## 4 Results and Discussion
### SIFs and T-Stress Concerning Frictional Contact  
The purpose of this section is to study the effect of friction on stress intensity factors and T-stresses at the onset of fracture. Figure 6 clearly shows the dimensionless \(K\) and \(T\), calculated by the WFM and MFEM, versus the loading angle with different friction  
\begin{table}
\begin{tabular}{c c c} \hline  & **Categories** & **Values** \\ \hline \multirow{2}{*}{Geometry characters} & \(R\) (mm) & 50 \\  & \(B\) (mm) & 30 \\ \hline \multirow{3}{*}{Material properties} & Compressive strength \(\sigma_{c}\) & 28 \\  & (MPa) & 3.81 \\ \hline \multirow{3}{*}{Material properties} & Poisson’s ratio \(v\) & 0.21 \\  & Young’s modulus \(E\) (GPa) & 15 \\ \hline \end{tabular}
\end{table}
Table 1: CCBD model parameters for simulation.  
Figure 4: Process of the simulation [19].  
Figure 5: Numerical model of a central crack Brazilian disc.  
coefficients (\(\mu\) = 0, 0.2, 0.4, 0.6 and 0.8) and relative lengths of the crack (\(\alpha\) = 0.3, 0.5 and 0.7). By comparison, the results between MFEM and WFM have strong consistency with a maximum error within 3%, which shows that the proposed MFEM is reliable.  
Figure 6: The normalized SIFs and T-stresses for various friction coefficients. (**a**) \(\mu\) = 0, (**b**) 0.2, (**c**) 0.4, (**d**) 0.6, (**e**) 0.8.  
First, some variation rules of the stress intensity factors and T-stress with different loading angles and relative crack lengths can be observed, which is similar to the research conclusions of some scholars [14; 16; 20; 25]. Meanwhile, when the crack is closed, the setting the frictional contact makes the normalized mode I SIF \(K_{1}\)* = 0, while the \(K_{1}\)* and T* remain the same trends no matter what the friction coefficient is. These results indicate that the SIF of mode I and T-stress are indeed independent of friction, which is consistent with the conclusions predicted by WFM.  
However, it can be seen from the Figure 6, the friction has a significant influence on the mode II stress intensity factor when the crack is closed. In addition, as increasing the friction coefficient \(\mu\), the \(K_{\rm II}\)* decreases obviously when relative length of crack \(\alpha\) and loading angle \(\beta\) have the certain values. For example, with the friction coefficient increases from 0 to 0.8, for the loading angle of \(60^{\circ}\), the \(K_{\rm II}\)* decreases 120.7% when \(\alpha\) = 0.5. In addition, with the larger \(\alpha\) and \(\beta\), the effect of friction is more significant as the amount of increase in friction coefficient is constant. For example, as the friction coefficient increases from 0 to 0.6, the \(K_{\rm II}\)* decreases 14.6% when \(\alpha\) = 0.3 and \(\beta\) = \(40^{\circ}\), while the \(K_{\rm II}\)* decreases 58.5% for \(\alpha\) = 0.5 and \(\beta\) = \(50^{\circ}\). It is well known that the crack is in a viscous state when the \(K_{\rm II}\)* is less than 0 [50]. Figure 6 also shows that when \(\mu\) is equal to 0 (the crack surface is smooth enough), the crack would not be in a sticky state from beginning to end with the loading angle increases from 0 to \(90^{\circ}\). However, with a larger friction coefficient, the loading angle corresponding to \(K_{\rm II}\)* = 0 decreases gradually, which indicates that the crack is in a viscous state earlier under the condition. At the same time, the larger loading angle and relative crack length will make this phenomenon more obvious.  
In general, the foregoing results show that the friction has a significant effect on the mode II SIF after the crack is closed, which indicates that it is necessary to consider influence of the friction in the real contact state, but has no influence on the stress intensity factor of mode I and T-stress.