The Temperature Distribution in Plasma-Sprayed Thermal-Barrier Coatings During Crack Propagation and Coalescence

A Finite-Element Model (FEM) for thermal-barrier coatings was employed to elaborate the temperature distribution on yttria-stabilized zirconia (YSZ) free surface during cracks coalescing, then the influence of sintering of YSZ induced by heat-transfer overlapping on energy release rate was quantificationally evaluated. A three-dimensional model including three layers was fabricated. Two types of cracks, with and without depth variations in YSZ coating, were introduced into the model, respectively. The temperature rise of YSZ coating over the crack is independent of each other at the beginning of crack propagation. As crack distance shortens, the independent temperature-rise regions begin to overlap, while maximum temperature is still located at the crack center before crack coalescence. The critical distance that the regions of temperature rise, just overlapping, is the sum of half lengths of two coalescing cracks (i.e., a1 + a2), which is independent of cracking path. The maximum temperature in YSZ sharply increases once cracks coalesce. Compared with one delamination crack, the effective energy-release rate induced by heat-transfer overlapping increases in the range of 0.2%–15%, depending on crack length and crack distance, which is on some level comparable to that of deterioration of thermal expansion misfit induced by temperature jump between crack faces.


Introduction
The thermal barrier coatings (TBC) are fabricated via the ceramic material coated on the superalloy to protect the superalloy components and increase gas temperature, thereby significantly increasing the efficiency of the gas turbine [1][2][3][4].Commercial TBC include four layers: ceramic topcoat, thermally grown oxide (TGO), bond coat, and superalloy substrate.The ceramic topcoat provides the thermal insulation and is typically made of 7%-8% yttria-stabilized zirconia (YSZ).MCrAlY is usually used as bond coat to reduce the thermal expansion misfit between YSZ and substrate.In addition, the bond coat also protects the substrate from hot corrosion and oxidation.TGO is not present in as-deposited TBC systems, which generates during TBC operation due to the high temperature oxidation of the bond coat.
The temperature difference between the YSZ free surface and the inside surface of the metal component is an objective reality, depending on the service environment of the TBC.It has been demonstrated that the temperature across YSZ decreases rather uniformly [2,[5][6][7] during the

The Model of TBC Systems
A cylindrical three-dimensional Finite-Element Model (FEM) was fabricated to evaluate the temperature distribution in TBC with coalescing cracks, as shown in Figure 1a.The model has been described in detail in our previous report [6,7] and will therefore be outlined here only shortly.The FEM included three layers: substrate, bond coat, and YSZ coating; the associated thicknesses of the three layers were 3.0 mm, 150 µm, and 250 µm, respectively.Penny-cracks with two different types were introduced into the model, respectively.The first type was that the cracks with an identical depth in YSZ coating were introduced into the model to illustrate the change of temperature distribution with inplane cracks coalescing, as shown in Figure 1b.All crack lengths (2a) in this model were fixed to be Coatings 2018, 8, 311 3 of 11 2.0 mm.Associated crack widths (δ) were all 20 µm, depending on the ratio of crack length versus its width [33].The various distances (d 1 ) between adjacent edges of two cracks ranged from "0a" to "3.0a" were designed.Associated unit of the distance is millimeters (mm).Additionally, the cracks in the model were located in the YSZ coating with distance (D) to interface of 20 µm in term of location depth.In order to clarify the influence of cracking paths on temperature distribution, the second-type model with three penny-shaped cracks of different depths was conducted into YSZ coating.The middle crack located in YSZ coating with distance to interface of 20 µm, and, correspondingly, the depths of the other two cracks changed along YSZ thickness, as shown in Figure 1c.The distance (d 2 ) between middle crack and side crack along YSZ thickness increased from 0 to 150 µm.Meanwhile, the inplane distance between edge of central crack and that of side crack (d) was 2.0 mm.0 to 150 μm.Meanwhile, the inplane distance between edge of central crack and that of side crack (d) was 2.0 mm.

