Metallurgical Transactions A

, Volume 24, Issue 4, pp 795–808 | Cite as

Mathematical model of phase transformations and elastoplastic stress in the water spray quenching of steel bars

  • Y. Nagasaka
  • J. K. Brimacombe
  • E. B. Hawbolt
  • I. V. Samarasekera
  • B. Hernandez-Morales
  • S. E. Chidiac
Transformations
Received:

Abstract

A mathematical model, based on the finite-element technique and incorporating thermo-elasto-plastic behavior during the water spray quenching of steel, has been developed. In the model, the kinetics of diffusion-dependent phase transformation and martensitic transformation have been coupled with the transient heat flow to predict the microstructural evolution of the steel. Furthermore, an elasto-plastic constitutive relation has been applied to calculate internal stresses resulting from phase changes as well as temperature variation. The computer code has been verified for internal consistency with previously published results for pure iron bars. The model has been applied to the water spray quenching of two grades of steel bars, 1035 carbon and nickel-chromium alloyed steel; the calculated temperature, hardness, distortion, and residual stresses in the bars agreed well with experimental measurements. The results show that the phase changes occurring during this process affect the internal stresses significantly and must be included in the thermomechanical model.

Keywords

Ferrite Austenite Martensite Residual Stress Metallurgical Transaction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

b

kinetic constant in Avrami equation, s-1

B

nozzle diameter, mm

[B]

strain-displacement matrix, m-1

C

specific heat, J kg-1 °C-1

[C]

heat capacitance matrix, J °C-1

{d}

nodal displacement vector, m

[D]

material properties matrix, MPa

[Dp] [D]

matrix for plasticity, MPa

[D] [D]

matrix for elasticity, MPa

f

von Mises yield function, MPa2

{f1},{f2}

elements of force vector, MPa {f3}, {f4}

{f}

heat flow vector, W

h

heat-transfer coefficient, W m-2 °C-1

H

nozzle to object distance, mm

k

thermal conductivity, W m-1 K-1

Ki

empirical constant in Eq. [10], MPa mm mm-1

[K]

conductance matrix, W °C-1

[K]

stiffness matrix, MPa m-1

Ms

martensite start temperature, °C

n

kinetic constant in Avrami equation

n

vector normal to the surface of a boundary, m

[N]

shape function

Nu

Nusselt number

Pr

Prandtl number

Q

heat released per unit time and volume, Wm-3

Re

Reynolds number

r

radial coordinate, m

R

radius, m

rmin

fraction of ΔT to allow only one element to yield

Se

boundary of a finite element, m

t

time, s

tAvCCT

transformation start time under continuous cooling, s

Δt

time step, s

ΔH

enthalpy of transformation, J kg-1

T

temperature, °C

Tc

ambient temperature, °C

T0

initial temperature, °C

U

velocity, m s-1

Ve

volume of a finite element, m3

X

fraction transformed

z

axial coordinate, m

α

thermal expansion coefficient, m m-1 °C-1

β

transformation expansion coefficient, m m-1

λ

positive constant defined by Eq. [B9]

ξi

volume fraction of phasei

ρ

density, kg m-3

Φ

vector of nodal temperatures, °C

σ

stress, MPa

θ

parameter used to discretize Eq. [A1]

{gs’}

deviatoric stress, MPa

{dσ}

stress increment, MPa

{dε}

total strain increment, m m-1

{dεo}

initial strain increment, m m-1

{dεe}

elastic strain increment, m m-1

{dεth}

thermal strain increment, m m-1

{dεtr}

transformation strain increment, m m-1

{dεp}

plastic strain increment, m m-1

{dεtp}

transformation plasticity strain increment, m m-1

{dεth’}

elastic strain increment due to variations in mechanical properties as function of temperature, mm-1

{dεtr’}

elastic strain increment due to variations in mechanical properties as function of phase composition, m m-1

