Coupled Thermo-Mechanical Modeling of Crack-Induced Stress Fields in Thermal Barrier Coatings with Varying Crack Geometries

Abstract

Under service conditions, randomly distributed cracks in the top coat (TC) layer of thermal barrier coatings (TBCs) lead to local stress concentrations, which serve as the primary drivers of crack propagation and coating delamination. This study systematically analyzes the influence of crack defects on the thermal stress distribution in TBCs, based on their microstructural characteristics, using a multi-physics-coupled finite element model. Numerical analysis of crack characteristics reveals that crack length significantly influences the stress distribution in the coatings, with the maximum tensile stress at the crack tip increasing from 104.02 to 238.51 MPa as the crack half-length extends from 400 to 1000 μm. Shorter cracks induce lower tensile stresses, thereby retarding crack propagation and delaying coating delamination. Crack depth also influences the stress distribution, with the maximum tensile stress decreasing from 205.88 to 101.65 MPa as the crack is buried deeper, from 50 to 200 μm, indicating a more stable stress state less prone to propagation in deeper cracks. For inclined cracks, increasing the inclination angle induces a shift in stress from tensile to compressive, with larger inclination angles exhibiting greater stability. Accordingly, this study proposes a laser scribing strategy to mitigate crack-tip stress concentration, which is validated through comparison with two-dimensional crack models. Laser scribing shortens crack length by interrupting crack continuity, relieves localized thermal expansion strain, effectively suppresses crack growth, and significantly enhances the crack resistance and thermal shock stability of the coating.

1. Introduction

Turbine blades, as critical components of aero-engines, operate under extremely harsh environmental conditions for prolonged periods [1]. Thermal barrier coatings (TBCs) serve as protective layers and are subjected to combined thermal, mechanical, and chemical stresses during service, eventually leading to failure and delamination from the substrate [2,3]. Premature coating delamination presents a critical barrier to safe application, making the extension of TBC service life vital for ensuring engine reliability and safety [4,5]. At elevated temperatures, failure mechanisms primarily arise from stress mismatches between mechanical loads and thermal cycling, which readily induce crack initiation and propagation at the coating surface and interface, ultimately resulting in delamination and failure [6,7]. Therefore, comprehensive analysis of the failure mechanisms of TBCs under complex service environments is essential for improving their longevity and overall reliability.

The spalling failure of TBCs resulting from crack propagation under high-temperature service conditions has become a major obstacle to their engineering applications [8]. Stress concentration at crack tips during thermal cycling is widely regarded as the primary mechanism responsible for TBC spallation [9,10]. Numerous studies have demonstrated that crack propagation occurs when localized stresses exceed the material’s yield strength, with primary contributing factors including thermal stresses from mismatched coefficients of thermal expansion, thermally induced growth stresses in the thermally grown oxide (TGO) layer, and phase transformation stresses [11,12,13]. Kyaw et al. [14] employed finite element analysis to investigate stress mismatches within a TBC system, showing that maximum tensile stress concentrated at the peak of the bond coat (BC) layer and the valley of the top coat (TC) layer, with significant thermal mismatch stresses developing during the cooling stage, thereby promoting crack propagation [15]. As TGO thickness increases, it becomes a key factor in driving interfacial crack initiation and propagation [16]. To accurately simulate the initiation and propagation of cracks in TBCs under multi-physics coupling conditions, the extended finite element method and the cohesive zone model (CZM) are widely employed to model multi-crack interactions and interface delamination. Previous studies have demonstrated that XFEM is effective in capturing the evolution paths of internal (bulk) cracks, while CZM is well-suited for characterizing the interfacial crack driving mechanisms. The integrated application of these two methods enables a comprehensive analysis of the effects of TGO thickness evolution, interface roughness, and thermal stress cycling on the early-stage spallation behavior of TBC systems [17,18,19].

