Scattering of elastic waves by a circular hole in the unsaturated soil

Abstract

The scattering of elastic waves by a circular hole embedded in the unsaturated soil is investigated in this study. To this end, the mass balance equations, the momentum balance equations, the Biot-type constitutive relations, and the saturation-capillary-pressure relation for the unsaturated soil are used to formulate the differential equations for the independent variables of the unsaturated soil, namely, the displacement of the solid skeleton and the pressures of the pore water and gas. Based on the aforementioned differential equations for the independent variables, the Helmholtz equations for the scalar and vector potentials of the unsaturated soil are established. With the Helmholtz equations for the potentials, the scattered waves of the circular hole are obtained based on the wave function expansion method. By using the boundary conditions of the circular hole, the unknown coefficients in the scattered waves are obtained, and the scattered wave field of the circular hole is thus determined. The Helmholtz equations for the potentials and the potential-variable relations for the unsaturated soil obtained in this study may facilitate solving various dynamic problems of the unsaturated soil. Numerical results of this paper suggest that the stress and pore pressure concentration phenomena caused by low frequency incident waves are more pronounced than those caused by the corresponding static load. Also, for the incident P1 wave, with increasing saturation, the stress concentration phenomenon becomes less obvious, while that for the pore pressures becomes more pronounced.