Strain-Induced Cracking Behavior of Coating / Substrate Systems and Strain Tolerant Design for Thick Coatings

: The life span for a coating attached to its substrate is basic support for their desired protective function. Therefore, it is necessary to ﬁnd out the causes responsible for the failure of coatings during service. This paper developed a ﬁnite element model to investigate the cracking behavior of plasma-sprayed ceramic coatings induced by the mismatch strain of thermal expansion between coating and substrate. Crack propagation a ﬀ ected by coating thicknesses was realized by the virtual crack closure technique (VCCT). The residual stresses ( σ 22 and σ 12 ) and the strain energy release rate (SERR) induced at the tip of pre-crack in ceramic coatings are calculated. Results show that the σ 22 and σ 12 at the tip of the pre-crack increases continuously with the thickening ceramic coatings. The SERRs at the tip of the pre-crack in top-coat (TC) were increased with the thickness of ceramic coatings, resulting in the propagation of cracks. The crack length increases with the thickening of ceramic coatings. The crack propagation and coalescence lead to coating spallation, which is one of the main failure modes for plasma sprayed ceramic coatings during service. Given that, strain tolerant design was developed by inserting vertical pores in coatings. It was found that the SERRs were decreased with the increase in the number of vertical pores, as well as their depth. Moreover, the coatings with vertical pores appear to be crack-resistant, in particular for the thicker coatings. This suggests that the strain tolerant design is helpful to extend the life span of thick coatings, which makes a fundamental contribution to the design and preparation of advanced protective coatings in future applications.


Introduction
Ceramic coatings are often used to protect metallic substrate for desired functions, in particular for high-temperature-related fields, such as thermal barrier coatings (TBCs) and radar-absorbing coatings [1][2][3]. The coating thickness can be 200 µm-3 mm [4,5]. As one of the widely-used ceramic protective coatings, TBCs are advanced materials system generally used for many years to protect the metallic components of the gas turbine engine-such as nozzles, turbine blades, combustors, and vanes that undergo hot gases-thus the temperature of the substrate surface can be significantly lowered [6][7][8][9][10]. TBCs with a thickness of 200 µm-1 mm serve to insulate the engine's components from high and large-time heat loads [11]. As a result, the TBCs are used to attain greater functioning temperatures [12], which are highly needed for higher efficiency [13] and thrust-weight ratios. Therefore, the demand to further understand the short life span of TBCs and to suggest some reasonable coating optimization methods. The failure tendency and failure modes of TBCs are often judged by the magnitude of maximum stress and deriving force. The research methods are mostly static. Crack growth/fracture in which layers are dissociated from one another, or material is damaged due to external load. With the build out of finite element modeling (FEM) [58], many methods such as: virtual crack closure technique (VCCT), cohesive zone model (CZM), extended finite element method (XFEM), and conventional finite element method (CFEM) can be used to simulate the behavior of crack propagation [59][60][61][62]. Among various methods, VCCT has been used prominently to investigate the dynamic failure behavior of coatings. Because VCCT is very suitable to calculate the SERR (or J integral) during the crack propagation based on the thought that the necessary energy when the crack propagates a tiny displacement is equal to the work of making the crack closed [62].
This study mainly focuses on two aspects. On one hand, the dominant parameters on the spallation of APS ceramic coatings were investigated by a finite element model, to understand the cracking propagation during service. On the other hand, a strain-tolerant design for APS ceramic coatings was proposed to enhance its crack-resistance, to extend the life span for thick coatings.