Three-dimensional numerical simulations analyzing peak and trough effects at the TC/BC interface indicate that, although variations in TGO thickness have limited impact on radial and circumferential stresses at the BC interface, elevated TGO growth stresses tend to induce early cracking, whereas the TGO creep effect may delay crack formation at the TC/TGO interface [20,21]. At the microscale, the surface roughness and porosity of the TC significantly influence the distribution of residual stresses in the TGO during thermal cycling. Studies have shown that surface roughness during heating induces localized stress concentrations that promote horizontal crack initiation, while during cooling, porosity shifts stress concentration from the TC/TGO interface to nearby pores, accelerating horizontal crack propagation within the TC as TGO thickness increases [22,23]. To investigate the effects of TGO growth on thermal-cycling-induced stress states and crack propagation behavior, various layered structural models have been developed in recent years to quantitatively simulate the continuous propagation, coalescence, and exfoliation of multiple cracks, incorporating inhomogeneous transverse growth strains, fracture mechanics, and extended finite element methods [24,25]. Considering lateral growth strain within the TGO, tensile and shear stresses in the TC accumulate periodically, and localized stress concentrations become the dominant factor promoting early crack propagation, while the continued interlocking growth of multiple cracks further exacerbates overall spallation. Additionally, the distribution of vertical and periodic surface cracks significantly affects TBC system failure behavior. Studies employing TBC models with multiple vertical cracks have systematically investigated the effects of crack spacing, distribution uniformity, and coating thickness on interfacial stress, crack driving forces, and dynamic propagation behavior, revealing that optimizing vertical crack spacing or surface crack density can suppress crack propagation, mitigate interfacial cracking, and thereby enhance overall TBC durability [26].

Therefore, a comprehensive analysis of stress field distribution in cracked regions during thermal cycling is of critical theoretical importance for elucidating the failure mechanisms of TBCs. Accurate characterization of stress field intensity in cracked TBC systems not only aids in interpreting failure phenomena observed during thermal cycling tests or actual service but also establishes a robust theoretical basis for optimizing TBC structural design. Currently, research on surface cracks in TC layers remains limited, and the quantitative characterization of local stress distribution in crack-containing TBC systems is still inadequate. Accordingly, this study employs the finite element method to systematically analyze the stress field distribution and deformation response of crack-containing TC layers under thermal cycling and investigates the influence of crack structural parameters on local stress evolution. Based on the simulation results, a strategy to alleviate crack-tip stress concentration and suppress crack propagation is proposed and validated through numerical simulations. This study aims to provide a robust theoretical foundation and technical guidance for optimizing the structural design and extending the service life of TBCs.

2. Materials and Methods

2.1. Structural Characterization and Crack Geometry Extraction

In order to better observe the crack structure within the TC layer, X-ray CT scanning experiments were conducted on the thermal-barrier-coated ceramic insulation layer using a nano-CT system manufactured by Zeiss—one of the most advanced nano-scale CT systems. This equipment offers a detail resolution of up to 200 nm and a voxel resolution of 0.3 μm. During the experiments, a magnification of approximately 130× was selected for X-ray projection imaging, yielding an imaging resolution of 0.65 μm. The corresponding transillumination parameters are presented below:

(1)

Tube voltage: 150 kV;

(2)

Tube current: 15 μA;

(3)

Integration time: 6000 ms;

(4)

Magnification: 130×;

(5)

Focus-to-sample distance: 11.4 mm;

(6)

Focus-to-detector distance: 12 mm;

(7)

Number of projections: 1801 over 360°.

The TC layer scan data were imported into Avizo software to perform threshold segmentation of pores and cracks, enabling the visualization of internal pore and crack structures. The blue-rendered areas in Figure 1 represent pores and cracks, illustrating the complexity of the coating’s internal structure. The internal cracks in the TC layer of this thermal barrier coating primarily exhibit a long, stripe-like distribution pattern and are randomly located at varying depths within the TC layer, with no fixed orientation. These long, stripe-like cracks are associated with localized thermal stress concentrations within the coating or stress mismatches caused by differences in material thermal expansion, which can lead to brittle fracture in the coating.

Thus, crack morphology and distribution not only reflect the mechanical behavior of coatings under high-temperature conditions but also serve as key indicators in failure analysis. Detailed analysis enables the identification of crack formation mechanisms and clarifies their specific morphology and distribution, providing a foundation for subsequent investigations into the coating’s thermal insulation performance, service life, and failure mechanisms. These insights can guide targeted design optimization, enhance coating reliability and stability, and support the performance improvement and practical application of thermal barrier coatings.

2.2. Finite Element Modeling

2.2.1. Geometric Model and Boundary Conditions

The thermal barrier coating structure model constructed in this study consists of a ceramic layer (TC), a thermally grown oxide layer (TGO), a bonding layer (BC), and a high-temperature alloy substrate (SUB). To simplify the calculation, the model was treated as a two-dimensional planar model, and its geometry is shown in Figure 2a. The overall width of the model is 3 mm, and the thicknesses of each layer are 250 or 800 μm for the TC layer, 150 μm for the BC layer, 5 μm for the TGO layer, and 3 mm for the SUB layer. The layers are bonded to form an integrated structure, with the interfacial roughness between layers neglected and the assumption that the layers are tightly adhered to each other, ensuring efficient heat and stress transfer within the thermal barrier coating.

To investigate crack behavior in the TC layer, the following modeling assumptions are applied:

(1)

The dynamic growth of the TGO layer is neglected.

