Crack Initiation Criteria in EBC under Thermal Stress

For design of multi-layered environmental barrier coatings (EBCs), it is essential to assure mechanical reliability against interface crack initiation and propagation induced by thermal stress owing to a misfit of the coefficients of thermal expansion between the coating layers and SiC/SiC substrate. We conducted finite element method (FEM) analyses to evaluate energy release rate (ERR) for interface cracks and performed experiment to obtain interface fracture toughness to assess mechanical reliability of an EBC with a function of thermal barrier (T/EBC; SiC/SiAlON/mullite/Yb-silicate gradient composition layer/Yb2SiO5 with porous segment structure) on an SiC/SiC substrate under thermal stress due to cooling in fabrication process. Our FEM analysis revealed that a thinner SiAlON layer and a thicker mullite layer are most suitable to reduce ERRs for crack initiation at the SiC/SiAlON, SiAlON/mullite and mullite/Yb2Si2O7 interfaces. Interface fracture tests of the T/EBC with layer thicknesses within the proposed range exhibited fracture at the SiC/SiAlON and SiAlON/mullite interfaces. We also estimated the approximate fracture toughness for the SiC/SiAlON and SiAlON/mullite interfaces and lower limit of fracture toughness for the mullite/Yb2Si2O7 interface. Comparison between ERR and fracture toughness indicates that the fabricated T/EBC possesses sufficient mechanical reliability against interface crack initiation and propagation.


Introduction
Silicon carbide (SiC) fiber reinforced SiC matrix (SiC/SiC) composites are one-third lighter and have approximate 100-200 K higher heat resistance than current heat-resistant super alloys (Nickel-based super alloys) [1][2][3]. Application of SiC/SiC to advanced hot-section components in airplane engines is expected for improving fuel consumption and curbing emission of carbon dioxide [2,3]. SiC/SiC composites react with oxygen to form silica, which hinders degradation of the composites at high temperature over 1373-1473 K [4,5]. In the condition of high temperature and humidity (i.e., combustion gas environment), however, the silica reacts with water vapor to form silicic acid (Si(OH 4 )) gas [4,5], leading to wall thinning of the composites. Thus, application of the SiC/SiC composites to the hot-section components inevitably requires environmental barrier coatings (EBCs).
To achieve superior environmental shielding performance and thermomechanical durability, EBCs consist of several layers with different shielding functions [6,7]. For using EBCs at about 1673 K, which is the durable temperature of the heat-resistant SiC fiber, it is of immense importance to select a material thermal stress that occurs by the difference in CTEs of the coating layers and the substrate, the Ybsilicate dense layer is formed with gradient composition to have YbDS and YbMS on the substrate and top sides, respectively. Hereafter it is referred to as the Yb-silicate gradient composition layer. The topmost YbMS layer with a porous segment structure (topcoat) is employed for thermal shock resistance so as to relax strain due to rapid temperature change in the thickness direction associated with start and stop of the engine.  In general, fracture of brittle materials follows the Griffith theory; i.e., a crack initiates or propagates when the energy release rate (ERR) exceeds fracture toughness for the corresponding fracture mode. While fracture toughness is a material property that should be evaluated in experiments, ERR can be calculated by numerical simulations through, for example, finite element method (FEM) calculations. Unlike conventional TBCs having simple structures, the mechanical behavior of T/EBC is expected to be complicated due to its complex structure consisting of multiple layers on a substrate. Numerical simulation can be an efficient tool to evaluate ERR even in such a complex structure.
In designing T/EBC structure with sufficient mechanical reliability, we must optimize a wide range of parameters, including conditions of multi-layer deposition processes and structural dimensions such as layer thicknesses. Layer thicknesses should affect much the mechanical state and play a crucial role in the mechanical stability of T/EBC under thermomechanical loading, which the component should undergo in operation. It is of importance to understand how such structural parameters influence the mechanical reliability. The aim of this study is to propose the condition of layer thicknesses for preventing interface crack initiation and propagation during the cooling process in the fabrication of the T/EBC. We perform FEM analysis for a T/EBC model with varying thicknesses of the coating layers to investigate the dependence of the layer thicknesses on ERR of crack initiation at interfaces, where thermal stress concentrates significantly. The analysis result facilitates the determination of layer thicknesses to prevent interface crack initiation. Then, the T/EBC is fabricated within the proposed thicknesses of the coating layers. We calculate ERR for interface crack initiation and propagation by conducting FEM analysis to a model of the fabricated T/EBC. Comparing the ERRs and fracture toughness obtained by interface fracture tests, we assess the validity of the proposed layer thicknesses.

Fabrication Procedure of T/EBC
In this study, the substrate was SiC/SiC (50 mm square and 3 mm in thickness) whose matrix was formed by chemical vapor infiltration and subsequent melt infiltration. Before T/EBC deposition, a SiC layer with an adequate thickness was deposited by chemical vapor deposition in order to smoothen the surface to be coated.
Each constituent layer of T/EBC shown in Figure 1 is a complex oxide or an oxynitride containing multiple cations. When these compounds are heated to a high temperature, the composition of vapor containing cation differs substantially from that of compounds in solid phase (incongruent vaporization). Thus, deposition methods accompanying the melting process of raw materials, such as plasma spraying, bring about the incongruent vaporization of the materials during deposition, resulting in significantly deviated compositions of coating layers.
In this study, therefore, dual electron beam-physical vapor deposition (EB-PVD) was employed to prepare layers of compounds which were evaporated incongruently. Two target raw materials located in the deposition chamber were evaporated using two electron guns regulated individually, which enabled independent control of vapor pressures of gases generated from the targets. Compound layers with arbitrary compositions were deposited on a substrate placed at a proper distance from the targets. During deposition, small amount of source gas was introduced in the coating chamber, and the coated surface was heated up to about 1473 K by irradiation of a direct diode laser (wavelength of 915 nm) to promote full crystallization and densification of each layer. After the deposition finished, the specimen was cooled from the deposition temperature to room temperature at the average cooling rate of 20 K/min. Table 1 lists the target materials (Ingots A and B) and source gases used in the deposition of the layers. During deposition of the Yb-silicate gradient composition layer, the ratio of two electron beam powers irradiated to two targets was gradually changed with coating time to control the composition toward the thickness direction. The YbMS topcoat was deposited on a rotating sample for out-of-surface to form the porous segmented structure owing to the shadowing effect. The cross-sectional microstructure of T/EBCs was observed using a scanning electron microscope (SEM; SU8000 SEM, HITACHI, Tokyo, Japan) with energy dispersive spectroscopy (EDS).