Geometry of Model
The TBC system, which is made up of TC, BC, and SUB, was used to investigate the crack propagation in TC. In this paper, the FEM software package ABAQUS (Version: 6.14-1) [63] is adopted to carry out numerical calculations. Figure 1a shows the whole geometry of the model which is used for finite element analysis (FEA). The length of the model is chosen as 10 mm and the thickness of the substrate and BC are chosen as 5 mm and 150 µm. The effect of TC thickness on crack propagation was investigated here, therefore, the thicknesses of TC were selected to be different. The pre-crack along the coating surface is inserted in the model, as shown in Figure 1a, in this model, a is the crack length and b is the vertical distance from TC/BC interface. In detail, a refers to 100 µm and b is 50 µm. In this study, models with and without vertical pores were investigated comparatively. For the former case, vertical pores were inserted in TC for strain tolerant design. In this model, c is the width of vertical pores, e is the distance between two vertical pores, and d is the depth of vertical pores. In this study, models with and without vertical pores were investigated comparatively. For the former case, vertical pores were inserted in TC for strain tolerant design. In this model, c is the width of vertical pores, e is the distance between two vertical pores, and d is the depth of vertical pores. Where h tc , h bc , and h sub are the thickness of TC, BC, and Sub, respectively. The width of vertical pores Coatings 2020, 10, 1066 4 of 19 is 5 µm. The influence of the number of vertical pores and depth of vertical pores on crack propagation is also performed here. The model is periodic symmetry; therefore, a representative unit-cell was used for simulation. This horizontal pre-crack shows a series of cracks. Once, this horizontal pre-crack propagates to the left side of the model, it will intersect with the crack in the neighboring region and then coating spallation will occur. The residual stresses will be calculating again due to the effect of crack. The virtual crack closure technique (VCCT) [64], which is based on the fracture mechanics concepts, is used to calculate the strain energy release rate (SERR) at the crack tip. The regular quadrilateral element is used to mesh the model, and the overall mesh model is represented in Figure 1b. The mesh improvement is accomplished near the crack region and interface because of our great interest in this region. The refine mesh in this region is vividly presented in Figure 1c. At the crack tip, the SERR will not be affected by the size of the mesh because the mesh grid is dense enough.

Boundary Conditions
The current finite element model is based on the following hypotheses: In this model, (1) TC layer aside with a pre-existing crack and all other layers are homogeneous and isotropic. The geometric morphology at the BC/TC interference is flat; (2) The mechanical properties such as the Poisson ratio and Young's modulus of all layers are listed in Table 1 [65][66][67][68]; (3) The SUB and BC layer behaved as elastic-plastic material, and the TC layer was regarded as elastic behavior [69]. Table 1 shows the properties of the materials used for SUB, BC, and TC. It is worth noting that the properties of the materials were based on typical materials used for TBC. Since the representative unit a from the coating sample, periodic boundary conditions were used in this study. The periodicity boundary constraint which can be understood by utilizing a multi-point coupling (MPC), is applied on the right edge of the model shown in Figure 1b. Thus, all the nodes located on the edge have similar displacement along direction 1. Simultaneously, all these nodes can move freely along the y-direction. On the left edge of the model, a strain of 0.2% is applied to model the mismatch strain between coating and substrate. Additionally, nodes of the bottom boundary of model are restricted to move along the y-direction, which can stop the incident of rigid-body displacement.

