FE Realization of Thermo-Visco-Plastic Constitutive Models using VUMAT in ABAQUS/Explicit Program

Transcription

1 COMPUTATIONAL MECHANIC ICM2007, Jul 0-August 1, 2007, Bing, China 2007 Tsinghua Univrsit Prss & ringr FE Ralization of Thrmo-Visco-Plastic Constitutiv Modls using VUMAT in ABAQU/Exlicit Program C. Y. Gao* Dartmnt of Mchanics, Zhjiang Univrsit, Hangzhou, China Abstract ABAQU softwar has rovidd man kinds of lmnt ts and matrial modls in its libraris for usrs. But with th raid dvlomnt of industrial tchnolog th matrial modls in th currnt matrial librar of ABAQU cannot wll dscrib som ractical roblms such as ultra-high-sd cutting. Howvr, ABAQU rovids th abilit of allowing usrs to introduc thir slf-dfind matrials into th main rogram b usr subroutin (UMAT or VUMAT) so as to gt mor accurat simulation rsults for thir scific roblms. In this ar, th mthod of how to raliz a usr-dfind gnralizd matrial modl in ABAQU/Exlicit rogram b th xlicit usr matrials subroutin VUMAT is rsntd in dtails. Comard with UMAT in ABAQU/tandard, VUMAT adots xlicit intgration algorithm (.g. Forward/Backward Eulr mthods) for strss udating. It dosn t nd itrations and Jacobian and so maks th comutational fficinc nhancd gratl. K words: FE simulation, ABAQU, VUMAT, constitutiv modl INTRODUCTION Although thr ar man lmnt ts and matrial modls in th libraris of ABAQU softwar. But som ractical roblms such as ultra-high-sd cutting cannot b wll dscribd b using th matrial modls includd in th currnt matrial librar of ABAQU. Fortunatl, ABAQU has rovidd th rogram-rdvlomnt abilit for usrs b allowing usrs to introduc thir slf-dfind matrials into th main rogram b usr subroutin (UMAT or VUMAT) so as to gt mor accurat simulation rsults for thir scific roblms. In this ar, th mthod of how to raliz a usr-dfind gnralizd matrial modl in ABAQU/Exlicit rogram b th xlicit usr matrials subroutin VUMAT is rsntd in dtails. Th xamls of usr subroutin for isotroic and kinmatic hardning modls hav bn givn in ABAQU manual. But for mor comlicatd constitutiv modls which tak th rat hardning ffct and th thrmal softning ffct into account togthr with th strain hardning ffct, th rquird usr subroutins ar still not full invstigatd and rmains difficult for most usrs. Among thos comlicatd constitutiv modls, th most oularl-usd on is Johnson-Cook modl. Johnson and Cook (1985) roosd a uniaxial constitutiv rlationshi dscribing th von Miss ild strss as follows: n = ( A + Bε )(1 + C ln & ε )(1 T ) (1) * m * whr A, B, C, n and m ar matrial constants; T = ( T Troom) /( Tmlt Troom ), and T is th absolut tmratur, ε is th quivalnt lastic strain, ε& is th lastic strain rat. Th matrial constants ar dtrmind from tnsion or torsion straining tsts. Th modl has succssfull dscribd Talor Clindr Imact Tst for a varit of matrials b codd in som FE rograms (for xaml, th Lagrangian matrial 62

