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Interpreting load ratio dependence of near-threshold fatigue crack growth by a new crack closure model
International Journal of Pressure Vessels and Piping 110 (2013) 9e13
Contents lists available at SciVerse ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
Interpreting load ratio dependence of near-threshold fatigue crack growth by a new crack closure model Ming-Liang Zhu, Fu-Zhen Xuan*, Shan-Tung Tu Key Laboratory of Pressure System and Safety, MOE, School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China
a b s t r a c t Keywords: Fatigue crack growth Crack closure Near-threshold fatigue Load ratio Crack driving force
It was still short of generalized fatigue crack growth (FCG) models in the near-threshold regime due to its complex influencing factors. The near-threshold FCG behaviour of a rotor steel 25Cr2Ni2MoV at different load ratios was investigated experimentally, and the FCG driving mechanism was theoretically analysed based on equivalent driving force model. It was found that the crack growth process was determined by combined effects of equivalent driving force at constant amplitude loading and crack closure. A new crack closure model was proposed by considering the influences of load ratio and FCG rate, which could successfully interpret the effect of load ratio on FCG. The correlation of the crack closure model with the transition of driving forces in crack advance was beneficial to unify crack closure theory and crack growth driving parameters in the near-threshold regime. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Understanding the fatigue performance of materials and thus developing a service-life prediction method is a hot issue of structural integrity assessment. In practice, some components have complex loading conditions. For example, in rotating components, the rotors or blades undergo mechanical, thermal, residual and randomly vibrating stresses [1]. On one hand, both constant and variable amplitude fatigue need to be investigated. On the other hand, the load patterns in the components are featured by various stress ratios (the ratio of minimum to maximum loads, Pmin/Pmax), making the integrity assessment complicated. Therefore, it is critical to understand the fatigue mechanisms of materials at different load ratios. During the past several years, the load ratio dependence of fatigue behaviour has been widely investigated. Walker [2] developed an equation including the load ratio parameter, and it was always used in the Paris regime. Elber [3] proposed the crack closure concept in 1970s, since then the influence of the load ratio on long crack growth has been explained by an effective stress intensity range DKeff (DKeff ¼ Kmax Kc) approach based on the plasticity-induced crack closure mechanism. Subsequently the popular NASGRO [4] formula has been developed and incorporated
* Corresponding author. E-mail address: [email protected] (F.-Z. Xuan). 0308-0161/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijpvp.2013.04.015
into damage tolerant design process. In addition, models for correlating fatigue threshold with load ratio have been established in many materials and documented by Kwofie [5] and Bulloch [6]. Until now, it has been generally accepted that load ratio can affect fatigue threshold, but whether crack closure dominates nearthreshold fatigue crack growth (FCG) is still an open issue [7,8]. In fact, the key point in the crack closure concept is the determination of crack opening load. Newman Jr. [9] and Schijve [10] deduced equations to correlate crack closure level with stress ratio, load level and specimen constraint under constant amplitude loading procedure. However, it would be improper to transfer these kinds of crack closure equations into describing FCG in the loadshedding procedure in near-threshold regime. This has been verified by Yamada et al. [11e13] who found that the crack opening load was tended to increase with the decreasing of load level. Unfortunately, little has been reported in crack closure models for FCG in near-threshold regime. Another approach to load ratio dependence of FCG is from the perspective of crack growth driving force. Sadananda and Vasudevan [14e16] argued that both DK and Kmax parameters controlled FCG. While Kwofie [5,17] thought it was DK and Km that drive crack growth under cyclic loadings, and attempted to generalize da/dNe DK curve at different load ratios. However, whether crack closure plays a role in the transition of crack growth driving parameter is not clear yet. Therefore, in the present work, near-threshold FCG behaviour of a rotor steel 25Cr2Ni2MoV was investigated at various stress ratios.
10
M.-L. Zhu et al. / International Journal of Pressure Vessels and Piping 110 (2013) 9e13
Table 1 Tensile properties of the rotor steel 25Cr2Ni2MoV. Material
Elongation, Reduction Yield strength, Ultimate tensile strength, Rm (MPa) A (%) of area, Z (%) Rp0.2 (MPa)
25Cr2Ni2MoV 767.68
864.29
15.94
62.15
Both DKeff approach and crack growth driving force model were utilized to correlate FCG. A new crack closure model was proposed based on the influences of load ratio and FCG rate. Finally, the mechanisms of the load ratio and FCG rate dependences of crack closure were discussed from the view point of driving force transition behaviour in FCG. 2. Material and methods A rotor steel 25Cr2Ni2MoV was used in the experiment. The mechanical properties of the steel are listed in Table 1. A loadshedding procedure was carried out by using compact tensile (CT) specimens according to ASTM E647. The shape and dimensions of CT specimen are shown in Fig. 1. All the specimens were precracked and fatigue tests were performed at load ratios of 0.1, 0.3, 0.5, 0.7 and 0.9. During the test, the value of load was reduced continuously at a ratio of 5% until the crack growth ceased. The increment of crack propagation at each load level must be 4e6 times larger than the radius of crack tip plasticity zone yielded at the previous load level. Crack length was measured by an optical magnified microscopy with a resolution of 0.01 mm. 3. Results Fig. 2 shows the relationship between FCG rate da/dN and the range of stress intensity factor DK at different load ratios. It can be observed that da/dN changes almost linearly with DK at all load ratios when da/dN is above 106 mm/cyc. In the load-shedding process, the value of da/dN reduces accordingly. However, for da/ dN below 106 mm/cyc, da/dN varies nonlinearly with DK, and DK reduces slowly. The larger the load ratio, the higher the da/dN. Fatigue in this region is often referred to as near-threshold regime. In this work, DK at da/dN of 107 mm/cyc is defined as the fatigue threshold DKth. The threshold values at different load ratios are presented in Fig. 3. With R increasing, DKth decreases rapidly first and then becomes stable when R is higher than 0.7. By polynomial fitting, the relationship between DKth and R is illustrated in Eq. (1).
