APPARENT AND PHYSICALLY BASED CONSTITUTIVE ANALYSES FOR HOT DEFORMATION OF AUSTENITE IN 35Mn2 STEEL

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1 49 6 Vol49 No ACTA METALLURGICA SINICA Jun 213 pp º Ã 35Mn2 ³Í Ê Ü 1) ĐÛ 1,2) 1) Æ Ý 2) 1) Å±± ± ±, 183 2) Å± Û¼± ¼», 183 Ð Ê µ ¼ 3 Æ ² Ù, ÛÎ 35Mn2 Æ ²µÛ ºÐ Î Ç ĐÂ ¹Ù ² ¾ ÜÜĐ ², Ù ¼ ÌÃ 278 kj/mol, Æ ÆÌÃ (27 kj/mol) ÔÐ, ÊÐ Ê Æ Â Ð Â Ø ² ¾ ÄÅ «ß ÜÜĐ ², Ç Æ«ß ² ¼ Â¹, Đ ¼Đ µû 991, Æ ØÖ¹ 419%, Ê Đ ¼Đ ¾, Ç ¼ Â¹ Æ ÈÏÔ ²Ì, Young s Ì Æ ÆÛ Õ µû, Ç Æ Ì Å ĐÂ¹ È µû, Ë Õ Ï ÜÜĐ ², ØÐß Ü, Ð ÕÌÊ Â ¾ 35Mn2,, ²µÛ Ô Â TG1117 ¹ÌÇ A Ñ (213) APPARENT AND PHYSICALLY BASED CONSTITUTIVE ANALYSES FOR HOT DEFORMATION OF AUSTENITE IN 35Mn2 STEEL WEI Hailian 1), LIU Guoquan 1,2), XIAO Xiang 1), ZHANG Minghe 2) 1) School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 183 2) State Key Laboratory for Advanced Metals and Materials, University of Science and Technology Beijing, Beijing 183 Correspondent: LIU Guoquan, professor, Tel: (1) , gliu@ustbeducn Supported by National Natural Science Foundation of China (No517119) and National High Technology Research and Development Program of China (No213AA3161) Manuscript received , in revised form ABSTRACT The constitutive relationships of a 35Mn2 steel during hot compression testing were systematically investigated using three methods The first method is a conventional hyperbolic sine equation with peak stress dependent constants, the activation energy Q determined by this method is about 278 kj/mol, very close to the austenite lattice self diffusion activation energy (27 kj/mol), indicating the rate controlling mechanism is dislocation climb controlled by diffusion The second method is a developed hyperbolic sine equation with strain dependent constants, comparing with experimental results, the correlation coefficient and average relative error of predicted and measured values are 991 and 419%, respectively, indicating that the developed equations can give an accurate estimate of the flow stress for the experimental steel The third method is a physically based approach accounting for the dependence of the Young s modulus and the self diffusion coefficient of austenite on temperature, which is also capable of representing the flow stress of the material as a function of the deformation conditions, but the fitting precision by this method is lower than by the conventional hyperbolic sine equation, and through modification, the fitting precision of the physically based approach is improved in this work KEY WORDS 35Mn2 steel, hot deformation, constitutive relationship * ½ ²ÈÜ Ï ½ ÞÒ Ï 213AA3161 ¹Ï : , ¹Ï «: ÕÙ : ¼,, 1988, DOI: 13724/SPJ

