AN IMPROVED MULTIAXIAL STRESS-STRAIN CORRECTION MODEL FOR ELASTIC FE POSTPROCESSING. H. Lang 1, K. Dreßler 1

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1 AN IMPROVED MULTIAXIAL STRESS-STRAIN CORRECTION MODEL FOR ELASTIC FE POSTPROCESSING H. Lag 1, K. Drßlr 1 1 Frauhofr Istitut für Tcho- ud Wirtschaftsmathmatik Frauhofr Platz 1, Kaisrslautr, Grmay ABSTRACT I this papr, th modl of Köttg, Barky ad Soci, which corrcts th lastic strss ad strai tsor historis at otchs of a mtallic spcim udr o-proportioal loadig, is improvd. It ca b usd i coctio with ay multiaxial σ -ε -law of icrmtal plasticity. For th corrctio modl, w itroduc a costrait for th strai compots that gos back to th work of Hoffma ad Sgr. Paramtr idtificatio for th improvd modl is prformd by Automatic Diffrtiatio ad a stablishd last squars algorithm. Th rsults agr accuratly both with trasit FE computatios ad otch strai masurmts. KEYWORDS Jiag s Modl of Elastoplasticity, Strss-strai corrctio, Paramtr Idtificatio, Automatic Diffrtiatio, Last-Squars Optimizatio, Colma-Li Algorithm INTRODUCTION I ordr to giv a rliabl liftim prdictio for a lastoplastic mtallic body, which is subjctd to xtrior loads, good kowldg of th local strsss ad strais ovr tim is cssary. Numrical aalysis may b applid, cf. Figur 1. Figur 1: Possibl ways i fatigu aalysis I LCF-aalysis, th xpsiv trasit lastoplastic boudary valu problm with a p appropriat lastoplastic costitutiv matrial law ɺ σ ( x, t) = C ( x, t) ɺ ε ( x, t) is solvd i ordr

2 to rciv th ral lastoplastic strsss σ ad strais ε. Th followig laws ar wll kow ad joy grat popularity: Liar kimatic (plus isotropic) hardig [10, 11], Mroz- Garud [11], Armstrog-Frdrick [11], Chaboch [11], Jiag-Shitoglu [07], ad may mor I HCF-aalysis, th much chapr liar lastic boudary valu problm with Hook s costitutiv law σ ( x, t) = C( x) ε ( x, t) is solvd to rciv th lastic strsss ad strais ε. If th locatios, whr th lastic strss σ crosss th yild strss - ad whr it thus ovrstimats th lastoplastic strss σ -, ar small ad if th rspos of th body is liarly lastic almost vrywhr, local corrctios may b applid bfor projctio oto scalar quatitis is do. Hr multiaxial Nubr approachs, cf. Glika t al. [01, 03, 12], which ar basd o Nubr s rgy rlatio ε σ =ɶ ε ɶ σ (for ach compot), ij ij ij ij ad psudo paramtr approachs, cf. Köttg t al. ad Hrtl [05, 08, 09], hav b suggstd. But so far, computatioal rsults for th citd multiaxial corrctio approachs diffr xcssivly from trasit lastoplastic FE solutios or masurmts. I this papr, w propos th followig two suggstios for Köttg s approach, which turd out to rsult i sigificat improvmt of xistig rsults: First, w impos Sgr s strai ratio rlatio o th corrctio modl, scod, w prform paramtr ssitivity aalysis by Automatic Diffrtiatio (AD). Morovr, it is possibl to giv aalytical rror stimats (of global atur) for th corrctd strsss ad strais ad a ral mathmatical justificatio for Köttg s approach i th cas of liar kimatic hardig matrial, cf. [10]. Th modifid corrctio modl lis somwhr btw HCF ad LCF i Figur 1: It is fast ough for HCF, th aalytical stimats i [10] sur a small rror for th rag of LCF. THE ORIGINAL STRESS-STRAIN CORRECTION MODEL OF KÖTTGEN Th modl w cosidr is th psudo paramtr modl of Köttg t al. [05, 08, 09], which is calld th σ -approach thri. W simply am it Köttg s modl. Th iput σ ( t) at a fixd poit x of th body ad a st of (fictiv) psudo or lastic paramtrs govr th movmt of a (fictiv) lastic yild surfac i a (fictiv) lastic strss spac with a costitutiv lastoplasticity law, which is appropriat for th matrial. Figur 2: Th lastic strss spac (lft-had) ad th corrctig strss spac (right-had) i coctio with Jiag s costitutiv modl Th movmts i th corrctig strss spac with a corrctig yild surfac ar simultaous to th movmts i th lastic strss spac with th sam yild surfac ormal ( t ). Hr th ral matrial paramtrs togthr with th costitutiv matrial (th

