Phase-Field Modeling for Dynamic Recrystallization

Transcription

1 0 (0000) 0 0 Plas lav ths spac mpty Phas-Fld Modlg for Dyamc Rcrystallzato T. Takak *, A. Yamaaka, Y. Tomta 3 Faculty of Martm Sccs, Kob Uvrsty, 5--, Fukamam, Hgashada, Kob, , Japa (Emal : takak@martm.kob-u.ac.p) Graduat School of Scc ad Tchology, Kob Uvrsty, -, Rokkoda, Nada, Kob, , Japa (Emal : yamaaka@sold.mch.kob-u.ac.p) 3 Faculty of Egrg, Kob Uvrsty, -, Rokkoda, Nada, Kob, , Japa (Emal : tomta@mch.kob-u.ac.p) Abstract Dyamc rcrystallzato (DRX) xhbts vry complcatd phoma cludg both hardg du to th accumulato of dslocatos ad softg du to th uclato ad growth of rcrystallzd gras. I othr words, th mchacal bhavour durg th DRX procss s closly rlatd to th voluto of mcrostructurs ad dslocatos. I ths study, th phas-fld modl, whch abls th smulato of mcrostructural varato durg th DRX procss, s dvlopd by gralzg th mult-phas-fld mthod proposd by Stbach t al. Th hardg du to th accumulato of dslocatos ad th uclato crtra of rcrystallzato ar xprssd thortcally by mployg th mthod proposd by Guo t al. I ordr to cofrm th basc prformac of th dvlopd modl, a smulato of sgl-gra growth s prformd. As a rsult, th charactrstc varato of th gra boudary mgrato rat du to th dslocato-dsty chag ad a good agrmt wth th thortcal rsult wr obsrvd. Furthrmor, th DRX procss cludg uclato s smulatd for a rgular-hxagoal gra structur. It s cofrmd that a typcal strss stra curv wth multpl paks ca b rproducd by th prdcto of mcrostructural voluto. Kywords Phas-Fld Mthod, Modlg, Smulato, Dyamc Rcrystallzato, Mcrostructur Evoluto, Dslocato. * Corrspodg Author AES-Advacd Egrg Solutos (Ottawa, Caada) All rghts ar rsrvd

2 T. Takak, A. Yamaaka, Y. Tomta Itroducto It s wll kow that wh a low-to-mdum stackg fault rgy (SFE) mtal s dformd udr a hgh-tmpratur vromt, dyamc rcrystallzato (DRX) occurs [, ]. DRX s dstgushd from statc rcrystallzato (SRX), whch occurs post-dformato aalg. Although both rcrystallzato procsss hav may commo charactrstcs, th most dffrt pot s that for DRX, th dslocato dsty th rcrystallzd gras crass wth cotuous dformato, whl for SRX, t s costat. Sc DRX xhbts complcatd phoma such as hardg du to dformato ad softg du to rcrystallzd gra growth, or th smultaous voluto of dslocatos ad mcrostructurs ovr tm, th costructo of a umrcal modl that abls th valuato of mcrostructural voluto s sstal, addto to xprmtal obsrvato. As a umrcal modl for DRX, th cllular automato (CA) mthod [3-9] has b succssfully appld to th mcrostructural vstgato of th DRX procss. Guo t al. [7-9] hav proposd a smulato mthod that coupls th CA mthod to a thortcal modl, whch th hardg by th accumulato of dslocatos s xprssd by quatos ad th softg by gra boudary mgrato du to rcrystallzato s smulatd by th CA mthod. Howvr, sc th CA mthod dscrbs th stats trms of dscrt varabls, t s problmatc wh appld to modlg curvatur-drv growth [0]. Th phas-fld mthod was frst usd to modl ddrt formato [, ], ad ovr th past dcad, t has b appld to varous problms of matral scc. Th phas-fld mthod ca asly rproduc complcatd shaps ad morphologs wthout trackg th locato of th trfac, sc th trfac mgrato s dscrbd by th tm voluto of a addtoal ordr paramtr, or th phas fld. I ths modl, th tm ad scal ca b tratd as th ral valus, ad th ffct of curvatur s xplctly cludd. I ths study, w stablsh a phas-fld modl for th DRX procss, whch th mult-phas-fld mthod proposd by Stbach t al. [4] s gralzd to smulat th gra boudary mgrato drv by stord rgy, or th softg procss. Th hardg procss s modld by th quatos mployd by Guo t al. [6, 7]. Fgur. Schmatc strss-stra curvs udr hot workg. Dyamc Rcrystallzato Fgur schmatcally llustrats typcal strss-stra curvs udr hot workg. I mtals wth a hgh-sfe, such as Al ad -F, th strss crass mootocally wth crasg stra, bcaus th dyamc rcovry s rapd ad th dslocato dsty dos ot achv a crtcal valu that orgats th uclato of rcrystallzato. O th othr had, th low- or mdum-sfe mtals, such as Cu, N ad γ-f, xhbt DRX. Th strss-stra curv for th low- or mdum-sfe mtals s charactrzd by th Zr-Holloma paramtr Z, dfd by th followg quato: Z = ε& xp( Q RT ), () whr ε& s th stra rat, T s th dformato tmpratur, Q s th actvato rgy ad R s th gas costat. As show Fg. (b), udr th codtos of low Z,.., low ε& or hgh T, th strss-stra curv has

