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Crack patching: Predicting fatigue crack growth
Theoretical and Applied Fracture Mechanics 45 (2006) 79–91 www.elsevier.com/locate/tafmec
Crack patching: Predicting fatigue crack growth R. Jones a
a,b,*
, K. Krishnapillai a, S. Pitt
c
DSTO Centre for Structural Mechanics, Department of Mechanical Engineering, Monash University, P.O. Box 31, Victoria 3800, Australia b Rail CRC, Department of Mechanical Engineering, Monash University, P.O. Box 31, Wellington Road, Clayton, Victoria 3800, Australia c Air Vehicles Division, Defence Science and Technology Organisation, 506 Lorimer Street, Fishermans Bend 3207, Australia
Abstract This paper examines the crack growth history of a range of test specimens, and cracks repaired with a composite patch, and shows that in the low to mid range DK region there is a near linear relationship between the log of the crack length and the number of cycles. A simple methodology for predicting crack growth is then presented and validated by comparison with a range of experimental studies. 2006 Published by Elsevier Ltd. PACS: 81.40.Np Keywords: Composite repairs; Fatigue; Crack growth modelling
1. Introduction In the 1950s, crack growth relation [1] with a simple log linear relationship was proposed lnðaÞ ¼ kN þ a0
ð1Þ
which can also be written as da=dN ¼ ka
ð2Þ
* Corresponding author. Address: Rail CRC, Department of Mechanical Engineering, Monash University, P.O. Box 31, Wellington Road, Clayton, Victoria 3800, Australia. Tel.: +61 3 990 53809; fax: +61 3 990 51825. E-mail address: [email protected] (R. Jones).
0167-8442/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.tafmec.2006.02.001
where N is the ‘‘fatigue life’’, k is a parameter that is geometry and load dependent, a0 is the initial flawlike size and k ¼ f ðrÞ
ð3Þ
For constant amplitude loading it was found [1,2] that k could be expressed as k ¼ bðDrÞ
3
ð4Þ
where b was problem dependent. Researchers at the Australian Defence Science and Technology Organisation (DSTO) have subsequently presented a wealth of experimental data [2–12] that has shown the applicability of this law for crack growth in the 7050 aluminium alloy, and also for steels, used in F/A-18 as well as for cracking in Macchi aircraft
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in service with the RAAF. Other researchers [13,14] have also commented upon the apparent exponential rate of crack growth at small crack lengths. Indeed, the review of [14] revealed that this approximation holds for a large cross section of problems with flaws that range in size from 50 lm to several mm’s. The work in [15] summarises a range of experimental studies on the fatigue performance of fuselage lap joints, revealed that multi-site damage in fuselage lap joints also conforms to the Frost–Dugdale crack growth law. This phenomenon can be seen in the results of the early Foster–Miller tests on panels representative of a Boeing civil transport aircraft performed at the Full-Scale Aircraft Structural Test Evaluation and Research (FASTER) facility at the Federal Aviation Administration (FAA) William J. Hughes Technical Center [16,17] as part of the FAA Aging Aircraft research program. The resultant crack growth data is summarized in [17], and some of the results are adapted and presented in Fig. 1, where we see that to a first approximation crack growth conforms to the Frost– Dugdale law. (The slope of the line varies as a result of the slightly different stress states at the various fastener locations.) Here we have adopted the terminology used in [17], which presents what is termed the fastest growing crack as well as crack growth from a number of other rivets. Note that in this figure the rivet radius (r) has been added to the crack depth (a). The explanation for this behaviour together with a predictive methodology that is based on Eq. (2) is presented in [17].
Representative tests on test coupons containing multi-site damage were also reported in [18] using the USAF wide panel test specimen geometry. The skin was a 1.6 mm thick 2024-T3 aluminium alloy with a 7075-T6 doubler and longeron, tested at 103.4 MPa with a bypass ratio = 0.66, bending ratio = 0.36 and a bearing stress ratio = 2.09. Crack growth data for cracks at ten different fasteners, with a radius (r) of 2.4 mm, are presented in Fig. 2 which is taken from the work in [15]. (Here the terminology used in [18] was as follows: 7A8L refers to cracking in row A fastener number 7A8 where L refers to the left tip, and R refers to the right tip.) Fig. 2 again shows see a near linear relationship between ln(a + r) and the number of cycles. The recent work in [19] revealed that the growth of cracks under composite repairs also conforms to the Frost–Dugdale law and that for medium crack lengths, up to say 10–20 mm, the effect of the patch is dominantly due to the reduction in the local stress field. It is important to note that [19] stressed that for composite repairs to long cracks, which for composite repairs to thin plates means crack lengths greater than approximately 40 mm’s, the crack growth rate can be described using a Paris like crack growth law. These findings together with the realisation that ‘‘in the threshold regime, there is something missing either in the (crack closure) model’’, see [20], led to the conjecture [14,19,21] that in the low DK region, i.e. Region I, the crack growth rate can be expressed in the form
Crack Length a + r (mm)
100
Fastest Rivet 71 Rivet 52 Rivet 31 Rivet 12
10
1 3
4
5
6
7
8
Number of cycles x 104
Fig. 1. Crack growth data in the FAA curved panel tests from [16,17].
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81
10.0
Crack length a + r (mm)
7A8L 7A16R 7A7R 7A17L 7A13R 7A12R 7A6L 7A14R 7A15R 7A6R 7A17R 7A7L 7A14L 7A16L
1.0 0
0.2
0.4
0.6
0.8
1
Number of cycles x 10
1.2
1.4
1.6
5
Fig. 2. Measured crack growth rates, adapted from [15,18].
ð1m=2Þ
da=dN ¼ CðaÞ
ðDK eff Þ
m
ð5Þ
where C, and m are constants, and DKeff is an effective stress intensity factor. This relationship follows the form proposed in [22,23] and contains the Frost–Dugdale law as a special case, viz, m = 3. The ability of this variant of the ‘‘fractal/incomplete self-similarity’’ growth law to model cracks growing under constant amplitude loading was studied in [21]. As outlined above it is well know that Paris like laws and closure based crack growth models cannot predict the growth of flaws from near micron sized flaws. However, the works in [24] have recently shown how fractal based laws overcome this deficiency and presented an illustrative example in which crack growth from a near micron sized corrosion pit in a full scale test articles was accurately predicted in [24] also used this approach to predict the entire crack growth history in the 1969 General Dynamics, now Lockheed, F-111 wing test. Indeed, [24] also shows that when using a fractal variant of the unified crack growth model, as developed in [25,26] c ð1bÞ ð2cÞ=2c da=dN ¼ C DK btot ðK max;tot Þ a ð6Þ the entire crack growth history can be predicted for constant amplitude tests on a (wide) range of aluminium and titanium alloys tested under a range of R values. Here C, b, and c are material constants. In the present paper we continue this development by showing that for composite repairs to small cracks, which for repairs to thin plates is typically
less than 25 mm, crack growth is governed by Eq. (5) rather than a Paris like crack growth law. 2. Crack growth in composite repairs to cracked metal structures Externally bonded composite patches have proved to be an effective method of repairing cracked, or damaged, structural components, see Table 1 from [26], and a typical composite repair to a rib stiffened panel is shown in Fig. 3. Composite repairs act in two fashions: (1) They reduce the net section stresses in the (cracked) structure. (2) The fibres bridging the crack restrict the opening of the crack faces. It is widely accepted that for small cracks the reduction in the net section stresses is the primary mechanism reducing the stress intensity factor, and thereby crack growth. There are very few experimental studies where the stress intensity factor under a patched crack has been directly measured. Most work has inferred the stress intensity factor from crack growth studies. To the best of the authors knowledge the work in [27], which is summarised in [28] Section 6.3.1 on pp. 126–128, is the only study that has directly measured the stress intensity factor and related it to the stress in the uncracked specimen. In this work X-ray back reflection [27] is used to determine the stress intensity
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Table 1 Australian bonded patch repairs, from [28] Aircraft
Problem
Remarks on bond durability
C130
Stress corrosion cracked stiffeners in wing, aluminium alloy 7075 Fatigue cracking in lower wing skin, AU4SG aluminium alloy
Over 20 years of service. No bond durability problems where bonding carried out as specified 180 wings repaired/reinforced. Eight bond durability problems over 8 years. Failures were associated with adhesive voiding caused by extreme humidity at the repair station in Malaysia No bond durability failures to steel, or aluminium wing skin, surface over 10 years
Mirage III
F-111-C
F-111-C F-111-C F-111-C
F-111-C C-141 (USAF) P-3C Boeing 747
Sea King helicopter F-111-C
Cracking in stiffeer runout #2 on upper surface of the D6ac steel wing pivot fittings Stress corrosion cracking in weapon bay longeron flange, aluminium alloy 7075-T6 Stress corrosion cracking in longeron adjacent to refuel receptacle, 7049-T6 Metal-to-metal and sandwich structure repairs. RAAF adopted GB+S and changed to FM 300 adhesive in 1992 Pork-chop panel (lower fuselage). Panels rebuilt after repeated in-service failures Fatigue cracking in wing riser weep holes, 7075-T6 Full depth corrosion damage in horizontal tail, aluminium alloy 7075-T6 Simulated repairs to several regions including fuselage lap-joint, wing leading edge, trailing edge flap and engine thrust reverser cowl Fatigue crack in frame Fatigue cracking in lower wing skin at fuel flow hole under forward auxiliary spar
Over 10 aircraft repaired with no bond durability problems in 8 years Over ten aircraft repaired. No bond durability problems in 8 years No bond durability failures in over 7 years
Repeat rebuild rate reduced from 95% to zero. No bond durability failures in 7 years No bond durability problems around 5 years No bond durability problems after >10 years of service FAA Aging Aircraft demonstrator repairs; 37,000 flying hours, 7020 landings over an 11 year period with no significant bond durability problems Operated in am offshore ship-borne environment for 4 years with no problems No bond durability problems in over 2.5 years service
Fig. 3. View of a boron epoxy repair to a stiffened panel showing both the patched surface and underside of the test configuration. Note that in this case the stiffener is also cracked.
