Abstract
The specimen-size effect in very-high-cycle fatigue (VHCF) of additively manufactured (AM) metals arises from the statistical nature of defect-controlled crack initiation. In this work, the ultrasonic VHCF behavior of powder bed fusion–laser beam–fabricated (PBF-LBed) AlSi10Mg was examined using axially loaded specimens with different gauge sizes, representing distinct control volumes. A pronounced negative size effect on VHCF strength was observed, despite identical nominal stress amplitudes. Fatigue cracks consistently initiate from internal and subsurface lack-of-fusion defects surrounded by fine granular areas (FGAs), indicating that the size effect results from statistical scaling of critical defect populations rather than a change in initiation mechanism. Weibull-based probabilistic S–N curves accurately describe fatigue life scatter, and a control-volume-based weakest-link model provides physically consistent and conservative descriptions of size-dependent VHCF behavior. These results demonstrate that the VHCF size effect in PBF-LBed AlSi10Mg is a statistical-mechanical phenomenon governed by defect populations within the stressed volume, offering a mechanics-informed basis for fatigue life assessment and size extrapolation of AM components.
Highlights
A clear specimen-size effect is observed in VHCF of PBF-LBed AlSi10Mg.
The size effect is governed by statistical activation of defect-controlled crack initiation within the control volume.
Weibull-based probabilistic S–N curves capture fatigue life scatter across different specimen sizes.
A control-volume-based weakest-link model enables prediction of size-dependent VHCF behavior.
1 Introduction
The development of additive manufacturing (AM) technologies, particularly powder bed fusion–laser beam (PBF-LB) (BS EN ISO/ASTM 52900:2021, 2022), has revolutionized the fabrication of metallic components with complex geometries, enabling applications across aerospace, automotive, and biomedical industries (King et al., 2015; DebRoy et al., 2018; Zenou et al., 2018; Gu et al., 2021; Leary, 2020; Qian et al., 2020a; Du et al., 2021; Gibson et al., 2021; Long et al., 2022; Su et al., 2024; Liu Z. Z. et al., 2024; Chua et al., 2024; Du et al., 2024; Xu et al., 2025). Aluminum alloys such as AlSi10Mg processed via PBF-LB exhibit distinctive microstructural features and mechanical properties (Romano et al., 2018; Marşavina et al., 2021; Sanaei and Fatemi, 2021; Araújo et al., 2022; Behvar et al., 2023; Matušů et al., 2024; Javidrad et al., 2024; Yi et al., 2024). However, their very-high-cycle fatigue (VHCF) behavior, typically referring to failure beyond 107 cycles (Naito et al., 1983; Atrens et al., 1983; Bathias and Paris, 2005; Sakai, 2009; Pan et al., 2018; Pan and Hong, 2019; Chang et al., 2019; Sun et al., 2019; Su et al., 2017; Pan et al., 2020a; Chang et al., 2020; Pan et al., 2021; Gao et al., 2025a; Cong et al., 2021; Zhou et al., 2022; Gao et al., 2025b; Pan et al., 2024a; Pan et al., 2024b), remains insufficiently understood (Qian et al., 2020b; Tridello et al., 2022; Tusher and Ince, 2023; Pan et al., 2024c; Pan and Hong, 2024; Gao et al., 2024). In VHCF, fatigue failure often originates from internal defects (e.g., lack-of-fusion (LoF) pores and inclusions) (Su et al., 2025a; Su et al., 2025b), and therefore the role of specimen geometry and size becomes particularly important because it alters the probability of defect activation within the highly stressed region under cyclic loading (Sanaei and Fatemi, 2021; Murakami; Peng et al., 2022; Hu et al., 2023; Afazov et al., 2023; Karakas et al., 2023).
Ultrasonic fatigue testing (Bathias and Paris, 2005; Pan et al., 2020b; Avateffazeli and Haghshenas, 2022; Liu L. et al., 2024), operating at 20 kHz, enables efficient interrogation of VHCF behavior by accelerating cycle accumulation and facilitating mechanistic studies in life regimes that are difficult to access using conventional-frequency tests. For AM materials where internal defects are inherent, the stressed volume (often described as a control volume (Murakami, 2002)) becomes a key statistical factor: larger highly stressed volumes are more likely to contain a critical defect, thereby reducing fatigue strength. Consistently, in conventional metals, fatigue strength decreases with increasing specimen size due to the classical “size effect” (Furuya, 2008; Furuya, 2011; Montagnoli et al., 2023; Tao et al., 2024; Pan et al., 2025). For AM alloys, this effect is further complicated by defect variability and microstructural heterogeneity, motivating a systematic investigation that goes beyond deterministic S–N descriptions (Sanaei and Fatemi, 2021; Murakami, 2002; Peng et al., 2022; Hu et al., 2023; Afazov et al., 2023; Karakas et al., 2023).
Current VHCF studies on AM materials often emphasize defect characterization and deterministic stress–life (S–N) relations (Suresh, 1998; Schijve, 2009; Wöhler, 1858; Wöhlerb, 1860; Wӧhler, 1867), while the stochastic nature of defect-controlled failure remains under-addressed in many practical assessments. More importantly, fractographic evidence in VHCF commonly shows that crack initiation at internal defects is accompanied by a surrounding fine granular area (FGA), suggesting that a defect should not always be treated as an “equivalent initial crack” from the outset. Instead, the FGA can be viewed as an incubation zone that mediates the transition from a defect-controlled damage site to a mechanically propagating short crack under near-threshold cyclic loading (Naito et al., 1983; Atrens et al., 1983; Bathias and Paris, 2005; Sakai, 2009; Pan et al., 2018; Pan and Hong, 2019; Chang et al., 2019; Sun et al., 2019; Su et al., 2017; Pan et al., 2020a; Chang et al., 2020; Pan et al., 2021; Gao et al., 2025a; Cong et al., 2021; Zhou et al., 2022; Gao et al., 2025b; Pan et al., 2024a; Pan et al., 2024b). Specimen size affects VHCF performance not only by changing the number of defects sampled, but also by increasing the likelihood of activating a defect–incubation pair capable of reaching the condition for stable crack propagation within the control volume (Murakami, 2002; Furuya, 2008; Furuya, 2011; Montagnoli et al., 2023; Tao et al., 2024; Pan et al., 2025). Therefore, a mechanistically consistent description should couple defect statistics, control-volume effects, and probabilistic S–N representation to interpret the size dependence of VHCF behavior.
