## 2 Methodology and Model
### Fracture Characteristic  
By fully relaxing the entire slab, the bond lengths and interface distance of the three terminated interfaces were obtained, and the results are collected in Table 1.  
Figure 2: Stress–strain curves of the three terminated c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(1\(\overline{1}\)02) interfaces. The letter A represents the area under the stress–strain curve. The bond lengths in the interface region for the three terminated interfaces listed in Table 1 are divided according to whether the bond is parallel to or across the interface. The ranges of the bond lengths are observed parallel to the interfaces. The ranges on the Al\({}_{2}\)O\({}_{3}\) side are mostly limited by bulk values (1.856 and 1.969 A), while the ranges on the ZrO\({}_{2}\) side are shorter than the bulk values. The results show that no obvious tendency is found in the variation in the stacking sequence of ZrO\({}_{2}\). The Al cations near the interfaces tend to bond with the O anions in c-ZrO\({}_{2}\) (except the Zr-terminated interface), and the O anions in \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) near the interfaces tend to bond with the Zr cations. Moreover, the cation-cation interaction (Al-Zr) is found in the O-terminated and Zr-terminated interfaces, adding stability to the structure. In order to better understand the failure process of the c-ZrO\({}_{2}\)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\) interfaces under the tensile loading, we investigated the details of the interfacial atomic bonds. Figure 3 shows the interfacial bond length differentials in the interface region during the tension loading process, and the evolution of the interfacial atomic bonds during the tension process is plotted in Figure 4.  
For the O-terminated interface shown in Figure 3a, the bond lengths of the Al-O bond parallel to the interface are almost unchanged, while the atomic bond lengths of the other bonds increase gradually. The bond lengths of the Zr-O bond parallel to the interface begin to shrink when the strain exceeds 8%, and the bond lengths of the Al-O, Zr-O, and Zr-Al bonds across the interface increase rapidly. When reaching the critical strain of 9%, the atomic bond breaking occurs at the Al-O, Zr-O, and Zr-Al bonds across the interface break. Combined with Figure 2, the stress gradually decreases after the critical strain of 9%, indicating that the failure mode of the interface is similar to the ductile fracture. The Zr-terminated interface has a similar fracture process, as shown in Figure 3c, and the atomic bond breaking takes place at the Al-O, Zr-O, and Zr-Al bonds across the interface. While for the 2O-terminated interface shown in Figure 3b, when exceeding the critical strain of 6%, the bond lengths of the Al-O and Zr-Al bonds across the interface increase rapidly  
Figure 3: Interfacial bond length differentials for c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(1\(\bar{1}\)02) interfaces during the stretching process. (**a**) O-terminated, (**b**) 2O-terminated, and (**c**) Zr-terminated. The vertical line indicates the critical strain. Pictorial showing the interfacial bonds is shown in the inset panel. The green, grey, and red spheres represent Zr, Al, and O atoms, respectively. and the stresses drop abruptly (Figure 2), demonstrating that the brittle fracture occurs at this interface. Figure 4 shows the evolution of the atomic bonds for the c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(1\(\overline{1}\)02) interfaces under tensile loading. For the O-terminated interface, as seen in Figure 4a, the weaker Zr-Al bonds break at first when the strain reaches 2%. With the increase of the tensile load, the Zr-O bond across the interface breaks. After that, one of the Al-O bonds breaks with the increasing tensile load. Finally, the other Al-O bonds across the ZrO\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\) interfaces also break when the strain reaches the critical strain of 9%.