R. Jones et al. / Theoretical and Applied Fracture Mechanics 45 (2006) 79–91
head in an F/A-18 centre barrel (CB) test undertaken as part of the Australian Defence Science and Technology Organisation (DSTO) Flaw IdeNtification through the Application of Loads (FINAL) program utilising ex-service Canadian Forces (CF) and US Navy (USN) CBs loaded using a modified mini-FALSTAFF spectrum, see [34]. In this case one of the cracks grew from a small 3.2 lm corrosion pit. In each case the trend lines reveal that da/dB is proportional to ap, where the power p varies between 0.88 and 1.03, and as such the experimental results are in good agreement with the predicted behaviour (i.e. da/dB / a) given the likely experimental error in these measurements. It will be shown that this crack growth law also holds for cracks repaired with a composite repair.
factors in a 1.5 mm thick 7075-T6 aluminium alloy specimens patched on both surfaces with a 0.49 mm thick graphite epoxy laminate. Both uncracked and centre cracked, with total crack lengths (2a) of 10, 30 and 40 mm, specimens were examined. The uncracked specimens were used to determine the (residual) stress in the aluminium after patching As a result of this study it was found [27,28] that in this case the experimentally determined stress intensity factor for a patched plate where the stress field in the skin under the patch is rT is given by p K ¼ rT ðpaÞ
ð7Þ
and that crack bridging effects were negligible. The recent review paper [19] revealed that (1) For composite repairs to through cracks in thin sheets the growth of small to medium length cracks, that have low to mid range DK’s, follows the law proposed by Frost and Dugdale [1,2]. (2) Whenever (1) is valid and bending effects are negligible then the effect of the patch is primarily due to the reduction of the net section stress.
2.1. Experimental crack growth rates This section presents the results of a range of fatigue tests to illustrate that a (near) linear relationship between the crack growth rate and crack length is also seen for crack growth under composite repairs. Case 1: Presented in [35] is the results for a 5 mm crack in a 160 mm wide and 3.14 mm thick 2024-T3 aluminium alloy specimen patched with a seven ply (0.889 mm thick) semicircular uni-directional composite patch with a radius of 80 mm. The specimen was subjected to constant amplitude fatigue testing with rmax = 138 MPa and R = 0.1, see [35], Fig. 6.23 p. 155. Bending, due to neutral axis offset effects was
It has been shown [1,2,29–32] that the crack growth rate was proportional to the crack length, and that this relationship holds for cracks that have length scales ranging from 50 lm to several mm’s. This relationship [33] also holds for crack growth under complex load histories. This is illustrated in Fig. 4, where we plot the crack growth rate per block against the crack depth for cracking in a bulk-
Crack growth rate da/dB (mm/block)
83
0.1 y = 0.106 x y = 0.056 x
0.96
0.88
0.01
CB1 Y488 C4
0.001
CB1 Y488 C3
y = 0.091 x 0.0001 0.001
CB1 Y488 C1
1.03
0.01
CB1 Y488 C2
0.1
1
Crack length a (mm)
Fig. 4. Growth rate per block versus a in the FINAL fatigue test program, from [34].
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eliminated by testing two specimens joined back to back. The resultant crack growth rate data, from [35], are shown in Fig. 5. This figure supports the hypothesis of a linear relationship between the crack growth rate da/dN and crack length (a). Case 2: Raizenne et al. [36] presented crack growth data for spectrum loading for the same test configuration outlined in Case 1. The results of this study are summarised in Fig. 6, where we again see that the crack growth rate per block da/dB is essentially a linear function of crack length. Case 3: Compared in [37] is the efficiency of glare and boron epoxy patches, and the results are shown in Fig. 7. In this case tests were
performed on centre cracked panels and only one side of the panel was patched. Here we again see that the crack growth rate da/dN is essentially a linear function of a. Case 4: Studied in [38] is the behaviour of both boron epoxy and glare patches to a centre cracked 7075-T6 aluminium panel. The panel thicknesses studied were 1.6, 2.5, and 4.0 mm. The panels were 180 mm wide and 600 mm long, and contained a 25 mm long initial fatigue crack. Single sided patches 75 mm wide and 190 mm long were applied to all panels. The panels were tested under constant amplitude loading with rmax = 120 MPa and rmin = 6 MPa.
0.0001
y = 9.0210-6 x1.05
0.00001 1
10
100
Crack length a (mm)
Fig. 5. Crack growth rate versus crack length, from Baker and Jones [35].
1
Crack growth rate da/dB (mm/block)
Crack growth rate da/dn (mm/cycle)
0.001
y = 0.012 x
1.10
0.1
0.01 1.00
10.00
100.00
Crack length a (mm)
Fig. 6. Crack growth rate versus crack depth, from Raizenne et al. [36].
R. Jones et al. / Theoretical and Applied Fracture Mechanics 45 (2006) 79–91
85
Crack growth rate da/dN (mm/cycle)
0.0001 Boron Glare -7
1.02
y = 1.2010 x 0.00001 y = 5.18 10-7 x0.98
0.000001 10
100
Crack length a (mm)
Crack growth rate da/dn (mm/cycle)
Fig. 7. Crack growth rate (da/dN) versus crack length, from Fredell and Guijt [37].
0.01 Glare Boron epoxy spec b1 Boron epoxy spec b1
0.001
-5
-5
0.99
y = 3.6010 x
1.00
y = 2.81 x 10 x
-5
1.02
y = 1.9910 x
0.0001 10
100
Crack length a (mm)
Fig. 8. Crack growth rate versus crack length for a centre cracked 4 mm thick 7075 aluminium plate with both glare and boron epoxy patches, from Gujit et al. [38].
The results for the 4 mm thick panels repaired using both boron epoxy and glare patches are presented in Fig. 8. For the 4 mm plate the boron epoxy patch and the glare patches had stiffness ratios of 1.22 and 1.13 respectively. Here the stiffness ratio was defined as Epatchtpatch/ Eplatetplate. This example again shows that da/dN is essentially a linear function of a.
