus ˆ( C ) oly dds o h srchig ar U C ad is idd of h roaio ar R of F (Objciviy). ˆ( C ) o ay raslad ad/or road rfrc cofiguraio is h sam a ay im (Isoroy). From quaio (.7) ad alyig, h sadard Colma-Noll rocdur lads o a cosiuiv quaio xrssd i rms of marials variabls (.8). 6

62 S ˆ C J l( J) J r( C) l( J) 6 ( ) J C J C C l( J) 3 C I C C (.8) l( J) 3 ( ) J r( ) J 3 or i rms of saial variabls(.9) T FSF J 5 (.9) l( J) l( J) 3 ( 3 ) i J dv( b) J J Th mai raso o us hyrlasiciy isad of small srai lasiciy is du o h larg dislacm of h ool durig h cuig rocss..5. Workic Cosiuiv Modl. Durig mal cuig, h workic udrgos larg dformaios, hrfor is cssary o iclud marial ad gomrical o-liariis i h cosiuiv modl. Exrimal rsuls hav show ha h workic srss-srai rlaioshi is affcd by h srai ra ad mraur durig lasic dformaio. For h sam valu of srai, (i) h srss is highr for highr srai ra ad (ii) h srss is lowr for highr mraurs. I his scio, w cosidr h formulaio of hrmo-lasiciy a fii srais wih isoroic hardig, followig h aroach roosd by Simo al. [, 4, 5]. Th srai ad srai ra hardig ad h hrmal sofig ar akig io accou wih h followig hardig laws: (i) Voc[6] ad Simo al. [7] (ii) Johso ad Cook [8] ad (iii) Bakr [9]..5.. J hrmo-lasiciy a fii srais Th formulaio of h cosiuiv quaios is basd o wo basic assumios: Th srss rsos is isoroic. Thrfor, h fr rgy is idd of h oriaio of h rfrc cofiguraio. Furhrmor, w assum ha h lasic flow is isochoric (sadard assumio i mal lasiciy). I mas d( F ) d( C ) J (.3) d( F) d( F ) J Wih hs wo assumios, w rocd o ouli h govrig quaios of h modl..5.. Mulilicaiv dcomosiio of h dformaio gradi 6

63 Th mai hyohsis o fii srai lasolasiciy is h mulilicaiv dcomosiio of h dformaio gradi, io lasic ad lasic ars (, ) F X F ( X, ) F ( X, ) (.3) This assumio admis h xisc of a local usrssd irmdia cofiguraio. Followig h sadard covios i coiuum mchaics rlaiv o h rfrc lacm of h body, h righ Cauchy-Gr sors ar dfid T C : F F (.3) Emaaig from dcomosiio (.3), h lasic ar of h Cauchy-Gr sor is dfid as T C : F F (.33) I h sam way, h oal ad h lasic Gr Lagrag srai sor ar dfid as ad E ( C ) (.34) ( E C ) (.35) whr do h symmric ui sor i h rfrc cofiguraio. Similarly, associad wih h curr cofiguraio ar h Eulria sors T b FF (.36) ad T b F F (.37) calld h oal ad lasic lf Cauchy-Gr sors, rscivly. Th ivrs of h lf Cauchy Gr is calld Figr dformaio sor : ( T T c b FF ) F F (.38) ad Th ulria srais sor aks h form T T c : b ( F F ) F F (.39) ad ( c ) (.4) ( c ) (.4) whr is h symmric ui sor i h curr cofiguraio As a cosquc h followig rlaioshis mrgs T T T T T T b F F FF F F F( F F ) F FC F (.4) To simlify our oaio, w us h sam symbol for h ui sor i boh h rfrc ad h curr cofiguraios. 63

64 .5..3 Th Li drivaiv for h sor b Th Li drivaiv for h sor b is dfid as T T L T v b F F b F F FC F (.43) Th Li drivaiv of b sor is xacly h ush forward of h im drivaiv of h ullback of h saial sor b. Mor iformaio abou ushforward ad ull-back oraios is giv i rfrcs [3, ] Dviaoric-Volumric dcomosiio of h dformaio gradi Th sli i oliar hory basd o h dformaio gradi aks h followig mulilicaiv form. L F do h volum rsrvig ar of h dformaio gradi. Thrfor d( F ). Also, rcall ha J : d( F ) givs h volum chag. Th s ad F J 3F d( F ) (.44) J 3 F F (.45) Associad wih F ad F w dfi h volumric rsrvig ar of h righ Cauchy-Gr sor as 3 3 C J C J F F (.46) ad h volumric rsrvig ar of h Lagragia srss sor is giv by h sadard xrssio ( ) E C (.47) Similarly, h volumric rsrvig ar of h lasic lf Cauchy-Gr sor is.5..5 Srss rsos. Hyrlasic rsos b J 3C J 3 b (.48) Cosis wih h assumio of isoroy, w characriz h srss rsos by a sord rgy of h form ˆ ˆ ˆ ˆ( ) (, ) ( ) ˆ ( ) ˆ T M J U J W b K(, ) (.49) Th lasic ar of h fr rgy is ucould io volumric/dviaoric rsos dscribd by h fucios UJ ˆ( ) ad Wˆ( b ), rscivly. Th fucio ˆ(, ) M J dscribs h hrmo-mchaical coulig du o hrmal xasio ad rovids h oial for h associad lasic srucural roy, 64

65 whil h fucio T ˆ( ) is h oial for h urly hrmal roy. Th fucio K ˆ(, ) is a oliar fucio of h quival lasic srai ad mraur which dscribs h isoroic srai hardig via h rlaio (, ) K. To mak mars as cocr as ossibl, w cosidr h followig xlici forms[, ] Uˆ( J ) l ( J ) Ŵ( b )= r( b ) 3 r( C ) 3 (.5), Tˆ( ) c ( ) l( ) l( J) ˆM(J, )=-3 ( ) J whr,, c ad ca b irrd as h shar modulus, h bulk modulus, h ha caaciy ad h hrmal xasio coffici, rscivly. Som rmarks ca b mad abou h srucur of h fr rgy fucio: (i) h srucur of h fr rgy is rsricd o mraur idd marial roris; (ii) h hrmolasic fr rgy is dcould from h lasic coribuio K'(, ) associad wih h hardig variabl (his assumio is moivad by h xrimal obsrvaio ha h laic srucur rmais uaffcd by h lasic dformaio)[]; (iii) Th fucios UJ ˆ( ) ad Wˆ( b ) graliz h liar isoroic lasic modl. (iv) Th fucio K ˆ(, ) rrss h visibl (macroscoic) lasic dformaios ha ar h rsul of microscoic dislocaio (crysallograhic dfcs i h crysal srucur) moio ad mulilicaio. Grally, h marial xhibis high srgh if hr ar ihr high lvls of dislocaios or o dislocaios. I addiio, h fucio K ˆ(, ) rrss h yild srss dcrasig as h grai siz is icrasd []. Also, ˆ(, ) K rrss h dcras i dislocaio dsiy du o h haig of h marial abov is criical mraur (hrmal sofig). Thr ar four mai srghig mchaisms for mals, ach is a mhod o rv dislocaio moio ad roagaio, or mak i rgically ufavorabl for h dislocaio o mov (work hardig, solid soluio srghig, rciiaio hardig ad grai boudary srghig). 65

66 I addiio, hr ar ohr facors ha affc h sha ad h magiud of h hardig oial amog hm [3]: (i) marial comosiio, (ii) rvious ha ram, (iii) h y of crysal srucur ad (iv) rior hisory of lasic dformaio. Diffr hardig oials ha rrs h work hardig homo hav b roosd i h liraur, which rflc som of h srai hardig ars obsrvd i h xrims. Amog hm h followig: Voc ad Simo hardig oial Voc [6] rsd ad Simo [] alid h followig oial dscribig isoroic hardig: ˆ(, ) ( ) ( ) ( ) ( K h H ) for (.5) H ( ) for whr is h sauraio xo ad h fucios h ( ), ( ), ( ) dscrib liar hrmal sofig ( ) ( )( ( )) ( ) ( )( ( )) h h( ) h( )( ( )) h (.5) whr ( ) is h iiial yild srss, ( ) is h fial sauraio hardig srss, h ( ) is h liar hardig modulus, all obaid a h rfrc mraur, whil ad h ar h flow srss sofig ad hardig sofig aramr, rscivly. Th abov oial allows us o sudy marials xhibiig a combiaio of liar ad sauraio- y hardig. Johso ad Cook Johso ad Cook [8] rs a oial o dscrib isoroic hardig i mals subjcd o larg srais, high srais ras ad high mraurs m ( ) K(, ) B (.53) whr is h quival lasic srai, is h cosa work hardig xo, B is h cosa srgh coffici, is h rfrc mraur ad ml is h rfrc ml mraur ad m is a owr ml 66

67 coffici of h hrmal sofig rm. Th ddc of hardig oial o mraur rrss h sofig of h marial wh is mraur is icrasd ad i is icludd i a mulilicaiv mar i h hardig oial. For mos mals,..5. I is imora o rmark, ha if, m ad ml h, h Johso ad Cook hardig oial dgras i h liar como of h Voc hardig oial. is a rasoabl aroximaio for havily rsraid mals. Bäkr Bäkr [9] rss a oial o dscrib isoroic hardig i a iaium alloy whr ( ) K(, ) B( ) (.54) is h quival lasic srai, is a mraur dd work hardig xo, B is a mraur dd srgh coffici ad m m B B x ad x (.55) ml whr B is h iiial srgh coffici, is h iiial work hardig xo, ml is h rfrc ml mraur ad m is a owr coffici of h hrmal sofig rm. Th hardig oial rsd i [9] icluds h hrmal sofig hrough h dgradaio wih h mraur of h srgh coffici ad h work hardig xo Yild codiio Exrm codiios of srai, srai ra ad mraur ar courd durig cuig. Srai valus i h rag -, srai ras ris o valus as high as 6 ad mraurs i h rag of -ºC. I umrical aalysis of mal cuig rocsss, accura flow srss modls ar cosidrd xrmly cssary o rrs work marial cosiuiv bhavior udr high srai ra dformaio codiios. W cosidr h classical Miss-Hubbr yild codiios, xrssd i rms of Kirchhoff srss sor, for h cas of ra idd lasiciy (,, ) dv( ) ( y K '(, )) 3 dv( ) ( y 3 ) ad for ra dd lasiciy ml (.56) 67

68 f(,, ) dv( ) ( y )( g( )) 3 (,, ) ( y ) g( ) 3 if (,, ) (.57) whr..dos h flow srss, y y dos h flow srss a, K'(, ) h isoroic oliar hardig modulus, h isoroic hardig a, ( g ) h srai ra hardig modulus ad h hardig aramr. Th xrssios ( y ) ad g ( ) dds o h hardig law usd. Numrous mirical ad smi-mirical flow srss modls hav b roosd. Th followig mraur, srai ad srai-ra dd modls rovid a samlig of h modls i curr us: (i) Simo[], (ii) Johso ad Cook [8] ad (iii) Bäkr [9] Simo flow modl y y if y ( ) H ( K )( x( )) y y ( w ( )) H H( w ( )) K K ( w ( ) if if h h (.58) This modl dscribs h srai hardig ad hrmal sofig for mos sls i mraur rag 3K ad 4K []. Commo valus of marials cosa of h Simo yild fucio ar show i Tabl. y w H 45 MPa. 93K 9.4 MPa w h K if. 75 MPa 9.93 Tabl. Simo yild fucio. Marial roris. 68

69 quival srss (MPa) x x 8 x 4 x quival srai Figur 6. Variaio of Simo yild fucio i rms of srais ad mraurs. Figur 6 shows h flow srss-srai curv for h Simo modl a diffr mraurs for h marial roris rsd i Tabl.Th ffc of h hrmal sofig is show i Figur 6 Johso ad Cook flow modl Johso-Cook s cosiuiv quaio is commoly usd o modl h hrmovisco-lasic bhavior of workic marial i umrical simulaios of h chi formaio rocss. Th hr ky marial rsoss ar srai hardig, srai-ra ffcs, ad hrmal sofig. Ths hr ffcs ar combid, i a mulilicaiv mar, as is show i h followig lis * m ( y )( g( )) A B( ) ( ) C l( ) * ml for for for ml if if ml (.59) (.6) (.6) 69

70 whr h firs brackd rm rrss h srai hardig of h yild srss, h x rm rrss h sofig of h yild srss du o local hrmal ffcs, ad h fial brackd rm modls h icras i h yild srss a lvad srai ras. is h quival lasic srai, is h dimsiolss srai ra, * is h homogous mraur, is h rfrc mraur ad ml is h rfrc ml mraur. A is h iiial yild srss ad B ad rrs h ffc of srai hardig. C is dimsiolss srai ra hardig coffici ad m is a owr coffici of h hrmal sofig rm. Viscous ffcs ar ak io accou as soo as h quival viscolasic srai-ra,, bcoms highr ha h hrshold. Srai-ra ssiiviy is h govrd by h viscolasic aramr,c. I h sam way, hrmal sofig is modld as soo as h mraur,, bcoms highr ha a rfrc mraur. Commo valus of marials cosa of h Johso-Cook yild fucio ar show i Tabl. A B C m 79 MPa 5 MPa /s Tabl. Paramrs of h Johso-Cook cosiuiv-law[8] Figur 7 shows h flow srss-srai curvs for h Johso-Cook modl a diffr mraurs (a) ad a diffr srais ras (b), for h marial roris rsd i Tabl. Th ffc of h hrmal sofig is show i Figur 6(a) ad h ffc of srai hardig is show i Figur 7(b). Bäkr flow modl A marial modl ha caurs h mai ffcs i chi formaio a high cuig sds. Tha is o say, his modl allows us o sudy h rasiio from coiuous o srrad chi as h ool sd is icrasd. Bäkr yild srss is rsd i h followig quaios: whr ad ( y )( g( )) B( ) C l( ) if if (.6) ar srai ad srai ra, h mraur, B ad h mraur-dd marial aramrs, ad C is dimsiolss srai ra 7

71 hardig coffici. Viscous ffcs ar ak io accou as soo as h quival viscolasic srai-ra,, bcoms highr ha h hrshold. quival srss (MPa) quival srai (a) quival srss (MPa) x x 8 x 4 x quival srai (b) Figur 7. Flow srss srai curvs for Johso Cook law a a srai ra of s - for (a) diffr srai ras ad (b) diffr mraurs 7

72 quival srss (MPa) quival srai (a) 4 quival srss (MPa) x 3 x 4 x quival srai (b) Figur 8. Flow srss srai curvs for Bäkr law a a srai ra of s- for (a) diffr srai ras ad (b) diffr mraurs. Th mraur dd coffici has h form 7

73 m m B B x( ) x( ) (.63) ml whr B is h iiial srgh coffici, is h iiial work hardig xo, ml is h rfrc ml mraur ad m is a owr coffici of h hrmal sofig rm. Commo valus of marials cosa of h Bäkr yild fucio ar show i Tabl 3. B C ml 6 MPa /s 85K Tabl 3. Paramrs of h Bäkr cosiuiv-law[9] Figur 8 shows h flow srss-srai curvs for h Bäkr modl a diffr mraurs (a) ad a diffr srais ras (b), for h marial roris rsd i Tabl 3. Th ffc of h hrmal sofig is show i Figur 8(a) ad h ffc of srai hardig is show i Figur 8 (b) Th associa flow rul Th fucioal form of h corrsodig associa flow rul is uiquly drmid by h ricil of maximum lasic dissiaio, giv h sord rgy fucio (.49) ad h yild fucio (.57). For h Miss-Hubbr yild fucio (.57) ad h fr rgy fucio (.49), Simo [4, 5] shows ha h flow rul aks h form basd o h ricil of maximum lasic dissiaio. A dail rocdur abou how o g h flow rul is show i h followig lis: Du o h rsricio o isoroy imlid by h rmolasic domai, h fucioal form of h iral rgy fucio ca b wri as ˆ( b,, ) wih (.64) Th fr rgy xrssd i rms of h iral rgy via Lgdr rasformaio ˆ( b,, ) ˆ ( b,, ) (.65) Exloiig h Scod Law of Thrmodyamics, cosiuiv quaios cosis wih h assumd fr rgy ar drivd, as follows D d (.66) ˆ d W diffria h fr rgy fucio (.49) wih rsc o im ml ˆ ˆ ˆ ˆ b b Takig h driva of b wih rsc o im (.67) 73

74 b FF FC F FC F F F FC F (.68) Usig h dfiiio of h saial vlociy gradi l FF ad h Li drivaiv of h lasic lf Cauchy Gr sor Lv b. Th im driva of b is wri as b lb b l L v b (.69) Isrig quaio (.69) io quaio (.67), h drivaiv of h fr rgy fucio (.67) bcoms ˆ ˆ ˆ ˆ ( lb Lv b ) (.7) b By isrig h rlaio d sym l io (.7) ad usig h Lgdr Trasformaio (.65), h dissiaio iqualiy bcoms D ˆ ˆ ˆ ˆ b d L vb b b (.7) By dmadig ha (.7) hold for all admissibl rocsss, h Kirchhoff srss sor is obaid by h gral xrssio: ˆ b b W T F F (.7) C ( l( J)) l( J ) J -3 ( ) dv b J J Th hydrosaic ad dviaoric ars of h Kirchhoff srss sor ar : -3 ( l( J)) ( ) l( J ) J (.73) s : dv( b ) ad h roy cosiuiv quaio ˆ (.74) Tˆ( ) ˆ M(, J ) Kˆ(, ) Th dissiaio iqualiy bcoms D ˆ ˆ vb b (.75).5..8 Evoluio quaios ad maximum lasic dissiaio. Now, w d o dfi h voluio quaios for h iral variabls i h modl i ordr o coml h cosiuiv hory of lasiciy a fii srais. 74

75 Basd o h hrmomchaical ricil of maximum dissiaio, h roblm is o fid (,, ) such ha h dissiaio fucio (.75) aais a maximum subjc o h cosrai (,, ) (ra-idd lasiciy), rscribd irmdia cofiguraio ( b is fixd) ad rscribd ras ( Lb v,, ). Th roblm ca b rformulad as a cosraid miimizaio of h gaiv of h dissiaio (,, ) arg mi ( D) (,, ) (.76) ˆ ˆ arg mi L vb (,, ) b Bu h roblm ca b xrssd as a ucosraid miimizaio roblm by iroducig a Lagragia fucioal L (,, ; ) D(,, ) (,, ) Lv bb (,, ) Th soluio o h roblm is giv by L (,, ; ) Lv bb (,, ) L (,, ; ) (,, ) (.77) (.78) L (,, ; ) (,, ) whr h cosiscy aramr is h Lagrag mulilir saisfyig h Kuh Tuckr codiios (,, ) (,, ) (.79) I is imora o rmark ha h Kuh Tuckr codiios ar quival o h loadig-uloadig codiios. I summary, h voluio quaios of h iral variabls ar L b v (,, ) b (,, ) (,, ) From xrssios (.43) ad (.4), h Li drivaiv of h lasic lf Cauchy- Gr sor ca b xrssd i marial dscriio as dv s C f( dv( ),, ) C C C dv s 3 ( 3 y ) (.8) (.8) 75

76 Usig h scific cosiuiv quaios ad dcomosig b io is shrical ad dviaoric ars, h xac flow rul (.8) bcoms s 3 3 Lv b J J r( b ) (.8) 3 Th firs rm i (.8) ca b glcd i mos mals, bcaus his rm is of h ordr of h flow srss ovr h shar modulus, which for mal lasiciy, is 3 of h ordr of []. W arriv a h modifid flow rul I quaio (.83) w hav usd ha F 3 Lv b J r( b ) 3 FC F r( b ) 3 i. Fr rgy fucio ˆ Tˆ( ) Mˆ (, J ) Uˆ ( J ) Wˆ ( b ) Kˆ (, ) ii. Kirchhoff srss J dv( b ) ( l( J)) : -3 ( ) l( J ) J s : dv( b ) iii. Vo Miss yild cririo J 3 F (,, ) dv ( ) ( y ) 3 iv. Evoluio quaios 3 Lv b J r( b ) 3 3 ( ) 3 y : Tˆ ( ) : ˆ M(, J ) Kˆ(, ) (.83) Box. Could hrmomchaical J flow hory. Ra idd lasiciy Th voluio quaios (.8) for h ra idd hory ca b asily xdd o icorora ra-mraur-dd rsos. Cosidr a rgularizd dissiaio fucio D ddig o a rgularizaio aramr C (, ), dfid as C 76

77 DC D(,, ) h( z ) (.84) whr z ( (,, ), C ) (.85) I quaio (.84), hz s a diffriabl fucio saisfyig h codiios (a) hz ( ), for all z R, ad (b) hz ( ) if ad oly if z. Th rgularizaio fucio hz () dds o h isoroic hardig law usd (Johso Cook, Bäkr). Th ricil of maximum dissiaio lads o a flow rul idical o (.8) wih h Kuh-Tuckr codiios rlacd by h cosiuiv quaio hz () 3 (,, ), C( ) y (.86), ohrwis I, coclusio, h viscolasic cosiuiv modl is a aly rgularizaio of h ra-idd modl, ad h soluio of h viscolasic roblm covrgs o h soluio of h ra idd roblm as h aly (fluidiy) aramr C i. Fr rgy fucio ˆ Tˆ( ) Mˆ (, J ) Uˆ ( J ) Wˆ ( b ) Kˆ (, ) ii. Kirchhoff srss J dv( b ) ( l( J)) : -3 ( ) l( J ) J s : dv( b ) iii. Plasic mulilir 3 (,, ) C( y ) iv. Evoluio quaios 3 Lv b J r( b ) 3 3 ( ) 3 y : Tˆ ( ) : ˆ M(, J ) Kˆ(, ) Box. Could hrmomchaical J flow hory. Ra dd lasiciy 77

