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Prediction of short fatigue crack propagation behaviour by characterization of both plasticity and roughness induced crack closures
International Journal of Fatigue 24 (2002) 529–536 www.elsevier.com/locate/ijfatigue
Prediction of short fatigue crack propagation behaviour by characterization of both plasticity and roughness induced crack closures X.-P. Zhang
a,*
, J.-C. Li b, C.H. Wang c, L. Ye a, Y.-W. Mai
a
a
Department of Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia National Facility of Dynamic Testing and Research Civil Engineering, University of Technology, Sydney, Australia Aeronautical and Maritime Research Laboratory Defence Science and Technology Organization, Melbourne, Australia b
c
Received 17 March 2000; received in revised form 16 April 2001; accepted 10 August 2001
Abstract A new fatigue crack growth model was proposed to characterize short fatigue crack growth behaviour, through combining the mechanisms of both plasticity induced crack closure and fracture surface roughness induced crack closure. The results obtained using this proposed model show good agreement with the analytical predictions of Budiansky–Hutchinson’s complex functionanalytical solution, and are reasonably close to the experimental data acquired by other researchers. In addition, the results of short fatigue crack growth rate versus stress intensity factor range predicted by the present model can well characterize the experimental results for a 2024-T3 structural aluminium alloy under two different constant-amplitude cyclic loads, and the predictions from the present model also show the reasonable improvement compared to Newman’s plasticity induced closure model. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Short fatigue crack; Crack growth rate; Plasticity induced crack closure; Surface roughness induced crack closure; Combined model
1. Introduction So far, the majority of fatigue crack growth data, if available, were obtained by using specimens with a through-thickness long crack under constant amplitude loading. In particular, most experimental research rarely simulate the cracking configurations experienced in normal services, where failure generally occurs by propagation of subcritical cracks. The subcritical cracks range from several microns to a few hundred microns. This type of crack is usually called short crack or small crack, although short cracks and small cracks may be different in a strict definition or size. Short cracks are very difficult to detect using current in-situ non-destructive testing (NDT) methods. Typical NDT resolution is not sufficient for detection of short cracks. In addition, along with the improved capability of detecting crack, it has been found
* Corresponding author. Tel.: +61-2-9351-7146; fax: +61-2-93513760. E-mail address: [email protected] (X.-P. Zhang).
that the actual initiation stage, from a micro-defect to a short crack occurs during very few loading cycles; in contrast the lifetime in the short crack growth stage, which by traditional concepts should still be defined as the initiation stage, takes up a fairly large percentage in the total fatigue lifetime of components or structures. That implies fatigue cracking failure is probably dominated by the short crack propagation and the lifetime for short crack growth would be the most important part in the total fatigue lifetime. The short crack effect and characterization of short fatigue crack growth are, therefore, quite critical for accurate lifetime prediction of components and structures since anomalously fast crack growth rates are often observed at stress intensity factors well below the threshold where long fatigue cracks are generally presumed dormant [1–10]. As is commonly accepted, crack closure is one of the most important factors resulting in the anomaly of short crack growth and the distinction of growth behaviour from long cracks [1,2,4,7,9]. It is agreed that plasticityinduced closure is the primary mechanism of crack closure arising from the development of residual plastic
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X.-P. Zhang et al. / International Journal of Fatigue 24 (2002) 529–536
stretch in the crack-wake. However, it has been recognized, both theoretically and experimentally, that using only the continuum-based plasticity induced closure analysis could not explain some experimental phenomena, and the total crack closure should include both plasticity- and surface roughness-induced closures. Studies demonstrated that fracture surface roughness induced closure plays an important role in crack growth process, in particular at low crack growth rate regime (da/dN⬍10⫺6 mm/cycle) and predominantly in plane strain condition [11]. For example, the investigation on short fatigue crack closure for a near-alpha IMI 834-Ti alloy with in-situ SEM studies [12] showed that crack closure increased significantly soon after the short crack propagated through the first grain boundary. Furthermore, it was also shown that in the subsequent crack growth, the level of crack closure appeared to decrease as the crack length increased. It seems appropriate to rationalize this in terms of the crack path tortuosity and changes in fracture morphology. It can be envisaged that because of the zig-zag shape of crack growth path the idealized plasticity induced closure, which is based on a flat crack surface, should be modified. The FEM model with only plasticity induced crack closure also predicted lower closure levels than those obtained from in-situ SEM investigation [13]. In other words, it means that roughness induced closure plays a very important role in this case, due to crack deflection. In practices, it is neither pure plasticity induced closure nor pure roughness induced closure but a combination of both closure mechanisms that dominates the early stage of fatigue crack growth [13,14]. On this point, for most long cracks, this does not seem particularly important, because scales of the deflection of crack growth path, compared to the size of crack opening, are too small to be considered. For short cracks it is, however, another situation. Normally, a small crack length means small crack opening, but the small length does not dominate the size of the crack deflection which is usually controlled by microstructures. In this case, the size of the crack deflection is always comparable to the size of the crack opening. Therefore, the crack deflection can not be neglected, in other words, the idealized flat crack growth path is not appropriate to describing short crack features, and the crack growth path deflection is strongly affected by microstructural features of materials. This work developed a new predictive model for short fatigue crack growth accounting simultaneously for both mechanisms of plasticity and crack surface roughness induced crack closures. The results obtained by using the proposed model were compared with the analytical predictions of Budiansky–Hutchinson’s complex function-analytical solution and the experimental data obtained by other researchers. In addition, the results predicted by the present model have also been compared with the experimental results for a 2024-T3 structural
aluminium alloy under two different constant-amplitude cyclic loads.