This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    T. Fukuda, T. Takayama, N. Hamasaka, K. Okawa, K. Tsuda, and H. Ikeda:J. Jpn. Soc. for Heat Treatment, 1989, vol. 29, pp. 296–301.Google Scholar
  2. 2.
    T. Hara:Tetsu-to-Hagané, 1963, vol. 49, pp. 1669–1775.Google Scholar
  3. 3.
    K. Inoue, K. Haraguchi, and S. Kimura:Trans. ISIJ, 1978, vol. 18 (1), pp. 11–15.Google Scholar
  4. 4.
    Y. Toshioka:Tetsu-to-Hagané, 1976, vol. 62, pp. 1756–66.Google Scholar
  5. 5.
    H. Fujio, T. Aida, and U. Matsumoto:Bull. JSME, 1977, vol. 20, pp. 1051–57.Google Scholar
  6. 6.
    T. Inoue and K. Tanaka:J. Soc. Mater. Sci. Jpn., 1972, vol. 22, pp. 218–23.Google Scholar
  7. 7.
    K. Okamura and H. Kawashima:J. Jpn. Soc. for Heat Treatment, 1987, vol. 28, pp. 141–48.Google Scholar
  8. 8.
    B. Hildenwall and T. Ericsson: Report of the Institute of Technology No. R-098, Linköping University, Linköping, Sweden, 1977.Google Scholar
  9. 9.
    T. Inoue and Z. Wang:Mater. Sci. Technol., 1985, vol. 1 (10), pp. 845–50.Google Scholar
  10. 10.
    Z.G. Wang and T. Inoue:J. Soc. Mater. Sci. Jpn., 1983, vol. 32, pp. 991–96.Google Scholar
  11. 11.
    T. Kishino, S. Nagaki, and T. Inoue:J. Soc. Mater. Sci. Jpn., 1979, vol. 28, pp. 861–67.Google Scholar
  12. 12.
    S. Denis, A. Simon, and G. Beck:Trans. ISIJ, 1982, vol. 22 (7), pp. 504–13.Google Scholar
  13. 13.
    K. Miyao, Z.G. Wang, and T. Inoue:J. Soc. Mater. Sci. Jpn., 1986, vol. 35, pp. 1352–57.Google Scholar
  14. 14.
    S. Sjöstrom:Mater. Sci. Technol., 1985, vol. 1 (10), pp. 823–29.Google Scholar
  15. 15.
    F. Abbasi and A.J. Fletcher:Mater. Sci. Technol., 1985, vol. 1 (10), pp. 830–37.Google Scholar
  16. 16.
    S. Denis, S. Sjöström, and A. Simon:Metall. Trans. A, 1987, vol. 18A, pp. 1203–12.Google Scholar
  17. 17.
    A. Johnson and R.F. Mehl:Trans. AIME, 1939, vol. 135, pp. 416–58.Google Scholar
  18. 18.
    M. Avrami:J. Chem. Phys., 1939, vol. 7, pp. 1103–12.CrossRefGoogle Scholar
  19. 19.
    M. Avrami:J. Chem. Phys., 1940, vol. 8, pp. 212–24.CrossRefGoogle Scholar
  20. 20.
    M. Avrami:J. Chem. Phys., 1941, vol. 9, pp. 177–84.CrossRefGoogle Scholar
  21. 21.
    D.P. Koistinen and R.E. Marburger:Acta Metall., 1957, vol. 7 (1), pp. 59–60.Google Scholar
  22. 22.
    G.W. Greenwood and R.W. Johnson:Proc. R. Soc., 1965, vol. 238A, pp. 403–22.Google Scholar
  23. 23.
    C.L. Magee and M.W. Paxton:Trans. AIME, 1968, vol. 242, pp. 1741–49.Google Scholar
  24. 24.
    F. Kreith and W.Z. Black:Basic Heat Transfer, Harper and Row, New York, NY, 1980.Google Scholar
  25. 25.
    O.C. Zienkiewicz:The Finite Element Method, 3rd ed., McGraw-Hill Book Co. Ltd., London, 1977.Google Scholar
  26. 26.
    G. Yagawa and N. Miyazaki:Thermal Stress, Creep, Heat Conduction Analysis by Finite Element Method, Science Co., Tokyo, Japan, 1985.Google Scholar