78 .6 Fricioal coac cosrais Figur 9. Evoluio of fricioal coac bw h ool ad h workic Wih rfrc o Figur 9, a h sag of dformaio rocss corrsodig o h dformaio maigs () ad () of h workic ad ool bodis, rscivly, h ga, saraig a marial oi () X o () from h ool boudary, is dfid i h saial dscriio by () () () () gn ( X ) ( X ) (.87) () () Whr X is h marial oi o dfid by h closd oi rojcio, which is giv by () () () () () () ( X ) arg mi ( X ) ( X ) (.88) X () () Th ui ormal vcor is dfid i a sadard way as h ouward ui () () () () () ormal o ( X ) a x ( X ). Assum ha coac has b sablishd bw h ool ad h workic, ha is g N, hr is o coac amog hm g N ad fially, hr is raio wh g N.6. Normal bhavior I h dfiiio of h ormal coac bhavior, i is assumd ha raio bw h wo bodis is admissibl. I addiio, a liar rlaioshi bw h ormal coac forc ad h raio is osulad rsulig i h followig cosiuiv quaio gn if gn P N (.89) ohrwis 78

79 .6. Tagial bhavior Fricio occurs a h ool chi irfac udr xrm codiios of mraur, rssur ad srai. Is mchaism is o wll udrsood y, mos of h umrical sudis of machiig rocss, i has b usual o idaliz i by a Coulomb y fricio. P P (.9) T Somims, is usual o us h followig cosiuiv law for h agial forc a h ool chi irfac P g (.9) T whr T is h agial aly facor ad g T is h agial rlaiv dislacm. r = ad P N = N T T.5 si g(gt )( gt) _, -.5 = x -4 - = x - = x _ g T Figur. Rgularizaio of h Coulomb fricio law Th, h Coulomb fricio law ca b r wri as TgT if PT PN sick PT P sig( g ) ohrwis sli N T (.9) So of, a Coulomb's law rgularizaio is usd, which is mos robus ad siml wh i is imlmd usig h fii lm mhod. This rgularizaio is rsd i h followig lis PT rsig( gt )( gt ) P N (.93) 79

80 whr quaio (.93) rrss i comariso wih quaio (.9) a smooh rasiio from sick o sli. I quaio (.93) as ds o zro h rgularizaio aroachs h Coulomb law. This rgularizaio rcivs h am of Noro-Hoff fricio law. I Figur, h fucio sig( g )( g ) is lod agais g T for diffr valus of h aramr. I ca b s ha as ds o zro, h fucio sig( g )( g ) ds o h fucio sig( g ), T showig ha i h limi h rgularizaio is xacly, h Coulomb fricio law..6.3 Ha rasfr a h ool chi irfac I wha follows w formula h cosiuiv rlaioshis for h ha flux Q ad h dissiaio D fric a h ool chi irfac. Th ha flux i h coac zo also ds a cosiuiv quaio for is drmiaio. W assum h followig srucur for h cosiuiv quaio for h ha flux: T () () () () () () N N Q Q h(,, P )( ) h(,, P ) g (.94) () () whr ar h mraurs of boh coac surfacs ad h(,, P ) h () () () () hrmal coducac coffici. is h mraur a x ( X ), () () whr X is h marial oi o dfid by h closd oi rojcio () () (.88). Th ha rasfr coffici h(,, P N ) dds uo h surfac mraurs ad h coac rssur. Du o h chical imossibiliy of obaiig rfcly la surfacs, h ral coac ara is always limid ad corrsods o a sris of sos. Drmiaio of h ru coac ara is fudamal for h modlig of hrmal homa. Thrfor, ha xchag a h ool chi irfac is ossibl by ha coducio hrough h sos, ha coducio hrough h gas coaid i h caviis ad radiaio bw micro-caviy surfacs. Th assumio ha h miod mchaisms ac i aralll is wll accd, lads o h followig rlaioshi for h hrmal coducac coffici () () () () () () () () N s N g N r N h(,, P ) h (,, P ) h (,, P ) h (,, P ) (.95) whr h s is h hrmal coducac coffici du o coducio hrough h sos, h g is h hrmal coducac coffici du o coducio hrough h gas coaid i h caviis ad r h is h hrmal coducac coffici du o ha rasfr by radiaio bw micro-caviy surfacs. Th coac rssur is h facor of mos ifluc o coac hrmal coducac coffici. As coac rssur grows, coac coducac grows. This is aribud o h fac ha h coac surfac bw h bodis icrass as h coac rssur icrass. Wh h coac rssur is T T N T 8

81 icrasd o such a x ha h ral coac ara of coac is a larg orio of h aar coac ara, i is o logr ossibl for h ral coac ara o icras roorioaly o h load. Coac bw h ool ad h work surfac is so arly coml ovr a larg ar of h oal ara, such ha h hrmal coducac coffici is idd of h coac rssur. For simliciy, i his work, w will assum ha h hrmal coducac coffici is idd of h mraurs of h bodis i coac. Such ha quaio (.94) is simlifid as () () Q h( ) hg Q (.96) Wh wo bodis, say a cuig ool ad a workic, ar i coac, h ra a which ha is grad, is shard bw h wo bodis. This ha mus b aoriod bw h ool ad h chi. Th ha rig h ool ad h workic is giv by [4] () () () c k () fric fric fric () () () () () () R D D c k c k () () () c k () fric fric fric () () () () () () R D D c k c k (.97) whr h ha graio du o fricio a h ool chi irfac is giv by Dfric PT gt rsig( gt )( gt ) PN g T (.98) Ad whr, c ad k rrs hrmal roris amly mass dsiy, coduciviy ad scific ha rscivly..7 Variaioal Formulaio. Wak Form of h IBVP Icludig Fricioal Coac Cosrais W dfi h s of admissibl dislacms ad admissibl mraurs of as h s of all sufficily rgular dislacm ad mraur fucios ha saisfy h ssial boudary codiio, dod hr rscivly as U ( i) ( i) ( i) ( i) ( i) : ( ) R : d( F) ad ( i) ( i) ( i) ( i) ( i) ( i) : ( ) R : ad (.99) Th saial vrsio of h virual work ricil sas ha h body is i quilibrium if, ad oly if, is Cauchy srss saisfis h quaio. Th wak form of h momum balac quaio (.7) (b) ca b jusifid by akig h ( i) ( i ) L ir roduc of (.7) (b) wih ay V, ad, makig us of h divrgc horm, ladig o h followig xrssio: 8

82 ( i) ( i) S ( ) ( ) ( i) s ( i) i ( i) ( i) ( i) ( i) i ( i) ( i) : ds ( b v ) ds d ( i) ( i) ( i) ( i) ( i) d V (.) whr V is h sac of virual dislacms of ( i) ( i) ( i) ( i) V ( ) R : (.) Th dyamic wak form of h rgy balac quaios o h body ca b ( i) ( i) obaid by akig h L ir roduc of (.7) (c) wih ay T, ad, makig us of h divrgc horm, ladig o h followig xrssio: ( i) ( i) ( i) ( i) ( i) ( i) ( i) ds q ds ( i) ( i) S S ( i) ( i) q S ( i) ( i) ( i) ( i) ( i) ( i) DidS ( q ) d q ( i) ( i) ( i) ( i) ( i) ( q ) d whr T is h sac of virual mraurs of such ha T (.). For simliciy h L ir roduc will b rrsd as, ad wih a sligh abus i oaio,,, ad, roduc o h boudaris, ad, rscivly. As a cosquc, quaios (.) ad (.) ca b wri as dy, mch q ( i) s ( i) ( i) ( i) ( i) ( i), ( i) ( i) ( i) ( i),, ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i),, q, D ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) q q will do h L ir i, q, q sa, mch b v (.3) Doig by G ad G h dyamic ad quasi-saic wak forms of h momum balac quaios, rscivly, xcludig fricioal coac c, mch coribuio, ad by G h fricioal coac coribuio o h wak form of h momum balac quaios, rscivly dfid as G G G ( i) ( i) ( i) ( i) dy, mch Gsa, mch v ( i) s ( i) ( i) ( i) ( i) ( i) sa, mch, b, ( i) ( i) c, mch, (.4) 8

83 Ad doig by dy, hrm G ad sa, hrm G h dyamic ad quasi-saic wak forms of h rgy balac quaios, rscivly, xcludig hrmal fricioal c, hrm coac coribuio,, ad by G h hrmal fricioal coac coribuio o h wak form of h rgy balac quaios, rscivly dfid as ( i) ( i) ( i) ( i) ( i) dy, hrm, sa, hrm ( i) ( i) ( i) ( i) ( i) ( i) ( i) sa, hrm, q, Di, q G G G G ( i) ( i) c, hrm, q q (.5) Th wak form of h momum balac ad rgy quaios for body ca b xrssd i shor oaio as G G ( i) ( i) dy, mch Gc, mch ( i) ( i) ( i) ( i) V, T ( i) ( i) dy, hrm Gc, hrm (.6) For h coac roblm bw h workic ad h ool, h momum balac ad rgy quaios ak h form ( i) ( i) Gdy, mch Gc, mch i i ( i) ( i) Gdy, hrm Gc, hrm i i ( i) ( i) ( i) ( i) V, T Th lockig roblms. A mixd dislacm-rssur formulaio. (.7) I is wll kow ha ur dislacm formulaios ar o suiabl for roblms i which h cosiuiv bhavior xhibi icomrssibiliy sic hy d o lockig. Lockig mas, i his cocio, ha h cosrai codiios du o icomrssibiliy which ar rlad o h ur volumric mod (i h lasic cas h codiio is d( F ) ad for lasic flow h codiio d( F ) d( C ) holds) cao b saisfid. Thus, his bhavior is also calld volum lockig. As lockig is rs i h machiig roblm, w ado a mixd formulaio i h momum balac quaio of h workic. Iroducig a rssur/dviaoric dcomosiio of h Cauchy srss sor, h sadard xrssio of h quilibrium quaio bcoms G () () () () () dy, mch Gsa, mch v () () () s () () () () () sa, mch ( ), b, G dv G () () () c, mch, () (.8) Th rssur fild () i h variaioal quaio (.8) is a addiioal variabl drmid by h followig. Th variaioal quaio h rrss h wak form of h rssur cosiuiv quaio. 83

84 () () () () com, mch, volu, mch () () () ( l( J )) () volu, mch l( ) 3 ( () ) G q G (.9) G J q J Takig io accou h mixd formulaio usd a h workic, h momum ad rgy balac quaios for h coac roblm bw h ool ad h workic (.7) ak h form whr ( i) ( i) Gdy, mch Gc, mch i i ( i) ( i) Gdy, hrm Gc, hrm i i () () () () Gcom, mch q Gvolu, mch, () () () ( l( J )) () () volu, mch l( ) 3 ( () ) G J q q Q J Sabilizaio via h Polyomial Prssur Projcio(PPP) (.) (.) Mixd formulaios hav o fulfill addiioal mahmaical codiios, which guara is sabiliy. This codiio is kow as BB-codiio, amd afr is ivors Babuska ad Brzzi. Liar riagl fii lms do o saisfy BBcodiio; cosquly, a sabilizaio of h rssur fild is dd. I our aroach w us a sabilizd formulaio basd i h so-calld Polyomial Prssur Proycio(PPP) rsd ad alid o Soks quaio i [5, 6]. Th mhod is obaid by modificaio of h mixd variaioal quaio by usig local L olyomial rssur rojcios. Alicaio of rssur rojcios i cojucio wih miimizaio of h rssur-dislacm mismach limias icosiscy of qual-ordr aroximaios ad lads o a sabl variaioal formulaio. Ulik ohr sabilizaio mhods, h Polyomial rssur rojcio (PPP) dos o rquir scificaio of a sabilizaio aramr or calculaio of highr-ordr drivaivs. I addiio, PPP ca b imlmd a h lm lvl ad rducs o a siml modificaio of h wak coiuiy quaio (icomrssibiliy cosrai). I his work, w xd h PPP o solid mchaics roblms ivolvig larg srais. () () Giv a fucio L, h L rojcio oraor : L P is dfid by () () () () ( i) () roj, mch ( ) G q ds q P (.) S () () whr is h bs aroximaio of i h sac of olyomials of ordr ( P ). 84

85 To sabiliz h mixd form (.8)-(.9), w add h rojcio oraor o quaio (.9) () () () () () ( i) sab, mch ( ) ( ) () G q q ds (.3) S whr is h sabilizaio aramr ad s h workic shar modulus. Alicaio of h rojcio oraor o h rssur s ad rial fucios srvs o rmov h aroximaio icosiscy rs for qual-ordr dislacm ad rssur sacs. Th rol of h form G () sab, mch is o furhr aliz rssur variaio away from h rag of h divrgc oraor. Takig io accou h mixd formulaio ad h olyomial rssur sabilizaio rms o dal wih h icomrssibiliy homa i h workic, h momum ad rgy balac quaios for h coac roblm bw h ool ad h workic (.7) ak h form whr ( i) ( i) Gdy, mch Gc, mch i i ( i) ( i) Gdy, hrm Gc, hrm i i () () () () () Gcom, mch q Gvolu, mch Gsab, mch, () () () ( l( J )) () volu, mch l( ) 3 ( () ) G J q J () () () () () ( i) sab, mch ( ) ( ) () S () () () () ( i) roj, mch ( ) G q q ds G q ds S () () () (.4) (.5) q Q,, q P Th s of govrig quaios for h dislacm, rssur ad mraur variabls is comld by addig h followig cosrai quaio o h s of govrig quaios. () I aricular, for wo iracig bodis ad (), h fricioal coac mchaical ad hrmal coribuios o h wak form of h momum ad () () () () rgy balac quaios, a h saial coac ois x ( X ) () () () () ad x ( X ), a ay im I (,) () () c, mch : c, mch c, mch (,) () () c, hrm : c, hrm c, hrm G G G G G G (.6) Th wak form of h quilibrium codiios a h ool chi irfac giv by (.4), ca b xrssd as 85

86 Q () () () () () () () () () () hc, Qhc, () (), (.7) Usig (.7) h mchaical ad hrmal coac coribuio o h wak form of h momum ad rgy balac quaios ak h siml form G (,) () () () () () () () c, mch :,,, () () () (,) () () () () () () () c, hrm : hc, hc, hc, G Q Q Q () () ().8 Numrical Igraio Algorihm (.8).8. Th icrmal boudary valu roblm. Fii lm discrizaio ( i) lm ( i) Cosidr a saial discrizaio io a disjoi collcio of () o-ovrlaig lms wih characrisic siz h. Th fii lm mhod for umrical soluio of roblm (.) cosiss of ( i) ( i) ( i) ( i) () () rlacig h fucioal ss U, V,, T ad P, Q wih discr ( ), ( ), subss i h i h ( i), h ( i), h (), h h (), h U, V,, T ad P, Q grad by a fii ( i) ( i) lm discrizaio h of h domai. L a ( X ) b a gric fild dfid ovr h domai of h lm. Th fii lm irolaio of h fild a wihi lm of body i is obaid as od ( i), h ( i) ( i) ( i), ( i) j j j whr j a is h valu of a a od j, ad a ( X ) a N ( X ) (.9) ( i), j N is h sha fucio such ha is valu is a h od j ad zro a ay ohr od of h lm. Th irolad fucio, ow dfid ovr h aroximad domai is giv by whr j o i ( i), h ( i) ( i) ( i) ( i) j j j a ( X ) a N ( X ) (.) N is a icwis olyomial fucio h global sha fucio associad wih h global od j ad o i is h oal umbr of odal ois i h fii lm msh. Wih h iroducio of h abov irolaio rocdur, w gra h fii dimsioal ss 86

87 ad U o i ( i), h : ( i), h ( i) ( i) ( i) ( i) ( X ) j Nj ( X ) : ( i), h ( i) j ( i), h o i ( i), h : ( i), h ( i) ( i) ( i) ( i) ( i), h ( X ) j N j ( X ) : ( i) j ( i), h (), h (), h () o i P : ( X ) V T j () () () j N j X ( ) o i ( i), h ( i), h ( i) ( i) ( i) ( i), h j Nj X j ( ) : o i ( i), h ( i), h ( i) ( i) ( i) ( i), h j Nj X j ( i), h ( ) : ( i), h (.) (.) o i (), h (), h () () ( i) (), h j j ( X ) : j Q q q N q Th fii lm aroximaio o h coiuum variaioal roblm (.) ( i) ( i) ( i) ( i) is h obaid by rlacig h fucioal ss U, V,, T ad () () ( ), ( ), P, Q wih discr subss i h i h ( i), h ( i), h (), (), U, V,, T ad h h P, Q. To driv h discrizd form of (.) i is covi o iroduc h sadard marix oaios ha follow ad whr ( i) ( i) (),., q ( i) ( i) () ( i), h ( i), h (), h N N N ( i) ( i) ( i) ( i) () () ( i), h ( i), g ( i) ( i), h N N ( i), g ( i) (), h (), g () N q (), h (.3) (.4),, q ar h vcor of odal dislacms, odal mraurs, odal rssurs, virual dislacms, virual mraurs ad virual rssur, rscivly. I is also covi o iroduc h global srai-dislacm marix, which i wo dimsios has h forma 87

88 g g g N, N, Noi, u g g g N, N, No i, g g g g g g N, N, N, N, No i, Noi, B (.5) ad h global gradi-mraur marix For h vcor fild N, N, Noi, B (.6) N N N,, o i, ( i), h ( i) ( i) u N, h mulilicaio of B by a global vcor of odal dislacm givs a array of srais ad for h fild ( i), h ( i) ( i) N h mulilicaio of B by a global vcor of odal mraurs givs h arrays of h mraur gradis. Fially, h array of Cauchy srss como is dfid as,, T (.7) ( i) ( i) ( i) ( i) Wih h abov oaio a had, h rlacm of U, V,, T ad () () ( ), ( ), P, Q wih i h i h, ( i), h ( i), h (), (), U V,, T ad h h P, Q. i (.) ad sic (.) quaios ar saisfid for all vcors ( i ) ( i, ), q (), h discr courar of (.) is giv by h quaios S u ( i), g T ( i) ( i) ( i), g T ( i) ( i) ( i) ( i) ds b v ds ( ) ( ) ( ) ( i), g T ( i) ( i) ( i), g T ( i) ( i) ( ) d ( ) d ( i) ( i) (.8) () S ( i) ( i) S S () S ( i) ( i) q S ( i) ( i) ( i), g ( i), g T ( i) ( i) ( i), g T ( i) ( i) c ( ) ds ( ) q ds ( i), g T ( i) ( i) ( i), g T ( i) ( i) DidS q d q ( ) ( ) ( ) ( i), g T ( i) ( i) ( ) ( q ) d ( i), g ( i), g T ( ) ds ( i), g T () ( l( J )) ( ) l( J ) 3 ( () ) ds J () (.9) (.3) Th fii lm discr boudary valu roblm is h formulad as follows. Fid h vcor of global dislacms, global rssurs ad global mraurs, such ha 88

89 whr F ( i),i, mch ( i) ( i) ( i), x, mch ( i), c, mch ( i), dy, mch ( i) F ( u, ) F F F ( u ) ( i),i, hrm ( i) ( i), x, hrm ( i), c, hrm ( i), dy, hrm ( i) F ( ) F F F ( ) F (), rss, mch (), volu, mch F F F ( i),i, mch ( i) ( i) u ( i), g T ( i) F ( u, ) ( ) F S ( i), x, mch ( i), g T ( i) ( i) F ( i), c, mch ( i), g T ( i) ( i) ( i), dy, mch ( i) ( i) ( i) ( ) ( ) F ( u ) v ( i),i, hrm ( i) ( i), g T ( i) F ( ) ( ) q S ( i), g T ( ) ds ( i) ( i) ids d d ds S ( i), x, hrm ( i), g T ( i) ( i) ( ) ( q ) d q F ( i), c, hrm ( i), g T ( i) ( i) ( ) ( q ) d ( i), dy, hrm ( i) ( i) ( i) ( i), g () () S q D ( ) c ( (), rss, mch (), g (), g T () ( ) ds F (), volu, mch (), g T () S S S ds ( i), g T ( i) ) () ds ( l( J )) ( ) l( J ) 3 ( () ) ds J () (.3) (.3) (.33) (.34) I acual fii lm comuaios, h abov forc vcors ar obaid as h assmblis of lm vcors. Th fii lm assmbly oraor, lm A, imlis ha ach como of h global forc associad wih a aricular global od is obaid as h sum of h corrsodig coribuios from h lm forc vcors of all lms ha shar ha global od. I his work, h lm forc vcors ar valuad usig Gaussia quadraurs. Th sadard lm sha fucio of h hr-odd liar riagl ar dfid as 89