(2)

Residual stress generated during coating preparation is not considered.

(3)

Phase transition-induced thermal effects are excluded from the calculations.

(4)

Interfaces between layers are fully bonded with no relative displacement.

To simplify the calculation, the model is treated as a two-dimensional planar geometry. Interfacial roughness is neglected, and perfect adhesion between layers is assumed to ensure efficient thermal and mechanical continuity across the TBC.

As shown in Figure 2b, symmetric boundary conditions along the y-axis were applied to the left edge of the two-dimensional model, while displacements at the bottom surface (in the y-direction) and the right boundary (in the x-direction) were restricted [27,28]. In the finite element simulation, the service-related thermal cycling conditions were divided into three stages, and periodic thermal boundary conditions were applied to the upper surface of the coating.

Specifically, in each thermal cycle, the upper surface was rapidly heated from 20 °C to 1425 °C within 5 min, held at 1425 °C for 50 min, and then cooled back to 20 °C over 5 min, completing one full thermal cycle per hour. A similar thermal cycling profile was applied to the bottom surface of the substrate, with the maximum temperature limited to 875 °C. Figure 3 presents the temperature–time profile for a single thermal cycle.

2.2.2. Grid Division and Material Parameters

The model was discretized using CPE4T four-node plane strain elements. Due to the pronounced stress concentration and strain gradient near the crack tip, a local mesh refinement strategy was employed to enhance numerical accuracy and capture detailed mechanical responses in this critical region. Additionally, the axial direction of the TGO layer was also refined due to its significantly smaller thickness compared to the substrate. A mesh and time-step independence study was conducted for the transient simulations, in which the variation in maximum von Mises stress near the crack tip was compared under various mesh densities and refinement levels. The results of the mesh independence analysis are summarized in

Table 1. Verification of grid and time step independence.

ElementSize/μm	Grid Size Near Crack/μm	Number of Elements	Maximum Mises/GPa	Time Step Range/s	ElementSize/μm	Grid Size Near Crack/μm	Number of Elements
10	0.2–0.3	51966	2.68	10–20–30	10	0.2–0.3	51966
15	0.3–0.4	24834	2.69	20–30–40	15	0.3–0.4	24834
25	0.4–0.5	10345	2.67	30–40–50	25	0.4–0.5	10345
45	0.5–0.6	4299	2.69	40–50–60	45	0.5–0.6	4299

, indicating that mesh independence is achieved when the mesh size near the crack tip is below 0.8 μm, the maximum mesh size in the matrix and other regions does not exceed 50 μm, and the time step remains below 50 s. Based on this validation, the final simulation employed a global mesh size of 15 μm, a refined mesh of 0.2–0.3 μm near the crack tip, and a time step of 20–30 s, as shown in the meshing schematic in Figure 4.

In finite element modeling of TBC systems, the choice of material parameters is critical for elucidating failure mechanisms. Assuming isotropy and material homogenization, each layer is treated as an isotropic, homogeneous medium, and a multilayer constitutive model is constructed for the TBC/TGO/SUB system.

For the TC layer, typically composed of yttria-stabilized zirconia (YSZ), and the SUB layer, commonly made of LaMgAl₁₁O₁₉-based high-temperature alloy, both exhibit predominantly linear elastic behavior under conventional service conditions. Therefore, Hooke’s law is employed to characterize the stress–strain relationship. The tensor form is given by

$${\sigma }_{ij}=C_{ijkl}\cdot {\epsilon }_{kl}$$

(1)

Assuming material isotropy, the equation simplifies to the above form, where

$$\sigma =\lambda \cdot tr(\epsilon )\cdot I+2\mu \cdot \epsilon$$

(2)

where $\sigma$ and $\epsilon$ denote the stress and infinitesimal strain tensors, respectively; $tr(\epsilon )$ is the trace of the strain tensor; $I$ is the second-order identity tensor; $\lambda$ and $\mu$ are the Lamé constants, which are related to the Young’s modulus $E$ and Poisson’s ratio $\nu$ by

$$\lambda ={\displaystyle \frac{E\cdot \nu }{(1+\nu )(1-2\nu )}}\begin{array}{cc},& \mu \end{array}={\displaystyle \frac{E}{2(1+\nu )}}$$

(3)

