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Conduction degradation in anisotropic multi-cracked materials

  • S. GiordanoEmail author
  • P. L. Palla
Regular Article

Abstract

The electrical and thermal conduction properties of disordered solids and the possible degradation processes induced by the generation of cracks are central issues in the field of the heterogeneous materials. However, most of the existing theories are unable to consider an arbitrary density of cracks. We obtained an exact result for the fields induced within an elliptic anisotropic inhomogeneity embedded in a different anisotropic (two-dimensional) conductor. Then, we applied it to show that the degradation process strongly depends on the statistical orientational distribution of defects: in particular we theoretically prove that parallel cracks lead to the power law decay log σ ∼ − log N while random oriented cracks lead to the exponential law decay log σ ∼ −N (where σ is the effective conductivity of a region with a large number N of defects), as recently predicted by numerical findings.

Keywords

Solid State and Materials 

References

  1. 1.
    G.W. Milton, The Theory of Composites (Cambridge University Press, Cambridge, 2002)Google Scholar
  2. 2.
    S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer-Verlag, New York, 2002)Google Scholar
  3. 3.
    M. Sahimi, Heterogeneous Materials I, Linear Transport and Optical Properties (Springer-Verlag, New York, 2003)Google Scholar
  4. 4.
    M. Sahimi, Heterogeneous Materials II, Nonlinear and Breakdown Properties and Atomistic Modeling (Springer-Verlag, New York, 2003)Google Scholar
  5. 5.
    L.J. Walpole, Adv. Appl. Mech. 11, 169 (1981)CrossRefGoogle Scholar
  6. 6.
    Heterogeneous Media: Micromechanics, Modelling, Methods, and Simulations, edited by K.Z. Markov, L. Preziozi (Birkhauser, Boston, 2000)Google Scholar
  7. 7.
    L.H.K. Van Beek, Prog. Dielectr. 7, 71 (1967)Google Scholar
  8. 8.
    Selected Papers on Linear Optical Composite Materials, edited by A. Lakhtakia (SPIE Milestone Series, Bellingham, 1996)Google Scholar
  9. 9.
    S. Giordano, M.I. Saba, L. Colombo, Eur. Phys. J. B 76, 261 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    A.P. Chatterjee, J. Phys.: Condens. Matter 23, 155104 (2011) ADSCrossRefGoogle Scholar
  11. 11.
    T. Pointer, E. Liu, J.A. Hudson, Geophys. J. Int. 142, 199 (2000) ADSCrossRefGoogle Scholar
  12. 12.
    S.R. Tod, Geophys. J. Int. 146, 249 (2001) ADSCrossRefGoogle Scholar
  13. 13.
    R.K. Nalla, J.H. Kinney, R.O. Ritchie, Biomaterials 24, 3955 (2003) CrossRefGoogle Scholar
  14. 14.
    N.A. Danova, S.A. Colopy, C.L. Radtke, V.L. Kalscheur, M.D. Markel, R. Vanderby, R.P. McCabe, A.J. Escarcega, P. Muir, Bone (N.Y.) 33, 197 (2003)CrossRefGoogle Scholar
  15. 15.
    C. Brosseau, A. Beroual, Progr. Mater. Sci. 48, 374 (2003)CrossRefGoogle Scholar
  16. 16.
    M. Kachanov, Appl. Mech. Rev. 45, 305 (1992)ADSCrossRefGoogle Scholar
  17. 17.
    M. Kachanov, Adv. Appl. Mech. 30, 259 (1994)CrossRefGoogle Scholar
  18. 18.
    S. Giordano, A. Mattoni, L. Colombo, Rev. Comp. Chem. 27, 1 (2011)Google Scholar
