2.1 Fatigue failures in presence of defects: General model for the PSN curve

For a safe design of components subjected to fatigue loads, an appropriate experimental assessment of the material response with respect to the number of cycles is fundamental. For this reason, the S–N curves, that model the variation of the fatigue life, or the number of cycles to failure, with respect to the applied stress amplitude, are widely adopted. In a probabilistic framework, the statistical distribution of the fatigue life can be also attained, since the fatigue phenomenon is intrinsically characterized by a large scatter that must be carefully taken into account when components are to be designed. International standards prescribe how to analyze the fatigue test results and to assess the distribution of the fatigue life up to the High Cycle Fatigue (HCF) regime.²⁰ Moreover, in order to model the fatigue response of parts also in the Very High Cycle Fatigue (VHCF) regime, different models have been proposed in the literature.^21-25 If the fatigue failure originates from manufacturing defects, the influence of the defect size on the fatigue life distribution should be also considered.^{17, 26}

Starting from the models proposed by Paolino et al,^{22, 26} a general model for the fatigue life in the HCF region and in the VHCF region can be defined, permitting the estimation of the so called “duplex PSN curves.”^{23, 24} It involves a first linear decreasing trend in the HCF region, a transition stress, and a second linear decreasing trend in the VHCF regime ending with a horizontal asymptote (i.e., the fatigue limit). Moreover, the dependency between the fatigue life and the defect size distribution is also included in the model.

Equation 1 reports the general statistical model for the fatigue life in presence of defects in the HCF and VHCF regimes.

(1)

in which F_Y(y; x) is the cumulative distribution function (cdf) of the fatigue life Y (i.e., Y = log₁₀(N_f), being N_f the number of cycles to failure) for a given logarithm of applied stress amplitude x (i.e., x = log₁₀(s_a), being s_a the applied stress amplitude), is the cdf of the transition stress between HCF and VHCF region, and F_Y|HCF(y; x) and F_Y|VHCF(y; x) are the marginal cdf of the fatigue life in the HCF region and in the VHCF region, according to Paolino et al.²⁶ The transition stress is assumed to follow a normal distribution, that is, , with constant mean and standard deviation (Φ(·) is the cdf of a standardized Normal distribution).

The marginal cdf of the fatigue life in the VHCF region (or due to a second failure mode) is given by

(2)

being the cdf of the conditional fatigue life (i.e., the fatigue life conditioned to the defect size), the cdf of the fatigue limit, and the truncated probability density function (pdf) of the defect size (assumed as the projected area of the defect in a direction perpendicular to the load direction, according to Murakami¹⁷). is assumed to be normally distributed, with mean depending on x and and constant standard deviation σ_Y,V. is also assumed to be normally distributed, with mean dependent on the material Vickers hardness and on and constant standard deviation For more details on the model, the reader is referred to Paolino et al.²⁶ According to Murakami,¹⁷ the defect size is assumed to follow a Largest Extreme Value Distribution (LEVD). In particular, a truncated (to 0) distribution of the LEVD is assumed, since the defect size cannot be negative.²⁷ According to Equation 2, the marginal distribution of the fatigue life does not depend on the defect size.

The expression for the marginal cdf of the fatigue life in the HCF region (or due to a first failure mode) is the same of Equation 2, but in this case, the distribution of the fatigue limit is obviously not considered (i.e., )). It is worth to note that the distribution of the fatigue life and of the defect size is different for the HCF region and for the VHCF region (i.e., the constant parameters of the distribution are different for the HCF region and the VHCF region).

Equations 1 and 2 permit to model the so-called “duplex S-N curve,” according to Wei et al. and Kikuchi et al.^{23, 24} The general model can be also modified to consider simpler cases. For example, if the fatigue limit is not present, the model in Equation 1 is modified by considering . On the other hand, if the experimental data follow one linear trend ending with fatigue limit, in Equation 1 is equal to 0.

