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Fatigue propagation threshold of short cracks under constant amplitude loading
International Journal of Fatigue 25 (2003) 1319–1326 www.elsevier.com/locate/ijfatigue
Fatigue propagation threshold of short cracks under constant amplitude loading Mirco D. Chapetti ∗ INTEMA, University of Mar del Plata-CONICET, J.B. Justo 4302, B7608FDQ Mar del Plata, Argentina Received 31 May 2002; received in revised form 25 November 2002; accepted 20 February 2003
Abstract In this study a threshold for fatigue crack propagation as a function of crack length is defined from a depth given by the position d of the strongest microstructural barrier to crack propagation, which defines the plain fatigue limit. The material threshold is estimated from the plain fatigue limit ⌬seR, the position d of the strongest microstructural barrier and the threshold for long cracks, ⌬KthR. The threshold for eight different materials for which experimental results can be obtained from the literature was estimated. Good agreement was observed in all cases. Some quantitative analyses of the fatigue propagation behavior of short cracks are carried out and discussed. 2003 Elsevier Ltd. All rights reserved. Keywords: Short cracks; Fatigue crack growth threshold; Non-propagating cracks
1. Introduction The initiation and early propagation of fatigue cracks are strongly influenced by the microstructure and the grain size, and they seem to be related to the fatigue limit of metals [1–6]. Both, the fatigue limit and the high cycle fatigue resistance depend on the effective resistance of the microstructural barriers that has to be overcome by the cracks. Each of these barriers has a characteristic dimension and a critical stress range associated with its resistance to crack propagation. The plain fatigue limit is determined by the strongest microstructural barrier, since that resistance is generally greater than the resistance to crack nucleation. On the other hand, above the fatigue limit and for a given stress range, each barrier has an associated number of cycles that is necessary to propagate the crack. The effect of crack size on fatigue crack propagation threshold can be described conveniently by means of the Kitagawa–Takahashi plot relating the threshold stress with the crack size, as shown in Fig. 1. If the strongest microstructural barrier to fatigue crack propagation is
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placed at a given distance d from the material surface, the crack is non-damaging with respect to the plain fatigue limit up to a crack size a = d [2,4–6]. For a microstructurally short crack (MSC, the crack length is of the order of the microstructural dimensions) initiated from a plain surface the fatigue limit at a given stress ratio R, ⌬seR, defines the critical nominal stress range needed for continued crack growth (microstructural threshold). If the applied stress range ⌬s is smaller than ⌬seR, cracks (included in the MSC regime) are arrested at microstructural barriers placed at depths smaller than d. On the other hand, for long cracks (LC) the fatigue crack propagation threshold decreases with increasing crack size [1–8]. The threshold for LC (whose length is greater than that at which crack closure is fully developed) is defined in terms of the threshold value of the stress intensity factor range, ⌬KthR, thus LC in constant amplitude loading can only grow by fatigue if the applied stress intensity factor range ⌬K is greater than ⌬KthR. In the physically short crack (PSC) regime, a transition between the microstructurally short and long crack regimes (MSC and LC), the threshold is below ⌬seR and ⌬KthR. Although the influence of microstructure is still important, the development of crack closure governs the
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Nomenclature a C∗ and d D ⌬s ⌬seR ⌬sn ⌬sth ⌬K ⌬KC ⌬KCR ⌬KdR ⌬Kth ⌬KthR ⌬KthReff k M R r
crack length m environmentally sensitive material constants position from the surface of the strongest microstructural barrier notch length applied stress range plain fatigue limit nominal applied stress range threshold stress range for crack propagation applied stress intensity factor range extrinsic component of ⌬Kth extrinsic component of ⌬KthR microstructural threshold fatigue crack propagation threshold fatigue crack propagation threshold for long cracks effective threshold for long cracks (without crack closure) material constant that takes into account the development of ⌬KC microstructural characteristic dimension stress ratio (minimum stress/maximum stress) notch root radius
gation threshold to be known as a function of crack length. In this study an expression for estimating the threshold for fatigue crack propagation as a function of crack length is obtained by using only the plain fatigue limit, ⌬seR, the threshold for long crack, ⌬KthR, and the microstructural characteristic dimension (e.g. grain size).
2. Crack growth threshold
Fig. 1. Kitagawa–Takahashi type diagram showing the threshold between propagation and non-propagating cracks; after Ref. [6].
threshold level in this regime. In order to denote both parameters, the plain fatigue limit (⌬seR) and the propagation threshold for LC (⌬KthR) depend on the stress ratio for a given material, the sub-index R is included. The difference between the total applied driving force defined by the applied stress intensity factor range for a given geometrical and loading configuration, ⌬K, and the threshold for crack propagation, ⌬Kth, defines the effective driving force applied to the crack. By knowing the crack growth rate as a function of the effective driving force for a given material, it is possible to estimate the fatigue crack propagation behavior in the given conditions. This concept is the base of the Resistance-Curve Method [5,7,9], which needs the variation of the propa-
It is well known that the plain fatigue limit is defined by the ability of the first grain boundary to arrest a microcrack (see Fig. 1). This is a material-based limit (depending on the microstructural characteristic dimension, M) as Miller has pointed out [6]. In recent works carried out by Chapetti and coworkers [4,10], the position and the effective resistance of microstructural barriers and their relation with the fatigue limit were analyzed and modeled, and additional evidences that the strongest microstructural barrier defines the fatigue limit of plain and blunt-notched specimens in steels were obtained. An intrinsic resistance to MSC propagation is defined by using the position d of the strongest microstructural barrier and the plain fatigue limit ⌬seR. This intrinsic resistance is considered to be a microstructural threshold for crack propagation and is then defined as follows ⌬KdR ⫽ Y⌬seR冑πd,
(1)
where Y is the geometrical correction factor. In most cases the nucleated microstructurally short surface
M.D. Chapetti / International Journal of Fatigue 25 (2003) 1319–1326
cracks are considered semicircular [1–10], and the value of Y would then be 0.65. Because the plain fatigue limit depends on the stress ratio R, the microstructural threshold also does. The value of d is usually given by the microstructural characteristic dimension, M (e.g. ferrite grain size, bainite or martensite lath length, etc. [4,6,10]). Then, for the LC regime (where crack closure has built up to a steady state level), a total extrinsic component ⌬KCR can be defined as the difference between the mechanical threshold for LC, ⌬KthR, and the microstructural one, ⌬KdR ⌬KCR ⫽ ⌬KthR⫺⌬KdR.
