Sage Journals: Discover world-class research

Abstract

In this work, fatigue crack propagation tests were carried out at both macroscopic and microscopic scales. The propagation process was monitored using the digital image correlation (DIC) technique, which provided data on the fatigue crack growth rate and the unified constraint parameter A_p (a parameter can represent both geometric constraint material constraint), and the correlation between the fatigue crack propagation rate and constraint was studied. Results demonstrate that a linear relationship between the fatigue crack propagation rate and the unified constraint parameter A_p can be established for specimens at both scales, indicating a clear correlation between crack growth rate and structural constraint. Moreover, this correlation has been verified across different materials. In addition, based on the observed linear relationship, a new criterion for crack propagation was developed.

Keywords

fatigue crack propagation rate unified constraint parameter linear relationship crack propagation criterion macroscopic and microscopic

Highlights

The relationship between the fatigue crack propagation rate and the unified constraint parameter was established experimentally.

The relationship is irrelevant to the material, the stress level, and also irrelevant to the macro or micro conditions.

Based on the relationship, a new crack propagation criterion was established.

Introduction

Fatigue is an important factor affecting the life and reliability of structures. According to statistics, more than 80% of engineering structure failures are related to fatigue.^1–3 To achieve long-life and high-reliability structural design, it is necessary to conduct in-depth research on fatigue issues.

The fatigue crack propagation rate is widely considered a primary indicator of fatigue performance. In the past, people have focused their research on the fatigue crack propagation rate and obtained a series of fatigue crack propagation models. Paris et al.⁴ summarized the power-law law describing the fatigue crack propagation rate, which is known as the Paris formula. To more accurately describe the fatigue crack propagation behavior, Forman et al. considered the influence of average stress on the crack propagation rate da/dN and ΔK, and proposed the Forman formula by modifying the Paris formula.⁵ Paris formula and Forman formula have been widely used in macroscopic fatigue crack propagation. However, the actual fatigue crack propagation process is a multi-scale development process from micro to macro, and Paris formula and Forman formula are not applicable to describe the short crack fatigue propagation law.^6,7

To this end, Deng et al.⁸ investigated the fatigue crack propagation mechanism of 304 stainless steel under varying stress levels. Results indicate that short cracks exhibit significantly higher propagation rates than long cracks, accompanied by pronounced fluctuations and deceleration phenomena. This behavior is primarily impeded by grain boundaries and annealed twin boundaries, with annealed twin boundaries exhibiting the strongest crack-inhibiting effect. As stress levels increase, microstructural sensitivity diminishes, fluctuations decrease, and behavior approaches that of long cracks. Based on these findings, an effective driving force model was proposed, along with an empirical formula to describe the fatigue crack propagation behavior of short cracks. Ye et al.⁹ investigated the fatigue crack propagation mechanisms of microcracks, mesoscale physical cracks, and macroscopic long cracks, proposing a multiscale fatigue crack propagation model. This model explicitly divides the entire fatigue life into the three consecutive stages mentioned above and achieves precise prediction of material fatigue failure behavior by accumulating the propagation life of each stage.

In addition, it is well known that constraint can affect the propagation of fatigue cracks. Constraint is closely related to the crack-tip stress triaxiality, and can be regarded as the resistance of a structure against plastic deformation.¹⁰ Under small-scale yielding, the stress field at the crack tip is generally controlled by a single fracture mechanics parameter ‘K_I’ or ‘J’, whereas under large-scale yielding, the relationship between ‘K_I’ or ‘J’ and the stress field at the crack tip gets lost, and this phenomenon is called the loss of constraint. Generally, the constraint effect caused by crack size, specimen or structural geometry, and loading mode is referred to as geometric constraint, while the constraint effect resulting from material property mismatch is called material constraint. Geometric constraint can be further divided into in-plane constraint and out-of-plane constraint. The in-plane constraint is influenced by specimen dimensions in the direction of crack propagation, such as the length of the unbroken ligament, while the out-of-plane constraint is affected by specimen/structural dimensions in the direction parallel to the crack front, such as the specimen thickness. The loading mode can be either in-plane or out-of-plane, thus affecting both in-plane and out-of-plane constraints. Geometric constraint and material constraint are collectively referred to as unified constraint.

The influence of constraints on fatigue crack propagation has been extensively studied, yet consensus on its mechanisms remains elusive. Varfolomeev et al.¹¹ investigated the influence of constraints on fatigue crack propagation through experimental and numerical simulations, confirming that variations in specimen geometry significantly alter crack propagation behavior. Tong¹² argued that negative T-stress (low constraint) enlarges the plastic zone at the crack tip. This leads to a stronger crack closure effect, which in turn decelerates crack propagation. In contrast, an opposing view is held by Hutař et al.^13,14 They introduced T-stress as a key parameter to quantify crack tip constraint effect, identifying variations in constraint levels as the fundamental physical mechanism underlying the pronounced differences in fatigue crack propagation rates and threshold values observed across geometries. Their work consistently associated high constraint levels with reduced fatigue crack propagation rates. Conversely, positive T-stress (high constraint) maintains or reduces the plastic zone size, resulting in a higher propagation rate. Furthermore, the specific regime in which the constraint effect is operative is also debated. Seitl and Hutař¹⁵ and Wang et al.,¹⁶ through combined experimental and modeling approaches, concluded that constraints influence behavior both in the Paris regime and the near-threshold zone, the findings of Hutař et al.¹⁴ suggest that the effect is confined solely to the near-threshold region.

