3.1.1 Crack modeling

The advantage of these formulations is that they describe the smooth transition between short cracks $\to$ long cracks, which corresponds to $\mathrm{n}\to \infty$ for $\sqrt{\mathit{\text{area}}}>10\sqrt{{\mathit{\text{area}}}_0}.$⁷⁸

Titanium alloys

As briefly mentioned in Sections 1 and 2, the $\sqrt{\mathit{\text{area}}}$ parameter model can be used to evaluate how a small defect affects the fatigue limit.³⁹ According to a study by Murakami, the $\sqrt{\mathit{\text{area}}}$ parameter model can be used to forecast the fatigue limit for machined specimens with artificial surface roughness. The maximum height values in that literature ranged from 20.5 to 74 μm while in some other papers, it could be bigger than the mentioned range. For instance, in a study by Nakatani et al.⁷⁹ on the AM'd Ti-6Al-4V, the maximum amount reported by area measurement is 88 μm. As a result, it is important to confirm the application range of the “ $\sqrt{\mathit{\text{area}}}$ parameter model.” Some writers⁴¹ explored the relationship between $∆K$ and $\sqrt{\mathit{\text{area}}}$ as depicted in Figure 8 and conducted fatigue tests on specimens without HIP that were made by EBM and DMLS following Nakatani's work.⁷⁹ A crack with any shape was given a $∆K$ value, which was determined using Equation 15. Here, $∆\sigma$ is the stress range (MPa), and $\sqrt{\mathit{\text{area}}}$ is the square root of the projection area of the defect observed at the crack initiation site ( $\mathit{\mu m}$). As a result, the threshold $∆k$ could be computed using the $\sqrt{\mathit{\text{area}}}$ defect model; in Figure 8, Masuo et al.⁴¹ have published the threshold $∆k$ of Ti-6Al-4V AM'd alloy depending on defect size.

Murakami³⁹ presented the following evaluation strategy for periodic surface notches as artificial surface roughness, where a represents the vertical root-to-peak distance and b represents the horizontal peak-to-peak distance.

$$\frac{\sqrt{{\mathit{\text{area}}}_R}}{2b}=2.97\left(\frac{a}{2b}\right)-3.51{\left(\frac{a}{2b}\right)}^2-9.74{\left(\frac{a}{2b}\right)}^3,\text{for}\frac{a}{2b}\le 0.195$$ (22)

$$\frac{\sqrt{{\mathit{\text{area}}}_R}}{2b}\approx 0.38,\text{for}\frac{a}{2b}\ge 0.195$$

Thus, using Murakami's model for Ti-6Al-4V alloy, it would be possible to determine the area of the effective defect ( $\surd {\mathit{\text{area}}}_R$) based on the roughness contour and then by substituting the $\surd {\mathit{\text{area}}}_R$ in Equation 15, the $∆k$ threshold would be determined as Equation 23.

$$∆k=0.65\times {\sigma }_w\sqrt{\pi \sqrt{{\mathit{\text{area}}}_R}}$$ (23)

$${\sigma }_{w:\mathit{\text{is the fatigue limit}}}$$

The fatigue strength of the AM'd Ti-6Al-4V was lower than the CM'd Ti-6Al-4V at both RT and ET, according to the difference in materials at each temperature, as shown in Figure 5. But at ET, the difference between the AM'd Ti-6Al-4V and the CM'd Ti-6Al-4V was a little less significant than it was at RT. Focusing on the impact of temperature on the same material, both showed a decrease in the fatigue limit from RT to ET; however, the AM'd Ti-6Al-4V decrease was less significant than the CM'd one. For AM'd Ti-6Al-4V, the ratio of the change to the RT fatigue limit (300 MPa) was 17%, and the difference between the RT and ET fatigue limits was 50 MPa (=300–250 MPa). The difference for the CM'd material was 150 MPa (=625–475 MPa), with a 24% ratio to the fatigue limit at RT (625 MPa). This shows that the CM'd Ti-6Al-4V fatigue properties were somewhat sensitive to temperature, but the AM'd ones were rather insensitive to temperature. The mechanism of the fatigue failure in the AM'd Ti-6Al-4V is attributed to this. The defect is the primary component of the AM'd materials. The increase in ductility of the material, which is often shown by the decrease in hardness, causes the stress concentration sensitivity to be less sensitive at ET.⁵⁰

The low fatigue strength of the AM'd Ti-6Al-4V is attributed to the initial defects that are unavoidably created during the AM process based on the results of the fatigue tests and the inspection of the fracture surface. Murakami proposed Equation 5, which is empirically demonstrated to correspond well with experimental results, taking into account the aforementioned and determining constant coefficients in the equation to fit the experimental results. The applicability of Equation 5 at both RT and ET was reported in the limited papers. Hardness (HV) and the largest defect size inside the sample's risk volume ( $\sqrt{\mathit{\text{area}}})$ are the two quantities needed for Equation 5. Using the extreme value statistics from Murakami's study, it is possible to calculate the $\sqrt{\mathit{\text{area}}}$.

Based on the S-N data (Figure 5) of AM'd Ti-6Al-4V in the IT-RBF test by Kakiuchi et al.⁵⁰ and Masuo et al.,⁴¹ Figure 9 depicts the estimation for the $\sqrt{\mathit{\text{area}}}$ using the extreme value statistics mentioned in Section 1 and 2. The greatest defect size in the AM'd Ti-6Al-4V fatigue specimen was calculated from this method to be 85.1 μm. Using hardness at each temperature and the area obtained as shown in Figure 9, Equation 5 projected fatigue limits at both ET and RT. The outcomes of the anticipated fatigue limit values are displayed in Table 1. The ratio of the difference to the experimental value was 7%, and there was a discrepancy of 21 MPa between the anticipated value and the experimental value. Similarly, for the predicted and estimated ET fatigue limits, the ratio of the difference to the experimental value was 2%, and there was a 6-MPa difference between the anticipated value and the observed value. The error ratio was under 10% at both temperatures, and the predicted and experimental fatigue limits closely matched each other. This demonstrates unequivocally that Murakami's Equation 5 applies to both the fatigue limit prediction at RT and ET if hardness is evaluated at the same temperature.

