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Comprehensive form of solution for Lamb’s dynamic problem expressed by Green’s functions

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Abstract

By the analysis for the vectors of a wave field in the cylindrical coordinate and Sommerfeld’s identity as well as Green’s functions of Stokes’ solution pertaining the conventional elastic dynamic equation, the results of Green’s function in an infinite space of an axisymmetric coordinate are shown in this paper. After employing a supplementary influence field and the boundary conditions in the free surface of a semi-space, the authors obtain the solutions of Green’s function for Lamb’s dynamic problem. Besides, the vertical displacement u zz and the radial displacement u rz can match Lamb’s previous results, and the solutions of the linear expansion source u rr and the linear torsional source u θθ are also given in the paper. The authors reveal that Green’s function of Stokes’ solution in the semi-space is a comprehensive form of solution expressing the dynamic Lamb’s problem for various situations. It may benefit the investigation of deepening and development of Lamb’s problems and solution for pertinent dynamic problems conveniently.

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References

  1. Lamb, H. On the propagation of tremors over the surface of an elastic solid. Philosophical Transactions of the Royal Society, 203, 1–42 (1904)

    Article  Google Scholar 

  2. Erigen, A. C. and Suhubi, E. S. Elastic Dynamics Linear Theory, Academic Press, New York (1975)

    Google Scholar 

  3. Pekeris, C. L. The seismic surface pulse. Proceedings of the National Academy of Sciences of the United States of America, 41(7) 469–480 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  4. Wang, Y. S. Exact solution for the dynamic vertical surface displacement of the elastic half-space under vertical harmonic point load. Acta Mechanica Sinica, 12(4) 386–391 (1980)

    Google Scholar 

  5. Paul, S. On the displacement produced in a porous elastic half-space by an impulsive line load (non-dissipative case). Pure and Applied Geophysics, 114(4) 605–614 (1976)

    Article  Google Scholar 

  6. Paul, S. On the disturbance produced in a semi-infinite poroelastic medium by a surface load. Pure and Applied Geophysics, 114(4) 615–627 (1976)

    Article  Google Scholar 

  7. Reissner, E. Freie und erzwungene torsionss-chwingungen des elastischen halbraumes. Journal of Applied Physics, 8, 229–245 (1937)

    MATH  Google Scholar 

  8. Reissner, E. and Sagoci, H. F. Forced torsional oscillations of an elastic half-space. Journal of Applied Physics, 15(9) 652–662 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bycroft, G. N. Forced vibrations of a rigid circular plate on a semi-infinite elastic space and on an elastic stratum. Philosophical Transactions of the Royal Society, 248, 327–368 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rahman, M. The Reissner-Sagoci problem for a half-space under buried torsional forces. International Journal of Solid and Structures, 37, 1119–1132 (1998)

    Article  Google Scholar 

  11. Kontoni, D. P. N., Beskos, D. E., and Manolis, G. D. Uniform half-plane elastic-dynamic problems by an approximate boundary element method. Soil Dynamics and Earthquake Engineering, 6(4) 227–238 (1987)

    Article  Google Scholar 

  12. Manolis, G. D. Elastic wave scattering around cavities in inhomogeneous continua by the BEM. Journal of Sound and Vibration, 266(2) 281–305 (2003)

    Article  MATH  Google Scholar 

  13. Dominguez, J. and Abascal, R. Dynamics of Foundations in Topics in Boundary Element Research, Springer-Verlag, Berlin (1987)

    Google Scholar 

  14. Padro, L. A., Azna, J. J., and Maeso, O. BEM-FEM coupling model for the dynamic analysis of piles and pile groups. Engineering Analysis with Boundary Elements, 31(6) 473–484 (2007)

    Article  Google Scholar 

  15. Ding, B. Y., Ding, C. H., and Meng, F. L. The Green function on two-phase saturated medium by concentrated force (in Chinese). Acta Mechanica Sinica, 33(2) 234–241 (2001)

    Google Scholar 

  16. Ding, B. Y., Meng, F. L., and Hu, M. Y. The source vector and static displacement field by elastic dislocation on the two-phase saturated medium. Acta Seismologica Sinica, 23(3) 239–245 (2001)

    Google Scholar 

  17. Ding, B. Y., Song, X. C., and Yuan, J. H. Solution for displacement response of saturated soil by a concentrated load in tunnel of rectangular section. Engineering Mechanics, 26(6) 153–157 (2009)

    Google Scholar 

  18. Ding, B. Y., Dang, G. H., and Yuan, J. H. Green function of the saturated soil tunnel concentrated loads calculation of vibration displacement response. Journal of Vibration and Shock, 28(11) 110–114 (2009)

    Google Scholar 

  19. Ding, B. Y., Fan, L. B., and Wu, J. H. The Green function and wave field on two-phase saturated medium by concentrated force. Chinese Journal of Geophysical, 42(6) 800–808 (1999)

    Google Scholar 

  20. Ding, B. Y., Ding, C. H., Chen, Y., and Tao, H. B. Green function on two-phase saturated medium by concentrated force in two-dimensional displacement field. Applied Mathematics and Mechanics (English Edition) 25(8) 951–956 (2004) DOI 10.1007/BF02438804

    Article  MATH  Google Scholar 

  21. Ding, B. Y., Yuan, J. H., and Pan, X. D. The abstracted and integrated Green functions and OPP of BEM in soil dynamics. Science in China, Series G, 39(2) 284–292 (2009)

    Google Scholar 

  22. Zheng, P., Ding, B. Y., Zhao, S. X., and Ding, D. Dynamic response of a multilayered poroelastic half-space to harmonic surface tractions. Transport in Porous Media, 98(21) 229–249 (2013)

    Article  MathSciNet  Google Scholar 

  23. Ding, B. Y. and Yuan, J. H. Dynamic Green’s functions of a two-phase saturated medium subjected to concentrated force. International Journal of Solids and Structures, 48, 2288–2303 (2011)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Civil Engineering, Zhejiang University of Technology, Hangzhou, 310014, P. R. China

    Bo-yang Ding  (丁伯阳), Ting Xu  (徐 庭), Jun Chen  (陈 军) & Ming Ou-Yang  (欧阳明)

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  1. Bo-yang Ding  (丁伯阳)
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  2. Ting Xu  (徐 庭)
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  3. Jun Chen  (陈 军)
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  4. Ming Ou-Yang  (欧阳明)
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Corresponding author

Correspondence to Bo-yang Ding  (丁伯阳).

Additional information

Project supported by the National Natural Science Foundation of China (No. 11172268)

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Ding, By., Xu, T., Chen, J. et al. Comprehensive form of solution for Lamb’s dynamic problem expressed by Green’s functions. Appl. Math. Mech.-Engl. Ed. 34, 1543–1552 (2013). https://doi.org/10.1007/s10483-013-1766-8

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  • DOI: https://doi.org/10.1007/s10483-013-1766-8

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