For the TGO layer and the BC layer, typically consisting of NiCoCrAlY alloy, inelastic deformation behavior must be considered. Long-term thermal cycling experiments have shown that when the operating temperature exceeds the critical threshold of 600 °C, creep relaxation can reduce interfacial stress amplitudes by over 58%. This time–temperature-dependent degradation mechanism critically influences the kinetics of coating delamination failure [29,30]. Accordingly, both plasticity and creep effects are incorporated into the numerical simulations. The plastic response is characterized using a kinematic hardening model expressed as

$${\sigma }_{ij}=C_{ijkl}({\epsilon }_{kl}-{\epsilon }_{kl}^{pl}){\begin{array}{cc},& \stackrel{·}{\alpha }\end{array}}_{ij}=C\cdot {\stackrel{·}{\epsilon }}_{kl}^{pl}$$

(4)

where ${\epsilon }_{kl}^{pl}$ is the plastic strain rate tensor, ${\alpha }_{ij}$ denotes the back stress tensor, and $C$ is a function describing the evolution of the yield surface governed by the following hardening modulus.

At elevated temperatures, time-dependent deformation becomes dominant, and creep is modeled using the Norton power-law equation:

$${\dot{\epsilon }}_{cr}=B{\sigma }^n$$

(5)

where $\sigma$ is the applied stress, MPa; n is the creep exponent; B is the creep pre-factor, MPa⁻ⁿ·s⁻¹; and ${\dot{\epsilon }}_{cr}$ is the creep strain rate, s⁻¹.

The creep parameters for the TBC system are summarized in

Table 2. Physical parameters of thermal barrier coatings [31,32].

Material	Temperature /°C	Elastic Modulus /GPa	Poisson’s Ratio	Coefficient of Thermal Expansion/10−5°C−1	Density /kg·m−3	Specific Heat/J·kg−1°C−1	Conductivity/W·m−1K−1
LaMgAl11O19	20	28.83	0.23	0.83	3321	578.4	1.53
	200	25.47	/	0.95	/	805.4	1.18
	400	22.11	/	1.05	/	913.2	0.82
	600	18.75	/	1.10	/	1007.9	0.65
	800	15.37	/	1.15	/	1055.3	0.52
	1000	12.01	/	1.20	/	1089.6	0.41
	1200	8.65	/	1.30	/	1094.5	0.32
YSZ	20	48	0.1	1.04	5280	450	1.80
	200	47	/	1.05	/	491	1.76
	500	43	/	1.07	/	532	1.75
	700	39	/	1.08	/	573	1.72
	1100	25	/	1.09	/	615	1.69
	1200	22	/	1.10	/	656	1.67
TGO	20	3978	0.27	0.8	1000	/	10
	200	390	/	0.82	/	/	7.8
	400	380	/	0.84	/	/	6.0
	1000	325	/	0.93	/	/	4.4
NiCoCrAlY	25	225	0.30	1.20	7320	501	4.30
	400	186	/	1.39	/	592	6.40
	600	166	/	1.48	/	670	8.00
	800	147	/	1.55	/	781	10.20

/ indicates a fixed number that is not significantly affected by temperature. Where not indicated, it means that this parameter is not significantly affected by temperature and is negligible.

. The elastic modulus, Poisson’s ratio, and thermal expansion coefficients of each layer are presented in

Table 3. Plasticity parameters of bonded and oxidized layers [33].

Material	Temperature/°C	Stress/MPa	Plastic Strain	A (10−11·MPa−n·s−1)
NiCoCrAlY	25	1000	0	/
	400	2500	0.23	/
	600	2200	0.3	/
	800	375	0.022	/
	900	60	0.02	/
	1000	19	0.01	/
	1100	/	/	1.35
TGO	1100	/	/	5.0299

/ indicates a fixed number that is not significantly affected by temperature. Where not indicated, it means that this parameter is not significantly affected by temperature and is negligible.

, while the plasticity parameters of the bond coat are provided in

Table 4. Creep parameters of materials in layers of thermal barrier coatings [34].

Material	B/MPa−n·s−1	N	Temperature/°C
TC	1.8 × (10−11–10−5)	1	1000
TGO	7.3 × (10−12–10−4)	1	1000
BC	6.54 × 10−19	4.57	≤600
	2.2 × 10−12	2.99	700
	1.84 × 10−7	1.55	800
	2.15 × 10−8	2.45	≥850
SUB	4.85 × 10−36	1	10
	2.25 × 10−9	3	1200

. By carefully assigning these material parameters, the constructed computational model reliably captures the mechanical behavior of the TBC system under complex thermo-mechanical loading.