Because the fatigue crack propagation rate is generally measured using standard specimens in laboratories¹⁷ without considering the difference in constraint levels between actual structures and laboratory specimens,^18–20 when the fatigue crack propagation rate of high-constraint specimens is used to evaluate the structural integrity of low-constraint structures, conservative or non-conservative evaluation results are obtained. To conduct a more accurate integrity assessment of structures, many researchers have carried out a lot of research in the past decades, and promoted the constraint parameters from the initial single parameter (K_I,²¹CTOD,²²J^23,24) to the two-parameter (K-T,²⁵J-T,²⁶J-Q,²⁷J-A₂^28–30, J-T_Z^31–34) and three-parameter (K-T-T_Z^35,36, J-Q-T_Z,³⁷J-Q-M³⁸). With the increase in the number of constraint parameters, the characterization of the constraint effect becomes more accurate; however, it causes trouble in calculations. Furthermore, some unified constraint parameters (β_T,³⁹φ,⁴⁰A_p^41,42, A_d,⁴³A_d*^44–46) have been gradually developed. In recent researches, these unified parameters have been applied in structural integrity assessment.^47–50 However, it must be pointed out that most of these parameters are focused on monotonic loading, the effect of constraints on the fatigue crack propagation still need further research, especially, a correlation between the constraint and fatigue crack propagation still needs to be established.

Recently, the authors^51,52 investigated the relationship between fatigue and constraint, and based on the unified constraint parameter A_p, the correlation of the fatigue crack growth rate, creep-fatigue crack growth rate with unified constraint parameter was established separately. However, these relationships were derived from finite element simulations. The authors have long sought to experimentally validate these correlations and further apply them in practice, which is the primary motivation for the present study.

In present study, fatigue crack propagation tests were carried out at both macroscopic and microscopic scales. The main objective of this study is to experimentally validate the relationship between the fatigue crack propagation rate and the constraint at both macro and micro scales. The propagation process was monitored using the digital image correlation (DIC) technique, which provided data on the fatigue crack growth rate and the unified constraint parameter, and the correlation between the fatigue crack propagation rate and the constraint was studied.

Experimental method

Material

Nickel-based GH4169 superalloy were selected in this study, the chemical composition of GH4169 superalloy is listed in Table 1. The Young’s modulus of GH4169 superalloy is 216 GPa, the yield strength is 1276 MPa, the ultimate tensile strength is 1420 MPa, and the elongation is 17.52%.

Table 1.

Chemical composition of the nickel-based GH4169 superalloy (wt %).

C	S	P	Mn	Fe	Cr	Mo	Nb	Al	Ti	B	Ni
0.023	0.001	0.012	0.03	18.75	17.86	2.98	5.38	0.57	0.99	0.0048	Bal.

Macroscopic and microscopic specimens

Both macroscopic and microscopic specimens (the initial crack lies at the macro- or micro-scale) with different constraints were used, as shown in Figure 1. For macroscopic specimens (Figure 1(a) and (b)), the constraint of the specimens was changed by varying the initial crack length (a/W = 0.2, 0.3, 0.5, 0.7, and B = 5 mm). Specimens with a/W = 0.2 and 0.3 are low-constraint specimens, while specimens with a/W = 0.5 and 0.7 are high-constraint specimens. For microscopic specimens (Figure 1(c) and (d)), the constraint of the specimens was changed by varying the specimen thickness (B = 0.5, 0.7, and 0.9 mm, a = 75 μm). Specimen with B = 0.5 mm is low-constraint specimen, while specimens with B = 0.7 and 0.9 mm are high-constraint specimens.

Figure 1.

Schematic diagram (a) and physical image (b) of macroscopic specimen, schematic diagram (c) and physical image (d) of microscopic specimen.

For macroscopic specimens, the specimen notch is made by wire cutting, with a kerf width of 0.2 mm and a notch radius of 0.1 mm; for microscopic specimens, the specimen notch is produced by laser, with a spot diameter of 30 μm and a notch radius of 15 μm.

Fatigue crack propagation experiment

Fatigue crack propagation experiments were conducted on an INSTRON 8801 fatigue crack propagation test platform (Figure 2). For macroscopic specimens, two groups of experiments with different stress levels (maximum stress = 600 and 800 MPa, the dimensionless value of the ratio of maximum stress to yield stress is about 0.47 and 0.63, respectively) were carried out. For microscopic specimens, one group of experiments with only one stress level (maximum stress = 800 MPa) was carried out. The stress ratio was 0.1, and the cyclic waveform was sinusoidal. In addition, the strain field at the crack tip during fatigue crack propagation was measured in real-time by DIC.

Figure 2.

Fatigue crack propagation test platform.

