Analytical solution for Fredlund-Hasan unsaturated consolidation using mode superposition method

This note presents an exact analytical solution of Fredlund-Hasan consolidation for unsaturated soils under an arbitrary loading using the mode superposition method. Air pressure and water pressure are expressed in a series of products of Eigen functions with respect to depth and normal coordinates with respect to time, respectively. A pressure function substitutes for the related homogeneous problem. Eigen functions can be first obtained based on two pairs of drainage condition. Then, the nonhomogeneous governing equations of unsaturated soils under constant loading, exponential loading, and sine loading are considered. After matrix algebraic manipulation, the normal coordinates can be obtained based on the initial conditions that compose the final solutions of the Fredlund-Hasan unsaturated consolidation. This solution can avoid inconvenient Laplace transform and inverse Laplace transform. Through comparisons with a classic example, the validation of the proposed analytical solution is verified against other analytical solutions and the finite difference method solutions.