2 732 Ú» Å 49 35Mn2 ± Ë È [1], Ñ È É ÓÜ, Ù Ú Ó, ¹ 35Mn2 ÙÑ Ð ÅÍ, ³ ±¹ ³ Å, ÔÞ, ÍÇ Ý¾ ³ [2 15] Ø, ÍÇ Î ¹³ Ï ³ [2 13] : Z = εexp( Q RT ) (1) ε = A 1 σ n1 exp( Q RT ) (2) ε = A 2 exp(βσ)exp( Q RT ) (3) ε = A[sinh(ασ)] n exp( Q RT ) (4), ε ±Ã À, s 1 ; A, A 1, A 2, α, β, n, n 1 ±³, α = β/n 1 ; R ±ÓÇ, 83145J/(mol K);Q ÍÄ, J/mol; T Ö, K;σ ± Ãº Ï ³, Ö Ã ÀÙ³ Å«È Zener Hollomon Z ( (1)), Ç ³ ( (2)) ²È ασ <8, ³ ( (3)) ²È ασ >12, Ý ÝÝ ³ ( (4)) ²È Ó ÃºĐ, É Z Ãº Ü [2,3] 3 ( (2) (4)) É ± Û, Ñ ÄÍ Ë ÃºÙÃ Ã Õ¹ º È ÁÚÃº ÐÀÃº ÃºÆÎ Ã ÙÃ Ãº ³ [7,9,16] [5,6,8,1 13] Ã Ù ¹³ Å«, ÉÃ Ð ¹³ ÝÝ ³, È ³ Ãº, ÝÝ ³ ĐÝ ¹ [17 19], Ï ³ Ù³ É ³ Ø³, Ú ³ ± ¹ Ý ± [14,15,17 19] Ã Õ±Í, Ø Õ±Í É ±Â, ¾µÍ µç Ü Ö Ü, (5) : ε/d(t) = B[sinh(α σ/e(t))] 5 (5) Ñ, B α ±³ ; D(T) = D exp(q sd /(RT)), D ±, Q sd ± ÍÄ ; E(T) É Young s Í Ö Ü D, Q sd E(T) É Frost Ashby [2] ÐÐ 3 Ð È ¹ ³ ³ÁÚ µè 1 Ï ³ Ú ¹³, ÐÀÃº ÃºÆÎ Ã ÙÃ Ã º Ü µè ÝÝ ³ Í É ÃºÝ, ØÙ 3 Ð É ÁÚÃ º Ü, ÖÙ Young s Í µç Ü Å«, ÅÍ Ð Õ±É Ì Á Ç ³ Ç Î 1 ÆÖ 2 Ð, ÐØ 3 ÐØ Ò¹, É Õ± ³Í º ÃÈ ¹ ±, µèê ½ 3 ÐÜÏ¹ ÝØ³ (35Mn2 ) ³ 1 Ë ½ À³ («,%) : C 36, Si 27, Mn 142, S 45, P 53, Al 1, N 45, Fe Ð ½ Ù ÆÃ ½ 5 kg, Đ Ð 2 mm ³ ³ min, ÞÀĐ Ð Đ Ð ³ 8 mm, 15 mm, Gleeble 15 Í ½Ë ÈÊ Ð µçê Û Ö : 9, 95, 1, 15 11, Ã À 1, 1, 1 1 s 1 ÅÇ : Õ, Ù 1 K/s À 115, 3 min, 67 K/s À ß Ö, 3 s, ½, Ã 6 Ø À 2 Ë ÏÆ 21 Ò Å Ò É 1 Í É ½ ÙÃº ÙÃ Ý, ½ Ãº Ö Ò Ã À Ý ÍÃ À 1 s 1, Í Ö, ÃºÝ ÒÀ ØĐÁÚ, Ãº, Ã Ñ, Ãº Ò Ä ÐÀ, Ø Ë Å É, ½ ß ÒÀ ØĐ ( 1a) Í Ö 11, ½ ÃºÝ Ã À 1 1 s 1 ÒÀ Ø ĐÁÚ, ÝÍÃ À 1 1 s 1, Ãº Ý Ë ( 1b) 1a b Á, ½ Ãº Ã Ã À Ò Ö ÝÒ 22 ½ ÎÉÓ µ È É ÃºÙ» [21], ¾ ÇÛ È Ãº ³, È Ãº ½ ÍÄ ½ ÝÝ ³ Ù (2) (3) Þ Đ Ù, (6) (7): ln ε+ Q R (1 T ) = lna 1 +n 1 lnσ (6)

