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Lamb’s problem for solids of general anisotropy. (English) Zbl 0954.74523

Summary: A method to construct solutions for elastic waves generated in a half-space has been developed, based on representing the wave field by a superposition of time-transient plane waves. By the use of this method, the classical Lamb’s problem has been solved for a solid of general anisotropy. New expressions for the 2-D and 3-D solutions to Lamb’s problem in both the time-domain and the frequency-domain have been obtained. These expressions, which have been given in terms of integrals defined in a finite domain, have a simple structure which is attractive for numerical applications. The usefulness and computability of the integral expressions have been demonstrated by numerical examples.

MSC:

74J10 Bulk waves in solid mechanics
74E10 Anisotropy in solid mechanics
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[1] Abubakar, I., Disturbance due to a line source in a semi-infinite, transversely isotropic medium, Geophys.J., 6, 337-359 (1962) · Zbl 0114.46301
[2] Barnett, D. M.; Lothe, J., Surface (Rayleigh) waves in anisotropic elastic half-space: The surface impedance method, (Proc. Roy. Soc. London Ser, A 402 (1985)), 135-166 · Zbl 0587.73030
[3] Ben-Menahem, A.; Sena, A. G., The elastodynamic Green’s tensor in an anisotropic half-space, Geophys. J. Int., 102, 421-443 (1990) · Zbl 0714.73016
[4] Burridge, R., The direction in which Rayleigh waves may be propagated on crystals, Quart. J. Mech. Appl. Math., 23, 217-224 (1970) · Zbl 0219.73023
[5] Burridge, R., Lamb’s problem for an anisotropic half-space, Quart. J. Mech. Appl. Math., 24, 81-98 (1971) · Zbl 0232.73022
[6] Cagniard, L.; Flinn, E. A.; Dix, C. H., Reflection and Refraction of Progressive Seismic Waves (1962), McGraw-Hill: McGraw-Hill New York
[7] de Hoop, A. T., Modification of Cagniard’s method for solving seismic pulse problems, Appl. Sci. Res., B 8, 349-356 (1960) · Zbl 0100.44208
[8] Duff, G. F.D., The Cauchy’s problem for elastic wave in an anisotropic medium, Philos. Trans. Roy. Soc. London Ser, A 252, 249-273 (1960) · Zbl 0103.42502
[9] Garvin, W. W., Exact transient solution of the burried line source problem, (Proc. Roy. Soc. London Ser, A 234 (1956)), 528-541 · Zbl 0071.40101
[10] Gel’fand, I. M.; Vilenkin, N. Ya., (Generialized Functions, Vol. 5 (1966), Academic Press: Academic Press New York)
[11] Johnson, L. R., Green’s functions for Lamb’s problem, Geophys. J. Roy. Astrono. Soc., 37, 99-131 (1974) · Zbl 0298.73029
[12] Kraut, E. A., Advances in the theory of anisotropic elastic wave propagation, Rev. Geophys., 1, 401-448 (1963)
[13] Lamb, H., On the propagation of tremors over the surface of an elastic solid, Phil. Trans. Roy. Soc. Londonser, A 203, 1-42 (1904) · JFM 34.0859.02
[14] Musgrave, M. J.P., Crystals Acoustics (1970), Holden-Day: Holden-Day San Francisco, CA · Zbl 0201.27601
[15] Payton, R. G., Elastic Wave Propagation in Transversly Isotropic Media (1983), Martinus Nijhoff: Martinus Nijhoff The Hague · Zbl 0574.73023
[16] Spies, M., Elastic wave propagation in general transversely isotropic media. I. Green’s functions and elastodynamic holography, J. Acoust. Soc. Amer., 96, 2, 1144-1157 (1994), Pt. 1
[17] Tewary, V. K.; Fortunco, C. M., A computationally efficient representation for propagation of elastic waves in anisotropic solids, J. Acoust. Soc. Amer., 91, 4, 1888-1896 (1992), Pt. 1
[18] van der Hijden, J. H.M. T., Propagation of Transient Elastic Waves in Stratified Anisotropic Media (1987), North-Holland: North-Holland Amsterdam · Zbl 0667.73020
[19] Wang, C.-Y.; Achenbach, J. D., A new look at 2-D time-domain elastodynamic Green’s functions for general anisotropic solids, Wave Motion, 16, 389-405 (1992) · Zbl 0764.73021
[20] Wang, C.-Y.; Achenbach, J. D., Elastodynamic fundamental solutions for anisotropic solids, Geophys. J. Int., 118, 384 (1994)
[21] C.-Y. Wang and J.D. Achenbach and S. Hirose, “2-D time-domain BEM for scattering of elastic waves in anisotropic solids of general anisotropy”, Int. J. Solids and Structures; C.-Y. Wang and J.D. Achenbach and S. Hirose, “2-D time-domain BEM for scattering of elastic waves in anisotropic solids of general anisotropy”, Int. J. Solids and Structures · Zbl 0918.73307
[22] Willis, J. R., Self-similar problems in elastodynamics, Philos. Trans. Roy. Soc. London Ser, A 274, 435-491 (1973) · Zbl 0317.73062
[23] Willis, J. R., Arrivals associated with a class of self-similar problems in elastodynamics, (Math. Proc. Cambridge, Philos. Soc., 77 (1975)), 591-607 · Zbl 0313.73025
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