Metals and Materials International

, Volume 24, Issue 2, pp 265–281 | Cite as

Extended Hall–Petch Relationships for Yield, Cleavage and Intergranular Fracture Strengths of bcc Steel and Its Deformation and Fracture Behaviors

Article

Abstract

Extended Hall–Petch relationships for yield (\( \sigma_{y} \)), cleavage (\( \sigma_{\text{cl}} \)) and intergranular fracture (\( \sigma_{\text{ig}} ) \) strengths of pure iron have been established through the direct calculation of the proportional constant \( (k) \) and the estimation of the friction stress \( (\sigma_{0} ) \). The magnitude orders of \( k \) and \( \sigma_{0} \) are generally \( k_{y} < k_{\text{cl}} < k_{\text{ig}} \) and \( \sigma_{y0} < \sigma_{\text{cl0}} < \sigma_{\text{ig0}} \), respectively. Based on the Hall–Petch relationships, micro-yielding in a bcc steel occurs at the instance that the pile-up dislocations within a specific grain showing the Schmid factor of 0.5 propagate into the neighboring grain. The initial brittle crack is formed at the instance that the flow strength exceeds the brittle fracture strength. Once the brittle crack is formed, it grows catastrophically. Due to the smallest and \( k_{y} \) and \( \sigma_{\text{y0}} \), the cleavage and the intergranular fracture occur always after micro-yielding. The {100} cleavage fracture of the steel is due to the lowest theoretical {100} cleavage strength. Due to the thermal components included in cleavage and intergranular fracture strengths, they show also the temperature and strain rate dependence observed in yield strength. The increase in susceptibility to brittle fracture with decreasing temperature and increasing strain rate is due to the increase in dislocation density which causes the high work hardening rate.

Keywords

Extended Hall–Petch relationships Hall–Petch coefficient Friction stress Micro-yielding Ductile–brittle transition 

Notes

Acknowledgments

The authors are grateful to Mr. Y. S. Shin and Mr. H. J. Sung for mechanical tests and Mrs. J. H. Yoon for AES analyses.

