Steel Forming and Heat Treating Handbook - Antonio Gorni On Line

STEEL FORMING AND 

HEAT TREATING 

HANDBOOK 

Antonio Augusto Gorni 

São Vicente SP 

Brazil 

agorni@iron.com.br 

www.gorni.eng.br 

This Version: 20 February 2014

Gorni 

Steel Forming and Heat Treating Handbook 

i 

FOREWORD 

This is a compilation of some useful mathematical formulas, graphics and data in the area of forming, heat treatment and 

physical metallurgy of steels. The very first version arose about thirty years ago as a handwritten sheet with a few formulas. 

Afterwards it was converted to a digital format and eventually posted on-line, hoping that it could be also helpful worldwide. 

It must be noted that these formulas were compiled at random, generally in a need-to-know basis. So, this Handbook is in 

permanent construction and very far to be complete. Finally, the author thanks Seok-Jae Lee for his contribution. 

DISCLAIMER 

The formulas and information compiled in this text are provided 

without warranty of any kind. 

Use them at your own risk! 

However, any help regarding the correction of eventual mistakes 

is appreciated. 

A Non-Stop Work 

First On-Line Release: 04 April 2002



SUMMARY 

ii 

Austenite Formation Temperature ...........................................................................................................................1 

Austenite Grain Size ...............................................................................................................................................8 

Austenite No-Recrystallization Temperature ............................................................................................................9 

Austenite Solubility Products ................................................................................................................................ 11 

Austenite Solubilization Temperature .................................................................................................................... 16 

Austenite Transformation Temperatures: Ar 3 and Ar 1 ............................................................................................ 17 

Austenite Transformation Temperatures: Bs and Bf ............................................................................................... 27 

Austenite Transformation Temperatures: Ms and Mf .............................................................................................. 34 

Critical Diameter – Austenite Hardenability ........................................................................................................... 48 

Density of Bulk Steel at Ambient Temperature ...................................................................................................... 49 

Density of Bulk Steel at High Temperature ............................................................................................................ 51 

Density of Liquid Steel .......................................................................................................................................... 54 

Density of Microstructural Components at Ambient Temperature .......................................................................... 55 

Density of Microstructural Components at High Temperature ............................................................................... 59 

Dimensional Changes during Austenite Transformation ........................................................................................ 62 

Equivalent Carbon – H.A.Z. Hardenability ............................................................................................................. 65 

Equivalent Carbon – Hydrogen Assisted Cold Cracking .......................................................................................... 69 

Equivalent Carbon – Peritectic Point ...................................................................................................................... 74 

Fe-C Equilibrium Diagram .................................................................................................................................... 76 

Fe-C Equilibrium Equations in the Solidification and Eutectoid Range .................................................................. 78 

Ferrite Solubility Products ..................................................................................................................................... 80 

Hardness After Austenite Cooling .......................................................................................................................... 82 

Hardness After Tempering ..................................................................................................................................... 87 

Hardness After Welding ......................................................................................................................................... 88 

Hardness-Tensile Properties Equivalence .............................................................................................................. 90



iii 

Hot Strength of Steel ............................................................................................................................................. 92 

Jominy Curves .................................................................................................................................................... 104 

Lattice Parameters of Phases ............................................................................................................................... 106 

Liquid Steel Solubility Products ........................................................................................................................... 108 

Liquidus Temperature of Steels ........................................................................................................................... 109 

Poisson Ratio ...................................................................................................................................................... 110 

Precipitate Isothermal Solubilization Kinetics ...................................................................................................... 111 

Solidus Temperature of Steels ............................................................................................................................. 115 

Relationships Between Chemical Composition x Process x Microstructure x Properties ........................................ 116 

Schaeffler Diagram .............................................................................................................................................. 133 

Shear Modulus of Steel and its Phases ................................................................................................................ 134 

Sheet and Plate Cutting Force and Work ............................................................................................................. 136 

Specimen Orientation for Mechanical Testing ...................................................................................................... 139 

Thermal Properties of Steel.................................................................................................................................. 141 

Thermal Properties of Steel Scale ........................................................................................................................ 149 

Thermomechanical Processing of Steel ................................................................................................................ 150 

Time-Temperature Equivalency Parameters for Heat Treating .............................................................................. 152 

Welding Effects ................................................................................................................................................... 156 

Welding Pool Phenomena .................................................................................................................................... 157 

Young Modulus ................................................................................................................................................... 158 

Appendixes 

Useful Data and Constants ................................................................................................................................. 160 

Unit Conversions ................................................................................................................................................ 167 

General Statistical Formulas ............................................................................................................................... 169


- Austenite Formation Temperatures 

. Andrews 


1 

Ae 723 16.9 Ni 29.1 Si 6.38W 

10.7 Mn 16.9 Cr 290 As 

1 

 

Ae 

3 

910 203 

400 Ti 

C 

44.7 

Si 

15.2 

Ni 

31.5 Mo 104 V 

13,1 W 30.0 

Mn 11.0 Cr 

20.0 Cu 

700 

P 400 

Al 

120 

As 

 

Notation: 

Ae1: Lower Equilibrium Temperature Between Ferrite and Austenite [°C] 

Ae3: Upper Equilibrium Temperature Between Ferrite and Austenite [°C] 

Alloy Content: [weight %] 

Observations: 

- Both formulas are valid for low alloy steels with less than 0.6%C. 

Source: ANDREWS, K.W. Empirical Formulae for the Calculation of Some Transformation Temperatures. Journal of the 

Iron and Steel Institute, 203, Part 7, July 1965, 721-727. 

. Eldis 

Ae 712 17.8 Mn 19.1 Ni 20.1 Si 11.9 Cr 9. 8 Mo 

1 

 

Ae 871 254.4 C 14.2 Ni 51. 7 Si 

3 

 

Notation: 

Ae1: Lower Equilibrium Temperature Between Ferrite and Austenite [°C] 

Ae3: Upper Equilibrium Temperature Between Ferrite and Austenite [°C]




2 

Observations: 

- Both formulas were proposed by ELDIS for low alloy steels with less than 0.6%C. 

Source: BARRALIS, J. & MAEDER, G. Métallurgie Tome I: Métallurgie Physique. Collection Scientifique ENSAM, 

1982, 270 p. 

. Grange 

Ae1 1333 25 Mn 40 Si 42 Cr 26 Ni 

Ae3 1570 323 C 25 Mn 80 Si 3 Cr 32 Ni 

Notation: 

Ae1: Lower Equilibrium Temperature Between Ferrite and Austenite [°F] 

Ae3: Upper Equilibrium Temperature Between Ferrite and Austenite [°F] 


Source: GRANGE, R.A. Estimating Critical Ranges in Heat Treatment of Steels. Metal Progress, 70:4, April 1961, 73-75. 

. Hougardy 

Ac 739 22 C 7 Mn 2 Si 14 Cr 13 Mo 13 Ni 

1 

 

Ac 902 255 C 11 

Mn 19 Si 5 Cr 13 Mo 20 Ni 55V 

3 

 

Notation: 

Ac1: Lower Temperature of the Ferrite-Austenite Field During Heating [°C]



Ac3: Upper Temperature of the Ferrite-Austenite Field During Heating [°C] 


3 

Source: HOUGARDY, H.P. Werkstoffkunde Stahl Band 1: Grundlagen. Verlag Stahleisen GmbH, Düsseldorf, 1984, p. 

229. 

. Kariya 

Ac 754.83 32.25 C 17.76 

Mn 23.32 Si 17.3 

Cr 4.51 Mo 15. 62 V 

1 

 

Notation: 

Ac1: Lower Temperature of the Ferrite-Austenite Field During Heating [°C] 


Source: KARIYA, N. High Carbon Hot-Rolled Steel Sheet and Method for Production Thereof. European patent 

Application EP 2.103.697.A1, 23.09.2009, 15 p. 

. Kasatkin 

Ac 

1 

723 7.08 Mn 37.7 Si 18.1 Cr 44.2 Mo 8.95 Ni 50.1V 

21.7 Al 3.18 W 

297 S 830 N 11.5 

C Si 14.0 Mn Si 

3.10 Si Cr 57.9 C Mo 15.5 Mn Mo 5.28 C Ni 6.0 Mn Ni 6.77 Si Ni 0.80 Cr Ni 27.4 C V 30.8 Mo V 0.84 Cr 

2 

2 

2 

3.46 Mo 0.46 Ni 28 V 

2 

 

Observations: 

- Multiple Correlation Coefficient r = 0.96 

- Residual Mean-Square Deviation √do = 10.8°C



4 

Ac 

3 

912 370 C 27.4 Mn 27.3 Si 6.35 Cr 32.7 Ni 95.2 V 

900 B 16.2 C Mn 32.3 C Si 15.4 C Cr 48.0 C Ni 4.32 Si Cr 17.3 Si Mo 18.6 Si Ni 4.80 Mn Ni 40.5 Mo V 

174 C 

2 

2.46 Mn 

2 

6.86 Si 

2 

0.322 Cr 

2 

9.90 Mo 

2 

1.24 Ni 

190 Ti 72.0 Al 64.5 Nb 5.57 W 

2 

60.2 V 

2 

332 S 276 P 485 N 

 

Observations: 


- Residual Mean-Square Deviation √do = 14.5°C 

T 

188 370 C 7.93 Mn 26.8 Cr 33.0 Mo 23.5 Ni 52.5 V 

20.7 C Cr 6.26 Si Cr 64.2 C Mo 55.2 C Ni 10.8 Mn Ni 1.33 Cr 

194 Ti 47.8 Al 87.4 Nb 3.82 W 

2 

8.83 Mo 

2 

1.91 Ni 

2 

266 P 53.0 C Si 

37.8 V 

2 

Notation: 



ΔT: Intercritical Temperature Range [°C] 


Observations: 


- Residual Mean-Square Deviation √do = 16.8°C 

- These equations (Ac1, Ac3, ΔT) are valid within these composition limits: C ≤ 0.83%, Mn ≤ 2.0%, Si ≤ 1.0%, Cr ≤ 

2.0%, Mo ≤ 1.0%, Ni ≤ 3.0%, V ≤ 0.5%, W ≤ 1.0%, Ti ≤ 0.15%, Al ≤ 0.2%, Cu ≤ 1.0%, Nb ≤ 0.20%, P ≤ 0.040%, S 

≤ 0.040%, N ≤ 0.025%, B ≤ 0.010%. 

Source: KASATKIN, O.G. et al. Calculation Models for Determining the Critical Points of Steel. Metal Science and Heat 

Treatment, 26:1-2, January-February 1984, 27-31. 

. Lee



5 

A cm 

224.4 992.4 C 465.1 C 

7 

2 

46.7 Cr 

19.0 C Cr 6.1 Cr 

2 

7.6 Mn 10.0 Mo 6.8 Cr Mo 6.9 

Ni 3.7 C 

Ni 2.7 Cr 

Ni 

0.8 Ni 

2 

16. 

Si 

Notation: 

Acm: Upper Equilibrium Temperature Between Ferrite and Cementite [°C] 


Observations: 

- Equation valid for the following alloy range: 0.2% C 0.7%; Mn 1.5%; Si 0.3%; Ni 2.8%; Cr 1.5%; Mo 

0.6%. 

- Regression coefficient r² = 0.998837; precision interval: 3°C. 

Source: LEE, S.J. & LEE, Y.K. Thermodynamic Formula for the Acm Temperature of Low Alloy Steels. ISIJ 

International, 47:5, May 2007, 769-774. 

. Park 

Ac 955 350 C 25 Mn 51 Si 106 Nb 100 Ti 68 Al 11Cr 

33 Ni 16 

Cu 67 Mo 

3 

 

Notation: 



Observations: 

- Formula specifically developed for TRIP steels. 

Source: PARK, S.H. et al. Development of Ductile Ultra-High Strength Hot Rolled Steels. Posco Technical Report, 1996, 

50-128.



6 

. Roberts 

Ae 3 

910 25 Mn 11 Cr 20 Cu 60 Si 700 P 250 Al F n 

Notation: 

Ae3: Upper Equilibrium Temperature Between Ferrite and Austenite [°C] 


Fn: value defined according to the table below: 

C Fn 

0.05 24 

0.10 48 

0.15 64 

0.20 80 

0.25 93 

0.30 106 

0.35 117 

0.40 128 

Source: ROBERTS, W.L.: Flat Processing of Steel; Marcel Dekker Inc., New York, 1988. 

. Trzaska 

Ac 739 22.8 C 6.8 Mn 18.2 Si 11.7 Cr 15 

Ni 6.4 Mo 5V 

28 Cu 

1 

 

Ac 937.3 224.5 C 17 

Mn 34 Si 14 

Ni 21.6 Mo 41.8 V 20 Cu 

3 

 

Notation:



Ac3: Lower Temperature of the Ferrite-Austenite Field During Heating [°C] 



7 

Source: TRZASKA, J. et al. Modelling of CCT Diagrams for Engineering and Constructional Steels. Journal of Materials 

Processing Technology, 192-193, 2007, 504-510.


- Austenite Grain Size 

. Lee & Lee 


8 

d 

89098 3581 C 1211 Ni 1443 Cr 4031 Mo 

76671 exp 

t 

R T 

 

 

 

 

 

0.211 

Notation: 

d: Austenite Grain Size [μm] 

R: Universal Gas Constant, 8.314 J/mol.K 

T: Austenitizing Temperature [K] 

t: Austenitizing Time [s] 

Observations: 

- Equation valid under the following alloy range: 0.15% C 0.41%; 0.73% Mn 0.85; 0.20% Si 0.25%; Ni 

1.80%; Cr 1.45%; Mo 0.45. 

Source: LEE, S.J. et al.: Prediction of Austenite Grain Growth During Austenitization of Low Alloy Steels. Materials and 

Design, 29, 2008, 1840-1844.


- Austenite No-Recrystallization Temperature 


9 

. Boratto et al. 

Tnr 887 464 C ( 6445 Nb 644 Nb) ( 732 V 230 V ) 890 Ti 363 Al 357 Si 

Notation: 

Tnr: Temperature of No-Recrystallization [°C]. A temperature below which austenite recrystallization stops 

completely. 


Observations: 

- Equation valid under the following alloy range: 0.04% C 0.17%; 0.41% Mn 1.90; 0.15% Si 0.50%; 

0.002% Al 0.650; Nb 0.060%; V 0.120%; Ti 0.110%; Cr 0.67%; Ni 0.45. 

Source: BORATTO, F. et al.: Effect of Chemical Composition on Critical Temperatures of Microalloyed Steels. In: 

THERMEC ‘88. Proceedings. Iron and Steel Institute of Japan, Tokyo, 1988, p. 383-390. 

T pr 

. Bai et al. 

12 

174 logNb 

C 

N 1444 

14 

 

 

T 

nr 

T 

pr 

75 

Notation: 

Tpr: Temperature of Partial Recrystallization [°C]. Temperature below which full recrystallization between 

deformation steps is no longer possible.



Tnr: Temperature of No-Recrystallization [°C]. Temperature below which austenite recrystallization stops 

completely. 


Observations: 

- In the specific case of steels including Ti, N content in the first equation is the effective N obtained by 

subtracting the N combined with Ti from the total N in the steel. 

Source: BAI, D. et al.: Development of Discrete X80 Line Pipe Plate at SSAB Americas. In: International Symposium on 

the Recent Developments in Plate Steels. Proceedings. Association for Iron and Steel Technology, 

Warrendale, 2011, p. 13-22. 

10


- Austenite Solubility Products 

. General 


11 

m n 

( aM 

) ( a 

X 

) 

log 

10 

ks 

log 

10 

log 

10 

a 

M 

m 

X 

n 

m 

[ M ] [ X ] 

[ MX ] 

n 

Q 

 

 

RT 

 

2.303 

C 

 

2.303 

A 

B 

T 

Notation: 

MmXn: Precipitate Considered for Calculation 

Ai: Activity 

M, X: Alloy Contents [weight %] 

T: Temperature [K] 

C: Constant 

A, B: Constants of the Solubility Product, given in the table below: 

Precipitate A B Source 

7060 1.55 Narita 

AlN 6770 1.03 Irvine 

7750 1.80 Ashby 

BN 13970 5.24 Fountain 

Cr23C6 7375 5.36 Ashby 

Mo2C 7375 5.00 Ashby 

NbC 

7900 3.42 Narita 

7290 3.04 Meyer 

9290 4.37 Johansen



5860 1.54 Meyer 

NbCN 

6770 2.26 Ashby 

NbC0.87 

NbN 

7520 3.11 Ashby 

7700 3.18 Mori 

8500 2.80 Narita 

10230 4.04 Smith 

10800 3.70 Ashby 

TiC 7000 2.75 Irvine 

TiN 

15020 3.82 Narita 

8000 0.32 Ashby 

VC 9500 6.72 Narita 

V4C3 8000 5.36 Ashby 

VN 

8330 3.46 Irvine 

7070 2.27 Irvine 

12 

Observations: 

- aAmBn is equal to one if the precipitate is pure. 

- aAmBn 1 if there is co-precipitation with another element. 

- The product [M] m [X] n (that is, ks) defines the graphical boundary of solubilization in a graph [M] x [X]. 

Sources: 

- ASHBY, M.F. & EASTERLING, K.E. A First Report on Diagrams for Grain Growth in Welds. Acta Metallurgica, 

30, 1982, 1969-1978.



13 

- FOUNTAIN, R. & CHIPMAN, J. Solubility and Precipitation of Vanadium Nitride in Alpha and Gamma Iron. 

Transactions of the AIME, Dec. 1958, 737-739 

- GLADMAN, T. The Physical Metallurgy of Microalloyed Steels. The Institute of Materials, London, 1997, 363 p. 

- IRVINE, K.J. et al. Grain-Refined C-Mn Steels. Journal of the Iron and Steel Institute, 205:2, Feb. 1967, 

161-182. 

- JOHANSEN, T.G. et al. The Solubility of Niobium (Columbium) Carbide in Gamma Iron. Transactions of the 

Metallurgical Society of AIME, 239:10, October 1967, 1651-1654. 

- NARITA, K.et al. Physical Chemistry of Ti/Zr, V/Nb/Ta and Rare Elements in Steel. Transactions of the ISIJ, 

15:5, May 1975, 145-151. 

- SMITH, R.P. Transactions of the AIME, 224, 1962, 190. 

- Values compiled by Rajindra Clement Ratnapuli and Fúlvio Siciliano from assorted references when not 

specified above. 

. Irvine 

12 

log[ Nb] C N . 

 

 

 

2 26 

14 

 

6770 

T 

Notation: 


Alloy Content: [weight %]



Source: IRVINE, K.J. et al. Grain-Refined C-Mn Steels. Journal of the Iron and Steel Institute, 205:2, Feb. 1967, 

161-182. 

14 

. Mori 

log [ Nb] 

[ N] 

0.65 

[ C] 

0.24 

10400 

4.09 

T 

Notation: 



Source: MORI, T. et al.: Thermodynamic Behaviors of Niobium-Carbide-Nitride and Sulfide in Steel. Tetsu-to-Hagané, 

51:11, 1965, 2031-2011. 

. Siciliano 

. . 

12 

[ Mn] [ Si] 

log[ Nb] C N . 

 

 

 

838 0 246 1730 0 594 6440 

2 26 

14 

T 

Notation: 



Source: SICILIANO JR., F..: Mathematical Modeling of the Hot Strip Rolling of Nb Microalloyed Steels. Ph.D. Thesis, 

McGill University, February 1999, 165 p. 

. Dong



15 

 

log [ Nb] 

 

C 

 

 

12 

14 

 

N 

 

 

1371 [ Mn] 

923 [ Si] 

8049 

3.14 0.35 [ Si] 

0.91[ Mn] 

 

T 

Notation: 



Source: DONG, J.X. et al.: Effect of Silicon on the Kinetics of Nb(C,N) Precipitation during the Hot Working of Nb-bearing 

Steels. ISIJ International, 40:6, June 2000, 613-618. 

. Irvine 

log [ V ] 

 

N 

 

3.46 0.12 Mn 

8330 

T 

Notation: 



Source: ASHBY, M.F. & EASTERLING, K.E. A First Report on Diagrams for Grain Growth in Welds. Acta Metallurgica, 

30, 1982, 1969-1978.


