- Email: [email protected]
Crack patching revisited
Composite Structures 76 (2006) 218–223 www.elsevier.com/locate/compstruct
Crack patching revisited R. Jones a
a,*
, S. Pitt
b
DSTO Centre of Expertise for Structural Mechanics, Department of Mechanical Engineering, Monash University, P.O. Box 31, Monash University, Vic. 3800, Australia b Air Vehicles Division, Defence Science and Technology Organisation, 506 Lorimer Street, Fishermans Bend 3207, Australia Available online 23 August 2006
Abstract This paper presents the generic relationship between crack length and fatigue life for cracks repaired with an externally bonded composite patch. We then show that the crack growth rate follows the generalised Frost–Dugdale growth law. A simple methodology for predicting crack growth is also presented and validated by comparison with experimental data. 2006 Published by Elsevier Ltd. Keywords: Composite repairs; Crack growth; Frost-Dugdale; Crack patching
1. Introduction The science of fatigue crack growth has traditionally revolved around the relationship between the stress intensity factor range, DK and the crack growth rate, da/dN. Paris et al. [1] were the first to link da/dN with the maximum stress intensity factor Kmax. Liu [2] was the first to suggest the crack growth rate, da/dN was a function of the stress intensity factor range, DK. A similar relationship was subsequently proposed by Paris and Erdogan [3]. Here the results of a series of constant-amplitude crack growth tests were used to express the crack growth rate da/dN (where N is the number of fatigue cycles, and a is the crack depth, or length, at time N) as a function of DK on a log– log scale. Plotting the data in this fashion revealed a region of growth where a linear relation between log (da/dN) and log (DK) appeared to exist. This led to the well-known Paris equation, viz: da=dN ¼ CDK
m
ð1Þ
where C and m are experimentally obtained constants. For many commonly used structural materials the value of m has been found to lie between 2.5 and 4.0. This equation
*
Corresponding author. E-mail address: [email protected] (R. Jones).
0263-8223/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.compstruct.2006.06.037
is a simple model (curve fit) of the central region (region II or the ‘Paris’ region) of results of fatigue crack growth experiments [3]. In the mid 1970s Pearson [4] showed that fatigue crack growth laws determined for macroscopic crack growth data could not be used to predict the growth of small sub-millimetre cracks, and that the constants in the crack growth law were a function of the size of the crack. He also revealed that this inconsistency was not due to crack-tip plasticity effects. At this stage it is important to note that the Paris equation was not the first law proposed to describe crack growth. Frost and Dugdale [5] had earlier (1958) reported that crack growth under constant amplitude loading could be described via a simple log-linear relationship, viz: LnðaÞ ¼ kN þ Lnða0 Þ or a ¼ a0 ekN
ð2Þ
which gives as the crack growth rate equation: da=dN ¼ ka
ð3Þ
where k is a parameter that is material, geometry and load dependent, N is the ‘‘fatigue life’’, and a0 is the initial crack-like flaw size (depth of the crack at the start of loading). For constant amplitude loading Frost and Dugdale [5,6] found that k could be expressed as: k ¼ /ðDrÞ
3
ð4Þ
R. Jones, S. Pitt / Composite Structures 76 (2006) 218–223
where / only depends on the nature of the loading, and the geometry of the structure. Using this cubic stress dependency we can estimate the growth rates at one stress level (r2) from a knowledge of those of another stress level (r1) given the initial crack size a0 for the new crack and the value of k for the original crack growth, viz: 3
a2 ¼ a02 eðr2 =r1 Þ k1 N
ð5Þ
Several predictions made using Eq. (5) are presented in the following sections. Liu [6] subsequently confirmed the log linearity reported in [5], and in his review paper [7] he highlighted that for short cracks da/dN was proportional to the crack length. Indeed, this finding has been highlighted by a large number of researchers, viz: Harkegard et al. [8], Nisitani et al. [9], Kawagoishi et al. [10], Caton et al. [11], Murakamia and Miller [12], Polak and Zezulka [13] Tomkins [14], and Zhang [15]. Whereas these studies only dealt with constant amplitude fatigue the review paper by Barter et al. [16] revealed that this log linear relationship holds for a large cross section of problems under both spectrum and constant amplitude loading. The paper by Molent et al. [17] subsequently summarised an extensive range of experimental studies to reveal that Eq. (3) also applied to full scale structures under complex spectrum loading. The same conclusion was highlighted by Molent, Sun and Green in their compendium of F/A-18 fatigue crack growth data [18]. This work, which examined more than 300 different cracks in various F/A-18 related full-scale fatigue tests and the associated coupon test programs, revealed a near log-linear relationship holds for crack growth from starting lengths of near microns up to lengths of 5 mm or more. Jones et al. [19] revealed that the growth of cracks under composite repairs also conforms to the Frost–Dugdale law, and that for small to medium crack lengths, i.e., from 0.1 to 20 mm, the effect of the patch is dominantly due to the reduction in the local stress field. Jones et al. [20] subsequently revealed how the crack growth rate in a plate repaired with a composite patch could be predicted using a growth law of the form da=dN ¼ Ca1c=2 ððDK tot Þ
1p
K pmaxt Þ
c
219
confirms the work of Wnuk and Yavari [22] and provides a sound base for linking the fractal/non-self-similar based growth law, as given in Eq. (6), to the two parameter growth law developed in [23] and the concepts proposed by Spagnoli [21]. The present paper presents a range of new examples to further support the findings presented in [19,20] that crack growth in composite repairs follows the generalised Frost– Dugdale law, i.e., Eq. (6). 2. Crack growth under complex load spectra Let us consider crack growth under a repetitive block of loads. Let us further assume that each block consists of a spectrum with n cycles that have peak stresses of ri, i = 1, . . . , n, with the associated cyclic ranges being Dri, i = 1, n. In this case the crack growth per block, da/dB, can be written as n X Ca1c=2 ððDK tot i Þ1P K pmaxt i Þc ð7Þ da=dB ¼ i¼1
We can also assume that the crack growth is such that to a first approximation the crack length a can be considered to be a constant in any given block. In accordance with common engineering practice for the ith cycle in this block DKi can be expressed as: pffiffiffiffiffiffi ð8Þ DK i ¼ bDri pa Substituting Eq. (8) into Eq. (7) and noting that within the block b can be assumed to be constant gives: " # n X p c p c þ ð1pÞ da=dB ¼ C ððDri Þ ðri Þ Þ ðb pÞ a ð9Þ i¼1
Drþ i
where is the tensile component of the ith load cycle. This yields as an equivalent block crack growth law da=dB ¼ kac=21 DK ref
ð10Þ
where
ð6Þ
where DKtot corresponds to the tensile part of the load cycle, i.e., the positive or tensile component, of DK, and c and p are material parameters. Eq. (6) is a generalization of the crack growth law first presented in [16]. This relationship follows the form proposed by Spagnoli [21] for non-self similar growth and for the growth of a fractal crack. Wnuk and Yavari [22] revealed how a fractal crack can be related to a flaw with a finite notch radius. In this context Noroozi et al. [23] derived a two parameter crack growth model by assuming that at each stage it could be treated as a flaw with a finite notch radius. Jones et al. [24] subsequently revealed how Eq. (6) can also be derived, for physically small cracks, from the two parameter crack growth model as developed in [23]. This finding both
Fig. 1. View of a boron epoxy repair to a stiffened panel showing both the patched surface and underside of the panel. Note that in this case the stiffener is also cracked, from [27].
