An Axisymmetric Inverse Approach for Cold Forging Modeling

Transcription

1 An Axisymmtic Invs Aoach fo Cold Foging Modling Ali Halouani, uming Li, Boussad Abbès, ing-qiao Guo Abstact his a snts th fomulation of an axi-symmtic lmnt basd on an fficint mthod calld th Invs Aoach (I.A.) fo th numical modling of cold foging ocss. In contast to th classical incmntal mthods, th Invs Aoach xloits th nown sha of th final at and xcuts th calculation fom th final at to th initial billt. h assumtions of th ootional loading and th simlifid tool actions ma th I.A. calculation vy fast. h mtal s incomssibility is nsud by th nalty mthod. h comaison with Abaqus shows th fficincy and limitations of th I.A. which will b a good tool fo th liminay fom dsign. Indx ms Cold foging ocss, lag stains, intgatd constitutiv law, axi-symmtical lmnt, Invs Aoach. I. INRODUCION In a cold foging ocss, th mtal is lastically dfomd und th tool action. h foging ocss allows not only to chang th billt s sha but also to imov th mtal otis bcaus it fins th mtal gain siz. Fogd ats a oftn usd fo high fomanc and high liability alications wh th stngth and th human safty a cucially imotant. h numical modlling lays an imotant ol in th tool dsign fo th foging ocss. Many sach gous wo on th fowad mthod o on th bacwad tacing mthod fo th foging simulation and otimization [4]-[7]. Vy advancd wos hav bn don by Chnot, Foumnt t al. fom CEMEF in Fanc and th cosonding softwa FORGE is lagly usd in th foging industy. wo simlifid mthods calld Invs Aoach (I.A.) and Psudo Invs Aoach (P.I.A.) hav bn dvlod by Batoz, Guo t al. [8], [9] fo th sht foming modling. hy a lss accuat but much fast than classical incmntal aoachs. Manuscit civd Mach 18, 1. his wo was suotd in at by Fnch stat and by Chamagn-Adnn Rgion und th Pojct OPOMEF thi sonso and financial suot a gatfully acnowldgd. A. Halouani is with MAN/GRESPI, Univsity of Rims Chamagn-Adnn, F51687 Rims, Fanc (-mail: ali.halouani@tudiant.univ-ims.f).. Li is with MAN/GRESPI, Univsity of Rims Chamagn-Adnn, F51687 Rims, Fanc (-mail: yuming.li@univ-ims.f). B. Abbès is with MAN/GRESPI, Univsity of Rims Chamagn-Adnn, F51687 Rims, Fanc (cosonding autho, hon: ; fax: ; mail: boussad.abbs@univ-ims.f)..q Guo is with MAN/GRESPI, Univsity of Rims Chamagn-Adnn, F51687 Rims, Fanc (mail: yq.guo@univ-ims.f). h aim of th snt wo is to study th fasibility of th I.A. fo th cold foging modling. h fomulation of an axi-symmtic lmnt basd on th I.A. is dvlod fo th liminay fom dsign and otimization. In this study, fistly, w snt th basic ida of th I.A. and th main sts of modlling. hn, w tal about th fomulation of an axi-symmtic lmnt basd on th I.A.: th incil of vitual wo in lag dfomation, th lag logaithmic stains, th intgatd constitutiv law, th tchniqu to nsu th incomssibility of th mtal and th tatmnt of th bounday conditions. An xaml will b sntd to show th fficincy and limitations of th snt I.A. fo th foging ocss modlling. II. OULINE OF HE INVERSE APPROACH h Invs Aoach is basd on th nowldg of th final at sha. h diction of th tajctois of all matial oints fom th initial billt to th nown final at is don in on st by comaing dictly th initial and final configuations. wo basic assumtions a usd in this study: th assumtion of ootional loading (fo cold foging) givs an intgatd constitutiv law without considing th stain ath and th visco-lasticity, th assumtion of contact btwn th at and tools allows on to lac th tool actions by nodal focs without contact tatmnt. hs two assumtions ma th I.A. calculation vy fast. h I.A. ocdu is caid out as follows (Fig. 1): 1) h finit lmnt msh is catd on th nown final at. ) As an initial solution, th nods at th at contou a mad on th contou of th initial billt and a lina solution allows dtmining th ositions of th oth nods (intnal nods) in th initial billt. 3) h lag stains a calculatd by using th Cauchy-Gn lft tnso btwn th two mshs, th stsss a obtaind by using an intgatd constitutiv law. 4) An imlicit Nwton Rahson algoithm is usd to mov th nods in th initial billt in od to satisfy quilibium in th final at.

