Complex function method for the scattering of harmonic plane waves by an arbitrary-shaped cavity in an unsaturated medium

Abstract

General formulations of the complex function method for the scattering of harmonic waves by an arbitrary-shaped cavity in the unsaturated medium are constructed. In light of the momentum and the mass balance equations, the wave fields of the unsaturated medium with unknown coefficients are solved by Fredlund’s theory and Helmholtz’s decomposition. Complex-valued expressions for the displacements and the total stresses of the solid skeleton, the pore pressures, and the seepages of the unsaturated medium are obtained by the conformal transformation technique to map the arbitrary cavity onto a unit circle in the complex plane. The boundary value problem results in a series of infinite algebraic equations with unknown coefficients by considering the stress, displacement, and permeable conditions along the cavity. A parametric study for the scattering of the incident P1 waves around a rectangular cavity in an unsaturated medium is performed to investigate the dynamic stress concentrations, displacements, and pore pressures. Numerical results show that the saturation degree of the unsaturated medium has a considerable influence on the dynamic response of the cavity by the incident P1 waves. With the increase of the saturation degree, the dynamic stress concentration around the rectangular cavity decreases, and the equivalent pore pressure increases. For obliquely incident P1 waves, the shielding effect of the rectangular cavity is obvious.