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Modeling R-dependence of near-threshold fatigue crack growth by combining crack closure and exponential mean stress model
International Journal of Fatigue 122 (2019) 93–105
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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Modeling R-dependence of near-threshold fatigue crack growth by combining crack closure and exponential mean stress model Samuel Kwofiea, Ming-Liang Zhub,
T
⁎
a
Department of Materials Engineering, College of Engineering, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Key Laboratory of Pressure Systems and Safety, Ministry of Education, School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China
b
A R T I C LE I N FO
A B S T R A C T
Keywords: Fatigue crack growth Stress ratio Mean stress Crack closure Fatigue threshold
A previously proposed equivalent driving force model was modified to account for crack closure for modeling stress ratio, R, dependence of fatigue crack growth in near-threshold regime. A new fatigue crack growth driving force model was established in the form of ΔK⋅f(R), where the f(R) includes both crack closure and mean stress contributions, and well informs the shift from dominance of stress intensity factor range ΔK to maximum stress intensity factor Kmax with increasing R. The model was validated widely with sound agreement between modeling results and experimental data.
1. Introduction
elastic fracture mechanics is applicable, has been widely investigated both experimentally [2–7] and analytically [8,9]. It is generally accepted that an increase of R results in higher FCG rate at a given ΔK and in lower ΔKth in near-threshold regime [10–13]. Mechanistic explanations for such behavior have focused on either the presence of crack closure at lower R [2], or the occurrence of static fracture modes at higher R [14]. Models have been proposed to describe the observed crack growth behavior, with those of Elber [15] and Walker [16] being the most widely known. During the past several decades, great progress has been achieved on crack closure, such as the work by Newman [17], McClung et al. [18], and Schijve [19]. A complete historic review of the research and development of R-dependence of crack closure has been reported by one of the current authors [13]. Quite recently, an equivalent driving force (EDF) approach [20], which is an exponential mean stress model, was proposed by one of the current authors for correlation of FCG data at different R. The EDF model was expressed as,
An important aspect of engineering design is the design against fatigue while component is in service. For fatigue-prone bodies where crack initiation stage constitutes a small fraction of total life of engineering materials, the “damage tolerant” approach to design is often utilized. In this approach, prediction of remaining fatigue life is based on modeling of crack propagation of a flaw (or crack) whose size can be detected or inspected by non-destructive techniques. According to Paris-Erdogan, the rule of fatigue crack growth (FCG) can be represented as [1],
da/dN = C(ΔK) m
(1)
where da/dN is fatigue crack growth rate, a is crack size, N is cyclic number, ΔK is the stress intensity factor range, and C and m are material constants, and are dependent on loading and environmental conditions. Theoretically, a log–log plot of da/dN against ΔK is expected to yield a straight line graph with slope m and intercept of log(C). However, such plot of experimental data yields a sigmoidal curve showing three regimes of crack propagation: (i) near-threshold regime with ΔK close to ΔKth, where ΔKth is the threshold stress intensity factor range below which crack growth does not occur; (ii) intermediate regime, where crack growth is stable and follows the Paris-Erdogan law; and (iii) unstable regime, where the maximum stress intensity factor Kmax approaches the fracture toughness, Kc. Furthermore, the parameters C and m vary with stress ratio, R. The influence of R on FCG behavior of long cracks, where linear ⁎
1+R ⎞ ⎞ ΔK eq = ΔKRexp ⎛α ⎛ ⎝ ⎝ 1 − R ⎠⎠
(2a)
where ΔKeq is the equivalent form of the applied ΔK, ΔKR, at corresponding R, and α is mean stress sensitivity factor. At a fixed rate of crack growth, ΔKeq remains constant, and the ΔKR (applied ΔK), is thus derived as Eq. (2b) or Eq. (2c).
1+R ⎞ ⎞ ΔKR = ΔK eqexp ⎛−α ⎛ ⎝ ⎝ 1 − R ⎠⎠
Corresponding author. E-mail address: [email protected] (M.-L. Zhu).
https://doi.org/10.1016/j.ijfatigue.2019.01.006 Received 18 July 2018; Received in revised form 16 December 2018; Accepted 14 January 2019 Available online 17 January 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.
(2b)
International Journal of Fatigue 122 (2019) 93–105
S. Kwofie, M.-L. Zhu
1+R ⎞ In(ΔKR) = In(ΔK eq) − α ⎛ ⎝1 − R ⎠
1+R ⎞ ⎞ ΔK eq = (Kmax − K op)exp ⎛α ⎛ ⎝ ⎝ 1 − R ⎠⎠
(2c)
According to the EDF model, a plot of experimental data of ln(ΔKR) 1+R versus 1 − R for −1 < R < 1 was expected to yield a straight line with a slope value of -α and y-axis intercept ln(ΔKeq) being ln(ΔK−1), where ΔK-1 is the stress intensity factor range at R = −1. However, plots of crack growth data of Al- and Ti-alloys at constant growth rates between 1 × 10−7 mm/cycle and 1 × 10−5 mm/cycle, yielded two straight lines with different slopes, resulting in two different values of α, and the two lines intersected at R = Rc ≈ 1/3, as reported in [20]. Zhu et al. [10] applied the EDF model to correlate FCG data of 25Cr2Ni2MoV rotor steel, and found that Rc was not a constant but dependent on the level of crack growth rate, which was associated with crack closure. Zhu et al. [10] also found that the EDF model accurately described the R-dependence of FCG data in higher da/dN region, but was unable to explain the effect of R on FCG behavior in the nearthreshold regime, where FCG is much lower. The contribution Zhu et al. has achieved is the finding and modeling of the dependence of crack closure on both R and FCG level in the near-threshold regime. Recently, Zhu et al. [13] proposed a combined crack growth driving force and crack closure approach to interpret the R-dependence of FCG in both Paris and near-threshold regimes, and the newly developed FCG model had sound predictive capability of FCG behavior in various Cr-Mo-V steels. This indicates the robustness of combination approach to R-dependence of FCG, and perhaps a general approach to modeling R-effect on long crack FCG behavior. In terms of the EDF model [20], when applied, differentiates FCG at lower and higher R, as observed from the Rc, and this may be explained in terms of varied crack closure roles or influences below and above Rc. However, such crack closure roles or influences are not explicitly included in and represented by the exponential mean stress model [20]. This drives us to revisit the EDF model to account for crack closure effects for modification, in a different way with that proposed before [10,11,13], and to make it more relevant to explain the R-dependence of FCG behavior in both intermediate and near-threshold regimes. Therefore, in this work, the original model of Eq. (2a) was modified to incorporate crack closure effect in the near-threshold FCG regime to develop a more general FCG model. A relation between stress intensity range at threshold condition, ΔKthR, and R was derived and compared with published data of various ferrous and non-ferrous alloys. In addition, it was shown that the modified model was capable of collapsing FCG data of different R onto a master curve at R = 0. This is useful to predict FCG behavior at different R in the near-threshold regime as well as the Paris-Erdogan regime.
From Eqs. (3) and (4), ΔK eq , is thus redefined in terms of Eq. (5), for crack closure (5i) and closure-free (5ii) cases, as follows,
( )
ΔK eq =
( ( ) ), ( ( ) ),
⎧ (Kmax − K op)exp α ⎪ ⎨ (K ⎪ max − Kmin )exp α ⎩
1+R 1−R
1+R 1−R
for Kmin < K op (i) ⎫ ⎪ for Kmin ≥ K op (ii) ⎬ ⎪ ⎭
(5a)
Or
ΔK eq
(
K
) ( ( )) ( ( ))
⎧ Kmax 1 − op exp α 1 + R , for Kmin/ Kmax < K op/ Kmax (i) ⎫ ⎪ ⎪ Kmax 1−R = ⎨ K (1 − R)exp α 1 + R , for K /K ⎬ K / K (ii) ≥ min max op max ⎪ max ⎪ 1−R ⎩ ⎭ (5b) Now let R = Rc be the transition from closure to closure-free condiK tion, and assuming op remain constant for Kmin < K op at a particular Kmax da/dN level, then Eqs. (5)(i) and (5)(ii) would be equivalent as shown in Eq. (6a), from which a relation is derived as Eq. (6b). Note here the Kop is the crack opening level when R is Rc, and Eq. (6b) is a sound establishment only when R is Rc.
K op ⎞ 1 + Rc ⎞ 1 + Rc ⎞ ⎞ = Kmax (1 − Rc )exp ⎛α ⎛ ⎞ exp ⎛α ⎛ Kmax ⎛1 − Kmax ⎠ ⎝ ⎝ 1 − Rc ⎠ ⎠ ⎝ ⎝ 1 − Rc ⎠ ⎠ ⎝ ⎜
⎟
(6a)
K op Kmax
= Rc
(6b)
Hence, from Eqs. (5b) and (6b), the ΔK eq is rearranged as,
ΔK eq
( ( ) ), ( ( ) ),
⎧ Kmax (1 − Rc )exp α ⎪ = ⎨ K (1 − R)exp α ⎪ max ⎩
1+R 1−R
1+R 1−R
Kop Kmax
≤ Rc (i) ⎫ ⎪ for R ≥ Rc (ii) ⎬ ⎪ ⎭
for R <
(7a)
or
ΔK eq
(
) ( ( ) ), ( ( ))
⎧ ΔKR 1 − Rc exp α 1 + R ⎪ 1−R 1−R = 1+R ⎨ ΔKRexp α 1 − R , ⎪ ⎩
for R < Rc (i) ⎫ ⎪ for R ≥ Rc (ii) ⎬ ⎪ ⎭
(7b)
Note in Eq. (7a), the value of Kop is normally within R and Rc, while Kmax here it is assumed to be Rc for simplicity based on an almost constant Kmax for R less than Rc, and also due to the difficulty of obtaining Kop ΔK values at a particular R and da/dN level. As Kmax = 1 − RR , Eq. (7b) is rewritten as Eq. (7c), where ΔK eff is the effective stress intensity factor range due to crack closure,
2. Modeling R-dependence of FCG behavior 2.1. A general driving force model To account for crack closure as well as transition of crack growth behavior from closure-affected to closure-free behavior, Eq. (2) needs to be modified. Firstly, Eq. (2) can be written as Eq. (3), where ΔKR = Kmax − Kmin.