TABLE 1. Experimental and predicted fatigue limit based on Murakami's model and extreme value statistics of AM'd Ti-6A-4V IT-RBF.⁵⁰

Item	RT	ET
Vickers hardness	351	256
Experimental fatigue limit	300 MPa	250 MPa
Predicted fatigue limit	321 MPa	256 MPa

In the extensive work that Beretta and Romano⁸¹ have done about Kitagawa diagrams, the fatigue limit for wrought and AM'd Ti-6Al-4V based on the size of defects has been determined. Even though the studies that were used did not focus on high-temperature RBF, they do provide a good viewpoint on the trend of material fatigue limits based on the size of defects. Unfortunately, there are relatively few resources for AM'd metals in IT-RBF. As a result, by referring to the work published by Beretta and Romano,⁸¹ it is possible to forecast that the behavior of AM'd metals in high temperatures will follow the same pattern.

The Kitagawa diagram can explain the significant amount of scatter that is present across the various fatigue datasets and provides a very excellent overview of the trend in the fatigue parameters concerning the defect size for both materials (as seen in Figure 10). In addition, the outcomes of the AM procedures are extremely comparable to those of conventional manufacturing techniques (such as casting and forging), and in certain cases, they are even superior.

When applying the El-Haddad formulation, which was presented in Equations 19 and 20, and taking into account a microstructural length of roughly 200 μm for Ti-6Al-4V, it is possible to provide an accurate description of the mean trend of the diagram. The few pieces of data that can be found in the scientific literature on the size below which the defects are considered to be harmless⁸² are in agreement with the Kitagawa diagrams that can be found in Figure 10.

About Murakami's model (Equation 15), Figure 10 illustrates the range of defect sizes across which a slope of 1:3 best characterizes the $∆k_{\mathit{th}}$ behavior. In the case of Ti-6Al-4V, the range for this is between 80- and 600- $\mathrm{\mu m}$ area. It is important to note that the scatter along the $∆k_{\mathit{th}}$ the diagram is less than along the $∆{\sigma }_w$ diagram as a direct result of the accurate calculation of the SIF using Equation 15 for the various data points. The strong influence that defects have on AM parts is supported by other data found in the scholarly literature.

Nickel alloys

Radhakrishnan et al.⁶² recently researched the behavior of AM'd Inconel 718 in IT-RBF. Figure 11A is a representative image that was obtained by X-ray micro-computed tomography (XRMCT) that shows the porosity distribution in three dimensions in the bulk material. The size distribution of the pores is depicted in Figure 11B, along with their respective aspect ratios ( $\lambda$). An aspect ratio is defined as the ratio of the smallest ellipsoid diameter (2a) to the largest ellipsoid diameter (2c). The majority of the pores have (2c) that varies between 8.1 and 80  $\mathrm{\mu m}$ with ( $\lambda$) range of 0.7–0.9. The presence of an irregular morphology leading to a value of ( $\lambda$) less than 0.5 in pores that are larger than 80  $\mathrm{\mu m}$ is evidence that the pores are lack of fusion (LOF). In Figure 11A,B, the pores that have (2c) that are greater than 100  $\mathrm{\mu m}$ are anticipated to be important for fatigue life are highlighted.

Figure 5 shows the stress amplitude ( ${\sigma }_a$) versus number of cycles to failure ( $N_f$) plots for the results of the HCF tests performed at RT and ET. At RT, ${\sigma }_f$ equals 325 MPa, or roughly one-third of ${\sigma }_u$ (UTS strength), Solberg et al.⁸⁵ reported $\frac{{\sigma }_f}{{\sigma }_u}~0.26$ for runout cycles $2\times {10}^6$ in unnotched conditions. Similar to this, Wan et al.⁸⁶ linked a low ${\sigma }_f~240$ MPa to the existence of large, localized pores created during the LB-PBF process. However, utilizing ultra-high frequency (1 kHz) resonance-based fatigue testing, Yang et al.⁸⁷ reported substantially higher ${\sigma }_f~480$ MPa. The ${\sigma }_f$ achieved in the work by Rahakrishnan et al.⁶² and those previously published are significantly lower than 480 MPa, which may be due to the fact that metals' ${\sigma }_f$ can be increased by fatigue loading at extremely high frequencies, especially if tests are conducted in an oxidizing atmosphere.⁶² The S-N curve shown in Figure 6 reveals that ${\sigma }_f$ at ET is approximately 250 MPa. For comparison, there is only one reported experimental data on the ${\sigma }_f$ of AM Inconel 718 at high temperatures (about 600°C).⁸⁸ ${\sigma }_f$ in Figure 6 is only 30 MPa lower than σf of peak-aged DED Inconel 718 specimens that has been reported by Yu et al.⁸⁸ Using ultrasonic resonance-based fatigue testing (20 kHz) at 650°C, ${\sigma }_f$ of 450 MPa was found for the peak-aged LB-PBF Inconel 718 specimens. In the study by Radhakrishnan et al.,⁶² the observed decrease in ${\sigma }_f$ at ET relative to RT coincides with the decrease in ${\sigma }_u$; therefore, $\frac{{\sigma }_f}{{\sigma }_u}$ stays constant at 0.29. A lower strain hardening exponent (n = 0.19 at RT as opposed to 0.15 at 600°C) may make the beginning of short fatigue cracks (SFCs) at high stress concentration sites like pores easier, even when ${\sigma }_y$ (yield stress) does not decrease considerably at ET.

$∆k$ (as the driving force for crack growth) controls the propagation rate of fatigue cracks. The crack may be stopped by plasticity, roughness, oxidation-induced closure, or a combination of some or all of those mechanisms if $∆k$ is less than a critical threshold value, which is typically attained for a crack development rate ( $\frac{\mathit{da}}{\mathit{dN}}$) of ${10}^{-10}m/\mathit{\text{cycle}}$. This suggests that even in the stress-life strategy, where the components already have preexisting defects that make crack nucleation simple, the proximity of crack tip $∆k$ to $∆k_{\mathit{th}}$ might result in an extended fatigue life (such as the LOF pores in the alloy examined here). Figure 12 provides schematic representations of applied bending fatigue load concerning pores.

