Statistics Toolbox for Damage and Fracture

  • A. Hansen
  • S. Roux
Part of the International Centre for Mechanical Sciences book series (CISM, volume 410)


We review a number of statistical models dealing with damage and fracture of heterogeneous materials. A choice is made to emphasize the models which provide tools and concept of relevance to the field, rather than proposing a single model.


Fractal Dimension Fiber Bundle Critical Exponent Percolation Threshold Multifractal Spectrum 


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  1. Bak P., (1996) “How Nature works: the science of self-organized criticality” (Copernicus, New York)Google Scholar
  2. Bak P., Tang C. and Wiesenfeld K., (1987) Phys. Rev. Lett. 59, 381MathSciNetCrossRefADSGoogle Scholar
  3. Barabási, A.L. and Stanley, H.E., (1995) “Fractal growth equations” ( Cambridge University Press, Cambridge )Google Scholar
  4. Batrouni G.G. and Hansen A. (1988) J. Stat. Phys. 52, 747MathSciNetCrossRefMATHADSGoogle Scholar
  5. Batrouni G.G. and Hansen A. (1998) Phys. Rev. Lett. 80, 325CrossRefADSGoogle Scholar
  6. Batrouni G.G., Hansen A. and Nelkin M., (1987) J. Physique 48, 771CrossRefGoogle Scholar
  7. Boffa J.M., Allain C., Chertcoff R., Hulin J.P., Plouraboué F. and Roux S., (1999) Eur. Phys. J. B7, 179CrossRefADSGoogle Scholar
  8. Bouchaud E., (1997) J. Phys. Cond. Mat. 9, 4319CrossRefADSGoogle Scholar
  9. Bouchaud E., Lapasset G. and Planes J., (1990) Europhys. Lett. 13, 73CrossRefADSGoogle Scholar
  10. Bouchaud J.P., Bouchaud E., Lapasset G. and Planes J., (1993) Phys. Rev. Lett. 71, 2240CrossRefADSGoogle Scholar
  11. Brown S.R. and Scholz C.H., (1985) J. Geophys. Res. 90, 1 2575Google Scholar
  12. Burridge, R. and Knopoff, L., (1967) Bull. Seism. Soc. Am. 57, 341Google Scholar
  13. Carroll, W.F., “A Primer for Finite Elements in Elastic Structures” (1999) ( Wiley, New York )Google Scholar
  14. Coniglio A., (1981) Phys. Rev. Lett. 46, 250CrossRefADSGoogle Scholar
  15. Daguier P., Henaux S., Bouchaud E. and Creuzet F., (1996) Phys. Rev. E53, 5637ADSGoogle Scholar
  16. Daguier P., Nghiem B., Bouchaud E. and Creuzet F., (1997) Phys. Rev. Lett. 78, 1062CrossRefADSGoogle Scholar
  17. Daniels H.E., (1945) Proc. Roy. Soc. London Ser. A183, 404MathSciNetADSGoogle Scholar
  18. Daniels H.E., (1989) Adv. Appl. Prob. 21, 315MathSciNetCrossRefMATHGoogle Scholar
  19. Daniels H.E. and Skyrme, H.T.R., (1985) Adv. Appl. Prob. 17, 85MathSciNetCrossRefMATHGoogle Scholar
  20. David, H.A., (1981) “Order Statistics” ( Wiley, New York )MATHGoogle Scholar
  21. de Arcangelis L. and Herrmann, H.J. (1989) Phys. Rev. B39, 2678CrossRefGoogle Scholar
  22. de Arcangelis L., Redner S. and Coniglio A., (1985) Phys. Rev. B31, 4725CrossRefGoogle Scholar
  23. Delaplace A., (1998) unpublished.Google Scholar
  24. Delaplace A., Roux S. and Pijaudier-Cabot G., (1999a) Int. J. Solids Struct. 36, 1403MathSciNetCrossRefMATHGoogle Scholar
  25. Delaplace A., Roux S. and Pijaudier-Cabot G., (1999b) Int. J. Fracture (to appear)Google Scholar
  26. Delaplace A., Schmittbuhl J. and Mâloy K.J., (1999c) preprintGoogle Scholar
  27. Ding, E.J. and Lu, Y.N., (1993a) Phys. Rev. Lett. 70, 3627CrossRefADSGoogle Scholar
  28. Ding, E.J. and Lu, Y.N., (1993b) Commun. Theor. Phys. 19, 283Google Scholar