DKth ¼ 6:238 1:679,R 14:6,R2 þ 12:369,R3
13.75
4. Discussion 4.1. da/dN correlation with DKeff Crack closure plays important roles in near-threshold FCG. Usually, the crack closure concept, which takes actual crack growth driving force at crack tip into account, can explain the load ratio dependence of FCG behaviour in Paris regime. For example, Forth [18] developed an effective driving force model for crack propagation. As shown in Eq. (2), DKeff is dependent on DK and R.
DKeff ¼ DK 0:7 1:1R2 þ 0:4R3
. ð1 RÞ
(1)
50 30
30
62.5
12
30°
3
12.5
Fig. 1. Shape and dimensions of compact tension specimen used in fatigue threshold tests.
(2)
In this work, da/dN is also correlated with DKeff, and corresponding results are illustrated in Fig. 4. It can be observed that fatigue data at all R are converged into a single line when da/dN is larger than 2 106 mm/cyc. Therefore, in this region, FCG is independent of load ratio by using DKeff. However, the stress ratio influence cannot be neglected since da/dN below 2 106 mm/cyc scatter at various R. It is therefore concluded that DKeff can only explain the load ratio dependence in the region where FCG rate is higher. The reason is that Eq. (2) is based on plasticity induced crack closure under constant amplitude loading. It would definitely be unsuitable to the
12.5
13.75
Fig. 2. Relationship between da/dN and DK at different load ratios.
Fig. 3. Fatigue threshold versus load ratio.
M.-L. Zhu et al. / International Journal of Pressure Vessels and Piping 110 (2013) 9e13
Fig. 4. Relationship between da/dN and DKeff at different load ratios.
Fig. 5. The relationship between ln(DKR) and (1 þ R)/(1 R) at different load ratios.
case in which the load is reduced continuously, i.e., the nearthreshold fatigue regime. It can be deduced from Eq. (2) that the coefficient of crack closure (U ¼ DKeff/DK) is constant at each R, therefore, other parameters such as load levels are not considered. In the near-threshold fatigue regime, new crack closure models are required in order to unify da/dN curves for various load ratios. 4.2. The equivalent driving force method in FCG Due to the limitations of DKeff method, Kwofie [17] proposed an equivalent driving force model in FCG. In Kwofie’s model, fatigue damage is thought to be caused by both stress amplitude sa and mean stress sm. Eq. (3) shows the equivalent fatigue damage induced by load levels at R s 1 and the fatigue damage induced by load amplitude sar at R ¼ 1. This model is further transformed to equivalent fatigue damage at R s 1 and R ¼ 0, as shown in Eq. (4).
sar ¼ sa exp a
sm sa
DK0 ¼ DKR exp a
2R 1R
11
5. Crack closure model In this work, it is assumed that both crack closure and the equivalent driving force drive the growth of fatigue crack in the near-threshold regime. As for the results shown in Fig. 6, it is believed that Kwofie’s model eliminates the mean stress effect at high da/dN region. It is therefore deduced that the fatigue data differences in Fig. 6 will be solely resulted from the influence of crack closure. As mentioned before, the crack closure model of Forth [18] is not effective since U is a constant for one specific load ratio. Prior to propose a new crack closure model, two assumptions are made as follows: The crack opening load in the near-threshold fatigue regime is dependent on both load ratio R and FCG rate da/dN. The degree of crack closure is negligible at higher load ratios, i.e., R > 0.9.
