## 4 Results and Discussion
### The Influence of Initial Crack Length on Crack Propagation  
The impact of the initial crack length on the crack growth behavior is examined in this section. The crack extension during flame thermal shock loading for a prefabricated crack length of 2 mm is displayed in Figure 5. The graph demonstrates that the trend of crack extension continues to extend along the upper edge of the applied flame thermal shock at prefabricated crack angles of 0\({}^{\circ}\), 26.6\({}^{\circ}\), and 45\({}^{\circ}\), which is the same as that in Section 4.1. However, the crack propagation direction changes when the crack angle is between 63.4\({}^{\circ}\) and 90\({}^{\circ}\). In this section, the differences in the crack propagation rate, stress distribution, and temperature gradient caused by variations in the length of pre-existing cracks are explored.  
Crack propagation, x-direction stress distribution, and temperature distribution diagrams for two different lengths of pre-existing cracks (1 mm and 2 mm) are displayed in  
Figure 4: The initial crack angle of 0\({}^{\circ}\): (**a1–a3**) phase-field evolution; (**b1–b3**) x-direction stress (MPa); (**c1–c3**) temperature distribution (\({}^{\circ}\)C).  
Figure 5: Crack propagation diagram under the influence of different prefabricated crack lengths: (a) 1 mm; (b) 2 mm.  
Figure 6a, b and c, respectively, to visually demonstrate the influence of different lengths of pre-existing cracks on crack propagation under flame thermal shock loading.  
As exhibited in Figure 6, the crack growth rate rises with the crack length, particularly close to the upper tip of the flame thermal shock load. A temperature gradient develops between the upper edge and the interior of the ceramic under the impact of thermal shock loading. When compared to the crack length of 1 mm, the longer crack length is more prone to being affected by the temperature gradient earlier. Therefore, the extension rate of the upper tip of the crack is greatly increased. However, the temperature gradient within the ceramic tends to stay stable as long as the flame thermal shock load is maintained. The crack propagation rate slows down in the latter two stages, which is consistent with the findings in Section 4.1. Nevertheless, the propagation rate of the lower tip decreases as the length of the prefabricated crack at the lower tip of the crack increases. The temperature gradient change is not obvious because the distance between the lower tip of the crack and the upper surface of the flame thermal shock load rises. It can be concluded from the comparative analysis that the upper crack tip is more likely to initiate a crack than the lower crack tip. Additionally, the maximum growth pace of the upper crack tip increases with the distance from the heated surface. On the contrary, the maximum propagation speed of the lower crack tip decreases with this distance.  
The x-direction stress and temperature distribution for different lengths of pre-cracks are illustrated in Figures 7 and 8. The variation in crack length does not cause significant temperature changes, except for slight differences around the crack tip. Overall, the length of the crack has little effect on the temperature distribution. Similarly, the stress distribution seems to be mostly constant over the length of the crack, with only slight differences near the crack tip.  
Figure 6: Crack propagation diagram for different prefabricated crack lengths at different time: (**a**) 0.21 s, (**b**) 0.34 s and (**c**) 1 s.