## 1 Introduction
For instance, [PERSON] et al. [11; 16] and [PERSON] et al. [18] accurately calculated the stress intensity factors and T-stresses for CCBD under concentrated force and confining pressure through conventional finite element method (FEM). [PERSON] et al. [19] and [PERSON] et al. [20] calculated those fracture parameters under different confining pressure coefficients based on the improved finite element method. Furthermore, various methods were reported for the fracture propagation under complex conditions. [PERSON] et al. [21] developed a technique of automatic local meshing based on finite element method to simulate mixed-mode fracture growth automatically. [PERSON] et al. [4] and [PERSON] et al. [22] combined finite element method and MTS criterion to simulate the crack propagation from pure I to compression-shear fracture of Brazilian disk specimens. These results had a high matching degree with the experimental outcome. Meanwhile, numerical methods such as generalized particle dynamics, unconventional state-based peridynamics and phase field method, combining MTS and Mohr-Coulomb criterion, would simulate crack growth of different modes fracture [23; 24; 25]. [PERSON] et al. [26] also proposed a convenient numerical method introducing an enrichment function to deal with complex practical problems, which was called as extended finite element method (XFEM). Based on XFEM, [PERSON] et al. [27] combined traditional MTS criterion and LATIN method to explore the propagation of crack concerning frictional contact. The results showed that the simulated extension path was consistent with the experimental observations of [PERSON] and [PERSON] [5]. However, in the simulation of two-dimensional crack propagation problem, XFEM usually adopts first-order elements with weak analytical performance [28], which might result in inaccurate outcome when simulating compression-shear fracture. Moreover, there are few reports on fracture parameters and propagation of crack, considering the effect of friction for CCBD specimens through the general numerical methods. Therefore, prompted by an improved numerical method proposed by [PERSON] et al. [19], a modified finite element method (MFEM) is developed. The MFEM involving the behavior of the frictional contact is based on FEM using user defined subroutine. Additionally, the purpose of this study is to investigate the influence of friction on fracture parameters and propagation of crack for CCBD specimens under radial compression forces when the crack surface is in contact, which is usually difficult to be studied through experimental tests. The framework of this investigation is briefly summarized as follows. Section 2 mainly elaborates the principle of the interaction integral for MFEM and the generalized MTS (GMTS) criterion. Section 3 mainly illustrates when the crack surface is in contact, the analytical formula of SIFs and T-stresses concerning the friction by using the WFM for CCBD. Section 4 introduces some calculation examples for CCBD specimens about the effects of friction on fracture parameters (mainly SIFs and T-stresses) and crack propagation. Section 5 summarizes the conclusion of the whole work.