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.

## Introduction

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 [1]. The crack growth rates *da*/*dN* are expressed as a function of stress intensity factor range Δ*K*_{I} 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 Δ*K*_{th} 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 Δ*K*_{I} is defined as fatigue crack growth threshold Δ*K*_{th} [2].

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

*da*/

*dN*as a function of the stress intensity factor range Δ

*K*

_{I}for ferritic steels in air and water environments [1]. The relation between fatigue crack growth rate and stress intensity factor range is given by

where *n* is the slope of the log (*da*/*dN*) versus log (Δ*K*_{I}) line and is given by *n *=* *3.07 in air environment, and *C*_{0} is a scaling constant. The scaling constant in air environment is expressed as *C*_{0} = 3.78 × 10^{−9}*S*, 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* = *K*_{min}/*K*_{max}, where *K*_{max} and *K*_{min} are the maximum and minimum stress intensity factors associated with cyclic stress range of maximum stress *σ*_{max} and minimum stress *σ*_{min}, respectively. The units of Δ*K*_{I}, *K*_{max} and *K*_{min} are MPa√m, and *da*/*dN* is given by mm/cycle in this paper.

The stress intensity factor range Δ*K*_{I} is simply defined by Δ*K*_{I} = *K*_{max} − *K*_{min} for 0 ≤ *R* <1.0. However, for the case of *R *<* *0, the stress intensity factor range Δ*K*_{I} is complicated in an air environment. The Δ*K*_{I} under negative *R* ratio depends on the crack depth *a* and the flow stress *σ _{f}*. The flow stress is given by

*σ*= 0.5(

_{f}*σ*

_{y}+

*σ*

_{U}), where

*σ*is the yield strength and

_{y}*σ*is the ultimate tensile strength in units of MPa and

_{U}*a*is in units of

*m*. For −2 ≤

*R*<

*0 and*

*K*

_{max}−

*K*

_{min}≤ 1.12 σ

*√π*

_{f}*a*, the scaling parameter is

*S*=

*1, and*

*ΔK*

_{I}=

*K*

_{max}. For the case of

*R*< −2 and

*K*

_{max}−

*K*

_{min}≤ 1.12 σ

*√π*

_{f}*a*,

*S*=

*1 and*

*ΔK*

_{I}= (1 −

*R*)

*K*

_{max}/3. When

*R*<

*0 and*

*K*

_{max}−

*K*

_{min}> 1.12 σ

*√π*

_{f}*a*,

*S*=

*1 and*

*ΔK*

_{I}=

*K*

_{max}−

*K*

_{min}. Therefore, the stress intensity factor ranges

*ΔK*

_{I}under negative

*R*ratios are categorized as

*ΔK*

_{I}=

*K*

_{max},

*ΔK*

_{I}= (1 −

*R*)

*K*

_{max}/3 or

*ΔK*

_{I}=

*K*

_{max}−

*K*

_{min}, depending on the stress ratio, crack depth, and flow stress.

*ΔK*

_{I}<

*ΔK*

_{th}, the scaling constant is

*C*=

*0 and*

*ΔK*

_{th}is the threshold

*ΔK*

_{I}value below which fatigue crack growth rate is negligible. The threshold

*ΔK*

_{th}in Appendix A-4300 of the ASME Code Section XI is given as follows;

The threshold *ΔK*_{th} in units of MPa√m is a function of stress ratio *R*. Figure 1 shows the relationship between *ΔK*_{th} and *R* expressed by Eq. (2). The threshold *ΔK*_{th} increases with decreasing *R* ratio for 0 ≤ *R *<* *1 and *ΔK*_{th} is a constant for *R *<* *0. The threshold of *ΔK*_{th} = 5.5 MPa√m in Eq. (2) cannot be used itself for practical purposes.