3 T. Takak, A. Yamaaka, Y. Tomta 3 multpl paks at low stra, ad udr hgh Z,.., hghε& or low T, t xhbts a sgl pak. For both cass, wh th strss attas a crtcal ~ σ s = μb k k. (5) valu σ c whch s somwhat lss tha th maxmum strss σ max, th uclato of rcrystallzato s gratd. Sc th dslocato dsty of th rcrystallzd gra s cosdrably smallr tha that of th dformd matral, softg occurs wth th growth of th w gra. Fally, th stady-stat strss σ s s achvd. From th gradt of th strss-stra curv at σ = 0, k ca b dtrmd. Th stady-stat strss σ s s rlatd to th stra rat ε& ad tmpratur T as = m Q act ε& Aσ s xp, (6) RT whr A ad m ar costats ad Q act s th actvato rgy. Wh σ s s dtrmd from Eq. 6, k ca b obtad from Eq. 5. k = μb k σ s (7) ~ 3 Thortcal Modl W mploy th modl of dslocato voluto ad uclato usd Rfs. [6] ad [7]. 3. Dslocato voluto modl Th varato of th dslocato dsty ρ wth rspct to th stra ε s gv by dρ = k ρ k ρ. () dε Hr, th frst trm of th rght had sd xprsss th work hardg, ad k s a costat that rprsts hardg. Th scod trm s th dyamc rcovry trm, ad k s a fucto of tmpratur T ad stra rat ε&. Th flow strss σ s rlatd to th dslocato dsty as follows: σ = μb ~ ρ, (3) whr s a dslocato tracto coffct of aroud 0.5, μ s th shar modulus, ad b ~ s th magtud of th Burgrs vctor. I our umrcal modl, th local dslocato voluto s xprssd by Eq., ad th macroscopc tru strss s calculatd from Eq. 3 by chagg ρ to a avrag dslocato dsty ρ av ovr a umrcal rgo. Th followg formulatos ar prformd ordr to dtfy k ad k Eq.. From Eqs. ad 3, w ca obta th followg quato, dσ ~ σ = μb k, (4) dε σ s whr σ s s th stady-stat strss ad s xprssd as 3. Nuclato modl Sc th uclato orgats at hgh-agl boudars, t s assumd that th uclato of DRX oly occurs at th orgal ad rcrystallzd gra boudars. I ths cas, th uclato mchasm rducs to bulgg uclato. Cosdrg that th uclato orgats wh th dslocato dsty rachs a crtcal valu ρ c, ρ c ca b calculatd o th bass of th bulgg mchasm as follows [3]: 0γε& ρc = ~ 3b LMτ 3, (8) whr γ s th gra boudary rgy, L s th ma fr path of th dslocato, M s th gra boudary moblty ad τ s th dslocato l ~ rgy calculatd by τ = cμb, whch c s a costat of th ordr of 0.5. Th ma fr path L s calculatd from costat of about 0. L = K ρ, whr K s a 4 Phas-Fld Modl 4. Phas-fld quatos for DRX W gralz th mult-phas-fld modl proposd by Stbach t al. [4] to smulat th DRX phomo, whch th rcrystallzd gra boudary mgrato s drv by th stord rgy. W cosdr a systm cotag N dffrt