1+R ⎞ ⎞ ΔK eq = (Kmax − Kmin )exp ⎛α ⎛ ⎝ ⎝ 1 − R ⎠⎠
(4)
1+R ⎞ ⎞ ΔK eq = ΔK eff exp ⎛α ⎛ ⎝ ⎝ 1 − R ⎠⎠
(7c)
Where (3)
ΔK eff =
where ΔK eq is the equivalent stress intensity range due to combined effects of applied stress intensity factor range ΔKR , and load ratio, R. Normally Eq. (3) is valid only when Kmin ≥ K op where Kop is the stress intensity factor at which crack is fully open. In the case of compressive stresses where Kmin < 0, assuming crack is closed and Kmin does not contribute to crack growth. Furthermore, it is known that crack closure may also occur when Kmin is positive but less than Kop. Therefore for all cases when Kmin < K op , K op replaces Kmin in Eq. (3) to yield Eq. (4).
(
)
⎧ ΔKR 11−−Rc , for R < Rc (i) ⎫ R ⎨ Δ , for R ≥ Rc (ii) ⎬ K R ⎩ ⎭
(7d)
Essentially, Eq. (7d(i)) describes the effective stress intensity factor range in the R-regime where crack closure influences FCG whereas Eq. (7d(ii)) describes the effective stress intensity factor range in the Rregime where crack-closure influence is minimal. From Eq. (7d) we obtain Eq. (7e) which has a form very similar to the traditional crack closure coefficient. Based on Eq. (7e), Eq. (7c) can be rearranged as Eq. (8), and a new expression f(R) is defined as Eq. (8c). 94
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S. Kwofie, M.-L. Zhu
Fig. 1. Left: Typical plot of f(R) versus R, showing the combined effects of crack closure and mean stress (Rc = 0.72, α = 0.001); Right: plot of U(R) versus R, showing the case when growth is insensitive to mean stress (α = 0 or f(R) = U(R)), but influenced solely by crack closure.
U (R) =
ΔK eff ΔKR
=
⎧ ⎨ ⎩
(
1 − Rc 1−R
),
1,
for − 1 ≤ R ≤ Rc (i) ⎫ for Rc ≤ R ≤ 1 (ii) ⎬ ⎭
1+R ⎞ ⎞ ΔK eq = ΔKR U (R)exp ⎛α ⎛ ⎝ ⎝ 1 − R ⎠⎠
ΔK eq = ΔKR f (R) = ΔK 0 f (0)
The Eq. (10a) is then combined with Eqs. (7e) and (8c), and yields Eq. (10b).
(7e)
( (
(10b)
(8b) At threshold condition, Eq. (10b) still holds and hence,
where
1+R ⎞ ⎞ f (R) = U (R)exp ⎛α ⎛ ⎝ ⎝ 1 − R ⎠⎠
( (
ΔKR f (R) = ΔKRef f (Ref )
(8c)
(11)
where ΔKth0 is the threshold stress intensity factor range at R = 0, and ΔKthR is the threshold stress intensity factor range, at any R. From Eq. (11) the effect of R on Kmax − th is deduced as:
Kmax − thR (1 − R) =
( (
⎧ Kmax − th0 (1 − R)exp −α 1 2−RR
) )
2R ⎨K max − th0 (1 − R c )exp − α 1 − R ⎩
for − 1 ≤ R < R c (i) ⎫ for R c ≤ R < 1
(ii) ⎬ ⎭
(12a)
or
(
)
⎧ exp −α 1 2−RR , for − 1 ≤ R < R c (i) ⎫ Kmax − thR = R R 1 − 2 c ⎨ Kmax − th0 exp −α 1 − R , for R c ≤ R < 1 (ii) ⎬ ⎩ 1−R ⎭
(
)
(12b)
where Kmax-thR is the Kmax-th value at any R, while Kmax-th0 is the Kmax-th value at R = 0. With known values of Rc and α, Eqs. (11) and (12b) can be employed to describe the R-dependence of ΔKthR and Kmax-thR of a given material at the threshold condition. In fact, from Eqs. (11) and (12b), Kmax − thR and ΔKthR can be well correlated as:
(9a)
Or
f (Ref ) f (R)
) )
⎧ (1 − R)exp −α 1 2−RR , for − 1 ≤ R < Rc (i) ⎫ ΔKthR = ⎨ (1 − Rc )exp −α 2R , for Rc ≤ R < 1 (ii) ⎬ ΔKth0 1−R ⎩ ⎭
It is indicated in Eq. (8) that the ΔKeq is represented as a combined form of ΔKR and f(R), and the f(R) includes both crack closure and mean stress effects. A plot of f(R) as a function of R for −1 ≤ R < 1 is shown in Fig. 1(left). Here the Rc and α are selected as 0.72 and 0.001, respectively. It is observed that for R ≤ Rc, where crack closure occurs, the value of f(R) is less than one, while for Rc < R < 1, where FCG is least affected by crack closure, the value of f(R) > 1. Note that for α = 0, f (R) = U (R) , which is the case when crack growth is insensitive to mean stress. A plot of U(R) versus R is shown in Fig. 1(right), where U < 1 for R < Rc and U = 1, for R ≥ Rc. It is observed from Fig. 1 that for the R-regime where crack closure effect is dominant in fatigue crack growth, the role of mean-stress in fatigue cracking is secondary. As an equivalent fatigue driving force, ΔK eq should be equal at all R, as a result, Eq.(8b) is written as Eq. (9), Where “Ref” refers to Reference stress ratio. It is inferred that
ΔKR = ΔKRef
) )
⎧ (1 − R)exp −α 1 2−RR , for − 1 ≤ R < Rc (i) ⎫ f (0) ΔKR = = ΔK 0 f (R) ⎨ (1 − Rc )exp −α 2R , for Rc ≤ R < 1 (ii) ⎬ 1−R ⎩ ⎭
(8a)
Or
ΔK eq = ΔKR f (R)
(10a)
Kmax − th0
(9b)
ΔKth0
Kmax − thR 1 ΔKthR = Kmax − th0 1 − R ΔKth0
(12c)
2.2. FCG threshold model 2.3. FCG model At any given FCG rate including threshold conditions, ΔK eq remains constant for all R. Now let Ref = 0, and assuming Rc > 0, then a combination of Eqs. (8b) and (9a) yields,
From Eq. (9b), the crack growth rate at the “Reference stress ratio’’ is expressed as 95
International Journal of Fatigue 122 (2019) 93–105
S. Kwofie, M.-L. Zhu
ΔK 0 = ΔKR
f (R) f (0)
(at equal da/dN values)
(14c)
For each R, ΔK 0 is calculated according to Eq. (14c) and the corda responding growth rate dN is known. With known value of n, log-log
( )
R
( ) da
plot of dN (determined according to Eq. (14b)) versus ΔK 0 (de0 termined according to Eq. (14c)), is expected to collapse data onto the crack growth curve at R = 0. Furthermore, from crack growth curve at R = 0, the crack growth rate in case of ΔKR and R, can also be estimated using Eq. (14b) and Eq. (14c). This can be achieved by calculating ΔK 0 according to Eq. (14c) and reading off the corresponding growth rate da from the crack growth curve at R = 0, which is then used in Eq. dN
( )
0
(14b) to determine
( ). da dN R
2.4. Determination of parameters The constant values of Rc and α can be determined from fatigue threshold data of ΔKthR at various R. Fatigue threshold data can be derived from several specimens using the load-shedding procedure based on ASTM E647, as reported in [11,12]. These data can be used in Eq. (11) to determine the constants α and Rc. As for this method, the Rc can be directly estimated when plotting ΔKthR as a function of R or by a ΔKthR-KmaxR plot and finding out the transition point, as indicated in [10,13]. When Rc is known, the value of α can be calculated using Eq. (11(ii)) or estimated by fitting all the data point based on Eq. (11(i)). Alternately, a minimum of three selected data points can be used with Eq. (11) for the determination of the constants. These include ΔKth0, ΔKth-1 and ΔKth0.8, at R = 0, R = −1 and R = 0.8, respectively. These data when input into Eq. (11) enables the determination of α and Rc, as indicated in Eqs. (15a) and (15b). The choice of R = 0.8 is based on the assumption that Rc is often less than 0.8, which is consistent with most threshold data found in literature. With the Rc and α known, then the general form of FCG model and fatigue threshold model can be fully applied to predict the R-dependence of FCG behavior.
Fig. 2. (a) A Log-Log plot of da/dN versus ΔKRef from data of 25Cr2Ni2MoV steel [10], illustrating how (da/dN)Ref and (da/dN)R are extracted for determination of the parameter, n, as shown by A, B, C, and D points; (b) A LogLog plot of (da/dN)Ref/(da/dN)R versus f(R)/f(Ref) for a linear fitting with a slope of n = 2. n
f (R) ⎞ ⎛ da ⎞ = C (ΔKR )m ⎛⎜ ⎟ ⎝ dN ⎠Ref ⎝ f (Ref ) ⎠
where n ≠ m, and n is a constant independent of R but depends on material and loading conditions. Eq. (13) can be viewed as modification to the Paris-Erdogan law of Eq. (1) taking into account the effect of R via the function f(R). Combining Eqs. (13) and (1) yields Eq. (14a), n
where
( )
da dN R
= C (ΔKR
)m
(14a)
represents the crack growth behavior at R. The
parameter n is also different from m in terms of the method of determination, i.e., m is a fitted value in the Paris regime while n is a fitted
( ) ( ) ⎞⎠ versus
da / value using log-log plot of ⎛ dN Ref ⎝ (14a). Thus, at given crack growth rate,
da dN R
f (R) f (Ref )
based on Eq.
( ) , the role of Eq. (9b) is to da dN R
translate ΔKR due to stress ratio R, horizontally, to a position of ΔKRef at the reference stress ratio. In case of R = Ref, simply,
( )
da dN Ref
=
( ) . In da dN R
the case where ‘‘Ref’’ = 0, Eq. (14a) and Eq. (9b) can be rewritten as Eqs. (14b) and (14c), respectively.
