## 3 The WFM Concerning Frictional Contact  
It is assumed that the studied Brazilian disc obeys the assumptions of small deformation, isotropic and linear elasticity. According to the research of [PERSON] [1] and [PERSON] [11, 14],  
Figure 3: Schematic of the crack propagation angle and loading angle. the expressions of the stress intensity factors and T-stress for CCBD under diametrical forces without involving the friction can be obtained. \[K_{\text{I}}^{\prime}=\int_{0}^{a}h_{1}(x,a)\sigma_{\theta}(x)dx \tag{15}\]  
\[K_{\text{II}}^{\prime}=\int_{0}^{a}h_{2}(x,a)\tau_{\theta}(x)dx \tag{16}\]  
\[T^{\prime}=\int_{0}^{a}h_{3}(x,a)\sigma_{\theta}(x)dx-\sigma_{\theta}\bigg{|}_{ r=a}+\sigma_{r}\bigg{|}_{r=a} \tag{17}\]  
where \(h_{1}(x,a)\), \(h_{2}(x,a)\), \(h_{3}(x,a)\), obtained by the boundary configuration method, are the weight function independent of the loading condition. \(\sigma_{r}\), \(\sigma_{\theta}\) and \(\tau_{r\theta}\) are the stress components for the uncracked disk. In addition, the detailed formulas mentioned above can be found in the Refs. [43; 44]. When the crack changes from pure mode I to mixed mode, the stress intensity factors and T-stress of CCBD can be further written as follows:  
\[K_{\text{I}}^{\prime}=\sigma\sqrt{\pi a}\bigg{[}f_{11}^{\prime}+2\sum_{i=1}^{ n}A_{1i}f_{1i}\alpha^{2(i-1)}\bigg{]} \tag{18}\]  
\[K_{\text{II}}^{\prime}=2\sigma\sqrt{\pi a}\sum_{i=1}^{n}A_{2i}f_{2i}\alpha^{2 (i-1)} \tag{19}\]  
\[T^{\prime}=\sigma f_{1}+2\sigma\sum_{i=1}^{n}\left(A_{2i}(\theta)f_{i}-A_{2i}( \theta)-A_{1i}(\theta)\right)\alpha^{2(i-1)} \tag{20}\]  
where \(\sigma\) is the normal stress, where \(\sigma=P/(\pi BR)\). The relative length of crack \(\alpha\) equals \(a/R\). \(f_{ji}\) and \(A_{ji}\) (\(j=1\), \(2\) and \(i=1\), \(2\),..., \(n\)) are the coefficients [1]. However, the situation is much more complicated when the crack changes from pure mode II to compression-shear state so that the crack surface is closed. At this time, with the increase of the loading angle, the boundary of the crack surface may overlap or even cross, which would cause the normal pressure and friction on the crack surface [2]. In addition, when the crack surface is closed, CCBD is equivalent to a crack-free disk. The central crack of the CCBD is assumed to be a linear crack that is in contact under radial forces. The shear stress of crack surface changes due to the friction generated by the contact [45]. Assuming that the friction coefficient is \(\mu\), through the Coulomb friction law, the changed shear stress \({\tau_{r\theta}}^{\prime}\) can be obtained as follows [46]:  
\[{\tau_{r\theta}}^{\prime}=\tau_{r\theta}+\mu\sigma_{\theta} \tag{21}\]  
where shear stress \(\tau_{r\theta}\) should be greater than the friction, otherwise the crack would be in the state of cementation. Meanwhile, the formula for the mode II stress intensity factor considering friction can be further derived from Equation (16) as bellow  
\[K_{\text{II}}^{\prime}=\int_{0}^{a}h_{2}(x,a){\tau_{r\theta}}^{\prime}(x)dx \tag{22}\]  
According to the above analysis, the friction generated by contact will not affect the stress components (\(\sigma_{r}\) and \(\sigma_{\theta}\)) except the shear stress (\(\tau_{r\theta}\)). Therefore, the formulas for mode I stress intensity factor (\(K_{\text{I}}\)) and T-stress (\(T\)), involving the behavior of the friction, are consistent with those without considering the friction. The compression-shear crack surfaces cannot be intercalated due to the consideration of contact. Therefore, the \(K_{I}\) in compression-shear fracture is taken as 0 [17]. Therefore, the mode I SIF concerning the frictional contact can be sorted into the following form  
\[K_{\rm I}=\left\{ \begin{array}{cc}K_{\rm I}^{\prime},&(K_{\rm I}^{\prime}>0)\\ 0,&(K_{\rm I}^{\prime}\leq 0)\end{array} \right. \tag{23}\]  
Then, the dimensionless intensity factors and T-stress can be respectively written as bellow:  
\[K_{\rm I}^{*}=\left\{