  19. 19.
    C.K. Chao, R.C. Chang, J. Appl. Phys. 72, 2598 (1992) ADSCrossRefGoogle Scholar
  20. 20.
    V.I. Kushch, A.S. Sangani, Proc. R. Soc. Lond. A 456, 683 (2000) ADSzbMATHCrossRefGoogle Scholar
  21. 21.
    I. Sevostianov, Int. J. Eng. Sci. 44, 513 (2006)CrossRefGoogle Scholar
  22. 22.
    S. Giordano, L. Colombo, Phys. Rev. Lett. 98, 055503 (2007) ADSCrossRefGoogle Scholar
  23. 23.
    S. Giordano, L. Colombo, Phys. Rev. B 77, 054106 (2008) ADSCrossRefGoogle Scholar
  24. 24.
    E.A. Brener, H. Müller-Krumbhaar, R. Spatschek, Phys. Rev. Lett. 86, 1291 (2001) ADSCrossRefGoogle Scholar
  25. 25.
    R. Spatschek, C. Gugenberger, E.A. Brener, Phys. Rev. B 80, 144106 (2009) ADSCrossRefGoogle Scholar
  26. 26.
    R. Spatschek, C. Müller-Gugenberger, E. Brener, B. Nestler, Phys. Rev. E 75, 066111 (2007) MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    R. Spatschek, M. Hartmann, E. Brener, H. Müller-Krumbhaar, K. Kassner, Phys. Rev. Lett. 96, 015502 (2006) ADSCrossRefGoogle Scholar
  28. 28.
    S. Giordano, P.L. Palla, J. Phys. A Math. Theor. 41, 415205 (2008) MathSciNetCrossRefGoogle Scholar
  29. 29.
    J.D. Eshelby, Proc. R. Soc. Lond. A 241, 376 (1957) MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. 30.
    J.D. Eshelby, Proc. R. Soc. Lond. A 252, 561 (1959) MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. 31.
    T. Mura, Micromechanics of Defects in Solids (Kluwer Academic, Dordrecht, 1987)Google Scholar
  32. 32.
    R.J. Asaro, V.A. Lubarda, Mechanics of Solids and Materials (Cambridge University Press, Cambridge 2006) Google Scholar
  33. 33.
    A.J. Duncan, T.G. Mackay, A. Lakhtakia, J. Phys. A Math. Theor. 42, 165402 (2009) MathSciNetADSCrossRefGoogle Scholar
  34. 34.
    W.N. Zou, Q.S. Zheng, Q.C. He, Proc. R. Soc. A 467, 607 (2011) MathSciNetADSzbMATHCrossRefGoogle Scholar
  35. 35.
    C.K. Chao, M.H. Shen, J. Appl. Phys. 73, 7129 (1993) ADSCrossRefGoogle Scholar
  36. 36.
    I. Sevostianov, Int. J. Eng. Sci. 44, 513 (2006)CrossRefGoogle Scholar
  37. 37.
    L.V. Gibiansky, S. Torquato, Proc. R. Soc. A 353, 243 (1995) zbMATHGoogle Scholar
  38. 38.
    M. Kachanov, I. Sevostianov, B. Shafiro, J. Mech. Phys. Solids 49, 1 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  39. 39.
    M.P. Lutz, R.W. Zimmerman, Int. J. Solids Struct. 42, 429 (2005)zbMATHCrossRefGoogle Scholar
  40. 40.
    J.A. Stratton, Electromagnetic theory (Mc Graw Hill, New York, 1941)Google Scholar
  41. 41.
    L.D. Landau, E.M. Lifshitz, Electrodynamics of continuous media (Pergamon Press, London, 1984) Google Scholar
  42. 42.
    B. Michel, W.S. Weiglhofer, Archiv für Elektronik und Übertragungstechnik 51, 219 (1997)Google Scholar
  43. 43.
    B. Michel, W.S. Weiglhofer, Archiv für Elektronik und Übertragungstechnik 52, 31 (1998)Google Scholar
  44. 44.
    W.S. Weiglhofer, A. Lakhtakia, B. Michel, Microw. Opt. Technol. Lett. 15, 263 (1997)CrossRefGoogle Scholar
  45. 45.
    A. Lakhtakia, W.S. Weiglhofer, Int. J. Electronics 87, 1401 (2000) CrossRefGoogle Scholar
  46. 46.
    S. Giordano, P.L. Palla, L. Colombo, Europhys. Lett. 83, 66003 (2008) ADSCrossRefGoogle Scholar
