Numerical equivalent inclusion method: a new computational method for analyzing stress fields in and around inclusions of various shapes

The present study has attempted to develop a new computational method for the elastic stress analysis of inclusions based on the equivalent inclusion method. The proposed method can avoid the complexity of mathematics required for the analysis of non-uniform eigenstrain distributions within inclusions having various shapes. The paper is focused on the formulation for two-dimensional case. The fundamental integral equation is shown first to have a kernel with the 1/r-singularity. The two-dimensional equations are then discretized by using the triangle polar coordinates. It is shown that the adoption of this coordinate system can eliminate the singularity. Eigenstrain distributions within inclusions having various shapes were calculated by the present method in order to obtain stress distributions within them as well as those in the vicinity of the matrix–inclusion interfaces. The shapes of the inclusions described here are ellipse, circle, triangle and rectangle. The results obtained by the present method were compared and showed good agreements with those obtained by the theories and/or by the FEM analyses except for the sharp corner points of the triangular inclusions where the outside normal vectors can not be determined uniquely and thus the stress becomes singular.