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Reflection–transmission matrix model for the axisymmetric deformation of a layered transversely isotropic saturated soil

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Abstract

The reflection–transmission matrix (RTM) method for the layered transversely isotropic saturated soil (TISS) undergoing axisymmetric consolidation is developed in this study. To this aim, a system of partial differential equations is established based on the governing equations for the TISS first. By means of the Hankel–Laplace integral transform method, the system of partial differential equations is reduced to a system of ordinary differential equations. By using the general solution to the system of ordinary differential equations, the static wave vector corresponding to the state vector of the TISS is introduced, and the transfer matrices for the state and wave vectors are defined. The RTMs for the aforementioned static wave vector of the layered TISS are then established. By means of the RTMs for the layered TISS, solutions for the layered TISS subjected to external sources are derived. Numerical results due to the proposed RTM method are compared with some existing results, validating the proposed RTM method. Parametric analyses are conducted to illustrate the influence of some parameters on the consolidation of the layered TISS.

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References

  1. Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)

    Article  MATH  Google Scholar 

  2. Senjuntichai, T., Rajapakse, R.K.N.D.: Exact stiffness method for quasi-statics of a multi-layered poroelastic medium. Int. J. Solids Struct. 32, 1535–1553 (1995)

    Article  MATH  Google Scholar 

  3. Wang, J.G., Fang, S.: The state vector solution of axisymmetric Biot’s consolidation problems for multilayered poroelastic media. Mech. Res. Commun. 28(6), 671–677 (2001)

    Article  MATH  Google Scholar 

  4. Ai, Z.Y., Wu, C., Han, J.: Transfer matrix solutions for three-dimensional consolidation of a multi-layered soil with compressible constituents. Int. J. Eng. Sci. 46, 1111–1119 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ai, Z.Y., Cheng, Y.C., Zeng, W.Z.: Analytical layer-element solution to axisymmetric consolidation of multilayered soils. Comput. Geotech. 38, 227–232 (2011)

    Article  Google Scholar 

  6. Christian, J.T., Boehmer, J.W.: Plane strain consolidation by finite elements. J. Soil Mech. Found. 96, 1435–1457 (1970)

    Google Scholar 

  7. Chiou, Y.J., Chi, S.Y.: Boundary element analysis of Biot consolidation in layered elastic soils. Int. J. Numer. Anal. Methods Geomech. 18, 377–396 (1994)

    Article  MATH  Google Scholar 

  8. Booker, J.R., Small, J.C.: Finite layer analysis of consolidation. Int. J. Numer. Anal. Methods Geomech. 6, 151–171 (1982)

    Article  MATH  Google Scholar 

  9. Booker, J.R., Small, J.C.: Finite layer analysis of consolidation II. Int. J. Numer. Anal. Methods Geomech. 6, 173–194 (1982)

    Article  MATH  Google Scholar 

  10. Gaspar, F.J., Lisbona, F.J., Vabishchevich, P.N.: A finite difference analysis of Biot’s consolidation model. Appl. Numer. Math. 44, 487–506 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, S.L., Chen, L.Z., Zhang, L.M.: The axisymmetric consolidation of a semi-infinite transversely isotropic saturated soil. Int. J. Numer. Anal. Methods Geomech. 29, 1249–1270 (2005)

    Article  MATH  Google Scholar 

  12. Chen, S.L., Zhang, L.M., Chen, L.Z.: Consolidation of a finite transversely isotropic soil layer on a rough impervious base. J. Eng. Mech. 131, 1279–1290 (2005)

    Article  Google Scholar 

  13. Ai, Z.Y., Wang, Q.S.: Transfer matrix solutions for axisymmetric consolidation of multilayered transversely isotropic soils. Rock Soil Mech. 30, 921–925 (2009)

    Google Scholar 

  14. Chen, G.J., Zhao, X.H., Yu, L.: Transferring matrix method to solve axisymmetric problems of layered cross-anisotropic elastic body. Chin. J. Geotech. Eng. 20, 522–527 (1998)

    Google Scholar 

  15. Chen, G.J., Zhao, X.H.: Solutions for consolidation axisymmetric problems of cross-anisotropic layered subsoils. China Civil Eng. J. 32, 14–20 (1999)

