A NEW ONE PARAMETER KINETICS MODEL OF DYNAMIC RECRYSTALLIZATION AND GRAIN SIZE PREDICATION

Õ 48 Õ 12 Vol.48 No.12 212 Û 12 Õ 151 1519 Í ACTA METALLURGICA SINICA Dec. 212 pp.151 1519 Æ È ÒÕ Þ Đ ÕÜÌÏ Ê ³ 1) µ²¹ 1) ½ 1) ¼ º 2) 1) ĐÔ CAD Ñ Á ¼, 23 2), Õ ÄÅËÏ, ÆÂ Ô Avrami Æ Ú ¾, ÀÂÏ º Ñ ¼Å ¾, Â È Ë È Ø. ¹ Gleeble 15 Ù Æ, Ú ¾ Ñ Ä AZ31B  Æ, Õ ¼ÅË Â ¾, Ö Æ, Æ Â ¾. ϲ «ÑÛ ¼ ÄÅ ¾ ²ÝÓ, Ö Î ¾, «ÑÛ Ú, ÙÖ Æ Î ßÀ «Ï, Ç Â «ÝÓÚ Æ«. Í, Å, ¾, AZ31B Ø TG146 Å Ä Ð A Å Ã 412 1961(212)12 151 1 A NEW ONE PARAMETER KINETICS MODEL OF DYNAMIC RECRYSTALLIZATION AND GRAIN SIZE PREDICATION LIU Juan 1), LI Juqiang 1), CUI Zhenshan 1), RUAN Liqun 2) 1) National Engineering Research Center of Die & Mold CAD, Shanghai Jiao Tong University, Shanghai 23 2) Department of Mechanical Engineering, Kumamoto University, Kumamoto, Japan Correspondent: CUI Zhenshan, professor, Tel: (21)6282765, E-mail: cuizs29@sjtu.edu.cn Supported by National Science and Technology Major Project (No.212ZX412 11) and National Natural Science Foundation of China (No.59511) Manuscript received 212 8 16, in revised form 212 9 3 ABSTRACT Dynamic recrystallization (DRX) is considered as one of the most important microstructural evolution mechanisms to obtain fine metallurgical structures, eliminate defects and improve mechanical properties of products. Although the DRX kinetics models proposed by researchers have some differences in parameters and forms, they are all based on the Avrami functioescribing the relationship betweeynamically recrystallized volume fraction and strain or time. Avrami equation is in the form of exponential function and the kinetics curve of DRX exhibits different when the exponent is assumed to be different (between 1 and 2). Under these conditions, however, the exponential function cannot exactly describe the slow rapid slow property of the development speed of DRX process. By introducing the velocity of development of DRX process, which is referred to as the variation of the dynamically recrystallized volume fraction with strain, namely, the first partial derivative of the dynamically recrystallized volume fraction to strain, a new kinetics model