## Threshold Δ*K*_{th} Under Negative *R* Ratio

The threshold *ΔK*_{th} is not well defined in the ASME Code Section XI, although the stress intensity factor ranges *ΔK*_{I} against fatigue crack growth rates of *da*/*dN*-*ΔK*_{I} are clearly defined, as mentioned above. The definition of the threshold under positive stress ratio *R* is, needless to say, *ΔK*_{th} = *K*_{max} − *K*_{min}. However, under negative stress ratio *R*, the threshold *ΔK*_{th} is supposed to be *ΔK*_{th} = *K*_{max}, *ΔK*_{th} = (1 − *R*) *K*_{max}/3 or *ΔK*_{th} = *K*_{max} − *K*_{min}, 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 *ΔK*_{th} as *K*_{max} - *K*_{min} ≤ 1.12 σ* _{f}* √π

*a*. Therefore, the thresholds under negative

*R*ratio are

*ΔK*

_{th}=

*K*

_{max}for −2 ≤

*R*<

*0 and*

*ΔK*

_{th}= (1 −

*R*)

*K*

_{max}/3 for

*R*< −2. The

*ΔK*

_{I}for

*R*< −2 is rewritten as

*ΔK*

_{I}= (1 −

*R*)

*K*

_{max}/3 = (

*K*

_{max}−

*K*

_{min})/3. Therefore,

*ΔK*

_{th}for

*R*< −2 is

*ΔK*

_{th}= (

*K*

_{max}−

*K*

_{min})/3. Then, the threshold

*ΔK*

_{th}shown in Fig. 1 can be used for practical purposes.

The threshold given by *ΔK*_{th} = 5.5 MPa√m for *R *<* *0 can be expressed by *K*_{max} using stress ratio *R*. The relationship between *K*_{max} at threshold and negative stress ratio *R* is illustrated in Fig. 2, together with *ΔK*_{th} at the positive *R* ratio. The *K*_{max} in Fig. 2 is equivalent to *ΔK*_{th} in Fig. 1. In the case of −2 ≤ *R *<* *0, it is a constant value of *K*_{max} = 5.5 MPa√m. When *R* < −2, the *K*_{max} decreases from 5.5 MPa√m at *R* = −2.0 to 2.75 MPa√m at *R* = −5.0 (see *K*_{max} in Table 1). The relation of *K*_{max} 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 *ΔK*_{I} derived from fatigue crack growth rates *da*/*dN* − *ΔK*_{I}. 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 *K*_{max} against *ΔK*_{th} under negative *R* ratio is inferred from the definition by ASTM (American Society of Testing and Materials) E647 [2]. The ASTM E647 recommends to use *ΔK*_{I} = *K*_{max} − *K*_{min} for 0 ≤ *R* and *ΔK*_{I} = *K*_{max} 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 *ΔK*_{I} = *K*_{max} 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

*K*

_{I}and

*K*

_{max}, crack closure under negative

*R*ratio is discussed herein. The stress intensity factor at crack closure is slightly lower than that at crack opening during fatigue crack growth [3]. Then, the crack closure and opening are treated as the same in this paper. Effective stress intensity factor range

*ΔK*

_{eff}represents the stress intensity factor range while crack tip is opening by the applied load cycle. The effective stress intensity factor range

*ΔK*

_{eff}is generally given as,

where *K*_{op} 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 [4] was based on only 0 ≤ *R*, and the curve of *U*_{EASON} was beyond the range of applicability. The Kurihara's model [5] 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 [6] was obtained by experimental data for 4340 steels. The Schijve's model was developed from tests of 2024 aluminum [7]. It is shown that crack closure *U* for all models decreases with decreasing *R* ratio.

*C*

_{0}, that is

*U*

_{ASME}= 0.6667/(1 −

*R*) for −2 ≤

*R*<

*0 [8]. Crack closure*

*U*

_{ASME}is shown to be applicable to extend less than

*R*= −2 [9]. Effective stress intensity factor ranges under negative

*R*ratios are obtained by the following crack closure

*U*.