(2)
The total extrinsic component of the threshold corresponds to a steady state level and is built up after a given crack length that depends on the material (a = l in Fig. 1). For a given material, ⌬KCR is a constant value that depends on the stress ratio R. In order to obtain an expression to estimate the fatigue crack propagation threshold as a function of crack length, the development of the extrinsic component (⌬KC) has to be defined. The expression proposed by McEvily and Minakawa [7] to model the development of the crack closure component of the stress intensity factor range is used here to estimate ⌬KC ⌬KC ⫽ ⌬KCR(1⫺e⫺ka),
(3)
where the material constant k defines the development of the extrinsic component ⌬KC for each stress ratio, and a is the crack length expressed in mm. According to expressions (2) and (3), ⌬KCR is not equal to the closure component for LC because ⌬KdR is not equal to the effective threshold (without the crack closure component) for LC, ⌬KthReff. However, the definition of the extrinsic component of the threshold for LC does not change the actual total threshold, given by ⌬KthR. Besides, the author believes that the fatigue propagation threshold for microstructural short cracks is not given by ⌬KthReff because a local crack driving force should also be considered [11–14]. The local crack driving force is provided by the strain energy stored in the form of internal stress fields generated by cyclic plastic deformation and is related with the surface strain concentration [12,13]. The value of the local crack driving force is initially high and rapidly drops as crack grows [12,13], and its influence on short crack propagation behavior extends to a few microstructural characteristic dimensions, M (e.g. grain size) [14]. The material threshold for crack propagation as a function of the crack length, ⌬Kth, is then defined as ⌬Kth ⫽ ⌬KdR ⫹ ⌬KC ⫽ Y⌬sth冑πa
aⱖd.
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Fig. 2 shows schematically the threshold curve given by expression (5). For a crack length a = d, ⌬Kth = ⌬KdR, and ⌬Kth tends to ⌬KthR for LC. In terms of the threshold stress, we get ⌬sth ⫽
⌬KdR ⫹ (⌬KthR⫺⌬KdR)[1⫺e⫺k(a⫺d)]
⌬sth ⫽ ⌬seR
Y冑πa
a ⬍ d,
aⱖd, (6a) (6b)
where ⌬KthR and ⌬seR (and so ⌬KdR) are all functions of the stress ratio R. Finally, an expression for the material parameter k, that defines the development of the extrinsic component of ⌬Kth, ⌬KC, should be obtained. We can use the fact that the plain fatigue limit is given by the resistance of the strongest microstructural barrier, and that the threshold stress for crack propagation decreases as the crack length increases for a crack length a ⬎ d (see Fig. 1). By using this concept it is possible to obtain a maximum value for the parameter k. The limit would be given by the following condition: the slope of the threshold stress as a function of crack length is equal to zero at a crack length a = d, that is to say, the threshold stress curve is tangent to an horizontal line defined by ⌬s = ⌬seR. By deriving Eq. (6a), equating to zero and solving for k, it is easy to arrive at the following expression kⱕ
⌬KdR 1 . 2d (⌬KthR⫺⌬KdR)
(7)
In order to check the ability of expression (7) to describe the development of the extrinsic component, the threshold stress given by expressions (6a) and (7) was analyzed for actual value ranges of d, ⌬KdR and ⌬KthR. In some cases the upper limit of the value of k is not enough to assure that the threshold stress decreases monotonically as the crack length increases. This problem arises when the total extrinsic component (⌬KCR) is greater enough than the microstructural threshold (⌬KdR) and/or for relatively high d values. Then, the value of k should be smaller than that given by expression (7). Experimental short crack propagation threshold data for
(4)
From expressions (2)–(4), we finally get ⌬Kth ⫽ ⌬KdR ⫹ (⌬KthR⫺⌬KdR)1⫺e⫺k(a⫺d)
aⱖd, (5)
Fig. 2. Defined fatigue crack propagation threshold as a function of crack length, given by expression (5).
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two steels with the same composition but different grain sizes were obtained from the literature [15] (steel S20C, M = 7.8 and 55 µm, see Table 1), in order to define a constant fitting parameter in expression (7). A value of k equal to one half of its upper limit given by expression (7) gives a threshold for crack propagation in good agreement with the experimental results (see first two graphs in Fig. 3). Then, the following expression is finally defined to estimate the material parameter k ⌬KdR 1 ⌬KdR 1 . ⫽ k⫽ 4d (⌬KthR⫺⌬KdR) 4d ⌬KCR
(8)
Fig. 3 shows the estimated ⌬Kth given by expressions (5), (1) and (8) for eight materials, including the two steels used to define expression (8), other four steels, electrolytic copper and 2023-T3 aluminum alloy (see Table 1). Experimental results obtained from the literature are also shown [15–18]. Solid lines correspond to the estimated threshold given by expression (5) by considering semicircular surface cracks for the definition of ⌬KdR (Y = 0.65 in Eq. (1)). In accordance with author’s analysis, through thickness cracks (Y = 1.12) should be considered for the materials from reference [16]. Estimated thresholds for these cases are shown by dashed curves. Very good agreement can be seen in all cases. Table 1 shows values of ⌬seR, ⌬KthR, M, d, ⌬KdR, ⌬KCR and k for the materials whose thresholds are shown in Fig. 3, as well as for some others for which experimental results of the threshold are not available [15–21]. It should be noted that in this paper ⌬K is defined as the full difference between the maximum and minimum values of K and not the tensile part alone. The material parameter k is proportional to ⌬KdR/⌬KCR in expression (8). This is in accordance with the general trend observed for steels. Low carbon steels
show relatively low k values defined by a small relation between the intrinsic and the extrinsic components and a relatively large grain size. On the other hand, high strength steels show relatively high k values given by a relatively high ⌬KdR/⌬KCR relation and small microstructural characteristic dimensions. Taking into account the defined parameter k, expression (5) can be rewritten as follows ⌬KC ⌬Kth⫺⌬KdR ⫽ ⌬KthR⫺⌬KdR ⌬KCR
冉
⫽ 1⫺exp ⫺
(9)
冊
1 ⌬KdR (a / d⫺1) 4 ⌬KCR
aⱖd.