of DRX was proposed in comparison with the classical kinetics model of DRX. The new model represents the characteristics of DRX: the dynamically recrystallized volume fraction equals zero when the strain is smaller than the critical strain, and the maximum of the dynamically recrystallized volume fraction equals 1; once the strain exceeds the critical strain, the dynamically recrystallized volume fraction slowly increases first, and then rapidly increases, at last slowly increases. Consequently, the new kinetics model is in agreement with the development law of DRX process and includes few parameters which have clear * Ò» º Ó 212ZX412 11 Ò» Ó 59511 : 212 8 16, : 212 9 3 ÁÆ «: É ³, Þ, 1976 ½, Â ß DOI: 1.3724/SP.J.137.212.486

Õ 12 È ²Ñ : ¹»ÄÐ Ð Þ ÍÙ¾ 1511 physical meaning and are easy to determine. By conducting Gleeble 15 thermomechanical simulation compression tests at the temperatures ranging from 523 to 673 K and at the strain rates 1, 1,.1 and 1 s 1, the kinetics model for Mg alloy AZ31B characterized by DRX for instance was built and parameters were determined. Microscopic examination shows that the experimental results are in good agreement with the predicted values, which validates the accuracy of the new kinetics model. Then combined with grain size of DRX model, the kinetic model built under steady state conditions was rewritten as superimposed step form to apply in the prediction of grain size under unsteady state conditions. The simulated data accord with the experimental results by means of quantitative metallography, which verified the rationality of the superimposed prediction method. KEY WORDS dynamic recrystallization (DRX), dynamically recrystallized volume fraction, kinetics model, magnesium alloy AZ31B ² Ë ÐÐß Ï, Ç Ð, Ð Ò Ð. à Рµ, ˼±Ã Рƾ Î., Đ ¹»ËÐ ½ Æ ², ²Â ÐÐ Ò. Ï ÐÅÅ, ß ½ ÅÆ, ÛÆ, Í Ð ±ÅÅ. [1 9] ß Ã Õ Ø ÅÐĐ ÅÆ Û Ð ½ Æ. Æ Ó «ØÙ Ü ÈĐ, Ò ÐµÎ Æ Š[1]. ÐÕ ÒÐ ßÌ ¹, Æ Õ Å¾Í, Ó Å¾ Ì ÈÆ, Ì, Í ËÀ ² Ì, Õ Í ÐÆ Å. Æ Ó, ÓÐ Ôµ ¾ Đ, ½ ÆÂ, ÆÐĐ ± Đ, Ð ½ ÅÆ» [11 13]. Barnett Ò [14] à à Mg 3Al 1Zn Ï ½ Ð, ½ [15] AZ61 Æ ÐĐ ² Ã. µâà ¹ Ò ÐĐ Đ ¼±, ÕÉ, ß Ã ± Đ Ð Ò ½ÆÐ Æ AZ31B Đ Đ. Áº ³ÞÔ Ð ÒÜ Ð ½ ÅÆ Ð Û, Ç, Ç Ð. 1 ÑÔ Ý Ù 1.1 ÓÖ ß «Sellars Ò [1,2] Áà ½ ÅÆ Ð Ý, Á Ú ÇÐ Ä, ß Ã ÐĐ Û, Õ Avrami ÁÃĐ Æ Û, : f drx = 1 exp(.693( t t.5 )) (1), f drx Đ Æ, t.5 ž 5% Đ Ð Ö, t Ö. Sellars Ö º Æ.693, Ð Ò Á Ò Ð. Mc- Queen Ò [3] ß ÐĐ Đ Ì Ö ÁÐ. º Å [4] º Yada ß Ã Đ ÆÛ, : f drx = 1 exp(.693( ε εc ε.5 ) 2 ) (2) ε.5 = 1.144 1 3 d.28 ε 5 exp( 642 T ), ε c ÇÅ, ε.5 ž 5% Đ Ð, d Ï, T, ε Ì. Sellars Yada Ø, ³ Ø, Ò Ì, Ö Ð. Ð Õ Yada º Ã, ¼ Ô Õ ÔµÆ Ú, µ Ò Ý²Ü Marc, Superform Deform Ò Ã Yada Ð ½ ÅÆÐ. Sellars, Yada Ð ε.5 Ð, ž 5% Đ Ð, ±Ù Á, Õ. Ê Yada Ð Æ 2, Sellar Ð Æ 1. Ü, Kim Ò [5,6] ÇÐ Ä ß Ã Ð Đ ÆÛ, : f drx = 1 exp( ( ε εc ε ) m ) m = 1.12(Z/A) 8 (3), ε ²Æ Ð ; Z Ð Ì, Zener Hollomon ½Æ; A ½ Æ. Kim º ε Ð Yada Ð ε.5 ß, ³ Û Đ Ð Ð ¾ Ê, Đ ÐÌ, Á Ý,, ÖÁ Ó Ý, È Kim Ð ε Yada Ð ε.5 Ð, Ð Đ Ì Ð, Ì ²Æ Ð. Kim Ò Æ.693, Ð ÆÌ Z Ð Æ, Ê

1512 Á Õ 48 ½ÆÐ Sellars ÆÃ. : Kopp Ò [7] º ß Đ ÆÛ f drx = 1 exp( k d ( ε εc ε.5 ε c )) ε.5 = k 1 Z n1 (4), k d, k 1 n 1 ½Æ. Kopp Yada Ã, Æ k d Ð Æ.693, 1 Ê Æ Ð 2 Ê Æ, Ò ε.5 ε c Ð ε.5. Laasraoui Jonas [8] Serajzadeh Ò [9] ß Ã Ø ÒÜ Ø Ð Đ, ¼ Ù f drx = 1 exp( k d ( ε ε c ε p ) ) (5), ε p ; k d ½Æ. Yada, É ε p Ð ε.5, µ Ð k d Ì Z Ð Æ Ì ε T Ð Æ, µ ØÐ ÒÜ k d Æ, ɹ Ô Đ ÅÆ Ð. ¼ Á, Ò ÐĐ ÆÛ Õ Avrami, ØÐ Ð Æ Æ ε.5 ½ÆÐ. ½Æ Ò Ð ÒÜ, Õ Kopp Ð ºÍà Sellars Yada Æ.693 (5) Æ Ð, Å Ð Kopp ÛÐ Õ Avrami ÐĐ Đ. f drx = 1 exp(.693( ε ε c ε.5 ε c ) ) (6) 1.2 ÐÚ ³ Đ Ð Ù: ÇÅ Đ Æ, 1; ÇÅ Á, Đ Ð Æ É Ô, Á Ì Ô, ÁÓÉ Ô, Ź à ÐĐ ÆÛ : ε εc (1 ε.5 εc ) f drx = [1+k v ] 1 (7), k v Đ Ì Ò»Ð Æ, ¼ Ë Ð Ï Ð. ε c.2, ε.5.5, ³ (6) ÂÁ Ø ÐĐ Đ, Ý 1a ; ε c.2, ³ (7) ÂÁ Ø k v ε.5 ÐĐ Đ, Ý 1b. ÐÝ 1b ¼ Á,» Đ ÐĐ ¾Î. (7) ¼, Ð½Æ k v ± 1..8.6 (a).4 =1.2 =2 =3 =4 1..8.6.4.2 (b) k v.5 1,.4 15,.5 15,.6 2,.7 1 ¾ Fig.1 Kinetics curves of DRX based on modified Kopp s model (a) and the new kinetics model (b) Ð : ε ³Õ ε c, f drx Õ ; ε Ð Õ ε c, f drx Õ 1; ε ÒÕ ε.5,» ÀÒÕ.5. ³ºÍĐ ÐÌ, (7) Æ, ¼ Đ ÐÌ Æ: v drx = f drx ε = lnk v k (1 ε.5 ε c ε εc (1 ε.5 εc ) ε εc ε.5 εc ) v (1+k v ) 2 (8) ε c.2, ε.5.5, (6) Æ, ÂÁ Ø, Õ Avrami ÐĐ Ì, Ý 2a ; ε c.2, ³ (8)  Á Ø k v ε.5 ÐĐ Đ, Ý 2b. ÐÝ 2b ¼ Á,» Đ ÐÅÅ,. Ý 1a Ý 2a Ð Æ Õ 2 Ð Ð. Sellars Kopp º Ã Æ 1 Ð, =1, Đ Đ Á Û Ð, Đ Ì ÙÐ, ¾. Yada º Ã Æ 2 Ð, ÐÝ 1a Ý 2a ¼ Á, Đ ÐĐ ¾ ¾Î, =2, Đ Ì Ð Ì ε.5 Å.

