SUMMARYBased on the Fredlund consolidation theory of unsaturated soil, exact solutions of the governing equations for one-dimensional consolidation of single-layer unsaturated soil are presented, in which the water permeability and air transmission are assumed to be constants. The general solution of two coupled homogeneous governing equations is first obtained. This general solution is expressed in terms of two functions 1 and 2 , where 1 and 2 , respectively, satisfy two second-order partial differential equations, which are in the same form. Using the method of separation of variables, the two partial differential equations are solved and exact solutions for three typical homogeneous boundary conditions are obtained. To obtain exact solutions of nonhomogeneous governing equations with three typical nonhomogeneous boundary conditions, the nonhomogeneous boundary conditions are first transformed into homogeneous boundary conditions. Then according to the method of undetermined coefficients and exact solutions of homogenous governing equations, the series form exact solutions are put forward. The validity of the proposed exact solutions is verified against other analytical solutions in the literature.
Abstract:Based on consolidation equations proposed for unsaturated soil, an analytical solution for 1D consolidation of an unsaturated single-layer soil with nonhomogeneous mixed boundary condition is developed. The mixed boundary condition can be used for special applications, such as tests occur in laboratory. The analytical solution is obtained by assuming all material parameters remain constant during consolidation. In the derivation of the analytical solution, the nonhomogeneous boundary condition is first transformed into a homogeneous boundary condition. Then, the eigenfunction and eigenvalue are derived according to the consolidation equations and the new boundary condition. Finally, using the method of undetermined coefficients and the orthogonal relation of the eigenfunction, the analytical solution for the new boundary condition is obtained. The present method is applicable to various types of boundary conditions. Several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with mixed boundary condition.
SUMMARYBased on the Biot theory, the exact solutions for one-dimensional transient response of single layer of fluid-saturated porous media and semi-infinite media are developed, in which the fluid and solid particles are assumed to be compressible and the inertial, viscous and mechanical couplings are taken into account. First, the control equations in terms of the solid displacement u and a relative displacement w are expressed in matrix form. For problems of single layer under homogeneous boundary conditions, the eigen-values and the eigen-functions are obtained by means of the variable separation method, and the displacement vector u is put forward using the searching method. In the case of nonhomogeneous boundary conditions, the boundary conditions are first homogenized, and the displacement field is constructed basing upon the eigen-functions. Making use of the orthogonality of eigen-functions, a series of ordinary differential equations with respect to dimensionless time and their corresponding initial conditions are obtained. Those differential equations are solved by the state-space method, and the series solutions for three typical nonhomogeneous boundary conditions are developed. For semi-infinite media, the exact solutions in integral form for two kinds of nonhomogeneous boundary conditions are presented by applying the cosine and sine transforms to the basic equations. Finally, three examples are studied to illustrate the validity of the solutions, and to assess the influence of the dynamic permeability coefficient and the fluid inertia to the transient response of porous media.
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