The study on interactions between inclusions and singularities such as a point force and/or dislocation has been given considerable attention for many years by researchers, since their interactions with dislocations play an important role in determining the mechanical properties of materials and these solutions can be used as Green's functions for the practical problem, e.g., crack-inclusions interaction problem.
In this study, bimaterial containing an in-plane or an out-of-plane singularity embedded in the inclusion or in the unbounded matrix is first analyzed by using analytic continuation. Next, the series forms of solutions for the trimaterial with two concentric circular inclusions having an identical singularity are found based on an alternating technique using the solution for the bimaterial case. The sum of the first three or four terms of solutions derived provides an excellent approximation for most of material combinations. As an example for application, a trimaterial having a straight crack in an infinite matrix under out-of-plane loading is considered, and the stress intensity factor of the crack-tip closer to the inclusions is calculated to show the effect of the material properties and geometric parameters.