Õ 12 È ²Ñ : ¹»ÄÐ Ð Þ ÍÙ¾ 1513 Velocity of DRX v drx Velocity of DRX v drx 6 5 4 3 2 1 (a) =1 =2 =3 =4 6 (b) k v.5 5 1,.4 15,.5 15,.6 4 2,.7 3 2 1 2 ¾ ËÏ Fig.2 Schematic of velocity curves based on modified True stress, MPa Kopp s model (a) and the new kinetics model (b) 2 18 16 14 12 1 8 6 4 2 (a) 25 o C 3 o C 35 o C 4 o C Õ 2, ÆÐ Ô, Đ Ì, Đ Ì Ð Õ ε.5 Å., Avrami,»ÐĐ Đ Ò Æ Ù: (1) ÐÕ Æ ÆÐ ³, Avrami ³ÕÇÅ Õ Ð,» ¼ ÇÅ Ð Æ. (2) ε.5 Ò ÆÃÐ, Æ 5% Ð. (3) ½Æ k v ÃĐ Å¾Ð Đ Ì Ð ³. 4)» Ð ½Æ º,  ε c ε.5 Z ½Æ Ì ß, Ò k v, Ð (6) ºÃ ± ½Æ. 2 Ñ Æ AZ31B(Mg 3Al 1Zn(³Á Æ, %)), Ê 1 mm, 17 mm. Gleeble 15 Ú Ç Ò, 1., Ø Ì Ð Ý 3. ÆË ÒÐ Ë Ò: (1), ÐÕÔµ Æ«Ô, ± Ô Ñ±. (2) Ì Ô Ì Ó, ÐÕÔµ Æ Ô¼, Ë. (3) Õ, ÐÕĐ Ð²Æ ÕÔµ Æ Ð«Æ, Ë True stress, MPa 2 (b) 18 16 14 12 1 8 6 4 2 25 o C 3 o C 35 o C 4 o C True stress, MPa 2 (c) 18 16 14 12 1 8 6 4 2 25 o C 3 o C 35 o C 4 o C True stress, MPa 2 (d) 18 16 14 12 1 8 6 4 2 25 o C 3 o C 3 AZ31B Ê [16] Fig.3 The stress strain curves of magnesium alloy AZ31B [16] 35 o C 4 o C (a) ε=1 s 1 (b) ε=1 s 1 (c) ε=.1 s 1 (d) ε=1 s 1

I A 1514 Y e l p ~. (4) Y < e l p, &N sg N K~G, S ee? a. Aw3l$n a h l+ MS h 9. 4 ad AZ31B q 48 ( y 4 C 5 #t 5nx l E 5_. [,I 5 C e T X a B EW n 5 f!: Y < 5 l 3!, B EW h p; Y < e T X 4 "s4k D (a), 3 (b), 35 (c) and 4 Fig.4 OM images of magnesium alloy AZ31B at the temperatures of 25 (d) (ε =1 s 1, ε=1.) 5 ad AZ31B d SW"s4k D Fig.5 OM images of magnesium alloy AZ31B at the strain rates of 1 s 1 (a), 1 s 1 (b),.1 s 1 (c) and 1 s 1 (d) (T =35, ε=1.)

Õ 12 È ²Ñ : ¹»ÄÐ Ð Þ ÍÙ¾ 1515 Ð Ô, Ï Ù³. 3 Æ Bergstrom [17] º ÏĐ Æ, É Ø. f drx = σ σ p σ s σ p (9), σ p, σ s, σ ½Ð. Ð (7) Æ 1 f drx f drx ε εc = k 1 ε.5 εc v (1) ln( 1 f drx f drx ) = lnk v (1 ε ε c ε.5 ε c ) (11) ÐÕ «Ò ε c ε.5, Ö ε c =(.6.85)ε p, Å.8. À Ö Ú lnz ln(ε p ) lnz ln(ε.5 ) л,, Z= εexp(q/(rt)), Q= 158732.3 kj/mol [16], ε c ε.5 Ð [18], Ý 6a b. ÁÖ Ú Ç Ð ln((1 f drx )/f drx ) 1 (ε ε c )/(ε.5 ε c ), ¼ k v, Ý 6c, Ð (12). ε εc (1 ε f drx = (1+15.5 εc ) ) 1 ε c = 834 Z 75 ε.5 = 426 Z 781 (12),Z= εexp(q/(rt)),q=158732.3kj/mol [16], R Æ. ³ (12), β Æ AZ31B Ø Ø Ì ÐĐ Đ Ý 7. à Ð, ÁÆ Ç Ç. Ý 8 Æ AZ31B Ð Ï, Ï 22 µm. 3 Ì.1 s 1, Ø ÁÐ Æ AZ31B ½ Ý 9. ÐݼÚ, Á.15, Ò º Å¾Đ ; Á Ô.3, ž Đ ; ÁÐ Ô,, ÏÙº, Á.5, µ žÃĐ ; Á.8, Þ Đ Ð. ln( p ) (a) -.5-1. -1.5-2. -2.5-3. -3.5 Experiment data Linear fit -4. 