    Google Scholar 

  16. Kennett, B.L.N.: Reflections, rays and reverberations. Bull. Seismol. Soc. Am. 64, 1685–1696 (1974)

    Google Scholar 

  17. Kennett, B.L.N.: Seismic Wave Propagation in a Stratified Medium. Cambridge University Press, Cambridge (1983)

    MATH  Google Scholar 

  18. Lu, J.F., Hanyga, A.: Fundamental solution for a layered porous half space subject to a vertical point force or a point fluid source. Comput. Mech. 35, 376–391 (2005)

    Article  MATH  Google Scholar 

  19. Xu, B., Lu, J.F., Wang, J.H.: Dynamic response of a layered water-saturated half space to a moving load. Comput. Geotech. 35, 1–10 (2008)

    Article  Google Scholar 

  20. Xie, X.B., Yao, Z.X.: A generalized reflection-transmission coefficient matrix method to calculate static displacement field of a dislocation source in a stratified half-space. Chin. J. Geophys. 32, 191–205 (1989)

    Google Scholar 

  21. He, Y.M., Wang, W.M., Yao, Z.X.: Static deformation due to shear and tensile faults in a layered half-space. Bull. Seismol. Soc. Am. 93, 2253–2263 (2003)

    Article  Google Scholar 

  22. Kuvshinov, B.N.: Reflectivity method for geomechanical equilibria. Geophys. J. Int. 170, 567–579 (2007)

    Article  Google Scholar 

  23. Zhu, L., Rivera, L.: A note on the dynamic and static displacements from a point source in multilayered media. Geophys. J. Int. 148, 619–627 (2002)

    Article  Google Scholar 

  24. Carcione, J.M.: Wave Fields in Real Media-Wave Propagation in Anisotropic. Anelastic and Porous Media. Elsevier Science, Amsterdam (2007)

    Google Scholar 

  25. Sneddon, I.N.: The Use of Integral Transforms. McGraw-Hill, New York (1972)

    MATH  Google Scholar 

  26. Swift, R.J., Wirkus, S.A.: A Course in Ordinary Differential Equations. Taylor & Francis Group, LLC, UK (2006)

    MATH  Google Scholar 

  27. Piessens, R.: Bibliography on numerical inversion of the Laplace transform and applications. J. Comput. Appl. Math. 1, 115–128 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  28. Stehfest, H.: Numerical inversion of Laplace transforms. Commun. ACM 13, 47–49 (1970)

    Article  Google Scholar 

  29. Rajapakse, R.K.N.D., Senjuntichai, T.: Fundamental solutions for a poroelastic half-space with compressible constituents. J. Appl. Mech. 60, 847–856 (1993)

    Article  MATH  Google Scholar 

  30. Puswewala, U.G.A., Rajapakse, R.K.N.D.: Axisymmetric fundamental solutions for a completely saturated porous elastic solid. Int. J. Eng. Sci. 26, 419–436 (1988)

    Article  MATH  Google Scholar 

  31. Schapery, R.A.: Approximate methods of transform inversion for viscoelastic stress analysis. Proc. Fourth US Natl. Congr. Appl. Mech. 2, 1075–1085 (1962)

    MathSciNet  Google Scholar 

  32. Pan, E.: Static Green’s functions in multilayered half spaces. Appl. Math. Model. 21, 509–521 (1997)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Civil Engineering, Jiangsu University, Zhenjiang, 212013, Jiangsu, People’s Republic of China

    Jian-Fei Lu, Yu-Qi Liu & Meng-Qin Shi

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  1. Jian-Fei Lu
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  2. Yu-Qi Liu
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  3. Meng-Qin Shi
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Correspondence to Jian-Fei Lu.

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Lu, JF., Liu, YQ. & Shi, MQ. Reflection–transmission matrix model for the axisymmetric deformation of a layered transversely isotropic saturated soil. Acta Mech 227, 2181–2205 (2016). https://doi.org/10.1007/s00707-016-1619-0

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  • DOI: https://doi.org/10.1007/s00707-016-1619-0

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