2.3. Simulation Methodology

In this study, numerical simulations of the thermo-mechanical coupling behavior of a TBC system under cyclic thermal loading were conducted using the standard finite element module in ANSYS. A two-dimensional axisymmetric model was adopted, and PLANE223 high-order coupled-field elements were selected for mesh discretization. Each PLANE223 element is an 8-node plane strain element with both temperature and displacement degrees of freedom, capable of accurately capturing thermo-mechanical coupling effects in the multilayer structure, particularly near the crack tip. Transient analysis based on a directly coupled thermo-mechanical approach was employed to capture the time-dependent evolution of stress fields within the coating, with emphasis on the crack tip region. A strong coupling between temperature and stress fields was implemented using iterative solving techniques based on the maximum stress criterion, focusing on the influence of crack geometry on local stress distribution near the tip. The parametric model was developed using APDL scripting to enable flexible adjustment of geometric variables such as crack length, location, and inclination angle. Three key parameters—crack length, burial depth, and inclination angle—were selected to represent the principal geometric features of open cracks. This numerical approach provides an efficient analytical tool and theoretical basis for investigating the influence of crack defects on the extension mechanism of TBCs under complex thermo-mechanical loading.

3. Results and Discussion

3.1. Temperature Field Calculation Results

Under thermal cyclic loading, a pronounced nonuniform temperature gradient develops within the TBC system. The transient temperature distribution of the TBC system is illustrated in Figure 5a, showing a distinct temperature difference between the upper and lower crack surfaces at the high-temperature steady state (300 s), as shown in Figure 5b, with a temperature difference of approximately 50 °C across the crack region. This indicates significant thermal nonuniformity within the crack region, with local temperature gradients reaching magnitudes on the order of 10² °C/mm. As the core insulating layer, the temperature distribution in the top coat (TC) significantly influences the stress state and deformation behavior of the entire coating system, with thermally induced stresses due to temperature gradients acting as the primary mechanical drivers of coating failure.

3.2. Effect of Crack Characteristics on Local Stress Distribution

Failure in TBC systems is primarily initiated by the nucleation and propagation of transverse cracks, governed by the combined effect of tensile stress in the coating thickness direction (σ_yy) and interfacial shear stress (σ_xy). Among these, tensile stress in the y-direction, induced by thermal expansion near cracks, acts as the primary driving factor for crack propagation.

To investigate stress distribution at the crack tip, a thermo-mechanical coupling model was developed to systematically analyze how various crack geometries affect the normal stress component σ₂₂ under service conditions, and to examine its intrinsic correlation with coating failure mechanisms. Here, σ₂₂ represents the normal stress in the y-direction: positive values correspond to tensile stress (material under tension), while negative values represent compressive stress (material under compression).

Figure 6 presents the time-dependent evolution of the normal stress component σ₂₂ over a single thermal cycle for a crack length of 400 μm and a depth of 125 μm in the TC layer, clearly delineating the stress response stages under coupled thermo-mechanical loading. During the heating phase (0–300 s), σ₂₂ values at monitoring points gradually increase with temperature, exhibiting an initial rise, followed by a temporary drop and subsequent increase. This trend is governed by variations in the internal temperature gradient during the coupled thermo-mechanical process. At the onset of heating, a significant temperature gradient between the surface and interior induces uneven thermal expansion and localized stress concentration. As heating continues, the internal temperature becomes more uniform, leading to reduced thermal mismatch and a temporary stress decrease. The initial stress rise is attributed to the mismatch in coefficients of thermal expansion between the TC surface and the substrate, with the surface expanding more rapidly and inducing stress accumulation. As the system approaches thermal equilibrium, thermal mismatch reduces and the stress stabilizes. The stress evolution near the crack tip during heating is driven by localized thermal expansion and temperature field nonuniformity. During the constant-temperature phase (300–3300 s), the stress remains stable, indicating that thermal equilibrium has been achieved. In the cooling phase (3300–3600 s), the stress gradually decreases. Initially, stress relaxation occurs rapidly due to faster contraction of the cooler surface. As the temperature continues to decrease, the stress reduction rate slows and eventually stabilizes.

3.2.1. Effect of Crack Length on Stress Distribution

With prolonged service time, crack initiation and propagation become the most prominent degradation mechanisms in TBCs. To systematically investigate the effect of crack length on the thermo-mechanical behavior and durability of the coating, this study employs a layered coating model and uses numerical simulation to analyze stress field distributions for crack lengths ranging from 400 to 1000 μm. Due to the symmetry of the model, the simulated crack length was set to one-half of the actual physical length.

As illustrated in Figure 7, increasing crack length significantly alters the stress distribution within the coating and leads to pronounced stress concentration near the crack tip. The maximum stress consistently occurs at the crack tip and is accompanied by a steep stress gradient, indicating that this region plays a critical role in stress evolution and crack-driven failure mechanisms.