The Boundary Conditions and Heat Load
The roughness on both surfaces of substrate and bond coat was neglected during calculation.The thermal-expansion increments among layers were different, inducing local inplane slipping between layers.The local slipping in this model was ignored.Meanwhile, the radiation and convection within crack were ignored [7] and the boundary of model was supposed to be adiabatic.On the basis of the result that the high-temperature stage can be considered as a steady-state heat transfer, and the heat-flux density can be calculated by the simplified heat-transfer formula.The calculated heat-flux density (q, 0.66 × 10 6 W/m 2 ) was loaded on the YSZ free surface.The thermal properties of YSZ coating exhibit anisotropy because of its typical laminar structure [1,2,[10][11][12][13][14][15][22][23][24][25]34].As a result, the thermal conductivity of plasma-sprayed YSZ coating along thickness is different from that of inplane.The thermal conductivity of the bond coat in the case corresponds to the MCrAlTaY (Metco Amdry 997) and that of the substrate corresponds to the nickel-based superalloy (Inconel 738).The thermal conductivity of layers used in the model is shown in Table 1 [7].Table 1.The thermal conductivity used in the model [7].

Parameters
Values (W•m −

The Temperature Distribution in YSZ Coating with Coalescing Cracks
Figure 2 shows the temperature distribution on the YSZ free surface depending on the distance of cracks.The results show that the regions of temperature rose over the cracks, independent of each other when two cracks had enough distance.It means that the elevated temperatures corresponding to the delamination crack were independent from each other at the beginning of crack propagation.

The Boundary Conditions and Heat Load
The roughness on both surfaces of substrate and bond coat was neglected during calculation.The thermal-expansion increments among layers were different, inducing local inplane slipping between layers.The local slipping in this model was ignored.Meanwhile, the radiation and convection within crack were ignored [7] and the boundary of model was supposed to be adiabatic.On the basis of the result that the high-temperature stage can be considered as a steady-state heat transfer, and the heat-flux density can be calculated by the simplified heat-transfer formula.The calculated heat-flux density (q, 0.66 × 10 6 W/m 2 ) was loaded on the YSZ free surface.The thermal properties of YSZ coating exhibit anisotropy because of its typical laminar structure [1,2,[10][11][12][13][14][15][22][23][24][25]34].As a result, the thermal conductivity of plasma-sprayed YSZ coating along thickness is different from that of inplane.The thermal conductivity of the bond coat in the case corresponds to the MCrAlTaY (Metco Amdry 997) and that of the substrate corresponds to the nickel-based superalloy (Inconel 738).The thermal conductivity of layers used in the model is shown in Table 1 [7].
Table 1.The thermal conductivity used in the model [7].