2 dnamic cod EPIC-2 and ABAQU). But for th cas of vr high strain rats (such as ultra-high-sd cutting), it cannot rovid satisfactor flow strss rsults whn comard with xrimntal data. Th intrinsic rason ma b that this t of constitutiv rlationshi is stablishd url on th mirical summar to a lot of xrimntal rsults and thus lack of microscoic hsical basis. Thr is also anothr t of constitutiv modls stablishd on th hsical mchanism of lastic dformation of crstal matrials. Zrilli and Armstrong (1987) dvlod a constitutiv modl that accounts for strain, strain rat and tmratur dndnc in a could mannr b dislocation mchanics. It is drivd basd on th conct of thrmal activation analsis for ovrcoming local obstacls of dislocation motions. And th roosd two diffrnt forms for th two diffrnt classs of mtals: bod-cntrd-cubic (b.c.c.) and fac-cntrd-cubic (f.c.c.) bcaus of thir diffrnt crstal structur and dislocation charactristics. For th f.c.c. mtals, th constitutiv rlation is ( c c ln & ε ) ] = c1 + c2 ε x[ 4 T (2) Th matrial constants ar dirctl rlatd with scific hsical charactristic aramtrs, that is to sa, th can b hsicall intrrtd. In ractic, th ar usuall dtrmind b numrical fitting of xrimntal data of macroscoic lastic dformation of matrials for convninc. This modl has bn usd widl in matrial dnamic comutational cod. Rcntl w hav mad som imortant imrovmnts on this modl in virtu of th high-strain-rat xrimntal data of OFHC Cu. To vrif our nw rlation, w nd to raliz it in a FE rogram. o, w will tr to raliz a nw thrmo-visco-lastic constitutiv modl in ABAQU/Exlicit rogram b using VUMAT subroutin in this ar. Th gnral form of this t of matrial modls can b writtn as = f ( ε, & ε, T ) () In fact, on can introduc an constitutiv rlation of this t into th ABAQU/Exlicit rogram in light of th numrical tchniqus rsntd blow. ALGORITHM OF VUMAT 1. Elasto-lastic formulation of VUMAT If w adot isotroic hardning law for strain hardning, th basic govrning quations of th gnral constitutiv modl ar almost th sam as thos of isotroic hardning lasticit modl xct that thir ild strsss hav diffrnt xrssions. For th lastic art, th gnralizd Hook s law is obd and th strss should b xrssd in Grn-Naghdi rat form in a corotational framwork, i.. & = λδ & ε + 2μ & ε (4) Th intgration form is Δ = λδ Δε + 2 μδε (5) For th lastic art, Miss ild critrion and lastic flow law ar still usabl [1]. Th ild function is 2 ( ε, & ε, T ) = 0 (6) whr = δ /. Th lastic flow law is & ε 2 & = ε (7) whr & ε 2 = & ε & ε, and ε ( = t ε& dt ) is dfind as quivalnt lastic strain Diffrnc btwn VUMAT and UMAT Comard with th imlicit algorithm [2] usd b UMAT in 624

3 ABAQU/tandard, VUMAT in ABAQU/Exlicit adots th xlicit tim intgration algorithm (lik Forward/Backward Eulr mthods) []. Th nw strss udating mthod dosn t nd larg numbrs of itrations as wll as th Jacobian of lastic-lastic tnsor, so th comutational fficinc is nhancd gratl in ABAQU/Exlicit rogram. But it rquirs to dfin a stabl limit dtrmind b minimum charactristic lmnt lngth and lastic wav sd of th matrial (i.. Δ t max = L / Cd ). Bsids, th tim incrmnt cannot b rdfind in th rogram. Th VUMAT subroutin has a vctorizd form bcaus its intrfac asss th stat data for blocks of matrial oints to th subroutin on ach call. It uss a two-stat architctur---- th initial valus ar in th old arras, th nw valus must b ut in th nw arras. olid lmnts for lasticit ar formulatd in trms of th Jaumann rat of strss in UMAT but in trms of th Grn-Naghdi rat of strss in a corotational framwork whn usd with a VUMAT.. Intgration rocdur of VUMAT Th intgration rocdur is basd on th xlicit tim intgration algorithm. First, th lastic rdicting strss is gnratd using von Miss strss of url lastic bhavior 2 r r r = (8a) r = + 2 μδε + λδ Δε (8b) old TART of ubroutin VUMAT ubroutin Intrfac (stat data transfr) NO If inlastic dformation? Calculat nw lastic strss YE Call ubroutin VUild ( ) Calculat th lastic rdicting strss If ild? (Miss ild Critrion) YE Calculat th ild strss with usr constitutiv rlation: = f ( ε, & ε, T ) Calculat th incrmnt of quivalnt lastic strain NO Udat th strss Udat th strss (nw strss = rdicting strss) Udat stat variabls, scific intrnal nrg, inlastic nrg END of ubroutin VUMAT (Rturn to main rogram) Figur 1: Th flowchart of VUMAT subroutin 625

