Exact solutions to one-dimensional transient response of incompressible fluid-saturated single-layer porous media

doi:10.1007/s10483-013-1654-7

Applied Mathematics and Mechanics (English Edition) ›› 2013, Vol. 34 ›› Issue (1): 75-84.doi: https://doi.org/10.1007/s10483-013-1654-7

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Exact solutions to one-dimensional transient response of incompressible fluid-saturated single-layer porous media

SHAN Zhen-Dong^1,2, LING Dao-Cheng¹, DING Hao-Jiang¹

1. Ministry of Education Key Laboratory of Soft Soils and Geoenvironmental Engineering, Zhejiang University, Hangzhou 310058, P. R. China;
2. Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, P. R. China

Received:2011-11-12 Revised:2012-11-11 Online:2013-01-03 Published:2013-01-03
Contact: Dao-sheng LING E-mail:dsling@zju.edu.cn
Supported by:
the Earthquake Administration Foundation for Seismological Researches of China (No. 200808022), the National Natural Science Foundation of China (Nos. 50778163 and 50708095), and the National Basic Research Program of China (No. 2007CB714200)

Abstract

Abstract: Based on the Biot theory of porous media, the exact solutions to one dimensional transient response of incompressible saturated single-layer porous media under four types of boundary conditions are developed. In the procedure, a relation between the solid displacement u and the relative displacement w is derived, and the well-posed initial conditions and boundary conditions are proposed. The derivation of the solution for one type of boundary condition is then illustrated in detail. The exact solutions for the other three types of boundary conditions are given directly. The propagation of the compressional wave is investigated through numerical examples. It is verified that only one type of compressional wave exists in the incompressible saturated porous media.

Key words: transient response, incompressible porous medium, exact solution, saturated, nonlinear system, random vibration, equivalent nonlinearization method, mean square response

SHAN Zhen-Dong;LING Dao-Cheng;DING Hao-Jiang. Exact solutions to one-dimensional transient response of incompressible fluid-saturated single-layer porous media. Applied Mathematics and Mechanics (English Edition), 2013, 34(1): 75-84.

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References

[1] Biot, M. A. Theory of propagation of elastic waves in a fluid-saturated porous solid, I. low-frequency range. Journal of the Acoustical Society of America, 28(2), 168–178 (1956)

[2] Biot, M. A. Theory of propagation of elastic waves in a fluid-saturated porous solid, II. higher frequency range. Journal of the Acoustical Society of America, 28(2), 179–191 (1956)

[3] Biot, M. A. Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics, 33(4), 1482–1498 (1962)

[4] De Boer, R. Theory of Porous Media, Springer, Berlin (2000)

[5] Bowen, R. Incompressible porous media models by use of the theory of mixtures. International Journal of Engineering Science, 18(9), 1129–1148 (1980)

[6] Bowen, R. Compressible porous media models by use of the theory of mixtures. International Journal of Engineering Science, 20(6), 697–735 (1982)

[7] Schanz, M. and Diebels, S. A comparative study of Biot’s theory and the linear theory of porous media for wave propagation problems. Acta Mechanica, 161(3-4), 213–235 (2003)

[8] Schanz, M. Poroelastodynamics: linear models, analytical solutions, and numerical methods. Applied Mechanics Reviews, 62(3), 1–15 (2009)

[9] Carcione, J. M., Morency, C., and Santos, J. E. Computational poroelasticity— a review. Geophysics, 75, 229–243 (2010)

[10] Garg, S. K., Nayfeh, A. H., and Good, A. J. Compressional waves in fluid-saturated elastic porous media. Journal of Applied Physics, 45(5), 1968–1974 (1974)

[11] Simon, B. R., Zienkiewicz, O. C., and Paul, D. K. An analytical solution for the transient response of saturated porous elastic solids. International Journal for Numerical and Analytical Methods in Geomechanics, 8(4), 381–398 (1984)

[12] Gajo, A. and Mongiovi, L. An analytical solution for the transient response of saturated linear elastic porous media. International Journal for Numerical and Analytical Methods in Geomechanics, 19(6), 399–413 (1995)

[13] Schanz, M. and Cheng, A. H. D. Transient wave propagation in a one-dimensional poroelastic column. Acta Mechanica, 145, 1–8 (2000)

[14] Shan, Z. D., Ling, D. S., and Ding, H. J. Exact solutions for one dimensional transient response of fluid-saturated porous media. International Journal for Numerical and Analytical Methods in Geomechanics, 35(4), 461–479 (2011)

[15] De Boer, R., Ehlers, W. G., and Liu, Z. F. One-dimensional transient wave propagation in fluid-saturated incompressible porous media. Archive of Applied Mechanics, 63(1), 59–72 (1993)

[16] Nakhle, H. A. Partial Differential Equations with Fourier Series and Boundary Value Problems, Person Prentice Hall, New Jersey (2004)

[17] Lubich, C. Convolution quadrature and discretized operational calculus I. Numerische Mathematik, 52, 129–145 (1988)

Exact solutions to one-dimensional transient response of incompressible fluid-saturated single-layer porous media

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