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Fatigue crack growth discrepancies with stress ratio
Theoretical and Applied Fracture Mechanics 51 (2009) 1–10
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Fatigue crack growth discrepancies with stress ratio R. Jones a,*, B. Farahmand b, C.A. Rodopoulos c,d a
Department of Mechanical and Aerospace Engineering, Monash University, P.O. Box 31, Victoria 3800, Australia TASS Inc., 115th Ave NEKirkland, WA 98034, USA Laboratory of Technology and Strength of Materials, Department of Mechanical Engineering and Aeronautics, University of Patras, Panepistimioupolis-Rion, 26500, Patras, Greece d Materials and Engineering Research Centre, Sheffield Hallam University, Howard Street, Sheffield S1 1WB, UK b c
a r t i c l e
i n f o
Article history: Available online 23 January 2009 Keywords: Ti–6Al–4V Crack growth R ratio Frost–Dugdale Crack closure
a b s t r a c t This paper examines cracking in D6ac and 4340 steel along with Mil Annealed and STOA Ti–6Al–4V and finds that the data implies that in the Paris Region (Region II) of the crack growth curve there is only a minimal R ratio dependency. Presented is a theoretical basis for explaining this behaviour and suggest alternative ways for characterising crack growth prediction through the use of the Generalised Frost– Dugdale crack growth law. The Fatigue Damage Map method is then used to explain the physics behind this behaviour. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The accurate estimation of the fatigue lives of metallic structural components in highly demanding environments is still a challenge for both the designer and the fleet manager. Military aircraft are often required to achieve long lives under demanding operational conditions consisting of highly complex and variable spectra. As a result there has been an increasing use of titanium in primary structural members due to its high strength, light weight, and good fatigue and fracture toughness properties. Indeed, the bulkheads in the F-22, the Super Hornet, the Swiss F/A-18, and the Joint Strike Fighter are made of titanium. In the F-22, titanium accounts for 36%, by weight, of all structural materials used in the aircraft. Until recently it had been thought that fatigue crack growth in high strength aerospace steels and titanium was well understood. However, as concluded in the review of fatigue crack growth under variable amplitude loading presented in Ref. [1], viz: Experimental results also suggest that the underlying causes of load interaction phenomena are not necessarily similar for different groups of metals, e.g. steels and Al and Ti alloys. In this context it should be noted that it was recently reported [2,3] that crack growth data, obtained for D6ac and 4340 steels, obtained using CT specimens tested under the ASTM constant R load reducing technique, exhibited an apparently anomalous behaviour
* Corresponding author. E-mail address: [email protected] (R. Jones). 0167-8442/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2009.01.004
in that whilst there was no R ratio dependence, and hence no closure, in the Paris Region (Region II) the da/dN versus DK relationship appeared to be R ratio dependent in Region I, see Fig. 1. This behaviour, i.e. the da/dN versus DK relationship appearing to be R ratio dependent in Region I but showing no R ratio dependence and hence no closure in the Paris Region, is also evident in data presented in Ref. [4] which studied cracking in QIN (HY80) steel, see Fig. 2. Near threshold, or Region I, crack growth data for solution treated and over-aged (STOA) condition Ti–6Al–4V obtained using CT specimens tested under the ASTM constant R load reducing technique has recently been presented in Ref. [5]. Here the Region I data also revealed a fanning of the da/dN versus DK curves due to an apparent R ratio effect that clearly dissipated as you approach Region II. Similarly crack growth data was presented in Ref. [6], see Figs. 5 and 6 in Ref. [6], for bimodal Ti–6Al–4V. In this work it was shown that whilst the da/dN versus DK relationship was strongly R ratio dependent in Region I the R ratio dependency in Region II appeared to be relatively small. As such the crack growth data presented in Refs. [5,6] resemble the crack growth data presented in Refs. [2–4] in that the R ratio effect dissipates as you approach the Paris Region. The present paper investigates this phenomenon and shows that NASA, Northrop–Grumman and the NASGRO crack growth data reveal that the R ratio effect on crack growth in Mil Annealed Ti–6Al–4V is quite small. A similar conclusion, i.e. that the R ratio effect dissipates as you approach the Paris Region, is reached for STOA Ti–6Al–4V. We also attempt to identify the reasoning behind this behaviour via the Fatigue Damage Map approach and show that the intrinsic fatigue damage locus (IFDL) associated with materials with a weak
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R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10
Fig. 1. Fatigue crack growth data. Plot reproduced from Ref. [2]. Fig. 3. The da/dN versus K relationship for R ratio’s of 0.0, 0.25, 0.43, 0.66, and 0.85.
Table 1 LT test configurations. Test frequency, Hz Ct3-5-tl Ct3-10b-lt Ct3-12-lt Ct3-25-lt Ct3-27-lt Ct3-29-lt Ct3-46-lt Ct3-47-lt
Constant Constant Constant Constant Constant Constant Constant Constant
p Kmax (=15 MPa m) test R = 0.3, load increasing R = 0.9, load increasing R = 0.7, load increasing R = 0.9, load increasing R = 0.3, load increasing R = 0.1, load increasing R = 0.8, load increasing
18 20 20 20 22 10 20 10
Fig. 2. Fatigue crack growth data from QIN (HY80) steel. Plot reproduced from Ref. [4].
R ratio dependency differs markedly from that associated with materials which exhibit a strong R ratio dependency. 2. Cracking in Ti–6Al–4V It is commonly believed that, like many aircraft quality aluminium alloys, crack growth in Mil Annealed Ti–6Al–4V shows a marked R ratio effect with a significant increase in the R ratio resulting in a significant increase the crack growth rate. One common way to account for R ratio effects is via the Walker crack growth law [7], viz:
da=dN ¼ CðDK ð1pÞ K pmax Þm ¼ CðDK=ð1 RÞp Þm
ð1Þ
where C, p, and m are experimentally determined constants and R = Kmin/Kmax. The values of p given in Ref. [8], Table 11.1, for 2024-T3 and 7075-T6 aluminium alloys are 0.32 and 0.36, respec-
Fig. 4. The generalised Frost–Dugdale representation of crack growth in D6a steel, from Ref. [10].
R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10
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Fig. 6. Crack growth in the Ti–6Al–4V centre cracked panel tests reported in Ref. [14].
Fig. 5. Crack growth in Ti–6Al–4V.
tively. In this paper, we will show that for Mil Annealed Ti–6Al–4V the Kmax dependency, and hence the R ratio dependency, is much smaller with a value of (typically) p = 0.08. To this end we will first examine Hudson’s early work on cracking in Mil Annealed Ti–6Al–4V [9]. This study analysed 203.2 mm wide, 610 mm long, and 1.27 mm thick centre cracked panels, with an initial (total) crack length of 3.81 mm, tested under constant amplitude loading with a mean stress of 172.35 MPa and R ratio’s of 0, 0.25, 0.45, 0.66, and 0.85, which correspond to Dr’s (=rmax rmin) of 344.7, 206.8, 137.9, 68.9, and 27.6 MPa, respectively. The associated da/dN versus DK/(1 R)0.08 relationship for these tests is shown in Fig. 3. Here we see that the constants associated with Eq. (1), i.e. Walkers law, for Ti–6AL–4V are C = 8 1011, m = 2.6 and p = 0.08. With a value of p 0.08 the da/dN versus DK relationship appears to be relatively independent of the R ratio. At this point it should be noted that it has recently been shown [10] that crack growth in the D6ac steel CT specimens tested under both constant Kmax and constant R ratio load increasing tests, see Table 1, conformed to the Generalised Frost–Dugdale law1 [11]:
da=dN ¼ C að1c=2Þ ðDK=ð1 RÞp Þc =ð1 K max =K c Þ ðda=dNÞ0
weak R ratio dependency, and that this relationship holds over three orders of magnitude. Interestingly this value of c compares well with that of c = 2.6 obtained in Ref. [13] for Mil Annealed Ti–6Al–4V tested under spectrum loading, and with the exponent m = 2.6, obtained from Fig. 1, in the Walker equation. Now use these constants in conjunction with Eq. (2) to predict the crack length history in the test data presented in Ref. [14]. This work presented the crack histories for 304.8 914.4 4.064 (thick) mm Ti–6Al–4V panels containing a centrally located crack. Two sets of constant amplitude tests were performed. One test had rmax = 172.35 MPa, and rmin = 8.62 MPa, i.e. R = 0.05, whilst the other had rmax = 258.5 MPa, and rmin = 8.62 MPa, i.e. R = 0.033. Fig. 6 shows that there is good agreement between the measured and predicted crack length histories. Note that in this case we used the values of C, c, p, and Kc as derived from Hudson’s data and set da/dN0 = 0. As a result of these two examples we see that for the five different R ratio’s, which range from R = 0 to R = 0.85, the da/dN versus
ð2Þ
where C, p, and c are material constants and the term (da/dN)0 reflects both the fatigue threshold and the nature of the notch (defect/ discontinuity) from which cracking initiates. In this study [10] it was found that Eq. (2) holds over five orders of magnitude, viz 108 < da/dN < 103 mm/cycle, see Fig. 4. Over this range of da/dN it was found that C = 8.12 1012, c = 2.6, and p 0.05, see Fig. 4, and that over this range the effect of Kmax approaching Kc could effectively be neglected. With such a low value of p (0.05) the R ratio dependency is very weak. Fig. 5 shows that a similar phenomena holds for Mil Annealed Ti–6Al–4V in that cracking also appears to conform the Generalized Frost–Dugdale law [10] with C 2.5 1011, c = 2.5, p p = 0.08 and Kc = 100 MPa m, so that crack growth has a very 1 It has also been shown [12] that a large cross-section of rail steels conforms to the Generalised Frost–Dugdale law and that this relationship also holds over five orders of magnitude.
