Exact solutions for one‐dimensional consolidation of single‐layer unsaturated soil

Abstract

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.

Analytical solution for 1D consolidation of unsaturated soil with mixed boundary condition

“…The method described represents an extension of the method of Shan et al (2012) and can be used to solve various kinds of boundary conditions. An analytical solution for an unsaturated single-layer soil subjected to an arbitrary load was also obtained.…”

“…The method of Shan et al (2012) can address problems with three types of boundary conditions, but cannot solve problems with mixed boundary conditions. By incorporating some improvements to the method of Shan et al (2012), the present method can be used to solve various kinds of boundary value problems.…”

“…By incorporating some improvements to the method of Shan et al (2012), the present method can be used to solve various kinds of boundary value problems. The solution procedure is illustrated below.…”

“…Qin et al (2010) obtained an analytical solution for unsaturated single-layer soil subjected to a gradually-increasing load using the same method and employing the same boundary condition as Qin et al (2008). Shan et al (2012) used the separation of variables method to give exact solutions for unsaturated single-layer soil subjected to an arbitrary load and with three types of boundary conditions. These analytical solutions are for unsaturated single-layer soil with boundaries that are permeable or impermeable to both air and water phases.…”

“…As an extension of Shan et al (2012), this paper presents an analytical method which can be used to solve various kinds of boundary value problems. As an example, an analytical solution for an unsaturated single-layer soil with mixed boundary condition is presented.…”