(3)
Therefore, the crack opening load KO(R) at a load ratio of R can be written as,
(4)
KoðRÞ ¼ DKðRÞ DKð0:9Þ þ KminðRÞ
where DKR is the applied stress intensity range at load ratio R, a is the mean-stress-sensitivity factor of the material. This factor, which indicates how sensitive the crack growth behaviour of the material is to change in R, can be determined from the slope of ln(DKR) with (1 þ R)/(1 R). In this work, a is the slope of ln(DKR) with (1 þ R)/ (1 R) at a da/dN of 2.5 106 mm/cyc, as depicted in Fig. 5. According to Eq. (4), DK at various load ratios in Fig. 2 is changed to their equivalent driving force DK0 at R ¼ 0. It can be observed in Fig. 6 that da/dN at R of 0.1, 0.3 and 0.5 is tended to merge with each other, particularly when da/dN is above 1 106 mm/cyc. Similar results can be found at R of 0.7 and 0.9. However, fatigue data at lower and higher R are still discrepant, and da/dN scatters in the near-threshold regime. The discrepancies of fatigue data at lower and higher R are influenced by the a value, and are essentially related to the relative importance of stress amplitude and mean stress under higher and lower load ratios, as argued by Kwofie [17]. In Ref. [17], FCG is believed to be controlled by mean stress at higher R while it is stress amplitude dominated in lower R. It is therefore concluded that the equivalent driving force model is helpful to eliminate the fatigue data differences at higher or lower load ratios in higher da/dN region, but it cannot explain the effect of load ratio on FCG behaviour in the near-threshold regime.
(5)
where DK(R) and DK(0.9) are the range of stress intensity factor at a load ratio of R and 0.9, respectively. Here, the values of DK(R) and DK(0.9) are at the same level of da/dN, and R is less than 0.9 (R 0.9).
Fig. 6. The relationship of da/dN with equivalent DK0.
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M.-L. Zhu et al. / International Journal of Pressure Vessels and Piping 110 (2013) 9e13
From Eq. (5), the crack closure coefficient U(R) can be obtained at one da/dN,
UðRÞ ¼ ¼
UðRÞ ¼
ΔKeff Kmax Ko ¼ ΔK R Kmax Kmin R Kmax ðRÞ ΔKðRÞ þ ΔKð0:9Þ KminðRÞ ΔKðRÞ
DKð0:9Þ DKðRÞ
(6)
(7)
In this work, U(R) is calculated based on FCG data in Fig. 6 and Eq. (7), and is shown as a function of da/dN at different load ratios in Fig. 7. It is apparent that U(R) reduces gradually with the decreasing of da/dN at each load ratio. The reduction of U(R) indicates the increase of crack closure level. This means FCG shows an increasing trend of crack closure when approaching fatigue threshold in the load-shedding process, which agrees well with the higher crack opening load measured by Yamada et al. [11e13]. It is also worth noting that U(R) is smaller at lower load ratios at the same da/dN. In addition, U(R) differences between higher and lower load ratios are tended to increase with the decrease of da/dN. Nevertheless, U(R) values at a load ratio of 0.7 are considerably larger than those at lower load ratios such as 0.1, 0.3 and 0.5, which are close to each other. Therefore, the crack closure is serious when the load ratio is lower. In the double-logarithm axes as indicated in Fig. 7, the relationship between U(R) and da/dN is polynomially fitted into the form of Eq. (8). Here, U(R) is shown as a function of load ratio R and da/dN. In case of R equalling 0.1 and 0.7, the corresponding U(R)s are presented in Eqs. (9) and (10), respectively.
UðRÞ
2 3 da da da ¼ AðRÞ þ BðRÞ , þ DðRÞ , þ CðRÞ , dN dN dN
2 da da Uð0:1Þ ¼ 0:368 þ 93; 221$ þ 6:9 1010 $ dN dN 3 da 1:83 1016 $ dN
Fig. 7. Crack closure coefficient as a function of da/dN at various load ratios.
Fig. 8. The relationship between ln(DKR) and (1 þ R)/(1 R) at various da/dN.
2 da da 4:58 1010 $ dN dN 3 da þ 9:96 1015 $ dN
Uð0:7Þ ¼ 0:898 þ 106; 716$
(10)
The U(R) in Eq. (8) is a revised form of crack closure model, and is dependent on both load ratio and FCG rate. The significance for introducing FCG rate as a variable is that FCG rate is a resultant parameter that can include the influences of load level and specimen constraint on crack opening load. It is therefore believed that the proposed crack closure model can generalize da/dNeDK curves at various load ratios in both near-threshold and Paris regimes. 6. Transition of FCG driving parameter
(8)
(9)
As mentioned above, the U parameter varied between higher and lower load ratios, which is critical to load ratio dependence of FCG. Recently, the mechanism of FCG has also been interpreted by crack growth controlling parameters, such as the double-parameter model (DK and Kmax) proposed by Sadananda and Vasudevan [14e 16]. Whereas Kwofie [17] thought it was DK that dominated FCG at a load ratio below the transitional load ratio Rc, above which FCG is controlled by mean stress intensity factor Km.
Fig. 9. The relationship between DK and Kmax at various da/dN.
