Experimental Validation of 2D Skeletal Point Method for Creep-Fatigue-Interaction Life Assessment in Perforated Plate Specimens Under Uniaxial Load

Abstract

Geometric discontinuities in aero-engine turbine blades generate multiple stress concentrations along the airfoil, rendering life prediction exceptionally challenging. While conventional skeletal point method (SPM) offers reasonable accuracy in predicting creep-fatigue-interaction (CFI) life for simple structural specimens, they prove inadequate for geometries with poor symmetry. This study introduces a novel two-dimensional skeletal point method (2D SPM) to analyze stress evolution characteristics, identify representative stresses, and predict CFI life in complex structures. Leveraging the film-cooling hole (FCH) features of a representative turbine blade, three perforated plate specimens were designed, manufactured, and subjected to CFI testing. Failure analysis confirmed crack initiation at hole-edge stress concentration zones, followed by inward propagation. Specimen fracture surfaces exhibited predominantly ductile dimpling features, with multi-origin fatigue characteristics observed only near hole-edges, collectively indicating creep-damage-dominated failure mechanisms. Five life prediction methodologies were comparatively evaluated. The results demonstrate that the 2D-SPM achieved the highest accuracy (all predictions within twofold scatter bands), followed by the conventional SPM (also within twofold scatter bands). The nominal stress method showed moderate accuracy (within fivefold scatter bands), while both hot point method and TCD methods proved unsuitable for creep-fatigue scenarios with significant stress evolution.

1. Introduction

The performance and reliability of aero-engine hot-section components, particularly turbine blades, directly determine engine thrust and efficiency. In the pursuit of higher thermal efficiency, the turbine inlet temperature has been continuously elevated, now far exceeding the melting point of nickel-based superalloys [1]. Consequently, blades operate under extreme temperatures, complex mechanical loads, and severe thermal cycling, making the issue of high-temperature creep-fatigue interaction increasingly prominent [2]. To ensure blade durability in such harsh environments, active thermal protection techniques such as film-cooling are widely employed. This technology involves introducing discrete holes or slots on the component surface to inject cooler air from the compressor into the high-temperature mainstream gas at specific angles, thereby forming a protective cool-air film over the surface [3]. This film layer isolates the hot gas from the substrate material, significantly reducing surface temperature through convective and adiabatic shielding, allowing reliable operation in gas environments well above the material’s melting point [4]. However, the machining of film-cooling holes (FCHs) creates unavoidable geometric discontinuities on the blade, which become critical stress concentration zones and introduce complex three-dimensional multiaxial stress states. This complicates the failure mechanisms in these regions far beyond those observed in standard material specimens, severely constraining the structural integrity and life prediction accuracy of the blades [5].

Numerous theories and methods have been developed for predicting the creep and fatigue life of superalloys. Commonly used creep life analysis approaches include the Omega method [6], Theta method [7], and Wilshire method [8]. Fatigue life prediction methods often involve the Larson–Miller parameter [9], the Monkman–Grant relationship, and continuum damage mechanics approaches [10]. However, these methods are difficult to apply directly to actual blade structures with FCH, primarily because standard specimens are typically under simple uniaxial stress states, whereas geometric discontinuities such as film holes and cooling channels exhibit significant multiaxial stress concentrations. Therefore, directly applying standard specimen life models to predict the creep-fatigue life of holed structures often leads to considerable inaccuracies.

To accurately assess the life of structures with FCHs, it is essential to develop analytical methods capable of comprehensively considering multiaxial stresses, material anisotropy, and CFI effects. Conventional life prediction methodologies exhibit fundamental limitations in addressing these complexities, particularly regarding stress redistribution effects during prolonged dwell times. The widely implemented nominal stress approach utilizes cross-sectional average stresses, inherently neglecting stress gradients at discontinuities. While computationally efficient, its disregard for local stress concentration renders it unsuitable for precision-critical components. Hot point methods, focusing exclusively on peak stresses at singular critical locations, demonstrate two key deficiencies in CFI regimes: (a) theinability to capture multi-site damage initiation documented in polycrystalline superalloys, and (b) neglect of stress relaxation-driven damage migration. Due to its convenience, the traditional Skeletal Point Method (SPM) is widely used in the rupture lifetime of notched bars [11,12,13,14] and perforated plates [15]. However, the reliance on predefined symmetric paths fundamentally restricts its application to asymmetric configurations like film-hole clusters, where stress evolution along irregular boundaries cannot be characterized unidimensionally. Theory of Critical Distances (TCD), pioneered by Neuber for notch fatigue analysis, employs characteristic length parameters to average stresses around stress concentrators [16]. Though effective for room-temperature fatigue, its underlying assumption of static stress fields directly contradicts time-dependent stress redistribution in CFI. However, when applied to components with intricate geometries, the aforementioned methods share a common limitation: they are unable to derive a representative equivalent stress over the entire cross-section that can be reliably used for predicting the CFI life of the structure.

