Analytical Solution for the Transient Response of a One-Dimensional Viscoelastic Unsaturated Porous Medium

The Laplace transform method is widely used for solving viscoelastic dynamic problems. However, the numerical inversion process of this method suffers from instability, especially for long-duration response issues. Moreover, the solutions obtained via the Laplace transform method are often semi-analytical. In this study, addressing the one-dimensional transient response problem of unsaturated viscoelastic porous media, an analytical solution is proposed based on the finite Fourier transform method. First, leveraging the symmetry of the problem, the solution of a typical stress-displacement boundary problem is transformed into the solution of a stress-stress boundary problem. Second, by applying finite Fourier sine and cosine transforms, the original second-order viscoelastic partial differential governing equations are transformed into a system of first-order ordinary differential equations in the frequency domain for solution, and the analytical solution in the frequency domain is provided using the state-space method. Finally, the finite Fourier inverse transform method is used to present the analytical solution in series form for the original problem in the time domain. The accuracy of the proposed method is validated through comparison with the Laplace solution method and existing elastic solution results. The analysis of numerical examples shows that the damping coefficient $\eta$ has a significant influence on the $P_3$ wave velocity. As the damping coefficient $\eta$ increases, the wave phase generally tends to decrease. With the increase in saturation $S_w$, the velocities of $P_1$ and $P_2$ waves increase, while the velocity of the $P_3$ wave decreases.