Figure 1 shows a typical example of a duplex probabilistic S–N (PSN) curve with the statistical models considered for modeling the fatigue life. The expressions for the mean of the finite fatigue life distribution in the HCF and in VHCF region and of the fatigue limit, introduced in Equations 1 and 2, are also reported. The constant parameters involved in the models for the finite fatigue life in the HCF region and for the finite and infinite (i.e., the fatigue limit) life in the VHCF region are estimated separately, according to the procedure described in Paolino et al.²⁶ Given these coefficients, the mean and the standard deviation of the transition stress distribution are estimated by maximizing the likelihood function of the model in Equation 1, so that both failures and run-out specimens can be considered. Alternatively, all the unknown parameters can be estimated by maximizing the likelihood function of the complete model. In this case, however, the number of parameters to be estimated is large, and the choice of the initial values for the parameters to be estimated can be difficult (e.g., by using the function fminsearch of MATLAB® that adopts the Nelder–Mead simplex algorithm). From Equation 1, the α quantile of the PSN curve can be estimated by substituting F_Y(y; x) with α and numerically solving Equation 1 with respect to x for different values of y. Therefore, for each value of y in the selected range of number of cycles to failure, the x value of the αth quantile curves is obtained by numerically solving Equation 1, since a closed-form solution does not exist. From a practical point of view, y can be iteratively varied in the range of interest to obtain the corresponding x_α for the the αth quantile curves. Given y and x_α, the α quantile of the PSN curve can be built on an S–N plot. Generally, the authors use the fzero function in a MATLAB® code to define the PSN curves.

2.2 Fatigue failures in presence of defects: Statistical methodology

The number of AM specimens for fatigue tests is generally smaller than that available for traditionally built specimens, due to the high cost of the AM processes and powders. Therefore, the information contained in the experimental dataset should be “maximized,” by properly accounting also for the large scatter of fatigue data, mainly associated with the random distribution of the defect size. In this respect, if the fatigue tests are carried out at different stress levels and, for each level, the number of results is limited, it might be difficult to analyze the fatigue data (or to assess the influence of a specific factor). The analyses, on the contrary, could be more effective if all the data are gathered together at a defined number of cycles to failure (or stress amplitude) which is considered as representative of the fatigue life: In this way, also the scatter associated with the experimental data can be properly taken into account.

In order to “shift” all the experimental failures to a defined number of cycles to failures (in the following, denoted as reference number of cycles, N_ref) or to a defined stress amplitude (in the following, denoted as reference stress amplitude, s_ref), Equation 1 can be exploited. Figure 2 shows graphically the procedure for assessing, for each experimental failure, the corresponding failure at N_ref or s_ref. As a representative example, the case of a PSN curve with a decreasing trend and a fatigue limit is shown.

According to Figure 2, a specific PSN curve passes through each experimental datapoint: It is worth to note that this is true only if the parameters of the distribution of the fatigue life are properly estimated and are close to the actual values. For example, the ith failure (square gray marker in Figure 2, with coordinates (y_ith; x_ith)) is crossed by the PSN curve characterized by the failure probability α_th (i.e., F_Y(y; x) = α_th). By following the PSN curve with α_th failure probability, it is therefore possible to assess for the i_th failure the corresponding failure at N_ref or s_ref (the gray diamond marker in Figure 2 is the i_th failure at N_ref = 10⁶ cycles). By repeating this procedure for all the experimental failures, it is possible to gather all the data and to properly compare them with statistical methodologies, like the analysis of variance (ANOVA) or the hypothesis test. In this way, the scatter associated with the experimental data can be properly taken into account, since the influence of a factor can be “hidden” or overestimated due to the experimental large scatter.

For the sake of clarity, in order to shift all the experimental data, the unknown parameters of the model in Equation 1 (or of a simplified model, depending on the experimental dataset) are to be estimated according to the procedure described in Section 2.1. Given the unknown parameters, the model for the fatigue life is completely defined. For the i_th failure (gray square marker in Figure 2), the logarithm of the number of cycles to failure and of the applied stress amplitude (y_ith and x_ith, respectively) is known and can be inserted in Equation 1, which can be solved to find the α_th PSN curve (blue curve in Figure 2) that crosses the i_th failure. In other words, each point on the α_th curve permits to assess the stress amplitude characterized by a probability of failure equal to α_th. Therefore, given α_th, the stress amplitude for the i_th failure shifted at a larger number of cycles to failure (e.g., in Figure 2, green diamond marker) can be easily obtained by numerically solving Equation 1.