90 N N N ( i),( ) ( i),( ) ( i),( ) 3 (, ) (, ) (, ) (.35) This lms sha fucios ar usd o discriz h dislacm, rssur ad mraur filds. Fii lms, which ar drivd from mixd mhods, hav o fulfill addiioal mahmaical codiios, which guara h sabiliy of h lm formulaio. This codiio is kow as BB-codiio, amd afr is ivors Babuska ad Brzzi. Th FEM soluio of h variabls i h (icomrssibl) solid (workic) domai imlis solvig h momum ad icomrssibiliy quaios. This is o such as siml roblm as h icomrssibiliy codiio limis h choic of h FE aroximaios for h dislacm ad rssur o ovrcom h wll-kow div-sabiliy codiio. I our work, w us a sabilizd mixd FEM basd o h Polyomial Prssur Projcio (PPP) aroach, which allows for a liar aroximaio for h dislacm, mraur ad rssur variabls. Th discr courar of rojcio oraor for liar riagl fii lms is giv by (.36), whr w hav usd ha h rssur rojcio is cosa ad discoiuous amog fii lms (), sab, mch,( ) (),( ) T (),( ) (),( ) T (),( ) (),( ) (),( ) ds F ( ) (.36) () (),( ) S To sabiliz h mixd form, w add h discr rojcio oraor (.36) o quaio(.34). Fially, h rsulig s of discrizd quaios ha w d o solv o modl mal cuig has h followig form: Momum ( i),i, mch ( i), x, mch ( i), c, mch ( i), dy, mch m ( i) F F F F M a Ergy ( i),i, hrm ( i), x, hrm ( i), c, hrm ( i), dy, hrm ( i) F F F F M Imcomrssibiliy (), rss, mch (), volu, mch (), sab, mch F F F (.37).8. Th icrmal boudary valu roblm. Tim discrizaio.8.. Imlici could algorihm (Moolihic schm) For simliciy, a ariio of h im domai I :, T io N im ss, of h sam lgh is cosidrd. L us focus o h im s, whr. Th alicaio of a imlici backward-eulr im 9

91 igraio schm o h roblm (dislacm, vlociis ad mraurs) (.37) yilds h followig algorihm dfid by h iiial codiios Could sysm of quaios Momum ( i), dy, mch F ( v ) ( i),i, mch ( i) ( i) ( i) ( i) ( i) ( i) ( i), x, mch ( i) ( i), c, mch F ( u ) F ( ( u,, ; ( u, ))) F ( u ) Icomrssibiliy M M G F Ergy sab () (), volu, mch 9 () () () u ( J (, )) ( i), dy, hrm F ( ) ( i),i, hrm ( i) ( i) ( i) ( i) ( i) ( i) ( i), x, hrm ( q( ); Di(, ; (, )) ( i), c, hrm F u u F F Uda odal variabls ( i) ( i) ( i) ( i) () () () ( i) ( i) v v v u u v Box 3. Imlici could soluio schm. Th s of quaios rsd i Box 3, show a simulaous soluio schm of h could sysms of quaios whr h mraur varis durig h mchaical s ad h cofiguraio varis durig h hrmal s. A firs glac, h simulaous soluio is h obvious o, bu a dh aalysis show ha is a comuaioally isiv rocdur []. Th moolihic schm is ucodiioally sabl du o is fully imlici characr. Th diffr im scals associad wih h hrmal ad mchaical filds suggsd ha a ffciv umrical igraio of h could roblm should ak advaag of hs diffr im scals. O of h ffciv igraio schms is h so-calld saggrd algorihms, whrby h roblm is ariiod io svral smallr sub-roblms ha ar solvd squially (sliig ach im s i svral sudo-im ss). Mos of h im, his chiqu is scially araciv from a comuaioal viwoi sic h larg ad o symmric sysm ha rsul from a simulaous soluio schm is rlacd by a much smallr, subsysm. Basd o h global oraor sli for fii dformaio lasiciy rsd i quaios(.7), (.) ad (.4), a formal sli of h roblm io a mchaical has wih h mraur hld cosa, followd by a hrmal has a a fixd cofiguraio is rsd i h followig lis:.8.. Isohrmal sli

92 Th followig lis rs a summary of h isohrmal sli, dvlod i []. L ad b h iiial ad fial im s. L b h im icrm Momum quaio for fixd iiial mraur ( i), dy, mch ( i),* F ( v ) ( i),i, mch ( i) ( i),* ( i),* ( i),* ( i) ( i),* ( i),* ( i), x, mch ( i),* ( i), c, mch ( i),* F ( u ) F ( ( u,, ; ( u, ))) F ( u ) Icomrssibiliy sab (),* (), volu, mch (),* (),* (),* M M F ( J ( u, )) G Uda odal variabls ( i),* ( i),* ( i),* ( i) ( i) (),* (),* (),* v v v u u v Ergy quaio a udad fixd cofiguraio ( i), dy, hrm ( i),* F ( ) ( i),i, hrm ( i),* ( i),* ( i),* ( i),* ( i) ( i),* ( i),* ( q( ); Di (, ; (, )) ( i), x, hrm ( i), c, hrm F u u F F Uda odal variabls ( i),* ( i),* Box 4. Imlici isohrmal sli. Th abov algorihm is basd o h alicaio of a imlici backward-eulr diffrc formula o h momum quaio for fixd iiial mraur (mraur a rvious im s) ad h alicaio of a imlici backward- Eulr diffrc schm o h rgy quaio a fixd cofiguraio (cofiguraio obaid as a soluio of h mchaical roblm). Th soluio of h balac of momum quaio for fixd iiial mraur, ( i),* ( i),*, givs a uda of h rimary variabls u ad a firs uda of h iral variabls of h form,, +,, b b (.38) Alog wih a icrmal valu of h cosiscy aramr saisfyig h Kuh-Tuckr codiios ad dod by. Th soluio of h balac of rgy wih iiial codiios ( i),* ( i),* ( i),*,, u ad iiial iral variabls b,, givs a uda of h rimary variabl 9

93 ( i),* ad a scod uda of h iral lasic variabls (a fixd cofiguraio) of h form,, +,, b b (.39) Alog wih a icrmal valu of h cosiscy aramr saisfyig h Kuh-Tuckr codiios ad dod by. I gral, as a cosquc,, +,, b b. I summary, h isohrmal sli solvs h mchaical roblm wih a rdicd valu of mraur qual o h mraur of h las covrgd im s ad, h, solvs h hrmal roblm usig h cofiguraio obaid as a soluio of h mchaical roblm. A full Nwo-Rahso schm is usd for h soluio of h o-liar sysm; h cssary liarizaio will b rsd lar i his char. Th wll-kow rsricio o codiioal sabiliy is h crucial limiaio of h isohrmal aroach, which of bcoms criical for srogly could roblms. Howvr, his rsricio is o sigifica for mal lasiciy[]. Armro ad Simo [7] rovid h suffici codiios for sabiliy of h isohrmal sli: ( i) ( i) ( i) ( i) ( i) c c K K h m h m ( i) ( i) ( i) ( i) (.4) ( i) ( i ) whr, ar h Lamé cosa, m h hrmal xasio ( i) ( i) coffici,, c h dsiy ad h scific ha ad h,, K ar h miimum lm siz of h msh, h maximum allowd im s, ad a cosa. I cas whr h mchaical iria ca b cosidrd gligibl, Armro ad Simo [7] rovid h suffici codiios for sabiliy of h isohrmal sli: ( i) ( i) ( i) ( i) ( i) m E c c m ( i) ( i) ( i) ( i) ( i) h E k h k E c (.4) ( i) ( i) whr m h hrmal xasio coffici, E h lasic modulus,, c h dsiy ad h scific ha ad h, ar h miimum lm siz of h msh ad h allowd im s. Prvious rsricios dmosras ha algorihms basd o h isohrmal sli ar o suiabl for srogly could roblms, sic h sabiliy rsricio hrasd i rms of h Coura umbr bcoms icrasigly rsriciv h highr h coulig (icras i h hrmal xasio coffici). Th umrical simulaio of mal cuig ca b cosidrd a wakly could roblm (h hrmal xasio coffici of mals is usually small), as a rsul, h isohrmal sli will rform wll i mos of h umrical simulaios of mal cuig rsd i his work. Th sabiliy rsricio of h isohrmal 93

94 sli is circumvd usig a isroic sli, i which o mus solv firs a mchaical roblm a cosa roy (simas h mraur chag i h mchaical roblm), followd by a hrmal ha coducio roblm a cosa (fixd) cofiguraio [7] Isohrmal IMPL-EX sli Momum quaio for fixd iiial mraur (lasic roblm wih shar modulus chagig from lm o lm ) ( i), dy, mch ( i),** ( i),i, mch ( i) ( i),** ( i),** ( i),** ( i ( ) ( (,, ; ( i), x, mch ( i),** ( i), c, mch ( i),** F ( u ) F ( u ) ) F v F u )) Icomrssibiliy sab (),** (), volu, mch (),** (),* (),* M M F ( J ( u, )) G Uda odal variabls ( i),** ( i),** ( i (),** (),** (),** u u v ),** Ergy quaio a udad fixd cofiguraio ( hrmal roblm wih mraur dd xral ha sourc) ( i), dy, hrm ( i),** F ( ) ( i),i, hrm ( i),** ( i),** ( i),** ( i) F ( q( ); Di ( u, ; ( i), x, hrm ( i), c, hrm F F Uda odal variabls ( i),** ( i),** Cosiuiv quaio ad uda iral variabls (Plasic algorihm) ( i),** ( i),** + f ( b,, ) (( u, ),( b,, )) Box 5. Isohrmal IMPL-EX sli Th isohrmal schm rsd i [] dcouls h hrmo mchaical roblm i wo mor siml roblms, bu, y, h mchaical roblm is could wih h voluio quaios of iral variabls ad h hrmal roblm is could wih h voluio quaios of h iral variabls, boh of hm ar could hrough h lasic mulilir. Th abov raso, suggss dcoulig h roblm i h followig hr siml roblms: (i) a lasic roblm wih shar modulus chagig from lm o lm, (ii) a hrmal roblm wih a mraur dd lasic ha sourc ad (iii) a rlaxaio rocss affcig h srss ad h iral variabls a h igraio ois. I his work, w rs a w saggrd algorihm, which is basd o h isohrmal sli rsd i [] ad h IMPL-EX igraio schm of h cosiuiv quaios rsd i [8]. Usig h igrdis rsd abov, )) 94

95 h soluio of h could sysm of ODE (.37) could b dcould i h hr siml roblms miod i h rvious aragrah. I addiio, h lasic ad h hrmal roblms uda h iral variabls accordig o a rdicd lasic mulilir (xlici), whil h cosiuiv quaios lav h dislacms, vlociis ad mraurs uchagd (imlici). For simliciy, a ariio of h im domai I :, T io N im ss, of h sam lgh is cosidrd. L us focus o h im s, whr. A imlici backward-eulr diffrc formula is alid o h momum quaio ad o h rgy quaio. I firs s h xraolaio of h lasic mulilir is do. Th, h srsss ar comud via a imlici backward-eulr igraio of (.8) ad h balac of momum (.3) is solvd imlicily rovidig h odal dislacm ad rssur for fixd iiial mraur. Th soluio of h balac of momum rovidd a fixd iiial mraur ad a xraolad valu of h iral variabl cosiu a o-liar sysm of quaio du o gomrical oliariis, which has o b iraivly, solvd raig uil covrgc is achivd. Th soluio of h balac of momum quaio for fixd iiial ( i),** ( i),**, mraur, givs a uda of h rimary variabls u ad a firs uda of h iral variabls of h form,, +,, b b (.4) Th, i h scod s, h soluio of h balac of rgy wih iiial codiios ( i),** ( i),**,, u, iiial iral variabls b,, ad h xraolaio of h lasic mulilir ( i),** givs a uda of h rimary variabl ad a scod uda of h iral lasic variabls (a fixd cofiguraio) of h form,, +,, b b (.43) ( i),** ( i),** ( i),** Fially, i h hird s, h valus of u,, rmai fixd, ad a imlici backward-eulr igraio of h cosiuiv modl (.8) is do usig as iiial iral variabls b,, uda of h iral variabls of h form,, +,,. Giv, as a cosquc a fially b b (.44) Th s of iral variabls obaid a h d of his im s, will b h s of iral variabls usd as h sarig oi i h x s of h fracioal s mhod roosd i his work. As summary abou h isohrmal IMPL-EX sli is show i Box 5. 95

96 I is irsig o o ha h boudary valus of h momum quaio ar icludd i h lasic quaios wih shar modulus chagig from lm o lm ad h boudary valus of h balac of rgy ar imosd o h hrmal roblm wih mraur dd lasic ha sourc. I addiio, h lasic algorihm cosiss of a collcio of sysms of ordiary diffrial quaios, ach o of which rais o a diffr igraio oi. A full Nwo-Rahso schm is usd for h soluio of h o-liar sysm; h cssary liarizaio will b rsd lar i his char Smi-xlici igraio schm Th alicaio of a forward-eulr im igraio schm o h mchaical ad h hrmal roblm yilds h followig algorihm Momum ( i), dy, mch m ( i) F M v ( i),i, mch ( i) ( i) ( i) ( i), x, mch ( i) ( i), c, mch ( i) F ( ( u, )) F ( u ) F ( u ) Uda odal vlociis ad dislacms ( i) ( i) ( i) m ( i), dy, mch u ( i) ( i) u v v v v v M F Ergy ( i), dy, hrm ( i) F M ( i),i, hrm ( i) ( i) ( i) ( i) ( i) ( q( ); Di(, ; (, ))) ( i), x, hrm ( i), c, hrm F u u F F Uda odal mraurs ( i) ( i) ( i) ( i), dy, hrm M F Icomrssibiliy (), rss, mch () (), volu, mch () () ( i) (), sab, mch () J F ( ) F (( u, )) F ( ) Box 6. Smi-xlici igraio schm Th mai ss i h soluio of could hrmo-mchaical roblm ar rsd i h followig lis. Firs, a mchaical s is ak basd o h curr disribuio of mraurs, ad h vlociis ad dislacm ar udad usig a xlici forward-eulr. Scod, h ha grad is rasfrrd o h hrmal roblm ad h mraurs ar udad usig a forward-eulr algorihm. Th, h rsulig mraurs ar rasfrrd o h mchaical roblm ad icororad io h hrmal sofig cosiuiv modl. Fially, h odal rssur is udad imlicily i ordr o dal wih h icomrssibiliy cosrai imosd by J lasiciy. Exlici schm ar codiioally sabl, i mas ha h im s usd i h simulaios should b lss or qual ha a giv criical im s, h criical im s corrsod o h im ha aks o a wav o ravl hrough h small fii lm of h msh. I cas of a lasic marial, h criical im s 96

97 dds o h msh siz, h lasic modulus, h Poisso raio, ad h dsiy of h marial. h c (.45) 3 whr h, v ar h miimum lm siz of h msh ad h maximum allowd im s, ad,,, ar h bulk modulus, h Poisso raio ad h marial dsiy, rscivly. Th rsricio imosd o h im s by h xlici schms, allows hikig ha for umrical simulaio ha ivolvs log riod of comuig im or low sds imlici schms ar mor favorabl i comariso wih xlici schms. O h corary, wh vlociis ar high ad h coac codiios ar comlx, is cssary o dcras h im s usd i imlici formulaios, so i his cas xlici formulaios aar as a ffciv ad a ffici ool, wih a irsig comuig im. I h x char, som cuig xamls will show a comariso bw imlici isohrmal sli, imlici isohrmal IMPL-EX sli ad h smi-xlici schm..8.3 Workic cosiuiv law: im discrizaio Th roblm of igraig umrically h iiial-valu ODE quaios rrsd by (.8) i cojucio wih h codiio (.79) is h focus of his scio Imlici Backward-Eulr L C,, do h iiial sa a im, ad assum ha h dformaio gradi ad mraur fild F, a im ar rscribd. L us focus o h im s, whr. Usig a imlici ucodiioally sabl schm o (.83) ad h scalar quaios of (.8) givs F ( C C ) F r( b ) 3 (.46) 3 ( y ) 3 Th righ had sid of quaio (.46) i rms of saial variabls bcoms b,, ( F bf r b ) (.47) 3 alog wih h followig courar of h loadig-uloadig codiios: 97

98 f f (,, ) (,, ) whr h yild codiio is dfid by h Miss criria (.48) f (,, ) dv ( ) ( y, ) (.49) 3 A closd form soluio of hs quaios is obaid by dfiig h hrmolasic sa by h rlaios, rial,, rial, rial dv( b ) b F C F F b F s (.5) rial rial (, ( f s y )) 3 W obsrv ha h rial sa is drmid solly i rms of h iiial codiios b,, ad h giv icrmal dformaio gradi F,. W rmark ha his sa may o, corrsod o ay acual sa, ulss h icrmal rocss is lasic. A aalysis of quaio (.5) rvals wo alraiv siuaios: Firs, w cosidr h cas for which rial f (.5) I follows ha h rial sa is admissibl i h ss ha, rial,, rial s rial, rial dv( b ) b b F C F F b F s s ad saisfy. Th srss srai rlaioshi. Th flow rul ad h hardig law wih 3. Th Kuh-Tuckr codiios, sic rial f (,, ) f (.5) (.53) saisfy (.48). rial Nx, w cosidr h cas for which f. Clary, h rial sa cao b, rial a soluio o h icrmal roblm sic, b, violas h cosrai codiio f (,, ). As a rsul, w rquir ha rial so ha o obai. s s. 98

99 To summariz our rsuls, h coclusio ha a icrmal rocss for giv icrmal dformaio gradi is lasic or lasic is draw solly o h basis of h rial sa accordig o h cririo lasic s rial f (.54) lasic s Hr w xami h algorihmic roblm for a icrmal lasic rocss characrizd by h codiios rial f f (,, ) (.55) ad (.56) Th objciv is o drmi h soluio ( b,,, s, ) o h roblm(.47),(.48) ad (.49). To accomlish his w xrss h Kirchhoff srss sor s i rms of rial s ad as follows dv b s ( ) dv( F, bf, ) r( b ) (.57) 3 rial s r( b ) 3 Th uda of Kirchhoff srss sor ad h sor b d h drmiaio of h rac of b. By akig h rac of quaio (.47) ad usig (.5) w coclud ha, rial r( b ) r( b ) (.58) Th subsiuig (.58) i (.47) w g,, b rial ( b r b ) 3 (.59) ad usig h hyrlasic rlaioshis yilds s rial, rial s r( b ) 3 (.6) 99

100 Thrmolasic rial sa: Giv iiial daa b,, ad h curr valus F,, s From (.57) ad h dfiiio s s, h ormal is drmid i rms of h rial srss s rial, l, J 3, F F ad ( ( )) 3 rial rial s y, f rial, IF f s b,, b rial,, ad EXIT ELSE, rial Cosiscy aramr: S r( b ) ad 3 comu by solvig g( ) rial, rial f r( b ) 3 (, ( y )) 3 (, ( y )) 3 Rur ma: S rial rial s s ad rform h uda s s b 3 3 rial, rial r( ) 3 y, ( ( )) 3 Uda h irmdia cofiguraio by h closd form formula,, b rial ( rial b r b ) 3 END Box 7. Rur maig algorihm for flow hory. Isoroic Hardig

101 s s b 3 s b s 3 rial rial, rial r( ), rial rial rial r( ) rial rial rial s s rial (.6) By akig h do roduc of (.57) wih ad usig (.49), w obai h followig scalar oliar quaios ha drmis h cosiscy aramr : rial, rial g( ) s r( b ) ( y, ( )) 3 3 rial, rial f r( b ) ( y, ( )) (.6) 3 3 (, ( y )) 3 Equaio (.6) is ffcivly solvd by a local Nwo iraiv rocdur sic g ( ) is a covx fucio for h isoroic hardig fucios usd i his work, ad h covrgc of h Nwo-Rahso is guarad. Oc is drmid from (.6) h irmdia cofiguraio, h hardig variabl ad lasic roy ar udad from (.46) IMPLEX igraio schm Th IMPLEX (Imlici-EXlici) adod hri is h o iord by Olivr. al. [8], origially cocivd for addrssig h roblm of robusss ad sabiliy arisig i h umrical simulaio of marial failur. Th ssc of h mhod is o solv xlicily for som variabls, i h ss ha h valus a h bgiig of h icrm ar rsumd kow, ad imlici for ohr variabls, wih h rimary moivaio o hac h scral roris of h algorihmic ag moduli. Howvr, our rimary moivaio of usig IMPLEX is o rduc h quaio solvig ffor associad o h soluio of h fully imlici schm. Th xlici igraio of som variabls i h could hrmomchaical J flow hory ad hrfor, h us of xraolad valus i h balac of momum ad rgy, allow us o solv a could hrmo-mchaical roblm as a squc of hr ucould roblms. Firs, a lasic roblm wih shar modulus chagig from lm o lm; scod, a hrmal roblm wih a mraur dd lasic ha sourc ad fially, a rlaxaio rocss affcig h srss ad h iral variabls a h igraio ois. I is imora, o rmark, ha h mchaical ad hrmal roblm ar solvd usig

102 a IMPLEX igraio schm of h J lasiciy modl, whil rlaxaio ss calculas srsss ad iral variabls usig h imlici Back-Eulr im igraio rsd i h rvious scio. Th argums i suor of IMPLEX igraio schm i h umrical simulaio of mal cuig wr alrady u forward abov. Hr w simly choos h variabl o b rad xlicily ad driv h srss uda algorihm Giv iiial daa b,, ad h curr valus F,,, Exlici xraolaio 3 3 L F J 3F ad s,,, rial T rial, rial s dv b b f b f, rial 3 Comu h srsss ad h lasic roy : S r( b ) 3 ad rial rial 4 Comu h Plasic Powr s s. Prform h uda rial s s 3 y, ( ( )) ( ) 3 D mc y Box 8. IMPLEX xlici sag for hrmo-laso-lasic modls. arisig from his choic. By dfiiio, h quival lasic srai is a moooically icrasig fucio of im,. For his raso, i is a logical cadida o b rad xlicily, sic is voluio ca b rdicd mor accuraly ha ohr variabls xhibiig o-moooic bhavior. Th followig aalysis ursus, o dvlo a xrssio for xlicily udaig h quival lasic srai a usig