To overcome the aforementioned limitations, this paper develops a Two-Dimensional Skeletal Point Method (2D-SPM). The main innovations are: (a) being capable of handling structures with complex geometries; (b) providing a mathematical description of the skeletal point locations; and (c) offering an effective engineering solution for the strength design and life assessment of turbine blades in aero-engines, which is critical for ensuring structural reliability under complex operating conditions.

2. Materials and Methods

2.1. Material Properties

DD32 is a second-generation nickel-based single-crystal superalloy. Its components are shown in

Table 1. Components of DD32 Single-Crystal Superalloy.

Ele	C	Cr	Ni	Co	W
wt%	0.12–0.18	4.3–5.6	-	8–10	7.7–9.5
Ele	Mo	Al	Nb	Ta	Re
wt%	0.8–1.4	5.6–6.3	1.4–1.8	3.5–4.5	3.5–4.5

, and its elastic properties are shown in

Table 2. Elastic Properties of DD32 Single-Crystal Superalloy.

θ/°C	20	800	900	1000
E/GPa	140	111	104	95.8
G/GPa	124	101	96.9	92.0
μ	0.352	0.438	0.444	0.449

[17].

The lasting properties of DD32 on orientation [001] can be expressed by thermal-mechanical parameter synthesis equation (Larson–Miller equation):

$$\mathrm{l}\mathrm{g}\sigma =-5.48394+0.96917\times P-0.03443\times P^2+3.5474\times {10}^{-4}\times P^3$$

(1)

where $P=T\times \left(20+\mathrm{l}\mathrm{g}{t}_f\right)\times {10}^{-3}$.

2.2. Specimens and CFI Tests

This study addresses the FCHs characteristics of a specific turbine blade body. Three distinct types of specimens with simulated FCHs were designed and fabricated, as illustrated in Figure 1. The crystal orientation is consistent with the blade, [001] parallel to the loading direction. The load employed a trapezoidal waveform with a dwell time of 7.5 min, as depicted in Figure 2a.

The tests were carried out on a QBR-100 electronic creep testing machine, as shown in Figure 2b. The fixture system is depicted in Figure 2c, with the rod manufactured from DZ22 superalloy. The furnace can operate in a range of 200 °C to 1200 °C, employing three thermocouples to monitor the temperature. Thermo monitoring and the control device are shown in Figure 2d. The test matrix is presented in

Table 3. Test Matrix and Results of Film-Cooling Simulation Specimens.

	Type	Mean Stress /MPa	Load/kN	Temperature/°C	Result
					Cycle	Hour
1	Leading Edge	279	2.27	961	1024	131
2		279	2.27	961	972	125
3	Front Concave	500	5.63	900	663	83
4	Middle Concave	642	4.73	845	2283	307
5		642	4.73	845	2489	319

.

2.3. Specimen Simulation

Three finite element (FE) models were generated for the three perforated plates, as illustrated in Figure 3. The FE models consisted of 86,000, 6700, and 101,075 elements and 127,975, 88,876, and 21,575 nodes respectively. The structured mesh configuration at critical stress concentration zones ensures the accuracy of the simulation.

The load and boundary conditions were configured as follows (see Figure 4): A uniform load was applied to one end of the specimen, generating a nominal stress of 279 MPa, 500 MPa, and 642 MPa (2.27 kN, 5.63 kN, and 4.73 kN), respectively. At the opposite end, displacement boundary conditions were imposed along the loading direction. The model maintained a uniform temperature at 961 °C, 900 °C, and 845 °C, respectively.

A visco-plastic constitutive model was employed in the simulations, which was developed in previous work by our team [18,19]. The total strain tensor can be decomposed as follows:

$${\mathit{\epsilon }}_{\mathrm{t}}={\mathit{\epsilon }}_{\mathrm{e}}+{\mathit{\epsilon }}_{\mathrm{i}\mathrm{n}}$$

(2)

where ${\mathit{\epsilon }}_{\mathrm{t}}$ stands for the total strain tensor, ${\mathit{\epsilon }}_{\mathrm{e}}$ stands for the elastic strain tensor, and ${\mathit{\epsilon }}_{\mathrm{i}\mathrm{n}}$ stands for the inelastic strain tensor. The stress–strain relationship can be expressed as:

$$\mathit{\sigma }=\mathit{C}\mathbf{:}{\mathit{\epsilon }}_{\mathrm{e}}$$

(3)

where $\mathit{\sigma }$ stands for the stress tensor, “$\mathbf{:}$” represents double-dot product operation, and $\mathit{C}$ is the elastic matrix, which can be expressed as:

$$\left[C\right]=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{12}& 0& 0& 0\\ C_{12}& C_{11}& C_{12}& 0& 0& 0\\ C_{12}& C_{12}& C_{11}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{44}& 0\\ 0& 0& 0& 0& 0& C_{44}\end{array}\right]$$