3 6» : ÇÎÓ 35Mn2 ³Å ± Ø 733 ln ε+ Q R (1 T ) = lna 2 +βσ (7) Í Ö Ð, µè Ð (8) (9): n 1 = [ ln ε lnσ ] T (8) β = [ ln ε σ ] T (9) (8) (9), n 1 β Ë Â» É ln ε lnσ p ln ε σ p À, 2 É 2a b Í Ö n 1 β, n 1 = , β = 61718, α=β/n 1 = 1 Ù (4) Þ ĐÙ, (1): ln ε+ Q R (1 ) = lna+nln[sinh(ασ)] (1) T Í Ö Ð, µè Ð (11): ln ε n = [ ln[sinh(ασ)] ] T (11) ln[sinh(ασ p )] ln ε ( 3a), Ú Í Ö n ÍÃ À Ð, µè Ð (12): Q = Rn[ ln[sinh(ασ)] ] ε (12) (1/T) ln[sinh(ασ p )] 1/T ( 3b), Ú ÍÃ À Ñ À , µè (12) Ú ½ ÍÄ kj/mol (1) (4), Zener Hollomon Z Ãº Ü Ë Â»É lnz ln[sinh(ασ p )], 4 É 4 ½ ÝÝ ³ : Z = εexp(277791/(rt))= (sinh(1σ p )) (13) ÍÄ Q ± Ú³ Ö, Arrhenius ±Â, ÍÄ È Î ¹Ë Æ Ô [22,23] [7,24 26] ¹ o C 95 o C 1 o C 15 o C 11 o C s -1 1 s -1 1 s -1 1 s ¼ Ì È ß Â¹Ü Fig1 Flow curves obtained at different deformation conditions 1 s 1 and different temperatures 11 and different strain rates ln(, s -1 ) o C 95 o C 1 o C 15 o C 11 o C p ln( p ) 2 ĐÂ¹ Â µû Fig2 Relationships between strain rate and peak stress ln ε σ p ln(, s -1 ) ln ε lnσ p 9 o C 95 o C 1 o C 15 o C 11 o C

4 734 Ú» Å 49 ln[sinh( p )] o C 95 o C 1 o C 11 o C 15 o C ln[sinh( p )] s -1 1 s -1 1 s -1 1 s ln(, s -1 ) 1/T, K -1 3 ĐÂ¹ Â Ï Õ µû Fig3 Relationships between ln[sinh(ασ p)] and ln ε, and ln[sinh(ασ p)] and 1/T lnz lnz=449557ln[sinh( p )] R 2 = ln[sinh( )] p 4 ln[sinh(ασ p)] lnz µû Fig4 Relationship between lnz and ln[sinh(ασ p)] ZK6Mg Ý Mg 3Al Ý Ti 6Al 4V V Ti Ý Ý¾³ À Ë, ß ÍÄ ³ ÍÄ Õ, Ð³ Ø É À Ë ± Ã ½ ÍÄ kJ/mol, µç ÍÄ (27 kj/mol) Õ [17,23], Ë ½ É À Ë ± Ã 23 À ÎÉÓ µ È, Ã ÙÃ Ã» 1 (14) Ý Table 1 Values of the coefficients in Eq(14) º, Ý ÄÍ Í Ã Ã Í, Ã Ù ½ Ãº Å«, È Ð Ü ÍÃ ÙÃ Ãº ³, Ã ³ Q, n, α lna Ã Þ 5 8, Û 5 ³ Q, n, α lna Ã ÜÈ 5 Û ( (14)), ÑÜ 1, Ø 5, ÅÍ α(ε) = α +α 1 ε+α 2 ε 2 +α 3 ε 3 +α 4 ε 4 +α 5 ε 5 n(ε) = N +N 1 ε+n 2 ε 2 +N 3 ε 3 +N 4 ε 4 +N 5 ε 5 lna(ε) = A +A 1 ε+a 2 ε 2 +A 3 ε 3 +A 4 ε 4 +A 5 ε 5 Q(ε) = Q +Q 1 ε+q 2 ε 2 +Q 3 ε 3 +Q 4 ε 4 +Q 5 ε 5 (14) ½Þ ³ ½ Ãº Ð, µ È ³ ÍÃ Í Ö Ã À Ãº 32, ½ Ù, 6 ½ Ç É Ð½Þ ³ Ð Ö, Á È Ù º ½Þ 7 ³ Ãº ½, α Value n Value lna Value Q Value α 2115 N A Q α N A Q α N A Q α N A Q α N A Q α N A Q

5 6» : ÇÎÓ 35Mn2 ³Å ± Ø th order polynomial fit n 5th order polynomial fit n lna (c) lna 5th order polynomial fit Q, 1 5 J/mol 32 (d) Q 31 5th order polynomial fit Ú µ Â ε µû Fig5 Relationships between α, n, lna (c), Q (d) and the true strain ε obtained by polynomial fit of the experimental steel 1 s -1 Experimental Predicted 9 o C 95 o C 1 o C 15 o C 11 o C s -1 Experimental Predicted 9 o C 95 o C 1 o C 15 o C 11 o C s -1 Experimental Predicted 9 o C 95 o C 1 o C 15 o C 11 o C s -1 Experimental Predicted 9 o C 95 o C 1 o C 15 o C 11 o C ¼ Ì È ß Â¹Đ ¼Đ Ø Fig6 Comparisons between predicted and measured flow stress curves of the experimental steel under different deformation conditions