2. A combined model of short fatigue crack growth accounting for both mechanisms of plasticity and roughness induced crack closures An improved short fatigue crack growth model should include both mechanisms of plasticity- and fracture surface roughness-induced crack closures. It should not only consider the mix of both crack closures, but should also recover their limiting functions, i.e. should also recover either pure plasticity induced crack closure or pure roughness induced crack closure. Due to the existence of the plastic wake and the fracture surface roughness as shown in Fig. 1, the onset of crack opening (dop) or equivalently, the total crack closure, is given by: dop⫽dpop⫹drop cos q
(1)
where dpop and drop are equivalent crack surface displacements due to plasticity-induced crack closure and fracture surface roughness induced-crack closure, respectively. As shown in Fig. 1, it is assumed that there exists the idealized plastic residual wake and the fracture surface asperity is in regular triangular form. At the cracktip, the crack-opening displacement can be expressed by Dugdale’s model [15] as: d⫽K 2I/(Esy).
(2)
Thus, the crack opening stress intensity factor, Kop, can be obtained from the models of Dugdale and Suresh– Ritchie [16] and Eq. (2). That is: Kop ⫽ Kmax
冪
dop ⫽ dmax
冪
dpop+drop cos q dmax
(3)
and cos q⫽
h
冑h +(w/2) 2
⫽
2
1
冪
冉 冊
1 1+ 2h/w
2
⫽
2g
冑4g +1 2
(4)
where g is defined as the fracture surface roughness factor, g=h/w, and w denotes the mean base width of fracture surface asperities, and h is the average height of asperities. g actually is the ratio of the average height to the mean base width of asperities. Using Dugdale’s model again and substituting Eq. (4) into Eq. (3), thus we have
Kop⫽
冪
(Kop)2+(K rop)2
2g
冑4g +1 2
(5)
where K pop and K rop can be calculated by modifying New-
X.-P. Zhang et al. / International Journal of Fatigue 24 (2002) 529–536
Fig. 1.
Illustration of a deflected crack at (a) full opening and (b) commencement of opening.
man’s numerical analytical method [17,18], and the Suresh–Ritchie’s model, respectively. K pop is dependent on plane strain or plane stress, cyclic stress ratio R and applied to yield stress ratio s/sy. In Suresh–Ritchie’s model, K rop can be estimated by K rop⫽Kmax
冪1+2gc 2gc
冪
(K pop)2+(Kmax)2
4g2c
(1+2gc)冑4g2+1
Kop⫽rKK pop
.
(7)
(8)
where
冪
Kmax−Kop Kmax−Kop ⫽ Kmax−Kmin (1−R)Kmax
1− ⫽
冉 冊冑
1+
K rop K pop
2
2g 2
4g +1
.
(9)
Clearly, Kop for the two limiting cases of pure plasticityinduced crack closure and pure surface roughnessinduced crack closure can be recovered easily from Eq. (7) or Eq. (8). Also, in this crack closure model, the effective stress intensity factor range ratio, or the effective crack driving force, U, is given by:
(10a)
冪冉 冊
4g2c K pop 2 + Kmax (1+2gc)冑4g2+1 1−R
or
冪
1− U⫽
We rewrite Eq. (5), then have
rK ⫽
U⫽
(6)
where c is the mixture ratio of mode II to mode I displacements, i.e. c=uII/uI. It does not depend on the state of stress at the crack-tip, that is, plane strain or plane stress. Combining Eqs. (5) and (6), then gives
Kop⫽
531
4g2c [1−(1−R)Up]2+ (1+2gc)冑4g2+1 1−R
(10b)
where
冤
冪冉 冊 冥
Up⫽ 1⫺
K pop Kmax
2
/(1⫺R)
(10c)
and Up denotes the effective crack driving force arisen from pure plasticity-induced crack closure. Since there is no crack growth path deflection when the crack grows in the first grain (assuming that the mean grain size is dg), there should be no crack surface roughness induced closure, then we have: Kop⫽K pop for a⬍dg.
(11)
Furthermore, when the crack grows up to a relatively
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X.-P. Zhang et al. / International Journal of Fatigue 24 (2002) 529–536
large length, the size of plastic zone at crack tip is fairly large, consequently K pop would be very high and much larger than K rop, i.e. rK→1 [referring to Eq. (9)], this actually means the roughness induced crack closure would lose its effect, then only plasticity induced closure exists, by Eqs. (7)–(9), we have Kop⬵K pop
for adg.