4 T. Takak, A. Yamaaka, Y. Tomta 4 gras,,... N. Phas fld taks a valu of sd th th gra, ad s 0 sd othr gras, ad 0 < < at th gra boudary. s ot a dpdt varabl ad must satsfy followg codto: N =. (9) = Hr, w us th fr-rgy fuctoal F = V f dv, N N a β = + W + f dv V β ββ = β = + (0) whr a β ad W β ar th gradt coffct ad th barrr hght, rspctvly, whch ar rlatd to th trfac rgy γ ad th trfac thckss δ. f s th bulk fr-rgy dsty ad a fucto of,,... N,.., f f ( L ) =,, N. Now, w df a stp fucto σ a as L0 < < σ =. () 0Llswhr Usg ths stp fucto, th umbr of locally prst phass s xprssd as = N = ( x, t) σ. () Usg Eq., Eq. 9 rducs to = =. (3) By troducg th Lagrag multplr λ,,,... N ca b tratd as dpdt varabls. Γ = F + λ dv (4) V = Sc s a o-cosrvd ordr paramtr, th tm voluto quato of ca b xprssd as follows: o = λ, (5) δ o whr dots th tm voluto wthout cosdrg a spcfc tm scal. Furthrmor, a trfac fld ψ s wly dfd as follows: ψ = ( < ), (6) whr ψ = ψ. Substtutg Eq. 6 to Eq. 3, w obta = ψ +. (7) = By th dfto of th trfac fld ψ, or usg Eq. 6, th tm voluto of ψ bcoms o o o ψ = = +. (8) δ δ W ca s from Eq. 8 that th tm voluto of ψ s dpdt of th Lagrag multplr λ. From Eq. 7, o o ψ = =. (9) Substtutg Eq. 8 to Eq. 9, w hav o =. (0) = δ δ Th fuctoal drvatv /δ s calculatd as f f = δ. () ak f = Wkk + k + k= ( k ) Takg to accout Eq. ad th moblty M, th tm voluto quato of s obtad as & = = = M δ δ M = k = ( Wk W k ) k + ( ak a k ) f f k + () Hr, w slct f f 8 = π ΔE, (3) whr ΔE s th dffrc bulk fr-rgy dsty btw gras ad. Th factor 8/π s dd to satsfy 8 ( ) = w obta π d. Fally, 0

5 T. Takak, A. Yamaaka, Y. Tomta 5 ( Wk W k ) k + ( ak a k ) k & M k= =. = 8 ΔE π (4) 4. Phas-fld paramtrs Th paramtrs a, W ad M Eq. 4 ar rlatd to th matral costats. Lt us cosdr th cas of N = ad =. For =, =, a = a = a, W = W = W, M = M ad ΔE = ΔE, th tm voluto quato of = s wrtt as & M 8 = a W ( ) + ( ) ΔE. π (5) I sam way, from Eq. 0, w obta a F = ( ) + W( ) + f dv. (6) I a o-dmsoal problm, Eqs. 5 ad 6 rspctvly rduc to & M 8 = a W ( ) + ( ) ΔE x π (7) a F = + W( ) + f dv. (8) x From Eq. 7, th phas-fld profl udr th qulbrum codto ca b obtad by sttg & = 0 ad ΔE = 0 as follows: W = s x, (9) a whr = / at x = 0. Solvg Eq. 9 for x, a x = s ( + ). (30) W Sc th trfac rgo s 0 < <, th trfac thckss δ s calculatd as aπ δ =. (3) W Th gra boudary rgy γ s obtad from Eq. 8 by sttg f = 0 as a W γ = π. (3) 4 Cosdrg th codto that a gra boudary wth th qulbrum phas-fld profl mgrats wth a costat vlocty V, w obta followg quato from Eq. 7: 4a V = M ΔE. (33) π W Comparg Eq. 33 ad th gra boudary moblty M, 4a M = M. (34) π W From Eqs. 3, 3 ad 34, th phas-fld paramtrs ar rlatd to th matral paramtrs as follows: 4γ W =, a = δγ δ π π, M = M. (35) 4δ Gralzg Eq. 35 to th mult gra problm, γ W 4 =, a = δγ δ π π, M = M. 4δ (36) whr a = a, W = W, M = M, γ = γ ad M = M, ad w st δ = δ. Th drvg forc of th mgrato of th rcrystallzd gra boudary s xprssd by th dslocato dsty dffrc btw gras ad, or Δ E = τ ( ρ ρ ), (37) whr ΔE = ΔE. Th dslocato dsty crass wth cotuous dformato by Eq.. Wh uclato occurs, th dslocato dsty th w gra s st to a tal valu ρ 0. W assum that th dslocato dsty a gra s uform,.., ρ s uformly dstrbutd th rgo of > 0 ad ρ = ρ 0 wh = 0. 5 Numrcal Rsults O th bass of Rfs. [5] ad [7], w mploy th followg matral paramtrs for OFHC coppr: = 0.5, μ = 4. GPa, b ~ = 0.56 m, A = , m = 7.58, Q act = 6 KJ/mol, R = 8.34 J/mol K, k = /m, γ = 0.65