- Austenite Solubilization Temperature 


16 

0 

T ( C) 

 

d 

B log 

10 

A 

( a 

A 

) 

m 

( a 

B 

) 

n 

273 

Notation: 

AmBn: Precipitate considered for calculation 

Td: Solubilization temperature [°C] 

ax: Alloy content [weight %] 

A, B: Constants of the solubility product, given in the table at the topic Austenite Solubilization Products.



17 

- Austenite Transformation Temperatures: Ar3 and Ar1 

. Blás 

Ar 903 328 C 102 

Mn 116 Nb 0. 909 v 

3 

 

Notation: 

Ar3: Ferrite Start Temperature [°C] 

Alloy Amount: [weight %] 

v: Cooling Rate [°C/s] 

Observations: 

- This formula was determined using temperature data got from samples cooled directly from hot rolling 

experiments. Thus it includes the effects of hot forming over austenite decomposition. 

- Useful range: 0.024-0.068% C, 0.27-0.39% Mn, 0.004-0.054% Al, 0.000-0.094% Nb, 0.0019-0.0072% N, 1.0- 

35°C/s 

- r = 0.934; Standard Error of Deviation = 5°C 

Source: BLÁS, J.G. et al.: Influência da Composição Química e da Velocidade de Resfriamento sobre o Ponto Ar3 em Aços 

de Baixo C Microligados ao Nb. In: 44° Congresso Anual da Associação Brasileira de Metais, ABM, São 

Paulo, vol. 1, Outubro 1989, p 11-29. 

. Choquet 

Ar3 902 527 C 62 Mn 60 Si 

Notation: 


Alloy Amount: [weight %]



Observations: 

- This formula was determined using data got from samples cooled directly from hot rolling experiments. Thus it 

includes the effects of hot forming over austenite decomposition. 

Source: CHOQUET, P. et al.: Mathematical Model for Predictions of Austenite and Ferrite Microstructures in Hot Rolling 

Processes. IRSID Report, St. Germain-en-Laye, 1985. 7 p. 

18 

. Kariya 

Ar 910 203 C 30 Mn 44.7 Si 11Cr 

31.5 Mo 15. 2 Ni 

3 

 

Notation: 



Source: KARIYA, N. High Carbon Hot-Rolled Steel Sheet and Method for Production Thereof. European Patent 

Application EP 2.103.697.A1, 23.09.2009, 15 p. 

. Mintz 

First Proposal: 

2 

Ar 833.6 190.6 C 67.4 Mn 1522 S 2296 Nti 1532 

Nb 7.91 d 

1/ 0. 117 CR 

3 

 

N 

ti 

 

N 

t 

 

Ti 

3.5 

Notation:




Nt: Total Nitrogen Content [weight %] 

d = Austenite Grain Diameter [mm] 

CR = Cooling Rate [°C/min] 


19 

Observations: 

- This formula was determined using temperature data got from non-hot deformed samples. 

- Useful range: 0.04-0.75% C, 0.30-1.60% Mn, 0.02-0.49% Si, 0.014-0.085% Al, 0.00-0.31% Nb, 0.004-0.008% 

N, 0.003-0.032% S, d: 0.070-0.950 mm, CR: 25-200°C/min 

- r = 0.949; Standard Error of Deviation = 15.9°C. The coefficients for Nti (82.2%), d (87.4%) and CR (92.8%) were 

not significant for a 95% minimum confidence level. 

- The unexpected positive effect of S can be associated to the enhanced nucleation of ferrite at sulphides. 

Second Proposal: 

Ar 868 181 C 75.8 Mn 1086 S 3799 Nti 1767 

Nb 0. 0933 CR 

3 

 

N 

ti 

 

N 

t 

 

Ti 

3.5 

Notation: 



Nt: Total Nitrogen Content [weight %] 

CR = Cooling Rate [°C/min] 

Observations: 

- This formula was determined using temperature data got from non-hot deformed samples and includes TRIP 

steels.



- Useful range: 0.04-0.75% C, 0.31-2.52% Mn, 0.01-1.22% Si, 0.00-1.55% Al, 0.000-0.042% Nb, 0.0012-0.014% 

N, 0.002-0.110% P, 0.001-0.032% S, d: 0.1-1.0 mm, CR: 10-200°C/min 

- r = 0.939; Standard Error of Deviation = 18.1°C. 

Source: MINTZ, B. et al. Regression Equation for Ar3 Temperature for Coarse Grained as Cast Steels. Ironmaking and 

Steelmaking, 38:3, March 2011, 197-203. 

20 

. Ouchi 

Ar3 910 310 C 80 Mn 20 Cu 15 Cr 55 Ni 80 Mo 0, 35 ( h 8) 

Notation: 



h: Plate Thickness [mm] 

Observations: 

- This formula was determined using temperature data got from samples of Nb microalloyed steels cooled directly 

from hot rolling experiments. Thus it includes the effects of hot forming over austenite decomposition. 

Source: OUCHI, C. et al. The Effect of Hot Rolling Condition and Chemical Composition on the Onset Temperature of 

Gamma-Alpha Transformation After Hot Rolling. Transactions of the ISIJ, March 1982, 214-222. 

. Pickering 

Ar 910 230 C 21 Mn 15 Ni 32 Mo 45 Si 13W 

104 V 

3 

 

Notation: 

Ar3: Ferrite Start Temperature [°C]



Observations: 

- Applicable to Plain C Steels. 


21 

Source: PICKERING, F.B.: Steels: Metallurgical Principles. In: Encyclopedia of Materials Science and Engineering, 

vol. 6, The MIT Press, Cambridge, 1986. 

. Santos 

Ar3 874.44 512.0465 C 40.915 Mn 23.075 Si 567.126 C² 

199.551 

C Mn 265.797 C Si 4.148 A 1.03 

CR 11.334 

ln CR 

 

 

A 

0.002 

2 ln 

 

ln 

d 

2 

 

 

Notation: 



A: Austenite Grain Size [ASTM units] 

CR: Continuous Cooling Rate [°C/s] 

dγ: Austenite Grain Size [μm] 

Observations: 

- Equation fitted with data from 94 points; r² = 0.9888 

- Applicable to plain C Steels. 

Source: SANTOS, A.A.: Previsão das Temperaturas Críticas de Decomposição da Austenita em Ferrita e Perlita Durante 

Resfriamento Contínuo. In: 41° Seminário de Laminação – Processos e Produtos Laminados e Revestidos, 

Associação Brasileira de Metalurgia e Materiais, Joinville, 2004, 10 p.



22 

. Sekine 

Ar 868 396 C 68.1 Mn 24.6 Si 36.1 Ni 24.8 Cr 20. 7 Cu 

3 

 

Notation: 



Observations: 



Source: TAMURA, I. et al.: Thermomechanical Processing of High-Strength Low-Alloy Steels. Butterworths, 

London, 1988, 248 p. 

. Shiga 

Ar 910 273 C 74 Mn 56 Ni 16 

Cr 9 Mo 5 Cu 

3 

 

Notation: 



Observations: 


experiments. Thus it includes the effects of hot forming over austenite decomposition.



Source: SHIGA, C. et al.: Development of Large Diameter High Strength Line Pipes for Low Temperature Use. Kawasaki 

Steel Technical Report, December 1981, 97-109. 

23 

. Proprietary #1 

Ar 879.4 516.1 C 65.7 Mn 38.0 Si 274. 7 P 

3 

 

Notation: 




Ar 901 325 C 92 Mn 33 Si 287 P 40 Al 20 Cr 

3 

 

Notation: 



Observations: 

- The previous conditioning of the steel samples that supplied data for the deduction of this formula is unknown. 

Source: Unknown. 


Ar 706.4 350.4 C 118. 2 Mn 

1


Notation: 

Ar1: Ferrite End Temperature [°C] 



24 

Observations: 

- The previous conditioning of the steel samples that supplied data for the deduction of this formula is unknown. 

- Samples cooled at 20°C/s. 

Source: Unknown. 

. Yuan 

Non-Deformed Austenite: 

D 

370 exp 

 

0.1 

2 

Ar3 325 CR 5649 Nb 78194 Nb 1019 

6.7 

 

Notation: 



Nb: Niobium content [weight %] 

Observations: 

- This formula was determined using temperature data got from non-hot deformed samples. 

- Base steel: 0.11% C, 1.20% Mn, 0.20% Si, 0.005% N. Useful range: 0.000-0.038% Nb, CR: 0.5-30°C/s. 

Deformed Austenite:



25 

D 

1 

370 exp 

 

0.1 

2 

 

Ar 

3 

198 CR 6646 Nb 2327 Nb 66 

830 

6.7 

 

 

 

t 

0.05 

 

 

Notation: 



Nb: Niobium content [weight %] 

t0.05: Nb(CN) Precipitation Start Time [s] 

Δ: Residual Strain in Austenite 

Observations: 



- Base steel: 0.11% C, 1.20% Mn, 0.20% Si, 0.005% N. Useful range: 0.000-0.038% Nb, CR: 0.5-30°C/s. See 

reference for details about the calculation of t0.05 and Δ, which requires external models. 

Source: YUAN, X.Q. et al.: The Onset Temperatures of to -Phase Transformation in Hot Deformed and Non-Deformed 

Nb Micro-Alloyed Steels. ISIJ International, 46:4, Apr. 2006, 579-585. 

. Zhao 

2 

M a 

820 603.76 C 247.13 C 66.24 Mn 55.72 Ni 

2 

3 

0.165 Co 0.00255 Co 28. 01 Ru 

Notation: 

Ma: Massive Ferrite Start Temperature [°C] 


3.97 Ni 

2 

0.151 Ni 

3 

31.10 Cr 2.348 Cr 

2 

24.29 Mo 31.88 Cu 0.196 Co



Source: ZHAO, J.: Continuous Cooling Transformations in Steels. Materials Science and Technology, 8:11, November 

1992, 997-1002. 

26



27 

- Austenite Transformation Temperatures: Bs and Bf 

. Bodnar 

B s 

844 597 C 63 Mn 16 

Ni 78 Cr 

Notation: 

Bs: Bainite Start Temperature [°F] 


Source: ZHAO, Z. et al. A New Empirical Formula for the Bainite Upper Temperature Limit of Steel. Journal of Materials 

Science, 36, 2001, 5045-5056. 

. Kirkaldy 

B s 

656 57.7 C 35 Mn 75 Si 15.3 

Ni 34 Cr 41. 2 Mo 

Notation: 

Bs: Bainite Start Temperature [°C] 


Observations: 

- This is a modification of Steven & Haynes’ formula using isothermal transformation diagrams determined for 

low and high alloy steels produced by U.S. Steel. 

Source: KIRKALDY, J.S. et al. Prediction of Microstructure and Hardenability in Low Alloy Steels. In: Phase 

Transformations in Ferrous Alloys, AIME, Philadelphia, 1983, 125-148. 

. Kunitake & Okada



28 

B s 

732 202 C 85 Mn 216 Si 37 Ni 47 Cr 39 Mo 

Notation: 



Observations: 

- These authors concluded that the measured Bs temperature for steels with a greater ni or Cr content is much 

higher than that predicted by Steven & Haynes. 

- Reliable chemical composition range: 0.11-0.56% C, 0.34-1.49% Mn, 0.14-0.40% Si, 0.07-1.99% Mo, 0.14- 

4.80% Cr, 0.23-4.33% Ni. 

Source: KUNITAKE, T. & OKADA, Y. The Estimation of Bainite Transformation Temperatures in Steels by Empirical 

Formulas. Tetsu-to-Hagané, 84:2, February 1998, 137-141. 

. Lee 

2 

B s 

984.4 361.9 C 261.9 C 28.3 Mn 43. 7 Si 

Notation: 

Bs: Bainite Start Temperature [K] 


Observations: 

- Formula specifically developed for TRIP steels. 

Source: LEE, J.K. et al. Prediction of Tensile Deformation Behaviour of Formable Hot Rolled Steels. Posco Technical 

Research Laboratories Report, Pohang, 1999.



29 

. Lee 

B s 

 

2 

2 

745 110 

C 59 Mn 39 Ni 68 Cr 106 

Mo 17 Mn Ni 6 Cr 29 Mo 

Notation: 



Observations: 

- This formula is based in the equations of Steven & Haynes, Kirkaldy and Kunitake & Okada, as well data from 

many time-temperature diagrams for several steels, including low alloy steels and steels with Ni and Cr 

contents up to 4.5%, which were published in the Atlas of Time-Temperature Diagrams for Iron and Steels 

by G.F. Vander Voort through ASM International, Metals Park, in 1991. 


4.48% Cr, 0.00-4.34% Ni. 

Source: LEE, Y.K. et al. Empirical Formula of Isothermal Bainite Start Temperature of Steels. Journal of Materials 

Science Letters, 21:16, 2002, 1253-122. 

. Li 

B s 

637 58 C 35 Mn 15 

Ni 34 Cr 41 Mo 

Notation: 



Observations:



- This formula is a modification of Kirkaldy’s Bs equation. 

- It assumes that Si amount is constant and equal to 0.25%, as most low alloy steels exhibit a content of this 

alloy element in this order of magnitude. 


0.98% Cr, 0.02-3.04% Ni, 0.05-0.11% Cu. 

Source: LI, M. et al. A Computational Model for the Prediction of Steel Hardenability. Metallurgical and Materials 

Transactions B, 29:6, June 1998, 661-672. 

30 

. Steven & Haynes 

B s 

830 270 C 90 Mn 37 Ni 70 Cr 83 Mo 

B 

50 

B s 

50 

B 

100 

B s 

120 

Notation: 



Bx: Temperature Required for the Formation of x% of Bainite [°C] 

Observations: 

- Reliable chemical composition range: 0.10-0.55% C, 0.2-1.7% Mn, 0.0-1.0% Mo, 0.0-3.5% Cr, 0.0-5.0% Ni. 

Source: STEVEN, W. & HAYNES, A.G. The Temperature of Formation of Martensite and Bainite in Low Alloy Steels. 

Journal of the Iron and Steel Institute, 183, 1956, 349-359. 

. Suehiro



31 

Bs 718 425 C 42. 

5 Mn 

Notation: 



Source: SUEHIRO, M. et al. A Kinetic Model for Phase Transformation of Low C Steels during Continuous Cooling. Tetsuto-Hagané, 

73:8, June 1987, 1026-1033. 

. Takada 

B s 

1336 1446 

C 62.3 Mn 36.5 Si 47.8 Cr 160 

V 77. 5 Mo 

Notation: 

Bs: Bainite Start Temperature [K] 


Observations: 

- Formula developed specifically for forging steels. 

- Reliable chemical composition range: 0.11-0.40% C, 0.50-2.52% Mn, 0.31-1.26% Si, 0.20-1.96% Cr. 

Source: TAKADA, H. Alloy Designing of High Strength Bainite Steels for Hot Forging. Tetsu-to-Hagané, 88:9, September 

2002, 534-538. 

. van Bohemen 

B s 

839 86 Mn 23 Si 67 Cr 33 Ni 75 Mo 270 1 exp( 1.33 

C)


Notation: 




32 

Observation: 

- Factors multiplying substitutional elements are less than 10% different from the factors found by Steven and 

Haynes. 

- Standard error of estimate σ = 13°C; correlation coefficient R² = 0.97. 

Source: van Bohemen, S.M.C. Bainite and Martensite Start Temperature Calculated with Exponential Carbon 

Dependence. Materials Science and Technology, 28:4, April 2012, 487-495. 

. Wang & Cao 

B 36. 6 

s 

Mn eq 

Mn eq 

Mn 3.43 

Mo 0. 56 Ni 

Notation: 



Source: WANG, S. & KAO, P. The Effect of Alloying Elements on the Structure and Mechanical Properties of ULCB Steels. 

J. of Materials Science, 28, 1993, 5169-75. 

. Zhao



33 

2 

B s 

720 585.63 C 126.60 C 91.68 Mn 7.82 Mn² 

0.3378 Mn³ 

66.34 Ni 6.06 Ni² 

0.232 Ni³ 

31.66 Cr 2.17 Cr² 

 

42.37 Mo 9.16 Co 0.1255 Co² 

0.000284 Co³ 

36.02 Cu 46. 15 Ru 

Source: ZHAO, J.: Continuous Cooling Transformations in Steels. Materials Science and Technology, 8:11, Nov. 1992, 

997-1002. 

B s 

630 45 Mn 40 V 35 Si 30 Cr 25 Mo 20 Ni 15W 

Source: ZHAO, Z. et al. A New Empirical Formula for the Bainite Upper Temperature Limit of Steel. Journal of Materials 

Science, 36, 2001, 5045-5056. 

Notation: 





34 

- Austenite Transformation Temperatures: MS and Mf 

. Andrews 

M s 

539 423 C 30.4 Mn 17.7 Ni 12.1 Cr 11.0 Si 7. 0 Mo 

2 

M s 

512 453 C 16.9 Ni 9.5 Mo 217 C 71.5 C Mn 15 Cr 67. 6 C Cr 

Notation: 

Ms: Martensite Start Temperature [°C] 


Observations: 

- Formula valid for low alloy steels with less than 0.6%C, 4.9% Mn, 5.0% Cr, 5.0% Ni and 5.4% Mo. 

Source: ANDREWS, K.W. Empirical Formulae for the Calculation of Some Transformation Temperatures. Journal of the 

Iron and Steel Institute, 203, Part 7, July 1965, 721-727. 

. Capdevila 

M s 

764.2 

302.6 C 30.6 Mn 16.6 Ni 8.9 Cr 2.4 Mo 11.3 

Cu 8.58 Co 7.4 W 14. 

5 Si 

Notation: 

Ms: Martensite Start Temperature [K] 


Observations:



- Equation valid for steels with chemical composition between the following limits: 0.001 ≤ C ≤ 1.65, Mn ≤ 3.76, 

Si ≤ 3.40, Cr ≤ 17.9, Ni ≤ 27.2, Mo ≤ 5.10, V ≤ 4.55, Co ≤ 30.0, Al ≤ 1.10, W ≤ 12.9, Cu ≤ 0.98, Nb ≤ 0.23, B ≤ 

0,0010, 0.0001 ≤ N ≤ 0.060. 

Source: CAPDEVILA, C. et al. Determination of Ms Temperature in Steels: A Bayesian Neural Network Model. ISIJ 

International, 42:8, August 2002, 894-902. 

35 

M s 

. Carapella 

496.1 (1 0.344 C) 

(1 0.051 Mn) 

(1 0.018 Si) 

(1 0.025 Ni) 

(1 0.039 Cr) 

(1 0.016 Mo) 

(1 0.010 W) 

(1 0.067 Co) 

Notation: 

Ms: Start Temperature of the Martensitic Transformation [°C] 


Source: CARAPELLA, L.A. Computing A 11 or Ms (Transformation Temperature on Quenching), Metal Progress, 46, 1944, 

108. 

. Eichelman & Hull 

M s 

8.9 

Ni) 

33.3 (1.33 Mn) 

27.8 (0.47 Si) 

1666.7 (0.068 C N 17. 8 

41.7 (14.6 Cr) 

5.6 

 

Notation: 



Observations: 

- Equation valid for 18-8 stainless steels.



Source: EICHELMAN, G.H. & HULL, F.C. The Effects of Composition on the Temperature of Spontaneous Transformation 

of Austenite to Martensite in 18-8 Stainless Steels. Transactions of the American Society for Metals, 45, 

1953, p. 77-104. 

36 

. Eldis 

M s 

531 391.2 C 43.3 Mn 21.8 Ni 16. 2 Cr 

Notation: 



Observations: 

- Equation valid for steels with chemical composition between the following limits: 0.10~0.80% C; 0.35~1.80% 

Mn; < 1.50% Si; < 0.90% Mo; < 1.50% Cr; < 4.50% Ni. 


1982, 270 p. 

. Finkler & Schirra 

C 

0.86 N 

0.15 ( Nb Zr) 

0.066 ( Ta Hf ) 33 

Mn 17 Cr 17 Ni 21 Mo 39 V W 

M s 

635 474 

11 

Notation: 



Observations: 

- Equation valid for high temperature martensitic steels with 8,0 to 14% Cr.