(12)
This means that for very long cracks, plasticity induced crack closure is the dominant mechanism. Fig. 2 shows the results of the influence of surface roughness factor (g) and mode II sliding level (c) at crack tip on short fatigue crack growth rates predicted by the present model. In Fig. 2, it is clearly seen that the rougher fracture surface (i.e. larger g value) results in large decrease of da/dN due to the increase of surface roughness induced closure. And the higher level of mode II shear sliding at crack tip (i.e. larger c value) has the similar effect. The above combined mechanisms of both plasticityand roughness-induced crack closures can also be obtained by combining Budiansky–Hutchinson’s small scale yielding model [19] and Suresh–Ritchie’s model. Budiansky–Hutchinson presented a complex functionanalytical solution to characterize fatigue crack growth with crack closure behaviour using ⌬Keff. In light of the Budiansky–Hutchinson’s results, still referring to Fig. 1, the crack closure stress intensity factor, Kcl, is given by Kcl 1 ⫽1⫺ Kmax 冑0.54
冪1−d
dR
(13)
fracture surface roughness and that it is only a function of stress ratio R [21], the effective plastic stretch is dR⫽dpR⫹drR
(14)
p R
r R
where d and d are, respectively, the plastic stretch pertaining to plasticity-induced closure and surface roughness contribution. Using the Suresh–Ritchie and Dugdale models again, then Eq. (13) becomes: 1 Kcl ⫽1⫺ Kmax 冑0.54
冪1− 1+2gc − d
dpR
2gc
(15)
max
which recovers the Budiansky–Hutchinson analytical result for c=0 (i.e. there exists only pure mode I deformation) or g=0 (i.e. flat fracture surface). Accordingly, the effective stress intensity factor range ratio, U, is given by U⫽
1 1 Kmax−Kcl ⫽ (1−R)Kmax 1−R 冑0.54
冪1− d
dR
.
(16)
max
In the absence of fracture surface asperities and noting Eqs. (14) and (16), we have 1
U⫽ p
· 冑0.54
冪
1−
dpR dmax (17)
1−R
where Up is the effective stress intensity factor range ratio for pure plasticity-induced crack closure. The crack closure stress intensity factor can now be derived from Eqs. (15) and (17). That is,
max
where dR and dmax are the total plastic stretch and the maximum crack-tip opening displacement at maximum load. It should be indicated that generally Kop and Kcl will be identical [20], except in the contact region yielding occurs then Kop may be lower than Kcl. Assuming that the plastic stretch is not coupled to the
冦
Kcl⫽Kmax 1⫺
冧
0.54[(1−R)U ] − . 1+2gc 冑0.54 冪 1
p 2
2gc
(18)
Consequently, the effective stress intensity factor range ratio Eq. (16) becomes U⫽
冪[U ] − 1+2gc 0.54(1−R) . p 2
2gc
1
2
(19)
It should be noted hereto we consider Kcl to be identical to Kop, since the difference between them is generally less than the scatter related to the crack closure measurements. 3. Results and discussion 3.1. Comparisons of present model, Budiansky– Hutchinson’s analytical solutions and experimental data Fig. 2. Influence of surface roughness factor and shear sliding level on short fatigue crack growth rates predicted by the present model.
Comparisons of crack closure and effective crack driving force under plane stress condition between the
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533
Fig. 3. Comparisons of crack closure between current numerical results and analytic solutions based on Budiansky–Hutchinson’s analyses (both for plane stress condition).
Fig. 5. Comparisons of theoretical predictions and experimental data for plane strain.
present model [by Eqs. (15) and (16)] and the model based on Budiansky–Hutchinson’s analytic solutions are shown in Figs. 3 and 4, respectively. It can be seen that the latter agrees quite well with the predictions from the present model. And exactly saying, predictions from the present model show a bit higher crack closure level (Kop/Kmax) and lower effective crack driving force (U) than Budiansky–Hutchinson’s analytic solutions where only plasticity induced crack closure is taken into account. Moreover, crack closures estimated using the present model were also compared with other researchers’ experimental data [22,23] obtained from 10 mm thick specimens of a structural aluminum alloy 2024 in T3 and T6 heat treatment conditions, respectively, as shown in Fig. 5. By the present model, the numerical predictions under different stress ratio R were carried out in plane strain condition using Eq. (7) for s/sy=0.15 and 0.77 and g=0.5, c=0.2 from fracture surface profile measurements [16]. It can be seen that the predictions are reasonably close to the test data at high values of stress ratio R.
In addition, the present model has also been used to characterize the experimental results obtained by this work for a structural aluminum alloy 2024-T3 (solution heat-treatment plus cold-worked, with a thickness of 5 mm). The 2024-T3 aluminium alloy had a typical recrystallized pancake-shaped grains with the approximate grain scale of 150, 75, and 50 µm in rolling orientation, width and thickness directions, respectively. Its mechanical properties are: yielding strength sy=394 MPa, ultimate tension strength sUTS=442 MPa, and the percentage prolongation d5 =7.5%. All specimens had a geometry of 5 mm thick, 50 mm wide and 300 mm long. The specimens were double-edge-notched with a round notch root radius of 3 mm The longitudinal axis of all specimens was parallel to the rolling orientation of 2024-T3 aluminium alloy sheet. Before test, notch surfaces for all specimens were well polished in longitudinal direction to prevent scratched or roughness in the crack initiating and advancing orientation. The fatigue tests were carried out in a 50 kN-capacity MTS 880 electro-hydraulic servo-controlled test system with a frequency of 15 Hz. Constant amplitude loading mode was used with maximum stresses of 78 and 98 MPa, respectively, and keeping a constant stress ratio of 0.1 (i.e. R=0.1). The measurement of short fatigue crack length was conducted in surfaces and edges of the notches simultaneously. In early growing stage of short cracks, the crack length was monitored and measured by replication technique. When cracks grew up to a certain length (longer than 300 µm), an optical microscope with magnification of 100 times was used easily to monitor and measure the crack length. The results of fatigue crack length versus number of cycles under two different constant amplitude stresses (78 and 98 MPa, respectively) are shown in Fig. 6. The experimental results of fatigue crack growth rate da/dN versus ⌬K are shown in Figs. 7 and 8. In Figs. 7 and 8, the model developed by this work was employed to characterize the fatigue crack growth behaviour, in
Fig. 4. Comparison of effective crack driving force between numerical results from present model and results from Budiansky–Hutchinson’s analytic solution.