ERR Calculation by FEM Analysis
For ensuring the mechanical reliability of T/EBC in the fabrication procedure, we need to determine its layer thicknesses with which ERR for interface crack initiation due to the thermal stress, G th init , can be reduced so that interface crack is not likely to initiate. In general, it is difficult to fabricate T/EBC without any initial interface crack such as microcrack. Thus, under the proposed layer thicknesses for preventing interface crack initiation, evaluating the behavior of interface crack propagation is also required. This means that we need to calculate ERR for interface crack propagation due to the thermal stress, G th prop , and compare with it and interface fracture toughness. In this study, in order to investigate the effect of layer thicknesses on G th init , we conducted two-step FEM analyses using a T/EBC model with varying layer thickness; (1) Thermal stress analysis: FEM analysis for evaluating the thermal stress state after the cooling process and (2) Interface crack introduction analysis: FEM analysis for calculating ERR by introducing interface crack in the thermal stress state that was obtained in Step (1). Then, for obtaining G th prop , we conducted the two-step FEM analyses to the T/EBC model with the optimized layer thickness to prevent interface crack initiation. ABAQUS (ver.6.14.6, Dassault Systemes, Vélizy-Villacoublay, France) was used in the calculations.

FEM Model of T/EBC
The state of thermal stress in T/EBC is affected by the thicknesses of the coating layers. In particular, the difference in CTEs of the SiAlON and mullite layers is large compared to those at other interfaces (CTEs are explained in Section 2.2.2), and thus the thicknesses of the SiAlON and mullite layers are expected to significantly affect the state of thermal stress after cooling. We examined the simulation models mimicking T/EBC with various thicknesses of the SiAlON and mullite layers as shown in Figure 2. Note that the FEM model used in this analysis is a model of the vicinity of interface edge of T/EBC deposited sample, which is indicated by a blue rectangle in Figure 2.
2). The SiC/SiC substrate was treated as an in-plane isotropic material owing to the orientation of SiC fibers. T/EBC structure was represented by a 2-D simulation model, where displacements of the left and bottom edges of the model along the x and y directions, respectively, (shown in Figure 2) were constrained. A plane strain state was assumed to the z direction. In this analysis, the thicknesses of the SiAlON and mullite layers were changed (5, 15 and 25 µm), while the thicknesses of other layers were fixed as below; the SiC, Yb-silicate gradient composition and topcoat YbMS layers were 25, 100 (20 µm times five layers) and 200 µm, respectively. The thickness and width of the SiC/SiC substrate were 3000 and 4000 µm, respectively. The regions near all interfaces, where strong stress concentration is expected, were divided into a finer mesh (0.5 µm × 0.5 µm).

Thermal Stress Analysis
Since the experimental temperature is decreased gradually during the cooling process in fabrication of the T/EBC as explained in Section 2.1, the temperature dependence of the elastic properties (Young' modulus and Poisson's ratio) and the thermal property (CTE) must be taken into account for the thermal stress analysis. In addition, it is necessary to take account of the effect of creep on the mechanical state for the conditions of the experimental temperature and the cooling rate in the present deposition process. Thus, in this study, a series of the thermal stress analyses was carried out to evaluate the thermal stress distribution in T/EBC, where the creep effect was considered. Thermal stress is accumulated in T/EBC throughout the cooling process; i.e., stress is assumed to be null at the high temperature in the deposition process, and we evaluate the thermal stress state in T/EBC after cooling to room temperature from the high temperature. The thermal stress is dominated by stress in the x-axis direction in Figure 2 mainly, and thus the loading parallel to the interface, in other words, mode II-rich thermal loading is applied to T/EBC. The temperature condition for this analysis was determined from the experimental condition of the cooling process after deposition of T/EBC (cooling from a high temperature in the deposition process to room temperature at cooling rate of 20 K/min). A uniform temperature distribution in T/EBC was assumed during the cooling process. Although the surface temperature of the coated layers is up to 1473 K in the deposition process as described in Section 2.1, in this analysis we set the deposition temperature (the initial temperature) to be 1673 K, which is an assumed operation temperature of T/EBC and provides an estimation on the safer side. In addition, the cooling rate of 20 K/min can cause a considerable effect of creep during the cooling process. In general, it is known that creep emerges significantly at temperatures higher than half the melting point [15]. Thus, in this analysis, the stress was set to be null at the initial state of 1673 K, and the development of stress distribution was calculated at a cooling rate of 20 K/min while creep was considered within the temperature range of 1673-1173 K. Creep was assumed to be negligible for SiC and SiC/SiC substrate. In this study, the Yb-silicate gradient composition layer was modeled with five layers with different ratios of the YbDS and YbMS as follows; YbDS:YbMS = 100:0, 80:20, 50:50, 20:80, and 0:100. For computational efficiency and simplicity, the segmented YbMS layer was regarded as a homogenized material with an anisotropic (in-plane isotropic) mechanical property equivalent to the segmented layer [14]; hereafter it is referred to as the YbMS anisotropic homogeneous layer ( Figure 2). The SiC/SiC substrate was treated as an in-plane isotropic material owing to the orientation of SiC fibers. T/EBC structure was represented by a 2-D simulation model, where displacements of the left and bottom edges of the model along the x and y directions, respectively, (shown in Figure 2) were constrained. A plane strain state was assumed to the z direction. In this analysis, the thicknesses of the SiAlON and mullite layers were changed (5, 15 and 25 µm), while the thicknesses of other layers were fixed as below; the SiC, Yb-silicate gradient composition and topcoat YbMS layers were 25, 100 (20 µm times five layers) and 200 µm, respectively. The thickness and width of the SiC/SiC substrate were 3000 and 4000 µm, respectively. The regions near all interfaces, where strong stress concentration is expected, were divided into a finer mesh (0.5 µm × 0.5 µm).