The Temperature Distribution in YSZ Coating with Coalescing Cracks
Figure 2 shows the temperature distribution on the YSZ free surface depending on the distance of cracks.The results show that the regions of temperature rose over the cracks, independent of each other when two cracks had enough distance.It means that the elevated temperatures corresponding to the delamination crack were independent from each other at the beginning of crack propagation.The area of the elevated temperature over the crack followed the crack shape.As a result, the region of the elevated temperature could be considered as a union of numerous circular isothermals, and the maximum temperature corresponded to the center of the crack.As crack distance shortened, the regions of temperature rose corresponding to the cracks contacting and overlapping each other.As a result, the isothermals no longer perfectly followed the crack shape, as shown in Figure 2. The results indicate that the independent elevated-temperature regions started to overlap due to crack propagation.Consequently, compared with the other crack-free regions, the sintering rate of YSZ between cracks increased although this region was also crack-free.The architecture (i.e., microcrack healing [6,9,10,13] and bonding of intersplat during thermal spraying [10,35]) induced by sintering caused the stiffening of YSZ.Associated energy-release rate increased, depending on increasing elastic modulus [6,11,16,36].In addition, the corresponding thermal conductivity of YSZ in the region increased, reducing the thermal protection offered to the bond coat [37,38].Moreover, the YSZ coating between cracks partly sintered before crack coalescence due to the temperature overlap.Thus, vertical cracks induced by sintering [6] would generate rapidly in this region once two cracks coalesced.Therefore, the temperature overlap partly affected the durability of TBC via the increasing sintering rate, and will therefore be discussed in detail in below section.The area of the elevated temperature over the crack followed the crack shape.As a result, the region of the elevated temperature could be considered as a union of numerous circular isothermals, and the maximum temperature corresponded to the center of the crack.As crack distance shortened, the regions of temperature rose corresponding to the cracks contacting and overlapping each other.As a result, the isothermals no longer perfectly followed the crack shape, as shown in Figure 2. The results indicate that the independent elevated-temperature regions started to overlap due to crack propagation.Consequently, compared with the other crack-free regions, the sintering rate of YSZ between cracks increased although this region was also crack-free.The architecture (i.e., microcrack healing [6,9,10,13] and bonding of intersplat during thermal spraying [10,35]) induced by sintering caused the stiffening of YSZ.Associated energy-release rate increased, depending on increasing elastic modulus [6,11,16,36].In addition, the corresponding thermal conductivity of YSZ in the region increased, reducing the thermal protection offered to the bond coat [37,38].Moreover, the YSZ coating between cracks partly sintered before crack coalescence due to the temperature overlap.Thus, vertical cracks induced by sintering [6] would generate rapidly in this region once two cracks coalesced.Therefore, the temperature overlap partly affected the durability of TBC via the increasing sintering rate, and will therefore be discussed in detail in below section.
In order to address how the overlap affects temperature distribution, the temperature profiles along crack lengths were examined, as shown in Figure 3.In spite of no recognizable expression in the temperature field shown in Figure 2b, the temperature fields over the cracks did indeed have a role on each other.The majority heat transfer across the YSZ had to make a detour from the crack tip, because of the much lower thermal conductivity of the air in the delamination crack.This heat transfer overlaps when distance between two cracks ranges in a critical value, inducing a temperature rise in the YSZ between cracks.The increasing temperature between cracks induced an increasing driving force for YSZ sintering, then improved the sintering rate.It is interesting that the maximum temperature on the YSZ free surface still corresponded to the crack center, and the maximum temperature had no significant change with shortening crack distance.It means that YSZ coating over the center of a small crack prioritized sintering before cracks coalesced.This result can explain why a fraction of small vertical cracks sometimes did not distribute around the center of the large coalesced crack [6].In order to address how the overlap affects temperature distribution, the temperature profiles along crack lengths were examined, as shown in Figure 3.In spite of no recognizable expression in the temperature field shown in Figure 2b, the temperature fields over the cracks did indeed have a role on each other.The majority heat transfer across the YSZ had to make a detour from the crack tip, because of the much lower thermal conductivity of the air in the delamination crack.This heat transfer overlaps when distance between two cracks ranges in a critical value, inducing a temperature rise in the Coatings 2018, 8, 311 5 of 11 YSZ between cracks.The increasing temperature between cracks induced an increasing driving force for YSZ sintering, then improved the sintering rate.It is interesting