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.

摘要

通过研究一组完整的两个位移势, 推导了指数梯度横向各向同性基体涂层系统弹性静力学格林函数的解析式。 以线积分的形式表示三维点荷载和片荷载格林函数的应力和位移。 该公式基于傅里叶级数和汉克尔变换, 包括一组完整的转换应力-位移和位移-位势关系。 为了便于说明, 格林函数被简化为特殊情况, 如指数级半空间和均匀双层半空间格林函数。 由于被积函数的复杂性, 对积分进行了数值计算, 并且针对积分的数值计算, 提出了一个强有力有效的方法, 该方法考虑了积分奇点存在的情况。 对现有的均匀双层各向同性和各向同性的半空间数值解进行比较, 以确定本方案的准确性。 通过典型的数值分析的例子, 展示了指数梯度双层半空间格林函数的一般特征, 据此可以识别材料性能变化程度。