(4)

where $C_{11}=E∕\left(1+\nu \right)\left(1-2\nu \right)+E∕\left(1+\nu \right)$, $C_{12}=E∕\left(1+\nu \right)\left(1-2\nu \right)$, $C_{44}=G$. $E$, $G$ and $\nu$ stands for elastic modulus, shear modulus, and Poisson ratio, respectively.

The visco-plastic flow equation can be written as

$${\dot{\epsilon }}_{ij}^{in}={\displaystyle \frac{3}{2}}{⟨{\displaystyle \frac{f}{k}}⟩}^n{\displaystyle \frac{M_{ijkl}}{{\left(I-D\right)}_{klmn}}}{\displaystyle \frac{\left({\sigma }_{mn}-X_{mn}\right)}{{\sigma }_{eq}}}$$

(5)

where $f$ stands for the yield function:

$$f={\sigma }_{eq}-R-k$$

(6)

The equivalent stress ${\sigma }_{eq}$ can be expressed as:

$${\sigma }_{eq}=\sqrt{{\displaystyle \frac{3}{2}}{\left({\sigma }_{ij}-X_{ij}\right)M}_{ijkl}^{-1}\left({\sigma }_{kl}-X_{kl}\right)}$$

(7)

$M$ is the anisotropic matrix, which can be expressed as:

$$M=\left[\begin{array}{cccccc}{M}_{11}& 0& 0& 0& 0& 0\\ 0& M_{11}& 0& 0& 0& 0\\ 0& 0& M_{11}& 0& 0& 0\\ 0& 0& 0& M_{44}& 0& 0\\ 0& 0& 0& 0& M_{44}& 0\\ 0& 0& 0& 0& 0& M_{44}\end{array}\right]$$

(8)

$D$ is the damage matrix, which can be expressed as:

$$D=\left[\begin{array}{cccccc}{D}_{11}& 0& 0& 0& 0& 0\\ 0& D_{11}& 0& 0& 0& 0\\ 0& 0& D_{11}& 0& 0& 0\\ 0& 0& 0& D_{44}& 0& 0\\ 0& 0& 0& 0& D_{44}& 0\\ 0& 0& 0& 0& 0& D_{44}\end{array}\right]$$

(9)

where $D_{11}$ and $D_{44}$ represent the damage caused by normal stress and shear stress, respectively. The evolution rates are expressed as:

$${\dot{D}}_{11}=D_n{⟨{\displaystyle \frac{{\overline{\sigma }}_{\mathrm{e}\mathrm{q}}-R-k_0}{K_n}}⟩}^{n_n}{\displaystyle \frac{1}{{\left(1-D\right)}^{k_n}}}$$

(10)

$${\dot{D}}_{44}=D_t{⟨{\displaystyle \frac{{\overline{\sigma }}_{\mathrm{e}\mathrm{q}}-R-k_0}{K_t}}⟩}^{n_t}{\displaystyle \frac{1}{{\left(1-D\right)}^{k_t}}}$$

(11)

$X$ stands for the back stress tensor, and can be expressed as:

$${\dot{X}}_{ij}=N_{ijkl}{\dot{\epsilon }}_{kl}^{in}-Q_{ijkl}{X}_{kl}\dot{p}-P_{ijkl}{J}_2{\left(X\right)}^r{X}_{ij}$$

(12)

$R$ stands for the isotropic hardening scalar, whose evolution rate can be expressed as:

$$\dot{R}=b\left(W-R\right)\dot{p}$$

(13)

where $\dot{p}$ stands for the evolution rate of the equivalent plastic strain

$$\dot{p}=\sqrt{{\displaystyle \frac{2}{3}}{\dot{\epsilon }}_{ij}^{\ast \mathrm{i}\mathrm{n}}{M}_i^{-1}{\dot{\epsilon }}_{kl}^{\ast \mathrm{i}\mathrm{n}}}$$

(14)

$J_2\left(X\right)$ means the second invariant of back stress tensor.

