## 1 Introduction  
The thermal barrier coating (TBC) is a high-temperature protection technology widely used in gas turbine engines to protect hot components, such as combustors, nozzles, vanes, and turbine blades [1; 2; 3; 4; 5; 6; 7]. The TBC enables the engines to achieve greater efficiency by insulating the engine components from prolonged thermal loads. Typical TBC systems include a nickel-based superalloy substrate, a NiCrAlY or NiCoCrAlY bond coat (BC), a thermally grown oxide (TGO), and a Yttria-stabilized zirconia (YSZ) ceramic top coat (TC) [8]. The function of the BC is to prevent oxidation corrosion of the substrate and alleviate thermal expansion mismatch (TEM). The TC is the heat insulation layer between the hot gas and substrate. Since the TBC is usually operated in harsh environments, premature failure is inevitable. Therefore, understanding the failure mechanism of the TBC is crucial to help us improve and optimize the TBC system to meet engineering requirements. During the service of the TBC, the growth of TGO [9; 10], phase transformation [11; 12], infiltration of calcium-magnetism-alumina-silicate (CMAS) [13; 14], residual stress [15; 16], and high-temperature sintering [17; 18] affect the life of the TBC and eventually lead to the TBC's spallation and delamination. Among them, the growth of TGO is the key factor. With the increase of TGO's thickness, the thermal expansion mismatch (TEM) occurs between the TC/TGO/BC and eventually leads to a TBC fracture at the TGO/BC and TC/TGO interfaces [19]. Therefore, it is of great significance to investigate the mechanical properties and failure mechanisms of these two interfaces. It is still a challenge to obtain the failure mechanism of the interface directly by experiments due to the lack of experimental conditions. Instead, first-principle calculations can reveal the basic physical properties of interface structures, such as electronic, atomic structure, and mechanical properties, and have been applied to study the failure mechanism of interfaces [20; 21; 22; 23; 24]. [PERSON] et al. [21] found that thetensile strength of the Al-terminated Cu/Al\({}_{2}\)O\({}_{3}\) is weaker than the O-terminated interface, and the mechanical properties of the Al-terminated interface are mainly dominated by the Cu-Al electronic hybridization interactions. [PERSON] et al. [22] concluded that the tensile strength and failure mechanism of the Ni/Al\({}_{2}\)O\({}_{3}\) interface were strongly influenced by the termination of \(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(0001). The failure of the Al-terminated O-site interface occurred at the interface under lower stress, while the failure of the O-terminated Al-site interface occurred within the Ni layer. [PERSON] et al. [23] found that the ideal tensile strength and adsorption work of the O-terminated \(\gamma\)-Fe/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\) interface are 3.5 times and 5 times those of the Al-terminated interface, respectively. The fracture mode of the Al-terminated interface tends to be a brittle fracture, while the O-terminated interface is more like a ductile fracture. [PERSON] et al. [24] found that Ti-terminated TiSiC\({}_{2}\) and O-terminated Al\({}_{2}\)O\({}_{3}\) are the most stable of their respective crystals. The higher work of adhesion of the TiSiC\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\) interface is due to the powerful attraction between the Ti and O atoms on the interface. Our research group has examined the mechanical strength and fracture characteristics of the Ni/Al\({}_{2}\)O\({}_{3}\) interface under tensile, shear, and mixed mode loadings by first-principle calculations [25; 26; 27; 28]. In general, the first-principle tensile simulation is an accurate and effective method to investigate the tensile strength and failure mechanism of the interface. However, few studies have considered the TC/TGO interface. [PERSON] et al. [29] concluded that the adhesion of the ZrO\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\) interfaces is sensitive to the atomic layer's thickness.