220
R. Jones, S. Pitt / Composite Structures 76 (2006) 218–223
pffiffiffiffiffiffi DK ref ¼ bDrref pa
ð11Þ
Here Drref is the maximum stress seen in the block, and k is a function of the load spectra, viz: " # n X ð1pÞ p c k¼C ððDrþ ðri Þ =Drref Þ ð12Þ i Þ i¼1
For small cracks where the crack length is small with respect to the dimensions of component we often find with b is approximately constant. In this case Eq. (9) shows that the crack growth rate per block will be proportional to the crack length. This result was first observed in [17,20] (see Fig 1). 3. Crack growth in composite repairs Externally bonded composite patches have proved to be an effective method of repairing cracked, or damaged, structural components and a typical composite repair to a rib stiffened panel is shown in Fig. 2. 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 by Baker et al. [25], which is summarised in Baker [26] Section 6.3.1 on pp. 126–128, is one of only two studies that have directly measured the stress intensity factor and related it to the stress in the uncracked specimen. In this work Baker et al. used X-ray back reflection to determine the stress intensity 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 panels, with total crack lengths (2a) of 10, 30 and 40 mm, were examined. As a result of this study it was found [25,26] 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 ð13Þ K ¼ rT ðpaÞ and that crack bridging effects were negligible. This finding contradicts the conclusions subsequently published by Aktepe and Baker [28], who presented the results for a 10 and 30 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 75 mm, see Fig. 2. Aktepe and Baker used both strain gauges and a K gauge, all placed a distance of 6 mm from the crack tip, to determine the stress intensity factor. They reported that the measured K was in good agreement with the theoretical value proposed by Rose [29]. At this point it is important to note that the stress field for a centre cracked (infinite) plate subject to a remote uniform stress r can be written [30] in the form: KI h h 3h rx ¼ pffiffiffiffiffiffiffi cos 1 sin sin r ð14Þ 2 2 2 2pr KI h h 3h 1 þ sin sin ry ¼ pffiffiffiffiffiffiffi cos ð15Þ 2 2 2 2pr KI h h 3h rxy ¼ pffiffiffiffiffiffiffi sin cos cos 2 2 2 2pr
ð16Þ
where p K I ¼ r pa
ð17Þ
Note the 2nd term in the expression for rx. This has a significant effect on both the crack tip stress and strain fields. For example along the line h = 0, i.e., directly in front of the crack, we see that p rx ¼ rð ða=2rÞ 1Þ ð18Þ
Fig. 2. Geometry used to estimate the stress intensity factor, from [28].
This shows that the ratio of the contribution ofpthe singular p term to the total value is equal to (a/2r)/( (a/2r) 1). Thus if we take a = 10 mm and r = 6 mm, as per Aktepe and Baker [28] then this ratio is 10.5. In other words for these values of a and r the non singular and the singular
R. Jones, S. Pitt / Composite Structures 76 (2006) 218–223
rx þ ry ¼ jðey þ ex þ ez Þ ¼ ð1 þ mÞjðey þ ex Þ p ¼ ð2K 1 = ð2prÞ cosðh=2Þ rÞ p ¼ rð ð2a=rÞ 1Þ
ð19Þ
In this instance the ratio of the contribution of the singular term to the total value is 2.2 which again shows that the non singular term cannot be ignored. Unfortunately, the relative contributions are problem dependent. We do not have the equivalent analytical solution, that includes the higher order term, for the edge notch problem studied by Aktepe and Baker [28]. Nevertheless on the basis of the simple example presented above the estimates of K obtained in [28] by ignoring the contribution of the non-singular terms are highly questionable. As a result the claim that this test substantiated Rose’s theory does not follow. Indeed, it contradicts the experimental findings presented by Baker et al. in [25] and is contrary to the experimental crack growth data presented in [19,20,24] where a linear relationship is seen for (patched) crack lengths ranging from 5 mm to 20 mm. Another problem inherent in the experimental tests performed in [28] is that prior to testing the crack was first grown to its test length. It was then patched and measurements were taken. This is equivalent to testing at one stress level and then changing the load to a greatly reduced level, which is known to result in a very significant retardation of the crack growth rate and the associated apparent stress intensity factor. As such it would have been preferable to grow the crack a reasonable distance, to avoid the added complication of retardation due to a prior large overload, before attempting to measure the stress intensity factor. Let us now summarise, the papers by Jones et al. [19,20,24] have shown 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 generalised law Frost and Dugdale as given in Eq. (6). (2) Whenever (1) is valid and bending effects are negligible then, as found in the earlier work of Baker et al. [25], the effect of the patch is primarily due to the reduction of the net section stress. At this stage it should be noted that Jones et al. [19] were not the first to note that for crack growth under composite repairs the crack growth rate is proportional to the crack length. This honour goes to the USAF study summarised by Naboulsi and Mall [31]. At this stage it should be noted that Jones et al. [20] stressed that for composite repairs to long cracks, which for composite repairs to thin plates
means crack lengths greater than approximately 40 mm, the stress intensity factor becomes constant, as proposed by Rose [29]. Furthermore, for mid range DK’s, which for aluminium alloys correspond to values greater than p approximately 15–20 MPa m, the crack growth rate can be described using a Paris like crack growth law, see [20,24] for more details. Let us next examine a range of new examples to further support these findings. Example 1. The F111 lower wing skin repair: Baker et al. [32] presented the results of a comprehensive repairsubstantiation program which was undertaken for a safety-critical bonded repair to an F-111C aircraft (A8145) in service with the Royal Australian Air Force. This repair followed the discovery of a 48 mm long, tip-to-tip, crack on the lower wing skin of aircraft A8-145. The cracking occurred in an area lying below the forward auxiliary spar, at the spar station FASS 281.28, where the thickness of the integral stiffener reduces from approximately 8 mm to a nominal skin thickness of approximately 4 mm. The repair substantiation program presented in [32] included fatigue tests on both patched and unpatched panels and representative wing box specimens. These specimens were tested under a load spectra that was representative of in-service usage. The flat panel specimens had a working area of approximately 300 mm · 190 mm and were intended to simulate the wing skin including the geometrical features seen in the aircraft. The box specimens consisted of two flat panels attached by three spars to form a boxlike structure, approximately 900 mm · 430 mm · 65 mm, that simulated the wing. This structure allowed the full compressive loads of the load spectrum to be applied. In the panel specimens these compressive loads were truncated. The resultant crack growth history, which is taken from Fig. 6 in [32], is shown in Fig. 3. Here we see that crack growth in both the patched panels and the patched box tests support the hypothesis of a approximately log-linear relationship.
