Point Loads in Cross‐Anisotropic, Layered Halfspaces
Publication: Journal of Engineering Mechanics
Volume 115, Issue 3
Abstract
The airn in this paper is to present an explicit solution for the Green's functions associated with static and dynamic loads and dislocations acting on, or within, elastic cross‐anisotropic halfspaces and full spaces. The methodology employed represents a generalization of a procedure described by Kausel and Peek in 1982 for dynamic loads acting within isotropic strata of finite depth. In essence, the method consists of a discretization of the medium in the direction of layering and an idealization of the underlying halfspace, in the case of dynamic loads, in terms of paraaxial approximations. For static loads, on the other hand, the halfspace can be modeled exactly. The Green's functions are obtained, as before, with an integral transform evaluated in closed form. As a result, no numerical integrations are necessary, which constitutes a significant advantage over other numerical procedures currently available for this problem. Since the resulting equations for displacements are expressed in the spatial domain, they can be used directly as kernels in integral representations of problems in elastodynamics, such as seismic sources, soil‐structure interaction, and scattering of waves.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Aki, K., and Larner, K. L. (1970). “Surface motion of a layered medium having an irregular interface due to incident plane SH waves.” J. Geophys. Res., 75(5), 933–954.
2.
Alekseyev, A. S., and Mikhaylenko, B. G. (1976/1977). “Solution of Lamb's problem for a vertically inhomogeneous elastic halfspace.” Akad. Nauk Izu. Physics of the Solid Earth, 12(1), 748–755.
3.
Anderson, D. L. (1961). “Elastic wave propagation in layered anisotropic media.” J. Geophys. Res., 66(9), 2953–2963.
4.
Apsel, R. J. (1979). “Dynamic Green's functions for layered media and applications to boundary value problems,” thesis presented to the University of California, at San Diego, Calif., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
5.
Bahar, L. Y. (1972). “Transfer matrix approach to layered systems.” J. Engrg. Mech. Div., ASCE, 98(5), 1159–1172.
6.
Bahar, L. Y. (1977). “Transfer matrix approach to two‐dimensional elastodynamics of layered media.” Computing methods in geophysical mechanics, ASME, AMD, Vol. 25, New York, N.Y.
7.
Barden, L. (1963). “Stresses and displacements in a cross‐anisotropic soil.” Geotechnique, 13(1), 198–210.
8.
Biot, M. A. (1963). “Continuum dynamics of elastic plates and multilayered solids under initial stress.” J. Math. Mech., 12, 793–810.
9.
Bouchon, M., and Bard, P. Y. (1980). “The seismic response of sediment‐filled valleys.” BSSA, 70(4), 1263–1286.
10.
Clayton, R., and Engquist, B. (1977). “Absorbing boundary conditions for acoustic and elastic wave equations.” Bull. Seism. Soc. Am., 67(6), 1529–1540.
11.
Crampin, S., Chesnokov, E. M., and Hipkin, R. G. (1984). “Seismic anisotropy—the state of the art.” Geophys. J., R. Astr. Soc., 76(1), 1–16.
12.
Dunkin, J. W. (1965). “Computation of modal solutions in layered elastic media at high frequencies.” Bull. Seism. Soc. Am., 55(2), 335–348.
13.
Engquist, B., and Majda, A. (1977). “Absorbing boundary conditions for the numerical simulation of waves.” Mathematics of Computation, 31(139), 629–651.
14.
Eringen, A. C., and Suhubi, S. (1975). Elastodynamics. Vol. 2, Academic Press, New York, N.Y.
15.
Fredholm, I. (1900). “Sur les equations de l'equilibre d'um corps solide elastique.” Acta Mathematica, 23(1) (in French).
16.
Gerrard, C. M. (1976). Stresses and displacements in layered cross‐anisotropic elastic systems.” Proc. Fifth Australia‐New Zealand Conf. on Soil Mechanics and Foundation Engineering, 187–197.
17.
Gilbert, F., and Backus, G. E. (1966). “Propagator matrices in elastic wave and vibration problems.” Geophysics, XXXI(2), 326–332.
18.
Harkrider, D. G. (1964). “Surface waves in multilayered elastic media, Rayleigh and Love waves from buried sources in a multilayered elastic halfspace.” Bull. Seism. Soc. Am., 54(2), 627–679.
19.
Haskell, N. A. (1953). “The dispersion of surface waves on multilayered media.” Bull. Seism. Soc. Am., (43) 17–34.
20.
Herrmann, R. B. (1977). “Research study of earthquake generated SH waves in the near‐field and near‐regional field.” Report DACW39‐76‐C‐0058, Waterways Experiment, Vicksburg, Miss.
21.
Johnson, L. R. (1974). “Green's function for Lamb's problem.” Geophys. J., R. Astr. Soc., 37(1), 99–131.
22.
Kausel, E. (1981). “An explicit solution for the Green's functkms for dynamic loads in layered media.” Technical Report R81‐13, Department of Civil Engineering, M.I.T., Cambridge, Mass.
23.
