Semianalytical Solution for the Transient Response of Multilayered Unsaturated Viscoelastic Porous Media

Taking into account the stratification and viscoelastic properties of unsaturated porous media, the Poynting–Thomson viscoelastic model is integrated into the Biot-type wave equations tailored for unsaturated porous media, thereby formulating a set of one-dimensional viscoelastic governing equations that capture the transient response characteristics of unsaturated porous media. Based on the correspondence principle between elasticity and viscoelasticity, expressions for the solid-phase skeleton, particle, and shear modulus in the Laplace domain are derived. Subsequently, using the state-space method and Hamilton–Cayley theorem, the general solutions for the transient response of a single layer are derived. By combining boundary conditions, continuity conditions, and the transfer matrix method, the analytical solutions for the transient response results of any layer within the Laplace domain are obtained. Finally, the time-domain solutions are then obtained via the numerical inverse Laplace transform. The correctness of the proposed solutions is verified by comparing it with existing solutions. Analytical examples demonstrate that an increase in the dash-pot viscosity coefficient corresponds to a reduction in the response amplitudes of pore pressure and solid-phase displacement, indicating a notable effect of the dash-pot viscosity coefficient on wave propagation delay. Additionally, saturation levels are found to have a substantial impact on both wave velocity and hysteresis characteristics.