From a mechanics perspective, the specimen-size effect in VHCF of AM metals remains insufficiently understood, particularly in defect-dominated regimes where crack initiation is governed by near-threshold cyclic damage accumulation. While previous studies have documented size-dependent fatigue strength and internal defect-controlled failure in AM alloys (Du et al., 2024; Sanaei and Fatemi, 2021; Behvar et al., 2023; Javidrad et al., 2024; Yi et al., 2024; Tridello et al., 2022; Pan et al., 2024c; Pan and Hong, 2024), a unified interpretation that links specimen geometry, control volume, defect statistics, and probabilistic fatigue response is still lacking. In VHCF, the stressed volume becomes a mechanically relevant domain in which the interaction between cyclic stress fields and defect populations determines the probability of activating a critical crack incubation site. Consequently, fatigue behavior cannot be described solely by nominal stress–life relations but must be interpreted within a statistical-mechanical framework incorporating weakest-link concepts. The present study addresses this issue by systematically investigating the ultrasonic VHCF behavior of PBF-LBed AlSi10Mg specimens with different control volumes, combining experimental characterization with probabilistic S–N analysis and control-volume-based weakest-link modeling, with the aim of elucidating the mechanical origin of size effects and establishing a framework for fatigue life extrapolation across specimen sizes.
2 Materials and methods
2.1 PBF-LBed AlSi10Mg fabrication
Table 1 lists the nominal chemical composition of the pre-alloyed aluminum powder in weight percent. As indicated, the powder corresponds to an AlSi10Mg alloy, and the AM material produced by the PBF-LB process retains the same AlSi10Mg composition.
TABLE 1
| Al | Si | Mg | Fe | Ti | Others |
|---|---|---|---|---|---|
| Remainder | 9.75 | 0.22 | 0.092 | 0.011 | <0.01 |
Chemical composition of the pre-alloyed AlSi10Mg powder (wt%).
The AlSi10Mg aluminum alloy examined in this work was fabricated using an SLM 280 system. A stripe-like scanning strategy was adopted, with a layer-to-layer rotation angle () of 66.7° to avoid spatial periodicity breaking spatial periodicity, as illustrated in Figure 1a. Five key process parameters governing the additive manufacturing process were set up accordingly, namely, laser power (370 W), layer thickness (0.03 mm), scanning speed (1,300 mm/s), hatch spacing (0.19 mm), and the volumetric laser energy density (50 J/mm3), defined as .
FIGURE 1
2.2 Specimen design and ultrasonic fatigue testing
All the fatigue specimens were extracted from the as-built blocks along the building direction (BD) without heat treatment, as shown in Figure 1b. Prior to fatigue testing, the gauge sections of all specimens were mechanically ground and polished using the same procedure. The specimens were first ground with 800-grit SiC paper and subsequently polished using 1,200-, 1,500-, and 2000-grit papers to remove deep scratches. Final polishing was performed using polishing cloth and abrasive paste to obtain a bright surface. Although the surface roughness was not quantitatively measured, the polishing procedure produced mirror-like gauge surfaces, ensuring that fatigue crack initiation was predominantly governed by internal defects rather than surface imperfections. The relative density of the specimens was not quantitatively measured in the present study. However, all specimens were fabricated under identical processing conditions from the same batch, ensuring consistency in defect population across different specimens. This implies that the observed VHCF behavior is primarily governed by statistical variations in defect population within the control volume rather than differences in bulk material density.
For specimens of different sizes, the concept of a control volume (or critical volume) is commonly used to describe the region most susceptible to fatigue crack initiation. Rather than considering the entire loaded domain, the control volume is typically defined using a stress-based threshold, denoted as , representing the material volume where the local stress exceeds a fraction of the maximum stress. In this study, is adopted, corresponding to regions with (Tridello et al., 2021), which is a commonly adopted approximation and does not affect the qualitative interpretation of the size effect. The stress distribution was determined based on the analytical solution for the stress field in the minimum cross-section of the specimen under ultrasonic axial loading. Accordingly, the region corresponds to the highly stressed volume near the specimen center, where fatigue crack initiation is most likely to occur. The choice of such a high-stress threshold is physically motivated by the strong nonlinearity of fatigue damage with respect to stress amplitude. According to Basquin-type relationships, fatigue life is highly sensitive to stress variations, particularly in the VHCF regime. As a result, regions where stress amplitude is significantly lower than the maximum contribute negligibly to fatigue damage accumulation. Therefore, the effective damage zone is dominated by the high-stress tail of the stress distribution, and the definition of provides a physically meaningful approximation of the fatigue-relevant volume, which allows the control volume to be determined in a reproducible manner for specimens with different geometries.
Based on the control volume method, three groups (Group-1, Group-2, and Group-3) of ultrasonic fatigue specimens were designed to investigate size effects, with minimum cross-sectional diameters of 3 mm, 6 mm, and 12 mm, respectively. The specimen geometries are shown in Figures 2a–c. All specimens satisfy the 20 kHz resonance condition required for ultrasonic loading, in accordance with the simplified one-dimensional wave equation for variable cross-section specimens. The corresponding control volumes were calculated to be 31 mm3, 225 mm3, and 1,512 mm3, respectively. Figure 2d illustrates the variation of the control volume as a function of the minimum cross-sectional size for the three specimen groups. As the minimum cross-sectional diameter increases, both the minimum cross-sectional area and the associated control volume increase monotonically. Notably, the increase in control volume is significantly more pronounced than that of the cross-sectional area, indicating that both the effective area and volume susceptible to fatigue crack initiation expand with increasing specimen size.