103 valus obaid i rvious im ss by a imlici Backward-Eulr igraio rocdur. L us cosidr, h Taylor xasio of h quival lasic srai a aroud : ( ) O( ) Nx, h Taylor xasio is carrid ou a aroud, yildig ( ) O( ) (.63) (.64) Th sadard xlici diffrc schm is obaid rucaig h rmaidr rms O( ). Th abov xlici diffrc quaio rss a icovic ha sur ha h yild codiio is o forcd a ad as a rsul, i is ossibl for h soluio, ovr may im ss, o drif away from h yild surfac. I ordr o avoid ha his drif from h yild surfac grows ubouddly, Olivr al. roos o aroxima h driva i (.64) usig h drivaiv aarig i (.63). Hc, rucaig h rmso i quaio (.63), o gs ( ) (.65) Th abov quaio is a Backward-Eulr igraio of h quival lasic srai, i h ss ha h quival lasic srai a,, is obaid by a xrssio ha uss a drivaiv valuad a. As a rsul, ad ar obaid a ims ad usig h imlici schm rsd i h rvious scio. From (.65), w ca dduc ha (.66) Fially, isrig h abov xrssio io (.64), ad rucaig h rmaidr rms, yilds ( ) (.67) This xrssio cosius a xlici xraolaio of h quival lasic srai a i rms of h imlici valus comud a ad. No ha 3

104 h IMPLEX algorihm is a mulis mhod, sic wo ois ar usd o advac h soluio i im o oi. Th algorihmic lasic mulilir rsulig from his xraolaio rads: 3 () 3 ( ) (.68) 3 Exrssio (.68) rvals ha h lasic or lasic aur of h rsos rdicd by h IMPLEX igraio schm a is dicad by h rsos comud imlicily a. This may giv ris o ovrshoos ad oscillaios i h rasiios from lasic o ilasic ad vic vrsa. Now, ss 3 ad 4 i Box 7 ca b ursud i rms of xraolad lasic mulilir yildig h IMPLEX igrad valus of h rmaiig variabls,. s ad Thos IMPLEX rsuls ar h subsiud i Box 5 o fulfill h momum ad rgy quaios. Th IMPLEX xlici sag for boh cass is summarizd i Box Algorihmic cosiuiv sor Th ulima goal i h umrical simulaio of mal cuig rocsss is o solv a iiial boudary valu roblm (IBVP) for h dislacm ad mraur filds. Th umrical soluio of his roblm rlis o h saial discrizaio, via a Galrki fii lm, of h momum ad rgy quaios ad a im discrizaio of h dislacm, vlociy ad mraur filds. I cas of a imlici discrizaio h rsos i obaid by solvig a squc of liarizd roblms. Th horis udrlyig h saial ad moral discrizaio wr rsd i scios.8. ad.8.. Th liarizaio of h wak form of h momum ad rgy quaio ar o addrss i his work. W rfr h radr o [3, ] for furhr dails. I h followig lis, w rovid a xrssio for h algorihmic ag moduli, which is a ky asc i h liarizaio of h wak form of h momum quaio. I addiio, w rovid a liarizaio of h lasic owr rlva i h liarizaio of h wak form of h rgy quaio. Algorihmic cosiuiv sor: imlici igraio schm Th xrssio for h ag moduli for h imlici srss udad algorihm will b rsd i h followig lis 4

105 S C rial Adv N DEV( N ) 3N N (.69) whr h cofficis, ad 3 ar dfid by h xrssios S rial S d 3 d rial y (.7) 3 Ad, whr 3 rial S rial A dv is giv by rial S d 3 d rial S rial dv C 3 3 J C C C C A C C I whr 3 I h oraor has h followig como form y 3 rial S (.7), ijkl C, ikc, jl C, ilc, jk (.7) I is imora o rmark ha, h cosis dviaoric ag modulus is osymmrical. Th las oi o coml h drivaio of h cosis ag modulus is o calcula h drivaivs of h isoroic hardig fucio usd i his work wih rsc o h lasic mulilir. Th followig quaios rs h drivaivs of h Voc ad Simo, Johso Cook ad Bäkr isoroic hardig fucios. I cas of h Johso Cook ad Bäkr modls h drivaiv dds o h quival lasic srai ra as showd i quaios (.59) ad (.6). Firs, i cas of h Voc ad Simo modl h drivaiv aks h followig valu ( ) y H( w ( )) h if h ( K ( w ( ) K ( w ( )) x( ) (.73) 5

106 Scod, i cas of h Johso Cook h drivaiv aks h followig valu if h lasic srai ra is lss ha h hrshold srai ra If m y A B ml m B( ) ml ( ) ( ( ) ) (.74) I addiio, i cas h lasic srai ra is grar ha h hrshold srai ra h drivaiv aks h followig valu If m y C l( ) B( ) ml m ( ( ) ) ml ( ) A B C (.75) Fially, h drivaiv aks h followig form i cas of h Bäkr modl If ( ) If y K ml ( ) x( )( ) x( ) ml ml K x( ) x( )( ) y C K ml ml l( ) x( ) ( ) x( ) ml x( ) (.76) (.77) x( ) K x( )( ) ml C ml I quaios (.76) ad (.77) w hav usd a gx ( ) gx ( ) gx () a l( a ) (.78) x x Sic, h srss uda formula is cas i rms of saial quaiis; i is covi o rasform h marial algorihmic ag moduli (.69) io h saial cofiguraio via a ull-forward oraio as follows ml 6

107 s a F F F F A dv dv rial s 3 s dv( ) (.79) (.8) Algorihmic cosiuiv sor: IMPLEX igraio schm Th drivaio of h algorihmic ag moduli for h IMPLEX srss uda algorihm follows a similar rocdur o ha usd for h imlici schm. Th osymmrical xrssio for h cosis dviaoric lasolasic modul for h IMPLEX srss uda schm is giv by S rial Adv Adv N DEV( N ) 3N N (.8) C whr h cofficis, ad 3 ar dfid by h xrssios rial rial 3 S rial S 3 S (.8) rial ad A dv is giv by quaio (.7). As was said abov, a comariso of h cofficis of quaio (.8) ad quaio (.7) shows ha h algorihmic ag modulus is simlr i IMPLEX schm ha i imlici schm. Also, quaio (.8) shows ha h ag moduli of h IMPLEX schm is idd of h isoroic hardig fucio usd, by h abov raso h ask of imlmig a w hardig fucio isid h IMPLEX schm is simlr ha i h imlici schm. Sic, h srss uda formula is cas i rms of saial quaiis; i is covi o rasform h marial algorihmic ag moduli (.8) io h saial cofiguraio via a ull-forward oraio as follows s adv F F F F Adv (.83) rial s 3 s Liarizaio of h algorihmic dissiaio dv( ) (.84) 7

108 I h sam way, h soluio of h mchaical roblm usig a imlici igraio schm rquirs h algorihmic lasolasic ag moduli, h soluio of h hrmal roblm rquirs h liarizaio of h algorihmic dissiaio. Th mchaical dissiaio (.75) ha coms from h fr rgy (.49) dds oly o h iiial flow srss.this faur, howvr, is o cosis wih h xrimal obsrvaio o mals which suggs ha ar of h work hardig ossss a dissiaiv characr. I ordr o accommoda h xrimal obsrvaios iroducd abov io h homological hrmolasic cosiuiv modl, a addiioal dissiaio hyohsis cocrig h amou of mchaical dissiaio mus b iroducd. I racic, his is accomlishd by assumig ha h mchaical dissiaio is a fracio of h oal lasic owr. whr.85,.95 y D mch ( y ) 3 (.85), is a cosa dissiaio facor chos i h rag of Liarizaio of h algorihmic dissiaio: imlici igraio schm A imlici Backward-Eulr im discrizaio of h lasic dissiaio is show i h x quaio ( ) D mch 3 y (.86) Th drivaiv of h dissiaio wih rsc o h mraur is giv by h followig xrssio D ( ) y mc a 3 b 3 whr h cofficis a ad b ar giv by h xrssios ( y ) a 3 ( y ) b Th rms a ad b dds o h yild fucios y (.87) (.88). Th rm b has b calculad i h rvious scio. Thrfor, i is oly cssary o calcula h drivaiv of h yild fucios wih rsc o h mraur fild, as is show i h followig lis Firs, h driva wih rsc o mraur of h Simo ad Voc yild fucio is 8

109 y y K w H Kif( x( )) wh ( ) ( x( )) (.89) Scod, h driva wih rsc o mraur of h Johso-Cook yild fucio is If If m y ( ( ) ) ml ml ( ) m A B (.9) ( ) ( ) C l( ) y y (.9) Fially, h drivaiv wih rsc o mraur of Bäkr yild fucios is giv by If ( ) If x( ) ( ) l( ) y ml ml K x( ) ml x( ) ( ) ml x( ) ml ( ) ( ) ml y y K C l( ) (.9) (.93) From quaios(.9), (.9), (.9) ad (.93), i is imora o o ha h liarizaio of h lasic dissiaio is icwis dfid. Liarizaio of h algorihmic dissiaio: IMPLEX igraio schm Sarig from h xraolad valu of h lasic mulilir, h lasic dissiaio a could b wri as ( ) D mch 3 y (.94) As h xraolad valu of h lasic mulilir is hld cosa durig h im icrm, h liarizaio of h IMPLEX dissiaio is giv by D mc ( y ) (.95) 3 A comariso of quaios (.95) ad (.87) shows how siml i is o liariz h lasic dissiaio i cas of usig IMPLEX. 9

110 Th drivaiv of h yild fucio wih rsc o h mraur fild for ach of h modls usd i his work hav b rsd i quaios (.89), (.9),(.9),(.9) ad (.93). Usig h cofficis iroducd i quaio(.88), h liarizaio i cas of IMPLEX is simlifid as D mc a (.96).8.4 Discrizaio of h fricioal coac modl.8.4. Coac bw h workic ad a rigid ool Th uros of his scio is o dscrib a aroach for modlig h ool as a Figur. Rigid ool discrizaio rigid body. Th ool is modld as hr surfacs movig a cosa sd (ool sd), ach of hs surfacs rrsig h rak fac, h ool i ad h flak fac, rscivly (Figur ). Mahmaically, a workic aricl has rad h ool if i saisfis h iqualiis rsd i h followig quaio

111 T O O L f y cg( )( y R cos C ) C Rs( ) r, j, i, i, j ( y, i C, i ) ( y, j C, j ) R f a y, j, j y, i, i ff y, j a( )( y, i C, i Rs, j C C 3 ( )) C R cos( ) (.97) whr R is h ool radius, h rak agl, h flak agl, C h cr of h ool i ad ( y, i, y, j ) ar all ois ha li wihi or o h surfac of h ool. A workic aricl is i coac wih h ool if i saisfis ay of h qualiis rsd i h abov quaio. Ohrwis, h aricl is o i coac wih h ool. Nx, w dscrib h rocdur o calcula h coac forcs bw h ool ad h workic usig a sragy of ga alizaio (raio) rsd i scio.3.4 Giv h s of aricls ha hav rad h ool, calcula h aricl rojcio o ach of h surfacs usig h quaios rsd i Tabl 4. Closs oi rojcio Rak fac x x k k k, i, i k k, j x, j x ( C R) x k k, i, j, i, j, i x mx m ( R cos( ) C ) mc mrs( ) m m x mx m( R cos( ) C ) C Rs( ) k k, j, i, i, j, j m Tool i k x C k k x C x R C Flak fac x x k k x k k, i x, i k, j, j x ( C R) k k, i, j, i, j, i x x ( ( C Rs( )) C R cos( )) x x ( ( C Rs( )) C R cos( )) k k, j, i, i, j, j Tabl 4. Closs oi rojcio of a workic aricl.

112 Rak fac Normal cos,s x Tool i C k R Flak s, cos fac k Ga k k k k g x x Rak fac Tool i Flak fac Tabl 5. Normal vcor o h surfac of h ool ad ga dfiiio Coac forc Coac siffss marix k k k c, g F Th calcula h surfac ormal a h closd oi rojcio usig quaios rsd i Tabl 5 ad calcula h disac bw a aricl ad is closd oi rojcio (ga) for ach of h surfacs. For ach aricl, slc h surfac havig h miimum ga. Tha is o say, ach aricl has b associad wih h disac ha has rad io h ool ad h ormal o h surfac o which i has rad. I h las s, a coac forc roorioal o h ga i h dircio o h surfac ormal is alid o h aricl (Tabl 6). I h global imlici schm dvlod i his work, h liarizaio of h coac forc is cssary o sur quadraic covrgc; hrfor, i is also rsd i Tabl 6. Th mahmaical rocdur o driv h quaios rsd i Tabl 4, Tabl 5 ad Tabl 6 is basd o h soluio of h miimal disac roblm bw h ool surfac discrizd i hr surfacs ad h workic aricl. Th miimal disac roblm is rsd i h followig quaio k k arg mi f ( y ) x x y (.98) k R k k g k k C x C Tabl 6. Coac forc ad coac siffss agais a rigid ool usig a aly aroach. whr y is h uda osiio of h workic aricl ad k x is is closs oi rojcio. A daild soluio of h roblm (.98) for ach of h hr surfacs ha dscrib h ool is o rsd bcaus i is a log bu a siml calculaio. Th osiio of h surfac ool dds o h cr of h ool i, as is show i quaio(.97). Likwis, h cr of h ool i dds o ool vlociy ad im hrough h x quaio x

113 C C ; V (.99) I cas of h xlici igraio schm, quaios rsd i Tabl 4 rforms h corrcio of h osiio of h aricl usig a miimum disac cririo Coac bw h workic ad a lasic ool Th od-o-sgm (NTS) coac lm is o of h mos commoly usd discrizaio i larg dformaio fii lm simulaio of coac () () roblms. Cosidr ha a workic aricls wih coordia x s ad s mraur coms io coac wih h masr (ool) sgm dfid by h () () odal coordias x, x ad odal mraurs surfac coordia alog h masr surfac, w hav ad (), (). By iroducig h () () () () x ( ) x ( x x ) (.) () () () () ( ) (.) whr x () () ad () ()ar liar irolaios of h masr surfac ad h masr surfac mraur rscivly. Th ormalizd ag vcor of h masr sgm ca b asily comud as Figur. Nod-o-sgm coac lm. 3

114 () x () () () x x ˆ () ( ) a (.) l l () () whr l x x is h lgh of h masr sgm. Th ui ormal o h masr sgm ca b comud as () () 3 a (.3) Th fii lm discrizaio of quaio (.87) ca b wri as N () () () () s ( ) g x x x (.4) () whr is h rojcio of h workic aricl x s oo h masr sgm. Mahmaically, h rojcio is wri as ( () () () xs x ) a (.5) l Th rm g N i quaio (.4) is also kow as h rlaiv dislacm i h dircio ormal o h ool. Th cosiuiv rlaioshis for h ormal forc P N ar giv i quaio (.89) ad is discr vrsio is giv by PN g N (.6) Now, w iroduc h discr rlaiv agial vlociy by ( ) l gt (.7) Ad h discr rlaiv dislacm g ( ) l (.8) T whr is h rojcio a h bgiig ad is h rojcio a h d of h im s. Iroducig quaio (.7) i quaio (.93), h discr courar of h Noro-Hoff fricio law is wri as PT gn ( gt ) sig( g T ) (.9) Th fii lm discrizaio of h hrmal ga rrsd i quaio (.94) is giv by () s () () g ( ) (.) Q From (.96) ad usig (.) h discr coac ha flux is wri as () s ad h discr fricioal dissiaio () () Q h ( ) (.) l D sig P g fric r ( ) N T (.) 4

115 Th coribuios of h hrmo-mchaical coac i h mchaical ad hrmal wak forms (.4) ak h form (,) c, mch N N T T (,) c, hrm Q fric D G P g P g G Q g D g For h discr slav aricl h mchaical variaios ar giv by g g N T l () () () () s ( ) () () whr w hav s s x s ad Ad h hrmal variaios whr ad () () () () N () () () s ( ) a ( ) Q () () i x i () () () s ( ) () () () s g v v v g v (( ) v v ) D ar h orios of h flux fric rscivly. Mahmaical xrssios for ad g l (.3) (.4) (.5) D rig bodis () ad () ar giv by quaio () () () () (.97). I quaio (.5) w hav s v s s ad v i i. For a global algorihmic ram usig Nwo s mhod (imlici ad IMPLEX igraio schms), w hav o liariz quaios (.3). Exlici xrssios for h ag siffss marix associad o a ormal forc giv by quaio (.6) wr firs drivd by Wriggrs ad Simo i [9]. For comlss, h liarizaio of h coac coribuio will b discussd hr for h cas of aly mhods (quaio (.6)). Wihi h aly aroach, h sarig quaio for h NTS-lm is h coribuio of h ormal coac forc o h virual work ricil giv by h x quaio PN gn gn g N (.6) Th liarizaio of his xrssios yilds P g g g g g (.7) whr ad N N N N N N g N is giv by h xrssio () () () () g u ( ) u u (.8) N g N aks h form s 5

116 () () () () () () () gn us ( ) u u a ( ) l gns () () () () () () ( u u ) ( ) (.9) l () () () () () () () ( u u ) ( s ( ) a ) l whr w hav usd h liarizaio of h ormalizd ag vcor of h masr sgm, () () () () () u u ( ) ( ) a (.) l h liarizaio of h ui ormal o h masr sgm, () () () () () u u ( ) ( ) a (.) l h liarizaio of h masr sgm lgh () () () l ( u u ) a (.) ad h liarizaio of h rojcio of h slav od oo h masr sgm () () () () gns () () () us ( ) u u a ( u u ) (.3) l l Iroducig ad w ca say h marix form of wih u s u u u () () s s () () s () (),, T ( ) a, T a, s () a () () s () () a a s s N ( ), N s () () () s () () s s (.4) (.5) (.6) g N T gn s K u s (.7) K T T gn T s s s os s os l N T T N l N N (.8) 6

117 I cas of h aly mhod This yilds h siffss marix T T N N s s os s s P g N N gn K u (.9) K Ns T Ns s os gn K N N K T gn T T g (.3) N T N snos N sts Ts Nos N snos l l Th oi of darur i h cas of fricioal coac usig Noro-Hoff fricio law is h coribuio of h fricio law o h virual work ricil giv by h x quaio ( ) l P g g g Liarizaio of h abov xrssio yilds T T N T g P g g g T T T N T Th liarizaio of h ormal ga g g g N N N g g g T T T g N l ( ) g T g T l g T (.3) (.3), h liarizaio of h rojcio of h slav od oo h masr sgm, h liarizaio of h curr masr sgm lgh l has b rsd i quaios (.8),(.3) ad (.). Th liarizaio of h variaio g T gt l l (.33) Th liarizaio of h variaio is obaid afr som algbra 7

118 () u ( ) ( ) s u u a a l ( u u ) a ( ) a l () () () () () () () () () () () () () s () () () () () () () ( u u ) s ( ) u () s ( ) u () l u () () () () () ( ) l gn () () () () () () ( u 3 u ) ( ) a l () () () ( u u ) a () () () gn ( 3 ) l Th roduc bw h liarizaio of h variaio ad l is qual o l (.34) () () () () () () () l s ( ) a ( u u ) a l (.35) g N () () () () () () ( ) ( u u ) a l Isrig quaios (.34) ad (.35) i quaio(.33), h liarizaio of h variaio g T could b wri as () () () () () () () ( u u ) gt s ( ) l () () () () us ( ) u u () () () ( ) l () () () () () () () us ( ) u u a ( ) a (.36) l () () () a ( u u ) () () () gn ( ) l gn () () () () () () ( u u ) ( ) a l I marix form, h liarizaio of h variaio of g T is wri as follows wih K sl T sl T s s g K u (.37) N N N N T T l g T T T s s s s s s N T T N st s T snos l (.38) 8

119 Furhrmor, xrssig h roducs gn gt, g T ad lg T i marix form T T gn T gn gt s Ts Ns N sns us l T gn T Ts Ts N sts T l l gt s u s (.39) gn T ( gn ) T T sns N 3 sns l l T T gn T l gt s Ts T s N st s us l Usig xrssio i (.39), quaio umbr (.3) ca b xrssd as follows wih T NH ( PT gt ) s K u s (.4) NH gt T gn T K Ts Ns N sns l g T T gt l l gn g g g g T N T s s N sts N T ( N ) T T sns N 3 sns T T N T N Ts T s N st s l g l l (.4) gt gn K sl Th abov quaio rrss h ag siffss marix associad o h Noro-Hoff fricio law. No ha his marix is usymmrical which corrsods o h o-associaiviy of h Noro-Hoff fricio law. Somims, h raio or ga g N is much lowr ha h masr sid lgh l, ha is, g N l. I h abov siuaio, mos of h rms of h siffss marix (.4) ar gligibl. I his cas, h siffss marix ca b simlifid as NH gt T gt T K Ts Ns gn Ts T s (.4) Mos of h ims, i h umrical simulaio of mal cuig, h ool advac r im s ar much smallr ha h characrisic msh siz of h ool. Thrfor, mos of h umrical simulaio of mal cuig carrid ou i his work has b carrid ou usig h siffss marix (.4). Th siffss marix rsd i quaios (.4) will b usd oly i cas ha h ool advac r 9

120 im s is comarabl wih h siz of h ool lm or wh covrgc roblms xis. I h sam way ha h mchaical roblm ivolvs h liarizaio of h coribuio of ormal ad fricio forcs o h virual work ricil wih rsc o dislacm, h hrmal roblm ivolvs h liarizaio of h coribuio of h coac ha flux ad fricioal dissiaio o h virual hrmal work wih rsc o mraur. Th, h followig lis rs h liarizaio of h virual hrmal work wih rsc o mraur for h NTS coac lm (,) c, hrm G Q g (.43) I (.43) h discr coac ha flux Q follows from (.), whras h discr fricioal dissiaio D is giv by quaio (.). Th liarizaio fric of h discr coac ha flux aks h form Q h, ( ), (.44) I marix form, h hrmal siffss marix associad o h ha rasfr a h ool chi irfac is giv by h followig xrssio ( ) Q K h ( ) ( ) ( ) (.45) ( ).9 Mshig i h Paricl Fii Elm Mhod (PFEM) Th iiial dvloms of h Paricl Fii Elm Mhod (PFEM) ook lac i h fild of fluid mchaics [], bcaus PFEM facilias rackig ad modlig of fr surfacs. Lar o, h Paricl Fii Elm (PFEM) was alid i a variy of simulaio roblms: fluid srucur iracio wih rigid bodis, rosio rocsss, mixig rocsss, could hrmo-viscous rocsss ad hrmal diffusio roblms [, ]. Firs alicaios of PFEM o solid mchaics wr do i roblms ivolvig larg srais ad roaios, muli body coacs ad craio of w surfacs (rivig, owdr fillig ad machiig) [3]. I his work, w xdd h Paricl Fii Elm Mhod o h umrical simulaio of mal cuig rocsss. Th Paricl Fii Elm Mhod is basd o hr mai igrdis: ) h Dlauay riagulaio, ) h -sha mhod ad 3) h fii lm mhod. Th fii lm mhod was xlaid arlir i his char. Th followig lis will mak a brif iroducio o h Dlauay riagulaio ad -sha mhod, as wll as w will jusify hir us.