M.-L. Zhu et al. / International Journal of Pressure Vessels and Piping 110 (2013) 9e13
Fig. 8 shows the relationship between ln(DKR) and (1 þ R)/ (1 R) at various da/dN. The turning point in the curve stands for the transition of fatigue load ratio Rc. It can be found that the value of Rc is not a constant and increases with the decreasing of da/dN. For example, Rc at a da/dN of 2.5 106 mm/cyc is 0.53, and is then increased to 0.72 at a da/dN of 1.0 107 mm/cyc. Similarly, the increasing trend of Rc is also observed in Fig. 9. It is worth noting that the transition of crack growth driving parameter transfers from DK to Kmax at a load ratio of 0.5 at 2.5 106 mm/ cyc, and is finally increased to a load ratio of 0.7 at 1.0 107 mm/ cyc. It is argued that FCG is controlled by DK at lower load ratios. In this case, the parameter DK will be seriously influenced by crack closure, because DKapp will be reduced to DKeff. Whereas crack closure has a negligible effect on the parameter of Kmax. Thus, the degree of crack closure is lower at higher load ratios. And the load ratio dependence of crack closure can be interpreted through the transition of crack growth driving force from DK to Kmax. It is therefore concluded that the parallel approaches of crack closure model and crack growth driving force method are essentially unified theories. In Figs. 8 and 9, the enhance of the transition load ratio Rc would lead to the increasing range of load ratios within which FCG is driven by DK. Therefore, the influence of crack closure will be extended to extremely lower da/dN, which in turn rationalizes the crack closure behaviour at higher load ratios. 7. Conclusions In this work, the near-threshold FCG behaviour of a rotor steel 25Cr2Ni2MoV was investigated at different load ratios. The load ratio dependence of FCG mechanism was discussed based on crack closure model and FCG driving force method. The main conclusions are listed as follows: (1) Increasing the load ratio lowered fatigue threshold while enhanced the FCG rate in the near-threshold regime. Both the effective driving force model and the equivalent driving force method were insufficient to explain the load ratio dependence of near-threshold FCG behaviour. (2) FCG process was determined by combined effects of equivalent driving force at constant amplitude loading and crack closure. A new crack closure model was proposed by taking the load ratio and FCG rate into account, which could successfully interpret the influence of load ratio on FCG.
13
(3) The dependences of crack closure on the load ratio and FCG rate were further related with the transition of driving forces in crack advance, which were beneficial to unify crack closure theory and crack growth driving parameters in the nearthreshold regime. Acknowledgements The authors are grateful for the supports provided by National Natural Science Foundation of China (51205131) and the Natural Science Foundation of Shanghai (12ZR1442900). References [1] Wang W-Z, Xuan F-Z, Zhu K-L, Tu S-T. Failure analysis of the final stage blade in steam turbine. Eng Fail Anal 2007;14:632e41. [2] Walker K. The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7075-T6 aluminum. ASTM STP 462. American Society for Testing and Materials; 1970. [3] Elber W. The significance of fatigue crack closure. In: Damage tolerance in aircraft structures, ASTM STP 486, Canada 1970. p. 230e42. [4] NASGRO reference manual e version 4.02. NASA Johnson Space Center and Southwest Research Institute; 2002. [5] Kwofie S. Equivalent stress approach to prediction the effect of stress ratio on fatigue threshold stress intensity range. Int J Fatigue 2004;26:299e303. [6] Bulloch JH. Fatigue threshold in steels-Mean stress and microstructure influences. Int J Press Vessel Pip 1994;58:103e27. [7] McEvily AJ, Ritchie RO. Crack closure and the fatigue-crack propagation threshold as a function of load ratio. Fatigue Fract Mater Struct 1998;21:847e 55. [8] Hans-Joachim G, Johanna S. Influence of Kmax and R on fatigue crack growth e a 3D-model. Adv Eng Mater 2010;12:283e7. [9] Newman Jr JC. A crack opening stress equation for fatigue crack growth. Int J Fract 1984;24:R131e5. [10] Schijve J. Some formulas for the crack opening stress level. Eng Fract Mech 1981;14:461e5. [11] Yamada Y, Newman Jr JC. Crack closure under high load-ratio conditions for Inconel-718 near threshold behavior. Eng Fract Mech 2009;76:209e20. [12] Yamada Y, Newman Jr JC. Crack-closure behavior of 2324-T39 aluminum alloy near-threshold conditions for high load ratio and constant Kmax tests. Int J Fatigue 2009;31:1780e7. [13] Yamada Y, Newman Jr JC. Crack closure under high load ratio and Kmax test conditions. Procedia Eng 2010;2:71e82. [14] Sadananda K. Factors governing near-threshold fatigue crack growth. In: 2nd International conference on fatigue and fatigue thresholds. Birmingham, England: Engineering Materials Advisory Services Ltd; 1984. p. 543e53. [15] Sadananda K, Vasudevan AK. Fatigue crack growth mechanisms in steels. Int J Fatigue 2003;25:899e914. [16] Sadananda K, Vasudevan AK. Fatigue crack growth behavior of titanium alloys. Int J Fatigue 2005;27:1255e66. [17] Kwofie S, Rahbar N. An equivalent driving force model for crack growth prediction under different stress ratios. Int J Fatigue 2011;33:1199e204. [18] Forth SC, Newman Jr JC, Forman RG. On generating fatigue crack growth thresholds. Int J Fatigue 2003;25:9e15.