From a practical point of view, the proposed methodology can be applied only if the constant coefficients involved in Equation 1 are properly estimated. Therefore, for each experimental failure, the stress amplitude, the number of cycles to failure, and the defect originating failure should be known. By properly modifying Equation 1, the proposed procedure can be also applied without knowing the defect sizes. However, in this case, the information concerning to the influence of defects is missed.

3.1 Titanium alloys

The influence of defects on the fatigue response up to N_f = 2 · 10⁶ cycles of a grade 23 ELI TA6V alloy used in medical and aeronautical applications was investigated in Le et al.²⁸ Standard dog bone specimens for the fatigue tests were obtained after a machining process of samples (200 μm removed) produced with an SLM280 HL machine. Before the machining process, the specimens were heat-treated (annealing at 850 ° C for 2 h and furnace cooling) to remove residual stresses. The fatigue tests were carried out with R equal to 0.1, and the effect of the building orientation was investigated by producing specimens in the XY direction, at 45° with respect to the building platform and in the Z direction. The S–N plot reported in Le et al.²⁸ depicts the maximum stress, normalized with respect to the average value of the ultimate stress in the three building directions, with respect to the number of cycles to failure: In the following analysis, the normalized data are considered. Moreover, two batches for the XY building orientation, three batches for the 45° building orientation and two batches for the Z building orientation were considered (i.e., all batches were obtained with the same process parameters) to investigate the fatigue response scatter. For each building orientation, Bx refers to the x batch (e.g., B1 refers to the first batch).

For the analysis of the experimental data, the model involving the presence of the fatigue limit at the end of the curve was considered for all the dataset, according to Le et al,²⁸ apart from the batch B3 (45° building orientation) for which a clear fatigue limit cannot be estimated from the experimental data. Figure 3A plots, for each building orientation and batch, the fatigue strength at the reference N_ref = 10⁷ cycles (i.e., all the experimental failures have been shifted to N_ref = 10⁷ cycles). For each batch, the estimated fatigue strengths at 10⁷ cycles are reported within the round brackets; whereas, the σ_max0,n in Le et al.²⁸ (i.e., the horizontal asymptote of the curve) is reported within the square brackets. In Figure 3B, the data points at N_ref = 10⁷ cycles have been grouped to assess the influence of the building orientation on the fatigue response.

According to Figure 3A and in agreement with the analysis in Le et al,²⁸ the scatter associated with the batches is large, even if the same process parameters were considered. Indeed the defect sizes vary among the batches, thus inducing significantly different fatigue responses. The normalized fatigue strength at 10⁷ cycles estimated with the model reported in Equation 1 (median value reported in round brackets) is close to the fatigue strength reported in Tab. 3 in Le et al.²⁸ (square brackets in the Figure), thus confirming that the proposed model is appropriate in assessing the influence of defects on the fatigue life. In addition, by shifting all the experimental failures at 10⁷ cycles, it is possible to statistically compare failures occurred at different number of cycles, adding more strength to the analysis. For example, for each building orientation, an ANOVA analysis has been carried out to assess the influence of the batch variability on the fatigue response. The fatigue strength of the batches varies significantly for each building orientation, as the p value for each ANOVA is equal to 0. By shifting and grouping all the data together, it has been possible to assess the batch variability given the large scatter associated with the experimental failures. Therefore, in agreement with the conclusions of Le et al,²⁸ this analysis confirms that the variability between different batches can be large, even if the specimens are produced with the same process parameters.

In Figure 3B, the effect of the building direction is investigated by grouping all the data for each building orientation. It is worth to note that the large scatter could hide the effect of this factor: However, since the same process parameters were used for manufacturing all the specimens, all the data coming from different batches must be considered together. Otherwise, the scatter associated with the AM process is neglected, obtaining less reliable results and missing important available information. As highlighted by the authors, a clear tendency between the specimens built in the XY directions and at 45° is not observed: Indeed, by carrying out an ANOVA between these two building orientations, the p value is 6.2%, larger than a typical 5% significance level, confirming the obtained results. Moreover, the specimens built in the Z directions are characterized by the smallest fatigue strength. However, if all the experimental data are considered together and an ANOVA is carried out, the building orientation is confirmed to be a statistically significant factor, with p value equal to 0. A linear interpolation between the fatigue strength and the building angle (0°corresponds to the XY building direction and 90° corresponds to the Z building direction) has been also carried out in Figure 3B: The slope of the curve is negative, and the 95% confidence interval for the slope does not contain the zero value, proving that the fatigue strength significantly decreases with the building angle. It can be concluded that, according to Le et al,²⁸ the building orientation strongly affects the fatigue response, with the fatigue strength being larger for XY and the 45° and decreasing significantly for the Z building direction.

