Advertisement

Field-theoretical description of the formation of a crack tip process zone

  • Alexei Boulbitch
  • Alexander L. Korzhenevskii
Open Access
Regular Article

Abstract

The crack tip process zone is regarded as a region where the solid physical properties are altered due to high stress. They are controlled by the solid degrees of freedom existing within the zone and vanishing outside, and can be divided into two classes: (1) zones always existing at the tip and (2) those emerging as soon as certain conditions are met. We focus on the zones of the second kind and argue that they can be described analogously to phase transitions taking place locally. We report both a numerical and an analytical solution for the process zone. We find that the zone can only exist within a limited domain of the dynamic phase diagram, at one side of the phase transition line. We describe this domain and establish its dependence on the crack velocity. We show the existence of a critical crack velocity above which the zone cannot exist.

Keywords

Solid State and Materials 

References

  1. 1.
    J. Fineberg, M. Marder, Phys. Rep. 313, 1 (1999)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    H. Gao, J. Mech. Phys. Solids 44, 1453 (1996)ADSCrossRefGoogle Scholar
  3. 3.
    M.J. Buehler, F.F. Abraham, H. Gao, Nature 426, 141 (2003)ADSCrossRefGoogle Scholar
  4. 4.
    M.J. Buechler, H. Gao, Nature 439, 307 (2006)ADSCrossRefGoogle Scholar
  5. 5.
    A.Y. Grosberg, A.R. Khokhlov, Statistical Physics of Macromolecules (AIP Press, New York, 1994)Google Scholar
  6. 6.
    E. Bouchbinder, A. Livne, E. Bouchbinder, J. Fineberg, Phys. Rev. Lett. 101, 264302 (2008)ADSCrossRefGoogle Scholar
  7. 7.
    A. Livne, E. Bouchbinder, J. Fineberg, Phys. Rev. Lett. 101, 264301 (2008)ADSCrossRefGoogle Scholar
  8. 8.
    L.I. Slepyan, Sov. Phys. Dokl. 26, 538 (1981)ADSGoogle Scholar
  9. 9.
    M. Marder, S. Gross, J. Mech. Phys. Solids 43, 1 (1995)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    M.J. Fineberg, E. Bouchbinder, Int. J. Fract. 196, 33 (2015)CrossRefGoogle Scholar
  11. 11.
    D.A. Kessler, H. Levine, Phys. Rev. E 59, 5154 (1999)ADSCrossRefGoogle Scholar
  12. 12.
    D.A. Kessler, H. Levine, Phys. Rev. E 63, 016118 (2000)ADSCrossRefGoogle Scholar
  13. 13.
    I.S. Aranson, V.A. Kalatsky, V.M. Vinokur, Phys. Rev. Lett. 85, 118 (2000)ADSCrossRefGoogle Scholar
  14. 14.
    A. Karma, D.A. Kessler, H. Levine, Phys. Rev. Lett. 87, 045501 (2001)ADSCrossRefGoogle Scholar
  15. 15.
    L.O. Eastgate et al., Phys. Rev. E 65, 036117 (2002)ADSCrossRefGoogle Scholar
  16. 16.
    A. Karma, A.E. Lobkovsky, Phys. Rev. Lett. 92, 245510 (2004)ADSCrossRefGoogle Scholar
  17. 17.
    H. Henry, H. Levine, Phys. Rev. Lett. 93, 105504 (2004)ADSCrossRefGoogle Scholar
  18. 18.
    A. Livne, O. Ben-David, J. Fineberg, Phys. Rev. Lett. 98, 124301 (2007)ADSCrossRefGoogle Scholar
  19. 19.
    K. Ravi-Chandar, W. Knauss, Int. J. Fract. 26, 65 (1984)CrossRefGoogle Scholar
  20. 20.
    J. Fineberg, S.P. Gross, M. Marder, H.L. Swinney, Phys. Rev. Lett. 67, 457 (1991)ADSCrossRefGoogle Scholar
  21. 21.