Fig. 7. Hudson and NASGRO R ratio dependency of Mil Annealed Ti–6Al–4V.
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R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10
Fig. 10. Summary of the ASTM constant R ratio load reducing tests for (STOA) Ti– 6Al–4V, from Ref. [5].
Fig. 8. Northrop–Grumman, Hudson and NASGRO R ratio dependency of Mil Annealed Ti–6Al–4V.
DK relationship is essentially independent of the R ratio. We also see that the crack growth conforms to the Generalised Frost–Dugdale growth law and that the crack length histories presented in Ref. [14] can be predicted using the constants obtained from the data presented in Ref. [9]. This study has revealed that, in Region II, the R ratio dependency of Mil Annealed Ti–6Al–4V is quite small, i.e. p 0.08. To further illustrate this finding Fig. 7 presents the Region II crack growth data previously presented in Fig. 3 together with the associated materials response obtained from NASGRO, and Walkers law, i.e. Eq. (1), with the constants as determined above, viz: C = 8 1011, m = 2.6, and p = 0.08. Here we see that, in Region II, when presented in this fashion Hudson’s data [9] and the NASGRO data essentially coincide. Fig. 8 shows the crack growth behaviour obtained by Northrop–Grumman [15] superimposed over both Hudson’s and the NASGRO results. Note that for any given value of R the Northrop–Grumman test data has a reasonably large amount of scatter. Consequently, so as to not hide/obscure the
Hudson and the NASGRO data it has been shown on a separate figure. Despite this scatter it is clear that all three sets of data are in good agreement. At this point it should be noted that when da/dN is less than approximately 106 mm/cycle Hudson’s data [9] and the NASGRO data diverge. It is unclear if this is a real phenomenon or is due to similitude being lost in Region I. This relatively small R ratio dependency (p 0.08) in the Paris Region is also evident in the experimental results presented in Ref. [16], see Fig. 9, for Mil Annealed Ti–6Al–4V specimens with cracks growing from an initial 0.35 mm deep edge slit in a 10 mm thick dogbone specimen. In this study, the specimens were tested in a salt solution (3.5% NaCl by mass) and growth was terminated at a maximum crack length of approximately 2.5 mm, see Ref. [16] for more details. This phenomena is, i.e. a fanning in Region I and a greatly reduced R ratio effect in Region II, was also observed for the (STOA) Ti–6Al–4V specimens tested at 177 °C (350 F) in Ref. [17] who stated: An obvious R ratio effect on the fatigue crack growth rate was observed in the near threshold regime even at such high ratios. However, as DK increased the influence of R on da/dN decreased. The same trend was also observed at RT. To further evaluate this phenomena we will examine the results presented in Ref. [5] in which they used ASTM compact tension specimens with several different widths, viz: 51 mm and 76 mm, together with the ASTM constant R load reducing method to obtain da/dN versus DK data for (STOA) Ti–6Al–4V for R ratio’s ranging from 0.1, 0.4, to 0.7, see Fig. 10. If as stated in Ref. [17] crack growth in Region II is tending to become relatively R ratio independent for (STOA) Ti–6Al–4V then it should follow that any fanning in the Region I data should dissipate as you approach the Paris Region. Fig. 10 confirms this. 3. Applying the Fatigue Damage Map method
Fig. 9. The da/dN versus K relationship for R ratio’s of 0.01, 0.5, and 0.7, for Mil Annealed Ti–6Al–4V, from Ref. [16].
In this section, we will use the Fatigue Damage Map Method (FDM) to identify the reasoning behind the weak R ratio dependency seen in high strength aerospace materials. The details of the method can be found in Refs. [18–21]. The application of the FDM approach for the selected materials and comparison to the experimental results presented above is shown in Figs. 11–13. Here FDM Ver. 1.08 was used to predict the two thresholds of the material and test configuration. At this stage it is important once again
R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10
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Fig. 11a. A snap shot from FDM Ver. 1.08 (stress range crack length) for Ti–6Al–4V at R = 0 for a through-thickness crack.
Fig. 11b. Comparison of experimental results and threshold predictions from the FDM for Ti–6Al–4V at the selected stress ratios. The different line colours correspond to specific R ratios. The full line represents the 1st threshold and the dashed line represents the 2nd threshold.
to repeat the rationale underpinning the FDM approach. Fatigue damage represents the interaction between two physical mechanisms. The first is the creation and growth of crack tip plasticity. The second involves the creation of a new fracture surface and the propagation of the crack tip. The first of course does not necessitate the second as a crack may arrest. During the initial stages of fatigue, the rates of these mechanisms may be non-proportional. To elaborate on this issue it is important to consider the following. During loading the stress field ahead of the crack tip will activate a dislocation source, which in turn will start emitting dislocations along a slip plane. The number of dislocations is a function of the maximum (local) stress rmax. Upon unloading, a portion of dislocations will be recovered by the source, while others will remain trapped due to lattice friction. During the second loading cycle, the new dislocations will have to overcome those trapped from the first cycle and hence their density at the trapping sites will be further increased as will their length in comparison to the end of crack tip plastic zone (i.e. the furthest dislocation). During the Nth cycle the difference between dislocation trapping sites
and that of the furthest will be so small as to provide near rmax independency, i.e. Dr dependency. At this point proportionality between the crack tip and crack tip plasticity rate will be established. It is the same point at which steady-state, also referred to as Stage II, conditions can apply. Therefore it is necessary to understand that at the beginning of crack growth there is a transitional period which depends mostly on rmax, the crack length (a), R and the material itself. To evaluate the effect of the crack length it is important to recall from Ref. [22] that: (a) when a fatigue crack can naturally initiate it will generally appear at a surface or surfaces (e.g. a corner crack); (b) because of its small perimeter the number of grain sampled by its crack tip plasticity could be small enough that a polycrystalline behaviour is not to established. As such, the corresponding flow resistance will have value smaller that that of its bulk property. As a result more dislocations will be emitted at the same rmax. Of course the issue is more complex since grain boundaries can restrict the expansion of dislocation movement. The rational outlined above is the main reason why, when short and long cracks are plotted under the same DK values, the first will often appear to have higher growth rates. The effect of R ratio is quite straightforward, since high R values will reduce the distance between the trapping sites and that of the furthest faster. The material itself is quite important in this process considering that either the ratio between the fatigue limit and the cyclic yield stress (for short cracks) or the bulk flow resistance (for long through-thickness cracks) controls, in principle, the lattice friction and hence controls this transitional period. The phenomenon of the transitional period in terms of number of cycles (da/dN) or in terms of DK is similar and in the case of naturally initiated cracks helps explain the behaviour of short cracks, while for through-thickness long cracks it cracks helps explain the near threshold behaviour. The 1st threshold represents the natural ability of the material to arrest cracks not longer than the average grain size, at a corresponding stress no larger than the fatigue limit. Here we use the average grain size since the experimental data come from fine grain material and through-thickness cracks. As a result crack tip plasticity is under immediate polycrystalline behaviour. A detailed analysis of the effect of polycrystalline behaviour of fatigue cracks can be found in Refs. [22,23]. The second threshold is taken from a point where the gradient of the Dr-a curve saturates to a value approaching the classical LEFM -0.5 (singularity) and hence steady-state growth conditions (i.e. Stage II crack growth) hold. The range between the first and
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Fig. 12a. Snap shot from FDM Ver. 1.08 (stress range crack length) for 4340 at R = 0.5 for a through-thickness crack.
Fig. 12b. Comparison between experimental results and threshold predictions from the FDM for 4340 steel at the selected stress ratios.
Fig. 13b. Comparison between experimental results and thresholds predictions from the FDM for D6ac steel at the selected stress ratios.
Fig. 13a. Snap shot from FDM Ver. 1.08 for D6ac at R = 0.5 and considering a through-thickness crack.
