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pure and applied geophysics

, Volume 131, Issue 4, pp 551–576 | Cite as

Predicting the overall properties of composite materials with small-scale inclusions or cracks

  • J. A. Hudson
  • L. Knopoff
Article

Abstract

This paper will summarise the present state of knowledge concerning the elastic and dissipative properties of composite materials in the long wavelength or static approximation. In this case the material, although containing numerous inclusions or cracks or other types of microstructure, can be regarded as a continuum. Established results are listed for the elastic parameters following a review of approximate and exact methods of their derivation.

Key words

Composite materials elastic waves cracked materials 

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References

  1. Aki, K., andChouet, B. (1975),Origin of Coda Waves: Source Attenuation and Scattering Effects, J. Geophys. Res.80, 3322–3342.Google Scholar
  2. Anderson, D. L., Minster, B., andCole, D. (1974),The Effect of Oriented Cracks on Seismic Velocities, J. Geophys. Res.79, 4011–4015.Google Scholar
  3. Batchelor, G. K., andGreen, J. T. (1972),The Determination of the Bulk Stress in a Suspension of Spherical Particles to Order c 2, J. Fluid Mech.56, 401–427.Google Scholar
  4. Beran, M. J., andMcCoy, J. J. (1970a),Mean-field Variations in a Statistical Sample of Heterogeneous Linearly Elastic Solids, Int. J. Solids Structures6, 1035–1054.Google Scholar
  5. Beran, M. J., andMcCoy, J. J. (1970b),The Use of Strain Gradient Theory for Analysis of Random Media, Int. J. Solids Structures6, 1267–1275.Google Scholar
  6. Bhatia, A. B. (1959),Scattering of High-frequency Sound Waves in Polycrystalline Materials, J. Acoust. Soc. Amer.31, 16–23.Google Scholar
  7. Biot, M. A. (1956),The Theory of Propagation of Elastic Waves in a Fluid-saturated Porous Solid, J. Acoust. Soc. Amer.28, 168–191.Google Scholar
  8. Bonilla, L. L., andKeller, J. B. (1985),Acousto-elastic Effect and Wave Propagation in Heterogeneous, Weakly Anisotropic Materials, J. Mech. Phys. Solids33, 241–262.Google Scholar
  9. Budiansky, B. (1965),On the Elastic Moduli of Some Heterogeneous Materials, J. Mech. Phys. Solids13, 223–227.Google Scholar
  10. Budiansky, B., andO'Connell, R. J. (1979),Comment on ‘Elastic Moduli of Two Component Systems' by A. K. Chatterjee, A. K. Mal and L. Knopoff, J. Geophys. Res.84, 5687–5688.Google Scholar
  11. Burridge, R., andKeller, J. B. (1981),Poro-elasticity Equations Derived from Microstructure, J. Acoust. Soc. Amer.70, 1140–1146.Google Scholar
  12. Chatterjee, A. K., Mal, A. K., andKnopoff, L. (1978),Elastic Moduli of Two Component Systems, J. Geophys. Res.83, 1785–1792.Google Scholar
  13. Eshelby, J. D. (1957),The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems, Proc. Roy. Soc. A241, 376–396.Google Scholar
  14. Garbin, H. D., andKnopoff, L. (1973),The Compressional Modulus of a Material Permeated by a Random Distribution of Circular Cracks, Qu. Appl. Math.30, 453–464.Google Scholar
  15. Hashin, Z. (1962),The Elastic Moduli of Heterogeneous Materials, J. Appl. Mech.29, 143–150.Google Scholar
  16. Hashin, Z., andShtrikman, S. (1962),A Variational Approach to the Theory of the Elastic Behaviour of Polycrystals, J. Mech. Phys. Solids10, 343–352.Google Scholar
  17. Hershey, A. V. (1954),The Elasticity of an Isotropic Aggregate of Anisotropic Cubic Crystals, J. Appl. Mech.21, 236–240.Google Scholar
  18. Hill, R. (1952),The Elastic Behaviour of a Crystalline Aggregate, Proc. Phys. Soc. London, A65, 349–354.Google Scholar
  19. Hill, R. (1963),Elastic Properties of Reinforced Solids: Some Theoretical Principles, J. Mech. Phys. Solids11, 357–372.Google Scholar