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X.-P. Zhang et al. / International Journal of Fatigue 24 (2002) 529–536
Fig. 6. Fatigue crack length versus number of cycles for a 2024-T3 aluminium alloy under different applied stresses (smax=78 and 98 MPa). Fig. 8. Short crack growth rate versus ⌬K under constant-amplitude stress of 78 MPa, and comparisons with present model and Newman model.
induced closure when the crack grows up to a relatively large length. But for the major stage of short fatigue crack growth, the prediction from the present model shows a reasonable improvement compared to Newman’s numerical method, as fracture surface roughness induced crack closure has been taken into account for this structural aluminium alloy that has coarse grained microstructures. 3.2. Effect of crack-tip shear sliding on crack closure and crack driving force
Fig. 7. Short crack growth rate versus ⌬K under constant-amplitude stress of 98 MPa, and comparisons with present model and Newman model.
particular at initial and early stages of crack propagation. Predictions by using modifying Newman’s numerical method are also shown in Figs. 7 and 8 for comparison. As mentioned above, these predictions (by modifying Newman’s method) only accounted for pure plasticity induced crack closure. For convenience, experimental data for long fatigue crack growth rates in 2024-T3 and 2024-T6 aluminium alloys [24,25] are given in Tables 1 and 2 for comparison. In Figs. 7 and 8, it can be seen clearly that the prediction from the present model shows a good coincidence with the results of Newman’s model when the crack is very short (less than 80 µm) and when the crack has grown up to fairly longer (a few millimeters long). This may imply that there is almost no fracture surface roughness induced closure at much earlier stage of the crack growth, and also much less influence of roughness
Crack surface roughness induced crack closure is strongly attributed to the crack path deflections caused by the heterogeneous material microstructures and the shear sliding of the crack-tip. This effect is predominant at low R, where the minimum crack-tip opening displacement is significantly smaller than the asperity height of the fracture surface. Fig. 9 shows the influence of crack-tip shear sliding on the plane stress effective crack driving force U calculated from Eq. (10) for different intensity of crack surface roughness. Clearly, U decreases with increasing shear sliding level c, since a larger shear sliding gives a stronger engagement of asperities on the fracture surfaces leading to more severe crack closure. In the limit, g=0, there is no surface roughness induced crack closure for flat fracture surfaces (except plasticity induced crack closure) at any level of crack-tip shear sliding, c.
4. Conclusions 1. A combined model of short fatigue crack closure accounting for both mechanisms of plasticity- and
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535
Table 1 Long crack growth data in 2024-T3 aluminium alloy [24] da/dN (m/cycle,×10⫺11) ⌬K (MPa√m)
5.01 3.63
126 3.55
19.9 3.70
39.8 3.80
63.1 4.17
126 4.57
199 5.01
316 5.25
501 5.62
1000 6.63
Table 2 Long crack growth data in 2024-T6 aluminium alloy [25] da/dN (m/cycle) ⌬K (MPa√m)
9.93×10⫺13 0.707
1.13×10⫺10 3.36
1.60×10⫺9 1.87
4.08×10⫺9 2.54
1.56×10⫺8 4.01
4.90×10⫺8 6.40
1.00×10⫺7 9.00
References
Fig. 9. Influence of mode II sliding on crack driving force at short crack length (a=0.10 mm) in plane stress condition.
roughness-induced crack closures was developed to characterize short crack growth behaviour. 2. The results of crack closure and effective crack driving force estimated by numerical simulation with the present model agreed well with available test data and those analytic solutions obtained by the Budiansky– Hutchinson model. 3. The predicted results of short fatigue crack growth rates versus stress intensity factor range obtained from the present combined model show a reasonable improvement compared to Newman’s plasticity induced closure crack growth model. 4. Crack-tip shear sliding also enhanced crack closure and reduced the effective crack driving force. The higher the shear sliding the more was the crack closure.
Acknowledgements This work was supported in part by an Australian Research Council Large Project Grant awarded to YWM, XPZ and CHW, and in part by the Airframe and Engine Division, Aeronautical and Maritime Research Laboratory, Defence Science and Technology Organization of Australia.
