Effect of R load ratio on fatigue crack growth resistance of steels used in automotive applications: experimental results and use of performance prediction models

This research consisted in comparing the fatigue crack growth (FCG) performance of four HSLA/ AHSS steels used in automotive applications and with different microstructures, and the application of some prediction models for the da/dN versus ΔK traditional sigmoidal curve as a function of the R load ratio. FCG tests were carried out on C(T) test specimens with R-ratios varying between 0.03 and 0.7. Using the original and empirical methodology proposed by Paris and Erdogan to describe the da/dN-ΔK relationship, the results showed significant differences in function of microstructure, and a deleterious effect of R-ratio increase on the crack growth rate. In order to check existing methodologies based on physical considerations for predicting the fatigue behavior of materials and the effect of the R-ratio mainly in the fatigue threshold ΔK th region, the well-known crack closure model proposed by Elber, an approach using two parameters as a driving force for the crack growth proposed by Vasudevan and co-authors and a combination of these two models recently proposed by Zhu and co-authors were compared. The manifestation of crack closure and its qualitatively expected dependence on the R-ratio were verified for the studied steels, but the Elber model was not able to provide a master curve that accurately summarized the effect of the R-ratio on the sigmoidal fatigue curve of steels. The combined use of two critical thresholds, ΔK th * and K max *, for predicting fatigue crack growth according to the Vasudevan model also did not provide accurate results in evaluating the effect of the R-ratio. Regardless of the verified dispersions, there is a connection between the two-parameter methodology and crack closure, hence the model by Zhu and co-authors could be a promising alternative. However, this model also showed significant dispersions and was unable to create a master curve to adequately predict the effect of R-ratio on crack growth. Thus, it can be concluded that this research topic is still open, requiring a more in-depth phenomenological knowledge to predict the effect of the R-ratio on FCG.


Introduction
One of the main challenges of the automotive industry in recent decades is the reduction of the vehicles weight, aiming the new requirements of energy saving and environmental restrictions.Therefore, automobile manufacturers look for materials with conflicting high properties, for example, mechanical tensile strength, formability and fracture toughness, without the need to increase its thickness [1][2][3].Steels are commonly used because they meet these characteristics and are in constant development.Ferrite-Bainite (FB) and Ferrite-Martensite (FM) are examples of dual-phase advanced high strength steels (AHSS) widely used in this sector 1 as a substitute for traditional Ferrite-Pearlite (FP) high strength low alloy steels (HSLA).
Due to the type of mechanical loads involved in the use of materials for the automotive industry, research on fatigue crack growth (FCG) has received great attention of the scientific community [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].Studies carried out to estimate the life of a material under fatigue from concepts of Fracture Mechanics provide sigmoidal logarithmical da/dN versus ΔK curves, where da/dN is the FCG rate and ΔK (= K max -K min ) is the applied driving force for FCG.One of the first relationships proposed to characterize these curves is due to Paris and Erdogan [21], and is presented in Equation 1.This is the well-known "Paris equation", which shows a linear trend between da/dN and ΔK, with C and m being constants of the material.
2/15 Tecnol Metal Mater Min.2023;20:e2767 The traditional Paris equation can be then modified to correlate FCG data with R values, as shown in Equation 5.In this equation, C and m are not the same constants of Equation 1. Elber measured the closure stress intensity in a 2024-T3 aluminum alloy at various load levels and R-ratios, and obtained an empirical relationship, Equation 6, then extended for subsequent researchers to various other materials.This method obviously requires the experimental crack closure determination, represented by K cl , and the knowledge of an expression that correlates the crack closure with the R-ratio.
( ) ( ) ( ) For the highest R ratios, closure was not observed, so ΔK = ΔK eff .When data at lower R ratios are corrected for closure, the R-ratio effect disappears and all data exhibit the same behavior.
The Elber methodology has generated several adaptations to describe the FCG over the past few decades.For example, according to Hudak and Davidson [37], if ΔK th is related to R this relationship leads to Equation 7, where R* is the R-ratio above which closure no longer exerts an influence, and ΔK th * is the ΔK eff for this extreme situation (ΔK th * = U ΔK th ).Equation 7predicts that the threshold stress intensity range varies linearly with R below R* and is constant at higher R ratios.This equation considers that the threshold consists of two components: an extrinsic component that is a function of the R-ratio and is controlled by crack closure (R ≤ R*) and an intrinsic component that is a material property independent of crack closure (R > R*).Equation 7 also predicts that for R = 0, we have: ΔK th = K cl +ΔK th *.