Contents lists available at SciVerse ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
Interpreting load ratio dependence of near-threshold fatigue crack growth by a new crack closure model Ming-Liang Zhu, Fu-Zhen Xuan*, Shan-Tung Tu Key Laboratory of Pressure System and Safety, MOE, School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China
a b s t r a c t Keywords: Fatigue crack growth Crack closure Near-threshold fatigue Load ratio Crack driving force
It was still short of generalized fatigue crack growth (FCG) models in the near-threshold regime due to its complex influencing factors. The near-threshold FCG behaviour of a rotor steel 25Cr2Ni2MoV at different load ratios was investigated experimentally, and the FCG driving mechanism was theoretically analysed based on equivalent driving force model. It was found that the crack growth process was determined by combined effects of equivalent driving force at constant amplitude loading and crack closure. A new crack closure model was proposed by considering the influences of load ratio and FCG rate, which could successfully interpret the effect of load ratio on FCG. The correlation of the crack closure model with the transition of driving forces in crack advance was beneficial to unify crack closure theory and crack growth driving parameters in the near-threshold regime. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Understanding the fatigue performance of materials and thus developing a service-life prediction method is a hot issue of structural integrity assessment. In practice, some components have complex loading conditions. For example, in rotating components, the rotors or blades undergo mechanical, thermal, residual and randomly vibrating stresses [1]. On one hand, both constant and variable amplitude fatigue need to be investigated. On the other hand, the load patterns in the components are featured by various stress ratios (the ratio of minimum to maximum loads, Pmin/Pmax), making the integrity assessment complicated. Therefore, it is critical to understand the fatigue mechanisms of materials at different load ratios. During the past several years, the load ratio dependence of fatigue behaviour has been widely investigated. Walker [2] developed an equation including the load ratio parameter, and it was always used in the Paris regime. Elber [3] proposed the crack closure concept in 1970s, since then the influence of the load ratio on long crack growth has been explained by an effective stress intensity range DKeff (DKeff ¼ Kmax Kc) approach based on the plasticity-induced crack closure mechanism. Subsequently the popular NASGRO [4] formula has been developed and incorporated
* Corresponding author. E-mail address: [email protected] (F.-Z. Xuan). 0308-0161/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijpvp.2013.04.015
into damage tolerant design process. In addition, models for correlating fatigue threshold with load ratio have been established in many materials and documented by Kwofie [5] and Bulloch [6]. Until now, it has been generally accepted that load ratio can affect fatigue threshold, but whether crack closure dominates nearthreshold fatigue crack growth (FCG) is still an open issue [7,8]. In fact, the key point in the crack closure concept is the determination of crack opening load. Newman Jr. [9] and Schijve [10] deduced equations to correlate crack closure level with stress ratio, load level and specimen constraint under constant amplitude loading procedure. However, it would be improper to transfer these kinds of crack closure equations into describing FCG in the loadshedding procedure in near-threshold regime. This has been verified by Yamada et al. [11e13] who found that the crack opening load was tended to increase with the decreasing of load level. Unfortunately, little has been reported in crack closure models for FCG in near-threshold regime. Another approach to load ratio dependence of FCG is from the perspective of crack growth driving force. Sadananda and Vasudevan [14e16] argued that both DK and Kmax parameters controlled FCG. While Kwofie [5,17] thought it was DK and Km that drive crack growth under cyclic loadings, and attempted to generalize da/dNe DK curve at different load ratios. However, whether crack closure plays a role in the transition of crack growth driving parameter is not clear yet. Therefore, in the present work, near-threshold FCG behaviour of a rotor steel 25Cr2Ni2MoV was investigated at various stress ratios.
10
M.-L. Zhu et al. / International Journal of Pressure Vessels and Piping 110 (2013) 9e13
Table 1 Tensile properties of the rotor steel 25Cr2Ni2MoV. Material
Elongation, Reduction Yield strength, Ultimate tensile strength, Rm (MPa) A (%) of area, Z (%) Rp0.2 (MPa)
25Cr2Ni2MoV 767.68
864.29
15.94
62.15
Both DKeff approach and crack growth driving force model were utilized to correlate FCG. A new crack closure model was proposed based on the influences of load ratio and FCG rate. Finally, the mechanisms of the load ratio and FCG rate dependences of crack closure were discussed from the view point of driving force transition behaviour in FCG. 2. Material and methods A rotor steel 25Cr2Ni2MoV was used in the experiment. The mechanical properties of the steel are listed in Table 1. A loadshedding procedure was carried out by using compact tensile (CT) specimens according to ASTM E647. The shape and dimensions of CT specimen are shown in Fig. 1. All the specimens were precracked and fatigue tests were performed at load ratios of 0.1, 0.3, 0.5, 0.7 and 0.9. During the test, the value of load was reduced continuously at a ratio of 5% until the crack growth ceased. The increment of crack propagation at each load level must be 4e6 times larger than the radius of crack tip plasticity zone yielded at the previous load level. Crack length was measured by an optical magnified microscopy with a resolution of 0.01 mm. 3. Results Fig. 2 shows the relationship between FCG rate da/dN and the range of stress intensity factor DK at different load ratios. It can be observed that da/dN changes almost linearly with DK at all load ratios when da/dN is above 106 mm/cyc. In the load-shedding process, the value of da/dN reduces accordingly. However, for da/ dN below 106 mm/cyc, da/dN varies nonlinearly with DK, and DK reduces slowly. The larger the load ratio, the higher the da/dN. Fatigue in this region is often referred to as near-threshold regime. In this work, DK at da/dN of 107 mm/cyc is defined as the fatigue threshold DKth. The threshold values at different load ratios are presented in Fig. 3. With R increasing, DKth decreases rapidly first and then becomes stable when R is higher than 0.7. By polynomial fitting, the relationship between DKth and R is illustrated in Eq. (1).