Matrix $\mathit{N}$, $\mathit{Q}$, $\mathit{P}$ reflects the anisotropic evolution of back stress, and can be expressed as

$$\left[N\right]=\left[\begin{array}{cccccc}{N}_{11}& 0& 0& 0& 0& 0\\ 0& N_{11}& 0& 0& 0& 0\\ 0& 0& N_{11}& 0& 0& 0\\ 0& 0& 0& N_{44}& 0& 0\\ 0& 0& 0& 0& N_{44}& 0\\ 0& 0& 0& 0& 0& N_{44}\end{array}\right]$$

(15)

$$\left[Q\right]=\left[\begin{array}{cccccc}{Q}_{11}& 0& 0& 0& 0& 0\\ 0& Q_{11}& 0& 0& 0& 0\\ 0& 0& Q_{11}& 0& 0& 0\\ 0& 0& 0& Q_{44}& 0& 0\\ 0& 0& 0& 0& Q_{44}& 0\\ 0& 0& 0& 0& 0& Q_{44}\end{array}\right]$$

(16)

$$\left[P\right]=\left[\begin{array}{cccccc}{P}_{11}& 0& 0& 0& 0& 0\\ 0& P_{11}& 0& 0& 0& 0\\ 0& 0& P_{11}& 0& 0& 0\\ 0& 0& 0& P_{44}& 0& 0\\ 0& 0& 0& 0& P_{44}& 0\\ 0& 0& 0& 0& 0& P_{44}\end{array}\right]$$

(17)

The visco-plastic model is coupled with creep damage to simulate the tertiary stage of creep. The creep damage can be written as follows:

$${\dot{d}}_{\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{p}}={\left({\displaystyle \frac{{\sigma }_{ea}}{K_d}}\right)}^{n_d}{\left(1-d_{\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{p}}\right)}^{-k_d}$$

(18)

Other variants not mentioned above are all temperature-dependent material parameters, whose values are listed in

Table 4. Material parameters of the visco-plastic constitutive model at different temperatures.

Temperature/°C	800	845	961
E/MPa	91,350	89,955	80,549
ν	0.438	0.4407	0.4462875
G/MPa	62,180	65,732	70,121
K	821	884	1239
n	6.584	6.008	4.2853
N11	5730	6283	8338
N44	7036	5182	1486
Q11	85	66	23
Q44	799	728	749
P11	0	0	0
P44	0	0	0
r	1	1	1
k0	632	575	440
Dn	0	0	0
Kn	10,000	10,000	10,000
nn	1	1	1
kn	1	1	1
Dt	0	0	0
Kt	10,000	10,000	10,000
nt	1	1	1
kt	1	1	1
M11	1	1	1
M44	1.025	0.93815	0.99975
b	226	169	133
W	46	51	68

.

2.4. Framework of 2-Dimensional Skeletal Point Method (2D-SPM)

To characterize the stress evolution near stress concentration zones, deform simulation is first conducted based with a visco-plastic constitutive model. Stress redistribution across the entire structure occurs as the creep time progresses: stress relaxation manifests in the initially high-stress regions, gradually stabilizing to a constant level; concurrently, stresses in low-stress regions increase and stabilize. Ultimately, all stresses converge to a uniform magnitude, typically in the vicinity of nominal stress.

Creep-fatigue simulations on perforated plates (Figure 5a) revealed converging stress profiles at a unique spatial locus. This critical intersection defines: (a) the stress relaxation zone (interior to the converging point): continuous stress reduction over simulated duration; the (b) skeletal point (SP): a unique spatial position where stress remains time-invariant (∂σ/∂t = 0), serving as the stress-stability marker; and (c) the stress elevation zone (SP to exterior): stress increase toward nominal levels.

The traditional skeletal point method (SPM) analyzes stress distributions along a predefined one-dimensional path to locate the SP, rendering it effective only for highly symmetric geometries such as single-hole plates (uniform thickness, Figure 5a) and notched round bars (axisymmetric profiles, Figure 5b). However, for geometrically irregular structures (e.g., turbine blades), SPM faces critical limitations, because of the path selection ambiguity: no rational criterion exists to define a representative stress evaluation path. Consequently, conventional SPM fails to deliver reliable life assessment for complex components.

This work advances the traditional one-dimensional skeletal point method (SPM), by proposing a novel Two-Dimensional Skeletal Point Method (2D-SPM) for uniaxially loaded complex structures.

As illustrated in Figure 6, for any cross-section of uniaxially loaded complex structures exhibiting stress concentrations, this method initiates by selecting arbitrary stress concentration points A_i along the boundaries with geometric discontinuties, and reference points B_j on the outer contour. A unique skeletal point S_k exists along each radial line connecting A_i to B_j, where the stress maintains time-invariance (∂σ/∂t = 0). Critically, within the segment spanning A_i to S_k, stress exhibits monotonic decay, signifying local relaxation effects, while the interval from S_k to B_j demonstrates progressive stress elevation indicative of energy redistribution phenomena.

