Response of an elastic half-space with power-law nonhomogeneity to static loads. (English) Zbl 1161.74350
Summary: In this paper a series of problems for an isotropic elastic half-space with power-law nonhomogeneity are considered. The action of surface vertical and horizontal forces applied to the half-space is studied. A part of the paper deals with the case of zero-valued surface shear modulus (for positive values of the power determining the nonhomogeneity). This condition leads to simple solutions for two-dimensional (2D) case when radial distribution of stresses exists for surface loads concentrated along an infinite line. Corresponding results for the three-dimensional (3D) case are constructed on the basis of the relationships between 2D and 3D solutions developed in the paper. A more complicated case, in which the shear modulus at the surface of the half-space differs from zero, is treated using fundamental solutions of the differential equations for Fourier-Bessel transformations of displacements. In the paper the fundamental solutions are built in the following two forms: (a) a combination of functions expressing displacements of the half-space under the action of vertical and horizontal forces in the case of zero surface shear modulus, and (b) a representation of the fundamental solutions using confluent hypergeometric functions. The results of numerical calculation given in the paper relate to Green functions for the surface vertical and horizontal point forces.
References:
| [1] | Ter Mkrtichian L.N. (1961). Some problems in the theory of elasticity of non-homogeneous elastic media. J. Appl. Math. Mech. (Prikl. Math. Mekh.) 25: 1667–1675 · Zbl 0107.18202 · doi:10.1016/0021-8928(62)90144-2 |
| [2] | Szefer G. (1976). A problem of a non-homogeneous semi-infinite elastic body (in Polish). Rozpr. Ing. 15: 19–48 · Zbl 0159.26801 |
| [3] | Wilson J.T. (1942). Surface waves in a heterogeneous medium. Bull. Seism. Soc. Am. 32: 297–304 · Zbl 0063.08275 |
| [4] | Muravskii G. (1997). Time-harmonic problem for a non-homogeneous half-space with exponentially varying shear modulus. Int. J. Solids Struct. 34: 3119–3139 · Zbl 0942.74584 · doi:10.1016/S0020-7683(96)00226-0 |
| [5] | Selvadurai A.P.S., Singh B.M. and Vrbik J. (1986). A Reissner–Sagoci problem for a non-homogeneous elastic solid. J. Elast. 16: 383–391 · Zbl 0602.73011 · doi:10.1007/BF00041763 |
| [6] | Selvadurai, A.P.S.: On the indentation of a non-homogeneous elastic geomaterial: Analytical and computational estimates. In: Numerical Models in Geomechanics–NUMOG V. Balkema, Rotterdam (1995) |
| [7] | Selvadurai A.P.S. (1996). The settlement of a rigid circular foundation resting on a half-space exhibiting a near surface elastic non-homogeneity. Int. J. Numer. Anal. Meth. Geomech. 20: 351–364 · Zbl 0859.73071 · doi:10.1002/(SICI)1096-9853(199605)20:5<351::AID-NAG830>3.0.CO;2-L |
| [8] | Vrettos C. (1998). The Boussinesq problem for soils with bounded non-homogeneity. Int. J. Numer. Anal. Meth. Geomech. 22: 655–669 · Zbl 0911.73056 · doi:10.1002/(SICI)1096-9853(199808)22:8<655::AID-NAG938>3.0.CO;2-R |
| [9] | Rao C.R. (1974). Rayleigh waves in a half-space with bounded variation in density and rigidity. Bull. Seism. Soc. Am. 64: 1263–1274 |
| [10] | Rao C.R.A. and Goda M.A.A. (1978). Generalization of Lamb’s problem to a class of inhomogeneous elastic half-spaces. Proc. R. Soc. Lond. A 359: 93–110 · Zbl 0369.73013 · doi:10.1098/rspa.1978.0033 |
| [11] | Vrettos C. (1990). In-plane vibration of soil deposits with variable shear modulus: I Surface waves. Int. J. Numer. Anal. Meth. Geomech. 14: 209–222 · Zbl 0702.73051 · doi:10.1002/nag.1610140304 |