Figure 8 illustrates the evolution of normal stress (σ₂₂) along the Y-direction at the crack tip under varying crack lengths. A clear positive correlation is observed between crack length and crack-tip stress: as the crack length increases from 400 to 1000 μm, σ₂₂ at the crack tip increases significantly, accompanied by an increasingly pronounced stress concentration zone. The maximum σ₂₂ increases from 104.02 to 238.51 MPa, indicating that longer cracks lead to stronger stress amplification.

This increase in crack-tip stress is primarily due to the enhanced thermal stress response induced by longer cracks during thermal expansion. The crack tip consistently acts as the primary stress concentration site, and this effect is particularly pronounced under thermally coupled loading, where crack length strongly influences stress sensitivity. Notably, during the heating and cooling phases, the crack-tip stress increases significantly with crack length. This behavior is attributed to differences in thermal expansion and the presence of thermal gradients, which are exacerbated by longer cracks. In particular, peak temperatures result in a sharp increase in crack-tip stress. During cooling, the stress gradually decreases; however, coatings with longer cracks retain elevated stress levels, indicating that crack length also affects the rate of stress relaxation.

Thus, increasing crack length leads to higher stress concentration at the crack tip, driven by thermal expansion and gradient effects. The thermal stress response becomes more severe for longer cracks, especially during heating and cooling, thereby increasing the risk of fracture.

3.2.2. Effect of Crack Depth on Stress Distribution

Crack burial depth, defined as the vertical distance from the coating surface to the crack, significantly affects stress distribution at the crack tip. To examine this effect, simulations were performed under varying crack burial depths, with a fixed crack length of 600 μm and burial depths ranging from 50 to 200 μm within the TC layer. Figure 9 presents the layered coating model and the corresponding stress field distributions for different crack depths. The crack position along the thickness direction significantly alters the overall stress field within the coating. In all cases, maximum stress occurs at the crack tip; however, the influence of stress concentration on the surrounding material varies with crack depth. Specifically, shallower cracks closer to the coating surface induce broader nonuniformity in the stress field, indicating that burial depth plays a key role in modulating stress transfer and local mechanical behavior in the TBC system.

Figure 10 and Figure 11 illustrate the time-dependent evolution of crack tip σ₂₂ under thermal loading for various burial depths from the free surface. During the heating phase, σ₂₂ rises rapidly at each depth, with the rate of increase varying based on crack burial depth. For shallow cracks, proximity to the free surface results in strong thermal stress concentration at the tip, reaching up to 205.89 MPa. Cracks near the surface are more directly exposed to thermal cycling, intensifying stress responses.

With increasing depth, the crack tip becomes more insulated from surface effects, resulting in a reduced σ₂₂ value of 101.65 MPa. Deeper cracks are less influenced by external thermal loads, and the associated stress transfer paths are more complex, which weakens the concentration effect. During thermal expansion, surface regions experience stronger thermal gradients due to direct heat exposure, leading to higher stresses in shallower cracks. In contrast, deeper cracks exhibit lower thermal gradients and a reduced stress response.

Shallow cracks allow thermal stresses to accumulate and transfer more rapidly to the tip, whereas deeper cracks, embedded within the coating, experience reduced thermal expansion effects. Therefore, crack burial depth significantly affects crack-tip stress behavior. Shallower cracks experience higher σ₂₂ values and pose a greater risk of stress-driven failure. This trend is highly relevant for coating design, as burial depth must be considered when evaluating durability and resistance to crack propagation.

3.2.3. Effect of Crack Inclination on Stress Distribution

To systematically investigate the influence of crack inclination on the stress field within the coating, a 600 μm crack with inclination angles ranging from 10° to 80° was positioned at the center of the simulation domain, and the stress field distribution in the x–y cross-section was analyzed. As shown in Figure 12 and Figure 13, variations in crack inclination significantly alter the coating’s stress distribution characteristics, with the maximum stress consistently localized at the crack tip. Pronounced stress field inhomogeneity is observed in the vicinity of the central crack region. Figure 13 compares the temporal evolution of σ₂₂ at both crack tips for different inclination angles, showing that the inclined crack tip closer to the coating surface exhibits a higher stress concentration, with correspondingly greater tensile stress in the cross-section.

Figure 14 illustrates the temporal evolution of σ₂₂ at the crack tip for various inclination angles. Crack inclination angle significantly affects crack-tip stress, showing a clear fluctuation trend as the angle increases. At 20°, the crack tip exhibits the highest σ₂₂ value of 582.80 MPa, indicating the strongest stress concentration. With increasing inclination angle, the crack-tip stress gradually decreases, with a sharp transition at 40°, where the stress direction reverses from tensile to compressive, reaching −62.04 MPa. Beyond 50°, the compressive stress intensifies further, reaching a minimum of −6786.86 MPa at 80°. This suggests that larger inclination angles enhance stress relief at the crack tip and significantly alter the stress direction.