Through systematic variation of (A_i, B_j) pair combinations, an infinite set of skeletal points S₁, S₂, …} is generated. Sequentially connecting these points forms a continuous closed curve Γ_S with explicit physical zoning functions: the interior region constitutes a Stress Relaxation Zone where stress diminishes temporally (∂σ/∂t < 0); while the exterior region represents a Stress Elevation Zone characterized by temporally increasing stress (∂σ/∂t > 0).

Following the acquisition of the boundary Γ_S, the stress magnitude at each discretized point along this contour is extracted through finite element post-processing. Subsequently, a line integral is performed along the closed contour Γ_S to obtain the characteristic boundary-averaged stress:

$$\overline{\sigma }={\displaystyle \frac{1}{L}}{\oint }_L\sigma \left(x\right)\mathrm{d}x$$

(19)

where $L$ denotes the total arc length of the contour Γ_S. This metric serves as a critical input for life assessment algorithms targeting geometrically complex components.

Through finite element simulations of specimens under creep-fatigue loading, stress relaxation zone boundaries are determined. For single-hole plates introducing a sole geometric discontinuity (i.e., one circular hole), stress concentration exclusively occurs around the film cooling hole. Consequently, the stress relaxation zone exhibits circumferential distribution near the hole perimeter, as visualized in Figure 7a. Similarly, for notched cylindrical bars, stress concentration regions localize at the notch tip, resulting in an annular stress relaxation zone encompassing the notched interface (Figure 7b).

Through FE simulation, the critical section and stress relaxation zone boundary of the specimen are determined. The average stress along this boundary is then calculated according to Equation (1), serving as the characteristic stress of the specimen.

To clarify, 2D-SPM and other creep model address different levels of analysis: creep model (e.g., Norton–Bailey model) describes material behavior, while the 2D-SPM bridges the gap between material data and structural life prediction under complex geometries. They have totally different objectives and outputs. (i) The Norton–Bailey model and its variants (e.g., the Theta projection method and Wilshire model) are material-level constitutive models, which describes the evolution of creep strain over time under a given uniaxial or multiaxial stress state. They do not account for structural geometry or stress redistribution. (ii) In contrast, the proposed 2D-SPM is a structural life assessment tool. Its purpose is to extract a characteristic stress from a complex geometry (via the stress relaxation zone boundary ΓS), and then use that stress to predict the CFI rupture life of the entire component. The 2D-SPM does not replace material creep laws; rather, it provides a rational method to reduce a multiaxial, non-uniform stress field to a representative scalar stress that can be used with existing uniaxial life prediction equations (e.g., Larson–Miller type correlations).

3. Results

3.1. CFI Life Prediction by 2D-SPM

Through FE simulation, the boundaries of the stress relaxation zones on the critical cross-sections of each specimen were determined, as well as the corresponding stress distribution curves along them, as illustrated in Figure 8.

For each specimen, a total of 42 skeletal points were used to construct the closed contour Γ_S. The average spacing between adjacent skeletal points is less than 0.05 mm, which ensures a sufficiently smooth representation of the contour given the characteristic dimensions of the specimens and the finite element mesh size. The chosen resolution (42 points) captures the stress variation along Γ_S with high fidelity, as confirmed by the smoothness of the resulting stress distribution curves. The predicted lives by the Two-Dimensional Skeletal Point Method are shown in

Table 5. Prediction of Perforated Plate CFI Life via Two-Dimensional Skeletal Point Method.

Specimen Type	Characteristic Stress/MPa	Temperature/°C	CFI Life/h	Predicted Life/h	Prediction/Test
Leading Edge Simulator	341	961	128	130	1.02
Front Concave Simulator	512	900	83	74	0.89
Middle Concave Simulator	562	845	313	360	1.15

.

3.2. FEM Simulation

(1)

Leading Edge Simulator

The simulation computed the deformation behavior of perforated plate specimens over 500,000 s (≈139 h), with contour plots of stress/strain fields at discrete time intervals, as presented in Figure 9.

The stress contour plots reveals that stress concentration zones develop at both sides of the holes, while reduced stress regions emerge along the upper and lower boundaries. As creep time progresses, the stress concentration intensity notably diminishes, concurrently elevating the minimum stress levels around the perforation edges. Significant stress interaction exists between adjacent perforations throughout all loading stages, whereas no substantial stress coupling occurs between discrete exhaust FCH.

(2)

Front Concave Simulator

Simulation captured the deformation evolution of perforated plate specimens over a 50-h duration, with Figure 10 presenting the stress and strain distribution contour plots at sequential time intervals.

Stress distribution contours reveal distinct concentration zones at the lateral perforation edges (left/right), contrasting with low stress regions along the upper/lower boundaries. With progressive creep exposure, stress concentrations attenuate significantly while minimum stress levels progressively elevate around the perforation periphery.