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R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10 Table 2 Material properties used for the application of the FDM. Values from several references. Material
Ti–6Al–4V
4340 steel
2024-T351
D6AC steel
Yield stress (MPa) Ultimate tensile strength (MPa) Fatigue limit at R = 0 Cyclic yield stress (MPa) Elastic modulus (GPa) Shear modulus (GPa) Fracture toughness or tearing failure (MPam1/2) Average grain size (lm) Burger’s vector Elongation to failure (%)
960 1035 620 1007 120 45 131–150 for sheet 12–16 2.8910 10–12
1410 1500 870 1450 198–205 72–80 90–101 for sheet 10–19 2.4810 11–12
320 520 200 450 72400 28000 137 52 2.8610 15–17
1379 1483 1105 1404 210–263 74–90 90–120 for sheet 2–10 2.4810 7–11
Fig. 14a. Snap shot from FDM Ver. 1.08 (stress range versus crack length) for 4340 at R = 0 and selection of the eight points.
Fig. 14b. Identification of the eight propagation rate points from FDM Ver. 1.08 (crack growth versus crack length) for 4340 at R = 0 and considering a through-thickness crack. Since propagation is not possible at rates below the Burger’s vector, point A is found within the low threshold region.
second threshold for pre-cracked through-thickness specimens indicates the range for low near threshold propagation. The parameters/inputs for the application of the FDM for the selected materials are shown in Table 2 (in all cases a crack correction factor of 1.2 was used). In 2006, the FDM was enhanced to enable it to predict the crack growth rates corresponding to each area. A detailed explanation of the rational involved is given in Ref. [24]. Fig. 14, shows a typical procedure used in this predictive process. In this approach, eight
points need to be identified, see Ref. [24]. Point A represents the 1st threshold corresponding to a crack length equal to the average grain size. Point B corresponds to the steady-state 2nd threshold (where the LEFM singularity is met). Point C identifies the end of the near threshold region at high stress levels (here there is large crack tip plasticity and the classical LEFM stress singularity is not met) with a crack length equal to the average grain size. Point D represents the end of Stage II crack growth at high stress levels (large crack tip plasticity and the LEFM singularity is not met).
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R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10
Point E is the 2nd end of the near threshold where singularity is met. Point F represents the end of steady-state threshold where the LEFM singularity is met. Point G is the beginning of Stage II crack growth with singularity at high stress levels. Point H is the end of the region where we have knowledge of the above points, see Fig. 14b. Knowing these points, the corresponding growth rates can now be extracted, as shown in Fig. 14b. This procedure delivers a locus of DK v’s da/dN which is a material property, see Ref. [24]. Latter on, it will be shown that this locus can encapsulate every possible loading condition. A typical example, and comparison, is shown in Fig. 15. Fig. 16 shows the loci for R = 0, 0.5, and 0.7. From here it can be seen that there exist a common area where the propagation rates of all three stress ratios can co-exist. In addition the near threshold area given by the closed locale A–B–D–E for R = 0, or A–B–E for R = 0.5, disappears at R = 0.7. Similar condition exists for the D6ac and Ti–6Al–4V.
Fig. 15a. The FDM locus for 4340 steel at R = 0 compared with the experimental results presented in Ref. [2]. The near threshold area is given by A–B–E–D.
Fig. 15b. The FDM locus for 4340 steel at R = 0.5 compared with the experimental results presented in Ref. [2]. The enclosed FDM shows the disappearance of near threshold conditions. The (near threshold) area A–D–B is calculated to disappear at R = 0.8.
Fig. 16. The 4340 crack growth loci produced via the FDM points for R = 0, 0.5, and 0.7. The experimental results from Fig. 3 are plotted in order to establish the Stage II independency with respect to R. It is worth noting that at R = 0.7 only points A, B, H, and F are maintained.
4. Explaining R independency in Stage II crack growth To identify the reasoning behind the propensity of the materials in question to exhibit a weak R ratio dependency during Stage II crack growth it is appropriate to contrast their behaviour with a material with a known strong R ratio dependency. To this end Fig. 17 presents a plot, similar to that presented in Fig. 16, for 2024-T351 aluminium alloy. Comparing Figs. 16 and 17 reveals the following: (a) 2024-T351 shows a distinct reduction of the range defining the 1st and 2nd threshold as the R ratio increases. In contrast 4340 steel shows that at R = 0.5 and R = 0.7 they both share approximately similar values; (b) 2024-T351 exhibits a well defined near threshold condition even at R = 0.7, whilst 4340 steel appears to immediately produce Stage II crack growth with an established LEFM stress singularity; and (c) for 4340 the area where the propagation rates of all three R ratios can coincide is much larger than that for 2024-T351, see Figs. 18(a) and (b), and 19. To further elaborate these findings and identify whether they hold any physical meaning it is necessary to establish the difference between a fine and coarse grain material. In previous section, it was mentioned that the transition from rmax to Dr dependency represents the cumulative, with loading cycles or crack length, increase of the dislocation density on the slip plane towards the end of it. It was also stated that as the stress ratio increases the above phenomenon will saturate faster. In a quite simplistic way it is rational to say that if the dislocation density along the crack tip plasticity is approximately the same, flow resistance is also uniform and the classical LEFM singularity can easily be attained. From
Fig. 17. The crack growth loci for R = 0, 0.5, and 0.7 for 2024-T351 sheet. The FDM inputs are given in Table 2.
R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10
which the dislocation density is seeking saturation, is inversely proportional to flow resistance. Hence, it is hypothesized that the selected high strength materials will require less cumulative time, or crack length, compared to the aluminium alloy. The reader can easily identify the difference by comparing the propagation rates p at DK = 10 MPa m. The difference between the 1st and 2nd threshold solely depend on the grain size. Here it is important to note two issues: (a) in general fine grain materials will emit a significantly smaller number of dislocations than their counterparts. Hence a singularity will be, by-default, achieved faster and (b) again by-default BCC and HCP fine grain materials will reach faster polycrystalline behaviour at threshold conditions than coarse grain FCC materials. Bear in mind that polycrystalline behaviour for BCC structures is found at a Schmidt factor of 2, while, for FCCs a Taylor factor of 3.07 is needed.
10-5
4340
da/dN (m/c)
10-6
10-7
Overlaping Area
10-8
10-9
10-10 1
10
100
Range of stress intensity factor ΔΚ (MPam1/2)
5. A non-dimensional approach to fatigue crack growth
10-5
It has recently been shown [25] that for centre cracks, and surface flaws, in a large panel, where the material follows the Generalised Frost–Dugdale law, subjected to repeated block loading such that there are a large number of blocks to failure then the crack length history can be approximated by
2024-T351
da/dN (m/c)
10-6
10-7
ðB Bi Þ=ðBf Bi Þ ¼ 1 lnða=af Þ= lnðai =af Þ; Overlaping Area
10-8
10-9
10-10 1
9
10
100
Range of stress intensity factor ΔΚ (MPam1/2) Fig. 18. (a) Overlapping area for 4340 steel and (b) for 2024-T351 aluminium alloy.
Fig. 19. Response of D6ac surface flaw specimens and centre cracked Mil Annealed Ti–6Al–4V specimens under representative flight load spectra.