100.00
5.47E-04x
y = 4.90E+01e 2 R = 0.95
a (mm)
terms both (significantly) contribute to the crack tip stress field. This flows over to the crack tip strain field. The K gauge effectively measures the bulk stress. For this particular problem we see that along on h = 0 we have
221
2.52E-04x
y = 3.92E+01e 2 R = 0.98
Patched Panels Test 1 Patched Panels Test 2 Patched Box Test 1 Patched Box Test 2
10.00 0
500
1000
1500
2000
Flights
Fig. 3. Crack growth in both box and panel tests under F111 block loading, adapted from [32].
R. Jones, S. Pitt / Composite Structures 76 (2006) 218–223
Example 2. Fredell [34] presented specimen test results for a series of patched and unpatched panels tested under a representative C-5A load spectra and presented results for both filtered and unfiltered (full) spectra. These results are analysed in Fig. 4 where we see that the crack growth curves are essentially log linear. Fig. 4 also presents the crack growth histories for both patched tests predicted using Eq. (5) together with the assumption that the effect of the patch is primarily due to the reduction of the net section stress. In both cases the predicted crack growth histories are in good agreement with the experimental data.
100
y = 22.5e
2.08E-06x
2
a (mm)
222
R = 0.988 y = 19.6e
9.30E-07x
2
R = 0.995 Specimen 2, patched at 20 mm Specimen 1, Unpatched 20mm
10 0
200000
400000
600000
800000
1000000
1200000
Cycles
Fig. 6. Crack growth in a simulated frigate repair, adapted from [35].
Example 3. Guijt and Verhoeven [31] presented results for a simulated C-5A repair using Glare 2 patches on an 7075 aluminium alloy panel. The resultant crack growth histories are presented in Fig. 5, where we again see a near log linear relationship, together with the crack growth history, for the unpatched panels, predicted using Eq. (5) and the associated patched results. Here we have used Eq. (5) together with the assumption that the effect of the patch is primarily due to the reduction of the net section stress. From this figure we see that crack growth was log-linear and that the predicted crack growth history was in good agreement with the test data. Unpatched Filtered Patched Filtered Unpatched Full Patched Full Predicted Patched Full Predicted Patched Filtered
100
a (mm)
2.67E-02x
y = 12.5e
1.80E-02x
y = 12.7e
2
R =1.0
2
R = 1.01
3.59E-03x
y = 12.0e 2
R = 0.90
Example 4. As part of a collaborative programme sponsored by the UK and French navies and Lloyd’s Register of Shipping QinetiQ performed an initial study to demonstrate that carbon-fibre composite patching can control crack growth and extend fatigue life ship steels. As part of this program Dalzel-Job et al. [35] studied graphite epoxy composite repairs to cracks in a Lloyd’s grade A ship plate. The specimens were 140 mm wide by 15 mm thick by 900 mm long. A 20 mm long spark-eroded notch was used to initiate fatigue cracking at the plate centreline. The patch consisted of a 24 mm thick carbon-epoxy pre-preg, laid up to extend across the whole width of the plate. The full thickness region of the patch covered 466 mm of the plate length and was bounded by 142 mm tapered areas at either end, see [35] for more details. The patched and unpatched specimens were tested under constant amplitude loading with a remote stress of 100 MPa and an R ratio of 1. The crack growth history is presented in Fig. 6 where we again see a near linear relationship between the log of the crack length and the number of cycles.
4.16E-03x
y = 11.7e
10 0
10
20
30
40
50
60
70
4. Conclusion
Blocks
Fig. 4. Crack growth in panel tested under a both filtered and full C-5A load spectra, adapted from [33].
C5a patched
A number of researchers have proposed crack growth laws whereby the crack growth rate was proportional to the crack length. Naboulsi and Mall [31], Jones et al. [19,20,24] have shown that this relationship also holds for composite repairs to cracked metallic structures. This observation led to the development of a growth law of the form
Unpatched Predicted
da=dn ¼ Ca1m=2 ðDKÞm
a (mm)
100
that resembles that proposed by Spagnoli [21] for non selfsimilar/fractal crack growth and can be derived from the two parameter crack growth law of Noroozi et al. [23]. The present paper has presented a range of examples that further supports these findings. We have also presented a simple method for predicting the patched crack growth history from a knowledge of the unpatched crack growth history.