The crucial defects in Figure 11 of LB-PBF Inconel 718 XRMCT results are the LOF pores with $\lambda <0.5$. For instance, Figure 13 displays the fractography of a failed specimen with a LOF pore that has dimensions of $2\mathrm{c}~288\mathrm{\mu m}$ and $2\mathrm{a}~178\mathrm{\mu m}$ at RT RBF test. As schematically shown in Figure 12, these LOF pores are near the surface, with their principal axes extending at an angle of $\phi$ with the specimen's central axis. $∆k$ under cyclic loading is determined by linear elastic fracture mechanics (LEFM), which is as follows:

$$∆k=Y.∆\sigma .\sqrt{\mathit{\pi a}}$$ (24)

The assumption of $Y=0.65$ generally be used to justify the Murakami model's estimate of the fatigue limit from surface defects Equation 24 is the same definition as Equation 15. Therefore, attention must be made to compute form factor values more accurately when sharp defect, such as a LOF pore, is present. The Irwine⁸⁹ model could be applied as follows to develop the SIF factor around an ellipsoid defect:

$$∆k_{\mathit{\text{Irwin}}}=\frac{∆\sigma .\left(R-d\right).\sqrt{\mathit{\pi a}}}{\Phi .R}\left({\mathit{Sin}}^2\phi +{\lambda }^2{C\mathrm{o}s}^2\phi \right)$$ (25)

$$\Phi ={\int }_0^{\pi /2}\sqrt{{\mathit{Sin}}^2\phi +{\lambda }^2{\mathit{Cos}}^2\phi }\mathit{d\phi }$$ (26)

where d is the distance between the pore and the surface, R is the radius of the hourglass specimen, and $\mathrm{\Phi }$ is an elliptical integral calculated between 0° and 90° for a pore with an aspect ratio ( $\lambda$). Since the highest stress concentration would always be on the surface, the Irwin definition provides adequate adjustment for RB conditions. Based on this equation, the SIF would decrease as the defect's distance from the surface increased. The direction of the crack's propagation and the shape of the inclusion are two additional factors in this equation, meaning that as the angle ( $\phi$) is increased, the crack's initiation and propagation may not always occur at the inclusion's tip. In other words, the direction that allows the pore to become more symmetrical (i.e., $\lambda \to 1$⁸⁹) is the direction in which the fatigue crack will nucleate and expand. Additionally, the ideal direction of the crack propagation under fully reversed fatigue loading, such as RBF, is often toward the center of the sample, or the mode-I direction. The same phenomenon was modeled by Schönbauer et al.⁹⁰ based on El-Haddad short and long crack model.⁷⁷ They demonstrated that the start of cracks from surface voids would always be a function of SIF around the voids and that it would typically occur from the mouths of voids rather than the tip of those.

3.2.1 Nickel alloys

It can be considered that the phenomena that occur in nickel-based alloys are comparable to those that occur in titanium and aluminum, even though nickel-based alloys have unique properties such as rafting, the carbide effect, and gamma phase behavior. Because chromium is particularly sensitive to the reaction with oxygen and carbon, the behavior of Inconel alloys as Ni-Cr-based alloys is very intriguing when they are heated to high temperatures. The structure of Inconel 718 is a face-centered cubic (FCC) solid solution of nickel combined with iron, chromium, and molybdenum. The precipitate phases that occur in Inconel 718 are denoted by the symbols γ′ and γ″, respectively. Fine-scale precipitates like γ′ and γ″ allow for high strengths to be maintained even at high temperatures. γ′ is a metastable phase with a body-centered tetragonal structure of Ni₃Nb, whereas γ″ is an ordered FCC structure of Ni₃(Al,Ti).⁹¹

The grain size of the superalloy is determined by the grain boundary precipitation phase. At high temperatures (1290–1830°C), the metastable γ″ phase transforms the γ′ phase, which results in the formation of thin plates that elongate into the stable phase of orthorhombic Ni₃Nb. Precipitated at the grain boundary are boron, carbon, and zirconium, all of which have atomic radii that are significantly lower than that of nickel.⁴⁹ In materials that contain many phases and allotropies, the fatigue behavior and, as a consequence, the S-N curve will look different at RT and higher temperatures depending on the history of the material's heat treatment.

The majority of the carbides found in Ni-based superalloys are MC carbides that are rich in titanium and molybdenum: M₆C carbides that contain Ni, Co, Mo, and Cr, as well as M₂₃C₆ carbides that are rich in Cr. The eutectic MC carbides that are created during the casting solidification of the superalloys often have the shape of irregular blocks and have incoherent interfaces with the matrix.⁹² In addition, the MC carbides tend to break down at high temperatures (more than 800°C) as MC + γ → (M6C, M₂₃C₆) + γ′.

In most cases, the two carbides M₆C and M₂₃C₆ will precipitate at the borders of the grains. Grain boundary carbides that have the right size and the right shape can effectively prevent the sliding of grain borders, which in turn helps to improve the strength of the alloys. However, continuous carbide chains have the potential to encourage the development and propagation of microcracks, which can ultimately result in an early failure of the alloys.

As can be seen in Figure 14A, the source of the crack was situated beneath the surface.⁶² The crack propagation resulted in the formation of a significant number of fatigue stria patterns. As illustrated in Figure 14, instantaneous fracture occurred when the crack propagation went beyond a particular extent. This was caused by a sharp increase in stress in some parts of the alloy, which led to the crack. It was found (Figure 14B) that the cracks in the alloy were most prevalent at the eutectic boundary and the trifurcate grain boundaries of the wrought alloy. In addition, it was discovered by Yu et al.⁷¹ that there was a significant quantity of carbide chains in the area surrounding the fractures in wrought materials, and we expect the same phenomenon in AM'd materials. This finding suggested that the continuous distribution of carbides along the grain boundaries played a role in the beginning and spread of the cracks.

Even though Radhakrishnan et al.⁶² did not mention the presence of carbides in AM'd Inconel 718 IT-RBF, we anticipate precipitates to be present at the material's grain boundaries given the quick solidification process of AM. As a result, the main crack initiation sites in RBF would be these carbides and surface defects as well. It could be a good hypothesis to consider that in high-temperature RBF, the crack will initiate from inclusions and porosities, especially those that are surrounded by carbide chains (or at least connected to brittle carbides). Considering the stress gradient from the surface to the neutral center of the specimen in the RBF test, as well as the effect of porosities and inclusions in the crack initiation sites, these carbide chains in the surface might operate as localized brittleness points for dislocations to accumulate, which will accelerate crack initiation from inclusions and porosities at high temperatures.

Titanium alloys

It has been proven that even in heavily porous structures, in the RBF test, the fracture began from the surface or beneath the surface.⁹⁶ This information can be found in the study that Horke et al.⁹⁷ have conducted on the effect of sintering conditions on the fatigue behavior of titanium alloy. The fracture surfaces of both AM'd and wrought materials have been shown in Figure 16. In both cases, the crack starts near the surface because of surface inclusions and porosities. Similar results in the AM'd specimen have been found by Masuo et al.,⁴¹ and they refer to this figure and the fact that the porosities and inclusions in the surface (and beneath the surface) act as the fracture starting locations. They investigated two different kinds of AM'd Ti alloys using EBM and DMLS to study their fatigue behavior and found that surface inclusion is where cracks begin to form.