3 sam as i th lastic strss spac) uiquly dtrmi th volutio. Output of th pl corrctig strss spac is th corrctd strss σɶ ( t) (ad th corrctd plastic strai εɶ ( t), togthr with Hook s law l 1 ɶ ε ( t) = C ɶ σ ( t) yildig th corrctd total strai ( ) l pl ɶ ε t = ɶ ε ( t) + ɶ ε ( t) ). For illustratio cf. Figur 2, for dtails th radr is rfrrd to [10] or th appdix i [08]. For liar kimatic hardig matrial, it is possibl to writ dow th bst choic for th psudo paramtrs i a abstract fashio, cf. [10]. For mor complx matrial laws, th bst choic of psudo paramtrs is still a op qustio, but for practic it is sufficit to dtrmi thm by last-squars-fittig. THE MODIFIED CORRECTION MODEL W suggst to modify Köttg s modl [08, 09, 10] by th itroductio of Sgr s strai-ratio costrait for sharply otchd axls [06]. This rsults i ε11( t) ε11( t) = ɶ, (1.1) ε 22( t) ɶ ε 22( t) whr 1 ad 2 dot th local coordiat systm dirctios, which ar tagtial to th otch root: 1 is th logitudial dirctio, 2 is th circumfrtial dirctio, as it is displayd i Figur 3. Figur 3: Choic of local coordiat systm Rlatio (1.1) costituts a hard algbraic costrait (with giv lft-had lastic strai ratio) o th diffrtial quatios of th Köttg modl, thus, it bcoms diffrtial-algbraic. A appropriat prdictor-corrctor algorithm for its itgratio is giv i [10]. W giv two rasos, why to impos rlatio (1.1) o th corrctio modl. First, it is provd i [10], that it is ihrtly impossibl for th origial corrctio modl to approximat ach compot of th pla strss tsor for uiaxial ad biaxial loadig. Th raso is o of th mai drawbacks of Köttg s modl, amly that th diffrc of th corrctd strss dviator dv σɶ ad th corrctd backstrss αɶ is coupld i a simultaous mar to th diffrc of th lastic strss dviator dv σ ad th (fictiv) lastic backstrss α. I.. th tsors dv σ α ad dvɶ σ ɶ α ar colliar at ay tim, s th dashd tsors i Figur 2. Equivaltly, th yild surfac ormals i both strss spacs coicid. For biaxial loadig, th lastic strss is trappd i a pla, cosqutly so is th corrctd strss. Thrfor, it is clar that th xistt rsults for th pla strss tsor [05, 08, 09] caot b accurat i all compots. Scod, th algbraic costrait (1.1) is motivatd by Sgr s obsrvatio that i sharply otchd rgios, th rlatio ε11( t) ε11( t) (1.2) ε ( t) ε ( t) 22 22

4 for th lastic FE ad th lastoplastic FE strai holds approximatly, cf. [06]. So it is vry atural that th corrctd total strai should satisfy th sam rlatio. Obsrvatio (1.2) is purly mpirical, a rigorous mathmatical justificatio is still a op qustio. (For th cas of combid axial tsio ad circumfrtial torsio, th lastic ratio at th lft-had sid i (1.2) is a costat, idpdt of t.) Rul (1.1) givs additioal iformatio ito th corrctio modl, ad it forcs th corrctd strss tsor σɶ to brak out of its pla as it is show i Figur 4. Figur 4: Elastic FE strss, Elastoplastic FE strss, origial modl ad improvd modl PARAMETER IDENTIFICATION Th d to dtrmi a hug amout of psudo paramtrs is a drawback both of th origial ad of th improvd corrctio modl. This is a big problm spcially for costitutiv modls, whr th umbr of matrial paramtrs is alrady larg. For th modl of Jiag ad Shitoglu [07] for xampl, th umbr of matrial paramtrs that hav to b idtifid lis somwhr btw 40 ad 80. For th corrctio modls cosidrd, thr is a d for additioal psudo paramtrs. For th oliar larg-scal paramtr idtificatio, w us th wll-stablishd trust-rgio algorithm of Colma ad Li [02]. Th