At low-inclination angles, the crack tip experiences strong thermal expansion mismatch and steep temperature gradients, resulting in localized stress concentration. As the angle increases, the orientation disperses the stress more effectively, reducing its magnitude. Therefore, crack inclination plays a key role in regulating crack-tip stress, with larger angles effectively reducing stress concentration and lowering the risk of crack propagation.

In summary, crack inclination significantly influences the time-dependent evolution of σ₂₂ at the crack tip. Larger inclination angles are associated with higher initial stress sensitivity, lower steady-state stress levels, and more pronounced fluctuations during unloading. These findings highlight the critical role of crack inclination in shaping local stress field evolution and emphasize the importance of considering inclination effects when modeling and assessing the structural integrity and crack growth behavior of TBC systems.

3.3. Optimization Analysis of Surface Scoring Strategy

As demonstrated in the preceding analysis, short cracks induce relatively weak stress–strain responses [35]. In contrast, longer cracks tend to exhibit significant stress concentration at the crack tip, which increases the risk of crack propagation and coating failure. Based on the principles of stress relief and concentration mitigation, converting a long crack into multiple shorter segments can effectively reduce peak stress levels at the crack tip [36]. Laser scribing technology, as illustrated in Figure 15, offers a practical means of segmenting long cracks. To evaluate the effectiveness of this approach in modulating the stress–strain behavior of the coating, a comparative analysis must be performed using a finite element model incorporating segmented cracks.

3.3.1. Analysis of Crack Stress Response at Different Lengths

To better illustrate the stress distribution at the crack tip, the post-processing results display only the stress distribution at the TC layer crack. Figure 16 shows that, after laser scribing, the location of the maximum internal stress shifts and is no longer concentrated at the crack tip. This observation highlights the limitation of using the overall stress maximum as a benchmark for evaluating crack-tip stress, as done in earlier sections. Accordingly, a fixed node at the crack tip is selected as a consistent observation point, and the crack-tip stress is evaluated before and after laser treatment by analyzing its time-dependent variation at this node. This approach enables a quantitative assessment of the stress mitigation effect of laser scribing at the crack tip.

As shown in Figure 17, crack-tip stress after laser scoring exhibits non-monotonic fluctuation with respect to crack length. During the heating phase, crack-tip stress increases with rising temperature and thermal expansion mismatch. Increasing crack length intensifies stress concentration. However, laser scoring mitigates this effect by segmenting the crack, reducing stress concentration and resulting in fluctuating rather than monotonically increasing crack-tip stress. Notably, at a crack length of 1000 μm, the stress is comparable to that at 400 μm and closely aligns with values at 600 and 800 μm. During the constant-temperature phase, crack-tip stress stabilizes, and laser scoring results in smaller stress variations due to shortened effective crack length. In the cooling phase, the crack-tip stress decreases gradually.

Importantly, laser scoring induces a notable reversal of stress direction at the crack tip—from tensile to compressive. Although the absolute stress magnitude changes modestly, this reversal can effectively suppress crack propagation. By reversing the stress direction, laser scoring reduces the influence of tensile loading at the crack tip, thereby enhancing the coating’s crack resistance and thermal shock stability.

3.3.2. Analysis of Crack Stress Response at Different Locations

As shown in Figure 18 and Figure 19, crack depth exhibits a contrasting influence on crack-tip stress after indentation compared to the unmodified case. In the unscored model, crack-tip stress typically decreases with increasing crack depth, whereas in the presence of indentation, crack-tip stress increases with depth. At depths of 175 and 200 μm, the stress increases significantly, reaching −534.47 and −577.73 MPa, respectively, while shallower cracks (e.g., 50 μm) exhibit a lower compressive stress of −211.55 MPa. This suggests that indentation modifies the crack stress distribution pattern, enhancing stress concentration at greater depths during thermal cycling.

Further analysis shows that indentation also alters the stress direction at the crack tip. As crack depth increases, the stress shifts from tensile to compressive, with deeper cracks exhibiting stronger compressive effects. The increasing magnitude of negative stress indicates that the crack tip is subjected to higher compressive loading. This stress reversal inhibits crack propagation, as compressive stress counteracts tensile-driven crack opening and thus delays crack growth.