0.0273x
y = 12.485e 2 R = 1.0
y = 11.7e
ð20Þ
0.00428x
2
R = 0.99
10 0
10
20
30
40
50
60
Blocks
Fig. 5. Crack growth under a representative C-5A load spectra, adapted from [31].
References [1] Paris PC, Gomez MP, Anderson WE. A rational analytic theory of fatigue. Trend Eng 1961;13:9–14.
R. Jones, S. Pitt / Composite Structures 76 (2006) 218–223 [2] Liu HW. Fatigue crack propagation and applied stress range. ASME Trans, J Basic Eng 1963;85D(1):116–22. [3] Paris PC, Erdogan F. A critical analysis of crack propagation laws. ASME Trans, J Basic Eng 1963;85D(4):528. [4] Pearson S. Initiation of fatigue cracks in commercial aluminium alloys and the subsequent propagation of very short cracks. Eng Fract Mech 1975;7:235–47. [5] Frost NE, Dugdale DS. The propagation of fatigue cracks in test specimens. J Mech Phys Solids 1958;6:92–110. [6] Liu HW. Crack propagation in thin metal sheet under repeated loading. ASME Trans, J Basic Eng 1961:23–32. [7] Liu HW. A review of fatigue crack growth analyses. Theory Appl Fract Mech 1991;16:91–108. [8] Harkegard G, Denk J, Stark K. Growth of naturally initiated fatigue cracks in ferritic gas turbine rotor steels. Int J Fatigue 2005:1–12. [9] Nisitani H, Goto M, Kawagoishi N. A small-crack growth law and its related phenomena. Eng Fract Mech 1992;41(4):499–513. [10] Kawagoishi N, Chen Q, Nisitani H. Significance of the small crack growth law and its practical application. Metall Mater Trans A 2000;31A:2005–13. [11] Caton MJ, Jones JW, Boileau JM, Allison JE. The effect of solidification rate on the growth of small fatigue cracks in a cast 319-type aluminum alloy. Metall Mater Trans A 1999;30A:3055–68. [12] Murakamia Y, Miller KJ. What is fatigue damage? A view point from the observation of low cycle fatigue process. Int J Fatigue 2005:1–15. [13] Polak J, Zezulka P. Short crack growth and fatigue life in austeniticferritic duplex stainless steel. Fatigue Fract Eng Mater Struct 2005;28: 923–35. [14] Tomkins B. Fatigue crack propagation—an analysis. Phil Mag 1968;18:1041–66. [15] Zhang JZ. A shear band decohesion model for small fatigue cracks growth in an ultra-fine grain aluminium alloy. Eng Fract Mech 2000;65:665–81. [16] Barter S, Molent L, Goldsmith N, Jones R. An experimental evaluation of fatigue crack growth. J Eng Fract Mech 2005;12(1):99–128. [17] Molent L, Jones R, Barter S, Pitt S. Developments in crack growth assessment, Int J Fatigue, in press. [18] Molent L, Sun Q, Green AJ. The F/A-18 fatigue crack growth data compendium, DSTO-TR-1677. Australia, Melbourne: Platform Sciences Laboratory; 2005. [19] Jones R, Barter SA, Molent L, Pitt S. Crack patching: an experimental evaluation of fatigue crack growth. J Compos Struct 2004;67/2:229–38. [20] Jones R, Krishnapilla K, Pitt S. Crack Patching: predicting fatigue crack growth, Theory Appl Fract Mech, in press.
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[21] Spagnoli A. Self-similarity and fractals in the Paris range of fatigue crack growth. Mech Mater 2005;37:519–29. [22] Wnuk MP, Yavari A. A correspondence principle for fractal and classical cracks. Eng Fract Mech 2005;72:2744–57. [23] Noroozi AH, Glinka G, Lambert S. A two parameter driving force for fatigue crack growth analysis. Int J Fatigue 2005;27: 1277–96. [24] Jones R, Barter S, Molent L, Pitt S. A multi-scale approach to crack growth. In: Sih GC, editor. Multiscaling in molecular and continuum mechanics: biology electronics and material science. Springer; 2006. [25] Baker AA, Hawkes GA, Lumley EJ. Proceedings 2nd International Conference on Composite Materials. Toronto: Metallurgical Society of AIME; 1978. [26] Baker A. Crack patching: experimental studies, practical applications. In: Baker A, Jones R, editors. Bonded repair of aircraft structure. The Hague: Martinus Nijhoff Publishers; 1988. p. 107–73 [chapter 6]. [27] Jones R, Whittingham B, Marshall IH. Bonded repairs to rib stiffened wing skins. J Compos Struct 2002;57:453–8. [28] Aktepe B, Baker A. Validation of stress intensity estimations in patched panels. In: Baker A, Rose LRF, Jones R, editors. Advances in the bonded composite repair of metallic aircraft structure. Elsevier Applied Science Publishers; 2002, ISBN 0-08-042699-9 [chapter 18]. [29] Rose LRF. Theoretical analysis of crack patching. In: Baker AA, Jones R, editors. Bonded repair of aircraft structures. Martinus Nijhoff; 1988 [chapter 6]. [30] Sih GC. On the Westergard method of crack analysis. Int J Fract 1966;2:628–31. [31] Naboulsi S, Mall S. Fatigue crack growth analysis of adhesively repaired panel using perfectly and imperfectly bonded composite patches. Theory Appl Fract Mech 1997;28:13–28. [32] Baker AA, Rose LRF, Walker KF, Wilson ES. Repair substantiation for a bonded composite repair to F111 lower wing skin. Appl Compos Mater 1999;6:251–67. [33] Fredell R, Guijt C. GLARE patching efficiency studies. In: Baker A, Rose LRF, Jones R, editors. Advances in the bonded composite repair of metallic aircraft structure. Elsevier Applied Science Publishers; 2002, ISBN 0-08-042699-9 [chapter 14]. [34] Guijt C, Verhoeven S. C-5A fuselage crown cracking. In: Baker A, Rose LRF, Jones R, editors. Advances in the bonded com posite repair of metallic aircraft structure. Elsevier Applied Science Publishers; 2002, ISBN 0-08-042699-9 [chapter 31]. [35] Dalzel-Job J, Sumpter JDG, Livingstone F. Repairing steel ships with composite patches. In: Ship repair and conversion technology; 4th Quarter; 2003.