As a result, the fracture and crack behavior on a macroscopic scale is quite straightforward to understand. On the other hand, the behavior of fatigue at the microscopic scale is dependent on the behavior of the microstructure under high temperatures. Both Burtscher et al.⁹⁸ and Wu et al.³³ have demonstrated that in Ti-alloys with a high amount of Al (40%) the $\raisebox{1ex}{${\alpha }_2$}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right.$ colonies and β, γ phases play a significant effect in the start and propagation orientation of cracks. The microstructure is mostly made up of $\raisebox{1ex}{${\alpha }_2$}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right.$ lamellar colonies, as seen in Figure 17. These colonies are bordered by ${\beta }_0$ (bright contrast) and $\gamma$ (dark contrast) phase mixtures. The following is a condensed version of the sequence of phase transitions that this alloy experiences, as well as its two transition temperatures:

$$\begin{array}{c}\beta \stackrel{1400-1410°C}{\Rightarrow }\beta +\alpha \stackrel{1225-1230°C}{\Rightarrow }\alpha +\beta +\gamma \stackrel{.}{\Rightarrow }{\alpha }_2+{\beta }_0+\gamma \\ \beta \mathit{\text{transition}}\gamma \mathit{\text{solvation}}\end{array}$$

In the AM process, the process is not in an equilibrium situation; as a result, it has some residual phases, which were discussed earlier. This can be shown by referring to the Ti-Al phase diagram.⁹⁹ Intermetallic phases such as Ti₃Al, TiAl, and others would be produced at temperatures distinct from one another as the temperature rose and fell. In higher temperatures, it has more time and energy; therefore, the dissolved aluminum would have to precipitate on the border of the grain in phases with a large amount of Al to reduce its structural distortion energy. Also, Yang et al.¹² demonstrated that as the temperature rose, the phase underwent recrystallization at the grain's boundary. This occurred because the border of the grain is the center of energy when the temperature is high, and all of the dissolved atoms attempt to immigrate there. Therefore, Al concentration would be different in two phases of the lamellar structure of $\raisebox{1ex}{${\alpha }_2$}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right.$ colonies. As it has been stated previously, a difference in the structure parameters (coherency) could be a center of stress intensity as well as the site where cracks first start to form. Burtscher et al.⁹⁸ demonstrated that the lamellar colonies $\raisebox{1ex}{${\alpha }_2$}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right.$ are the site where the crack first begins, and the micro-crack would progress along the lamellar interface until it reached the eutectic grain boundary (Figure 17A), which is the interface between three distinct phases.

In a different study by Nakatani et al.⁷⁹ and Kakiuchi et al.,⁵⁰ they reported that in the RBF test of an AM'd titanium alloy (Ti-6Al-4V), crack initiation had been caused by surface defects (porosities beneath the surface and inclusions). Accordingly, from the perspective of AM, surface defects are the sites where cracks begin to form, whereas, from the perspective of wrought metals, some colonies and secondary phase boundaries may be initiation sites. Since there is competition and sometimes intensification of lamellar colonies microstructural crack initiation sites (Figure 17A) and porosities sites (Figure 17B), it is impossible to precisely categorize the crack mechanism in AM'd titanium alloys with colonies microstructure in high-temperature RBF tests. However, referring to the discussed reports, generally speaking, it would be a good assumption that in high-temperature RBF tests, the crack would initiate from the surface (or beneath the surface) from porosities and inclusions, especially those that have an interface with lamellar colony structures.

Similar results have been published by Wu et al.³³ and could be seen in Figure 18A that shows that the crack initiation in the tensile mode (perpendicular to the crack mode) has been initiated from colonies and grain boundaries. A schematic of the structure that is identically comparable to the one reported by Yang et al.¹² could be found in Figure 18B. The behavior of the micro-cracks and the growth of the cracks followed the theory that was presented.

According to the schematic in Figure 18B, porosities and inclusions, particularly those that are located in the colony microstructure of eutectic points in the grain boundary, are more preferred sites for crack initiation in AM'd titanium alloys. The starting crack from preferred locations in the surface may spread in the interlamellar direction in transgranular mode, according to Burtscher et al.⁹⁸ mechanisms for crack propagation in RBF tests of titanium alloys in wrought materials. Therefore, if the crack were to develop in a transgranular manner, a lamellar structure might be preferred. As a result, in AM'd titanium alloys, internal lamellar structures in grains would be crucial for crack propagation direction, while the involvement of colony microstructure would be crucial for crack initiation in the presence of surface inclusions and porosities.

Aluminum alloys

Crack initiation, crack propagation, and final fracture are the three regions that make up a fracture surface in aluminum alloys, just as they do in other kinds of materials. Although in most RBF papers the fatigue behavior is only investigated based on S-N curves and crack behavior based on fracture mechanics, here, it would be discussed a little more based on phase diagrams and allotropies during elevated temperatures. In Figure 19, a portion of the phase diagram for aluminum as well as other materials that are primary solutions has been displayed. Let us look at the 7075-T7351 aluminum alloy that Yang et al.¹⁰⁰ researched and found to have a chemical composition (in wt.%) of 5.89 zinc, 2.48 magnesium, 1.59 copper, 0.22 chromium, 0.06 iron, 0.02 titanium, and the remaining weight in Al. Concerning Figure 19, the representation of our material in each diagram is represented by a green line. Following the AM process, the first solid nucleation will have the largest amount of Al since the cooling rate of the melt pool is high. Subsequently, as the solidification layer thickens, the percentage of impurities (soluble metals, Mg, Zn, etc.) will increase.

As a result, throughout the AM process, the melting point will always be a function of the solidification face, and after the process is complete, the microscopic structure will have a modest segregation and impurity profile.^101-103 Assuming that the melt pool is in the melting zone at 680 $°\mathrm{C}$ and that rapid cooling has taken place, it could be predicted that the final thin layer of the solidified crust will have close to 10% copper (Figure 19A). The final thin layer probably will have close to 12% zinc, and the final thin layer will have close to 0.1% magnesium (Figure 19C if we neglect the interaction effect of the activity of these elements). These predictions are based on the assumptions that the melting zone is at 680°C and that rapid cooling has taken place. In light of the preceding, concerning the equilibrium phase diagram, it can be deduced that under conditions of non-equilibrium solidification (such as the AM process), the material structure will have the concentration variety of a solid solution. We make an approximation of the ultimate structure by referring to two different phase diagrams, but this is not a completely accurate method because the impurities influence one another's activities. Therefore, in the three-phase diagram shown in Figure 19D, the intersection of the green lines represents the alloy composition, and the intersection of the red lines represents the eutectics that follow thin layers in the solidification process. It would appear that the final structure will contain both intermetallic phases, such as AlMg₄Zn₁₁ and Al₆Mg₁₁Zn₁₁, as well as precipitants containing Fe. In most cases, after homogenizing and performing any other kinds of heat treatment, it would be reheated and given some time to allow the composition to return to normal. Figure 20 is a schematic of the effect of solidification segregation and passing heat processing on final AM products. The fact that the specimen undergoing the RBF test at a high temperature will go through heat treatment once again is demonstrated in Figure 20.