The influence of defects and manufacturing processes on the fatigue response of Ti6Al4V specimens has been also investigated in Masuo et al.²⁹ Dog bone specimens were produced through a Direct Metal Laser Sintering (DMLS) and an Electron Beam Melting (EBM) process and subjected to different types of post-treatments. Rotating bending fatigue tests were carried out up to 10⁷ cycles (runout number of cycles).

The experimental data obtained on specimens subjected to a surface polishing process (with #600 emery paper) and not subjected to the hot isostatic process (HIP) process are considered in the following analysis. The influence of the manufacturing process, DMLS and EBM, on the fatigue response is investigated. A model involving a decreasing linear trend and a fatigue limit was considered for the fatigue life. Figure 4A compares the cdfs of the fatigue strength at 10⁷ cycles; in Figure 4B, the empirical cumulative distribution function (ecdf) of the difference between the fatigue strengths at 10⁷ cycles obtained by testing the specimens produced through the DMLS and the EBM processes is plotted. In order to compute the ecdf for each manufacturing process, 1000 random values of the fatigue strength at 10⁷ cycles have been simulated according to Equation 1 and by considering the parameters estimated according to the procedure reported in Section 2.2.

According to Figure 4A, the median fatigue strength at 10⁷ cycles for the DMLS is larger (365 MPa for the DMLS specimens and 230 MPa for the EBM specimens) and close to the value obtained in Masuo et al.²⁹ (fatigue limit equal to 370 MPa for the DMLS specimens and in the range [240 − 260] MPa for the EBM specimens). In order to assess if the fatigue strength of the DMLS specimens is larger, the ecdf of the difference between the fatigue strength has been computed and plotted in Figure 4B. According to Figure 4B, the difference is not statistically significant for a 5% significance level (ecdf equal to −2.4). However, a zero difference corresponds to an ecdf value equal to 5.26%, slightly larger than 5%. Therefore, due to this limited difference, it can be concluded that the fatigue response of specimens produced through the DMLS process is better than that of the specimens produced through EBM, in agreement with the results in Masuo et al.²⁹

In Günther et al,³⁰ the VHCF response of a Ti6Al4V alloy produced through different AM processes and the effect of post-treatments have been experimentally assessed. In particular, raw cylinders were produced through an SLM and an EBM processes and then machined to obtain the specimens for the ultrasonic (loading frequency of 19 kHz) fully reversed tension–compression fatigue tests. The specimens produced with the SLM process were subjected to different treatments: The specimens within the batch SLM-1b were subjected to a heat treatment involving a heating temperature of 800 ° C, whereas the specimens in the batch SLM-2 were subjected to hot isostatic pressing (HIP) at 920 ° C for 2 h at 1000 bar in an argon atmosphere. The EBM specimens were not subjected to posttreatments.

Figure 5 compares the pdf of the VHCF strength at N_ref = 3 · 10⁹ cycles for the three tested batches in Günther et al.³⁰ A monotonic decreasing trend without fatigue limit was considered to model the fatigue life.

According to Figure 5, the production process and the subsequent post-treatments significantly affect the VHCF response of the investigated Ti6Al4V specimens, in agreement with the analyses in Günther et al.³⁰ In particular, the VHCF response at N_ref = 3 · 10⁹ is about the same for the SLM-1b specimens (produced through SLM and heat-treated at 800 ° C) and the specimens produced through EBM. On the other hand, an HIP process permits to enhance the VHCF response. The trend is clear in Figure 5 and is further confirmed through an ANOVA of the experimental failures shifted at N_ref = 3 · 10⁹ cycles. The difference in fatigue strength is statistically significant by comparing the batches SLM-2 and the batches SLM-1b and EBM (p value close to 0), whereas it is not significant by comparing the VHCF strengths for the EBM and the SLM-1b batches (p value of 47.4%).

