Three-dimensional dynamic Green’s functions in transversely isotropic tri-materials

An analytical derivation of the elastodynamic fundamental solutions for a transversely isotropic tri-material full-space is presented by means of a complete representation using two displacement potentials. The complete set of three-dimensional point-load, patch-load, and ring-load Green’s functions for stresses and displacements are given, for the first time, in the complex-plane line-integral representations. The formulation includes a complete set of transformed stress-potential and displacement-potential relations in the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For the numerical computation of the integrals, a robust and effective methodology is laid out. Selected numerical results for point-load and patch-load Green’s functions are presented to portray the dependence of the response on layering, the frequency of excitation, and type of loading.