Referring to Figure 20, when the material is subjected to an IT-RBF test, it would heat up again and have time (for example, 10⁶ cycles until fracture), so as a result of the presence of temperature and mechanical stress, atoms can migrate as well as the time and energy to do so. Because the solid solution has a profile concentration during the AM'd sample, the material will attempt to reject the saturated solid solutions impurities to lower the lattice distortion energy. As a result, depending on the phase diagram, the impurities will migrate to the grain boundaries as the temperature is raised. According to the phase diagram (D) in Figure 19, GP zones (MgZn₂) can form in Al-7075 when the temperature in the RBF test is increased. The presence of these GP zones in the Al alloy has been associated with resistance against fatigue propagation as well as an improvement in the way fatigue behaves when exposed to high temperatures.^{104, 105}

The microstructural properties that are pertinent for fatigue loading conditions must be discovered to explore the fatigue behavior of AM'd alloys for elevated-temperature applications. The stages of fatigue damage buildup, crack initiation, and small and long fatigue crack propagation of a specimen or component's fatigue lifetime can be affected by diverse microstructural properties in conflicting ways.¹¹⁰ For instance, in AM'd metallic materials, fine grains and other microstructural characteristics might contribute, positively or negatively, to crack initiation and the endurance limit as well as crack propagation, especially in the near-threshold regime.¹¹¹ The heterogeneous microstructural characteristics produced by AM offer a variety of features that localize strain and act as precursors to the onset of fatigue crack.¹¹² These features/sites are illustrated in Figure 21 and summarized below as specified by Michi et al.¹¹³:

• Solidification and gas porosities, typical fatigue crack initiation features in cast alloys,¹¹⁴ play significant roles in defining the fatigue behavior of AM'd alloys at increased temperatures. Awd et al.¹¹⁵ demonstrated that increased porosity in Scalmalloy generated by DED techniques caused a 30% decrease in fatigue strength compared to LPBF-processed material. Moreover, modeling of fatigue by Haridas et al.¹¹⁶ in AM Al–Cu–Mg–Zr showed that the amount and size of porosity strongly influenced the fatigue life, with a distribution of larger irregular-shaped porosities decreasing component lifetime. AM-induced keyhole porosity, if present, could also create potential challenges for high-temperature alloys containing volatile Mg, Zn, and/or Mn, whose vaporizations are expected to increase the tendency for keyholing.¹¹³
• Inadequate solidification of metal powder can lead to the creation of agglomerates, which can become stress raisers that cause fatigue cracks.¹¹⁶
• When the common strain localizing features such as porosity is eliminated by post-processing treatments such as hot isostatic pressing, other microstructural defect precursors that form during cyclic loading of AM'd alloys, such as persistent slip bands, are predicted to initiate fatigue cracks.¹¹²

Even though we do not know much about IT-RBF of AM'd metals, Figure 21, developed by the authors, may schematically show the contributing factors to the IT-RBF failure in AM'd materials. Inclusions and porosities, which ideally wouldn't be present in wrought materials, could be extra areas where cracks begin to form in AM'd materials at RT. All surface defects would be potential crack initiation sites, especially under high temperatures where grain boundary slippage, atomic migration, phase allotropes, and other phenomena would lead to the emergence of new preferred crack initiation sites. This schematic may show a general mechanism for how temperature affects crack initiation sites by increasing porosities and slipping grain boundaries, but depending on the material, there may be other parallel phenomena that also have a relationship between temperature and crack initiation sites, such as the carbides effect in nickel superalloys, intermetallic phases in aluminum alloys, phase allotropes in titanium alloys, and so on.

According to the research of Radhakrishnan et al.⁹⁵ on Inconel, Kakiuchi et al.⁵⁰ on Ti-6Al-4V, and the hypothesis stated in Equation 13, it is conceivable that Murakami's model might include the extreme value statistics technique with high-temperature parameters to predict fatigue. The $\sqrt{\mathit{\text{area}}}$ for an AM'd sample could be calculated at RT using the extreme value statistic. A hardness test at various temperatures would be used to estimate the hardness-temperature relation for Equation 13. Using this strategy, we took into account the fact that, while it would affect the grain size, the temperature did not affect the size of the surface defects. The hardness function, which depends on temperature and is more easily quantifiable than the fatigue test, would take into account the only effect of temperature on the fatigue test.¹¹⁷ The Arrhenius equation typically describes the relationship between hardness and temperature, and a high-temperature nano-indentation test can be used to assess the precise amount of that relationship. Utilizing Table 1 and the Arrhenius equation for hardness, we were able to approach the fatigue limit at high temperatures using experimental data. The fatigue limit estimation would therefore be performed using the conventional Murakami's model (even though it has been developed for steel) based on defect size data (at RT) and the hardness-temperature function. Figure 22 is a summary of the mentioned method for fatigue limit estimation and validation of experimental data from the IT-RBF test of Ti-6Al-4V (Table 1).

Even though this approach has been shown to accurately predict the Ti-6Al-4V fatigue limit at high temperatures, further testing is still required for AM'd metals. In our opinion, AM'd metals' fatigue behavior at high temperatures would entirely match Murakami's equation's general form, but for the most accurate estimation, some of the equation's constants would need to be changed by data fitting. Also, we employed the conventional estimation of the hardness-temperature equation in our first hypothesis, even though it would be more accurate if it were to be developed based on additional experimental data fitting. In reality, we are optimistic that by adapting Murakami's technique for steel alloys at RT to elevated temperatures, we will be able to do so.

4.1.1 S-N data behavior

The two factors that have the greatest impact on fatigue life are temperature and surface (mentioned before), especially in the RBT. Figure 23 presents the S-N curve that was determined in the investigation for untreated and carburized 316L specimens tested at various temperatures. The effect of temperature and surface treatment on S-N curves is made abundantly clear by these data, which were recorded by RBF under ITF conditions.