(
) ( ) In a different point of view, Vasudevan et al. [38][39][40][41][42] have proposed a two-parameter driving force approach where ΔK and K max are both required to FCG, without invoking crack closure.In this methodology, two critical intrinsic thresholds, ΔK th * and K max *, need to be reached simultaneously to start the FCG.They point out that their approach is analogous to the "Goodman diagram" for S-N curves, where fatigue life is a function of both stress amplitude and mean stress.Once it is recognized that there are two driving forces for fatigue crack growth, it becomes more appropriated to Two deviations are observed in this linear trend.When da/dN approaches to zero, it is possible to determine the threshold stress intensity amplitude, ΔK th , below which there is no crack growth (near-threshold regime).At high ΔK levels, K max approaches to the fracture toughness K c of the material.
Since the pioneering work of Paris and Erdogan, the effect of R load ratio (or stress ratio), which is defined as the ratio of minimum to maximum load (R = P min /P max ), on FCG has been investigated by several researchers.An important procedure is to adapt the cited Paris equation to a relation with a form of Equation 2, where f(R) is an analytical equation including R as a variable.
The R-effect on FCG is indeed a valuable matter in designing of materials and structures for engineering applications.
The aim of the research on R-effect is rather simple, that is, to collapse all FCG data at various R into one single "master curve" and find the rules that governing the changing of fatigue strength with R.However, the pathway to the issue is not unique; hence several efforts are based on different points of view.
The concept of crack closure proposed by Elber [35,36] has been reasonably successful in correlating FCG data at various R-ratios.Crack closure results in a decrease in the fatigue driving force ΔK due to, for example, roughness and oxidation of the fracture surface in the near-threshold region (ΔK th ) or plasticity at the crack tip in the validity region of the Paris equation.The driving force ΔK is then replaced by an effective stress intensity amplitude, ΔK eff , according to Equation 3. K cl is the stress intensity factor where crack closure occurs and K max is the maximum stress intensity factor.Elber also introduced an effective stress intensity ratio, Equation 4. Effect of R load ratio on fatigue crack growth resistance of steels used in automotive applications: experimental results and use of performance prediction models represent fatigue crack growth rate, da/dN, in terms of a three-dimensional (3D) plot as a function of ΔK, and K max .However, a more convenient 2D representation of 3D behavior can be obtained if cuts perpendicular to the da/dN axis are taken.This methodology generates an "L-shaped" threshold curve that can be considered as a fatigue map: it represents an interplay of the two driving forces needed to enforce a given crack growth rate.As the two threshold values from the model of Vasudevan change with the da/dN crack growth rate, then a crack growth "trajectory map" can be defined following the changes of ΔK and K max as a function of da/dN.These maps then reflect the material resistance to crack growth and changes in the trajectory (more than one L-shape curve for the same da/dN) correspond to changes in the mechanism that controls the FCG.
According to Anderson [20], it is interesting to note that when the "L-shaped" curve is plotted as ΔK th versus R, the resulting trend is similar to the crack closure theory that can be expressed mathematically by Equation 7, where [20,37].Therefore, the existence of thresholds for both K max and ΔK can be considered wholly consistent with the closure argument, although Vasudevan and his colleagues insist on disagreeing with this reasoning.They insist that the two parameters ΔK th * and K max * are intrinsic characteristics of the material, and that the FCG does not occur simply because of an extrinsic factor like the crack closure.
In a recent research by Zhu et al. [43,44], the application of these two approaches previously mentioned independently to R-effect in the near-threshold fatigue regime (Elber and Vasudevan) is observed to be either shortage of accuracy or full of uncertainties, as a result of the complexities and difficulties in both experimental measurement and modeling key parameters in the case of the procedure used to generate the near-threshold region.One possible solution to this issue is to take both the two approaches into account by using a suitable merging technique considering both ΔK eff and K max parameters.
Zhu et al. [43,44] investigated the R-effect on FCG, combined the crack closure concept and the two-parameters approach method.They also considered a new crack opening equation, without involving crack closure measurement, and an equation relating the crack growth rate da/dN to the driving force ΔK in the near-threshold regime (ΔK th ), based on several values for the R-ratio.This new model assumes the existence of the crack closure, however only influencing the FCG below a certain value for the R.In other words, there is a particular value of R, here called R c (the same physical meaning of R* of the crack closure model), above which the crack closure will have no influence on the FCG.A general FCG equation in the form of the Paris equation, da/dN = C[A(R)ΔK B(R) ] m , was put forward for the near-threshold FCG, which was reasonably used to predict the R dependence of FCG curve and fatigue threshold, with the C and m constants valid only for this region.This FCG equation was further validated using FCG data in the open literatures and believed to be eligible and suitable to predict fatigue thresholds in a variety of Cr-Mo-V steels.
The methodology proposed by Zhu et al. [43,44] requires an FCG test with a relatively high R value (reference value), when there is no influence of crack closure.The generated curve da/dN versus ΔK in the threshold region gives reference values for the driving force ΔK ref .From these reference data, values of (da/dN) R and (ΔK) R are obtained for any value of R, using Equation 8 and Equation 9, respectively, and the material-specific analytical equations A and B including R as a variable, provided by Equations 10 and 11.
Considering the importance of an adequate selection of steels for the current context of the automobile industry, which searches for the use of materials with reduced weight and increased mechanical properties, and also considering the importance of having an accurate prediction model of fatigue life of these materials, this research consisted in comparing the fatigue crack growth performance of four HSLA/AHSS steels, with different microstructures (ferrite/ pearlite, ferrite/pearlite/bainite, ferrite/bainite and ferrite/ martensite), and the application of some prediction models for da/dN versus ΔK curve as a function of the R load ratio mainly in the region of the threshold ΔK th .

