## 4 Results and Discussion  
### The Influence of Initial Crack Angle on Crack Propagation  
Figure 3 presents the simulation results of ceramic specimens with various pre-existing crack angles (\(0^{\circ}\), \(26.6^{\circ}\), \(45^{\circ}\), \(63.4^{\circ}\), and \(90^{\circ}\)) under flame thermal shock. It is clear that the initiation and propagation of initial cracks at five different angles exhibit remarkably similar processes under flame thermal shock. The x-direction stress distribution in the ceramic plate is impacted by the change in the temperature gradient when the upper edge of the ceramic plate experiences flame thermal shock. The ends of the cracks develop into a wing-shaped pattern, and the cracks grow outward along the width direction of the sample. As shown in Figure 3, the simulation results agree well with the experimental results in reference [49]. It is indicated that the proposed phase-field model can simulate the crack patten under hot thermal shock. And it is interesting to remark that there is a difference between the hot thermal shock and the cold shock [47; 48]. When the ceramic is subjected to cold shock, the cracks will initiate form the cold surface, and expand inward. However,  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{E}\) & \(\mathbf{v}\) & \(\mathbf{f_{t}}\) & \(\mathbf{G_{f}}\) & \(\mathbf{T_{0}}\) & \(\mathbf{q_{0}}\) \\ \hline
370 GPa & 0.22 & 30 MPa & 22.47 N/m & 20 \({}^{\circ}\)C & \(2,000,000\left(\mathrm{W/m^{2}}\right)\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material parameters [49].  
Figure 2: (**a**) Geometry model and boundary conditions of ceramic plate, (**b**) finite element mesh.  
the cracks tend to propagate from the inside under hot shock. This difference is due to the stress field in the ceramic specimen. It should be noted that the change in the prefabricated crack angle will have some slight implications on the distribution of stress, while the effect on the temperature gradient is insignificant.  
The crack propagation process in different periods is explored by taking an initial crack angle of 0\({}^{\circ}\) as an example. Figure 4 shows the phase-field evolution stages, the x-direction stress distribution, and the temperature gradient changes of the ceramic plate. During the period of 0-0.21 s, the temperature near the heating surface rises rapidly due to the loading of the upper edge due to the thermal shock of the flame, generating an obvious temperature gradient. As a result, the crack forms, and the stress near the crack tip increases significantly, driving the crack to propagate towards the upper edge. In the period of 0.21-0.34 s, the flame thermal shock load remains loaded on the ceramic plate, causing a larger temperature gradient and a stronger driving force for crack propagation. The crack propagation rate rises dramatically at this stage because of the influence of the temperature gradient. Although the top edge of the ceramic plate is still continuously subjected to flame thermal shock during the period of 0.34-1.0 s, the temperature gradient tends to be stable throughout this period. Therefore, the crack propagation rate exhibits a trend of initially increasing, and then, progressively decreasing.  
Figure 3: Evolution of thermal shock cracks in prefabricated cracked ceramic plates from five angles. (**a**) Damage; (**b**) x-direction stress distribution (MPa); (**c**) temperature distribution (\({}^{\circ}\)C); and (**d**) the results of reference [49].