FIGURE 2
In a laboratory air environment at room temperature, ultrasonic fatigue tests were performed, using USF-2000 testing machine at 20 kHz (Figure 3a), on all the three specimen groups (twelve specimens each group) under fully reversed axial tension–compression loading (). Figure 3b illustrates the whole ultrasonic fatigue testing scheme. The loading direction is along the specimen axis and is therefore parallel to the BD. All loaded specimens were tested either to failure, defined by final fracture or by a sufficiently large and/or rapid drop in resonance frequency, indicating the formation of a fatigue crack or substantial internal damage, or to run out. For specimens that did not fail up to cycles, the test was terminated and fatigue failure was considered not to have occurred within the investigated life regime. During ultrasonic loading, forced cooling with cold air was required to prevent excessive temperature rise caused by cyclic thermo-mechanical dissipation, which could otherwise reduce the apparent fatigue resistance. The cold air was generated using a vortex tube supplied with compressed air from an air compressor, as shown in Figure 3A. The surface temperature was monitored using an infrared thermographic camera focused on the minimum cross-section of the specimen. Throughout the tests, the temperature rise was maintained below approximately 10 °C above ambient temperature (typically below 35 °C), indicating that thermal effects are not the dominant factor governing fatigue behavior. Furthermore, to evaluate potential size effect on self-heating, specimens with different minimum diameters were compared under identical loading conditions. No significant difference in stabilized temperature was observed among the three specimen sizes, indicating comparable thermal conditions during testing.
FIGURE 3
It should be noted that some specimens did not fully separate after the ultrasonic test was stopped and required additional tensile loading to achieve complete fracture. In such cases, the recorded cycle count slightly underestimates the true fatigue life, as the remaining unbroken ligament would contribute to an additional crack-growth life. However, in the VHCF regime this contribution is typically small and can be neglected.
2.3 Internal defect characterization
To characterize the distribution and statistical features of the as-built printing defects in the three specimen groups, high-energy synchrotron X-ray computed tomography (XCT) was performed on the PBF-LBed AlSi10Mg alloy (Chen et al., 2012). These XCT samples were extracted from the same build and fabricated under identical processing conditions as the fatigue specimens, ensuring direct comparability of the defect population and its statistical characteristics. Cylindrical specimens with dimensions of 5 mm in diameter and 5 mm in height were extracted for XCT analysis. The scans were performed using an X-ray energy of 26 keV, with an exposure time of 0.5 s per projection. A full rotation scan was conducted with an angular increment of 0.25°, resulting in approximately 1,440 projections. The reconstructed spatial resolution was approximately 3.25 μm. The projection data were reconstructed into cross-sectional slices using PITRE software, followed by three-dimensional reconstruction and defect analysis using AVIZO. Internal defects were identified based on grayscale contrast between pores and the surrounding matrix using a global thresholding approach. The equivalent defect diameter , given as , was calculated based on the defect volume , and additional geometric descriptors such as sphericity were also evaluated. The analyzed volumes are considered representative of the material condition, and the defect statistics were obtained from a large population of internal defects.
Figure 4 illustrates the three-dimensional characterization and statistical quantification of internal defects. As shown by the three-dimensional CT-based defect reconstruction of the specimen in Figures 4a,b, the defects are relatively uniformly distributed throughout the material. Figure 4c presents the statistical distribution of the characteristic defect size, defined as the equivalent diameter of the defect projection area perpendicular to the loading axis. The results indicate that defects with equivalent diameters smaller than 30 μm account for 87.87% of the total population, while more than 97.54% of the defects have equivalent diameters below 50 μm. These findings confirm that most defects in the material are small-sized. As the defect size increases, the probability of defect occurrence decreases progressively. The quantitative defect statistics presented here provide essential input for the subsequent analysis of size effects.
FIGURE 4
3 Results
3.1 S-N data
Figure 5 shows the relationship between fatigue resistance, expressed in terms of stress amplitude, and fatigue life for the PBF-LBed AlSi10Mg specimens with three different control volumes. Specimens marked with arrows represent runouts that survived loading cycles without fatigue failure. The fatigue strength at cycles was directly determined from experimental results and taken as the highest stress level at which no failure occurred. No extrapolation based on S–N fitting was used for this determination. For Group-1, the maximum number of cycles to failure reached , corresponding to a fatigue strength of approximately 112 MPa at cycles, while the fatigue limit at cycles is close to 100 MPa. For Group-2, the longest fatigue life to failure was cycles, with a fatigue strength of about 97 MPa at cycles and an estimated fatigue limit of approximately 89 MPa at cycles. For Group-3, fatigue failure occurred up to cycles, corresponding to a fatigue strength of roughly 92 MPa at cycles and a fatigue limit of about 81 MPa at cycles.
FIGURE 5
Based on these results, it can be inferred that both HCF and VHCF behavior exhibit a pronounced specimen-size dependence with increasing specimen size, or equivalently, increasing control volume, indicating a clear size effect. This size effect manifests as a negative correlation between fatigue performance and specimen size, whereby fatigue resistance deteriorates as the specimen size increases. In the present study, the HCF/VHCF strengths of PBF-LBed AlSi10Mg consistently show a negative size effect, with larger control volumes leading to lower fatigue strength. Specifically, the fatigue strength of the second specimen group is reduced by approximately 10% compared with that of the first group, while the fatigue strength of the third group decreases by nearly 20% relative to the first group.
Further examination of the S–N curves reveals that the negative size effect is more pronounced in the VHCF regime (– cycles) than in the HCF regime (– cycles). Because VHCF behavior is more strongly governed by internal as-built defects than HCF behavior, and the influence of such defects becomes increasingly dominant with increasing number of loading cycles.
3.2 Fractography
After completion of the fatigue tests, all failed specimens were subjected to comprehensive and systematic SEM analysis of their fracture morphology. Crack initiation is predominantly located within the highly stressed central region, including both internal and subsurface defects, which further validates the rationality of the control volume definition adopted in this study. SEM observations reveal that lack-of-fusion (LoF) defects remained the dominant crack initiation sites under ultrasonic loading conditions. Figure 6 presents representative defect-induced crack initiation morphologies observed in VHCF regimes. All fracture surfaces in the present study exhibit a single crack initiation site, with no evidence of multiple crack origins.
