Asymmetric Green’s functions for exponentially graded transversely isotropic substrate–coating system

Abstract

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 substrate–coating system is presented. Three-dimensional point–load and patch–load Green’s functions for stresses and displacements are given in line-integral representations. The formulation includes a complete set of transformed stress–potential and displacement–potential relations, with utilizing Fourier series and Hankel transforms. As illustrations, the present Green’s functions are degenerated to the special cases such as an exponentially graded half-space and a homogeneous two-layered half-space Green’s functions. Because of complicated integrand functions, the integrals are evaluated numerically and for numerical computation of the integrals, a robust and effective methodology is laid out which gives the necessary account of the presence of singularities of integration. Comparisons of the existing numerical solutions for homogeneous two-layered isotropic and transversely isotropic half-spaces are made to confirm the accuracy of the present solutions. Some typical numerical examples are also given to show the general features of the exponentially graded two-layered half-space Green’s functions that the effect of degree of variation of material properties will be recognized.

Keywords

functionally graded material / transversely isotropic / bi-material / Green’s function / coating-substrate / displacement potential

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