Materials and methodology
Four hot rolled low carbon steels were received from a Brazilian company specializing in the stamping of automotive components.The materials were identified as S1, S2, S3 and S4.Their chemical composition, determined by optical emission spectrometry (OES), is presented in Table 1.S1 is a HSLA modified AISI 1010 grade steel, with a ferrite-pearlite microstructure.S2 is a HSLA niobium microalloyed steel, with ferrite-pearlite microstructure.S3 and S4 are AHSS dual-phase ferrite-bainite and ferrite martensite steels, respectively, being the first a niobium microalloyed and the second a high chromium steel.
The samples used for microstructural examination were cut, ground, polished and etched with a special solution (picral and after Na-metabisulfite) developed to reveal microconstituents of high strength steels [45].The samples were examined using a LEICA light optical microscope (LOM) and a TESCAN VEGA3 scanning electron microscope 4/15 Tecnol Metal Mater Min.2023;20:e2767 (SEM).The corresponding ferrite grain size of each steel was measured according to the standard ASTM E112 [46].
Tensile properties of the steels were obtained according to the standard ASTM E8 M [47] on a 100kN 5882 Instron electro-mechanical system interfaced to a computer for machine control and data acquisition.Three specimens are considered for each steel.The Vickers hardness was obtained according to the standard ASTM E92 [48].
All the FCG tests were performed on a 100kN 8802 Instron servo-hydraulic materials testing system interfaced to a computer for machine control and data acquisition, at room temperature in atmospheric air.The tests were conducted under a sinusoidal waveform with a frequency of 35Hz, in accordance with the recommendations of ASTM E647 [49].
Compact tension C(T) specimens oriented in T-L direction were considered for each steel, with 3mm thickness, 50mm width and a fatigue pre-crack length of 3mm to have an initial crack size to width ratio (a/W) of 0.26.Crack opening displacement (COD) measurements were performed using a COD extensometer and crack size evaluation was performed using the compliance method.Crack size curves in function of the number of cycles have been obtained and transformed to crack growth rate curves (da/dN) as a function of cyclic stress intensity factor (ΔK).Five R-ratio are considered: R = 0.03, 0.1, 0.4, 0.5 and 0.7.The fatigue threshold value ΔK th was defined as the stress intensity factor range at which the fatigue crack growth rate decreased to below 1x10 -7 mm/cycle.This value was estimated by a K-decreasing procedure (load-shedding process), limiting the normalized K-gradient, C = (1/K).(dK/da) to -0.08mm -1 .ΔK th was obtained from the best-fit straight line from a linear regression of log da/dN versus log ΔK between growth rates of 10 -6 and 10 -7 mm/cycle.
Closure measurements for the application of the Elber model [35,36] were calculated for many crack sizes at the threshold region through the compliance technique, where a change of linearity was formed in the applied load versus COD curve.With K cl calculated, Equation 5 was applied in order to obtain the "master curve" da/dN versus ΔK eff for the four studied steels.Equations of the type of Equation 6were obtained with the experimental results, to know the dependence of the effective stress intensity ratio with the R-ratio.This methodology served to find a particular value for the R-ratio, above which the crack closure is not observed, an important result to be used in the subsequent models analyzed.
The values of ΔK and K max used in the Vasudevan et al. [38][39][40][41][42] methodology for the FGC rate of 1x10 -7 mm/cycles were plotted in graphs for the identification of the critical values that allow the FCG.In this way, the values of ΔK th * and K* max, th were obtained for all studied steels.Equation 12was used to increase the amount of data at the rate of 1x10 -7 mm cycle, where the parameter K 0 represents the ΔK th value for R = 0 and γ is an adjustment constant between 0 and 1 [50].
( ) Still exploring the concepts adopted by Vasudevan and with the obtained value of R*, graphs of ΔK versus R were traced to verify the validity of Equation 7and to analysis the possible existence of a connection between this model and crack closure.
With the adopted value of R c (R c = R*), the methodology of Zhu et al. [43,44] was applied.For six specific da/dN values, graphs were plotted between lnΔK and (2R/(1-R)) for different R and the slope of each line (α) was taken.When R is less than R c , α is called α 1 , otherwise α is equal to α 2 .For each test performed with different R, the Paris equation was applied to the region of ΔK th with the correction for ΔK according to Equation 13 and the constants C 0 and m 0 were obtained.From the parameters C 0 , m 0 , α 1 and α 2 , the crack closure coefficient U (R) is obtained by Equation 14 and later simplified in Equation 15.
With this information it can use Equation 8 to Equation 11, to predict the fatigue crack growth in the nearthreshold regime for different values of R.
3 Results and discussion