Laser scoring not only altered the stress distribution at the crack tip but also enhanced the coating’s crack resistance by reducing effective crack length and reversing stress direction. After laser scoring, crack depth and crack-tip stress exhibit a positive correlation—particularly for deeper cracks, which show both higher compressive stress and a pronounced shift in stress direction, thereby contributing to a reduction in crack propagation rate. As an effective crack intervention, laser scoring contributes to the improved reliability and extended service life of thermal barrier coatings by optimizing local stress fields and mitigating crack growth.

3.3.3. Analysis of Crack Stress Response at Different Inclination Angles

After laser scoring, cracks with varying inclination angles exhibit distinct stress distribution patterns. As shown in Figure 20 and Figure 21, when the inclination angle is small (0°–20°), crack-tip stress concentration remains low, and laser scoring effectively suppresses stress in this region. As the inclination angle increases to 30°–50°, local stress at the crack tip intensifies, and the stress concentration zone shifts progressively along the crack propagation direction. Although laser scoring partially mitigates the crack extension tendency in this range, it also enlarges the high-stress region, thereby increasing the risk of crack growth. At inclination angles above 60°, stress concentration becomes highly pronounced, with high-stress zones extending significantly along the crack path. This suggests that under thermal loading, high-inclination cracks are prone to unstable propagation. Laser scoring can locally redistribute stress but cannot fully eliminate the driving force for crack advancement in such cases.

Figure 22 demonstrates that crack inclination significantly influences crack-tip stress evolution, including both magnitude and direction. Compared to unmodified cracks, those treated with laser scoring exhibit a distinct stress evolution pattern. Laser scoring induces a reversal in stress polarity, shifting crack-tip stress from compressive (negative) to tensile (positive). This effect is particularly evident at higher inclination angles, where stress relief becomes more pronounced. With increasing inclination, tensile stress is effectively reduced and replaced by compressive stress, which helps suppress crack propagation. During the heating phase, cracks with smaller inclination angles exhibit higher tensile stress, whereas during cooling, cracks with higher inclination demonstrate smoother stress relaxation and shift toward compressive stress. As crack inclination increases, the peak stress at the crack tip progressively decreases. For inclination angles of 30° and above, crack-tip stress rapidly declines and eventually becomes compressive, indicating that higher inclination effectively reduces stress concentration. Following laser scoring, thermal stress is redistributed and no longer concentrated exclusively at the crack tip. Increasing inclination further promotes a more uniform stress distribution, with smoother stress relief—especially in high-inclination cracks. This demonstrates that scoring improves stress uniformity and mitigates localized stress buildup.

Overall, laser scoring is more effective in adjusting stress fields for low- to medium-inclination cracks, whereas its ability to suppress crack propagation diminishes at high-inclination angles. Crack inclination thus becomes a critical factor governing stress concentration and crack growth tendency.

4. Conclusions

This study conducted a systematic investigation of the thermal stress field and cracking behavior in TBCs. By integrating numerical simulation, CT-based experimental analysis, and a multi-physics coupling approach, a thermo-mechanical model incorporating coating microstructure was developed and used to quantitatively analyze stress distribution under varying crack geometries. The main conclusions are summarized as follows:

(1)

Crack length, depth, and inclination angle significantly influence crack-tip stress. Under thermo-mechanical loading, thermal expansion inhomogeneity leads to strong stress concentration at the crack tip. Specifically, as the crack length increases from 400 to 1000 μm, the maximum tensile stress rises from 104.02 to 238.51 MPa. In contrast, increasing crack depth results in reduced crack-tip stress, indicating a negative correlation.

(2)

When crack depth increases from 50 to 200 μm, the maximum crack-tip stress decreases from 205.89 to 101.65 MPa. Shallow cracks are less constrained by surrounding material, resulting in higher local stress concentration, whereas deeper cracks are more constrained, promoting a more uniform stress field. Crack inclination also significantly affects stress behavior. As inclination increases, the crack-tip stress decreases, and the stress direction transitions from tensile to compressive.

(3)

When the inclination angle reaches 40°, the stress drops sharply and changes from tensile to compressive. With further increases in inclination, the compressive stress intensifies and becomes more uniformly distributed, indicating that higher inclination angles help mitigate stress concentration and reduce the risk of crack propagation.

(4)

Laser scoring treatment was shown to effectively reduce crack-tip stress by shortening the effective crack length and modifying the local stress direction. At a crack length of 1000 μm, the post-scoring stress was comparable to that of a 400 μm crack, confirming the scoring’s regulatory effect. Additionally, scoring reversed the stress direction from tensile to compressive, suppressing crack extension.

These results provide a theoretical foundation for optimizing the crack-resistance design of TBCs. They highlight the importance of simultaneously controlling crack geometry—length, depth, and inclination—and applying localized scoring strategies to redistribute stress and extend coating service life under high-temperature cyclic loading.