Thresholds of fatigue crack growth rates are important characteristics for fatigue crack growth assessment for the integrity of structural components. ASME Code Section XI provides fatigue crack growth thresholds for ferritic steels in air and water environments. The threshold is given as a constant value under a negative stress ratio. However, the thresholds are not clearly defined in the range of negative stress ratios. The definition seems to be maximum stress intensity factors. Besides, the thresholds expressed by the maximum stress intensity factors decrease with decreasing stress ratios. This means that the thresholds under negative stress ratios become unconservative assessments. The objective of this paper is to discuss the definition of fatigue crack growth threshold and to propose the threshold equation for the ASME Code Section XI, based on experimental data obtained from a literature survey.
Reference fatigue crack growth rates for ferritic steels are provided by Appendix A in ASME (American Society of Mechanical Engineers) Boiler and Pressure Vessel Code Section XI . The crack growth rates da/dN are expressed as a function of stress intensity factor range ΔKI and the growth rates are affected by stress ratio R that is the minimum stress divided by the maximum stress over the load cycle. The ASME Code Section XI also provides the fatigue crack growth threshold ΔKth as a function of stress ratio R for ferritic steels in air and water environments in Appendix A.
The thresholds of fatigue crack growth rates are very important for flaw evaluations to determine whether the components with the detected flaws are required or not required to be repaired or replaced. However, the thresholds of the fatigue crack growth rates are not easy to obtain experimentally. This is because the fatigue crack growth rates are significantly small, and it takes a huge time to determine the threshold values. When the fatigue crack growth rate is at the order of da/dN = 10−8 mm/cycle, fatigue crack growth rate is negligible. The stress intensity factor range ΔKI is defined as fatigue crack growth threshold ΔKth .
In these circumstances, almost all thresholds were taken under positive stress ratio, which means cyclic tensile and tensile stresses. Experimental data of fatigue crack growth thresholds under tensile and compression stresses are sparse, particularly, thresholds under large compression stress are few, because of specimen's buckling. Therefore, thresholds were not clearly described under the negative stress ratio.
The objective of this paper is to introduce the thresholds of the fatigue crack growth for ferritic steels provided by Appendix A of the ASME Code Section XI and to demonstrate that the thresholds are not well defined in the range of negative stress ratio. The definition of the thresholds is discussed based on experimental data. In addition, thresholds of effective stress intensity factor ranges are analyzed using fatigue crack closure. Finally, the fatigue crack growth thresholds under negative and positive stress ratios are proposed for ferritic steels in terms of a simple equation for the ASME Code Section XI.
ASME Code Section XI Fatigue Crack Growth Rates
where n is the slope of the log (da/dN) versus log (ΔKI) line and is given by n = 3.07 in air environment, and C0 is a scaling constant. The scaling constant in air environment is expressed as C0 = 3.78 × 10−9S, where S is the scaling parameter to account for the stress ratio R and is given by S = 25.72(2.88 − R)−3.07. The stress ratio is expressed as R = Kmin/Kmax, where Kmax and Kmin are the maximum and minimum stress intensity factors associated with cyclic stress range of maximum stress σmax and minimum stress σmin, respectively. The units of ΔKI, Kmax and Kmin are MPa√m, and da/dN is given by mm/cycle in this paper.
The stress intensity factor range ΔKI is simply defined by ΔKI = Kmax − Kmin for 0 ≤ R <1.0. However, for the case of R < 0, the stress intensity factor range ΔKI is complicated in an air environment. The ΔKI under negative R ratio depends on the crack depth a and the flow stress σf. The flow stress is given by σf = 0.5(σy + σU), where σy is the yield strength and σU is the ultimate tensile strength in units of MPa and a is in units of m. For −2 ≤ R < 0 and Kmax − Kmin ≤ 1.12 σf √πa, the scaling parameter is S = 1, and ΔKI = Kmax. For the case of R < −2 and Kmax − Kmin ≤ 1.12 σf √πa, S = 1 and ΔKI = (1 − R) Kmax/3. When R < 0 and Kmax − Kmin > 1.12 σf √πa, S = 1 and ΔKI = Kmax − Kmin. Therefore, the stress intensity factor ranges ΔKI under negative R ratios are categorized as ΔKI = Kmax, ΔKI = (1 − R) Kmax/3 or ΔKI = Kmax − Kmin, depending on the stress ratio, crack depth, and flow stress.
