Skip to main content

Advertisement

We’d like to understand how you use our websites in order to improve them. Register your interest.

Elastodynamic response of an anisotropic medium due to a line-load

Abstract

Two-dimensional elastodynamic displacements and stresses for a monoclinic solid have been obtained in relatively simple form by the use of the eigenvalue method, following Laplace and Fourier transforms. The main aim of this paper is to present a straightforward analytical eigenvalue method for a monoclinic solid which avoids the cumbersome nature of the problem and is convenient for numerical computation. The use of matrix notation avoids unwieldy mathematical expressions. A particular case of normal line-load acting in an orthotropic solid is discussed in detail. The corresponding deformation in time-domain is obtained numerically. The variations of elastodynamic displacements and stresses for an anisotropic medium with the horizontal distance have been shown graphically. It has been found that anisotropy is affecting the trend of distribution curves significantly.

References

  1. Achenbach, J. D., Wave Propagation in Elastic Solids, North-Holland-Elsevier, Amsterdam, 1973.

  2. Atanackovic, T. M. and A. Guran, Theory of Elasticity for Scientist and Engineers, Birkhauser Boston, 2000.

  3. Bonafede, M. and E. Rivalta, The tensile dislocation problem in a layered elastic medium, Geophys. J. Int., 136, 341–356, 1999a.

  4. Bonafede, M. and E. Rivalta, On tensile cracks close to and across the interface between two welded elastic half-spaces, Geophys. J. Int., 138, 410–434, 1999b.

  5. Buchwald, V. T., Elastic waves in anisotropic media, Proc. R. Soc. Lond., A253, 563–580, 1959.

  6. Burridge, R., The singularity on the plane lids of the wave surface of elastic media with cubic symmetry, Q. J. Mech. Appl. Math., 20, 40–56, 1967.

  7. Debnath, L. and Loknath, Integral Transforms and Their Application, CRC Press Inc., New York, 1995.

  8. Duff, G. F. D., The Cauchy problem for elastic waves in an anisotropic medium, Phil. Trans. R. Soc. Lond., A252, 249–273, 1960.

  9. Garg, N. R., D. K. Madan, and R. K. Sharma, Two-dimensional deformation of an orthotropic elastic medium due to seismic sources, Phys. Earth Planet. Inter., 94, 43–62, 1996.

  10. Garg, N. R., R. Kumar, A. Goel, and A. Miglani, Plane strain deformation of an orthotropic elastic medium using an eigenvalue approach, Earth Planets Space, 55(1), 2003.

  11. Hoing, G. and U. Hirdes, A method for the numerical inversion of the Laplace transform, J. Comp. Appl. Math., 10, 113–132, 1984.

  12. Karabolis, D. L. and D. E. Beskos, Dynamic response of 3-D rigid surface foundations by time domain boundary element method, Earthquake Eng. Structure Dyn., 12, 73–93, 1984.

  13. Lamb, H., On the propagation of tremors over the surface of an elastic solid, Phil. Trans. Roy. Soc. Am., 203, 1–42, 1904.

  14. Lighthill, M. J., Studies on magneto-hydrodynamic waves and other anisotropic wave motions, Phil. Trans. R. Soc. Lond., A252, 397–430, 1960.

  15. Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944.

  16. Maruyama, T., On the force equivalents of dynamical elastic dislocations with reference to the earthquake mechanism, Bull. Earth. Res. Inst. (Tokyo), 41, 467–486, 1963.

  17. Maruyama, T., On two-dimensional elastic dislocations in an infinite and semi-infinite medium, Bull. Earthq. Res. Inst., 44, 811–871, 1966.

  18. Mase, G. T. and G. E. Mase, Continuum Mechanics for Engineers, CRC Press LLC., New York, 1999.

  19. Mura, T., Micromechanics of Defects in Solids, 2nd edition, Kluwer Academic Publishers, London, 1987.

  20. Niwa, Y., T. Fukui, S. Kato, and K. Fujiki, An application of the integral equation method to two-dimensional elastodynamics, in Theoretical and Applied Mechanics, vol. 28, University of Tokyo Press, Tokyo, pp. 281–290, 1980.

  21. Okada, Y., Surface deformation due to inclined shear and tensile faults in a homogenous isotropic half space, Bull. Seismol. Soc. Am., 75, 1135–1154, 1985.

  22. Okada, Y., Internal deformation due to shear and tensile faults in a halfspace, Bull. Seismol. Soc. Am., 82, 1018–1040, 1992.

  23. Pan, E., Static response of a transversely istropic and layered half space to general dislocation sources, Phys. Earth Planet. Inter., 58, 103–117, 1989.

  24. Payton, R. G., Elastic Wave Propogation in Transversely Isotropic Media, Matrtinus Nijhoff Publishers, The Hague, 1983.

  25. Piersanti, A., G. Spada, R. Sabadini, and M. Bonafede, Global post-seismic deformation, Geophys. J. Int., 120, 544–566, 1995.

  26. Press, W. H., S. A. Teukolsky, W. T. Velleling, and B. P. Flannery, Numerical Recipes in Fortran, Cambridge University Press, Cambridge, 1986.

  27. Ross, S. L., Differential Equations, 3rd edition, John Wiley and Sons Inc., 1984.

  28. Sneddon, I. N., Fourier Transforms, McGraw-Hill, New York, 1951.

  29. Stokes, G. G., On the dynamical theory of diffraction, Trans. Camb. Phil. Soc., 9, 1–62, 1849.

  30. Tverdokhlebov, A. and J. Rose, On Green’s function for elastic waves in anisotropic media, J. Acoust. Soc. Am., 83, 118–121, 1988.

  31. Wang, C. Y. and J. D. Achenbach, A new look at 2-D time-domain elastodynamic Green’s functions for general anisotropic solids, Wave Motion, 16, 389–405, 1992.

  32. Wang, C. Y. and J. D. Achenbach, A new method to obtain 3-D Green’s functions for anisotropic solids, Wave Motion, 18, 273–289, 1993.

  33. Wang, C. Y. and J. D. Achenbach, Elastodynamic fundamental solutions for anisotropic solids, Geophys. J. Int., 118, 384–392, 1994.

  34. Yeatts, F. R., Elastic radiation from a point source in an anisotropic medium, Phys. Rev. B, 29, 1674–1684, 1984.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to N. R. Garg.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Garg, N.R., Goel, A., Miglani, A. et al. Elastodynamic response of an anisotropic medium due to a line-load. Earth Planet Sp 56, 407–417 (2004). https://doi.org/10.1186/BF03352494

Download citation

Key words

  • Monoclinic
  • orthotropic
  • eignevalue method
  • two-dimensional deformation integral transforms
  • elastodynamic