37 

Source: FINKLER, H. & SCHIRRA, M. Transformation Behavior of High Temperature Martensitic Steels with 8 to 14% 

Chromium. Steel Research, 67:8, August 1986, p. 328-336. 

. Grange & Stewart 

M s 

537.8 

361.1 C 38.9 ( Mn Cr) 

19.4 

Ni 27. 8 Mo 

Notation: 



Source: GRANGE, R.A. & STEWART, H.M. The Temperature Range of Martensite Formation. Transactions of the 

AIME, 167, 1946, 467-490. 

. Hougardy 

M 

s 

0.495 M 

sjh 

0.00095 M 

2 

sjh 

40 

Notation: 

Ms: Corrected Martensite Start Temperature [°C] 

Msjh: Martensite Temperature Start According to Jaffe & Hollomon [°C] 


Observation: 

- Correction of Jaffe & Hollomon equation considering several other similar equations already published.


 

V 1 

exp k ( M T) 

M 

s 

q 

 


38 

k 

 

3 

4 

6 

2 

8 

3 

11 

4 

0.36 

10 

0.10 10 

M 

s 

0.34 10 

M 

s 

0.32 10 

M 

s 

0.52 10 

M 

s 

q 

 

2 

4 

2 

8 

3 

2.08 

0.76 10 

M 

s 

0.16 10 

M 

s 

0.90 10 

M 

s 

Notation: 

VM: Volume Fraction of Martensite 

Ms: Temperature at Which 1% Martensite Forms [ºC] 

T: Temperature [ºC] 

Source: HOUGARDY, H.P. Description and Control of Transformations in Technical Applications. Steel: A Handbook for 

Materials Research and Engineering – Volume 1: Fundamentals, Springer-Verlag, Berlin, 1992, 167-200. 

. Jaffe & Hollomon 

M S 

550 350 C 40 Mn 35V 

20 Cr 17 

Ni 10 

Cu 10 

Mo 8W 

15 Co 30 Al 

Notation: 

Ms: Start Temperature of the Martensitic Transformation [°C] 


Observation: 

- Hougardy proposed a correction to this formula. 

Source: GRANGE, R.A. & STEWART, H.M. Hardenability and Quench Cracking. Transactions of the AIME, 167, 1946, 

617-646.



39 

. Krauss 

M s 

561 474 C 33 Mn 17 Cr 17 Ni 21 Mo 

Notation: 



Source: KRAUSS, G. Principles of Heat Treatment and Processing of Steels, ASM International, 1990, p. 43-87. 

. Kunitake 

M s 

560.5 

407.3 C 37.8 Mn 14.8 Cr 19.5 Ni 4.5 Mo 7.3 Si 20. 5 Cu 

Notation: 



Source: KUNITAKE, T. Prediction of Ac1, Ac3 and Ms Temperatures by Empirical Formulas. Heat Treating (Japan), 

41, 2001, p. 164-168. 

M 

. Lee & Van Tyne 

 

V 1 

exp K ( M T) 

LV 

s 

n LV 

K LV 

0.0231 

0.0105 C 0.0017 Ni 0.0074 Cr 0. 0193 Mo



40 

2 

n LV 

1.4304 1.1836 

C 0.7527 C 0.0258 Ni 0.0739 Cr 0. 3108 Mo 

Notation: 

VM: Volume Fraction of Martensite 

Ms: Martensite Start Temperature [K] 



Observation: 

- Start Temperature of Martensitic Transformation Calculated According to Capdevila. 

Source: LEE, S.J. & VAN TYNE, C.J. A Kinetics Model for Martensite Transformation in Plain C and Low-Alloyed Steels. 

Metallurgical and Materials Transactions A, 43A:12, February 2012, 423-427. 

. Li 

M s 

540 420 C 35 Mn 12 Cr 20 Ni 21 Mo 10.5 

Si 10.5 

W 20 Al 140 V 

Notation: 



Source: LI, C. et al. Computation of Ms Temperature in Carbon Equivalence Method. Journal of Liaoning 

Technology University, 17, 1998, 293-298. 

. Liu 

. C < 0.05%



41 

M s 

550 350 C 45 Mn 30 Cr 20 Ni 16 

Mo 5 Si 8W 

6 Co 15 Al 35 ( V Nb Zr Ti) 

. C > 0.05% 

M s 

525 350 ( C 0.05) 45 Mn 30 Cr 20 Ni 16 

Mo 5 Si 8W 

6 Co 15 Al 35 ( V Nb Zr Ti) 

Notation: 



Source: LIU, C. et al. A New Empirical Formula for Ms Temperature in Pure Iron and Ultra Low Carbon Alloy 

Steels. Journal of Materials Processing Technology, 113:1-3, 2001, 556-562. 

. Mahieu 

M s 

539 423 C 30.4 Mn 7.5 Si 30. 0 Al 

Notation: 



Observation: 

- Equation valid for TRIP steels with 0.91 Al 1.73%. Apparently it is a development of the Andrews formula. 

Source: MAHIEU, J. et al. Phase Transformation and Mechanical Properties of Si-free CMnAl Transformation-Induced 

Plasticity-Aided Steel. Metallurgical and Materials Transactions A, 33A:8, August 2002, 2573-2580.


. Mikula & Wojnar 


42 

M s 

635.02 

549.82 C 85.441 Mn 68.967 Si 18.07 

Cr 30,965 Ni 69.301 Mo 6.603 V 420.26 Nb 553.8 Ti 1746. 

5 B 

Notation: 



Source: LIS, A.K. & LIS, J. High Strength Hot Rolled and Aged Microalloyed 5% Ni Steel. Journal of Achievements in 

Materials and Manufacturing Engineering, 18:1-2, September-October 2006, 37-42. 

M s 

. Nehrenberg 

498.9 

300 C 33.3 Mn 22.2 Cr 16.7 

Ni 11.1( 

Si Mo) 

Notation: 



Source: NEHRENBERG, A.E. In: Contribution to Discussion on Grange and Stewart. Transactions of the AIME, 167, 

1946, 494-498. 

M s 

. Payson & Savage 

498.9 

316.7 C 33.3 Mn 27.8 Cr 16.7 

Ni 11.1( 

Si Mo W) 

Notation: 





43 

Source: PAYSON, P. & SAVAGE, C.H. Martensite Reactions in Alloy Steels. Transactions A.S.M., 33, 1944, 261-280. 

M s 

. Rowland & Lyle 

498.9 

333.3 C 33.3 Mn 27.8 Cr 16.7 

Ni 11.1( 

Si Mo W) 

Notation: 



Source: ROWLAND, E.S. & LYLE, S.R. The Application of Ms Points to Case Depth Measurement. Transactions A.S.M., 

37, 1946, 27-47. 

. Steven & Haynes 

M s 

561.1 

473.9 C 33 Mn 16.7 ( Cr Ni) 

21. 1 Mo 

M 

10 

M 

s 

18 

M 

50 

M 

s 

85 

M 

90 

M 

s 

185 

M 

100 

M 

s 

387 

Notation: 

Ms: Martensite Start Temperature [°F]



Mx: Temperature Required for the Formation of x% of Martensite [°F] 

44 

Source: STEVEN, W. & HAYNES, A.G. The Temperature of Formation of Martensite and Bainite in Low Alloy Steels. 

Journal of the Iron and Steel Institute, 183, 1956, 349-359. 

M s 

. Sverdlin-Ness 

520 320 C 50 Mn 30 Cr 20 ( Ni Mo) 

5 ( Cu Si) 

Notation: 



Source: SVERDLIN, A.V. & NESS, A.R. The Effects of Alloying Elements on the Heat Treatment of Steel. In: Steel Heat 

Treatment Handbook, Marcel Dekker, New York, 1997, p. 45-91. 

. Tamura 

M s 

550 361 C 39 Mn 20 Cr 17 

V 17 

Ni 10 

Cu 5 ( Mo W) 

15 

Co 30 Al 

Notation: 



Source: TAMURA, I. Steel Material Study on the Strength. Nikkan Kogyo Shinbun Ltd., Tokyo, 1970, 40. 

. van Bohemen

f 


 

1 exp 

( T T) 

m 

KM 

 


45 

T KM 

462 273 C 26 Mn 13 

Cr 16 

Ni 30 Mo 

 

m 

0.0224 

0.0107 C 0.0007Mn 

0.00012 Cr 0.00005 Ni 0. 0001 Mo 

* 

Notation: 

f: Volume Fraction of Martensite as a Function of Undercooling Below TKM Temperature (TKM – T) 

T: Temperature [°C] 

TKM: Theoretical Martensite Start Temperature [°C] 


Observation: 

- Formula based from Koistinen and Marburger equation. 

- αm: Standard error of estimate σ = 0.0014 K -1 ; correlation coefficient R² = 0.79. 

Source: VAN BOHEMEN, S.M.C and SIETSMA, J. Effect of Composition on Kinetics of Athermal Martensite Formation in 

Plain Carbon Steels. Materials Science and Technology, 25:8, August 2009, 1009-1012. 

M s 

565 31 Mn 13 

Si 10 

Cr 18 

Ni 12 

Mo 600 1 exp( 0.96 

C) 

 

 

Notation: 



Observation: 

- Standard error of estimate σ = 13°C; correlation coefficient R² = 0.95. 

f 

 

1 exp 

( M T) 

m 

s



46 

 

m 

27.2 

0.14 Mn 0.21 Si 0.11Cr 

0.08 Ni 0.05 Mo 19.8 

1 exp( 1.56 

C) 

 

 

Notation: 

f: Volume Fraction of Martensite as a Function of Undercooling Below Ms Temperature (Ms – T) 




Observation: 

- Formula based from Koistinen and Marburger equation. 

- αm: Standard error of estimate σ = 0.005 K -1 ; correlation coefficient R² = 0.97. 

Source: VAN BOHEMEN, S.M.C. Bainite and Martensite Start Temperature Calculated with Exponential Carbon 

Dependence. Materials Science and Technology, 28:4, April 2012, 487-495. 

. Zhao 

M TM 

s 

420 208.33 C 33.428 Mn 1.296 Mn² 

0.02167 Mn³ 

16.08 Ni 0.7817 Ni² 

0.02464Ni³ 

2.473 Cr 30.00 Mo 12.86 Co 

0.2654 Co² 

0.001547 Co³ 

7.18 Cu 72.65 N 43.36 N 

2 

16.28 Ru 1.72 Ru 

2 

0.08117 Ru 

3 

M LM 

s 

540 356.25 C 47.59 Mn 2.25 Mn² 

0.0415 Mn³ 

24.56 Ni 1.36 Ni² 

0.0384 Ni³ 

17.82 Cr 1.42 Cr 

21.87 Co 0.468 Co² 

0.00296 Co³ 

16.52 Cu 260.64 N 17.66 Ru 

2 

17.50 Mo 

Notation: 

Ms TM : Twinned Martensite Start Temperature [°C] 

Ms LM : Lath Martensite Start Temperature [°C] 




Observation: 

- If Ms LM is higher than Ms TM , both lath martensite and twinned martensite can be present in steel. 

- However, if Ms LM is lower than Ms TM , only twinned martensite can exist. This condition is fulfilled for some 

steels above a critical composition, which can be determined setting Ms LM = Ms TM . 

Source: ZHAO, J.: Continuous Cooling Transformations in Steels. Materials Science and Technology, 8:11, Nov. 1992, 

997-1002. 

47


- Critical Diameter - Austenite Hardenability 


48 

. Dearden & O’Neill 

D i 

 

6 exp 7.1 

C 

 

 

Mn 

5.87 

 

Mo Cr 

 

3.13 6.28 

Si 

 

18 

Ni 

 

15 

Notation: 

Di: Critical Diameter [mm] 


Source: DEARDEN, J & O’NEIL, H.: A Guide to the Selection and Welding of Low Alloy Structural Steels. Transactions 

of the Institute of Welding, 3, Oct. 1940, 203-214.



49 

- Density of Bulk Steel at Ambient Temperature 

. Austenitic Steels 

 

 

 

(1.231 Fe 3.178 C 

sol 

1 

1.307 

Mn 2.436 Si 1.431 

Cr 1.205 

Cu 1.018 

Mo 1.137 

Ni 1.890 

Ti 1.111 

Co 2.186 N 2.032 TiC) .10 

6 

Notation: 

: Austenite Density [kg/m³] 

Alloy/TiC Content: [weight %] 

Observations: 

- Csol is the content of this element not bound in TiC. 

- Density calculated at 20°C. 

Source: BOHNENKAMP, U. et al.: Evaluation of the Density of Steels. Steel Research, 71:3, March 2000, 88-93. 

. Ferritic Steels 

 

 

1 

(1.270 Fe 1.380 

C 1.524 Mn 2.381 Si 1.384 

Cr 0.8477 Cu 1.076 Mo 1.370 Ni 2.012 V 

4.046 S) 

.10 

6 

Notation: 

: Ferrite Density [kg/m³] 


Observations: 

- C is considered insoluble in ferrite (that is, all C has gone to cementite). 

- The solubilities of the other alloy elements in cementite are zero. 

- Density calculated at 20°C.



50 

Source: BOHNENKAMP, U. et al.: Evaluation of the Density of Steels. Steel Research, 71:3, March 2000, 88-93. 

. Density of Fe-C Alloys in Heterogeneous Phase Mixtures 

1 

 

Steel 

f1 

f 

2 

f 

3 

f 

n 

... 

 

1 

2 

3 

n 

Notation: 

Steel: Steel Density [kg/m³] 

fi: Fraction of the phase i in the microstructure 

ρi: Density of the phase i 

Source: JABLONKA, A.: Thermomechanical Properties of Iron and Iron-Carbon Alloys: Density and Thermal Contraction, 

Steel Research, 62:1, September 1991, 24-33.


- Density of Bulk Steel at High Temperature 


51 

. BISRA 

ρ 

T 

1008 1023 1040 1524 

0 7861 7863 7858 7854 

15 7856 7859 7854 7849 

50 7847 7849 7845 7840 

100 7832 7834 7832 7826 

150 7816 7819 7817 7811 

200 7800 7803 7801 7794 

250 7783 7787 7784 7777 

300 7765 7770 7766 7760 

350 7748 7753 7748 7742 

400 7730 7736 7730 7723 

450 7711 7718 7711 7704 

500 7792 7699 7692 7685 

550 7673 7679 7672 7666 

600 7653 7659 7652 7646 

650 7632 7635 7628 7622 

700 7613 7617 7613 7605 

750 7594 7620 7624 7615 

800 7582 7624 7643 7641 

850 7589 7625 7617 7614 

900 7600 7600 7590 7590



52 

950 7572 7574 7564 7561 

1000 7543 7548 7538 7532 

1050 7515 7522 7512 7503 

1100 7488 7496 7486 7474 

Notation: 

ρ: Density of steel [kg/m³] 


Observations: 

- Chemical composition of the steels [wt %]: 

Steel C Mn Si P S Cu 

1008 0.08 0.31 0.08 0.029 0.050 - 

1023 0.23 0.64 0.11 0.034 0.034 0.13 

1040 0.42 0.64 0.11 0.031 0.029 0.12 

1524 0.23 1.51 0.12 0.037 0.038 0,11 

Source: Physical Constants of Some Commercial Steels at Elevated Temperatures, BISRA/Butterworths Scientific 

Publications, London, 1953, 1-38. 

. Picquè 

5 

2 

7875.96 

0.297 T 5.62 10 

T (T ≤ Ar3) 

8099.79 

0.506 T (T > Ar3) 

Notation:


ρ: Density of steel [kg/m³] 



53 

Observation: 

- Formulas specific for a 0.16% C, 0.5% Mn steel 

Source: PICQUÉ, B. Experimental Study and Numerical Simulation of Iron Oxide Scales Behavior in Hot Rolling. Doctor 

Thesis, École de Mines de Paris, 2004, p. 247.


- Density of Liquid Steel 

. Jablonka 


54 

 

Steel 

( 8319.49 0.835 T) 

(1 0.01C) 

Notation: 

Liquid Iron: Liquid Iron Density [kg/m³] 

T: Temperature, [°C] 

C: Carbon content [weight %] 


Steel Research, 62:1, September 1991, 24-33. 

. Yaws 

 

Liquid Iron 

1.9946 

 

0.22457 

T 

0.7 

1 

 

9340 

Notation: 

Liquid Iron: Liquid Iron Density [g/ml] 

T: Absolute Temperature, [K] 

Source: YAWS, C.L.: Liquid Density of the Elements. Chemical Engineering, November 2007, 44-46.



- Density of Microstructural Constituents at Ambient Temperature 

55 

. Common Phases and Constituents 

Notation: 

C: Carbon Content [weight %] 

Phase/Constituent 

C 

[weight %] 

Specific Volume 

[cm³/g] at 20°C 

Austenite 

0.00 ~ 2.00 0.1212 + 0.0033 C 

Martensite 

0.00 ~ 2.00 0.1271 + 0.0025 C 

Ferrite 0.00 ~ 0.02 0.1271 

Cementite (Fe3C) 6.7 0.2 0.130 0.001 

Carbide 8.5 0.7 0.140 0.002 

Graphite 100 0.451 

Ferrite + Cementite 

0.00 ~ 2.00 0.1271 + 0.0005 C 

Low C Martensite + Carbide 0.25 ~ 2.00 0.1277 + 0.0015 (C – 0.25) 

Ferrite + Carbide 

0.00 ~ 2.00 0.1271 + 0.0015 C 

Source: THELNING, K.E.: Steel and its Heat Treatment – Bofors Handbook. Butterworths, London, 1981, 570 p. 

. Density and Molar Volume of Microalloy Carbides and Nitrides 

Compound Structure Molecular 

Mass 

Lattice 

Parameter 

[nm] 

Molecules per 

Unit Cell 

Density 

[g/cm³] 

Molar Volume 

[cm³/mol] 

NbC FCC 105 0.4462 4 7.84 13.39 

NbN FCC 107 0.4387 4 8.41 12.72 

VC FCC 63 0.4154 4 5.83 10.81 

VN FCC 65 0.4118 4 6.18 10.52



TiC FCC 60 0.4313 4 4.89 12.27 

TiN FCC 62 0.4233 4 5.42 11.44 

AlN CPH 41 

c = 0.4965 

a = 0.311 

6 3.27 12.54 

-Fe FCC 56 0.357 4 8.15 6.85 

-Fe BCC 56 0.286 2 7.85 7.11 

56 

Observations: 

- Data based on room temperature lattice parameters. 

Source: GLADMAN, T. The Physical Metallurgy of Microalloyed Steels. The Institute of Materials, London, 1997, 363 p. 

. Density and Molar Volume of Precipitates and Metals 

Compound Density 

[kg/m³] 

Molar Volume 

[cm³/mol] 

NbCN 9291 12,80 

ZrC 6,572 - 

ZrN 7,30 - 

Mn 7470 7,35 

Si 2330 12,06 

Cr 7140 7,23 

Cu 8920 7,11 

C 2267 5,29 

N - 13,54 

Steel 7850 7,00 

Observations: 

- Data based on room temperature lattice parameters.


Sources: 


57 

- SAN MARTIN, D. et al. Estudio y Modelización de la Influencia de las Partículas de Segunda Fase sobre el 

Crecimiento de Grano Austenítico en um Acero Microaleado com Niobio. Revista de Metalurgia – Madrid, 42:2, 

Marzo-Abril 2006, 128-137. 

- Web Elements (www.webelements.com) 

- NISHIZAWA, T. Thermodynamics of Microstructure Control by Particle Dispersion. ISIJ International, 40:12, 

December 2000, 1269-1274. 

- ADRIAN, H. Thermodynamical Model for Precipitation of Carbonitrides in HSLA Steels Containing Up to Three 

Microalloying Elements with or without Additions of Aluminum. Materials Science and Tecnology, 8:5, May 

1992, 406-420. 