3. Predicting crack growth Having established that the crack growth rate is often a linear function of the crack length let us evaluate the conjecture presented in [14,19,21] that in the low to mid range DK region, i.e. Region I, the crack growth rate is not governed by a Paris like equation but rather by an expression of the form
ð1m=2Þ
da=dN ¼ CðaÞ
ðDKÞ
m
ð8Þ
where C, and m are constants. The ability of this variant of the ‘‘fractal/incomplete self-similarity’’ growth law to model cracks growing under constant amplitude loading was studied in [21]. In this section we investigate its applicability to predict crack growth under composite repairs to 2024-T3 and 7075-T6 aluminium alloys. Here we will assume that, as shown in [19], when bending effects are negligible then the effect of the patch on the stress intensity factor is primarily due to the reduction of the net section stress and that crack growth is governed by Eq. (8). It should be noted that the works in [39–41] have shown that for composite repairs to edge cracks the effect of the repair is to lower the free surface correction factor from 1.12 to approximately 1, i.e. the stress intensity factor can be approximated as
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p K ¼ rp ðpaÞ
ð9Þ
To validate this value of C we considered the crack growth results presented in [43], this work was performed as the FAA Aging Aircraft program. The work in [43] also presented the crack growth history for a 12.7 mm edge crack in a 1.8 mm thick and 100 mm wide 2024-T3 aluminium alloy specimen subjected to constant amplitude fatigue tests with Drmax = 143 MPa and R = 0.18. The predicted, using m = 3 and C = 7.1 · 1012, and measured crack growth histories are presented in Fig. 10, and are again in good agreement. Having determined the values of C and m predict the growth of cracks in 2024-T3 aluminium specimens repaired using a bonded composite repair. Repair Case (1) In this example, presented in Fig. 13.7 on p. 385 [44], Baker presented the test results for a 10 mm crack in 160 mm wide and 3.14 mm thick 2024-T3 aluminium alloy specimen patched with a seven ply (0.889 mm thick) semi-circular uni-directional composite patch with a radius of 80 mm. The specimen was subjected to constant
where rp is the stress under the patch in the uncracked configuration. 3.1. Crack growth in repaired 2024-T3 aluminium alloy specimens To determine the constants m and C for 2024-T3 aluminium we considered the centre cracked panel results presented by Zuo et al. [42]. As in [1,2], and more recently the work in [14] has set m 3 for aluminium alloys. The specimen tested in [42] was 152 mm wide and 2.54 mm thick and was tested run under constant load amplitude conditions with a maximum stress of 60.45 MPa, and R = 0.2. The experimental results and the predicted crack growth history for m = 3 and C = 7.1 · 1012 are presented in Fig. 9. Here a ispin mm, da/dN is in mm/cycle, and DK is in MPa mm. From this it is seen that the predicted and measured crack growth histories are in good agreement.
100
Crack length a (mm)
y = 8.42 e0.0447x
10 Centre crack Zou et al. Predicted
1 0
5
10
15
20
Number of cycles x 104
Fig. 9. Experimental and predicted crack growth history for a centre cracked panel, from [42].
Crack length a (mm)
100
Roach Unpatched Predicted
10 0
1
2
3
4
5
6
7
8
Number of cycles x 1000
Fig. 10. Experimental and predicted crack growth history for an edge cracked panel, adapted from [43].
R. Jones et al. / Theoretical and Applied Fracture Mechanics 45 (2006) 79–91
amplitude loading with a Dr of 76 MPa with the R values increasing from 0.1 to 0.55, 0.64 and finally 0.75 as the test continued. The resultant crack growth data is plotted in Fig. 11, where it can be seen that as postulated in [44] crack growth was relatively insensitive to the R ratio and that the data was essentially log linear. A good agreement is obtained between the measured and predicted crack growth history. Fatigue tests with R = 0.1 and various rmax levels were also reported in [44], see Fig. 13.6 on p. 383. The measured crack growth data for the cases when rmax = 160 and 80 MPa, and there were no inbuilt disbonds, and the predicted crack growth data predicted are shown in Fig. 12. Here again it is seen that the predicted and measured data are in good agreement and that the data is essentially log linear.
87
Presented in [44] are the results for a seven ply repair, with dimensions as per Case (1) presented above, at various temperatures 20, 60 and 80 C, see Fig. 13.8 on p. 386, subjected to constant amplitude fatigue testing with rmax = 138 MPa and R = 0.1. He concluded that the crack length (a) versus cycles relationship was essentially the same over this temperature range and the data was inconsistent with the current crack growth models. The measured crack growth history is shown in Fig. 13 together with the crack growth data for the associated room temperature test presented in Case (1) and the predicted crack growth history. It is again seen that good agreement is obtained between the predicted and the measured crack growth data.
Crack length a (mm)
100
Experimental data Predicted
10 1
2
3
4
5
6
7
8
9
10
Number of cycles x 105
Fig. 11. a versus number of cycles for the edge cracked 2024-T3 aluminium plate, repaired with a seven ply boron epoxy patch, under variable R ratio from Baker [44].
100
Baker 160 MPa
Crack length a (mm)
Baker 80 MPa Predicted 160 MPa Predicted 80 MPa
10 1
11
21
31
41
51
61
71
81
91
Number of cycles x 104
Fig. 12. Crack length versus number of cycles for the edge cracked 2024-T3 aluminium plate, repaired with a seven ply boron epoxy patch, with rmax equal to 160 and 80 MPa, from Baker [44].
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R. Jones et al. / Theoretical and Applied Fracture Mechanics 45 (2006) 79–91 100
Crack length a (mm)
Temperature effects Predicted Baker room temp test
10 0
2
4
6
8
10
12
Number of cycles x 104
Fig. 13. Effect of variable temperature on crack growth for the edge cracked 2024-T3 aluminium plate, repaired with a seven ply boron epoxy patch, with rmax equal to 138 MPa, from Baker [44].
3.2. Crack growth in repaired 7075-T6 aluminium alloy specimens Consider crack growth in 7075-T6 aluminium alloy specimens repaired with an externally bonded composite patch. Before we can attempt to predict crack growth we need the constants C and m in the crack growth model. Used the crack growth data in [38] for a centre cracked 7075-T6 aluminium panel, see Fig. 14, to determine these constants. The panels were 180 mm wide, 600 mm long and 1.6 mm thick, and contained a 25 mm initial fatigue crack. They were tested under constant amplitude loading with rmax = 120 MPa and rmin = 6 MPa. The exper-
imental and predicted crack growth histories, using m = 3 and C = 3.22 · 1011, are shown in Fig. 14 where it can be seen that the agreement is quite good. Presented in [39–41] are the results of a USAF study into the composite repair of rib stiffened panels. Two plate thicknesses, 1.6 and 4.2 mm, with two initial crack sizes, 20 and 40 mm tip to tip, were tested. The patch thickness for the 1.6 mm and the 4.2 mm thick skin was 0.508 mm and 1.52 mm respectively. The loads applied to the specimens were chosen to give a peak stress in the skin, when the stiffener was intact, of 120 MPa, see [39–41]. The specimens were fatigued under constant ampli-
100 Experimental
Crack length a (mm)
Predicted
10 0
1
2
3
4
5
Number of cycles x 103
Fig. 14. Experimental and predicted crack growth histories for a centre cracked 1.6 mm thick 7075-T6 aluminium plate, from Guijt et al. [38].
R. Jones et al. / Theoretical and Applied Fracture Mechanics 45 (2006) 79–91
100
89
Crack length a (mm)
Specimen 9, 40 mm crack Specimen 10, 20 mm crack Specimen 15, 20 mm crack Predicted
y = 11.4 e 0.277x
y = 19.7 e 0.246x
y = 10.4 e 0.3148x
10 0
1
2
3
4
5
Number of cycles x 104
Fig. 15. Crack length versus number of cycles for a centre cracked 4.2 mm thick rib stiffened 7075 aluminium plate repaired with a boron epoxy patches, from Jones et al. [39].