5 lattr is implmtd.g. i th Optimizatio-Toolbox of MATLAB. It ca b usd for curv fittig i th last-squars ss, i.. 2 ɶ σ ( t) σ ( t) mi ad 2 ɶ ε ( t) ε ( t ) mi. Costraits, as for xampl th positivity of som paramtrs, ca b tak ito accout as wll. It is a gradit basd algorithm, thrfor o has to valuat th partial drivativs of th corrctd strsss rsp. strais with rspct to th paramtrs, i.. ɶ σ ( t ) / p ad ɶ ε ( t ) / p. I ordr to do so, it is vry ffctiv ad covit to us th Automatic i Diffrtiatio (AD) tchiqu [04], which computs th partial drivativs aalytically xact i paralll to th itgratio of th diffrtial-algbraic quatios of th corrctio modl. i RESULTS AND CONCLUSION A first xampl, which illustrats th improvmt of th corrctio modl ad whr w targt at lastoplastic FE computatios, has alrady b giv i th forgoig sctio, cf. Figur 4. Figur 5: Masurmts, origial corrctio modl by Köttg [08, 09] with paramtrs from Hrtl [05], improvd modl with paramtrs via AD, cf. [10]. Six diffrt loadig paths ar displayd.

6 I a scod xampl, w compar th improvd modl with strai masurmts at th otch root of a axl, mad of S460N stl. Hr w cotiu th work of Hrtl [05]. As th matrial xhibits th ratchttig ffct, all computatios hav b prformd with Jiag s [07] costitutiv modl, whr 82 paramtrs hav b idtifid by AD ad th Colma-Li algorithm. Th rsults hav b obtaid aftr 20 Colma-Li itratios. Compard to simpl ssitivity computatio via Fiit Diffrcs ( Extrior Diffrtiatio ), a spd-up of fiv has b obtaid with th taplss AD-forward mod. Th sstial thig from th physical poit of viw is, that our improvd vrsio of Köttg s modl togthr with Jiag s modl ad appropriat paramtrs is i fact abl to rproduc th complx trasit ad highly oliar lastoplasticity ffcts accuratly, basd o a fw liar lastic FE uit load cass, avoidig full trasit oliar lastoplastic FE computatios. (As a poitwis postprocssig it is clarly much fastr tha lastoplastic FE.) Th rsults for th circular loadig path (lft colum, middl li i Figur 5) ar ot compltly satisfactory; thy could b improvd by th us of Dörig s costitutiv law, which taks out-of-phas phoma bttr ito accout. REFERENCES [01] A. Bucyski, G. Glika: A aalysis of lasto-plastic strais ad strsss i otchd bodis subjctd to cyclic o-proportioal loadig paths. Biaxial/Multiaxial Fatigu ad Fractur (2000) pp [02] T. F. Colma, Y. Li: A itrior trust rgio approach for oliar miimizatio subjct to bouds. SIAM Joural o Optimizatio 6 (1996) pp [03] G. Glika, A. Bucyski, A. Ruggri: Elastic-plastic strss-strai aalysis of otchs udr o-proportioal loadig paths. Archivs of Mchaics 52 (2000) pp [04] A. Griwak: Evaluatig Drivativs. Pricipls ad Tchiqus of Automatic Diffrtiatio. SIAM Frotirs i Applid Mathmatics 19 (2000). [05] O. Hrtl: Witrtwicklug ud Vrifizirug is Nährugsvrfahrs zur Brchug vo Krbbaspruchug bi mhrachsigr ichtproportioalr Schwigblastug. Mastr s thsis, Bauhaus-Uivrsität Wimar (2003). [06] M. Hoffma, T. Sgr: Mhrachsig Krbbaspruchug bi proportioalr Blastug. Kostruktio 38 H2 (1986) pp [07] Y. Jiag, H. Shitoglu: Modlig of cyclic ratchttig plasticty. Parts I/II. Trasactios of th ASME 63 (1996) pp [08] V. B. Köttg, M. E. Barky, D. F. Soci: Psudo strss ad psudo strai basd approachs to multiaxial otch aalysis. Fatigu ad Fractur of Egirig Matrials ad Structurs 18/9 (1995) pp [09] V. B. Köttg, M. E. Barky, D. F. Soci: Structural strss-strai aalysis for oproportioal loadig suitabl for FEM-postprocssig. Itratioal Procdigs i Fatigu Dsig, Hlsiki (1995). [10] H. Lag: Th diffrc of th solutios of th lastic ad lastoplastic boudary valu problms ad a approach to multiaxial strss-strai corrctio. Dissrtatio, TU Kaisrslautr (2007). [11] J. Lmaitr, J.-L. Chaboch: Mchaics of solid matrials. Cambridg Uivrsity Prss (1998). [12] A. Moftakhar, A. Bucyski, G. Glika: Calculatio of lasto-plastic strais ad strsss i otchs udr multiaxial loadig. Itratioal Joural of Fractur 70 (1995) pp Cotact addrss: holgr.lag@itwm.frauhofr.d