Thermal Stress Analysis
Since the experimental temperature is decreased gradually during the cooling process in fabrication of the T/EBC as explained in Section 2.1, the temperature dependence of the elastic properties (Young' modulus and Poisson's ratio) and the thermal property (CTE) must be taken into account for the thermal stress analysis. In addition, it is necessary to take account of the effect of creep on the mechanical state for the conditions of the experimental temperature and the cooling rate in the present deposition process. Thus, in this study, a series of the thermal stress analyses was carried out to evaluate the thermal stress distribution in T/EBC, where the creep effect was considered. Thermal stress is accumulated in T/EBC throughout the cooling process; i.e., stress is assumed to be null at the high temperature in the deposition process, and we evaluate the thermal stress state in T/EBC after cooling to room temperature from the high temperature. The thermal stress is dominated by stress in the x-axis direction in Figure 2 mainly, and thus the loading parallel to the interface, in other words, mode II-rich thermal loading is applied to T/EBC. The temperature condition for this analysis was determined from the experimental condition of the cooling process after deposition of T/EBC (cooling from a high temperature in the deposition process to room temperature at cooling rate of 20 K/min). A uniform temperature distribution in T/EBC was assumed during the cooling process. Although the surface temperature of the coated layers is up to 1473 K in the deposition process as described in Section 2.1, in this analysis we set the deposition temperature (the initial temperature) to be 1673 K, which is an assumed operation temperature of T/EBC and provides an estimation on the safer side. In addition, the cooling rate of 20 K/min can cause a considerable effect of creep during the cooling process. In general, it is known that creep emerges significantly at temperatures higher than half the melting point [15]. Thus, in this analysis, the stress was set to be null at the initial state of 1673 K, and the development of stress distribution was calculated at a cooling rate of 20 K/min while creep was considered within the temperature range of 1673-1173 K. Creep was assumed to be negligible for SiC and SiC/SiC substrate. Since creep was not taken into account for all materials (i.e., the analysis is independent of time) below 1173 K, the system was cooled immediately from 1173 to 303 K.
As we consider a wide temperature range, we adopted temperature-dependent material properties; Young's modulus, E, Poisson's ratio, ν, CTE (reference temperature: 1673 K), α', and creep properties (creep coefficient, C, and creep index, n) listed in Tables 2-8, respectively. The elastic properties (i.e., E and ν) of the YbMS anisotropic homogeneous layer and the SiC/SiC substrate were regarded as anisotropic from the microstructure and the fiber orientation, respectively. Also, E and ν of the mullite layer were regarded as anisotropic on the basis of the experimental result of the in-plane and out-of-plane loading tests on the deposited mullite layer (along the x and y axes in Figure 2, respectively). The other layers were dealt with as isotropic materials. CTE and the creep properties of each layer and substrate were assumed to be isotropic. The temperature-dependent material properties were calculated from experimental measurements and literature (see Appendix A).

Interface Crack Introduction Analysis
ERR owing to the thermal stress, G th , is defined as the reduction in strain energy, Π, stored in a structure per increased crack area. When the crack area, A, increases to A + dA, ERR is evaluated as follows: Thus, it is necessary to obtain the strain energies stored in T/EBC with different crack lengths to evaluate G th init = G th A→0 and G th prop . We carried out FEM analyses for T/EBC models with various crack length under the stress condition obtained from the thermal stress analysis, and the strain energy in T/EBC was evaluated as a function of crack length.
The geometry, the boundary conditions and the mesh sizes of the model in this analysis were set in the same way as in the thermal stress analysis. The initial stress condition was derived from the stress distribution of the uncracked T/EBC model obtained by the thermal stress analysis. The temperature was kept constant at 303 K (temperature after the cooling process) during the analyses, and thus we used the material properties E and ν at 303 K shown in Tables 2-6. Interface cracks were introduced from the right edge of the simulation model shown in Figure 2. According to a preliminary FEM analysis, the SiC/SiAlON, SiAlON/mullite and mullite/YbDS interfaces were chosen as the objective interfaces to evaluate G th init and G th prop (the detail of the preliminary analysis is found in Appendix B).

Evaluation of ERR for Interface Crack Initiation and Propagation
We evaluated G th init and G th prop from the strain energy in T/EBC with different crack lengths obtained by the FEM analyses in Section 2.2.3 in the following ways.
Evaluation of G th init G th init is defined as ERR in the limit when A approaches 0, as follows: Note that G th init is equal to the first-order coefficient of the Taylor expansion of Π(A) around A = 0 with the inverted sign. Therefore, we approximated Π(A) as a polynomial from the FEM analyses for the T/EBC models with several different crack lengths, and G th init was evaluated as the first-order coefficient of the polynomial with the inverted sign. Two cases are possible according to the sign of G th init : for G th init > 0, crack initiation occurs if the interface fracture toughness, Γ, is smaller than G th init ; for G th init < 0, no crack initiation is expected. From the result of the preliminary analysis (Appendix B), we examined crack length a = 2.5, 5.0 and 7.5 µm to evaluate G th init at the objective interfaces.
Evaluation of G th prop ERR when a crack propagates from A to A + ∆A is given by numerical differentiation as follows: From the result of the preliminary analysis (Appendix B), the a for evaluation of G th prop was determined as 2.5, 5.0, 7.5, 10, 50, 100, 250, ..., 3500 µm in increments of 250 µm between 250 and 3500 µm.

Interface Fracture Test
Interfacial toughness was measured by an interface fracture test. The test method was a modification of the one designed for EBCs on SiC/SiC composites having a weak inter-laminar strength [16]. Figure 3 shows schematic of the test set-up. The edge of the coating was compressively loaded by the plate with a constant crosshead speed of 0.024 mm/min till the coating was delaminated from the substrate; thus, the mode II-rich loading at the interface was achieved with little loading to the substrate in the inter-laminar shear direction. The shear stress ought not to concentrate at only an interface between SiC/SiC substrate and T/EBC (SiC/SiAlON interface); i.e., it occurs along some interfaces in T/EBC. The pushing load, P, and the crosshead displacement, u, were measured during the test. The coating near the loading point was observed in the cross-sectional direction of the specimen by optical microscopy to determine when a crack was initiated at the loaded edge and the corresponding load, . The number of the tests was six.
The energy release rate, , associated with the initiation of an interface crack at the loaded edge under an applied load of is given by Definition of parameters and details of deriving Equation (4) are shown in Appendix C. The is defined as fracture toughness for the interface crack initiation, . Note that, is a nominal fracture toughness value of interface crack initiation including the effect of thermal residual stress.
After the tests, four of the specimens were mounted in resin and their cross-sections of the delaminated coating and substrate were observed; the cross-section was polished to mirror finish. SEM observation (VE-7800, Keyence corporation, Osaka, Japan) and energy dispersive X-ray (EDX) analysis (EX74120 attached to field emission SEM (JSM6500F, JEOL Ltd., Tokyo, Japan)) of the cross A block of specimen with a height, L, of 4 mm and a width, w, of 3 mm was prepared using a diamond blade saw in such a way that one of the fiber directions in the woven fabric of the SiC/SiC substrate was oriented to the longitudinal side of the specimen. The surface was as cut without polishing to avoid the coating delamination during the processing. The cut surface was smooth enough for us to distinguish the interface in the cross-sectional observation by conventional optical microscopy.
The specimen was set in a steel jig like a horizontal die, the upper part of which was adjustable. The specimen was put in the die and slid horizontally from the back side by a punch as indicated by the dashed arrow 1 in Figure 3 so that only the coating protruded from the die while the substrate was buried. Then the upper part was adjusted in the vertical direction (the dashed arrow 2 in Figure 3) to fix the specimen. The jig with the fixed specimen was placed under a universal testing machine (load cell capacity: 500 N, EZ-LX, Shimadzu Corporation, Kyoto, Japan) having a steel pushing plate with a thickness of 0.5 mm and a width of 5 mm.
The edge of the coating was compressively loaded by the plate with a constant crosshead speed of 0.024 mm/min till the coating was delaminated from the substrate; thus, the mode II-rich loading at the interface was achieved with little loading to the substrate in the inter-laminar shear direction. The shear stress ought not to concentrate at only an interface between SiC/SiC substrate and T/EBC (SiC/SiAlON interface); i.e., it occurs along some interfaces in T/EBC. The pushing load, P, and the crosshead displacement, u, were measured during the test. The coating near the loading point was observed in the cross-sectional direction of the specimen by optical microscopy to determine when a crack was initiated at the loaded edge and the corresponding load, P init . The number of the tests was six.
The energy release rate, G init , associated with the initiation of an interface crack at the loaded edge under an applied load of P init is given by Definition of parameters and details of deriving Equation (4) are shown in Appendix C. The G init is defined as fracture toughness for the interface crack initiation, Γ init . Note that, Γ init is a nominal fracture toughness value of interface crack initiation including the effect of thermal residual stress.
After the tests, four of the specimens were mounted in resin and their cross-sections of the delaminated coating and substrate were observed; the cross-section was polished to mirror finish. SEM observation (VE-7800, Keyence corporation, Osaka, Japan) and energy dispersive X-ray (EDX) analysis (EX74120 attached to field emission SEM (JSM6500F, JEOL Ltd., Tokyo, Japan)) of the cross section were done to identify the fractured interface. The remaining two specimens were served for the analyses of fracture surfaces; optical and SEM observations and EDX analysis were done. The ERR for interface crack initiation at the SiAlON/mullite interface, G th init, SA/Mu , depends on the thicknesses of h SA and h Mu as shown in Figure 4b, which is found to be expressed as the following equation within the examined range of h SA and h Mu (5-25 µm):

