## 4 Results and Discussion
### Crack Initiation and Propagation Concerning Frictional Contact
#### 4.2.2 Crack Propagation Trajectory  
In this section, the dynamic effect would not be taken into account. First, this section compares the fracture trajectories obtained by MFEM and the experimental results [22], as shown in Figure 8. The strong consistency of the two growth paths confirms the effectiveness of the MFEM. Due to the satisfactory accuracy and versatility of the MFEM, proved by the previous results, the loading angle \(\beta\) takes the values of \(30^{\circ}\) and \(45^{\circ}\)for the sake of convenience. Except for the loading angle, the settings of other parameters are consistent with Section 4.2.2. Figures 9 and 10 clearly illustrate the propagation trajectories, simulated by the MFEM, versus loading angle \(\beta\) with different friction coefficients (\(\mu=0\), \(0.3\) and \(0.6\)) and crack relative lengths (\(\alpha=0.3\) and \(0.5\)). From the results of the above figures, the friction has a certain effect on propagation of the crack. Meanwhile, in order to facilitate observation and comparison, crack propagation paths of the same crack length are overlapped. It can be found that, in spite of the setting of the contact, when the friction coefficient is \(0\), the crack would eventually deviate from the original crack line and extend towards the loading direction. Under this condition, the increase of relative crack length and loading angle will increase the deviation of the crack, which could not change the type of crack. According to the previous research [8], it can be found that this kind of crack is a tensile wing crack. However, as the friction increases, the crack propagation is significantly affected. Figure 7: The crack propagation angle \(\theta_{0}\) versus the loading angle \(\beta\) with different relative crack lengths \(\alpha\). (**a**) \(\alpha=0.3\), (**b**) \(\alpha=0.5\). Figure 8: Comparison of simulation results and experimental results for crack paths in CCBD specimens for \(\alpha=0.3\). (**a**) \(\beta=0^{\circ}\), (**b**) \(15^{\circ}\), (**c**) \(30^{\circ}\), (**d**) \(45^{\circ}\), (**e**) \(60^{\circ}\). Figure 10: The propagation trajectories simulated by the MFEM under different friction coefficients for \(\alpha=0.5\). Figure 9: The propagation trajectories simulated by the MFEM under different friction coefficients for \(\alpha=0.3\). Considering the symmetry of the CCBD, the propagation path extending from the right tip of the central crack is taken as the research object. In addition, the effects of different friction coefficients on growth trajectories of the crack are shown in the Figures 9 and 10. The crack propagation path is greatly affected, which is manifested that the crack propagation path deviate upward. At the same time, as increasing the friction coefficient, the amount of the upward deviation increases and the curvature of the path decreases. In addition, there are some influences on crack type due to the change of friction coefficient. It can be found that when the loading angle \(\beta\) is equal to \(30^{\circ}\), no matter what the friction coefficient is, the crack propagation path of the CCBD sample does not change significantly, which makes only the tensile wing crack occur. However, as \(\beta=60^{\circ}\), tensile crack or mixed tensile-shear crack would begin to occur. For example, when \(\beta=45^{\circ}\) and \(\mu=0.3\), the crack propagation path of the crack is like a straight line, without bending and extends in the direction of loading, which is called the tensile crack. Similarly, under the condition of this loading angle, mixed tensile-shear crack can be observed when \(\mu=0.6\) and \(\alpha=0.5\). According to the previous literature [50], when the values of \(\alpha\) and \(\beta\) for the CCBD are small, the effect of friction is not obvious due to the incomplete contact of the crack surface. In addition, these conclusions can be embodied in the Figures 9 and 10.