Analytical Solutions for Consolidation of Unsaturated Soil with Time-Dependent Boundary Conditions
Publication: International Journal of Geomechanics
Volume 21, Issue 9
Abstract
Unsaturated soil is widely distributed in geotechnical projects, of which the consolidation and settlement are of great importance. Based on the consolidation theory of unsaturated soil, proposed by Fredlund and Hansan, this paper deals with one-dimensional consolidation of single-layer unsaturated soil with six types of boundary conditions, which can be denoted by arbitrary specified time-dependent functions. The analytical solutions in the time domain are provided directly, by homogenization of nonhomogeneous boundary conditions and the eigenfunction expansion method. Until then, the solutions for excess pore-water pressure and excess pore-air pressure in the single-layer unsaturated soil with all possible Dirichlet or Neumann boundary conditions are provided. Finally, the analytical solutions in this paper are validated against published results, and responses of excess pore-water pressure and excess pore-air pressure in the soil layer subjected to different time-dependent boundary conditions are investigated.
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Acknowledgments
This research was supported by the Basic Science Program for Multiphase Evolution in Hypergravity of the National Natural Science Foundation of China (Grant No. 51988101).
Notation
The following symbols are used in this paper:
- Ca
- interactive constant associated with the air phase;
- Cw
- interactive constant associated with the water phase;
- coefficient of volume change with respect to the air phase;
- coefficient of volume change with respect to the water phase;
- consolidation parameter with respect to the air phase;
- consolidation parameter with respect to the water phase;
- g
- acceleration due to gravity;
- H
- thickness of the soil layer;
- ka
- coefficient of air permeability, obtained by the permeability test;
- kw
- coefficient of water permeability, obtained by the permeability test;
- M
- molecular mass of air;
- coefficient of air volume change caused by the change of the net-normal stress, obtained by triaxial test, determined as the slope of the (σ − ua) plot when d(ua − uw) is zero;
- coefficient of air volume change caused by the change of unit matric suction, obtained by triaxial test, determined as the slope of the (ua − uw) plot when d(σ − ua) is zero;
- coefficient of water volume change caused by the change of net-normal stress, obtained by triaxial test, determined as the slope of the (σ − ua) plot when d(ua − uw) is zero;
- coefficient of water volume change caused by the change of unit matric suction, obtained by triaxial test, determined as the slope of the (ua − uw) plot when d(σ − ua) is zero;
- n
- porosity of the soil before loading, obtained by the porosity test;
- q(t)
- external time-dependent load applied on the top surface of the soil;
- R
- universal gas constant;
- S
- degree of saturation;
- T
- absolute temperature;
- Tk(t)
- corresponding function of time;
- t
- elapsed time;
- u
- excess pore pressure vector;
- ua
- excess pore-air pressure;
- absolute pore-air pressure;
- atmospheric pressure;
- ud
- function specified to satisfy the homogeneous boundary conditions;
- us
- function specified to satisfy the time-dependent boundary conditions;
- uw
- excess pore-water pressure;
- x
- investigated depth;
- kth eigenfunction;
- γw
- unit weight of water; and
- ωk
- kth positive root of the characteristic equation.
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History
Received: Oct 27, 2020
Accepted: May 10, 2021
Published online: Jul 5, 2021
Published in print: Sep 1, 2021
Discussion open until: Dec 5, 2021
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