Layer Thickness Condition for Preventing the Objective Interface Crack Initiation
with the coefficients, c kl , listed in Table 9. Figure 5 shows the contour chart of G th init, SA/Mu as a function of h SA and h Mu shown in Equation (5). The level of G th init, SA/Mu is indicated by the depth of color (violet); a deeper-colored region corresponds to a lower G th init, SA/Mu and thus a better combination of layer thicknesses. This result suggests that one of the SiAlON or mullite layers must be thick and the other be thin in order to decrease G th init, SA/Mu and hinder crack initiation at the SiAlON/mullite interface.
Therefore, a thick mullite layer is necessary to decrease , / , i.e., to suppress crack initiation at the mullite/YbDS interface.
From the above results, it is found that a thin SiAlON layer and a thick mullite layer are required for reducing and preventing crack initiation at the SiC/SiAlON, SiAlON/mullite and mullite/YbDS interfaces, where strong stress concentration is expected during the cooling process in fabrication of T/EBC as shown in Figure 1.

Phase Structure of T/EBC
In Section 3.1, we found that thinning the SiAlON layer together with thickening the mullite layer was effective in suppressing interface crack formation during cooling where a large thermal stress is expected, i.e., the SiC/SiAlON, SiAlON/mullite and mullite/YbDS interfaces. In this study, therefore, the thickness of the SiAlON layer and the mullite layer were determined to be 5 and 20 µm, As shown in Figure 4c, it is found that a thin mullite layer (h Mu = 5 µm) results in a rise in the ERR for interface crack initiation at the mullite/YbDS interface, G th init, Mu/YD , with increasing h SA , while a thicker mullite layer (h Mu = 15 and 25 µm) reduces G th init, Mu/YD to almost zero regardless of h SA . Therefore, a thick mullite layer is necessary to decrease G th init, Mu/YD , i.e., to suppress crack initiation at the mullite/YbDS interface.
From the above results, it is found that a thin SiAlON layer and a thick mullite layer are required for reducing G th init and preventing crack initiation at the SiC/SiAlON, SiAlON/mullite and mullite/YbDS interfaces, where strong stress concentration is expected during the cooling process in fabrication of T/EBC as shown in Figure 1.

Phase Structure of T/EBC
In Section 3.1, we found that thinning the SiAlON layer together with thickening the mullite layer was effective in suppressing interface crack formation during cooling where a large thermal stress is expected, i.e., the SiC/SiAlON, SiAlON/mullite and mullite/YbDS interfaces. In this study, therefore, the thickness of the SiAlON layer and the mullite layer were determined to be 5 and 20 µm, respectively, and T/EBC was prepared by the dual EB-PVD process described in Section 2.1. We confirmed no delamination at all the interfaces of the fabricated T/EBC. Cross-sectional SEM images and elemental mapping images of T/EBC were shown in Figure 6a,b, respectively, where we found the thicknesses of constituent layers to be; SiAlON: about 5 µm, mullite: about 20 µm, Yb-silicate gradient composition layer: about 100 µm, and porous segmented YbMS layer: about 200 µm. That is, T/EBC was successfully fabricated nearly as designed.  (ℎ , ℎ ) interpolated with Equation (5).

Phase Structure of T/EBC
In Section 3.1, we found that thinning the SiAlON layer together with thickening the mullite layer was effective in suppressing interface crack formation during cooling where a large thermal stress is expected, i.e., the SiC/SiAlON, SiAlON/mullite and mullite/YbDS interfaces. In this study, therefore, the thickness of the SiAlON layer and the mullite layer were determined to be 5 and 20 µm, respectively, and T/EBC was prepared by the dual EB-PVD process described in Section 2.1. We confirmed no delamination at all the interfaces of the fabricated T/EBC. Cross-sectional SEM images and elemental mapping images of T/EBC were shown in Figure 6a

Behavior of Interface Crack Initiation and Propagation in T/EBC
To investigate the behavior of crack initiation and propagation during the cooling process in fabrication of T/EBC shown in Section 3.2, calculated ERRs ( , ) must be compared with interface fracture toughness obtained by experiment. Thus, we carried out the interface fracture