In this work, the VIC-3D DIC system provided by Correlated Solutions, Inc (CSI) was used. The system is equipped with a Basler ace acA4112-30 μm camera and a Schneider 50 mm lens. The camera has a resolution of 4096 × 3000 pixels, the resolution is 106.4 pixels per millimeter, and a photo is taken every 0.1 s. Such a configuration not only directly avoids the pseudo-strain errors associated with the single-camera vertical imaging in traditional 2D-DIC but also utilizes the principle of triangulation to reconstruct the three-dimensional coordinates of the sample surface. This enables accurate quantification of out-of-plane displacements and surface depressions related to the crack tip plastic zone, effectively eliminating systematic measurement errors caused by pseudo-strain and ensuring the accuracy of the experimental results.

In the DIC analysis, the selected subset size was 31 × 31 pixels, with a step size of 5 pixels and a strain window size of 13 × 13 data points. The main procedure used for applying the speckle pattern is as follows: Speckle pattern fabrication—creating artificial speckles with high grayscale contrast on the specimen. System calibration—establishing the spatial coordinate system defined by the two cameras, and determining the internal camera parameters (such as image scale, focal length, image center, lens distortion, etc.) and the external system parameters (such as stereo angle, camera spacing, distance from the cameras to the test object, etc.). Deformation acquisition—capturing clear images of the deformation process of the measured object.

For microscopic specimens, due to their small size, they cannot be effectively clamped by the built-in clamps of the fatigue testing machine. To solve this problem, the authors designed a special collet, as shown in Figure 3. These clamps are highly adaptable, capable of tightly gripping microscale specimens of various thicknesses, while also ensuring the stability of the gripping process.

Figure 3.

The clamping state of the microscopic specimen during the test.

Experimental results

Macroscopic specimens

a - N curve of the macroscopic specimens

Figure 4 presents a comparison of the a - N curves for all macroscopic specimens under different constraints, tested at 600 and 800 MPa until fracture (the crack length was measured through the software ‘Image - J’). It can be found that with the increase of the constraint (initial crack length), the fatigue life tended to decrease, which precisely reflects the correctness of the results. At a stress level of 600 MPa (Figure 4(a)), as the a/W ratio increases from 0.3 to 0.7, the number of cycles to failure of the specimen significantly decreases from 19,279 to 7440, with the entire a - N curve exhibiting a notable leftward shift as constraint increases. At a stress level of 800 MPa (Figure 4(b)), the accumulation of fatigue damage accelerates. The fatigue life at 800 MPa was significantly lower than that at 600 MPa. The life corresponding to a/W = 0.3 is only 7255 cycles, while that at a/W = 0.7 further decreases to 3382 cycles. The change of the a - N curves at low constraints was more sensitive than that at high constraints. In the low constraint range (a/W = 0.2–0.5), the life drastically decreases with increasing constraint. In the high constraint range (a/W = 0.5–0.7), the a - N curve eventually stabilizes. It should be pointed out that at 600 MPa, the authors also conducted test on specimen with a/W = 0.2. However, due to its low constraint, the crack growth rate was extremely slow, and the crack extension was minimal, preventing the extraction of valid data.

Figure 4.

a– N curves of the macroscopic specimens with different constraints at (a) 600 MPa and (b) 800 MPa.

Strain field at the crack tip

Figure 5 displays the evolution process of the crack tip strain (PEEQ) field during fatigue crack propagation in the macroscopic specimen with the initial crack length a/W = 0.3 at 800 MPa (the image recognition and post processing methods were used to manually identify the crack tip position of each image, and marked it by the white dot). The entire propagation process is well demonstrated. Additionally, it can be seen from Figure 5 that with the increase in the number of cycles, the crack propagation length and the crack tip strain field gradually increased and then increased rapidly before fracture. Throughout the entire propagation process, the crack deflected downward, which may be attributed to minor clearances or misalignments in the testing machine’s fixtures, pins, and connections. It should be noted that this does not affect the final result.

Figure 5.

Evolution process of the crack tip strain field during fatigue crack propagation in the macroscopic specimen with a/W = 0.3 at 800 MPa.

The crack tip strain fields of the macroscopic specimens with different constraints at the 3000th cycle under 800 MPa are presented in Figure 6. The crack tip strain fields of the macroscopic specimens just before fracture under 800 MPa are exhibited in Figure 7. It is noticeable from Figure 6 that at the same cycle, the strain field at the crack tip increased gradually with the increase of constraint, especially for the macroscopic specimen with a/W = 0.7, the crack started to propagate. It is evident from Figure 7 that the strain field at the crack tip decreased gradually with the increase of a/W just before fracture, indicating that under a high constraint, a smaller accumulation of strain was required to cause the macroscopic specimen to fracture. Upon closer inspection, the cracks in all specimens in Figure 7 deflected downward, which may be attributed to minor clearances or misalignments in the testing machine’s fixtures, pins, and connections. However, this crack deflection is not expected to affect the conclusions drawn in this study.

Figure 6.

Comparison of the crack tip strain fields of the macroscopic specimens with different constraints at the 3000th cycle under 800 MPa.

Figure 7.

Comparison of the crack tip strain fields of the macroscopic specimens with different constraints just before fracture under 800 MPa.