121 Figur 3. Rmshig ss usig h sadard PFEM. (a) Disribuio of ois; (b) Dlauay riagulaio; (c) Dlauay riagulaio; wih sha.[] A Dlauay riagulaio for a s P of ois i a la is a riagulaio DT( P ) such ha o oi i P is isid h circumcircl of ay riagl i DT( P ). Dlauay riagulaios maximiz h miimum agl of all h agls of h riagls i h riagulaio; hy d o avoid skiy riagls. To avoid global rmshig ad irolaio from msh o msh, i h Paricl Fii Elm Mhod (PFEM) msh qualiy is imrovd by rformig a rriagulaio of h domai which cosiss of r-comuig lm cociviy usig a Dlauay riagulaio whr h curr osiio of h aricls (i.. of h msh ods) is k fixd. Msh disorio is allviad i h siri of h Paricl Mhod (PFEM), bcaus whvr h msh qualiy is o logr saisfacory, accordig o som criria associad o lm disorio, h cociviy of xisig ods is rcomud usig a Dlauay riagulaio. This choic has som imora imlicaio, h Dlauay riagulaio gras h covx figur of miimum ara which closs all h ois ad which may b o coformal wih h xral boudaris. A ossibiliy o ovrcom his roblm is o coul h Dlauay riagulaio wih h so-calld -sha mhod. Th mai ida of h -sha mhod cosiss i rmovig h ucssary (oo larg ad oo disord) riagls from h msh usig a cririo basd o h msh disorio. For vry riagl of h msh h radius R of h circumcircl of h lm is comud. Morovr, a yical dimsio of h msh h is calculad as h dim dim b x j i bi, i x j (.46)

122 Giv h iiial oi s Figur 4. Paricl fii lm (PFEM) flowchar i comuaioal fluid mchaics roblms. whr b is h umbr of ighbor aricls, which ar dfid as h aricls coaid i h sam simlics as aricl j ad dim is h dimsio of h roblm. Th idx of a lm disorio h All h riagls ha do o saisfy h codiio: R is comud as: (.47) (.48) ar rmovd from h msh, whr is a aramr o b assigd o h basis of h gomric accuracy rquird for h cosidrd roblm. Th -sha mhod ca gra aricls ha do o blog o ay lm.

123 Thr ar svral oios for h ram of isolad aricls. Th firs o is l hm mov frly, bcaus hy ca b rjoid o h body i fuur im ss. Th scod o is rmov h aricl from h aalysis. This is rcommdd wh aricls do o coribu aymor o h mchaical roris of h domai. A xaml of h rmshig schm usig PFEM is show i Figur 7. I h Lagragia aroach, h aricls mov bcaus of h mdium flow ad i may ha ha aricls cocra i sam rgios of h domai ad, o h corary, i ohr rgios h umbr of aricls bcoms oo low o obai a accura soluio. To ovrcom hs difficulis PFEM adds ad rmovs aricls. If h disac d ods bw wo ods is dods h, o of h ods is rmovd. If h radius r of a lm circumshr is r h, a w od is addd a h cr of h circumshr. Th flow variabls i h w od ar liarly irolad from ha of h lm ods, ad h assigd marial roris ar h os of h lms. Th PFEM cosiss of h followig ss. Th iiial aalysis domai is filld wih a s of ois rfrrd o as aricls which ar dowd wih iiial vlociy, rssur ad osiio. Th accuracy of h umrical soluio is clarly dd o h cosidrd umbr of aricls. A iiial fii lm msh is grad usig h aricls as ods hrough a Dlauay riagulaio ad xral boudaris ar idifid by mas of h -sha chiqu. As log as msh disorio is accabl, solv h o-liar Lagragia form of h govrig quaios fidig vlociy ad rssur a vry od of h msh. A flowchar soluio schm usig PFEM alid o fluid mchaics roblms is rsd i Figur 4. I h PFEM, h siz of ach im s is assumd small ough o avoid rmshig durig h iraios for h soluio of h o-liar quaios i h im s islf. Msh disorio is chckd oly a covrgc..9. Mshig i h Paricl Fii Elm Mhod: umrical simulaio of mal cuig rocss Th sadard PFEM rss som waksss wh alid i orhogoal cuig simulaio. For xaml, h xral surfac grad usig -sha may affc h mass cosrvaio (Figur 5 ad Figur 6), h chi sha ad somims gras a uhysical wldig of h workic ad h chi (Figur 6). To dal wih his roblm, i his work w roos h us of a cosraid Dlauay algorihm. Furhrmor, addiio ad rmoio of aricls ar h ricial ools, which w mloy for sidsig h difficulis, associad wih dformaio-iducd lm disorio, ad for rsolvig h diffr scals of h soluio. Th isrio of ods is basd o h quidisribuio of lasic owr, such ha, lms xcdig h rscribd olrac TOL 3

124 ( i),( ) S mch () D ds TOL (.49) ar argd for rfim. Hr, D is giv by quaio (.86) ad mch () ds dos h domai of h lm. A aricl is isrd i h gauss oi of h fii lm..5.4 alha sha Cosraid Dlauay Workic ara alha sha valu (a) (b) Figur 5. Waksss of h alha sha mhod i umrical simulaio of machiig (a) workic volum ddcy o h alha sha valu (b) workic sha usig a alha sha=. 4

125 Th rmoio o ods is basd o a Zikiwicz ad Zhu [4, 5] rror simaor dfid by h followig xrssio * (.5) * whr is h rcovrd quival lasic srai ad max is h maximum quival lasic srai. A aricl is rmovd if ad oly if, h rror i all h lms blogig o h aricl is lss ha a giv olrac ad h siz of h lms is lss ha a maximum valu h max. max.35 alha sha Cosraid Dlauay.3 Workic ara alha sha valu (a) (b) Figur 6.Waksss of h alha sha mhod i umrical simulaio of machiig (a) workic volum ddcy o h alha sha valu (b) uhysical wldig of h workic ad h chi du o h alha sha. 5

126 I h umrical simulaio of mal cuig rocss, dsi h coiuous Dlauay riagulaio, lms aris wih uaccabl asc raios; for his raso, h msh is also subjcd o lalacia smoohig. Lalacia smoohig is a algorihm o smooh a msh. For ach od i a msh, a w osiio is chos basd o h osiio of ighbors ad h od is movd hr. I h cas ha a msh is oologically a rcagular grid h his oraio roducs h Lalacia of h msh. Mor formally, h smoohig oraio may b dscribd r-od as: N xj x i (.5) N j Whr N is h umbr of adjac vrics o od i ad x i is h w osiio for od i. All h iformaio cssary i subsqu im ss has ow o b rasfrrd o h w msh, i icluds odal iformaio lik dislacms, mraurs, rssur ad Gauss oi iformaio lik iral variabls. Th mai faur of h PFEM is ha fii lms ar oly a ma for h comuaio, bu hy ar o cssarily k. This mas ha h msh ha dfis h domai for a iiial oi s ad boudary cosrai cosraid dlauay h riagls ha li isid of h domai uda of h aricl osiio Figur 7. Rmshig ss i h umrical simulaio of mal cuig. 6

127 giv im s ca chag for h x im s. This is h raso for assigig all h roblm variabls o h aricls. Cosquly, h hisory of h coiua rmais i h aricls. Thy bcom h saial rfrc ad all iformaio of h aalysis is k o hm. Th sadard PFEM[3] rasfr iral variabls usig h rasfr oraors rsd i [6] or is icrmal vrsio rsd i [7]. Alhough h schm rsd i [7] dcras h umrical diffusio of h sa filds i comariso wih h schm rsd i [6], i boh schms, h rasfrrd variabls viola h cosiuiv quaio, graig as a rsul a uhysical srig back of h machid surfac. (a) (b) Figur 8. Prdicd chi sha usig diffr rasfr oraors (a) smoohig, (b) rojcio cuig forc (N/mm) 5 5 rojcio smoohig ool dislacm(mm) Figur 9. Prdicd cuig forcs usig diffr rasfr oraors. 7

128 Giv h iiial oi s Cosraid Dlauay Triagulaio Idify h riagls ha li isid of h domai Soluio of h balac of momum ad ha quaio FALSE k=k+ NO Chck covrgc YES Mov h ods o h w osiio = + Plasic Powr>TOL msh disorio>tol FALSE TRUE TRUE Add ad rmov aricls Figur 3. Flowchar of h soluio schm i h umrical simulaio of mal cuig rocsss. Figur 8 shows h rdicd chi sha a h sam im s usig h smoohig ad h rojcio rasfr oraors. A comariso shows ha h smoohig rasfr oraor rdic a uhysical srig back du o xcssiv umrical diffusio, du o h uhysical srig back h smoohig rasfr oraor is uabl o rdic h rsidual srsss ovr h machid surfac. Furhrmor, Figur 8 shows ha h bdig of h chi is mor rooucd i h umrical simulaio usig h smoohig ha i h rojcio rasfr oraor. Figur 9 shows ha h rdicd cuig forc usig h rojcio oraor rachs a sady sa, whil, h smoohig oraor rdics a cuig 8

129 forc ha dcrass as h ool movs. Du o h rvious rasos, h rojcio oraor from o msh o aohr usig formr ad w igraio ois will b usd i all h umrical simulaio of mal cuig rsd i his work. Alhough, h rojcio braks h hilosohy of aricl basd mhods lik PFEM ad icras h comuaioal cos i comariso wih h smoohig oraor, h uhysical rsuls rdicd wih h smoohig oraor jusifis usig h rojcor oraor. Th rocdur w follow i his work i ordr o ada h msh is h followig:. Uda h aricl osiios. Add aricls i hir Gauss oi if hir lasic owr is grar ha a giv valu. 3. Basd o curvaur iformaio ad lasic owr rfi h boudary ha dscribs h workic. 4. Rmov aricls if rror simaors i lasic srais ar lss or qual ha a giv valu. W rmov a aricl if i all h old fii lms joid o a aricl h rror is lss or qual o a giv olrac. 5. Dvlo Dlauay Triagulaio cosraid by h rfid old msh boudary ad dl h riagls ousid h boudary. 6. Esimas msh qualiy. If msh qualiy is lss or qual ha a giv olrac, dvlo a Lalacia smoohig of h udad aricl osiios. Fid smoohd aricls i h w msh. Trasfr aricl iformaio (dislacm, rssur, mraur) usig sha fucios 7. Calcula h global coordias of h gauss ois of h w riagulaio. 8. Fid which riagl of h old riagulaios coais ach of h Gauss ois of h w riagulaio. 9. Usig h iformaio of 4, uda h iral variabls of h w riagulaio. This s uss ha h Gauss iformaio of fii lm of h w msh is h Gauss iformaio of h closs fii lm of h old msh. I is imora o rmark ha s 4 ad 6 ar oioal. Th mai advaag of h roosd sragy is: i is o cssary o cra a coml w msh; w oly ada h msh wih h addiio ad rmoio of aricls ad imrov msh qualiy usig Dlauay riagulaio. shows a summary abou h rmshig schm usd i h umrical simulaio of mal cuig. Fially, Figur 3 shows h soluio schm of a mal cuig roblm usig h aricl fii lm mhod (PFEM). 9

130 Rfrcs [] M. Crmosi, A. Fragi, ad U. Prgo, "A Lagragia fii lm aroach for h aalysis of fluid srucur iracio roblms," Iraioal Joural for Numrical Mhods i Egirig, vol. 84,. 6-63,. [] J. C. Simo ad C. Mih, "Associaiv could hrmolasiciy a fii srais: Formulaio, umrical aalysis ad imlmaio," Comur Mhods i Alid Mchaics ad Egirig, vol , 99. [3] J. Bo ad R. W. N, Noliar Coiuum Mchaics for Fii Elm Aalysis: Cambridg Uivrsiy Prss, 997. [4] J. C. Simo, "A framwork for fii srai lasolasiciy basd o maximum lasic dissiaio ad h mulilicaiv dcomosiio: ar I. coiuum formulaio," Comur Mhods i Alid Mchaics ad Egirig archiv, vol. 666,. 99-9, 988. [5] J. C. Simo, "A framwork for fii srai lasolasiciy basd o maximum lasic dissiaio ad h mulilicaiv dcomosiio. Par II: Comuaioal ascs," Comur Mhods i Alid Mchaics ad Egirig, vol. 68,. -3, 988. [6] E. Voc, "A racical srai hardig fucio.," Mallurgia, 955. [7] J. C. Simo ad T. J. R. Hughs., Comuaioal Ilasiciy. Nw York: Srigr-Vrlag, 998. [8] G. H. Johso ad W. H. Cook, "A cosiuiv modl ad daa for mals subjcd o larg srais high srai ras ad high mraurs," Procdigs of h 7h symosium o ballisics, 983. [9] M. Bäkr, "Fii lm simulaio of high-sd cuig forcs " Joural of Marials Procssig Tchology, vol. 76,. 7 6, 6. [] T. Blyschko, W. K. Liu, ad B. Mora, Noliar Fii Elm for Coiua ad Srucurs.: Wily,. [] M. Čaađija ad J. Brić, "Associaiv could hrmolasiciy a fii srai wih mraur-dd marial aramrs," Iraioal Joural of Plasiciy, vol., , 4. [] J. Lublir, Plasiciy Thory: Dovr Publicaios, 8. [3] W. D. Callisr ad D. G. Rhwisch, Marials Scic ad Egirig: A Iroducio: WILEY,. [4] G. S. Skho ad J. L. Cho, "Numrical simulaio of coiuous chi formaio durig o-sady orhogoal cuig simulaio," Egirig Comuaios, vol., 993. [5] P. B. Bochv, C. R. Dohrma, ad M. D. Guzburgr, "Sabilizaio of Low-Ordr Mixd Fii Elms for h Soks Equaios," SIAM Joural o Numrical Aalysis, vol. 44,. 8-, 8. 3

131 [6] C. R. Dohrma ad P. B. Bochv, "A sabilizd fii lm mhod for h Soks roblm basd o olyomial rssur rojcios," Iraioal Joural for Numrical Mhods i Fluids, vol. 46,. 83, 4. [7] F. Armro ad J. C. Simo, "A w ucodiioally sabl fracioal s mhod for o-liar could hrmomchaical roblms," Iraioal Joural for Numrical Mhods i Egirig, vol. 35, , 99. [8] J. Olivr, A. E. Hus, ad J. C. Ca, "A imlici/xlici igraio schm o icras comuabiliy of o-liar marial ad coac/fricio roblms," Comur Mhods i Alid Mchaics ad Egirig, vol. 97, , 8. [9] P. Wriggrs ad J. C. Simo, "A o o ag siffss for fully oliar coac roblms," Commuicaios i Alid Numrical Mhods, vol.,. 99-3, 985. [] S. R. Idlsoh, E. Oña, ad F. D. Pi, "Th aricl fii lm mhod: a owrful ool o solv icomrssibl flows wih frsurfacs ad brakig wavs," Iraioal Joural for Numrical Mhods i Egirig, vol. 6, , 4. [] E. Oña, S. R. Idlsoh, M. A. Cligua, ad R. Rossi, "Advacs i h aricl fii lm mhod for h aalysis of fluid mulibody iracio ad bd rosio i fr surfac flows," Comur Mhods i Alid Mchaics ad Egirig, vol. 97, , 8. [] E. Oña, M. A. Cligua, ad S. R. Idlsoh, "Modlig bd rosio i fr surfac flows by h aricl fii lm mhod," ACTA GEOTECHNICA, vol , 6. [3] J. Olivr, J. C. Ca, R. Wylr, C. Gozálz, ad J. Hradz, "Paricl Fii Elm Mhods i Solid Mchaics Problms," Comuaioal Mhods i Alid Scics, vol. 7,. 87-3, 7. [4] O. C. Zikiwicz ad J. Z. Zhu, "Th surcovrg ach rcovry ad a osriori rror simas. Par : Th rcovry chiqu," Iraioal Joural for Numrical Mhods i Egirig, vol. 33, , 99. [5] O. C. Zikiwicz ad J. Z. Zhu, "Th surcovrg ach rcovry ad a osriori rror simas. Par : Error simas ad adaiviy," Iraioal Joural for Numrical Mhods i Egirig, vol. 33, , 99. [6] D. Prić, C. Hochard, M. Duko, ad D. R. J. Ow, "Trasfr oraors for volvig mshs i small srai laso-lasiciy," Comur Mhods i Alid Mchaics ad Egirig, vol. 37, , 996. [7] J. M. Carboll, "Modlig of Groud Excavaio wih h Paricl Fii Elm Mhod,". 3

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133 Char 3 3 Numrical modlig of mal cuig rocsss usig PFEM This char rss umrical simulaios usig h formulaio roosd i his work. Firs of all, umrical simulaios of wo bchmark (Cook Mmbra ad Taylor imac s) ss ar rs ad validad qualiaivly ad quaiaivly wih soluios rord i h liraur. Furhrmor, a racio s validas h lockig fr lm y roosd i his work i hrmo-mchaical roblms. Fially, h roosd formulaio is usd i h umrical simulaio of coiuous ad srrad chi formaio..3 o corr dilacm(m) T Q QP TP OSS TP PPP Elms r sid Figur 3. Pla srai Cook`s Problm: covrgc of diffr formulaios for icomrssibl lasiciy. T sadard dislacm for riagular lms, Q sadard dislacm for quadrilarals lms, QP mixd ma dilaaio/rssur aroach for quadrilaral lms, TP OSS mixd formulaio for liar riagls usig orhogoal sub grid scal as a sabilizaio sragy, TP PPP mixd formulaio for liar riagls usig Polyomial rssur rojcio. 33

134 o corr dislacm(m).5 TP OSS TP PPP msh siz Figur 3. Pla srai Cook`s Problm: covrgc of diffr formulaios for J- Plasiciy. TP OSS mixd formulaio for liar riagls usig orhogoal sub grid scal as a sabilizaio sragy, TP PPP mixd formulaio for liar riagls usig Polyomial rssur rojcio. 3. Pla srai Cook s Mmbra roblm Th Cook Mmbra roblm is a bdig domiad xaml ha has b usd by may auhors as a rfrc s o chck hir lm formulaio. Hr i will b usd o valida h roosd formulaio i icomrssibl lasiciy ad lasiciy. Th rsuls of our formulaio will b comard agais QP fii lm ad a mixd fii lm usig Orhogoal Subgrid Scal as a sabilizaio sragy. Th roblm cosiss i a ard al, clamd o o sid ad subjcd o a sharig load a h fr d. I ordr o s h covrgc bhavior of diffr formulaios, h roblm has b discrizd io 6 6,4 4 ad 4 4. Th followig marials roris ar assumd: Youg`s Modulus = 7, Poisso`s raio =.4999 ad alid forc =. Figur 3 shows h bhavior of boh quadrilaral ad riagular fii lms i cas of arly icomrssibl lasiciy. Th figur shows h oor rformac of h Q ad T sadard lms wihi h cox of arly icomrssibl lasiciy, du o a xrm lockig. Furhrmor, h figur 34

135 6 Prssur(Pa). y(m) x(m) (a) Orhogoal Sub grid Scal c= Prssur(Pa). -. y(m) x(m) (b) Polyomial Prssur Projcio c= Figur 33. Prssur fild for mixd formulaio usig Orhogoal Sub Grid Scal ad Polyomial Prssur Projcio as sabilizaio sragis ad J-Plasiciy. shows ha h roosd formulaio covrgs similarly o OSS bu a low comuaioal cos. I is imora o rmark ha i Polyomial rssur rojcio sragy h sabilizaio aramr is msh siz idd ad ha h sabilizaio rms addd o h mixd formulaio ar lmary dd. I 35

