Abstract
The dynamic responses of a poroelastic half-space to an internal point load and fluid source are investigated in the frequency domain in this paper. By virtue of a method of displacement potentials, the 3D general solutions of homogeneous wave equations and fundamental singular solutions of inhomogeneous wave equations are derived, respectively, in the frequency domain. The mirror-image technique is then applied to construct the dynamic Green’s functions for a poroelastic half-space. Explicit analytical solutions for displacement fields and pore pressure are obtained in terms of semi-infinite Hankel-type integrals with respect to the horizontal wavenumber. In two limiting cases, the solutions presented in this study are shown to reduce to known counterparts of elastodynamics and those of Lamb’s problem, thus ensuring the validity of our result.
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References
Burridge R., Vargas C.: The fundamental solution in dynamic poroelasticity. Geophys. J. R. Astr. Soc. 58, 61–90 (1979)
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)
Bonnet G.: Basic singular solutions for a poroelastic medium in the dynamic range. J. Acoust. Soc. Am. 82, 1758–1762 (1987)
Cheng A.D., Badmus T., Beskos D.E.: Integral equation for dynamic poroelasticity in frequency domain with BEM solution. J. Eng. Mech. ASCE. 117, 1136–1157 (1991)
Chen J.: Time domain fundamental solution to Biot’ s complete equations of dynamic poroelasticity. Part I: two-dimensional solution. Int. J. Solids. Struct. 31, 1447–1490 (1994)
Chen J.: Time domain fundamental solution to Biot’s complete equations of dynamic poroelasticity. Part II: three-dimensional solution. Int. J. Solids. Struct. 31, 169–202 (1994)
Philippacopoulos A.: Spectral Green’s dyadic for point source in poroelastic media. J. Eng. Mech. ASCE. 124, 24–31 (1998)
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)
Schanz M.: Poroelastodynamics: linear models, analytical solutions, and numerical methods. Appl. Mech. Rev. 63, 1–15 (2009)
Senjuntichai T., Rajapakse R.: Dynamic Green’s functions of homogeneous poroelastic half-plane. J. Eng. Mech. ASCE. 120, 2381–2404 (1994)
Pekeris C.L.: The seismic buried pulse. Proc. Natl. Acad. Sci. 41, 629–639 (1955)
Philippacopoulos A.: Buried point source in a poroelastic half-space. J. Eng. Mech. ASCE. 123, 860–869 (1997)
Jin B., Liu H.: Dynamic response of a poroelastic half space to horizontal buried loading. Int. J. Solids. Struct. 38, 8053–8064 (2001)
Philippacopoulos A.: Lamb’s problem for fluid-saturated, porous media. Bull. Seismol. Soc. Am. 78, 908–923 (1988)
Nowacki W.: Dynamic Problems of Thermoelasticity. Noordhoff, Leyden (1975)
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)
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)
Biot M.A.: Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33, 1482–1498 (1962)
Biot M.A.: Generalized theory of acoustic propagation in porous dissipative media. J. Acoust. Soc. Am. 34, 1254–1264 (1962)
Rice J., Cleary M.P.: Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev. Geophys. Space Phys. 14, 227–241 (1976)
Eringen A.C., Suhubi E.S.: Elastodynamics. vol. II. Linear Theory. Academic, New York (1975)
Watson G.: A Treatise on the Theory of Bessel Functions. Macmillan, New York (1944)
Chave A.: Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48, 1671–1686 (1983)
Pan E.: Static Green’s functions in multilayered half-spaces. Appl. Math. Model. 21, 509–521 (1997)
Bouchon M., Aki K.: Discrete wavenumber representation of seismic source wave fields. Bull. seism. Soc. Am. 67, 259–277 (1977)
Apsel R., Luco J.: On the green functions for a layered half-space. Part II. Bull. seism. Soc. Am. 73, 931–951 (1983)
Hisada Y.: An efficient method for computing Green’s functions for a layered half-space with sources and receivers at close depths. Bull. seism. Soc. Am. 84, 1457–1472 (1994)
Hisada Y.: An efficient method for computing Green’s functions for a layered half-space with sources and receivers at close depths. Part II. Bull. Seism. Soc. Am. 85, 1080–1093 (1995)
Kim Y.K., Kingsbury H.B.: Dynamic characterization of poroelastic materials. Exp. Mec. 19, 252–258 (1979)
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Zheng, P., Zhao, SX. & Ding, D. Dynamic Green’s functions for a poroelastic half-space. Acta Mech 224, 17–39 (2013). https://doi.org/10.1007/s00707-012-0720-2
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DOI: https://doi.org/10.1007/s00707-012-0720-2