Modeling Tool Used for Crack Propagation
When the crack growth is simulated in the TC layer, another crack surface must be made once a predetermined criterion is met during the thermal cycling. In ABAQUS [63], by utilizing the "debond" method, the growth of crack can be effectively captured. This crack propagation tool enables the crack to propagate along a fixed way. The virtual crack closure technique (VCCT) method, based on the principles of linear elastic fracture mechanics (LEFM) that was proposed by Rybicki and Kanninen [64], is used to calculate strain energy release rate (SERR) referred to as the crack driving force. With this technique, the components of SERR G I and G II can be effectively obtained.
The VCCT technique is one of the efficient and most popular tools to find the mode-dependent components of SERR G I and G II by using nodal force and displacement. A relatively stiff spring is placed between the node pair at the crack-tip to extract the internal nodal forces, while the node pair behind the crack-tip is utilized to extract the information for displacement openings [35]. The schematic Coatings 2020, 10, 1066 5 of 19 diagram of VCCT for four node elements is shown in Figure 2. The components of SERR G I and G II can be expressed as [35] where G I and G II are the SERR for Mode I and Mode II respectively, and F ν,3,4, and F u,3,4 are the components of nodal force at crack tip. The v i and u i are the components of nodal displacement behind the crack tip and B is the thickness in third-direction and its value for the two-dimension (2D) model normally equals 1, with unit thickness. In this work, the criterion of crack-propagation for a top-coat layer can be examined by a power law [71], which is expressed as where G equiv and G equivc are the equivalents and critical equivalent SERR, G IC and G IIC are the critical SERR for mode I and II respectively, the value, 1.0, is chosen for both a m and a n . The crack propagation occurs when the parameter f becomes 1.0. When the value 1.0 is chosen for power exponents (a m and a n ) [35,70], the power law stated as Once the G equiv reaches the G equivc calculated by using the user specified mode-max criterion, the nodes of the crack tip will debond [63]. The fracture toughness of the TC layer reported in previous articles [72][73][74] is in the range of 5 to 15 J/m 2 . In current research, we presumed G IC = G IIC = 15 J/m 2 [35,73] for TC layer. It is noted that the value of critical SERR is surmised to be equivalent for mode (I and II) due to the deficiency of relevant mode-dependent experimental data.
Coatings 2020, 10, x FOR PEER REVIEW 5 of 19 schematic diagram of VCCT for four node elements is shown in Figure 2. The components of SERR GI and GII can be expressed as [35] (1) where GI and GII are the SERR for Mode I and Mode II respectively, and Fν,₃,₄, and Fᵤ,₃,₄ are the components of nodal force at crack tip. The vi and ui are the components of nodal displacement behind the crack tip and B is the thickness in third-direction and its value for the two-dimension (2D) model normally equals 1, with unit thickness. In this work, the criterion of crack-propagation for a top-coat layer can be examined by a power law [71], which is expressed as where Gequiv and Gequivc are the equivalents and critical equivalent SERR, GIC and GIIC are the critical SERR for mode I and II respectively, the value, 1.0, is chosen for both am and an. The crack propagation occurs when the parameter f becomes 1.0. When the value 1.0 is chosen for power exponents (am and an) [35,70], the power law stated as

Gequiv/Gequivc = (GI/GIc) + (GII/GIIc)
Once the Gequiv reaches the Gequivc calculated by using the user specified mode-max criterion, the nodes of the crack tip will debond [63]. The fracture toughness of the TC layer reported in previous articles [72][73][74] is in the range of 5 to 15 J/m 2 . In current research, we presumed GIC = GIIC = 15 J/m 2 [35,73] for TC layer. It is noted that the value of critical SERR is surmised to be equivalent for mode (I and II) due to the deficiency of relevant mode-dependent experimental data.

Stress Distribution in TC with and without Vertical Pores
To investigate the effect of coating thickness on SERR of pre-crack, a very large critical energy release rate (Gequivc) was used for cases with and without vertical pores, the stress distribution in TC layer are primarily investigated to achieve some important data at the tip of pre-crack. Generally, the residual stresses developed in ceramic coating due to the difference of thermal expansion coefficient of elements, which cause the degradation of coatings. The formation of the cracks is usually attributed to the quenching stress and thermal mismatch stress. The detailed of residual stresses can be seen our published paper [4]. The black horizontal line in Figure 3 is the crack path and the point where stress