3.2 Aluminum alloys

In this section, the experimental results on aluminum alloys available in the literature are analyzed. In Qian et al,³¹ ultrasonic fatigue tests were carried out on AlSi10Mg specimens produced through an SLM process to investigate the effect of the stress ratio and of the building direction up to 10⁹ cycles. Fatigue tests were carried out at a load frequency of 20 kHz on specimens produced in the XY and in Z direction and at stress ratio R equal to −1, 0 and 0.5.

For each experimental failure, the fatigue strengths at N_ref = 10⁹ cycles were estimated by considering a monotonic decreasing trend for the fatigue life. Figure 6A compares the VHCF strength at N_ref = 10⁹ cycles for the investigated building orientation and stress ratio. In Figure 6B,C the Haigh diagrams were estimated by considering a Goodman model passing through the 0.1% fatigue strength for a stress ratio equal to −1 and the tensile properties reported in the paper.³¹ In Figure 6D, the effect of the building orientation on the VHCF strength for different stress ratios is shown: For each stress ratio, the p value associated to the ANOVA, carried out to assess the influence of the building orientation, is reported.

According to Figure 6A and as for the specimens produced with traditional manufacturing processes, the stress ratio has a significant influence on the VHCF strength for both the building orientation, with the fatigue strength decreasing as the stress ratio increases. This is confirmed by analyzing the Haigh diagram estimated in Figure 6B,C for the XY and the Z building orientation, respectively. Indeed, the Haigh diagram estimated by considering the 0.1% fatigue strength for R = −1 is in agreement with the experimental results, even if some failures occurred within the “safe region.” This could be due to the large scatter of the experimental data, associated with the random distribution of the defect size and to the limited number of experimental failures. However, this analysis provides a qualitative indication on the effect of the stress ratio, and this was possible thanks to the possibility of gathering all the experimental failures with the proposed methodology. A detailed analysis on the effect of the building orientation and on possible interactions between the stress ratio and the building orientation has been also carried out, according to Figure 6D. For each stress ratio, the experimental failures for the different building directions have been compared. It is found that the building orientation significantly affects the VHCF response for a stress ratio equal to −1 and 0.5 (p value close to 0), whereas it has no significant effects for a stress ratio of 0. An ANOVA considering all the experimental data at 10⁹ cycles has been also carried out. As expected, the building orientation and the stress ratio significantly influence the VHCF response, as the p value is equal to 0. The interaction between the building orientation and the stress ratio has also been found to be significant (p value of about 0.8%). Indeed, depending on the stress ratio, the building orientation has a different effect on the VHCF response. This result can be due to the small number of available experimental results, and further analysis with a larger dataset should be carried out. For example, if the same ANOVA is carried out without considering the experimental results obtained with tests at a stress ratio of 0.5, the p value for the interactions increases to 5.1%.

In Tang and Pistorius,³² the effect of the building direction and of the hatch spacing on the fatigue response in the HCF region of an AlSi10Mg alloy was investigated. Standard dog bone specimens were produced through an SLM process and tested at a stress amplitude of 80 and of 100 MPa (stress ratio of 0.1). The specimens were built in the XY and in the Z direction: For each building orientation, three batches of specimens were produced with three hatch spacing (0.16, 0.19, and 0.22 mm). After the AM process, the specimens were heat-treated at 300 ° C for 2 h, to minimize the residual stresses, and thereafter machined to the final geometry.

A linear decreasing trend (without fatigue limit) was considered for modeling the fatigue life: Thereafter, all the experimental failures have been shifted to 2 · 10⁶ cycles. Figure 7A,B shows the fatigue strengths at 2 · 10⁶ cycles for the investigated building direction and hatch spacing, with the former focusing on the effect of the hatch spacing and the latter on the effect of the building orientation. Figure 7C,D compares the pdf of the fatigue strength for the XY and Z building orientation, respectively.

According to Figure 7A,C,D, the hatch spacing has a significant influence on the fatigue response, with the fatigue strength decreasing with the hatch spacing for both the building orientation, in agreement with the analyses carried out by Tang and Pistorius.³² However, differently from Tang and Pistorius,³² the proposed method permits to gather failures that occurred at different stress amplitudes, thus increasing the statistical significance of the analysis.

