Abstract: In this paper, a single crack problem and a periodic crackantiplane problem of functionally graded materials (FGMs) arestudied. An elementary solution is obtained, which represents the influencecaused by a point dislocation placed at a point t on the real axis. TheFourier transform method is used to derive the elementary solution. Afterusing the obtained elementary solution, the singular integral equation isformulated for the periodic crack problem. In the solution of the singularintegral equation, the influence at the center crack caused by the manyremote cracks is considered approximately. Finally, numerical results arepresented, and the influence caused by the materials property \alpha is addressed.