[1] Pearson S. Investigation of fatigue cracks in commercial aluminium alloys and subsequent propagation of very short fatigue cracks. Engineering Fracture Mechanics 1975;7:235–47. [2] Miller KJ. Short crack problem. Fatigue of Engineering Materials and Structures 1982;5:223–32. [3] Miller KJ. Initiation and growth rates of short fatigue cracks. In: Fundamentals of Deformation and Fracture, Eshelby Memorial Symposium. Cambridge: Cambridge University Press, 1985:477–500. [4] Tanaka K. Mechanisms and mechanics of short fatigue crack propagation. JSME International Journal (Bulletin of the JSME) 1987;30(259):1–13. [5] Newman JC Jr. Fracture mechanics parameters for small fatigue cracks. In: Larsen JM, Allison JE, editors. Small-crack test methods, ASTM STP 1149, 1992. p. 6–33. [6] Wang CH, Miller KJ. Short fatigue crack growth under mean stress, uniaxial loading. Fatigue and Fracture of Engineering Materials and Structures 1993;16(2):181–98. [7] Newman JC Jr. Fatigue-life prediction methodology using a crack-closure model. Journal of Engineering Materials and Technology 1995;117:433–9. [8] Wang CH. Effect of stress ratio on short fatigue crack growth. Journal of Engineering Materials and Technology 1996;118:362–6. [9] Newman JC Jr. The merging of fatigue and fracture mechanics concepts: a historical perspective. Progress in Aerospace Sciences 1998;34:347–90. [10] Ritchie RO. Small crack growth and the fatigue of traditional and advanced materials, Plenary Lecture. In: Proceedings of Fatigue’99, Beijing, China, 1999. p. 3–14. [11] Baverjee S. Crack closure in fatigue—a review, Part I: mechanisms and prediction. Transactions of the Indian Institute of Metals 1985;38(2):167–86. [12] Dowson AL, Halliday MD, Beevers CJ. In-situ SEM studies of short crack growth and crack closure in a near-alpha Ti alloy. In: Conf. IRC’92, Birmingham, England, 1992. [13] Zhang JZ, Halliday MD, Poole P, Bowen P. Crack closure in small fatigue cracks—a comparison of finite element predictions with in-situ scanning electron microscope measurement. Fatigue and Fracture of Engineering Materials and Structures 1997;20(9):1279–93. [14] Li JC, Taylor D. A new model of short crack closure. In: Advances in Fracture Research, Proceedings of the 9th International Conference on Fracture, Sydney, Australia, 1997. Vol. 3, p. 14431–50. [15] Dugdale DS. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids 1960;8:100–4. [16] Suresh S, Ritchie RO. A geometric model for fatigue crack clos-
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Prediction of short fatigue crack propagation behaviour by characterization of both plasticity and roughness induced crack closures X.-P. Zhang
a,*
, J.-C. Li b, C.H. Wang c, L. Ye a, Y.-W. Mai
a
a
Department of Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia National Facility of Dynamic Testing and Research Civil Engineering, University of Technology, Sydney, Australia Aeronautical and Maritime Research Laboratory Defence Science and Technology Organization, Melbourne, Australia b
c
Received 17 March 2000; received in revised form 16 April 2001; accepted 10 August 2001
Abstract A new fatigue crack growth model was proposed to characterize short fatigue crack growth behaviour, through combining the mechanisms of both plasticity induced crack closure and fracture surface roughness induced crack closure. The results obtained using this proposed model show good agreement with the analytical predictions of Budiansky–Hutchinson’s complex functionanalytical solution, and are reasonably close to the experimental data acquired by other researchers. In addition, the results of short fatigue crack growth rate versus stress intensity factor range predicted by the present model can well characterize the experimental results for a 2024-T3 structural aluminium alloy under two different constant-amplitude cyclic loads, and the predictions from the present model also show the reasonable improvement compared to Newman’s plasticity induced closure model. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Short fatigue crack; Crack growth rate; Plasticity induced crack closure; Surface roughness induced crack closure; Combined model
1. Introduction So far, the majority of fatigue crack growth data, if available, were obtained by using specimens with a through-thickness long crack under constant amplitude loading. In particular, most experimental research rarely simulate the cracking configurations experienced in normal services, where failure generally occurs by propagation of subcritical cracks. The subcritical cracks range from several microns to a few hundred microns. This type of crack is usually called short crack or small crack, although short cracks and small cracks may be different in a strict definition or size. Short cracks are very difficult to detect using current in-situ non-destructive testing (NDT) methods. Typical NDT resolution is not sufficient for detection of short cracks. In addition, along with the improved capability of detecting crack, it has been found
* Corresponding author. Tel.: +61-2-9351-7146; fax: +61-2-93513760. E-mail address: [email protected] (X.-P. Zhang).
that the actual initiation stage, from a micro-defect to a short crack occurs during very few loading cycles; in contrast the lifetime in the short crack growth stage, which by traditional concepts should still be defined as the initiation stage, takes up a fairly large percentage in the total fatigue lifetime of components or structures. That implies fatigue cracking failure is probably dominated by the short crack propagation and the lifetime for short crack growth would be the most important part in the total fatigue lifetime. The short crack effect and characterization of short fatigue crack growth are, therefore, quite critical for accurate lifetime prediction of components and structures since anomalously fast crack growth rates are often observed at stress intensity factors well below the threshold where long fatigue cracks are generally presumed dormant [1–10]. As is commonly accepted, crack closure is one of the most important factors resulting in the anomaly of short crack growth and the distinction of growth behaviour from long cracks [1,2,4,7,9]. It is agreed that plasticityinduced closure is the primary mechanism of crack closure arising from the development of residual plastic
0142-1123/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 1 ) 0 0 1 6 1 - X
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stretch in the crack-wake. However, it has been recognized, both theoretically and experimentally, that using only the continuum-based plasticity induced closure analysis could not explain some experimental phenomena, and the total crack closure should include both plasticity- and surface roughness-induced closures. Studies demonstrated that fracture surface roughness induced closure plays an important role in crack growth process, in particular at low crack growth rate regime (da/dN⬍10⫺6 mm/cycle) and predominantly in plane strain condition [11]. For example, the investigation on short fatigue crack closure for a near-alpha IMI 834-Ti alloy with in-situ SEM studies [12] showed that crack closure increased significantly soon after the short crack propagated through the first grain boundary. Furthermore, it was also shown that in the subsequent crack growth, the level of crack closure appeared to decrease as the crack length increased. It seems appropriate to rationalize this in terms of the crack path tortuosity and changes in fracture morphology. It can be envisaged that because of the zig-zag shape of crack growth path the idealized plasticity induced closure, which is based on a flat crack surface, should be modified. The FEM model with only plasticity induced crack closure also predicted lower closure levels than those obtained from in-situ SEM investigation [13]. In other words, it means that roughness induced closure plays a very important role in this case, due to crack deflection. In practices, it is neither pure plasticity induced closure nor pure roughness induced closure but a combination of both closure mechanisms that dominates the early stage of fatigue crack growth [13,14]. On this point, for most long cracks, this does not seem particularly important, because scales of the deflection of crack growth path, compared to the size of crack opening, are too small to be considered. For short cracks it is, however, another situation. Normally, a small crack length means small crack opening, but the small length does not dominate the size of the crack deflection which is usually controlled by microstructures. In this case, the size of the crack deflection is always comparable to the size of the crack opening. Therefore, the crack deflection can not be neglected, in other words, the idealized flat crack growth path is not appropriate to describing short crack features, and the crack growth path deflection is strongly affected by microstructural features of materials. This work developed a new predictive model for short fatigue crack growth accounting simultaneously for both mechanisms of plasticity and crack surface roughness induced crack closures. The results obtained by using the proposed model were compared with the analytical predictions of Budiansky–Hutchinson’s complex function-analytical solution and the experimental data obtained by other researchers. In addition, the results predicted by the present model have also been compared with the experimental results for a 2024-T3 structural
aluminium alloy under two different constant-amplitude cyclic loads.