The most intriguing element of the RBF investigation is the surface, both as a concentration of the highest stress that can be measured with the RBF test and as an indication of crack start caused by defect distribution. One of the examples of an unexpected fracture surface in simple steel alloys is shown in Figure 24, which depicts the RBF test carried out at RT. Because the material in this specimen has been subjected to a surface hardening technique (laser cladding), the crack that developed in the specimen started in two separate places before it finally broke in the middle of the specimen.^{118, 119} In Figure 24, the red arrows point out the fusion line discontinuity as well as the oxide layer that has become trapped in the melt pool. Referring to the previous section, these inclusions are the stress concentration zone. Furthermore, according to Murakami's model, this defect may affect fatigue strength because of the $\sqrt{\mathit{\text{area}}}$ parameter.

4.1.2 Carburizing effect

Another study conducted by Liu et al.²⁸ on carburized 316L austenite stainless steel indicated that the improvement in fatigue endurance brought about by carburizing was close to 15% even when the material was kept at RT. Figure 25 illustrates the distribution of carbon content, residual stress, and nano hardness in the depth of the specimen surface. Carburizing causes the surface of the specimen to become harder than the inside. Because of the different carbon solutions in the solid state of the structure, the residual stress would increase as the depth increased, even though the carbon content and hardness were shown to decrease with increasing depth (referring to diffusion Fick's law, the carbon content would decrease). Because of this, it is anticipated that the surface's fatigue qualities would improve as the hardness of the surface increased, thanks to Murakami's model. Furthermore, as the residual compressive stress increases, the fatigue limit is expected to improve. These two assumptions are accurate up until the point when inclusion is present; once inclusion is present, the behavior of the material becomes competitive in terms of stress concentration and local yielding. Results have demonstrated that an increase in carburizing time and temperature, concerning Fick's low, will cause a rise in the diffusion depth of carbon, which, in turn, will cause an increase in the probability that impurities will precipitate. Because there is a significant amount of chromium at high temperatures, chromium carbides have the potential to form. However, because chromium carbides have a low coherency with the base material, the stress concentration around them would eventually reach the threshold stress for the initiation of cracks. This is especially correct in austenite 316L stainless steel. Figure 26 displays the scanning electron microscopy (SEM) of the fracture surfaces of both untreated and carburized 316L stainless steel. It is abundantly visible that carburizing caused all of the cracks to start forming beneath the surface and at a distance of more than one hundred microns from the surface (exactly after the carburized layer Figure 25). According to the findings of the EDS analysis, the commencement of the crack in the carburized sample did not take place due to the presence of any carbides; rather, it took place due to the presence of oxide impurities. Because of this data, it could be deduced that the stress concentration around the carbides is lower than that surrounding the oxides.¹²⁰ To put it another way, we can suppose that the coherence of the carbides is higher than that of the oxides.¹²¹

The fatigue strength of carburized specimens has been improved by the presence of carburized case at the surface region. In addition to this, when the high compressive residual stress is generated in the carburized case, the tension stress that is self-balanced with the compressive residual stress is generated at the substrate near the boundary between the carburized case and the substrate. This is because the tensile stress is the result of the carburized case being subjected to high compressive residual stress. After the superposition of residual stress and applied stress, the area of the subsurface that is closest to the boundary between the carburized case and the substrate experienced the greatest amount of stress (for an illustration of this, see Figure 27, the dashed line of stress distribution). This is where the fatigue crack will begin to form. In addition, it has been reported that at RT, there is a significant difference in the elastic–plastic properties of the base material and the carburized case. This difference in properties may be responsible for the strain inconsistency that occurs near the boundary during the fatigue loading process. It is clear from looking at Figure 27 that the process of fracture for the untreated specimen at the increased temperature is the same as the mechanism for the specimen at the ambient temperature. Even still, the region of highest stress appears to be close to the interface between the carburized casing and the substrate. The local fatigue strength is readily exceeded by the stress when it is applied at a reasonably high level, which results in the onset of fatigue cracks near the boundary as well. Because the carburized layer has a strengthening effect on the whole specimen, it is difficult to surpass the local fatigue strength of the material even when it is subjected to the maximum amount of stress. This is because the applied stress is relatively low. However, there are inclusions in the materials, and Figure 26 demonstrates that these inclusions are more brittle than the base material. At higher temperatures, the difference in mechanical properties between the carburized case and the substrate narrows, while the difference in mechanical properties between the substrate material and the inclusion widens. This creates a stress concentrator and results in lower local fatigue strength at the location of the inclusion. Because of this, cracks caused by fatigue start at the internal stress concentrators, which are brittle inclusions, even when there is a little amount of external stress and a high temperature.

4.1.3 Effect of microstructure

Inclusions and defects can have a significant impact on the behavior of fatigue. From both Murakami's and fracture mechanics' points of view, inclusions are the stress intensity site that can cause the crack initiation threshold strength to be exceeded. This means that the inclusions may exert their impact in a variety of different ways. When viewed from a metallurgical perspective, inclusions can be thought of as a point of incoherence in the structure, which causes some distortion in the way energy is distributed throughout the structure. The resistance of metals to fatigue can be adversely affected by any kind of deformation in the structure's configuration. Impurities such as oxides, carbides, nitrides, and other similar compounds caused a second phase to form in the base metal. This indicates that the impurities distorted the configuration and energy of the metal structure. Because of this, depending on the viewpoint and the application, the impacts of inclusion on fatigue would be explained, and efforts would be made to optimize them. Some of these configuration discontinuities generate beneficial effects, while others make the properties of the metal worse. There is a complex relationship between crack initiation, crack propagation, and the effects of defects on the material.

4.1.4 Porosities-vacancies-voids

Murakami demonstrated that the size and shape of an inclusion (regardless of whether it was a porous or second phase) affected the fatigue behavior of the material. That is, the Murakami model considers the net area of inclusion to be an effective parameter on fatigue, whereas fracture mechanics consider stress intensity around porosity to be an effecting parameter on fatigue strength.³⁸

In the other investigation, conducted by An et al.,¹²² the oxide inclusion and the titanium nitride inclusion in SAE52100 bearing steel were found to be the places where the fracture first began. The crack nucleation site around the inclusion is seen in Figure 28. Porosities can be found in a variety of areas on the fracture surface. It was previously mentioned that the porosities created during the manufacturing process of the material can serve as an initiation site for the crack nucleation process. Additionally, porosities can be found in the area of the final fracture. Porosities have the potential to develop in this scenario because of the micro void coalescence (MVC) mechanism that occurs during fracture.^{17, 123, 124} Figure 29 illustrates some of the final voids that can be found in the coarsened final fracture area of bearing steel.¹²³