    A. Livne, G. Cohen, J. Fineberg, Phys. Rev. Lett. 94, 224301 (2005)ADSCrossRefGoogle Scholar
  22. 22.
    B.T. Goldman, G. Cohen, J. Fineberg, Phys. Rev. Lett. 114, 054301 (2015)ADSCrossRefGoogle Scholar
  23. 23.
    S.J. Pennycook, Ultramicroscopy 123, 28 (2012)CrossRefGoogle Scholar
  24. 24.
    S.J. Wang et al., Nat. Commun. 67 45 (2014)Google Scholar
  25. 25.
    S.W. Robertson et al., Acta Mater. 55, 6197 (2007)CrossRefGoogle Scholar
  26. 26.
    I. Roth et al., in ESOMAT 2009 – 8th European Symposium on Martensitic Transformations, edited by P. Sittner, V. Paidar, H. Seiner (EDP Sciences, Les Ulis, France, 2009)Google Scholar
  27. 27.
    S. Daly et al., Acta Mater. 55, 6322 (2007)CrossRefGoogle Scholar
  28. 28.
    X. Tan et al., Acta Mater. 62, 114 (2014)CrossRefGoogle Scholar
  29. 29.
    F. Meschke et al., J. Am. Ceram. Soc. 83, 353 (2000)CrossRefGoogle Scholar
  30. 30.
    Y.H. Lu et al., Intermetallics 10, 823 (2002)CrossRefGoogle Scholar
  31. 31.
    E. Sgambittera, C. Maletta, F. Furgiuele, Sripta Mat. 101, 64 (2015)CrossRefGoogle Scholar
  32. 32.
    P.M. Kelly, L.R.F. Rose, Progr. Mater. Sci. 47, 463 (2002)CrossRefGoogle Scholar
  33. 33.
    S.D. Antolovich, D. Fahr, Eng. Fracture Mech. 4, 133 (1972)CrossRefGoogle Scholar
  34. 34.
    E. Hornbogen, Acta Metall. 26, 147 (1978)CrossRefGoogle Scholar
  35. 35.
    S.K. Hann, J.D. Gates, J. Mater. Sci. 32, 1249 (1997)ADSCrossRefGoogle Scholar
  36. 36.
    E.C. Oliver et al., Appl. Phys. A 74, S1143 (2002)ADSCrossRefGoogle Scholar
  37. 37.
    Z. Khan, M. Ahmed, J. Mater. Eng. Perform. 5, 201 (1996)CrossRefGoogle Scholar
  38. 38.
    M.K. Banerjee, N.R. Bandyopadhyay, J. Mazumder, in Processing and Fabrication of Advanced Materials VI, edited by K.A. Khor, T.S. Srivatsan, J.J. Moore (IOM Communications, London, 1998), Vols. 1, 2Google Scholar
  39. 39.
    H. Oettel, U. Martin, Int. J. Mat. Res. 97, 1642 (2006)CrossRefGoogle Scholar
  40. 40.
    A.L. McKelvey, R.O. Ritchie, Phil. Mag. A 80, 1759 (2000)ADSCrossRefGoogle Scholar
  41. 41.
    A.L. McKelvey, R.O. Ritchie, Metal. Mater. Trans. A 32, 731 (2001)CrossRefGoogle Scholar
  42. 42.
    H.F. Lopez, Mater. Lett. 51, 144 (2001)CrossRefGoogle Scholar
  43. 43.
    K. Kimura, T. Asaoka, K. Funami, in Proc. Int. Conf. on Thermomech. Proc. of Steels and Other Materials, edited by T. Chandra, T. Sakai (Minerals, Metals and Materials Soc., Warrendale, PA, 1997), Vols. I, II, p. 1675Google Scholar
  44. 44.
    X. Wang, Z. Yue, in Fracture and Damage Mechanics V, edited by M.H. Aliabadi, Q. Li, L. Li, F.G. Buchholz, (2006), Vols. 324–325, p. 919, Parts 1 and 2Google Scholar
  45. 45.