Fig. 4 shows plots given by this expression for different values of ⌬KdR/⌬KCR, as well as all experimental results shown in Fig. 3. The extrinsic component of the experimental results was estimated by subtracting the microstructural threshold (⌬KdR), calculated with ⌬seR and d in Table 1, from the experimental threshold (⌬Kth) data (given in Fig. 3). For most of the analyzed materials the extrinsic component of the threshold is fully developed at a depth of about 20–30 times the microstructural characteristic dimension, except for the S20C steel with a grain size of 0.0078 mm and for copper (50– 60 times M). It is also possible to find examples for which ⌬KC is fully developed for a crack length of 10 times the microstructural characteristic dimension or less, as for the case of the JIS SUJ2 steel shown in Table 1. Finally, we can say that expressions (1), (5), and (8) define a material threshold for fatigue crack propagation that can be estimated from the plain fatigue limit ⌬seR, the position of the strongest microstructural barrier d, and the mechanical threshold for LC, ⌬KthR. These are
Table 1 Experimental and estimated data for several materials Material
0.42Ca 2.25Cr1Moa S20Ca S20Ca SM41B(S)a SM41B(L)a JIS SUJ2 2024-T3a Coppera Inconel 718 Ti–6Al–4V Ti–6Al–4V a
Reference
[17] [18] [15] [15] [16] [16] [19] [16] [18] [20] [21] [21]
R
⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 0.1 0.8
⌬seR (MPa) ⌬KthR (MPa m1/2)
M (mm)
d
⌬KdR (MPa m1/2) Eq. (1) Y = 0.65
450 500 470 326 396 326 2400 300 146 920 450 160
0.018 0.025 0.0078 0.055 0.014 0.064 0.01 0.027 0.05 0.01 0.02 0.02
M M M M M M M M 2M M M M
2.2 2.9 1.5 2.8 1.7 3 8.7 1.8 1.7 3.3 2.3 0.8
5.9 9.1 10.4 12.4 10.2 12.3 12 6.4 5 11.5 4.3 2.4
Experimental results of the threshold are shown in Fig. 3.
⌬KCR (MPa m1/2) Eq. (3)
3.7 6.2 8.9 9.6 8.5 9.4 3.3 4.6 3.3 8.2 2 1.6
⌬KdR/⌬KCR
0.59 0.46 0.17 0.29 0.20 0.32 2.64 0.39 0.51 0.40 1.15 0.50
k (mm⫺1) Eq. (8)
8.2 4.6 5.4 1.3 3.6 1.25 66 3.6 1.3 10 14.4 6.3
M.D. Chapetti / International Journal of Fatigue 25 (2003) 1319–1326
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Fig. 3. Estimated fatigue crack propagation threshold as a function of crack length for eight materials (see Table 1). Experimental results from the literature are also shown.
mechanical and microstructural parameters, which can be obtained through standardized mechanical tests and simple microstructural analyses. The defined threshold as a function of crack length also allows definition of a crack initiation period as the number of fatigue cycle necessary to initiate a MSC of length d. This assumption is indicated in Figs. 2 and 7.
3. Application example and discussions
Fig. 4. Extrinsic component of the fatigue crack propagation threshold as a function of crack length.
The difference between the total applied driving force and the material threshold for crack propagation defines the effective driving force applied to the crack. The high cyclic fatigue crack propagation behavior can be estimated for a given crack length range if the crack growth rate as a function of the effective driving force is known for a given material. The initial crack length is given by
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the position of the strongest microstructural barrier if the material were free of cracks or crack like flaws. The final crack length is defined according to critical parameters related with the loading conditions of the analyzed components. In the case of smooth specimens or components, the stress can be considered constant for any crack length, equal to the nominal applied stress. The following general expression can be used to estimate the applied driving force as a function of crack length [22]: ⌬K ⫽ Y⌬sn冑πa,
(10)
where ⌬sn is the nominal applied stress range. The crack aspect ratio as a function of crack length has to be defined for the combination of component geometry and loading conditions, which allows definition of the value of the parameter Y as a function of crack length. In the case of notches, an expression for the stress concentration as a function of crack length is needed, that is to say, the normalized applied stress distribution defined by the notch has to be known. Approximate local stress intensity factor range for small cracks at notches that are in elastic stress fields in terms of the cracks length, a, notch root radius, r, nominal stress range, ⌬sn, and elastic stress-concentration factor, kt, is given by the following expression obtained by Luka´ s et al. [18] ⌬K ⫽
kt
冑1 ⫹ 4.5(a / r)
Y⌬sn冑πa.
(11)
Quantitative analysis of fatigue crack growth requires a constitutive relationship of general validity be established between the rate of fatigue crack growth, da/dN, and some function of the range of the applied stress intensity factor, ⌬K (crack driving force). Besides, it has to take into account the threshold for the whole crack length range, including the short crack regime where the fatigue crack propagation threshold is a function of crack length. Among others, the following relationship meets these requirements [23] da ⫽ C∗(⌬K⫺⌬Kth)m, dN
(12)
where C∗ and m are environmentally sensitive material constants obtained from long crack fatigue behavior and ⌬Kth is the crack growth threshold as a function of a and R and represents the effective resistance of the material to fatigue crack propagation. The fatigue crack propagation life for a given crack length range and a given material can be obtained by integrating expression (12) and using expression (5) for the threshold of the material (⌬Kth), and a proper expression for the applied driving force (⌬K) given by the analyzed loading and geometrical configuration. Fig. 5 shows a da/dN–⌬K plot with some experimental data corresponding to short cracks propagating
Fig. 5. da/dN–⌬K plot showing experimental data from short cracks propagating from notches in JIS SM41B(S) steel [16] (see also Table 1). Threshold and Paris region crack propagation behaviors for LC, as well as the estimated propagation behavior of surface cracks in smooth material under ⌬sn = 400 MPa, are also shown.