6 736 Ú» Å 49 Predicted flow stress R 2 = Experimental flow stress 7 ²µÛ Â¹Đ ¼Đ µ Fig7 Correlation between the experimental and predicted flow stress data from the constitutive equation Ü 991, Ø Ó Í, Ø³Á Ù º δ: δ = 1 N N i=1 σ e σ p σ e (15), σ e Ãº ½, σ p µè ³, N ½ Ú (N=32) δ 419% Ú ËµÈ ³ ½ ÃºÅÍÓ Ö, È Ð ½ Ãº 24 Ä µ È 241 Þ ÐØÕÚ ÅÙ Frost Ashby [2], γ Fe ± C Mn Õ ³, γ Fe È ½ Ú, É D(T) E(T) Ä : D(T) = D exp( Q RT ) = exp( 27 ) (16) RT E(T) = E (1 T M dgt 3 ) = G dt T M (1 91 T ) (17) Ñ, E G È ³ 3 K Young s Í ÖÍ, T M ±³ (5), Í 2 ß B α ÐÐ Ú α, Mirzadeh [14] Ã Ê È Ð Ø Ð, (2) (3) Æ À (18) (19): ε/d(t) = B (σ p /E(T)) n 1 (18) ε/d(t) = B exp(β σ p /E(T)) (19) α µè α = β /n 1 Ú, Ý n 1 β ln( /D(T)) ln( /D(T)) ln( p /E(T)) /E(T) p 8 ÇÙ n 1 β Ü Fig8 Plots used for determining the values of n 1 and β for calculation of α É ln( ε/d(t)) ln(σ p /E(T)) ln( ε/d(t)) σ p /E(T) À 8 ln( ε/d(t)) ln(σ p /E(T))( 8a) ln( ε/d(t)) σ p /E(T) ( 8b), Â»Ú n 1 β , α = (5), ( ε/d(t)) 1/5 sinh(α σ p /E(T)) À È B, Ø B 1/5 = ( 9), ½ É Õ± ³ : ε exp(27/(rt)) = [sinh( σ p /E(T))] 5 (2) 242 Þ ÐØÕÚ ÅØ«, É Õ± ³Í Young s Í µç Ü Ö Ü, Ó, ÌÑ Ö ÄÍ Ï ÝÝ ³Í Ö Ø± É Õ± ³Í ± ±Â, Ð n=5 ÍÄ (Q sd ) È (5) [14], ĐÝ É ±Íº, Ø ±Î º È ÑÄ, ½ Ë (5)

