## 3 Results and Discussion  
### Shear Strength Parameters  
Figure 2 shows the shear stress-shear displacement curves for the c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(1\(\overline{1}\)02) interface. The peak shear stresses are 5.04 and 5.69 GPa at the critical shear displacements; and \(\delta_{c}\), 1.43 A, and 1.85 A along the \(<\)1\(\overline{1}\)01\(>\) and \(<\)1\(\overline{1}\)\(\overline{2}\)0\(>\) slip directions of the O- model (see Figure 2a,b), respectively. In addition, for the 2O- model, the peak shear stresses along the \(<\)1\(\overline{1}\)01\(>\) and \(<\)1\(\overline{1}\)\(\overline{2}\)0\(>\) directions are 3.66 and 4.52 GPa at 1.43 and 2.24 A (see Figure 2c,d), respectively. Furthermore, for the Zr- model shown in Figure 2e,f, the peak shear stresses for the Zr- model are 2.94 and 3.95 GPa at 1.23 and 2.24 A in the \(<\)1\(\overline{1}\)01\(>\) and \(<\)1\(\overline{1}\)\(\overline{2}\)0\(>\) directions, respectively. Compared with the 2O- and Zr- shear deformation models, the peak shear stress in the O- model is greater in the same slip direction. This implies that the termination of c-ZrO\({}_{2}\)(001) has a significant influence on the adhesive strength of the interface. In addition, the peak shear stress in the \(<\)1\(\overline{1}\)\(\overline{2}\)0\(>\) direction is higher than that in the \(<\)1\(\overline{1}\)01\(>\) direction for all three models. This indicates that the \(<\)1\(\overline{1}\)00\(>\) direction has a stronger shear resistance than the \(<\)1\(\overline{1}\)01\(>\) direction in the interface slip process. In Figure 2a-f, the maximum shear stress values in region II are 3.13, 5.11, 2.65, 2.98, 2.51, and 2.59 GPa, respectively. The maximum shear stress value in region I is larger than that in region II in the same model. The shear moduli of the \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) single crystals obtained from the calculations and experiments are 168 GPa and 163 GPa, respectively [28,29]. The shear moduli of the c-ZrO\({}_{2}\) single crystals obtained by calculation and experiment are 106 and 81 GPa, respectively [30,31]. The maximum theoretical shear modulus of the supercell is estimated to be 38.7 GPa. Compared with the shear moduli of the \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) and c-ZrO\({}_{2}\) single crystals, this value is much smaller. These results indicate that the c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(1\(\overline{1}\)02) interfaces are weaker than those of bulk \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) and c-ZrO\({}_{2}\). Furthermore, this implies that the c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(1\(\overline{1}\)02) interfaces are not simply complements of bulk \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) and c-ZrO\({}_{2}\). The energy per unit area of the slip plane \(\Phi\) is defined as \(\Phi=\int\tau d\delta\); \(\tau\) and \(\delta\) are the shear stress and displacement, respectively. The maximum value of \(\Phi\) is termed the unstable stacking energy \(\gamma_{us}\)[32], which is the area between \(\delta\) = 0 and the first \(\delta=\delta_{t}\) at which \(\tau\) = 0 again. It is an important parameter for characterizing the interfacial shear strength. The mechanical properties of ZrO\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\) interfaces can be more comprehensively understood using the ideal shear strength combined with the unstable stacking energy. As shown in Figure 3, the unstable stacking energy \(\gamma_{us}\) of the O- model is larger than that of the other two models in the same slip direction; whereas the unstable stacking energy \(\gamma_{us}\) in the  
Figure 1: Schematic diagram of the c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(1\(\overline{1}\)02) interfacial shear simulation. The red, pink, and blue spheres represent O, Al, and Zr atoms, respectively. To better demonstrate the system, the model has been duplicated in the horizontal direction. <11\(\overline{2}\)0> direction is higher than that in the <1\(\overline{1}\)01> direction for all three models.