In rock material, the smooth crack mostly becomes kinked crack during its growth due to the anisotropy of rock material. For the kinked crack, the outer normal direction of the crack surface at the kinked line is not unique, so the continuous element is not feasible to discrete the crack surface near the kinked line. Therefore four types of new discontinuous element are employed for the kinked crack. A dual boundary element method (DBEM) based on the method suited for the smooth crack problem is proposed to deal with the kinked crack problem. The numerical verification shows that the results calculated with the present method are in very good agreement with those with the primary method for the smooth crack problem. Furthermore, the proposed method is used to analyze the kinked crack in transversely isotropic rock material of infinite domain. The stress intensity factor (SIF) is obtained. The rectangular smooth crack in transversely isotropic rock material bends. The crack opening of both crack surfaces at two sides of the kinked line has mutual inhibiting effect due to the crack surface subjected to uniformly-distributed normal load. The dip angle between the crack surface and the isotropy plane changes as the crack surface bends gradually. It is found that the fracture behavior of the kinked crack is also affected by the anisotropy of the rock material.