  27. 27.
    E.B. Hawbolt, B. Chau, and J.K. Brimacombe:Metall. Trans. A, 1983, vol. 14A, pp. 1803–15.Google Scholar
  28. 28.
    British Steel Corporation:Atlas of Continuous Cooling Transformation Diagrams for Engineering Steels, British Steel Corporation, Sheffield, United Kingdom, 1977.Google Scholar
  29. 29.
    P.C. Campbell: Ph.D. Thesis, The University of British Columbia, Vancouver, 1989.Google Scholar
  30. 30.
    E.B. Hawbolt, B. Chau, and J.K. Brimacombe: Sixth Research Report of AISI, 1984.Google Scholar
  31. 31.
    P.K. Agarwal and J.K. Brimacombe:Metall. Trans. B, 1981, vol. 12B, pp. 121–33.Google Scholar
  32. 32.
    Y. Yamada and N. Yoshimura:Int. J. Mech. Sci., 1968, vol. 10, pp. 343–54.CrossRefGoogle Scholar
  33. 33.
    S. Ueda and T. Yamakawa:J. Jpn. Welding Soc., 1973, vol. 42, pp. 567–77.Google Scholar
  34. 34.
    W. Mitter, F.G. Rammerstofer, O. Grundler, and G. Wiedner:Mater. Sci. Technol., 1985, vol. 1 (10), pp. 793–97.Google Scholar
  35. 35.
    H. Bühler, H. Buchholz, and E.H. Schulz:Arch. Eisenhüttenwes., 1932, vol. 5, p. 413.Google Scholar
  36. 36.
    G. Sachs:Z. Metallkd., 1927, vol. 19, pp. 352–57.Google Scholar
  37. 37.
    British Iron and Steel Research Association:Atlas of Isothermal Transformation Diagrams of B.S. En. Steels, 2nd ed., Iron and Steel Institute, London, 1956.Google Scholar
  38. 38.
    J. Giusti: Ph.D. Thesis, University of Paris VI, 1981.Google Scholar
  39. 39.
    J.B. Leblond: Framatome Internal Report No. TM/C DC/80.066, Société Framatome, Saint-Marcel, France, 1980.Google Scholar
  40. 40.
    J.B. Leblond, G. Mottet, J. Devaux, and J.-C. Devaux:Mater. Sci. Technol., 1985, vol. 1, pp. 815–22.Google Scholar
  41. 41.
    British Iron and Steel Research Association:Physical Constants of Some Commercial Steels at Elevated Temperatures, Butterworth’s Scientific Publications, London, 1953.Google Scholar
  42. 42.
    The Japan Society of Mechanical Engineers:The Data of Heat Transfer, 1975, p. 111.Google Scholar
  43. 43.
    F. Wever and A. Rose:Stahl Eisen, 1954, vol. 74, pp. 748–55.Google Scholar
  44. 44.
    J.K. Brimacombe:Metall. Trans. B, 1989, vol. 20B, pp. 291–313.Google Scholar

Copyright information

© The Minerals, Metals and Materials Society, and ASM International 1993

Authors and Affiliations

  • Y. Nagasaka
    • 1
  • J. K. Brimacombe
    • 2
    • 3
  • E. B. Hawbolt
    • 3
  • I. V. Samarasekera
    • 3
  • B. Hernandez-Morales
    • 3
  • S. E. Chidiac
    • 4
  1. 1.Manufacturing Engineering Research DepartmentKomatsu Ltd.OsakaJapan
  2. 2.Materials Process EngineeringJapan
  3. 3.Centre for Metallurgical Process EngineeringThe University of British ColumbiaVancouverCanada
  4. 4.Institute for Research in ConstructionNational Research Council of CanadaOttawaCanada

Personalised recommendations