The building orientation also influences the fatigue response, especially for a hatch spacing of 0.19 and 0.22 mm. In order to investigate if the interaction between the building direction and the hatch spacing is significant, an ANOVA considering all the experimental data and the factors “hatch distance” and “building orientation” has been carried out. The results of the ANOVA are reported in Table 1: the sum of squares (SS), the degrees of freedom (DOF), the mean squares (MS), and the estimated p values are reported.

According to Table 1, the hatch distance and the building orientation significantly affect the fatigue response of the investigated AlSi10Mg alloy, as the p value is close to 0. Furthermore, the interaction is statistically significant with p value equal to 3.5%. In general, for larger hatch distances, the effect of the building orientation seems to be larger, as shown in Figure 7A,B.

3.3 Maraging steel

In Meneghetti et al,³³ the influence of defects on the fatigue response of a maraging steel was experimentally assessed. In particular, the specimens were produced in the XY and in the Z directions and tested without performing heat treatments, after a polishing process with progressively finer emery paper from grade 80 up to grade 800. The specimens were subjected to fully reversed (R =  − 1) axial fatigue tests with a servo-hydraulic testing machine.

According to Meneghetti et al,³³ a linear decreasing trend was considered for modeling the fatigue life. Figure 8A reports the experimental dataset in an SN plot, together with the estimated median and 1% PSN curves for the two investigated building orientation. Figure 8B,C compares the estimated fatigue strengths at N_ref = 30,000 and N_ref = 2 · 10⁶ cycles, respectively.

According to Figure 8A, the slope of the PSN curves is significantly different, being larger for the PSN curves estimated for the specimens produced in the XY direction, in agreement with the analyses carried out in Meneghetti et al.³³ Accordingly, the effect of the building orientation depends on the number of cycles to failures, as proved Figure 8B,C. The fatigue response at 30,000 cycles (Figure 8B) for the two building orientations is similar (p value of 11% through an ANOVA); whereas, it is significantly different for a number of cycles to failure N_ref = 2 · 10⁶ (p value of 0). The dependency between the fatigue response and the building orientation can be also demonstrated with an ANOVA by considering all the experimental failures at 30,000 cycles and at 2 · 10⁶ cycles (factors “building orientation” and “number of cycles to failure”). The building orientation and the number of cycles to failure have been found to significantly affect the fatigue response, as expected (p value equal to 0); the interaction is also found to be statistically significant, with a p value equal to 0.4%, thus confirming that the effect of the building orientation depends on the considered number of cycles to failure, as highlighted by the Meneghetti et al.³³ Even in this case, the proposed methodology confirms, in a statistical framework, the results obtained by the authors.

3.4 H13 steel

The fatigue response of an H13 steel produced through an SLM process was experimentally assessed in Pellizzari et al.³⁴ The specimens were built in the XY directions, at 45°, and in the Z direction. After the manufacturing process, the specimens were double-tempered at 650 ° C for 2 h in a vacuum furnace. The specimens were thereafter subjected to a turning and a milling process, with a removal of more than 0.1-mm material. Fully reversed axial fatigue tests were carried out.

The fatigue life has been modeled by considering a monotonic decreasing trend for the specimens built in the XY direction and at 45°, whereas a model including also the fatigue limit at the end of the curve has been considered for the specimens built in the Z direction. The authors warn on the risk of extrapolating experimental data for the HCF region, due to the small number of experimental data (especially for the specimens in the Z direction). However, in the present paper, the fatigue response in the HCF is investigated, and the analyses are qualitatively carried out by shifting the experimental data at a number of cycles to failures N_ref = 3 · 10⁶, corresponding to the runout number of cycles.

Figure 9 plots the fatigue strengths at 3 · 10⁶ cycles for the experimental failures at the three investigated building directions.