Behavior of Interface Crack Initiation and Propagation in T/EBC
To investigate the behavior of crack initiation and propagation during the cooling process in fabrication of T/EBC shown in Section 3.2, calculated ERRs (G th init , G th prop ) must be compared with interface fracture toughness obtained by experiment. Thus, we carried out the interface fracture toughness tests (Section 2.3) to obtain Γ, which is to be compared with G th init and G th prop with h SA = 5 µm and h Mu = 20 µm.    toughness tests (Section 2.3) to obtain , which is to be compared with and with hSA = 5 µm and hMu = 20 µm. Table 10 lists of the objective interface cracks for the T/EBC model with hSA = 5 µm and hMu = 20 µm. From this result, the SiAlON/mullite interface is found to possess the highest of the objective interfaces.  Figure 7 shows the relationships between and a at the objective interfaces in the T/EBC with hSA = 5 µm and hMu = 20 µm. In the -a relationships at all the objective interfaces, we observe three stages with increasing a: (A) shows a relatively rapid increase; (B) decreases; (C) increases again.  The reason for the decrease in at Stage B is explained by the distribution of out-of-plane thermal stress in the vicinity of the interface edge. Figure 8a shows the σy distributions in the coating layers along the y axis, which are obtained at the positions rx distant from the right interface edge in the x direction (rx = 0, 5, 10, 20, 30 and 40 µm). The highest level of σy was observed at the interface edge (rx = 0 µm), and the σy distribution reduces immediately to a negligible level within rx = 30 µm. Thus, the effect of σy on the coating layers is significant but extremely localized in the vicinity of interface edge, while that is negligible inside the simulation model. Note that the effect of σx on is marginal for sufficiently small a because we find σx ~ 0 at the interface edge (rx = 0 µm) directly from the balance of stress component in the x direction. Figure 8b shows the σx distributions in the coating layers along the y axis at various rx. These results indicate that σy has a dominating effect on for very short cracks. The mechanism of the second rise in at Stage C is attributed to the increase in the level of σx inside the simulation model as shown in Figure 8b. Therefore, the threestage behavior in the -a relationship is explained as follows: in the beginning, rises as a crack propagates owing to the strong σy near the interface edge (Stage A); then decreases because of a steep drop in σy inside the model (Stage B); for a sufficiently long crack, is mainly affected by σx and rises again with increasing σx (Stage C). The reason for the decrease in G th prop at Stage B is explained by the distribution of out-of-plane thermal stress in the vicinity of the interface edge. Figure 8a shows the σ y distributions in the coating layers along the y axis, which are obtained at the positions r x distant from the right interface edge in the x direction (r x = 0, 5, 10, 20, 30 and 40 µm). The highest level of σ y was observed at the interface edge (r x = 0 µm), and the σ y distribution reduces immediately to a negligible level within r x = 30 µm. Thus, the effect of σ y on the coating layers is significant but extremely localized in the vicinity of interface edge, while that is negligible inside the simulation model. Note that the effect of σ x on G th prop is marginal for sufficiently small a because we find σ x~0 at the interface edge (r x = 0 µm) directly from the balance of stress component in the x direction. Figure 8b shows the σ x distributions in the coating layers along the y axis at various r x . These results indicate that σ y has a dominating effect on G th prop for very short cracks. The mechanism of the second rise in G th prop at Stage C is attributed to the increase in the level of σ x inside the simulation model as shown in Figure 8b. Therefore, the three-stage behavior in the G th prop -a relationship is explained as follows: in the beginning, G th prop rises as a crack propagates owing to the strong σ y near the interface edge (Stage A); then G th prop decreases because of a steep drop in σ y inside the model (Stage B); for a sufficiently long crack, G th prop is mainly affected by σ x and rises again with increasing σ x (Stage C).  The reason for the decrease in at Stage B is explained by the distribution of out-of-plane thermal stress in the vicinity of the interface edge. Figure 8a shows the σy distributions in the coating layers along the y axis, which are obtained at the positions rx distant from the right interface edge in the x direction (rx = 0, 5, 10, 20, 30 and 40 µm). The highest level of σy was observed at the interface edge (rx = 0 µm), and the σy distribution reduces immediately to a negligible level within rx = 30 µm. Thus, the effect of σy on the coating layers is significant but extremely localized in the vicinity of interface edge, while that is negligible inside the simulation model. Note that the effect of σx on is marginal for sufficiently small a because we find σx ~ 0 at the interface edge (rx = 0 µm) directly from the balance of stress component in the x direction. Figure 8b shows the σx distributions in the coating layers along the y axis at various rx. These results indicate that σy has a dominating effect on for very short cracks. The mechanism of the second rise in at Stage C is attributed to the increase in the level of σx inside the simulation model as shown in Figure 8b. Therefore, the threestage behavior in the -a relationship is explained as follows: in the beginning, rises as a crack propagates owing to the strong σy near the interface edge (Stage A); then decreases because of a steep drop in σy inside the model (Stage B); for a sufficiently long crack, is mainly affected by σx and rises again with increasing σx (Stage C).  Figure 9 shows a typical P-u curve during the tests. By the cross-sectional optical microscopy, the onset of interface fracture near the loaded edge was identified at Point A in P-u curve before the maximum load. The load at Point A is defined as . The crack then propagated down through the interface to the midway till the load reached the maximum; finally it grew unstably to the complete delamination at the maximum load. Buckling or compressive fracture of the coating was not observed during the test. The measured values for respective tests are listed in Table 11.  Examples of the cross-sectional SEM images of the specimens after the tests were shown in Figure 10. At the loaded edge where the interface crack started to propagate, fracture occurred below  Figure 9 shows a typical P-u curve during the tests. By the cross-sectional optical microscopy, the onset of interface fracture near the loaded edge was identified at Point A in P-u curve before the maximum load. The load at Point A is defined as P init . The crack then propagated down through the interface to the midway till the load reached the maximum; finally it grew unstably to the complete delamination at the maximum load. Buckling or compressive fracture of the coating was not observed during the test. The measured P init values for respective tests are listed in Table 11.  Figure 9 shows a typical P-u curve during the tests. By the cross-sectional optical microscopy, the onset of interface fracture near the loaded edge was identified at Point A in P-u curve before the maximum load. The load at Point A is defined as . The crack then propagated down through the interface to the midway till the load reached the maximum; finally it grew unstably to the complete delamination at the maximum load. Buckling or compressive fracture of the coating was not observed during the test. The measured values for respective tests are listed in Table 11.  Examples of the cross-sectional SEM images of the specimens after the tests were shown in Figure 10. At the loaded edge where the interface crack started to propagate, fracture occurred below  Examples of the cross-sectional SEM images of the specimens after the tests were shown in Figure 10. At the loaded edge where the interface crack started to propagate, fracture occurred below or above the SiAlON layer, i.e., either at the interface between the SiC and SiAlON layers (Figure 10a) or at the interface between the SiAlON and mullite layers (Figure 10b). The crack path was along either of the two interfaces, and it was switched to each other by crack kinking across the SiAlON layer. However, crack propagation within a layer, such as a crack passing through the inside of layer parallel to the coating plane, was not observed.