Crack patching revisited R. Jones a
a,*
, S. Pitt
b
DSTO Centre of Expertise for Structural Mechanics, Department of Mechanical Engineering, Monash University, P.O. Box 31, Monash University, Vic. 3800, Australia b Air Vehicles Division, Defence Science and Technology Organisation, 506 Lorimer Street, Fishermans Bend 3207, Australia Available online 23 August 2006
Abstract This paper presents the generic relationship between crack length and fatigue life for cracks repaired with an externally bonded composite patch. We then show that the crack growth rate follows the generalised Frost–Dugdale growth law. A simple methodology for predicting crack growth is also presented and validated by comparison with experimental data. 2006 Published by Elsevier Ltd. Keywords: Composite repairs; Crack growth; Frost-Dugdale; Crack patching
1. Introduction The science of fatigue crack growth has traditionally revolved around the relationship between the stress intensity factor range, DK and the crack growth rate, da/dN. Paris et al. [1] were the first to link da/dN with the maximum stress intensity factor Kmax. Liu [2] was the first to suggest the crack growth rate, da/dN was a function of the stress intensity factor range, DK. A similar relationship was subsequently proposed by Paris and Erdogan [3]. Here the results of a series of constant-amplitude crack growth tests were used to express the crack growth rate da/dN (where N is the number of fatigue cycles, and a is the crack depth, or length, at time N) as a function of DK on a log– log scale. Plotting the data in this fashion revealed a region of growth where a linear relation between log (da/dN) and log (DK) appeared to exist. This led to the well-known Paris equation, viz: da=dN ¼ CDK
m
ð1Þ
where C and m are experimentally obtained constants. For many commonly used structural materials the value of m has been found to lie between 2.5 and 4.0. This equation
*
Corresponding author. E-mail address: [email protected] (R. Jones).
0263-8223/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.compstruct.2006.06.037
is a simple model (curve fit) of the central region (region II or the ‘Paris’ region) of results of fatigue crack growth experiments [3]. In the mid 1970s Pearson [4] showed that fatigue crack growth laws determined for macroscopic crack growth data could not be used to predict the growth of small sub-millimetre cracks, and that the constants in the crack growth law were a function of the size of the crack. He also revealed that this inconsistency was not due to crack-tip plasticity effects. At this stage it is important to note that the Paris equation was not the first law proposed to describe crack growth. Frost and Dugdale [5] had earlier (1958) reported that crack growth under constant amplitude loading could be described via a simple log-linear relationship, viz: LnðaÞ ¼ kN þ Lnða0 Þ or a ¼ a0 ekN
ð2Þ
which gives as the crack growth rate equation: da=dN ¼ ka
ð3Þ
where k is a parameter that is material, geometry and load dependent, N is the ‘‘fatigue life’’, and a0 is the initial crack-like flaw size (depth of the crack at the start of loading). For constant amplitude loading Frost and Dugdale [5,6] found that k could be expressed as: k ¼ /ðDrÞ
3
ð4Þ
R. Jones, S. Pitt / Composite Structures 76 (2006) 218–223
where / only depends on the nature of the loading, and the geometry of the structure. Using this cubic stress dependency we can estimate the growth rates at one stress level (r2) from a knowledge of those of another stress level (r1) given the initial crack size a0 for the new crack and the value of k for the original crack growth, viz: 3
a2 ¼ a02 eðr2 =r1 Þ k1 N
ð5Þ
Several predictions made using Eq. (5) are presented in the following sections. Liu [6] subsequently confirmed the log linearity reported in [5], and in his review paper [7] he highlighted that for short cracks da/dN was proportional to the crack length. Indeed, this finding has been highlighted by a large number of researchers, viz: Harkegard et al. [8], Nisitani et al. [9], Kawagoishi et al. [10], Caton et al. [11], Murakamia and Miller [12], Polak and Zezulka [13] Tomkins [14], and Zhang [15]. Whereas these studies only dealt with constant amplitude fatigue the review paper by Barter et al. [16] revealed that this log linear relationship holds for a large cross section of problems under both spectrum and constant amplitude loading. The paper by Molent et al. [17] subsequently summarised an extensive range of experimental studies to reveal that Eq. (3) also applied to full scale structures under complex spectrum loading. The same conclusion was highlighted by Molent, Sun and Green in their compendium of F/A-18 fatigue crack growth data [18]. This work, which examined more than 300 different cracks in various F/A-18 related full-scale fatigue tests and the associated coupon test programs, revealed a near log-linear relationship holds for crack growth from starting lengths of near microns up to lengths of 5 mm or more. Jones et al. [19] revealed that the growth of cracks under composite repairs also conforms to the Frost–Dugdale law, and that for small to medium crack lengths, i.e., from 0.1 to 20 mm, the effect of the patch is dominantly due to the reduction in the local stress field. Jones et al. [20] subsequently revealed how the crack growth rate in a plate repaired with a composite patch could be predicted using a growth law of the form da=dN ¼ Ca1c=2 ððDK tot Þ
1p
K pmaxt Þ
c
219
confirms the work of Wnuk and Yavari [22] and provides a sound base for linking the fractal/non-self-similar based growth law, as given in Eq. (6), to the two parameter growth law developed in [23] and the concepts proposed by Spagnoli [21]. The present paper presents a range of new examples to further support the findings presented in [19,20] that crack growth in composite repairs follows the generalised Frost– Dugdale law, i.e., Eq. (6). 2. Crack growth under complex load spectra Let us consider crack growth under a repetitive block of loads. Let us further assume that each block consists of a spectrum with n cycles that have peak stresses of ri, i = 1, . . . , n, with the associated cyclic ranges being Dri, i = 1, n. In this case the crack growth per block, da/dB, can be written as n X Ca1c=2 ððDK tot i Þ1P K pmaxt i Þc ð7Þ da=dB ¼ i¼1
We can also assume that the crack growth is such that to a first approximation the crack length a can be considered to be a constant in any given block. In accordance with common engineering practice for the ith cycle in this block DKi can be expressed as: pffiffiffiffiffiffi ð8Þ DK i ¼ bDri pa Substituting Eq. (8) into Eq. (7) and noting that within the block b can be assumed to be constant gives: " # n X p c p c þ ð1pÞ da=dB ¼ C ððDri Þ ðri Þ Þ ðb pÞ a ð9Þ i¼1
Drþ i
where is the tensile component of the ith load cycle. This yields as an equivalent block crack growth law da=dB ¼ kac=21 DK ref
ð10Þ
where
ð6Þ
where DKtot corresponds to the tensile part of the load cycle, i.e., the positive or tensile component, of DK, and c and p are material parameters. Eq. (6) is a generalization of the crack growth law first presented in [16]. This relationship follows the form proposed by Spagnoli [21] for non-self similar growth and for the growth of a fractal crack. Wnuk and Yavari [22] revealed how a fractal crack can be related to a flaw with a finite notch radius. In this context Noroozi et al. [23] derived a two parameter crack growth model by assuming that at each stage it could be treated as a flaw with a finite notch radius. Jones et al. [24] subsequently revealed how Eq. (6) can also be derived, for physically small cracks, from the two parameter crack growth model as developed in [23]. This finding both
Fig. 1. View of a boron epoxy repair to a stiffened panel showing both the patched surface and underside of the panel. Note that in this case the stiffener is also cracked, from [27].
