Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-20T08:24:26.118Z Has data issue: false hasContentIssue false
  • English
  • Français

Overall properties of a cracked solid

Published online by Cambridge University Press:  24 October 2008

J. A. Hudson
J. A. Hudson
Affiliation:
University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

The differential-integral equation of motion for the mean wave in a solid material containing embedded cavities or inclusions is derived. It consists of a series of terms of ascending powers of the scattering operator, and is here truncated after the third term. This implies the second-order interactions between scatterers are included but those of the third order are not.

The formulae are specialized to the case of thin cracks, either aligned in a single direction or randomly oriented. Expressions for the overall elastic constants are derived for the case of long wavelengths. These expressions are accurate to the second order in the number density of scatterers.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 2 , September 1980 , pp. 371 - 384
Copyright
Copyright © Cambridge Philosophical Society 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Anderson, D. L., Minster, B. and Cole, D.The effect of oriented cracks on seismic velocities. J. Geophys. Res. 79 (1974), 4011–4015.CrossRefGoogle Scholar
(2)Bristow, J. R.Micro-cracks and the static and dynamic constants of annealed and heavily cold-worked metals. Brit. J. Appl. Phys. 11 (1960), 81–85.CrossRefGoogle Scholar
(3)Budiansky, B. and O'Connell, R. J.Elastic moduli of a cracked solid. Int. J. Solids Struct. 12 (1976), 81–97.CrossRefGoogle Scholar
(4)Chatterjee, A. K., Mal, A. K. and Knopoff, L.Elastic moduli of two-component systems. J. Geophys. Res. 83 (1978), 1785–1792. See also the discussion in J. Geophys. Res. 84 (1979), 5687–5690.CrossRefGoogle Scholar
(5)Eason, G., Fulton, J. and Sneddon, I. N.The generation of waves in an infinite elastic solid by variable body forces. Phil. Trans. Roy. Soc. A248 (1956), 575–607.Google Scholar
(6)Foldy, L. L.The multiple scattering of waves. Phys. Rev. (2) 67 (1945), 107–119.CrossRefGoogle Scholar
(7)Garbin, H. D. and Knopoff, L.The compressional modulus of a material permeated by a random distribution of circular cracks. Quart. Appl. Math. 30 (1973), 453–464.CrossRefGoogle Scholar
(8)Garbin, H. D. and Knopoff, L.The shear modulus of a material permeated by a random distribution of free circular cracks. Quart. Appl. Math. 33 (1975), 296–300.CrossRefGoogle Scholar
(9)Garbin, H. D. and Knopoff, L.Elastic moduli of a medium with liquid-filled cracks. Quart. Appl. Math. 33 (1975), 301–303.CrossRefGoogle Scholar
(10)Hoenig, A.Elastic moduli of a non-randomly oriented cracked body. Int. J. Solids Struct. 15 (1979), 137–154.CrossRefGoogle Scholar
(11)Keller, J. B.Stochastic equations and wave propagation in random media. Proc. Symp. Appl. Math. 16 (1964), 145–170.CrossRefGoogle Scholar
(12)Lax, M.Multiple scattering of waves. Phys. Rev. (2) 85 (1952), 621–629.CrossRefGoogle Scholar
(13)Walsh, J. B.New analysis of attenuation in partially melted rock. J. Geophys. Res. 74 (1969), 4333–4337.CrossRefGoogle Scholar
(14)Watt, J. P., Davies, G. F. and O'Connell, R. J.The elastic properties of composite materials. Rev. Geophys. Space Phys. 14 (1976), 541–563.CrossRefGoogle Scholar
(15)Willis, J. R. and Acton, J. R.The overall elastic moduli of a dilute suspension of spheres. Qu. J. Mech. Appl. Math. 29 (1976), 163–177.CrossRefGoogle Scholar