References

  1. 1.
    N.J. Petch, J. Iron Steel Inst. 174, 25 (1953)Google Scholar
  2. 2.
    M. Etou, S. Fukushima, T. Sasaki, Y. Haraguchi, K. Miyata, M. Wakita, T. Tomida, N. Imai, M. Yoshida, Y. Okada, ISIJ Int. 48, 1142 (2008)CrossRefGoogle Scholar
  3. 3.
    A. Cracknell, N.J. Petch, Acta Metall. 3, 186 (1955)CrossRefGoogle Scholar
  4. 4.
    W. Sylwestrowicz, E.O. Hall, Proc. Phys. Soc. B 64, 495 (1951)CrossRefGoogle Scholar
  5. 5.
    E.O. Hall, Proc. Phys. Soc. B 64, 747 (1951)CrossRefGoogle Scholar
  6. 6.
    J. Heslop, N.J. Petch, Philos. Mag. 3, 1128 (1958)CrossRefGoogle Scholar
  7. 7.
    J. Harding, Acta Metall. 17, 949 (1969)CrossRefGoogle Scholar
  8. 8.
    J. Heslop, N.J. Petch, Philos. Mag. 1, 866 (1956)CrossRefGoogle Scholar
  9. 9.
    N.J. Petch, Philos. Mag. 3, 1089 (1958)CrossRefGoogle Scholar
  10. 10.
    V.F. Moiseev, V.I. Irefilov, Phys. Status Solidi 18, 881 (1966)CrossRefGoogle Scholar
  11. 11.
    J.D.S. Sumpter, J.S. Kent, Mar. Struct. 17, 575 (2004)CrossRefGoogle Scholar
  12. 12.
    K. Felkins, H.P. Leighly, A. Jankovic, JOM 50, 12 (1998)CrossRefGoogle Scholar
  13. 13.
    A. Kelly, W.R. Tyson, A.H. Cottrell, Philos. Mag. 15, 567 (1967)CrossRefGoogle Scholar
  14. 14.
    J.R. Rice, R. Thomson, Philos. Mag. 29, 73 (1974)CrossRefGoogle Scholar
  15. 15.
    P.B. Hirsch, S.G. Roberts, J. Samuels, Proc. R. Soc. Lond. A 421, 25 (1989)CrossRefGoogle Scholar
  16. 16.
    L. Vitos, Computational Quantum Mechanics for Materials Engineers (Springer, New York, 2007)Google Scholar
  17. 17.
    P. Soven, Phys. Rev. 156, 809 (1967)CrossRefGoogle Scholar
  18. 18.
    C. Kresse, J. Hafner, Phys. Rev. B 47, 558 (1993)CrossRefGoogle Scholar
  19. 19.
    G. Kresse, J. Furthmuller, Phys. Rev. B 54, 11169 (1996)CrossRefGoogle Scholar
  20. 20.
    G. Kresse, D. Joubert, Phys. Rev. B 59, 1758 (1999)CrossRefGoogle Scholar
  21. 21.
    E. Smith, J.T. Barnby, Met. Sci. 1, 56 (1967)CrossRefGoogle Scholar
  22. 22.
    A.N. Stroh, Proc. R. Soc. A 223, 404 (1954)CrossRefGoogle Scholar
  23. 23.
    N.T. Barrett, The Principles of Engineering Materials (Prentice-Hall, Englewood Cliffs, 1973)Google Scholar
  24. 24.
    F.R.N. Nabarro, Mater. Sci. Eng. A 234, 67 (1997)CrossRefGoogle Scholar
  25. 25.
    Y. Kamimura, K. Edagawa, S. Takeuchi, Acta Mater. 61, 294 (2013)CrossRefGoogle Scholar
  26. 26.
    E. Orowan, Rep. Prog. Phys. 12, 185 (1949)CrossRefGoogle Scholar
  27. 27.
    W. Köster, H. Franz, Metall. Rev. 6, 1 (1961)Google Scholar
  28. 28.
    M.P. Seah, E.D. Hondros, Proc. R. Soc. Lond. A 335, 191 (1973)CrossRefGoogle Scholar
  29. 29.
    S.A. Kim, W.L. Johnson, Mater. Sci. Eng. A 452–453, 633 (2007)CrossRefGoogle Scholar
  30. 30.
    N.H. Heo, J.W. Nam, Y.-U. Heo, S.-J. Kim, Acta Mater. 61, 4022 (2013)CrossRefGoogle Scholar
  31. 31.
    J.W. Morris Jr., C.S. Lee, Z. Guo, ISIJ Int. 43, 410 (2003)CrossRefGoogle Scholar
  32. 32.
    S. Morito, T. Ogawa, T. Furuhara, T. Maki, ISIJ Int. 46, 91 (2005)CrossRefGoogle Scholar