FIGURE 6
Crack initiation sites are classified as internal or subsurface based on their location relative to the specimen surface. For Group-1, as shown in Figures 6a,b, the crack initiates near the surface, adjacent to a small AM defect within an oval FGA. Among the eleven specimens, five cases exhibit internal initiation, while six cases show subsurface initiation. For all the subsurface cases, the measured defect depths from the surface are all above 12 μm, which validates the control-volume framework adopted in this study. The shallow control volume of this small-diameter specimen reduces the probability of critical internal flaws, leading to a transition from subsurface to internal crack initiation. In contrast, Group-2, as shown in Figures 6c,d, exhibits clear evidence of internal crack initiation, with a well-developed FGA encompassing the crack initiation site. All the eleven cases showed internal initiation. The FGA morphology is indicative of crack nucleation under low cyclic plastic strain, likely induced by a internal defect acting as a stress concentrator under high-frequency cyclic loading. Group-3, as shown in Figures 6e,f, further reinforces this trend: the crack initiates from an internal defect within the specimen, for all the ten cases. The associated FGA spans a wider area, reflecting a larger plastic zone resulting from both the larger defect size and increased control volume.
Although the initiation location differs among the three groups, with partial cases in Group 1 showing subsurface initiation and Groups 2–3 exhibiting internal initiation, the underlying crack-initiation mechanism can still be categorized within the same mechanism class. In all cases, fatigue cracks nucleate from LoF defects and are associated with the formation of FGAs. It is acknowledged that subsurface initiation may be influenced by reduced stress triaxiality and possible environmental effects, compared with fully internal initiation. However, the dominant crack-initiation mechanism remains defect-controlled and FGA-mediated.
4 Discussion
4.1 Critical defect statistics
Although the location and morphology of crack-initiating defects also exert some influence, defect size, characterized by the equivalent defect diameter, remains the dominant controlling factor. For simplicity, characterization focuses exclusively on the quantitative analysis of critical defect size. The equivalent diameters of all critical defects responsible for crack initiation were measured using ImageJ (Schneider et al., 2012; Thermo Fisher Scientific, 2023). Figure 7 shows the defect equivalent diameter distribution along with fatigue life. No clear one-to-one correlation between defect equivalent diameter and fatigue life is observed within each specimen group, and a significant scatter is present. This indicates that the size effect is primarily governed by the statistical scaling of defect populations rather than a direct correlation between defect size and fatigue life for individual specimens. This observation is consistent with the weakest-link mechanism, in which failure is controlled by the extreme value of defect populations within the control volume. Some statistics based on the data are summarized in Table 2, which gives a clear comparison between the three specimen groups.
FIGURE 7
TABLE 2
| Group | Counts | Min | Max | Mean | STD |
|---|---|---|---|---|---|
| Group-1 | 11 | 41 | 115 | 87.0 | 19.6 |
| Group-2 | 11 | 75 | 157 | 109.4 | 22.5 |
| Group-3 | 10 | 82 | 203 | 141.1 | 37.6 |
Statistics of the defect equivalent diameters (unit: μm).
These results indicate that the increase in the equivalent diameter of crack-initiating critical defects is the primary origin of the size effect observed in VHCF performance of PBF-LBed AlSi10Mg. As the specimen size increases (i.e., as the control volume becomes larger), the total number of defects contained within the control volume increases, thereby raising the probability of encountering large defects. Consequently, fracture surfaces of larger specimens tend to exhibit larger critical defects compared with those of smaller specimens. In other words, when a sufficiently large number of small specimens are tested, some of them may exhibit internal crack initiation from defects comparable in size to those observed in large specimens. Although the present analysis is based on fracture-origin defects rather than full-field XCT data, these defects correspond to the critical population that directly governs fatigue failure in the VHCF regime. In addition, the standard deviation (STD) of the defect equivalent diameter increases with specimen size, indicating a broader distribution of critical defects. This can be attributed to the increase in control volume, which leads to sampling of a larger defect population and a higher probability of encountering extreme defect sizes. As a result, not only the average defect size increases, but also the statistical scatter becomes more pronounced, reflecting the stochastic nature of defect-controlled crack initiation in the VHCF regime.
4.2 Weibull-based probabilistic S–N curves
A probabilistic approach based on the Weibull distribution was then employed to describe the scatter in fatigue life of PBF-LBed AlSi10Mg. Subsequently, a probabilistic control-volume model was used to extrapolate the probabilistic S–N (P–S–N) curves of larger specimens from those obtained for smaller specimens. Runouts were treated as right-censored observations in statistical analysis. In the Basquin fitting, only failure data were used to estimate the stress–life relationship, while runouts were excluded from the regression. In the Weibull P–S–N fitting, both failures and runouts were incorporated through a maximum likelihood estimation (MLE) framework with right-censoring.
For the commonly adopted survival probability of 50%, the stress–life relationship can be expressed by the following Basquin equation:where denotes the applied stress amplitude, and represents fatigue life. The material parameters can be determined by fitting the experimental S–N data, as shown in Figure 8, using the least-squares method. Through this fitting procedure, the corresponding Weibull parameters and for the three specimen groups can be obtained, as shown in Table 3.
FIGURE 8
TABLE 3
| Group | m | C |
|---|---|---|
| Group-1 | 26.35 | 6.91 × 1058 |
| Group-2 | 24.89 | 2.88 × 1053 |
| Group-3 | 19.22 | 6.56 × 1043 |
Basquin parameters obtained from S–N fitting.
The probabilistic control volume model proposed by Sun et al. (Sun et al., 2016) is adopted for fatigue life prediction. This approach assumes that a large specimen can be regarded as being composed of small reference specimens , each representing a sufficiently small control volume. According to the weakest-link theory (WLT), the fatigue performance of specimen is governed by the weakest element among the constituent specimens. It is further assumed that the fatigue life of the reference specimen follows a Weibull distribution:where is the scale parameter, is the shape parameter, and is the location parameter. The variation of with specimen size reflects the transition from defect-sensitive behavior to a more statistically averaged weakest-link response as the control volume increases.
Therefore, the fatigue life of specimen (defined as the minimum fatigue life among the reference specimens ):
At the same survival probability:
Similarly, for specimen with a control volume of , the fatigue life of specimen and that of reference specimen satisfy the following relationship:
Considering that the control volume of the reference specimen is relatively small, it follows that:
If fatigue life follows a two-parameter Weibull distribution, the term can be cut off and the relationship between fatigue lives of specimens and with different control volumes can be obtained as:
It should be noted that fitting Weibull-based P–S–N curves ideally requires multiple fatigue life data points obtained under identical stress levels. However, due to experimental cost constraints, the number of available data points at a single stress level is often limited. To address this issue, the Basquin relationship was first employed to convert the fatigue lives obtained at different stress amplitudes into equivalent fatigue lives at a common reference stress level . The transformation of fatigue life data from other stress levels to the reference stress level can be expressed as follows:
After transforming the fatigue life data, the fatigue results obtained for the three specimen groups were fitted using a Weibull distribution to construct the corresponding P–S–N curves. The fitted parameters are summarized in Table 4.