from notches in a JIS SM41B steel, obtained by Akiniwa et al. [16] (this material is also included in Fig. 3 and Table 1). Threshold and Paris region crack propagation behaviors for LC are also shown [16]. The figure shows the propagation behavior of a surface crack in smooth material for a nominal stress range equal to 400 MPa (just above the plain fatigue limit of the material), estimated by integrating Eq. (12) from a crack length equal to d, using expressions (5) and (10), and taking the parameter Y equal to 1.12 to consider through thickness cracks. Fig. 5 reveals that a short crack propagates faster than a long crack at a given ⌬K, and still propagates in the region below ⌬KthR (threshold for LC). When the crack length exceeds 1 mm the propagation rate of the short crack comes closer to that of the long crack. The dashed line shows the microstructural threshold, ⌬KdR, which represents the minimum ⌬K for crack propagation. It is actually possible to observe fatigue propagation of MSCs at ⌬K ⬍ ⌬KdR, but any propagation at a ⬍ d is considered part of the crack initiation period. Fig. 6 shows the estimated fatigue propagation behavior of short cracks initiated at three notches with different geometries for the same material. Two of those notches have the same stress-concentration factor, kt (equal to 3), but different notch root radius, r = 3 and 0.5 mm. The third notch is the same as the one used by Tanaka and Akiniwa [24] to obtain the experimental results shown in Fig. 5: D = 3 mm, r = 0.16 mm and kt = 8.48. For the r = 3 mm notch only the nominal applied stress range equal to 140 MPa is considered. This stress level is just 10 MPa above the estimated
M.D. Chapetti / International Journal of Fatigue 25 (2003) 1319–1326
Fig. 6. Estimated fatigue crack propagation behavior of short cracks for three different notched SM41B(S) steel specimens.
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r = 3 mm and kt = 2.65 analyzed experimentally by Akiniwa et al. [16] is 155 MPa, and mechanically short non-propagating cracks were not found. In the case of the r = 0.5 mm notch and ⌬sn = 140 MPa the estimated stress distribution can also drive the initiated crack of length d but only to a depth of about 0.09 mm, at which the stress level becomes smaller than the threshold stress. By increasing the applied stress level, the crack can propagate further. The fatigue limit is estimated to be about 180 MPa, for which a nonpropagation crack length as long as 0.5 mm can be observed. Fig. 7 shows the stress distribution for ⌬sn = 200 MPa. It can be seen that the crack can propagate without arresting, since the applied stress is greater than the threshold stress for any crack length. The estimated fatigue limit associated to this notch is 190 MPa, 60 MPa greater than the estimated one for r = 3 mm notch. The difference is due to the notch size effect. Different notch geometries define different stress gradients in the short crack regime. Included in Fig. 7 is the stress distribution estimated using expression (11) for the sharp notch (kt = 8.48) and ⌬sn = 130 MPa. It is worth noting that in this case the stress level near the surface is quite greater than twice the yield stress of the material (560 MPa [16]). However, the threshold condition defining the fatigue limit is located at about 0.5 mm from the surface, where the stress level is below the plain fatigue limit. Another important feature that should be considered is the influence of the notch length (D) on fatigue crack propagation threshold for sharp notches.
4. Summary and conclusions
Fig. 7. Kitagawa–Takahashi type diagram showing the estimated threshold stress for fatigue crack propagation for SM41B(S) steel [16]. The applied stress distributions given by expression (11) are also shown for three different notch geometries.
fatigue limit for the same notch geometry (equal to 130 MPa), for which mechanically short non-propagating cracks cannot be formed. This fact can be analyzed in Fig. 7, which shows a Kitagawa–Takahashi type diagram with the threshold stress for the propagation of fatigue crack estimated with expressions (6a) and (6b), and the applied stress distributions estimated by using expression (11) for the three notch geometries. The stress distribution corresponding to r = 3 mm notch and ⌬sn = 140 MPa propagates the initiated crack of length d without arresting, since the applied stress is greater than the threshold stress for any crack length. It is important to note that the fatigue limit for the notch with
A microstructural threshold for crack propagation (⌬KdR) is defined by the plain fatigue limit (⌬seR) and the position (d) of the strongest microstructural barrier. A total extrinsic threshold to crack propagation (⌬KCR) is then defined by the difference between the crack propagation threshold for LC (⌬KthR) and the microstructural threshold (⌬KdR). The development of the extrinsic component is considered to be exponential and a development parameter k is defined as a function of the same microstructural and mechanical parameter used to define the material threshold for crack propagation. The difference between the total applied driving force and the material threshold for crack propagation defines the effective driving force applied to the crack. If crack growth rate as a function of this effective driving force is known for a given material, the fatigue crack propagation behavior in a given crack length range can be estimated. The initial crack length is given by the position of the strongest microstructural barrier if the material were free of cracks or crack like flaws. The final
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crack length is defined according to critical parameters related with the loading conditions of the analyzed components. The methodology presented in this paper, as well as the defined fatigue crack propagation threshold, is based on the experimental evidence that the fatigue limit of smooth and notched components represents the threshold stress for the propagation of initiated cracks, so that fatigue limit depends on the effective resistance of the microstructural barriers that have to be overcome by initiated cracks. The expression defined to estimate the material threshold for crack propagation as a function of the crack length allows definition of a crack initiation period as the number of load cycles necessary to initiate a crack of depth d, from which the crack propagation behavior can be analyzed. The definition of a crack initiation period of a component is some times ambiguous, and depends on the defects that are formed during manufacturing, fabrication, assembly, repair and maintenance. It is also intimately tied to the sensitivity of the crack detection technology. However, in any case an analytical methodology is required to predict the growth kinetics of cracks from a given initial size, and in many of those cases part of the material short crack regime has to be included. The presented model is simple and can be applied to analyses based on elastic stress distributions to study some interesting fatigue phenomena related with small fatigue cracks, including the mechanism of non-propagation crack development, notch size effect and fatigue notch sensitivity. The application of the model is limited to crack behavior evaluation near the fatigue crack growth threshold. Further analyses have to be carried out to evaluate the applicability and limitations of the methodology to study those cases where important plastic strain behavior is involved. The defined threshold as a function of crack length was estimated for eight different materials, including six steels, one aluminum alloy and one copper, and compared with experimental results from literature. Good agreement between estimated and experimental threshold was observed in all cases. Quantitative analyses of the high cycle fatigue propagation behavior (near threshold) of short cracks in notches were carried out and results are also in good agreement with experimental results from literature.
Acknowledgements The author wishes to express his thanks for funding provided by Consejo Nacional de Investigaciones Cientı´ficas y Te´ cnicas (CONICET), and by Agencia Nacional de Promocio´ n Cientı´fica y Tecnolo´ gica (PICT’98 12-04585/6), Argentina.