4 r Th rdicting strss is obtaind basd on a strss-comnsation mthod [4]. If th lastic rdictor is largr than th currnt ild strss, lastic flow occurs. Thn w calculat th quivalnt lastic strain incrmnt. To avoid local itration, it is aroximatl calculatd xlicitl as th following quation [5] r Δε = ( ) /(μ + h) (9) hr h = d / dε stands for th hardning at th bginning of th incrmnt. This aroximation will b rasonabl givn that th siz of th tim incrmnt usd in an ABAQU/Exlicit analsis is gnrall vr small. Finall, th intgration rocdur of th VUMAT subroutin is illustratd b th blow flowchart: INTERPRETATION OF VUMAT CODING Th Fortran cod of VUMAT subroutin can b rogrammd in trms of th intgration rocdurs abov. om k tis for th coding of VUMAT ar discussd blow. 1) inc w adot isotroic hardning in our gnral constitutiv modl, th main rogram of ABAQU Inut fil is almost th sam as that of isotroic hardning lasticit modl if for th idntical xaml. Thr ar onl svral sntncs ndd to b modifid. Th rlatd sntncs in th main rogram of th isotroic hardning lasticit modl ar listd blow on th lft; and th corrsonding modifid sntncs in th main rogram of th usr-dfind modl ar shown blow on th right: *OLID ECTION, ELET=_lst1, MATERIAL=Cu *MATERIAL, NAME=Cu *DENITY ρ, *ELATIC E, ν *PLATIC, HARDENING=KINEMATIC //tabl of data dscribing th lastic hardning curv 626 *OLID ECTION, ELET=_lst1, MATERIAL=vumat *MATERIAL, NAME= vumat *UER MATERIAL, CONTANT = n E, ν, *DEPVAR m, *DENITY ρ, Hr n mans th numbr of usr-dfind matrial rortis which corrsonds to th variabl nros dfind in th intrfac of VUMAT, and m mans th numbr of usr-dfind stat variabls associatd with th matrial t which corrsonds to th variabl nstatv dfind in th intrfac of VUMAT. Th ar two imortant aramtrs in th rogram. 2) Usr subroutin VUMAT is calld for blocks of matrial oints at ach incrmnt. Whn th subroutin is calld, it is automaticall rovidd with th data of stat (including strss and solution-dndnt stat variabls) b th main rogram at th start of th incrmnt. Th first art of VUMAT is calld intrfac which is in standard format and cannot b modifid. Its function is to dfin all variabls usd and transfr th variabl data btwn th main rogram and th VUMAT subroutin. Th data ar assd in and out b a grou of arras. Th main dimnsion of ths arras is qual to nblock, which is th numbr of matrial oints to b rocssd in this call to VUMAT. Each ntr in an arra corrsonds to a singl matrial oint. All matrial oints in th sam block hav th sam matrial nam and blong to th sam lmnt t. Comlt dscritions of all ths variabl aramtrs ar rovidd in Chatr 25 of ABAQU Analsis Usr's Manual [6]. For th gnral constitutiv modl, thr ar thr solution-dndnt stat variabls (lastic strain, strain rat and tmratur), which nd to b dfind b usrs and udatd in ach incrmnt. W can sciall dclar thm as thr slf-dfind stat variabls lik: TATE(*,1) = quivalnt lastic strain TATE(*,2) = lastic strain rat