| [12] | Vrettos C. (1990). In-plane vibration of soil deposits with variable shear modulus: II Line load. Int. J. Numer. Anal. Meth. Geomech. 14: 649–662 · Zbl 0724.73198 · doi:10.1002/nag.1610140905 |
| [13] | Vrettos C. (1991). Forced anti-plane vibrations at the surface of an inhomogeneous half-space. Soil Dyn. Earthquake Eng. 10: 230–235 · doi:10.1016/0267-7261(91)90016-S |
| [14] | Vrettos C. (1991). Time-harmonic Boussinesq problem for a continuously non-homogeneous soil. Earthq. Eng. Struct. Dyn. 20: 961–977 · doi:10.1002/eqe.4290201006 |
| [15] | Muravskii G. (1997). On time-harmonic problem for non-homogeneous elastic half-space with shear modulus limited at infinite depth. Eur. J. Mech. A/Solids 16: 227–294 |
| [16] | Muravskii G.B. (2001). Mechanics of Non-Homogeneous and Anisotropic Foundations. Springer, Heidelberg · Zbl 1050.74002 |
| [17] | Hardin B.O. and Drnevich V.P. (1972). Shear modulus and damping in soils: Measurement and parameter effects. J. Soil Mech. Found. Div. ASCE 98: 603–625 |
| [18] | Hardin B.O. and Drnevich V.P. (1972). Shear modulus and damping in soils: Design equations and curves. J. Soil Mech. Found. Div. ASCE 98: 667–692 |
| [19] | Gazetas, G.: Foundation vibration. In: Fang, H.-Y. (ed.) Foundation Engineering Handbook, pp 553–593 (1991) |
| [20] | Rostovtsev N.A. and Khranevskaia I.E. (1971). The solution of the Boussinesq problem for a half space whose modulus of elasticity is a power function of the depth. J. Appl. Math. Mech. (Prikl. Math. Mekh.) 35: 1053–1061 |
| [21] | Booker J.R., Balaam N.P. and Davis E.H. (1985). The behavior of an elastic non-homogeneous half-space. Part I. Line and point loads. Int. J. Numer. Anal. Meth. Geomech. 9: 353–367 · Zbl 0569.73106 · doi:10.1002/nag.1610090405 |
| [22] | Booker J.R., Balaam N.P. and Davis E.H. (1985). The behavior of an elastic non-homogeneous half-space. Part II. Circular and strip footings. Int. J. Numer. Anal. Meth. Geomech. 9: 369–381 · Zbl 0569.73107 · doi:10.1002/nag.1610090406 |
| [23] | Gibson R.E. (1967). Some results concerning displacements and stresses in a non-homogeneous elastic half-space. Geotechnique 17: 58–67 · doi:10.1680/geot.1967.17.1.58 |
| [24] | Holl D.L. (1940). Stress transmission in earths. Proc. High Res. Board 20: 709–721 |
| [25] | Klein G.K. (1956). Consideration of non-homogeneity, discontinuity of deformation and other mechanical properties of soils for estimating construction on continues foundation (in Russian). Trudy MISI 14: 168–180 |
| [26] | Kassir M.K. (1972). Boussinesq problems for nonhomogeneous solid. J. Eng. Mech. Div. 98: 457–470 |
| [27] | Oner M. (1990). Vertical and horizontal deformation of an inhomogeneous elastic half-space. Int. J. Numer. Anal. Meth. Geomech. 14: 613–629 · doi:10.1002/nag.1610140903 |
| [28] | Kassir M.K. and Chuaprasert M.F. (1974). A rigid punch in contact with a nonhomogeneous elastic solid. J. Appl. Mech. ASME 41: 1019–1024 · Zbl 0294.73017 · doi:10.1115/1.3423426 |
| [29] | Chuaprasert M.F. and Kassir M.K. (1974). Displacements and stresses in non-homogeneous solid. J. Eng. Mech. Div. ASCE 100: 861–872 · Zbl 0294.73017 |
| [30] | Kassir M.K. and Sih G.C. (1975). Three-dimensional crack problems. Mech. Fract. 2: 382–409 · Zbl 0312.73112 |
| [31] | Jeon S.-P., Tanigawa Y. and Hata T. (1998). Axisymmetric problem of a non-homogeneous elastic layer. Arch. Appl. Mech. 68: 20–29 · Zbl 0920.73033 · doi:10.1007/s004190050143 |
