Analytical And Semi-analytical Solutions To One-Dimensional Consolidation In Unsaturated Soils

In this paper, a series of analytical and semi-analytical solutions to one-dimensional consolidation in unsaturated soil with finite thickness are obtained under several kinds of boundary conditions under the large-area uniform instantaneous loading and loading changed exponentially with time. The semi-analytical solution to one-dimensional consolidation for linearly elastic unsaturated soils is extended to that for the viscoelastic unsaturated soils.Main contents can be drawn as follows:(1) For the large-area uniform instantaneous loading, based on the Fredlund's one-dimensional consolidation theory for unsaturated soil, the transfer relations between the state vectors at top surface and any depth are gained by using the Laplace transform and Cayley-Hamilton mathematical methods to the Fredlund's governing equations of water and air, Darcy's law and Fick's law.(2) For the large-area loading changing exponentially with time, the governing equations of one-dimensional consolidation are derived based on Fredlund's one-dimensional consolidation theory for unsaturated soil. The transfer relation between the state vectors at the top surface and at arbitrary depth are gained by using the Laplace transform and Cayley-Hamilton mathematical methods to this governing equations of water and air, Darcy's law and Fick's law.(3) The excess pore-air pressure, the excess pore-water pressure and the settlement in the Laplace-transformed domain are obtained by using the Laplace transform with the initial conditions and certain boundary conditions, in the cases of the large-area uniform instantaneous loading and loading changing exponentially with time.(4) For the large-area uniform instantaneous loading and loading changing exponentially with time, the analytical solutions of the excess pore-air pressure, the excess pore-water pressure and the soil settlement are obtained by performing the inverse Laplace transforms under the boundary conditions of the top surface permeable to water and air, and the bottom impermeable to water and air. A typical example result is given to show the changes in the excess pore-air pressure, the excess pore-water pressure and the soil settlement rate with time under different air-water coefficient rates.(5) For the large-area uniform instantaneous loading and loading changing exponentially with time, a computer program using the method of Crump and F. Durbin is developed to obtain the semi-analytical solutions of the excess pore-air pressure, the excess pore-water pressure and the soil settlement rate under several kinds of boundary conditions. At the same time the consolidation behavior for unsaturated soils is analyzed according to the semi- analytical solution, and the results prove that the method is of high precision. (6) For the large-area uniform instantaneous loading and loading changing exponentially with time, a computer program is developed to obtain the numerical solutions to one-dimensional consolidation in unsaturated soils by the finite difference method under several kinds of boundary conditions. Comparisons between the numerical and analytical results indicate that the results of the two methods are almost the same.(7) The analytical solution is compared with the Terzaghi solution for saturated soils and it is known that the two solutions are almost the same for saturated soils. Therefore, the analytical solution to one-dimensional consolidation in unsaturated soilis correct.(8) Unsaturated soil is assumed to obey Merchant viscoelastic model. By using Lee's matching law, the excess pore-air pressure, excess pore-water pressure and settlement in the Laplace-transformed domain are obtained for one-dimensional consolidation in unsaturated soils, under the large-area uniform instantaneous loading. A computer program using the method by Crump and Durbin is developed to obtain the semi-analytical solution of the excess pore-air pressure, excess pore-water pressure and settlement under the boundary conditions of the top surface permeable to water and air and the bottom impermeable to water and air. At the same time the consolidation behavior of the viscoelastic unsaturated soils is analyzed according to the semi-analytical solution. The changes in the excess pore-air pressure, excess pore-water pressure and settlement of the viscoelastic unsaturated soil layer with thetime under different air-water coefficients k_a / k_w and E_l ,ηin the Merchantviscoelastic model are analyzed.In this paper, some linear assumptions associated with the transport processes and constitutive equations were made, which provided useful initial approximations for unsaturated soil consolidation, and the linear models require relatively less data and it becomes possible for the models to be solved by means of analytical methods. Such analytical solutions are often used to gain a quick insight into the physics of a complex coupled system, and the influence of various material properties and boundary conditions can be investigated. Therefore, developing simplified analytical solutions is always valuable when dealing with complex problems of consolidation in unsaturated soils. The analytical solutions have a high academic value to the studies on unsaturated soils consolidation mechanism and consolidation characteristics, and also the engineering value for solving practical problems.