. Relationship Between Lattice Parameter and Density 

 

( a 

n M 

10 

10 ) 3 

N 

Notation: 

: Density [kg/m³] 

n: Number of atoms per unit cell (depends on crystalline structure): 

. Cubic body-centered: 2 

. Cubic face-centered: 4 

M: Molecular mass [kg/mol]: 

. Pure Fe: 0.055847 

. Cementite: 0.179552 

a: Lattice Parameter [Å] 

N: Avogrado’s number: 6.023 . 10 23



58 





- Density of Microstructural Constituents at High Temperature 

59 

. Fink 

 

T 20C 

 

 

 

T 20C 

 

 

0.47 T 

0.33 T 

Notation: 

T : Austenite Density at Temperature T [kg/m³] 

 

20°C : Austenite Density at 20°C [kg/m³] 

T : Ferrite Density at Temperature T [kg/m³] 

 

20°C : Ferrite Density at 20°C [kg/m³] 

T: Temperature[°C] 

Source: FINK, K. et al.: Physikalische Eigenschaften von Stählen, insbesondere von warmfesten Stählen. 

Thyssenforschung, 2:2, 1970, 65-80. 

 

T 

 

 

T 

 

T 

 

 

. Jablonka 

5 

2 

2 

7875.96 

0.297 T 5.62 10 

T (1 2.6210 

C) 

 

8099.79 

0.506 T (1 1.4610 

2 C) 

5 

2 

2 

7875.96 

0.297 T 5.62 10 

T (1 2.6210 

C) 

 

T 

 

Fe3C 

 

10 

2 

4 

2 

7686.45 

6.63 10 

T 3.12 T



60 

Notation: 

T : Delta Ferrite Density at Temperature T [kg/m³] 

T : Austenite Density at Temperature T [kg/m³] 

T : Ferrite Density at Temperature T [kg/m³] 

Fe3C T : Cementite Density at Temperature T [kg/m³] 

T: Temperature[°C] 

C: Carbon content [weight %] 

Observations: 

- Carbon content in ferrite is limited to 0.02% maximum. 



. Molar Volume of Austenite as Function of Temperature 

 

6 

5 

V M 

6.688726 10 

exp 7.3097 10 

Notation: 

 

T 

VM : Molar Volume of Austenite, [m³/mol] 


Source: FERNÁNDEZ, D.M.S.M. Modelización de la Cinética de Austenización y Crecimiento de Grano Austenítico en 

Aceros Ferrítico-Perlíticos. Tesis Doctoral, Centro Nacional de Investigaciones Metalúrgicas, Madrid, Julio 

2003, 258 p. 

. Relationship Between Density and Thermal Expansion



61 

 

th 

 

3 

( 

T 

0 

) 

( 

T) 

1 

Notation: 

th : Thermal Expansion/Contraction 

ρ(T0): Density at lower/higher T0 

ρ(T): Densotu at temperature T 

T0: Reference temperature 




- Dimensional Changes during Austenite Transformation 


62 

L 

C 

 

 

L 

. During Cooling 

 

2 

5 

3 

3 

2 

6 

9 

2 

1.232 

10 

1.347 

10 

T 6.544 10 

C 

5.829 10 

C 

9.766 10 

C 

T 2.379 10 

T 

C0 (1 X 

 

X 

A 

A 

) C 

tr 

Notation: 

ΔLγ→α: Specimen Length Change During Austenite Transformation [μm]; 

L: Specimen Original Length [μm] 


Cγ: Carbon Content in Austenite [wt %] 

C0: Initial Carbon Content [wt %] 

Ctr: Carbon Content of Transformed Phases [wt %] 

XA: Untransformed Austenite Volume Fraction 

Source: PARK, S.H. Microstructural Evolution of Hot Rolled TRIP Steels During Cooling Control. In: 40th Mechanical 

Working and Steel Processing Conference, ISS/AIME, Pittsburgh, October 1998, 283-291. 

. After General Heat Treating 

Transformation 

V 

[%] 

l 

[mm/mm]



Spheroidized Pearlite Austenite -4.64 + 2.21 C -0.0155 + 0.0074 C 

Austenite Martensite 4.64 – 0.53 C 0.0155 – 0.0018 C 

Spheroidized Pearlite Martensite 1.68 C 0.0056 C 

Austenite Lower Bainite 4.64 – 1.43 C 0.0155 – 0.0048 C 

Spheroidized Pearlite Lower Bainite 0.78 C 0.0026 C 

Austenite Upper Bainite 4.14 – 2.21 C 0.0155 – 0.0074 C 

Spheroidized Pearlite Upper Bainite 0 (Zero) 0 

63 

Notation: 

- C: Carbon Content [weight %]. 

Sources: 

- THELNING, K.E.: Steel and its Heat Treatment – Bofors Handbook. Butterworths, London, 1981, 570 p. 

- KRAUSS, G. Steel: Processing, Structure and Performance. ASM International, Metals Park, 2005, 420 p. 

. After Quenching 

V 

V 

100 

VC 

V 

 

100 

A 

 

1.68 C 

 

M 

VA 

( 4.64 

2.21 C 

100 

A 

) 

Notation: 

- V/V: Volumetric Change after Quenching [%] 

- VC: Non-solubilized Cementite Volumetric Fraction [%] 

- VA: Austenite Volumetric Fraction [%] 

- 100 – VC – VA: Martensite Volumetric Fraction [%] 

- CM: Carbon Content Solubilized in Martensite [weight %]



- CA: Carbon Content Solubilized in Austenite [weight %] 

64 

Source: THELNING, K.E.: Steel and its Heat Treatment – Bofors Handbook. Butterworths, London, 1981, 570 p.



65 

- Equivalent Carbon – H.A.Z. Hardenability 


C 

EQ _ Dearden 

C 

 

Mn 

6 

 

Mo Cr V 

 

4 5 

 

Cu 

 

13 

Ni 

 

15 

P 

2 

Notation: 

CEQ_Dearden: Equivalent Carbon (Dearden) [%] 


Source: DEARDEN, J & O’NEIL, H.: A Guide to the Selection and Welding of Low Alloy Structural Steels. Transactions 

of the Institute of Welding, 3, October 1940, 203-214. 

. Bastien 

C 

EQ _ Bastien 

C 

Mn 

4,4 

Mo Cr 

 

7,7 15,4 

 

Ni 

10,3 

ln( CR 

) 13,9 10,6 

m 

C EQ _ Bastien 

Notation: 

CEQ_Bastien: Equivalent Carbon (Bastien) [%] 


CRm: Critical Cooling Rate at 700°C [°C/s], that is, minimum cooling rate that produces a fully martensitic 

structure) 

Source: BASTIEN, P.G.: Metal Construction and British Welding Journal, 49, 1970, 9.



66 

. IIW - International Institute of Welding 

C 

EQ _ IIW 

C 

 

Mn 

6 

Cr 

 

 

Mo V 

5 

 

Cu Ni 

15 

Notation: 

CEQ_IIW: Equivalent Carbon (IIW) [%] 


Source: HEISTERKAMP, F. et al.: Metallurgical Concept And Full-Scale Testing of High Toughness, H2S Resistant 

0.03%C - 0.10%Nb Steel. C.B.M.M. Report, São Paulo, February 1993. 

. Kihara 

C 

EQ _ Kihara 

C 

 

Mn 

6 

Mo Cr 

 

4 5 

V 

 

14 

Ni 

 

40 

Si 

24 

Notation: 

CEQ_Kihara: Equivalent Carbon (Kihara) [%] 


Source: KIHARA, H. et al. Technical Report of JRIM, 1, 1959, 93. 

. Shinozaki 

C 

EQ _ FBW 

C 

 

Mn 

5 

Si Cr 

 

15 9 

V 

7 Nb (1 10C) 

 

(50 C 1) 

3 

1.3 Ti (1 5 C) 

 

Mo (1 6 C) 

29 B (11 C 1) 

2



Notation: 

CEQ_FBW: Equivalent Carbon Designed Specifically for Flash Butt Welding [%] 


67 

Source: SHINOZAKI, M. et al.: Effects of Chemical Composition and Structure of Hot Rolled High Strength Steel Sheets on 

the Formability of Flash Butt Welded Joints. Kawasaki Steel Technical Report, 6, Sept. 1982, 21-30. 

. Stout 

C 

EQ _ Stout 

1000 C 

 

 

 

Mn 

6 

Cr Mo Ni Cu 

 

10 20 40 

Notation: 

CEQ_Stout: Equivalent Carbon (Kihara) [%] 


Source: STOUT, R.D. et al. Welding Journal Research Supplement, 55, 1976, 89s-94s. 

. Yurioka 

C 

EQ _ Yurioka 

C 

 

Mn 

6 

Mo Cr 

 

4 8 

 

Ni 

12 

 

Si 

24 

 

Cu 

15 

log( t ) 10,6 

_ 

4,8 

m 

C EQ 

Yurioka 

Notation: 

CEQ_Yurioka: Equivalent Carbon (Yurioka) [%] 




tm: Critical Cooling Time from 800 to 500°C [s] (that is, maximum cooling time that produces a fully martensitic 

structure) 

Source: YURIOKA, N. et al.: Metal Construction, 19, 1987, 217R. 

68



69 

- Equivalent Carbon – Hydrogen Assisted Cold Cracking 

. Bersch & Koch (Hoesch) 

C 

EQ _ Bersch 

C 

 

Mn 

Si 

Cr 

Mo V Cu 

20 

 

Ni 

Notation: 

CEQ_Bersch: Equivalent Carbon for Pipeline Steels [%] 


Observations: 

- Formula deduced for pipeline steels 

Source: 

- BERSCH, B. et al. Weldability of Pipe Steels for Low Operating Temperatures. 3R International, 1, 1977. 

- PATCHETT, B.M. et al.: Casti Metals Blue Book: Welding Filler Metals. Casti Publishing Corp., Edmonton, 

February 1993, 608 p. (CD Edition). 

. DNV 

C 

EQ _ DNV 

C 

 

Mn 

10 

 

Si 

24 

Ni Cu 

 

40 

Cr 

 

5 

V 

 

10 

Mo 

4 

Notation: 

CEQ_DNV: Equivalent Carbon (DNV) [%] 




Source: HANNERZ, N.E.: The Influence of Si on the Weldability of Mild and High Tensile Structural Steels. IIW 

Document IX-1169-80, 1980. 

70 

. Graville 

C 

EQ _ HSLA 

C 

 

Mn 

16 

Ni 

 

50 

Cr 

 

23 

Mo 

 

7 

Nb V 

 

5 9 

Notation: 

CEQ_HSLA: Equivalent Carbon (Uwer & Graville) [%] 


Observations: 


Source: GRAVILLE, B.A.: In: Proc. Conf. on Welding of HSLA Structural Steels, ASM, Materials Park, 1976. 

. Ito & Bessyo (I) 

P C Si Mn Cu Cr Ni Mo V 

cm 

5 B 

30 20 60 15 10 

Notation: 

Pcm: Cracking Parameter [%] 


Observations: 

- Formula deduced for pipeline steels with C < 0.15% 

- This is the most popular formula for this kind of material.


Sources: 


- Equation valid under the following conditions: 0.07% C 0.22%; 0.40% Mn 1.40%; Si 0.60%; V 

0.12%; Cr 1.20%; Ni 1.20%; Cu 0.50%, Mo 0.7%, B 0,005%. 

- ITO, Y. et al.: Journal of the Japan Welding Society., 37, 1968, 983. 

- ITO, Y. & BESSYO, K. Weldability Formula of High Strength Steels Related to Heat-Affected-Zone Cracking. The 

Sumitomo Search, 1, 1969, 59-70. 

71 

. Ito & Bessyo (II) 

P Si Mn Cu Cr Mo V d H 

c 

C 

 

30 20 15 10 600 60 

Notation: 

Pc: Cracking Parameter [%] 

Alloy Content: [weight %], except 

H: Hydrogen amount in the weld metal, [cm³/100 g] 

d: Plate Thickness, [mm] 

Source: ITO, Y. & BESSYO, K.: Weldability Formula of High Strength Steels. I.I.W. Document IX-576-68. 

. Mannesmann 

C 

EQ _ PLS 

C 

 

Si 

25 

Mn Cu 

 

16 

 

Cr 

20 

 

Ni 

60 

 

Mo 

40 

V 

 

15 

Notation:



72 

CEQ_PLS: Equivalent Carbon for Pipeline Steels [%] 


Observations: 


- A version of this formula divides V by 10 

Sources: 

- DUREN, C. & NIEDEROFF, K.: In: Proc. on Welding and Performance of Pipeline, TWI, London, 1986. 

- HEISTERKAMP, F. et al.: Metallurgical Concept And Full-Scale Testing of High Toughness, H2S Resistant 

0.03%C - 0.10%Nb Steel. C.B.M.M. Report, São Paulo, February 1993. 

. Uwer & Hohne 

C 

EQ _ Uwer 

C 

 

Mn 

10 

 

Cu 

20 

Ni Cr 

 

40 20 

Mo 

10 

Notation: 

CEQ_Uwer: Equivalent Carbon (Uwer & Hohne) [%] 


Source: UWER, D. & HOHNE, H.: Determination of Suitable Minimum Preheating Temperature for the Cold-Crack-Free 

Welding of Steels. IIW Document IX-1631-91, 1991. 

. Yurioka



73 

Mn Si Cr Mo V Cu Ni Nb 

C EQ _ Yurioka 

C A( 

C) 

 

5 

6 24 5 15 20 5 

A ( C) 

0.75 0.25 tanh 20 ( C 

 

0.12) 

Notation: 

CEQ_Yurioka: Equivalent Carbon for Pipeline Steels [%] 


Observations: 

- Formula for C-Mn and microalloyed pipeline steels 

- This formula combines Carbon Equivalent equations from IIW and Pcm 

Sources: 

- YURIOKA, N.: Physical Metallurgy of Steel Weldability. ISIJ International, 41:6, June 2001, 566-570. 

 

B 

 

- PATCHETT, B.M. et al.: Casti Metals Blue Book: Welding Filler Metals. Casti Publishing Corp., Edmonton, 

February 1993, 608 p. (CD Edition).


- Equivalent Carbon – Peritectic Point 

. Mills 


74 

C P _ Mills 

C 0.02 Mn 0.04 Ni 0.1 Si 0.04 Cr 0. 1 Mo 

Notation: 

CP_Mills: Equivalent Carbon for Peritectic Point [%] 


Source: XU, J. et al. Effect of Elements on Peritectic Reaction in Molten Steel Based on Thermodynamic Analysis. ISIJ 

International, 52:12, October 2012, 1856-1861. 

C 

P _ Inf 

. Miyake et al. 

f 

1 

0.10 

C Sup 

f 

P _ 

2 

0.05 

f 

1 

0.0828 Si 0.0195 Mn 0.07398 Al 0.04614 Ni 0.02447 Cr 0.01851 Mo 0.090 

f 

2 

0.2187 Si 0.03291 

Mn 0.2017 Al 0.06715 Ni 0.04776 Cr 0.04601 Mo 0.173 

Notation: 

CP_Inf: Lower Bound of Carbon Peritectic Content Range [%] 

CP_Sup: Upper Bound of Carbon Peritectic Content Range[%] 


Observations:



- Alloy elements contents are assumed to be 4.0% or less, excluding 0%. 

75 

Source: MIYAKE, T. et al. Method of Continuous Casting of High-Aluminum Steel and Mold Powder. U.S. Patent n° US 

8,146,649 B2, April 3, 2012, 13 p. 

. Wolf 

C P _ Wolf 

C 0.04 Mn 0.1 Ni 0.7 N 0.14 Si 0.04 Cr 0.1 Mo 0. 4 Ti 

Notation: 

CP_Wolf: Equivalent Carbon for Peritectic Point [%] 


Source: WOLF, M.M. Estimation of Crack Susceptibility for New Steel Grades. In: 1 st European Conference on 

Continuous Casting, Florence, 1991, 2.489-2.499.


- Fe-C Equilibrium Diagram 


76



77 

Source: ASM’s Heat Treating One-Minute Mentor, 

http://www.asminternational.org/pdf/HTSRefCharts/Vol4p4Fig1.pdf.



78 

- Fe-C Equilibrium Equations in the Solidification and Eutectoid Range 

. Liquidus of -iron 

% C 

1536 T 

 

79 

. Liquidus of -iron 

% C 

1525 T 

 

56.0198 

3 

1.1284 10 

( T 1525) 

 

56.0198 

2 

Notation: 

%C: coordinate in the Fe-C diagram [%] 


. Solidus of -iron 

% C 

1536 T 

 

460 

. Solidus of -iron 

1525 T 

% C 

185


. Start of transformation of -iron (on cooling) 


79 

% C 

1392 

T 

1140 

. End of transformation of -iron (on cooling) 

% C 

1392 

T 

624.3749 

. A3 Line 

% C 

 

4 

911 T 2.818 10 

( T 911) 

 

484.785 484.785 

2 

5 

2.8574 10 

( T 911) 

 

484.785 

3 

. Acm Line 

% C 

0.8 

4 

T 723 7.7917 10 

( T 723) 

 

453.7137 453.7137 

2 




- Ferrite Solubility Products 


80 

. General 

log 10 

( a 

A 

) 

a 

m 

A 

( a 

m B n 

B 

) 

n 

 

A 

B 

T 

Notation: 

AmBn: Precipitate Considered for Calculation 

ax: Alloy Content [weight %] 



Precipitate A B Source 

AlN 9595 2.65 Kunze & Reichert 

BN 13560 4.53 Fountain & Chipman 

MnS 8400 2.77 Ivanov 

NbC 10990 4.62 Kunze 

NbN 10650 3.87 Kunze 

TiN 17640 6.17 Kunze 

VC 12265 8.05 Taylor 

VN 7830 2.45 Froberg 

VN 8120 2.48 Roberts & Sandbert 

ZrN 18160 5.24 Kunze 

Observations:





81 

Sources: 

- TAYLOR, K.A. et al. Scripta Metallurgica, 32, 1995, 7. 

- FROBERG, M.G. & GRAF, H. Stahl und Eisen, 80, 1960, 539. 

- KUNZE, J. Nitrogen and Carbon in Iron and Steels – Thermodynamics. Akademie Verlag, Berlin, 1991, p. 

192. 

- FOUNTAIN, R.W. & CHIPMAN, J. In: Transactions of the Metallurgical Society of AIME, 224, 1964, 599. 

- KUNZE, J. & REICHERT, J. Neue Hütte, 26, 1981, 23. 

- ROBERTS, W. & SANDBERG, A. Report IM 1489. Institute for Metallurgical Research, Stockholm, 1990. 

- IVANOV, B.S. et al. Stahl, 8, 1996, 52.


- Hardness After Austenite Cooling 

. Blondeau 


82 

HV 

 

f 

FP 

HV 

FP 

 

f 

B 

HV 

B 

 

f 

M 

HV 

M 

HV 

HV 

HV 

42 223 C 53 Si 30 Mn 7 Cr 19 Mo 12.6 Ni (10 19 

Si 8 Cr 4 Ni 130 V) 

log 

FP 

v r 

323 185 C 330 Si 153 Mn 144 Cr 191 Mo 65 Ni (89 53 C 55 Si 22 Mn 20 Cr 33 Mo 10 

Ni) 

log 

B 

v r 

127 949 C 27 Si 11 Mn 16 Cr 8 Ni 21 log 

M 

v r 

 

 

 

 

 

 

log( v 1 

) 9.81 4.62 C 1.05 Mn 0.54 Ni 0.5 Cr 0.66 Mo 0. 00183 

log( v 2 

) 10.17 

3.80 C 1.07 Mn 0.70 Ni 0.57 Cr 1.58 Mo 0. 0032 

PA 

PA 

log( v 3 

) 6.36 0.43 C 0.49 Mn 0.78 Ni 0.27 Cr 0.38 Mo 2 Mo 0. 0019 

PA 

1 

PA 

T 

 

4.58 log( t) 

 

H 

 

 

1 

Notation: 

HV: Global Hardness [Vickers] 

fFP: Fraction of Ferrite-Pearlite in Microstructure 

fB: Fraction of Bainite in Microstructure 

fM: Fraction of Martensite in Microstructure 

HVFP: Hardness of Ferrite-Pearlite [Vickers] 

HVB: Hardness of Bainite [Vickers]



HVM: Hardness of Martensite [Vickers] 


vr: Applied Cooling Rate at 700°C [°C/h] 

v1: Critical Cooling Rate at 700°C for Martensitic Quenching [°C/h] 

v2: Critical Cooling Rate at 700°C for Bainitic Quenching [°C/h] 

v3: Critical Cooling Rate at 700°C for Annealing [°C/h] 

PA: Austenitization Parameter [K] 

T: Austenitization Temperature [K] 

t: Austenitization Soaking Time [h] 

H: Austenitization Activation Energy: 240 kJ/mol for C Steels; 418 kJ/mol if Mo 0.04% 

83 

Observations: 

- Limits for Austenitization: 1073 K (800°C) ≤ T ≤ 1373 K (1100°C) and t ≤ 1 h. 