  33. 33.
    S. Morito, H. Yoshida, T. Maki, X. Huang, Mater. Sci. Eng. A 438–440, 237 (2006)CrossRefGoogle Scholar
  34. 34.
    J. Hidalgo, M.J. Santofimia, Metall. Mater. Trans. A 47, 5288 (2016)CrossRefGoogle Scholar
  35. 35.
    W.C. Leslie, Metall. Trans. 3, 5 (1972)CrossRefGoogle Scholar
  36. 36.
    G. Wand, S. Schonecker, S. Hertzman, Q.-M. Hu, B. Johansson, Phys. Rev. B 91, 224203 (2015)CrossRefGoogle Scholar
  37. 37.
    M.P. Seah, Acta Metall. 28, 955 (1980)CrossRefGoogle Scholar
  38. 38.
    R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K. Kelley, Selected Values of the Thermodynamic Properties of Elements and Selected Values of the Thermodynamic Properties of Binary Alloys (ASM, Metals Park, 1973)Google Scholar
  39. 39.
    N.H. Heo, Age Hardening and DuctileBrittleDuctile Transition in FeMnNiX Alloys. Ph.D. thesis (1993)Google Scholar
  40. 40.
    D. McLean, Grain Boundaries in Metals (Oxford University Press, London, 1957)Google Scholar
  41. 41.
    N.H. Heo, Acta Mater. 44, 2015 (1996)Google Scholar
  42. 42.
    J.W. Morris Jr., Z. Guo, C.R. Krenn, Y.-H. Kim, ISIJ Int. 41, 599 (2001)CrossRefGoogle Scholar
  43. 43.
    R.E. Mistler, R.L. Coble, J. Appl. Phys. 45, 1507 (1974)CrossRefGoogle Scholar
  44. 44.
    A. Inoue, H. Nitta, Y. Iijima, Acta Mater. 55, 5910 (2007)CrossRefGoogle Scholar
  45. 45.
    J. Benito, J. Jorba, J.M. Manero, A. Roca, Metall. Mater. Trans. A 36, 3317 (2005)CrossRefGoogle Scholar
  46. 46.
    H.M. Ledbetter, S.A. Kim, Mater. Sci. Eng. A 101, 87 (1988)Google Scholar
  47. 47.
    J.A. Rayne, B.S. Chandrasekhar, Phys. Rev. 122, 1714 (1961)CrossRefGoogle Scholar
  48. 48.
    R. Armstrong, I. Codd, R.M. Douthwaite, N.J. Petch, Philos. Mag. 7, 45 (1962)CrossRefGoogle Scholar
  49. 49.
    N. Hansen, Acta Metall. 25, 863 (1977)CrossRefGoogle Scholar
  50. 50.
    N. Hansen, B. Ralph, Acta Metall. 30, 411 (1982)CrossRefGoogle Scholar
  51. 51.
    N.H. Heo, Y.-U. Heo, S.-J. Kim, ISIJ Int. 56, 1096 (2016)Google Scholar
  52. 52.
    C.J. McMahon Jr., M. Cohen, Acta Metall. 13, 591 (1965)CrossRefGoogle Scholar
  53. 53.
    N.H. Heo, K.H. Chai, J.G. Na, Acta Mater. 48, 2901 (2000)CrossRefGoogle Scholar
  54. 54.
    J.M. Howe, Interfaces in Materials (Wiley, New York, 1997)Google Scholar
  55. 55.
    E.O. Hall, Proc. Phys. Soc. B 64, 742 (1951)CrossRefGoogle Scholar
  56. 56.
    W.C. Leslie, The Physical Metallurgy of Steels (McGraw-Hill, New York, 1982)Google Scholar
  57. 57.
    J. Daming, W. Yinong, H. Bande, L. Tingquan, J. Mater. Sci. Lett. 15, 1597 (1996)Google Scholar
  58. 58.
    S.S. Hecker, D.L. Rohr, D.F. Stein, Metall. Trans. A 9, 481 (1978)CrossRefGoogle Scholar
  59. 59.
    R.L. Tobler, D. Meyn, Metall. Trans. A 19, 1626 (1988)CrossRefGoogle Scholar
  60. 60.
    J.R. Low Jr., Relation of Properties to Microstructure (American Society for Metals, Cleveland, 1954)Google Scholar
  61. 61.