2. A combined model of short fatigue crack growth accounting for both mechanisms of plasticity and roughness induced crack closures An improved short fatigue crack growth model should include both mechanisms of plasticity- and fracture surface roughness-induced crack closures. It should not only consider the mix of both crack closures, but should also recover their limiting functions, i.e. should also recover either pure plasticity induced crack closure or pure roughness induced crack closure. Due to the existence of the plastic wake and the fracture surface roughness as shown in Fig. 1, the onset of crack opening (dop) or equivalently, the total crack closure, is given by: dop⫽dpop⫹drop cos q
(1)
where dpop and drop are equivalent crack surface displacements due to plasticity-induced crack closure and fracture surface roughness induced-crack closure, respectively. As shown in Fig. 1, it is assumed that there exists the idealized plastic residual wake and the fracture surface asperity is in regular triangular form. At the cracktip, the crack-opening displacement can be expressed by Dugdale’s model [15] as: d⫽K 2I/(Esy).
(2)
Thus, the crack opening stress intensity factor, Kop, can be obtained from the models of Dugdale and Suresh– Ritchie [16] and Eq. (2). That is: Kop ⫽ Kmax
冪
dop ⫽ dmax
冪
dpop+drop cos q dmax
(3)
and cos q⫽
h
冑h +(w/2) 2
⫽
2
1
冪
冉 冊
1 1+ 2h/w
2
⫽
2g
冑4g +1 2
(4)
where g is defined as the fracture surface roughness factor, g=h/w, and w denotes the mean base width of fracture surface asperities, and h is the average height of asperities. g actually is the ratio of the average height to the mean base width of asperities. Using Dugdale’s model again and substituting Eq. (4) into Eq. (3), thus we have
Kop⫽
冪
(Kop)2+(K rop)2
2g
冑4g +1 2
(5)
where K pop and K rop can be calculated by modifying New-
X.-P. Zhang et al. / International Journal of Fatigue 24 (2002) 529–536
Fig. 1.
Illustration of a deflected crack at (a) full opening and (b) commencement of opening.
man’s numerical analytical method [17,18], and the Suresh–Ritchie’s model, respectively. K pop is dependent on plane strain or plane stress, cyclic stress ratio R and applied to yield stress ratio s/sy. In Suresh–Ritchie’s model, K rop can be estimated by K rop⫽Kmax
冪1+2gc 2gc
冪
(K pop)2+(Kmax)2
4g2c
(1+2gc)冑4g2+1
Kop⫽rKK pop
.
(7)
(8)
where
冪
Kmax−Kop Kmax−Kop ⫽ Kmax−Kmin (1−R)Kmax
1− ⫽
冉 冊冑
1+
K rop K pop
2
2g 2
4g +1
.
(9)
Clearly, Kop for the two limiting cases of pure plasticityinduced crack closure and pure surface roughnessinduced crack closure can be recovered easily from Eq. (7) or Eq. (8). Also, in this crack closure model, the effective stress intensity factor range ratio, or the effective crack driving force, U, is given by:
(10a)
冪冉 冊
4g2c K pop 2 + Kmax (1+2gc)冑4g2+1 1−R
or
冪
1− U⫽
We rewrite Eq. (5), then have
rK ⫽
U⫽
(6)
where c is the mixture ratio of mode II to mode I displacements, i.e. c=uII/uI. It does not depend on the state of stress at the crack-tip, that is, plane strain or plane stress. Combining Eqs. (5) and (6), then gives
Kop⫽
531
4g2c [1−(1−R)Up]2+ (1+2gc)冑4g2+1 1−R
(10b)
where
冤
冪冉 冊 冥
Up⫽ 1⫺
K pop Kmax
2
/(1⫺R)
(10c)
and Up denotes the effective crack driving force arisen from pure plasticity-induced crack closure. Since there is no crack growth path deflection when the crack grows in the first grain (assuming that the mean grain size is dg), there should be no crack surface roughness induced closure, then we have: Kop⫽K pop for a⬍dg.