Kausel, E. (1986). “Wave propagation in anisotropic layered media.” Int. J. Numer. Methods Engrg., 23, Aug., 1567–1578.
24.
Kausel, E., and Roesset, J. M. (1981). “Stiffness matrices for layered soils.” Bull. Seism. Soc. Am., 71(6), 1743–1761.
25.
Kausel, E., and Peek, R. (1982a). “Dynamic loads in the interior of a layered stratum: an explicit solution.” Bull. Seism. Soc. Am., 72(5), 1459–1481
(see also Errata, BSSA, 74, Aug., 1984, 1508).
26.
Kausel, E., and Peek, R. (1982b). “Boundary integral method for stratified soils,” Technical Report R82‐50, Department of Civil Engineering, M.I.T., Cambridge, Mass.
27.
Kausel, E., and Seale, S. H. (1987). “Static loads in layered halfspaces.” J. Appl. Mech., ASME, 54(2), June, 403–408.
28.
Kausel, E., and Seale, S. H. (1989). “Dynamic and static impedances of cross‐anisotropic halfspaces.” Soil Dynamics and Earthquake Engrg., Apr.
29.
Knopoff, L. (1964). “A matrix method for elastic wave problems,” Bull. Seism. Soc. Am., 54(1), 431–438.
30.
Kröner, E. (1953). “Das Fundamentalintegral der Anisotropen Elastischen Differentialgleichungen.” Zeitschrift für Physik, 136, 402 (in German).
31.
Lindman, E. L. (1975). “Free‐space boundary conditions for time dependent wave equation.” J. Computational Physics, 18, 66–78.
32.
Love, A. E. H. (1944). A treatise on the mathematical theory of elasticity. Dover Press, New York, N.Y., 160.
33.
Mindlin, R. D. (1936). “Force at a point in the interior of a semi‐infinite solid.” J. Phys., 79, May, 195–202.
34.
Mooney, H. M. (1974). “Some numerical solutions for Lamb's problems.” BSSA, 64(2), 473–491.
35.
Pan, Y.‐C., and Chou, T.‐W. (1976). “Point force solution for an infinite transversely isotropic solid.” J. Appl. Mech., 43(1), 608–612.
36.
Pan, Y.‐C., and Chou, T.‐W. (1979). “Green's function solutions for semi‐infinite transversely isotropic materials.” Int. J. Engrg. Sci., 17(5), 545–551.
37.
Pao, Y.‐H. (1983). “Elastic waves in solids.” J. Appl. Mech., ASME, 50(1), 1152–1164.
38.
Pekeris, C. (1955a). “The seismic surface pulse.” Proc., Nat. Acad. of Sci., 41, 469.
39.
Pekeris, C. (1955b). “The seismic buried pulse.” Proc., Nat. Acad. of Sci., 41, 629.
40.
Pekeris, C., and Lifson, H. (1957). “Motion on the surface of a uniform half‐space produced by a buried pulse.” J. Acoust. Soc. Am., 29, 1233.
41.
Pestel, E. C., and Leckie, F. A. (1963). Matrix methods in elastomechanics. McGraw–Hill, New York, N.Y.
42.
Phinney, R. A. (1965). “Theoretical calculation of the spectrum of first arrivals in layered elastic mediums.” J. Geophys. Res., 70(20), 5107–5123.
43.
Pinney, E. (1954). “Surface motion due to a point source in a semi‐infinite elastic medium.” Bull. Seism. Soc. Am., 44(1), 571–596.
44.
Poulos, H. G., and Davis, E. H. (1974). Elastic solutions for soil and rock mechanics. John Wiley & Sons, New York, N.Y.
45.
Seale, S. H. (1975). “Dynamic loads in layered halfspaces,” thesis presented to the Department of Civil Engineering, Massachusetts Institute of Technology, at Cambridge Mass., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
46.
Tajimi, H. (1980). “A contribution to theoretical prediction of dynamic stiffness of surface foundations.” Proc., 7th World Conf. on Earthquake Engineering, Istanbul, Turkey, Vol. 5, 105–112.
47.
Thomson, W. T. (1950). “Transmission of elastic waves through a stratified solid medium.” J. Appl. Phys., 21(1), 89–93.
48.
Waas, G. (1980). “Dynamisch Belastete Fundamente auf Geschichtetemn Baugrund.” VDI Berichte, 381, 185–189 (in German).
49.
Waas, G., Riggs, H. R., and Werkle, H. (1985). “Displacement solutions for dynamic loads in transversely‐isotropic stratified media.” Earthquake Eng. Struct. Dyn., 13, 173–193.
50.
Whittaker, W. L., and Christiano, P. (1979). “Dynamic response of flexible plates bearing on an elastic half‐space.” Technical Report RP‐125‐9‐79, Department of Civil Engineering, Carnegie‐Mellon University, Pittsburgh, Pa.
Information & Authors
Information
Published In
Copyright
Copyright © 1989 ASCE.
History
Published online: Mar 1, 1989
Published in print: Mar 1989
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.