Geodesic
Seepage consolidation under space deformation of elastic half-space
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 161 (2019) no. 3, pp. 355-364 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The process of seepage consolidation of elastic saturated half-space under the action of an arbitrary normal load on its surface was investigated in the case of space deformation. The fluid and skeleton grains were assumed to be incompressible. The purpose of the study was to obtain analytical formulas for the main characteristics of consolidation. To fulfill this purpose, a mathematical model of consolidation with the use of the compatibility equation was proposed. The sum of the effective normal stresses was found as a solution to the first boundary value problem for the heat equation in a half-space. Then the first boundary value problem for an auxiliary function satisfying the Laplace equation was solved. This made it possible to obtain explicit expressions for the fluid pressure and the sum of the total normal stresses. Finally, the surface settlement of the half-space was determined. To illustrate the proposed approach, examples of determining the characteristics of consolidation with load on the surface of a half-space, load over the areas of a circle and a square were given. The maximum settlement of the square centre was determined. The obtained results can be used as tests when applying numerical methods for solving the problems of seepage consolidation.
Mots-clés : consolidation
Keywords: elastic semi-space, load, pressure. 
@article{UZKU_2019_161_3_a2,
     author = {A. V. Kosterin and E. V. Skvortsov},
     title = {Seepage consolidation under space deformation of elastic half-space},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {355--364},
     year = {2019},
     volume = {161},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2019_161_3_a2/}
}
TY  - JOUR
AU  - A. V. Kosterin
AU  - E. V. Skvortsov
TI  - Seepage consolidation under space deformation of elastic half-space
JO  - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
PY  - 2019
SP  - 355
EP  - 364
VL  - 161
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/UZKU_2019_161_3_a2/
LA  - ru
ID  - UZKU_2019_161_3_a2
ER  - 
%0 Journal Article
%A A. V. Kosterin
%A E. V. Skvortsov
%T Seepage consolidation under space deformation of elastic half-space
%J Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
%D 2019
%P 355-364
%V 161
%N 3
%U http://geodesic.mathdoc.fr/item/UZKU_2019_161_3_a2/
%G ru
%F UZKU_2019_161_3_a2
A. V. Kosterin; E. V. Skvortsov. Seepage consolidation under space deformation of elastic half-space. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 161 (2019) no. 3, pp. 355-364. http://geodesic.mathdoc.fr/item/UZKU_2019_161_3_a2/

[1] K. Tertsagi, Soil Mechanics Theory, Gosstroiizdat, M., 1961, 507 pp. (In Russian)

[2] N. M. Gersevanov, Principles of Soil Dynamics, Gosstroiizdat, M.–L., 1937, 241 pp. (In Russian)

[3] V. A. Florin, Theory of Soil Consolidation, Gosstroiizdat, M.–L., 1948, 248 pp. (In Russian)

[4] V. A. Florin, Principles of Soil Mechanics, v. 1, Gosstroiizdat, M.–L., 1959, 357 pp. (In Russian)

[5] M. A. Biot, “General theory of three-dimensional consolidation”, J. Appl. Phys., 12:2 (1941), 155–164 | DOI | Zbl

[6] M. A. Biot, “Consolidation settlement under rectangular load distribution”, J. Appl. Phys., 12:5 (1941), 426–430 | DOI | Zbl

[7] M. A. Biot, “General solutions of the equations of elasticity and consolidation for a porous materials”, J. Appl. Mech., 23:1 (1956), 91–96 | MR | Zbl

[8] V. N. Nikolaevskii, K. S. Basniev, A. T. Gorbunov, G. A. Zotov, The Mechanics of Saturated Porous Media, Nedra, M., 1970, 335 pp. (In Russian)

[9] V. N. Nikolaevskii, Mechanics of Porous and Fissured Media, Nedra, M., 1984, 232 pp. (In Russian)

[10] J. Bear, M. Y. Corapcioglu, Fundamentals of Transport Phenomena in Porous Media, Martinus Nijhoff Publ., Dordrecht, 1984, 1003 pp.

[11] O. Coussy, Mechanics and Physics of Porous Solids, John Wiley and Sons, London, 2010, 300 pp.

[12] R. L. Shiffman, “A bibliography of consolidation”, Fundamentals of Transport Phenomena in Porous Media, eds. Bear J., Corapcioglu M. Y., Martinus Nijhoff Publ., Dordrecht, 1984, 617–669 | DOI

[13] A. P. S. Selvadurai, “The analytical method in geomechanics”, Appl. Mech. Rev., 60:3 (2007), 87–106 | DOI

[14] A. V. Kosterin, E. V. Skvortsov, “Seepage consolidation of elastic half-space under an axisymmetric load”, Fluid Dyn., 49:5 (2014), 627–633 | DOI | MR | Zbl

[15] A. V. Kosterin, E. V. Skvortsov, “Seepage consolidation under plane deformation of elastic half-space”, Fluid Dyn., 53:2 (2018), 270–276 | DOI | DOI | MR | Zbl

[16] S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1951, 506 pp. | MR | Zbl

[17] A. G. Egorov, A. V. Kosterin, E. V. Skvortsov, Consolidation and Acoustic Waves in Saturated Porous Media, Izd. Kazan. Univ., Kazan, 1990, 102 pp. (In Russian)

[18] K. L. Johnson, Contact Mechanics, Cambridge Univ. Press, Cambridge, 1985, 452 pp. | Zbl

[19] E. Detornay, A. H. D. Cheng, “Fundamentals of poroelasticity”: Hudson J. A., Comprehensive Rock Engineering: Principles, Practice and Projects, v. 2, Pergamon Press, Oxford, UK, 1993, 113–171

[20] A. D. Polyanin, Handbook of Linear Equations of Mathematical Physics, Fizmatlit, M., 2001, 576 pp. (In Russian)

[21] A. V. Kosterin, E. V. Skvortsov, “The elastic semi-space consolidation under the normal load on a square area”, Russ. Math., 60:10 (2016), 64–66 | DOI | MR | Zbl

[22] A. V. Kosterin, E. V. Skvortsov, “Analytical formula for integral whose kernel containing error function”, Lobachevskii J. Math., 37:3 (2016), 266–267 | DOI | MR | Zbl

[23] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series. Special Functions, Nauka, Fizmatlit, 1983, 750 pp. (In Russian) | MR