Vacancy creation in crystalline structures is an additional phenomenon that frequently occurs in studies of high-temperature fatigue.¹²⁵ The rise in temperature causes an increase in these vacancies, and their movement and accumulation at grain borders, where they cause macro-voids, also contribute to this increase (in porosities). This process is the primary cause of the formation of porosity in the area surrounding the ultimate fracture when the temperature is high.¹²⁶

4.1.5 Inclusions

The fracture mechanics and the Murakami model both have a fair description of the influence that the shape and size of inclusions have on fatigue. On the other hand, the second phase has a different effect on the mechanisms of fatigue and cracking. It has been noted that certain inclusions, such as second-phase precipitates, can affect fatigue properties. In the situations that were discussed in the previous sections, it was demonstrated that carbides, nitrides, bromides, and other elements in steel alloys all displayed various effects. Crack initiation and propagation can be affected by certain lamellar structures that evolved as components of secondary phases,^{12, 15} certain allotropies such as GP zones in aluminum alloys,^{104, 127} or the gamma phase in titanium alloys,³³ and the perlite structure of steel. The phases, grain boundaries, structure and dislocation slippage system, and inclusion coherency all have very complex correlations with one another. In the investigation of the fatigue of steel alloys, the secondary phase effects are not separately significant when compared with memory-shaped alloys and Ti, Ni, and Al alloys. This is because there are not many phases of allotropy or very strong ductile inclusion in common steels, and the majority of the effect of inclusion is studied based on the hardness and $\sqrt{\mathit{\text{area}}}$ model. Comparatively, the memory-shaped alloys and Ti, Ni, and Al alloys are more sensitive to secondary phases than steel alloys.

4.2.1 S-N data behavior

It has been demonstrated by Horke et al.⁹⁷ that the fatigue strength and limit of Ti-6Al-4V alloy decreased by close to 30% in standard casting samples as the temperature increased from 25°C to 450°C. Additionally, the fatigue behavior of metal injection molded (MIM) components made of the same alloy and subjected to higher temperatures were examined. The findings of the S-N curves of the cast specimen and the MIM manufactured specimen at increased temperature are given in Figure 30.

Wang et al.³⁴ found the same outcomes in their investigation of the effects of high temperature on the Ti-6Al-4V alloy. They demonstrated that an increase in temperature on an untreated specimen, as well as on a specimen that had been rolled or subjected to some other form of surface treatment, resulted in a decrease in the fatigue strength, and every S-N curve demonstrated a different fatigue limit (Figure 30).

The normalized fatigue strength data by UTS of titanium alloys at various temperatures are shown in Figure 31 of Heneff's comprehensive review paper.¹⁵ The diagram demonstrates that both of the fatigue strength and UTS reduce simultaneously until $700°\mathrm{C}$, after this temperature the fatigue strength decreases dramatically more than UTS. This occurrence happened because of phase allotropy in high temperatures. Wu et al.³³ demonstrated that the lamellar phase of $\raisebox{1ex}{${\alpha }_2$}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right.$ colonies and the $\beta$, γ phase are responsible for the reduction in fatigue life with increasing temperatures, and similar results have been reported by other authors.^{12, 128}

4.2.2 Effect of microstructure

Masuo et al.⁴¹ have shown that the Murakami model may be utilized for titanium alloys with an estimation precision of more than 90% for the fatigue limit. In the research carried out by Tajiri et al.,⁴³ the author improved the Murakami model by boosting the computation of the effective magnitude of defects in the material. They demonstrated that the algorithm developed by Murakami treats both of the defects as if they were a single defect whenever the distance between defects I and II is less than the area of either defect I or defect II. This occurs when the distance between the two defects is less than the area of either defect I or defect II. They upgraded the $\sqrt{\text{area}}$ estimation, and they deemed defects I and II to be independent defects for the commencement of cracks whenever the distance between defect a and defect b in areas I and II was higher than the $\sqrt{\mathit{\text{area}}}$ of one of the areas (Figure 3).

A comparison has been made using a summary of various experiments that have been reported, which can be found in Table 2 as evidence of the microstructure effects of lamellar colonies in Ti alloys. According to Table 2, the fatigue limit does not appear to be significantly affected by variations in temperature, which is not what one would anticipate. It is concluded the effect of temperature on fatigue is a driver of changes in microstructure. It would appear that the microstructure would have a varied lamellar structure and size depending on the temperature. As a result, the microstructure will vary as a direct result of the temperature, which will affect fatigue behavior.

According to Table 2, fully lamellar (FL) structures exhibited significant shifts in their fatigue limits when the temperature rose from RT to 800 $°\mathrm{C}$, whereas nearly lamellar (NL) structures did not demonstrate any significant shifts in their fatigue limits. This phenomenon demonstrates the effect of the microstructure type and its effect on the initiation and propagation of cracks, both of which have been discussed in various parts of this state of the art.

The graphical comparison shown in Figure 32 is based on the presented data in Table 2, which indicates that as the temperature rose, the FL structure exhibited a higher level of temperature sensitivity. In other words, it is possible to make the hypothesis that a change occurred in the microstructure of these alloys as the temperature increased to roughly 800 $°\mathrm{C}$ (referring to Figure 31). Therefore, Murakami's model, which has been applied to other alloys (instead of steel), should be anticipated to have a modifier coefficient that depends on the features of the microstructure. Since the requirement of some prediction models based on the microstructural features of the materials is important for fatigue strength prediction, this quotation might serve as a guide for subsequent research.

4.3.1 S-N data behavior

The fatigue behavior of aluminum alloys when subjected to high temperatures is shown to have a very close association with phase allotropies.¹³⁴ The fatigue behavior of this material, in addition to its other mechanical properties, is significantly influenced by the supersaturated solid solutions (SSSS) allotropies that exist in this alloy.¹³⁵ Additionally, in this alloy, a heat treatment applied to the samples before the fatigue testing can have a significant impact on the alloy's fatigue life. In the study that was done by Tian et al.,¹⁰⁴ they demonstrated that the fatigue behavior of an alloy under 420°C is better than 350°C by using a re-aging process. This is contrary to what would be expected, which is for the structural properties to decrease as the temperature rises, and for the fatigue behavior to become more apparent. They were able to demonstrate that the nano-phase was stable at the high temperature of treatment by demonstrating that the phase continued to exist both before and after the fatigue test. In the context of alloys based on the Al-Cu system, the precipitation sequence is typically understood to go as SSSS → GP zones → θ″ → θ′ → Al₂Cu.¹³⁶

Strzelecki and Tomaszewski¹⁶ investigated Philip and Esin,^{59, 61} Lee et al.,⁶⁰ Esin,⁶¹ Manson,¹³⁸ and Manson and Muralidharan¹³⁹ and their four-point model for finding the best relationship between the axial fatigue test and RBF. Some researchers attempted to find a relation between the results of conventional fatigue tests and those of RBF tests. They indicated that the Lee technique is the best model for describing the relation between axial and rotational fatigue, but they also mentioned that the data on RBF is more scattered than the data from axial tests that can provide us a sign concerning aberrant phenomena regarding fatigue behavior due to certain conditions such as the temperature effect and re-aging phenomena while being tested.¹⁰⁰ In a different study conducted by Kruml et al.,¹⁴⁰ the fatigue behavior of a TiAl alloy containing 48% aluminum was studied at an elevated temperature. The authors employed the Basquin model, the Coffin–Monson model, and a combination model to describe the S-N curve. When the effect of Nb on fatigue behavior is studied, it shows an improvement in fatigue behavior, which could be because of the microstructure allotropies that have been mentioned in Table 2 before.