Comparison among the steels performance
Figure 1 shows the microstructures of the four studied steels.The S1 steel contains a microstructure composed of Effect of R load ratio on fatigue crack growth resistance of steels used in automotive applications: experimental results and use of performance prediction models equiaxial and relatively coarse grains of ferrite and colonies of fine pearlite.The microstructure of the S2 steel consists of elongated and finer grains of ferrite and pearlite colonies.A small volume fraction of bainite is also observed.A very refined structure is observed in S3 steel, constituted by elongated grains of ferrite and bainite.An equiaxial and refined microstructure is observed in S4 steel, formed by martensite islands surrounded by a ferritic matrix.These different microstructures result from the different chemical compositions and also from the different thermo-mechanical rolling procedures applied in the steels [1][2][3].
The ferritic grain size of each material and the volume fractions of the phases/constituents are shown in Table 2.
It is confirmed that all steels are ferritic with small amounts of other phases/ constituents, and that S3 steel has the most refined structure among the studied steels.The values found for volume fraction of the phases/constituents agree with values commonly used in steels for the automotive industry [1][2][3].It is interesting to note that up to a certain value of martensite volume fraction, usually 50%, the value of the threshold ΔK th will tend to increase, according to literature data 4-19 , improving the fatigue resistance of the material.However, these high values of martensite cannot be industrially considered, due to operational losses in the stamping stage for the manufacture of automotive components.
Table 3 summarizes the main tensile results and also hardness values.The values of tensile strength, strainhardening coefficient and strain to fracture are in accordance with the classification for HSLA and AHSS steels used in the automotive industry [1,51].
It can be seen an inverse relation between the mechanical strength and the ductility of the steels, due the presence of different phases/constituents, elements in solid solution, precipitates and ferritic grain size [52][53][54][55].
The S2, S3 and S4 steels presented a similar fracture energy (U parameter estimated by the area below the tensile curves, and shown in Table 3), superior to the S1 steel.The S4 steel has a lower yield strength than the other steels but, because of a higher strain-hardening capacity, reaches the higher ultimate tensile strength.The strain hardening and high strength of DP steels are a result of martensite formation.Hard martensite regions in ferrite provide dispersion strengthening according to the rule of mixtures, i.e., the more martensite, the greater the strengthening, but also introduce  high densities of dislocations into the ferrite around the martensite.In the ferrite adjacent to the martensite there is a very high density of dislocations.These dislocations are generated by the shear and volume changes associated with the transformation of austenite to martensite.Dislocation densities are much lower in the ferrite removed from the martensite islands.The dislocations around the martensite are not pinned and account for the absence of the discontinuous yielding (no sharp yield point) exhibited in mild steels and HSLA steels.The martensite-induced dislocations move at low stresses, creating low yield strengths, and interact to produce high rates of strain hardening [56][57][58][59][60]. Independently of the differences in mechanical properties of the steels, all tensile specimens presented a ductile behavior, with operation of the mechanism of nucleation, growth and coalescence of microcavities.The dimples had the smallest size in S3 steel, due to its smaller grain size.
The sigmoidal da/dN versus ΔK curves for the four steels, with R = 0.1, are presented in Figure 2. It is possible to see the three different regions of FCG commonly found in metals.When comparing the fatigue behavior of the steels, it is observed that they have similar behavior in the linear Region II (Paris's equation), since this region is less sensitive to chemical composition and microstructure.In region I and region III, there is a significant different behavior among the steels, since the FCG in these regions is strongly influenced by the chemical composition and the microstructure of the material [5,6,[9][10][11]15].The S3 presented the highest value of ΔK th , having, then, the best fatigue performance among the materials, followed by S4 steel.The best performance of AHSS steels is reported in the literature 3 as a consequence of the presence of the dual-phase microstructure, i.e., bainite or martensite in a ferritic matrix.In this case, the FCG in the ferrite will create tortuosity and bifurcation when the crack encounters the hardest phase, the driving force decreases and the FCG rate decreases.
Figure 3 shows the fracture surfaces of fatigued specimens for R = 0.1.For each steel considered, the fracture surface was analyzed for two da/dN crack growth rates, 1×10 -7 and 5×10 -5 mm/cycle, which correspond, respectively, to the beginning of region I (crack growth threshold) and region II (linear region between da/dN and ΔK).For all steels the fatigue fracture surface presented at the near-threshold region a predominant transgranular fracture mode, with the "hill-and-valley" type appearance and shear facets, with an associated zig-zag path primarily through the ferrite.Such fractures show high linear roughness and high crack deflection angles.At higher growth rates, characteristics of region II, fracture surfaces remain transgranular, but with evidence of striations.These fracture mechanisms did not change with the change in the value of R. All these fractographic characteristics are found in many materials [6,15,17,22,25].
The deleterious effect of increased R-ratio on the ΔK th is showed in Table 4 for all the studied steels, as expected and reported in the literature (researches cited in the introduction of this article) for several materials.When considering the performance of each steel, S3 and S4 presents better behavior in the values of ΔK th .The presence of bainite/martensite in these microstructures justifies again such behavior [9,11,15].
From the set of mechanical properties evaluated, it can be concluded that the two AHSS grade steels are superior to the two HSLA steels.The following applied models should therefore predict this performance.It is observed that the curves for different values of R tend to overlap when ΔK is "corrected" to ΔK eff , generating a "master curve" for the material.However, it is important to observe that it exists a certain dispersion (33% maximum) in an attempt to trace the "master curve" in the near-threshold region.This dispersion can be related to the experimental determination of the crack closure, that is affected by existing   different threshold testing methods and crack growth shielding mechanisms.It can also be argued that the crack closure could not be the unique agent controlling the FCG in this region, according to the ideas of Vasudevan and co-authors.This subject will be later.