Microscopic specimens

a - N curve of the microscopic specimens

Figure 8 shows that the a - N curves of the microscopic specimens with different constraints at 800 MPa. From the results, it can be seen that the fatigue life of the specimen with B = 0.5 mm (low constraint) is longer. While the fatigue life of the specimens with B = 0.7 and 0.9 mm (high constraint) is similar, mainly due to their similar level of crack-tip constraint. It should be noted that, likely due to the relatively high load, the crack propagated rapidly in the initial stage, and therefore the phase of microscopic crack propagation was not adequately captured.

Figure 8.

a–N curves of the microscopic specimens with different constraints at 800 MPa.

Strain field at the crack tip

Figure 9 displays the evolution process of the crack tip strain field during fatigue crack propagation in the microscopic specimen with B = 0.7 mm. The same with macroscopic specimen, with the increase in the number of cycles, the crack propagation length and the crack tip strain field gradually increased and then increased rapidly before fracture.

Figure 9.

Evolution process of the crack tip strain field during fatigue crack propagation in the microscopic specimen with B = 0.7 mm.

The crack tip strain fields of the microscopic specimens with different constraints (different specimen thicknesses) at the 8150th cycle are presented in Figure 10. It is noticeable from Figure 10 that at the same cycle, the strain field at the crack tip increased gradually with the increase of B, especially for the microscopic specimen with B = 0.7 and 0.9 mm, there is a significant difference with the microscopic specimen with B = 0.5 mm. The strain field at the crack tip is larger and the crack has propagated.

Figure 10.

Comparison of the crack tip strain fields of the microscopic specimens with different constraints at the 8150th cycle.

The relationship between the fatigue crack propagation rate and the unified constraint parameter

Macroscopic specimens

A strain value of 0.00392 (PEEQ = 0.00392) was selected in the DIC image, and the area of the region surrounded by it was calculated to obtain the strain field area (the area was calculated through the software ‘Image - J’). In addition, the area of the region surrounded by PEEQ = 0.00392 in the specimen with a/W = 0.5 was taken as the reference value. Then, a ratio of the strain field area to the reference value at the same PEEQ (i.e. the unified constraint parameter A_p^41,42) can be calculated. A_p is expressed as^41,42:

A_{p} = A_{PEEQ} / A_{ref}

(1)

Where A_PEEQ is the area surrounded by the PEEQ isoline ahead of the crack tip—for example, the area surrounded by PEEQ = 0.00392 isoline in different specimens; and A_ref is the reference area surrounded by the PEEQ isolines in a reference specimen—for example, the area surrounded by PEEQ = 0.00392 isoline in the specimen with a/W = 0.5. The reference value of A_ref corresponding to the reference specimen is given in the following figures.

It should be noted that fatigue crack propagation involves both elastic and plastic components, each playing distinct yet indispensable roles at different levels. The elastic component provides the driving force for crack propagation, while the plastic component carries out the physical process of crack advancement. For this reason, when selecting DIC images, the images corresponding to the maximum stress were consistently chosen. This approach, with the aim of accounting for both elastic and plastic deformation, also best reflects the most critical stress state of the crack. In addition, at this maximum stress point, the load is sufficiently high to fully overcome crack closure effects. Analyzing data at the maximum stress point effectively avoids interference from crack closure phenomena and allows for the acquisition of strain fields that most accurately represent the true crack tip driving force.⁵³

The crack propagation length can be measured from the DIC image directly, and the fatigue crack propagation rate can be calculated. The relationship between the fatigue crack propagation rate and the unified constraint parameter in each specimen was shown in Figure 11.

Figure 11.

Linear relationship between the fatigue crack propagation rate and the unified constraint parameter for macroscopic specimens with different initial crack lengths a/W under different load level: 600 MPa (a) and 800 MPa (b).

Figure 11 shows that the fatigue crack propagation rate was approximately linearly related to $\sqrt{A_{p}}$ under different constraints and stress levels. Compared with low constraint (a/W = 0.2 and 0.3), the slope of the correlation line is higher under high constraint (a/W = 0.5 and 0.7); the slope of the correlation line is higher under high load (800 MPa) than under low load (600 MPa).

To further verify the influence of the strain value on the results, the strain values of 0.00814 and 0.0109 were selected for the specimen with a/W = 0.5 at 800 MPa to calculate A_p. The relationship between the fatigue crack propagation rate and the unified constraint parameter for different strain values is displayed in Figure 12. When different strain values were selected, the same linear relationship was detected between the fatigue crack propagation rate and the unified constraint parameter, indicating that this relationship was irrelevant to the strain value and could be flexibly selected according to the actual size of the crack tip strain field under different loading conditions and constraints.

Figure 12.

The relationship between the fatigue crack propagation rate and the $\sqrt{A_{p}}$ under different strain values in macroscopic specimens.