DKth ¼ 6:238 1:679,R 14:6,R2 þ 12:369,R3
13.75
4. Discussion 4.1. da/dN correlation with DKeff Crack closure plays important roles in near-threshold FCG. Usually, the crack closure concept, which takes actual crack growth driving force at crack tip into account, can explain the load ratio dependence of FCG behaviour in Paris regime. For example, Forth [18] developed an effective driving force model for crack propagation. As shown in Eq. (2), DKeff is dependent on DK and R.
DKeff ¼ DK 0:7 1:1R2 þ 0:4R3
. ð1 RÞ
(1)
50 30
30
62.5
12
30°
3
12.5
Fig. 1. Shape and dimensions of compact tension specimen used in fatigue threshold tests.
(2)
In this work, da/dN is also correlated with DKeff, and corresponding results are illustrated in Fig. 4. It can be observed that fatigue data at all R are converged into a single line when da/dN is larger than 2 106 mm/cyc. Therefore, in this region, FCG is independent of load ratio by using DKeff. However, the stress ratio influence cannot be neglected since da/dN below 2 106 mm/cyc scatter at various R. It is therefore concluded that DKeff can only explain the load ratio dependence in the region where FCG rate is higher. The reason is that Eq. (2) is based on plasticity induced crack closure under constant amplitude loading. It would definitely be unsuitable to the
12.5
13.75
Fig. 2. Relationship between da/dN and DK at different load ratios.
Fig. 3. Fatigue threshold versus load ratio.
M.-L. Zhu et al. / International Journal of Pressure Vessels and Piping 110 (2013) 9e13
Fig. 4. Relationship between da/dN and DKeff at different load ratios.
Fig. 5. The relationship between ln(DKR) and (1 þ R)/(1 R) at different load ratios.
case in which the load is reduced continuously, i.e., the nearthreshold fatigue regime. It can be deduced from Eq. (2) that the coefficient of crack closure (U ¼ DKeff/DK) is constant at each R, therefore, other parameters such as load levels are not considered. In the near-threshold fatigue regime, new crack closure models are required in order to unify da/dN curves for various load ratios. 4.2. The equivalent driving force method in FCG Due to the limitations of DKeff method, Kwofie [17] proposed an equivalent driving force model in FCG. In Kwofie’s model, fatigue damage is thought to be caused by both stress amplitude sa and mean stress sm. Eq. (3) shows the equivalent fatigue damage induced by load levels at R s 1 and the fatigue damage induced by load amplitude sar at R ¼ 1. This model is further transformed to equivalent fatigue damage at R s 1 and R ¼ 0, as shown in Eq. (4).
sar ¼ sa exp a
sm sa
DK0 ¼ DKR exp a
2R 1R
11
5. Crack closure model In this work, it is assumed that both crack closure and the equivalent driving force drive the growth of fatigue crack in the near-threshold regime. As for the results shown in Fig. 6, it is believed that Kwofie’s model eliminates the mean stress effect at high da/dN region. It is therefore deduced that the fatigue data differences in Fig. 6 will be solely resulted from the influence of crack closure. As mentioned before, the crack closure model of Forth [18] is not effective since U is a constant for one specific load ratio. Prior to propose a new crack closure model, two assumptions are made as follows: The crack opening load in the near-threshold fatigue regime is dependent on both load ratio R and FCG rate da/dN. The degree of crack closure is negligible at higher load ratios, i.e., R > 0.9.
