## 1 Introduction  
Crack defects are often distributed in brittle media, including concretes and rocks. Under the action of complex external loads, these defects cause local cracking and form macroscopic cracks, resulting in the decrease of strength and stiffness of the medium, which may reproduce and extend to complete penetration. In order to keep the integrity and safety of structures, the study on crack initiation and growth is of guiding significance. Meanwhile, the stress intensity factors (SIFs) and T-stresses, as important fracture parameters of the linear elastic fracture mechanics (LEFM) used to evaluate the fracture performance of brittle materials, also have important practical significance in engineering fracture problems. Additionally, the central crack Brazilian disc has been widely used due to its convenience in changing the fracture mode of crack [1, 2, 3]. In any case, rough brittle materials with many closed cracks are subjected to the effect of the friction in actual underground engineering such as tunnel operation and gas exploration. Under these conditions, the fracture parameters, crack initiation and propagation of brittle materials under compressive-shear loading would be greatly affected. Therefore, in order to better understand the failure mechanisms and propagation behavior concerning the effect of friction of brittle materials such as rocks, some experimental tests were performed in this area. [PERSON] et al. [4] carried out composite fracture loading by using the cracked chevron notched Brazilian disk (CCNBD) specimens made of three kinds of rock materials. In the test, a displacement gauge was used to measure the displacement of radial compression in the crack tip. Moreover, this experimental result showed that in the case of compression-shear fracture, both the initiation angle and the initiation point of the crack would be affected. [PERSON] and [PERSON] [5] described compression-shear experiments of uniaxial compression concerning frictional contact. In order to control friction, the brass sheets or thin Teflon sheets were inserted into the crack. Their results showed that the specimen failure was caused by the axial splitting rather than shear failure and the crack propagated in the direction of the applied load, which was also demonstrated by the maximum tangential stress (MTS) criterion. Meanwhile, [PERSON] [6] and [PERSON] et al. [7] found a similar phenomenon through a range of experiments. Furthermore, compression-shear experiments were reported as well on the law of crack propagation [8; 9; 10]. Although the experimental results could provide guidance in other directions, their implementation was usually limited by external conditions such as time-consuming and complex processes. Additionally, the cracks were not easy to contact due to the crack width [4] and some physical parameters such as friction coefficient could not be controlled easily. Therefore, theoretical method has attracted growing interest because of its convenience and reliability. In the past few years, on the basis of previous reports by [PERSON] [1] and [PERSON] [11], [PERSON] [12] and [PERSON] [13] analyzed the stress field inside the centrally cracked Brazilian disk (CCBD) specimen under different loading conditions when the crack surface contacted through the theoretical study. Furthermore, an analytical formula was obtained for calculating the Mode II SIFs in CCBD specimens under different loading conditions by using the weight function method (WFM). Meanwhile, prompted by some theoretical researches on T-stress [14; 15], [PERSON] [16] proved that the friction had no effect on T-stress in CCBD specimens subjected to diametric forces and confining pressure. [PERSON] et al. [17] discussed the effect of roughness between crack surfaces and friction coefficient on the dimensionless Mode II stress intensity factor and crack initiation angle using Distributed Dislocation Technique (DDT) and MTS criterion. Currently, due to the complex boundary conditions, numerical approaches are increasingly used.