(at equal ΔK 0 values)
ΔKth0.8 = (1 − R c )exp(−8α ) ΔKth0
(15b)
3. Model validation As mentioned above, the newly developed model can be employed to correlate FCG data of various R using Eqs. (14b) and (14c), and to predict the R-dependence of fatigue thresholds using Eqs. (11) and
n
f (R) ⎞ ⎛ da ⎞ = ⎛ da ⎞ ⎛⎜ ⎟ ⎝ dN ⎠0 ⎝ dN ⎠R ⎝ f (0) ⎠
(15a)
As for the determination of n values, a log-log plot of da/dN versus ΔKRef at reference stress ratio was carried out at first, as illustrated in Fig. 2a using crack growth data of 25Cr2Ni2MoV steel [10] as an example. Note the reference stress ratio is 0.1, thus the da/dN data at R of 0.1 are original without any changes, whereas those da/dN data at R other than 0.1 need to be transformed into ΔKRef based on Eq. (9b). For simplicity, one parallel line is drawn at da/dN of 8 × 10−7 mm/cycles which is determined as (da/dN)R. The next step is plotting vertical lines representing equal ΔKRef, and the four intersected points at A, B, C, and D on the reference crack growth curve are defined as (da/dN)Ref. In this way, (da/dN)Ref and (da/dN)R are extracted and the ratio values of (da/ dN)Ref/(da/dN)R are calculated at the four points. Finally, a Log-Log plot of (da/dN)Ref/(da/dN)R versus f(R)/f(Ref) is conducted for a linear fitting with a slope of n = 2, based on Eq. (14a), as depicted in Fig. 2b. Note that the determination of (da/dN)Ref and (da/dN)R may have some errors due to unavoidable interpolation on the FCG data. Nevertheless, it is observed that (da/dN)Ref/(da/dN)R values are almost similar at various ΔKRef for the same f(R)/f(Ref), implying a convincing n value with limited, small and acceptable error.
(13)
f (R) ⎞ ⎛ da ⎞ = ⎛ da ⎞ ⎛⎜ values) ⎟ (at equal ΔKRef ⎝ dN ⎠Ref ⎝ dN ⎠R ⎝ f (Ref ) ⎠
ΔKth − 1 = 2exp(α ) ΔKth0
(14b) 96
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S. Kwofie, M.-L. Zhu
Fig. 6. Comparison of fatigue thresholds at various R between modeling results and experimental data of STOA Ti-6Al-4V [20]. Fig. 3. Comparison of fatigue thresholds at various R between modeling results and experimental data of 25Cr2Ni2MoV steel [10].
Fig. 4. Comparison of fatigue thresholds at various R between modeling results and experimental data of Ti-6Al-4V alloy [6].
Fig. 7. Comparison of fatigue thresholds at various R between modeling results and experimental data of Al-2024-T3 [21].
Fig. 8. Comparison of fatigue thresholds at various R between modeling results and experimental data of Electrolytic Tough Pitch (ETP) Copper (quarter-hard) [22].
Fig. 5. Comparison of fatigue thresholds at various R between modeling results and experimental data of 7075-T7351 Aluminium Alloy [19].
(12b). As mentioned above, in the case of FCG data, and assuming Ref = 0, based on Eq. (10b), f(R)/f(0) is thus defined as Eq. (16).
( (
) )
1 2R f (R) ⎧ 1 − R exp α 1 − R , for − 1 ≤ R < R c (i) ⎫ = ⎨ 1 exp α 2R , for R ≤ R < 1 (ii) ⎬ f (0) c 1−R ⎩ 1 − Rc ⎭
Essentially, Eqs. (14b) and (14c) translate data points of (ΔKR , da on the FCG curve at R, to (ΔK 0 , dN ) on the FCG curve of
( )
( )
da ) dN R
R = 0. A log–log plot of
( )
da versus dN 0
0
ΔK 0 is expected to collapse crack growth data of different R onto the FCG curve at R = 0. Thus, Eqs. (14b) and (14c) and the fatigue threshold models of Eqs. (11) and (12b) are used to validate the FCG data of various steels and alloys in open
(16) 97
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S. Kwofie, M.-L. Zhu
Copper (quarter-hard) (Fig. 8) [24], and 2195-Al-Li Alloy (Fig. 9) [25]. Similarly, FCG behavior from modeling using Eqs. (14b) and (14c), and experiments are compared, and the corresponding results are shown in Figs. 10–16. The left parts of Figs. 10–16 present the conventional log-log plots of da/dN versus ΔKR according to the Paris-Erdogan law of Eq. (1), while the right parts of Figs. 10–16 illustrates a parallel plot of (da/dN)0 (determined from Eq.(14b) versus ΔK0 (determined in terms of Eq.(14c)). The parameter n, determined as illustrated in Fig. 2, is listed in Table 1, for the respective materials. It is worth noting that all data points at various R collapse onto one single master curve, especially, in the near-threshold regime where the original disparities among different R disappear. The model was further verified by using FCG data of two welded joints of 25Cr2Ni2MoV and 30Cr2Ni4MoV steels [11,12]. As illustrated in Figs. 17–21, the perfect agreement between experimental data and modeling further indicates the robustness of the model developed in this work.
Fig. 9. Comparison of fatigue thresholds at various R between modeling results and experimental data of 2195 Aluminium-Lithium-Alloy [23].
4. Discussion
literatures. Comparison of modeled results with experimental data is shown in Figs. 3–16 for different materials. The parameters α and Rc as well as ΔKth0 for various materials were estimated from available data and are listed in Table 1. Figs. 3–9 show threshold data of ΔKth (experimental) and Kmax-th (calculated) as a function of R, compared with modeling results of Eqs. (11) and (12b), respectively. It is found that agreement between experimental data and modeling results is very good. Note that the y-axis denotes normalized fatigue threshold value, i.e., the ratio of fatigue threshold at any given R with threshold value at R of 0, ΔKthR/ΔKth0 and Kmax-thR/Kmax-th0. It is apparent that the new developed fatigue threshold model has good predictability as good agreement is achieved in various steels and alloys, i.e., 25Cr2Ni2MoV rotor steel (Fig. 3) [10], Ti-6Al-4V (Fig. 4) [6], Al-7075-T7351 (Fig. 5) [21], STOA Ti-6Al-4V, (Fig. 6) [22], Al-2024-T3 (Fig. 7) [23], Electrolytic Tough Pitch (ETP)
4.1. Significance of modeling parameters The good predictability of the model is largely dependent on how the parameters are determined. In this work, Rc, is estimated based on fatigue threshold values at the da/dN of 1 × 10−7 mm/cycle, as described in Section 2.3. Fatigue threshold could be determined at da/dN of 1 × 10−8 mm/cycle if sufficient data were at hand at low da/dN level. This is the parameter required for the model when crack closure influences are incorporated into the original exponential mean stress model. Note that the fatigue threshold values are achieved from loadshedding procedure based on ASTM E647. The parameter α, which physically means mean stress sensitivity factor, is unique for a given material, loading condition and environment, and remains constant for all R, as shown in the present model. This is different from the initial
Fig. 10. Plots of da/dN versus ΔK according to Eq. (1) (left) and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of 25Cr2Ni2MoV steel [10]. 98
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S. Kwofie, M.-L. Zhu
Fig. 11. Plots of da/dN versus ΔK according to Eq. (1) (left) and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of Ti-6Al4 V alloy [6].
Fig. 12. Plots of da/dN versus ΔK according to Eq. (1) (left) and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of 7075T7351 Aluminium Alloy [19].
EDF model [20], where the α value is varied at higher and lower R. The parameter of α controls how strong the mean stress will influence the FCG behavior, and rationalizes the one α-value in the new model due to the mean stress being active only in high R. As can be seen from Table 1, α is more related to microstructures and material types, and the α value of Ti-6Al-4V could be 4–10 times higher than steels and other alloys. Though the current model is validated using the same materials
from which the parameters are determined, unlike the original model, the newly developed one in this work focuses on modifying the effectiveness of the stress intensity factor range by taking into account the role of crack closure in the near-threshold FCG regime. The agreement between modeling results and experimental data presented here further verify that crack closure cannot be neglected in FCG modeling. The new model is thus more mechanistically representative of the FCG behavior, a reason that underlines the good predictability. The ΔK⋅f(R) can be 99
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Fig. 13. Plots of da/dN versus ΔK according to Eq. (1) (left) and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of STOA Ti-6Al-4V [20].
Fig. 14. Plots of da/dN versus ΔK according to Eq. (1) (left) and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of 2024T3 Aluminium Alloy [21].
4.2. Significance of ΔK versus Kmax
regarded as the driving force parameter for FCG in the near-threshold and Paris regimes (see Figs. 10–16), similar to the one proposed in [13]. Furthermore, the new model is developed based on the concept of combining mean stress effect and crack closure roles, which verifies the effectiveness of the combined approach proposed by Zhu et al. [13].
Though ΔKR⋅f(R) is the driving force for fatigue cracking, one may wonder the significance of ΔK versus Kmax in FCG behavior under the influence of R. From the very definition of R, Eq. (17), there comes a fixed correlation of ΔKthR/Kmax-thR with R at fatigue threshold 100
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Fig. 15. Plots of da/dN versus ΔK according to Eq. (1) (left), and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of Electrolytic Tough Pitch (ETP) Copper (quarter-hard) [22].