Interface Fracture Toughness of T/EBC
Coatings 2019, 9, 697 16 of 25 or above the SiAlON layer, i.e., either at the interface between the SiC and SiAlON layers (Figure 10a) or at the interface between the SiAlON and mullite layers (Figure 10b). The crack path was along either of the two interfaces, and it was switched to each other by crack kinking across the SiAlON layer. However, crack propagation within a layer, such as a crack passing through the inside of layer parallel to the coating plane, was not observed.  Figure 11a shows a fracture surface near the loaded edge after the tests observed by optical microscopy. The fracture surface was classified into three according to their colors: the largest black area denoted by Region A, the gray area by Region B, and the small white area by Region C as illustrated in Figure 11a. EDX maps of a selected area in the fracture surface (indicated by a dashed rectangle in Figure 11a) are shown in Figure 11b. Mapping of element distributions shows the concentration of Si in Region A. In Region B the existence of Si, Al and N was detected. Region C was characterized by Yb concentration. Apparently Al also looks rich in Region C, but it is due to misdetection because the Mα line of Yb is very close to Kα line of Al which was used for the mapping of Al distribution. These results suggest that Region A with the largest area corresponds to the fracture surface created by the interface fracture between the SiC and SiAlON layers observed in Figure 10a; Region B showing the second largest area corresponds to the surface created by the SiAlON/mullite interface fracture (Figure 10b). We could not observe the fractured interface corresponding to Region C in the cross-sectional SEM observation of the T/EBC and substrate ( Figure  10) because of the limited area of Region C. To determine the exact location of the fracture surface in Region C, we cut after the tests one of the substrate in the cross-section including Region C. The result is shown in Figure 12 suggesting that the fracture occurred in the Yb-silicate gradient composition layer.
(a) (b)  Figure 11a shows a fracture surface near the loaded edge after the tests observed by optical microscopy. The fracture surface was classified into three according to their colors: the largest black area denoted by Region A, the gray area by Region B, and the small white area by Region C as illustrated in Figure 11a. EDX maps of a selected area in the fracture surface (indicated by a dashed rectangle in Figure 11a) are shown in Figure 11b. Mapping of element distributions shows the concentration of Si in Region A. In Region B the existence of Si, Al and N was detected. Region C was characterized by Yb concentration. Apparently Al also looks rich in Region C, but it is due to misdetection because the M α line of Yb is very close to K α line of Al which was used for the mapping of Al distribution. These results suggest that Region A with the largest area corresponds to the fracture surface created by the interface fracture between the SiC and SiAlON layers observed in Figure 10a; Region B showing the second largest area corresponds to the surface created by the SiAlON/mullite interface fracture (Figure 10b). We could not observe the fractured interface corresponding to Region C in the cross-sectional SEM observation of the T/EBC and substrate ( Figure 10) because of the limited area of Region C. To determine the exact location of the fracture surface in Region C, we cut after the tests one of the substrate in the cross-section including Region C. The result is shown in Figure 12 suggesting that the fracture occurred in the Yb-silicate gradient composition layer.
To calculate the line fractions of Regions A, B and C along the loaded edge, the fracture surfaces near the edge were analyzed using image processing software (Image J 1.48v, National Institute of Health, Bethesda, MD, USA) ( Table 12). As shown in Table 12, fracture surface was dominantly composed of Regions A and B; the ratio of Region C was relatively small. Thus the crack initiation at the edge was supposed to occur primarily either at the interface between the SiC and SiAlON layers or at the interface between the SiAlON and mullite layers.
SiAlON/mullite interface fracture (Figure 10b). We could not observe the fractured interface corresponding to Region C in the cross-sectional SEM observation of the T/EBC and substrate ( Figure  10) because of the limited area of Region C. To determine the exact location of the fracture surface in Region C, we cut after the tests one of the substrate in the cross-section including Region C. The result is shown in Figure 12 suggesting that the fracture occurred in the Yb-silicate gradient composition layer.  To calculate the line fractions of Regions A, B and C along the loaded edge, the fracture surfaces near the edge were analyzed using image processing software (Image J 1.48v, National Institute of Health, Bethesda, MD, USA) ( Table 12). As shown in Table 12, fracture surface was dominantly composed of Regions A and B; the ratio of Region C was relatively small. Thus the crack initiation at the edge was supposed to occur primarily either at the interface between the SiC and SiAlON layers or at the interface between the SiAlON and mullite layers.