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Fatigue propagation threshold of short cracks under constant amplitude loading Mirco D. Chapetti ∗ INTEMA, University of Mar del Plata-CONICET, J.B. Justo 4302, B7608FDQ Mar del Plata, Argentina Received 31 May 2002; received in revised form 25 November 2002; accepted 20 February 2003
Abstract In this study a threshold for fatigue crack propagation as a function of crack length is defined from a depth given by the position d of the strongest microstructural barrier to crack propagation, which defines the plain fatigue limit. The material threshold is estimated from the plain fatigue limit ⌬seR, the position d of the strongest microstructural barrier and the threshold for long cracks, ⌬KthR. The threshold for eight different materials for which experimental results can be obtained from the literature was estimated. Good agreement was observed in all cases. Some quantitative analyses of the fatigue propagation behavior of short cracks are carried out and discussed. 2003 Elsevier Ltd. All rights reserved. Keywords: Short cracks; Fatigue crack growth threshold; Non-propagating cracks
1. Introduction The initiation and early propagation of fatigue cracks are strongly influenced by the microstructure and the grain size, and they seem to be related to the fatigue limit of metals [1–6]. Both, the fatigue limit and the high cycle fatigue resistance depend on the effective resistance of the microstructural barriers that has to be overcome by the cracks. Each of these barriers has a characteristic dimension and a critical stress range associated with its resistance to crack propagation. The plain fatigue limit is determined by the strongest microstructural barrier, since that resistance is generally greater than the resistance to crack nucleation. On the other hand, above the fatigue limit and for a given stress range, each barrier has an associated number of cycles that is necessary to propagate the crack. The effect of crack size on fatigue crack propagation threshold can be described conveniently by means of the Kitagawa–Takahashi plot relating the threshold stress with the crack size, as shown in Fig. 1. If the strongest microstructural barrier to fatigue crack propagation is
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placed at a given distance d from the material surface, the crack is non-damaging with respect to the plain fatigue limit up to a crack size a = d [2,4–6]. For a microstructurally short crack (MSC, the crack length is of the order of the microstructural dimensions) initiated from a plain surface the fatigue limit at a given stress ratio R, ⌬seR, defines the critical nominal stress range needed for continued crack growth (microstructural threshold). If the applied stress range ⌬s is smaller than ⌬seR, cracks (included in the MSC regime) are arrested at microstructural barriers placed at depths smaller than d. On the other hand, for long cracks (LC) the fatigue crack propagation threshold decreases with increasing crack size [1–8]. The threshold for LC (whose length is greater than that at which crack closure is fully developed) is defined in terms of the threshold value of the stress intensity factor range, ⌬KthR, thus LC in constant amplitude loading can only grow by fatigue if the applied stress intensity factor range ⌬K is greater than ⌬KthR. In the physically short crack (PSC) regime, a transition between the microstructurally short and long crack regimes (MSC and LC), the threshold is below ⌬seR and ⌬KthR. Although the influence of microstructure is still important, the development of crack closure governs the
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Nomenclature a C∗ and d D ⌬s ⌬seR ⌬sn ⌬sth ⌬K ⌬KC ⌬KCR ⌬KdR ⌬Kth ⌬KthR ⌬KthReff k M R r
crack length m environmentally sensitive material constants position from the surface of the strongest microstructural barrier notch length applied stress range plain fatigue limit nominal applied stress range threshold stress range for crack propagation applied stress intensity factor range extrinsic component of ⌬Kth extrinsic component of ⌬KthR microstructural threshold fatigue crack propagation threshold fatigue crack propagation threshold for long cracks effective threshold for long cracks (without crack closure) material constant that takes into account the development of ⌬KC microstructural characteristic dimension stress ratio (minimum stress/maximum stress) notch root radius
gation threshold to be known as a function of crack length. In this study an expression for estimating the threshold for fatigue crack propagation as a function of crack length is obtained by using only the plain fatigue limit, ⌬seR, the threshold for long crack, ⌬KthR, and the microstructural characteristic dimension (e.g. grain size).
2. Crack growth threshold
Fig. 1. Kitagawa–Takahashi type diagram showing the threshold between propagation and non-propagating cracks; after Ref. [6].
threshold level in this regime. In order to denote both parameters, the plain fatigue limit (⌬seR) and the propagation threshold for LC (⌬KthR) depend on the stress ratio for a given material, the sub-index R is included. The difference between the total applied driving force defined by the applied stress intensity factor range for a given geometrical and loading configuration, ⌬K, and the threshold for crack propagation, ⌬Kth, defines the effective driving force applied to the crack. By knowing the crack growth rate as a function of the effective driving force for a given material, it is possible to estimate the fatigue crack propagation behavior in the given conditions. This concept is the base of the Resistance-Curve Method [5,7,9], which needs the variation of the propa-
It is well known that the plain fatigue limit is defined by the ability of the first grain boundary to arrest a microcrack (see Fig. 1). This is a material-based limit (depending on the microstructural characteristic dimension, M) as Miller has pointed out [6]. In recent works carried out by Chapetti and coworkers [4,10], the position and the effective resistance of microstructural barriers and their relation with the fatigue limit were analyzed and modeled, and additional evidences that the strongest microstructural barrier defines the fatigue limit of plain and blunt-notched specimens in steels were obtained. An intrinsic resistance to MSC propagation is defined by using the position d of the strongest microstructural barrier and the plain fatigue limit ⌬seR. This intrinsic resistance is considered to be a microstructural threshold for crack propagation and is then defined as follows ⌬KdR ⫽ Y⌬seR冑πd,
(1)
where Y is the geometrical correction factor. In most cases the nucleated microstructurally short surface
M.D. Chapetti / International Journal of Fatigue 25 (2003) 1319–1326
cracks are considered semicircular [1–10], and the value of Y would then be 0.65. Because the plain fatigue limit depends on the stress ratio R, the microstructural threshold also does. The value of d is usually given by the microstructural characteristic dimension, M (e.g. ferrite grain size, bainite or martensite lath length, etc. [4,6,10]). Then, for the LC regime (where crack closure has built up to a steady state level), a total extrinsic component ⌬KCR can be defined as the difference between the mechanical threshold for LC, ⌬KthR, and the microstructural one, ⌬KdR ⌬KCR ⫽ ⌬KthR⫺⌬KdR.