220
R. Jones, S. Pitt / Composite Structures 76 (2006) 218–223
pffiffiffiffiffiffi DK ref ¼ bDrref pa
ð11Þ
Here Drref is the maximum stress seen in the block, and k is a function of the load spectra, viz: " # n X ð1pÞ p c k¼C ððDrþ ðri Þ =Drref Þ ð12Þ i Þ i¼1
For small cracks where the crack length is small with respect to the dimensions of component we often find with b is approximately constant. In this case Eq. (9) shows that the crack growth rate per block will be proportional to the crack length. This result was first observed in [17,20] (see Fig 1). 3. Crack growth in composite repairs Externally bonded composite patches have proved to be an effective method of repairing cracked, or damaged, structural components and a typical composite repair to a rib stiffened panel is shown in Fig. 2. 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 by Baker et al. [25], which is summarised in Baker [26] Section 6.3.1 on pp. 126–128, is one of only two studies that have directly measured the stress intensity factor and related it to the stress in the uncracked specimen. In this work Baker et al. used X-ray back reflection to determine the stress intensity 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 panels, with total crack lengths (2a) of 10, 30 and 40 mm, were examined. As a result of this study it was found [25,26] 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 ð13Þ K ¼ rT ðpaÞ and that crack bridging effects were negligible. This finding contradicts the conclusions subsequently published by Aktepe and Baker [28], who presented the results for a 10 and 30 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 75 mm, see Fig. 2. Aktepe and Baker used both strain gauges and a K gauge, all placed a distance of 6 mm from the crack tip, to determine the stress intensity factor. They reported that the measured K was in good agreement with the theoretical value proposed by Rose [29]. At this point it is important to note that the stress field for a centre cracked (infinite) plate subject to a remote uniform stress r can be written [30] in the form: KI h h 3h rx ¼ pffiffiffiffiffiffiffi cos 1 sin sin r ð14Þ 2 2 2 2pr KI h h 3h 1 þ sin sin ry ¼ pffiffiffiffiffiffiffi cos ð15Þ 2 2 2 2pr KI h h 3h rxy ¼ pffiffiffiffiffiffiffi sin cos cos 2 2 2 2pr
ð16Þ
where p K I ¼ r pa
ð17Þ
Note the 2nd term in the expression for rx. This has a significant effect on both the crack tip stress and strain fields. For example along the line h = 0, i.e., directly in front of the crack, we see that p rx ¼ rð ða=2rÞ 1Þ ð18Þ
Fig. 2. Geometry used to estimate the stress intensity factor, from [28].
This shows that the ratio of the contribution ofpthe singular p term to the total value is equal to (a/2r)/( (a/2r) 1). Thus if we take a = 10 mm and r = 6 mm, as per Aktepe and Baker [28] then this ratio is 10.5. In other words for these values of a and r the non singular and the singular
R. Jones, S. Pitt / Composite Structures 76 (2006) 218–223
rx þ ry ¼ jðey þ ex þ ez Þ ¼ ð1 þ mÞjðey þ ex Þ p ¼ ð2K 1 = ð2prÞ cosðh=2Þ rÞ p ¼ rð ð2a=rÞ 1Þ
ð19Þ
In this instance the ratio of the contribution of the singular term to the total value is 2.2 which again shows that the non singular term cannot be ignored. Unfortunately, the relative contributions are problem dependent. We do not have the equivalent analytical solution, that includes the higher order term, for the edge notch problem studied by Aktepe and Baker [28]. Nevertheless on the basis of the simple example presented above the estimates of K obtained in [28] by ignoring the contribution of the non-singular terms are highly questionable. As a result the claim that this test substantiated Rose’s theory does not follow. Indeed, it contradicts the experimental findings presented by Baker et al. in [25] and is contrary to the experimental crack growth data presented in [19,20,24] where a linear relationship is seen for (patched) crack lengths ranging from 5 mm to 20 mm. Another problem inherent in the experimental tests performed in [28] is that prior to testing the crack was first grown to its test length. It was then patched and measurements were taken. This is equivalent to testing at one stress level and then changing the load to a greatly reduced level, which is known to result in a very significant retardation of the crack growth rate and the associated apparent stress intensity factor. As such it would have been preferable to grow the crack a reasonable distance, to avoid the added complication of retardation due to a prior large overload, before attempting to measure the stress intensity factor. Let us now summarise, the papers by Jones et al. [19,20,24] have shown 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 generalised law Frost and Dugdale as given in Eq. (6). (2) Whenever (1) is valid and bending effects are negligible then, as found in the earlier work of Baker et al. [25], the effect of the patch is primarily due to the reduction of the net section stress. At this stage it should be noted that Jones et al. [19] were not the first to note that for crack growth under composite repairs the crack growth rate is proportional to the crack length. This honour goes to the USAF study summarised by Naboulsi and Mall [31]. At this stage it should be noted that Jones et al. [20] stressed that for composite repairs to long cracks, which for composite repairs to thin plates
means crack lengths greater than approximately 40 mm, the stress intensity factor becomes constant, as proposed by Rose [29]. Furthermore, for mid range DK’s, which for aluminium alloys correspond to values greater than p approximately 15–20 MPa m, the crack growth rate can be described using a Paris like crack growth law, see [20,24] for more details. Let us next examine a range of new examples to further support these findings. Example 1. The F111 lower wing skin repair: Baker et al. [32] presented the results of a comprehensive repairsubstantiation program which was undertaken for a safety-critical bonded repair to an F-111C aircraft (A8145) in service with the Royal Australian Air Force. This repair followed the discovery of a 48 mm long, tip-to-tip, crack on the lower wing skin of aircraft A8-145. The cracking occurred in an area lying below the forward auxiliary spar, at the spar station FASS 281.28, where the thickness of the integral stiffener reduces from approximately 8 mm to a nominal skin thickness of approximately 4 mm. The repair substantiation program presented in [32] included fatigue tests on both patched and unpatched panels and representative wing box specimens. These specimens were tested under a load spectra that was representative of in-service usage. The flat panel specimens had a working area of approximately 300 mm · 190 mm and were intended to simulate the wing skin including the geometrical features seen in the aircraft. The box specimens consisted of two flat panels attached by three spars to form a boxlike structure, approximately 900 mm · 430 mm · 65 mm, that simulated the wing. This structure allowed the full compressive loads of the load spectrum to be applied. In the panel specimens these compressive loads were truncated. The resultant crack growth history, which is taken from Fig. 6 in [32], is shown in Fig. 3. Here we see that crack growth in both the patched panels and the patched box tests support the hypothesis of a approximately log-linear relationship.