Crack length (mm)
100
Specimen 2 Specimen 4 Predicted
10 0
1
2
3
4
Number of cycles x 104
Fig. 16. Crack length versus number of cycles for a centre cracked 1.25 mm thick rib stiffened 7075 aluminium plate repaired with a boron epoxy patches, from Jones et al. [39].
tude loading in a 500 kN servo-hydraulic MTS fatigue testing machine with rmax = 120 MPa and rmin = 6 MPa, i.e. R ratio R = rmin/rmax = 0.05. The results of this study are shown in Figs. 15 and 16. Refs. [39–41] reported that the effect of the stiffener was negligible and as such the stress intensity factor can be calculated using Eq. (9). From these figures we see that there is a near linear relationship between ln(a) and the number of cycles and that there is good agreement between the measured and predicted crack growth histories. At this stage it should be noted that Jones et al. [19] stressed that for composite repairs to long cracks, which for composite repairs to thin plates means crack lengths greater than approximately
40 mm’s, the stress intensity factor becomes constant. Furthermore, for mid range DK’s, which for aluminium alloys correspond p to values greater than approximately 15–20 MPa m, the crack growth rate can be described using a Paris like crack growth law. This can be seen in the results presented in [39– 41] for a 1.6 mm thick skin with a broken stiffener, see Fig. 17, where a near linear relationship prevails for the crack length versus the number of cycles relation. However, this paper has shown that for composite repairs to small cracks, which for repairs to thin plates are typically less than 25 mm, where DK is relatively small crack growth can be approximated using the Frost–Dugdale like growth law proposed in [19], i.e. Eq. (8).
90
R. Jones et al. / Theoretical and Applied Fracture Mechanics 45 (2006) 79–91 90
Crack length a (mm)
80 70 60 50 Specimen 3 Specimen 6
40 30 0
1
2
3
4
5
6
7
8
9
Number of cycles x 103
Fig. 17. Crack growth for a 1.25 mm thick skin specimen with a broken stiffener, specimens 6 and 3.
4. Conclusion A number of researchers have proposed crack growth laws whereby the crack growth rate was proportional to the crack length. It has been shown that such a relationship also holds for composite repairs to cracked metallic structures. Indeed, for composite repairs to small cracks, where DK is small, crack growth follows the fractal growth law proposed in [19], and not a Paris type law. References [1] N.E. Frost, D.S. Dugdale, The propagation of fatigue cracks in test specimens, Journal Mechanics and Physics of Solids 6 (1958) 92–110. [2] N.E. Frost, K.J. Marsh, L.P. Pook, Metal Fatigue, Clarendon Press, Oxford, 1974. [3] S. Barter, L. Molent, N. Landry, P. Klose, P. White, Overview of the F/A-18 aft fuselage combined manoeuvre and dynamic buffet fatigue and residual strength testing, in: Proceedings of International Aerospace Conference, Brisbane, Australia, 2004. [4] R.A. Pell, P.J. Mazeika, L. Molent, The comparison of complex load sequences tested at several stress levels by fractographic examination, in: Proceedings of International Committee on Aeronautical Fatigue, 17th Symposium, Stockholm, Sweden 9–11 June 1993, Engineering Failure Analysis, submitted for publication. [5] R.A. Pell, L. Molent, A.J. Green, The fractographical comparison of F/A-18 aluminium alloy 7050-T7451 bulkhead representative coupons tested under two fatigue load spectra at several stress levels, DSTO-TR-1547, Melbourne, Australia, February 2004. [6] N. Goldsmith, Fractographic examination of cracking in hole No. 2 of Mirage wing A3094, DSTO, Letter N86/ 79NTG, February 1981.
[7] S.A. Barter, Fatigue crack growth in 7050T7451 aluminium alloy thick section plate with a surface condition simulating some regions of F/A-18 structure, Defence Science and Technology Organisation, DSTO-TR-1458, Melbourne, Australia, July 2003. [8] G. Clark, Effects of cracks identified in Mirage wing 40LH on wing structural integrity, DSTO-TR-0771, 1998. [9] S.A Barter, B. Bishop, G. Clark, Defect assessment on F/A18 488 bulkhead tested at ARL, Defence Science and Technology Organisation, Aircraft Material Report, ARLMAT-TM-125, Melbourne, Australia, 1991. [10] P. White, S. Barter, L. Molent, Probabilistic fracture prediction based on aircraft specific fatigue test data, in: Proceedings of the 6th Joint FAA/DoD/NASA Aging Aircraft Conference, San Diego, September 16–19, 2002. [11] S.A. Barter, G. Clark, N.T. Goldsmith, Influence of initial defect conditions on structural fatigue in RAAF aircraft, in: Proceedings of International Committee on Aeronautical Fatigue, 17th Symposium, Stockholm, Sweden, 9–11 June 1993. [12] S.A. Barter, P.K. Sharp, G. Clark, The identification of fatigue-critical regions during fatigue testing of Macchi MB326H centre section lower booms, in: Proceedings of The Ninth International Conference on Fracture, Sydney, April 1997. [13] D.Y. Wang, An investigation of initial fatigue quality, in: Design of Fatigue and Fracture Resistant Structures, in: P.R. Abelkis, C.M. Hudson (Eds.), ASTM STP, vol. 761, American Society For Testing and Materials, 1982, pp. 191– 211. [14] S. Barter, L. Molent, N. Goldsmith, R. Jones, An experimental evaluation of fatigue crack growth, Journal Engineering Fracture Mechanics 12 (1) (2005) 99–128. [15] R. Jones, L. Molent, U.H. Tiong, K. Krishnapillai, Understanding crack growth in fuselage lap joints, Journal of Theoretical and Applied Fracture Mechanics, submitted for publication. [16] G. Samavedam, D. Hoadley, J. Davin, Test Facility for evaluation of structural integrity of stiffened and jointed aircraft curved panels, in: S.N. Atluri, S.G. Sampath, P.
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Tong (Eds.), Structural Integrity of Ageing Airplanes, Springer, Berlin, 2002, pp. 321–337. D.Y. Jeong, P. Tong, Onset of multiple site damage and widespread fatigue damage in aging airplanes, International Journal of Fracture 85 (1995) 185–200. S.A. Fawaz, Equivalent initial flaw size testing and analysis of transport aircraft skin splices, Fatigue Fracture Engineering Materials and Structures 26 (2003) 279–290. R. Jones, S.A. Barter, L. Molent, S. Pitt, Crack patching: an experimental evaluation of fatigue crack growth, Journal of Composite Structures 67/2 (2004) 229–238. J.C. Newman Jr., A. Brot, C. Matias, Crack-growth calculations in 7075-T7351 aluminum alloy under various load spectra using an improved crack-closure model, Engineering Fracture Mechanics 71 (2004) 2347–2363. R. Jones, S. Barter, L. Molent, S. Pitt, Crack growth at low K’s and the Frost–Dugdale law Special Issue in Honour of Professor G.C. Sih, Journal of Chinese Institute of Engineers 27 (6) (2004) 869–875. An. Carpinteri, An. Spagnoli, A fractal analysis of size effect on fatigue crack growth, International Journal of Fatigue 26 (2004) 125–133. An. Spagnoli, Self-similarity and fractals in the Paris range of fatigue crack growth, Mechanics of Materials 37 (2005) 519–529. R. Jones, L. Molent, S. Pitt, E. Siores, Recent developments in fatigue crack growth, in: E.E. Gdoutos (Ed.), Proceedings of the 16th European Conference on Fracture (ECF16), Alexandropoulos, Greece, July 2006. S. Dinda, D. Kujawski, Correlation and prediction of fatigue crack growth for different R-ratios using Kmax and DK· parameters, Engineering Fracture Mechanics 71 (2004) 1779–1790. G. Glinka, D. Kujawski, T. Tsakalakos, M. Croft, R. Holtz, K. Sadananda, Analysis of fatigue crack growth using two driving force parameters, in: Proceedings of The International Conference on Fatigue Damage of Structural Materials V, September 19–24, 2004, Hyannis, MA, USA. A.A. Baker, G.A. Hawkes, E.J. Lumley, Proceedings 2nd International Conference on Composite Materials, Metallurigcal Society of AIME, Toronto, 1978. A.A. Baker, Crack patching: experimental studies, practical applications, in: A. Baker, R. Jones (Eds.), Bonded Repair of Aircraft Structure, Martinus Nijhoff Publishers, The Hague, 1988, pp. 107–173 (Chapter 6). H. Nisitani, M. Goto, N. Kawagoishi, A small-crack growth law and its related phenomena, Engineering Fracture Mechanics 41 (4) (1992) 499–513. N. Kawagoishi, Q. Chen, H. Nisitani, Significance of the small crack growth law and its practical application, Metallurgical and Materials Transactions A 31A (2000) 2005–2013. M.J. Caton, J.W. Jones, J.M. Boileau, J.E. Allison, The effect of solidification rate on the growth of small fatigue