TABLE 4
| Group | k | λ |
|---|---|---|
| Group-1 | 0.780 | 5.26 × 109 |
| Group-2 | 0.817 | 3.07 × 109 |
| Group-3 | 1.001 | 1.98 × 108 |
Weibull parameters obtained from P–S–N fitting.
Figure 8 presents the Weibull-based P–S–N curves for the three specimen groups. As shown in the figure, the S–N data for all three groups fall entirely within the 1%–99% survival probability bounds. This result demonstrates that the Weibull-based probabilistic P–S–N curves are capable of accurately capturing the scatter in fatigue performance of PBF-LBed AlSi10Mg.
4.3 Probabilistic control volume modeling of size effects
Using the shape parameter obtained from the Weibull-based P–S–N fitting described above, predictions were carried out by substituting into Equation 1. Figure 9a shows the prediction of the fatigue behavior of the second specimen group based on the P–S–N curves of the first specimen group. It can be observed that only one data point lies above the 1% survival probability curve, indicating that the overall prediction accuracy is satisfactory. Figure 9b presents the prediction of the third specimen group using the first specimen group as the reference. In this case, one data point is also located above the 1% survival probability curve, suggesting that the prediction is slightly conservative. Figure 9c shows the prediction of the third specimen group based on the second specimen group, from which the predicted results are likewise in good agreement with the experimental data.
FIGURE 9
These results demonstrate that the probabilistic control volume model provides a physically consistent interpretation of the size-effect behavior of PBF-LBed AlSi10Mg. The model successfully captures the observed trends in fatigue performance across different specimen sizes, reflecting the influence of defect statistics within the control volume. It should be noted that the present formulation is primarily descriptive, as the Weibull parameters are calibrated for each specimen group. Therefore, the model is intended to explain the observed size effect rather than to provide fully predictive capability across different specimen sizes. This supports the view that the size effect in VHCF is governed by the statistical nature of defect populations rather than deterministic scaling laws.
4.4 Implications for VHCF design of AM components
The present results highlight that VHCF performance of AM components cannot be reliably assessed using fatigue data obtained from a single specimen size or geometry. From a mechanics standpoint, the effective fatigue resistance in the VHCF regime is governed not only by the applied nominal stress amplitude but also by the stressed control volume that statistically samples defect populations capable of initiating crack incubation under near-threshold cyclic loading.
The observed negative size effect implies that increasing component dimensions inherently elevates the probability of activating a critical internal defect, even when defect size distributions and manufacturing parameters remain unchanged. Consequently, fatigue design strategies based solely on nominal S–N curves derived from small laboratory specimens may significantly overestimate the VHCF strength of larger AM components. Instead, fatigue assessment should incorporate volumetric scaling through control volume concepts, whereby fatigue strength is treated as a probabilistic quantity rather than a deterministic material constant. Recent defect-based probabilistic frameworks combining Kitagawa–Murakami type relationships with statistical S–N modeling further support this perspective, providing a direct link between defect size, threshold stress intensity, and fatigue strength (Esposito et al., 2026). In addition, log-Weibull formulations with stress-dependent dispersion have been shown to better capture the evolution of fatigue scatter in the VHCF regime (Esposito et al., 2025). The probabilistic control volume framework employed in this study provides a mechanics-informed basis for VHCF design of AM components. By coupling weakest-link theory with Weibull-based fatigue statistics, this approach enables extrapolation of fatigue performance across component sizes while maintaining physical consistency with defect-controlled crack initiation mechanisms. Such a framework is particularly relevant for large AM parts subjected to long-life cyclic loading, where internal defects dominate failure behavior and traditional safety factors may prove inadequate.
5 Conclusion
This study investigated the specimen-size effect on the ultrasonic VHCF behavior of PBF-LBed AlSi10Mg by combining experimental characterization with probabilistic mechanics modeling. The main conclusions are summarized as follows:
The VHCF strength of the PBF-LBed AlSi10Mg exhibits a pronounced negative size effect with increasing specimen size or control volume. This behavior cannot be attributed solely to nominal stress levels but instead arises from the statistical amplification of critical internal defect activation within the mechanically relevant control volume under near-threshold cyclic loading.
Fatigue crack initiation in the VHCF regime consistently originates from internal and subsurface lack-of-fusion defects surrounded by FGAs. The observed size effect therefore reflects a statistical scaling of critical defect populations rather than a change in the underlying crack initiation mechanism.
Weibull-based probabilistic S–N curves, combined with weakest-link theory and control volume concepts, provide a physically consistent and generally conservative framework for describing fatigue scatter and size-dependent VHCF performance across different specimen geometries.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
ZM: Investigation, Resources, Methodology, Supervision, Writing – original draft. LL: Resources, Supervision, Funding acquisition, Conceptualization, Formal Analysis, Writing – original draft, Project administration. YD: Writing – original draft, Visualization, Validation, Data curation. HS: Visualization, Writing – original draft, Validation, Software, Writing – review and editing, Formal Analysis.
Funding
The author(s) declared that financial support was not received for this work and/or its publication.
Conflict of interest
Author ZM were employed by Vanadium & Titanium Research (Beijing Branch) of Pansteel Research Institute Co., Ltd.
Author ZM were employed by Ansteel Beijing Research Institute Co., Ltd.
Author DY were employed by CCCC First Highway Consultants Co., LTD
The remaining author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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The author(s) declared that generative AI was not used in the creation of this manuscript.