that the maximum temperature on the YSZ free surface still corresponded to the crack center, and the maximum temperature had no significant change with shortening crack distance.It means that YSZ coating over the center of a small crack prioritized sintering before cracks coalesced.This result can explain why a fraction of small vertical cracks sometimes did not distribute around the center of the large coalesced crack [6]. Figure 4 shows the minimum temperatures between two cracks with different distances.The results show that the minimum temperature was identical to the service temperature of as-sprayed YSZ when cracks were distant, as the distance shortened, and the temperature increased.In this paper, the critical distance is defined as the distance that the temperatures over cracks were perfectly independent from each other.Thus, the critical distance is appropriately equal to the crack length when the lengths of two cracks are comparable.Actually, the two crack lengths are various before coalescence.Thus, it is more significant to clarify the critical distance between two cracks with different lengths.The critical distance deduced via Figure 4 can be separated into two parts; the dimension of each temperature-rise area is about the half of the crack length.Thus, the critical distance is the sum of the half-lengths of two coalescing cracks (i.e., a1 + a2) when cracks with different lengths propagate to each other.The sintering rate of YSZ between cracks increases with crack propagating, resulting in the elastic modulus increasing.On the basis of the data handling in the report [39], the evolution of elastic modulus, E, depended on sintering temperature, can be described by following formula: where T represents the sintering temperature, and a, b, c are the fitting constants dependent on YSZ initial microstructure, sintering time and stress state, etc., which are in the range of 16-21, 8 × 10 −50 -10 −47 , and 15-16, respectively.The constant a presents the elastic modulus of as-sprayed YSZ coating via the test method in Reference [39]. Figure 4 shows the minimum temperatures between two cracks with different distances.The results show that the minimum temperature was identical to the service temperature of as-sprayed YSZ when cracks were distant, as the distance shortened, and the temperature increased.In this paper, the critical distance is defined as the distance that the temperatures over cracks were perfectly independent from each other.Thus, the critical distance is appropriately equal to the crack length when the lengths of two cracks are comparable.Actually, the two crack lengths are various before coalescence.Thus, it is more significant to clarify the critical distance between two cracks with different lengths.The critical distance deduced via Figure 4 can be separated into two parts; the dimension of each temperature-rise area is about the half of the crack length.Thus, the critical distance is the sum of the half-lengths of two coalescing cracks (i.e., a 1 + a 2 ) when cracks with different lengths propagate to each other.Figure 4 shows the minimum temperatures between two cracks with different distances.The results show that the minimum temperature was identical to the service temperature of as-sprayed YSZ when cracks were distant, as the distance shortened, and the temperature increased.In this paper, the critical distance is defined as the distance that the temperatures over cracks were perfectly independent from each other.Thus, the critical distance is appropriately equal to the crack length when the lengths of two cracks are comparable.Actually, the two crack lengths are various before coalescence.Thus, it is more significant to clarify the critical distance between two cracks with different lengths.The critical distance deduced via Figure 4 can be separated into two parts; the dimension of each temperature-rise area is about the half of the crack length.Thus, the critical distance is the sum of the half-lengths of two coalescing cracks (i.e., a1 + a2) when cracks with different lengths propagate to each other.The sintering rate of YSZ between cracks increases with crack propagating, resulting in the elastic modulus increasing.On the basis of the data handling in the report [39], the evolution of elastic modulus, E, depended on sintering temperature, can be described by following formula: where T represents the sintering temperature, and a, b, c are the fitting constants dependent on YSZ initial microstructure, sintering time and stress state, etc., which are in the range of 16-21, 8 × 10 −50 -10 −47 , and 15-16, respectively.The constant a presents the elastic modulus of as-sprayed YSZ coating The sintering rate of YSZ between cracks increases with crack propagating, resulting in the elastic modulus increasing.On the basis of the data handling in the report [39], the evolution of elastic modulus, E, depended on sintering temperature, can be described by following formula: where T represents the sintering temperature, and a, b, c are the fitting constants dependent on YSZ initial microstructure, sintering time and stress state, etc., which are in the range of 16-21, 8 × 10 −50 -10 −47 , and 15-16, respectively.The constant a presents the elastic modulus of as-sprayed YSZ coating via the test method in Reference [39].