Fig. 16. Plots of da/dN versus ΔK according to Eq. (1) (left), and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of 2195 Aluminium-Lithium-Alloy [23].
increases sharply with ΔKthR varying little, which indicates that Kmax is driving fatigue cracking at the near-threshold regime. As a result, the parameter of Rc in this model denotes a transition point where the significance of ΔKR transits to Kmax. This accords well with the general finding that cyclic loading at lower R involves the “fatigue” type of cracking while loading at high R involves the “static” type of cracking.
conditions, as shown in Eq. (18). It is apparent that an increase in R results in a decrease of ΔKthR/Kmax-thR, which implies either a decreasing ΔKthR and/or an increasing Kmax-thR. It is observed from Figs. 3–9 that in case of R < Rc, where crack closure is active, ΔKthR decreases strongly while Kmax-thR is almost stable, which indicates that ΔKthR is driving the fatigue cracking behavior at the near-threshold regime. Whereas for R > Rc, the crack closure is not active, and Kmax-
thR
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Table 1 Parameters employed for prediction of FCG and fatigue threshold. Materials 25Cr2Ni2MoV steel [10] Ti-6Al-4V [6] Al-7075-T7351 [19] STOA Ti-6Al-4V [20] Al-2024-T3 [21] EPT COPPER (1/4-Hard) [22] 2195 Al-Li-Alloy [23] Welded 25Cr2Ni2MoV steel [12] (Notch in weld metal) Welded 25Cr2Ni2MoV steel [12] (Notch in HAZ) 30Cr2Ni4MoV [11] (As received, notch in weld metal) 30Cr2Ni4MoV [11] (Aged, notch in base metal)
da/dN (mm/cycle) −7
1 × 10 1 × 10−7 1 × 10−7 1 × 10−7 1 × 10−7 1 × 10−7 1 × 10−7
Rc
α
ΔKth0 (MPa-m1/2)
n
0.72 0.4 0.53 0.4 0.5 0.4 0.5 0.72 0.72 0.72 0.72
0.005 0.04 0.01 0.01 0.008 0.03 0.01 0.04 0.05 0.01 0.01
6.8 12.0 3 5.2 3.5 6.5 2.6
2.0 2 1.2 2 2 2 1.2 2.3 2.3 2.3 2.0
Fig. 17. Crack growth data of 25Cr2Ni2MoV steel welded joint with notch in weld metal: original experimental data (left) and present modeling results (right) [12].
R=
Kmin K ΔK − ΔK = max =1− Kmax Kmax Kmax
ΔKthR +R=1 Kmax − thR
4.3. Significance of the f(R) model (17) Conventionally, the ΔKeff is expressed as Eq. (19), where the crack closure coefficient U is determined as Eq. (20). The U can be regarded as one specific case of the f(R) in case there is little mean stress effect. Crack closure and mean stress oppose each other in terms of their contribution to crack growth. Whereas crack closure tends to decrease the applied stress intensity range, mean stress amplifies the applied stress intensity range, to present a full view of driving force for fatigue cracking. The overall effect would depend on which of the two dominates the exchanges. For f(R) < 1, crack closure dominates and suppresses mean stress effects, which is the case within the threshold regime. For f(R) > 1, the crack closure effect is minimal, and mean stress effects dominantly control crack growth. Therefore, f(R), which is defined in terms of Eq. (8) is more representative of FCG behavior due to both crack closure and mean stress effects.
(18)
In the fatigue damage point of view, in a higher R regime, where fatigue process is represented by higher mean stress with lower stress amplitude, the role of high mean stress is to keep crack open while localized monotonic and cyclic crack tip plasticity drives crack growth. In a lower R regime, the fatigue loading is represented as lower mean stress and higher stress amplitude, the lower mean stress cannot keep the crack always open while the cyclic crack tip plasticity drives crack growth slowly. It is thus inferred that the mean stress level is indicative of whether crack surface can touch each other, whereas the stress amplitude denotes the different modes of crack growth. As a result, the R effect is not equivalent to mean stress contribution, as was often anticipated, whether the crack can grow or how it will be grown should also be taken into account. This all accords with and rationalizes the physics of fatigue crack growth as denoted by the f(R) model.
ΔK eff = Kmax − K op U=
ΔK eff ΔKR
=
(19)
K op ⎞ 1 ⎛ 1− 1−R⎝ Kmax ⎠ ⎜
⎟
(20)
In addition, note that when α = 0, f(R) = U(R), and this indicates 102
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Fig. 18. Crack growth data of 25Cr2Ni2MoV steel welded joint with notch in heat-affected zone: original experimental data (left) and present modelling results (right) [12].
Fig. 19. Crack growth data of welded 30Cr2Ni4MoV steel in as-received condition (weld metal notched): original experimental data (left) and present modelling results (right) [11].
combining mean stress and crack closure effects in this work is different from the one proposed by Zhu et al. in [10,13] where the mean stress and crack closure are added for R-dependence interpretation of FCG, while in this work, these two aspects are multiplied as shown by the f (R).
that crack growth is insensitive to mean stress and that crack closure solely influences crack growth (see Fig. 1(right)). It can be observed from Fig. 1(left) that with the increasing of R, the f(R) values are gradually increased due to combined effects of crack closure and mean stress, with f(R) < 1, in the closure-dominated regime, while for f (R) > 1, the fatigue cracking is within closure-free and mean-stressdominated FCG regime. Finally, it is worth noting that the way of
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Fig. 20. Crack growth data of 30Cr2Ni4MoV steel welded joint under long-term aging condition with notch in weld metal: original experimental data (left) and present modeling results (right) [11].
Fig. 21. Crack growth data of 30Cr2Ni4MoV steel welded joint under long-term aging condition with notch in base metal: original experimental data (left) and present modeling results (right) [11].
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5. Conclusions
Acta Mater 1997;45:2777–87. [7] Ritchie RO, Boyce BL, Campbell JP, Roder O, Thompson AW, Milligan WW. Thresholds for high-cycle fatigue in a turbine engine Ti–6Al–4V alloy. Int J Fatigue 1999;21:653–62. [8] Chiang CR. Threshold stress intensity factor of fatigue cracks. Eng Fract Mech 1994;49:29–33. [9] Kardomateas GA, Carlson RL. Predicting the effects of load ratio on the fatigue crack growth rate and fatigue threshold. Fatigue Fract Eng Mater Struct 1998;21:411–23. [10] Zhu M-L, Xuan F-Z, Tu S-T. Interpreting load ratio dependence of near-threshold fatigue crack growth by a new crack closure model. Int J Press Vessels Pip 2013;110:9–13. [11] Du Y-N, Zhu M-L, Liu X, Xuan F-Z. Effect of long-term aging on near-threshold fatigue crack growth of Ni–Cr–Mo–V steel welds. J Mater Process Technol 2015;226:228–37. [12] Du Y-N, Zhu M-L, Xuan F-Z. Transitional behavior of fatigue crack growth in welded joint of 25Cr2Ni2MoV steel. Eng Fract Mech 2015;144:1–15. [13] Zhu M-L, Xuan F-Z, Tu S-T. Effect of load ratio on fatigue crack growth in the nearthreshold regime: a literature review, and a combined crack closure and driving force approach. Eng Fract Mech 2015;141:57–77. [14] Ritchie RO, Knott JF. Mechanisms of fatigue crack growth in low alloy steel. Acta Metall 1973;21:639–48. [15] Elber W. The significance of fatigue crack closure. In: Damage tolerance in aircraft structures, ASTM STP 486, Canada; 1970. p. 230–42. [16] Walker K. The effect of stress ratio during crack propagation and fatigue for 2024T3 and 7075-T6 aluminum. In: Effects of environment and complex load history on fatigue life: a Symposium, ASTM STP 462, Lutherville-Timonium, MD; 1970. p. 1–14. [17] Newman JC. A crack opening stress equation for fatigue crack growth. Int J Fract 1984;24:R131–5. [18] McClung RC, Chell G, Lee Y-D, Russell D, Orient G. Development of a practical methodology for elastic-plastic and fully plastic fatigue crack growth, NASA/CR1999-209428; 1999. [19] Schijve J. Fatigue of structures and materials. Springer; 2001. [20] Kwofie S, Rahbar N. An equivalent driving force model for crack growth prediction under different stress ratios. Int J Fatigue 2011;33:1199–204. [21] Newman JC, Brot A, Matias C. Crack-growth calculations in 7075–T7351 aluminum alloy under various load spectra using an improved crack-closure model. Eng Fract Mech 2004;71:2347–63. [22] Boyce BL, Ritchie RO. Effect of load ratio and maximum stress intensity on the fatigue threshold in Ti–6Al–4V. Eng Fract Mech 2001;68:129–47. [23] Miller M, Gallagher J. An analysis of several fatigue crack growth rate (FCGR) descriptions. Fatigue crack growth measurement and data analysis. ASTM International; 1981. [24] Liaw PK, Leax TR, Williams RS, Peck MG. Near-threshold fatigue crack growth behavior in copper. Metall Trans A 1982;13:1607–18. [25] Chen DL, Chaturvedi MC. Near-threshold fatigue crack growth behavior of 2195 aluminum-lithium-alloy—prediction of crack propagation direction and influence of stress ratio. Metall Mater Trans A 2000;31:1531–41.
In this work, an exponential mean stress model is combined with crack closure for developing a more general form of FCG model for interpreting the R-dependence of FCG behavior in both Paris and nearthreshold regimes. A more general model for FCG and fatigue threshold prediction was developed, and was validated widely by using published data of various steels and alloys in open literatures. The model could collapse FCG data at various R onto one single master curve, and good agreements were achieved between modeling results and experimental data of fatigue threshold. All these have implied the feasibility of combining crack closure approach with mean stress effects for a sound interpretation of R-dependence of FCG in both Paris and near-threshold regimes. The new approach has also provided information about the transition of crack growth driving force with varying R, and has informed of an updated concept that R-dependence of FCG is not equivalent to the mean stress effect. Acknowledgement Funding support by the Ghanaian Government in terms of Book and Research Allowance is greatly appreciated. Ming-Liang Zhu would like to thank the support by National Natural Science Foundation of China (No. 51575182) and the 111 Project (No. B13020). References [1] Paris P, Erdogan F. A Critical Analysis of Crack Propagation Laws. J Basic Eng 1963;85:528–33. [2] Schmidt RA, Paris PC. Threshold for fatigue crack propagation and the effects of load ratio and frequency. Progress in flaw growth and fracture toughness testing, ASTM STP536. Philadelphia: ASTM; 1973. p. 79–94. [3] Ritchie RO. Near-threshold fatigue crack propagation in ultra-high strength steel: influence of load ratio and cyclic strength. J Eng Mater Technol 1977;99:195–204. [4] Liaw PK, Lea TR, Logsdon WA. Near-threshold fatigue crack growth behavior in metals. Acta Metall 1983;31:1581–7. [5] Taylor D. A compendium of fatigue thresholds and growth rates. West Midlands, UK: Engineering Materials Advisory Services Ltd.; 1985. [6] Dubey S, Soboyejo ABO, Soboyejo WO. An investigation of the effects of stress ratio and crack closure on the micro-mechanisms of fatigue crack growth in Ti-6Al-4V.