(2)
The total extrinsic component of the threshold corresponds to a steady state level and is built up after a given crack length that depends on the material (a = l in Fig. 1). For a given material, ⌬KCR is a constant value that depends on the stress ratio R. In order to obtain an expression to estimate the fatigue crack propagation threshold as a function of crack length, the development of the extrinsic component (⌬KC) has to be defined. The expression proposed by McEvily and Minakawa [7] to model the development of the crack closure component of the stress intensity factor range is used here to estimate ⌬KC ⌬KC ⫽ ⌬KCR(1⫺e⫺ka),
(3)
where the material constant k defines the development of the extrinsic component ⌬KC for each stress ratio, and a is the crack length expressed in mm. According to expressions (2) and (3), ⌬KCR is not equal to the closure component for LC because ⌬KdR is not equal to the effective threshold (without the crack closure component) for LC, ⌬KthReff. However, the definition of the extrinsic component of the threshold for LC does not change the actual total threshold, given by ⌬KthR. Besides, the author believes that the fatigue propagation threshold for microstructural short cracks is not given by ⌬KthReff because a local crack driving force should also be considered [11–14]. The local crack driving force is provided by the strain energy stored in the form of internal stress fields generated by cyclic plastic deformation and is related with the surface strain concentration [12,13]. The value of the local crack driving force is initially high and rapidly drops as crack grows [12,13], and its influence on short crack propagation behavior extends to a few microstructural characteristic dimensions, M (e.g. grain size) [14]. The material threshold for crack propagation as a function of the crack length, ⌬Kth, is then defined as ⌬Kth ⫽ ⌬KdR ⫹ ⌬KC ⫽ Y⌬sth冑πa
aⱖd.
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Fig. 2 shows schematically the threshold curve given by expression (5). For a crack length a = d, ⌬Kth = ⌬KdR, and ⌬Kth tends to ⌬KthR for LC. In terms of the threshold stress, we get ⌬sth ⫽
⌬KdR ⫹ (⌬KthR⫺⌬KdR)[1⫺e⫺k(a⫺d)]
⌬sth ⫽ ⌬seR
Y冑πa
a ⬍ d,
aⱖd, (6a) (6b)
where ⌬KthR and ⌬seR (and so ⌬KdR) are all functions of the stress ratio R. Finally, an expression for the material parameter k, that defines the development of the extrinsic component of ⌬Kth, ⌬KC, should be obtained. We can use the fact that the plain fatigue limit is given by the resistance of the strongest microstructural barrier, and that the threshold stress for crack propagation decreases as the crack length increases for a crack length a ⬎ d (see Fig. 1). By using this concept it is possible to obtain a maximum value for the parameter k. The limit would be given by the following condition: the slope of the threshold stress as a function of crack length is equal to zero at a crack length a = d, that is to say, the threshold stress curve is tangent to an horizontal line defined by ⌬s = ⌬seR. By deriving Eq. (6a), equating to zero and solving for k, it is easy to arrive at the following expression kⱕ
⌬KdR 1 . 2d (⌬KthR⫺⌬KdR)
(7)
In order to check the ability of expression (7) to describe the development of the extrinsic component, the threshold stress given by expressions (6a) and (7) was analyzed for actual value ranges of d, ⌬KdR and ⌬KthR. In some cases the upper limit of the value of k is not enough to assure that the threshold stress decreases monotonically as the crack length increases. This problem arises when the total extrinsic component (⌬KCR) is greater enough than the microstructural threshold (⌬KdR) and/or for relatively high d values. Then, the value of k should be smaller than that given by expression (7). Experimental short crack propagation threshold data for
(4)
From expressions (2)–(4), we finally get ⌬Kth ⫽ ⌬KdR ⫹ (⌬KthR⫺⌬KdR)1⫺e⫺k(a⫺d)
aⱖd, (5)
Fig. 2. Defined fatigue crack propagation threshold as a function of crack length, given by expression (5).
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two steels with the same composition but different grain sizes were obtained from the literature [15] (steel S20C, M = 7.8 and 55 µm, see Table 1), in order to define a constant fitting parameter in expression (7). A value of k equal to one half of its upper limit given by expression (7) gives a threshold for crack propagation in good agreement with the experimental results (see first two graphs in Fig. 3). Then, the following expression is finally defined to estimate the material parameter k ⌬KdR 1 ⌬KdR 1 . ⫽ k⫽ 4d (⌬KthR⫺⌬KdR) 4d ⌬KCR
(8)
Fig. 3 shows the estimated ⌬Kth given by expressions (5), (1) and (8) for eight materials, including the two steels used to define expression (8), other four steels, electrolytic copper and 2023-T3 aluminum alloy (see Table 1). Experimental results obtained from the literature are also shown [15–18]. Solid lines correspond to the estimated threshold given by expression (5) by considering semicircular surface cracks for the definition of ⌬KdR (Y = 0.65 in Eq. (1)). In accordance with author’s analysis, through thickness cracks (Y = 1.12) should be considered for the materials from reference [16]. Estimated thresholds for these cases are shown by dashed curves. Very good agreement can be seen in all cases. Table 1 shows values of ⌬seR, ⌬KthR, M, d, ⌬KdR, ⌬KCR and k for the materials whose thresholds are shown in Fig. 3, as well as for some others for which experimental results of the threshold are not available [15–21]. It should be noted that in this paper ⌬K is defined as the full difference between the maximum and minimum values of K and not the tensile part alone. The material parameter k is proportional to ⌬KdR/⌬KCR in expression (8). This is in accordance with the general trend observed for steels. Low carbon steels
show relatively low k values defined by a small relation between the intrinsic and the extrinsic components and a relatively large grain size. On the other hand, high strength steels show relatively high k values given by a relatively high ⌬KdR/⌬KCR relation and small microstructural characteristic dimensions. Taking into account the defined parameter k, expression (5) can be rewritten as follows ⌬KC ⌬Kth⫺⌬KdR ⫽ ⌬KthR⫺⌬KdR ⌬KCR
冉
⫽ 1⫺exp ⫺
(9)
冊
1 ⌬KdR (a / d⫺1) 4 ⌬KCR
aⱖd.