the above, it is easily understood that under the same DK conditions the size of crack tip plasticity, and hence the size over
ð3Þ
where ai and af, which should generally be less than approximately 0.8 time the critical crack length acr, are the initial and final crack sizes, and Bi and Bf are the corresponding number of blocks, or flight numbers, and B is the number of blocks at crack length a. Here it should be noted that as the crack size approaches its critical length we need to account for Kmax approaching its fracture toughness, see Refs. [25,26]. However, this effect tends to be small and, as shown in Ref. [25] general only affects the region a/acr > 0.8. A similar deviation from Eq. (3) occurs as the crack length approaches the initial discontinuity size. In this case the discontinuity is influenced by the geometry of the starting notch/defect and as such is not (yet) acting as a crack of length a. To overcome this Molent et al. [27] suggested that a more accurate representation is obtained by using an equivalent pre-crack size (EPS) rather than the size of the initial discontinuity. As a result the slope of the (B Bi)/(Bf Bi) versus /(a) (=1 ln(a/af)/ln(ai/af)) curves sometimes differs slightly from that suggested by Eq. (3). Nevertheless, this approach predicts that, for tests on centre cracked panels under repeated block loading, for the majority of the fatigue life we should see a near linear relationship between the number of load blocks and the log of the crack length. In Section 2, we saw that crack growth in Mil Annealed Ti–6Al– 4V appears to conform to Generalised Frost–Dugdale law. It should thus follow that crack growth in centre cracked panels under repeated block loading should (approximately) conform to Eq. (3). To evaluate this prediction Fig. 19 presents the test results given in Ref. [15] for crack growth in a 152.4 mm wide and 7.34 mm thick centre cracked panel subjected to a fighter load spectra with a peak remote stress of 710 MPa. For comparison Fig. 19 also presents the results of a USAF [28] investigation, performed as part of the F-111 certification study, into cracking in surface flawed D6ac plate specimens under block loading, where one block represented 200 flight hours, representing the mission spectrum for the critical location in the F-111 wing pivot fitting. Fig. 19 presents the results for two specific specimens (P5I9 and P5I10) where both the initial and the final flaw shapes were very close to a semi-circular surface flaw. The spectrum used in specimen test P5I9 represented a tension–compression spectra whilst the spectrum used in specimen test P5I10 represented a tension–tension load spectra. The UASF test specimens were 406.4 mm long, 96.52 mm wide and
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R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10
Table 3 Test configuration and crack lengths. Test
Crack geom
Material
Spectra
ai (mm)
af (mm)
acr (mm)
USAF test 1 USAF test 2
Surface flaw Ibid
D6ac D6ac
2.92 2.92
5.94 5.59
6.99 5.79
Grumman test
Centre cracked panel
Mil Annealed Ti– 6AL–4V
Representative flight spectra, tension loads only Representative flight spectra, tension and compression loads Representative fighter aircraft spectra
0.562
0.949
0.949
7.62 mm thick and contained a 2.921 mm semi-circular surface flaw. The 7.62 mm thickness was representative of the critical location in the Wing Pivot Fitting location, see Ref. [28]. The values of ai, af, acr, Bi, Bf and the number of blocks to failure in each tests are given in Table 3. From this figure it is seen that, as predicted, there was a near linear relationship between the log of the crack length and the number of blocks/cycles. However, it should be noted that differences in the slopes reflects the uncertainties in the initial flaw sizes. 6. Conclusion Mil Annealed Ti–6Al–4V and (STOA) Ti–6Al–4V both appear to exhibit a similar behaviour to that documented in Ref. [2] and shown in Ref. [3] in that the R ratio dependency in the Paris Region is relatively small. Given the extensive use of titanium in the F-22, the Super Hornet, and the Joint Strike Fighter this finding, and the fact that crack growth appears to conform to the Generalised Frost–Dugdale law, should be further investigated. Similarly the fanning in Region I needs further investigation to assess whether it is a real phenomena or merely indicative of the fact that similitude may not be applicable in Region I. The concept and consequent application of the Fatigue Damage Map revealed that the basic material properties incorporate specific tendencies towards the propensity of each material to exhibit different degrees of fatigue damage stages. Indeed, we have seen that this method can pinpoint the changes in the above stages in the DK versus da/dN relationship. The selection of the limit cases, both in terms of a maximum and a minimum propensity, creates a closed form locus. For the selected stress ratio’s, this locus is a unique material property. We shall name this area as the intrinsic fatigue damage locus (IFDL). A knowledge of the IFDL allows the designer to estimate the physical fatigue limits of the material, independently of the test configuration used. In this exercise, the IFDL was determined for several stress ratios and for each of the materials in question. The large size of the overlapping regions of the IFDL’s indicates the strong potential of the materials to exhibit R ratio independency. Comparison with the IFDL for 2024-T351 aluminium alloy illustrates the differences in the IFDL between materials with a strong R ratio dependency and those with a weak R ratio dependency. References [1] M. Skorupa, Load interaction effects during fatigue crack growth under variable amplitude loading – a literature review. Part II: qualitative interpretation, Fatigue Fract. Eng. Mater. Struct. 22 (1999) 905–926. [2] S.C. Forth, M.A. James, W.M. Johnston, J.C. Newman Jr., Anomolous fatigue crack growth phenomena in high-strength steel, in: Proceedings of the International Congress on Fracture, Italy, 2007.
Bi
Bf
Blocks to failure
42 80
69 108
72 111
253
306
306
[3] S.C. Forth, W.M. Johnston, B.R. Seshadri, The effect of the laboratory specimen on fatigue crack growth rate, in: Proceedings of the 16th European Conference on Fracture, Alexandropoulos, Greece, July 2006. [4] M.N. James, J.F. Knott, An assessment of crack closure and the extent of the short crack regime in QlN (HY80) steel, Fatigue Fract. Eng. Mater. Struct. 8 (2) (1985) 177–191. [5] J.J. Ruschau, J.C. Newman Jr., Improved test method for very low fatigue-crackgrowth-rate data, in: Proceedings of the 2008 American Helicopter Society (AHS), US, 2008. [6] B.L. Boyce, R.O. Ritchie, Effect of load ratio and maximum stress intensity on the fatigue threshold in Ti–6Al–4V, Eng. Fract. Mech. 68 (2001) 129–147. [7] E.K. Walker, The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7076-T6 aluminum, in: Effect of Environment and Complex Load History on Fatigue Life, ASTM STP 462, American Society for Testing and Materials, Philadelphia, 1970, pp. 1–14. [8] N.E. Dowling, Mechanical Behaviour of Materials, Prentice Hall Publishers, New Jersey, USA, 1993. [9] C.M. Hudson, Fatigue Crack Propagation in Several Titanium and One Superalloy Stainless-Steel Alloys, NASA TN D-2331, October 1964. [10] R. Jones, S.C. Forth, Cracking in D6ac steel, Theor. Appl. Fract. Mech., submitted for publication. [11] R. Jones, L. Molent, S. Pitt, Crack growth from small flaws, Int. J. Fatigue 29 (2007) 1658–1667. [12] R. Jones, B. Chen, S. Pitt, Similitude: cracking in steels, Theor. Appl. Fract. Mech. 48 (2) (2007) 161–168. [13] W. Zhuang, S. Barter, L. Molent, Flight by flight fatigue crack growth life assessment, Int. J. Fatigue 29 (9–11) (2007) 1647. [14] T.R. Porter, Method of analysis and prediction for variable amplitude fatigue crack growth, Eng. Fract. Mech. 4 (1972) 717–736. [15] P.D. Bell, M. Creager, Crack Growth Analysis for Arbitrary Spectrum Loading, Volume I – Results and Discussion, Final Report: June 1972–October 1974, Technical Report AFFDL-TR-74-129. [16] M.R. Bache, W.J. Evans, The fatigue crack propagation resistance of Ti–6Al–4V under aqueous saline environments, Int. J. Fatigue 23 (2001) S319–S323. [17] J. Ding, R. Hall, J. Byrne, Effects of stress ratio and temperature on fatigue crack growth in a Ti–6Al–4V alloy, Int. J. Fatigue 27 (2005) 1551–1558. [18] S.A. Curtis, J. Solis Romero, E.R. de los Rios, C.A. Rodopoulos, A. Levers, Predicting the interfaces between fatigue crack growth regimes in 7150-t651 aluminium alloy using the fatigue damage map, Mater. Sci. Eng. A A344 (2003) 79–85. [19] C.A. Rodopoulos, E.R. de los Rios, J.R. Yates, A fatigue damage map for the 2024T3 aluminium alloy, Fatigue Fract. Eng. Mater. Struct. 26 (7) (2003) 569–576. [20] C.A. Rodopoulos, J.-H. Choi, E.R. de los Rios, J.R. Yates, Stress ratio and the fatigue damage map – part I: modelling, Int. J. of Fatigue 26 (2004) 739–746. [21] C.A. Rodopoulos, J.-H. Choi, E.R. de los Rios, J.R. Yates, Stress ratio and the fatigue damage map – part II the 2024-t351 aluminium alloy, Int. J. Fatigue 26 (2004) 747–752. [22] C.A. Rodopoulos, G. Chliveros, Fatigue damage in polycrystals – part 1: the numbers two and three, Theor. Appl. Fract. Mech. 49 (1) (2008) 61–67. [23] C.A. Rodopoulos, G. Chliveros, Fatigue damage in polycrystals – part 2: intrinsic scatter of fatigue life, Theor. Appl. Fract. Mech. 49 (1) (2008) 77–97. [24] C.A. Rodopoulos, Predicting the evolution of fatigue damage using the fatigue damage map method, Theor. Appl. Fract. Mech. 45 (2006) 252–265. [25] R. Jones, D. Peng, S. Pitt, Thoughts on the physical processes underpinning crack growth, Fatigue Fract. Eng. Mater. Struct. (2009) (invited paper). [26] R. Jones, S. Pitt, D. Peng, An equivalent block approach to crack growth, multiscale fatigue crack initiation and propagation of engineering materials: structural integrity and microstructural worthiness, in: G.C. Sih (Ed.), Springer press, 2008, ISBN: 978-1-4020-8519 (June). [27] L. Molent, Q. Sun, A.J. Green, Characterisation of equivalent initial flaw sizes in 7050 aluminium alloy, Fatigue Fract. Eng. Mater. Struct. 29 (2006) 916–937. 56. [28] H.A. Wood, T.L. Haglage, Crack Propagation Test Results for Variable Amplitude Spectrum Loading in Surface Flawed D6AC Steel 1971, Wright Patterson AFB, FBR-TM-71-2.