Application of the Elber model
Table 5 shows a comparison between values of ΔK th and ΔK eff .It is observed a reduction in the value of ΔK th that occurs when it is corrected for ΔK eff due to the effect of crack closure.It was observed that the crack closure U coefficient varies with R following the form of a linear equation, similar to Equation 6.Table 6 presents the intercept and the slope of the equations for all steels.
To better analyze the conjecture of curves overlapping in the Elber model, it was performed a linear fit to log(da/dn) = C' + m log(ΔK eff ).The results are shown in Figure 5.It was also plotted the confidence bands and projection bands.A confidence level of 95% was defined, which is usual in this kind of analysis.Projection bands are used to confirm the basic assumption of curves overlapping.
For S1 sample, the coefficient of determination R 2 is ~ 0.95, which represents a good fitting.It is observed that most of the adjusted points to ΔK eff are included within the prediction bands, reinforcing the conjecture of overlap.The residuals analysis shows a distribution which can be interpreted either a signal of outliers or the existence of a changing behavior (discussed below in this text).For S2 it is observed the same behavior as that of S1: most of "normalized" points are within the prediction bands and the residuals histogram shows the presence of outliers or a changing behavior.This assumption will be better discussed below.However, for S3 and S4, most of the normalized points are outside the prediction bands, as one can see in the Figure 5.Although the coefficient of determination R 2 is close to 0.9 for both cases, the results do not support the basic assumption of measurements overlapping.In both cases, the dispersion of data is greater than in previous S1 and S2 cases, as one can observe in the residual plots.