- Equations Valid for the Following Chemical Composition Range: 0.10% < C < 0.50%, Mn < 2.0%, Si < 1.0%, Ni 

≤ 4.0%, Cr < 3.0%, Mo < 1.0%, Cu < 0.5%, V < 0.2%, 0.010% < Al < 0.050% and Mn + Ni + Cr + Mo < 5.0%. 

Source: BLONDEAU, R. et al.: Mathematical Model for the Calculation of Mechanical Properties of Low-Alloy Steel 

Products: A Few Examples of its Application. In: 16 th International Heat Treatment Conference – Heat 

Treating ‘76, The Metals Society Stratford-upon-Avon, 1976, 189-200. 

. Lorenz 

HV 

 

Si Mn Cu Cr Ni Mo V 

2019 C 

(1 0.5 log t8 / 5 

) 0.3 66 (1 0.8 log t8 / 5 

) 

11 8 9 5 17 6 3 

 

 

 

 

Notation: 

HV: Maximum Hardness for a Martensitic-Bainitic HAZ Microstructure [Vickers, 10 kg Load] 


t8/5: Cooling Time Between 800°C and 500°C [s]



Source: LORENZ, K. et al.: Evaluation of Large Diameter Pipe Steel Weldability by Means of the Carbon Equivalent. In: 

International Conference on Steels for Linepipe and Fittings, Metals Society, London, Oct. 1981, 322-332. 

84 

. Murry 

HV 

HV 

max 

 

0 

HV 

 

max 

HV0 

exp 

n n 

t 

t0 

 

 

0.6 U t 

n 

 

0.04 

t0 

C exp 2.3 

 

0 

 

i 

Ci 

 

d 

 

n C exp 2.3 

 

0 

 

0.03 0.09 

 

d C 

 

 

0.43 

i 

Ci 

 

 

 

U 

1 

 

p exp 

0.35 

ln 

 

 

t 

 

t M 

2 

 

 

 

t M 

t 

0 

12 

1 

n 

 

p 

 

 

HV 

C 

t 

0 

 

 

 

3.1 

 

exp 2.3 

 

0 

 

 

 

i 

C i 

 

700 

HV 

300 9 

 

0 

HVmin 

.055 

max 

lg 

0 t 

300



85 

HVmin 63 

i 

0.75 

i 

Z i 

Notation: 

HVmax: Maximum Hardness for a Martensitic Structure 

C: Carbon [weight %] 

Ci: Alloy Content [weight %] 

d: Mean Austenite Grain Size [mm] 

t 700 300: Cooling time between 700°C and 300°C [s] 

HVmin: Minimum Hardness at Equilibrium Calculated According to the Substitution Solid Solution Hardening 

Effect According Lacy and Gensamer 

Zi: Concentration of the Element i in solid solution with ferrite at equilibrium [at %] 

μi: Action Coefficient of the Element i, as described in the following table: 

Element Mn Si P Ni Cr Mo V W 

μi 14.27 22.42 61.16 12.44 2.84 19.57 8.15 22.42 

General Constants: α0 = 0.89, β0 = 1.22, γ0 = 1.82 

Constants: αi, βi and γi according to the alloy element: 

Observations: 

Element i αi βi γi 

Mn 0.39 0.94 1.40 

Si 0.20 0.15 0.80 

Ni 0.22 0.40 0.12 

Cr 067 0.09 2.40 

Mo 0.17 0.72 0.79 

V 0.20 0.50 0.90 

Nb 0.40 1.20 1.00



- Equations Valid for the Following Chemical Composition Range: 0.05% ≤ C ≤ 0.80%, 0.50% ≤ Mn ≤ 2.5%, 

0.15% ≤ Si ≤ 0.35%, Ni ≤ 1.0%, Cr ≤ 1.5%, Mo ≤ 0.5%, V ≤ 0.1%, Nb ≤ 0.040%. 

Source: MURRY, G.: Transformations Dans les Aciers, Document M 1 115, Techniques de l'Ingénieur, Paris, 1985, 

54 p. 

86


- Hardness After Tempering 

. Spies 


87 

HB 2.84 

HRC 75 C 0.78 Si 14.24 Mn 14.77 Cr 128.22 Mo 54.0 V 0.55 T 435.66 

Notation: 

HB: Brinell Hardness After Hardening and Tempering 

HRC: Rockwell Hardness (C Scale) After Hardening 


T: Tempering Temperature [°C] 

Observations: 

- This equation is valid within the following ranges: HRC: 20~65; C: 0.20~0.54%; Mn: 0.50~1,90%; Si: 

0.17~1.40%; Cr: 0.03~1.20%; T: 500~650°C. 

Source: SPIES, H.J. et al.: Möglichkeiten der Optimierung der Auswahl vergütbarer Baustähle durch 

Berechnung der Härt-und-vergütbarkeit. Neue Hütte, 8:22, 1977, 443-445.


- Hardness After Welding 


88 


HV 

max 

1200 C EQ _ 

Dearden 

200 

C 

EQ _ Dearden 

C 

 

Mn 

6 

 

Mo CrV 

 

4 5 

 

Cu 

 

13 

Ni 

 

15 

P 

2 

Notation: 

CEQ_Dearden: Equivalent Carbon (Dearden) [%] 


HVmax = Maximum Hardness [Vickers] 

Observations: 

- This equation calculates maximum hardness after welding. 

Source: DEARDEN, J & O’NEIL, H.: Trans. Int. Weld., 3, 1940, 203. 

. Shinozaki et al. 

HV 78 331 

C EQ _ FBW 

C 

EQ _ FBW 

C 

 

Mn 

5 

Si Cr 

 

15 9 

V 

7 Nb (1 10C) 

 

(50 C 1) 

3 

1.3 Ti (1 5 C) 

 

Mo (1 6 C) 

29 B (11 C 1) 

2 

Notation: 

CEQ_FBW: Equivalent Carbon Designed Specifically for Flash Butt Welding [%] 



HV: Hardness at the Welding Interface [Vickers] 


89 

Source: SHINOZAKI, M. et al.: Effects of Chemical Composition and Structure of Hot Rolled High Strength Steel Sheets on 

the Formability of Flash Butt Welded Joints. Kawasaki Steel Technical Report, 6, Sept. 1982, 21-30.


- Hardness-Tensile Properties Equivalence 


90



91 

Source: ASM Handbook – Mechanical Testing and Evaluation. ASM International, vol. 8, Metals Park, 2000, 275.


- Hot Strength of Steel 

. Solid Solution Effect 


92



93 

Alloy Element 

[%/atomic %] [%/weight %] 

Mn 2.1 2.13 

Si 3.8 7.52 

Cr 1.8 1.93 

Mo 21 12.31 

Cu 0 0 

Ni 0 0 

Nb 210 127.06 

V 29 31.76 

Ti 63 59.76 

Observations: 

- Plot generated from data available in the original source. 

Source: TAMURA, I. et al.: Thermomechanical Processing of High Strength Low Alloy Steels. Butterworths, 

London, 1988, 248 p. 

. Misaka 

2 

0, 13 

2851 

2968 C C 

0,21 d 

 

2 1120 

exp 0.126 

1.75 C 0.594 C 

 

T 

Notation: 

: Steel Mean Flow Stress [kgf/mm²] 

 

 

 

 

 

 

dt


C: C content [weight %] 

T: Absolute Temperature [K] 

: True Strain 

έ: Strain Rate [s -1 ] 


94 

Observations: 

- The mean flow stress calculated by this equation is given in effective (von Mises) units, as it was determined 

under plane strain conditions. 

- Equation valid for the following parameter range: C 1.20%; 750 έ 1200°C; 0.5; 20 έ 200 s -1 . 

Source: MISAKA, Y. et al. Formulatization of Mean Resistance to Deformation of Plain C Steels at Elevated Temperature. 

Journal of the Japan Society for the Technology of Plasticity, 8:79, 1967-1968, 414-422. 

. Misaka Reloaded 

 

f 

2 

0, 13 

2851 

2968 C C 

0,21 d 

 

2 1120 

g exp 0.126 

1.75 C 0.594 C 

 

T 

 

 

 

 

 

 

dt 

f 0,916 

0,18 Mn 0,389 V 0,191 Mo 0, 004 Ni 

If T, expressed in Celsius degrees, is between Ar3 and Ar1, then g must be calculated according to the formula below. 

Otherwise g is equal to unity. 

g 0,7893 

0, 769 C 

Ar 974,76 734, 65 C 

3 

 

Ar 876,81 336, 26 C 

1



95 

Notation: 


f: Effect of alloy elements on Mean Flow Stress. 

g: Softening Factor Due to Intercritical Deformation 

C: Carbon Content [weight %] 


: True Strain 

t: Time [s] 

Mn: Manganese Content [weight %] 

V: Vanadium Content [weight %] 

Mo: Molybdenum Content [weight %] 

Ni: Nickel Content [weight %] 

Ar3: Temperature of Start Austenite Transformation in Proeutectoid Ferrite [°C] 

Ar1: Temperature of Finish Austenite Transformation in Proeutectoid Ferrite [°C] 

Observations: 



Source: MISAKA, Y. et al. Estimation of Rolling Force in Computer Controlled Hot Rolling of Plates and Strip - Theme III: 

Mathematical Model for Estimating Deformation Resistance in Hot Rolling of Steels. Tetsu-to-Hagané, 67:2, 

1981, A53-A56. 

. Senuma & Yada 

a 

 

(1 X ) 

n 

dyn 

s 

X 

dyn



96 

 

n 

c (1 e 

b 

b 

 

) 

e 

 

0 

b 

 

b 9850 

0.315 

e 

8000 

 

T 

(Senuma 1984) 

b 6227 

0.28 

e 

7500 

 

T 

(Wang & Tseng) 

c 1.0 

10 

11 

(Senuma 1984) 

 

c 8.5 10 

10 1 

1 

D0 

 

 

 

(Yada & Senuma) 

8000 

 

4 

T 

 

c 

4.76 10 

e (Senuma) 

2500 

 

T 

0.05 e (Wang & Tseng 1996) 

c 

If ε εc: 

X dyn 

1 e 

 

 

2 

( 

 

C ) 

0.693 

 

0.5


2 

9 0.2 (1613 

T) 

 

3.8242 

10 

 

 

 

 

s 

 

 

290 


97 

. According to Senuma (1984): 

5 

0.28 0.05 

1.144 

D0 

 

0 .5 

10 

e 

6420 

 

T 

. According to Wang & Tseng: 

2 

0.28 0.03 

1.07 

D0 

 

 

0.5 

10 

e 

2650 

 

T 

8670 

 

0.27 

T 

D dyn 

22600 e 

If Xdyn > 0.95: 

. According to Yada & Senuma: 

D 

p 

D 

dyn 

( D 

pd 

D 

dyn 

 

) 1 

e 

 

8000 

 

0.1 

e 

T 

295 

t 

 

 

 

. According to Wang & Tseng:



98 

D 

p 

D 

dyn 

1.1 ( D 

pd 

D 

dyn 

 

) 1 

e 

 

8000 

 

0.1 

e 

T 

295 

t 

 

 

 

D pd 

5380 e 

6840 

 

T 

If ε < εc: 

D 

st 

 

5 

 

S 0. 6 

V 

 

 

 

24 (0.4914 e 0.155 e 0.1433 e 

S V 

 

D 

0 

3 ) 

X 

st 

1 e 

 

 

2 

( t t 

 

s ) 

0.693 

 

t0.5 

 


t 

0.5 

0.286 10 

 

S 

v 

7 

 

0.2 

 

2 

e 

18000 

 

T 

. According to Wang & Tseng:



99 

t 

0.5 

 

2.2 10 

S 

v 

12 

 

0.2 

 

2 

e 

30000 

 

T 

t 4 t 

0.95 

. 322 

0.5 

If Xst < 0.95 (i.e., tip < t0.95): 

D 

u 

D 

st 

( 0.2 0.8 X 

st) 

If Xst 0.95 and tip t0.95: 


D 

g 

 

D 

2 

st 

1.44 

10 

12 

t 

g 

e 

63800 

 

T 

. According to Wang & Tseng: 

D 

g 

 

D 

2 

st 

1.44 

10 

12 

t 

g 

e 

32100 

 

T 

t 

g 

t 

ip 

 

t 0.95



100 

In case of multipass hot rolling: 

 

R 

(1 X 

t 

dyn 

) (1 X 

st 

) X 

s 

dyn 

( D 

( D 

pd 

pd 

D 

D 

p 

dyn 

) 

) 

e 

t 

n 

8000 

 

 

T 

90e 

 

ta 

 

 

0.7 

 

 

 

 

( 

b c 

0 

) 

ln 

r 

b c 

 

( 

) 

 

 

r 

 

b 

Notation: 


ρ: Dislocation Density [cm - ²] 

ρ0: Initial Dislocation Density [cm - ²] 

ρn: Dislocation Density in the Dynamically Recovered Region [cm - ²] 

ρs: Dislocation Density in the Dynamically Recristallized Region [cm - ²] 

ρr: Dislocation Density After Deformation/Static Recovery [cm - ²] 

ρt: Remaining Dislocation Density in the Dynamically Recovered Region [cm - ²] 

Xdyn: Fraction of Dynamic Recrystallization 

Xst: Fraction of Static Recrystallization 


: True Strain 

εc: Critical Strain for the Onset of Dynamic Recrystallization 

ε0.5: Strain Required for 50% Dynamic Recrystallization 

εr: Residual Strain After One Pass of Hot Rolling 

̇: Strain Rate [s -1 ]



D0: Grain Size Before Deformation [μm] 

Ddyn: Dynamically Recrystallized Grain Size [μm] 

Dp: Transition Grain Size from Ddyn to Dpd at a time t after deformation [μm] 

Dpd: Grain Size resulted from driving force due to the decrease of dislocation density [μm] 

Dst: Statically Recrystallized Grain Size [μm] 

Du: Mixed Grain Size Due to Incomplete Static Recrystallization [μm] 

Dg: Grain Size After Complete Static Recrystallization Plus Growth [μm] 

Sv: Nucleation Site Area [μm -1 ] 

t: Time [s] 

ts: Incubation Time for Static Recrystallization [s] 

t0.5: Time Required for 50% Static Recrystallization [s] 

t0.95: Time Required for 95% Static Recrystallization [s] 

tip: Time Interval Between Successive Rolling Passes [s] 

tg: Time Available for Grain Growth [s] 

ta: Time After Deformation [s] 

101 

Observations: 

- This model is very interesting as it links hot strength with microstructural evolution. 



- The value of constant a depends on steel composition. For instance: 

. 0.00175 MN/m² (Senuma 1984: 0.08-0.81% C, 0.62-1.14% Mn, 0.20-0.24% Si) 

. 0.00165 MN/m² (Yada & Senuma: 0.05-0.40% C, 0.00-1.00% Mn, 0.00-0.50% Si) 

. 0.00180 MN/m² (Wang & Tseng: 0.05-0.81% C, 0.20-1.50% Mn, 0.01-0.50% Si) 

- Suggested value for ρ0: 1 x 10 -8 cm -2 (Wang & Tseng) 

- ts can be assumed as being zero as it is negligibly short for the deformation conditions of hot flat rolling (Wang 

& Tseng). 

- If εr from the former pass is greater than 0, then it must be added to value of ε of the next pass. 

Sources:



- SENUMA, T. & YADA, H. Microstructure Evolution of Plain Carbon Steels. 7 th Riso International Symposium 

on Metallurgy and Materials Science, Riso National Laboratory, Roskilde, 1986, p. 547-552. 

- WANG, S.R. & TSENG, A.A. Macro- and Micro-Modeling of Hot Rolling of Steel Coupled by a Micro Constitutive 

Relationship. Iron and Steelmaker, September 1996, 49-61. 

- SENUMA, T. et al. Structure of Austenite of Carbon Steels in High Speed Hot Working Process. Tetsu-to- 

Hagané, 70:15, November 1984, 2112-2119. 

- YADA, H. & SENUMA, T. Resistance to Hot Deformation of Steels. Journal of the Japan Society for 

Technology of Plasticity, 27:300, 1986, 34-9. 

- YANAGIMOTO, J. & LIU, J. Incremental Formulation for the Prediction of Microstructural Change in Multi-Pass 

Hot Forming. ISIJ International, 39:2, February 1999, 171-175. 

102 

. Shida 

Calculation algorithm expressed in Visual Basic: 

Function Shida(C, T, Def, VelDef) 

Dim nShida, Td, g, Tx, mShida, SigF As Single 

nShida = 0.41 – 0.07 * C 

Td = 0.95 * (C + 0.41) / (C + 0.32) 

T = (T + 273) / 1000 

If T >= Td Then 

g = 1 

Tx = T 

mShida = (-0.019 * C + 0.126) * T + (0.075 * C – 0.05) 

Else



103 

g = 30 * (C + 0.9) * (T – 0.95 * (C + 0.49) / (C + 0.42)) ^ 2 + (C + 0.06) / (C + 0.09) 

Tx = Td 

mShida = (0.081 * C – 0.154) * T + (-0.019 * C + 0.207) + 0.027 / (C + 0.32) 

End If 

SigF = 0.28 * g * Exp(5 / Tx – 0.01 / (C + 0.05)) 

Shida = 2 / Sqr(3) * SigF * (1.3 * (Def / 0.2) ^ nShida – 0.3 * (Def / 0.2)) * _ 

(VelDef / 10) ^ mShida 

End Function 

Notation: 

Shida: Steel Mean Flow Stress [kgf/mm²] 



Def: True Strain 

VelDef: Strain Rate [s -1 ] 

Observation: 

- The mean flow stress calculated is this algorithm is already expressed in effective (von Mises) units, that is, 

corrected for plane strain conditions, as it is multiplied by 2/√3. 

- Equation valid for the following parameter range: C 1.20%; 700 έ 1200°C; 0.7; 0.1 έ 100 s -1 . 

- The effect of some alloy elements over hot strength can be considered by Shida equation. In this case carbon 

content must be replaced by an equivalent carbon (Ceq) content, which formula is described below: 

C eq 

C 

Mn Cr V Nb 

 

6 12 

Sources: 

where Mn is the manganese content, Cr is the chromium content, V is the vanadium content and Nb is the 

niobium content, all expressed as weight percent.



104 

- SHIDA, S. Empirical Formula of Flow Stress of C Steels - Resistance to Deformation of C Steels at Elevated 

Temperature. Journal of the Japan Society for Technology of Plasticity, 10:103, 1969, 610-7. 

- LENARD, J.G. et al. : Mathematical and Physical Simulation of the Properties of Hot Rolled Products. 

Elsevier, Amsterdam, 1999, 248 p.


- Jominy Curves 

. Just 

- d < 6,4 mm 


105 

J d 

60 C 20 

- 6,4 ≤ d < 39,7 mm 

J d 

98 

C 

0.00992 d 

2 

C 

20 Cr 6.4 Ni 19 Mn 34 Mo 28V 

19.05 

d 

1.80 d 

7 

- C < 0.28% and 6,4 mm ≤ d < 39,7 mm 

J d 

87 C 14 Cr 5.3 Ni 16 Mn 29 Mo 16.8 

d 1.39 d 22 

- C > 0.29% and 6,4 mm ≤ d < 39,7 mm 

J d 

78 C 22 Cr 6.9 Ni 21 Mn 33 Mo 16.1 

d 1.17 d 18 

Observation: 

- Equations valid for the following chemical composition range: 0.10% ≤ C ≤ 0.60%, 0.45% ≤ Mn ≤ 1.75%, 0.15% 

≤ Si ≤ 1.95%, Ni ≤ 5.0%, Cr ≤ 1.55%, Mo ≤ 0.52% and V ≤ 0.2%. 