Fig. 4 shows plots given by this expression for different values of ⌬KdR/⌬KCR, as well as all experimental results shown in Fig. 3. The extrinsic component of the experimental results was estimated by subtracting the microstructural threshold (⌬KdR), calculated with ⌬seR and d in Table 1, from the experimental threshold (⌬Kth) data (given in Fig. 3). For most of the analyzed materials the extrinsic component of the threshold is fully developed at a depth of about 20–30 times the microstructural characteristic dimension, except for the S20C steel with a grain size of 0.0078 mm and for copper (50– 60 times M). It is also possible to find examples for which ⌬KC is fully developed for a crack length of 10 times the microstructural characteristic dimension or less, as for the case of the JIS SUJ2 steel shown in Table 1. Finally, we can say that expressions (1), (5), and (8) define a material threshold for fatigue crack propagation that can be estimated from the plain fatigue limit ⌬seR, the position of the strongest microstructural barrier d, and the mechanical threshold for LC, ⌬KthR. These are
Table 1 Experimental and estimated data for several materials Material
0.42Ca 2.25Cr1Moa S20Ca S20Ca SM41B(S)a SM41B(L)a JIS SUJ2 2024-T3a Coppera Inconel 718 Ti–6Al–4V Ti–6Al–4V a
Reference
[17] [18] [15] [15] [16] [16] [19] [16] [18] [20] [21] [21]
R
⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 0.1 0.8
⌬seR (MPa) ⌬KthR (MPa m1/2)
M (mm)
d
⌬KdR (MPa m1/2) Eq. (1) Y = 0.65
450 500 470 326 396 326 2400 300 146 920 450 160
0.018 0.025 0.0078 0.055 0.014 0.064 0.01 0.027 0.05 0.01 0.02 0.02
M M M M M M M M 2M M M M
2.2 2.9 1.5 2.8 1.7 3 8.7 1.8 1.7 3.3 2.3 0.8
5.9 9.1 10.4 12.4 10.2 12.3 12 6.4 5 11.5 4.3 2.4
Experimental results of the threshold are shown in Fig. 3.
⌬KCR (MPa m1/2) Eq. (3)
3.7 6.2 8.9 9.6 8.5 9.4 3.3 4.6 3.3 8.2 2 1.6
⌬KdR/⌬KCR
0.59 0.46 0.17 0.29 0.20 0.32 2.64 0.39 0.51 0.40 1.15 0.50
k (mm⫺1) Eq. (8)
8.2 4.6 5.4 1.3 3.6 1.25 66 3.6 1.3 10 14.4 6.3
M.D. Chapetti / International Journal of Fatigue 25 (2003) 1319–1326
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Fig. 3. Estimated fatigue crack propagation threshold as a function of crack length for eight materials (see Table 1). Experimental results from the literature are also shown.
mechanical and microstructural parameters, which can be obtained through standardized mechanical tests and simple microstructural analyses. The defined threshold as a function of crack length also allows definition of a crack initiation period as the number of fatigue cycle necessary to initiate a MSC of length d. This assumption is indicated in Figs. 2 and 7.
3. Application example and discussions
Fig. 4. Extrinsic component of the fatigue crack propagation threshold as a function of crack length.
The difference between the total applied driving force and the material threshold for crack propagation defines the effective driving force applied to the crack. The high cyclic fatigue crack propagation behavior can be estimated for a given crack length range if the crack growth rate as a function of the effective driving force is known for a given material. The initial crack length is given by
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the position of the strongest microstructural barrier if the material were free of cracks or crack like flaws. The final crack length is defined according to critical parameters related with the loading conditions of the analyzed components. In the case of smooth specimens or components, the stress can be considered constant for any crack length, equal to the nominal applied stress. The following general expression can be used to estimate the applied driving force as a function of crack length [22]: ⌬K ⫽ Y⌬sn冑πa,
(10)
where ⌬sn is the nominal applied stress range. The crack aspect ratio as a function of crack length has to be defined for the combination of component geometry and loading conditions, which allows definition of the value of the parameter Y as a function of crack length. In the case of notches, an expression for the stress concentration as a function of crack length is needed, that is to say, the normalized applied stress distribution defined by the notch has to be known. Approximate local stress intensity factor range for small cracks at notches that are in elastic stress fields in terms of the cracks length, a, notch root radius, r, nominal stress range, ⌬sn, and elastic stress-concentration factor, kt, is given by the following expression obtained by Luka´ s et al. [18] ⌬K ⫽
kt
冑1 ⫹ 4.5(a / r)
Y⌬sn冑πa.
(11)
Quantitative analysis of fatigue crack growth requires a constitutive relationship of general validity be established between the rate of fatigue crack growth, da/dN, and some function of the range of the applied stress intensity factor, ⌬K (crack driving force). Besides, it has to take into account the threshold for the whole crack length range, including the short crack regime where the fatigue crack propagation threshold is a function of crack length. Among others, the following relationship meets these requirements [23] da ⫽ C∗(⌬K⫺⌬Kth)m, dN
(12)
where C∗ and m are environmentally sensitive material constants obtained from long crack fatigue behavior and ⌬Kth is the crack growth threshold as a function of a and R and represents the effective resistance of the material to fatigue crack propagation. The fatigue crack propagation life for a given crack length range and a given material can be obtained by integrating expression (12) and using expression (5) for the threshold of the material (⌬Kth), and a proper expression for the applied driving force (⌬K) given by the analyzed loading and geometrical configuration. Fig. 5 shows a da/dN–⌬K plot with some experimental data corresponding to short cracks propagating
Fig. 5. da/dN–⌬K plot showing experimental data from short cracks propagating from notches in JIS SM41B(S) steel [16] (see also Table 1). Threshold and Paris region crack propagation behaviors for LC, as well as the estimated propagation behavior of surface cracks in smooth material under ⌬sn = 400 MPa, are also shown.