Microscopic specimens

For microscopic specimens, a strain value of 0.00387 (PEEQ = 0.00387) was selected, the A_p values and crack propagation rates of the specimens with different constraints (different specimen thicknesses B) were also calculated, and the relationship between the fatigue crack propagation rate and the unified constraint parameter in microscopic specimens were shown in Figure 13. Figure 13 shows that the linear relationship between the fatigue crack propagation rate and the unified constraint parameter was also exist for microscopic specimens. Compared with low constraint (B = 0.5 mm), the slope of the correlation line is higher under high constraint (B = 0.7 and 0.9 mm). When B = 0.7 and 0.9 mm, the slopes of the two specimens are similar, which indicates that the constraint states of the two specimens are basically the same.

Figure 13.

Linear relationship between the fatigue crack propagation rate and the unified constraint parameter for microscopic specimens with different specimen thicknesses B under 800 MPa.

The strain values of 0.00238, 0.00338 and 0.00387 were selected for the microscopic specimen with B = 0.5 mm to calculate A_p to verify the influence of the strain value on the results. The relationship between the fatigue crack propagation rate and the unified constraint parameter for different strain values is shown in Figure 14. The same with macroscopic specimen (Figure 12), this relationship was irrelevant to the strain value.

Figure 14.

The relationship between the fatigue crack propagation rate and the $\sqrt{A_{p}}$ under different strain values in microscopic specimens.

Overall, the linear relationship between the fatigue crack propagation rate and the unified constraint parameter exists in both macroscopic and microscopic specimens. The existence of the relationship indicates that the fatigue crack propagation rate was strongly correlated to the area of the region surrounded by the strain field ahead of the crack tip. Chikh et al.⁵⁴ tried to introduce the length of plastic zone as a mechanical parameter into the fatigue crack propagation criterion. The results of this study propound that the area of the region surrounded by the strain field ahead of the crack tip might be more suitable to be introduced as a mechanical parameter into the fatigue crack propagation criterion.

In addition, since only the initial crack is at the microscopic scale, the whole propagation remains fully continuum and geometry-controlled, without linking to local crystallography or heterogeneity, and does not address grain orientation, slip activity, phase morphology, or crack–microstructure interactions.^55,56 This diverges from the traditional understanding of crack propagation at the microscopic scale.

Different materials

To further verify the applicability of the linear relationship, the relationship between the fatigue crack propagation rate and the unified constraint parameter for different materials was established based on previously available data,^57–62 as shown in Figure 15. Figure 15 shows that the fatigue crack propagation rate was also approximately linearly related to the unified constraint parameter irrespective of the material. This indicates that the linear relationship is widely present and can be applied to different materials, which is helpful for its engineering applications. It should be noted that for different materials, this linear relationship exists; however, it has not yet been normalized across materials.

Figure 15.

Linear relationship between the fatigue crack propagation rate and the $\sqrt{A_{p}}$ for different materials.

A New crack propagation criterion

Based on the correlation lines in Figures 11 and 13, the relationship between da/dN and $\sqrt{A_{p}}$ can be derived. Utilizing this relationship, a new fatigue crack propagation criterion can be established, as defined in equation (2):

d a / d N = c_{1} \sqrt{A_{p}} - c_{2}

(2)

where c₁ and c₂ are parameters related material and constraint. With this criterion, during the finite element simulations, the $\sqrt{A_{p}}$ at the crack tip can be extracted in real time. According to the number of cycles N at that moment, the corresponding fatigue crack propagation length a can be obtained, thereby realizing the control of the crack propagation length. This criterion can be coded, and the flowchart of its implementation process is shown in Figure 16.

Figure 16.

Flowchart of the new crack propagation criterion.

Specifically, the process begins by establishing a finite element model containing an initial crack and inputting key material parameters and criterion constants. Subsequently, cyclic loading is applied by invoking the DLOAD subroutine, initiating incremental iterative calculations. Within the current incremental step, the stress-strain state is updated using a time-integration method, and the USDFLD subroutine is called to compute and monitor field variable data at integration points around the crack tip in real time. Once the local PEEQ reaches a predefined critical threshold, the program accurately calculates the area of the plastic influence zone A_PEEQ enclosed by the threshold contour through numerical integration of the integration points. This area data is then converted into a unified constraint parameter A_p and substituted into the newly established linear criterion to determine the current crack growth rate. Finally, crack initiation and propagation are dynamically simulated by triggering element failure and reassembling the global stiffness matrix. This process is iteratively carried out in subsequent cycles until the entire fatigue propagation life and the a - N curve are predicted.

Based on this criterion, the finite element simulations of the crack propagation process were conducted for the typical macroscopic specimen with a/W = 0.5 under 800 MPa of Figure 11(b). A comparison between the tested result and the simulated result is presented in Figure 17. It can be observed that the a - N curve and crack propagation length obtained based on the new criterion are in good agreement with the tested results, which fully demonstrates that this criterion can be applied to simulate the fatigue crack propagation process.

Figure 17.

A comparison between the tested result and the simulated result: a-N curve (a) and crack propagation length at the 3735th cycle (b).

As is well known, fatigue crack growth is governed by cyclic plastic dissipation, non-proportional strain accumulation, and microstructural barriers.^63,64 The lack of a physical crack-driving mechanism is a drawback of the unified constraint parameter A_p and also a shortcoming of this criterion. Nonetheless, the unified constraint parameter A_p has its advantage in that it does not require excessive consideration of details. Instead, it reflects crack propagation simply through the area surrounded by the PEEQ isoline, providing good simplification. This criterion shares the same advantage.