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cracks in a Cast 319-Type aluminium alloy, Metallurgical and Materials Transactions A 30A (1999) 3055–3068. Y. Murakamia, K.J. Miller, What is fatigue damage? A view point from the observation of low cycle fatigue process, International Journal of Fatigue (2005) 1–15. L. Molent, R. Jones, S. Barter, S. Pitt, Recent developments in fatigue crack growth assessment, International Journal of Fatigue, in press. L. Molent, B. Dixon, S. Barter, Flaw identification through the application of loading (FINAL), in: Proceedings Structural Integrity and Fracture Conference, Brisbane, Australia, September 2004. A. Baker, R. Jones, Bonded Repair of Aircraft Structure, Martinus Nijhoff Publishers, The Hague, 1988. M.D. Raizenne, J.B.R. Heath, T. Benak, 1988 TTCP PTP-4 Collaborative test program—variable amplitude loading of thin metallic materials repaired with composite patches, Laboratory Technical Report, LTR-ST-1662, National Aeronautical Establishment, Ottawa, Canada. R. Fredell, C. Guijt, GLARE patching efficiency studies, in: A. Baker, L.R.F. Rose, R. Jones (Eds.), Advances in the Bonded Composite Repair of Metallic Aircraft Structure, Elsevier Applied Science Publishers, 2002, ISBN 0-08042699-9 (Chapter 14). C.B. Guijt, S. Verhoeven, J.M. Greer, R.M. van Galen, A new approach to manipulate thermal stresses in Bonded repairs, in: Proceedings 6th Joint FAA/DoD/NASA Conference on Aging Aircraft September 16–19, 2002, San Francisco, USA. R. Jones, B. Whittingham, I.H. Marshall, Bonded repairs to rib stiffened wing skins, Journal of Composite Structures 57 (2002) 453–458. T. Ting, R. Jones, W.K. Chiu, I.H. Marshall, J. Greer, Composite repairs to rib stiffened panels, Journal of Composite Structures 47 (1999) 737–744. R. Jones, Numerical analysis and design, in: A. Baker, L.R.F. Rose, R. Jones (Eds.), Advances in the Bonded Composite Repair of Metallic Aircraft Structure, Elsevier Applied Science Publishers, 2002, ISBN 0-08-042699-9 (Chapter 9). J.Z. Zuo, Al.Th. Kermanidis, Sp.G. Pantelakis, Strain energy density prediction of fatigue crack growth from hole of aging aircraft structures, Theoretical and Applied Fracture Mechanics 38 (2002) 37–51. D. Roach, Damage tolerance assessment of bonded composite doubler repairs for commercial aircraft applications, in: A. Baker, L.R.F. Rose, R. Jones (Eds.), Advances in the Bonded Composite Repair of Metallic Aircraft Structure, Elsevier Applied Science Publishers, 2002, ISBN 0-08042699-9 (Chapter 17). A.A. Baker, Boron epoxy patching efficiency studies, in: A. Baker, L.R.F. Rose, R. Jones (Eds.), Advances in the Bonded Composite Repair of Metallic Aircraft Structure, Elsevier Applied Science Publishers, 2002, ISBN 0-08042699-9 (Chapter 13).
Crack patching: Predicting fatigue crack growth R. Jones a
a,b,*
, K. Krishnapillai a, S. Pitt
c
DSTO Centre for Structural Mechanics, Department of Mechanical Engineering, Monash University, P.O. Box 31, Victoria 3800, Australia b Rail CRC, Department of Mechanical Engineering, Monash University, P.O. Box 31, Wellington Road, Clayton, Victoria 3800, Australia c Air Vehicles Division, Defence Science and Technology Organisation, 506 Lorimer Street, Fishermans Bend 3207, Australia
Abstract This paper examines the crack growth history of a range of test specimens, and cracks repaired with a composite patch, and shows that in the low to mid range DK region there is a near linear relationship between the log of the crack length and the number of cycles. A simple methodology for predicting crack growth is then presented and validated by comparison with a range of experimental studies. 2006 Published by Elsevier Ltd. PACS: 81.40.Np Keywords: Composite repairs; Fatigue; Crack growth modelling
1. Introduction In the 1950s, crack growth relation [1] with a simple log linear relationship was proposed lnðaÞ ¼ kN þ a0
ð1Þ
which can also be written as da=dN ¼ ka
ð2Þ
* Corresponding author. Address: Rail CRC, Department of Mechanical Engineering, Monash University, P.O. Box 31, Wellington Road, Clayton, Victoria 3800, Australia. Tel.: +61 3 990 53809; fax: +61 3 990 51825. E-mail address: [email protected] (R. Jones).
0167-8442/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.tafmec.2006.02.001
where N is the ‘‘fatigue life’’, k is a parameter that is geometry and load dependent, a0 is the initial flawlike size and k ¼ f ðrÞ
ð3Þ
For constant amplitude loading it was found [1,2] that k could be expressed as k ¼ bðDrÞ
3
ð4Þ
where b was problem dependent. Researchers at the Australian Defence Science and Technology Organisation (DSTO) have subsequently presented a wealth of experimental data [2–12] that has shown the applicability of this law for crack growth in the 7050 aluminium alloy, and also for steels, used in F/A-18 as well as for cracking in Macchi aircraft
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in service with the RAAF. Other researchers [13,14] have also commented upon the apparent exponential rate of crack growth at small crack lengths. Indeed, the review of [14] revealed that this approximation holds for a large cross section of problems with flaws that range in size from 50 lm to several mm’s. The work in [15] summarises a range of experimental studies on the fatigue performance of fuselage lap joints, revealed that multi-site damage in fuselage lap joints also conforms to the Frost–Dugdale crack growth law. This phenomenon can be seen in the results of the early Foster–Miller tests on panels representative of a Boeing civil transport aircraft performed at the Full-Scale Aircraft Structural Test Evaluation and Research (FASTER) facility at the Federal Aviation Administration (FAA) William J. Hughes Technical Center [16,17] as part of the FAA Aging Aircraft research program. The resultant crack growth data is summarized in [17], and some of the results are adapted and presented in Fig. 1, where we see that to a first approximation crack growth conforms to the Frost– Dugdale law. (The slope of the line varies as a result of the slightly different stress states at the various fastener locations.) Here we have adopted the terminology used in [17], which presents what is termed the fastest growing crack as well as crack growth from a number of other rivets. Note that in this figure the rivet radius (r) has been added to the crack depth (a). The explanation for this behaviour together with a predictive methodology that is based on Eq. (2) is presented in [17].
Representative tests on test coupons containing multi-site damage were also reported in [18] using the USAF wide panel test specimen geometry. The skin was a 1.6 mm thick 2024-T3 aluminium alloy with a 7075-T6 doubler and longeron, tested at 103.4 MPa with a bypass ratio = 0.66, bending ratio = 0.36 and a bearing stress ratio = 2.09. Crack growth data for cracks at ten different fasteners, with a radius (r) of 2.4 mm, are presented in Fig. 2 which is taken from the work in [15]. (Here the terminology used in [18] was as follows: 7A8L refers to cracking in row A fastener number 7A8 where L refers to the left tip, and R refers to the right tip.) Fig. 2 again shows see a near linear relationship between ln(a + r) and the number of cycles. The recent work in [19] revealed that the growth of cracks under composite repairs also conforms to the Frost–Dugdale law and that for medium crack lengths, up to say 10–20 mm, the effect of the patch is dominantly due to the reduction in the local stress field. It is important to note that [19] stressed that for composite repairs to long cracks, which for composite repairs to thin plates means crack lengths greater than approximately 40 mm’s, the crack growth rate can be described using a Paris like crack growth law. These findings together with the realisation that ‘‘in the threshold regime, there is something missing either in the (crack closure) model’’, see [20], led to the conjecture [14,19,21] that in the low DK region, i.e. Region I, the crack growth rate can be expressed in the form
Crack Length a + r (mm)
100
Fastest Rivet 71 Rivet 52 Rivet 31 Rivet 12
10
1 3
4
5
6
7
8
Number of cycles x 104
Fig. 1. Crack growth data in the FAA curved panel tests from [16,17].