from notches in a JIS SM41B steel, obtained by Akiniwa et al. [16] (this material is also included in Fig. 3 and Table 1). Threshold and Paris region crack propagation behaviors for LC are also shown [16]. The figure shows the propagation behavior of a surface crack in smooth material for a nominal stress range equal to 400 MPa (just above the plain fatigue limit of the material), estimated by integrating Eq. (12) from a crack length equal to d, using expressions (5) and (10), and taking the parameter Y equal to 1.12 to consider through thickness cracks. Fig. 5 reveals that a short crack propagates faster than a long crack at a given ⌬K, and still propagates in the region below ⌬KthR (threshold for LC). When the crack length exceeds 1 mm the propagation rate of the short crack comes closer to that of the long crack. The dashed line shows the microstructural threshold, ⌬KdR, which represents the minimum ⌬K for crack propagation. It is actually possible to observe fatigue propagation of MSCs at ⌬K ⬍ ⌬KdR, but any propagation at a ⬍ d is considered part of the crack initiation period. Fig. 6 shows the estimated fatigue propagation behavior of short cracks initiated at three notches with different geometries for the same material. Two of those notches have the same stress-concentration factor, kt (equal to 3), but different notch root radius, r = 3 and 0.5 mm. The third notch is the same as the one used by Tanaka and Akiniwa [24] to obtain the experimental results shown in Fig. 5: D = 3 mm, r = 0.16 mm and kt = 8.48. For the r = 3 mm notch only the nominal applied stress range equal to 140 MPa is considered. This stress level is just 10 MPa above the estimated
M.D. Chapetti / International Journal of Fatigue 25 (2003) 1319–1326
Fig. 6. Estimated fatigue crack propagation behavior of short cracks for three different notched SM41B(S) steel specimens.
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r = 3 mm and kt = 2.65 analyzed experimentally by Akiniwa et al. [16] is 155 MPa, and mechanically short non-propagating cracks were not found. In the case of the r = 0.5 mm notch and ⌬sn = 140 MPa the estimated stress distribution can also drive the initiated crack of length d but only to a depth of about 0.09 mm, at which the stress level becomes smaller than the threshold stress. By increasing the applied stress level, the crack can propagate further. The fatigue limit is estimated to be about 180 MPa, for which a nonpropagation crack length as long as 0.5 mm can be observed. Fig. 7 shows the stress distribution for ⌬sn = 200 MPa. It can be seen that the crack can propagate without arresting, since the applied stress is greater than the threshold stress for any crack length. The estimated fatigue limit associated to this notch is 190 MPa, 60 MPa greater than the estimated one for r = 3 mm notch. The difference is due to the notch size effect. Different notch geometries define different stress gradients in the short crack regime. Included in Fig. 7 is the stress distribution estimated using expression (11) for the sharp notch (kt = 8.48) and ⌬sn = 130 MPa. It is worth noting that in this case the stress level near the surface is quite greater than twice the yield stress of the material (560 MPa [16]). However, the threshold condition defining the fatigue limit is located at about 0.5 mm from the surface, where the stress level is below the plain fatigue limit. Another important feature that should be considered is the influence of the notch length (D) on fatigue crack propagation threshold for sharp notches.
4. Summary and conclusions
Fig. 7. Kitagawa–Takahashi type diagram showing the estimated threshold stress for fatigue crack propagation for SM41B(S) steel [16]. The applied stress distributions given by expression (11) are also shown for three different notch geometries.
fatigue limit for the same notch geometry (equal to 130 MPa), for which mechanically short non-propagating cracks cannot be formed. This fact can be analyzed in Fig. 7, which shows a Kitagawa–Takahashi type diagram with the threshold stress for the propagation of fatigue crack estimated with expressions (6a) and (6b), and the applied stress distributions estimated by using expression (11) for the three notch geometries. The stress distribution corresponding to r = 3 mm notch and ⌬sn = 140 MPa propagates the initiated crack of length d without arresting, since the applied stress is greater than the threshold stress for any crack length. It is important to note that the fatigue limit for the notch with
A microstructural threshold for crack propagation (⌬KdR) is defined by the plain fatigue limit (⌬seR) and the position (d) of the strongest microstructural barrier. A total extrinsic threshold to crack propagation (⌬KCR) is then defined by the difference between the crack propagation threshold for LC (⌬KthR) and the microstructural threshold (⌬KdR). The development of the extrinsic component is considered to be exponential and a development parameter k is defined as a function of the same microstructural and mechanical parameter used to define the material threshold for crack propagation. The difference between the total applied driving force and the material threshold for crack propagation defines the effective driving force applied to the crack. If crack growth rate as a function of this effective driving force is known for a given material, the fatigue crack propagation behavior in a given crack length range can be estimated. The initial crack length is given by the position of the strongest microstructural barrier if the material were free of cracks or crack like flaws. The final
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crack length is defined according to critical parameters related with the loading conditions of the analyzed components. The methodology presented in this paper, as well as the defined fatigue crack propagation threshold, is based on the experimental evidence that the fatigue limit of smooth and notched components represents the threshold stress for the propagation of initiated cracks, so that fatigue limit depends on the effective resistance of the microstructural barriers that have to be overcome by initiated cracks. The expression defined to estimate the material threshold for crack propagation as a function of the crack length allows definition of a crack initiation period as the number of load cycles necessary to initiate a crack of depth d, from which the crack propagation behavior can be analyzed. The definition of a crack initiation period of a component is some times ambiguous, and depends on the defects that are formed during manufacturing, fabrication, assembly, repair and maintenance. It is also intimately tied to the sensitivity of the crack detection technology. However, in any case an analytical methodology is required to predict the growth kinetics of cracks from a given initial size, and in many of those cases part of the material short crack regime has to be included. The presented model is simple and can be applied to analyses based on elastic stress distributions to study some interesting fatigue phenomena related with small fatigue cracks, including the mechanism of non-propagation crack development, notch size effect and fatigue notch sensitivity. The application of the model is limited to crack behavior evaluation near the fatigue crack growth threshold. Further analyses have to be carried out to evaluate the applicability and limitations of the methodology to study those cases where important plastic strain behavior is involved. The defined threshold as a function of crack length was estimated for eight different materials, including six steels, one aluminum alloy and one copper, and compared with experimental results from literature. Good agreement between estimated and experimental threshold was observed in all cases. Quantitative analyses of the high cycle fatigue propagation behavior (near threshold) of short cracks in notches were carried out and results are also in good agreement with experimental results from literature.
Acknowledgements The author wishes to express his thanks for funding provided by Consejo Nacional de Investigaciones Cientı´ficas y Te´ cnicas (CONICET), and by Agencia Nacional de Promocio´ n Cientı´fica y Tecnolo´ gica (PICT’98 12-04585/6), Argentina.
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