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Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Fatigue crack growth discrepancies with stress ratio R. Jones a,*, B. Farahmand b, C.A. Rodopoulos c,d a
Department of Mechanical and Aerospace Engineering, Monash University, P.O. Box 31, Victoria 3800, Australia TASS Inc., 115th Ave NEKirkland, WA 98034, USA Laboratory of Technology and Strength of Materials, Department of Mechanical Engineering and Aeronautics, University of Patras, Panepistimioupolis-Rion, 26500, Patras, Greece d Materials and Engineering Research Centre, Sheffield Hallam University, Howard Street, Sheffield S1 1WB, UK b c
a r t i c l e
i n f o
Article history: Available online 23 January 2009 Keywords: Ti–6Al–4V Crack growth R ratio Frost–Dugdale Crack closure
a b s t r a c t This paper examines cracking in D6ac and 4340 steel along with Mil Annealed and STOA Ti–6Al–4V and finds that the data implies that in the Paris Region (Region II) of the crack growth curve there is only a minimal R ratio dependency. Presented is a theoretical basis for explaining this behaviour and suggest alternative ways for characterising crack growth prediction through the use of the Generalised Frost– Dugdale crack growth law. The Fatigue Damage Map method is then used to explain the physics behind this behaviour. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The accurate estimation of the fatigue lives of metallic structural components in highly demanding environments is still a challenge for both the designer and the fleet manager. Military aircraft are often required to achieve long lives under demanding operational conditions consisting of highly complex and variable spectra. As a result there has been an increasing use of titanium in primary structural members due to its high strength, light weight, and good fatigue and fracture toughness properties. Indeed, the bulkheads in the F-22, the Super Hornet, the Swiss F/A-18, and the Joint Strike Fighter are made of titanium. In the F-22, titanium accounts for 36%, by weight, of all structural materials used in the aircraft. Until recently it had been thought that fatigue crack growth in high strength aerospace steels and titanium was well understood. However, as concluded in the review of fatigue crack growth under variable amplitude loading presented in Ref. [1], viz: Experimental results also suggest that the underlying causes of load interaction phenomena are not necessarily similar for different groups of metals, e.g. steels and Al and Ti alloys. In this context it should be noted that it was recently reported [2,3] that crack growth data, obtained for D6ac and 4340 steels, obtained using CT specimens tested under the ASTM constant R load reducing technique, exhibited an apparently anomalous behaviour
* Corresponding author. E-mail address: [email protected] (R. Jones). 0167-8442/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2009.01.004
in that whilst there was no R ratio dependence, and hence no closure, in the Paris Region (Region II) the da/dN versus DK relationship appeared to be R ratio dependent in Region I, see Fig. 1. This behaviour, i.e. the da/dN versus DK relationship appearing to be R ratio dependent in Region I but showing no R ratio dependence and hence no closure in the Paris Region, is also evident in data presented in Ref. [4] which studied cracking in QIN (HY80) steel, see Fig. 2. Near threshold, or Region I, crack growth data for solution treated and over-aged (STOA) condition Ti–6Al–4V obtained using CT specimens tested under the ASTM constant R load reducing technique has recently been presented in Ref. [5]. Here the Region I data also revealed a fanning of the da/dN versus DK curves due to an apparent R ratio effect that clearly dissipated as you approach Region II. Similarly crack growth data was presented in Ref. [6], see Figs. 5 and 6 in Ref. [6], for bimodal Ti–6Al–4V. In this work it was shown that whilst the da/dN versus DK relationship was strongly R ratio dependent in Region I the R ratio dependency in Region II appeared to be relatively small. As such the crack growth data presented in Refs. [5,6] resemble the crack growth data presented in Refs. [2–4] in that the R ratio effect dissipates as you approach the Paris Region. The present paper investigates this phenomenon and shows that NASA, Northrop–Grumman and the NASGRO crack growth data reveal that the R ratio effect on crack growth in Mil Annealed Ti–6Al–4V is quite small. A similar conclusion, i.e. that the R ratio effect dissipates as you approach the Paris Region, is reached for STOA Ti–6Al–4V. We also attempt to identify the reasoning behind this behaviour via the Fatigue Damage Map approach and show that the intrinsic fatigue damage locus (IFDL) associated with materials with a weak
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R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10
Fig. 1. Fatigue crack growth data. Plot reproduced from Ref. [2]. Fig. 3. The da/dN versus K relationship for R ratio’s of 0.0, 0.25, 0.43, 0.66, and 0.85.
Table 1 LT test configurations. Test frequency, Hz Ct3-5-tl Ct3-10b-lt Ct3-12-lt Ct3-25-lt Ct3-27-lt Ct3-29-lt Ct3-46-lt Ct3-47-lt
Constant Constant Constant Constant Constant Constant Constant Constant
p Kmax (=15 MPa m) test R = 0.3, load increasing R = 0.9, load increasing R = 0.7, load increasing R = 0.9, load increasing R = 0.3, load increasing R = 0.1, load increasing R = 0.8, load increasing
18 20 20 20 22 10 20 10
Fig. 2. Fatigue crack growth data from QIN (HY80) steel. Plot reproduced from Ref. [4].
R ratio dependency differs markedly from that associated with materials which exhibit a strong R ratio dependency. 2. Cracking in Ti–6Al–4V It is commonly believed that, like many aircraft quality aluminium alloys, crack growth in Mil Annealed Ti–6Al–4V shows a marked R ratio effect with a significant increase in the R ratio resulting in a significant increase the crack growth rate. One common way to account for R ratio effects is via the Walker crack growth law [7], viz:
da=dN ¼ CðDK ð1pÞ K pmax Þm ¼ CðDK=ð1 RÞp Þm
ð1Þ
where C, p, and m are experimentally determined constants and R = Kmin/Kmax. The values of p given in Ref. [8], Table 11.1, for 2024-T3 and 7075-T6 aluminium alloys are 0.32 and 0.36, respec-
Fig. 4. The generalised Frost–Dugdale representation of crack growth in D6a steel, from Ref. [10].
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Fig. 6. Crack growth in the Ti–6Al–4V centre cracked panel tests reported in Ref. [14].
Fig. 5. Crack growth in Ti–6Al–4V.
tively. In this paper, we will show that for Mil Annealed Ti–6Al–4V the Kmax dependency, and hence the R ratio dependency, is much smaller with a value of (typically) p = 0.08. To this end we will first examine Hudson’s early work on cracking in Mil Annealed Ti–6Al–4V [9]. This study analysed 203.2 mm wide, 610 mm long, and 1.27 mm thick centre cracked panels, with an initial (total) crack length of 3.81 mm, tested under constant amplitude loading with a mean stress of 172.35 MPa and R ratio’s of 0, 0.25, 0.45, 0.66, and 0.85, which correspond to Dr’s (=rmax rmin) of 344.7, 206.8, 137.9, 68.9, and 27.6 MPa, respectively. The associated da/dN versus DK/(1 R)0.08 relationship for these tests is shown in Fig. 3. Here we see that the constants associated with Eq. (1), i.e. Walkers law, for Ti–6AL–4V are C = 8 1011, m = 2.6 and p = 0.08. With a value of p 0.08 the da/dN versus DK relationship appears to be relatively independent of the R ratio. At this point it should be noted that it has recently been shown [10] that crack growth in the D6ac steel CT specimens tested under both constant Kmax and constant R ratio load increasing tests, see Table 1, conformed to the Generalised Frost–Dugdale law1 [11]:
da=dN ¼ C að1c=2Þ ðDK=ð1 RÞp Þc =ð1 K max =K c Þ ðda=dNÞ0
weak R ratio dependency, and that this relationship holds over three orders of magnitude. Interestingly this value of c compares well with that of c = 2.6 obtained in Ref. [13] for Mil Annealed Ti–6Al–4V tested under spectrum loading, and with the exponent m = 2.6, obtained from Fig. 1, in the Walker equation. Now use these constants in conjunction with Eq. (2) to predict the crack length history in the test data presented in Ref. [14]. This work presented the crack histories for 304.8 914.4 4.064 (thick) mm Ti–6Al–4V panels containing a centrally located crack. Two sets of constant amplitude tests were performed. One test had rmax = 172.35 MPa, and rmin = 8.62 MPa, i.e. R = 0.05, whilst the other had rmax = 258.5 MPa, and rmin = 8.62 MPa, i.e. R = 0.033. Fig. 6 shows that there is good agreement between the measured and predicted crack length histories. Note that in this case we used the values of C, c, p, and Kc as derived from Hudson’s data and set da/dN0 = 0. As a result of these two examples we see that for the five different R ratio’s, which range from R = 0 to R = 0.85, the da/dN versus
ð2Þ
where C, p, and c are material constants and the term (da/dN)0 reflects both the fatigue threshold and the nature of the notch (defect/ discontinuity) from which cracking initiates. In this study [10] it was found that Eq. (2) holds over five orders of magnitude, viz 108 < da/dN < 103 mm/cycle, see Fig. 4. Over this range of da/dN it was found that C = 8.12 1012, c = 2.6, and p 0.05, see Fig. 4, and that over this range the effect of Kmax approaching Kc could effectively be neglected. With such a low value of p (0.05) the R ratio dependency is very weak. Fig. 5 shows that a similar phenomena holds for Mil Annealed Ti–6Al–4V in that cracking also appears to conform the Generalized Frost–Dugdale law [10] with C 2.5 1011, c = 2.5, p p = 0.08 and Kc = 100 MPa m, so that crack growth has a very 1 It has also been shown [12] that a large cross-section of rail steels conforms to the Generalised Frost–Dugdale law and that this relationship also holds over five orders of magnitude.