136 shows ha our roosal allows gig similar rsuls o h OSS sragy bu a low comuaioal cos. Th sabilizaio aramr usd i PPP ad OSS was. Nx xamls ivolvs Cook`s Mmbra bu J-lasiciy ad h followig assumd marials roris: Youg`s Modulus = 7, Poisso`s raio =.4999, yild srss =.43 hardig modulus =.35 ad kimaic hardig modulus.5 ad a alid forc of.8 i 5 icrms. Figur 3 shows a comariso of h o corr dislacm for h mixd fii lm usig OSS ad PPP as sabilizaio sragis. Figur 3 shows ha h covrgc bhavior of wo formulaios is rally similar. As w say i cas of lasic bhavior, PPP is siml o imlm ad do o d ad xra calculaio lik h rojcd rssur gradi i OSS. Figur 33 rss rssur coour a h sam im (fial) of h dformaio rocss. A smooh coour fild ca b idifid i boh mixd formulaios. A h sam im, h rdicd rsuls ar vry similar quaiaivly. 3. Taylor imac s Th roblm cosiss of h imac of a cylidrical bar wih iiial vlociy of 7m/s io a rigid wall. Th bar has a iiial lgh of 3.4 mm ad a iiial radius of 3. mm. Marial roris of h bar ar yical of cor: dsiy kg / m, Youg`s modulus E.7 MPa, Poisso`s raio v.35, iiial yild srss Y 4MPa ad hardig modulus H MPa. A riod of 8 s has b aalyzd. Our rsuls rs a D axisymmric simulaio W will comar qualiaivly ad quaiaivly h rsuls rdicd wih h roosd formulaio wih h rsuls rdicd wih Caracrisic Bas Sli (CBS)[] ad Avrag Nodal Prssur (ANP)[]. Th fial gomry of h bar is i good agrm wih h rsuls obaid i h liraur. W obaid similar rsuls o Zikiwicz wih CBS, Bo wih Avrag Nodal Prssur formulaio ad D Michli wih his rcly dvlod formulaio for xlici dyamics. Th fial radius usig h roosd formulaio givs a fial radius of 7.6. Tabl shows h comariso of h fial radius obaid wih our Figur 34. Prssur fild 8 s afr h imac for h roosd formulaio. 36

137 Figur 35. Equival lasic srai disribuio 8 s afr h imac for h roosd formulaio. formulaios wih h rsuls rsd i h liraur. D Formulaio Michli[3] CBS[] ANP[] This work Fial radius 7,7 7,7 6,99 7,6 Tabl 7. Fial radius of h rod afr h imac obaid wih D Michli formulaio, CBS formulaio, ANP formulaio ad h roosd formulaio of his work. Figur 34 ad Figur 35 show h umrical rsuls of h rssur ad ffciv lasic srai disribuio usig h formulaio roosd i his work. Th valus for h quival lasic srai ad rssur fild obaid wih h rs formulaio coicid wll wih hos giv by FIC ad CBS formulaio. 3.3 Thrmo-mchaical racio s W cosidr a rcagular scim i la srai submid o uiform racio forcs. Th scim cosidrd i h simulaio has a widh of.866 mm ad a lgh of mm. Figur 36 dsis h msh of h iiial cofiguraio. Th bar is assumd isulad alog is laral fac, whil h mraur is hld cosa ad qual o 73K o h ur ad lowr facs. Th oal valu of imosd dislacm is icrasd o 5mm alid i qual im ss, wih a ra of icras of mm/s. Th chos valus of hrmo mchaical roris of h scim ar giv i Tabl, hy corrsod o sl. W cosidr h sourc rm i h rgy quaio dfid as a fracio of h lasic work, i his xaml w us a facor of.9. Du o h symmry of h soluio, oly o quarr of h scim is discrizd, imosig h corrsodig symmry boudary codiios. As a soluio sragy w us a mixd liar dislacm-liar rssur fii lm ad h Polyomial Prssur Projcio as a sabilizaio sragy. 37

138 Figur 36. Plai srai arly adiabaic shar badig. Iiial cofiguraio. Elasic Modulus 69 MPa Poisso,3 Dsiy 78 Kg/m 3 Yild Srss Sauraio Srss Srai Hardig Modulus 45 MPa 75 MPa 9,4 MPa Hardig Exo 6,93 Ha Coduciviy Ha Caaciy Thrmal xasio Thrmal sofig modulus 45 N/sK,499mm /s K -5 (/K), (/K) Tabl 8. Pla srai arly adiabaic shar badig-marials roris. Th simulaios ar rformd udr quasisaic codiios wih h isohrmal imlici sli roosd by Simo [4], rsd i h rvious char ad h isohrmal IMPLEX sli roosd i his work. Figur 37 shows h mraur ad vo Miss fild a h fial cofiguraio. Figur 38 shows h load/dislacm obaid wih h roosd formulaio. Rsuls rsd i Figur 37 ad Figur 38 hav b obaid usig h imlici isohrmal sli rsd i rvious char. Furhrmor, Figur 38 shows a comariso wih h rsuls rsd by Ibrahimbhovic [5] usig a four od lm wih icomaibl mods ad Bi ad Movahhdy [6] usig a Arbirary Lagragia Eulria formulaio. Th rdicd forcs ar similar 38

139 durig h srai hardig ar of h forc dislacm curv, bu i h sofig brach of h forc dislacm curv h rdicd forcs ar diffr for h hr formulaios. Our formulaio rdics h smallr forc amog h formulaios availabl i h liraur, showig ha our roosd formulaio dos o lock i sofig. Tmraur(K) vo Miss(MPa) Figur 37. Pla srai arly adiabaic shar badig. Tmraur ad vo Miss srss fild a 5 mm. Th load dislacm curv obaid usig h isohrmal IMPLEX sli roosd i his work is rsd i Figur 39. Th oal valu of imosd dislacm is icrasd o 5mm ad alid i -5-- qual im ss o aalyz h ovrshoos ad oscillaios i h rasiios from lasic o ilasic sa. Th rsuls rsd i Figur 39 show ha h ovrshoo dcrass by icrasig h umbr of im ss usd. Usig im ss, h uhysical ovrsho rdic by h isohrmal IMPLEX sli is gligibl, alhough, h rsuls rdicd wih 5 im ss ca b cosidrabl saisfacory, akig io accou ha w idify h ovrshoo as a uhysical rdicio ha coms from h igraio schm. Th comuig im d o solv h hrmo-mchaical racio s usig h isohrmal IMPLEX sli is grar ha h comuig im dd by h isohrmal imlici sli, cosidrig ha i boh cass w ar gig h sam accuracy. Nvrhlss, h isohrmal IMPLEX sli will b cosidrd as a 39

140 alraiv i h umrical modlig of mal machiig, bcaus i mal cuig h siz of h im ss is rsricd by h coac codiios bw h ool ad h workic, i such a way ha for h im ss of irs, boh schms giv rsuls wih similar accuracy. I h rvious siuaio, h IMPLEX is a br choic bcaus is ds lss comuig im r im s i comariso wih h imlici sli. A s of xamls will b rsd i scio 3.6 ha show h advaags of h IMPLEX schm i h umrical simulaio of mal cuig rocsss. 8 Load(N/m) 6 4 Ibrahimbgovic al. mm/s Bi al. mm/s This work mm/s Dislacm(mm) Figur 38. Pla srai arly adiabaic shar badig. Load/dislacm curv 3.4 Machiig sl usig a ra idd yild fucio Our firs alicaio cocrs h cuig of a rcagular block of a commo sl of lgh 7mm ad widh 3.6mm a a vlociy of 3.33 m/s, a cuig dh of.5 mm, a rak agl of º, ad a ool radius of.5 mm. Marial bhavior is giv by a Simo law ha aks io accou hrmal sofig (Tabl 8). Coduciviy ad scific ha dos o dd o mraur, w cosidr hm cosa. Th followig assumios ar mad: Firs, h ool is suosd o b rigid ad fricio is glcd. Furhrmor, h hrmal xchag bw h ar ad h ool ar also glcd. Th iria of h ar is glcd. Imlici dyamics was usd. Tim ss wr of 8.5 which cssias 4

141 of 4.5 ss for a ool ravl of.6 mm. Th assumio ha h ool is rigid is rasoabl, sic h dformaio of h ool is scodary rlaivly o h xcssiv lasic dformaio of h workic. forc(n/mm) ss 5 ss ss ss ool dislacm(mm) Figur 39. Pla srai arly adiabaic shar badig. Load/dislacm curv. Isohrmal IMPLEX sli. Tmraur, ffciv lasic srai ra ad vo Miss coours ar rsd i Figur 4. Vo Miss srss show i Figur 4, dmosras ha rlaivly high srsss aris i h rimary shar zo ad a h ool chi irfac. Th localizaio of his zo agrs wih simlifid modls, bu i diffrs i ha h maximum srss is o cofid o oly a la of xcssiv shar. I is also imora o o h dvlom of rsidual srsss a ad blow h surfac of h roducd w surfac ad i h ur ar of h chi, scially ar h ool-chi irfac whr uloadig du o curlig of h chi occurrd. Th ffciv srai ra i h rimary ad h scodary shar zo is of h ordr of 5 ad has is highs valu clos o h ool i. Fially, mraur disribuio is show i h workic. Tmraur rachs is ak o h ool i ad o h machid surfac. Figur 4 dics cuig ad hrus forcs alid o h ool, obaid from h simulaios. Nvrhlss, h rdicd chi is coiuous; h cuig ad hrus forc dos o rach a sady sa du o h srog ddcy of h yild hardig fucio o h liar hardig modulus. 4

142 (a) (b) (c) Figur 4. Coiuous chi formaio usig a ra idd yild fucio: (a) vo Miss (MPa) ; (B) srai ra(/s); (c) mraur(k) Th coac lgh bw h ool ad h workic, h dformd chi hickss ad h shar agl ar.6 mm,.5 mm ad º. 4

143 4 forc(n/mm) cuig forc hrus forc ool dislacm(mm) Figur 4. Cuig ad hrus forc vs. cuig ool dislacm for a ra idd yild fucio 3.5 Machiig a AISI 434 usig diffr rak agls Th scod alicaio cocrs h cuig of a rcagular block of high srgh AISI 434 sl of lgh 7mm ad widh 3.6mm, a a vlociy of 3.33 m/s, a cuig dh of. mm ad a rak agl of º ad 6º. Marial bhavior is giv by a Johso Cook law ha aks io accou hrmal sofig ad srai ra hardig (Tabl ). Coduciviy ad scific ha dos o dd o mraur, w cosidr hm cosa. Th followig assumios ar mad: Firs, h ool is suosd o b rigid ad fricio is glcd. Furhrmor, h hrmal xchag bw h ar ad h ool ar also glcd. Th iria of h ar is glcd. Imlici dyamics was usd. Tim 8 4 ss wr of.5 which cssias of.5 ss for a ool ravl of. mm. Oly isrio of aricls was usd i his xaml. 43

144 (a) (b) (c) Figur 4. Coiuous chi formaio usig h ra dd Johso Cook hardig law ad a rak agl of 6º : (a) vo Miss (MPa) ; (B) srai ra(/s); (c) mraur(k) For a ool rak agl of 6º, dformaio is largly cofid o h rimary shar zo ad o h boudary layr adjac o h ool (Figur 4). No shar localizaio occurs ad a coiuous chi formaio is rdicd. A yical disribuio of mraur fild wihi h workmarial is show i Figur 4. Highs mraurs ar obsrvd o h ousid surfac of h chi currly i coac wih h rak fac. Tmraur i h dircio of h shar la is foud o vary from high of abou 78 K ar h cuig dg o abou 5K ar h umachid fr surfac. Also mraur alog h rak fac chags 44

145 from 78K ar h cuig dg o 75K a h oi whr h coac bw h ool ad h chi com o a d. 5 forc(n/mm) 5 cuig forc hrus forc ool dislacm(mm) Figur 43. Cuig ad hrus forc vs. ool dislacm for a ra dd Johso Cook hardig law ad a rak agl of 6º Th largs accumulad lasic srais occur wihi h boudary layr adjac o h ool. I his rgio, h flow of h marial is faciliad by hrmal sofig ad h lasic srais aai valus u o 4. Srais i h chi irior rmais wihi -.5 rag uo xi from h rimary shar zo. Figur 43 shows h horizoal ad vrical rdicd cuig forcs. I is foud ha horizoal cuig forc ris quickly o a valu of 7N r mm of widh of cu wihi a shor disac of.5mm. Th as h chi hickss ad cuig mraurs i h dformig zo sabiliz, h horizoal cuig forc holds o a cosa valu of 7N/mm. Th sady sa vrical forc como, also kow as hrus forc was foud o avrag aroud 4 N r mm widh of cu. Th coac lgh bw h ool ad h workic, h dformd chi hickss ad h shar agl ar.5 mm,.6 mm ad 3º. For a ool rak agl of º, dformaio is largly cofid o h rimary shar zo ad o h boudary layr adjac o h ool (Figur 44). No shar localizaio occurs ad a coiuous chi formaio is rdicd. Th maximum umbr of aricls is ar 335. A yical disribuio of mraur fild wihi h workmarial is show i Figur 44. Highs mraurs ar obsrvd o h ousid surfac of h chi currly i coac wih h rak fac. Tmraur i h dircio of h shar la is foud o vary from high 45

146 of abou 79 K ar h cuig dg o abou 5K ar h umachid fr surfac. Also mraur alog h rak fac chags from 79K ar h cuig dg o 76K a h oi whr h coac bw h ool ad h chi com o a d. (a) (b) (c) Figur 44. Coiuous chi formaio usig h ra dd Johso Cook hardig law ad a rak agl of º : (a) vo Miss (MPa) ; (B) srai ra(/s); (c) mraur(k) Th largs accumulad lasic srais occur wihi h boudary layr adjac o h ool. I his rgio, h flow of h marial is faciliad by hrmal sofig ad h lasic srais aai valus u o 3,5. Srais i h chi irior rmais wihi - rag uo xi from h rimary shar zo. Figur 45 shows h horizoal ad vrical rdicd cuig forcs. I is foud ha horizoal cuig forc ris quickly o a valu of 8N r mm of widh of cu 46

147 wihi a shor disac of.5mm. Th as h chi hickss ad cuig mraurs i h dformig zo sabiliz, h horizoal cuig forc hld o a cosa valu of 8N/mm. Th sady sa vrical forc como, also kow as hrus forc was foud o avrag aroud 5 N r mm widh of cu. Th coac lgh bw h ool ad h workic, h dformd chi hickss ad h shar agl ar.6 mm,.67 mm ad 8º. 5 forc(n/mm) 5 cuig forc hrus forc ool dislacm(mm) Figur 45. Cuig ad hrus forc vs. cuig ool dislacm for a ra dd Johso Cook hardig law ad a rak agl of º A comariso of h rdicd cuig ad hrus forcs for wo diffr rak agls (º ad 6º) shows ha a icras i rak agl imlis a dcras i forcs, du o hrmal sofig homa is mor localizd wh h rak agl is icrasd. Furhrmor, wih a rak agl of h 6º h coac lgh is rducd ad h shar agl is icrasd du o h fasr curlig of h chi. Fially, h dformd chi hickss is rducd du o a icras i h rak agl. 47

148 3.6 Imlici, IMPLEX or xlici im igraio schms i h umrical simulaio of mal cuig rocsss? 4 cuig forc (N/mm) IMPLICIT smi-xlici IMPLEX ool dislacm(mm) º forc(n/mm) IMPLICIT smi-xlici IMPLEX ool dislacm(mm) 6º Figur 46. Prdicd cuig forcs usig imlici, smi-xlici ad IMPLEX im igraio schms for diffr rak agls (Simo yild fucio) 48

149 5 forc(n/mm) 5 IMPLICIT smi-xlici IMPLEX ool dislacm(mm) º cuig forc (N/mm) IMPLICIT smi-xlici IMPLEX ool dislacm(mm) 6º Figur 47. Prdicd cuig forcs usig imlici, smi-xlici ad IMPLEX im igraio schms for diffr rak agls (Johso Cook yild fucio) As o of h mai objcivs of his PhD hsis is dcras h comuig im ha is cssary o carry ou h umrical simulaio of a yical orhogoal cuig ss. W rs hr a comariso of a imlici, IMPLEX ad smixlici im igraio schms i rms of rdicd cuig forcs ad comuig im. Th roosd s is h sam rsd i scio for 49

150 AISI 434 sl ad for commo sl wih h yild fucio roosd by Simo. Cuig sd, udformd chi hickss, rak ad rlas agls ad marial roris ar h sam as usd i scio AISI 434 Rak agl 6º Simo 6º Tim igraio schm Tool dislacm (mm) Max umbr of aricls Comuig Tim im (hours) icrms Smi-xlici 45 4, Imlici 5,5,4 335 IMPLEX 3,75,4 353 Smi-xlici 39,5 96, Imlici 5,7,4 335 IMPLEX 3,5, Smi-xlici 47 9, Imlici 6,5,4 335 IMPLEX 3,8, Smi-xlici 44 98, Imlici 5,5,4 34 IMPLEX 4,4 354 Tabl 9. Numrical simulaios of orhogoal cuig rocsss usig imlici, IMPLEX ad xlici im igraios schms: Comuig im Th followig assumios ar mad: Firs, h ool is suosd o b rigid ad fricio is glcd. Furhrmor, h hrmal xchag bw h ar ad h ool ar also glcd. Th iria of h ar is glcd i imlici dyamics ad h IMPLEX aroachs. I cas of smi-xlici dyamics, im ss wr o h ordr of 3 which 6 cssias of ss ad i cas of imlici ad IMPLEX schms, im 8 4 ss wr of.5 which cssias of ss for a ool ravl of.4mm. Figur 46 ad Figur 47 allow us o rmark h followig oics abou h im igraios schms: Prdicd cuig forcs usig imlici im igraio schms hav lss ois ha IMPLEX ad smi-xlici umrical schms. This characrisic rrss o of h advaags of imlici im igraios schms ovr ohr schms usually rord i h liraur. Horizoal ad vrical cuig forcs rdicd for boh schms ar qualiaivly similar, alhough i imlici ad IMPLEX schm w glc irial forcs. W ca coclud ha i roosd cuig codiios irial rms ar gligibl. 5

151 Smi-xlici Imlici IMPLEX Smi-xlici Imlici IMPLEX Smi-xlici Imlici IMPLEX Smi-xlici Imlici IMPLEX , 39,5 47, 44, 3 5,5 3,75 5,7 3,5 6,5 3,8 5,5 4, 6º 6º AISI 434 Simo Figur 48. Numrical simulaios of orhogoal cuig rocsss usig imlici, IMPLEX ad xlici im igraios schms: Comuig im Th ois i imlici rdicd forcs is du o saial discrizaio ad rcocio ad daa rasfr of hisorical variabls. I Tabl 9 ad Figur 48, w ca obsrv h comuig im o rdic h chi formaio rocss i a orhogoal cuig s usig ach of h im igraio schms rsd i his work. Mor or lss as Tabl 9 shows, a imlici schm ds mor or lss 7.8 ims lss calculaio im ha a smixlici schm for a ool dislacm of.4 mm ad h IMPLEX schm ds.5 ims lss comuig ha a smi-xlici schm. A addiio of aricls ad a rcocio of hm is do ach 5 icrms i imlici ad 7 IMPLEX schms ad ach 6.5 i smi-xlici schm, as a cosquc a h d of h simulaio all schms will had mor or lss h sam umbr of aricls. Du o his raso, h comariso of boh schms dos o dd o h umbr of aricls. I is imora o rmark ha h IMPLEX ad h imlici schm will b mor ffici i rms of comuig im ha h xlici schms a low cuig sds as h rsuls rsd i his work show, bu, as h cuig sd is icrasd h fficicy dcras u o h oi i which xlici schms ar mor ffici ha h ohr schms rsd i Char. I coclusio, h 5

152 mos aroria igraio schm should b slcd accordig o h cuig codiios of irs. IMPLEX igraio schm rrss a major s owards h sadardizaio of umrical mhods as dsig ools i h mal machiig idusry, bcaus h calculaio im is a facor ha maks urachabl is us. 3.7 Machiig a iaium alloy (Ti6Al4V) a diffr cuig sds. Th ffc o cuig forcs ad chi shas Physical aramrs Dsiy 44 kg/m 3 Elasic aramrs Poisso,3 Elasic modulus 5GPa Thrmal Paramrs Ha Coduciviy a ºC 6,785 N/sK Ha Coduciviy a 85ºC 4,375 N/sK Ha Caaciy a ºC,499mm /s K Ha Caaciy a 85ºC,7569mm/sK Ilasic aramrs Marial Cosa C,3 Marial Cosa K 6 MPa Marial Cosa,339 Marial Cosa T m 85K Marial Cosa m Tabl. Thrmo-mchaical ad marial aramrs for Ti6Al4V Th hird alicaio cocrs h cuig of a rcagular block of Ti6Al4V alloy of lgh µm ad widh 6 µm, a cuig dh of 35 µm, a rak agl of º ad a ool radius of µm. Th cuig sd has b varid bw. a m/s. Marial bhavior is giv by a modifid Johso-Cook law (Bäkr law) wih h marials roris show i Tabl. Th coduciviy ad scific ha dd liarly i h mraur. Th ool has b assumd o b mchaically rigid, h fricio ad h hrmal xchag bw h workic ad h ool ar glcd. Th soluio schm usd i h rs xaml is basd o h isohrmal imlici schm ad h rmshig is basd o h aricl fii lm mhod (PFEM) rsd i char. Isrio ad rmoio of aricls is usd i his xaml o sav comuig im ad i ordr o imrov h localizaio homo. Marial saraio i fro of 5