9
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81
10.0
Crack length a + r (mm)
7A8L 7A16R 7A7R 7A17L 7A13R 7A12R 7A6L 7A14R 7A15R 7A6R 7A17R 7A7L 7A14L 7A16L
1.0 0
0.2
0.4
0.6
0.8
1
Number of cycles x 10
1.2
1.4
1.6
5
Fig. 2. Measured crack growth rates, adapted from [15,18].
ð1m=2Þ
da=dN ¼ CðaÞ
ðDK eff Þ
m
ð5Þ
where C, and m are constants, and DKeff is an effective stress intensity factor. This relationship follows the form proposed in [22,23] and contains the Frost–Dugdale law as a special case, viz, m = 3. The ability of this variant of the ‘‘fractal/incomplete self-similarity’’ growth law to model cracks growing under constant amplitude loading was studied in [21]. As outlined above it is well know that Paris like laws and closure based crack growth models cannot predict the growth of flaws from near micron sized flaws. However, the works in [24] have recently shown how fractal based laws overcome this deficiency and presented an illustrative example in which crack growth from a near micron sized corrosion pit in a full scale test articles was accurately predicted in [24] also used this approach to predict the entire crack growth history in the 1969 General Dynamics, now Lockheed, F-111 wing test. Indeed, [24] also shows that when using a fractal variant of the unified crack growth model, as developed in [25,26] c ð1bÞ ð2cÞ=2c da=dN ¼ C DK btot ðK max;tot Þ a ð6Þ the entire crack growth history can be predicted for constant amplitude tests on a (wide) range of aluminium and titanium alloys tested under a range of R values. Here C, b, and c are material constants. In the present paper we continue this development by showing that for composite repairs to small cracks, which for repairs to thin plates is typically
less than 25 mm, crack growth is governed by Eq. (5) rather than a Paris like crack growth law. 2. Crack growth in composite repairs to cracked metal structures Externally bonded composite patches have proved to be an effective method of repairing cracked, or damaged, structural components, see Table 1 from [26], and a typical composite repair to a rib stiffened panel is shown in Fig. 3. Composite repairs act in two fashions: (1) They reduce the net section stresses in the (cracked) structure. (2) The fibres bridging the crack restrict the opening of the crack faces. It is widely accepted that for small cracks the reduction in the net section stresses is the primary mechanism reducing the stress intensity factor, and thereby crack growth. There are very few experimental studies where the stress intensity factor under a patched crack has been directly measured. Most work has inferred the stress intensity factor from crack growth studies. To the best of the authors knowledge the work in [27], which is summarised in [28] Section 6.3.1 on pp. 126–128, is the only study that has directly measured the stress intensity factor and related it to the stress in the uncracked specimen. In this work X-ray back reflection [27] is used to determine the stress intensity
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Table 1 Australian bonded patch repairs, from [28] Aircraft
Problem
Remarks on bond durability
C130
Stress corrosion cracked stiffeners in wing, aluminium alloy 7075 Fatigue cracking in lower wing skin, AU4SG aluminium alloy
Over 20 years of service. No bond durability problems where bonding carried out as specified 180 wings repaired/reinforced. Eight bond durability problems over 8 years. Failures were associated with adhesive voiding caused by extreme humidity at the repair station in Malaysia No bond durability failures to steel, or aluminium wing skin, surface over 10 years
Mirage III
F-111-C
F-111-C F-111-C F-111-C
F-111-C C-141 (USAF) P-3C Boeing 747
Sea King helicopter F-111-C
Cracking in stiffeer runout #2 on upper surface of the D6ac steel wing pivot fittings Stress corrosion cracking in weapon bay longeron flange, aluminium alloy 7075-T6 Stress corrosion cracking in longeron adjacent to refuel receptacle, 7049-T6 Metal-to-metal and sandwich structure repairs. RAAF adopted GB+S and changed to FM 300 adhesive in 1992 Pork-chop panel (lower fuselage). Panels rebuilt after repeated in-service failures Fatigue cracking in wing riser weep holes, 7075-T6 Full depth corrosion damage in horizontal tail, aluminium alloy 7075-T6 Simulated repairs to several regions including fuselage lap-joint, wing leading edge, trailing edge flap and engine thrust reverser cowl Fatigue crack in frame Fatigue cracking in lower wing skin at fuel flow hole under forward auxiliary spar
Over 10 aircraft repaired with no bond durability problems in 8 years Over ten aircraft repaired. No bond durability problems in 8 years No bond durability failures in over 7 years
Repeat rebuild rate reduced from 95% to zero. No bond durability failures in 7 years No bond durability problems around 5 years No bond durability problems after >10 years of service FAA Aging Aircraft demonstrator repairs; 37,000 flying hours, 7020 landings over an 11 year period with no significant bond durability problems Operated in am offshore ship-borne environment for 4 years with no problems No bond durability problems in over 2.5 years service
Fig. 3. View of a boron epoxy repair to a stiffened panel showing both the patched surface and underside of the test configuration. Note that in this case the stiffener is also cracked.
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head in an F/A-18 centre barrel (CB) test undertaken as part of the Australian Defence Science and Technology Organisation (DSTO) Flaw IdeNtification through the Application of Loads (FINAL) program utilising ex-service Canadian Forces (CF) and US Navy (USN) CBs loaded using a modified mini-FALSTAFF spectrum, see [34]. In this case one of the cracks grew from a small 3.2 lm corrosion pit. In each case the trend lines reveal that da/dB is proportional to ap, where the power p varies between 0.88 and 1.03, and as such the experimental results are in good agreement with the predicted behaviour (i.e. da/dB / a) given the likely experimental error in these measurements. It will be shown that this crack growth law also holds for cracks repaired with a composite repair.
factors in a 1.5 mm thick 7075-T6 aluminium alloy specimens patched on both surfaces with a 0.49 mm thick graphite epoxy laminate. Both uncracked and centre cracked, with total crack lengths (2a) of 10, 30 and 40 mm, specimens were examined. The uncracked specimens were used to determine the (residual) stress in the aluminium after patching As a result of this study it was found [27,28] that in this case the experimentally determined stress intensity factor for a patched plate where the stress field in the skin under the patch is rT is given by p K ¼ rT ðpaÞ
ð7Þ
and that crack bridging effects were negligible. The recent review paper [19] revealed that (1) For composite repairs to through cracks in thin sheets the growth of small to medium length cracks, that have low to mid range DK’s, follows the law proposed by Frost and Dugdale [1,2]. (2) Whenever (1) is valid and bending effects are negligible then the effect of the patch is primarily due to the reduction of the net section stress.
2.1. Experimental crack growth rates This section presents the results of a range of fatigue tests to illustrate that a (near) linear relationship between the crack growth rate and crack length is also seen for crack growth under composite repairs. Case 1: Presented in [35] is the results for a 5 mm crack in a 160 mm wide and 3.14 mm thick 2024-T3 aluminium alloy specimen patched with a seven ply (0.889 mm thick) semicircular uni-directional composite patch with a radius of 80 mm. The specimen was subjected to constant amplitude fatigue testing with rmax = 138 MPa and R = 0.1, see [35], Fig. 6.23 p. 155. Bending, due to neutral axis offset effects was
It has been shown [1,2,29–32] that the crack growth rate was proportional to the crack length, and that this relationship holds for cracks that have length scales ranging from 50 lm to several mm’s. This relationship [33] also holds for crack growth under complex load histories. This is illustrated in Fig. 4, where we plot the crack growth rate per block against the crack depth for cracking in a bulk-
Crack growth rate da/dB (mm/block)
83
0.1 y = 0.106 x y = 0.056 x
0.96
0.88
0.01
CB1 Y488 C4
0.001
CB1 Y488 C3
y = 0.091 x 0.0001 0.001
CB1 Y488 C1
1.03
0.01
CB1 Y488 C2
0.1
1
Crack length a (mm)
Fig. 4. Growth rate per block versus a in the FINAL fatigue test program, from [34].
