Fatigue Crack Growth Thresholds at Negative Stress Ratio for Ferritic Steels in ASME Code Section XI

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 ΔK_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 ΔK_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 = 10⁻⁸mm/cycle, fatigue crack growth rate is negligible. The stress intensity factor range ΔK_I is defined as fatigue crack growth threshold ΔK_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.

where n is the slope of the log (da/dN) versus log (ΔK_I) 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 = 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 ΔK_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 ΔK_I is simply defined by ΔK_I = K_max − K_min for 0 ≤ R <1.0. However, for the case of R < 0, the stress intensity factor range ΔK_I is complicated in an air environment. The ΔK_I under negative R ratio depends on the crack depth a and the flow stress σ_f. The flow stress is given by σ_f = 0.5(σ_y + σ_U), where σ_y is the yield strength and σ_U is the ultimate tensile strength in units of MPa and a is in units of m. For −2 ≤ R < 0 and K_max − K_min ≤ 1.12 σ_f √πa, 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_f √πa, S = 1 and ΔK_I = (1 − R) K_max/3. When R < 0 and K_max − K_min > 1.12 σ_f √πa, 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.

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.

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 √πa. 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.

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.

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_f 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 σ_max/σ_f = 0.7, crack closure is calculated as U = 0.388. At a small stress of σ_max/σ_f = 0.05, 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_f = 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 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.

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 = 10⁻⁸mm/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 = 10⁻⁸mm/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.

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.

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 − 0.8R) given by the ASME Code Section XI is also shown in Fig. 6. Although the equation of ΔK_th = 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 ΔK_th = 5.5(1 − 0.8R) is a lower bound of the experimental data, as shown in Fig. 6. Conclusively, the equation of ΔK_th = 5.5(1 − 0.8R) is applicable to negative R ratio.

The results calculated by Eq. (6) are fairly in good agreement with the lower bound of the experimental data, as shown in Fig. 6.

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 − 0.8R) 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.

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).