The elastic modulus of YSZ corresponding to the different temperatures could be evaluated via Equation (1).Thus, in order to evaluate the increasing degree of energy-release rate merely induced by heat-transfer overlapping, the elastic modulus in crack-free regions with different distances to crack tip with and without heat-transfer overlapping were evaluated.The minimum temperatures between cracks were selected to calculate elastic modulus, i.e., the corresponding distance to crack tip is the half of the distance between cracks.Correspondingly, the second typical temperatures used in the case derive from the results of the delamination cracks [6].It is worth pointing out that these two typical temperatures from the regions that are identical distances to crack tips.
The fracture toughness of YSZ coating increases with increasing sintering degree of YSZ [6,40].Thus, both cracking driving force and cracking resisting force increases with the sintering of YSZ.The influence of sintering of YSZ on durability of TBC is a fighting process between driving force and resisting force.The driving force was quantitatively evaluated in this case.The elastic modulus evaluated in the case was used to calculate the strain energy-release rate.The strain energy-release rate, G h , is introduced to provide a quantitative description of driving force [11,16,41]: where h is the YSZ thickness, ε is the strain, υ refers to the Poisson's ratio, ∆α is the thermal expansion coefficient between YSZ and substrate, and ∆T is the temperature range.On the basis of elastic-modulus calculation, the strain energy release rate was calculated merely in term of elastic modulus induced by YSZ sintering.Figure 5 shows the range of increasing rate of energy-release rate in different crack-free regions before and after heat-transfer overlapping.The increment of strain energy-release rate can be neglected when the distance to crack tip exceeds the half of the crack length, 1.0a.The increasing rate of energy-release in the region with a distance to the crack tip of 1.0a is the range of 0.2%-1.0%when crack length is larger than 1.0 mm.As a crack propagates, i.e., the distance shortens, the strain energy-release rate in regions that heat transfer overlaps increases at various level.The increasing rate approaches 1.2%-28% when distance is 0.05a, depending on the crack length.However, the crack edge distance of 0.1a corresponding to the 0.05a in the case can not, indeed, keep for a long period because of the sudden coalescing.Thus, the increasing rate of energy release merely induced by increasing elastic modulus of YSZ is approximately 0.2%-15%, depending mainly on the crack length and the crack distance, as shown in Figure 5.In comparison, the additional thermal expansion misfit caused by temperature increment over the crack induces increment of the strain energy-release approach 10% when crack length is 1.0 mm [6].Thus, the effect of stiffening induced by sintering on strain energy-release rate is on some level comparable to that of deterioration of thermal-expansion misfit induced by a temperature jump, both of which affect strain energy-release rate dramatically and increase with increasing crack length.
crack length and the crack distance, as shown in Figure 5.In comparison, the additional thermal expansion misfit caused by temperature increment over the crack induces increment of the strain energy-release approach 10% when crack length is 1.0 mm [6].Thus, the effect of stiffening induced by sintering on strain energy-release rate is on some level comparable to that of deterioration of thermal-expansion misfit induced by a temperature jump, both of which affect strain energy-release rate dramatically and increase with increasing crack length.Figure 6 shows the temperature profiles of cracks before and after coalescence.The maximum temperature over the crack with a length of 2 mm in the case was about 1250 °C when two cracks just contacted each other.In comparison, the maximum temperature over the crack with a length of 4 mm exceeded 1400 °C.It signifies that the maximum temperature over the crack rose sharply accompanied with crack coalescence.As a result, crack coalescence dramatically improved the rate of sintering of YSZ coating.The lateral tension in YSZ coating induced by YSZ sintering [6,16]  Figure 6 shows the temperature profiles of cracks before and after coalescence.The maximum temperature over the crack with a length of 2 mm in the case was about 1250 • C when two cracks just contacted each other.In comparison, the maximum temperature over the crack with a length of 4 mm exceeded 1400 • C. It signifies that the maximum temperature over the crack rose sharply accompanied with crack coalescence.As a result, crack coalescence dramatically improved the rate of sintering of YSZ coating.The lateral tension in YSZ coating induced by YSZ sintering [6,16] accelerates delamination cracking and also induces the vertical cracking.The morphology of vertical cracks is fairly analogous to that of segmentation cracks [42][43][44], while their effect on the durability of TBC is the opposite.The oxidation rate of bond coat under the vertical crack dramatically accelerates, the mixed oxides deteriorated the durability of TBC [45][46][47] locally generated under the vertical crack.accelerates delamination cracking and also induces the vertical cracking.The morphology of vertical cracks is fairly analogous to that of segmentation cracks [42][43][44], while their effect on the durability of TBC is the opposite.The oxidation rate of bond coat under the vertical crack dramatically accelerates, the mixed oxides deteriorated the durability of TBC [45][46][47] locally generated under the vertical crack.