Crack closure depends on applied stress levels. When the applied cyclic load ranges are large, fatigue cracks tend to open. This means that the *K*_{op} in Fig. 3 approaches the *K*_{min}, and the stress intensity factor ranges *ΔK*_{I} approach *K*_{max} − *K*_{min}. In the reverse case, the *K*_{op} moves toward to the *K*_{max}, 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 [10]. In accordance with the Newman's model, crack closure is calculated as a function of σ_{max}/σ* _{f}*, where σ

*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*

_{f}*U*and applied stress under negative

*R*ratio calculated by the Newman's model. For example, at

*R*= −2, when stress level is σ

_{max}/σ

*= 0.7, crack closure is calculated as*

_{f}*U*=

*0.388. At a small stress of σ*

_{max}/σ

*= 0.05,*

_{f}*U*=

*0.214 in accordance with the Newman's model [10]. Crack closure is*

*U*=

*0.214 at low stress level at σ*

_{max}/σ

*= 0.05 is close to the value of*

_{f}*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

*K*

_{op}at crack opening are obtained easily.

Table 1 shows the crack closure *K*_{max}*, U*, *K*_{op} and *K*_{eff} (=*K*_{max} − *K*_{op}) 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 *K*_{eff} 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 *K*_{eff} decreases from 3.66 to 1.83 MPa√m. These *K*_{eff} are apparently small values compared with *ΔK*_{th} = 5.5 MPa√m. In addition, the *K*_{eff} 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 *ΔK*_{eff} are smaller than 5.5 MPa√m and the threshold of *ΔK*_{th} = 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 *K*_{max} and the *K*_{min}. While decreasing the applied cyclic loading under constant ratio of *K*_{min}/*K*_{max} in accordance with the ASTM E 647 fatigue test method [2], *ΔK*_{I} at the order of *da*/*dN* = 10^{−8 }mm/cycle is judged to be the threshold. Therefore, the values of *ΔK*_{th} are obtained as *K*_{max} − *K*_{min}, experimentally, even when *R* is negative.

Figure 5 shows fatigue crack growth rates for ferritic steel obtained by one of the authors [11]. 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 *ΔK*_{th} 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 *ΔK*_{th} for the low alloy steel are shown in Table 3, where the definition of *ΔK*_{th} is *K*_{max} − *K*_{min}. Using *ΔK*_{th} = *K*_{max} − *K*_{min} and *R* = *K*_{min}/*K*_{max}, the values of *K*_{max}, *K*_{min} are easily calculated. Looking at Table 3, it is shown that the experimental thresholds *ΔK*_{th} increase with a decreasing entire range of *R* ratio. In addition, the *K*_{max} and *K*_{min} calculated by *ΔK*_{th} reduce with decreasing *R* ratio for negative *R*. It is obvious that the *K*_{max} is not constant under negative *R* ratio. Conclusively, the threshold *ΔK*_{th} is determined by the full range of stress intensity factor of *ΔK*_{th} = *K*_{max} − *K*_{min}, experimentally. The definition of the threshold is suitable to simply use *ΔK*_{th} = *K*_{max} − *K*_{min} 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 *ΔK*_{th} for *R *<* *0 are larger than those at *R *=* * 0 Several steels show that thresholds *ΔK*_{th} at *R* = −1 are smaller than those at *R *=* *0, as shown in Table 4 by the asterisk *. The definition of *ΔK*_{th} for these cases is not clearly written. These values of the thresholds *ΔK*_{th} at *R* = −1 are seemed to be *K*_{max}. If the thresholds *ΔK*_{th} are *K*_{max} under negative R ratio, the values of *ΔK*_{th} are converted to the description of *K*_{max} − *K*_{min}. Taking low alloy steel shown in Table 4 by asterisk * for example, the threshold of *K*_{max} = 6.3 MPa√m at *R* = −1.0 becomes 12.6 MPa√m expressed by *K*_{max} − *K*_{min}. 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 [19]. Almost all codes do not clearly provide the definition of the thresholds *ΔK*_{th} = *K*_{max} or *ΔK*_{th} = *K*_{max} − *K*_{min} under negative *R* ratio, except BS 7910 [20]. In addition, many codes give constant values of *ΔK*_{th} under negative *R* ratio, except API 579 [21].

