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Dynamic 2.5-D Green’s function for a point load or a point fluid source in a layered poroelastic half-space. (English) Zbl 1403.76080
Summary: The complete dynamic two-and-a-half-dimensional (2.5-D) Green’s function for an internal point load or fluid source buried in a layered poroelastic half-space is derived and applied to the 2.5-D boundary element method (BEM) in this paper. Based on Biot’s theory, the general solutions are derived using the potential decomposition method and the Fourier transform. Utilizing the boundary conditions of the free surface, interfaces and bottom half-space, as well as the general solutions, the complete 2.5-D Green’s function for a layered poroelastic half-space is obtained using the transmission and reflection matrix (TRM) method. The solutions presented in this paper are free of numerical instability for the high frequency and large layer thickness. The proposed 2.5-D Green’s function is verified by comparison with the existing solutions. A case study of calculating vibrations from a semi-circular tunnel embedded in a layered poroelastic half-space is presented using the 2.5-D BEM along with the proposed 2.5-D Green’s function. The layer interfaces and the surface of the poroelastic half-space no longer have to be meshed, avoiding spurious reflections at mesh truncations. The boundary element mesh can be limited to the surface of the tunnel, significantly reducing the size of the boundary element mesh.

76M15 Boundary element methods applied to problems in fluid mechanics
65M80 Fundamental solutions, Green’s function methods, etc. for initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76S05 Flows in porous media; filtration; seepage
Full Text: DOI
[1] Papageorgiou, A. S.; Pei, D., A discrete wavenumber boundary element method for study of the 3-D response of 2-d scatterers, Earthq Eng Struct Dyn, 27, 619-638, (1998)
[2] Sheng, X.; Jones, C. J.C.; Thompson, D. J., Modelling ground vibration from railways using wavenumber finite- and boundary element methods, P R Soc Lond A Mat, 461, 2043-2070, (2005) · Zbl 1186.74051
[3] Lu, J. F.; Jeng, D. S.; Williams, S., A 2.5-D dynamic model for a saturated porous medium: part I. Green’s function, Int J Solids Struct, 45, 378-391, (2008) · Zbl 1167.74398
[4] Lu, J. F.; Jeng, D. S.; Williams, S., A 2.5-D dynamic model for a saturated porous medium. part II: boundary element method, Int J Solids Struct, 45, 359-377, (2008) · Zbl 1167.74397
[5] Tadeu, A.; Stanak, P.; Antonio, J.; Sladek, J.; Sladek, V., 2.5D elastic wave propagation in non-homogeneous media coupling the BEM and MLPG methods, Eng Anal Bound Elem, 53, 86-99, (2015) · Zbl 1403.74236
[6] Burridge, R.; Vargas, C. A., The fundamental solution in dynamic poroelasticity, Geophys J Int, 58, 61-90, (1979) · Zbl 0498.73018
[7] Norris, A. N., Radiation from a point source and scattering theory in a fluid-saturated porous medium, J Acoust Soc Am, 77, 2012-2023, (1985) · Zbl 0579.73107
[8] Zimmerman, C.; Stern, M., Boundary element solution of 3-D wave scatter problems in a poroelastic medium, Eng Anal Bound Elem, 12, 223-240, (1993)
[9] Cheng, A. H.D.; Badmus, D. E.; Beskos, D. E., Integral equations for dynamic poroelasticity in frequency domain with BEM solution, J Eng Mech, 117, 1136-1157, (1991)
[10] Philippacopoulos, A. J., Buried point source in a poroelastic half-space, J Eng Mech, 123, 860-869, (1997)
[11] Jin, B.; Liu, H., Dynamic response of a poroelastic half space to horizontal buried loading, Int J Solids Struct, 38, 8053-8064, (2001) · Zbl 1037.74025
