Potential Method for 3D Wave Propagation in a Poroelastic Medium and Its Applications to Lamb’s Problem for a Poroelastic Half-Space
Publication: International Journal of Geomechanics
Volume 16, Issue 2
Abstract
In cylindrical coordinates, a potential method is developed for three-dimensional (3D) wave propagation in a poroelastic medium. By using the proposed potential method, the wave propagation problems can be reduced to the determination of four scalar potentials governed by four scalar Helmholtz equations, representing the motions of , , , waves in the porous media, respectively. By the methods of separation of variables, the general solutions to those Helmholtz equations are found in cylindrical coordinates. Boundary value problems associated with a homogeneous poroelastic half-space loaded by surface tractions, that is, Lamb’s problem for a fluid-saturated medium is resolved using the obtained general solutions. It is shown that these potentials introduced in this research for 3D wave propagation problems can also be reduced to those reported by previous researchers for axisymmetric wave propagation in the fluid-saturated porous medium. Furthermore, numerical examples for the state–state and transient responses of the poroelastic half-space are provided.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This work was financially supported by the National Natural Science Foundation of China through Grant Nos. 11402150 and 51478435.
References
Aki, K., and Richards, P. G. (2002). Quantitative seismology, University Science Books, Sausalito, CA.
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.
Biot, M. A. (1956b). “Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range.” J. Acoust. Soc. Am., 28(2), 179–191.
Biot, M. A. (1962a). “Mechanics of deformation and acoustic propagation in porous media.” J. Appl. Phys., 33(4), 1482–1498.
Biot, M. A. (1962b). “Generalized theory of acoustic propagation in porous dissipative media.” J. Acoust. Soc. Am., 34(5), 1254–1264.
Biot, M. A., and Willis, D. G. (1957). “The elastic coefficients of the theory of consolidation.” J. Appl. Mech., 24(12), 594–601.
Bouchon, M., and Aki, K. (1977). “Discrete wave-number representation of seismic-source wave fields.” Bull. Seismol. Soc. Am., 67(2), 259–277.
Cheng, A. D., Badmus, T., and Beskos, D. E. (1991). “Integral equation for dynamic poroelasticity in frequency domain with BEM solution.” J. Eng. Mech., 1136–1157.
Huang, Y., and Zhang, Y. (2000). “Three-dimensional non-axisymmetric Lamb’s problem for saturated soil.” Sci. China Ser. E: Technol. Sci., 43(2), 183–193.
Johnson, D. L., Koplik, J., and Dashen, R. (1987). “Theory of dynamic permeability and tortuosity in fluid-saturated media.” J. Fluid Mech., 176(3), 379–402.
Norris, A. N. (1985). “Radiation from a point source and scattering theory in a fluid-saturated porous solid.” J. Acoust. Soc. Am., 77(6), 2012–2022.
Paul, S. (1976a). “On the displacements produced in a porous elastic half-space by an impulsive line load (non-dissipative case).” Pure Appl. Geophys., 114(4), 605–614.
Paul, S. (1976b). “On the disturbance produced in a semi-infinite poroelastic medium by a surface load.” Pure Appl. Geophys., 114(4), 615–627.
Philippacopoulos, A. (1988). “Lamb’s problem for fluid-saturated, porous media.” Bull. Seismol. Soc. Am., 78(2), 908–923.
Pride, S. R., Morgan, F. D., and Gangi, A. F. (1993). “Drag forces of porous-medium acoustics.” Phys. Rev. B: Condens. Matter, 47(9), 4964–4978.
Schanz, M. (2004). “Dynamic fundamental solutions for compressible and incompressible modeled poroelastic continua.” Int. J. Solids Struct., 41(15), 4047–4073.
Schanz, M. (2009). “Poroelastodynamics: Linear models, analytical solutions, and numerical methods.” Appl. Mech. Rev., 62(3), 030803-1–030803-15.
Senjuntichai, T., and Rajapakse, R. (1994). “Dynamic Green’s functions of homogeneous poroelastic half-plane.” J. Eng. Mech., 2381–2404.
Sharma, M. D. (1992). “Comments on ‘Lamb’s problem for fluid-saturated porous media’.” Bull. Seismol. Soc. Am., 82(5), 2263–2273.
Wang, H. (2000). Theory of linear poroelasticity with applications to geomechanics and hydrogeology, Princeton Univ. Press, Princeton, NJ.
Wenzlau, F., and Müller, T. M. (2009). “Finite-difference modeling of wave propagation and diffusion in poroelastic media.” Geophysics, 74(4), T55–T66.
Zheng, P., Ding, B., Zhao, S. X., and Ding, D. (2013a). “3D dynamic Green’s functions in a multilayered poroelastic half-space.” Appl. Math. Modell., 37(24), 10203–10219.
Zheng, P., Zhao, S. X., and Ding, D. (2013b). “Dynamic Green’s functions for a poroelastic half-space.” Acta Mech., 224(1), 17–39.
Zienkiewicz, O. C., Chang, C. T., and Bettess, P. (1980). “Drained, undrained, consolidating and dynamic behavior assumptions in soils.” Géotechnique, 30(4), 385–395.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 16 • Issue 2 • April 2016
Copyright
© 2015 American Society of Civil Engineers.
History
Received: Aug 29, 2014
Accepted: Apr 2, 2015
Published online: Jun 25, 2015
Discussion open until: Nov 25, 2015
Published in print: Apr 1, 2016
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.