- Equation Considering the Effect of Austenite Grain Size



106 

J d 

88 C 0.00553 C 19 Cr 6.3 Ni 16 Mn 35 Mo 5 Si 0.82 G 15.9 

d 1.33 d 2 

Observation: 

- Equation valid for the following conditions: 0.08% ≤ C ≤ 0.56%, 0.20% ≤ Mn ≤ 1.88, Si ≤ 3.80, Ni ≤ 8.94%, Cr ≤ 

1.97, Mo ≤ 0.53 and 1.5 ≤ Gγ ≤ 11. 

Notation: 

Jd: Hardness [Rockwell C] 


d: Distance from the Cooled End [mm] 

Gγ: Austenite Grain Size Index [mm] 

Source: JUST, E. New Formulas for Calculating Hardenability. Metal Progress, 96, November 1969, 87-88.


- Lattice Parameters of Phases 

. Ferrite 


107 

a 

2.8863[1 17.5 10 

6 

( T 

800)] 

Notation: 

aα: Ferrite Lattice Parameter [Ǻ] 


Observations: 

- 800 K < T < 1200 K 

. Austenite 

a 3.573 0.033 C 0.00095 Mn 0.0002 Ni 0.0006 Cr 0.0031 Mo 0. 0018V 

 

0 

 

Notation: 

aγ: Austenite Lattice Parameter [Ǻ] 


ξ: C [Atomic Fraction] 

Observations: 

- 1000 K < T < 1250 K 

- 0.0005 < ξ < 0.0365 

a 

 

(3.6306 0.78 ) [1 (24.9 50 ) 

10 

6 

( T 

1000)]


Notation: 

aγ: Austenite Lattice Parameter [Ǻ] 

Alloy Content: [Weight Percent] 

Observations: 

- 1000 K < T < 1250 K 


108 

. Cementite 

a 

4.5234 [1 (5.31110 

6 

1.942 

10 

9 

T 

9.655 10 

12 

T 

2 

) ( T 

293)] 

b 

5.0883[1 (5.31110 

6 

1.942 

10 

9 

T 

9.655 10 

12 

T 

2 

) ( T 

293)] 

c 

6.7426 [1 (5.31110 

6 

1.942 

10 

9 

T 

9.655 10 

12 

T 

2 

) ( T 

293)] 

Notation: 

aθ, bθ, cθ: Cementite Lattice Parameter [Ǻ] 


Observations: 

- 300 K < T < 1000 K 

Source: CABALLERO, F.G. et al. Modelling of Kinetics and Dilatometric Behaviour of Austenite Formation in a Lowcarbon 

Steel with a Ferrite Plus Pearlite Inicial Microstructure. Journal of Materials Science, 37, 2002, 3533- 

3540.


- Liquid Steel Solubility Products 


109 

. General 

log 

( a 

A 

) 

a 

m 

A 

( a 

m B n 

B 

) 

n 

 

A 

B 

T 

Notation: 

AmBn: Precipitate Considered for Calculation 

ax: Alloy Content [weight %] 



Precipitate A B 

MnS 8236 5.03 

TiN 16586 5.90 

TiS 8000 4.00 

ZrN 17000 6.38 

Observations: 



Source: Values compiled by Rajindra Clement Ratnapuli from assorted references.


- Liquidus Temperature of Steels 


110 

 

TLiq 1536 78 C 7, 6 Si 4, 9 Mn 34 P 30 S 5 Cu 31 , Ni 13 , Cr 3, 

6 Al 2 Mo 2V 18 Ti 

Notation: 

TLíq: Steel Melting Temperature [°C] 


Source: GUTHMANN, K. Günstige Giesstemperatur im Vergleich zum Erstarrungspunkt von Eisen- und Stahlschmelzen. 

Stahl und Eisen, 71(1951), 8, 399-402.


- Poisson Ratio 

. Definition 

: Poisson Ratio 

. Elastic Range: 0.3 

. Plastic Range: 0.5 


111 

Source: WILSON, A.D. Guidelines for Fabricating and Processing Plate Steel. Bethlehem-Lukens Plate Report, Burns 

Harbor, 2000, 97 p. 

. Fletcher 

Temperature 

[°C] 

600 0.327 

700 0.335 

800 0.344 

900 0.352 

1000 0.360 


Thesis, École de Mines de Paris, 2004, p. 243.


- Precipitate Isothermal Solubilization Kinetics 


112 

t 

 

r 

2 

0 

2 c D 

Notation: 

AmBn: Spheric precipitate considered for calculation 

t: Time for solubilization of the precipitate [s] 

r0: Radius of the precipitate [m], [cm] or [mm] 

c 

C 

C 

i 

p 

C 

m 

C 

i 

C 

 

C 

i 

p 

Cm: Solute concentration in the bulk metal [%] 

Ci: Solute concentration in the precipitate/matrix interface [%] 

C 

i 

A 

B 

T 

a 

B 

10 T: Temperature [K] 

A, B: Constants of the Solubility Product, given in the table at the topic Austenite Solubilization Products. 

aB: Alloy content [weight percent] 

Cp: Solute content in the precipitate [%] 

C 

p 

 

m M 

A 

m M n M 

A 

B 

Mx : Atomic mass of the element [g]



Cm: Solute content in a position far away from the precipitate [%] 

D: Solute Diffusion Coefficient [m²/s, cm²/s or mm²/s], calculated according to the general equation below: 

113 

D D 

0 

Q 

exp 

RT 

D0: Constant 

Q: Activation Energy for Diffusion [J] or [cal] 

R: Universal Gas Constant, 1.981 cal/mol.K 

R’: Universal Gas Constant, 8.314 J/mol.K 

Element Phase Equation Source 

Al 

Ferrite D [m 2 /s] = 0.30 * 10 -2 * Exp(-234500/R’ T) Pickering 

Austenite D [m 2 /s] = 0.49 * 10 -4 * Exp(-284100/R’ T) 

Austenite D [m 2 /s] = 2.10 * 10 -3 * Exp(-286000/R’ T) Borggren 

B Austenite D [m 2 /s] = 2 * 10 -4 * Exp(-87864/R’ T) 

C 

Ferrite 

D [cm 2 /s] = 0.02 * Exp(-20100/RT) 


Austenite D [m 2 /s] = 0.10 *10 -4 * Exp(-135700/R’ T) 

Cr 

Ferrite 

D [cm 2 /s] = 8.52 * Exp(-59900/RT)



114 

Austenite 

D [cm 2 /s] = 10.80 * Exp(-69700/RT) 

Fe 


Austenite D [m 2 /s] = 0.49 * 10 -4 * Exp(-284100/R’ T) Pickering 


Mn 

Austenite D [mm 2 /s] = 140 * Exp(-286000/R’ T) 

Austenite D [cm 2 /s] = 0.65 * Exp(-276000/R’ T) 


N 

Ferrite 

D [cm 2 /s] = 6.6 * 10 -3 * Exp(-18600/RT) 



Nb 

Austenite 

D [mm 2 /s] = 5.90 * 10 -4 * Exp(-343000/R’ T) Andersen 



P 

Austenite D [mm 2 /s] = 51 * Exp(-230120/R’ T) 

Austenite 

D [cm 2 /s] = 2.90 * Exp(-55000/RT)



Si Austenite D [m 2 /s] = 7.00 * 10 -4 * Exp(-286000/R’ T) Borggren 

Ti Austenite D [m 2 /s] =1.50 * 10 -5 * Exp(-251000/R’ T) Borggren 

115 

Ferrite 

D [cm 2 /s] = 3.92 * Exp(-57600/RT) 

V 


Austenite 

D [cm 2 /s] = 0.25 * Exp(-63100/RT) 


Sources: 

- ANDERSEN, I. & GRONG, O. Analytical Modelling of Grain Growth in Metals and Alloys in the Presence of 

Growing and Dissolving Precipitates – I. Normal Grain Growth. Acta Metallurgica and Materialia, 43:7, 1995, 

2673-2688. 

- GLADMAN, T. The Physical Metallurgy of Microalloyed Steels. The Institute of Materials, London, 1997, 363 p. 

- BORGGREN, U. e al. A Model for Particle Dissolution and Precipitation in HSLA Steels. Advanced Materials 

Research, 15-17, 2007, 714-719. 

- Information compiled by Rajindra Clement Ratnapuli from assorted references.


- Solidus Temperature of Steels 

 


T Sol 

1536 415,5 C 12,3 Si 6,8 Mn 124,5 P 183,9 S 4,3 Ni 1,4 Cr 4, 1 

Al 

116 

Notation: 

TSol: Steel Solidus Temperature [°C] 


Source: TAKEUCHI, E. & BRIMACOMBE, J.K. Effect of Oscillation-Mark Formation on the Surface Quality of 

Continuously Cast Steel Slabs. Metallurgical Transactions B, 16B, 9, 1985, 605-25.



117 

- Relationships Between Chemical Composition x Process x Microstructure x Properties 

. C-Mn Mild Steels 

YS 

17.4 

53.9 

32.3 Mn 83.2 Si 354.2 N 

sol 

 

d 

TS 

294.1 

27.7 Mn 83.2 Si 2.85 Pearl 

7.7 

d 

d 

15.4 

370 120 C 23.1 Mn 116 Si 554 P 143 Sn 1509 N 

sol 

d 

d 

 

0.28 

0.20 C 0.25 Mn 0.044 Si 0.039 Sn 1. 

2 

unif 

N sol 

1.40 

2.90 C 0.20 Mn 0.16 Si 2.2 S 3.9 P 0.25 Sn 

tot 

0.017 

d 

11.5 

50% 

ITT 19 

44 Si 700 N 

sol 

2.2 Pearl 

d 

Y 12.32 

19250 

Nsol 162 Mn 462 O 

Notation: 

YS: Yield Strength at 0.2% Real Strain [MPa] 

TS: Tensile Strength [MPa] 

d/d: Strain Hardening Coefficient at 0.2% Real Strain [1/MPa] 

unif: Uniform Elongation, Expressed as Real (Logarithmic) Strain



tot: Total Elongation, Expressed as Real (Logarithmic) Strain 

Pearl: Pearlite Fraction in Microstructure [%] 

50% ITT: Impact Transition Temperature for 50% Tough Fracture [°C] 

Y: Strain Ageing After 10 Days at Room Temperature [MPa] 


d: Grain Size [mm] 

118 

Source: PICKERING, F.B.: Physical Metallurgy and the Design of Steels. Allied Science Publishers, London, 1978, 

275 p. 

. C-Mn Steels Processed at a Hot Strip Mill 

d 

11.5 

2.2 (6 C Mn 30 P 35 S 23 Al 0.01 (723 Tcoil 

) 0.01 etot 

0.002 Tfin 

100 

Nsol) 

Pearl 

C eq 

0.06 

100 

0.78 

S 

0 

0. 1 

723 

T coil 

YS 

0.016 Pearl 

99.08 (38.2 

S 

0 

5.5 Mn 43 Si 114 P 45 S 31 

N 

12.6 

0.02 

d 

sol 

T fin 

) 

0.004 Pearl 

11.5 

TS 130.47 (19.8 

8.03 Mn 41.4 Si 57.7 P 69 S 262 N 

sol 

) 

S 

0 

d



0.12 

100 (0.000096 Pearl S0 0.05 Mn 4.23 P 4.36 S 2.37 Sn 1.16 N 

sol 

0.0006 T 

fin 

) 

d 

119 

Notation: 



: Total Elongation [%] 

d: Ferrite Grain Size [m] 


Tcoil: Coiling Temperature [°C] 

etot: Total Hot Rolling Conventional Strain [%] 

Tfin: Finishing Temperature [°C] 

Nsol: Solubilized (Free) Nitrogen [%] 

Pearl: Pearlite Fraction Present in Microstructure [%] 

S0: Pearlite Lamelar Spacing [mm] 

C eq 

C 

Mn 

 

6 

Si 

S 

24 

 

T T 

fin 

T 

fin 

coil 

Observations: 

- These equations are valid under the following conditions: Slab Reheating Temperature: 1250°C; Tfin: 

850~880°C; Tcoil: 615~650°C; Final Thickness: 1.8~4.0 mm; C: 0.08~0.18%; Mn: 0.40~1.00%; P < 0.020%; S 

< 0.020%; Si < 0.030%; Al: 0.020~0.050%; N: 0.0030~0.0090%. 

Source: ARTIGAS, A. et al.: Prediction de Propiedades Mecánicas y Microestructurales em Aceros Laminados em 

Caliente. Revista Metalurgica CENIM, 38, 2002, 339-347.



120 

. Hot/Cold Rolled and Annealed Mild Steel 

d 

d 

CR 

CR 

0.013 

0.28 d ( after 60% cold rolling and annealing ) 

HR 

0.011 

0.29 d ( after 70% cold rolling and annealing ) 

HR 

YS 

2.72 

3.37 

 

YS 28.16 

154 d 

HR 

d CR 

 

6.27 

78.5 

yield 

d CR 

n 

0.33 

 

0.01 

d CR 

Notation: 

dCR: Grain Size of Cold Rolled Strip [mm] 

dHR: Grain Size of Hot Rolled Strip [mm] 


yield: Yield Elongation [%] 

n: Strain Hardening Coefficient Measured during Tension Test 

Observations: 

- These equations are valid under the following conditions: C: 0.005~0,10%; Mn: 0.40%; P < 0.016%; S < 

0.026%; Si < 0.010%; Al: < 0.040%; N: 0.0020~0.0040%. 

- Cold rolled steel was box annealed at 700°C; the time of treatment, including heating of the samples, was 

equal to 32 hours, being followed by furnace cooling.



121 

Source: LANGENSCHEID, G. et al.: Untersuchungen über den Einflu der Korngröe des Warmbandes auf die 

Kaltbandeigenschaften. Hoesch Berichte aus Forschung und Entwicklung unserer Werke, 2, 1971, 64-70. 

. Mild Steel, Full Annealed 

5 

n 

10 

1 

d 

Notation: 

n: Strain Hardening Coefficient Measured during Tension Test 


Source: MORRISON, W.: The Effect of Grain Size on the Stress-Strain-Relationship in Low-Carbon Steel. Transactions 

of the ASM, 59, 1966, 824-845. 

. C-Mn Steels with Ferrite-Pearlite Structure (Grozier & Bucher) 

YS 

95.84 

40.68 Mn 70.40 Si 1.517 Pearl 

3.282 

d 

TS 

223.11 

56.74 Mn 101.97 

Si 4.323 Pearl 

2.344 

d 

Notation: 

YS: Yield Strength [MPa] 


Pearl: Pearlite Fraction in Microstructure [%]





122 

Observations: 

- These equations were fitted using at least 50 points of data. 

- Useful range: Mn: 0.00 ~ 1.60%; Si 0.00 ~ 0.80%; Pearl: 0 ~ 80%; d: 0.000252 ~ 0.002770 cm. 

- 95% confidence limits: yield strength, 26MPa; tensile strength, 52 MPa. 

- Eventually pearlite fraction can be calculated with the equation below: 

Pearl 10.7 

110.9 

C 11.3 Mn 48. 4 Si 

which was fitted used 32 points of date of ferritic-pearlitic hot rolled, air cooled and normalized steel , cooled 

in air with a mean cooling rate of 1°C/s at 760°C. Its useful range is 0.00~0.30% C; 0.00~1.80% Mn, 

0.00~0.25% Si and 0~40% Pearl. Its 95% confidence limit is 7%; correlation coefficient r is equal to 0.89. 

Source: GROZIER, J.D. & BUCHER, J.H.: Influence du Niobium et de l’Azote sur la Résistance des Aciers a Structure 

Ferrite-Perlite. Revue de Métallurgie, 63:11, Novembre 1966, 939-941. 

. C-Mn Steels with Ferrite-Pearlite Structure (Pickering) 

YS 

TS 

14.9 

246 4.15 Pearl 44.6 Mn 138 

Si 923 P 169 Sn 3754 N 

sol 

 

d 

492 3.38 Pearl 246 Mn 277 Si 2616 S 723 P 246 Cr 6616 N 

sol 

44.6 

d 

d 

15.4 

385 1.39 Pearl 111 Si 462 P 152 Sn 1369 N 

sol 

d 

d



123 

 

0.27 

0.016 Pearl 0.015 Mn 0.040 Si 0.043 Sn 1. 

0 

unif 

N sol 

1,30 

0.020 Pearl 0.30 Mn 0.20 Si 3.4 S 4.4 P 0.29 Sn 

tot 

0.015 

d 

6. 2 

T trans 

43 1.5 Pearl 37 Mn 

d 

Notation: 



d/d: Strain Hardening Coefficient at 0.2% Real Strain [1/MPa] 

unif: Uniform Elongation, Expressed as Real (Logarithmic) Strain 

tot: Total Elongation, Expressed as Real (Logarithmic) Strain 

Pearl: Pearlite Fraction in Microstructure [%] 

Ttrans: Fracture Appearance Transition Temperature [°C] 



Source: PICKERING, F.B.: The Effect of Composition and Microstructure on Ductility and Toughness; in: Towards 

Improved Ductility and Toughness, Climax Molybdenum Company, Tokyo, 1971, p. 9-32 

. Microalloyed Steels (Hodgson) 

YS 

 

62 .6 

26.1 Mn 60.2 Si 

759.0 P 212.9 Cu 3286.0 

N 

sol 

19.7 

 

d 

 

ppt



124 

TS 

164 .9 634.7 C 53.6 Mn 99.7 Si 

651.9 P 472.6 Ni 3339.4 

N 

sol 

11.0 

 

d 

 

ppt 

 

ppt 

57 log CR 700 V 7800 N 19 

sol 

Notation: 





ppt: Precipitation Strengthening [MPa], only for steels with V [MPa] 

CR: Cooling Rate [°C/s] 

Source: HODGSON, P.D. & GIBBS, R.K. A Mathematical Model to Predict the Mechanical Properties of Hot Rolled C-Mn 

and Microalloyed Steels. ISIJ International, 32:12, December 1992, 1329-1338. 

. Microalloyed Steels (Pickering) 

YS 

 

0 

37 Mn 83 Si 

2918 N 

sol 

15.1 

 

d 

ppt 

Notation: 


0: Friction Stress [MPa] 



ppt: Precipitation Strengthening [MPa], for steels with Nb, Ti and/or V, defined by the formula below [MPa]. 

Observations:



- The Friction Stress 0 value depends on the previous treatment of the material and can be found in the table 

below: 

125 

Condition 0 

[MPa] 

Mean 70 

Air Cooled 88 

Overaged 62 

- The effect of solid solution strengthening from another alloy elements solubilized in ferrite can be included in 

this equation, using the following linear coefficients: 

Element MPa/weight % 

Ni 33 

Cr -30 

P 680 

Cu 38 

Mo 11 

Sn 120 

C 5000 

N 5000 

- The precipitation strengthening contribution is calculated according to the Ashby-Orowan model. 

 

ppt 

5.9 

f x 

ln 

 

x 2.5 10 

4 

 

 

 

Notation: 

ppt: Precipitation Strengthening According to the Ashby-Orowan Model [MPa]



f: Volume Fraction of the Precipitate 

x: Mean Planar Intercept Diameter of the Precipitate [m] 

126 

Observations: 

- Relationship adequate for the calculation of the precipitation strengthening of quench-aged carbides and 

precipitate carbonitrides in Nb, V and Ti steels. 

- ppt can be calculated using a more simplified approach, multiplying the total content of the precipitating 

alloy by the factor B shown in the table below: 

Alloy and 

Precipitate 

Bmax 

[MPa/weight %] 

Bmin 

[MPa/weight %] 

Alloy Range 

[weight %] 

V as V4C3 1000 500 0,00 ~ 0,15 

V as VN 3000 1500 0,00 ~ 0,06 

Nb as Nb(CN) 3000 1500 0,00 ~ 0,05 

Ti as TiC 3000 1500 0,03 ~ 0,18 

Source: PICKERING, F.B. Some Aspects of the Relationships between the Mechanical Properties of Steels and their 

Microstructures. TISCO. Silver Jubilee Volume, Jan-Oct 1980, 105-132. 