Region Line Fraction
The energy release rate for the edge crack initiation along the SiC/SiAlON interface under (in Table 11) was calculated from Equations (A14)-(A18) and (4), regarding the SiC/SiC plate and the SiC layer as "substrate" and the rest of the layers as "coating" in Equation (A14); the mean value of 6.4 J/m 2 was obtained. When the SiAlON layer is counted as part of the substrate in addition to the SiC/SiC and SiC, the mean energy release rate for the crack along the SiAlON/mullite interface becomes 6.5 J/m 2 . These values of energy release rates are very close to each other because the thickness of the SiAlON layer lying between the two interfaces is quite small compared to the total thickness of the system. Under the assumption that the edge crack was initiated simultaneously through the edge (i.e., in the direction of depth) at regardless of the fractured interfaces, we can estimate that the SiC/SiAlON interface fracture and the SiAlON/mullite interface fracture both occur with significant proportions at almost the same energy release rates. This suggests that the interface toughness of these two interfaces are very close; both the interfaces should have an interface toughness of ~6.4 J/m 2 . The mean energy release rate at when the edge crack was initiated along the mullite/YbDS interface, which was supposed to have large thermal stress according to the FEM analysis, was calculated to be 6.8 J/m 2 . Since no fracture occurred at this interface at , we can expect that the toughness of the interface should be larger than 6.8 J/m 2 . The abovementioned results indicate that the approximate for the SiC/SiAlON and SiAlON/mullite interfaces are 6.4 J/m 2 Figure 12. Cross-sectional SEM observation including Region C. The energy release rate for the edge crack initiation along the SiC/SiAlON interface under P init (in Table 11) was calculated from Equations (A14)-(A18) and (4), regarding the SiC/SiC plate and the SiC layer as "substrate" and the rest of the layers as "coating" in Equation (A14); the mean value of 6.4 J/m 2 was obtained. When the SiAlON layer is counted as part of the substrate in addition to the SiC/SiC and SiC, the mean energy release rate for the crack along the SiAlON/mullite interface becomes 6.5 J/m 2 . These values of energy release rates are very close to each other because the thickness of the SiAlON layer lying between the two interfaces is quite small compared to the total thickness of the system. Under the assumption that the edge crack was initiated simultaneously through the edge (i.e., in the direction of depth) at P init regardless of the fractured interfaces, we can estimate that the SiC/SiAlON interface fracture and the SiAlON/mullite interface fracture both occur with significant proportions at almost the same energy release rates. This suggests that the interface toughness of these two interfaces are very close; both the interfaces should have an interface toughness of~6.4 J/m 2 . The mean energy release rate at P init when the edge crack was initiated along the mullite/YbDS interface, which was supposed to have large thermal stress according to the FEM analysis, was calculated to be 6.8 J/m 2 . Since no fracture occurred at this interface at P init , we can expect that the toughness of the interface should be larger than 6.8 J/m 2 . The abovementioned results indicate that the approximate Γ init for the SiC/SiAlON and SiAlON/mullite interfaces are 6.4 J/m 2 and the lower limit of Γ init for the mullite/YbDS interface is 6.8 J/m 2 . Table 13 shows comparison between G th init of the SiC/SiAlON, SiAlON/mullite and mullite/YbDS interface cracks and Γ init of the corresponding interfaces. G th init s of every objective interface crack are smaller than Γ init . This result suggests that the effect of thermal residual stress on the intrinsic fracture toughness for interface crack initiation is sufficiently small. Therefore, in this study, we regard the nominal value as the fracture toughness of interface crack initiation. Table 13 also suggests that the SiC/SiAlON, SiAlON/mullite and mullite/YbDS interface cracks are not likely to initiate by cooling in the T/EBC fabrication process.  Here, we assumed that fracture surfaces formed by interface crack initiation and propagation were the same and regarded Γ prop as equal to Γ init . As shown in Figure 13a,c, G th prop of the SiC/SiAlON and mullite/YbDS interface cracks are smaller than Γ prop of the corresponding interfaces. The results suggest that these interface cracks are not likely to propagate regardless of initial interface crack length. Figure 13b suggests that, if the T/EBC has an initial length of the SiAlON/mullite interface crack of about 1200 µm, the crack should propagate instantly because G th prop exceeds Γ prop at the crack length of 1200 µm. However, delamination along the SiAlON/mullite interface was not observed in the experiment (see Section 3.2). These results suggest that there was no initial crack exceeding the length threshold at the SiAlON/mullite interface.
The results shown in this section reveal that the T/EBC with the proposed layer thicknesses can be fabricated without delamination along interfaces by cooling in the fabrication process. This is confirmed by the result that no delamination along interface is observed as described in Section 3.2.

Future Prospects
In this study, we mainly focused on the mechanical reliability of T/EBC during the cooling process in fabrication. From the practical viewpoint, however, it is also essential to evaluate the mechanical reliability under an operating condition. In general, an EBC is exposed to a thermal cycle condition with a high humidity in operation, which can induce a microstructural change and chemical transformation of the coating layers through a reaction with heated oxygen and water vapor. Consequently, changes in the material properties of each layer due to microstructural change and chemical transformation are crucial to the mechanical state in T/EBC. Thus, it is necessary to evaluate a time dependent ERR by the FEM analysis where changes in material properties over time are incorporated. Furthermore, the interface fracture toughness changes presumably over time owing to microstructural change and chemical transformation, which should also be taken into account for a reliable design of T/EBC. Nevertheless, it is beyond the scope of this paper and will be considered in our future work. crack length of 1200 µm. However, delamination along the SiAlON/mullite interface was not observed in the experiment (see Section 3.2). These results suggest that there was no initial crack exceeding the length threshold at the SiAlON/mullite interface.
The results shown in this section reveal that the T/EBC with the proposed layer thicknesses can be fabricated without delamination along interfaces by cooling in the fabrication process. This is confirmed by the result that no delamination along interface is observed as described in Section 3.2.

Conclusions
To assess the mechanical reliability of T/EBC (SiC/SiAlON/mullite/Yb-silicate gradient composition layer/YbMS with porous segment structure) against interface crack initiation and propagation due to thermal stress induced in fabricated process, we carried out FEM analysis to evaluate ERR for interface cracks and performed experiment to obtain interface fracture toughness. Our FEM calculations revealed that G th init of the objective interfaces decreases by making the SiAlON layer thinner and the mullite layer thicker. We fabricated the T/EBC with layer thicknesses within the proposed range (h SA = 5 µm and h Mu = 20 µm) and confirmed no delamination along the interfaces. In the interface fracture test for the fabricated T/EBC, fracture surfaces were found at the SiC/SiAlON and SiAlON/mullite interfaces. We estimated the approximate fracture toughness for the SiC/SiAlON and SiAlON/mullite interfaces and minimum limit of fracture toughness for the mullite/YbDS interface. Comparison between the fracture toughness by the experiment and calculated G th init and G th prop indicated that the fabricated T/EBC possesses sufficient mechanical reliability against interface cracks.
Of course, the present study considers only the stress state just after the fabrication process and does not assure the mechanical reliability in operation under thermal cycle with a high humidity, which is regarded as our future work.

Coating Layer
Step 2 Step 3
The values of ν of the SiC, SiAlON and YbDS layers were considered to be identical to those of their bulk. The outline of the estimation procedure of ν of each layer at Step 1 is shown in Table A3.
Step 1: Measurement of ν of the bulk, ν bulk in some temperature range.
Step 2: Obtainment of ν in the temperature range from 303 to 1673 K by the assumption that ν bulk outside the temperature range in Step 1 was identical to that measured at adjacent temperature. Table A3. Estimation procedure of ν of SiC, SiAlON and YbDS layers (Step 1).

Coating Layer
Step 1

Specimen (Bulk) Measuring Method Temperature Range T [K]
SC Description in Table A1 JIS R1602 and ASTM C848 Description in Table A1  SA  Description in Table A1 Description in Table A1 Description in Table A1  YbDS Description in Table A1 JIS R1602 and ASTM C848 Description in Table A1 ν ij of the anisotropic mullite layer in the temperature range from 303 to 1673 K was estimated using ν ij of the bulk specimen described in Table A1 and E film of the mullite layer derived from the procedure shown in Table A2. ν of the YbMS used in this estimation was literature value [9] and temperature-independent. For the Yb-silicate compositional gradient layers, ν was obtained from Young's moduli and bulk moduli, B, of YbMS and YbDS by the following equations: The YbMS anisotropic homogeneous layer is characterized by anisotropic Young's modulus, E ani i , Poisson's ratio, ν ani ij , and shear modulus, G ani ij (i, j = x, y, z; i j). A method of calculations for these parameters will be discussed in Reference [14]. Shear modulus of mullite layer was estimated by Equation (A4).
Appendix A.2. CTE (Reference Temperature: 1673 K) α was calculated from CTE at the reference temperature of 303 K (α; evaluated as a function of temperature from experimental measurements) as follows: α of the SiC and SiAlON layers was considered to be identical to that of their bulk. The values of α of the sintered bodies and the wafer in the temperature range from 573 to 1673 K were measured based on JIS R1618. Then, α of the sintered bodies and the wafer at 323-1673 K was estimated by extrapolation of the relationship between α and T. α of the mullite, YbDS and YbMS layers at given temperature was estimated using measured values of film/substrate layered specimens deposited with the same conditions as in Section 2.1. The outline of the calculation procedure of α of each layer is described below; Step 1: Derivation of relationship between strain, ε, and T for each film/substrate layered specimen (described in Table A2), based on digital image correlation method.
Step 2: Approximation of the ε-T relationship derived in Step 1 by Equation (A6).
Step 3: Calculation of α in the temperature range from 323 to 1673 K using the approximation formula derived in Step 2.
The value of α of the Yb-silicate gradient composition layers was obtained by the law of mixture as follows [20]: In this study, α of the YbMS anisotropic homogeneous layer was regarded as identical to that of the YbMS dense layer.