| [32] | Jeon S.-P. and Tanigawa Y. (1998). Axisymmetrical elastic behaviour and stress intensity factor for a nonhomogeneous medium with penny-shaped crack. JSME Int. J. Ser. A 41: 457–464 |
| [33] | Aleksandrov, A., Soloviev, I.u.: Space problems of theory of elasticity (application of methods of complex argument functions) (in Russian). Nauka, Moscow (1978) |
| [34] | Lekhnitskii S.G. (1962). Radial distribution of stresses in a wedge and in a half-plane with variable modulus of elasticity. J. Appl. Math. Mech. (Prikl. Math. Mekh.) 26: 146–151 |
| [35] | Korenev B.G. (1957). Punch lying on an elastic half-space whose modulus of elasticity is a function of the depth (in Russian). Dokl. Akad. Nauk SSSR 112: 823–826 · Zbl 0087.19101 |
| [36] | Mossakovskii V.I. (1958). Pressure of a circular punch on an elastic half-space whose modulus of elasticity is an exponential function of the depth. Prikl. Mat. Mekh. 22: 123–125 |
| [37] | Rakov A.K. and Rvachev V.L. (1961). The contact problem of the theory of elasticity for a half-space whose modulus of elasticity is a power function of the depth. Dokl. Acad. Nauk. USSR 3: 286–290 |
| [38] | Popov G. (1963). Solution of contact problems of the theory of elasticity by the method of integral equations. J. Appl. Math. Mech. (Prikl. Math. Mekh.) 27: 1566–1573 · Zbl 0096.18902 |
| [39] | Popov G. (1967). Impression of a punch on a linerly deforming foundation taking into account friction forces. J. Appl. Math. Mech. (Prikl. Math. Mekh.) 31: 360–367 · Zbl 0152.42904 · doi:10.1016/0021-8928(67)90162-1 |
| [40] | Rostovtsev N.A. (1964). On certain solutions of an integral equation of the theory of a linearly deformable foundation. J. Appl. Math. Mech. (Prikl. Math. Mekh.) 28: 127–135 · Zbl 0151.36404 · doi:10.1016/0021-8928(64)90137-6 |
| [41] | Belik G.I. and Protsenko V.S. (1967). The contact problem of a half plane for which the modulus of elasticity of the material is expressed by a power function of the depth. Soviet Appl. Mech. 3: 80–82 · doi:10.1007/BF01262170 |
| [42] | Stachowicz, B., Szefer, G. (1966) On a contact problem for a non-homogeneous elastic halfplane (in Polish). Mech. Teor. Stos. 4, 83–96 |
| [43] | Stachowicz B. (1968). Determination of stresses under a punch in a non-homogeneous elastic semi-plane. Arch. Mech. Stos. 20: 669–687 · Zbl 0176.24201 |
| [44] | Protsenko V.S. (1967). Torsion of an elastic half-space whose modulus of elasticity varies by the power law. Soviet Appl. Mech. 3: 82–83 · doi:10.1007/BF00896798 |
| [45] | Popov G. (1959). Bending of an unbounded plate supported by an elastic half-space with a modulus of elasticity varying with depth. J. Appl. Math. Mech. (Prikl. Math. Mekh.) 23: 1566–1573 · Zbl 0096.18902 · doi:10.1016/0021-8928(59)90012-7 |
| [46] | Aleynikov S.M. (2006). Spatial Contact Problems in Geotechnics. Springer, Heidelberg · Zbl 1202.74002 |
| [47] | Snitko N.K. (1980). On action of a point force on a non-homogeneous elastic half-space (in Russian). Struct. Mech. Calculation Construction N 2: 76–78 |
| [48] | Awojobi A.O. and Gibson R.E. (1973). Plane strain and axially symmetric problems of a linearly non-homogeneous elastic half-space. Q. J. Mech. Appl. Math. 26: 285–302 · Zbl 0262.73025 · doi:10.1093/qjmam/26.3.285 |
| [49] | Awojobi A.O. (1974). The invariance of Gibson’s Law for a stratum on a frictionless base. Geotechnique 24: 359–366 · doi:10.1680/geot.1974.24.3.359 |
| [50] | Brown P.T. and Gibson R.E. (1972). Surface settlement of a deep elastic stratum whose modulus increases linearly with depth. Can. Geotech. J. 9: 467–473 · doi:10.1139/t72-045 |