. V-Ti-N Steels Processed by Recrystallization Controlled Rolling 

YS 

41.4 

575.20 C (27401 N 2) V 

eq 

ef 

419.5 

h 

f 

TS 

181.5 

74.1 

985.1 Ceq 

(31125 Nef 

39) V 

h 

f 

Notation: 

YS: Yield Strength at 0.2% Real Strain [MPa]




hf: Plate Thickness [mm] 

C Mn Cr Mo Ni Cu 

eq 

C 

 

6 5 15 


127 

N 

ef 

 

N 

tot 

 

Ti 

3.42 

Observations: 

- Formula Derived for Steels with Al Content over 0.010% and Si Content between 0.25 and 0.35%. 

- Precision of the Formulas: 40 MPa. 

Source: MITCHELL, P.S. et al.: Effect of Vanadium on Mechanical Properties and Weldability of Structural Steels. In: 

Low Carbon Steels for the 90’s. Proceedings. American Society for Metals/The Metallurgical Society, 

Pittsburgh, Oct. 1993. 

. Dual Phase Steels 

YS 

203 855 

1 

L 

TS 266 548 

1 1741 

L 

 

f 

d



128 

d 

1 

266 548 1741 

d 

L 

 

f 

d 

 

 

 

unif 

32 64 

1 

L 

 

Notation: 

LE: Yield Strength [MPa] 

LR: Tensile Strength [MPa] 

d/d: Strain Hardening Coefficient at Uniform Elongation [1/MPa] 

aunif: Uniform Elongation [%] 

L : Mean Ferritic Free Path [m] 

d : Mean Diameter of Martensite Islands [m] 

Sources: 

- GORNI, A.A. & BRANCHINI, O.L.G. Análise da Evolução do Encruamento de um Aço Bifásico. In: 4° Simpósio 

de Conformação Mecânica, EPUSP/UNICAMP/ABAL, São Paulo, Nov. 1990, 23-42. 

- GORNI, A.A. & BRANCHINI, O.L.G. Relações Microestrutura-Propriedades Mecânicas em um Aço Bifásico 

Laminado a Quente. In: 1º Seminário sobre Chapas Metálicas para a Indústria Automobilística, ABM/AEA, 

São Paulo, Set. 1992, 127-145. 

. Acicular Ferrite/Low Carbon Bainite Steels 

YS 

88 

37 Mn 83 Si 

2900 

N 

sol 

 

15.1 

d 

L 

 

disc 

ppt



TS 246 1900 C 230 ( Mn Cr) 

185 

Mo 90 W 125 Ni 65 Cu 385 ( V Ti) 

129 

ITT 

11.5 

19 44 Si 700 N 

sol 

0.26 ( 

disc 

 

ppt) 

 

d 

Notation: 



ITT: Impact Transition Temperature for 50% Tough Fracture [°C] 



dL: BainiteFerrite Lath Size [mm] 

disc: Strength Due to Dislocations [MPa] 

ppt: Precipitation Strengthening According to the Ashby-Orowan Model [MPa] 


d: Mean Spacing between High Angle Boundaries (“Packet” or Prior Austenite Grain Boundaries) 

 

disc 

b 

1.2 

10 

3 

( PICKERING ) or 8 10 

4 

( KEH ) 

 

ppt 

5.9 

f x 

ln 

 

x 2.5 10 

4 

 

 

 

Notation: 

: Empirical Constant 

: Shear Modulus [MPa] 

b: Burger’s Vector [cm] 

: Dislocation Density [lines/cm²] 

f: Volume Fraction of the Precipitate 

x: Mean Planar Intercept Diameter of the Precipitate [m]



130 

Sources: 

- PICKERING, F.B. Some Aspects of the Relationships between the Mechanical Properties of Steels and their 


- KEH, A.S., Work Hardening and Deformation Sub-Structure in Iron Single Crystals in Tension at 298K, 

Philosophical Magazine, 12:115, 1965, 9-30. 

. Medium C Steels 

3.8 

1 

 

3 

f 178 

 

 

63 Si N 

sol 

 

17.4 

YS 

3 f 35 58 Mn 

42 

d 

S 

 

 

 

 

 

0 

3.5 

1 

 

3 

f 720 

Si 

 

18.2 

TS 

3 f 246 1140 N 

sol 

 

97 

d 

S 

 

 

 

 

 

 

 

0 

11.5 5.6 13.3 

 

ITT f 

 

46 

762 

d 

S0 

p 

 

6 

1 

f 

 

335 3.48 10 

t 

48.7 Si N 

sol 

Notation: 



ITT: Impact Transition Temperature for 50% Tough Fracture [°C] 

f: Volume Fraction of Ferrite 

d: Ferrite Grain Size [mm] 




S0: Pearlite Lamelar Spacing [mm] 

p: Pearlite Colony Size [mm] 

t: Pearlitic Carbide Lamellar Thickness [mm] 


131 

Sources: 

- GLADMAN, T. e outros. Some Aspects of the Structure-Property Relationships in High Carbon Ferrite-Pearlite 

Steels. Journal of the Iron and Steel Institute, 210, Dec. 1972, 916-930. 

- PICKERING, F.B. Some Aspects of the Relationships between the Mechanical Properties of Steels and their 


. Si Non-Oriented Electrical Steels 

22.0 

YS 34.3 

258 P 34.2 Mn 52. 8 Si 

d 

11.2 

TS 183 506 P 48.7 Mn 109 Si 48.8 Al 2450 B 

d 

0.0412 

YR 0.424 

0.078 Si 0. 170 Al 

d 

Notation: 

YS: Lower Yield Strength [MPa] 


YR: Yield Ratio 

d: Ferrite Grain Size [mm]




132 

Observations: 

- These equations are valid under the following conditions: ULC Steel; Mn: 0.075~0.578%; P < 0.109%; S: 

0.003~0.004%; Si < 0.34%; Al: < 0.432%; N: 0.0014~0.0020%; B < 0.0030%. 

- Cold rolled steel was box annealed at 700°C; the time of treatment, including heating of the samples, was 

equal to 32 hours, being followed by furnace cooling. 

Source: PINOY, L et al. Influence of Composition and Hot Rolling Parameters on the Magnetic and Mechanical Properties 

of Fully Processed Non-Oriented Low-Si Electrical Steels. J. Phys. IV France, 8, 1998, Pr2-487/Pr2-490. 

PT 0.658 

0.474 Si 2.311 Al 25.99 O 12.51C 

123.7 Sinit 

130.2 S 

137.5 

N 5. 266 h 

Notation: 

PT: Core Loss [W/kg] 


h: Thickness [mm] 

Observations: 

- This equation ise valid under the following conditions: C: 0.002~0.040%; S: 0.004~0.015%; N: 

0.003~0.007%. 

- Negative effects of O and N are in direct contradiction with specific experimental results. 

- Adjusted Squared Multiple Correlation: 0.823; Residual Mean Square: 0.082 W²/kg² 

2 

h 

P T 

4.29 

66.4 C 0.0282 GBI 16.2 r 

Notation: 

PT: Total Core Loss at 15 KG [W/kg]



GBI: Number of Grain Boundary Intercepts per mm 

h: Thickness [mm] 

: Density [g/mm³] 

r: Resistivity [.mm] 


133 

Source: LYUDKOVSKY, G. et al. Non-Oriented Electrical Steels. Journal of Metals, January 1986, 18-26.


- Schaeffler Diagram 


134 

Source: Air Products Web Site 

(http://www.airproducts.com/maxx/software/UK/WeldingFaultFinder/wff22413.html).


- Shear Modulus of Steel and its Phases 

. Ferrite 


135 

( T 300) 

64000 1 

 

(-273°C < T < 300°C) 

2235 

 

 

 

( T 300) 

2235 

 

2 

64000 1 0.032 ( T 573) 

(300°C T < 700°C) 

 

 

 

( T 300) 

2235 

 

2 

2 

64000 1 0.032 ( T 573) 0.024 ( T 923) (700°C T < 770°C) 

( T 300) 

69200 1 

 

(770°C T < 911°C) 

1382 

. Austenite 

( T 300) 

81000 1 

 

1989 

(911°C T < 1392°C) 

. Delta Ferrite 

( T 300) 

39000 1 

 

(1392°C T < 1537°C) 

2514


Notation: 

: Shear Modulus [MPa] 



136 

Source: FROST, H.J. & ASHBY, M.F.: Pure Iron and Ferrous Alloys. In: Deformation-Mechanism Maps, The 

Plasticity and Creep of Metals and Ceramics. Pergamon Press, Cambridge, 1982.


- Sheet and Plate Cutting Force and Work 


137 

. Mesquita 

K c 

0. 88 TS 

F t P 

c 

K c 

Notation: 

Kc: Cutting Specific Pressure or Shear Stress [MPa] 


Fc: Cutting Force [N] 

t: Thickness [mm] 

P: Cutting Perimeter [mm] 

Source: MESQUITA, E.L.A. Conformação dos Aços Inoxidáveis. Manual da Acesita. Dezembro 1997, 39 p. 

. Tschaetsch 

F 

W t 

B 

a F t 

W 

1000 

F v 

P 

 

Notation: 

F: Cutting Force [N]



W: Width [mm] 

t: Thickness [mm] 

W: Cutting Work [N.mm] 

a: Mean Force/Maximum Force Ratio (≈ 0.6 for shearing) 

P: Cutting Power [W] 

v: Shear Speed [m/s] 

η: Machine Efficiency (≈ 0.7) 

τB: Shear Stress [MPa], as defined by the table or formulas described in the observation below. 

138 

Observations: 

- The value of τB can be calculated from the following equations, where C is the carbon weight content of steel: 

 

B 

223 550 C 

 

B 

249 786 C 

. Hot rolled or annealed steel (soft) – r² = 0.992, Standard Error of Deviation = 13 MPa: 

. Cold rolled steel (hard) – r² = 0.988, Standard Error of Deviation = 9 MPa: 

- These equations were fitted using the τB data available below, expressed in N/mm²: 

Steel C Mn Si Soft Hard 

St 12 0.10 max 0.50 max - 240 300 

St 13 0.10 max 0.50 max - 240 300 

St 14 0.08 max 0.40 max - 250 320



St 37 0.20 max 1.25 max 0.25 max 310 - 

St 42 0.25 max 1.25 max 0.25 max 400 - 

C10 0.08-0.13 0.30-0.60 0.30 max 280 340 

C20 0.18-0.23 0.30-0.70 0.30 max 320 380 

C30 0.27-0.34 0.50-0.80 0.10-0.40 400 500 

C60 0.57-0.65 0.60-0.90 0.15-0.35 550 720 

139 

Source: TSCHAETSCH, H. Shearing. Metal Forming Practice - Processes, Machines, Tools; Springer-Verlag, Berlin, 

2006, 218-40.


- Specimen Orientation for Mechanical Testing 


140



The first letter denotes the direction of the applied main tensile stress. 

141 

The second letter denotes the direction of crack propagation. 

Source: ASTM Standard E399-06. Standard Test Method for Linear-Elastic Plane-Strain Fracture Toughness KIC 

of Metallic Materials. ASTM International, West Conshohocken, 2006, 32 p.


- Thermal Properties of Steel 

. BISRA 


142 

T 

k 

1008 1023 1040 1524 

75 481.39 485.58 485,58 477,20 

125 502.32 506.51 502.32 493.95 

175 523.25 519.16 514.88 510.69 

225 544.18 531.62 527.44 527.44 

275 556.74 556.74 548.37 544.18 

325 569.30 537.48 569.30 565.11 

375 594.41 598.60 586.04 590.23 

425 623.71 623.71 611.16 615,34 

475 661.39 661.39 648.83 648.83 

525 694.88 703.25 690.69 694.88 

575 740.92 749.29 707.43 740.92 

625 753.48 78697 732.55 753.48 

675 858.13 845.57 770.22 837.20 

725 1138.59 1431.61 1582.31 1448.36 

775 958.59 950.22 602.78 820.46 

825 866.50 736.74 611.16 573.48 

875 648.83 648.83 615.34 581.85 

925 648.83 648.83 623.71 590.23 

975 657.20 648.83 623.71 598.60 

1025 657.20 648.83 632.09 606.97


1075 661.39 648.83 632.09 615.34 

1125 661.39 657.20 640.46 623.71 

1175 665.57 665.57 653.02 632.09 

1225 665.57 678.13 669.76 636.27 

1275 665.57 686.50 686.50 644.64 

T 

c 

1008 1023 1040 1524 

75 481.39 485.58 485.58 477.20 

125 502.32 506.51 502.32 493.95 

175 523.25 519.06 514.88 510.69 

225 544.18 531.92 527.44 527.44 

275 556.74 556.74 548.37 544.18 

325 569.30 573.48 569.30 565.11 

375 594.41 598.60 586.04 590.23 

425 623.71 623.71 611.16 615.34 

475 661.39 661.39 648.83 648.83 

525 694.88 703.23 690.69 694.88 

575 740.92 749.29 707.43 740.92 

625 753.48 786.97 732.55 753.48 

675 858.13 845.57 770.22 837.20 

725 1138.59 1431.61 1582.31 1448.36 

775 958.59 950.22 602.78 820.46 

825 866.50 736.74 611.16 573.48 

875 648.83 648.83 615.34 581.85 


143


925 648.83 648.83 623.71 590.23 

975 657.20 648.83 623.71 598.60 

1025 657.20 648.83 632.09 606.97 

1075 661.39 648.83 632.09 615.32 

1125 661.39 657.20 640.46 623.71 

1175 665.57 665.57 653.06 632.09 

1225 665.57 678.13 669.76 636.27 

1275 665.57 686.50 686.50 644.64 

T 

H 

1008 1023 1040 1524 

0 0 0 0 0 

25 11.459 11.564 11.613 11.407 

75 35.005 35.319 35.467 34.848 

125 59.598 60.121 60.164 59.127 

175 85.237 85.761 85.594 84.243 

225 111.923 112.028 111.652 110.196 

275 139.446 139.237 138.547 136.987 

325 167.597 167.492 167.489 164,719 

375 196.960 196.794 195,372 193.603 

425 227.143 227.352 225.302 223.742 

475 259.270 259.480 256.802 255.346 

525 293.177 293.596 290.290 288.939 

575 329.072 329.909 325.243 324.834 

625 366.432 368.316 361.242 362.194 


144



675 406.722 409.129 398.812 401.961 

725 456.640 466.059 457.625 459.100 

775 509.070 525.605 512.252 515.820 

825 554.697 567.779 542.601 550.668 

875 592.581 602.418 573.263 579.552 

925 625.022 634.859 604.240 608.854 

975 657.673 667.307 635.425 638.574 

1025 690.533 699.742 666.820 668.714 

1075 723.498 732.184 698.425 699.271 

1125 756.567 764.835 730.238 730.248 

1175 789.741 797.904 762.575 761.643 

1225 823.020 831.497 795.644 793.352 

1275 856.299 865.612 829.551 825.375 

145 

Notation: 

k: Thermal Conductivity [W/(m.°C)] 

c: Specific Heat Capacity [J/(kg.ºC)] 

H: Enthalpy [J/kg] 


Observations: 

- Chemical composition of the steels [wt %]: 

Steel C Mn Si P S Cu 

1008 0.08 0.31 0.08 0.029 0.050 - 

1023 0.23 0.64 0.11 0.034 0.034 0.13 

1040 0.42 0.64 0.11 0.031 0.029 0.12



146 

1524 0.23 1.51 0.12 0.037 0.038 0,11 

Source: Physical Constants of Some Commercial Steels at Elevated Temperatures, BISRA/Butterworths Scientific 

Publications, London, 1953, 1-38. 

. Chou 

k 

2 

5 

2 

80.91 

9.9269 10 

T 4.613 10 

T (T ≤ Ar3) 

3 

k 20.14 9.31310 

T (T > Ar3) 

9 

1.10948310 

c 4720.324 4.583364 T 

2 

T 

(800K < T < 1000K) 

c 11501.07 

12. 

476362 T 

(1000K < T < 1042K) 

c 34871.21 

32. 02658 T 

(1042K < T < 1060K) 

9 

5.21765710 

c 10068.18 

5.98686 T 

(1060K < T < 1184K) 

2 

T 

c 429.8495 

0. 1497802 T 

(1084K < T < 1665K) 

Notation: 

k: Thermal Conductivity [J/m.K.s] 

c: Specific Heat Capacity [J/kg.K] 

T: Temperature [K]



Source: SEREDYNSKI, F.: Performance Analysis and Optimization of the Plate-Rolling Process. In: Mathematical 

Process Models in Iron and Steelmaking. Proceedings. Iron and Steel Institute, Amsterdam, 1973. 

147 

. Krzyzanowski 

5 

c p 

422.7 48.66 exp 0.319 10 

 

T 

(T ≤ 700°C) 

c p 

 

T 

657.0 0.084 

1000 

 

 

24,6 

23.16 51.96 exp 2.02519 10 

3 

T 

(T > 700°C) 

 

7850 

6 

2 

1 

0.004 10 

T 3 

Notation: 

cp: Specific Heat [J/kg.°C] 


λ: Thermal Conductivity [J/m.°C.s] 

ρ: Density [kg/m³] 

Source: KRZYZANOWSKI, M. et al.: Finite Element Model of Steel oxide Failure During Tensile Testing Under Hot Rolling 

Conditions. Materials Science and Technology, 15:10, October 1999, 1191-1198. 

. Seredynski 

3 

k 58.6 10 

T 72.5 (T < 810°C)



148 

3 

k 10.75 

10 

T 16.8 (T > 810°C) 

7 

4 

D 0.15 10 

T 0.07825 10 

(700°C < T < 875°C) 

7 

4 

D 0.02667 10 

T 0.02966 10 

(T > 875°C) 

T T 

 

0.12491 

0.38012 

1.0948 

1000 1000 

Notation: 

k: Thermal Conductivity [J/m.°C.s] 

D: Thermal Diffusivity [m²/s] 

: Emissivity 


Observation: 

- Formulas specific for BS En 3 or SAE 1021 steel: 0.17-0.23% C; 0.60-0.90% Mn 

Source: SEREDYNSKI, F.: Performance Analysis and Optimization of the Plate-Rolling Process. In: Mathematical 

Process Models in Iron and Steelmaking. Proceedings. Iron and Steel Institute, Amsterdam, 1973. 

 

. Touloukian 

0,85 

[1 exp(42.68 0,02682 

Notation: 

: Emissivity 

T sup 

) 

0,0115 

]


Tsup: Superficial Temperature [K] 


149 

Source: HARDIN, R.A. et al.: A Transient Simulation and Dynamic Spray Cooling Control Model for Continuous Steel 

Casting. Metallurgical and Materials Transactions B, 34B:6, June 2003, 297-306.


- Thermal Properties of Steel Scale 

. Krzyzanowski 


150 

4 

k 1 

7.83310 

T 

(873K < T < 1473K) 

5 

c 674.969 0.297 T 4.36710 

T 

(600°C < T < 1100°C) 

 

E 240 1 4.7 10 

4 

 

 

T 25 

Notation: 

k: Conductivity [W/m.K] 

c: Specific Heat Capacity [J/kg.°C] 

E: Young’s Modulus [GPa] 


 

Source: KRZYZANOWSKI, M. et al.: Finite Element Model of Steel oxide Failure During Tensile Testing Under Hot Rolling 

Conditions. Materials Science and Technology, 15:10, October 1999, 1191-1198.


- Thermomechanical Processing of Steel 


151



Source: Requirements Concerning Materials and Welding. IACS – International Association of Classification 

Societies Requirement 1975, Revision 2, 2004, 228 p. 