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References
1
AfazovS.SerjoueiA.HickmanG. J.MahalR.GoyD.MitchellI. (2023). Defect-based fatigue model for additive manufacturing. Prog. Addit. Manuf.8, 1059–1066. 10.1007/s40964-022-00376-6
2
AraújoL. C.GabrielA. H. G.da FonsecaE. B.AvilaJ. A.JardiniA. L.JuniorR. S.et al (2022). Effects of build orientation and heat treatments on the tensile and fracture toughness properties of additively manufactured AlSi10Mg. Int. J. Mech. Sci.213, 106868. 10.1016/j.ijmecsci.2021.106868
3
AtrensA.HoffelnerW.DuerigT. W.AllisonJ. E. (1983). Subsurface crack initiation in high cycle fatigue in Ti6Al4V and in a typical martensitic stainless steel. Scr. Metall.17, 601–606. 10.1016/0036-9748(83)90385-x
4
AvateffazeliM.HaghshenasM. (2022). Ultrasonic fatigue of laser beam powder bed fused metals: a state-of-the-art review. Eng. Fail. Anal.134, 106015. 10.1016/j.engfailanal.2021.106015
5
BathiasC.ParisP. (2005). Gigacycle Fatigue in Mechanical Practice. New York, NY, USA: Marcel Dekker.
6
BehvarA.BertoF.HaghshenasM. (2023). A review on isothermal rotating bending fatigue failure: microstructural and lifetime modeling of wrought and additive manufactured alloys. Fatigue Fract. Eng. Mater. Struct.46, 3545–3595. 10.1111/ffe.14094
7
BS EN ISO/ASTM 52900:2021 (2022). Additive Manufacturing—General Principles—Fundamentals and Vocabulary. London, UK: BSI Standard Limited.
8
ChangY.ZhengL.PanX.HongY. (2019). Further investigation on microstructure refinement of internal crack initiation region in VHCF regime of high-strength steels. Frat. Integrità Strutt.13, 1–11. 10.3221/igf-esis.49.01
9
ChangY.PanX.ZhengL.HongY. (2020). Microstructure refinement and grain size distribution in crack initiation region of very-high-cycle fatigue regime for high-strength alloys. Int. J. Fatigue134, 105473. 10.1016/j.ijfatigue.2020.105473
10
ChenR. C.DreossiD.ManciniL.MenkR.RigonL.XiaoT. Q.et al (2012). PITRE: software for phase-sensitive X-ray image processing and tomography reconstruction. J. Synchrotron Radiat.19, 836–845. 10.1107/S0909049512029731
11
ChuaC.LiuY. T.WilliamsR. J.ChuaC. K.SingS. L. (2024). In-process and post-process strategies for part quality assessment in metal powder bed fusion: a review. J. Manuf. Syst.73, 75–105. 10.1016/j.jmsy.2024.01.004
12
CongT.QianG.ZhangG.WuS.PanX.DuL.et al (2021). Effects of inclusion size and stress ratio on the very-high-cycle fatigue behavior of pearlitic steel. Int. J. Fatigue142, 105958. 10.1016/j.ijfatigue.2020.105958
13
DebRoyT.WeiH. L.ZubackJ. S.MukherjeeT.ElmerJ. W.MilewskiJ. O.et al (2018). Additive manufacturing of metallic components—Process, structure and properties. Prog. Mater. Sci.92, 112–224. 10.1016/j.pmatsci.2017.10.001
14
DuL.PanX.QianG.ZhengL.HongY. (2021). Crack initiation mechanisms under two stress ratios up to very-high-cycle fatigue regime for a selective laser melted Ti-6Al-4V. Int. J. Fatigue149, 106294. 10.1016/j.ijfatigue.2021.106294
15
DuL.PanX.HongY. (2024). New insights into microstructure refinement in crack initiation region of very-high-cycle fatigue for SLM Ti-6Al-4V via precession electron diffraction. Materialia33, 102008. 10.1016/j.mtla.2024.102008
16
EspositoL.BrunoM.PentaF.RicciS.IannittiG. (2025). Investigating the impact of printing direction on fatigue behaviour of machined SLM AlSi10Mg through orientation analysis of idealized ellipsoidal defects. J. Alloys Compd.1016, 178912. 10.1016/j.jallcom.2025.178912
17
EspositoL.BrunoM.NaccaM. (2026). Enhanced weibull formulation for capturing fatigue life scatter in AM alloys supported by fractographic analysis. Procedia Struct. Integr.76, 50–58. 10.1016/j.prostr.2025.12.286
18
FuruyaY. (2008). Specimen size effects on gigacycle fatigue properties of high-strength steel under ultrasonic fatigue testing. Scr. Mater.58, 1014–1017. 10.1016/j.scriptamat.2008.01.039
19
FuruyaY. (2011). Notable size effects on very high cycle fatigue properties of high-strength steel. Mater. Sci. Eng. a-Struct.528, 5234–5240. 10.1016/j.msea.2011.03.082
20
GaoC.ZhangY.JiangJ.FuR.DuL.PanX. (2024). Research viewpoint on performance enhancement for very-high-cycle fatigue of Ti-6Al-4V alloys via laser-based powder bed fusion. Crystals14, 749. 10.3390/cryst14090749
21
GaoC.YuZ.ZhangY.FanT.ZhangB.SongH.et al (2025a). Experimental investigation on fatigue crack propagation in surface-hardened layer of high-speed train axles. Crystals15 (7), 638. 10.3390/cryst15070638
22
GaoC.ZhangY.FanT.WangJ.SongH.SuH. (2025b). Microstructure and mechanical properties of high-speed train wheels: a study of the rim and web. Crystals15 (8), 677. 10.3390/cryst15080677
23
GibsonI.RosenD.StuckerB.KhorasaniM. (2021). Additive Manufacturing Technologies. 3rd ed.Cham, Switzerland: Springer.
24
GuD.ShiX.PopraweR.BourellD.SetchiR.ZhuJ. (2021). Material-structure-performance integrated laser-metal additive manufacturing. Science372, eabg1487. 10.1126/science.abg1487
25
HuY.WuS.WuZ.WuW.QianW. (2023). Integrity of Additively Manufactured Advanced Materials and Structures. Beijing, China: National Defense Industry Press.
26
JavidradH.KocB.BayraktarH.SimsekU.GunaydinK. (2024). Fatigue performance of metal additive manufacturing: a comprehensive overview. Virtual Phys. prototy.19, e2302556. 10.1080/17452759.2024.2302556
27
KarakasO.KardesF. B.FotiP.BertoF. (2023). An overview of factors affecting high-cycle fatigue of additive manufacturing metals. Fatigue Fract. Eng. Mater. Struct.46, 1649–1668. 10.1111/ffe.13967
28
KingW. E.AndersonA. T.FerenczR. M.HodgeN. E.KamathC.KhairallahS. A.et al (2015). Laser powder bed fusion additive manufacturing of metals; physics, computational, and materials challenges. Appl. Phys. Rev.2, 041304. 10.1063/1.4937809
29
LearyM. (2020). Design for Additive Manufacturing. Amsterdam, Netherlands: Elsevier.