    G.M. Loughran, T.W. Shield, P.H. Leo, Int. J. Solids Struct. 40, 271 (2003)CrossRefGoogle Scholar
  46. 46.
    H. Qiu et al., Mater. Sci. Eng. A 579, 71 (2013)CrossRefGoogle Scholar
  47. 47.
    U.D. Hangen, G. Sauthoff, Intermetallics 7, 501 (1999)CrossRefGoogle Scholar
  48. 48.
    L.E. Tanner, D. Schryvers, S.M. Shapiro, Materials Sci. Eng. A 127, 205 (1990)CrossRefGoogle Scholar
  49. 49.
    A. Paradkar et al., Metall. Mater. Trans. 40A, 1604 (2009).CrossRefGoogle Scholar
  50. 50.
    M. Kerr et al., Scripta Mat. 62, 341 (2010)CrossRefGoogle Scholar
  51. 51.
    S.O. Kramarov, N.Y. Egorov, L.M. Katsnel’son, Sov. Phys. – Solid State 28, 1602 (1986)Google Scholar
  52. 52.
    A.A. Grekov et al., Ferroelectrics Lett. 8, 59 (1988)CrossRefGoogle Scholar
  53. 53.
    S. Lynch, R.M. McMeeking, Z. Suo, in Second International Conference on Intelligent Materials. ICIM’94, edited by C.A. Rogers, G.G. Wallace (Technomic Publishing Co., Lancaster, PA, USA, 1994), p. 856Google Scholar
  54. 54.
    G.G. Siu, W.G. Zeng, J. Mater. Sci. 28, 5875 (1993)ADSCrossRefGoogle Scholar
  55. 55.
    I. Birkby, R. Stevens, Key Eng. Mater. 122–124, 527 (1996)CrossRefGoogle Scholar
  56. 56.
    R.I. Todd, M.P.S. Saran, Transformation toughening, in Materials Processing Handbook, edited by J.R. Groza (CRC Press, Boca Raton, FL, USA, 2007), Vol. 20, pp. 1–20Google Scholar
  57. 57.
    J. Karger-Kocsis, J. Varga, J. Appl. Polym. Sci. 62, 291 (1996)CrossRefGoogle Scholar
  58. 58.
    J. Karger-Kocsis, J. Varga, G.W. Ehrenstein, J. Appl. Polym. Sci. 64, 2057 (1997)CrossRefGoogle Scholar
  59. 59.
    H.-J. Sue, J.D. Earls, R.E. Hefner Jr., J. Mater. Sci. 32, 4039 (1997)ADSCrossRefGoogle Scholar
  60. 60.
    H. Bai et al., J. Polym. Sci. B 47, 46 (2009)CrossRefGoogle Scholar
  61. 61.
    S.T. Kim et al., J. Mater. Sci. 33, 2421 (1998)ADSCrossRefGoogle Scholar
  62. 62.
    G.A. Maier et al., Macromolecules 38, 6099 (2005)ADSCrossRefGoogle Scholar
  63. 63.
    J.A. Horton, J.L. Wright, J.H. Schneibel, in Bulk Metallic Glasses, edited by W.L. Johnson, A. Inoue, C.T. Liu, (Oxford University Press, Oxford, 1999), Vol. 554, p. 185Google Scholar
  64. 64.
    J.A. Donovan, Nippon Gomu Kyokaishi 75, 239 (2002)CrossRefGoogle Scholar
  65. 65.
    S. Trabelsi, P.-A. Albouy, J. Rault, Macromolecules 35, 10054 (2002)ADSCrossRefGoogle Scholar
  66. 66.
    H.P. Zhang et al., Phys. Rev. Lett. 102, 245503 (2009)ADSCrossRefGoogle Scholar
  67. 67.
    J.-B. Le Cam, E. Toussaint, Macromolecules 43, 4708 (2010)ADSCrossRefGoogle Scholar
  68. 68.