2 Incmntal aoach: numous sts fom to C Invs aoach: 1 st fom C to convnint to dfin th stains in th local lmnt systm ( x,, z) : u, x x u cos w sin z w, z xz u, z w, x wh u and w a th vitual dislacmnts along x and z, th adial coodinat and is th inclination angl of th local fnc with sct to th global fnc ( ( ox, ) to ( o, )). (4) C Fig. 1 wo aoachs fo foging ocss modlling III. AXISMMERIC ELEMEN BASED ON I.A. A. Pincil of Vitual Wo (PVW) In th Invs Aoach, th final configuation is nown and tan as fnc configuation. h quilibium of th final at is xssd by th incil of vitual wo: W int W xt (1) W W lt lt with W int dv W xt v v u f dv x z xz x z z wh W int and W xt a th lmnt intnal and xtnal vitual wos, th Cauchy stsss, th vitual stains, () (3) u th vitual dislacmnts, f th volum focs. W not that: th vitual stains a infinitsimal, so thy a lina functions of vitual dislacmnts; whas, th abov Cauchy stsss a latd to th lag stains, so gnally thy a calculatd by an incmntal algoithm. In th snt study, a total mthod is oosd: 1) th dfomation gadint tnso and th Cauchy-Gn lft tnso a dfind; ) thn th incial longations and lag logaithmic stains a calculatd; 3) finally th Cauchy stss is calculatd by using an intgatd constitutiv law (Hncy-Miss). B. Vitual stain oato Fo an axi-symmtic oblm, th stain vcto is oftn dfind in th global cylindical coodinat systm (,, Z, Fig. ). In ou finit lmnt fomulation, it is mo h snt lmnt is an axi-symmtical CS lmnt with th nods and six dgs of fdom. h dislacmnts a intolatd linaly in tms of nodal dislacmnts: u N N N u 1 3 w N1 N N 3 un u1 w1 u w u3 w3 n (5) wh N i (x,z) a th lina intolation functions fo an lmnt. Substituting Eq. (5) into (4), w obtain th following vitual stain oato [B m ]: Bm un (6) with N1, x N, x N3, x N1cos N1sin Ncos Nsin N3cos N3sin Bm N1, z N, z N 3, z N1, z N1, x N, z N, x N3, z N3, x X Z, W, W M (,, Z), U, U, V Fig. Cylindical coodinat systm (,Z) C. Intnal foc vcto Substituting Eq. (6) into () givs th lmnt intnal foc vcto in th local fnc:

3 int Wint un B A m da un F (7) Sinc th stss vcto is not constant in an lmnt, gnally w should intoduc th fnc lmnt and us th numical intgation to calculat th intnal foc vcto in th local fnc. A ducd intgation mthod is oosd by Batoz and Dhatt []: th stsss a suosd lina in an lmnt and th baycnt is tan as singl intgation oint. hus th calculation of th intnal foc vcto bcoms vy siml: Fint A B (8) m m m 1 ( ) 3 D. Extnal foc vcto with m 1 3 In a foging ocss, th initial billt is submittd to a nomal ssu foc and a tangntial fiction foc on th contou. In th Invs Aoach, ths tool actions a simly sntd by som xtnal nodal focs at th final configuation to avoid th contact tatmnt. At a nod, th diction of th sultant foc n f can b dfind by th fiction con and th slid diction: 1 nf nt 1 wh n is th unit nomal vcto of th contou, t th unit vcto of th ojction of th nod dislacmnt on th tangnt diction of th contou, th fiction cofficint. h FE disctization allows on to stablish th following quations snting th quilibium on a nod : F ( P ) Fint Pn F Pn Z FZ i xt xt wh n n