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Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Modeling R-dependence of near-threshold fatigue crack growth by combining crack closure and exponential mean stress model Samuel Kwofiea, Ming-Liang Zhub,
T
⁎
a
Department of Materials Engineering, College of Engineering, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Key Laboratory of Pressure Systems and Safety, Ministry of Education, School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China
b
A R T I C LE I N FO
A B S T R A C T
Keywords: Fatigue crack growth Stress ratio Mean stress Crack closure Fatigue threshold
A previously proposed equivalent driving force model was modified to account for crack closure for modeling stress ratio, R, dependence of fatigue crack growth in near-threshold regime. A new fatigue crack growth driving force model was established in the form of ΔK⋅f(R), where the f(R) includes both crack closure and mean stress contributions, and well informs the shift from dominance of stress intensity factor range ΔK to maximum stress intensity factor Kmax with increasing R. The model was validated widely with sound agreement between modeling results and experimental data.
1. Introduction
elastic fracture mechanics is applicable, has been widely investigated both experimentally [2–7] and analytically [8,9]. It is generally accepted that an increase of R results in higher FCG rate at a given ΔK and in lower ΔKth in near-threshold regime [10–13]. Mechanistic explanations for such behavior have focused on either the presence of crack closure at lower R [2], or the occurrence of static fracture modes at higher R [14]. Models have been proposed to describe the observed crack growth behavior, with those of Elber [15] and Walker [16] being the most widely known. During the past several decades, great progress has been achieved on crack closure, such as the work by Newman [17], McClung et al. [18], and Schijve [19]. A complete historic review of the research and development of R-dependence of crack closure has been reported by one of the current authors [13]. Quite recently, an equivalent driving force (EDF) approach [20], which is an exponential mean stress model, was proposed by one of the current authors for correlation of FCG data at different R. The EDF model was expressed as,
An important aspect of engineering design is the design against fatigue while component is in service. For fatigue-prone bodies where crack initiation stage constitutes a small fraction of total life of engineering materials, the “damage tolerant” approach to design is often utilized. In this approach, prediction of remaining fatigue life is based on modeling of crack propagation of a flaw (or crack) whose size can be detected or inspected by non-destructive techniques. According to Paris-Erdogan, the rule of fatigue crack growth (FCG) can be represented as [1],
da/dN = C(ΔK) m
(1)
where da/dN is fatigue crack growth rate, a is crack size, N is cyclic number, ΔK is the stress intensity factor range, and C and m are material constants, and are dependent on loading and environmental conditions. Theoretically, a log–log plot of da/dN against ΔK is expected to yield a straight line graph with slope m and intercept of log(C). However, such plot of experimental data yields a sigmoidal curve showing three regimes of crack propagation: (i) near-threshold regime with ΔK close to ΔKth, where ΔKth is the threshold stress intensity factor range below which crack growth does not occur; (ii) intermediate regime, where crack growth is stable and follows the Paris-Erdogan law; and (iii) unstable regime, where the maximum stress intensity factor Kmax approaches the fracture toughness, Kc. Furthermore, the parameters C and m vary with stress ratio, R. The influence of R on FCG behavior of long cracks, where linear ⁎
1+R ⎞ ⎞ ΔK eq = ΔKRexp ⎛α ⎛ ⎝ ⎝ 1 − R ⎠⎠
(2a)
where ΔKeq is the equivalent form of the applied ΔK, ΔKR, at corresponding R, and α is mean stress sensitivity factor. At a fixed rate of crack growth, ΔKeq remains constant, and the ΔKR (applied ΔK), is thus derived as Eq. (2b) or Eq. (2c).
1+R ⎞ ⎞ ΔKR = ΔK eqexp ⎛−α ⎛ ⎝ ⎝ 1 − R ⎠⎠
Corresponding author. E-mail address: [email protected] (M.-L. Zhu).
https://doi.org/10.1016/j.ijfatigue.2019.01.006 Received 18 July 2018; Received in revised form 16 December 2018; Accepted 14 January 2019 Available online 17 January 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.
(2b)
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S. Kwofie, M.-L. Zhu
1+R ⎞ In(ΔKR) = In(ΔK eq) − α ⎛ ⎝1 − R ⎠
1+R ⎞ ⎞ ΔK eq = (Kmax − K op)exp ⎛α ⎛ ⎝ ⎝ 1 − R ⎠⎠
(2c)
According to the EDF model, a plot of experimental data of ln(ΔKR) 1+R versus 1 − R for −1 < R < 1 was expected to yield a straight line with a slope value of -α and y-axis intercept ln(ΔKeq) being ln(ΔK−1), where ΔK-1 is the stress intensity factor range at R = −1. However, plots of crack growth data of Al- and Ti-alloys at constant growth rates between 1 × 10−7 mm/cycle and 1 × 10−5 mm/cycle, yielded two straight lines with different slopes, resulting in two different values of α, and the two lines intersected at R = Rc ≈ 1/3, as reported in [20]. Zhu et al. [10] applied the EDF model to correlate FCG data of 25Cr2Ni2MoV rotor steel, and found that Rc was not a constant but dependent on the level of crack growth rate, which was associated with crack closure. Zhu et al. [10] also found that the EDF model accurately described the R-dependence of FCG data in higher da/dN region, but was unable to explain the effect of R on FCG behavior in the nearthreshold regime, where FCG is much lower. The contribution Zhu et al. has achieved is the finding and modeling of the dependence of crack closure on both R and FCG level in the near-threshold regime. Recently, Zhu et al. [13] proposed a combined crack growth driving force and crack closure approach to interpret the R-dependence of FCG in both Paris and near-threshold regimes, and the newly developed FCG model had sound predictive capability of FCG behavior in various Cr-Mo-V steels. This indicates the robustness of combination approach to R-dependence of FCG, and perhaps a general approach to modeling R-effect on long crack FCG behavior. In terms of the EDF model [20], when applied, differentiates FCG at lower and higher R, as observed from the Rc, and this may be explained in terms of varied crack closure roles or influences below and above Rc. However, such crack closure roles or influences are not explicitly included in and represented by the exponential mean stress model [20]. This drives us to revisit the EDF model to account for crack closure effects for modification, in a different way with that proposed before [10,11,13], and to make it more relevant to explain the R-dependence of FCG behavior in both intermediate and near-threshold regimes. Therefore, in this work, the original model of Eq. (2a) was modified to incorporate crack closure effect in the near-threshold FCG regime to develop a more general FCG model. A relation between stress intensity range at threshold condition, ΔKthR, and R was derived and compared with published data of various ferrous and non-ferrous alloys. In addition, it was shown that the modified model was capable of collapsing FCG data of different R onto a master curve at R = 0. This is useful to predict FCG behavior at different R in the near-threshold regime as well as the Paris-Erdogan regime.
From Eqs. (3) and (4), ΔK eq , is thus redefined in terms of Eq. (5), for crack closure (5i) and closure-free (5ii) cases, as follows,
( )
ΔK eq =
( ( ) ), ( ( ) ),
⎧ (Kmax − K op)exp α ⎪ ⎨ (K ⎪ max − Kmin )exp α ⎩
1+R 1−R
1+R 1−R
for Kmin < K op (i) ⎫ ⎪ for Kmin ≥ K op (ii) ⎬ ⎪ ⎭
(5a)
Or
ΔK eq
(
K
) ( ( )) ( ( ))
⎧ Kmax 1 − op exp α 1 + R , for Kmin/ Kmax < K op/ Kmax (i) ⎫ ⎪ ⎪ Kmax 1−R = ⎨ K (1 − R)exp α 1 + R , for K /K ⎬ K / K (ii) ≥ min max op max ⎪ max ⎪ 1−R ⎩ ⎭ (5b) Now let R = Rc be the transition from closure to closure-free condiK tion, and assuming op remain constant for Kmin < K op at a particular Kmax da/dN level, then Eqs. (5)(i) and (5)(ii) would be equivalent as shown in Eq. (6a), from which a relation is derived as Eq. (6b). Note here the Kop is the crack opening level when R is Rc, and Eq. (6b) is a sound establishment only when R is Rc.
K op ⎞ 1 + Rc ⎞ 1 + Rc ⎞ ⎞ = Kmax (1 − Rc )exp ⎛α ⎛ ⎞ exp ⎛α ⎛ Kmax ⎛1 − Kmax ⎠ ⎝ ⎝ 1 − Rc ⎠ ⎠ ⎝ ⎝ 1 − Rc ⎠ ⎠ ⎝ ⎜
⎟
(6a)
K op Kmax
= Rc
(6b)
Hence, from Eqs. (5b) and (6b), the ΔK eq is rearranged as,
ΔK eq
( ( ) ), ( ( ) ),
⎧ Kmax (1 − Rc )exp α ⎪ = ⎨ K (1 − R)exp α ⎪ max ⎩
1+R 1−R
1+R 1−R
Kop Kmax
≤ Rc (i) ⎫ ⎪ for R ≥ Rc (ii) ⎬ ⎪ ⎭
for R <
(7a)
or
ΔK eq
(
) ( ( ) ), ( ( ))
⎧ ΔKR 1 − Rc exp α 1 + R ⎪ 1−R 1−R = 1+R ⎨ ΔKRexp α 1 − R , ⎪ ⎩
for R < Rc (i) ⎫ ⎪ for R ≥ Rc (ii) ⎬ ⎪ ⎭
(7b)
Note in Eq. (7a), the value of Kop is normally within R and Rc, while Kmax here it is assumed to be Rc for simplicity based on an almost constant Kmax for R less than Rc, and also due to the difficulty of obtaining Kop ΔK values at a particular R and da/dN level. As Kmax = 1 − RR , Eq. (7b) is rewritten as Eq. (7c), where ΔK eff is the effective stress intensity factor range due to crack closure,
2. Modeling R-dependence of FCG behavior 2.1. A general driving force model To account for crack closure as well as transition of crack growth behavior from closure-affected to closure-free behavior, Eq. (2) needs to be modified. Firstly, Eq. (2) can be written as Eq. (3), where ΔKR = Kmax − Kmin.
