Fundamental solutions of a multi-layered transversely isotropic saturated half-space subjected to moving point forces and pore pressure

Abstract

The steady-state dynamic response of a multi-layered transversely isotropic (TI) saturated half-space due to point forces and pore pressure moving with a constant speed is investigated in this paper. To solve this problem, the dynamic stiffness method combined with the inverse Fourier transform is employed. First, the governing equations in terms of the displacement components and pore fluid pressure are solved in the transformed domain by employing the Fourier transform. Next, the exact three-dimensional (3D) dynamic stiffness matrices for the TI saturated layer, as well as the TI saturated half-space, are constructed, and the global dynamic matrix of the problem is formulated by assembling the dynamic matrices of the discrete layers and the underlying half-space. Finally, solutions in the frequency-wavenumber domain of the displacement, pore pressure and stress are obtained through the dynamic stiffness method. The result in the time-space domain is recovered by the Fourier synthesis of the frequency responses which, in turn, are obtained by numerical integration over on one horizontal wavenumber. The accuracy of the developed formulations is confirmed by comparison with existing solutions for an isotropic and saturated medium that is a special case of the more general problem addressed. Numerical results for both low and high source velocities are presented, and the effects of moving speed, material anisotropy, permeability, surface drainage condition and TI saturated layer on the dynamic response are analyzed. It is observed that the dynamic responses reach their peak values when the source velocity is equal to or approaches the phase velocities of SH-, qP1-, qP2- and qSV- in the horizontal direction and the phase velocity of qRayleigh waves. Material anisotropy is very important for the accurate assessment of the dynamic response due to the moving point forces and pore pressure in a TI saturated medium.