The Temperature Distribution Depended on Distance Between Cracks Across YSZ Coating
In order to clarify whether the relative depths of the cracks affect the critical distance that the temperatures over cracks are perfectly independent from each other, the influence of the distance between two cracks along YSZ thickness on temperature distribution was addressed, as shown in Figure 7.The inplane distance between cracks in the case was fixed to be the critical distance defined in the above section (i.e., a1 + a2).The results show no recognizable correlation between two elevated temperature regions over the cracks.It is in accordance with the result corresponding to the cracks in the absence of depth variation, as shown in Figure 2a.It indicates that the relative depths of the cracks maybe contribute little to the critical distance.A detailed discussion will be elaborated below.Compared with cracks in identical depth, the maximum temperature locates the center of the side crack, and the maximum temperature increases as the crack moves to the YSZ-coating free surface.Thus, the sintering rate of the YSZ coating increases as the cracking path moves to the YSZ-coating free surface.

The Temperature Distribution Depended on Distance Between Cracks Across YSZ Coating
In order to clarify whether the relative depths of the cracks affect the critical distance that the temperatures over cracks are perfectly independent from each other, the influence of the distance between two cracks along YSZ thickness on temperature distribution was addressed, as shown in Figure 7.The inplane distance between cracks in the case was fixed to be the critical distance defined in the above section (i.e., a 1 + a 2 ).The results show no recognizable correlation between two elevated temperature regions over the cracks.It is in accordance with the result corresponding to the cracks in the absence of depth variation, as shown in Figure 2a.It indicates that the relative depths of the cracks maybe contribute little to the critical distance.A detailed discussion will be elaborated below.Compared with cracks in identical depth, the maximum temperature locates the center of the side crack, and the maximum temperature increases as the crack moves to the YSZ-coating free surface.Thus, the sintering rate of the YSZ coating increases as the cracking path moves to the YSZ-coating free surface.
between two cracks along YSZ thickness on temperature distribution was addressed, as shown in Figure 7.The inplane distance between cracks in the case was fixed to be the critical distance defined in the above section (i.e., a1 + a2).The results show no recognizable correlation between two elevated temperature regions over the cracks.It is in accordance with the result corresponding to the cracks in the absence of depth variation, as shown in Figure 2a.It indicates that the relative depths of the cracks maybe contribute little to the critical distance.A detailed discussion will be elaborated below.Compared with cracks in identical depth, the maximum temperature locates the center of the side crack, and the maximum temperature increases as the crack moves to the YSZ-coating free surface.Thus, the sintering rate of the YSZ coating increases as the cracking path moves to the YSZ-coating free surface.The minimum temperature between two cracks in Figure 7 was plotted, as shown in Figure 8.The inplane distance between two cracks in this case was fixed to be the sum of halves of two crack lengths (i.e., a 1 + a 2 ).These minimum temperatures are all comparable.Thus, the critical distance that the temperature regions over cracks just overlap can be considered as the sum of the half-lengths of two cracks, which is independent of the relative depths of the cracks.The minimum temperature between two cracks in Figure 7 was plotted, as shown in Figure 8.The inplane distance between two cracks in this case was fixed to be the sum of halves of two crack lengths (i.e., a1 + a2).These minimum temperatures are all comparable.Thus, the critical distance that the temperature regions over cracks just overlap can be considered as the sum of the half-lengths of two cracks, which is independent of the relative depths of the cracks.

