**2. The origin and category of stress among the APS-prepared TBCs**

TBCs experiences complicated loads during the preparation process and service. Then, stress state in TBCs dynamically changes and influences the service properties and behaviors of TBCs.

#### **2.1 The origin and category of stress during preparation process**

In APS process, the molten ceramic feedstock impinge onto the surface of metal specimen, stack and form TBCs [12]. The coatings may experience violent temperature change in the process. The mismatch of thermal expansion coefficients between ceramics layer and superalloy may produce stress [13]. According to the mechanism of stress generation, the stress generating in preparation process mainly includes quenching stress and thermal mismatch stress, as shown in **Figure 3**.

The quenching stress is generated in the interaction process between molten ceramic feedstock and metal specimen. In this process, the feedstock after impact may form a lamellar structure and rapidly cool from the molten temperature to the specimen temperature in a short time. The lamellar structure shrinks sharply due to rapid cooling. The shrinkage of the newly formed lamellar structure is constrained by the specimen and the previous lamellar structure, resulting in stress in the coating [14]. The constraint of the specimen and the previous lamellar structure depends

**Figure 3.** *The stresses during preparation process of APS-prepared TBCs.*

mainly on the material of the lamellar structure, the temperature of the specimen, and the boundary shapes between the lamellar structures [15]. The existence of quenching stress has been proven by McPherson, and the value has been measured by Kuroda et al. It is found that the quenching stress is always tensile stress [16, 17]. Assuming that the shrinkage deformation of the lamellar structure is fully constrained, the resulting stress will be large, and much higher than the yield strength or fracture strength of the material. The maximum value of quenching stress given by the elastic mechanics analysis is about 1 GPa, while the values obtained from the experiments are mostly below 100 MPa. This is mainly due to the existence of many stress release mechanisms during the preparation of the coating. For example, a large number of microcracks are generated inside and at the boundary of the lamellar structure during the preparation process. These microcracks can act as important means of stress release, reducing stress below the fracture strength [13, 18].

The thermal mismatch stress generates during the cooling process after spraying. The thermal mismatch stress results from the mismatch of thermal expansion coefficients between the ceramic coating and the metal specimen [9, 19]. In the elastic range, the thermal mismatch stress can be estimated by:

$$
\Delta \mathbf{e}\_{th} = \int\_{T1}^{T1} (\mathbf{a}\_1 \langle T \rangle - \mathbf{a}\_2 \langle T \rangle) dT \tag{1}
$$

$$
\sigma\_{th}(T) = \Delta \varepsilon\_{th} E(T) \tag{2}
$$

where, Δ*α* (*T*) is the difference in thermal expansion coefficients between ceramic coating and metal material, and changes with temperature. *T*1 is the spray temperature. *T*2 is the certain temperature during the cooling process. *σ*th (*T*) is the thermal mismatch stress of coatings. *E* (*T*) is the elastic modulus of coating material. According to Eqs. (1) and (2), the thermal mismatch stress is related to the difference in thermal expansion coefficients, the spray temperature, and the elastic modulus of the coating material. The thermal mismatch stress increases with increasing the difference between spray and room temperature, the difference in the thermal expansion coefficients and the elastic modulus of the coating material [20]. Besides, the thermal mismatch stress may be tensile or compressive stress [21].
