zbMATH — the first resource for mathematics

Response of poroelastic halfspace to steady-state harmonic surface tractions. (English) Zbl 0601.73107
Traditionally, most formulations of dynamic halfspace problems have represented the material as either an elastic or a viscoelastic solid. Herein the counterpart of Lamb’s elastodynamic problem is reformulated and solved for a liquid-saturated poroelastic halfspace using Biot’s theory of poroelasticity. The responses of the solid and fluid phases are evaluated due to steady-state harmonic concentrated loads applied to each phase at the surface. The solutions are presented over a broad range of permeabilities and are compared to solutions to Lamb’s problem for equivalent drained and undrained solids. Methodology is then introduced by which these results are treated as Green functions for the solution of a mixed boundary-value problem, namely, the response of the poroelastic halfspace to steady-state harmonic vertical motion of a rigid, massless plate. It is observed that small differences exist among overall compliance functions for a drained solid, an undrained solid, and a liquid-saturated porous halfspace. However, use of the poroelastic model permits the distribution between effective skeletal normal stresses and fluid stresses to be determined.

74L10 Soil and rock mechanics
76S05 Flows in porous media; filtration; seepage
74B99 Elastic materials
74H99 Dynamical problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
Full Text: DOI
[1] Brutsaert, J. Geophys. Res. 69 pp 243– (1964)
[2] Biot, J. Acoust. Soc. Am. 28 pp 168– (1956)
[3] Biot, J. Acoust. Soc. Am. 34 pp 1254– (1962)
[4] Biot, J. Appl. Phys. 33 pp 1482– (1962)
[5] ’Study of elastic wave propagation and damping in granular materials’, Ph. D. thesis, Univ. of Florida (1961).
[6] Geertsma, Geophys. 27 pp 169– (1961)
[7] ’Propagation of compressional waves in a saturated soil’, In. Symp. Wave Propagation and Dynamic Properties of Earth Material, Albuquerque, New Mexico, 1967.
[8] Deresiewcz, Bull. Seismol. Soc. Am. 46 pp 599– (1960)
[9] Deresiewicz, Bull. Seismol. Soc. Am. 52 pp 595– (1962)
[10] Deresiewicz, Bull. Seismol. Soc. Am. 54 pp 409– (1964)
[11] Deresiewcz, Bull. Seismol. Soc. Am. 52 pp 627– (1962)
[12] Deresiewicz, Bull. Seismol. Soc. Am. 54 pp 1537– (1964)
[13] Deresiewicz, Bull. Seismol. Soc. Am. 57 pp 381– (1967)
[14] Jones, J. Appl. Mech. 91 pp 878– (1969)
[15] and , ’Dynamic impedance of a poroelastic subgrade’, Third ASCE-EMD Specialty Conf., Univ. of Texas, Austin, 1979.
[16] Zienkiewicz, Geotechn. 30 pp 385– (1980)
[17] Simon, Int. J. Numer. Anal. Methods Geomech. 8 pp 381– (1984)
[18] Burridge, Geophys. J. Roy. Astronom. Soc. 58 pp 61– (1979)
[19] Paul, J. Pure Appl. Geophys. 114 pp 605– (1976)
[20] Paul, J. Pure Appl. Geophys. 114 pp 615– (1976)
[21] Lamb, Phil. Trans. Roy. Soc. Lond. 203 pp 1– (1904)
[22] and , ’Offshore caissons on porous saturated soil’, Int. Conf. Recent Advances in Geotechnical Engineering and Solid Dynamics, University of Missouri, Rolla 1981.
[23] Ghaboussi, Proc. J. Eng. Mech Div., ASCE 98 pp 947– (1972)
[24] Zienkiewicz, Int. J. Num. Anal. Methods Geomech. 8 pp 71– (1984)
[25] Biot, J. Appl. Mech. 14 pp 954– (1957)
[26] Biot, J. Appl. Phys. 12 pp 155– (1941)
[27] ’Dynamic response of a saturated poroelastic halfspace to imposed surface loading’, Ph. D. thesis, Carnegie-Mellon Univ. (1982).
[28] Ghaboussi, J. Eng. Mech. Div., ASCE 98 pp 947– (1972)
[29] Wave Propagation in Elastic Solids, North-Holland, Amsterdam, (1973).
[30] ’The Use of Integral Transforms’, McGraw-Hill, New York, 1970.
[31] Longman, Math. Tables Other Aids Comp. 12 pp 205– (1958)
[32] Longman, Proc. Comb. Phil. Soc. 52 pp 764– (1956)
[33] Kim, Exp. Mech. 19 pp 252– (1979)
[34] Cleary, Int. J. Solids Struct. 14 pp 795– (1978)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.