1+R ⎞ ⎞ ΔK eq = (Kmax − Kmin )exp ⎛α ⎛ ⎝ ⎝ 1 − R ⎠⎠
(4)
1+R ⎞ ⎞ ΔK eq = ΔK eff exp ⎛α ⎛ ⎝ ⎝ 1 − R ⎠⎠
(7c)
Where (3)
ΔK eff =
where ΔK eq is the equivalent stress intensity range due to combined effects of applied stress intensity factor range ΔKR , and load ratio, R. Normally Eq. (3) is valid only when Kmin ≥ K op where Kop is the stress intensity factor at which crack is fully open. In the case of compressive stresses where Kmin < 0, assuming crack is closed and Kmin does not contribute to crack growth. Furthermore, it is known that crack closure may also occur when Kmin is positive but less than Kop. Therefore for all cases when Kmin < K op , K op replaces Kmin in Eq. (3) to yield Eq. (4).
(
)
⎧ ΔKR 11−−Rc , for R < Rc (i) ⎫ R ⎨ Δ , for R ≥ Rc (ii) ⎬ K R ⎩ ⎭
(7d)
Essentially, Eq. (7d(i)) describes the effective stress intensity factor range in the R-regime where crack closure influences FCG whereas Eq. (7d(ii)) describes the effective stress intensity factor range in the Rregime where crack-closure influence is minimal. From Eq. (7d) we obtain Eq. (7e) which has a form very similar to the traditional crack closure coefficient. Based on Eq. (7e), Eq. (7c) can be rearranged as Eq. (8), and a new expression f(R) is defined as Eq. (8c). 94
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S. Kwofie, M.-L. Zhu
Fig. 1. Left: Typical plot of f(R) versus R, showing the combined effects of crack closure and mean stress (Rc = 0.72, α = 0.001); Right: plot of U(R) versus R, showing the case when growth is insensitive to mean stress (α = 0 or f(R) = U(R)), but influenced solely by crack closure.
U (R) =
ΔK eff ΔKR
=
⎧ ⎨ ⎩
(
1 − Rc 1−R
),
1,
for − 1 ≤ R ≤ Rc (i) ⎫ for Rc ≤ R ≤ 1 (ii) ⎬ ⎭
1+R ⎞ ⎞ ΔK eq = ΔKR U (R)exp ⎛α ⎛ ⎝ ⎝ 1 − R ⎠⎠
ΔK eq = ΔKR f (R) = ΔK 0 f (0)
The Eq. (10a) is then combined with Eqs. (7e) and (8c), and yields Eq. (10b).
(7e)
( (
(10b)
(8b) At threshold condition, Eq. (10b) still holds and hence,
where
1+R ⎞ ⎞ f (R) = U (R)exp ⎛α ⎛ ⎝ ⎝ 1 − R ⎠⎠
( (
ΔKR f (R) = ΔKRef f (Ref )
(8c)
(11)
where ΔKth0 is the threshold stress intensity factor range at R = 0, and ΔKthR is the threshold stress intensity factor range, at any R. From Eq. (11) the effect of R on Kmax − th is deduced as:
Kmax − thR (1 − R) =
( (
⎧ Kmax − th0 (1 − R)exp −α 1 2−RR
) )
2R ⎨K max − th0 (1 − R c )exp − α 1 − R ⎩
for − 1 ≤ R < R c (i) ⎫ for R c ≤ R < 1
(ii) ⎬ ⎭
(12a)
or
(
)
⎧ exp −α 1 2−RR , for − 1 ≤ R < R c (i) ⎫ Kmax − thR = R R 1 − 2 c ⎨ Kmax − th0 exp −α 1 − R , for R c ≤ R < 1 (ii) ⎬ ⎩ 1−R ⎭
(
)
(12b)
where Kmax-thR is the Kmax-th value at any R, while Kmax-th0 is the Kmax-th value at R = 0. With known values of Rc and α, Eqs. (11) and (12b) can be employed to describe the R-dependence of ΔKthR and Kmax-thR of a given material at the threshold condition. In fact, from Eqs. (11) and (12b), Kmax − thR and ΔKthR can be well correlated as:
(9a)
Or
f (Ref ) f (R)
) )
⎧ (1 − R)exp −α 1 2−RR , for − 1 ≤ R < Rc (i) ⎫ ΔKthR = ⎨ (1 − Rc )exp −α 2R , for Rc ≤ R < 1 (ii) ⎬ ΔKth0 1−R ⎩ ⎭
It is indicated in Eq. (8) that the ΔKeq is represented as a combined form of ΔKR and f(R), and the f(R) includes both crack closure and mean stress effects. A plot of f(R) as a function of R for −1 ≤ R < 1 is shown in Fig. 1(left). Here the Rc and α are selected as 0.72 and 0.001, respectively. It is observed that for R ≤ Rc, where crack closure occurs, the value of f(R) is less than one, while for Rc < R < 1, where FCG is least affected by crack closure, the value of f(R) > 1. Note that for α = 0, f (R) = U (R) , which is the case when crack growth is insensitive to mean stress. A plot of U(R) versus R is shown in Fig. 1(right), where U < 1 for R < Rc and U = 1, for R ≥ Rc. It is observed from Fig. 1 that for the R-regime where crack closure effect is dominant in fatigue crack growth, the role of mean-stress in fatigue cracking is secondary. As an equivalent fatigue driving force, ΔK eq should be equal at all R, as a result, Eq.(8b) is written as Eq. (9), Where “Ref” refers to Reference stress ratio. It is inferred that
ΔKR = ΔKRef
) )
⎧ (1 − R)exp −α 1 2−RR , for − 1 ≤ R < Rc (i) ⎫ f (0) ΔKR = = ΔK 0 f (R) ⎨ (1 − Rc )exp −α 2R , for Rc ≤ R < 1 (ii) ⎬ 1−R ⎩ ⎭
(8a)
Or
ΔK eq = ΔKR f (R)
(10a)
Kmax − th0
(9b)
ΔKth0
Kmax − thR 1 ΔKthR = Kmax − th0 1 − R ΔKth0
(12c)
2.2. FCG threshold model 2.3. FCG model At any given FCG rate including threshold conditions, ΔK eq remains constant for all R. Now let Ref = 0, and assuming Rc > 0, then a combination of Eqs. (8b) and (9a) yields,
From Eq. (9b), the crack growth rate at the “Reference stress ratio’’ is expressed as 95
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ΔK 0 = ΔKR
f (R) f (0)
(at equal da/dN values)
(14c)
For each R, ΔK 0 is calculated according to Eq. (14c) and the corda responding growth rate dN is known. With known value of n, log-log
( )
R
( ) da
plot of dN (determined according to Eq. (14b)) versus ΔK 0 (de0 termined according to Eq. (14c)), is expected to collapse data onto the crack growth curve at R = 0. Furthermore, from crack growth curve at R = 0, the crack growth rate in case of ΔKR and R, can also be estimated using Eq. (14b) and Eq. (14c). This can be achieved by calculating ΔK 0 according to Eq. (14c) and reading off the corresponding growth rate da from the crack growth curve at R = 0, which is then used in Eq. dN
( )
0
(14b) to determine
( ). da dN R
2.4. Determination of parameters The constant values of Rc and α can be determined from fatigue threshold data of ΔKthR at various R. Fatigue threshold data can be derived from several specimens using the load-shedding procedure based on ASTM E647, as reported in [11,12]. These data can be used in Eq. (11) to determine the constants α and Rc. As for this method, the Rc can be directly estimated when plotting ΔKthR as a function of R or by a ΔKthR-KmaxR plot and finding out the transition point, as indicated in [10,13]. When Rc is known, the value of α can be calculated using Eq. (11(ii)) or estimated by fitting all the data point based on Eq. (11(i)). Alternately, a minimum of three selected data points can be used with Eq. (11) for the determination of the constants. These include ΔKth0, ΔKth-1 and ΔKth0.8, at R = 0, R = −1 and R = 0.8, respectively. These data when input into Eq. (11) enables the determination of α and Rc, as indicated in Eqs. (15a) and (15b). The choice of R = 0.8 is based on the assumption that Rc is often less than 0.8, which is consistent with most threshold data found in literature. With the Rc and α known, then the general form of FCG model and fatigue threshold model can be fully applied to predict the R-dependence of FCG behavior.
Fig. 2. (a) A Log-Log plot of da/dN versus ΔKRef from data of 25Cr2Ni2MoV steel [10], illustrating how (da/dN)Ref and (da/dN)R are extracted for determination of the parameter, n, as shown by A, B, C, and D points; (b) A LogLog plot of (da/dN)Ref/(da/dN)R versus f(R)/f(Ref) for a linear fitting with a slope of n = 2. n
f (R) ⎞ ⎛ da ⎞ = C (ΔKR )m ⎛⎜ ⎟ ⎝ dN ⎠Ref ⎝ f (Ref ) ⎠
where n ≠ m, and n is a constant independent of R but depends on material and loading conditions. Eq. (13) can be viewed as modification to the Paris-Erdogan law of Eq. (1) taking into account the effect of R via the function f(R). Combining Eqs. (13) and (1) yields Eq. (14a), n
where
( )
da dN R
= C (ΔKR
)m
(14a)
represents the crack growth behavior at R. The
parameter n is also different from m in terms of the method of determination, i.e., m is a fitted value in the Paris regime while n is a fitted
( ) ( ) ⎞⎠ versus
da / value using log-log plot of ⎛ dN Ref ⎝ (14a). Thus, at given crack growth rate,
da dN R
f (R) f (Ref )
based on Eq.
( ) , the role of Eq. (9b) is to da dN R
translate ΔKR due to stress ratio R, horizontally, to a position of ΔKRef at the reference stress ratio. In case of R = Ref, simply,
( )
da dN Ref
=
( ) . In da dN R
the case where ‘‘Ref’’ = 0, Eq. (14a) and Eq. (9b) can be rewritten as Eqs. (14b) and (14c), respectively.