According to Figure 9, the specimens built in the Z direction show the smallest fatigue response, in agreement with the analysis carried out by Pellizzari et al.³⁴ As discussed for the experimental results obtained in Le et al.²⁸ in Section 3.1, the difference between the fatigue response of specimens produced in the XY direction and at 45° is not significant (p value of 7.1%), but a trend between the building orientation can be qualitatively inferred from Figure 9 and can be confirmed by considering all the experimental failures in an ANOVA with the factor “building orientation.” In this case, the p value reduces to 0, proving the dependency of the fatigue strength on the building direction. However, as highlighted by the authors, additional analyses with a larger number of experimental data should be carried out.

3.5 Stainless steel

The influence of porosities and roughness on the fatigue response of a 316L stainless steel was investigated in Solberg et al.³⁵ Fatigue tests were carried out on as-built hourglass specimens produced with an SLM process and printed in the Z direction. The fatigue tests were carried out up to N_f = 2 · 10⁶ in load control and at a stress ratio R = 0.1.

Figure 10A plots the experimental dataset in an SN plot, which shows that a transition between internal and surface failures exists and that the two failure modes are characterized by a different slope. Therefore, the fatigue life has been modeled by considering two failure modes and a transition stress (Equation 1 and Paolino et al.²²). The estimated median, 10%, and 90% PSN curves are also shown in Figure 10A. In Figure 10B, the pdf of the transition stress is shown: The value of the transition stress found in Solberg et al.³⁵ is also reported.

According to Figure 10A, a duplex model is appropriate to model the fatigue life for the investigated 316L stainless steel, with the transition stress properly discriminating between the two failure modes. This is confirmed by analyzing Figure 10B: The value of the transition stress in Solberg et al,³⁵ equal to 124 MPa, is within the pdf for the transition stress estimated with the model in Equation 1. The duplex model in Figure 10B can be therefore reliably used to shift the experimental failures to the desired reference number of cycles.

3.6 Inconel

Yang et al.³⁶ investigated the fatigue response of a Nickel alloy Inconel IN718 in the HCF and in the VHCF region. Fully reversed ultrasonic fatigue tests up to 10⁹ cycles were carried out on hourglass specimens produced in the Z direction. The specimen shape was obtained through a machining process of as-built bars, followed by a mechanical polishing along the axial direction of the specimens. According to Yang et al,³⁶ two different failure modes were experimentally found: failures from surface defects in the HCF region and failures from internal defects in the VHCF region, both characterized by a different slope in a traditional SN plot and separated by a transition stress.

The fatigue life was therefore modeled by considering two different slopes for the two failure modes and a transition stress (model “two failure modes and transition stress” in Paolino et al.²²). The experimental data and the estimated median, 90%, and 10% PSN curves are shown in Figure 11A. In Figure 11B, the pdf of the transition stress is shown: The range of the transition stress found in Tang and Pistorius³² is also reported. Figure 11C compares the pdf of the fatigue life at a stress amplitude of 500 MPa to highlight the presence of a transition stress between the two failure modes. In particular, the pdfs are obtained by considering separately the experimental failures from surface defects and the experimental failures for internal defects (i.e., different linear models for the surface failures and for the internal failures).

According to Figure 11A, the model for the PSN curves is in agreement with the experimental data, with the median curve subdividing the experimental failures almost equally: six failures are above the median curve, whereas seven failures are below. Indeed, if two failure modes are found experimentally, the median curve must equally subdivide the whole experimental dataset, regardless of the failure mode (i.e., it should not necessarily equally subdivide surface failures and internal failures data). The transition stress, as well as the entire dataset, is characterized by a large scatter, and this is mainly due to the small number of experimental failures in this region. A second reason is also that the stress amplitude range for the surface failures and the internal failures partially overlaps, differently from the experimental dataset in Figure 10 where stress ranges for the two failure modes are clearly distinct. Figure 11A confirms that a model involving a transition stress between the two failure modes is more appropriate to describe the fatigue life in the HCF and in the VHCF region of the tested Inconel IN718. The model can be validated by considering Figure 11B: Indeed, the range for the transition stress in Yang et al.³⁶ is within the pdf estimated with Equation 1. Moreover, Figure 11C further confirms that surface and internal defects originate fatigue failures characterized by significantly different fatigue lives (the median value is equal to 2.1 · 10⁶ and 2.15 · 10⁸ cycles for the surface and internal failures, respectively) and that a transition stress should be considered to properly model the fatigue response of the tested IN718 and to discriminate between the HCF and the VHCF region.