n Z f int (9) (1) snts th diction of th sultant foc at th nod (Eq. 9), th intnsity of this foc P can b dtmind as follows: F P n nz F Z int h lmnt xtnal foc vcto is finally obtaind: F n F n n xt Z FZ n int Z (11) (1) h solid has an axis of volution ( o, Z ). Any oint M in th solid is dfind in th global fnc by its cylindical coodinats (, θ, Z, Fig. ). h dislacmnt fild is comosd of th adial, cicumfntial and vtical dislacmnts (U, V, W): U U V W (13) his allows us to obtain th diffntial of th dislacmnt fild: du ( du Vd) ( dv Ud) dw (14) Fo an axi-symmtic oblm, ach oint of th solid movs in its midian lan (V=) and th dislacmnt fild is indndnt of th cicumfntial coodinat ( ). In this cas, th dislacmnt gadint tnso in cylindical coodinats is ducd to: U, U,Z U U W, W,Z (15) In ou Invs Aoach in lag stain and lasticity, it is mo convnint to dfin th dfomation gadint tnso in th lmnt local systm: 1u,x u,z 1 ucos wsin u F I 1 l x w,x 1w,z (16) wh u is th dislacmnt vcto in th local fnc fom th initial osition vcto x to th final osition vcto x. F. Invs of th Cauchy-Gn lft tnso h invs of th Cauchy-Gn lft tnso in th local systm is dtmind by th following xssion: a b 1 1 B F F d l l (17) b c h tnso B dfind in th local fnc can b tansfomd to th incial fnc to obtain th th ignvalu ( 1,, 3 ), thn th th incial longations ( 1,, 3) : E. Lag logaithmic stains In th Invs Aoach, w us an intgatd constitutiv law which associats th total stains to th total stsss. hs lag stains a calculatd by th following sts: th invs of dfomation gadint tnso th invs of th Cauchy-Gn lft tnso th incial longations th logaithmic stains. h wod «invs» is usd to indicat that th nown final configuation is tan as fnc configuation, and th calculation is caid out fom th nown final at to th unnown initial billt B M M 1 ac ac with b 3 (18) (19)

4 M cos sin 1 sin cos () wh is th angl fom th local fnc to th incial fnc. Finally, th incial logaithmic stains a dfind by: 1 Log1 Log Log 3 3 (1) h assumtion of incomssibility of th mtal givs th volum stain null: () v 1 3 x z (3) 1 3 x z 1 G. Pnalization mthod fo mtal incomssibility h incomssibility of th mtal can b nsud by intoducing a Lagang multili o a nalization tm (Kobayashi and Altan, [5]). In ou Invs Aoach, w add a nalization tm in th Pincil of Vitual Wo: W dv u f dv K dv lt v v v v v (4) wh v is th volum stain, K is a gat ositiv facto which allows to annul v in th convgnc loo. Using th vitual stains oato (Eq. 6), w obtain a vcto of "quivalnt focs" lativ to th volum stain: 1 u B u B 1 1 v n m 63 n v so K dv u F v v n v v Fv ma K Bv x z m (5) (6) H. Intgatd constitutiv law (Hncy-Miss) In th snt study, th isotoic constitutiv law is adotd. h Von Miss cition of lasticity is xssd by: 1 P f ( ) (7) with ( x ) ( z ) ( z x ) 6 xz 1 wh is th quivalnt stss, th yild stss. h nomality law allows on to stablish th lation btwn th lastic stain at and th Cauchy stss using th lastic multili : f P (8) ( P ) (9) So 1 1 Using Eqs. (8) and (9), w obtain: P (3) h Hncy assumtion (ootional loading) allows to dictly intgating th lastic