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

^{1}

^{2}

^{*}

## Abstract

**:**

_{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.

## 1. Introduction

## 2. Model Development

#### 2.1. Geometry of Model

_{tc}, h

_{bc}, and h

_{sub}are the thickness of TC, BC, and Sub, respectively. The width of vertical pores 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.

#### 2.2. Boundary Conditions

#### 2.3. Modeling Tool Used for Crack Propagation

_{I}and G

_{II}can be effectively obtained.

_{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 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]

_{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

_{equiv}/G

_{equivc}= (G

_{I}/G

_{Ic})

^{am}+ (G

_{II}/G

_{IIc})

^{an}

_{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

_{equiv}/G

_{equivc}= (G

_{I}/G

_{Ic}) + (G

_{II}/G

_{IIc})

_{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.

## 3. Results and Discussion

#### 3.1. Stress Distribution in TC with and without Vertical Pores

_{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.

_{22}and σ

_{12}) at the crack tip with a very large critical energy release rate (G

_{equivc}), when one vertical pore is inserted into top-coat of thickness 3 mm. The values of residual stresses (σ

_{22}= 332.4 MPa and σ

_{12}= 110.3 MPa) are high when the depth of vertical pore is 0.5 mm and their values decreased with increasing the depth of vertical pore and its value (σ

_{22}= 119.6 MPa and σ

_{12}= 85.5 MPa) when the depth of vertical pore is maximum which is 2.5 mm. When two vertical pores are inserted into TC, the investigation of residual stresses (σ

_{22}and σ

_{12}) at the crack tip with very high “G

_{equivc}” as shown in Figure 5a,b. From Figure 5a,b, the values of σ

_{22}= 321 MPa and σ

_{12}= 109 MPa are very high when the depth of vertical pores is minimum (0.5 mm). The values of σ

_{22}= 85.5 MPa and σ

_{12}= 78 MPa decreased with increasing the depth of vertical pores up to 2.5 mm. 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].

_{22}and σ

_{12}) at the pre-crack tip with very high critical strain energy release rate for different thickness (3, 2 and 1 mm) of TC without, with one vertical pore and with two vertical pores. The values of σ

_{22}(224, 185, and 129.3 MPa) and σ

_{12}(136, 99.8, and 83 MPa) at the crack tip are very high in TC of a thickness (3, 2, and 1 mm) without vertical pores, respectively. Their values (σ

_{22}= 119.6, 132.6, and 121 MPa) and σ

_{12}= 85.5, 84, and 82.9 MPa) decreased when one vertical pore was inserted into TC and their value (σ

_{22}and σ

_{12}) decreased further from (119.6 to 85.5 MPa, 132.6 to 89.4 MPa depends, and 121 to 104 MPa) and from (85.5 to 78 MPa, 84 to76 MPa, and 82.9 to 75.6 MPa) when two vertical pores insert into TC of a thickness (3, 2, and 1 mm), respectively. The value of σ

_{22}and σ

_{12}further decreased by decreasing the thickness of TC.

_{22}at the crack tip is maximum (224 MPa) when the thickness of TC is 3 mm and the stress magnitude continuously decrease up to 56 MPa at the crack tip with decreasing the thickness of TC up to 0.3 mm as shown in Figure 9a. From 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.

_{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].

#### 3.2. Effect of General Feature on Cracking Driving Force

_{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.

#### 3.3. Investigation of Crack Propagation

#### 3.4. Strain Tolerant Design for Thick Coatings

_{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,c,e residual stresses σ

_{22}and σ

_{12}show a decreasing trend with increasing the depth of vertical pores when the crack densities are 0.1, 0.15, and 0.3 respectively. Here we noted that the maximum values of residual stresses σ

_{22}and σ

_{12}are 190 and 106.3 MPa, and minimum values 92.7 and 78.22 MPa respectively when the crack density is 0.1. Their values reduce with increasing the crack density and show the lowest values when the crack density is 0.3. Here SERR G

_{I}, G

_{II}, and G

_{t}also exhibit the decreasing trend with increasing the depth of vertical pores, when 0.1, 0.15, and 0.3 crack density as shown in 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.