Furthermore, this criterion is applicable to describing the entire crack propagation process, but it still has limitations for crack propagation at the microscopic scale, which is affected by factors such as grain size, grain orientation, and slip activity.⁶⁵

In summary, this study establishes the relationship between the fatigue crack propagation rate and the constraint through experimental methods. While previous research had only demonstrated this correlation via finite element simulations—leaving some doubt regarding its reliability—the experimental approach adopted here dispels such uncertainty and provides the most direct evidence for the existence of this relationship.

Furthermore, as mentioned in the abstract, previously developed constraint parameters have primarily focused on monotonic loading. The effect of constraints on fatigue crack propagation still requires further investigation, and a correlation between constraint and fatigue crack growth has yet to be firmly established. This study represents an extension of constraint parameters into the realm of fatigue, contributing to the development of fatigue-constraint theory.

Moreover, based on the established relationship, a new crack propagation criterion has been proposed and compared with experimental results. This criterion is grounded in the plastic zone—an aspect seldom explored in prior studies. As previously noted, only a few researchers have attempted to incorporate the length of the plastic zone as a mechanical parameter into fatigue crack propagation criteria. The findings of this study suggest that the area enclosed by the strain field ahead of the crack tip is more suitable for integration as a mechanical parameter in such criteria. This insight also holds significance for the advancement of fracture mechanics. However, it should be noted that for the criterion established in equation (2), any change in parameters—such as crack length, load, or thickness—results in a new straight line, which limits its applicability. In subsequent work, geometric normalization, load normalization and material normalization must be performed to enhance the general applicability of the criterion.

Conclusions

In this study, the DIC technique was adopted to investigate the fatigue crack propagation process in both macroscopic and microscopic specimens made of GH4169 alloy under different constraints. The fatigue crack propagation rate and the area of the region surrounded by strain field were calculated, and the relationship between the fatigue crack propagation rate and the unified constraint parameter was established. The main observations of this study are presented below.

(1) The relationship between the fatigue crack propagation rate and the unified constraint parameter was established experimentally.

(2) The linearly relationship is irrelevant to the PEEQ value and can be flexibly selected according to the actual size of the crack tip strain field under different load conditions and constraints.

(3) The linear relationship is irrespective of the material and stress level, and also irrespective of the macro or micro conditions.

(4) Based on the relationship, a new crack propagation criterion was established. The a - N curves and crack propagation length obtained based on the new criterion are in good agreement with the tested results, which fully demonstrates that this criterion can be applied to simulate the fatigue crack propagation process. However, the criterion has not been standardized. For broader application, normalization in terms of geometry, loading, and materials is required.

Footnotes

Appendix

Handling Editor: Chenhui Liang

ORCID iD

Jie Yang

Author contributions

Tong-Yi Wang: Experimental measurement, Formal analysis, Data curation, Writing – Original Draft. Shi-Fa Gu: Experimental measurement, Investigation. Jin-Yong Xiao: Validation, Writing – review & editing. Jie Yang: Conceptualization, Experimental measurement, Methodology, Writing – review & editing, Funding acquisition.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project is sponsored by National Natural Science Foundation of China (Grant No. 52375154), State Key Laboratory of Clean and Efficient Turbomachinery Power Equipment (DEC8300CG202516944A1228183).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Huang

Wang

, et al. Relations between metastable austenite and fatigue behavior of steels. Acta Metallurgica Sin 2020; 56(04): 459.

Wei

Liu

Tang

, et al. Investigation of novel type of cylindrical lithium-ion battery heat exchangers based on topology optimization. Energy 2024; 304: 131886.

Wang

Gong

, et al. Machine learning-based fatigue life prediction of metal materials: perspectives of physics-informed and data-driven hybrid methods. Eng Fract Mech 2023; 284: 109242.

Paris

Erdogan

. A critical analysis of crack propagation laws. J Basic Eng 1963; 85: 528–533.

Forman

Kearney

Engle

. Numerical analysis of crack propagation in cyclic-loaded structures. J Basic Eng 1967; 89: 459–463.

Pearson

. Initiation of fatigue cracks in commercial aluminium alloys and the subsequent propagation of very short cracks. Eng Fract Mech 1975; 7(2): 235–247.

Okazaki

Yada

Endoh

. Surface small crack growth behavior in low-cycle fatigue at elevated temperature and application limit of macroscopic crack growth law. Nucl Eng Des 1989; 111(1): 123–134.

Deng

Wang

, et al. Small fatigue crack growth mechanisms of 304 stainless steel under different stress levels. Int J Fatigue 2014; 64: 14–21.

Zhang

, et al. Fatigue life prediction of nickel-based GH4169 alloy on the basis of a multi-scale crack propagation approach. Eng Fract Mech 2018; 199: 29–40.

10.

Dodds

Fong Shih

Anderson

. Continuum and micromechanics treatment of constraint in fracture. Int J Fract 1993; 64: 101–133.

