Exact Solutions For One-Dimensional Transient Response Of Single-Layer Fluid-Saturated Porous Media

Based on the Biot Theoiy, the one-dimensional transient response of single-layer fluid-saturated porous media loaded arbitrarily at its top surface are studied thorouly, where the inertial, viscous and mechanical couplings are taken into account. Exact solutions are developed for both the Fully Dynamic Formulation and the Partly Dynamic Formulation.Firstly, dimensionless governing equations in matrix form for the compressible constituents are derived, with the boundary conditions homogenized. Then, the eigen-value problem for the corresponding nonviscous system is solved to get an orthogonal function base in spatial domain. Applying variation coeffient method and making use of the orthogonality of the base functions, a series of decoupled second-order ordinary differential equations together with its corresponding initial conditions are obtained in time domain. To get the solutions for displacement components, the second-order ordinary differential equations are solved by the state-space method.The total stress and fluid pressure can be determined directly by the constitutive equations.For the incompressible constituents, the relationship, which the displacement of solid particles and the relative displacement must satisfy, is derived from the continuity equation. By homogenizing the boundary conditions and using the method of separation of variables, the displacement of solid particles can be obtained. The total stress and fluid pressure can be determined by integrating the dynamic equilibrium equation of porous media and using the boundary conditions.For the Partly Dynamic Formulation, where the inertial term of relative displacement is not considered, the problem can be solved by making use of the same orthogonal function base as what is used in the Fully Dynamic Formulation. Finally, examples are studied to demonstrate the correctness and applicability of the presented solutions.