Abstract: In the case of arbitrary tractions on the side faces of the crack, a polynomial function of the radial coordinate can be employed to describe the side face loads in the scaled boundary finite element method (SBFEM). The SBFEM new shape function considering the side face loads is presented. The corresponding stiffness matrix and equivalent node load is derived together based on the SBFEM new shape function. The model of crack face contact using the SBFEM is proposed in this paper for the first time. Lagrange's multiplier method is used to establish the contact constraints of contact model between crack faces. The governing equations for the nonlinear surface contact problems in SBFEM is derived, including adhesion contact problems and sliding friction problems. The elements where the crack faces lie are divided into non crack tip elements and the crack tip element. For the former, the crack faces act as the boundary of the SBFEM element, the contact tractions on the boundary can be assigned to the nodes equivalently and the Lagrange's multiplier is applied for the point constraints. For the latter, the interpolation field of Lagrange's multiplier is constructed on the whole side faces. The Lagrange's multiplier is assumed to be linear along the side faces, the segment constraint approach is proposed to optimize the fulfilment of the contact constraints along the crack faces.By comparing the results of the calculation of analytical solution and software ABAQUS on three different numerical contact problems of crack faces, the accuracy and effectiveness of the proposed point-to-point and segment-to-segment contact model for fracture surfaces contact problems is verified in this paper.