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Transfer matrix solutions for three-dimensional consolidation of a multi-layered soil with compressible constituents. (English) Zbl 1213.74222

Summary: Transfer matrix solutions for three-dimensional consolidation of a multi-layered soil considering the compressibility of pore fluid are presented. The derivation of the solutions starts with the fundamental differential equations of Biot’s three-dimensional consolidation theory, takes into account the compressibility of pore fluid in the Cartesian coordinate system, and introduces the extended displacement functions. The relationship of displacements, stresses, excess pore water pressure, and flux between the ground surface \((z = 0)\) and an arbitrary depth \(z\) is established for Biot’s three-dimensional consolidation problem of a finite soil layer with compressible pore fluid by taking the Laplace transform with respect to \(t\) and the double Fourier transform with respect to \(x\) and \(y\), respectively. Based on this relationship of the transfer matrix, the continuity between layers, and the boundary conditions, the solutions for Biot’s three-dimensional consolidation problem of a multi-layered soil with compressible constituents in a Laplace-Fourier transform domain is obtained. The final solutions in the physical domain are obtained by inverting the Laplace-Fourier transforms. Numerical analysis is carried out by using a corresponding program based on the solutions developed in this study. This analysis demonstrates that the compressibility of pore fluid has a remarkable effect on the process of consolidation.

MSC:

74L10 Soil and rock mechanics
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[1] Biot, M. A., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 155-164 (1941) · JFM 67.0837.01
[2] McNamee, J.; Gibson, R. E., Displacement functions and linear transforms applied to diffusion through porous elastic media, Q. J. Mech. Appl. Math., 13, 1, 98-111 (1960) · Zbl 0097.42103
[3] McNamee, J.; Gibson, R. E., Plane strain and axially symmetric problem of the consolidation of a semi-infinite clay stratum, Q. J. Mech. Appl. Math., 13, 2, 210-227 (1960) · Zbl 0134.44604
[4] R.L. Schiffman, A.A. Fungaroli, Consolidation due to tangential loads, in: Proceedings of the Sixth International Conference on Soil Mechanics and Foundation Engineering, Montreal, Canada, vol. 1, 1965, pp. 188-192.; R.L. Schiffman, A.A. Fungaroli, Consolidation due to tangential loads, in: Proceedings of the Sixth International Conference on Soil Mechanics and Foundation Engineering, Montreal, Canada, vol. 1, 1965, pp. 188-192.
[5] Booker, J. R., Consolidation of a finite layer subject to surface loading, Int. J. Solids Struct., 10, 1053-1065 (1974)
[6] Booker, J. R.; Small, J. C., Finite layer analysis of consolidation I, Int. J. Numer. Anal. Methods Geomech., 6, 151-171 (1982) · Zbl 0482.73087
[7] Booker, J. R.; Small, J. C., Finite layer analysis of consolidation II, Int. J. Numer. Anal. Methods Geomech., 6, 173-194 (1982) · Zbl 0482.73088
[8] Booker, J. R.; Small, J. C., A method of computing the consolidation behavior of layered soils using direct numerical inversion of Laplace Transforms, Int. J. Numer. Anal. Methods Geomech., 11, 363-380 (1987) · Zbl 0612.73109
[9] Vardoulakis, I.; Harnpattanapanich, T., Numerical Laplace-Fourier Transform inversion technique for layered-soil consolidation problems: I. Fundamental solutions and validation, Int. J. Numer. Anal. Methods Geomech., 10, 4, 347-365 (1986) · Zbl 0589.73092
[10] Wang, J. G.; Fang, S. S., The state vector solution of axisymmetric Biot’s consolidation problems for multilayered poroelastic media, Mech. Res. Commun., 28, 6, 671-677 (2001) · Zbl 1026.74024
[11] Wang, J. G.; Fang, S. S., State space solution of non-axisymmetric Biot’s consolidation problems for multilayered poroelastic media, Int. J. Eng. Sci., 41, 1799-1813 (2003) · Zbl 1211.76122
[12] Z.Y. Ai, J. Han, A solution to plane strain consolidation of multi-layered soils, in: Luna, Hong, Ma, Huang (Eds.), Soil and Rock Behavior and Modeling, ASCE Geotechnical Special Publication, Proceedings of the GeoShanghai International Conference 2006, Shanghai, China, June 6-8, 2006, pp. 276-283.; Z.Y. Ai, J. Han, A solution to plane strain consolidation of multi-layered soils, in: Luna, Hong, Ma, Huang (Eds.), Soil and Rock Behavior and Modeling, ASCE Geotechnical Special Publication, Proceedings of the GeoShanghai International Conference 2006, Shanghai, China, June 6-8, 2006, pp. 276-283.
[13] Christian, J. T.; Boehmer, J. W., Plane strain consolidation by finite elements, J. Soil Mech. Found. Div. ASCE, 96, 4, 1435-1457 (1970)
[14] Cheng, A. H.-D.; Liggett, J. A., Boundary integral equation method for linear porous-elasticity with applications to soil consolidation, Int. J. Numer. Methods Eng., 20, 2, 255-278 (1984) · Zbl 0525.73124
[15] Yue, Z. Q.; Selvadurai, A. P.S.; Law, K. T., Excess pore water pressure in a poroelastic seabed saturated with a compressible fluid, Can. Geotech. J., 31, 989-1003 (1994)
[16] Senjuntichai, T.; Rajapakse, R. K.N. D., Exact stiffness method for quasi-statics of a multi-layered poroelastic medium, Int. J. Solids Struct., 32, 1535-1553 (1995) · Zbl 0921.73017
[17] Chen, G. J., Consolidation of multilayered half space with anisotropic permeability and compressible constituents, Int. J. Solids Struct., 41, 4567-4586 (2004) · Zbl 1133.74310
[18] Verruijt, A., Displacement functions in the theory of consolidation of thermoelasticity, J. Appl. Math. Phys., 22, 891-898 (1971) · Zbl 0235.76048
[19] Bahar, L. Y., Transfer matrix approach to layered systems, J. Eng. Mech. Div. ASCE, 98, 1159-1172 (1972)
[20] Ai, Z. Y.; Yue, Z. Q.; Tham, L. G.; Yang, M., Extended Sneddon and Muki solutions for multilayered elastic materials, Int. J. Eng. Sci., 40, 1453-1483 (2002) · Zbl 1211.74105
[21] Pan, E., Green’s functions in layered poroelastic half-space, Int. J. Numer. Anal. Methods Geomech., 23, 1631-1653 (1999) · Zbl 0962.74017
[22] Talbot, A., The accurate numerical inversion of Laplace transforms, J. Inst. Math. Appl., 23, 97-120 (1979) · Zbl 0406.65054
[23] Sneddon, I. N., The Use of Integral Transform (1972), McGraw-Hill: McGraw-Hill New York · Zbl 0265.73085
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