(3)
Therefore, the crack opening load KO(R) at a load ratio of R can be written as,
(4)
KoðRÞ ¼ DKðRÞ DKð0:9Þ þ KminðRÞ
where DKR is the applied stress intensity range at load ratio R, a is the mean-stress-sensitivity factor of the material. This factor, which indicates how sensitive the crack growth behaviour of the material is to change in R, can be determined from the slope of ln(DKR) with (1 þ R)/(1 R). In this work, a is the slope of ln(DKR) with (1 þ R)/ (1 R) at a da/dN of 2.5 106 mm/cyc, as depicted in Fig. 5. According to Eq. (4), DK at various load ratios in Fig. 2 is changed to their equivalent driving force DK0 at R ¼ 0. It can be observed in Fig. 6 that da/dN at R of 0.1, 0.3 and 0.5 is tended to merge with each other, particularly when da/dN is above 1 106 mm/cyc. Similar results can be found at R of 0.7 and 0.9. However, fatigue data at lower and higher R are still discrepant, and da/dN scatters in the near-threshold regime. The discrepancies of fatigue data at lower and higher R are influenced by the a value, and are essentially related to the relative importance of stress amplitude and mean stress under higher and lower load ratios, as argued by Kwofie [17]. In Ref. [17], FCG is believed to be controlled by mean stress at higher R while it is stress amplitude dominated in lower R. It is therefore concluded that the equivalent driving force model is helpful to eliminate the fatigue data differences at higher or lower load ratios in higher da/dN region, but it cannot explain the effect of load ratio on FCG behaviour in the near-threshold regime.
(5)
where DK(R) and DK(0.9) are the range of stress intensity factor at a load ratio of R and 0.9, respectively. Here, the values of DK(R) and DK(0.9) are at the same level of da/dN, and R is less than 0.9 (R 0.9).
Fig. 6. The relationship of da/dN with equivalent DK0.
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From Eq. (5), the crack closure coefficient U(R) can be obtained at one da/dN,
UðRÞ ¼ ¼
UðRÞ ¼
ΔKeff Kmax Ko ¼ ΔK R Kmax Kmin R Kmax ðRÞ ΔKðRÞ þ ΔKð0:9Þ KminðRÞ ΔKðRÞ
DKð0:9Þ DKðRÞ
(6)
(7)
In this work, U(R) is calculated based on FCG data in Fig. 6 and Eq. (7), and is shown as a function of da/dN at different load ratios in Fig. 7. It is apparent that U(R) reduces gradually with the decreasing of da/dN at each load ratio. The reduction of U(R) indicates the increase of crack closure level. This means FCG shows an increasing trend of crack closure when approaching fatigue threshold in the load-shedding process, which agrees well with the higher crack opening load measured by Yamada et al. [11e13]. It is also worth noting that U(R) is smaller at lower load ratios at the same da/dN. In addition, U(R) differences between higher and lower load ratios are tended to increase with the decrease of da/dN. Nevertheless, U(R) values at a load ratio of 0.7 are considerably larger than those at lower load ratios such as 0.1, 0.3 and 0.5, which are close to each other. Therefore, the crack closure is serious when the load ratio is lower. In the double-logarithm axes as indicated in Fig. 7, the relationship between U(R) and da/dN is polynomially fitted into the form of Eq. (8). Here, U(R) is shown as a function of load ratio R and da/dN. In case of R equalling 0.1 and 0.7, the corresponding U(R)s are presented in Eqs. (9) and (10), respectively.
UðRÞ
2 3 da da da ¼ AðRÞ þ BðRÞ , þ DðRÞ , þ CðRÞ , dN dN dN
2 da da Uð0:1Þ ¼ 0:368 þ 93; 221$ þ 6:9 1010 $ dN dN 3 da 1:83 1016 $ dN
Fig. 7. Crack closure coefficient as a function of da/dN at various load ratios.
Fig. 8. The relationship between ln(DKR) and (1 þ R)/(1 R) at various da/dN.
2 da da 4:58 1010 $ dN dN 3 da þ 9:96 1015 $ dN
Uð0:7Þ ¼ 0:898 þ 106; 716$
(10)
The U(R) in Eq. (8) is a revised form of crack closure model, and is dependent on both load ratio and FCG rate. The significance for introducing FCG rate as a variable is that FCG rate is a resultant parameter that can include the influences of load level and specimen constraint on crack opening load. It is therefore believed that the proposed crack closure model can generalize da/dNeDK curves at various load ratios in both near-threshold and Paris regimes. 6. Transition of FCG driving parameter
(8)
(9)
As mentioned above, the U parameter varied between higher and lower load ratios, which is critical to load ratio dependence of FCG. Recently, the mechanism of FCG has also been interpreted by crack growth controlling parameters, such as the double-parameter model (DK and Kmax) proposed by Sadananda and Vasudevan [14e 16]. Whereas Kwofie [17] thought it was DK that dominated FCG at a load ratio below the transitional load ratio Rc, above which FCG is controlled by mean stress intensity factor Km.
Fig. 9. The relationship between DK and Kmax at various da/dN.