  47. 47.
    S. Giordano, P.L. Palla, L. Colombo, Eur. Phys. J. B 68, 89 (2009)ADSCrossRefGoogle Scholar
  48. 48.
    H.L. Duan, J. Wang, Z.P. Huang, B.L. Karihaloo, Proc. R. Soc. A 461, 3335 (2005) MathSciNetADSzbMATHCrossRefGoogle Scholar
  49. 49.
    J. Wang, H.L. Duan, Z.P. Huang, B.L. Karihaloo, Proc. R. Soc. A 462, 1355 (2006) ADSzbMATHCrossRefGoogle Scholar
  50. 50.
    P.L. Palla, S. Giordano, L. Colombo, Phys. Rev. B 81, 214113 (2010) ADSCrossRefGoogle Scholar
  51. 51.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publication, New York, 1970)Google Scholar
  52. 52.
    I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, series and products (Academic Press, San Diego, 1965) Google Scholar
  53. 53.
    A. Lakhtakia, Int. J. Mod. Phys. C 3, 583 (1992)ADSCrossRefGoogle Scholar
  54. 54.
    J.J.H. Wang, Generalized Moment Methods in Electromagnetics (Wiley, New York, 1991)Google Scholar
  55. 55.
    J.C. Maxwell, A Treatise on Electricity and Magnetism (Clarendon, Oxford, 1881)Google Scholar
  56. 56.
    D.A.G. Bruggeman, Ann. Phys., Leipzig 24, 636 (1935)ADSCrossRefGoogle Scholar
  57. 57.
    H. Fricke, J. Phys. Chem. 57, 934 (1953)CrossRefGoogle Scholar
  58. 58.
    S. Giordano, J. Electrostat. 58, 59 (2003)CrossRefGoogle Scholar
  59. 59.
    Giordano, W. Rocchia, J. Appl. Phys. 98, 104101 (2005) ADSCrossRefGoogle Scholar
  60. 60.
    H. Hatta, M. Taya, Int. J. Eng. Sci. 24, 1159 (1986) zbMATHCrossRefGoogle Scholar
  61. 61.
    H. Hatta, M. Taya, J. Appl. Phys. 58, 2478 (1985) ADSCrossRefGoogle Scholar
  62. 62.
    H. Hatta, M. Taya, J. Appl. Phys. 59, 1851 (1986) ADSCrossRefGoogle Scholar
  63. 63.
    E.J. Garboczi, J.F. Douglas, Phys. Rev. E 53, 6169 (1996) ADSCrossRefGoogle Scholar
  64. 64.
    B. Shafiro, M. Kachanov, J. Appl. Phys. 87, 8561 (2000) ADSCrossRefGoogle Scholar
  65. 65.
    H.L. Duan, B.L. Karihaloo, J. Wang, X. Yi, Phys. Rev. B 73, 174203 (2006) ADSCrossRefGoogle Scholar
  66. 66.
    S. Giordano, W. Rocchia, J. Phys.: Condens. Matter 18, 10585 (2006) ADSCrossRefGoogle Scholar
  67. 67.
    A.V. Goncharenko, Phys. Rev. E 68, 041108 (2003) ADSCrossRefGoogle Scholar
  68. 68.
    S. Giordano, Eur. J. Mech. A Solids 22, 885 (2003)ADSzbMATHCrossRefGoogle Scholar
  69. 69.
    M. Kachanov, I. Sevostianov, Int. J. Solids Struct. 42, 309 (2005)zbMATHCrossRefGoogle Scholar
  70. 70.
    B. Michel, A. Lakhtakia, W.S. Weiglhofer, T.G. Mackay, Compos. Sci. Technol. 61, 13 (2001)CrossRefGoogle Scholar
  71. 71.
    A. Lakhtakia, Microw. Opt. Technol. Lett. 17, 276 (1998)CrossRefGoogle Scholar
  72. 72.
    S. Giordano, Int. J. Appl. Electr. Mech. 26, 1 (2007)Google Scholar
  73. 73.
    G.E. Pike, C.H. Seager, Phys. Rev. B 10, 1421 (1974) ADSCrossRefGoogle Scholar
  74. 74.
    S. Giordano, Physica A 375, 726 (2007) ADSCrossRefGoogle Scholar
  75. 75.
    S. Giordano, J. Eng. Mater. Technol. 129, 453 (2007) CrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Electronics, Microelectronic and Nanotechnology (IEMN UMR CNRS 8520)Villeneuve d’Ascq CedexFrance

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