    N. Saintier, G. Cailletaud, R. Piques, Mater. Sci. Eng. A 528, 1078 (2011)CrossRefGoogle Scholar
  69. 69.
    K. Tozawa et al., J. Electron Microscopy 48, 613 (1999)CrossRefGoogle Scholar
  70. 70.
    K.S. Watanabe et al., Radiation Effects and Defects in Solids 157, 101 (2002)ADSCrossRefGoogle Scholar
  71. 71.
    M. Nagumo et al., Scripta Materialia 49, 837 (2003)CrossRefGoogle Scholar
  72. 72.
    M. Takeda et al., J. Electron Microsc. 48, 609 (1999)CrossRefGoogle Scholar
  73. 73.
    K. Youssef, P. Kulshreshtha, G. Rozgonyi, Photovoltaics for the 21st Century 25, 49 (2010)Google Scholar
  74. 74.
    P.K. Kulshreshtha, K.M. Youssef, G. Rozgonyi, Solar Energy Mater. Solar Cells 96, 166 (2012)CrossRefGoogle Scholar
  75. 75.
    N. Nishiyama et al., Sci. Rep. 4, 6558 (2014)ADSCrossRefGoogle Scholar
  76. 76.
    K. Yoshida et al., Sci. Rep. 5, 10993 (2015)ADSCrossRefGoogle Scholar
  77. 77.
    C. Wünsche, E. Radlein, G.H. Frischat, Glass Sci. Tech. – Glastechnische Berichte 72, 49 (1999)Google Scholar
  78. 78.
    C.J. Gilbert, V. Schroeder, R.O. Ritchie, Metall. Mater. Trans. A30, 1739 (1999)CrossRefGoogle Scholar
  79. 79.
    C.J. Gilbert, V. Schroeder, R.O. Ritchie, in Bulk Metallic Glasses, edited by W.L. Johnson, A. Inoue, C.T. Liu (Material Res. Soc., Warrendale PA, 1999), Vol. 554, p. 343Google Scholar
  80. 80.
    I. Brough, R.N. Haward, G. Healey, A. Wood, Polymer 45, 3115 (2004)CrossRefGoogle Scholar
  81. 81.
    K. Nishimura, N. Miyazaki, Comp. Model. Eng. Sci. 2, 143 (2001)Google Scholar
  82. 82.
    Y.-F. Guo, D.-L. Zhao, Mater. Sci. Eng. A 448, 281 (2007)CrossRefGoogle Scholar
  83. 83.
    Y.-F. Guo, Y.-S. Wang, D.-L. Zhao, Acta Mater. 55, 401 (2007)CrossRefGoogle Scholar
  84. 84.
    A. Latapie, D. Farkas, Modell. Simul. Mater. Sci. Eng. 11, 745 (2003)ADSCrossRefGoogle Scholar
  85. 85.
    R. Matsumoto et al., Comp. Model. Eng. Sci. 9, 75 (2005)Google Scholar
  86. 86.
    I.R. Vatne et al., Mater. Sci. Eng. A 560, 306 (2013)CrossRefGoogle Scholar
  87. 87.
    M.J. Buehler et al., Phys. Rev. Lett. 99, 165502 (2007)ADSCrossRefGoogle Scholar
  88. 88.
    D. Sherman, M. Markovitz, O. Barkai, J. Mech. Phys. Solids 56, 376 (2008)ADSCrossRefGoogle Scholar
  89. 89.
    F. Atrash, D. Sherman, J. Mech. Phys. Solids 60, 844 (2012)ADSCrossRefGoogle Scholar
  90. 90.
    J.R. Kermode et al., Nature 455, 1224 (2008)ADSCrossRefGoogle Scholar
  91. 91.
    J. Mei et al., Int. J. Solids Struct. 48, 3054 (2011)CrossRefGoogle Scholar
  92. 92.
    M. Ruda, D. Farkas, G. Bertolino, Comp. Mater. Sci. 49, 743 (2010)CrossRefGoogle Scholar
  93. 93.
