Dynamic loading in a transversely isotropic and layered elastic half-space

By introducing the Cartesian system of vector functions, we investigate the responses of a layered transversely isotropic elastic half-space induced by a dynamic load. The load is vertically applied in the layered half-space. It can be either time-harmonic or horizontally moving with a given velocity. It is demonstrated that, in each layer, in terms of the Cartesian vector functions, we need only a 4 x 4 system of equations for the expansion coefficients of the displacement and traction components, instead of a 6 x 6 system as usually presented. Furthermore, utilizing the vector functions, both the time-harmonic and moving-load cases can be solved in the same way. For the layered case, the previously introduced dual-variable and position method is further used to propagate the solutions from one layer to the other, with unconditional stability. The fast Fourier transform (FFT) is applied to obtain accurate results in the space/time domain efficiently by the numerical inverse Fourier transform, properly developed within the framework of the discrete convolution-FFT (DC-FFT) algorithm. The unique formulation and its corresponding sophisticated algorithm are first validated against existing solutions, and then applied to calculate the displacements and stresses in a layered transversely isotropic half-space. Finally, a typical three-layer flexible pavement structure under a moving circular load is analyzed in detail. Numerical results clearly show the significant effects of both material anisotropy and layering. The present formulation is concise and efficient for the calculation of all the displacements and stresses, which can be applied to the involved engi-neering practice. The results can also serve as benchmarks for future research in this direction.