Pseudo Inverse Approach for cold forging processes and its comparison with Adaptive Incremental Approach

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1 èm Congrès Français d Mécaniqu Bsançon, 9 août au sptmbr 11 Psudo Invrs Approach for cold forging procsss and its comparison with Adaptiv Incrmntal Approach F.J. MENG a, A.HALOUANI b, C. LABERGERE a, Y.M. LI b, B. ABBES b, P. LAFON a, Y.Q. GUO b a. Institut Charls Dlaunay LASMIS, Univrsité d Tchnologi d Troys b. Group d Rchrch n Scincs Pour l Ingéniur, Univrsité d Rims Champagn Ardnn Abstract : Numrical simulation for highly non-linar forming procss rquirs fficint numrical mthods. In an optimization loop, th calculation cost is vry larg, so it is important to choos a good numrical simulation tool to rduc it. In this papr, w propos to simulat a D axisymmtric forging procss of a whl by using a psudo invrs approach and an adaptiv incrmntal approach. Ths two mthods will b compard in trms of computation tim and prdictability on th form of rough forging. Th comparison of th rsults will show th fficincy and limitations of th Psudo Invrs Approach. Résumé : La simulation numériqu ds procédés d mis n form st un problèm fortmnt non-linéair qui nécssit ds méthods numériqus prformants. Ls coûts ds calculs, souvnt importants, rndnt difficil l utilisation d un outil d simulation numériqu dans un boucl d optimisation. Nous proposons dans c papir d simulr un procédé D axisymétriqu d forgag d un rou n utilisant un approch psudo invrs t un approch MEF adaptativ incrémntal. Cs dux approchs sront nsuit comparés n trms d tmps d calcul t prédictibilité sur la form ds bruts d forg. La comparaison avc l approch adaptiv incrémntal montr l'fficacité t ls limits d l Approch Psudo Invrs. Kywords : Mtal Forming Procsss, Numrical Simulation, Psudo Invrs Approach, Adaptiv Incrmntal Approach 1 Introduction In this papr, w propos to simulat a D axisymmtric forging procss of a whl by using a psudo invrs approach and an adaptiv incrmntal approach. Ths two mthods us diffrnt matrial bhaviour modls and numrical tchnologis. Th adaptiv incrmntal approach (A.I.A.) is th classical mthod for th simulation of mtal forming procsss. It givs good strain and strss stimation, but is tim consuming. A nw approach calld Psudo Invrs Approach (P.I.A.) was proposd by Batoz, Guo t al. [1] for th sht forming modlling, which kps th rapidity of th Invrs Approach but givs good strss stimation owing to th loading history considration. By comparing th rsults of th two mthods, it can b found that th P.I.A. mthod is vry fficint but lss accurat than th incrmntal approach. Mthodology of th two numrical approachs.1 Psudo invrs approach A simplifid mthod calld Invrs Approach (I.A.) has bn dvlopd for th axisymmtrical cold forging modlling. Th approach is basd on th knowldg of th final part shap. Two basic assumptions ar usd: th assumption of proportional loading givs an intgratd constitutiv law without considring th strain path and th plastic flow, and th assumption of tool-billt actions allows on to rplac th tool actions by nodal forcs without contact tratmnt [-3]. Ths two assumptions mak th I.A. calculation vry fast. This approach givs fairly good strain stimation but poor strss stimation. Th P.I.A., basd on th traditional I.A., is dvlopd to improv th strss stimation. Th main 1