(3)

Middle Concave Simulator

The computational simulation characterizes the creep deformation evolution of perforated plate specimens over specified time intervals, with Figure 11 sequentially presenting stress and strain distribution contour plots at progressive creep durations.

The high-stress concentration factor (Kt ≈ 2.94) observed in the middle concave simulator is indeed higher than that of a standard circular hole in an infinite plate (Kt = 3 for a circular hole in infinite plate, but for a hole in a finite plate with moderate width, Kt typically less than 2.5). However, the geometry of the middle concave simulator is not a simple circular hole; it features a sharp re-entrant corner at the intersection between the hole and the specimen surface, with an included angle of approximately 28.5°. In contrast, the leading edge and front concave simulators have intersection angles of 60° and 45°, respectively. This acute angle significantly elevates the local stress concentration due to two mechanisms: (i) the geometric effect—a sharper corner reduces the load-bearing cross-section locally, leading to higher tensile stress; and (ii) the numerical artifact—the highly acute angle inevitably results in poor-quality finite elements (highly distorted or slender elements) in the meshing process, which may artificially amplify the computed peak stress despite our efforts to refine the mesh. Therefore, the reported maximum stress of 1890 MPa should be interpreted with caution; it may overestimate the true local stress to some extent.

As demonstrated by the stress distribution contours, the acute-angle flank (right side) of the oblique perforations constitutes a primary stress concentration zone, while adjacent upper and lower regions exhibit reduced stress states. Progressive creep exposure drives significant attenuation of stress concentrations over time. Although FCHs maintain triangular array configurations, prominent inter-hole stress concentrations persist, with substantial creep accumulation confirmed through strain distribution analysis.

4. Discussion

4.1. Failure Analysis

The appearance of fractured specimens is shown in Figure 12. After high-temperature testing, roughened oxides formed on the specimen surface, exhibiting black-gray and blue colors; residual white asbestos fibers were visible in the specimen gauge section (used during testing to fix thermocouples).

Necking occurred at the fractograph, with shear lips 45° inclined to the loading direction observed on the specimen edges. These features confirm typical ductile fracture characteristics, indicating significant creep deformation prior to fracture.

The fractographic morphology of the leading-edge film-cooling-hole specimen is presented in Figure 13a. The fracture surface exhibits overall roughness with fibrous texture characteristics; multiple creep cavities are visible (Figure 13b). This evidence supports the determination that the fracture mechanism is predominantly a creep-dominated ductile fracture.

The primary fracture surface exhibits multi-origin crack initiation characteristics, as illustrated in Figure 14. A detailed examination of Figure 14b identifies dual crack initiation sites between adjacent holes: cracks nucleating from stress concentration points at the hole periphery (lower region) coexist with linear initiation features at the inter-hole midpoint on the specimen surface (upper region). Figure 14c reveals a hole-edge initiated crack in the lower-right sector propagating radially outward, concurrent with stress concentration-induced crack initiation at the specimen corner (upper-left sector). Furthermore, Figure 14d confirms additional multi-origin signatures across the specimen surface.

The crack initiation sites are further corroborated by observations from the specimen side surfaces, as evidenced in Figure 15. Figure 15a,b depict crack nucleation induced by stress concentration near a single hole, where the primary crack deviates from the widest section of the hole periphery. This deviation indicates upward migration of the peak stress concentration zone along the hole edge. Figure 15c reveals multi-origin crack initiation on the specimen surface resulting from stress interference between adjacent holes. Figure 15d demonstrates localized stress concentration at the specimen boundary, where additional crack nucleation occurs.

The fractographic morphology of the mid-blade pressure surface film-cooling hole specimen is presented in Figure 16. The fracture surface exhibits overall roughness characterized by tear ridges (Figure 16a) and facet features (Figure 16b), coexisting with multiple creep cavities (Figure 16c,d). These morphological characteristics support the determination that the fracture mechanism is predominantly a creep-dominated ductile fracture, with additional quasi-cleavage fracture signatures present microscopically.

The primary fracture surface exhibits multi-origin crack initiation characteristics, as shown in Figure 17. Figure 17b displays a film-cooling hole on the fracture surface exhibiting linear initiation features along its periphery. Figure 17c,d represent crack initiation zones on the left and right sides of the primary fracture surface, respectively, both originating from localized stress concentration on the specimen surface.

Figure 18 presents the microscopic morphology on the specimen side surface. Figure 18a shows a film-cooling hole on the primary fracture surface where the crack path traverses the peak stress concentration zone at the hole periphery, confirming localized stress concentration as the dominant fracture initiator. Additional micro-cracks are observable on the specimen surface. Figure 18c demonstrates stress concentration induced by inter-hole stress interference between adjacent perforations, initiating multifocal cracks at the surface. Figure 18d reveals supplementary crack nucleation sites at the specimen boundary where localized stress concentration similarly develops.