(11)
Furthermore, when the crack grows up to a relatively
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large length, the size of plastic zone at crack tip is fairly large, consequently K pop would be very high and much larger than K rop, i.e. rK→1 [referring to Eq. (9)], this actually means the roughness induced crack closure would lose its effect, then only plasticity induced closure exists, by Eqs. (7)–(9), we have Kop⬵K pop
for adg.
(12)
This means that for very long cracks, plasticity induced crack closure is the dominant mechanism. Fig. 2 shows the results of the influence of surface roughness factor (g) and mode II sliding level (c) at crack tip on short fatigue crack growth rates predicted by the present model. In Fig. 2, it is clearly seen that the rougher fracture surface (i.e. larger g value) results in large decrease of da/dN due to the increase of surface roughness induced closure. And the higher level of mode II shear sliding at crack tip (i.e. larger c value) has the similar effect. The above combined mechanisms of both plasticityand roughness-induced crack closures can also be obtained by combining Budiansky–Hutchinson’s small scale yielding model [19] and Suresh–Ritchie’s model. Budiansky–Hutchinson presented a complex functionanalytical solution to characterize fatigue crack growth with crack closure behaviour using ⌬Keff. In light of the Budiansky–Hutchinson’s results, still referring to Fig. 1, the crack closure stress intensity factor, Kcl, is given by Kcl 1 ⫽1⫺ Kmax 冑0.54
冪1−d
dR
(13)
fracture surface roughness and that it is only a function of stress ratio R [21], the effective plastic stretch is dR⫽dpR⫹drR
(14)
p R
r R
where d and d are, respectively, the plastic stretch pertaining to plasticity-induced closure and surface roughness contribution. Using the Suresh–Ritchie and Dugdale models again, then Eq. (13) becomes: 1 Kcl ⫽1⫺ Kmax 冑0.54
冪1− 1+2gc − d
dpR
2gc
(15)
max
which recovers the Budiansky–Hutchinson analytical result for c=0 (i.e. there exists only pure mode I deformation) or g=0 (i.e. flat fracture surface). Accordingly, the effective stress intensity factor range ratio, U, is given by U⫽
1 1 Kmax−Kcl ⫽ (1−R)Kmax 1−R 冑0.54
冪1− d
dR
.
(16)
max
In the absence of fracture surface asperities and noting Eqs. (14) and (16), we have 1
U⫽ p
· 冑0.54
冪
1−
dpR dmax (17)
1−R
where Up is the effective stress intensity factor range ratio for pure plasticity-induced crack closure. The crack closure stress intensity factor can now be derived from Eqs. (15) and (17). That is,
max
where dR and dmax are the total plastic stretch and the maximum crack-tip opening displacement at maximum load. It should be indicated that generally Kop and Kcl will be identical [20], except in the contact region yielding occurs then Kop may be lower than Kcl. Assuming that the plastic stretch is not coupled to the
冦
Kcl⫽Kmax 1⫺
冧
0.54[(1−R)U ] − . 1+2gc 冑0.54 冪 1
p 2
2gc
(18)
Consequently, the effective stress intensity factor range ratio Eq. (16) becomes U⫽
冪[U ] − 1+2gc 0.54(1−R) . p 2
2gc
1
2
(19)
It should be noted hereto we consider Kcl to be identical to Kop, since the difference between them is generally less than the scatter related to the crack closure measurements. 3. Results and discussion 3.1. Comparisons of present model, Budiansky– Hutchinson’s analytical solutions and experimental data Fig. 2. Influence of surface roughness factor and shear sliding level on short fatigue crack growth rates predicted by the present model.
Comparisons of crack closure and effective crack driving force under plane stress condition between the
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533
Fig. 3. Comparisons of crack closure between current numerical results and analytic solutions based on Budiansky–Hutchinson’s analyses (both for plane stress condition).
Fig. 5. Comparisons of theoretical predictions and experimental data for plane strain.
present model [by Eqs. (15) and (16)] and the model based on Budiansky–Hutchinson’s analytic solutions are shown in Figs. 3 and 4, respectively. It can be seen that the latter agrees quite well with the predictions from the present model. And exactly saying, predictions from the present model show a bit higher crack closure level (Kop/Kmax) and lower effective crack driving force (U) than Budiansky–Hutchinson’s analytic solutions where only plasticity induced crack closure is taken into account. Moreover, crack closures estimated using the present model were also compared with other researchers’ experimental data [22,23] obtained from 10 mm thick specimens of a structural aluminum alloy 2024 in T3 and T6 heat treatment conditions, respectively, as shown in Fig. 5. By the present model, the numerical predictions under different stress ratio R were carried out in plane strain condition using Eq. (7) for s/sy=0.15 and 0.77 and g=0.5, c=0.2 from fracture surface profile measurements [16]. It can be seen that the predictions are reasonably close to the test data at high values of stress ratio R.