2 èm Congrès Français d Mécaniqu Bsançon, 9 août au sptmbr 11 dvlopmnts in th P.I.A. can b rsumd as follows: - Som intrmdiats configurations, without contact tratmnt, ar gnratd for som givn punch positions to considr th dformation history. - For ach intrmdiat configuration, th strain incrmnt is calculatd by th invrs mthod btwn th prvious and actual configurations. - Th mshs at th last and actual stps ar indpndnt. So at th bginning of th actual stp, a transfr of th strain and strss filds should b carrid out btwn ths two mshs for plastic intgration schm. - An fficint mthod for plastic intgration calld Dirct Scalar Algorithm is dvlopd: th quations in function of unknown strss vctors ar transformd into th scalar quations using th notion of th quivalnt strss; thus th plastic multiplir can b dirctly obtaind without itrations [4]. Th P.I.A. is a good numrical tool for th prliminary prform dsign and optimization for th forging procsss..1.1 Larg strain calculation In th I.A., th calculation of larg strains is don in on stp by dirctly comparing th initial billt C and th final part C and using th logarithmic strains [-3]. Th sam calculation is kpt in th P.I.A. but btwn two succssiv configurations. For an axi-symmtric problm, it is mor convnint to dfin th strains in th local lmnt systm ( x,, z). Th movmnt of a matrial point btwn two succssiv configurations is xprssd by x x u whr x and x ar th initial and final position vctors, u is th displacmnt vctor in th local rfrnc. Thn th invrs dformation gradint tnsor is dfind as follows: x u x x Th invrs Cauchy-Grn lft tnsor in th local rfrnc is dfind by: 1 d x d x I d x F d x L (1) Th ignvalus ( λ, λ, λ ) logarithmic strains: m x x x x B F F d d d B d m 1 T 1 () of th tnsor B 1 m givs th thr principal longations, thn th thr larg ln i i i 1,,3 (3) Finally, ths larg strains ar transformd into th lmnt local rfrnc by: L L x cos sin 1 1 z sin cos 3 xz sin cos sin cos whr is th angl from th local rfrnc to th principal strain rfrnc. (4).1. Intrmdiat configurations and fild transfr Th basic ida of th P.I.A is to us svral intrmdiat configurations and to xcut an invrs calculation btwn two succssiv configurations considring th initial strains and strsss in th last stp. Ths intrmdiat configurations ar dtrmind gomtrically thn corrctd by som itrations of quilibrium. Lt s considr th P.I.A. in two stps (Figur 1). For th configuration C 1, th two mshs ar indpndnt: M 1 1 is th msh cratd thn corrctd at th nd of stp 1, M 1 is th msh obtaind at ach itration of th stp. A transfr of th strain and strss filds should b carrid out btwn ths two mshs on C Simplifid tool actions In th forging procss, th contact tratmnt is oftn unstabl and tim consuming. In th P.I.A., to avoid th

3 èm Congrès Français d Mécaniqu Bsançon, 9 août au sptmbr 11 contact tratmnt, th tool actions ar simply rprsntd by som xtrnal nodal forcs. At a nod, th valu of th rsultant forc is unknown; th dirction of this forc nf can b dtrmind by th friction con and th slid dirction: 1 n f n t (5) 1 whr n is th unit normal vctor on th contour, t th unit vctor of th nod slid displacmnt in th tangnt dirction of th contour, th friction cofficint. In th itration loop, th two intrnal forcs at ach nod k ar known. Th lmnt intrnal forc vctor in th local rfrnc is thn givn by: T int Wint u B rda u F (6) * * n A m n Thus th quilibrium condition allows calculating th unknown intnsity of th rsultant tool action forc F : k k k k k P n r F r F xt F int k k k (7) P n Z F xt Z int k k T whr n n n rprsnts th dirction of th rsultant forc at th nod k. r Z f Initial msh on C A.I. (stp 1) Configuration C 1 at stp 1 Configuration C 1 at stp Msh M 1 Transfr of th strain and strss filds A.I. (stp ) Msh M 1 1 Configuration C at stp Figur 1. Transfr of Filds btwn Two Indpndnt Mshs. FEM adaptiv incrmntal approach Nowadays, in mtal forming procsss th classical FE numrical simulation is usd to avoid xpnsiv xprimnts and many rsarch works hav bn mad using this powrful tool [5]. In th laboratory LASMIS of Univrsity of Tchnology of Troys, xtnsiv works ar don sinc tn yars. A spcific numrical mthodology packag basd on FEM has bn dvlopd to solv lasto- plastic problms with ductil damag in larg strains, such as th numrical simulation of mtal forming procsss. In ordr to avoid larg msh distortions and achiv proprly th solution convrgnc, th mshing and r-mshing procdur ar also includd [6-8]...1 Elasto-plastic constitutiv quations