100.00
5.47E-04x
y = 4.90E+01e 2 R = 0.95
a (mm)
terms both (significantly) contribute to the crack tip stress field. This flows over to the crack tip strain field. The K gauge effectively measures the bulk stress. For this particular problem we see that along on h = 0 we have
221
2.52E-04x
y = 3.92E+01e 2 R = 0.98
Patched Panels Test 1 Patched Panels Test 2 Patched Box Test 1 Patched Box Test 2
10.00 0
500
1000
1500
2000
Flights
Fig. 3. Crack growth in both box and panel tests under F111 block loading, adapted from [32].
R. Jones, S. Pitt / Composite Structures 76 (2006) 218–223
Example 2. Fredell [34] presented specimen test results for a series of patched and unpatched panels tested under a representative C-5A load spectra and presented results for both filtered and unfiltered (full) spectra. These results are analysed in Fig. 4 where we see that the crack growth curves are essentially log linear. Fig. 4 also presents the crack growth histories for both patched tests predicted using Eq. (5) together with the assumption that the effect of the patch is primarily due to the reduction of the net section stress. In both cases the predicted crack growth histories are in good agreement with the experimental data.
100
y = 22.5e
2.08E-06x
2
a (mm)
222
R = 0.988 y = 19.6e
9.30E-07x
2
R = 0.995 Specimen 2, patched at 20 mm Specimen 1, Unpatched 20mm
10 0
200000
400000
600000
800000
1000000
1200000
Cycles
Fig. 6. Crack growth in a simulated frigate repair, adapted from [35].
Example 3. Guijt and Verhoeven [31] presented results for a simulated C-5A repair using Glare 2 patches on an 7075 aluminium alloy panel. The resultant crack growth histories are presented in Fig. 5, where we again see a near log linear relationship, together with the crack growth history, for the unpatched panels, predicted using Eq. (5) and the associated patched results. Here we have used Eq. (5) together with the assumption that the effect of the patch is primarily due to the reduction of the net section stress. From this figure we see that crack growth was log-linear and that the predicted crack growth history was in good agreement with the test data. Unpatched Filtered Patched Filtered Unpatched Full Patched Full Predicted Patched Full Predicted Patched Filtered
100
a (mm)
2.67E-02x
y = 12.5e
1.80E-02x
y = 12.7e
2
R =1.0
2
R = 1.01
3.59E-03x
y = 12.0e 2
R = 0.90
Example 4. As part of a collaborative programme sponsored by the UK and French navies and Lloyd’s Register of Shipping QinetiQ performed an initial study to demonstrate that carbon-fibre composite patching can control crack growth and extend fatigue life ship steels. As part of this program Dalzel-Job et al. [35] studied graphite epoxy composite repairs to cracks in a Lloyd’s grade A ship plate. The specimens were 140 mm wide by 15 mm thick by 900 mm long. A 20 mm long spark-eroded notch was used to initiate fatigue cracking at the plate centreline. The patch consisted of a 24 mm thick carbon-epoxy pre-preg, laid up to extend across the whole width of the plate. The full thickness region of the patch covered 466 mm of the plate length and was bounded by 142 mm tapered areas at either end, see [35] for more details. The patched and unpatched specimens were tested under constant amplitude loading with a remote stress of 100 MPa and an R ratio of 1. The crack growth history is presented in Fig. 6 where we again see a near linear relationship between the log of the crack length and the number of cycles.
4.16E-03x
y = 11.7e
10 0
10
20
30
40
50
60
70
4. Conclusion
Blocks
Fig. 4. Crack growth in panel tested under a both filtered and full C-5A load spectra, adapted from [33].
C5a patched
A number of researchers have proposed crack growth laws whereby the crack growth rate was proportional to the crack length. Naboulsi and Mall [31], Jones et al. [19,20,24] have shown that this relationship also holds for composite repairs to cracked metallic structures. This observation led to the development of a growth law of the form
Unpatched Predicted
da=dn ¼ Ca1m=2 ðDKÞm
a (mm)
100
that resembles that proposed by Spagnoli [21] for non selfsimilar/fractal crack growth and can be derived from the two parameter crack growth law of Noroozi et al. [23]. The present paper has presented a range of examples that further supports these findings. We have also presented a simple method for predicting the patched crack growth history from a knowledge of the unpatched crack growth history.
0.0273x
y = 12.485e 2 R = 1.0
y = 11.7e
ð20Þ
0.00428x
2
R = 0.99
10 0
10
20
30
40
50
60
Blocks
Fig. 5. Crack growth under a representative C-5A load spectra, adapted from [31].