stain at: t dt P 1 P ( 1 1 ) P (31) Hs Es E wh Es is th scant modulus of th uniaxial stss stain cuv. Adding th lastic stains, w obtain th total stain: ( C ( ) P) 1 1 (3) E E wh C is th lastic flxibility matix. Finally, th total Cauchy stsss a obtaind in tms of th total logaithmic stains as: H ( C( ) P) E E (33) s I. Bounday conditions on an igula contou In th snt Invs Aoach, th dislacmnts of th nods at th contou a suosd tangntial to th contou dfind by th tools. Dhatt and ouzot [1] oosd to stablish a contou fnc and imos th nomal dislacmnts null in this fnc. Considing a nod i on th contou of a msh. W stablish a contou fnc dfind by th tangntial and nomal dictions in which w imos th nomal ' dislacmnt V (tangnt dislacmnt U ). h ' i following matix allows on to tansfom ls dislacmnts btwn th contou fnc and global fnc: Ui cosi sin iu' i V i sini cos i V' i s i (34) It is mo convnint to tansfom th tangnt stiffnss matix and th vcto of sidual focs at th lmntay lvl. Aft th solution, th dislacmnts in th contou fncs should b -tansfomd into th global fnc. IV. NUMERICAL RESULS h validation of th simlifid Invs Aoach is don by using th cod ABAQUS /Exlicit. wo axi-symmtic ats a considd.

5 A. Numical sults of th at 1 h distibutions of th quivalnt lastic stain obtaind by th Invs Aoach and Abaqus a shown in Fig. 5. W not that th distibutions a simila and th maximal and minimal valus a in good agmnt. B. Numical sults of th at h scond axi-symmtic (Fig. 6) has a hoizontal lan of symmty. h half sction is mshd with 889 axi-symmtic tiangl lmnts (CAX3 of Abaqus, Fig. 7). h matial otis of th billt a: oung's modulus E=13 (MPa), Poisson's atio =.3, fiction cofficint =.15, Hollomon stss-stain cuv σ=14 ε. (MPa). h unch tavl is 3.7 (mm). Fig. 3 Gomty of th at 1 h gomty of th at 1 is shown in Fig. 3. In th simulation by th incmntal aoach, th initial billt is disctizd into 176 axi-symmtic tiangl lmnts (CAX3 of Abaqus). h tools (unch and di) a suosd igid and modld by analytic igid wi. On oint of th tool sufac was dfind as fnc oint. h tools dislacmnt was scifid by using this oint. A mast-slav contact aoach is usd in this simulation wh th tools a considd as th mast sufacs and th out sufac of th billt (sufac facing th tools) constituts th slav sufac. h matial of th billt is th lad whos otis a: oung's modulus E=17 (GPa), Poisson's atio =.4, fiction cofficint =.35, Hollomon stss-stain cuv σ=65.8 ε.7 (MPa). h unch is movd vtically. h total unch tavl is 3.5 (mm). In od to coma th two aoachs, w msh th billt (Fig. 4a) and us Abaqus to obtain th msh of th final at (Fig. 4b), thn w us this msh fo I.A. modling to obtain th msh of th initial billt (Fig. 4c). W not th msh of th initial billt obtaind by I.A. is vy simila to that of Abaqus. a. Invs Aoach b. Abaqus (Incmntal Aoach) Fig. 5 Equivalnt lastic stain obtaind by I.A. and Abaqus (4a) Initial billt Abaqus incmntal (4c) Initial billt Invs Aoach Fig. 6 Gomty of th at C (4b) Final t Fig. 4 Initial and final mshs of th at 1

