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Dynamic 2.5-D Green’s function for a poroelastic half-space. (English) Zbl 1403.74272
Summary: The dynamic two-and-a-half-dimensional (2.5-D) Green’s function for a poroelastic half-space subject to a point load and dilatation source is derived based on Biots theory, with the consideration of both a permeable surface and an impermeable surface. The governing differential equations for the 2.5-D Green’s function are established by applying the Fourier transform to the governing equations of the three-dimensional (3-D) Green’s function. The dynamic 2.5-D Green’s function is derived in a full-space using the potential decomposition and discrete wavenumber methods. The surface terms are introduced to fulfil the free-surface boundary conditions and thereby obtain the dynamic 2.5-D Green’s function for a poroelastic half-space with the permeable and impermeable surfaces. The half-space 2.5-D Green’s function is verified through comparison with the 2.5-D Green’s function regarding an elastodynamic half-space and the 3-D Green’s function for a poroelastic half-space. A numerical case is provided to compare between the full-space solutions and the half-space solutions with two different sets of free-surface boundary conditions. In addition, a case study of efficient calculation of vibration from a tunnel embedded in a poroelastic half-space is presented to show the application of the 2.5-D Green’s function in engineering problems.

74S15 Boundary element methods applied to problems in solid mechanics
65M38 Boundary element methods for initial value and initial-boundary value problems involving PDEs
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.)
Full Text: DOI
[1] Luco, J. E.; Wong, H. L.; De Barros, F. C.P., Three-dimensional response of a cylindrical canyon in a layered half-space, Earthq Eng Struct Dyn, 19, 6, 799-817, (1990)
[2] Papageorgiou, A.; 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)
[3] 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
[4] 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
[5] Lu, J.; Jeng, D.; 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
[6] Lu, J.; Jeng, D.; Williams, S., A 2.5-D dynamic model for a saturated porous medium. part II: boundary element method, Int J Solids Struct, 45, 2, 359-377, (2008) · Zbl 1167.74397
[7] Auersch, L., Response to harmonic wave excitation of finite or infinite elastic plates on a homogeneous or layered half-space, Comput Geotech, 51, 50-59, (2013)
[8] Burridge, R.; Vargas, C. A., The fundamental solution in dynamic poroelasticity, Geophys J Int, 58, 61-90, (1979) · Zbl 0498.73018
[9] Norris, A. N., Radiation from a point source and scattering theory in a fluid-saturated porous solid, J Acoust Soc Am, 77, 2012-2022, (1985) · Zbl 0579.73107
[10] Bonnet, G., Basic singular solutions for a poroelastic medium in the dynamic range, J Acoust Soc Am, 82, 1758-1762, (1987)
[11] Cheng, A. H.D.; Badmus, T.; Beskos, D. E., Integral equation for dynamic poroelasticity in frequency domain with BEM solution, J Eng Mech, 117, 1136-1157, (1991)
[12] 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)
[13] Senjuntichai, T.; Rajapakse, RKND., Dynamic green׳s functions of homogeneous poroelastic half-plane, J Eng Mech, 120, 2381-2404, (1994)
[14] Philippacopoulos, A. J., Buried point source in a poroelastic half-space, J Eng Mech, 123, 860-869, (1997)
[15] 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
[16] Zheng, P.; Zhao, S.; Ding, D., Dynamic green׳s functions for a poroelastic half-space, Acta Mech, 224, 1, 17-39, (2013) · Zbl 1401.74147
[17] Zheng, P.; Ding, B.; Zhao, S.; Ding, D., 3D dynamic green׳s functions in a multilayered poroelastic half-space, Appl Math Model, 37, 10203-10219, (2013) · Zbl 1449.76062
[18] Zeng, C.; Sun, H. L.; Cai, Y. Q., Analysis of three-dimensional dynamic response of a circular lining tunnel in saturated soil to harmonic loading, Rock Soil Mech, 35, 4, 1147-1156, (2014)
[19] Biot, M. A., Theory of propagation of elastic waves in a fluid-saturated porous solid. I. low-frequency range, J Acoust Soc Am, 28, 168-178, (1956)
[20] 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)
[21] Biot, M. A., Mechanics of deformation and acoustic propagation in porous media, J Appl Phys, 33, 1482-1498, (1962) · Zbl 0104.21401
[22] Biot, M. A., Generalized theory of acoustic propagation in porous dissipative media, J Acoust Soc Am, 34, 1254-1264, (1962)
[23] Biot, M. A.; Willis, D. G., The elastic coefficients of the theory of consolidation, J App Mech, 24, 79, 594-601, (1957)
[24] Hughes, D. S.; Cooke, C. E., The effect of pressure on the reduction of pore volume of consolidated sandstone, Geophysics, 18, 298-309, (1953)
[25] Detournay, E.; Cheng, A. H.D., Fundamentals of poroelasticity, (Hudson, J. A., Comprehensive rock engineering, (1993), Pergamon Oxford), 24-25
[26] Sneddon, I., Fourier transforms, (1951), McGraw-Hill New York
[27] Bouchon, M.; Aki, K., Discrete wave number representation of seismic source wave fields B, Seismol Soc Am, 67, 259-277, (1977)
[28] Bouchon, M., A review of the discrete wavenumber method, Pure Appl Geophys, 160, 445-465, (2003)
[29] Deresiewicz, H.; Skalak, R., On the uniquness in dynamic poroelasticity, Bull Seism Soc Am, 53, 783-788, (1963)
[30] Tadeu, A.; António, J.; Godinho, L., Green׳s function for two-and-a-half dimensional elastodynamic problems in a half-space, Comput Mech, 27, 484-491, (2001) · Zbl 0996.74049
[31] Hussein, M. F.M.; François, S.; Schevenels, M., The fictitious force method for efficient calculation of vibration from a tunnel embedded in a multi-layered half-space, J Sound Vib, 333, 25, 6996-7018, (2014)
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