(at equal ΔK 0 values)
ΔKth0.8 = (1 − R c )exp(−8α ) ΔKth0
(15b)
3. Model validation As mentioned above, the newly developed model can be employed to correlate FCG data of various R using Eqs. (14b) and (14c), and to predict the R-dependence of fatigue thresholds using Eqs. (11) and
n
f (R) ⎞ ⎛ da ⎞ = ⎛ da ⎞ ⎛⎜ ⎟ ⎝ dN ⎠0 ⎝ dN ⎠R ⎝ f (0) ⎠
(15a)
As for the determination of n values, a log-log plot of da/dN versus ΔKRef at reference stress ratio was carried out at first, as illustrated in Fig. 2a using crack growth data of 25Cr2Ni2MoV steel [10] as an example. Note the reference stress ratio is 0.1, thus the da/dN data at R of 0.1 are original without any changes, whereas those da/dN data at R other than 0.1 need to be transformed into ΔKRef based on Eq. (9b). For simplicity, one parallel line is drawn at da/dN of 8 × 10−7 mm/cycles which is determined as (da/dN)R. The next step is plotting vertical lines representing equal ΔKRef, and the four intersected points at A, B, C, and D on the reference crack growth curve are defined as (da/dN)Ref. In this way, (da/dN)Ref and (da/dN)R are extracted and the ratio values of (da/ dN)Ref/(da/dN)R are calculated at the four points. Finally, a Log-Log plot of (da/dN)Ref/(da/dN)R versus f(R)/f(Ref) is conducted for a linear fitting with a slope of n = 2, based on Eq. (14a), as depicted in Fig. 2b. Note that the determination of (da/dN)Ref and (da/dN)R may have some errors due to unavoidable interpolation on the FCG data. Nevertheless, it is observed that (da/dN)Ref/(da/dN)R values are almost similar at various ΔKRef for the same f(R)/f(Ref), implying a convincing n value with limited, small and acceptable error.
(13)
f (R) ⎞ ⎛ da ⎞ = ⎛ da ⎞ ⎛⎜ values) ⎟ (at equal ΔKRef ⎝ dN ⎠Ref ⎝ dN ⎠R ⎝ f (Ref ) ⎠
ΔKth − 1 = 2exp(α ) ΔKth0
(14b) 96
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Fig. 6. Comparison of fatigue thresholds at various R between modeling results and experimental data of STOA Ti-6Al-4V [20]. Fig. 3. Comparison of fatigue thresholds at various R between modeling results and experimental data of 25Cr2Ni2MoV steel [10].
Fig. 4. Comparison of fatigue thresholds at various R between modeling results and experimental data of Ti-6Al-4V alloy [6].
Fig. 7. Comparison of fatigue thresholds at various R between modeling results and experimental data of Al-2024-T3 [21].
Fig. 8. Comparison of fatigue thresholds at various R between modeling results and experimental data of Electrolytic Tough Pitch (ETP) Copper (quarter-hard) [22].
Fig. 5. Comparison of fatigue thresholds at various R between modeling results and experimental data of 7075-T7351 Aluminium Alloy [19].
(12b). As mentioned above, in the case of FCG data, and assuming Ref = 0, based on Eq. (10b), f(R)/f(0) is thus defined as Eq. (16).
( (
) )
1 2R f (R) ⎧ 1 − R exp α 1 − R , for − 1 ≤ R < R c (i) ⎫ = ⎨ 1 exp α 2R , for R ≤ R < 1 (ii) ⎬ f (0) c 1−R ⎩ 1 − Rc ⎭
Essentially, Eqs. (14b) and (14c) translate data points of (ΔKR , da on the FCG curve at R, to (ΔK 0 , dN ) on the FCG curve of
( )
( )
da ) dN R
R = 0. A log–log plot of
( )
da versus dN 0
0
ΔK 0 is expected to collapse crack growth data of different R onto the FCG curve at R = 0. Thus, Eqs. (14b) and (14c) and the fatigue threshold models of Eqs. (11) and (12b) are used to validate the FCG data of various steels and alloys in open
(16) 97
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Copper (quarter-hard) (Fig. 8) [24], and 2195-Al-Li Alloy (Fig. 9) [25]. Similarly, FCG behavior from modeling using Eqs. (14b) and (14c), and experiments are compared, and the corresponding results are shown in Figs. 10–16. The left parts of Figs. 10–16 present the conventional log-log plots of da/dN versus ΔKR according to the Paris-Erdogan law of Eq. (1), while the right parts of Figs. 10–16 illustrates a parallel plot of (da/dN)0 (determined from Eq.(14b) versus ΔK0 (determined in terms of Eq.(14c)). The parameter n, determined as illustrated in Fig. 2, is listed in Table 1, for the respective materials. It is worth noting that all data points at various R collapse onto one single master curve, especially, in the near-threshold regime where the original disparities among different R disappear. The model was further verified by using FCG data of two welded joints of 25Cr2Ni2MoV and 30Cr2Ni4MoV steels [11,12]. As illustrated in Figs. 17–21, the perfect agreement between experimental data and modeling further indicates the robustness of the model developed in this work.
Fig. 9. Comparison of fatigue thresholds at various R between modeling results and experimental data of 2195 Aluminium-Lithium-Alloy [23].
4. Discussion
literatures. Comparison of modeled results with experimental data is shown in Figs. 3–16 for different materials. The parameters α and Rc as well as ΔKth0 for various materials were estimated from available data and are listed in Table 1. Figs. 3–9 show threshold data of ΔKth (experimental) and Kmax-th (calculated) as a function of R, compared with modeling results of Eqs. (11) and (12b), respectively. It is found that agreement between experimental data and modeling results is very good. Note that the y-axis denotes normalized fatigue threshold value, i.e., the ratio of fatigue threshold at any given R with threshold value at R of 0, ΔKthR/ΔKth0 and Kmax-thR/Kmax-th0. It is apparent that the new developed fatigue threshold model has good predictability as good agreement is achieved in various steels and alloys, i.e., 25Cr2Ni2MoV rotor steel (Fig. 3) [10], Ti-6Al-4V (Fig. 4) [6], Al-7075-T7351 (Fig. 5) [21], STOA Ti-6Al-4V, (Fig. 6) [22], Al-2024-T3 (Fig. 7) [23], Electrolytic Tough Pitch (ETP)
4.1. Significance of modeling parameters The good predictability of the model is largely dependent on how the parameters are determined. In this work, Rc, is estimated based on fatigue threshold values at the da/dN of 1 × 10−7 mm/cycle, as described in Section 2.3. Fatigue threshold could be determined at da/dN of 1 × 10−8 mm/cycle if sufficient data were at hand at low da/dN level. This is the parameter required for the model when crack closure influences are incorporated into the original exponential mean stress model. Note that the fatigue threshold values are achieved from loadshedding procedure based on ASTM E647. The parameter α, which physically means mean stress sensitivity factor, is unique for a given material, loading condition and environment, and remains constant for all R, as shown in the present model. This is different from the initial
Fig. 10. Plots of da/dN versus ΔK according to Eq. (1) (left) and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of 25Cr2Ni2MoV steel [10]. 98
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Fig. 11. Plots of da/dN versus ΔK according to Eq. (1) (left) and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of Ti-6Al4 V alloy [6].
Fig. 12. Plots of da/dN versus ΔK according to Eq. (1) (left) and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of 7075T7351 Aluminium Alloy [19].
EDF model [20], where the α value is varied at higher and lower R. The parameter of α controls how strong the mean stress will influence the FCG behavior, and rationalizes the one α-value in the new model due to the mean stress being active only in high R. As can be seen from Table 1, α is more related to microstructures and material types, and the α value of Ti-6Al-4V could be 4–10 times higher than steels and other alloys. Though the current model is validated using the same materials
from which the parameters are determined, unlike the original model, the newly developed one in this work focuses on modifying the effectiveness of the stress intensity factor range by taking into account the role of crack closure in the near-threshold FCG regime. The agreement between modeling results and experimental data presented here further verify that crack closure cannot be neglected in FCG modeling. The new model is thus more mechanistically representative of the FCG behavior, a reason that underlines the good predictability. The ΔK⋅f(R) can be 99
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Fig. 13. Plots of da/dN versus ΔK according to Eq. (1) (left) and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of STOA Ti-6Al-4V [20].
Fig. 14. Plots of da/dN versus ΔK according to Eq. (1) (left) and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of 2024T3 Aluminium Alloy [21].
4.2. Significance of ΔK versus Kmax
regarded as the driving force parameter for FCG in the near-threshold and Paris regimes (see Figs. 10–16), similar to the one proposed in [13]. Furthermore, the new model is developed based on the concept of combining mean stress effect and crack closure roles, which verifies the effectiveness of the combined approach proposed by Zhu et al. [13].
Though ΔKR⋅f(R) is the driving force for fatigue cracking, one may wonder the significance of ΔK versus Kmax in FCG behavior under the influence of R. From the very definition of R, Eq. (17), there comes a fixed correlation of ΔKthR/Kmax-thR with R at fatigue threshold 100
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Fig. 15. Plots of da/dN versus ΔK according to Eq. (1) (left), and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of Electrolytic Tough Pitch (ETP) Copper (quarter-hard) [22].