M.-L. Zhu et al. / International Journal of Pressure Vessels and Piping 110 (2013) 9e13
Fig. 8 shows the relationship between ln(DKR) and (1 þ R)/ (1 R) at various da/dN. The turning point in the curve stands for the transition of fatigue load ratio Rc. It can be found that the value of Rc is not a constant and increases with the decreasing of da/dN. For example, Rc at a da/dN of 2.5 106 mm/cyc is 0.53, and is then increased to 0.72 at a da/dN of 1.0 107 mm/cyc. Similarly, the increasing trend of Rc is also observed in Fig. 9. It is worth noting that the transition of crack growth driving parameter transfers from DK to Kmax at a load ratio of 0.5 at 2.5 106 mm/ cyc, and is finally increased to a load ratio of 0.7 at 1.0 107 mm/ cyc. It is argued that FCG is controlled by DK at lower load ratios. In this case, the parameter DK will be seriously influenced by crack closure, because DKapp will be reduced to DKeff. Whereas crack closure has a negligible effect on the parameter of Kmax. Thus, the degree of crack closure is lower at higher load ratios. And the load ratio dependence of crack closure can be interpreted through the transition of crack growth driving force from DK to Kmax. It is therefore concluded that the parallel approaches of crack closure model and crack growth driving force method are essentially unified theories. In Figs. 8 and 9, the enhance of the transition load ratio Rc would lead to the increasing range of load ratios within which FCG is driven by DK. Therefore, the influence of crack closure will be extended to extremely lower da/dN, which in turn rationalizes the crack closure behaviour at higher load ratios. 7. Conclusions In this work, the near-threshold FCG behaviour of a rotor steel 25Cr2Ni2MoV was investigated at different load ratios. The load ratio dependence of FCG mechanism was discussed based on crack closure model and FCG driving force method. The main conclusions are listed as follows: (1) Increasing the load ratio lowered fatigue threshold while enhanced the FCG rate in the near-threshold regime. Both the effective driving force model and the equivalent driving force method were insufficient to explain the load ratio dependence of near-threshold FCG behaviour. (2) FCG process was determined by combined effects of equivalent driving force at constant amplitude loading and crack closure. A new crack closure model was proposed by taking the load ratio and FCG rate into account, which could successfully interpret the influence of load ratio on FCG.
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(3) The dependences of crack closure on the load ratio and FCG rate were further related with the transition of driving forces in crack advance, which were beneficial to unify crack closure theory and crack growth driving parameters in the nearthreshold regime. Acknowledgements The authors are grateful for the supports provided by National Natural Science Foundation of China (51205131) and the Natural Science Foundation of Shanghai (12ZR1442900). References [1] Wang W-Z, Xuan F-Z, Zhu K-L, Tu S-T. Failure analysis of the final stage blade in steam turbine. Eng Fail Anal 2007;14:632e41. [2] Walker K. The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7075-T6 aluminum. ASTM STP 462. American Society for Testing and Materials; 1970. [3] Elber W. The significance of fatigue crack closure. In: Damage tolerance in aircraft structures, ASTM STP 486, Canada 1970. p. 230e42. [4] NASGRO reference manual e version 4.02. NASA Johnson Space Center and Southwest Research Institute; 2002. [5] Kwofie S. Equivalent stress approach to prediction the effect of stress ratio on fatigue threshold stress intensity range. Int J Fatigue 2004;26:299e303. [6] Bulloch JH. Fatigue threshold in steels-Mean stress and microstructure influences. Int J Press Vessel Pip 1994;58:103e27. [7] McEvily AJ, Ritchie RO. Crack closure and the fatigue-crack propagation threshold as a function of load ratio. Fatigue Fract Mater Struct 1998;21:847e 55. [8] Hans-Joachim G, Johanna S. Influence of Kmax and R on fatigue crack growth e a 3D-model. Adv Eng Mater 2010;12:283e7. [9] Newman Jr JC. A crack opening stress equation for fatigue crack growth. Int J Fract 1984;24:R131e5. [10] Schijve J. Some formulas for the crack opening stress level. Eng Fract Mech 1981;14:461e5. [11] Yamada Y, Newman Jr JC. Crack closure under high load-ratio conditions for Inconel-718 near threshold behavior. Eng Fract Mech 2009;76:209e20. [12] Yamada Y, Newman Jr JC. Crack-closure behavior of 2324-T39 aluminum alloy near-threshold conditions for high load ratio and constant Kmax tests. Int J Fatigue 2009;31:1780e7. [13] Yamada Y, Newman Jr JC. Crack closure under high load ratio and Kmax test conditions. Procedia Eng 2010;2:71e82. [14] Sadananda K. Factors governing near-threshold fatigue crack growth. In: 2nd International conference on fatigue and fatigue thresholds. Birmingham, England: Engineering Materials Advisory Services Ltd; 1984. p. 543e53. [15] Sadananda K, Vasudevan AK. Fatigue crack growth mechanisms in steels. Int J Fatigue 2003;25:899e914. [16] Sadananda K, Vasudevan AK. Fatigue crack growth behavior of titanium alloys. Int J Fatigue 2005;27:1255e66. [17] Kwofie S, Rahbar N. An equivalent driving force model for crack growth prediction under different stress ratios. Int J Fatigue 2011;33:1199e204. [18] Forth SC, Newman Jr JC, Forman RG. On generating fatigue crack growth thresholds. Int J Fatigue 2003;25:9e15.