Dynamic Green's Functions of Homogeneous Poroelastic Half‐Plane
Publication: Journal of Engineering Mechanics
Volume 120, Issue 11
Abstract
This paper presents a comprehensive analytical and numerical treatment of two‐dimensional dynamic response of a dissipative poroelastic half‐plane under time‐harmonic internal loads and fluid sources. General solutions for poroelastodynamic equations corresponding to Biot's theory are obtained by using Fourier integral transforms in the x‐direction. These general solutions are used to solve boundary‐value problems corresponding to vertical and horizontal loads, and fluid sources applied at a finite depth below the surface of a poroelastic half‐plane. Explicit analytical solutions corresponding to above‐boundary‐value problems are presented. The solutions for poroelastic fields of a half‐plane subjected to internal excitations are expressed in terms of semiinfinite Fourier type integrals that can only be evaluated by numerical quadrature. The integration path is free from any singularities due to the dissipative nature of the elastic waves propagating in a poroelastic medium, and the Fourier integrals are evaluated by using an adaptive version of the trapezoidal rule. The accuracy of present numerical solutions are confirmed by comparison with existing solutions for ideal elasticity and poroelasticity. Selected numerical results are presented to portray the influence of the frequency of excitation, poroelastic material properties and types of loadings on the dynamic response of a poroelastic half‐plane. Green's functions presented in this paper can be used to solve a variety of elastodynamic boundary‐value problems and as the kernel functions in the boundary integral equation method.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Abramowitz, M., and Stegun, I. A. (1972). Handbook of mathematical functions. Dover, New York, N.Y.
2.
Achenbach, J. D. (1973). Wave propagation in elastic solids. North‐Holland, Amsterdam, The Netherlands.
3.
Apsel, R. J., and Luco, J. E. (1983). “On the Green's functions for a layered half space. Part II.” Bull. Seism. Soc. Am., 73(4), 931–951.
4.
Biot, M. A. (1941). “General theory of three‐dimensional consolidation.” J. Appl. Phys., 12(2), 155–164.
5.
Biot, M. A. (1956a). “Theory of propagation of elastic waves in a fluid‐saturated porous solid, I, low frequency range.” J. Acoust. Soc. Am., 28(2), 168–178.
6.
Biot, M. A. (1956b). “Theory of propagation of elastic waves in a fluid‐saturated porous solid, II, high frequency range.” J. Acoust. Soc. Am., 28(2), 179–191.
7.
Biot, M. A. (1962). “Mechanics of deformation and acoustic propagation in porous media.” J. Appl. Phys., 33(4), 1482–1498.
8.
Bonnet, G. (1987). “Basic singular solutions for a poroelastic medium in the dynamic range.” J. Acoust. Soc. Am., 82(5), 1758–1762.
9.
Bourbié, T., Coussy, O., and Zinszner, B. E. (1987). Acoustics of porous media. Gulf, Houston, Tex.
10.
Boutin, C., Bonnet, G., and Bard, P. Y. (1987). “Green functions and associated sources in infinite and stratified poroelastic media.” Geophysical Journal of the Royal Astronomical Society, London, U.K., 90, 521–550.
11.
Bowen, R. M. (1982). “Compressible porous media models by use of the theory of mixtures.” Int. J. Eng. Sci., 20(6), 697–735.
12.
Bowen, R. M., and Lockett, R. R. (1983). “Inertial effects in poroelasticity.” J. Appl. Mech., 50(2), 334–342.
13.
Burridge, R., and Vargas, C. A. (1979). “The fundamental solution in dynamic poroelasticity.” Geophysical Journal of the Royal Astronomical Society, London, U.K., 58, 61–90.
14.
Cheng, A. H.‐D., Badmus, T., and Beskos, D. E. (1991). “Integral equation for dynamic poroelasticity in frequency domain with BEM solution.” J. Engrg. Mech., ASCE, 117(5), 1136–1157.
15.
Deresiewicz, H., and Rice, J. T. (1962). “The effect of boundaries on wave propagation in a liquid‐filled porous solid: III. reflection of plane waves at a free plane boundary (general case).” Bull. Seism. Soc. Am., 52(3), 595–625.
16.
Deresiewicz, H. (1962). “The effect of boundaries on wave propagation in a liquid‐filled porous solid: IV. surface waves in a half‐space.” Bull. Seism. Soc. Am., 52(3), 627–638.
17.
Deresiewicz, H., and Skalak, R. (1963). “On uniqueness in dynamic poroelasticity.” Bull. Seism. Soc. Am., 53(4), 783–788.
18.
Dominguez, J. (1991). “An integral formulation for dynamic poroelasticity.” J. Appl. Mech., ASME, 58(2), 588–591.
19.
Dominguez, J. (1992). “Boundary element approach for dynamic poroelastic problems.” Int. J. Numer. Methods Engrg., 35(2), 307–324.