5 1 15 2 25 3 35 4 45 5 lnz ln(.5 ) -.5-1. -1.5-2. (b) -2.5 Experiment data Linear fit -3. 5 1 15 2 25 3 35 4 45 5 lnz ln((1-f drx )/f drx ) 6 4 2 (c) 25 o C, 1 s -1 25 o C,.1 s -1-2 3 o C,.1 s -1 3 o C, 1 s -1-4 35 o C, 1 s -1 35 o C, 1 s -1 Linear fit -6-2. -1.5-1. -.5.5 1. 1.5 2. 2.5 1-( - c )/(.5 - c ) 6 AZ31B ÆÅ Ù º Ü Fig.6 Relationships of lnz ln(ε p) (a), lnz ln(ε.5 ) (b) and ln((1 f drx )/f drx ) [1 (ε ε c)/ (ε.5 ε c)] (c)

1516 Á Õ 48 1. (a) 1. (b).8.6.4.2 25 o C 3 o C 35 o C 4 o C.8.6.4.2 1. (c) 1. (d).8.6.4.2.8.6.4.2 7 AZ31B Fig.7 Kinetics curves of DRX of magnesium alloy AZ31B at ε=1 s 1 (a), 1 s 1 (b),.1 s 1 (c) and 1 s 1 (d) ÐÕß Ð ½ ÅÆ Ò Ò Ì ÒÜ Ð, ¾ Ì Æ, É, Ð ÒÜ Ð ½ ÅÆ ÞÔ, ÒÜÐÆ Ú. ÐÖ i ½, ³ Á ε i ³Õ ½ ÇÅ Á ε i c, Đ Æ Á ; ³ Á ε i Õ ½ÇÅ Á ε i c, Đ Æ¼Ð (7) (8) Þ : 8 AZ31B Î Fig.8 The initial grain of magnesium alloy AZ31B Ç ÐĐ Æ Û Ð Ý 1. Ðݼ, º Ï ÐĐ Æ ÁÆ Ð. 4 ÔÛËÎ É º, 1 ± ØÐ Ö Á 5 Ê, Á  ÉÊ ÒÜ ÐĐ Ï D drx, lnz lnd drx ÀÝ Ú, Ý 11. ÐÉ, Đ Ï : D drx = 48.2583 Z.115 (13) fdrx i = ε i (1 f i 1 drx )lnk v f i 1 drx ε i.5 εi c (14), f i drx ³ÐĐ Æ Á; εi ³Ð Á; f i 1 drx ³ÐĐ Æ; εi.5 ½Å¾ 5% Đ Ð Á; ε i c ½ÇÅ Á. Đ Æ : f i drx = f i 1 drx + fi drx (15) Đ ÆÞÔ Õ.99, Ó ÞÔ, Đ Ã. ÏÐ Ï Õ Õ»: (1) Ï ¾ Ð ÒĐ

q 12 P & m : - j_/ L } _ } CB z i b CU u 2 ad 9 AZ31B 1517 4 3 4SW.1 s, "s 4H4k4^x 1 Fig.9 OM images of magnesium alloy AZ31B at the strain of.15 (a),.3 (b),.5 (c) and.7 (d) (T =3 ε =.1 s 1 ), 4. 1. Predicted by model.9 Experiment data Experiment data Linear fit 3.5.8 3. D drx.6 ln f drx.7.5 2.5.4.3 2..2.1 1.5.1.2.3.4.5.6.7.8.9 15 1. 2 25 3 Z 35 4 45 ~` ADVB Z k. ln 1 ad AZ31B ~` Mv3 B6N x Fig.1 Comparison of the predicted value by the model with the experimental data 11 Fig.11 Relationship betweeynamical recrystallization grain size with Z, lq 5 : B, J Y a > I l B9\:B. (2) a : B *, X 5 1 l a B } IX51l a B. 5:B*, ~* BEW < : Ddrx = q 2 f 2 + D 2 (1 f 2 Ddrx drx ) drx (16) =j ) jn y &x Marc, 7 6 G [: Bl N k v, '! 2 < h u 5 l 1/2 K %, a y :3 Q u 5 2 <a y, n y ~$L #y 12 12 m x}# -t $=^x Fig.12 Schematic of finite element meshes and reference position for sample