Stress Distribution in TC with and without Vertical Pores
To investigate the effect of coating thickness on SERR of pre-crack, a very large critical energy release rate (G equivc ) was used for cases with and without vertical pores, the stress distribution in TC layer are primarily investigated to achieve some important data at the tip of pre-crack. Generally, the residual stresses developed in ceramic coating due to the difference of thermal expansion coefficient of elements, which cause the degradation of coatings. The formation of the cracks is usually attributed to the quenching stress and thermal mismatch stress. The detailed of residual stresses can be seen our published paper [4]. The black horizontal line in Figure 3 is the crack path and the point where stress is maximum on the line is the pre-crack tip. Figure 3a,b demonstrate the residual stresses field including σ 22 and σ 12 at different thickness (3, 2, 1, 0,8, 0.5, and 0.3 mm) of TC. The residual stresses (σ 22 = 224 MPa and σ 12 = 136 MPa) are very high when the thickness of TC is 3 mm and their values gradually decrease (σ 22 = 224 to 56 MPa and σ 12 = 136 to 53.6 MPa) when decreasing the thickness of TC up to 0.3 mm. The continuous increase in stresses is necessary to initiate or propagate the crack [66,75]. It means that the crack is easier to extend with a larger thickness of TC.
Coatings 2020, 10, x FOR PEER REVIEW 6 of 19 is maximum on the line is the pre-crack tip. Figure 3a,b demonstrate the residual stresses field including σ22 and σ12 at different thickness (3, 2 [66,75]. It means that the crack is easier to extend with a larger thickness of TC.    These results indicate that the ceramic coating with vertical pores has a lower value of residual stresses compared to ceramic coating without vertical pores, as reported in the literature [4,76].           Figure 9b a similar trend can be observed for stress σ12. The continuous increase in stress is necessary to initiate or spread the crack [66]. The variation of stress σ22 and σ12 at the crack tip in Figure 9 is the function of TC thickness.     Figure 9a. From Figure 9b a similar trend can be observed for stress σ12. The continuous increase in stress is necessary to initiate or spread the crack [66]. The variation of stress σ22 and σ12 at the crack tip in Figure 9 is the function of TC thickness.     Figure 9a. From Figure 9b a similar trend can be observed for stress σ12. The continuous increase in stress is necessary to initiate or spread the crack [66]. The variation of stress σ22 and σ12 at the crack tip in Figure 9 is the function of TC thickness.   Figure 9b a similar trend can be observed for stress σ 12 . The continuous increase in stress is necessary to initiate or spread the crack [66]. The variation of stress σ 22 and σ 12 at the crack tip in Figure 9 is the function of TC thickness. Coatings 2020, 10, x FOR PEER REVIEW 9 of 19  Figure 10 shows the stress distribution in topcoat when one vertical pore and two vertical pores were inserted into 3, 2, and 1 mm TC thickness with different depth, along the crack path in TC. The stress (σ22, σ12) at the pre-crack tip is maximum when the depth of the vertical pore is minimum and the value of stress continuously decreases with increasing the depth of the vertical pore at same thickness (3, 2, and 1 mm) of TC and same number of vertical pores (1 and 2). The value of stress is higher in case of one vertical pore as compared to the cases of two vertical pores in all 3, 2, and 1 mm TC thickness, and at various depth of vertical pores. Here we also noted that the stress at the crack tip along the crack path is maximum when one vertical pore and two vertical pores at same depth, inserted into 3 mm thickness of topcoat. Its value has reduced at same number of vertical pores and same depth when the thickness of TC is 2 mm. The stress is minimum at the crack tip in case of one vertical pore and two vertical pores inserted into 1 mm thickness of TC. As a result, crack propagation or crack growth depends upon the thickness of TC and also depends upon the number of vertical pores in the top-coat and their depth in TC [76,77].  Figure 10 shows the stress distribution in topcoat when one vertical pore and two vertical pores were inserted into 3, 2, and 1 mm TC thickness with different depth, along the crack path in TC. The stress (σ 22 , σ 12 ) at the pre-crack tip is maximum when the depth of the vertical pore is minimum and the value of stress continuously decreases with increasing the depth of the vertical pore at same thickness (3, 2, and 1 mm) of TC and same number of vertical pores (1 and 2). The value of stress is higher in case of one vertical pore as compared to the cases of two vertical pores in all 3, 2, and 1 mm TC thickness, and at various depth of vertical pores. Here we also noted that the stress at the crack tip along the crack path is maximum when one vertical pore and two vertical pores at same depth, inserted into 3 mm thickness of topcoat. Its value has reduced at same number of vertical pores and same depth when the thickness of TC is 2 mm. The stress is minimum at the crack tip in case of one vertical pore and two vertical pores inserted into 1 mm thickness of TC. As a result, crack propagation or crack growth depends upon the thickness of TC and also depends upon the number of vertical pores in the top-coat and their depth in TC [76,77].  Figure 10 shows the stress distribution in topcoat when one vertical pore and two vertical pores were inserted into 3, 2, and 1 mm TC thickness with different depth, along the crack path in TC. The stress (σ22, σ12) at the pre-crack tip is maximum when the depth of the vertical pore is minimum and the value of stress continuously decreases with increasing the depth of the vertical pore at same thickness (3, 2, and 1 mm) of TC and same number of vertical pores (1 and 2). The value of stress is higher in case of one vertical pore as compared to the cases of two vertical pores in all 3, 2, and 1 mm TC thickness, and at various depth of vertical pores. Here we also noted that the stress at the crack tip along the crack path is maximum when one vertical pore and two vertical pores at same depth, inserted into 3 mm thickness of topcoat. Its value has reduced at same number of vertical pores and same depth when the thickness of TC is 2 mm. The stress is minimum at the crack tip in case of one vertical pore and two vertical pores inserted into 1 mm thickness of TC. As a result, crack propagation or crack growth depends upon the thickness of TC and also depends upon the number of vertical pores in the top-coat and their depth in TC [76,77].