  29. Duxbury, P.M., Beale, P.D. and Leath, P.L., (1986) Phys. Rev. Lett. 57, 1052CrossRefADSGoogle Scholar
  30. Duxbury, P.M. and Leath, P.L., (1994) Phys. Rev. B49, 12676CrossRefGoogle Scholar
  31. Ertas D. and Kardar M., (1992) Phys. Rev. Lett. 69, 929CrossRefADSGoogle Scholar
  32. Ertas D. and Kardar M., (1994) Phys. Rev. E49, R2532ADSGoogle Scholar
  33. Feder J., “Fractals” (1989) ( Plenum Press, New York )MATHGoogle Scholar
  34. Feng S. and Sen P.N., (1984) Phys. Rev. Lett. 52, 216CrossRefADSGoogle Scholar
  35. Forrest, B.M. and Tang, L.H., (1990) Phys. Rev. Lett. 64, 1405CrossRefADSGoogle Scholar
  36. Furuberg L., Feder J., Aharony A. and Jossang T., (1988) Phys. Rev. Lett. 61, 2117MathSciNetCrossRefADSGoogle Scholar
  37. Gao H. and Rice J.R., (1989) J. of Appl. Mech., 56, 828CrossRefMATHADSGoogle Scholar
  38. Grassberger P., (1999) Physica A262, 251MathSciNetCrossRefGoogle Scholar
  39. Gumbel, E.J., (1954) National Bureau of Standards Appl. Math. Series 33Google Scholar
  40. Gumbel, E.J., (1958) “Statistics of Extremes” (Columbia University Press, New York)Google Scholar
  41. Halpin-Healy, T. and Zhang, Y.C., (1995) Phys. Rep. 254, 215CrossRefADSGoogle Scholar
  42. Halsey T.C., Jensen M.H., Kadanoff L.P., Procaccia I. and Shraiman B.I., (1986) Phys. Rev. A33, 1141MathSciNetCrossRefMATHADSGoogle Scholar
  43. Hansen A., Enggy T. and MiXloy K.J., (1994) Fractals, 2, 527CrossRefMATHGoogle Scholar
  44. Hansen A. and Hemmer, P.C. (1994a) Phys. Lett. A184, 394CrossRefGoogle Scholar
  45. Hansen A. and Hemmer, P.C. (1994b) Trends in Statistical Physics 1, 213ADSGoogle Scholar
  46. Hansen A. and Hemmer, P.C. (1998) in “Probamat-21st Century,” edited by Frantziskonis, G. ( Kluwer, Dordrecht )Google Scholar
  47. Hansen A., Hinrichsen E.L. and Roux S., (1991) Phys. Rev. B43, 665CrossRefADSGoogle Scholar
  48. Hansen A., Hinrichsen E.L. and Roux S., (1991) Phys. Rev. Lett. 66, 2476CrossRefADSGoogle Scholar
  49. Hansen A., Plouraboué and Roux S., (1995) Fractals, 3, 91CrossRefMATHGoogle Scholar
  50. Harlow, D.G., (1985) Proc. Roy. Soc. Lond. Ser. A397, 211CrossRefADSGoogle Scholar
  51. Harlow, D.G. and Phoenix, S.L., (1991) J. Mech. Phys. Solids 39, 173CrossRefMATHADSGoogle Scholar
  52. Hemmer, P.C. and Hansen, A. (1992) ASME J. Appl. Mech. 59, 909CrossRefMATHADSGoogle Scholar
  53. Herrmann, H.J., Hansen, A. and Roux, S. (1989) Phys. Rev. B39, 637CrossRefADSGoogle Scholar
  54. Herrmann, H.J. and Roux, S., “Statistical Models for the Fracture of Disordered Media” (1990) ( North-Holland, Amsterdam )Google Scholar
  55. Kahng, B., Batrouni, G.G., Redner, S., de Arcangelis, L. and Herrmann, H.J. (1988) Phys. Rev. B. 37, 7625CrossRefADSGoogle Scholar
  56. Kantor Y. and Webman I., (1984) Phys. Rev. Lett. 52, 1891CrossRefADSGoogle Scholar
  57. Kardar, M., Parisi, G. and Zhang, Y.C., (1986) Phys. Rev. Lett. 56, 889CrossRefMATHADSGoogle Scholar
  58. Kloster, M., Hansen, A. and Hemmer, P.C., (1997) Phys. Rev. E56, 2615ADSGoogle Scholar
  59. Krajcinovic D., (2000) This book.Google Scholar
  60. Krug J., Kallabis H., Majumdar S.N., Cornell S.J., Bray A.J. and Sire C., (1997) Phys. Rev. E56, 2702ADSGoogle Scholar
  61. Malay K.J., Hansen A., Hinrichsen E.L. and Roux S., (1992) Phys. Rev. Lett. 68, 213CrossRefADSGoogle Scholar
  62. Mandelbrot B.B., (1974) J. Fluid Mech. 62, 331CrossRefMATHADSGoogle Scholar