Appendix A.3. Creep Properties
In this study, we assumed that the creep property of each layer follows the strain-hardening law expressed as where . ε c denotes creep strain rate.
As described in Section 2.2.2, we assumed that creep was negligible for the SiC layer. Creep properties of the SiAlON, mullite, YbDS and YbMS layers were estimated by the method described below; Step 1: Estimation of n at a given temperature and the average of n within the temperature range from the graph of the true stress and the strain rate derived from literature values and/or measured values. Noted that the average of n was used in the FEM analysis.
Step 2: Estimation of C at a given temperature using the average of n, true stress and strain rate derived in Step 1.
Step 3: Estimation of C in the temperature range from 1173 to 1673 K using C * and Q/R in Equation (A9) based on C versus inverse of T plot derived in Step 2.
The estimation method of the creep properties of each layer at Step 1 is shown in Table A4. For the Yb-silicate gradient composition layers, C and n were obtained as follows: The creep property of the YbMS anisotropic homogeneous layer was regarded as identical to that of the YbMS dense layer.

Coating Layer
Step 1

Appendix B. Preliminary FEM Analyses: Determination of Objective Interfaces and Crack Lengths
As was explained in Introduction, an interface crack in the T/EBC is likely to occur at interfaces with strong thermal stress concentration due to the difference in the CTEs from the SiC/SiC substrate. Here we determined the objective interfaces and crack lengths from the stress distribution obtained by a preliminary FEM analysis. The thermal stress analysis was carried out in the way shown in Section 2.2.2 for the uncracked T/EBC model with the SiAlON and mullite layers both being 25 µm thick. Figure A1 shows the distribution of σ x near the interface edge in the constituent layers of the T/EBC after the cooling process. Thermal stress is found to be concentrated particularly at the SiC/SiAlON, SiAlON/mullite and mullite/YbDS interfaces. Thus, we focused on crack initiation and propagation at these objective interfaces in this study.
From the balance of the stress component in the x direction, we deduce σ x~0 at the interface edge and thus a dominating effect of σ y on G th init . Therefore, we determined the crack lengths examined to evaluate G th init , based on the σ y distribution in the vicinity of the interface edge. Figure A2 shows the σ y distribution in the coating layers along the y axis at the positions r x distant from the interface edge (r x = 0, 5, 10, 60, 100, 300, 500 and 1000 µm). The highest level of σ y was observed at the interface edge (r x = 0 µm) while the σ y distribution reduces to a negligible level inside the model. On the basis of the results, we examined the crack lengths of a = 2.5, 5.0 and 7.5 µm to evaluate G th init . For evaluation of G th prop , the crack length a was varied as a = 2.5, 5.0, 7.5, 10, 50, 100, 250, ..., 3500 µm in increments of 250 µm between 250 and 3500 µm.      Figure A3 shows the loads and moments applied to the specimen. Pb and Mb ( = 1, 2) are the load and moment from surrounding jigs, respectively, and P is the applied load by the pushing plate. Equilibria of forces and moments are given by the following equations, respectively [16,23,24]:

Appendix C. Derivation Process of Energy Release Rate under Applying Mechanical Loading
and where hsub and hcoat are the thicknesses of the substrate and coating, respectively, and is the distance from the bottom of the substrate to the neutral axis of the specimen. When L is small compared to hsub, large P2 and M2 can be generated, but in this case these are negligible. The specimen is a multilayer composite of ten layers including the substrate with different Young's moduli, Ep, and thicknesses, hp ( = 1, 2, ⋯ ,10), so is given in a quite complicated form. Fortunately, in the loading condition for this test, the load applied to the coating is almost purely compressive parallel to the coating plane and the contribution of moment to the strain energy seems relatively small, thus the  Figure A3 shows the loads and moments applied to the specimen. P b and M b (b = 1, 2) are the load and moment from surrounding jigs, respectively, and P is the applied load by the pushing plate. Equilibria of forces and moments are given by the following equations, respectively [16,23,24]:

Appendix C. Derivation Process of Energy Release Rate under Applying Mechanical Loading
and where h sub and h coat are the thicknesses of the substrate and coating, respectively, and δ is the distance from the bottom of the substrate to the neutral axis of the specimen. When L is small compared to h sub , large P 2 and M 2 can be generated, but in this case these are negligible. The specimen is a multi-layer composite of ten layers including the substrate with different Young's moduli, E p , and thicknesses, h p (p = 1, 2, · · · , 10), so δ is given in a quite complicated form. Fortunately, in the loading condition for this test, the load applied to the coating is almost purely compressive parallel to the coating plane and the contribution of moment to the strain energy seems relatively small, thus the Young's modulus of the coating, E coat , and the substrate, E sub , may be simply approximated with the Voigt model in the rule of mixture, respectively, where s, t, h sub and h coat are varied according to the fractured interface. Thus δ can be obtained as the neutral axis position of a simple two-layer composite (single equivalent coating layer with E coat and substrate with E sub ): Energy release rate for the edge crack under an applied load of P is denoted by G and expressed as [24] Here, D is the dimensionless cross-sectional area and I is the moment of inertia of area written as I is again calculated assuming that the material system is a two-layer composite with a coating and substrate. Substituting Equations (A12), (A13) and P = P init into (A17) and ignoring P 2 and M 2 , Equation (4)  Here, is the dimensionless cross-sectional area and is the moment of inertia of area written as is again calculated assuming that the material system is a two-layer composite with a coating and substrate. Substituting Equations (A12), (A13) and = into (A17) and ignoring P2 and M2, Equation (4) is derived.