4.2. Contrast of Different Life Methods

Creep-fatigue life predictions were comparatively conducted for three specimen configurations using five distinct methodologies: the Nominal Stress Method, Hot Point Method, Skeletal Point Method, Theory of Critical Distances (TCD), and the proposed Two-Dimensional Skeletal Point Method.

(1)

Nominal Stress Method

The nominal stress method characterizes the entire specimen using the average stress over the minimum cross-section, which is then applied to the material’s life prediction equation. For single-hole straight perforated plates, the minimum cross-section corresponds to the plane containing the film-cooling hole axis. This nominal stress value is substituted into the thermal-mechanical parameter synthesis equation for DD32 single-crystal superalloy for life prediction, as defined by Equation (1).

Results by Nominal Stress Method are shown in

Table 6. Prediction of Perforated Plate CFI Life via Nominal Stress Method.

Specimen Type	Nominal Stress/MPa	Temperature/°C	CFI Life/h	Predicted Life/h	Prediction/Test
Leading Edge Simulator	279	961	128	497	3.79
Front Concave Simulator	500	900	83	90	1.08
Middle Concave Simulator	642	845	313	89	0.28

.

(2)

Hot Point Method

A “hot point” denotes the most critical failure-prone location within a structure while predicting life. The Hot Point Method employs the life expectancy at this specific point to represent the entire component’s durability, inherently yielding conservative prediction outcomes. For the perforated plate testing conducted under uniform temperature conditions, the hot point corresponds precisely to the maximum stress point on the specimen. Comparative life prediction results for the Hot Point Method are detailed in

Table 7. Prediction of Perforated Plate CFI Life via Hot Point Method.

Specimen Type	Maximum Stress/MPa	Temperature/°C	CFI Life/h	Predicted Life/h	Prediction/Test
Leading Edge Simulator	504	961	128	7	0.05
Front Concave Simulator	940	900	83	0.0001	0.00
Middle Concave Simulator	1890	845	313	0.0007	0.00

.

(3)

1-D Skeletal Point Method

The conventional 1-D Skeletal Point Method determines a specific material point in multiaxial stress fields, utilizing its stress state to predict multiaxial creep life based on uniaxial creep test data. Through finite element analysis of perforated plates under multiaxial creep conditions, a skeletal point is identified at the location where stress remains time-invariant near the hole edge. As demonstrated by the curve family in Figure 19, normalized stress distribution profiles converge at 11% normalized distance (approximately 0.313 mm from the hole edge). This convergence point is established as the skeletal position, with its characteristic stress of 520 MPa employed for life prediction. The resultant life assessment data are compiled in

Table 8. Prediction of Perforated Plate CFI Life via 1-D Skeletal Point Method.

Specimen Type	Skeletal Point Stress/MPa	Temperature/°C	CFI Life/h	Predicted Life/h	Prediction/Test
Leading Edge Simulator	315	961	128	223	1.74
Front Concave Simulator	515	900	83	70	0.84
Middle Concave Simulator	630	845	313	110	0.35

.

(4)

Theory of Critical Distance

The Theory of Critical Distances (TCD), initially established by Neuber in the mid-20th century, addresses notch effects by defining an effective stress for fatigue strength analysis as the average stress over a critical region surrounding a stress concentration “hot point”. The local stress processing formulation for TCD methodology is presented in Equation (20):

$${\sigma }_0={\displaystyle \frac{1}{L}}{\int }_0^L\sigma \left(r\right)\mathrm{d}r$$

(20)

In the Theory of Critical Distances (TCD) implementation, L represents the critical distance. For the three perforated plate specimens analyzed, the critical distance is defined as the radial span from the hole edge to the skeletal point, with values of L = 0.16, 0.11 and 0.05 mm, respectively.

The initial stress distribution $\sigma \left(r\right)$ is fitted using a polynomial function:

$$\sigma \left(r\right)={\sum }_{i=0}^n{a}_i{r}^i,n>2$$

(21)

For the single-hole plate life assessment, the polynomial order is set to n = 3. The average stress within twice the critical distance (2$L$) is computed accordingly, with the life prediction results presented in

Table 9. Prediction of Perforated Plate CFI Life via Theory of Critical Distance.

Specimen Type	Characteristic Stress/MPa	Temperature/°C	CFI Life/h	Predicted Life/h	Prediction/Test
Leading Edge Simulator	479	961	128	11	0.09
Front Concave Simulator	893	900	83	0.06	0.00
Middle Concave Simulator	1795	845	313	0.0008	0.00

.