The threshold ΔKth in units of MPa√m is a function of stress ratio R. Figure 1 shows the relationship between ΔKth and R expressed by Eq. (2). The threshold ΔKth increases with decreasing R ratio for 0 ≤ R < 1 and ΔKth is a constant for R < 0. The threshold of ΔKth = 5.5 MPa√m in Eq. (2) cannot be used itself for practical purposes.
Threshold ΔKth Under Negative R Ratio
The threshold ΔKth is not well defined in the ASME Code Section XI, although the stress intensity factor ranges ΔKI against fatigue crack growth rates of da/dN-ΔKI are clearly defined, as mentioned above. The definition of the threshold under positive stress ratio R is, needless to say, ΔKth = Kmax − Kmin. However, under negative stress ratio R, the threshold ΔKth is supposed to be ΔKth = Kmax, ΔKth = (1 − R) Kmax/3 or ΔKth = Kmax − Kmin, depending on the stress ratio, crack depth, and flow stress, as described before. Taking into account the stress level at fatigue threshold condition, applied cyclic stress ranges are very small, and it is reasonable to consider the condition of ΔKth as Kmax - Kmin ≤ 1.12 σf √πa. Therefore, the thresholds under negative R ratio are ΔKth = Kmax for −2 ≤ R < 0 and ΔKth = (1 − R) Kmax/3 for R < −2. The ΔKI for R < −2 is rewritten as ΔKI = (1 − R) Kmax/3 = (Kmax − Kmin)/3. Therefore, ΔKth for R < −2 is ΔKth = (Kmax − Kmin)/3. Then, the threshold ΔKth shown in Fig. 1 can be used for practical purposes.
The threshold given by ΔKth = 5.5 MPa√m for R < 0 can be expressed by Kmax using stress ratio R. The relationship between Kmax at threshold and negative stress ratio R is illustrated in Fig. 2, together with ΔKth at the positive R ratio. The Kmax in Fig. 2 is equivalent to ΔKth in Fig. 1. In the case of −2 ≤ R < 0, it is a constant value of Kmax = 5.5 MPa√m. When R < −2, the Kmax decreases from 5.5 MPa√m at R = −2.0 to 2.75 MPa√m at R = −5.0 (see Kmax in Table 1). The relation of Kmax and R line under negative R ratio shown in Fig. 2 is not written in Appendix A of the ASME Code Section XI and it is an implication by the definition of the stress intensity factor range ΔKI derived from fatigue crack growth rates da/dN − ΔKI. The threshold is not a smooth line against the stress ratio under the negative stress ratio region. From the view point of crack closure phenomenon under the region at negative stress ratio, there is no discrimination at R = −2.0, as mentioned later.
The description of Kmax against ΔKth under negative R ratio is inferred from the definition by ASTM (American Society of Testing and Materials) E647 . The ASTM E647 recommends to use ΔKI = Kmax − Kmin for 0 ≤ R and ΔKI = Kmax for R < 0. This is because, under tensile-compression stress, positive portion of the stress range is close to crack driving force. The ASTM E 647 also describes that crack closure is complicated and ΔKI = Kmax may be inappropriate, when crack closure is considered. Therefore, let us consider the crack closure at the threshold level of under negative R ratio.
Crack Closure Under Negative R Ratio
where Kop is the stress intensity factor at crack opening. Figure 3 illustrates a cyclic hysteresis loop and the stress intensity factor at crack opening. The crack closure ratio U is a key parameter in the fatigue crack growth process, particularly associated with compressive stress effect. It is well known by much experimental data that fatigue cracks close at positive loads during constant amplitude loading cycles.
Figure 4 shows four models of crack closure U in the literature. Note that the Eason model  was based on only 0 ≤ R, and the curve of UEASON was beyond the range of applicability. The Kurihara's model  was derived from experimental data, where the ferritic steels were subjected to cyclic tests at ambient temperature under the stress ratio of −5 ≤ R ≤ 0.8. The Heitmann's model  was obtained by experimental data for 4340 steels. The Schijve's model was developed from tests of 2024 aluminum . It is shown that crack closure U for all models decreases with decreasing R ratio.
Crack closure depends on applied stress levels. When the applied cyclic load ranges are large, fatigue cracks tend to open. This means that the Kop in Fig. 3 approaches the Kmin, and the stress intensity factor ranges ΔKI approach Kmax − Kmin. In the reverse case, the Kop moves toward to the Kmax, when the cyclic loading is small like threshold test stress level.