[12] Zheng, P.; Zhao, S. X.; Ding, D., Dynamic Green’s functions for a poroelastic half-space, Acta Mech, 224, 17-39, (2013) · Zbl 1401.74147
[13] Zhou, S. H.; He, C.; Di, H. G., Dynamic 2.5-D Green’s function for a poroelastic half-space, Eng Anal Bound Elem, 67, 96-107, (2016) · Zbl 1403.74272
[14] Thomson, W. T., Transmission of elastic waves through a stratified solid medium, J Appl Phys, 21, 89-93, (1950) · Zbl 0036.13304
[15] Haskell, N. A., The dispersion of surface waves on multi-layered media, Bull Seismol Soc Am, 43, 17-34, (1953)
[16] Dunkin, J. W., Computation of modal solution in layered, elastic media at high frequencies, Bull Seismol Soc Am, 55, 335-358, (1965)
[17] Luco, J. E.; Apsel, R. J., On the Green’s functions for a layered half-space: part I, Bull Seismol Soc Am, 73, 909-929, (1983)
[18] Apsel, R. J.; Luco, J. E., On the Green’s functions for a layered half-space: part II, Bull Seismol Soc Am, 73, 931-951, (1983)
[19] Wang, R., A simple orthonormalization method for stable efficient computation of Green’s function, Bull Seismol Soc Am, 89, 733-741, (1999)
[20] 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) · Zbl 1109.74323
[21] 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)
[22] Xu, B.; Lu, J. F.; Wang, J. H., Dynamic response of an infinite beam overlying a layered poroelastic half-space to moving loads, J Sound Vib, 306, 91-110, (2007)
[23] Zheng, P.; Ding, B. Y.; Zhao, S. X.; Ding, D., 3D dynamic Green’s functions in a multilayered poroelastic half-space, Appl Math Model, 37, 10203-10219, (2013) · Zbl 1449.76062
[24] Knopoff, L., A matrix method for elastic wave problems, Bull Seismol Soc Am, 54, 431-438, (1964)
[25] Degrande, G.; De Roeck, G.; van den Broeck, P.; Smeulders, D., Wave propagation in layered dry, saturated and unsaturated poroelastic media, Int J Solids Struct, 35, 4753-4778, (1998) · Zbl 0920.73043
[26] Lefeuve-Mesgouez, G.; Mesgouez, A., Three-dimensional dynamic response of a porous multilayered ground under moving loads of various distributions, Adv Eng Softw, 46, 75-84, (2012)
[27] Liu, Z. X.; Liang, J. W.; Wu, C. Q., Dynamic Green’s function for a three-dimensional concentrated load in the interior of a poroelastic layered half-space using a modified stiffness matrix method, Eng Anal Bound Elem, 60, 51-66, (2015) · Zbl 1403.74315
[28] Biot, M. A., Theory of propagation of elastic waves in a fluid-saturated porous solid, I, low frenquency range, J Acoust Soc Am, 28, 168-178, (1956)
[29] Biot, M. A., Theory of propagation of elastic waves in a fluid-saturated porous solid. II. higher frequency range, J Acoust Soc Am, 28, 179-191, (1956)
[30] Biot MA. Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 196; 33, 1482-1498. doi: 10.1063/1.1728759 · Zbl 0104.21401
[31] Biot, M. A., Generalized theory of acoustic propagation in porous dissipative media, J Appl Phys, 34, 1254-1264, (1962)
[32] Sneddon, I., Fourier transforms, (1951), McGraw-Hill New York
[33] Bouchon, M.; Aki, K., Discrete wave number representation of seismic source wave fields, Bull Seismol Soc Am, 67, 259-277, (1977)
[34] Bouchon, M., A review of the discrete wavenumber method, Pure Appl Geophys, 160, 445-465, (2003)
[35] Deresiewicz, H.; Skalak, R., On the uniquness in dynamic poroelasticity, Bull Seismol Soc Am, 53, 783-788, (1963)
[36] Hasheminejad, S. M.; Hosseini, H., Nonaxisymmetric interaction of a spherical radiator in a fluid-filled permeable borehole, Int J Solids Struct, 45, 1, 24-47, (2008) · Zbl 1167.74394
[37] He, C.; Zhou, S. H.; Di, H. G., A 2.5-D coupled FE-BE model for the dynamic interaction between saturated soil and longitudinally invariant structures, Comput Geotech, 82, 211-222, (2017)
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