Fig. 7. Hudson and NASGRO R ratio dependency of Mil Annealed Ti–6Al–4V.
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Fig. 10. Summary of the ASTM constant R ratio load reducing tests for (STOA) Ti– 6Al–4V, from Ref. [5].
Fig. 8. Northrop–Grumman, Hudson and NASGRO R ratio dependency of Mil Annealed Ti–6Al–4V.
DK relationship is essentially independent of the R ratio. We also see that the crack growth conforms to the Generalised Frost–Dugdale growth law and that the crack length histories presented in Ref. [14] can be predicted using the constants obtained from the data presented in Ref. [9]. This study has revealed that, in Region II, the R ratio dependency of Mil Annealed Ti–6Al–4V is quite small, i.e. p 0.08. To further illustrate this finding Fig. 7 presents the Region II crack growth data previously presented in Fig. 3 together with the associated materials response obtained from NASGRO, and Walkers law, i.e. Eq. (1), with the constants as determined above, viz: C = 8 1011, m = 2.6, and p = 0.08. Here we see that, in Region II, when presented in this fashion Hudson’s data [9] and the NASGRO data essentially coincide. Fig. 8 shows the crack growth behaviour obtained by Northrop–Grumman [15] superimposed over both Hudson’s and the NASGRO results. Note that for any given value of R the Northrop–Grumman test data has a reasonably large amount of scatter. Consequently, so as to not hide/obscure the
Hudson and the NASGRO data it has been shown on a separate figure. Despite this scatter it is clear that all three sets of data are in good agreement. At this point it should be noted that when da/dN is less than approximately 106 mm/cycle Hudson’s data [9] and the NASGRO data diverge. It is unclear if this is a real phenomenon or is due to similitude being lost in Region I. This relatively small R ratio dependency (p 0.08) in the Paris Region is also evident in the experimental results presented in Ref. [16], see Fig. 9, for Mil Annealed Ti–6Al–4V specimens with cracks growing from an initial 0.35 mm deep edge slit in a 10 mm thick dogbone specimen. In this study, the specimens were tested in a salt solution (3.5% NaCl by mass) and growth was terminated at a maximum crack length of approximately 2.5 mm, see Ref. [16] for more details. This phenomena is, i.e. a fanning in Region I and a greatly reduced R ratio effect in Region II, was also observed for the (STOA) Ti–6Al–4V specimens tested at 177 °C (350 F) in Ref. [17] who stated: An obvious R ratio effect on the fatigue crack growth rate was observed in the near threshold regime even at such high ratios. However, as DK increased the influence of R on da/dN decreased. The same trend was also observed at RT. To further evaluate this phenomena we will examine the results presented in Ref. [5] in which they used ASTM compact tension specimens with several different widths, viz: 51 mm and 76 mm, together with the ASTM constant R load reducing method to obtain da/dN versus DK data for (STOA) Ti–6Al–4V for R ratio’s ranging from 0.1, 0.4, to 0.7, see Fig. 10. If as stated in Ref. [17] crack growth in Region II is tending to become relatively R ratio independent for (STOA) Ti–6Al–4V then it should follow that any fanning in the Region I data should dissipate as you approach the Paris Region. Fig. 10 confirms this. 3. Applying the Fatigue Damage Map method
Fig. 9. The da/dN versus K relationship for R ratio’s of 0.01, 0.5, and 0.7, for Mil Annealed Ti–6Al–4V, from Ref. [16].
In this section, we will use the Fatigue Damage Map Method (FDM) to identify the reasoning behind the weak R ratio dependency seen in high strength aerospace materials. The details of the method can be found in Refs. [18–21]. The application of the FDM approach for the selected materials and comparison to the experimental results presented above is shown in Figs. 11–13. Here FDM Ver. 1.08 was used to predict the two thresholds of the material and test configuration. At this stage it is important once again
R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10
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Fig. 11a. A snap shot from FDM Ver. 1.08 (stress range crack length) for Ti–6Al–4V at R = 0 for a through-thickness crack.
Fig. 11b. Comparison of experimental results and threshold predictions from the FDM for Ti–6Al–4V at the selected stress ratios. The different line colours correspond to specific R ratios. The full line represents the 1st threshold and the dashed line represents the 2nd threshold.
to repeat the rationale underpinning the FDM approach. Fatigue damage represents the interaction between two physical mechanisms. The first is the creation and growth of crack tip plasticity. The second involves the creation of a new fracture surface and the propagation of the crack tip. The first of course does not necessitate the second as a crack may arrest. During the initial stages of fatigue, the rates of these mechanisms may be non-proportional. To elaborate on this issue it is important to consider the following. During loading the stress field ahead of the crack tip will activate a dislocation source, which in turn will start emitting dislocations along a slip plane. The number of dislocations is a function of the maximum (local) stress rmax. Upon unloading, a portion of dislocations will be recovered by the source, while others will remain trapped due to lattice friction. During the second loading cycle, the new dislocations will have to overcome those trapped from the first cycle and hence their density at the trapping sites will be further increased as will their length in comparison to the end of crack tip plastic zone (i.e. the furthest dislocation). During the Nth cycle the difference between dislocation trapping sites
and that of the furthest will be so small as to provide near rmax independency, i.e. Dr dependency. At this point proportionality between the crack tip and crack tip plasticity rate will be established. It is the same point at which steady-state, also referred to as Stage II, conditions can apply. Therefore it is necessary to understand that at the beginning of crack growth there is a transitional period which depends mostly on rmax, the crack length (a), R and the material itself. To evaluate the effect of the crack length it is important to recall from Ref. [22] that: (a) when a fatigue crack can naturally initiate it will generally appear at a surface or surfaces (e.g. a corner crack); (b) because of its small perimeter the number of grain sampled by its crack tip plasticity could be small enough that a polycrystalline behaviour is not to established. As such, the corresponding flow resistance will have value smaller that that of its bulk property. As a result more dislocations will be emitted at the same rmax. Of course the issue is more complex since grain boundaries can restrict the expansion of dislocation movement. The rational outlined above is the main reason why, when short and long cracks are plotted under the same DK values, the first will often appear to have higher growth rates. The effect of R ratio is quite straightforward, since high R values will reduce the distance between the trapping sites and that of the furthest faster. The material itself is quite important in this process considering that either the ratio between the fatigue limit and the cyclic yield stress (for short cracks) or the bulk flow resistance (for long through-thickness cracks) controls, in principle, the lattice friction and hence controls this transitional period. The phenomenon of the transitional period in terms of number of cycles (da/dN) or in terms of DK is similar and in the case of naturally initiated cracks helps explain the behaviour of short cracks, while for through-thickness long cracks it cracks helps explain the near threshold behaviour. The 1st threshold represents the natural ability of the material to arrest cracks not longer than the average grain size, at a corresponding stress no larger than the fatigue limit. Here we use the average grain size since the experimental data come from fine grain material and through-thickness cracks. As a result crack tip plasticity is under immediate polycrystalline behaviour. A detailed analysis of the effect of polycrystalline behaviour of fatigue cracks can be found in Refs. [22,23]. The second threshold is taken from a point where the gradient of the Dr-a curve saturates to a value approaching the classical LEFM -0.5 (singularity) and hence steady-state growth conditions (i.e. Stage II crack growth) hold. The range between the first and
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Fig. 12a. Snap shot from FDM Ver. 1.08 (stress range crack length) for 4340 at R = 0.5 for a through-thickness crack.
Fig. 12b. Comparison between experimental results and threshold predictions from the FDM for 4340 steel at the selected stress ratios.
Fig. 13b. Comparison between experimental results and thresholds predictions from the FDM for D6ac steel at the selected stress ratios.
Fig. 13a. Snap shot from FDM Ver. 1.08 for D6ac at R = 0.5 and considering a through-thickness crack.
