Fundamental solution for bonded materials with a free surface parallel to the interface. Part II: Solutions of concentrated force acting at the interior of the substrate and the case when the force acting at the interface

In this work, the exact analyses are presented for the plane problem of a coating material subjected to a concentrated force acting at the interior of the substrate and the case when the force at the interface. The stress functions are constructed as an infinite series form by utilizing the method of image. According to the orders of the image points from lower to higher, the terms in the stress functions series have the recursive relationships. For the case when the force acting at the substrate, the first two terms are the original stress functions for a homogenous infinite plane subjected to a concentrated force, which are known and simple. For the case when the force acting at the interface, the fundamental solution is obtained for two bonded dissimilar semi-infinite plane. The stress functions in this solution can be used as the first two terms for the problem considered in this paper. Therefore, all other terms can be derived by the recurrence equations explicitly. Also, through comparisons between the theoretical results and the numerical results by FEM, it is verified that the convergence rate of the solutions is very rapid. In most practical cases only the first several image points can ensure the solutions with satisfactory accuracy.