Three-dimensional dynamic ring load and point load Green's functions for continuously inhomogeneous viscoelastic transversely isotropic half-space

An analytical formulation is presented for three-dimensional Green's functions of continuously inhomogeneous linear viscoelastic transversely isotropic half-space subjected to either ring load or point load. It is assumed that the elastic moduli of the half-space vary in terms of depth as bounded exponentially functions, while the mass density is constant. The method of potential functions is used to partially decouple the governing equations, after which Fourier series expansion followed by Hankel integral transforms is applied to transform the partial differential equations to ordinary differential equations (ODEs) with variable coefficients. Then, Frobenius series method is employed to determine the potential functions and then the displacements and stresses in the transformed domain, which are used to evaluate these functions in physical domain. The validity of the formulations and numerical process is shown for several simplified cases comparing with the known solutions in the literature. Finally, the displacement and stress Green's functions are presented for several physical cases due to either unit ring load or unit point load. The results show that if the shear waves are produced in the interested direction, both inhomogeneity parameters and material damping may change the dynamic response of the half-space significantly, especially in high frequencies.