5 TATE(*,) = tmratur ) In th VUMAT subroutin, it will call anothr subroutin VUild to gt th valu of dnamic ild strss. Th subroutin VUild is to rcovr th hardning curv of ild strss vrsus lastic strain, which can b givn b a data tabl or an analtic dscrition. If th hardning curv is givn b a xlicit analtic xrssion, that is to sa, w can analticall dtrmin th gnral constitutiv rlationshi, th valu of ild strss can thn b gottn dirctl from th analtic rlation. Othrwis, if it is difficult to dtrmin th analtic xrssion of th gnral constitutiv rlationshi, but th strss-strain curvs at givn strain rats and tmraturs can b obtaind b dnamic matrial tst, w can inut th data airs of flow strss vrsus lastic strain into VUMAT b th command sntnc *UER MATERIAL, CONTANT = nros, and thn ass thm to th subroutin VUild b th arra ros(nros). Th command sntnc of calling subroutin VUild should b Call VUild (ildold, hard, stat(*,1), stat(*,2), stat(*,), ros(nros), nvalu) whr hard is hardning variabl (i.. h as shown bfor) including strain hardning, strain rat hardning and thrmal softning all togthr. nvalu (=nros/2-1) is th numbr of data airs (a air of data corrsonds to a saml oint on th hardning curv). 4) In th intrfac of th usr subroutin, th tmratur is assd into subroutin as rad-onl variabl and cannot b modifid (vn in a full could thrmal-strss analsis). If thr is no aaranc of adiabatic shar hnomnon, th dformation rocss can b tratd as isothrmal on, in which th tmratur is scifid in th *INITIAL CONDITION otion and ks constant throughout th whol analsis. If thr aars adiabatic matrial bhavior (convrsion of lastic work to dissiation hat), th tmraturs must b stord as a usr-dfind stat variabl and udatd in ach incrmnt. Th thortical valuation of tmratur incrmnt within a tim intrval is [7]: ΔT η = c ρ t + Δ t t dε hr c is th scific hat cofficint, and η is th convrting fficinc from lastic work to hat nrg. Th intgral on th right of th quation is just th incrmnt of lastic work, which is rlatd with th udating of inlastic scific nrg in th subroutin. Th thrmal rortis can b dfind using th following otions: *Conductivit, *cific Hat, *xansion, *Inlastic hat fraction. Not that thir units must us a uniform unit sstm rdfind b usrs for th rogram. (10) UMMARY Th following commnts should b hlful in th alication of th VUMAT subroutin: 1). Th usr subroutin is suitabl for larg-strain calculations bcaus th ncssar rotations of strss and strain has bn takn car of b ABAQU rogram. 2). Th usr subroutin can b usd with all lmnts that hav dislacmnt dgrs of frdom. For lan strss lmnts th strss comonnts to b dfind ar ; for lan strain and axismmtric lmnts th ar ; for thr-dimnsional lmnts th ar , 1. ). Bfor th usag of th usr subroutin, th singl lmnt tsting undr high-strain-rat loading (.g. high sd imact) must b carrid firstl to validat th coding [8]. Bcaus Johnson-Cook modl has alrad bn includd in th matrial librar of ABAQU, it is slctd to b ralizd b th cod of usr subroutin so that th numrical rsults can b vrifid b thos of ABAQU itslf. It has bn indicatd b xamls that th rogram with VUMAT subroutin for J-C modl ilds vr similar numrical rsults to thos of th rogram with otion *PLATIC, hardning=johnson Cook. 627

6 Acknowldgmnts Th tchnical suort of hanghai Agnc, ABAQU CHINA Inc. is gratfull acknowldgd. REFERENCE 1. Hibbit D, Karlsson B, ornson P. ABAQU Thor Manual (vrsion 6.5). Hibbit, Karlsson & ornson Inc., UA, Lu JF, Zhuang Z, Zhang F. Examl of ABAQU/tandard UMAT: Johnson-Cook constitutiv modl. Procding of ABAQU Usr Confrnc, Tsinghua Univ., , Bing, China.. Li HW, Yang H, un ZC. Exlicit incrmntal udat algorithm for modling crstal lastoviscolastic rsons in finit lmnt simulation, Trans. Nonfrrous Mtals ocit China, 2006; 16: Li HW, Yang H, Guo L, Li L, Guo L. Elastic-lastic constitutiv rlation with hbrid hardning and alication in cold ring rolling simulation. J. Mch. Eng., 2005; 41 (7): (in Chins). 5. Writing UMATs, VUMATs and UELs. Intrnal Tchniqu Rort of ABAQU, Hibbit, Karlsson & ornson Inc., UA, Hibbit D, Karlsson B, ornson P. ABAQU Analsis Usr s Manual (vrsion 6.5). Hibbit, Karlsson & ornson Inc., UA, Voiadjis GZ, Abd FH, Microstructural basd modls for bcc and fcc mtals with tmratur and strain rat dndnc. Mchanics of Matrials, 2005; 7: Zhuang Z, Zhang F, Cn. Nonlinar FE Analsis and Alication Examls in ABAQU. cinc Publishr, Bing, China,