Newman et al. had analyzed the effect of stress level on closure using nonlinear finite element analysis . In accordance with the Newman's model, crack closure is calculated as a function of σmax/σf, where σf is the flow stress taken to be the average of the yield stress and the ultimate tensile strength of the material. Table 2 tabulates the relationship between crack closure U and applied stress under negative R ratio calculated by the Newman's model. For example, at R = −2, when stress level is σmax/σf = 0.7, crack closure is calculated as U = 0.388. At a small stress of σmax/σf = 0.05, U = 0.214 in accordance with the Newman's model . Crack closure is U = 0.214 at low stress level at σmax/σf = 0.05 is close to the value of U = 0.222 at R = −2, obtained by Eq. (5). As the stress level at fatigue crack growth threshold is very small, Eq. (5) is applicable for threshold stress levels. Using Eqs. (4) and (5), stress intensity factors Kop at crack opening are obtained easily.
Table 1 shows the crack closure Kmax, U, Kop and Keff (=Kmax − Kop) under negative R ratio. For the case of −2 ≤ R < 0, crack closure U decreases from 0.667 to 0.222, and effective ranges of Keff are constant of 3.66 MPa√m, as shown in Table 1. For the case of R < −2, the U decreases from 0.222 to 0.111, and effective ranges of Keff decreases from 3.66 to 1.83 MPa√m. These Keff are apparently small values compared with ΔKth = 5.5 MPa√m. In addition, the Keff is not a smooth line at R = −2.0, although the crack closure U decreases with decreasing stress ratio R. It can be said that the effective stress intensity factor ranges ΔKeff are smaller than 5.5 MPa√m and the threshold of ΔKth = 5.5 MPa√m is not close to a crack driving force for R < 0.
Experimental Data of Fatigue Crack Growth Thresholds
It is necessary to understand how to obtain thresholds of fatigue crack growth rates under negative R ratio by experiments. Fatigue experts perform fatigue tests using cracked specimens setting at maximum tensile and minimum compression loads, which correspond to the Kmax and the Kmin. While decreasing the applied cyclic loading under constant ratio of Kmin/Kmax in accordance with the ASTM E 647 fatigue test method , ΔKI at the order of da/dN = 10−8 mm/cycle is judged to be the threshold. Therefore, the values of ΔKth are obtained as Kmax − Kmin, experimentally, even when R is negative.
Figure 5 shows fatigue crack growth rates for ferritic steel obtained by one of the authors . The test was conducted at room temperature in an air environment under the wide range of R = −5.0 to 0.12. The material is A 533 B low alloy steel weldment with yield stress of 484 MPa and ultimate tensile strength of 586 MPa. Fatigue crack growth rate da/dN decreases with decreasing R ratio. Besides, the thresholds ΔKth at around da/dN = 10−8 mm/cycle increase with decreasing R ratio. The author had also obtained thresholds for the same material at the same environment under R ratio of 0.23–0.92. These threshold values of ΔKth for the low alloy steel are shown in Table 3, where the definition of ΔKth is Kmax − Kmin. Using ΔKth = Kmax − Kmin and R = Kmin/Kmax, the values of Kmax, Kmin are easily calculated. Looking at Table 3, it is shown that the experimental thresholds ΔKth increase with a decreasing entire range of R ratio. In addition, the Kmax and Kmin calculated by ΔKth reduce with decreasing R ratio for negative R. It is obvious that the Kmax is not constant under negative R ratio. Conclusively, the threshold ΔKth is determined by the full range of stress intensity factor of ΔKth = Kmax − Kmin, experimentally. The definition of the threshold is suitable to simply use ΔKth = Kmax − Kmin in practical purposes.
Experimental data on thresholds of fatigue crack growth rates for ferritic steels exposed to air environment were collected by literature survey. Table 4 tabulates the fatigue threshold experimental data. There are many data taken at 0 ≤ R < 1.0. The data at R < 0 are also seen in Table 4. Almost all thresholds ΔKth for R < 0 are larger than those at R = 0 Several steels show that thresholds ΔKth at R = −1 are smaller than those at R = 0, as shown in Table 4 by the asterisk *. The definition of ΔKth for these cases is not clearly written. These values of the thresholds ΔKth at R = −1 are seemed to be Kmax. If the thresholds ΔKth are Kmax under negative R ratio, the values of ΔKth are converted to the description of Kmax − Kmin. Taking low alloy steel shown in Table 4 by asterisk * for example, the threshold of Kmax = 6.3 MPa√m at R = −1.0 becomes 12.6 MPa√m expressed by Kmax − Kmin. The value of 12.6 MPa√m is close to the threshold values of other steels.