In addition, the present model has also been used to characterize the experimental results obtained by this work for a structural aluminum alloy 2024-T3 (solution heat-treatment plus cold-worked, with a thickness of 5 mm). The 2024-T3 aluminium alloy had a typical recrystallized pancake-shaped grains with the approximate grain scale of 150, 75, and 50 µm in rolling orientation, width and thickness directions, respectively. Its mechanical properties are: yielding strength sy=394 MPa, ultimate tension strength sUTS=442 MPa, and the percentage prolongation d5 =7.5%. All specimens had a geometry of 5 mm thick, 50 mm wide and 300 mm long. The specimens were double-edge-notched with a round notch root radius of 3 mm The longitudinal axis of all specimens was parallel to the rolling orientation of 2024-T3 aluminium alloy sheet. Before test, notch surfaces for all specimens were well polished in longitudinal direction to prevent scratched or roughness in the crack initiating and advancing orientation. The fatigue tests were carried out in a 50 kN-capacity MTS 880 electro-hydraulic servo-controlled test system with a frequency of 15 Hz. Constant amplitude loading mode was used with maximum stresses of 78 and 98 MPa, respectively, and keeping a constant stress ratio of 0.1 (i.e. R=0.1). The measurement of short fatigue crack length was conducted in surfaces and edges of the notches simultaneously. In early growing stage of short cracks, the crack length was monitored and measured by replication technique. When cracks grew up to a certain length (longer than 300 µm), an optical microscope with magnification of 100 times was used easily to monitor and measure the crack length. The results of fatigue crack length versus number of cycles under two different constant amplitude stresses (78 and 98 MPa, respectively) are shown in Fig. 6. The experimental results of fatigue crack growth rate da/dN versus ⌬K are shown in Figs. 7 and 8. In Figs. 7 and 8, the model developed by this work was employed to characterize the fatigue crack growth behaviour, in
Fig. 4. Comparison of effective crack driving force between numerical results from present model and results from Budiansky–Hutchinson’s analytic solution.
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Fig. 6. Fatigue crack length versus number of cycles for a 2024-T3 aluminium alloy under different applied stresses (smax=78 and 98 MPa). Fig. 8. Short crack growth rate versus ⌬K under constant-amplitude stress of 78 MPa, and comparisons with present model and Newman model.
induced closure when the crack grows up to a relatively large length. But for the major stage of short fatigue crack growth, the prediction from the present model shows a reasonable improvement compared to Newman’s numerical method, as fracture surface roughness induced crack closure has been taken into account for this structural aluminium alloy that has coarse grained microstructures. 3.2. Effect of crack-tip shear sliding on crack closure and crack driving force
Fig. 7. Short crack growth rate versus ⌬K under constant-amplitude stress of 98 MPa, and comparisons with present model and Newman model.
particular at initial and early stages of crack propagation. Predictions by using modifying Newman’s numerical method are also shown in Figs. 7 and 8 for comparison. As mentioned above, these predictions (by modifying Newman’s method) only accounted for pure plasticity induced crack closure. For convenience, experimental data for long fatigue crack growth rates in 2024-T3 and 2024-T6 aluminium alloys [24,25] are given in Tables 1 and 2 for comparison. In Figs. 7 and 8, it can be seen clearly that the prediction from the present model shows a good coincidence with the results of Newman’s model when the crack is very short (less than 80 µm) and when the crack has grown up to fairly longer (a few millimeters long). This may imply that there is almost no fracture surface roughness induced closure at much earlier stage of the crack growth, and also much less influence of roughness
Crack surface roughness induced crack closure is strongly attributed to the crack path deflections caused by the heterogeneous material microstructures and the shear sliding of the crack-tip. This effect is predominant at low R, where the minimum crack-tip opening displacement is significantly smaller than the asperity height of the fracture surface. Fig. 9 shows the influence of crack-tip shear sliding on the plane stress effective crack driving force U calculated from Eq. (10) for different intensity of crack surface roughness. Clearly, U decreases with increasing shear sliding level c, since a larger shear sliding gives a stronger engagement of asperities on the fracture surfaces leading to more severe crack closure. In the limit, g=0, there is no surface roughness induced crack closure for flat fracture surfaces (except plasticity induced crack closure) at any level of crack-tip shear sliding, c.
4. Conclusions 1. A combined model of short fatigue crack closure accounting for both mechanisms of plasticity- and
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535
Table 1 Long crack growth data in 2024-T3 aluminium alloy [24] da/dN (m/cycle,×10⫺11) ⌬K (MPa√m)
5.01 3.63
126 3.55
19.9 3.70
39.8 3.80
63.1 4.17
126 4.57
199 5.01
316 5.25
501 5.62
1000 6.63
Table 2 Long crack growth data in 2024-T6 aluminium alloy [25] da/dN (m/cycle) ⌬K (MPa√m)
9.93×10⫺13 0.707
1.13×10⫺10 3.36
1.60×10⫺9 1.87
4.08×10⫺9 2.54
1.56×10⫺8 4.01
4.90×10⫺8 6.40
1.00×10⫺7 9.00
References
Fig. 9. Influence of mode II sliding on crack driving force at short crack length (a=0.10 mm) in plane stress condition.
roughness-induced crack closures was developed to characterize short crack growth behaviour. 2. The results of crack closure and effective crack driving force estimated by numerical simulation with the present model agreed well with available test data and those analytic solutions obtained by the Budiansky– Hutchinson model. 3. The predicted results of short fatigue crack growth rates versus stress intensity factor range obtained from the present combined model show a reasonable improvement compared to Newman’s plasticity induced closure crack growth model. 4. Crack-tip shear sliding also enhanced crack closure and reduced the effective crack driving force. The higher the shear sliding the more was the crack closure.
Acknowledgements This work was supported in part by an Australian Research Council Large Project Grant awarded to YWM, XPZ and CHW, and in part by the Airframe and Engine Division, Aeronautical and Maritime Research Laboratory, Defence Science and Technology Organization of Australia.
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