4.3.2 Effect of microstructure

Due to the low melting point and the high vapor pressure of aluminum, porosity is one of the most common defects found in aluminum alloys and especially in AM'd materials.¹⁴¹ As a result, throughout the manufacturing process, micro porosities in the solid will behave as crack-starting sites in fatigue tests. At RT, the development of pores does not take place during the fatigue test; nevertheless, the pores that were already there as a result of earlier processes will still have an effect. Porosities, also known as voids, form in a material subjected to a fatigue test when it is at an elevated temperature. Porosity can be controlled by temperature and the mechanism of diffusion; in addition, it has a relationship with the composition of the material. The effect of strontium on A356 Al alloy in fatigue behavior has been mentioned in Table 3, based on references. The researchers were able to demonstrate conclusively that the addition of strontium caused the microstructure dendrite to become larger and more rounded (Figure 34) form in both A356 and A356+T6 (heat-treated) specimens and that the addition of strontium caused porosities to become larger and more rounded. Figure 35 shows SEM images of fatigue fracture surfaces from different samples. It was feasible to distinguish between three separate regions: crack initiation, crack propagation, and fast fracture (final rupture). The presence of porosity near the sample surface can be seen in region 1 of Figure 35, which corresponds to the initiation and propagation of cracks (Figure 35A,B). This porosity has most likely served as a site for the nucleation and propagation of cracks, which ultimately resulted in early failure. Observations in nearby areas at higher magnifications revealed the presence of deformation bands and fatigue striations (Figure 35C). Porosity was discovered near the sample's surface, as shown in Figure 35A,B. Based on the findings made so far, it appears that the existence of porosity was the primary factor in the onset of early fatigue failure. It is possible to see cross-cycling marks in these locations, which are what distinguish the fatigue region from other parts.

It would appear, regarding Table 3, that strontium can, in general, increase fatigue properties; yet it demonstrates an opposite response when subjected to heat treatment. In Figures 34 and 36, strontium at high temperatures can improve the microstructure and turn the needle microstructure into a rounded microstructure. As a result, crack initiation will decrease due to a decrease in the amount of stress intensity in the microstructure. However, as demonstrated by the findings, without undergoing heat treatment, strontium is unable to demonstrate a significant change in its fatigue properties; this is the case even though increasing the eutectic area has a negative impact. In addition, as shown in Figure 36, the influence of temperature on the A356 alloy is significantly greater than the effect of Sr; hence, the effect of Sr on fatigue parameters can be disregarded.

Ceschini et al.¹³⁷ have shown that some intermetallic precipitates in the presence of Si in Al alloys can be crack initiation sites in a high-temperature fatigue test. Of course, the effect of inclusions in this alloy is dependent on the properties of the inclusions, and sometimes they have a positive and sometimes a negative effect on fatigue behavior. As it has been mentioned in the previous section, the influence of inclusions or, on the other hand, the second phase depends on the coherency between the base metal and the precipitates. The fracture surface SEM has shown that the crack started initiating from the inclusion that was close to the surface of the specimen. This is depicted in Figure 37. The propensity of magnesium to oxygen is greater than that of other impurities in aluminum alloys, which means that the sites at which cracks first begin to propagate are more likely to be dependent on secondary phases in the presence of magnesium when the temperature is high. The magnesium oxides are extremely brittle, and because the grain boundaries of the material at an elevated temperature are constantly being attacked by oxygen, the magnesium oxides are constantly produced in the crack tip.

$\epsilon :$ equivalent plastic strain.

${\dot{\epsilon }}^{*}:$ dimensionless plastic strain rate.

$T^{*}:$ homologous temperature.

$T_r:$ room temperature.

For the hypothesis made in Equation 30, the distance from the standard condition has been taken into account by the frequency and temperature coefficients, and the material history has been taken into account by the power n that should be calculated by data fitting. Reaching the yield point was used as the basis for determining the fatigue criterion. In Table 4, some of the findings from papers have been provided, and the results of our analysis can be seen in Figure 38. The subsequent study work will be devoting its attention to investigating this possibility. To use Equation 30, it will first refer to Table 4 and perform the calculations necessary to determine the temperature term and the frequency term as summarized in Table 5. The value of n can then be found by applying the logarithm to Equation 30. Table 5 presents the results of the calculations based on assertion Equation 30 that has been used to reference data in Table 4. The yield stress power, denoted by n, can range anywhere from 0.76 to 1.18. However, in the first phase, this n would be utilized for Equation 30 to try to estimate the existing data in Table 5. The correctness of n and the suggested model should be evaluated with tests.

Concerning Figure 39, the n variable in Equation 30 is more sensitive to the multiple factors of frequency and temperature. Even though the overall trend of n has a narrow range, this parameter can have a significant effect on the prediction of material fatigue. It could be classified by the n behavior into one of three areas (Figure 39 and Table 5), and for the most accurate assessment, it should strive to use each n for material with a heat treatment history that is comparable to the others.

For instance, the quantity of n that is considered average in area 1 is 0.874; consequently, if it were applied to Equation 30 for each test in area 1 or any other condition that is analogous, the anticipated fatigue limit would be close to the data that was collected. The same pattern of behavior can be observed in other domains. The most significant objective of this claim is to determine the generalized value of n for each of the many sorts of steel materials.

Figure 40 presents the results of a comparison between the anticipated fatigue life and empirical data. This comparison was generated using Equation 30, which was based on data collected from Table 5. As can be seen, the anticipated data scattering is not that wide and is relatively close to the actual data (especially in a range smaller than 700 MPa). This claim is capable of being developed, and once it is, it will be able to anticipate the fatigue limit at a variety of temperatures and frequencies by using yield stress as a basis.