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R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10 Table 2 Material properties used for the application of the FDM. Values from several references. Material
Ti–6Al–4V
4340 steel
2024-T351
D6AC steel
Yield stress (MPa) Ultimate tensile strength (MPa) Fatigue limit at R = 0 Cyclic yield stress (MPa) Elastic modulus (GPa) Shear modulus (GPa) Fracture toughness or tearing failure (MPam1/2) Average grain size (lm) Burger’s vector Elongation to failure (%)
960 1035 620 1007 120 45 131–150 for sheet 12–16 2.8910 10–12
1410 1500 870 1450 198–205 72–80 90–101 for sheet 10–19 2.4810 11–12
320 520 200 450 72400 28000 137 52 2.8610 15–17
1379 1483 1105 1404 210–263 74–90 90–120 for sheet 2–10 2.4810 7–11
Fig. 14a. Snap shot from FDM Ver. 1.08 (stress range versus crack length) for 4340 at R = 0 and selection of the eight points.
Fig. 14b. Identification of the eight propagation rate points from FDM Ver. 1.08 (crack growth versus crack length) for 4340 at R = 0 and considering a through-thickness crack. Since propagation is not possible at rates below the Burger’s vector, point A is found within the low threshold region.
second threshold for pre-cracked through-thickness specimens indicates the range for low near threshold propagation. The parameters/inputs for the application of the FDM for the selected materials are shown in Table 2 (in all cases a crack correction factor of 1.2 was used). In 2006, the FDM was enhanced to enable it to predict the crack growth rates corresponding to each area. A detailed explanation of the rational involved is given in Ref. [24]. Fig. 14, shows a typical procedure used in this predictive process. In this approach, eight
points need to be identified, see Ref. [24]. Point A represents the 1st threshold corresponding to a crack length equal to the average grain size. Point B corresponds to the steady-state 2nd threshold (where the LEFM singularity is met). Point C identifies the end of the near threshold region at high stress levels (here there is large crack tip plasticity and the classical LEFM stress singularity is not met) with a crack length equal to the average grain size. Point D represents the end of Stage II crack growth at high stress levels (large crack tip plasticity and the LEFM singularity is not met).
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Point E is the 2nd end of the near threshold where singularity is met. Point F represents the end of steady-state threshold where the LEFM singularity is met. Point G is the beginning of Stage II crack growth with singularity at high stress levels. Point H is the end of the region where we have knowledge of the above points, see Fig. 14b. Knowing these points, the corresponding growth rates can now be extracted, as shown in Fig. 14b. This procedure delivers a locus of DK v’s da/dN which is a material property, see Ref. [24]. Latter on, it will be shown that this locus can encapsulate every possible loading condition. A typical example, and comparison, is shown in Fig. 15. Fig. 16 shows the loci for R = 0, 0.5, and 0.7. From here it can be seen that there exist a common area where the propagation rates of all three stress ratios can co-exist. In addition the near threshold area given by the closed locale A–B–D–E for R = 0, or A–B–E for R = 0.5, disappears at R = 0.7. Similar condition exists for the D6ac and Ti–6Al–4V.
Fig. 15a. The FDM locus for 4340 steel at R = 0 compared with the experimental results presented in Ref. [2]. The near threshold area is given by A–B–E–D.
Fig. 15b. The FDM locus for 4340 steel at R = 0.5 compared with the experimental results presented in Ref. [2]. The enclosed FDM shows the disappearance of near threshold conditions. The (near threshold) area A–D–B is calculated to disappear at R = 0.8.
Fig. 16. The 4340 crack growth loci produced via the FDM points for R = 0, 0.5, and 0.7. The experimental results from Fig. 3 are plotted in order to establish the Stage II independency with respect to R. It is worth noting that at R = 0.7 only points A, B, H, and F are maintained.
4. Explaining R independency in Stage II crack growth To identify the reasoning behind the propensity of the materials in question to exhibit a weak R ratio dependency during Stage II crack growth it is appropriate to contrast their behaviour with a material with a known strong R ratio dependency. To this end Fig. 17 presents a plot, similar to that presented in Fig. 16, for 2024-T351 aluminium alloy. Comparing Figs. 16 and 17 reveals the following: (a) 2024-T351 shows a distinct reduction of the range defining the 1st and 2nd threshold as the R ratio increases. In contrast 4340 steel shows that at R = 0.5 and R = 0.7 they both share approximately similar values; (b) 2024-T351 exhibits a well defined near threshold condition even at R = 0.7, whilst 4340 steel appears to immediately produce Stage II crack growth with an established LEFM stress singularity; and (c) for 4340 the area where the propagation rates of all three R ratios can coincide is much larger than that for 2024-T351, see Figs. 18(a) and (b), and 19. To further elaborate these findings and identify whether they hold any physical meaning it is necessary to establish the difference between a fine and coarse grain material. In previous section, it was mentioned that the transition from rmax to Dr dependency represents the cumulative, with loading cycles or crack length, increase of the dislocation density on the slip plane towards the end of it. It was also stated that as the stress ratio increases the above phenomenon will saturate faster. In a quite simplistic way it is rational to say that if the dislocation density along the crack tip plasticity is approximately the same, flow resistance is also uniform and the classical LEFM singularity can easily be attained. From
Fig. 17. The crack growth loci for R = 0, 0.5, and 0.7 for 2024-T351 sheet. The FDM inputs are given in Table 2.
R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10
which the dislocation density is seeking saturation, is inversely proportional to flow resistance. Hence, it is hypothesized that the selected high strength materials will require less cumulative time, or crack length, compared to the aluminium alloy. The reader can easily identify the difference by comparing the propagation rates p at DK = 10 MPa m. The difference between the 1st and 2nd threshold solely depend on the grain size. Here it is important to note two issues: (a) in general fine grain materials will emit a significantly smaller number of dislocations than their counterparts. Hence a singularity will be, by-default, achieved faster and (b) again by-default BCC and HCP fine grain materials will reach faster polycrystalline behaviour at threshold conditions than coarse grain FCC materials. Bear in mind that polycrystalline behaviour for BCC structures is found at a Schmidt factor of 2, while, for FCCs a Taylor factor of 3.07 is needed.
10-5
4340
da/dN (m/c)
10-6
10-7
Overlaping Area
10-8
10-9
10-10 1
10
100
Range of stress intensity factor ΔΚ (MPam1/2)
5. A non-dimensional approach to fatigue crack growth
10-5
It has recently been shown [25] that for centre cracks, and surface flaws, in a large panel, where the material follows the Generalised Frost–Dugdale law, subjected to repeated block loading such that there are a large number of blocks to failure then the crack length history can be approximated by
2024-T351
da/dN (m/c)
10-6
10-7
ðB Bi Þ=ðBf Bi Þ ¼ 1 lnða=af Þ= lnðai =af Þ; Overlaping Area
10-8
10-9
10-10 1
9
10
100
Range of stress intensity factor ΔΚ (MPam1/2) Fig. 18. (a) Overlapping area for 4340 steel and (b) for 2024-T351 aluminium alloy.
Fig. 19. Response of D6ac surface flaw specimens and centre cracked Mil Annealed Ti–6Al–4V specimens under representative flight load spectra.