(5)

2-D Skeletal Point Method

Through finite element simulation, the stress relaxation zone boundaries on critical cross-sections of each specimen were determined (visually presented in Figure 20). The corresponding stress distribution curves along these boundaries are graphically demonstrated in Figure 21.

The predicted lives by the 2D-SPMare shown in Table 5.

The reversal in the relationship between the characteristic stress (2D-SPM) and the nominal stress across the three specimens can be understood by examining two competing factors: (i) the severity of stress concentration at the hole, and (ii) the extent of stress relaxation due to creep.

For the leading edge simulator (nominal stress 279 MPa, Kt ≈ 1.81) and the front concave simulator (nominal stress 500 MPa, Kt ≈ 1.88), the stress concentration factors are relatively moderate. Consequently, the stress relaxation near the hole is limited, and the skeletal point (or the contour ΓS) lies within a region where the stress remains elevated above the nominal value. Thus, the characteristic stress (341 MPa and 512 MPa) exceeds the nominal stress.

For the middle concave simulator, however, the geometry introduces an extremely sharp re-entrant corner (intersection angle ≈ 28.5°), leading to a very high local stress concentration (Kt ≈ 2.94, as discussed in Comment 3). Under such severe stress concentration, two phenomena occur: (i) the peak stress is extremely high (1890 MPa), which drives rapid creep relaxation in the immediate vicinity of the corner; and (ii) the skeletal point, defined as the locus of time-invariant stress, shifts outward to a region where the stress has already relaxed to a level substantially lower than the nominal stress. This explains why the characteristic stress for the middle concave simulator (562 MPa) is lower than its nominal stress (642 MPa).

A comparative assessment of the five life prediction methods reveals significant differences in their applicability to CFI life assessment of perforated plate specimens. The nominal stress method, while computationally straightforward, tends to overestimate or underestimate life due to its inherent neglect of local stress concentration and subsequent stress redistribution. The hot point method and TCD method both rely on initial stress fields and fail to capture the time-dependent stress relaxation and redistribution that are characteristic of CFI conditions, leading to severely non-conservative predictions. The conventional 1D-SPM improves prediction accuracy by identifying a time-invariant skeletal point along a predefined path; however, its reliance on symmetric geometries limits its applicability to complex configurations. In contrast, the proposed 2D skeletal point method (2D-SPM) introduces a physically meaningful stress relaxation zone boundary, which delineates the region where stress relaxes over time from the region where stress elevates. By integrating the stress distribution along this closed contour, the 2D-SPM yields a characteristic stress that inherently accounts for both stress concentration and redistribution. As summarized in

Table 10. Perforated Plates CFI Life and Predicted Life via Different Methods.

		Leading Edge Simulator	Front Concave Simulator	Middle Concave Simulator
Nominal Stress/MPa		279	500	642
Temperature/°C		961	900	845
CFI Life/h		128	83	313
Nominal Stress Method	Predicted Life/h	497	90	89
	Predicted/Test	3.79	1.08	0.28
Hot Point Method	Predicted Life/h	7	0.0001	0.0007
	Predicted/Test	0.05	0.00	0.00
1-D Skeletal Point Method	Predicted Life/h	223	10	110
	Predicted/Test	1.74	0.84	0.35
TCD Method	Predicted Life/h	11	0.06	0.0008
	Predicted/Test	0.09	0.00	0.00
2-D Skeletal Point Life	Predicted Life/h	130	74	360
	Predicted/Test	1.02	0.89	1.15

and illustrated in Figure 22, the 2D-SPM achieves the highest prediction accuracy, with all predictions falling within a twofold scatter band. These results demonstrate that the proposed method provides a reliable and practical engineering tool for the strength design and life assessment of aero-engine turbine blades under complex CFI loading conditions.

5. Conclusions

This study proposes a life prediction method for uniaxially loaded complex structures under CFI conditions, termed the two-dimensional skeletal point method (2D SPM). Perforated plate specimens were designed and subjected to creep-fatigue testing, with a comparative analysis of different life prediction methods yielding the following conclusions:

• Failure analysis of specimens indicated crack initiation at hole-edge stress concentration zones propagating inward. Fracture surfaces exhibited predominantly ductile features like dimples, with multi-origin fatigue characteristics observed only minimally near hole-edges, indicating creep-damage-dominated failure.
• Simulation calculations under test conditions determined stress distribution characteristics for all three specimen types.
• Specimen life analysis using the nominal stress method, hot point method, 1D-SPM, TCD method, and 2D-SPM demonstrated the highest accuracy for the 2D-SPM, followed by the conventional 1D-SPM. Both methods yielded predictions within twofold scatter bands. The nominal stress method showed moderate accuracy (within fivefold scatter bands), while the hot point and TCD methods proved unsuitable for creep-fatigue life prediction.