152



- Time-Temperature Equivalency Parameters for Heat Treating 

153 

. Anisothermal Austenitizing 

In this case the Austenitization Time-Temperature Equivalence Parameter in Terms of Grain Size, Pa, is the period of 

heating/cooling time between Tmax and Tmin, where 

Tmax: Maximum Temperature during the Austenitizing Treatment [°C]; 

Tmin: Temperature Calculated According to the Following Equation [°C]: 

T 

min 

T 

max 

 

R T 

H 

2 

max 

a 

Notation: 

R: Molar Gas Constant, 8.314 JK -1 mol -1 

Ha: Activation Energy of Austenitic Grain Coarsening, 460 kJmol -1 for low alloy steels 


1982, 270 p. 

. Isothermal Austenitizing 

P 

a 

 

1 

1 2, 

3 R 

 

Ta 

Ha 

 

log ta 

 

 

Notation: 

Pa: Austenitization Time-Temperature Equivalence Parameter in Terms of Grain Size [K]



154 

Ta: Austenitization Temperature [K] 


ta: Soaking time under Ta 

Ha: Activation Energy of Austenitic Grain Coarsening, 460 kJmol -1 for low alloy steels 


1982, 270 p. 

. Microstructural Banding Homogeneization 

t 

0.05 

 

0.3 l 

D 

0 

e 

2 

Q 

RT 

Notation: 

t0.05: Treatment Time Necessary to Achieve 5% Microstructure Banding [min] 

l: Mean Spacing between Bands [mm] 

Do: Diffusion Constant for the Alloy Element being Considered [cm²/s]: 

- P: 0.01 cm²/s 

- Mn: 0.16 cm²/s 

Q: Activation Energy for the Alloy Element Being Considered [cal/mol]: 

- P: 43700 cal/mol 

- Mn: 62500 cal/mol 


T: Austenitization Temperature [K] 

Source: YIMING, X. et al.: A Metallographic Investigation of Banding and Diffusion of Phosphorus in Steels. 

Microstructural Science, 20, 1993, 457-470.


. Tempering (Hollomon-Jaffe) 


155 

P T 

( c log t) 

c 21.53 

5. 8 C 

Notation: 

P: Hollomon-Jaffe Parameter [K] 

T: Tempering Temperature [K] 

c: Constant Characteristic of the Steel Being Tempered 

t: Soaking time under T [h] 

C: Carbon Content [wt%] 

Observations: 

- Other values for the constant c were proposed by several authors for carbon, microalloyed and low alloy 

steels: 

. 18 (Grange & Baughman) 

. 20 (Larson & Miller, Irvine et al., Thelning) 

Sources: 

- This expression was also deduced by Larson & Miller, which applied it to the study of metal creep. In that case 

c is equal to 20 and P is divided by 1000. Such relationship was also used for the study of hydrogen 

resistance and HAZ hardness of steels. 

- HOLLOMON, J.H. & JAFFE, L.D. Time-Temperature Relations in Tempering Steel. Transactions of the AIME, 

162, 1945, 223-249. 

- LARSON, F.R. & MILLER, J. A Time-Temperature Relationship for Rupture and Creep Stresses. Transactions of 

the American Society of Mechanical Engineers, 74, 1952, 765-775.



- GRANGE, R.A. & BAUGHMAN, R.W. Hardness of Tempered Martensite in Carbon and Low Alloy Steels. 

Transactions of the American Society for Metals, 68, 1956, 165-197. 

- THELNING, K.E.: Steel and its Heat Treatment – Bofors Handbook. Butterworths, London, 1981, 570 p. 

- IRVINE, K.J. et al. Grain-Refined C-Mn Steels. Journal of the Iron and Steel Institute, Feb. 1967, 161-182. 

156


- Welding Effects 


157 

. Weld Interface Cracking Susceptibility during Flash Butt Welding 

F eq 

 

( C 0.03) Si 

 

2 

2 

Mn 

 

10 

(4 Al) 

2 

3 Cr 

 

2 

2 

 

 

 

 

 

 

Notation: 

Feq: Weld Interface Cracking Susceptibility during Flash Butt Welding (No Crack = Zero) 


Source: MIZUI, M. et al.: Application of High-Strength Steel Sheets to Automotive Wheels. Nippon Steel Technical 

Report, 23, June 1984, 19-30. 

. Tensile Strength after Flash Butt Welding 

TS eq 

52 Mn Si Cr V 

C 

30 

5 7 9 2 

 

Notation: 

TSeq: Tensile Strength After Flash Butt Welding [kgf/mm²] 


Source: MIZUI, M. et al.: Application of High-Strength Steel Sheets to Automotive Wheels. Nippon Steel Technical 

Report, 23, June 1984, 19-30.


- Welding Pool Phenomena 


158 

Source: ROGER, F. Modeling Finds the Minimum Energy for the Best Weld. Comsol News, 2010, p. 55-58.


- Young Modulus 

. Definition 


159 

E 

2 G (1 ) 

Notation: 

E: Young Modulus 

G: Shear Modulus 

: Poisson Ratio 



Source: WILSON, A.D. Guidelines for Fabricating and Processing Plate Steel. Bethlehem-Lukens Plate Report, Burns 

Harbor, 2000, 97 p. 

. Steel, High Temperature: Pietrzyk 

 

T T 

E 2.07 

0.87438 10.0906 

 

 

1000 

1000 

 

2 

T 

14.48466 

1000 

 

3 

T 

6.20767 

1000 

 

4 

 

10 

 

5 

Notation: 

E: Young Modulus [MPa] 


Observation: 

- Valid for steel. No information available about the range of valid temperatures.




Thesis, École de Mines de Paris, 2004, p. 243. 

160 

. Steel, High Temperature: Tselikov 

E 

2 

2 

308250 42924 C 144000 

C 20525 Si 5289 Mn 12000 

P 174000 S 225,6 T 0,01379 T 

Notation: 

E: Young Modulus [kgf/cm²] 


Mn: Mn content [weight %] 

Si: Si content [weight %] 

P: P content [weight %] 

S: S content [weight %] 


Observation: 

- Valid for carbon, alloy and stainless steels between 20 and 900°C. 

Source: ROYZMAN, S.E. Thermal Stresses in Slab Solidification. Asia Steel, 1996, 158-162.



APENDIXES 

USEFUL DATA AND CONSTANTS 

161 

Fuels and Combustion Gases: 

- Density (Gas) 

. Natural Gas: 0.81 kg/Nm³ 

. Butane: 2.44 kg/Nm³ 

. Propane: 1.85 kg/Nm³ 

. Liquified Petroleum Gas (LPG): 2.29 kg/Nm³ 

. Air: 1.27 kg/Nm³ 

- Density (Liquid) 

. Butane: 0.58 kg/l 

. Propane: 0.51 kg/l 

. Liquified Petroleum Gas (LPG): 0.54 kg/Nm³ 

. Water: 1.00 kg/Nm³ 

- Heat Capacity in Function of Temperature 

. Heat Capacity [kcal/°C m³] = a + bT [°C]. Values of a and b for some gases are seen below: 

Gas a b 

C2H6 0.600 0.000540 

C3H8 0.871 0.001226 

CH4 0.380 0.000210 

CO 0.302 0.000022 

CO2 0.406 0.000090



H2 0.301 0.000200 

N2 0.302 0.000022 

O2 0.320 0.000059 

162 

- Net Heating Value 

. Acetylene (C2H2): 13412 Kcal/Nm³ 

. Basic Oxygen Steelmaking Off-Gas (OG Gas): 770 kcal/Nm³ 

. Blast Furnace Gas: 770 Kcal/Nm³ 

. Benzene (C6H6): 33,823 Kcal/Nm³ 

. Butane (C4H10): 29,560 Kcal/Nm³ 

. Butene/Buthylene (C4H8): 27,900 Kcal/Nm³ 

. Charcoal: 6,800 kcal/kg 

. Carbon Monoxide (CO): 3,019 Kcal/Nm³ 

. Coke Oven Gas: 4,400 Kcal/Nm³ 

. Diesel Oil: 10.200 kcal/kg 

. Electricity: 860 kcal/kW 

. Ethane (C2H6): 15,236 Kcal/Nm³ 

. Ethene/Ethylene (C2H4): 14,116 Kcal/Nm³ 

. Fuel Oil: 8,640~9,000 kcal/l or 9,600 ~ 10,000 kcal/kg 

. Hexane (C6H14): 41,132 Kcal/Nm³ 

. Hydrogen (H): 2,582 Kcal/Nm³ 

. Hydrogen Sulfide (H2S): 5,527 Kcal/Nm³ 

. i-Butane (C4H10): 28,317 Kcal/Nm³ 

. i-Pentane (C5H12): 34,794 Kcal/Nm³ 

. Liquified Petroleum Gas (LPG): 25,300 ~ 27,300 kcal/Nm³ 

. Methane (CH4): 8,557 Kcal/Nm³ 

. Natural Gas: 9,000 ~ 9,400 Kcal/Nm³ 

. Pentane (C5H12): 34,943 Kcal/Nm³ 

. Propane (C3H8): 21,809 Kcal/Nm³ 

. Propene/Propylene (C3H6): 20,550 Kcal/Nm³ 

. Toluene (C7H8): 40,182 Kcal/Nm³ 

. Xylene (C8H10): 46,733 Kcal/Nm³


. Wood: 2,500 kcal/kg 

- Typical Chemical Compositions 


163 

[% vol] N2 H2 CH4 C2H6 C3H8 CO CO2 O2 

Coke Oven Gas 3.09 61.55 24.54 0.42 0.06 8.04 0.00 0.26 

Natural Gas 1.83 ---- 87.91 7.08 1.91 ---- 0.59 0.00 

Mathematical Constants 

- e: 2.718281828 

- Pi: 3.141592654 

Physical Constants 

- Acceleration of gravity: g = 9.805 m/s² 

- Avogrado: NA = 6.022 x 10 23 1/mol 

- Boltzmann: k = 1.38065 x 10 -23 J/K 

- Ideal Gas Constant R: 

. 1.98717 cal/(K mol) 

. 82.056 cm³ atm/(K.mol) 

. 0.082056 l atm/(K.mol) 

. 8.31433 x 10 7 erg/(K.mol) 

. 8.31433 J/(K.mol) 

- Stefan-Boltzmann: σ = 5.6704 x 10 -8 W/(m² K²) 

Physical Properties of Scale (Iron Oxide)



164 

- Thermal Conductivity: 

. Industrial Scale: 3.0 W/(m.K) 

. Hematite (Fe2O3): 1.2 W/(m.K) 

. Magnetite (Fe3O4): 1.5 W/(m.K) 

. Wustite (FeO): 3.2 W/(m.K) 

- Density: 

. Industrial Scale: 

- 4.86 g/cm³ (Combustol) 

- 5.00 g/cm³ (Picqué) 

- 5.70 g/cm³ (Krzyzanowski) 

. Hematite (Fe2O3): 4.90 g/cm³ 

. Magnetite (Fe3O4): 5.00 g/cm³ 

. Wustite (FeO): 5.70 g/cm³ 

. Bulk, Porous Scale as Raw Material, Room Temperature: 

- 2.40~2.89 t/m³ 

- Stowage Factor: 0,38 m³/t 

- Hardness: 

. Hematite (Fe2O3): 1000 HV 

. Magnetite (Fe3O4): 320 ~ 500 HV 

. Wustite (FeO): 270 ~ 350 HV 

- Iron Content in Scale: 74.6% (Stoichiometric) 

- Linear Coefficient of Thermal Contraction: 

. Fe: 19 x 10 -6 m/°C 

. Wustite: 14 x 10 -6 m/°C 

- Melting Point: 

. Fayalite (2FeO.SiO2): 1177°C 

- Specific Heat: 

. Industrial Scale: 766 J/(kg.K) (600°C) 

. Hematite (Fe2O3): 980 J/(kg.K) 

. Magnetite (Fe3O4): 870 J/(kg.K) 

. Wustite (FeO): 725 J/(kg.K)


- Thermal Expansion Coefficient: 

. Ferrite (0 ~ 900°C): 15.3 x 10 -6 K -1 

. Hematite (Fe2O3): 

- 20 ~ 900°C: 14.9 x 10 -6 K -1 

- 100 ~ 300°C: 10.8 x 10 -6 K -1 

- 100 ~ 1000°C: 12.2 x 10 -6 K -1 

- 400 ~ 800°C: 13.0 x 10 -6 K -1 

. Magnetite (Fe3O4): 1.5 W/(m.K) 

- 25°C: 11.0 x 10 -6 K -1 

- 400°C: 14.0 x 10 -6 K -1 

- 550°C: 27.0 x 10 -6 K -1 

. Wustite (FeO): 

- 100 ~ 1000°C: 12.2 x 10 -6 K -1 

- 400 ~ 800°C: 15.0 x 10 -6 K -1 


165 

Physical Properties of Steel and its Microstructural Constituents 

- Densities: 

. Bulk Steel: 7850 kg/m³ 

. Ferrite (Fe ): 

- Caballero: 7882 kg/m³ 

- Jablonka: 7870 kg/m³ (20°C) 

. Cementite (Fe3C): 

- Caballero: 7687 kg/m³ 

- Jablonka: 7685 kg/m³ (20°C) 

. NbC: 7790 kg/m³ 

. VC: 5700 kg/m³ 

- Electrical Resistivity at 15.6°C: 17 x 10 -8 Ω.m 

- Emissivity of Polished Metal Surface: 

. 0.07 @ 38°C



. 0.10 @ 260°C 

. 0.14 @ 540°C 

- Emissivity of Oxidized Steel Plate at 15.6°C: 0.80 

- Heat Transfer Coefficient at Scale/Steel Interface: 30,000 W/m².K 

- Lattice Parameters (Ambient Temperature): 

. Ferrite (pure Fe): 2.866 Ǻ 

. Cementite: a = 4.5246 Ǻ, b = 5.0885 Ǻ, c = 6.7423 Ǻ 

- Linear Coefficient of Thermal Expansion: 

. Bulk: 11.7 x 10 -6 °C -1 

. Ferrite: 1.244 x 10 -5 °C -1 

. Austenite: 2.065 x 10 -5 l°C -1 

- Melting Point: 1300 ~ 1450°C 

- Modulus: 

. Bulk: 159,000 MPa 

. Shear: 83,000 MPa 

. Young: 207,000 MPa 

- Poisson’s Ratio: 



- Specific Heat: 0.12 cal/g.°C 

- Speed of Sound through Steel: 5,490 m/s 

- Thermal Conductivity at 15.6°C: 58.9 W/m.K 

- Volumetric Coefficient of Thermal Expansion: 35.1 x 10 -6 °C -1 

166 

Sources: 

- AL-OTAIBI, S. Recycling Steel Mill Scale as Fine Aggregate in Cement Mortars. European Journal of Scientific 

Research, 24:3, 2008, 332-338. 

- ANON.: Practical Data for Metallurgists. The Timken Company, Canton, September 2006, 130 p.



167 

- CABALLERO, F.G. et al. Modelling of Kinetics and Dilatometric Behaviour of Austenite Formation in a Low- 

Carbon Steel with a Ferrite Plus Pearlite Initial Microstructure. Journal of Materials Science, 37, 2002, 3533- 

3540. 

- HEIDEMANN, M. Fours Pits - Établissement de Bilans Thermiques Globaux et Étages dans les Temps. Rapport 

IRSID 77-02, Saint-Germain-en-Laye, Juillet 1976, 15 p. 

- JABLONKA, A.: Thermomechanical Properties of Iron and Iron-Carbon Alloys: Density and Thermal Contraction, 


- KRZYZANOWSKI, M. et al.: Finite Element Model of Steel oxide Failure During Tensile Testing Under Hot Rolling 

Conditions. Materials Science and Technology, 15:10, October 1999, 1191-1198. 

- METNUS, G.E. et al. Comparing CO2 Emissions and Energy Demands for Alternative Ironmaking Routes. Steel 

Times International, March 2006, 32-36. 

- PICQUÉ, B. Experimental Study and Numerical Simulation of Iron Oxide Scales Behavior in Hot Rolling. Doctor 

Thesis, École de Mines de Paris, 2004, p. 248. 

- SCHÜTZE, M. Protective Scales and Their Breakdown. John Wiley & Sons-The Institute of Corrosion, 

Chichester, 1997, 184 p. 

- WILSON, A.D. Guidelines for Fabricating and Processing Plate Steel. Bethlehem-Lukens Plate Report, Burns 

Harbor, 2000, 97 p.



UNIT CONVERSIONS 

168 

From Multiply by To 

A 10 -10 m 

bar 1.019716 kg/cm² 

BTU 1058.201058 J 

BTU 251.9958 cal 

cal 4.184 J 

F 5/9 (°F-32) °C 

ft 12 inch 

ft 0.30485126 m 

ft.lb 1.356 J ou N.m 

ft.lb 0.324 cal 

ft.lb 1.355748373 J 

ft.lb/s 1.355380862 W 

ft.lbf 1.355818 J or N.m 

ft.lbf 0.1382 kg 

ft² 92.90 x 10 -3 m² 

ft³ 0.02831685 m³ 

gallon 3.78541178 liters 

HP 0.7456999 kW 

HP 745.7121551 W 

in 25.4 mm 

in² 645.2 mm² 

in³ 16387.064 mm³ 

in-lb/in² 0.000175127 J/mm² 

J 9.45 x 10 -4 BTU 

J 0.2390 cal 

J 0.7376 ft.lb 


J 2.389 x 10 -7 th 

J 1 W.s 

J 2.777 x 10 -9 kWh 

Kcal/m² h °C 1,163 W/m² °C 

Kg 2.205 lb 

kgf 9.80665 N 

kgf.m 9.80665 J 

kgf/mm² 9.80665 MPa 

Kip 1000 lbf 

kN 224.8 lbf 

kN/mm 5.71 x 10 3 lbf/ft 

ksi 6.894757 MPa 

ksi 1000 psi 

ksi.√in 1.098901099 MPa.√m 

kW 1.341022 HP 

kW 0.860 th/h 

kW.h 3.6 x 10 6 J 

kW.h 3.412 x 10 3 BTU 

kW.h 8.6 x 10 5 cal 

lb 0.4535924 kg 

lb.in 0.1129815 J ou N.m 

lb/ft³ 0.016020506 g/cm³ 

lb/in³ 27.67783006 g/cm³ 

lbf 4.448222 N 

lbf/in² 1 psi 

long ton 1016.047 kg



M 10 10 A 

M 3.281 ft 

m² 10.76 ft² 

MBTU 1,000,000 BTU 

mm 0.0394 in 

mm² 1.550 x 10³ in² 

MMBTU 1,000,000 BTU 

MPa 1 N/mm² 

MPa 0.145 ksi 

MPa 145 lbf/in² 

MPa.m 910.06 psi.in 

MPa.m 0.920 Ksi.in 

N.m 1 J 

nm 10 -9 m 

oz 0.028352707 kg 

Pa 1 N/m² 



Pa 1.449 x 10 -4 psi 

Pa 1.020 x 10 -7 Kg/mm² 

pct (%) 10000 ppm 

ppm 0.0001 % 

psi 0.001 ksi 

psi 0.0068947573 MPa 

psi 0.0007030697 kgf/mm² 

Psi in 0.001098829 MPa m 

Rad 57.2958 ° 

Short Ton 907.1847 Kg 

Th 1 Mcal 

Th 4.186 x 10 6 J 

th/h 1.163 kW 

W 1 J/s 

W 0.001341 HP 

169



GENERAL STATISTICAL FORMULAS 

170 

- Correlation Coefficient 

r 

_ 

( Yest 

Y) 

_ 

2 

( Y Y) 

2 

Notation: 

r: Correlation Coefficient 

Y: Raw Data 

Yest: Estimated Data Calculated by the Fitted Equation 

Y _ : Mean of the Raw Data 

Source: SPIEGEL, M.R. Estatística, Editora McGraw-Hill do Brasil Ltda., São Paulo, 1976, 580 p. 

- Standard Error of Estimation 

 

( Y Y 

est 

n 

real 

2 

) 

Notation: 

: Standard Error of Estimation 

Yest: Estimated Data Calculated by the Fitted Equation 

Yreal: Real Data


n: Number of Points of Data 


171 

Source: SPIEGEL, M.R. Estatística, Editora McGraw-Hill do Brasil Ltda., São Paulo, 1976, 580 p.

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