7 6» : ÇÎÓ 35Mn2 ³Å ± Ø (Ãº n ), Ý (5), (21): ε/d(t) = B[sinh(α σ/e(t))] n (21) Í, (21) α α = (21), ln( ε/d(t)) ln(sinh(α σ p /E(T))) À Ö «È n lnb, 1 1, Ú n = 475, lnb = ³ (22) : εexp(27/(rt))= [sinh( σ p /E(T))] 475 (22) Ù 9 1, Đ n 475, Õ 5, Ì (21) ÖË (5) Ö, Ë Ù É Õ± ³ Ý± Í Ì ¹ É Õ± ³Í, ( /D(T)) 1/ R 2 = sinh( ' /E(T)) p 9 Â¹ 5 ² ( (5)) ( ε/d(t)) 1/5 sinh(α σ p/e(t)) Ü Fig9 Peak stresses presented according to the physically ln( /D(T)) based equation with the stress exponent of 5 (Eq(5)) R 2 = ln(sinh( ' /E(T))) p 1 ² ( (21)) ln( ε/d(t)) ln(sinh(α σ p/e(t))) Ü Fig1 Peak stresses presented according to the physically based equation with the stress exponent as a variable to be determined (Eq(21)) Ñ º ÃÈ, É ¹ 3 Æ (1) 35Mn2 ÝÝ ³ : Z = εexp(277791/(rt))= (sinh(1σ p )) ½ ÍÄ Ç 278 kj/mol, µç ÍÄ (27 kj/mol) Õ, Ë ½ À Ë ± Ã (2) µè ÝÝ Í, Ã Ù³ Å «, ½ ³ ÈÅ ³ ½ Ãº, ½ Ü 991, Ù º 419% ËØ ³ Ù ½ Ãº Ð (3) ÖÙ Young s Í µç Ü Å«, ½ É Õ± ³ : εexp(27/(rt))= [sinh( σ p /E(T))] 5 Í ÙØ ³ Ý, ß Ð 5 (Ãº n ), Ã Ö ¼Á [1] Wu Z F, Zhang D J, Pei R Y, Wang D C, Yan M L Oil Field Equipment, 2; 29(4): 3 (Ò, ÓÓ, «Ì, Æ, Æ Ê ¼Ê, 2; 29(4): 3) [2] Mirzadeh H, Cabrera J M, Prado J M, Najafizadeh A Mater Sci Eng, 211; A528: 3876 [3] MeysamiM,MousaviSAAAMater Sci Eng, 211; A528: 349 [4] Chen L, Wang L M, Du X J, Liu X Acta Metall Sin, 21; 46: 52 (, Å, ÔÆ, Æ Ü½²«, 21; 46: 52) [5] Sun C Y, Luan J D, Liu G, Li R, Zhang Q D Acta Metall Sin, 212; 48: 853 (, Á Ñ,, ² Å, ÓØÑ Ü½²«, 212; 48: 853) [6] Xiao X, Liu G Q, Hu B F, Zheng X, Wang L N, Chen S J, Ullah A Comp Mater Sci, 212; 62: 227 [7] Zhao H T, Liu G Q, Xu L Mater Sci Eng, 213; A559: 262 [8] Wu K, Liu G Q, Hu B F, Li F, Zhang Y W, Tao Y, Liu J T Mater Des, 211; 32: 1872 [9] Wei H L, Liu G Q, Xiao X, Zhao H T, Ding H, Kang R M Mater Sci Eng, 213; A564: 14 [1] Wang Z X, Liu X F, Xie J X Acta Metall Sin, 28; 44: 1378 ( Å,, Ü½²«, 28; 44: 1378) [11] Mirzadeh H, Najafizadeh A Mater Sci Eng, 21; A527: 116

8 738 Ú» Å 49 [12] Zhang H G, He Y, Liu X F, Xie J X Acta Metall Sin, 27; 43: 93 (Ó, Æ,, Ü½²«, 27; 43: 93) [13] Tan Y J, Pan Q L, He Y B, Li W B, Liu X Y, Fan X Acta Metall Sin, 29; 45: 887 (²ÀÑ, Ø,, ², Æº, 29; 45: 887) ± Ü½²«, [14] Mirzadeh H, Cabrera J M, Najafizadeh A Acta Mater, 211; 59: 6441 [15] El Wahabi M, Cabrera J M, Prado J M Mater Sci Eng, 23; A343: 116 [16] Lou Y, Li L X, Zhou J, Na L Mater Charact, 211; 62: 346 [17] Cabrera J M, Al Omar A, Jonas J J, Prado J M Metall Mater Trans, 1997; 28A: 2233 [18] Cabrera J M, Ponce J, Prado J M J Mater Process Technol, 23; : 43 [19] Cabrera J M, Jonas J J, Prado J M Mater Sci Technol, 1996; 12: 579 [2] Frost H J, Ashby M F Deformation Mechanism Maps: the Plasticity and Creep of Metals and Ceramics Oxford: Pergamon Press, 1982: 21 [21] Mirzadeh H, Najafizadeh A, Moazeney M Metall Mater Trans, 29; 4A: 295 [22] McQueen H J, Yue S, Ryan N D, Fry E J Mater Proc Technol, 1995; 53: 293 [23] Medina S F, Hernandez C A Acta Mater, 1996; 44: 137 [24] Galiyev A, Kaibyshev R, Gottstein G Acta Mater, 21; 49: 1199 [25] Srinivasan N, Prasad Y V R K, Rama Rao P Mater Sci Eng, 28; A476: 146 [26] Seshacharyulu T, Medeiros S C, Frazier W G, Prasad Y V R K Mater Sci Eng, 22; A325: 112 ( ß: Ö )

A CONSTITUTIVE MODEL FOR PARTICULATE-REINFORCED TITANIUM MATRIX COMPOSITES SUBJECTED TO HIGH STRAIN RATES AND HIGH TEMPERATURES

THERMAL SCIENCE, Year 203, Vol. 7, No. 5, pp. 36-367 36 A CONSTITUTIVE MODEL FOR PARTICULATE-REINFORCED TITANIUM MATRIX COMPOSITES SUBJECTED TO HIGH STRAIN RATES AND HIGH TEMPERATURES by Wei-Dong SONG