## 4. Conclusions

- 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. - 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. - 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.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Model of TBCs with the TC crack close to the interface: (

**a**) the physical geometry model; (

**b**) FE mesh and boundary conditions; and (

**c**) the refined meshes near the TC/BC interface.

**Figure 2.**Schematic diagram of the virtual crack closure technique (VCCT) used for the simulation of crack growth in the TC layer.

**Figure 3.**Variation of stress field: (

**a**) σ

_{22}and (

**b**) σ

_{12}at crack tip with different TC thickness.

**Figure 4.**Variation of stress field: (

**a**) σ

_{22}and (

**b**) σ

_{12}at crack tip for different depths of 3 mm thickness of TC with one vertical crack.

**Figure 5.**Variation of stress field: (

**a**) σ

_{22}and (

**b**) σ

_{12}at crack tip for different depths of 3 mm thickness of TC with two vertical cracks.

**Figure 6.**Variation of stress field: (

**a**) σ

_{22}and (

**b**) σ

_{12}at crack tip for 3 mm thickness of TC without, with one vertical pore, and with two vertical pores. The depth of vertical pores was 2.5 mm.

**Figure 7.**Variation of stress field: (

**a**) σ

_{22}and (

**b**) σ

_{12}at crack tip for 2 mm thickness of TC without, with one vertical pore, and with two vertical pores. The depth of vertical pores was 1.8 mm.

**Figure 8.**Variation of stress field: (

**a**) σ

_{22}and (

**b**) σ

_{12}at crack tip for 1 mm thickness of TC without, with one vertical pore, and with two vertical pores. The depth of vertical pores was 0.9 mm.

**Figure 9.**Variation of stresses (

**a**) σ

_{22}and (

**b**) σ

_{12}along crack path at different thickness of TC.

**Figure 10.**Stress distribution along crack line, with one and two vertical cracks in 3 mm thickness of TC (

**a**–

**d**), with one and two vertical cracks in 2 mm thickness of TC (

**e**–

**h**), with one and two vertical cracks in 1 mm thickness of TC (

**i**–

**l**).

**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**).

**Figure 13.**Plot of crack length as a function of depth of vertical pores for one vertical pore and two vertical pores.

**Figure 14.**Effect of crack density and crack depth on stress and SERR: stresses at density (

**a**) 0.1 mm, (

**c**) 0.15 mm, (

**e**) 0.3 mm and SERR at density (

**b**) 0.1 mm, (

**d**) 0.15 mm, (

**f**) 0.3 mm.

Layers | E (GPa) | ν | α (10^{−6}/K) |
---|---|---|---|

SUB | 210 | 0.3 | 9 |

BC | 200 | 0.3 | 13.6 |

TC | 50 | 0.15 | 14.8 |

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**MDPI and ACS Style**

Mehboob, G.; Xu, T.; Li, G.-R.; Hussain, S.; Mehboob, G.; Tahir, A.
Strain-Induced Cracking Behavior of Coating/Substrate Systems and Strain Tolerant Design for Thick Coatings. *Coatings* **2020**, *10*, 1066.
https://doi.org/10.3390/coatings10111066

**AMA Style**

Mehboob G, Xu T, Li G-R, Hussain S, Mehboob G, Tahir A.
Strain-Induced Cracking Behavior of Coating/Substrate Systems and Strain Tolerant Design for Thick Coatings. *Coatings*. 2020; 10(11):1066.
https://doi.org/10.3390/coatings10111066

**Chicago/Turabian Style**

Mehboob, Ghazanfar, Tong Xu, Guang-Rong Li, Shahnwaz Hussain, Gohar Mehboob, and Adnan Tahir.
2020. "Strain-Induced Cracking Behavior of Coating/Substrate Systems and Strain Tolerant Design for Thick Coatings" *Coatings* 10, no. 11: 1066.
https://doi.org/10.3390/coatings10111066