References [1] Paris PC, Gomez MP, Anderson WE. A rational analytic theory of fatigue. Trend Eng 1961;13:9–14.
R. Jones, S. Pitt / Composite Structures 76 (2006) 218–223 [2] Liu HW. Fatigue crack propagation and applied stress range. ASME Trans, J Basic Eng 1963;85D(1):116–22. [3] Paris PC, Erdogan F. A critical analysis of crack propagation laws. ASME Trans, J Basic Eng 1963;85D(4):528. [4] Pearson S. Initiation of fatigue cracks in commercial aluminium alloys and the subsequent propagation of very short cracks. Eng Fract Mech 1975;7:235–47. [5] Frost NE, Dugdale DS. The propagation of fatigue cracks in test specimens. J Mech Phys Solids 1958;6:92–110. [6] Liu HW. Crack propagation in thin metal sheet under repeated loading. ASME Trans, J Basic Eng 1961:23–32. [7] Liu HW. A review of fatigue crack growth analyses. Theory Appl Fract Mech 1991;16:91–108. [8] Harkegard G, Denk J, Stark K. Growth of naturally initiated fatigue cracks in ferritic gas turbine rotor steels. Int J Fatigue 2005:1–12. [9] Nisitani H, Goto M, Kawagoishi N. A small-crack growth law and its related phenomena. Eng Fract Mech 1992;41(4):499–513. [10] Kawagoishi N, Chen Q, Nisitani H. Significance of the small crack growth law and its practical application. Metall Mater Trans A 2000;31A:2005–13. [11] Caton MJ, Jones JW, Boileau JM, Allison JE. The effect of solidification rate on the growth of small fatigue cracks in a cast 319-type aluminum alloy. Metall Mater Trans A 1999;30A:3055–68. [12] Murakamia Y, Miller KJ. What is fatigue damage? A view point from the observation of low cycle fatigue process. Int J Fatigue 2005:1–15. [13] Polak J, Zezulka P. Short crack growth and fatigue life in austeniticferritic duplex stainless steel. Fatigue Fract Eng Mater Struct 2005;28: 923–35. [14] Tomkins B. Fatigue crack propagation—an analysis. Phil Mag 1968;18:1041–66. [15] Zhang JZ. A shear band decohesion model for small fatigue cracks growth in an ultra-fine grain aluminium alloy. Eng Fract Mech 2000;65:665–81. [16] Barter S, Molent L, Goldsmith N, Jones R. An experimental evaluation of fatigue crack growth. J Eng Fract Mech 2005;12(1):99–128. [17] Molent L, Jones R, Barter S, Pitt S. Developments in crack growth assessment, Int J Fatigue, in press. [18] Molent L, Sun Q, Green AJ. The F/A-18 fatigue crack growth data compendium, DSTO-TR-1677. Australia, Melbourne: Platform Sciences Laboratory; 2005. [19] Jones R, Barter SA, Molent L, Pitt S. Crack patching: an experimental evaluation of fatigue crack growth. J Compos Struct 2004;67/2:229–38. [20] Jones R, Krishnapilla K, Pitt S. Crack Patching: predicting fatigue crack growth, Theory Appl Fract Mech, in press.
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[21] Spagnoli A. Self-similarity and fractals in the Paris range of fatigue crack growth. Mech Mater 2005;37:519–29. [22] Wnuk MP, Yavari A. A correspondence principle for fractal and classical cracks. Eng Fract Mech 2005;72:2744–57. [23] Noroozi AH, Glinka G, Lambert S. A two parameter driving force for fatigue crack growth analysis. Int J Fatigue 2005;27: 1277–96. [24] Jones R, Barter S, Molent L, Pitt S. A multi-scale approach to crack growth. In: Sih GC, editor. Multiscaling in molecular and continuum mechanics: biology electronics and material science. Springer; 2006. [25] Baker AA, Hawkes GA, Lumley EJ. Proceedings 2nd International Conference on Composite Materials. Toronto: Metallurgical Society of AIME; 1978. [26] Baker A. Crack patching: experimental studies, practical applications. In: Baker A, Jones R, editors. Bonded repair of aircraft structure. The Hague: Martinus Nijhoff Publishers; 1988. p. 107–73 [chapter 6]. [27] Jones R, Whittingham B, Marshall IH. Bonded repairs to rib stiffened wing skins. J Compos Struct 2002;57:453–8. [28] Aktepe B, Baker A. Validation of stress intensity estimations in patched panels. In: Baker A, Rose LRF, Jones R, editors. Advances in the bonded composite repair of metallic aircraft structure. Elsevier Applied Science Publishers; 2002, ISBN 0-08-042699-9 [chapter 18]. [29] Rose LRF. Theoretical analysis of crack patching. In: Baker AA, Jones R, editors. Bonded repair of aircraft structures. Martinus Nijhoff; 1988 [chapter 6]. [30] Sih GC. On the Westergard method of crack analysis. Int J Fract 1966;2:628–31. [31] Naboulsi S, Mall S. Fatigue crack growth analysis of adhesively repaired panel using perfectly and imperfectly bonded composite patches. Theory Appl Fract Mech 1997;28:13–28. [32] Baker AA, Rose LRF, Walker KF, Wilson ES. Repair substantiation for a bonded composite repair to F111 lower wing skin. Appl Compos Mater 1999;6:251–67. [33] Fredell R, Guijt C. GLARE patching efficiency studies. In: Baker A, Rose LRF, Jones R, editors. Advances in the bonded composite repair of metallic aircraft structure. Elsevier Applied Science Publishers; 2002, ISBN 0-08-042699-9 [chapter 14]. [34] Guijt C, Verhoeven S. C-5A fuselage crown cracking. In: Baker A, Rose LRF, Jones R, editors. Advances in the bonded com posite repair of metallic aircraft structure. Elsevier Applied Science Publishers; 2002, ISBN 0-08-042699-9 [chapter 31]. [35] Dalzel-Job J, Sumpter JDG, Livingstone F. Repairing steel ships with composite patches. In: Ship repair and conversion technology; 4th Quarter; 2003.