Effect of General Feature on Cracking Driving Force
For the aim of simulating the cracking and coating delamination, the SERR values at the crack tip must be evaluated and compared with the fracture toughness of the ceramic TC layer [35]. Before the occurrence of propagation of crack, the typical results of changes in SERR with thickness of TC and the number of vertical pores into TC with different depths are plotted in Figure 11. G t represents the total SERR, which is the sum of SERR components G I and G II [66]. The SERRs continuously increase with increasing the thickness of TC as shown in Figure 11a. On the whole, the SERR G I , G II , and G t increase with increasing the thickness of TC. Here it needs to be noted that the crack driving force has the largest value (0.2432 N/mm) occur when the thickness of TC is 3 mm and the lowest value (0.0261 N/mm) occurs when the thickness of TC is 0.3 mm. When one vertical pore and two vertical pores with different depth inserted into TC of different thicknesses as shown as in Figure 10b-g. The variation of SERR G I , G II , and G t as a function of a depth of vertical pores, shows a decreasing trend with increasing the depth of vertical pores. It is also noted that the SERR G I , G II , and G t depend on the number of vertical pores and also depend upon the thickness of TC in which vertical pores are inserted. The lowest values of SERR G I = 0.036 N/mm, G II = 0.0234 N/mm, and G t = 0.0594 N/mm have noted for two vertical pores as compare to one vertical pore when the thickness of TC is 3 mm. It is also noted that the value of SERR G I , G II , and G t is lowest when two vertical pores are inserted into 1 mm thickness of TC as compare to 3 mm thickness, which means that the degradation of coating strongly depends upon the number of vertical pores and thickness of TC. This phenomenon can be explained easily. Due to vertical pores, the strain tolerance energy will enhance. Although the high strain tolerance energy is beneficial to the improvement of the thermal shock resistance, which controls the crack propagation [65,69]. Also, the stress σ 22 and σ 12 are largest when the thickness of TC is 3 mm and lowest when TC thickness is 0.3 mm and in case of vertical pores their value is lowest when two vertical pores were inserted in 1 mm TC thickness. Therefore, the crack deriving force is maximum when TC is 3 mm and decreases continuously with decreasing the thickness. Its value is also lowest in case of two vertical pores in 0.3 mm TC thickness. This indicates that the propagation or growth of crack depends upon the thickness of TC and also depends upon the number of vertical pores and depth of vertical pores.

