## 3 Results and Discussion
### Discussion  
In this study, shear simulations of the c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(1\(\overline{1}\)02) interface models along the \(<\)1\(\overline{1}\)01\(>\) and \(<\)1\(\overline{1}\)02\(>\) directions of the \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) lattice were performed using first-principles calculations. The O-\(<\)1\(\overline{1}\)00\(>\) model has the strongest shear resistance and the Zr-\(<\)1\(\overline{1}\)01\(>\) model is the most prone to slip. It indicates that the mechanical properties of c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\) (1\(\overline{1}\)02) interfaces are affected by the interfacial atomic configuration and shear direction. In the process of the ZrO\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\) interfacial shear deformation, the ZrO\({}_{2}\) atomic layers slide along the shear direction, accompanied by the shifting of Zr and Al atoms near the interface; moreover, shear failure generally occurs in the ZrO\({}_{2}\) atomic layers. Furthermore, the reconstruction of the interface structure has a significant influence on the mechanical strength of the interface region that resists sliding deformation. In the Ni(111)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\) (0001) interface, [PERSON] et al. [22] also observed the similar phenomenon. However, in the Al-terminated O-site interface, they observed that the maximum shear strength was along the Ni \(<\)110\(>\) direction; while the maximum unstable stacking energy was along the Ni \(<\)112\(>\) direction. This an intriguing observation; the reason for this phenomenon is the shifting of Al atoms near the interface during the sliding process. For our models, the shifting of Zr and Al atoms near the interface makes the sliding process more stable. In addition, the shear strength of the Ni/Al\({}_{2}\)O\({}_{3}\) interfaces and the ZrO\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\) interfaces are weak; indicating that shear failure occurs at any interface in TBC. We examined the ZrO\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\) interfaces using first-principle calculations to understand the tensile strength and failure properties [25]. In that work, the ideal tensile strengths and work of separation of the c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\) (1\(\overline{1}\)02) interfaces were obtained. Table 1 summarizes the ideal tensile strength, the work of separation, the ideal shear strength, and the unstable stacking energy. For all the models in Table 1, the O-\(<\)1\(\overline{1}\)0\(>\) model is the most stable model; having the maximum tensile strength and shear strength. The Zr-\(<\)1\(\overline{1}\)01\(>\) model is the most unstable model; it has the lowest tensile strength and shear strength. These properties are closely related to the strength of the bonds at the interface. Compared with the tensile strengths, the shear strengths of the same model are much smaller. It indicates that ZrO\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\) interfaces are more prone to shear failure than tensile failure. This is because of the different failure mechanisms between tensile failure and shear failure. Owing to the tensile failure of the interface, all the atomic bonds across the interface need to be broken; whereas shear failure only needs to be partially broken. Therefore, shear failure requires less stress and energy than tensile failure does. The cohesive zone model (CZM) is a widely accepted model in the elastic-plastic fracture mechanics community. The core task of developing a material-dependent CZM is to extract material parameters, such as the cohesive energy and maximum cohesive strength in the cohesive laws. In the continuum mechanics framework, there are many ways to develop cohesive zone models; in addition, more accurate cohesive zone models can be constructed based on first-principles results [34; 35; 36]. Figure 5 shows the shear stress-displacement curves fitted by the Fourier series; the fitting curve is basically consistent with the exponential cohesive zone model [37] in the inset.