    Y. Zhang et al., J. Nucl. Mater. 430, 96 (2012)ADSCrossRefGoogle Scholar
  94. 94.
    A. Falvo et al., J. Mater. Eng. Perform. 18, 679 (2009)CrossRefGoogle Scholar
  95. 95.
    F.F. Abraham et al., Phys. Rev. Lett. 73, 272 (1994)ADSCrossRefGoogle Scholar
  96. 96.
    S.J. Zhou et al., Phys. Rev. Lett. 76, 2318 (1996)ADSCrossRefGoogle Scholar
  97. 97.
    P. Gumbsch, S.J. Zhou, B.L. Holian, Phys. Rev. B 55, 3445 (1997)ADSCrossRefGoogle Scholar
  98. 98.
    T. Cramer, A. Wanner, P. Gumbsch, Phys. Rev. Lett. 85, 788 (2000)ADSCrossRefGoogle Scholar
  99. 99.
    I. Birkby, R. Stevens, Key Eng. Mater. 122–124, 527 (1996)CrossRefGoogle Scholar
  100. 100.
    R.I. Todd, M.P.S. Saran, Transformation toughening, in Materials Processing Handbook, edited by J.R. Groza (CRC Press, Boca Raton, Fla., USA, 2007), Vol. 20, pp. 1–20Google Scholar
  101. 101.
    B.L. Karihaloo, J.H. Andreasen, Mechanics of Transformation Toughening and Related Topics (Elsevier, Amsterdam, 1996)Google Scholar
  102. 102.
    B. Budiansky, J.W. Hutchinson, J.C. Lambropoulos, Int. J. Solids Struct. 19, 843 (1983)CrossRefGoogle Scholar
  103. 103.
    A.G. Evans, R.M. Cannon, Acta Metall. 34, 761 (1986)CrossRefGoogle Scholar
  104. 104.
    L.R.F. Rose, Proc. R. Soc. A 412, 169 (1987)ADSCrossRefGoogle Scholar
  105. 105.
    Q.P. Sun, K.C. Hwang, S.W. Yu, J. Mech. Phys. Solids 39, 507 (1991)ADSCrossRefGoogle Scholar
  106. 106.
    T. Baxevanis, A.F. Parrinello, D.C. Lagoudas, Int. J. Plasticity 50, 158 (2013)CrossRefGoogle Scholar
  107. 107.
    A.L. Roitburd, Sov. Phys. – Usp. 17, 326 (1974)ADSCrossRefGoogle Scholar
  108. 108.
    A.L. Roitburd, Sov. Phys. Solid State 26, 1229 (1984)Google Scholar
  109. 109.
    M.A. Grinfel’d, Lett. Appl. Eng. Sci. 19, 1031 (1981)Google Scholar
  110. 110.
    M.A. Grinfel’d, Izvestia, Earth Phys. 18, 28 (1982)Google Scholar
  111. 111.
    V.I. Levitas, I.B. Ozsoy, Int. J. Plasticity 25, 239 (2009)CrossRefGoogle Scholar
  112. 112.
    V.I. Levitas, I.B. Ozsoy, Int. J. Plasticity 25, 546 (2009)CrossRefGoogle Scholar
  113. 113.
    L.D. Landau, E.M. Lifshitz, Statistical Physics (Pergamon Press, Oxford, 1985)Google Scholar
  114. 114.
    V.M. Nabutovskii, B.Ya. Shapiro, Sov. Phys. J. Exp. Theor. Phys. 48, 480 (1978)ADSGoogle Scholar
  115. 115.
    A. Boulbitch, P. Toledano, Phys. Rev. Lett. 81, 838 (1998)ADSCrossRefGoogle Scholar
  116. 116.
    A. Boulbitch, A.L. Korzhenevskii, Phys. Rev. Lett. 107, 085505 (2011)ADSCrossRefGoogle Scholar
  117. 117.
    A. Boulbitch, A.L. Korzhenevskii, Phys. Rev. E 93, 063001 (2016)ADSCrossRefGoogle Scholar
  118. 118.