Conclusions
A three-dimensional FEM was established to evaluate the influence of multicracks on temperature distribution characteristic in plasma-sprayed thermal-barrier coatings.The main results from this model are outlined here.The temperature rises of the YSZ coating over the crack were independent from each other at the beginning of crack propagation.As cracks propagated, the independent elevated-temperature regions started to overlap.The critical distance of the regions of temperature rises, just overlapping, was the sum of half-lengths of two coalescing cracks (i.e., a1 + a2), which was independent of the cracking path in TBC.The maximum temperature was still located at the crack center before crack coalescence, in spite of heat-transfer overlapping between crack tips.The maximum temperature on the YSZ free surface sharply increased once cracks coalesced.

Conclusions
A three-dimensional FEM was established to evaluate the influence of multicracks on temperature distribution characteristic in plasma-sprayed thermal-barrier coatings.The main results from this model are outlined here.The temperature rises of the YSZ coating over the crack were independent from each other at the beginning of crack propagation.As cracks propagated, the independent

Figure 1 .
Figure 1.The schematic model of thermal-barrier coatings (TBC): (a) the penny-shaped cracks in a three-dimensional view; (b) cross-sectional view of the model included cracks with an identical depth; and (c) the model contained cracks with different depths.

Figure 1 .
Figure 1.The schematic model of thermal-barrier coatings (TBC): (a) the penny-shaped cracks in a three-dimensional view; (b) cross-sectional view of the model included cracks with an identical depth; and (c) the model contained cracks with different depths.

Figure 2 .
Figure 2. The distance of crack edges dependence of temperature distribution on the YSZ free surface: (a) 2.0a; (b) 1.0a; (c) 0.5a; (d) 0.3a; (e) 0.1a; and (f) 0 μm."a" is the half length of the crack.The lengths of three cracks are all 2.0 mm.

Figure 2 .
Figure 2. The distance of crack edges dependence of temperature distribution on the YSZ free surface: (a) 2.0a; (b) 1.0a; (c) 0.5a; (d) 0.3a; (e) 0.1a; and (f) 0 µm."a" is the half length of the crack.The lengths of three cracks are all 2.0 mm.

Coatings 2018, 8 , 11 Figure 3 .
Figure 3. Temperature profiles plotted as a function of the crack distance.

Figure 4 .
Figure 4.The minimum temperature between two cracks as a function of the distance between edges of two cracks."a" is the half length of the crack.

Figure 3 .
Figure 3. Temperature profiles plotted as a function of the crack distance.

Coatings 2018, 8 , 11 Figure 3 .
Figure 3. Temperature profiles plotted as a function of the crack distance.

Figure 4 .
Figure 4.The minimum temperature between two cracks as a function of the distance between edges of two cracks."a" is the half length of the crack.

Figure 4 .
Figure 4.The minimum temperature between two cracks as a function of the distance between edges of two cracks."a" is the half length of the crack.

Figure 5 .
Figure 5.The influence of heat-transfer overlapping on increasing-rate range of energy-release rate.

Figure 5 .
Figure 5.The influence of heat-transfer overlapping on increasing-rate range of energy-release rate.

Figure 6 .
Figure 6.The temperature profiles of cracks before and after coalescence.

Figure 6 .
Figure 6.The temperature profiles of cracks before and after coalescence.

Figure 7 .
Figure 7.The crack distance across the YSZ coating dependent of temperature distribution on the YSZ free surface: (a) 20 μm; (b) 50 μm; (c) 100 μm; and (d) 150 μm.The lengths of cracks and inplane distances between two cracks are all 2 mm.

Figure 7 .
Figure 7.The crack distance across the YSZ coating dependent of temperature distribution on the YSZ free surface: (a) 20 µm; (b) 50 µm; (c) 100 µm; and (d) 150 µm.The lengths of cracks and inplane distances between two cracks are all 2 mm.

Figure 8 .
Figure 8.The minimum temperature between two cracks as a function of the distance along YSZ thickness, the lengths of cracks, and inplane distances between two cracks are all 2 mm.

Figure 8 .
Figure 8.The minimum temperature between two cracks as a function of the distance along YSZ thickness, the lengths of cracks, and inplane distances between two cracks are all 2 mm.