Three-Dimensional Interfacial Green’s Function for Exponentially Graded Transversely Isotropic Bi-Materials

By virtue of a complete set of two displacement potentials, an analytical derivation of the elastostatic Green’s functions of an exponentially graded transversely isotropic bi-material full-space was presented. Three-dimensional point-load Green’s functions for stresses and displacements were given in line-integral representations. The formulation included a complete set of transformed stress-potential and displacement-potential relations, with the utilization of Fourier series and Hankel transform. As illustrations, the present Green’s functions were analytically degenerated into special cases, such as exponentially graded half-space and homogeneous full-space bi-material Green’s functions. Owing to the complicated integrand functions, the integrals were evaluated numerically, and in computing the integrals numerically, a robust and effective methodology was laid out which provided the necessary account of the presence of singularities of integration. Some typical numerical examples were also illustrated to demonstrate the general features of the exponentially graded bi-material Green’s functions which will be recognized by the effect of degree of variation of material properties.