Effect of General Feature on Cracking Driving Force
For the aim of simulating the cracking and coating delamination, the SERR values at the crack tip must be evaluated and compared with the fracture toughness of the ceramic TC layer [35]. Before the occurrence of propagation of crack, the typical results of changes in SERR with thickness of TC and the number of vertical pores into TC with different depths are plotted in Figure 11. Gt represents the total SERR, which is the sum of SERR components GI and GII [66]. The SERRs continuously increase with increasing the thickness of TC as shown in Figure 11a. On the whole, the SERR GI, GII, and Gt increase with increasing the thickness of TC. Here it needs to be noted that the crack driving force has the largest value (0.2432 N/mm) occur when the thickness of TC is 3 mm and the lowest value (0.0261 N/mm) occurs when the thickness of TC is 0.3 mm. When one vertical pore and two vertical pores with different depth inserted into TC of different thicknesses as shown as in Figure  10b-g. The variation of SERR Gı, Gıı, and Gt as a function of a depth of vertical pores, shows a decreasing trend with increasing the depth of vertical pores. It is also noted that the SERR Gı, Gıı, and Gt depend on the number of vertical pores and also depend upon the thickness of TC in which vertical pores are inserted. The lowest values of SERR Gı = 0.036 N/mm, Gıı = 0.0234 N/mm, and Gt = 0.0594 N/mm have noted for two vertical pores as compare to one vertical pore when the thickness of TC is 3 mm. It is also noted that the value of SERR Gı, Gıı, and Gt is lowest when two vertical pores are inserted into 1 mm thickness of TC as compare to 3 mm thickness, which means that the degradation of coating strongly depends upon the number of vertical pores and thickness of TC. This phenomenon can be explained easily. Due to vertical pores, the strain tolerance energy will enhance. Although the high strain tolerance energy is beneficial to the improvement of the thermal shock resistance, which controls the crack propagation [65,69]. Also, the stress σ22 and σ12 are largest when the thickness of TC is 3 mm and lowest when TC thickness is 0.3 mm and in case of vertical pores their value is lowest when two vertical pores were inserted in 1 mm TC thickness. Therefore, the crack deriving force is maximum when TC is 3 mm and decreases continuously with decreasing the thickness. Its value is also lowest in case of two vertical pores in 0.3 mm TC thickness. This indicates that the propagation or growth of crack depends upon the thickness of TC and also depends upon the number of vertical pores and depth of vertical pores.

Investigation of Crack Propagation
A pre-crack is inserted into the model. The growth of crack depends upon the thickness of the ceramic TC. The crack length continuously increases with increasing the thickness of ceramic TC. The variation of crack length with thickness of ceramic TC has been demonstrated in Figure 12. Here the normalized crack length a = a/L is used to survey the overall damage, which is defined by the ratio of total crack length "a" and length of model "L" [66]. The crack in TC is rapidly propagated toward the left into the model due to more and more stress exerted on the tip of crack with variation of TC thickness from 0.3 to 3 mm as shown in Figure 12. The one crack meets with a neighboring crack. The crack coalescence leads to the coating spallation, which is responsible for the ultimate failure of TBCs [35]. The growth of crack in TC can also be reduced by inserting vertical pores into the TC. Figure 13 exhibits that overall crack growth trend with the depth of vertical pores. Here we noted that the crack length (5.06005 mm) in case of without vertical pores is maximum. When one vertical pore of depth 0.1 mm is inserted into TC, the crack length, (5 mm) is reduced as compared to the case without vertical pore. The crack length is further reduced with increasing the depth of vertical pore and lowest value of crack length (3.25115 mm) when the depth of one vertical pore was 0.9 mm. When two vertical pores were inserted into TC, the crack length also exhibit decreasing trend with increasing the depth of vertical pores. The crack growth has minimum value (2.18438 mm) when two vertical pores of 0.9 mm depth were inserted into TC. From Figure 13, we noted that the growth of crack is reduced in case of two vertical pores as compare to one vertical pore. From these results, it is clear that the growth of crack depends upon TC thickness and also depend upon number of vertical pores and their depth. Figure 11. SERR 11 , SERR 12 , and SERR total at crack tip affected by, thickness of TC (a) and depth of vertical pores at different thicknesses of TC (b-g).