153 h ool has b modld by cosidrig h chi formaio rocss as a ur dformaio whr marial flows visco-lasically aroud h ool i. A addiioal ool is usd i ordr o avoid chi raio i h workic. A oal of 4 im ss wr dd i ordr o calcula ay of h chis show i Figur 49, Figur 5 ad Figur 5, h sadard comuig im was.5 o 9 hours o a comur ruig wih h rocssor Il Cor Ghz. Th xaml rsd i his scio has b ak from Bäkr [7] ad D Michli [8]. A scially aalig faur of high-sd cuig rocsss is ha h scific cuig forc for mos marials srogly dcrass wih icrasig h cuig sd. Th frquly obsrvd rasiio bw coiuous ad sgmd chi is rroducd by h modl Figur 49, Figur 5, Figur 5 show h mraur fild for sv diffr valus of cuig sd. A small cuig sd, coiuous chi ar formd wih icrasig h shar agl. Chi sgmaio is obsrvd a cuig sds highr ha 5m/s ad h sgmaio icrass wih icrasig h cuig sd. Plos of h cuig forc ar show i Figur 5 ad Figur 53. Th los ar disac-rsolvd, i such a way ha rsuls for diffr cuig sds ar comarabl. For coiuous chi, h cuig forc ds o a cosa valu, whras, for sgmd chi, h cuig forc oscillas aroud a ma valu. Th obsrvd dcras i h cuig forc a high cuig sd ca b hus xlaid as follows: icrasig h cuig sd causs a icras i h mraur. Alhough h srai ra icrass, causs a largr isohrmal flow srss, h mraur icras lads o hrmal sofig, so ha h ma flow srss is rducd. I coclusio, h simulaios show a srog dcras of h cuig forc wih icrasig cuig sd is maily a rsul of hrmal sofig which chags h ffciv srss-srai curv ad icras h shar agl. Figur 49, Figur 5 ad Figur 5 show aohr irsig homo: h widh of h shar zo i h coiuous chi bcoms smallr wih icrasig h cuig sd. As h icrasig mraur caus a dcras i hardig, h widh of h shar zo bcoms smallr so ha srai ras bcoms largr. 53

154 . m/s.m/s.5 m/s Figur 49.Comariso of chi morhologis of TiASl4V a diffr cuig sds 54

155 m/s m/s 5m/s Figur 5.Comariso of chi morhologis of TiASl4V a diffr cuig sds 55

156 m/s m/s Figur 5..Comariso of chi morhologis of TiASl4V a diffr cuig sds Adiabaic shar bad formaio rocss I Figur 54, w ca obsrv i dails h chi formaio of a adiabaic shar bad i our simulaio a m/s. Figur 54 shows h vo Miss srss isid a formig sgm. Subfigur (a) shows a sa whr o shar bad is arly fully dvlod ad dformaio occurs maily alog his srogly curvd bad. I subfigur (b) dformaio occurs i his bad, bu hr is also som dformaio i h rgio bhid his shar bad, ladig o a dammig of h marial. Dformaio cocraio bgis a h ool i, bu subfigur (c) shows ha a scod dformaio cocraio sars a h fr surfac bfor h shar bad is fully formd. Th cuig forc icras durig h firs sags ad rachs a maximum durig his has bcaus h wly formig shar bad. Alhough h dformaio cocras durig h sag of subfigurs (c) ad (d), his is o shar localizaio o h usual ss as h los of h vo Miss srss show. Durig his fac, h vo Miss srss isid h zo whr dformaio cocras is largr ha i h adjac zos, maily du o h highr srais ras i his rgio. I sag (), ru srai localizaio has bgu ar h ool i as h vo Miss srss is srogly rducd hr. This is corrlad wih a srog icras i h mraur fild. This figur also shows 56

157 ha h zo of dformaio cocraio a h fr surfac is o a localizd zo as h vo Miss srss is largr hr ha i h adjac zos. cuig forc (N/mm) m/s.m/s.5m/s m/s m/s ool dislacm(mm) Figur 5. Cuig forcs of TiASl4V a diffr cuig sds cuig forc (N/mm) m/s m/s m/s ool dislacm(mm) Figur 53. Cuig forcs of TiASl4V a diffr cuig sds 57

158 Thrfor, alhough hr ar wo zos of dformaio cocraio which afrwards joi, h zo of dformaio localizaio grows coiuously from h ool i. Subsquly localizaio has sard; o furhr dformaio aks lac i h dammd rgio bhid h shar bad. Afr dformaio localizaio has s i, h wo zos of dformaio cocraio joi ad h dformaio srogly localizs isid h shar bad (Subfigur (f)). (a) (b) (c) (d) () (f) Figur 54. Dvlom of h vo Miss srss isid a chi for a cuig sd of m/s 58

159 3.8 Orhogoal cuig of AISI 434 sl usig a dformabl ool. A friciolss aroach Th x xaml cocrs h cuig of a rcagular block of high srgh AISI 434 sl modld usig h Johso-Cook hardig law wih h ool cosidrd as a dformabl body. Th fricio ad h hrmal xchag bw h workic ad h ool ar glcd. Th marials ad rocsss aramrs ar collcd i Tabl. Coduciviy ad scific ha of h workic ad h ool dos o dd o mraur, w cosidr hm cosa. Workic dimsios wr ak as 3.5mm (lgh).5 mm (dh). Cuig sd was 3.33 m/s ad h d of cu was. mm. Cuig ool gomry was as follows Rak agl º Rlif agl = º (.5) radius of h cuig dg=.5mm Each im icrm was dsigd o corrsod o a ool ravl of mm. Simulaio was coiud uil h chi coms io coac wih h ool. Acually a ool ravl of. mm was covrd i imlici icrms. Tool Elasic Modulus 54 MPa Poisso, Workic Elasic Modulus 5 MPa Poisso,3 Marial Cosa A 79MPa Marial Cosa B 5 MPa Marial Cosa C,4 Marial Cosa,6 Marial Cosa m,3 Ha Coduciviy 5 N/sK Ha Caaciy,499mm /s K Thrmal xasio -5 (/K) Dsiy 7844 Kg/m 3 Tabl. Mchaical ad hrmal roris of AISI 434 sl Figur 55 shows h coours of quival srai ra ad h coours of quival srss. I is obsrvd ha h maximum valus of quival srai ra ad vo Misss occur ar h cuig dg ad ar of h ordr of 5 ad 8 MPa. I Figur 55 a comariso bw h rdicd forcs usig a dformabl ad a rigid ool is show, i ca b obsrvd ha h obaid cuig forc is slighly 59

160 lowr ad h fd forc slighly highr usig a dformabl ool, du o h chag i h local cuig dg gomry. vo Miss (MPa) Equival srai ra (/s) Figur 55. Miss quival srss ad quival srai fild usig a dformabl ool. A friciolss aroach Th coac lgh bw h ool ad h workic, h dformd chi hickss ad h shar agl ar.6 mm,.4 mm ad 3º. 3.9 Orhogoal cuig of AISI 434 sl usig a dformabl ool: Ha rasfr ad fricio bw h ool ad h workic Orhogoal dry machiig of AISI 434 sl by a high srgh sl ool has b simulad, a cuig sd of 3.33 m/s, a a fd of. mm, a rak agl of ad a clarac agl of º. Th fricio a h ool-chi irfac is modld usig h Noro-Hoff fricio law rsd i char wih fricioal 5 roris r 9 ad.th mchaical ad hrmal roris of h workic ad h ool ar h sam usd i scio

161 8 cuig forc (N/mm) 6 4 rigid ool dformabl ool ool dislacm(mm) fd forc (N/mm) 4 3 rigid ool dformabl ool ool dislacm(mm) Figur 56. Cuig ad fd forcs. A rigid vs. a dformabl ool. Th hrmal roris of h ool ar suosd o b machd o hos of h workic, givig a qual orio of fricioal ha allowd o h ool ad h chi. Th hrmal coducac coffici h usd was 9 W/(m K), i ordr o sur ha h hrmal ga a h ool-chi irfac will b gligibl. 6

162 Figur 57. Coiuous chi formaio: Dformabl ool, ad fricio ad ha rasfr a h ool chi irfac Figur 57 (a) shows h calculad disribuios of srai ra. Dformaio is cocrad, as xcd, ar h shar la ad alog h rak fac. Th srai ra rachs 35 (/s) i fro of h cuig dg. Figur 57(b) shows h mraur filds i h ool ad h workic. Th maximum mraur is 958 K ad aks lacs alog h rak fac. Figur 57(c) shows h lasic-srai disribuio, xrmly high valus, clos o 4, ar foud. Figur 57 (d) shows h vo Miss srss disribuio, h maximum srss aars i h ool, clos o h cuig dg. I h workic, h magiud of h srss riss i h dformaio zo bcaus of work hardig, bu i is limid o 5 MPa by hrmal sofig. Th coac lgh bw h ool ad h workic, h dformd chi hickss ad h shar agl ar.9 mm,.8 mm ad 9º. A comariso wih h friciolss cas of scio 3.8 is rsd i Figur 58. This comariso shows ha h cuig forc icrass 65N/mm whil h hrus forc icrass 35N/mm. Similarly, h coac lgh ad h udformd chi hickss icras wih fricio, bcaus fricio hidrs h roaioal movm of h chi. 6

163 5 cuig forc (N/mm) 5 5 friciolss Noro-Hoff fricio law ool dislacm(mm) 8 fd forc (N/mm) 6 4 friciolss Noro-Hoff fricio law ool dislacm(mm) Figur 58. Cuig ad hrus forcs: Dformabl ool, ad fricio ad ha rasfr a h ool chi irfac 63

164 3. Orhogoal cuig of 4CD4 sl: A xrimal comariso Marial Proris Coduciviy(N/sK) Scific Ha(mm/sK) Workic 4,6 a 373K 4,3 a 473K 37,7 a 673K 33, a 873K Tool(P) 5 Workic (4CD4) Thrmal xasio coffici (/K) 473(43K-473K) 59(63K-673K) 56(83K-873K) Tool(P) a 93K 4,5.- 6 a 673K Prcag of lasic owr io ha,9 Dsiy(Kg/m3) Workic (4CD4) 78 Tool(P) 6 Elasic Modulus (Ga) (4CD4) Poisso (4CD4),3 Plasiciy A(MPa) 598 Johso-Cook B(MPa) 768 Workic(4CD4) C,37 Tamb = 93K M,87 Tfus = 793K N,9 Thrmal Proris Thrmal Coducac (W/(mK)),E+8 Coac Mchaical Proris Pariio coffici,5 Fricio Coffici,3 Noro Hoff Coffici Prcag of fricio Ergy covrd io ha 6,E-5 Tabl. Mchaical ad hrmal roris of h workic ad h ool 64

165 I ordr o valida PFEM sragy, a cuig rocss of 4CD4 sl a 3m/mi, wih a ool radius of.4 mm, a rak agl of 6º ad a cuig dh is roosd. Marials ad coac roris usd ar h sam rsd i [9]. A summary of all h ius aramrs ar foud i Tabl. 8 Th im s usd durig h simulaio was. scods, as a rsul im ss wr dd for a ool dislacm of mm. Figur 59. Orhogoal cuig of 4CD4 sl Figur 59 shows h mraur fild afr a cuig lgh of mm. Th maximum ool mraur rachd is abou 86K. I is locad far from h cuig dg, ad aroximaly a a disac of.5 ims h udformd chi hickss. Th maximum vo Miss srss isid h chi-ic aks lac i h 65

166 rimary shar zo, whil h maximum vo Miss isid h ool is clos o h oi whr h ool loss h coac wih h machid surfac. h (mm) (mm) F c (N) F f (N) PFEM,6, Exrimal,49, Tabl 3. Exrimal ad umrical rsuls Daa abou xrimal rsuls hav b obaid from faa rord i h liraur[9]. Tabl 3 comars h umrical ad h xrimal cuig ad fd forcs rsuls obaid for h xaml rsd i his scio. I is obsrvd a good agrm bw h xrimal ad h umrical cuig forcs. Tabl 3 shows larg diffrcs bw xrimal ad umrical fd forcs. Rgardig h chi hickss a rlaivly good agrm was foud bw xrims ad umrical simulaios. Howvr, h ool-chi coac lgh masurd i h xrims is abou wo ims grar ha h lgh rdicd by h umrical simulaios. 66

167 Rfrcs [] J. Rojk, E. Oña, ad R. L. Taylor, "CBS-basd sabilizaio i xlici solid dyamics," Iraioal Joural for Numrical Mhods i Egirig, vol. 66, , 6. [] J. Bo ad A. J. Buro, "A siml avrag odal rssur rahdral lm for icomrssibl ad arly icomrssibl dyamic xlici alicaios," Commuicaios i Numrical Mhods i Egirig, vol. 4, , 998. [3] P. O. D Michli ad K. Moclli, "A w ffici xlici formulaio for liar rahdral lms o-ssiiv o volumric lockig for ifiisimal lasiciy ad ilasiciy," Iraioal Joural for Numrical Mhods i Egirig, vol. 79, , 9. [4] J. C. Simo ad C. Mih, "Associaiv could hrmolasiciy a fii srais: Formulaio, umrical aalysis ad imlmaio," Comur Mhods i Alid Mchaics ad Egirig, vol , 99. [5] A. Ibrahimbgovic ad L. Chorfi, "Covaria ricial axis formulaio of associad could hrmolasiciy a fii srais ad is umrical imlmaio," Iraioal Joural of Solids ad Srucurs, vol. 39, ,. [6] Y. Tadi Bi ad M. R. Movahhdy, "Cosis arbirary Lagragia Eulria formulaio for larg dformaio hrmo-mchaical aalysis," Marials & Dsig, vol. 3, ,. [7] M. Bäkr, "Fii lm simulaio of high-sd cuig forcs " Joural of Marials Procssig Tchology, vol. 76,. 7 6, 6. [8] P. O. D. Michli ad K. Moclli, "D high sd machiig simulaios usig a w xlici formulaio wih liar riagular lms," Iraioal Joural of Machiig ad Machiabiliy of Marials, vol. 9,. 66-8,. [9] P. J. Arrazola, "Modlisaio umriqu d la cou: ud d ssibili ds aramrs d r idificaio du from r ouilcoau," Docoral Thsis, L'Écol Cral d Nas, l'uivrsié d Nas, Frac, 3. 67

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169 69 Char 4 4 A Ssibiliy Aalysis o Gomric ad Cuig Codiios usig h Paricl Fii Elm Mhod (PFEM) Th objcivs of his char ar maily hr: Th firs o is o valida PFEM sragis as a ffici ool for umrical simulaio of mal cuig rocsss by a daild comariso (forcs, srsss, srais, mraur, c.) wih rsuls rovidd by commrcial fii lm sofwar (Abaqus, AdvaEdg, Dform) ad xrimal rsuls. Th scod is o carry ou a ssibiliy aalysis o gomric ad cuig codiios usig PFEM by mas of a Dsig of Exrims (DoE) mhodology. Ad h hird o is o idify h advaags ad drawbacks of PFEM ovr FEM ad mshlss sragis. Also, his char rss som advaags of PFEM ha dircly aly o h umrical simulaio of machiig rocsss: (i) allows h saraio of chi ad workic wihou usig a hysical or gomrical cririo (ii) rss gligibl umrical diffusio of sa variabls du o coiuous riagulaio, (iii) is a ffici umrical schm i comariso wih FEM. Th rsuls rsd i his char wr carrid ou durig h rsarch say a Modragó Uivrsiy udr h survisio of Profssor Pdro Arrazola. Bfor sarig wih h PFEM validaio, i h x scio, w will iroduc h hory of dsig of xrims. 4. Dsig of Exrims (DoE) Assum ha a crai aramr (rsos variabl) dds o svral idd variabls. I ordr o sudy h ffc of h variabls o h aramr, o has o gra daa ihr from xrims or from umrical simulaio. A ffici way of sudyig his ffc is hrough a ror dsig of xrims (DoE). Th mai objciv of h dsig of xrims is o obai a larg amou of iformaio wih a limid umbr of xrims. Th radiioal mhod of xrimaio is h o facor a a im mhod. I his mhod, oly o of svral variabls is chagd a a im, kig all ohr idd variabls cosa a som valus. Alhough his aroach is siml ad o gs ifrc bfor all xrims ar fiishd, i dos o rval h ffc of iracio amog variabls. Mos of h im, h ffc of o variabl o h dd aramr may b srogly iflucd by h valu of ohr idd variabls. This is calld h iracio ffc, which cao b simad rorly i h o facor a a im mhod. Thrfor, h radiioal mhod is cosidrd a iffici ad cosly aroach. I h full facorial mhod of xrimaio, ach idd variabl (facor) is dividd io diffr lvls. I som cass h variabls ak oly

170 discr valus ad hy d o b umbrs.g., rsc or absc of a lubrica i mal formig. O ca dcid o divid h rag io hr or mor lvls if mor umbr of xrims ca b coducd. Oc all h facors hav b dividd io a umbr of lvls, all ossibl combiaios of lvls ar cosidrd. Toal umbr of combiaios of facors is dd o h umbr of facors ad h lvls, as follows #xrims #facors #lvls. For xaml, if hr ar 7 7 facors a lvls, oal combiaios would b 8. Hc, i full facorial mhod, o would d o do 8 xrims. If h variabls ar dividd io 7 3 lvls, oal combiaios will b 3 87, a ormously high umbr. Thus, may ims, full facorial dsig is o rasoabl ad h fracioal facorial mhod is o b usd. To giv a xaml, firs cosidr h full facorial dsig for hr facors a wo lvls. Th firs lvl is rrsd by ad h scod lvl by +. Th 3 facorial dsig is show i Tabl 4. I his abl, colum AxB idicas h iracio ffc of facor AxB. Th lvl + idicas ha boh A ad B ar a h sam lvl ad idicas ha boh ar a diffr lvls. Similarly, AxBxC is h colum of iracio of hr facors. Exrim A B C AxB BxC CxA AxBxC Tabl 4. A full facorial dsig for hr facors a wo lvls. Wih a dsig of xrims, o ca fid h mai ad iracio ffcs of a facor. Th mai ffc idicas h idividual coribuio of h facors o h oal variabiliy ihr i h xrimal rsuls. For a wo lvl facor, h mai ffc is obaid as Effc of faco # xrims X i i i r (4.) # xrims whr Xi is h valu of h dd variabl i ach of h xrims ad h variabl lvl i aks h valus / ddig if X i is a rsos a lvl/lvl of h facor. Aar from h mai ffcs, o migh d o kow h ffc of iracio. This ffc is foud i a similar mar. I h orhogoal array, w ca mak h colums corrsodig o iracio of wo variabls ad h ra ha lvl 7

171 colum as corrsodig o a sara facor. For xaml, i Tabl 4, AxB, BxC, CxA ad AxBxC ar rad lik facors wih lvls + ad for fidig ou is ffc. Usually, h dsig of xrims is usd o sudy how som variabls lik h cuig sd, h fd ad h cuig dh ifluc h lifim of h ool. For kowig h ool lif i h machiig, w roos a DoE wih hr facors (cuig sd, h cuig dh ad h fd) a wo lvls. For ach facor, o lvl corrsods o h low valus ad h ohr lvl corrsods o h high valus. Tabl 5 rss h ool lif of a TiN coad carbid ool cuig a mdium carbo sl a diffr cuig aramrs. Exrim Cuig sd(m/mi) Fd (mm/rv) Cuig dh(mm) Tool lif(mi) Tabl 5. A Dsig of xrims o sudy h ddc of ool lif o cuig aramrs[] Usig quaios (4.) ad h daa rsd i Tabl 5, h mai ffcs of cuig sd, h fd ad h cuig dh ar giv by Effc of cuig sd Effc of fd Effc of cuig dh (4.) (4.3) (4.4) Thus, i is s ha i h giv rags of h rocss aramrs, h cuig sd has h maximum ffc o h ool lif followd by h fd ad dh of cu. Th gaiv valu of h aramrs idicas ha icrasig hs aramrs dcrass h ool lif. I h x scio, w will rs a orhogoal cuig simulaio of 4CD4 sl usig PFEM, his is h sam s rsd i scio 3. of his work. 7

172 4. Orhogoal cuig simulaio of 4CD4 sl usig h Paricl Fii Elm (PFEM) I ordr o valida PFEM sragy as a ffciv sragy o rdic chi formaio i orhogoal cuig rocss, i his work a cuig rocss of 4CD4 sl a 3m/mi, wih a ool radius of.4mm, rak agl of 6º ad cuig dh. mm is roosd. (a) (b) (c) (d) Figur 6. Comarisos of mraur filds giv by (a) PFEM, (b) Abaqus, (c) AdvaEdg ad (d) Dform. Th workic is cosidrd as a dformabl body i which h adaiv msh sragy is basd o PFEM. I h workic larg lasic ad lasic srais, ha coducio, ad ha graio du o lasiciy ar cosidrd. Th ool is cosidrd as a dformabl body i which a sadard fii lm mhod is usd. Small lasic srai ad ha rasfr du o coducio aks lac isid h ool. A riagl fii lm wih liar dislacm, rssur ad mraur abl o dal wih h icomrssibiliy cosrai imosd by h lasic homa was usd i h work-ic. I h ool a riagl fii lm wih liar dislacm ad mraur was usd. Th coac sragy bw h ool ad h workic is a aly od-osgm aroach. Th coac bw h ool ad h workic icluds ha rasfr du o coducio ad fricio. Ha rasfr du o covcio ad radiaio o h xral virom is cosidrd gligibl. 7

173 (a) (b) (c) (d) Figur 6. Comarisos of vo Miss filds giv by (a) PFEM, (b) Abaqus, (c) AdvaEdg ad (d) Dform. Th disac bw aricls i h rimary ad h scodary shar zos was slcd accordig o iformaio giv by fii lm rrors simaors ad o h ool radius siz. Numrical simulaio of chi formaio rquirs a hrmo-laso-visco-lasic law for h workic marial bhavior. I his work, h Johso-Cook yild fucio is usd. This marial law is usd i marials i which is yild srss dds o is quival srai, ra of srai ad mraur. Marials ad coac roris usd i h umrical simulaios ar h sam usd i h xaml 3. rsd i char 3 of his hsis. 73