6 T. Takak, A. Yamaaka, Y. Tomta 6 J/m, c = 0.5 ad K = 0. Th gra boudary moblty M s xprssd by ~ b δ = bd Q b M xp kt RT, (38) whr δ b s th charactrstc gra boudary thckss, D s th boudary slf-dffuso coffct, k s th Boltzma costat ad Q b s th boudary dffuso actvato rgy. W us δ b D = m 3 /s, k = J/K ad Q b = 04 KJ/mol [7]. Equato 4 s solvd by th ft dffrc mthod, ad th grd sz Δx = 5 μm ad th trfac thckss δ = 7Δx ar mployd. Th smulatos ar carrd out udr T = 775 K, ε& = 0-3 /s, ad ρ 0 = 0 9 /m. dformd matral ad rcrystallzd gra, ad th drvg prssur wth tm. It ca b obsrvd that th dslocato dsty th dformd matral crass slghtly, whras that th rcrystallzd gra crass rapdly from ρ 0. Cosqutly, th drvg prssur calculatd by th dffrc th dslocato dsts also rachs a pak. At th balac btw th drvg prssur ad th curvatur ffct, th pak valu of th mgrato vlocty s dtrmd. Th sold crcls Fg. dcat th thortcal valus calculatd from V = M(ΔE stor γ/r). It s, thrfor, cofrmd that th umrcal rsult prfctly agrs wth th thortcal valu. 5. Sgl-gra growth Durg th DRX procss, th drvg forc of gra boudary mgrato chags wth cotuous dformato. Hr, w prform fudamtal smulatos whch o rcrystallzd gra grows a dformd matral wth a crystal ortato, ad w compar umrcal ad thortcal rsults to clarfy th valdty of th proposd phas-fld modl for DRX. A uclus s placd at th org of th squar umrcal modl. Th radus of th uclus s st to R 0 = 5.5 μm, bcaus t must satsfy R 0 > γ/e c to avod th shrkg of th gra du to th curvatur ffct. Th crtcal dslocato dsty ρ c calculatd by Eq. 8 s /m, th, th stord rgy E c = 0. MPa. Fgur shows th varato of gra boudary mgrato vlocty wth tm. Th rsult dotd by a brok l s for a costat stord rgy E c. Ths corrspods to th SRX procss. I ths cas, th vlocty s slow at th bgg of th growth du to th curvatur ffct th gradually bcom fastr. O th othr had, th rsult for DRX dcats a pak vlocty at approxmatly 8 s. Fgur 3 shows th varato of dslocato dsts th Fgur. Varatos of gra boudary mgrato vlocty wth tm. Fgur 3. Varatos of dslocato dsts ad drvg prssur wth tm. 5. Mult gra uclato ad growth Th DRX smulato for a polycrystal mtal wth rgular hxagoal gras ad a ma gra sz s 78 μm s prformd hr. Th sz of th computatoal doma s μm