1518 Á Õ 48. ÚÐ 35, Ì.1/s. ½Æ : Æ AZ31B Ð Ú 1, Ç Æ Ú 2, 178 kg/m 3, Ê ¾ ÐÊ Æ 2 W/(m 2 ), Ê ÚÐÊ Æ 3 W/(m 2 ), µâå 8. Ê ÚÐ 35, Ê ÚÐе Æ.2, Ê Æ.9. Ë Å [16] ß Ð. Ý 13 ²ÁÃ Æ AZ31B Ø Å 1 AZ31B Table 1 Specific heat of magnesium alloy AZ31B T, 2 1 3 6 c, kj/(kg ) 1.4 1.42 1.148 1.414 Á Ð Ï ±. ÐݼÚ,, ÐÕ Å¾Đ, Ð Ï. Ã, Ð Ð Ï, ½ Ð Ï ³, ÐÕ Ø ØĐ Ð Ø, Ð Í Ï ±. Ý 12 Ð 3 ±Ù 3 ±½º ±, ÏÐ. Ý 14 3 ± Ø ±ÅÐ, A ÅР˳ÕÕ C Å, Ð B ÅÐ ¹ ÕÕ C ÅÐ. Ç Ð Ï Æ ÚÐ Ý 15, Ç Å 2 AZ31B Æ Å Table 2 Thermal conductivity of magnesium alloy AZ31B T, 2 5 1 15 2 25 λ, W/(m ) 76.9 83.9 87.3 92.4 97. 11.8 13 À É µ Î Fig.13 Distribution of grain size in the sample at the stroke distance of 3.79 mm (a), 5.69 mm (b) and 9.48 mm (c) 14 Ü 12 µ Fig.14 OM images of different areas in Fig.12 (a) position A (b) position B (c) position C

Õ 12 È ²Ñ : ¹»ÄÐ Ð Þ ÍÙ¾ 1519 Grain size, m 4 35 3 25 2 15 1 5 Calculate Experimental A B C Position 15 Ü 12 µ Ö Ù Î Fig.15 Simulated and measured grain size at different positions in Fig.12 Õ Ú, Ë,, Ã ß Ï Ð. 5 Ñ (1) Õ ÇÙÐ Ë Avrami ÐĐ Đ Ð, ÁûÐĐ Đ, Đ ÐÉ Ì É ÐÅÆ Ù, Ò «½Æº Ø ½Æ Ð Ù. (2) ß ÃĐ ÏÛ, РлÐĐ Đ Þ Õ ÒÜÐ Û Ð ³ÞÔ, Ð ÏÛ. (3) Û Ð ÒĐ Ð Å Æ AZ31B, Ö Gleeble Ç, ½ÆÌ, ß ÃĐ Đ, ÕÒ Ý Ú ÏÐÛ, Ö Æ, Ç Ã Ð. Ç Å [1] Sellars C M. Mater Sci Tech Ser, 199; 6: 172 [2] Sellars C M, Whiteman J A. Met Sci, 1979; 13(3 4): 187 [3] McQueen H J, Yue S, Ryan N D, Fry E. J Mater Process Tech, 1995; 53: 293 [4] Devadas C, Samarasekera I V, Hawbolt E B. Metall Trans, 1991; 22A: 335 [5] Kim S, Lee Y, Lee D, Yoo Y. Mater Sci Eng, 23; A355: 384 [6] Kim S, Yoo Y. Mater Sci Eng, 21; A311: 18 [7] Kopp R, Karnhausen K, de Souza M. Scand J Met, 1991; 2: 351 [8] Laasraoui A, Jonas J J. Metall Trans, 1991; 22A(7): 151 [9] Serajzadeh S, Taheri A K. Mech Res Commun, 23; 3: 87 [1] Avedesian M M, Baker H. United States of America: ASM International, 1999: 3 [11] McQueen H J. Mater Sci Eng, 24; A387 389: 23 [12] Tan J C, Tan M J. Mater Sci Eng, 23; A339: 124 [13] Mwembela A, Konopleva E B, McQueen H J. Scr Mater, 1997; 37: 1789 [14] Barnett M R, Beer A G, Atwell D, Oudin, A. Scr Mater, 24; 51: 19 [15] Zhou H T. PhD Thesis, Shanghai Jiao Tong University, 24 (¼. Õ, 24) [16] Liu J, Cui Z, Li C. Comp Mater Sci, 28; 41: 375 [17] Bergstrom Y. Mater Sci Eng, 1969/197; (5): 193 [18] Liu J, Cui Z, Ruan L. Mater Sci Eng, 211; A529: 3 (¾» ±: )