30
LiuZ. Z.ZhouQ. H.LiangX. K.WangX. B.LiG. C.VanmeenselK.et al (2024). Alloy design for laser powder bed fusion additive manufacturing: a critical review. Int. J. Extreme Manuf.6, 022002. 10.1088/2631-7990/ad1657
31
LiuL.WangS.MaY. (2024). Research of rotating and ultrasonic cycling fatigue in high-cycle regimes. J. Aeronaut. Mater.44, 142–150. 10.11868/j.issn.1005-5053.2023.000016
32
LongX.JiaQ.LiJ.ChongK.DuL.PanX.et al (2022). Mechanical properties and parameter optimization of TC4 alloy by additive manufacturing. China Surf. Eng.35, 215–223. 10.11933/j.issn.1007-9289.20210924002
33
MarşavinaL.StoiaD. I.EmanoilL. (2021). Fracture toughness in additive manufacturing by selective laser sintering: an overview. Mater. Des. process. Commun.3, e254. 10.1002/mdp2.254
34
MatušůM.DžuberováL.PapugaJ.RosenthalJ.ŠimotaJ.BeránekL. (2024). Fatigue analysis and heat treatment comparison of additively manufactured specimens from AlSi10Mg alloy. Int. J. Fatigue185, 108357. 10.1016/j.ijfatigue.2024.108357
35
MontagnoliF.InvernizziS.CarpinteriA. (2023). VHCF ultrasonic testson EN AW-6082 aluminum alloy samples over a wide dimensional range. Fatigue Fract. Eng. Mater. Struct.46 (8), 3099–3117. 10.1111/ffe.14056
36
MurakamiY. (2002). Metal Fatigue: Effect of Small Defects and Nonmetallic Inclusions. Oxford, UK: Elsevier.
37
NaitoT.UedaH.KikuchiM. (1983). Observation of fatigue fracture surface of carburized steel. J. Soc. Mater. Sc.32, 1162–1166. 10.2472/jsms.32.1162
38
PanX.HongY. (2019). High cycle and very high cycle fatigue behaviour of a titanium alloy with equiaxed microstructure under different mean stresses. Fatigue Fract. Eng. Mater. Struct.42, 1950–1964. 10.1111/ffe.13050
39
PanX.HongY. (2024). High-cycle and very-high-cycle fatigue of an additively manufactured aluminium alloy under axial cycling at ultrasonic and conventional frequencies. Int. J. Fatigue185, 108363. 10.1016/j.ijfatigue.2024.108363
40
PanX.SuH.SunC.HongY. (2018). The behavior of crack initiation and early growth in high-cycle and very-high-cycle fatigue regimes for a titanium alloy. Int. J. Fatigue115, 67–78. 10.1016/j.ijfatigue.2018.03.021
41
PanX.XuS.QianG.NikitinA.ShanyavskiyA.Palin-LucT.et al (2020a). The mechanism of internal fatigue-crack initiation and early growth in a titanium alloy with lamellar and equiaxed microstructure. Mater. Sci. Eng. A798, 140110. 10.1016/j.msea.2020.140110
42
PanX.QianG.WuS.FuY.HongY. (2020b). Internal crack characteristics in very-high-cycle fatigue of a gradient structured titanium alloy. Sci. Rep.10, 4742. 10.1038/s41598-020-61484-3
43
PanX.QianG.HongY. (2021). Nanograin formation in dimple ridges due to local severe-plastic-deformation during ductile fracture. Scr. Mater.194, 11363. 10.1016/j.scriptamat.2020.113631
44
PanX.SuH.LiuX.HongY. (2024a). Multi-scale fatigue failure features of titanium alloys with equiaxed or bimodal microstructures from low-cycle to very-high-cycle loading numbers. Mater. Sci. Eng. A890, 145906. 10.1016/j.msea.2023.145906
45
PanX.XuS.NikitinA.ShanyavskiyA.Palin-LucT.HongY. (2024b). Crack initiation induced nanograins and facets of a titanium alloy with lamellar and equiaxed microstructure in very-high-cycle fatigue. Mater. Lett.357, 135769. 10.1016/j.matlet.2023.135769
46
PanX.DuL.QianG.HongY. (2024c). Microstructure features induced by fatigue crack initiation up to very-high-cycle regime for an additively manufactured aluminium alloy. J. Mater. Sci. Technol.173, 247–260. 10.1016/j.jmst.2023.07.023
47
PanX.TaoZ.LongX. (2025). Extraordinary specimen-size effect on long-life fatigue of additively manufactured AlSi10Mg. Int. J. Mech. Sci.47, 1024–1035. 10.1016/j.ijmecsci.2025.110524
48
PengX.WuS.QianW.BaoJ.HuY.ZhanZ.et al (2022). The potency of defects on fatigue of additively manufactured metals. Int. J. Mech. Sci.221, 107185. 10.1016/j.ijmecsci.2022.107185
49
QianG.JianZ.PanX.BertoF. (2020a). In-situ investigation on fatigue behaviors of Ti-6Al-4V manufactured by selective laser melting. Int. J. Fatigue133, 105424. 10.1016/j.ijfatigue.2019.105424
50
QianG.JianZ.QianY.PanX.MaX.HongY. (2020b). Very-high-cycle fatigue behavior of AlSi10Mg manufactured by selective laser melting: effect of build orientation and mean stress. Int. J. Fatigue138, 105696. 10.1016/j.ijfatigue.2020.105696
51
RomanoS.Brückner-FoitA.BrandãoA.GumpingerJ.GhidiniT.BerettaS. (2018). Fatigue properties of AlSi10Mg obtained by additive manufacturing: defect-based modelling and prediction of fatigue strength. Eng. Fract. Mech.187, 165–189. 10.1016/j.engfracmech.2017.11.002
52
SakaiT. (2009). Review and prospects for current studies on very high cycle fatigue of metallic materials for machine structural use. J. Solid Mech. Mater. Eng.3, 425–439. 10.1299/jmmp.3.425
53
SanaeiN.FatemiA. (2021). Defects in additive manufactured metals and their effect on fatigue performance: a state-of-the-art review. Prog. Mater. Sci.117, 100724. 10.1016/j.pmatsci.2020.100724
54
SchijveJ. (2009). Fatigue of Structures and Materials. 2nd ed.Dordrecht, Germany: Springer.