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eliminated by testing two specimens joined back to back. The resultant crack growth rate data, from [35], are shown in Fig. 5. This figure supports the hypothesis of a linear relationship between the crack growth rate da/dN and crack length (a). Case 2: Raizenne et al. [36] presented crack growth data for spectrum loading for the same test configuration outlined in Case 1. The results of this study are summarised in Fig. 6, where we again see that the crack growth rate per block da/dB is essentially a linear function of crack length. Case 3: Compared in [37] is the efficiency of glare and boron epoxy patches, and the results are shown in Fig. 7. In this case tests were
performed on centre cracked panels and only one side of the panel was patched. Here we again see that the crack growth rate da/dN is essentially a linear function of a. Case 4: Studied in [38] is the behaviour of both boron epoxy and glare patches to a centre cracked 7075-T6 aluminium panel. The panel thicknesses studied were 1.6, 2.5, and 4.0 mm. The panels were 180 mm wide and 600 mm long, and contained a 25 mm long initial fatigue crack. Single sided patches 75 mm wide and 190 mm long were applied to all panels. The panels were tested under constant amplitude loading with rmax = 120 MPa and rmin = 6 MPa.
0.0001
y = 9.0210-6 x1.05
0.00001 1
10
100
Crack length a (mm)
Fig. 5. Crack growth rate versus crack length, from Baker and Jones [35].
1
Crack growth rate da/dB (mm/block)
Crack growth rate da/dn (mm/cycle)
0.001
y = 0.012 x
1.10
0.1
0.01 1.00
10.00
100.00
Crack length a (mm)
Fig. 6. Crack growth rate versus crack depth, from Raizenne et al. [36].
R. Jones et al. / Theoretical and Applied Fracture Mechanics 45 (2006) 79–91
85
Crack growth rate da/dN (mm/cycle)
0.0001 Boron Glare -7
1.02
y = 1.2010 x 0.00001 y = 5.18 10-7 x0.98
0.000001 10
100
Crack length a (mm)
Crack growth rate da/dn (mm/cycle)
Fig. 7. Crack growth rate (da/dN) versus crack length, from Fredell and Guijt [37].
0.01 Glare Boron epoxy spec b1 Boron epoxy spec b1
0.001
-5
-5
0.99
y = 3.6010 x
1.00
y = 2.81 x 10 x
-5
1.02
y = 1.9910 x
0.0001 10
100
Crack length a (mm)
Fig. 8. Crack growth rate versus crack length for a centre cracked 4 mm thick 7075 aluminium plate with both glare and boron epoxy patches, from Gujit et al. [38].
The results for the 4 mm thick panels repaired using both boron epoxy and glare patches are presented in Fig. 8. For the 4 mm plate the boron epoxy patch and the glare patches had stiffness ratios of 1.22 and 1.13 respectively. Here the stiffness ratio was defined as Epatchtpatch/ Eplatetplate. This example again shows that da/dN is essentially a linear function of a.
3. Predicting crack growth Having established that the crack growth rate is often a linear function of the crack length let us evaluate the conjecture presented in [14,19,21] that in the low to mid range DK region, i.e. Region I, the crack growth rate is not governed by a Paris like equation but rather by an expression of the form
ð1m=2Þ
da=dN ¼ CðaÞ
ðDKÞ
m
ð8Þ
where C, and m are constants. The ability of this variant of the ‘‘fractal/incomplete self-similarity’’ growth law to model cracks growing under constant amplitude loading was studied in [21]. In this section we investigate its applicability to predict crack growth under composite repairs to 2024-T3 and 7075-T6 aluminium alloys. Here we will assume that, as shown in [19], when bending effects are negligible then the effect of the patch on the stress intensity factor is primarily due to the reduction of the net section stress and that crack growth is governed by Eq. (8). It should be noted that the works in [39–41] have shown that for composite repairs to edge cracks the effect of the repair is to lower the free surface correction factor from 1.12 to approximately 1, i.e. the stress intensity factor can be approximated as
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p K ¼ rp ðpaÞ
ð9Þ
To validate this value of C we considered the crack growth results presented in [43], this work was performed as the FAA Aging Aircraft program. The work in [43] also presented the crack growth history for a 12.7 mm edge crack in a 1.8 mm thick and 100 mm wide 2024-T3 aluminium alloy specimen subjected to constant amplitude fatigue tests with Drmax = 143 MPa and R = 0.18. The predicted, using m = 3 and C = 7.1 · 1012, and measured crack growth histories are presented in Fig. 10, and are again in good agreement. Having determined the values of C and m predict the growth of cracks in 2024-T3 aluminium specimens repaired using a bonded composite repair. Repair Case (1) In this example, presented in Fig. 13.7 on p. 385 [44], Baker presented the test results for a 10 mm crack in 160 mm wide and 3.14 mm thick 2024-T3 aluminium alloy specimen patched with a seven ply (0.889 mm thick) semi-circular uni-directional composite patch with a radius of 80 mm. The specimen was subjected to constant
where rp is the stress under the patch in the uncracked configuration. 3.1. Crack growth in repaired 2024-T3 aluminium alloy specimens To determine the constants m and C for 2024-T3 aluminium we considered the centre cracked panel results presented by Zuo et al. [42]. As in [1,2], and more recently the work in [14] has set m 3 for aluminium alloys. The specimen tested in [42] was 152 mm wide and 2.54 mm thick and was tested run under constant load amplitude conditions with a maximum stress of 60.45 MPa, and R = 0.2. The experimental results and the predicted crack growth history for m = 3 and C = 7.1 · 1012 are presented in Fig. 9. Here a ispin mm, da/dN is in mm/cycle, and DK is in MPa mm. From this it is seen that the predicted and measured crack growth histories are in good agreement.
100
Crack length a (mm)
y = 8.42 e0.0447x
10 Centre crack Zou et al. Predicted
1 0
5
10
15
20
Number of cycles x 104
Fig. 9. Experimental and predicted crack growth history for a centre cracked panel, from [42].
Crack length a (mm)
100
Roach Unpatched Predicted
10 0
1
2
3
4
5
6
7
8
Number of cycles x 1000
Fig. 10. Experimental and predicted crack growth history for an edge cracked panel, adapted from [43].
R. Jones et al. / Theoretical and Applied Fracture Mechanics 45 (2006) 79–91
amplitude loading with a Dr of 76 MPa with the R values increasing from 0.1 to 0.55, 0.64 and finally 0.75 as the test continued. The resultant crack growth data is plotted in Fig. 11, where it can be seen that as postulated in [44] crack growth was relatively insensitive to the R ratio and that the data was essentially log linear. A good agreement is obtained between the measured and predicted crack growth history. Fatigue tests with R = 0.1 and various rmax levels were also reported in [44], see Fig. 13.6 on p. 383. The measured crack growth data for the cases when rmax = 160 and 80 MPa, and there were no inbuilt disbonds, and the predicted crack growth data predicted are shown in Fig. 12. Here again it is seen that the predicted and measured data are in good agreement and that the data is essentially log linear.
87
Presented in [44] are the results for a seven ply repair, with dimensions as per Case (1) presented above, at various temperatures 20, 60 and 80 C, see Fig. 13.8 on p. 386, subjected to constant amplitude fatigue testing with rmax = 138 MPa and R = 0.1. He concluded that the crack length (a) versus cycles relationship was essentially the same over this temperature range and the data was inconsistent with the current crack growth models. The measured crack growth history is shown in Fig. 13 together with the crack growth data for the associated room temperature test presented in Case (1) and the predicted crack growth history. It is again seen that good agreement is obtained between the predicted and the measured crack growth data.
Crack length a (mm)
100
Experimental data Predicted
10 1
2
3
4
5
6
7
8
9
10
Number of cycles x 105
Fig. 11. a versus number of cycles for the edge cracked 2024-T3 aluminium plate, repaired with a seven ply boron epoxy patch, under variable R ratio from Baker [44].
100
Baker 160 MPa
Crack length a (mm)
Baker 80 MPa Predicted 160 MPa Predicted 80 MPa
10 1
11
21
31
41
51
61
71
81
91
Number of cycles x 104
Fig. 12. Crack length versus number of cycles for the edge cracked 2024-T3 aluminium plate, repaired with a seven ply boron epoxy patch, with rmax equal to 160 and 80 MPa, from Baker [44].