Investigation of Crack Propagation
A pre-crack is inserted into the model. The growth of crack depends upon the thickness of the ceramic TC. The crack length continuously increases with increasing the thickness of ceramic TC. The variation of crack length with thickness of ceramic TC has been demonstrated in Figure 12.
Here the normalized crack length a = a/L is used to survey the overall damage, which is defined by the ratio of total crack length "a" and length of model "L" [66]. The crack in TC is rapidly propagated toward the left into the model due to more and more stress exerted on the tip of crack with variation of TC thickness from 0.3 to 3 mm as shown in Figure 12. The one crack meets with a neighboring crack. The crack coalescence leads to the coating spallation, which is responsible for the ultimate failure of TBCs [35]. The growth of crack in TC can also be reduced by inserting vertical pores into the TC. Figure 13 exhibits that overall crack growth trend with the depth of vertical pores. Here we noted that the crack length (5.06005 mm) in case of without vertical pores is maximum. When one vertical pore of depth 0.1 mm is inserted into TC, the crack length, (5 mm) is reduced as compared to the case without vertical pore. The crack length is further reduced with increasing the depth of vertical pore and lowest value of crack length (3.25115 mm) when the depth of one vertical pore was 0.9 mm. When two vertical pores were inserted into TC, the crack length also exhibit decreasing trend with increasing the depth of vertical pores. The crack growth has minimum value (2.18438 mm) when two vertical pores of 0.9 mm depth were inserted into TC. From Figure 13, we noted that the growth of crack is reduced in case of two vertical pores as compare to one vertical pore. From these results, it is clear that the growth of crack depends upon TC thickness and also depend upon number of vertical pores and their depth.

Strain Tolerant Design for Thick Coatings
Residual stresses σ22 and σ12, and SERR Gı, Gıı, and Gt with respect to depth of vertical pores at different crack density are plotted as shown in Figure 14. Crack density means the distance "e" between two vertical pores divided by TC thickness "b", crack density = e/b. The vertical pores in ceramic coatings can instead release strain energy and enhance the strain tolerance which should be one of the reasons to reduce crack growth and enhance the lifetime of coatings [4]. In Figure 14a Figure 14b,d,f. The overall values of SERR Gı, Gıı, and Gt are largest when the crack density is 0.1 and their values reduce with increasing the crack density and show the lowest values when crack density is 0.3. It is found from Figure 14 that the growth of crack also depends upon crack density. The growth of crack extended with reducing the crack density.
For TBCs, a thicker coating means larger thermal insulation, and a thicker coating is necessary for radar absorbing. Therefore, thick coatings have wide applications in engineering [69,78]. However, based on our investigation, a larger thickness often refers to a larger SERR at crack tip. This means a thicker coating is easier to be cracked or spalled. That is why these thicker coatings have a shorter life span, which is undesired for coating application. Based on our investigation, inserting vertical pores in thick coatings have a significant effect to reduce the SERR. The crack depth and density also have a distinct effect on the reduced degree of SERR. Therefore, to enhance crackresistance of thick coatings, inserting vertical pores with reasonable depth and density would be a feasible way. Another problem, how to insert vertical pores in APS ceramic coating. The dense-

Strain Tolerant Design for Thick Coatings
Residual stresses σ22 and σ12, and SERR Gı, Gıı, and Gt with respect to depth of vertical pores at different crack density are plotted as shown in Figure 14. Crack density means the distance "e" between two vertical pores divided by TC thickness "b", crack density = e/b. The vertical pores in ceramic coatings can instead release strain energy and enhance the strain tolerance which should be one of the reasons to reduce crack growth and enhance the lifetime of coatings [4]. In Figure 14a Figure 14b,d,f. The overall values of SERR Gı, Gıı, and Gt are largest when the crack density is 0.1 and their values reduce with increasing the crack density and show the lowest values when crack density is 0.3. It is found from Figure 14 that the growth of crack also depends upon crack density. The growth of crack extended with reducing the crack density.
For TBCs, a thicker coating means larger thermal insulation, and a thicker coating is necessary for radar absorbing. Therefore, thick coatings have wide applications in engineering [69,78]. However, based on our investigation, a larger thickness often refers to a larger SERR at crack tip. This means a thicker coating is easier to be cracked or spalled. That is why these thicker coatings have a shorter life span, which is undesired for coating application. Based on our investigation, inserting vertical pores in thick coatings have a significant effect to reduce the SERR. The crack depth and density also have a distinct effect on the reduced degree of SERR. Therefore, to enhance crackresistance of thick coatings, inserting vertical pores with reasonable depth and density would be a feasible way. Another problem, how to insert vertical pores in APS ceramic coating. The dense-