55
SchneiderC. A.RasbandW. S.EliceiriK. W. (2012). NIH image to ImageJ: 25 years of image analysis. Nat. Methods9, 671–675. 10.1038/nmeth.2089
56
SuH.LiuX.SunC. X.HongY. (2017). High-cycle and very-high-cycle fatigue behaviour of a titanium alloy with equiaxed microstructure under different mean stresses. Fatigue Fract. Eng. Mater. Struct.40, 979–993. 10.1111/ffe.12562
57
SuJ. L.JiangF. L.TengJ.ChenL. Q.YanM.RequenaG.et al (2024). Recent innovations in laser additive manufacturing of titanium alloys. Int. J. Extrem. Manuf.6, 032001. 10.1088/2631-7990/ad2545
58
SuH.LiuL.YangR.ChenZ.WangS.PengQ.et al (2025a). Fatigue performance of a wheel steel under rotating bending and ultrasonic axial cycling: role of elongated sulfides, effects of loading type and stress ratio. Int. J. Fatigue202, 109389. 10.1016/j.ijfatigue.2025.109389
59
SuH.LiY.PanX.KumarP.LongX. (2025b). Nanoscale grain formation at crack initiation region under very-high-cycle fatigue for Ti-6Al-4V manufactured by selective laser melting. Materialia44, 102619. 10.1016/j.mtla.2025.102619
60
SunC.ZhangX.LiuX.HongY. (2016). Effects of specimen size on fatigue life of metallic materials in high-cycle and very-high-cycle fatigue regimes. Fatigue Fract. Eng. Mater. Struct.39, 770–779. 10.1111/ffe.12415
61
SunC.SongQ.ZhouL.PanX. (2019). Characteristic of interior crack initiation and early growth for high cycle and very high cycle fatigue of a martensitic stainless steel. Mater. Sci. Eng. A758, 112–120. 10.1016/j.msea.2019.04.015
62
SureshS. (1998). Fatigue of Materials. 2nd ed.Cambridge, UK: Cambridge University Press.
63
TaoZ.WangZ.PanX.SuT.LongX.LiuB.et al (2024). A new probabilistic control volume scheme to interpret specimen size effect on fatigue life of additively manufactured titanium alloys. Int. J. Fatigue183, 108262. 10.1016/j.ijfatigue.2024.108262
64
Thermo Fisher Scientific (2023). AVIZO, Version 2023.1. Waltham, MA, USA: Thermo Fisher Scientific.
65
TridelloA.NiuttaC.BertoF.PaolinoD. (2021). Size-effect in very high cycle fatigue: a review. Int. J. Fatigue153, 106462. 10.1016/j.ijfatigue.2021.106462
66
TridelloA.FiocchiJ.BiffiC. A.RossettoM.TuissiA.PaolinoD. S. (2022). Effect of microstructure, residual stresses and building orientation on the fatigue response up to 109 cycles of an SLM AlSi10Mg alloy. Int. J. Fatigue160, 106825. 10.1016/j.ijfatigue.2022.106825
67
TusherM. M. H.InceA. (2023). A systematic review on high cycle and very high cycle fatigue behavior of laser powder bed fused (L-PBF) Al-Si alloys. Eng. Fail. Anal.154, 107667. 10.1016/j.engfailanal.2023.107667
68
WöhlerA. (1858). Bericht über die Versuche, welche auf der Königl. Niederschlesisch-Märkischen Eisenbahn mit Apparaten zum Messen der Biegung und Verdrehung von Eisenbahnwagen-Achsen während der Fahrt, angestellt wurden. Z. Bauwes.8, 642652.
69
WöhlerA. (1860). Versuche zur Ermittlung der auf die Eisenbahnwagen-Achsen einwirkenden Kräfte und der Widerstandsfähigkeit der Wagen-Achsen. Z. Bauwes. 10, 548616.
70
WӧhlerA. (1867). Wӧhler’s experiments on the strength of metals. Engineering4, 160161.
71
XuS.PanS.LiZ.LiS.HeX.PanX. (2025). Anisotropic tensile behavior and fracture characteristics of an additively manufactured nickel alloy without and with a heat treatment of solution aging. Mater. Sci. Eng. A927, 148015. 10.1016/j.msea.2025.148015
72
YiM.TangW.ZhuY.LiangC.TangZ.YinY.et al (2024). A holistic review on fatigue properties of additively manufactured metals. J. Mater. Process. Technol.329, 118425. 10.1016/j.jmatprotec.2024.118425
73
ZenouM.GraingerL. (2018). “Additive manufacturing of metallic materials,” in In Additive Manufacturing: Materials, Processes, Quantifications and Applications. Editors ZhangJ.JungY.-G. (Oxford, UK: Butterworth-Heinemann), 53–103.
74
ZhouY.SunJ.PanX.QianG.HongY. (2022). Microstructure evolution and very-high-cycle fatigue crack initiation behavior of a structural steel with two loading intermittence modes. Int. J. Fatigue161, 106904. 10.1016/j.ijfatigue.2022.106904
Summary
Keywords
control volume, defect-controlled crack initiation, probabilistic fatigue modeling, specimen-size effect, very-high-cycle fatigue (VHCF)
Citation
Ma Z, Liu L, Deng Y and Su H (2026) Ultrasonic very-high-cycle fatigue of PBF-LBed AlSi10Mg: effects of specimen size, control volume and defect statistics. Front. Mater. 13:1834842. doi: 10.3389/fmats.2026.1834842
Received
20 March 2026
Revised
27 April 2026
Accepted
15 May 2026
Published
09 June 2026
Volume
13 - 2026
Edited by
Alberto Sapora, Polytechnic University of Turin, Italy
Reviewed by
Luca Esposito, University of Naples Federico II, Italy
Liming Lei, Taihang Laboratory, China
Updates
Copyright
© 2026 Ma, Liu, Deng and Su.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Hang Su, hangsu2018@g.ucla.edu
† These authors have contributed equally to this work
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