Abstract: When dealing with discontinuous issues such as fracture and crack in geo-engineering, conventional finite element method (CFEM) need to refine mesh in the local zone including crack tip; furthermore, the mesh must be reconfigured and re-partitioned once the crack propagation happened. The extended finite element method (XFEM), based on the idea of partition of unit method, can successfully solve these problems easily, which may hardly be deposed by CFEM. Usually, contact problem of crack surface must be considered when studying some compound fracture. Based on the complementary theory, the complementary equation between relative displacement and contact force of the crack surface can be established, and solved by Newton's method without considering opening and closing iteration. Finally, as a numerical example, a plate with crack is compressed to indicate the effectiveness of this method. The results indicate that the method can prevent mesh penetration from one side of the crack into the other side effectively; and also obtain the stress distribution that is consistent with the actual on the crack surfaces.

Key words: extended finite element method (XFEM), complementary theory, contact problem