6 C (7a) Initial billt Abaqus incmntal (7b) Final at Invs Aoach (7c) Initial billt Fig. 7 Initial and final mshs of th at h msh of th initial billt and th msh of th final at obtaind by Abaqus a shown by Fig. 7a and 7b. hn w us th msh 7b fo th I.A. modling which givs th msh of th initial billt (Fig. 7c). A faily good agmnt is obsvd btwn th two mshs (Fig. 7a and 7c). Fig. 8 shows th distibutions of th quivalnt stain obtaind by th Invs Aoach and Abaqus incmntal aoach. Comaing th quivalnt stain distibutions obtaind by th both aoachs, on obsvs that th quivalnt lastic stain distibutions a quantitativly vy clos to ach oth. h maximum lastic quivalnt stains a sctivly.97 and 1.. h cntag o is asonably acctabl (5. %). Invs Aoach Abaqus (Incmntal Aoach) V. CONCLUSION A simlifid mthod calld Invs Aoach (I.A.) is dvlod fo th axi-symmtical cold foging modling. h aoach is basd on th nowldg of th sha of th final at. h assumtions of th ootional loading and th simlifid tool actions ma th calculation of th I.A. vy fast. h stain quivalnt sults obtaind by th Invs Aoach a lss accuat, but vy clos to thos obtaind by th Abaqus incmntal aoach. h Invs Aoachs is vy advantagous to quicly aliz th liminay fom dsign and otimiz th ocss aamts. Som limitations of th I.A. a obsvd. h assumtions on th constitutiv law and th tool actions a qustionabl; thy cannot ovid good stss stimation bcaus of nglcting th loading histoy. Fo comlx ats in vy lag dfomation, th mshing oation and a mo owful solution algoithm should b considd. h futu wo fo th I.A. in th fog alication is to imov th stss stimation. Rcntly, a nw aoach calld Psudo Invs Aoach has alady bn oosd by Guo t al. [9] fo th sht fo sht foming, which s th advantags of th I.A. but givs good stss stimation with th loading histoy considation, will b tid fo th fog alication. ACKNOWLEDGMEN h collaboation of ou atn th Univsity of chnology of oys is gatfully acnowldgd. REFERENCES [1] G. Dhatt, G. ouzot, Un ésntation d la méthod ds élémnts finis, dition, MALOINE S.A. Edito, [] J.L. Batoz, G. Dhatt, Modélisation ds stuctus a élémnts fini, Vol. 1, Solids élastiqus, Editu HERMES, Pais, 199. [3] J.L. Batoz, G. Dhatt, Modélisation ds stuctus a élémnts fini, Vol. 3, Coqus, Editu HERMES, Pais, 199. [4] Bohati C., Chnot, J.L., Finit lmnt fomulation fo non stady stat viscolastic dfomation, Int. J. Mth. Eng., 1, , [5] Kobayashi S., Oh S.I., Altan., Mtal foming and finit lmnt mthod, Oxfod Univsity Pss, [6] Zhao G., Wight E., Gandhi R.V., Foging fom dsign with sha comlxity contol in simulating bacwad dfomation, Int. J. Mach. Manuf., Vol 35-9, , [7] Foumnt L., Chung S.H., Dict and adjoint diffntiation mthods fo sha otimization in non-stady foming alication, Euoan Confnc on Comutational Mchanics, Cacow, 1. [8] Guo.Q., Batoz J.L., Dtaux, J.M., Duoux, P., Finit lmnt ocdus fo stain stimations of sht mtal foming ats, Int. J. fo Num. Mthods in Eng., Vol. 3, , 199. [9] Gati, W., Guo.Q., Nacu H., Batoz J.L., Aoch sudo invs ou stimation ds containts dans ls iècs mboutis axisymétiqus, Rvu Euoénn ds Elémnts Finis, Vol. 1, n 7-8, , 3. Fig. 8 Equivalnt lastic stain obtaind by I.A. and Abaqus