7 T. Takak, A. Yamaaka, Y. Tomta 7 ( grd) ad th umbr of orgal gras s sx. Prodc boudary codtos ar mployd at all boudars. Frst, a prlmary calculato s carrd out to crat th tal polycrystal structur. Fgur 4 shows tm slcs durg th prlmary calculato. Th sx ucl ar placd th calculatd postos ad grow sd a bas matral drv by a uform drvg forc. Sc th gra boudary rgs ar all dtcal, sx rgular-hxagoal gras ca b obtad. Th DRX smulato cosdrg uclato s prformd usg th cratd rgular-hxagoal gra structur. Th ucl ar gratd o th grd satsfyg th codtos ρ > ρ c ad = < 0.6. Th uclato rat s assumd to hardg du to th accumulato of dslocatos. As a rsult, t s cocludd that th dvlopd phas-fld modl ca smulat th DRX procss cludg th softg ad hardg. Fgur 4. Tm slcs durg computato to crat tal polycrystal structur. b. /s th computatoal ara. Hr, as mtod th prvous scto, th rqurd sz of th uclus s much largr tha that of a actual uclus. Thrfor, w us M = M /4 ad E stor = 4 E stor at all pots ad st th radus of th ucl to R 0 = 5Δx. Fgur 5 dmostrats th tm voluto of th mcrostructur. Wh th dslocato dsty rachs th crtcal valu ρ c, th rcrystallzd gras ar uclatd at th gra boudars ad grow toward th ctr of th orgal gras. Fgur 6 shows th varato of dslocato dsts wth th progrss of dformato. It ca b s that th dslocato dsty sd th rcrystallzd gras s lowr tha that th orgal gras ad crass wth th cotuous dformato. Th strss stra curvs ar llustratd Fg. 7. Th brok l shows th rsult obtad usg Eqs. ad 3 ad th sold l shows th prst rsult. Th op crcls Fg. 7 corrspod to thos Fgs. 5 ad 6. W ca obsrv typcal multpl paks causd by th softg du to th growth of rcrystallzd gras ad th Fgur 5. Mcrostructur volutos ε = (a) 0.0, (b) 0.4, (c) 0.6 ad (d) Fgur 6. Fgur 7. Varatos of dslocato dsty. Strss-stra curvs.

8 T. Takak, A. Yamaaka, Y. Tomta 8 6 Coclusos W hav stablshd a phas-fld modl for th DRX procss, whch th mult-phas-fld mthod proposd by Stbach t al. was gralzd to smulat th gra boudary mgrato drv by stord rgy. Th hardg was modld usg th thortcal quatos mployd by Guo t al. Th basc sgl-gra growth ad th DRX procss rsultg from th rgular hxagoal gra structur wr smulatd. It was cofrmd that th dvlopd phas-fld modl ca smulat th DRX procss. Ackowldgmt Ths rsarch was partally supportd by th Mstry of Educato, Cultur, Sports, Scc ad Tchology Grat--Ad for Sctfc Rsarch (B), , 007. Rfrcs [] Saka, T., Joas, J. J., (984) Dyamc rcrystallzato: Mchacal ad mcrostructural cosdratos, Acta Mtall., 3 (), [] Humphrys, F. J., Hathrly, M. (004) Rcrystallzato ad Rlatd Aalg Phoma, Elsvr. [3] Gotz, R. L., Stharama, V., (998) Modlg dyamc rcrystallzato usg cllular automata, Scr. Mat., 38 (3), [4] Gotz, R. L., (005) Partcl stmulatd uclato durg dyamc rcrystallzato usg a cllular automata modl, Scr. Mat., 5, [5] Gotz, R. L., (005) Partcl stmulatd uclato durg dyamc rcrystallzato usg a cllular automata modl, Scr. Mat., 5, [6] Kuglr, G., Turk, R., (004) Modlg th dyamc rcrystallzato udr mult-stag hot dformato, Acta Matr., 5, [7] Dg, R., Guo, Z. X., (00) Coupld quattatv smulato of mcrostructural voluto ad plastc flow durg dyamc rcrystallzato, Acta Matr., 49, [8] Dg, R., Guo, Z. X., (004), Mcrostructural voluto of a T-6Al-4V alloy durg b-phas procssg: xprmtal ad smulatv vstgatos, Matr. Sc. Eg. A, 365, [9] Qa, M., Guo, Z. X., (004) Cllular automata smulato of mcrostructural voluto durg dyamc rcrystallzato of a HY-00 stl, Matr. Sc. Eg. A, 365, [0] Modowk, M. A., (00) A rvw of mcrostructural computr modls usd to smulat gra growth ad rcrystallsato alumum alloys, J. Lght Mtals,, [] Kobayash, R., (993) Modlg ad umrcal smulatos of ddrtc crystal growth, Physca D, 63 (3-4) [] Warr, J.A., Bottgr, W.J., (995) Prdcto of ddrtc growth ad mcrosgrgato pattrs a bary alloy usg th phas-fld mthod, Acta Mtall., 43 (), [3] Robrts, W., Ahlblom, B., (978) A uclato crtro for dyamc rcrystallzato durg hot workg, Acta Mtal., 6, [4] Stbach, I., Pzzolla, F., (999) A gralzd fld mthod for multphas trasformatos usg trfac flds, Physca D, 34, [5] Blaz, L., Saka, T., Joas, J. J., (983) Effct of tal gra sz o dyamc rcrystallzato of coppr, Mtal Scc, 7,