## 2 Integral and Crack Propagation Criteria
### Criteria for Crack Propagation  
The crack initiation and propagation of brittle materials such as rocks and ceramics in engineering are often predicted by appropriate mixed mode fracture criterion. Currently, the common mixed mode fracture criteria have always been the focus of research, e.g., the maximum tangential stress criterion [35], the criterion of local symmetry [5; 36], the minimum strain energy density criterion [37] and the maximum energy release rate criterion [38]. Additionally, the related literatures [5; 22; 27] show that the MTS criterion or the local symmetry criterion could be used to accurately simulate the compression-shear mixed mode propagation process consistent with the experimental outcome under appropriate conditions, which confirms the accuracy of the MTS criterion and extends its applicable  
Figure 2: Schematic of the coordinate system for the crack tip.  
range of MTS fracture criterion. However, prompted by related researches [15, 16, 25, 39, 40, 41], T-stress has a significant effect on propagation angle of the crack, while the traditional MTS criterion only contains the singular term, which would inevitably make the generalized MTS criterion considering the influence of T-stress more accurate in predicting the propagation path. Therefore, the GMTS criterion would be adopted to predict the initiation and propagation trajectory of the crack concerning the frictional contact in this work.  
According to [PERSON] et al. and [PERSON] [16, 41], the tangential stress component of the linear elastic stress field around the crack tip can be written as:  
\[\sigma_{\theta\theta}=\frac{1}{\sqrt{2\pi r}}\cos\frac{\theta}{2}\left[K_{ \mathrm{I}}\cos^{2}\frac{\theta}{2}-\frac{3}{2}K_{\mathrm{II}}\sin\theta \right]\ +T\sin^{2}\theta \tag{10}\]  
where \(r\) and \(\theta\) are the polar coordinates of the crack tip.  
According to the theory of the criterion, the crack still propagates in the direction of the maximum tangential stress around the crack tip. In addition, the angle of the fracture initiation (Figure 3) can be obtained from the following forms  
\[\left.\frac{\partial\sigma_{\theta\theta}}{\partial\theta}\right|_{\theta= \theta_{0}}=0 \tag{11}\]  
\[\left[K_{\mathrm{I}}\sin\theta_{0}+K_{\mathrm{II}}(3\cos\theta_{0}-1)\right] -\frac{16}{3}T\sqrt{2\pi r_{c}}\cos\theta_{0}\sin\frac{\theta_{0}}{2}=0 \tag{12}\]  
where \(r_{c}\) is a material constant for brittle materials and represents a critical distance from the crack tip. In addition, \(r_{c}\) can be expressed as  
\[r_{c}=\frac{1}{2\pi}\left(\frac{K_{IC}}{\sigma_{\mathrm{I}}}\right)^{2} \tag{13}\]  
where \(K_{IC}\) is the fracture toughness of the pure mode I; \(\sigma_{t}\) is the tensile strength of the brittle material.  
Meanwhile, the equivalent stress intensity factor \(K_{\alpha\theta}\) can be adopted and the crack begins to expand when it is greater than or equal to \(K_{IC}\)[42]. The equivalent stress intensity factor can be expressed as below:  
\[K_{\alpha\theta}=\cos\frac{\theta_{0}}{2}\left[K_{\mathrm{I}}\cos^{2}\frac{ \theta_{0}}{2}-\frac{3}{2}K_{\mathrm{II}}\sin\theta_{0}\right]+\sqrt{2\pi r_{c }}T\sin^{2}\theta_{0} \tag{14}\]