Recommended Thresholds at Negative R Ratio for Codification
There are many fitness-for-service codes in the world and many codes provide fatigue crack growth thresholds for many materials, such as aluminum alloys, stainless steels, and ferritic steels, in various environments . Almost all codes do not clearly provide the definition of the thresholds ΔKth = Kmax or ΔKth = Kmax − Kmin under negative R ratio, except BS 7910 . In addition, many codes give constant values of ΔKth under negative R ratio, except API 579 .
Thresholds of fatigue crack growth rates are obtained as ΔKth = Kmax − Kmin under negative R ratio by experiments, as mentioned above. The threshold values under negative R ratio are larger than those under positive R ratio, as shown in Table 4. This means that the ΔKth under negative R ratio includes effective stress intensity factor due to crack driving force, and stress intensity factor due to compression stress that does not contribute to the crack driving force. Therefore, the ΔKth increases with decreasing stress ratio R for the case of R < 0.
From the other aspect of code user's standpoints, they know cyclic maximum and minimum stresses for the cracked components, when they need to evaluate fatigue crack growth problems. They can easily calculate maximum and minimum stress intensity factors and they do not want to take account of crack closures for complicated geometry components.
Therefore, the definition of recommended threshold of fatigue crack growth rate provided by codes and standards shall be ΔKth = Kmax − Kmin, instead of effective stress intensity factor range ΔKeff or maximum stress intensity factor Kmax under negative R ratio. It is a benefit that the definition of ΔKth = Kmax − Kmin is coherent in the entire range of stress ratio R.
Proposal of Fatigue Thresholds for Ferritic Steels for Codification
In order to establish fatigue crack growth rate thresholds for codification, a literature survey was conducted to collect thresholds for ferritic steels in an air environment. Table 4 shows the experimental data on thresholds ΔKth for various ferritic steels. The experimental data in Table 4 are plotted in Fig. 6 on the diagram of ΔKth − R relation, where the definition of the thresholds is ΔKth = Kmax − Kmin in the entire range of stress ratio R. The data on thresholds ΔKth at R = −1 seemed to be Kmax in Table 4 by asterisk * are not included in Fig. 6. The equation of ΔKth = 5.5(1 − 0.8R) given by the ASME Code Section XI is also shown in Fig. 6. Although the equation of ΔKth = 5.5(1 − 0.8R) given by Eq. (2) is applied for only 0 ≤ R < 1.0 in the ASME Code Section XI, it is pertinent to use for 0 < R. The equation can be extended to negative R ratio. The line of ΔKth = 5.5(1 − 0.8R) is a lower bound of the experimental data, as shown in Fig. 6. Conclusively, the equation of ΔKth = 5.5(1 − 0.8R) is applicable to negative R ratio.
Threshold ΔKth of fatigue crack growth rates for ferritic steels in an air environment is provided by a constant value of ΔKth = 5.5 MPa√m under negative stress ratio in Appendix A of the ASME Code Section XI. However, the definition of ΔKth under negative R ratio is not clearly defined. It seems to be ΔKth = Kmax = 5.5 MPa√m. However, the Kmax = 5.5 MPa√m is not close to the crack driving force, because effective stress intensity factors ΔKeff are apparently small compared with the Kmax. In addition, the expression on constant Kmax is less conservative, because Kmax decreases with decrease of R ratio.
From a literature survey on fatigue threshold data, almost all data were taken experimentally as ΔKth = Kmax − Kmin for the entire range of stress ratio, and the definition of ΔKth =Kmax − Kmin under negative R ratio is better to use from the practical view point of code users. In addition, the expression of ΔKth = Kmax − Kmin is coherent to the negative and positive ranges of R ratios. Based on the above analyses, it can be proposed to use the threshold of ΔKth equation for ferritic steels in air environment, as ΔKth = 5.5(1 − 0.8R) MPa√m for R < 0.8 and ΔKth = 2.0 MPa√m for 0.8 ≤ R for Appendix A-4300 in the ASME Code Section XI.
The authors wish to acknowledge useful discussions and precious advices by the Chair Gary Stevens and members of the Working Group on Flaw Evaluation Reference Curves in the ASME Code Section XI. A portion of this work was funded by the project No. CZ.02.1.01/0.0/0.0/17_048/0007373, financed by the European Union and from the state budget of the Czech Republic.
European Union (Funder ID: 10.13039/501100000780).