    A. Boulbitch, A.L. Korzhenevskii, Europhys. Lett. 112, 16003 (2015)CrossRefGoogle Scholar
  119. 119.
    V.I. Levitas, Int. J. Plasticity 16, 805 (2000)CrossRefGoogle Scholar
  120. 120.
    V.I. Levitas, Int. J. Plasticity 16, 851 (2000)CrossRefGoogle Scholar
  121. 121.
    A.V. Idesman, V.I. Levitas, E. Stein, Int. J. Plasticity 16, 893 (2000)CrossRefGoogle Scholar
  122. 122.
    C. Bjerken, A.R. Massih, Condensed Matter, arXiv:1110.1292 (2011)
  123. 123.
    E. Bouchbider et al., Rep. Prog. Phys. 77, 046501 (2014)ADSCrossRefGoogle Scholar
  124. 124.
    A. Karma, D.A. Kessler, H. Levine, Phys. Rev. Lett. 87, 045501 (2001)ADSCrossRefGoogle Scholar
  125. 125.
    V. Hakim, A. Karma, Phys. Rev. Lett. 95, 235501 (2005)ADSCrossRefGoogle Scholar
  126. 126.
    R. Spatschek, M. Hartmann, E. Brener, H. Mueller-Krumbaar, Phys. Rev. Lett. 96, 015502 (2006)ADSCrossRefGoogle Scholar
  127. 127.
    M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)ADSCrossRefGoogle Scholar
  128. 128.
    P. Toledano, V. Dmitriev, Reconstructive Phase Transitions, in Crystals and Quasicrystals (World Scientific, Singapore, 1996)Google Scholar
  129. 129.
    V. Dmitriev et al., Phys. Rev. Lett. 60, 1958 (1988)ADSCrossRefGoogle Scholar
  130. 130.
    V. Dmitriev et al., Phys. Rev. Lett. 62, 844 (1989)ADSCrossRefGoogle Scholar
  131. 131.
    S.R. Shenoy et al., Phys. Rev. B 60, R12537 (1999)ADSCrossRefGoogle Scholar
  132. 132.
    G.P. Cherepanov, Mechanics of Brittle Fracture (McGraw-Hill, New York, 1979)Google Scholar
  133. 133.
    L.D. Landau, E.M. Lifshitz, Theory of Elasticity (Pergamon Press, Oxford, 1986)Google Scholar
  134. 134.
    M.M. Vainberg, V.A. Trenogin, Theory of Branching of Solutions of Non-Linear Equations (Noordhoff, Leyden, 1974)Google Scholar
  135. 135.
    Wolfram Research, Inc., Mathematica, Version 10.0, Champaign, IL, 2014Google Scholar
  136. 136.
    V.P. Sakhnenko, V.M. Talanov, Sov. Phys. Solid State 21, 1401 (1979)Google Scholar
  137. 137.
    V.P. Sakhnenko, V.M. Talanov, Sov. Phys. Solid State 22, 458 (1980)Google Scholar
  138. 138.
    G.R. Barsch, T. Ohba, D.M. Hatch, Mater. Sci. Eng. A 273–275, 161 (1999)CrossRefGoogle Scholar
  139. 139.
    O. Shchyglo, U. Salman, A. Finel, Acta Mater. 60, 6784 (2012)CrossRefGoogle Scholar
  140. 140.
    E.Y. Tonkov, in High Pressure Phase Transformations: A Handbook (Gordon and Breach, Amsterdam, 1992), Vols. 1–3Google Scholar
  141. 141.
    M.A. Krivoglaz, Theory of X-ray and Thermal-Neutron Scattering by Real Crystals (Plenum Press, New York, 1969)Google Scholar
  142. 142.
    P.R. Okamoto, N.Q. Lam, S. Ohnuki, J. Electron Microsc. 48, 481 (1999)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.IEE S.A. ZAE WeiergewanConternLuxembourg
  2. 2.Institute for Problems of Mechanical EngineeringSt. PetersburgRussia

Personalised recommendations