Fig. 16. Plots of da/dN versus ΔK according to Eq. (1) (left), and da/dN × (f(R)/f(0))n versus ΔKRf(R)/f(0) according to Eq. (16) (right), based on FCG data of 2195 Aluminium-Lithium-Alloy [23].
increases sharply with ΔKthR varying little, which indicates that Kmax is driving fatigue cracking at the near-threshold regime. As a result, the parameter of Rc in this model denotes a transition point where the significance of ΔKR transits to Kmax. This accords well with the general finding that cyclic loading at lower R involves the “fatigue” type of cracking while loading at high R involves the “static” type of cracking.
conditions, as shown in Eq. (18). It is apparent that an increase in R results in a decrease of ΔKthR/Kmax-thR, which implies either a decreasing ΔKthR and/or an increasing Kmax-thR. It is observed from Figs. 3–9 that in case of R < Rc, where crack closure is active, ΔKthR decreases strongly while Kmax-thR is almost stable, which indicates that ΔKthR is driving the fatigue cracking behavior at the near-threshold regime. Whereas for R > Rc, the crack closure is not active, and Kmax-
thR
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Table 1 Parameters employed for prediction of FCG and fatigue threshold. Materials 25Cr2Ni2MoV steel [10] Ti-6Al-4V [6] Al-7075-T7351 [19] STOA Ti-6Al-4V [20] Al-2024-T3 [21] EPT COPPER (1/4-Hard) [22] 2195 Al-Li-Alloy [23] Welded 25Cr2Ni2MoV steel [12] (Notch in weld metal) Welded 25Cr2Ni2MoV steel [12] (Notch in HAZ) 30Cr2Ni4MoV [11] (As received, notch in weld metal) 30Cr2Ni4MoV [11] (Aged, notch in base metal)
da/dN (mm/cycle) −7
1 × 10 1 × 10−7 1 × 10−7 1 × 10−7 1 × 10−7 1 × 10−7 1 × 10−7
Rc
α
ΔKth0 (MPa-m1/2)
n
0.72 0.4 0.53 0.4 0.5 0.4 0.5 0.72 0.72 0.72 0.72
0.005 0.04 0.01 0.01 0.008 0.03 0.01 0.04 0.05 0.01 0.01
6.8 12.0 3 5.2 3.5 6.5 2.6
2.0 2 1.2 2 2 2 1.2 2.3 2.3 2.3 2.0
Fig. 17. Crack growth data of 25Cr2Ni2MoV steel welded joint with notch in weld metal: original experimental data (left) and present modeling results (right) [12].
R=
Kmin K ΔK − ΔK = max =1− Kmax Kmax Kmax
ΔKthR +R=1 Kmax − thR
4.3. Significance of the f(R) model (17) Conventionally, the ΔKeff is expressed as Eq. (19), where the crack closure coefficient U is determined as Eq. (20). The U can be regarded as one specific case of the f(R) in case there is little mean stress effect. Crack closure and mean stress oppose each other in terms of their contribution to crack growth. Whereas crack closure tends to decrease the applied stress intensity range, mean stress amplifies the applied stress intensity range, to present a full view of driving force for fatigue cracking. The overall effect would depend on which of the two dominates the exchanges. For f(R) < 1, crack closure dominates and suppresses mean stress effects, which is the case within the threshold regime. For f(R) > 1, the crack closure effect is minimal, and mean stress effects dominantly control crack growth. Therefore, f(R), which is defined in terms of Eq. (8) is more representative of FCG behavior due to both crack closure and mean stress effects.
(18)
In the fatigue damage point of view, in a higher R regime, where fatigue process is represented by higher mean stress with lower stress amplitude, the role of high mean stress is to keep crack open while localized monotonic and cyclic crack tip plasticity drives crack growth. In a lower R regime, the fatigue loading is represented as lower mean stress and higher stress amplitude, the lower mean stress cannot keep the crack always open while the cyclic crack tip plasticity drives crack growth slowly. It is thus inferred that the mean stress level is indicative of whether crack surface can touch each other, whereas the stress amplitude denotes the different modes of crack growth. As a result, the R effect is not equivalent to mean stress contribution, as was often anticipated, whether the crack can grow or how it will be grown should also be taken into account. This all accords with and rationalizes the physics of fatigue crack growth as denoted by the f(R) model.
ΔK eff = Kmax − K op U=
ΔK eff ΔKR
=
(19)
K op ⎞ 1 ⎛ 1− 1−R⎝ Kmax ⎠ ⎜
⎟
(20)
In addition, note that when α = 0, f(R) = U(R), and this indicates 102
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Fig. 18. Crack growth data of 25Cr2Ni2MoV steel welded joint with notch in heat-affected zone: original experimental data (left) and present modelling results (right) [12].
Fig. 19. Crack growth data of welded 30Cr2Ni4MoV steel in as-received condition (weld metal notched): original experimental data (left) and present modelling results (right) [11].
combining mean stress and crack closure effects in this work is different from the one proposed by Zhu et al. in [10,13] where the mean stress and crack closure are added for R-dependence interpretation of FCG, while in this work, these two aspects are multiplied as shown by the f (R).
that crack growth is insensitive to mean stress and that crack closure solely influences crack growth (see Fig. 1(right)). It can be observed from Fig. 1(left) that with the increasing of R, the f(R) values are gradually increased due to combined effects of crack closure and mean stress, with f(R) < 1, in the closure-dominated regime, while for f (R) > 1, the fatigue cracking is within closure-free and mean-stressdominated FCG regime. Finally, it is worth noting that the way of
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Fig. 20. Crack growth data of 30Cr2Ni4MoV steel welded joint under long-term aging condition with notch in weld metal: original experimental data (left) and present modeling results (right) [11].
Fig. 21. Crack growth data of 30Cr2Ni4MoV steel welded joint under long-term aging condition with notch in base metal: original experimental data (left) and present modeling results (right) [11].
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5. Conclusions
Acta Mater 1997;45:2777–87. [7] Ritchie RO, Boyce BL, Campbell JP, Roder O, Thompson AW, Milligan WW. Thresholds for high-cycle fatigue in a turbine engine Ti–6Al–4V alloy. Int J Fatigue 1999;21:653–62. [8] Chiang CR. Threshold stress intensity factor of fatigue cracks. Eng Fract Mech 1994;49:29–33. [9] Kardomateas GA, Carlson RL. Predicting the effects of load ratio on the fatigue crack growth rate and fatigue threshold. Fatigue Fract Eng Mater Struct 1998;21:411–23. [10] Zhu M-L, Xuan F-Z, Tu S-T. Interpreting load ratio dependence of near-threshold fatigue crack growth by a new crack closure model. Int J Press Vessels Pip 2013;110:9–13. [11] Du Y-N, Zhu M-L, Liu X, Xuan F-Z. Effect of long-term aging on near-threshold fatigue crack growth of Ni–Cr–Mo–V steel welds. J Mater Process Technol 2015;226:228–37. [12] Du Y-N, Zhu M-L, Xuan F-Z. Transitional behavior of fatigue crack growth in welded joint of 25Cr2Ni2MoV steel. Eng Fract Mech 2015;144:1–15. [13] Zhu M-L, Xuan F-Z, Tu S-T. Effect of load ratio on fatigue crack growth in the nearthreshold regime: a literature review, and a combined crack closure and driving force approach. Eng Fract Mech 2015;141:57–77. [14] Ritchie RO, Knott JF. Mechanisms of fatigue crack growth in low alloy steel. Acta Metall 1973;21:639–48. [15] Elber W. The significance of fatigue crack closure. In: Damage tolerance in aircraft structures, ASTM STP 486, Canada; 1970. p. 230–42. [16] Walker K. The effect of stress ratio during crack propagation and fatigue for 2024T3 and 7075-T6 aluminum. In: Effects of environment and complex load history on fatigue life: a Symposium, ASTM STP 462, Lutherville-Timonium, MD; 1970. p. 1–14. [17] Newman JC. A crack opening stress equation for fatigue crack growth. Int J Fract 1984;24:R131–5. [18] McClung RC, Chell G, Lee Y-D, Russell D, Orient G. Development of a practical methodology for elastic-plastic and fully plastic fatigue crack growth, NASA/CR1999-209428; 1999. [19] Schijve J. Fatigue of structures and materials. Springer; 2001. [20] Kwofie S, Rahbar N. An equivalent driving force model for crack growth prediction under different stress ratios. Int J Fatigue 2011;33:1199–204. [21] Newman JC, Brot A, Matias C. Crack-growth calculations in 7075–T7351 aluminum alloy under various load spectra using an improved crack-closure model. Eng Fract Mech 2004;71:2347–63. [22] Boyce BL, Ritchie RO. Effect of load ratio and maximum stress intensity on the fatigue threshold in Ti–6Al–4V. Eng Fract Mech 2001;68:129–47. [23] Miller M, Gallagher J. An analysis of several fatigue crack growth rate (FCGR) descriptions. Fatigue crack growth measurement and data analysis. ASTM International; 1981. [24] Liaw PK, Leax TR, Williams RS, Peck MG. Near-threshold fatigue crack growth behavior in copper. Metall Trans A 1982;13:1607–18. [25] Chen DL, Chaturvedi MC. Near-threshold fatigue crack growth behavior of 2195 aluminum-lithium-alloy—prediction of crack propagation direction and influence of stress ratio. Metall Mater Trans A 2000;31:1531–41.
In this work, an exponential mean stress model is combined with crack closure for developing a more general form of FCG model for interpreting the R-dependence of FCG behavior in both Paris and nearthreshold regimes. A more general model for FCG and fatigue threshold prediction was developed, and was validated widely by using published data of various steels and alloys in open literatures. The model could collapse FCG data at various R onto one single master curve, and good agreements were achieved between modeling results and experimental data of fatigue threshold. All these have implied the feasibility of combining crack closure approach with mean stress effects for a sound interpretation of R-dependence of FCG in both Paris and near-threshold regimes. The new approach has also provided information about the transition of crack growth driving force with varying R, and has informed of an updated concept that R-dependence of FCG is not equivalent to the mean stress effect. Acknowledgement Funding support by the Ghanaian Government in terms of Book and Research Allowance is greatly appreciated. Ming-Liang Zhu would like to thank the support by National Natural Science Foundation of China (No. 51575182) and the 111 Project (No. B13020). References [1] Paris P, Erdogan F. A Critical Analysis of Crack Propagation Laws. J Basic Eng 1963;85:528–33. [2] Schmidt RA, Paris PC. Threshold for fatigue crack propagation and the effects of load ratio and frequency. Progress in flaw growth and fracture toughness testing, ASTM STP536. Philadelphia: ASTM; 1973. p. 79–94. [3] Ritchie RO. Near-threshold fatigue crack propagation in ultra-high strength steel: influence of load ratio and cyclic strength. J Eng Mater Technol 1977;99:195–204. [4] Liaw PK, Lea TR, Logsdon WA. Near-threshold fatigue crack growth behavior in metals. Acta Metall 1983;31:1581–7. [5] Taylor D. A compendium of fatigue thresholds and growth rates. West Midlands, UK: Engineering Materials Advisory Services Ltd.; 1985. [6] Dubey S, Soboyejo ABO, Soboyejo WO. An investigation of the effects of stress ratio and crack closure on the micro-mechanisms of fatigue crack growth in Ti-6Al-4V.
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