To build th matrial bhaviour modl, w should considr coupld multi-physics (plastic strains, hardning and damag) in th forging procss. W us an advancd constitutiv quations accounting for mixd nonlinar isotropic and kinmatic hardning strongly coupld with ductil isotropic damag. Th coupling btwn th ductil damag and th lasto- plastic constitutiv quations is formulatd in th framwork of th thrmodynamics of irrvrsibl procsss togthr with th Continuum Damag Mchanics (CDM) thory [6-8]. In our study, to simplify th problm and for comparison with th P.I.A., th kinmatic hardning is nglctd and only thr pairs of th intrnal stat variabls ar usd: (, ) for th plastic flow; (r, R) for isotropic hardning and (D, Y) for ductil damag. Whn th currnt configuration contains som ductil damag, th concpt of ffctiv variabls basd on th hypothsis of total nrgy quivalnc is usd. 3

4 èm Congrès Français d Mécaniqu Bsançon, 9 août au sptmbr 11 Accordingly, th fully coupld constitutiv quations ar summarizd as following. 1 D : R 1 D Qr Stat rlation 1 v I II III Y 9v (1 D ) E (Damag) (1 D ) E H 1 1 D Qr Evolution quation J R f y 1 D 1 D (Von Miss plasticity critria) Dp n T with n 1 r br 1 D s Y Y D S 1 D J 3 dv : dv 3 1 D J dv Whr Q is th isotopic hardning modulus; b is th non-linr cofficint; S, s, β and Y ar th matrial cofficints dscribing ductil damag volution; n th outward normal tnsor to th yild surfac f=; y th limit yild strss... Numrical aspct Th fully coupld thrmo-mchanical constitutiv quations prsntd abov hav bn implmntd into Abaqus/Explicit FE cod using th VUMAT usr subroutin. A dynamic xplicit rsolution procdur has bn usd in ordr to solv th quilibrium problm [6-8]. Du to th larg dformations of th matrial, th msh is rapidly distortd. To avoid ths distortions, th rmshing procdur must b adaptd. By using an adaptiv rmshing tchniqu, th msh is rfind in damagd zons and magnifid in inactiv zons. Th rmshing opration is indispnsibl for gomtrically and physically complicatd non-linar forming procss. 3 Numrical application In this papr, a D axisymmtric forging procss is prsntd and th final shap of th whl is shown in Figur. Bcaus of symmtry conditions, only a quartr of th axisymmtric part was modld. A st of boundary conditions of symmtry wr usd: th lowr horizontal plan is fixd and th uppr tool is translatd with a constant vlocity. Th total uppr punch travl quals to 6mm. Figur. Shap of th final billt Th matrial of th whl is aluminium whos mchanical proprtis ar dfind as follows: Young's modulus E = 73 (MPa), Poisson's ratio =.3, friction cofficint btwn th tool-billt µ=.5. Th matrial bhaviour modls usd by th two simulation mthods ar a littl diffrnt. For th 4

5 èm Congrès Français d Mécaniqu Bsançon, 9 août au sptmbr 11 P.1783 P.I.A., th Hollomon strss-strain curv is usd: 17.68( ) MPa. By FEM adaptiv incrmntal approach, an advancd lasto-plastic constitutiv quation is usd by implmnting VUMAT usr subroutin in th calculation and th paramtrs using ar y =71(MPa), Q=3(MPa), b=3, S=1, s=1, =5 and Y =. Hr w don t tak into account th influnc of tmpratur. Although th two mthods us diffrnt matrial bhaviour modl, th final strss-strain curv of th matrial is almost th sam. 3.1 Analysis of th numrical simulation rsults Distribution of th quivalnt plastic strain and quivalnt strss Figur 3 shows th distributions of th quivalnt plastic strain obtaind by th P.I.A. and FEM adaptiv incrmntal approach. For th P.I.A., th billt is mshd with 1371 axisymmtric triangl lmnts. For th mthod of th FEM adaptiv rmshing procdur, th billt is mshd with a quadrilatral lmnt CAX4R and th whol procss includs 5 rmshing stps. Th initial numbr of th lmnt is 344 and th final numbr of th lmnt incras to 163; th maximum msh siz is 3mm and th minimum msh siz is.9mm. With th dformation incrasing, th msh is rfind du to th larg strain. This adaptiv rmshing tchnology avoids th lmnt distortion during th larg dformation mtal-forming procss. In Figur 3, w also obsrv that th strain distributions of th two mthods ar vry similar to ach othr. Both th maximum and minimum valus ar situatd in th sam locations. Th maximum plastic quivalnt strains obtaind by th P.I.A. and incrmntal approach ar rspctivly.919 and.91. a. Psudo invrs approach b. FEM adaptiv incrmntal approach Figur 3. Equivalnt plastic strain obtaind by P.I.A. and A.I.A. a. Psudo invrs approach b. FEM adaptiv incrmntal approach Figur 4. Equivalnt strss obtaind by P.I.A. and A.I.A. 5

