Dynamic 2.5D Green's functions for moving distributed loads acting on an inclined line in a multi-layered TI half-space

Dynamic two and a half dimensional (2.5D) Green's functions for a multi-layered transversely isotropic (TI) half-space are developed by using the dynamic stiffness method combining with the inverse Fourier transform. The 2.5D Green's functions correspond to solutions of uniformly distributed loads acting on part of a multi-layered TI half-space on a line which is inclined to the horizontal and moving along a horizontal straight line with a constant speed. Solutions in the frequency and wavenumber domains are first obtained, which are expressed as the summation of the responses restricted in the loaded layer and of the corresponding reaction forces. Results in the time and space domains are then recovered by Fourier synthesis of the frequency and wavenumber responses which in turn are obtained by numerical integration over on one horizontal wavenumber. The derived Green's functions are verified through comparison with the existing solutions for the isotropic medium that is a special case of the more general problem addressed. Parametric studies are performed in both the frequency and time domains, which show that dynamic responses are highly related to the TI parameters, the load frequency, the load speed and the TI layer. In addition, as an application example, these Green's functions combined with the indirect boundary element method (IBEM) are used to solve the 3D wave scattering of a 2D tunnel embedded in a multi-layered TI half-space. Comparison between the obtained surface displacement amplitudes with those of de Barros and Luco [12] for the isotropic case reinforces the validity and reliability of the presented formulations.