Strain Tolerant Design for Thick Coatings
Residual stresses σ 22 and σ 12 , and SERR G I , G II , and G t with respect to depth of vertical pores at different crack density are plotted as shown in Figure 14. Crack density means the distance "e" between two vertical pores divided by TC thickness "b", crack density = e/b. The vertical pores in ceramic coatings can instead release strain energy and enhance the strain tolerance which should be one of the reasons to reduce crack growth and enhance the lifetime of coatings [4]. In Figure 14a Figure 14b,d,f. The overall values of SERR G I , G II , and G t are largest when the crack density is 0.1 and their values reduce with increasing the crack density and show the lowest values when crack density is 0.3. It is found from Figure 14 that the growth of crack also depends upon crack density. The growth of crack extended with reducing the crack density.
For TBCs, a thicker coating means larger thermal insulation, and a thicker coating is necessary for radar absorbing. Therefore, thick coatings have wide applications in engineering [69,78]. However, based on our investigation, a larger thickness often refers to a larger SERR at crack tip. This means a thicker coating is easier to be cracked or spalled. That is why these thicker coatings have a shorter life span, which is undesired for coating application. Based on our investigation, inserting vertical pores in thick coatings have a significant effect to reduce the SERR. The crack depth and density also have a distinct effect on the reduced degree of SERR. Therefore, to enhance crack-resistance of thick coatings, inserting vertical pores with reasonable depth and density would be a feasible way. Another problem, how to insert vertical pores in APS ceramic coating. The dense-vertical cracked structure (DVC) method is used, the structure has to have a higher density than conventional TBC. As a result, the thermal conductivity should be higher, and the thermal insulation is weakened [8,79]. In future work, it is necessary to develop new ways to insert vertical pores in porous APS coatings. In this way, the function and life span can be simultaneously enhanced.
Coatings 2020, 10, x FOR PEER REVIEW 14 of 19 vertical cracked structure (DVC) method is used, the structure has to have a higher density than conventional TBC. As a result, the thermal conductivity should be higher, and the thermal insulation is weakened [8,79]. In future work, it is necessary to develop new ways to insert vertical pores in porous APS coatings. In this way, the function and life span can be simultaneously enhanced.

Conclusions
In this study, the strain-induced cracking behavior of plasma sprayed ceramic coatings were thoroughly investigated using a finite element model. The propagation of crack behavior in top-coat was studied with different thicknesses of TC. Additionally, the strain tolerant design was proposed by inserting vertical pores in coatings. The effect of the number of vertical pores and their depth on stresses and SERRs were also investigated to broadly understand cracking-resistant behavior. The main conclusions include: 1. In top-coat (TC), the maximum stress are mainly concentrated at the tip of crack, which may lead to incipient crack nucleate and can cause the crack propagation in TC. Besides, these stresses (σ22 and σ12) and SERR increase continuously with the thickening of TC. 2. Vertical pores can enhance the strain tolerance of the TCs. The values of stresses (σ22 and σ12) decrease when one vertical pore is inserted in TC as compare to without vertical pore and further

Conclusions
In this study, the strain-induced cracking behavior of plasma sprayed ceramic coatings were thoroughly investigated using a finite element model. The propagation of crack behavior in top-coat was studied with different thicknesses of TC. Additionally, the strain tolerant design was proposed by inserting vertical pores in coatings. The effect of the number of vertical pores and their depth on stresses and SERRs were also investigated to broadly understand cracking-resistant behavior. The main conclusions include:

1.
In top-coat (TC), the maximum stress are mainly concentrated at the tip of crack, which may lead to incipient crack nucleate and can cause the crack propagation in TC. Besides, these stresses (σ 22 and σ 12 ) and SERR increase continuously with the thickening of TC.

2.
Vertical pores can enhance the strain tolerance of the TCs. The values of stresses (σ 22 and σ 12 ) decrease when one vertical pore is inserted in TC as compare to without vertical pore and further decreased for two vertical pores. Their values also decreased with an increase in the depth of vertical pores.

3.
The values of SERRs for TBCs with vertical pores decrease compared to the TC without vertical pores. Their values also exhibit a decreasing trend with increasing the depth of vertical pores.
These results indicate that the TCs with vertical pores exhibits excellent cracking resistance. This would contribute to extending the life span of thick coatings.