the above, it is easily understood that under the same DK conditions the size of crack tip plasticity, and hence the size over
ð3Þ
where ai and af, which should generally be less than approximately 0.8 time the critical crack length acr, are the initial and final crack sizes, and Bi and Bf are the corresponding number of blocks, or flight numbers, and B is the number of blocks at crack length a. Here it should be noted that as the crack size approaches its critical length we need to account for Kmax approaching its fracture toughness, see Refs. [25,26]. However, this effect tends to be small and, as shown in Ref. [25] general only affects the region a/acr > 0.8. A similar deviation from Eq. (3) occurs as the crack length approaches the initial discontinuity size. In this case the discontinuity is influenced by the geometry of the starting notch/defect and as such is not (yet) acting as a crack of length a. To overcome this Molent et al. [27] suggested that a more accurate representation is obtained by using an equivalent pre-crack size (EPS) rather than the size of the initial discontinuity. As a result the slope of the (B Bi)/(Bf Bi) versus /(a) (=1 ln(a/af)/ln(ai/af)) curves sometimes differs slightly from that suggested by Eq. (3). Nevertheless, this approach predicts that, for tests on centre cracked panels under repeated block loading, for the majority of the fatigue life we should see a near linear relationship between the number of load blocks and the log of the crack length. In Section 2, we saw that crack growth in Mil Annealed Ti–6Al– 4V appears to conform to Generalised Frost–Dugdale law. It should thus follow that crack growth in centre cracked panels under repeated block loading should (approximately) conform to Eq. (3). To evaluate this prediction Fig. 19 presents the test results given in Ref. [15] for crack growth in a 152.4 mm wide and 7.34 mm thick centre cracked panel subjected to a fighter load spectra with a peak remote stress of 710 MPa. For comparison Fig. 19 also presents the results of a USAF [28] investigation, performed as part of the F-111 certification study, into cracking in surface flawed D6ac plate specimens under block loading, where one block represented 200 flight hours, representing the mission spectrum for the critical location in the F-111 wing pivot fitting. Fig. 19 presents the results for two specific specimens (P5I9 and P5I10) where both the initial and the final flaw shapes were very close to a semi-circular surface flaw. The spectrum used in specimen test P5I9 represented a tension–compression spectra whilst the spectrum used in specimen test P5I10 represented a tension–tension load spectra. The UASF test specimens were 406.4 mm long, 96.52 mm wide and
10
R. Jones et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 1–10
Table 3 Test configuration and crack lengths. Test
Crack geom
Material
Spectra
ai (mm)
af (mm)
acr (mm)
USAF test 1 USAF test 2
Surface flaw Ibid
D6ac D6ac
2.92 2.92
5.94 5.59
6.99 5.79
Grumman test
Centre cracked panel
Mil Annealed Ti– 6AL–4V
Representative flight spectra, tension loads only Representative flight spectra, tension and compression loads Representative fighter aircraft spectra
0.562
0.949
0.949
7.62 mm thick and contained a 2.921 mm semi-circular surface flaw. The 7.62 mm thickness was representative of the critical location in the Wing Pivot Fitting location, see Ref. [28]. The values of ai, af, acr, Bi, Bf and the number of blocks to failure in each tests are given in Table 3. From this figure it is seen that, as predicted, there was a near linear relationship between the log of the crack length and the number of blocks/cycles. However, it should be noted that differences in the slopes reflects the uncertainties in the initial flaw sizes. 6. Conclusion Mil Annealed Ti–6Al–4V and (STOA) Ti–6Al–4V both appear to exhibit a similar behaviour to that documented in Ref. [2] and shown in Ref. [3] in that the R ratio dependency in the Paris Region is relatively small. Given the extensive use of titanium in the F-22, the Super Hornet, and the Joint Strike Fighter this finding, and the fact that crack growth appears to conform to the Generalised Frost–Dugdale law, should be further investigated. Similarly the fanning in Region I needs further investigation to assess whether it is a real phenomena or merely indicative of the fact that similitude may not be applicable in Region I. The concept and consequent application of the Fatigue Damage Map revealed that the basic material properties incorporate specific tendencies towards the propensity of each material to exhibit different degrees of fatigue damage stages. Indeed, we have seen that this method can pinpoint the changes in the above stages in the DK versus da/dN relationship. The selection of the limit cases, both in terms of a maximum and a minimum propensity, creates a closed form locus. For the selected stress ratio’s, this locus is a unique material property. We shall name this area as the intrinsic fatigue damage locus (IFDL). A knowledge of the IFDL allows the designer to estimate the physical fatigue limits of the material, independently of the test configuration used. In this exercise, the IFDL was determined for several stress ratios and for each of the materials in question. The large size of the overlapping regions of the IFDL’s indicates the strong potential of the materials to exhibit R ratio independency. Comparison with the IFDL for 2024-T351 aluminium alloy illustrates the differences in the IFDL between materials with a strong R ratio dependency and those with a weak R ratio dependency. References [1] M. Skorupa, Load interaction effects during fatigue crack growth under variable amplitude loading – a literature review. Part II: qualitative interpretation, Fatigue Fract. Eng. Mater. Struct. 22 (1999) 905–926. [2] S.C. Forth, M.A. James, W.M. Johnston, J.C. Newman Jr., Anomolous fatigue crack growth phenomena in high-strength steel, in: Proceedings of the International Congress on Fracture, Italy, 2007.
Bi
Bf
Blocks to failure
42 80
69 108
72 111
253
306
306
[3] S.C. Forth, W.M. Johnston, B.R. Seshadri, The effect of the laboratory specimen on fatigue crack growth rate, in: Proceedings of the 16th European Conference on Fracture, Alexandropoulos, Greece, July 2006. [4] M.N. James, J.F. Knott, An assessment of crack closure and the extent of the short crack regime in QlN (HY80) steel, Fatigue Fract. Eng. Mater. Struct. 8 (2) (1985) 177–191. [5] J.J. Ruschau, J.C. Newman Jr., Improved test method for very low fatigue-crackgrowth-rate data, in: Proceedings of the 2008 American Helicopter Society (AHS), US, 2008. [6] B.L. Boyce, R.O. Ritchie, Effect of load ratio and maximum stress intensity on the fatigue threshold in Ti–6Al–4V, Eng. Fract. Mech. 68 (2001) 129–147. [7] E.K. Walker, The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7076-T6 aluminum, in: Effect of Environment and Complex Load History on Fatigue Life, ASTM STP 462, American Society for Testing and Materials, Philadelphia, 1970, pp. 1–14. [8] N.E. Dowling, Mechanical Behaviour of Materials, Prentice Hall Publishers, New Jersey, USA, 1993. [9] C.M. Hudson, Fatigue Crack Propagation in Several Titanium and One Superalloy Stainless-Steel Alloys, NASA TN D-2331, October 1964. [10] R. Jones, S.C. Forth, Cracking in D6ac steel, Theor. Appl. Fract. Mech., submitted for publication. [11] R. Jones, L. Molent, S. Pitt, Crack growth from small flaws, Int. J. Fatigue 29 (2007) 1658–1667. [12] R. Jones, B. Chen, S. Pitt, Similitude: cracking in steels, Theor. Appl. Fract. Mech. 48 (2) (2007) 161–168. [13] W. Zhuang, S. Barter, L. Molent, Flight by flight fatigue crack growth life assessment, Int. J. Fatigue 29 (9–11) (2007) 1647. [14] T.R. Porter, Method of analysis and prediction for variable amplitude fatigue crack growth, Eng. Fract. Mech. 4 (1972) 717–736. [15] P.D. Bell, M. Creager, Crack Growth Analysis for Arbitrary Spectrum Loading, Volume I – Results and Discussion, Final Report: June 1972–October 1974, Technical Report AFFDL-TR-74-129. [16] M.R. Bache, W.J. Evans, The fatigue crack propagation resistance of Ti–6Al–4V under aqueous saline environments, Int. J. Fatigue 23 (2001) S319–S323. [17] J. Ding, R. Hall, J. Byrne, Effects of stress ratio and temperature on fatigue crack growth in a Ti–6Al–4V alloy, Int. J. Fatigue 27 (2005) 1551–1558. [18] S.A. Curtis, J. Solis Romero, E.R. de los Rios, C.A. Rodopoulos, A. Levers, Predicting the interfaces between fatigue crack growth regimes in 7150-t651 aluminium alloy using the fatigue damage map, Mater. Sci. Eng. A A344 (2003) 79–85. [19] C.A. Rodopoulos, E.R. de los Rios, J.R. Yates, A fatigue damage map for the 2024T3 aluminium alloy, Fatigue Fract. Eng. Mater. Struct. 26 (7) (2003) 569–576. [20] C.A. Rodopoulos, J.-H. Choi, E.R. de los Rios, J.R. Yates, Stress ratio and the fatigue damage map – part I: modelling, Int. J. of Fatigue 26 (2004) 739–746. [21] C.A. Rodopoulos, J.-H. Choi, E.R. de los Rios, J.R. Yates, Stress ratio and the fatigue damage map – part II the 2024-t351 aluminium alloy, Int. J. Fatigue 26 (2004) 747–752. [22] C.A. Rodopoulos, G. Chliveros, Fatigue damage in polycrystals – part 1: the numbers two and three, Theor. Appl. Fract. Mech. 49 (1) (2008) 61–67. [23] C.A. Rodopoulos, G. Chliveros, Fatigue damage in polycrystals – part 2: intrinsic scatter of fatigue life, Theor. Appl. Fract. Mech. 49 (1) (2008) 77–97. [24] C.A. Rodopoulos, Predicting the evolution of fatigue damage using the fatigue damage map method, Theor. Appl. Fract. Mech. 45 (2006) 252–265. [25] R. Jones, D. Peng, S. Pitt, Thoughts on the physical processes underpinning crack growth, Fatigue Fract. Eng. Mater. Struct. (2009) (invited paper). [26] R. Jones, S. Pitt, D. Peng, An equivalent block approach to crack growth, multiscale fatigue crack initiation and propagation of engineering materials: structural integrity and microstructural worthiness, in: G.C. Sih (Ed.), Springer press, 2008, ISBN: 978-1-4020-8519 (June). [27] L. Molent, Q. Sun, A.J. Green, Characterisation of equivalent initial flaw sizes in 7050 aluminium alloy, Fatigue Fract. Eng. Mater. Struct. 29 (2006) 916–937. 56. [28] H.A. Wood, T.L. Haglage, Crack Propagation Test Results for Variable Amplitude Spectrum Loading in Surface Flawed D6AC Steel 1971, Wright Patterson AFB, FBR-TM-71-2.