20.
Erdélyi, A. (1954). Tables of integral transforms. Bateman manuscript project, Cal. Inst. of Tech., McGraw‐Hill, New York, N.Y.
21.
Eringen, A. C., and Suhubi, A. S. (1975). Elastodynamics. Vol. 2, Academic Press, New York, N.Y.
22.
Geertsma, J., and Smith, D. (1961). “Some aspect of elastic wave propagation in a fluid‐saturated porous solid.” Geophys., 26(2), 169–181.
23.
Halpern, M. R., and Christiano, P. (1986a). “Response of poroelastic halfspace to steady‐state harmonic surface tractions.” Int. J. Numer. Anal. Meth. Geomech., 10(6), 609–632.
24.
Halpern, M. R., and Christiano, P. (1986b). “Steady‐state harmonic response of a rigid plate baring on a liquid‐saturated poroelastic halfspace.” Earthquake Engrg. and Struct. Dynamics, 14(3), 439–454.
25.
Jones, J. P. (1961). “Rayleigh waves in porous elastic fluid saturated solid.” J. Acoust. Soc. Am., 33(7), 959–962.
26.
Karasudhi, P. (1990). Foundation of solid mechanics. Kluwer Academic Publishers, Dordrecht, The Netherlands.
27.
Kupradze, V. D., Gezelia, T. G., Basheleishvili, M. O., and Burchuladze, T. V. (1979). Three‐dimensional problems of the mathematical theory of elasticity and thermoelasticity. North‐Holland, Amsterdam, The Netherlands.
28.
Lamb, H. (1904). “On the propagation of tremors over the surface of an elastic solid.” Philosophical Transactions of the the Royal Society of London, Series A, 203, London, U.K., 1–42.
29.
Melan, E. (1932). “Der spanning zustand der durch eine einzelkraft im innern beanspruchten halbschiebe.” Zeitschrift für Angewandte Mathematik und Mechanik, Berlin, Germany, 12 (in German).
30.
Norris, A. N. (1985). “Radiation from a point source and scattering theory in a fluid‐saturated porous solid.” J. Acoust. Soc. Am., 77(6), 2012–2023.
31.
Nowacki, W. (1975). Dynamic problems of thermoelasticity. Noordhoff, Leyden, The Netherlands.
32.
Paul, S. (1976a). “On the displacement produced in a porous elastic half‐space by an impulsive line load (non‐dissipative case).” Pure Appl. Geophys., 114, 605–614.
33.
Paul, S. (1976b). “On the disturbance produced in a semi‐infinite poroelastic medium by a surface load.” Pure Appl. Geophys., 114, 615–627.
34.
Philippacopoulos, A. J. (1988a). “Lamb's problem for fluid‐saturated, porous media.” Bull. Seism. Soc. Am., 78(2), 908–923.
35.
Philippacopoulos, A. J. (1988b). “Waves in partially saturated medium due to surface loads.” J. Engrg. Mech., ASCE, 114(10), 1740–1759.
36.
Philippacopoulos, A. J. (1989). “Axisymmetric vibration of disk resting on saturated layered half‐space.” J. Engrg. Mech., ASCE, 115(10), 2301–2322.
37.
Poulos, H. G., and Davis, E. H. (1974). Elastic solutions for soil and rock mechanics. John Wiley & Sons, Inc., New York, N.Y.
38.
Rajapakse, R. K. N. D. (1990). “Response of an axially loaded elastic pile in a Gibson soil.” Géotechnique, London, England, 40, 237–249.
39.
Rajapakse, R. K. N. D., and Wang, Y. (1991). “Elastodynamic Green's functions of orthotropic half plane.” J. Engrg. Mech., ASCE, 117(3), 588–604.
40.
Rasolofosaon, P. N. J. (1991). “Plane acoustic waves in linear viscoelastic porous media: Energy, particle displacement, and physical interpretation.” J. Acoust. Soc. Am., 89(4), 1532–1550.
41.
Rice, J. R., and Cleary, M. P. (1976). “Some basic stress‐diffusion solutions for fluid saturated elastic porous media with compressible constituents.” Rev. Geophys. Space Phys., 14(2), 227–241.
42.
Skempton, A. W. (1954). “The pore pressure coefficients A and B.” Géotechnique, 4, 143–147.
43.
Sneddon, I. N. (1951). Fourier transforms. McGraw‐Hill, New York, N.Y.
44.
Wong, H. L., and Luco, J. E. (1986). “Dynamic interaction between rigid foundations in a layered half‐space.” Soil Dynamics and Earthquake Engrg., 5(3), 149–158.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 120 • Issue 11 • November 1994
Pages: 2381 - 2404
Copyright
Copyright © 1994 American Society of Civil Engineers.
History
Received: Sep 8, 1993
Published online: Nov 1, 1994
Published in print: Nov 1994
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.