6 èm Congrès Français d Mécaniqu Bsançon, 9 août au sptmbr 11 Figur 4 shows th distributions of th quivalnt strss obtaind by th P.I.A. and adaptiv incrmntal approach. W not that th two strss distributions ar similar. Th maximum valus ar vry clos to ach othr (167.8 and MPa), but th minimum valus ar fairly diffrnt Comparison of th CPU tim Th incrmntal approach is widly usd for th forging procss modlling and it givs good strain and strss stimation. But compard to th Psudo Invrs Approach, th incrmntal approach is vry tim consuming. Th total CPU tim usd for th simulation of th whl is 145 s by th incrmntal approach but only 3 s by P.I.A. (78% of tim saving). In traditional Invrs Approach, two basic assumptions ar usd: th assumption of proportional loading (for cold forging) givs an intgratd constitutiv law without considring th strain path nithr th viscoplasticity, and th assumption of simplifid tool-billt actions allows on to rplac th tool actions by nodal forcs without contact tratmnt. Ths two assumptions mak th I.A. calculation vry fast. Th P.I.A. (or multi-stp invrs approach) basd on th I.A. kps th I.A. s high fficincy but givs good strss stimation owing to th loading history considration. In th FEM adaptiv rmshing mthod, an advancd lasto-plastic matrial bhaviour modl and th fac to fac contract condition ar usd, so th whol procss is tim consuming. 4 Conclusion By comparing with th rsults of th two diffrnt mtal forming simulation mthods, w can find that th rsults of Psudo Invrs Approach hav a good agrmnt with th rsults of classical adaptiv incrmntal approach and th psudo Invrs Approach is provd to b vry tim saving. So in th cas of th complicat tools shap or vry larg dformation mtal forming procss, th adaptiv incrmntal approach should b usd to gt accurat rsults; in th cas of th prliminary prform dsign and optimization for th forging procss modlling, th P.I.A. provs to b vry usful and highly fficint. Rfrncs [1] W.Gati, Y.Q. Guo, H. Nacur, J.L. Batoz, Approch psudo invrs pour stimation ds contraints dans ls piècs mboutis axisymétriqus, Rvu Europénn ds Elémnts Finis, Vol. 1, n 7-8, pp , 3. [] A. Halouani, Y.M. Li, B. Abbès, Y.Q. Guo. An Axisymmtric Invrs Approach for Cold Forging Modlling, Enginring Lttrs, Vol. 18, Issu 4, pp , 1. [3] A. Halouani, Y.M. Li, B. Abbès, Y.Q. Guo. Simulation of axi-symmtrical forging procss by Invrs Approach, Matrials Scinc Forum Vols (11) pp [4] Y.M. Li, B. Abbs, Y.Q. Guo. Two Efficint Algorithms of Plastic Intgration for sht Forming Modlling, ASME J. Manufacturing Scinc and Tchnology, Vol. 19, p , [5] M. Issa M., K. Saanouni, C. Labrgèr, A. Rassinux, Prdiction of srratd chip formation in orthogonal mtal cutting by advancd adaptiv D numrical mthodology., Int. J. Machining and Machinability of Matrial,Vol 9., Nos ¾, 11 [6] Labrgr C., Rassinux A., Saanouni K., D adaptiv msh mthodology for th simulation of mtal forming procsss with damag, DOI 1.17/s z, articl in Prss, 11 [7] Labrgr C., Lstriz P., Saanouni K. Numrical dsign of xtrusion procss using finit thrmolastoviscoplasticity with damag. Prdiction of chvron shapd cracks, Ky Enginring Matrials, Vol 44, pags 65-7, 1 [8] J. Mariag, K. Saanouni, P. Lstriz, A. Chrouat, Numrical simulation of ductil damag in mtal forming procsss. A simpl prdictiv modl. 1.thortical and numrical aspcts, Int. J. Form. Procss. 5 ( 4) ()