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Lamb’s integral formulas of two-phase saturated medium for soil dynamic with drainage

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Abstract

When dynamic force is applied to a saturated porous soil, drainage is common. In this paper, the saturated porous soil with a two-phase saturated medium is simulated, and Lamb’s integral formulas with drainage and stress formulas for a two-phase saturated medium are given based on Biot’s equation and Betti’s theorem (the reciprocal theorem). According to the basic solution to Biot’s equation, Green’s function G ij and three terms of Green’s function G 4i , G i4, and G 44 of a two-phase saturated medium subject to a concentrated force on a spherical coordinate are presented. The displacement field with drainage, the magnitude of drainage, and the pore pressure of the center explosion source are obtained in computation. The results of the classical Sharpe’s solutions and the solutions of the two-phase saturated medium that decays to a single-phase medium are compared. Good agreement is observed.

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References

  1. Chen, J. Time domain function solution to Biot’s complete equations of dynamic poroelasticity, part II, two dimensional solution. International Journal of Solids and Structures 31(10), 1447–1490 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen, J. Time domain function solution to Biot’s complete equations of dynamic poroelasticity, part I, three dimensional solution. International Journal of Solids and Structures 31(2), 169–202 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Biot, M. A. Theory of propagation of elastic wave in a fluid-saturated soil. Journal of the Acoustical Society of America 28, 168–178 (1956)

    Article  MathSciNet  Google Scholar 

  4. Biot, M. A. Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics 33, 1482–1498 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cleary, M. P. Fundament solutions for a fluid-saturated porous solid. International Journal of Solids and Structures 13, 785–806 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  6. Burridge, R. and Vargas, C. A. The fundamental solutions in dynamic poroelasticity. Geophysical Journal of the Royal Astronomical Society 58(1), 61–90 (1979)

    MATH  Google Scholar 

  7. Norris, A. N. Radiation from a point source and scattering theory in a fluid saturated porous solid. Journal of the Acoustical Society of America 77, 2012–2023 (1985)

    Article  MATH  Google Scholar 

  8. Ding, B. Y., Song, X. C., and Yuan, J. H. The solution of Green function on fluid phase in two-phase saturated medium (in Chinese). Chinese Journal of Geophysics 52(7), 1858–1866 (2009)

    Google Scholar 

  9. Liu, Y. B., Li, Y. M., and Wu, R. S. Seismic wave propagation in transversely isotropic porous media (in Chinese). Chinese Journal of Geophysics 37(4), 499–514 (1994)

    Google Scholar 

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

    Google Scholar 

  11. 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 

  12. 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 14(3), 239–245 (2001)

    Article  Google Scholar 

  13. Eringen, A. C. and Suhubi, E. S. Elastic Dynamics, Vol. 2, Linear Theory, Academic Press, New York/San-Francisco/London, 76–138 (1975)

    Google Scholar 

  14. Miao, T. D., Zhu, J. J., and Ding, B. Y. Essay on constitutive relation of wave propagation in saturated porous media (in Chinese). Acta Mechanica Sinica 27(5), 536–543 (1995)

    Google Scholar 

  15. 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 (in Chinese). The Engineering Mechanics 30(3), 153–157 (2009)

    Google Scholar 

  16. Ding, B. Y., Dang, G. H., and Yuan, J. H. Calculation for displacement response of saturated soil by a concentrated load in tunnel of rectangular section (in Chinese). Journal of Vibration and Shock 28(11), 110–114 (2009)

    Google Scholar 

  17. 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 51(12), 1926–1937 (2008)

    Article  Google Scholar 

  18. Sharpe, J. A. The production of elastic waves by explosion pressure, I. Theory and empirical field observations. Geophysics 7(2). 144–152 (1942)

    Article  MathSciNet  Google Scholar 

  19. Ding, B. Y., Fan, L. B., and Meng, F. L. The displacement field of dislocation on the half-space of two-phase saturated medium (in Chinese). Chinese Journal of Geophysics 46(3), 408–414 (2003)

    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 

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Authors and Affiliations

  1. College of Civil Engineering and Architecture, Zhejiang University of Technology, Hangzhou, 310014, P. R. China

    Bo-yang Ding  (丁伯阳) & Jin-hua Yuan  (袁金华)

  2. Department of Civil Engineering, Zhejiang Agriculture and Forest University, Hangzhou, 313100, P. R. China

    Gai-hong Dang  (党改红)

Authors
  1. Bo-yang Ding  (丁伯阳)
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  2. Gai-hong Dang  (党改红)
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  3. Jin-hua Yuan  (袁金华)
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Corresponding author

Correspondence to Bo-yang Ding  (丁伯阳).

Additional information

Communicated by Chang-jun CHENG

Project supported by the National Natural Science Foundation of China (No. 10572129)

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Ding, By., Dang, Gh. & Yuan, Jh. Lamb’s integral formulas of two-phase saturated medium for soil dynamic with drainage. Appl. Math. Mech.-Engl. Ed. 31, 1113–1124 (2010). https://doi.org/10.1007/s10483-010-1347-9

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  • DOI: https://doi.org/10.1007/s10483-010-1347-9

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2000 Mathematics Subject Classification

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