  63. Mandelbrot B.B., Passoja D.E. and Paullay A.J., (1984) Nature 308, 721CrossRefADSGoogle Scholar
  64. Maslov S., (1995) Phys. Rev. Lett. 74, 562.CrossRefADSGoogle Scholar
  65. Milman V.Y., Blumenfeld R., Stelmashenko M.A. and Ball R.C., (1993) Phys. Rev. Lett. 71, 2240CrossRefGoogle Scholar
  66. Moukarzel C.F. and Duxbury P.M. (1999) Phys. Rev. E59, 2614ADSGoogle Scholar
  67. Moshev, V.V. and Evlampieva, S.E., (1997) Polym. Eng. and Sci. 37, 1348CrossRefGoogle Scholar
  68. Newman, W.I. and Gabrielov, A.M., (1991) Int. J. Fracture 50, 1Google Scholar
  69. Paczuski M., Maslov S. and Bak P., (1995) Phys. Rev. E53, 414.Google Scholar
  70. Peirce, F.T., (1926) J. Text. Ind. 17, 355Google Scholar
  71. Pijaudier-Cabot G., (2000) This book.Google Scholar
  72. Phoenix, S.L. and Taylor, H.M., (1973) Adv. Appl. Prob. 17, 85Google Scholar
  73. Roux S., (1986) J. Phys. A19, L351ADSGoogle Scholar
  74. Roux S. and Guyon, E., (1986) J. Physique Lett. 46, L999CrossRefGoogle Scholar
  75. Roux S., Hansen, A., Herrmann, H.J. and Guyon, E., (1988) J. Stat. Phys. 52, 237CrossRefADSGoogle Scholar
  76. Roux S. and Hansen A., (1990) EuroPhys. Lett. 11, 37CrossRefADSGoogle Scholar
  77. Roux S. and Hansen A., (1989) EuroPhys. Lett. 8, 729CrossRefADSGoogle Scholar
  78. Roux S. and Hansen A., (1992) J. Physique I I 2, 1007Google Scholar
  79. Roux S., Hansen, A. and Hinrichsen, E.L. (1991) J. Phys. A24, L295ADSGoogle Scholar
  80. Salençon, J., (1983) “Calcul a la Rupture et Analyse Limite” ( Presse de l’ ENPC, Paris )Google Scholar
  81. Schmittbuhl J., Roux S., Vilotte J.-P. and Mdlpy K.J., (1995a) Phys. Rev. Lett. 74, 1787CrossRefADSGoogle Scholar
  82. Schmittbuhl J., Vilotte J.P. and Roux S. (1995b) Phys. Rev. E51, 131ADSGoogle Scholar
  83. Schmittbuhl J. and Mdloy K.J., (1997) Phys. Rev. Lett. 78, 3888CrossRefADSGoogle Scholar
  84. Simonsen I., Hansen A. and Nes, O.M. (1998) Phys. Rev. E58, 2779ADSGoogle Scholar
  85. Smith, R.L. and Phoenix, S.L., (1981) ASME J. Appl. Mech. 48, 75MathSciNetCrossRefMATHADSGoogle Scholar
  86. Stauffer D. and Aharony A., “Introduction to percolation theory” (1992) ( Taylor and Francis, London )Google Scholar
  87. Tanguy A., Gounelle M. and Roux S. (1998) Phys. Rev. E58, 1577ADSGoogle Scholar
  88. van Kampen, N.G., (1992) “Stochastic processes in physics and chemistry” ( North-Holland, Amsterdam )Google Scholar
  89. van Mier J.G.M., (2000) This book.Google Scholar
  90. Wennberg, K.E., Batrouni, G.G., Hansen, A. and Horsrud, P. (1996) Transport in Porous Media 25, 247CrossRefGoogle Scholar
  91. Wilkinson D. and Willemsen J.F., (1983) J. Phys. A16, 3365MathSciNetADSGoogle Scholar
  92. Zapperi, S., Ray, P., Stanley, H.E. and Vespignani, A., (1997) Phys. Rev. Lett. 78, 1408CrossRefADSGoogle Scholar
  93. Zhang, S.D. and Ding, E.J., (1994) Phys. Lett. A193, 425CrossRefGoogle Scholar
  94. Zhou, Si.,. Lomdahl, P.S., Voter, A.F. and Holian, B.L. (1998) Eng. Fracture Mech. 61, 173Google Scholar

Copyright information

© Springer-Verlag Wien 2000

Authors and Affiliations

  • A. Hansen
    • 1
  • S. Roux
    • 2
  1. 1.Norvegian University of Science and TechnologyTrondheimNorway
  2. 2.Lab. Surface du Verre et InterfacesAubervilliersFrance

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