11.

Varfolomeev

Luke

Moroz

. Experimental and numerical investigations of fatigue crack growth in various specimen geometries. Procedia Eng 2010; 2(1): 1829–1837.

12.

Tong

. T-stress and its implications for crack growth. Eng Fract Mech 2002; 69(12): 1325–1337.

13.

Hutař

Seitl

Knésl

. Effect of constraint on fatigue crack propagation near threshold in medium carbon steel. Comput Mater Sci 2006; 37(1-2): 51–57.

14.

Hutař

Seitl

Kruml

. Effect of specimen geometry on fatigue crack propagation in threshold region. In: Proceedings of the 12th international conference on fracture, Ottawa, 2009.

15.

Seitl

Hutař

. Fatigue-crack propagation near a threshold region in the framework of two-parameter fracture mechanics. Mater Technol 2007; 41(3): 135.

16.

Wang

Zhu

, et al. A data-driven roadmap for creep-fatigue reliability assessment and its implementation in low-pressure turbine disk at elevated temperatures. Reliab Eng Syst Saf 2022; 225: 108523.

17.

ASTM. Standard test method for measurement of fatigue crack growth rates, E647-13A. ASTM International, 2014.

18.

Luke

Varfolomeev

Lütkepohl

, et al. Fatigue crack growth in railway axles: assessment concept and validation tests. Eng Fract Mech 2011; 78(5): 714–730.

19.

Zhang

, et al. Determination of the fatigue P-S-N curves – a critical review and improved backward statistical inference method. Int J Fatigue 2020; 139: 105789.

20.

Varfolomeev

Luke

. Consideration of fatigue crack growth aspects in the design and assessment of railway axles. In: Hütter

Zybell

(eds), Recent trends in fracture and damage mechanics. Springer, 2016.

21.

Irwin

. Fracture dynamics. Fract Met 1948; 147–166.

22.

Wells

. Application of fracture mechanics at and beyond general yield. Br Weld J 1963; 10: 563.

23.

Rice

. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 1968; 35(2): 379–386.

24.

Hutchinson

. Singular behaviour at the end of a tensile crack in a hardening material. J Mech Phys Solids 1968; 16(1): 13–31.

25.

Williams

. On the stress distribution at the base of a stationary crack. ASME J Appl Mech 1957; 24(1): 109–114.

26.

Betego

Hancock

. Two-parameter characterization of elastic-plastic crack-tip fields. J Appl Mech 1991; 58(1): 104–110.

27.

O’Dowd

Shih

. Family of crack-tip fields characterized by a triaxiality parameter—I. Structure of fields J Mech Phys Solids 1991; 39(8): 989–1015.

28.

Yang

. Higher order asymptotic crack-tip fields in a power-law hardening material. PhD Thesis, University of South Carolina, Columbia, South Carolina, 1993: 1.

29.

Yang

Chao

Sutton

. Higher order asymptotic crack tip fields in a power-law hardening material. Eng Fract Mech 1993; 45(1): 1–20.

30.

Chao

Yang

Sutton

. On the fracture of solids characterized by one or two parameters: theory and practice. J Mech Phys Solids 1994; 42(4): 629–647.

31.

Guo

. Elastoplastic three-dimensional crack border field—I. Singular structure of the field Eng Fract Mech 1993; 46(1): 93–104.

32.

Guo

. Elastoplastic three-dimensional crack border field—II. Asymptotic solution for the field. Eng Fract Mech 1993; 46(1): 105–113.

33.

Guo

. Elasto-plastic three-dimensional crack border field—III. Fracture parameters. Eng Fract Mech 1995; 51: 51.

34.

Guo

. Three-dimensional analyses of plastic constraint for through-thickness cracked bodies Eng Fract Mech 1999; 62: 383–407.

35.

Guo

. Recent advances in three-dimensional fracture mechanics. Key Eng Mater 2000; 183-187: 193–198.

36.

Guo

Pitt

Jones

. Three-dimensional strength assessment for damage tolerant structures. In: Proceedings of the international conference on strength theory, 1998.

37.

Zhao

Guo

She

. The in-plane and out-of-plane stress constraint factors and K−T−Tz description of stress field near the border of a semi-elliptical surface crack Int J Fatigue 2007; 29(3): 435–443.

38.

Burstow

Howard

Ainsworth

. The influence of constraint on crack tip stress fields in strength mismatched welded joints. J Mech Phys Solids 1998; 46: 845–872.

39.

Betegón

Peñuelas

. A constraint-based parameter for quantifying the crack tip stress fields in welded joints. Eng Fract Mech 2006; 73(13): 1865–1877.

40.

Mostafavi

Smith

Pavier

. Reduction of measured toughness due to out-of-plane constraint in ductile fracture of aluminium alloy specimens Fatigue Fract Eng Mater Struct 2010; 33(11): 724–739.

41.

Yang

Wang

Xuan

, et al. Unified characterisation of in-plane and out-of-plane constraint based on crack-tip equivalent plastic strain. Fatigue Fract Eng Mater Struct 2013; 36(6): 504.

42.