Application of the Vasudevan model
The fundamental threshold curves were constructed from the data of ΔK th and K max using experimental data for the four steels studied and predicted results obtained from Equation 12, as show in Figure 6.The data of all the materials result in curves with an L-shape, as was expected by this methodology.It is possible to define values of ΔK* and K* max where the FCG rate is close to zero.These two parameters are considered thresholds, ΔK th * and K* max,th , because the data were taken for da/dN = 1x10 -7 mm/cycle.Table 7 presents these values for each steel.It is important to observe that it also exists a certain dispersion of results.F-tests have been performed to compare the two datasets of each steel (results obtained experimentally with results calculated from the theoretical model).In all the cases, for a significance level of 0.5, the sets are statistically different.This dispersion may come from the use of Equation 12, which  Effect of R load ratio on fatigue crack growth resistance of steels used in automotive applications: experimental results and use of performance prediction models would require an adaptation.Another alternative would be to carry out a larger number of tests, using more R values to improve the Figure 7 shows ΔK th versus R curves obtained from experimental results.For all steels, R* = 0.7 was considered, according to the experimental results commented on the application of the Elber model.It is noted that these curves show a similar trend to the curves proposed by Hudak and Davidson [37] which consider crack closure as an extrinsic phenomenon that controls FCG.Thus, it is possible to argue that the existence of two thresholds for driving force, ΔK th and K max , are consistent with the crack closure effect.There is then a connection between the two parameters' methodology and the crack closure.However, when trying to complete Figure 7 with ΔK th values predicted by Equation 12, one can see that there is a reasonable dispersion of results.Thus, it is possible to confirm that this equation is not accurate to be used to represent the FCG.
The values of the ΔK th * and K* max,th thresholds for each material, taken from Figure 6 and according to Equation 7, are presented in Table 8.A considerable difference between these values and those presented by Table 7 (Vasudevan's model) is observed.By adopting Vasudevan and co-authors' model, it can already be seen that the two S3 and S4 AHSS steels would be the materials with the greatest resistance to fatigue cracking.
On the other hand, the analysis that considers the crack closure operation indicates that S4 steel is the most appropriate steel among the analyzed materials.Although the Vasudevan's model has the advantage of dispensing the experimental measurement of crack closure, this phenomenon was verified and its use in predicting fatigue behavior leads to a better precision in the selection of materials.
Therefore, it is always important to consider an analysis that takes crack closure into account.However,  another important conclusion taken from the results shown in Figure 7 is that, based on the significance of R*, the crack closure would cease to influence the FCG to a lower R value than expected.This fact seems to a positive point for the general considerations proposed by Vasudevan and co-authors.

Application of the Zhu model
For the application of this model, it was considered R c = 0.7, in the same way as in the previous models.The values of the coefficients of Equations 10 and 11 for each steel are presented in Table 9.According to Equation 8 and Equation 9, the predicted da/dN versus ΔK relationship is compared with the experimental data, as shown in Figure 8 for all steels.It can be observed that the predicted curves agree reasonably well with the experimental fatigue data in the near-threshold region with a maximum error not more than 12%.This indicates that the prediction of FCG behavior from R of 0.03 to 0.7 is promising based on the proposed merged approach.The fatigue thresholds ΔK th are also predicted when the value of ΔK th(0.7) in Equation 9is chosen at the da/dN of 1 x 10 -7 mm/cyc.As indicated in Figure 9 and Table 10 for all steels, the fatigue thresholds from experiments are quite similar to those predicted by the model in Equation 8and Equation 9.According to Zhu and co-authors' model  predictions, S4 steel has better performance in fatigue, followed by S3 for R = 0.1.However, it is important to note that some results a deviation and that there is not an adequate prediction for the other R values.In order to reduce these errors, the complex mathematical manipulation would require the adoption of a greater range of values for the R-ratio, to imply a greater precision in the determination of the A(R) and B(R) functions, Equation 10 and Equation 11.Another aspect to consider is the choice of a more representative equation between da/dN and ΔK for the threshold region than the Paris equation.
Using the Zhu concepts, the experimental results were compared with the results obtained from the theoretical model, discussed above.In most of the cases, for the several samples and for different values of R, the datasets produce different results when performing linear fitting (using a 0.03/0.1/0.5/0.7 6.0/9.5/5.3/4.0 7.5/6.6/6.2/4.0S4 0.03/0.1/0.4/0.7 7.7/7.0/6.9/5.7 6.9/7.0/6.6/5.7 Zhu and co-authors model.As an example, see Figure 10.It represents the S2 steel.For green dots, representing R = 0,1, prediction bands for linear fitting (violet lines at right) contains the points obtained from the theoretical model.However, in contrast with this, prediction lines for R = 0.5 (red dots) are very far from points obtained from theoretical model (represented by the red line at left).This is the usual situation obtained in the comparison between experimental and theoretical results.