174 (a) (b) (c) (d) Figur 6. Comarisos of quival lasic srai giv by (a) PFEM, (b) Abaqus, (c) AdvaEdg ad (d) Dform. As a cosquc of h rror simaors ad h ool radius siz, h aricl disac i h rimary ad h scodary shar zo is mor or lss 3.5 micromrs. Far away from h rimary shar zo h maximum aricl disac is mor or lss.5 mm. Th maximum umbr of aricls usd durig h simulaio was 386. Th umbr of ods usd i h ool was 6. Th im s usd durig h simulaio was cosa ad of h ordr of.-8 scods, maig ha for a ool dislacm of mm a a cuig sd of 3m/mi, i ds aroud ss. Th calculaio im was aroximaly 4 hours i h cas of a comur ruig wih 4Gb ad h followig rocssor: Il Cor Duo CPU Figur 6 (a) shows h mraur fild ovr h workic, chi ad ool ad afr a cuig lgh of mm. This is. milliscods of machiig im a h cuig sd of 3 m/mi. Th maximum ool mraur rachd is abou 86K. I is locad far from h cuig dg, ad aroximaly a h disac of h.5 ims h udformd chi hickss (). Morovr Figur 6 (a) shows h lasic srai fild ovr h workic ad h chi. Th mos sigifica lasic srai occurs ovr h machid surfac ad alog h surfac ha is coac wih h ool. Alog his w surfac, h lasic srais rachs a valu grar or qual ha. Also, Figur 6(a) shows vo Miss srss fild. Th maximum vo Miss srss, isid h chi-ic aks lacs i h rimary shar zo, whil h maximum vo Miss srss isid h ool is clos o h oi whr h ool loss h coac wih h machid 74

175 surfac, xacly h oi whr ool failur aks lac i ral machiig rocsss. Furhrmor, Figur 63 (a) shows h srai ra fild isid h workic. Th maximum valu of srai ra is 4 ad aks lac isid h rimary shar zo. As xcd, h srai ra dcrass raidly o a valu clos o zro i h limis of h rimary shar zo. Numrical rsuls shows ha i machiig, mraurs grar ha K, srais grar ha ad srai ras grar ha 5 aar. I ordr o fi umrical simulaios wih xrimal rsuls of machiig rocss, i is cssary o dvlo a xrimal su abl o characriz workic marial i xrm codiios ha ak lac i machiig rocsss. (a) (b) (c) (d) Figur 63. Comarisos of srai ra giv by (a) PFEM, (b) Abaqus, (c) AdvaEdg ad (d) Dform. I h x scio, w will rs a orhogoal cuig simulaio of 4CD4 sl usig PFEM ad a daild comariso (forcs, srsss, srais, mraur, c.) wih rsuls rovidd by commrcial fii lm sofwar (Abaqus, AdvaEdg, Dform) ad xrims. 4.3 Numrical ad Exrimal validaio of h PFEM sragy Daa abou xrimal rsuls, scially cu-ig forcs, fd forcs, chi hickss ad ool-chi coac lgh hav b obaid from daa rord i h liraur []. Validaio was carrid ou comarig umrical rsuls wih xrimal os ad umrical rsuls obaid from h commrcial sofwar Abaqus, Dform 75

176 ad AdvaEdg. I is imora o mio, som diffrcs ad similariis bw h formulaios, im igraios schms, boudary codiios, marials modls ad coac laws usd i ach of h umrical simulaios: PFEM, Dform ad Abaqus modl us Johso Cook yild fucio. (Tabl ) Abaqus, AdvaEdg ad Dform us Coulomb fricio law a h ool chi irfac, wih a fricio coffici of.3. (Tabl ) PFEM ad Dform us a Noro-Hoff fricio law wih a Noro Hoff cosa valu of 6-5. I AdvaEdg, workic marial bhavior is govrd by Marusich law. PFEM, Dform ad AdvaEdg cosidr h ool as a dformabl (P) whil Abaqus cosidr h ool as a rigid body. Furhrmor, PFEM, AdvaEdg ad Dform us a Lagragia dscriio of moio whil Abaqus uss a Arbirary Lagragia Eulria dscriio of moio. Also, PFEM ad Dform us imlici im igraio whil AdvaEdg ad Abaqus us xlici im igraio. Mor iformaio abou h cosiuiv modl, boudary codiios usd i h umrical modls dvlod i Abaqus ad AdvaEdg ar xlaid i dail i []. Program Formulaio Cosiuiv Modl Fricio Modl Procsss variabls (DoE) PFEM Abaqus Lagragia ALE Johso Cook Johso Cook Noro- Hoff Coulomb. Tmraur(Tool). Coac lgh 3. Tmraur(Chi) 4. Vo Miss (Chi) 5. Plasic srai AdvaEdg Lagragia Marusich Law Coulomb 6. Plasic srai ra 7. Chi hickss 8. Shar agl Dform Lagragia Johso Cook Noro- Hoff 9. Cuig forc. Fd forc Tabl 6. Formulaios, cosiuiv modls ad fricio modls usd i ach of h umrical simulaios. Tabl 6 rss a summary abou h formulaios, cosiuiv modls ad fricios modls usd i ach of h umrical simulaios. 76

177 Tabl 7 comars h umrical ad h xrimal cuig ad fd forcs rsuls obaid for h rfrc cuig s. I is obsrvd a good agrm bw h xrimal ad umrical cuig forcs rdicd by PFEM, Dform ad Abaqus. Isad, comarig xrimal cuig forcs wih AdvaEdg rsuls, highr diffrcs wr foud. Tabl 7 shows h larg diffrcs bw h xrimal ad umrical fd forcs. AdvaEdg rdics a fd forc ha is 34% grar, Dform 4% smallr, Abaqus % smallr, PFEM 38% smallr ha h xrimal fd forc. Rgardig o h chi hickss () a rlaivly qui good agrm was foud for all h rsuls. Howvr, h ool-chi coac lgh (h) masurd i h xrims is abou wo ims grar ha h lgh rdicd by h umrical simulaios. V.M. T (k) h (mm) (MPa) () (mm) F c (N) F f (N) PFEM 73,6 4 3, Abaqus 4, , AdvaEdg 44, , Dform 7,4 3,57, Exrimal -,49 - -, Tabl 7. Exrimal ad umrical rsuls (PFEM, Abaqus, AdvaEdg, Dform) Comarig rsuls for h maximum ool mraur i is obsrvd ha h largr diffrc occurs bw Dform-AdvaEdg, whil h smallr aks lac bw PFEM ad Dform. Th diffrcs ar du o marial modl ad fricio law usd i ach o of h sof-war (sad bfor). I h cas of h vo Miss srss h rsuls rdicd by PFEM, Dform, Abaqus ar rally similar, howvr h maximum vo Miss srss rdicd by Advadg is MPa grar ha h avrag srss rdicd by h ohr sofwar. Th diffrcs ad similariis amog h rdicd rsuls ar bcaus of hos xisig os bw PFEM, Abaqus, Dform ad AdvaEdg (sad bfor). Figur 6, Figur 6, Figur 6 ad Figur 63 show a comariso of h mraur fild, vo Miss srss fild, lasic srai fild ad srai ra fild. Th mraur filds rdicd by PFEM, Abaqus, Dform ad Advadg ar similar. Th vo Miss srsss ar similar for PFEM, Dform ad Abaqus, whil AdvaEdg shows a diffr fild ossibly du o h cosiuiv modl usd. Plasic srai filds rdicd by PFEM, AdvaEdg, Abaqus ad Dform ar similar; all h umrical simulaios show ha h mos sigifica lasic srais ak lac ovr h machid surfac ad ovr h surfac ha is i coac wih h ool. Thus, h umrical modl s u wih PFEM is cosidrd o b accura ough o carry ou a ssiiviy aalysis o rocss aramrs lik ool radius, rak agl, cuig vlociy ad cu-ig dh. 77

178 4.4 A Dsig of Exrims wih PFEM ad is comariso wih a DoE wih h commrcial sofwar (Abaqus, AdvaEdg ad Dform) (a) (b) (c) (d) () (f) (g) (h) Figur 64. Numrical ad Exrimal ffcs obaid afr ssibiliy aalysis. 78

179 This scio rss a DoE i ordr o sudy h ifluc of cuig codiios (lik ool radius, rak agl, ool vlociy ad udformd chi hickss) o ouu variabls lik h maximum ool mraur, h maximum chi mraur, h maximum workic vo Miss srss, h maximum workic srai ra, h ool-chi coac-lgh, h dformd chi hickss, h cuig forc ad h fd forc usig PFEM. A h sam im, w comar a DoE dvlod wih PFEM wih a DoE carrid ou usig Abaqus, Dform AdvaEdg ad xrimal rsuls (i hos cass whr rsuls ar availabl). Two diffr ool radius (.5mm/.5mm), wo diffr ool vlociis (5m/mi.3m/mi), wo rak agls (6º ad -6º) ad wo diffr udformd chi hickss (.5mm/.3mm) wr usd. As a rsul, 6 umrical simulaios wr carrid ou usig ach of h umrical ools, ad a oal of 64 umrical simulaios wr do. I is imora o rmark ha h chos valus rrs cuig codiios yically usd i idusrial alicaios. I Figur 64, V rrss h ool vlociy, rrss udformd chi hickss, g rrss h rak agl ad Rh rrss h ool radius. Figur 64 (a) shows ha PFEM rdics ha chi hickss has h gras (8%) ad rak agl has h smalls (-5.5%) ifluc o maxi-mum ool mraur. Furhrmor, h ffc of ool vlociy, chi hickss ad ool radius o ool mraur is similar i all h umrical simulaios. agl º. Figur 64 (b) shows h ffc of cuig codiios o coac lgh. PFEM rdics ha icrasig 6 ims h chi hickss imlis a icras of 5% i h coac lgh, h chi hickss big h mos sigifica cuig codiios o coac lgh. Th ffc of h rak agl rdicd by PFEM is oosi o ohr umrical simulaios ad xrims, ossibly du o h fricio law usd i h modl dvlod wih PFEM. Th ffc of h ool vlociy rdicd by PFEM, Abaqus ad AdvaEdg is diffr from h ffc giv by h xrims, suggsig ha somhig is missig i h modls of orhogoal cuig dvlod so far. Th ffc rdicd by Dform is clos o b gligibl. Mos of h umrical simulaios carrid ou i his work, dislay ha h ddcy of maxi-mum vo Miss, isid h rimary shar zo, o cuig codiios is lss ha %, ha is clos o b gligibl (Figur 64(c)). PFEM rdics ha h mos sigifica icras i srai ra i h rimary shar zo is du o a icras i h ool vlociy, whil h lss sigifica is h ffc of a icras i h rak agl (Figur 64 (d)). Th ffc of h ool vlociy is similar i all h umrical simulaios, h ffc of h chi hickss rdicd by Abaqus is oosi i comassio o ohr umrical simulaios ad h ffc of h ool radius is similar i all h umrical simulaios. Numrical simulaios dos o show a clar ar abou h ffc of h rak agl i srai ra i h rimary shar zo, ossibl du o is srog ddcy o msh siz ad fricio law usd a ool-chi irfac. 79

180 As xcd, PFEM rdics ha h mos sigifica ffc o dformd chi hickss is h udformd chi hickss (Figur 64()), qui similar o h ffc rdicd by h ohr umrical simulaios. All h umrical simulaios rdics a oosi ffc of h rak agl o dformd chi hickss, showig ha is cssary o imrov h fricio law usd a ool-chi irfac as suggsd by [3, 4]. PFEM ad Dform show a gligibl ffc of h ool radius o dformd chi hickss, h raso o ha bhavior is h coac law usd i h modl dvlod wih PFEM (Noro Hoff fricio law say ha fricio forc dds liarly o rlaiv vlociy, i mas clos o zro fricio forc wh rlaiv vlociy is clos o zro). Abaqus, AdvaEdg, Dform ad PEM rdic a dcras i h chi hickss icrasig cuig sd, whil xrims show ha icrasig ool vlociy has a gligibl ffc o chi hickss. Figur 64 (f) shows h ffcs of cuig codiios o shar agl. Th rsuls illusras ha h ffc of chi hickss o h shar agl, rdicd by umrical simulaios is smallr ha h rsuls giv by xrims. Th ffc of rak agl i shar agl giv by h xrims is oosi o h ffc rdicd by h umrical simulaios. AdvaEdg ad Dform rdic a corary ffc of ool radius o shar agl, du o h cosiuiv law usd. Mos of h umrical simulaios ad xrims rdic ha h ffc of h ool vlociy o shar agl is clos o b gligibl. Th big diffrcs bw xrims ad umrical simulaios idica ha is cssary o imrov h cosiu law o dscrib h bhavior of h workic marial ad o us a mor sohisica fricio law a chi ool ir-fac. Th ffcs o cuig forcs of h ool radius, h rak agl, h chi hickss ad h ool vlociy rdicd by h umrical simulaios ad xrims ar i fac similar, showig ha ool vlociy has a gligibl ffc whil icrasig 6 ims h chi hickss imlis icrasig 3% h cuig forc (Figur 64(g)). Furhrmor, accordig o PFEM, chagig ool radius imlis a largr icras ad icrasig rak agl imlis a smallr dcras of cuig forcs. Th diffrcs ar du o h fricio law usd i PFEM ad h smallr iraricl disac usd i PFEM i comariso wih h msh siz usd i ohr umrical simulaios. This idicas agai ha is cssary o imrov h workic cosiuiv law ad h fricio law usd a h chi-ool irfac i or-dr o fi h homology show by xrimal rsuls. PFEM, ohr umrical simulaios ad xrims rdic a icras i fd forcs du o a icras i ool radius ad udformd chi hickss; whil icrasig rak agl imlis dcrasig h fd forc (Figur 64(h)). All h umrical simulaios show ha h ool radius ad h rak agl ffc is mor ha wo ims h ffcs giv by h xrimal rsuls. For h cuig sds usd i his work, i-crasig ool vlociy ims imlis mor or lss h sam fd forcs. 8

181 4.5 Coclusios Th umrical simulaios rs PFEM as a romisig sragy o simula mal cuig rocsss, bcaus PFEM ovrcoms som disadvaags of umrical schms dvlod uil ow. For xaml, (i) allows h saraio of chi ad workic wihou usig a hysical or gomrical cririo, (ii) PFEM rducs h umrical diffusio du o r-mshig (rasi msh adaiviy is usd isad of rmshig, (iii) PFEM ds lss dgr of frdom ha usd i a umrical simulaio usig FEM. Furhrmor, PFEM rdics similar rsul o h ohr sofwar ad xrims as show i Figur 64 ad Tabl 7. Th comuig im dd by PFEM udr Malab rogrammig ad xlodig cod vcorizaio (iuiiv, cocis ad fasr rogrammig syl) is similar o FEM sofwar. I is xcd, ha PFEM udr high lvl rogrammig laguag ds lss comuig im ha sadard fii lm sofwar. 8

182 Rfrcs [] P. M. Dixi ad U. S. Dixi, Modlig of Mal Formig ad Machiig Procsss: by Fii Elm ad Sof Comuig Mhods, 8. [] P. J. Arrazola, "Modlisaio umriqu d la cou: ud d ssibili ds aramrs d r idificaio du from r ouilcoau," Docoral Thsis, L'Écol Cral d Nas, l'uivrsié d Nas, Frac, 3. [3] P. J. Arrazola, D. Ugar, ad X. Domíguz, "A w aroach for h fricio idificaio durig machiig hrough h us of fii lm modlig," Iraioal Joural of Machi Tools & Maufacur vol. 48, , 8. [4] P. J. Arrazola ad T. r. Özl, "Ivsigaios o h ffcs of fricio modlig i fii lm simulaio of machiig," Iraioal Joural of Mchaical Scics, vol. 5,. 3 4,. 8

183 Char 5 5 Cocludig rmarks Th rimary goal of his work was o xlor h ossibiliis of h umrical simulaio of chi formaio usig h aricl fii lm mhod (PFEM). A homological aroach was adod o mahmaically rrs h bhavior of h workic ad h ool. Rsarch ffor has b focusd o hr oics:. A sabilizd, mixd, dislacm-rssur formulaio for hrmo-laso-lasic solid mdia wih low ordr fii lms wih qual irolaio ordr for boh dislacm ad rssur filds. A rducio of h comuig im of a yical umrical simulaio of mal cuig rocsss basd o h corrc slcio of h im igraio schm of h hrmomchaical roblm, 3. Th dvlom of a mshig schm basd o h aricl fii lm mhod which icluds msh qualiy imrovm hrough Dlauay riagulaio, a iovaiv daa rasfr schm wih miimum umrical diffusio ad a ffici sragy o modl h saraio of h chi ad h ic. Th iovaiv ar of our modlig is maily cocd o h umrical simulaio of coiuous ad srrad chi formaio. 5. O h gral faurs of h roosd soluio schm I char w rsd h diffr umrical sragis availabl i h liraur o modl machiig rocsss. As discussd i char, h umrical modlig of machiig rocss ds of diffr igrdis icludig: h saial discrizaio of h workic ad h ool, h im discrizaio of h balac quaios, h coac sragy ha allows us o modl h rasfr of momum ad rgy a h ool-chi irfac, h coac sarchig algorihm, h cosiuiv modl of h workic ad h ool, h fricio law a h ool chi fac irfac ad h im igraio of h cosiuiv quaios. Th iroducio o h vocabulary ad h sa of h ar of h umrical simulaio of mal cuig rsd i char ar cosidrd a ssial radig for w rsarchrs as wll as rsarchrs irsd i xadig hir kowldg abou h umrical simulaio of mal cuig. Thrfor, h sa of ar is cosidrd a imora coribuio of his work. Th sa of ar rsd i char idifid hr igrdis ha ca b imrovd o oimiz ad o icras h robusss of h simulaio schms ha currly xiss, hs ar h followig: ) h fii lm discrizaio, ) h im discrizaio, ad 3) h mshig schm. 83

184 5.. O h mixd dislacm-rssur formulaio for hrmo-laso-lasic roblm. Workic lasic bhavior is cosidrd isochoric. This bhavior maks cssary h us of fii lms, which ar fr of volumric lockig. To discriz h workic domai w us a sabilizd, mixd, dislacmrssur formulaio for hrmo-laso-lasic solid mdia wih low ordr fii lms wih qual irolaio ordr for boh dislacm ad rssur filds which is a xsio ad validaio of h Polyomial Prssur Projcio o h fild of o-liar solid mchaics. Th iroducio of h sabilizd mixd dislacm-rssur fii lms has rovd crucial i avoidig h advrs ffcs of volumric lockig, xhibid yically by sadard ur dislacm fii lms. Th xsio of h Polyomial Prssur Projcio o h fild of o-liar solid mchaics ca b cosidrd also a origial coribuio of his work. 5.. O h im igraio schm of h could hrmomchaical roblm Du o h mulil lgh scals ivolvd i h umrical simulaio of mal cuig rocsss ad hrfor h subsaial amou of dgrs of frdom, h yical imlici ad xlici im igraio schms of h balac quaios ar im cosumig. W wr hus comlld o dvlo a alraiv, aarly ovl, mhod for dalig wih his roblm. Th roosd global igraio rocdur has h iuiiv flavor of a fracioal s mhod (FSM), sic i is basd o dcoulig of h balac quaios ad h voluio quaios for h iral variabls. Th algorihmic srucur udrlyig his mhodology has b discussd i a i-dh mar, lacig scial mhasis o h issu of covrgc wih dcrasig h im s owards h imlici/xlici soluio. Th iroducio of h IMPLEX (imlici-xlici) igraio schm has rovd crucial i dcrasig h comuig im u o 9 ims i som rrsaiv umrical simulaios of machiig rocsss i comariso wih h sadard imlici/xlici schms. Furhrmor, i has bcom vid ha, h IMPLEX rocdur offrs a ffici soluio o h rad-off bw robusss ad comuaioal im rquirms O h mshig schm usig h aricl fii lm mhod (PFEM) Th mshig schm i his work is roosd i h framwork of h aricl fii lm (PFEM). I his work, w add o h PFEM wo w igrdis: ) Th cosraid Dlauay riagulaio i ordr o imrov mass cosrvaio ad chi sha hrough h simulaio, ad ) Th isrio ad rmoio of aricls for rsolvig fi-scals faurs i h soluio. Also, w roosd a ovl rasfr oraor of h iral variabls ha allows us o 84

185 miimiz h rror du o umrical diffusio. Th mshig schm dvlod i his work is cosidrd a origial coribuio of his work. 5. O h simulaio chology Th simulaio ool dvlod i his hsis allowd us o sudy h ssibiliy of machiig o rocsss variabls lik h ool radius, h rak agl, h flak agl, h udformd chi hickss ad ool siffss. Also, h chology dvlod allows us o sima h cuig forc ad h fd forc, h coac lgh ad h dformd chi variabls ha giv us iformaio abou how o oimiz h cuig rocss. Likwis, h rasiio from coiuous o sgmd chi wih icrasig cuig sd is rdicd by our simulaio chology. 5.3 O lis of rsarch A oi ha clarly ds o b imrovd i fuur works is h hrdimsioal alicaios. Alhough h cocs ad idas dvlod hr aly o 3D as wll, h coac algorihm ad h rmshig dscribd i Char bcom much mor comlicad. I is also imora o ivsiga sragis o icras h im s i gral (.g. by igraig h aricls alog h rajcoris), aly aralll comuig chiqus wih domai dcomosiio mhods such ha rocsss lik drillig, urig, millig ca b simulad i rasoabl comuig ims. Th fricio law a h ool-chi irfac ad h workic cosiuiv law d furhr rsarch, i such a way ha h rdicd umrical rsuls fi wll h xrimal daa. 85