## 4 Discussion
In their work, the peak stress of the Al-terminated O-site model is 9.915 GPa at the strain of 11% under uniaxial tension, and the work of adhesion is 1.48 J/m\({}^{2}\). For the O-terminated Al-site model, the peak stress is 23.384 GPa at the strain of 14%, and the work of adhesion is 4.17 J/m\({}^{2}\). Then, the failure position of the Al-O model occurs at the interface, while the failure of the O-Al model occurs within the Ni layer. Their results indicate that the Al-O model is weakly adhered, while the O-Al model has much higher strength. Compared to their results, the bonding strength of the ZrO\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\) interfaces is weaker than that of the Ni/Al\({}_{2}\)O\({}_{3}\) interface. This indicates that the TBC is more prone to failure at the ZrO\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\) interfaces. However, considering the weak adhesion of the Al-O model, the failure of the TBC system may occur at any interface. The comprehensive consideration of the mechanical properties of the two interfaces will help us to understand the details of the failure process of the TBC and provide data for the optimization and improvement of the TBC system. In molecular dynamic simulations, interatomic interactions are described by atomic potential functions. However, the empirical potentials used often cannot accurately describe the interactions between dissimilar atoms, such as the Zr-Al, Al-O, and Zr-O bonds in this paper. [PERSON] et al. [40] creatively used atomic models to estimate the tension-shear coupling. According to their theory, the potential function is defined as:  
\[\Psi(\Delta_{r},\Delta_{\theta})=W_{sep}\bigg{\{}1-[1+(\Delta_{\theta}/L)]exp \bigg{(}(-\Delta_{\theta}/L)+\sin^{2}(\pi\Delta_{r}/b)\bigg{[}q+\bigg{(}\frac{ q-p}{1-p}\bigg{)}(\Delta_{\theta}/L)\bigg{]}exp(-\Delta_{\theta}/L)\bigg{)} \bigg{\}} \tag{1}\]Here, \(\Delta_{r}\) and \(\Delta_{\theta}\) are the shear displacement and tensile displacement, respectively. \(W_{spp}\) is the work of separation, \(L\) is the displacement value corresponding to the maximum tensile stress, \(b\) is the Burgers vector, and \(p\) and \(q\) are important parameters for measuring tension-shear coupling. For the pure tensile separation, the potential function can be simplified as [41]:  
\[\Psi(\Delta_{r}=0,\Delta_{\theta})=W_{sep}\{1-[1+(\Delta_{\theta}/L)]exp(- \Delta_{\theta}/L)\} \tag{2}\]  
Furthermore, the tensile stress is derived from Equation (2) as:  
\[\sigma(\Delta_{\theta})=\frac{\partial\Psi}{\partial\theta}=\frac{W_{sep}}{L }(\Delta_{\theta}/L)exp(-\Delta_{\theta}/L) \tag{3}\]  
In this work, the work of separation, \(W_{spp}\), and the theoretical tensile strength for the three interface terminations were calculated, and the value of the \(L\) can be obtained by fitting it to our first-principle results. Figure 5 shows the potential energy-displacement curve of the three terminated c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(1\(\overline{1}\)02) interfaces. Although the analytical curves do not exactly match the first-principle results, it is quite satisfactory in terms of the curve trend and maximum potential value. To further check the applicability of the above potential function, the tensile stress-displacement analytical curves of the three interface models based on Equation (3) are shown in Figure 6, which resemble, very much, the first-principle calculation results in shape. In addition, the analytical curves are in good agreement with the first-principle calculation results, both in terms of peak stress and curve area (\(W_{sp}\)), although there is a subtle difference between the displacements corresponding to the peak stress. This may be because the analytic expression does not take into account the relaxation effects. The reproduction of the peak stress and \(W_{sp}\) proves the accuracy of this potential function. It is thus believed that the potential function of c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(1\(\overline{1}\)02) constructed from our first-principle results is a reliable one.