Conclusions
• Considering the chemical compositions, microstructures and mechanical tensile strength, as well as the international specifications currently adopted for automotive steels, S1 and S2 can be considered as HSLA ferrite/pearlite steels, the first type AISI 1010 and the second microalloyed to niobium.S3 and S4 are AHSS ferrite/bainite and ferrite/martensite steels, respectively, the first microalloyed to niobium and the second with high chromium content.
• All steels presented the traditional sigmoidal curve of fatigue crack growth.The AHSS steels obtained the highest values for the threshold ΔK th , being then more resistant to fatigue cracking than the HSLA steels.All four steels showed the same sensitivity to R-ratio, that is, decrease of fatigue strength with increase of R.
The fracture mechanism was always the same, with tortuosity in the region close to the threshold and presence of striations at higher FCG rates.
• The manifestation of crack closure and its qualitatively expected dependence on the R-ratio were verified for the studied steels, but the Elber model was not able to provide a master curve that accurately summarized the effect of the R-ratio on the sigmoidal fatigue curve of steels.
• The combined use of two critical thresholds, ΔK th * and K max *, for predicting fatigue crack growth according to the Vasudevan model also did not provide accurate results in evaluating the effect of the R-ratio.
• The application of the methodology proposed by Zhu and coauthors, combining the two previous approaches, also provided a significant dispersion of results, once again failing to predict the effect of the R-ratio on the fatigue behavior of the steels.
• Considering that there is a connection between the two-parameter and crack closure models, the idea of Zhu and co-authors to link those methodologies is a promising alternative.However, experiments with a larger number of values for the R-ratio are needed, in order to have a more accurate expression among da/dN, ΔK and R.This research topic is still open, requiring a more in-depth phenomenological knowledge to predict the effect of the R ratio on fatigue crack growth.

Figure 4
Figure4shows the da/dN versus ΔK and da/dN versus ΔK eff curves using the Elber concept of crack closure for each steel studied.The deviation of the linearity of the curve P x COD was evaluated to verify the presence of crack closure in the performed tests.The existence of crack closure was precisely observed for R equal to 0.03, 0.1 and 0.4 and completely absent for R equal to 0.7.Thus, the value R = 0.7 was adopted for R* and R c used in the subsequent models.It's interesting to remember that, as remarked in the introduction of the article, R* (adaptation of Elber's model, proposed by Hudak and Davidson) has the same physical and numerical meaning as R c (proposed by Zhu and coauthors' model).It is observed that the curves for different values of R tend to overlap when ΔK is "corrected" to ΔK eff , generating a "master curve" for the material.However, it is important to observe that it exists a certain dispersion (33% maximum) in an attempt to trace the "master curve" in the near-threshold region.This dispersion can be related to the experimental determination of the crack closure, that is affected by existing

Figure 5 .
Figure 5. Linear Regression and residuals for linear regression.Elber's model.

Figure 8 .
Figure 8.Comparison of experimental and predicted da/dN x ΔK relationships in the near-threshold regime.

Figure 9 .
Figure 9. Predicted versus experimental fatigue thresholds for the studied steels and results from Zhu et al. [44] in a variety of Cr-Mo-V steels.

Table 1 .
Chemical composition of the studied steels (wt%)

Table 6 .
Parameters of linear equation of crack closure U coefficient (U = a 1 + b 1 R)

Table 7 .
Values of ΔK th * and K* max,th for each

Table 8 .
Values of ΔK th * and K* max,th taken form Figure6for each steel.

Table 9 .
Values of the parameters of Equations 10 and 11 for all studied steels

Table 10 .
Predicted versus experimental fatigue thresholds for the studied