Thresholds of fatigue crack growth rates are obtained as *ΔK*_{th} = *K*_{max} − *K*_{min} 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 *ΔK*_{th} 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 *ΔK*_{th} 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 *ΔK*_{th} = *K*_{max} − *K*_{min}, instead of effective stress intensity factor range *ΔK*_{eff} or maximum stress intensity factor *K*_{max} under negative *R* ratio. It is a benefit that the definition of *ΔK*_{th} = *K*_{max} − *K*_{min} 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 *ΔK*_{th} for various ferritic steels. The experimental data in Table 4 are plotted in Fig. 6 on the diagram of *ΔK*_{th} − *R* relation, where the definition of the thresholds is *ΔK*_{th} = *K*_{max} − *K*_{min} in the entire range of stress ratio *R*. The data on thresholds *ΔK*_{th} at *R* = −1 seemed to be *K*_{max} in Table 4 by asterisk * are not included in Fig. 6. The equation of *ΔK*_{th} = 5.5(1 − 0.8*R*) given by the ASME Code Section XI is also shown in Fig. 6. Although the equation of *ΔK*_{th} = 5.5(1 − 0.8*R*) 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 *ΔK*_{th} = 5.5(1 − 0.8*R*) is a lower bound of the experimental data, as shown in Fig. 6. Conclusively, the equation of *ΔK*_{th} = 5.5(1 − 0.8*R*) is applicable to negative *R* ratio.

*ΔK*

_{th}= 5.5(1 − 0.8

*R*) is applicable for extending to negative

*R*ratio, as mentioned above. When 0.8 <

*R*, the equation gives the threshold less than 2.0 MPa√m, which is smaller than the values of other fitness-for-service codes. All fitness-for-service codes give thresholds

*ΔK*

_{th}equal or greater than 2.0 MPa√m for ferritic steels. Therefore, the recommended

*ΔK*

_{th}for Appendix A in the ASME Code Section XI is

## Conclusion

Threshold *ΔK*_{th} of fatigue crack growth rates for ferritic steels in an air environment is provided by a constant value of *ΔK*_{th} = 5.5 MPa√m under negative stress ratio in Appendix A of the ASME Code Section XI. However, the definition of *ΔK*_{th} under negative *R* ratio is not clearly defined. It seems to be *ΔK*_{th} = *K*_{max} = 5.5 MPa√m. However, the *K*_{max} = 5.5 MPa√m is not close to the crack driving force, because effective stress intensity factors *ΔK*_{eff} are apparently small compared with the *K*_{max}. In addition, the expression on constant *K*_{max} is less conservative, because *K*_{max} decreases with decrease of *R* ratio.

From a literature survey on fatigue threshold data, almost all data were taken experimentally as *ΔK*_{th} = *K*_{max} − *K*_{min} for the entire range of stress ratio, and the definition of *ΔK*_{th} =*K*_{max} − *K*_{min} under negative *R* ratio is better to use from the practical view point of code users. In addition, the expression of *ΔK*_{th} = *K*_{max} − *K*_{min} 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 *ΔK*_{th} equation for ferritic steels in air environment, as *ΔK*_{th} = 5.5(1 − 0.8*R*) MPa√m for *R *<* *0.8 and *ΔK*_{th} = 2.0 MPa√m for 0.8 ≤ *R* for Appendix A-4300 in the ASME Code Section XI.

## Acknowledgment

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.

## Funding Data

European Union (Funder ID: 10.13039/501100000780).