  20. Hill, R. (1965),A Self-consistent Mechanic of Composite Materials, J. Mech. Phys. Solids.13, 213–222.Google Scholar
  21. Hudson, J. A. (1980),Overall Properties of a Cracked Solid, Math. Proc. Camb. Phil. Soc.88, 371–384.Google Scholar
  22. Hudson, J. A. (1981),Wave Speeds and Attenuation of Elastic Waves in Material Containing Cracks, Geophys. J. R. Astr. Soc.64, 133–150.Google Scholar
  23. Hudson, J. A. (1986),A Higher Order Approximation to the Wave Propagation Constants for a Cracked Solid, Geophys. J. R. Astr. Soc.87, 265–274.Google Scholar
  24. Hunter, S. C.,Mechanics of Continuous Media, 2nd ed. (Ellis Horwood, Chichester 1983).Google Scholar
  25. Keller, J. B. (1964),Stochastic Equations and Wave Propagation in Random Media, Proc. Symp. Appl. Math.16, 145–170.Google Scholar
  26. Kröner, E. (1958),Berechnung der elastischen Konstanten des Vielkrystalls aus den Konstanten des Einkrystalls, Z. Phys.151, 504–518.Google Scholar
  27. Kröner, E. (1975),Elastostatik statistisch aufgebauter Körper, Z. Angew. Math. Mech.55, T39-T43.Google Scholar
  28. McCoy, J. J. (1973),On the Dynamic Response of Disordered Composites, J. Appl. Mech.39, 511–517.Google Scholar
  29. Nur, A., andSimmong, G. (1969),The Effect of Saturation on Velocity in Low-porosity Rocks, Earth Planet. Sci. Lett.7, 183–193.Google Scholar
  30. O'Connell, R. J., andBudiansky, B. (1974),Seismic Velocities in Dry and Saturated Cracked Solids, J. Geophys. Res.79, 5412–5426.Google Scholar
  31. O'Connell, R. J., andBudiansky, B. (1977),Viscoelastic Properties of Fluid-saturated Cracked Solids, J. Geophys. Res.82, 5719–5735.Google Scholar
  32. Reuss, A. (1929),Berechnung der Fliessgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle, Z. Angew. Math. Mech.9, 49–58.Google Scholar
  33. Varadan, V. K., andVaradan, V. V. (1979),Frequency Dependence of Elastic (SH-) Wave Velocity and Attenuation in Anisotropic Two-phase Media, Wave Motion1, 53–63.Google Scholar
  34. Varatharajulu, V., andPao, Y.-H. (1976),Scattering Matrix for Elastic Waves, I. Theory, J. Acoust. Soc Amer.60, 556–566.Google Scholar
  35. Voigt, W.,Lehrbuch der Kristallphysik (Teubner, Leipzig 1928).Google Scholar
  36. Walsh, J. (1969),A New Analysis of Attenuation in Partially Melted Rocks, J. Geophys. Res.74, 4333–4337.Google Scholar
  37. Waterman, P. C. (1968),New Formulation of Acoustic Scattering, J. Acoust. Soc. Amer.45, 1417–1429.Google Scholar
  38. Waterman, P. C., andTruell, R. (1961),Multiple Scattering of Waves, J. Math. Phys.2, 512–537.Google Scholar
  39. Watt, J. P., Davies, G. F., andO'Connell, R. J. (1976),The Elastic Properties of Composite Materials, Rev. Geophys. Space Phys.14, 541–563.Google Scholar
  40. Willis, J. R. (1977),Bounds and Self-consistent Estimates of the Overall Properties of Anisotropic Composites, J. Mech. Phys. Solids25, 185–202.Google Scholar
  41. Willis, J. R., andActon, J. R. (1976),The Overall Elastic Moduli of a Dilute Suspension of Spheres, Quart. J. Mech. Appl. Math.29, 163–177.Google Scholar
  42. Wu, R.-S. (1985),Multiple Scattering and Energy Transfer of Seismic Waves—Separation of Scattering Effect from Intrinsic Attenuation—I. Theoretical Modelling, Geophys. J. R. Astr. Soc.82, 57–80.Google Scholar
  43. Wu, R.-S., andAki, K. (1985),Elastic Wave Scattering by a Random Medium and the Small-scale Inhomogeneities in the Lithosphere, J. Geophys. Res.90, 10,261–10,273.Google Scholar

Copyright information

© Birkhäuser Verlag 1989

Authors and Affiliations

  • J. A. Hudson
    • 1
  • L. Knopoff
    • 2
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK
  2. 2.Institute of Geophysics and Planetary PhysicsUniversity of CaliforniaLos AngelesUSA

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