Yang

Wang

Xuan

, et al. Unified correlation of in-plane and out-of-plane constraints with fracture toughness Fatigue Fract Eng Mater Struct 2014; 37(2): 132–145.

43.

Wang

Xuan

, et al. Unified constraint parameter based on crack-tip opening displacement. Eng Fract Mech 2018; 200: 175–188.

44.

Xiao

Wang

, et al. Engineering estimation method of unified constraint parameters for semi-elliptical surface cracks in plates. Eng Fract Mech 2020; 229: 106935.

45.

Xiao

Wang

Zhao

, et al. A general correlation between ductile fracture toughness and unified constraint parameter for different materials. Eng Fract Mech 2023; 285: 109296.

46.

Xiao

Yang

. A correlation between unified constraint parameter and J-R curves of different metallic materials. Eng Fract Mech 2025; 319: 110999.

47.

Xiao

Wang

, et al. Two-parameter fracture prediction for cracked plates under bending. Eng Fract Mech 2021; 255: 107974.

48.

Wang

, et al. Unified constraint-based FAD assessments for ductile fracture in cracked pipes. Int J Press Vessel Piping 2020; 185: 104132.

49.

Zheng

Wang

, et al. Integrity assessment considering unified constraint for reactor pressure vessel under pressurized thermal shock. Eng Fract Mech 2023; 290: 109508.

50.

Wang

, et al. Validation and application of a two-parameter J-A_d approach for fracture behaviour prediction Fatigue Fract Eng Mater Struct 2020; 43: 2998–3011.

51.

Tan

Yang

, et al. A new creep-fatigue crack growth model and a correlation of the creep-fatigue crack growth rate with unified constraint parameter. Int J Fatigue 2023; 166: 107248.

52.

Yang

Guo

, et al. Numerical modelling of a new FCP model and a correlation of the FCP rate with the constraint. Int J Fatigue 2022; 163: 107036.

53.

McClung

. The influence of applied stress, crack length, and stress intensity factor on crack closure. Metallurgical Trans A 1991; 22: 1559–1571.

54.

Ould Chikh

Imad

Benguediab

. Influence of the cyclic plastic zone size on the propagation of the fatigue crack in case of 12NC6 steel. Comput Mater Sci 2008; 43: 1010–1017.

55.

Yeratapally

Glavicic

Hardy

, et al. Microstructure based fatigue life prediction framework for polycrystalline nickel-base superalloys with emphasis on the role played by twin boundaries in crack initiation. Acta Mater 2016; 107: 152–167.

56.

Guo

Paramatmuni

, et al. Slip system-resolved GNDs and SEDs: a multi-scale framework for predicting crack nucleation in single-crystal metals. Acta Mater 2025; 288: 120853.

57.

Zheng

Wang

, et al. Crack propagation and debonding development of CFRP laminate strengthened high-strength steel plates under fatigue loadings. IOP Conf Ser Mater Sci Eng 2020; 768: 032047.

58.

Lupton

Zhu

, et al. In situ experimental study of near-tip strain evolution of fatigue cracks. Exp Mech 2015; 55(6): 1175–1185.

59.

Wang

Yang

, et al. Stress and strain analysis of crack tip of 2A12-T4 aluminum alloy friction stir welded joint. Hot Work Technol 2023; 9: 118.

60.

Zhang

. Study on low-cycle fatigue and fatigue crack growth of 304 austenitic stainless steel. PhD Thesis, Jiangsu University of Technology, Changzhou, 2021.

61.

Cai

, et al. In-situ experimental study on the fatigue crack propagation behavior of 7075-T6 and 2024-T3 aluminium alloys under variable amplitude loading and retardation model modification Theor Appl Fract Mech 2022; 119: 103372.

62.

. A study on multi-scale fatigue crack propagation and life prediction methodology on the basis of crack tip plasticity. PhD Thesis, East China University of Science and Technology, Shanghai, 2018.

63.

Wan

Dunne

FPE

. Microstructural fracture mechanics: stored energy density at fatigue cracks. J Mech Phys Solids 2021; 146: 104209.

64.

Guo

Peng

, et al. Beyond first-cycle damage: mechanistic drivers of fatigue crack nucleation in single crystals. J Mech Phys Solids 2026; 206: 106393.

65.

Karamitros

MacLachlan

Dunne

FPE

. Mechanistic fatigue in Ni-based superalloy single crystals: a study of crack paths and growth rates. J Mech Phys Solids 2022; 158: 104663.

Experimental verification of the linear relationship between fatigue crack propagation rate and constraint and a new crack propagation criterion

Abstract

Keywords

Highlights

Introduction

Experimental method

Material

Macroscopic and microscopic specimens

Fatigue crack propagation experiment

Experimental results

Macroscopic specimens

a - N curve of the macroscopic specimens

Strain field at the crack tip

Microscopic specimens

a - N curve of the microscopic specimens

Strain field at the crack tip

The relationship between the fatigue crack propagation rate and the unified constraint parameter

Macroscopic specimens

Microscopic specimens

Different materials

A New crack propagation criterion

Conclusions

Footnotes

Appendix

ORCID iD

Author contributions

Funding

Declaration of conflicting interests

References