| [51] | Gibson, R.E., Sills, G.C.: Some results concerning the plane deformation of a non-homogeneous elastic half-space. In: Proc Roscoe Mem Symp, pp 564–572. Cambridge University Press, Cambridge (1971) |
| [52] | Gibson R.E., Brown P.T. and Andrews K.R.F. (1971). Some results concerning displacements in a non-homogeneous elastic layer. Z. Ang. Math. Phys. 22: 855–864 · Zbl 0231.73041 · doi:10.1007/BF01591813 |
| [53] | Gibson R.E. (1974). The analytical methods in soil mechanics. Geotechnique 24: 115–140 · doi:10.1680/geot.1974.24.2.115 |
| [54] | Calladine C.R. and Greenwood J.A. (1978). Line and point loads on a non-homogeneous incompressible elastic half-space. Q. J. Mech. Appl. Math. 26: 507–529 · Zbl 0391.73011 · doi:10.1093/qjmam/31.4.507 |
| [55] | Rajapakse R.K.N.D. (1990). A vertical load in the interior of a non-homogeneous incompressible elastic halfspace. Q. J. Mech. Appl. Math. 43: 1–14 · Zbl 0698.73028 · doi:10.1093/qjmam/43.1.1 |
| [56] | Rajapakse R.K.N.D. and Selvadurai A.P.S. (1991). Responses of circular footing and anchor plates in non-homogeneous elastic soils. Int. J. Numer. Anal. Meth. Geomech. 15: 457–470 · doi:10.1002/nag.1610150702 |
| [57] | (1959). Non-homogeneity in Elasticity and Plasticity. Pergamon Press, Oxford · Zbl 0091.37902 |
| [58] | Golecki J.J. and Knops R.J. (1969). Introduction to a linear elastostatics with variable Poisson’s ratio. Acad. Gorn-Hutn w Krakowie 30: 81–92 |
| [59] | Selvadurai, A.P.S.: Elastic Analysis of Soil-Foundation Interaction; Developments in Geotechnical Engineering, vol 17. Elsevier, Amsterdam (1979) · Zbl 0404.73090 |
| [60] | Plevako V.P. (1971). On the theory of elasticity of non-homogeneous media. J. Appl. Math. Mech. (Prikl. Math. Mekh.) 35: 853–860 · Zbl 0252.73013 |
| [61] | Plevako V.P. (1972). On a possibility of using of harmonic functions for solving problems of theory of elasticity of non-homogeneous media. J. Appl. Math. Mech. (Prikl. Math. Mekh.) 36: 886–894 |
| [62] | Plevako V.P. (1973). Equilibrium of a nonhomogeneous half-plane under the action of forces to the boundary. J. Appl. Math. Mech. (Prikl. Math. Mekh.) 37: 858–866 · Zbl 0309.73017 · doi:10.1016/0021-8928(73)90015-4 |
| [63] | Plevako V.P. (1973). The deformation of a nonhomogeneous half-space under the action of a surface load. Soviet Appl. Mech. 9: 593–598 · Zbl 0305.73017 · doi:10.1007/BF00884183 |
| [64] | Rostovtsev N.A. (1964). On the theory of elasticity of non-homogeneous medium. J. Appl. Math. Mech. (Prikl. Math. Mekh.) 28: 745–757 · Zbl 0138.20803 · doi:10.1016/0021-8928(64)90060-7 |
| [65] | Waas G., Riggs H.R. and Werkle H. (1985). Displacements solutions for dynamic loads in transversely-isotropic stratified media. Earthq. Eng. Struct. Dyn. 13: 329–342 · doi:10.1002/eqe.4290130204 |
| [66] | Muravskii G. and Operstein V. (1996). Time-harmonic vibration of an incompressible linearly non-homogeneous half-space. Earthq. Eng. Struct. Dyn. 25: 1195–1209 · doi:10.1002/(SICI)1096-9845(199611)25:11<1195::AID-EQE607>3.0.CO;2-T |
| [67] | Muravskii G.B. (1996). Green function for an incompressible linearly non-homogeneous half-space. Arch. Appl. Mech. 67: 81–95 · Zbl 0885.73014 · doi:10.1007/BF00787142 |
| [68] | Muravskii G. (1997). Green functions for a compressible linearly inhomogeneous half-space. Arch. Appl. Mech. 67: 521–534 · Zbl 0903.73016 · doi:10.1007/s004190050136 |
| [69] | Gradsteyn I.S. and Ryzhik I.M. (1965). Table of Integrals, Series & Products. Academic, New York |
| [70] | Abramowitz M. and Stegun I.A. (1965). Handbook of Mathematical Functions. Dover, New York · Zbl 0171.38503 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