    C. Crussard, R. Borione, J. Plateau, Y. Morillon, F. Maratray, J. Iron Steel Inst. 183, 146 (1956)Google Scholar
  62. 62.
    D.F. Stein, J.R. Low Jr., A.U. Seybolt, Acta Metall. 11, 1253 (1963)CrossRefGoogle Scholar
  63. 63.
    D. Hull, Acta Metall. 9, 191 (1961)CrossRefGoogle Scholar
  64. 64.
    N.J. Petch, Philos. Mag. 1, 186 (1956)CrossRefGoogle Scholar
  65. 65.
    C. Zener, ASM 40, Fract. Metals 3 (1948)Google Scholar
  66. 66.
    A.H. Cottrell, Trans. AIME 212, 192 (1958)Google Scholar
  67. 67.
    J.P. Berry, J. Mech. Phys. Solids 8, 194 (1960)CrossRefGoogle Scholar
  68. 68.
    J.R. Willis, J. Mech. Phys. Solids 15, 151 (1967)CrossRefGoogle Scholar
  69. 69.
    J.P. Hirsh, J. Lothe, Theory of Dislocations (McGraw-Hill, New York, 1968)Google Scholar
  70. 70.
    B. Zhu, R.J. Asaro, P. Krysl, R. Bailey, Acta Mater. 53, 4825 (2005)CrossRefGoogle Scholar
  71. 71.
    W.A. Spitzig, A.S. Keh, Acta Metall. 18, 611 (1970)CrossRefGoogle Scholar
  72. 72.
    C. Jude-Esser, F. Grimpe, W. Dahl, ECF 10, 1017 (1994)Google Scholar
  73. 73.
    H. Dünnewald-Arfmann, M. Twickler, R. Twickler, W. Dahl, in Proceedings of ICSMA 8, vol. 2 (1988), p. 1063Google Scholar
  74. 74.
    J.W. Morris, Science 320, 1022 (2008)CrossRefGoogle Scholar
  75. 75.
    Y. Lan, H.J. Klaar, W. Dahl, Metall. Trans. A 23, 537 (1992)CrossRefGoogle Scholar
  76. 76.
    A. Lawley, H.L. Gaigher, Philos. Mag. 10, 15 (1964)CrossRefGoogle Scholar
  77. 77.
    A. Gilbert, B.A. Wilcox, G.T. Hahn, Philos. Mag. 12, 649 (1965)CrossRefGoogle Scholar
  78. 78.
    W.B. Morrison, W.C. Leslie, Metall. Trans. 4, 379 (1973)CrossRefGoogle Scholar
  79. 79.
    N. Nakada, M. Fujihara, T. Tsuchiyama, T. Takaki, ISIJ Int. 51, 1169 (2011)CrossRefGoogle Scholar
  80. 80.
    I. Samajdar, B. Verlinden, P. Van Houtte, D. Vanderschueren, Mater. Sci. Eng. A 238, 343 (1997)CrossRefGoogle Scholar
  81. 81.
    N.H. Heo, J.W. Yim, J. Korean Phys. Soc. 44, 1547 (2004)Google Scholar
  82. 82.
    R.A. Masumura, P.M. Hazzledine, P.K. Liaw, E.J. Lavernia, Acta Metall. 13, 4527 (1998)Google Scholar
  83. 83.
    J. Schiøtz, D.D. Di Tolla, K.W. Jacobsen, Nature 391, 561 (1998)CrossRefGoogle Scholar
  84. 84.
    H. Van Swygenhoven, Science 296, 66 (2002)CrossRefGoogle Scholar
  85. 85.
    J. Schiøtz, K.W. Jacobsen, Science 301, 1357 (2003)CrossRefGoogle Scholar
  86. 86.
    F.P. Buff, J. Chem. Phys. 19, 1591 (1951)CrossRefGoogle Scholar
  87. 87.
    K.P.D. Lagerlöf, J. Castaing, P. Pirouz, A.H. Heuer, Philos. Mag. 82A, 2841 (2002)CrossRefGoogle Scholar
  88. 88.
    J. Lian, B. Baudelet, Nanostruct. Mat. 2, 415 (1993)CrossRefGoogle Scholar

Copyright information

© The Korean Institute of Metals and Materials 2018

Authors and Affiliations

  1. 1.Graduate Institute of Ferrous TechnologyPohang University of Science and TechnologyPohangRepublic of Korea
  2. 2.Department of Materials Science and EngineeringSeoul National UniversitySeoulRepublic of Korea

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