88
R. Jones et al. / Theoretical and Applied Fracture Mechanics 45 (2006) 79–91 100
Crack length a (mm)
Temperature effects Predicted Baker room temp test
10 0
2
4
6
8
10
12
Number of cycles x 104
Fig. 13. Effect of variable temperature on crack growth for the edge cracked 2024-T3 aluminium plate, repaired with a seven ply boron epoxy patch, with rmax equal to 138 MPa, from Baker [44].
3.2. Crack growth in repaired 7075-T6 aluminium alloy specimens Consider crack growth in 7075-T6 aluminium alloy specimens repaired with an externally bonded composite patch. Before we can attempt to predict crack growth we need the constants C and m in the crack growth model. Used the crack growth data in [38] for a centre cracked 7075-T6 aluminium panel, see Fig. 14, to determine these constants. The panels were 180 mm wide, 600 mm long and 1.6 mm thick, and contained a 25 mm initial fatigue crack. They were tested under constant amplitude loading with rmax = 120 MPa and rmin = 6 MPa. The exper-
imental and predicted crack growth histories, using m = 3 and C = 3.22 · 1011, are shown in Fig. 14 where it can be seen that the agreement is quite good. Presented in [39–41] are the results of a USAF study into the composite repair of rib stiffened panels. Two plate thicknesses, 1.6 and 4.2 mm, with two initial crack sizes, 20 and 40 mm tip to tip, were tested. The patch thickness for the 1.6 mm and the 4.2 mm thick skin was 0.508 mm and 1.52 mm respectively. The loads applied to the specimens were chosen to give a peak stress in the skin, when the stiffener was intact, of 120 MPa, see [39–41]. The specimens were fatigued under constant ampli-
100 Experimental
Crack length a (mm)
Predicted
10 0
1
2
3
4
5
Number of cycles x 103
Fig. 14. Experimental and predicted crack growth histories for a centre cracked 1.6 mm thick 7075-T6 aluminium plate, from Guijt et al. [38].
R. Jones et al. / Theoretical and Applied Fracture Mechanics 45 (2006) 79–91
100
89
Crack length a (mm)
Specimen 9, 40 mm crack Specimen 10, 20 mm crack Specimen 15, 20 mm crack Predicted
y = 11.4 e 0.277x
y = 19.7 e 0.246x
y = 10.4 e 0.3148x
10 0
1
2
3
4
5
Number of cycles x 104
Fig. 15. Crack length versus number of cycles for a centre cracked 4.2 mm thick rib stiffened 7075 aluminium plate repaired with a boron epoxy patches, from Jones et al. [39].
Crack length (mm)
100
Specimen 2 Specimen 4 Predicted
10 0
1
2
3
4
Number of cycles x 104
Fig. 16. Crack length versus number of cycles for a centre cracked 1.25 mm thick rib stiffened 7075 aluminium plate repaired with a boron epoxy patches, from Jones et al. [39].
tude loading in a 500 kN servo-hydraulic MTS fatigue testing machine with rmax = 120 MPa and rmin = 6 MPa, i.e. R ratio R = rmin/rmax = 0.05. The results of this study are shown in Figs. 15 and 16. Refs. [39–41] reported that the effect of the stiffener was negligible and as such the stress intensity factor can be calculated using Eq. (9). From these figures we see that there is a near linear relationship between ln(a) and the number of cycles and that there is good agreement between the measured and predicted crack growth histories. At this stage it should be noted that Jones et al. [19] stressed that for composite repairs to long cracks, which for composite repairs to thin plates means crack lengths greater than approximately
40 mm’s, the stress intensity factor becomes constant. Furthermore, for mid range DK’s, which for aluminium alloys correspond p to values greater than approximately 15–20 MPa m, the crack growth rate can be described using a Paris like crack growth law. This can be seen in the results presented in [39– 41] for a 1.6 mm thick skin with a broken stiffener, see Fig. 17, where a near linear relationship prevails for the crack length versus the number of cycles relation. However, this paper has shown that for composite repairs to small cracks, which for repairs to thin plates are typically less than 25 mm, where DK is relatively small crack growth can be approximated using the Frost–Dugdale like growth law proposed in [19], i.e. Eq. (8).
90
R. Jones et al. / Theoretical and Applied Fracture Mechanics 45 (2006) 79–91 90
Crack length a (mm)
80 70 60 50 Specimen 3 Specimen 6
40 30 0
1
2
3
4
5
6
7
8
9
Number of cycles x 103
Fig. 17. Crack growth for a 1.25 mm thick skin specimen with a broken stiffener, specimens 6 and 3.
4. Conclusion A number of researchers have proposed crack growth laws whereby the crack growth rate was proportional to the crack length. It has been shown that such a relationship also holds for composite repairs to cracked metallic structures. Indeed, for composite repairs to small cracks, where DK is small, crack growth follows the fractal growth law proposed in [19], and not a Paris type law. References [1] N.E. Frost, D.S. Dugdale, The propagation of fatigue cracks in test specimens, Journal Mechanics and Physics of Solids 6 (1958) 92–110. [2] N.E. Frost, K.J. Marsh, L.P. Pook, Metal Fatigue, Clarendon Press, Oxford, 1974. [3] S. Barter, L. Molent, N. Landry, P. Klose, P. White, Overview of the F/A-18 aft fuselage combined manoeuvre and dynamic buffet fatigue and residual strength testing, in: Proceedings of International Aerospace Conference, Brisbane, Australia, 2004. [4] R.A. Pell, P.J. Mazeika, L. Molent, The comparison of complex load sequences tested at several stress levels by fractographic examination, in: Proceedings of International Committee on Aeronautical Fatigue, 17th Symposium, Stockholm, Sweden 9–11 June 1993, Engineering Failure Analysis, submitted for publication. [5] R.A. Pell, L. Molent, A.J. Green, The fractographical comparison of F/A-18 aluminium alloy 7050-T7451 bulkhead representative coupons tested under two fatigue load spectra at several stress levels, DSTO-TR-1547, Melbourne, Australia, February 2004. [6] N. Goldsmith, Fractographic examination of cracking in hole No. 2 of Mirage wing A3094, DSTO, Letter N86/ 79NTG, February 1981.
[7] S.A. Barter, Fatigue crack growth in 7050T7451 aluminium alloy thick section plate with a surface condition simulating some regions of F/A-18 structure, Defence Science and Technology Organisation, DSTO-TR-1458, Melbourne, Australia, July 2003. [8] G. Clark, Effects of cracks identified in Mirage wing 40LH on wing structural integrity, DSTO-TR-0771, 1998. [9] S.A Barter, B. Bishop, G. Clark, Defect assessment on F/A18 488 bulkhead tested at ARL, Defence Science and Technology Organisation, Aircraft Material Report, ARLMAT-TM-125, Melbourne, Australia, 1991. [10] P. White, S. Barter, L. Molent, Probabilistic fracture prediction based on aircraft specific fatigue test data, in: Proceedings of the 6th Joint FAA/DoD/NASA Aging Aircraft Conference, San Diego, September 16–19, 2002. [11] S.A. Barter, G. Clark, N.T. Goldsmith, Influence of initial defect conditions on structural fatigue in RAAF aircraft, in: Proceedings of International Committee on Aeronautical Fatigue, 17th Symposium, Stockholm, Sweden, 9–11 June 1993. [12] S.A. Barter, P.K. Sharp, G. Clark, The identification of fatigue-critical regions during fatigue testing of Macchi MB326H centre section lower booms, in: Proceedings of The Ninth International Conference on Fracture, Sydney, April 1997. [13] D.Y. Wang, An investigation of initial fatigue quality, in: Design of Fatigue and Fracture Resistant Structures, in: P.R. Abelkis, C.M. Hudson (Eds.), ASTM STP, vol. 761, American Society For Testing and Materials, 1982, pp. 191– 211. [14] S. Barter, L. Molent, N. Goldsmith, R. Jones, An experimental evaluation of fatigue crack growth, Journal Engineering Fracture Mechanics 12 (1) (2005) 99–128. [15] R. Jones, L. Molent, U.H. Tiong, K. Krishnapillai, Understanding crack growth in fuselage lap joints, Journal of Theoretical and Applied Fracture Mechanics, submitted for publication. [16] G. Samavedam, D. Hoadley, J. Davin, Test Facility for evaluation of structural integrity of stiffened and jointed aircraft curved panels, in: S.N. Atluri, S.G. Sampath, P.
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