Composite Structures 94 (2012) 1343–1351

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Composite Structures

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Application of the Hartman–Schijve equation to represent Mode I and Mode IIfatigue delamination growth in composites

R. Jones a,⇑, S. Pitt a, A.J. Bunner b, D. Hui c

a DSTO Centre of Expertise in Structural Mechanics, Department of Mechanical and Aerospace Engineering, Monash University, P.O. Box 31, Monash University, Victoria 3800, Australiab Mechanical Systems Engineering, Empa, Swiss Federal Laboratories for Materials Science and Technology, CH-8600 Duebendorf, Switzerlandc Composite Material Research Laboratory, Department of Mechanical Engineering, University of New Orleans, 2000 Lakeshore Drive, New Orleans, LA 70148, USA

a r t i c l e i n f o a b s t r a c t

Article history:Available online 27 November 2011

Keywords:Delamination growthFatigueHartman–SchijveExperimental testing

0263-8223/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruct.2011.11.030

⇑ Corresponding author.E-mail address: rhys.jones@eng.monash.edu.au (R

This paper discusses the potential of a variant of the Hartman–Schijve equation to represent both Modes Iand II constant amplitude delamination growth in composites. To this end we show that the delaminationgrowth rate da/dN can often be related to (D

pG � D

pGth)a, where a is approximately 2. As such the expo-

nent in this relationship is considerably lower than the exponent in ‘‘Paris like’’ power law representa-tions. We also show that this particular representation of delamination growth in composites issimilar to that seen for crack growth in metals.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Composite materials are now widely used in aircraft structuralcomponents. Australia’s current lead fighter the F/A-18 hasapproximately 39% of its external wetted surface fabricated fromadvanced composites. At Boeing, the state-of-the-art 787 passen-ger aircraft has approximately 50% composite parts on its majorcomponents. In the Joint Strike Fighter (JSF), a ‘‘next generation’’aircraft, a collaboration involving aerospace companies in theUSA, the UK, Australia, Canada, etc., advanced composites makeup approximately 50% of the aircraft. However, the primary struc-tural members in these aircraft are still metallic. A similar situationarises in the civilian sphere where skins, stabilators, fins, controlsurfaces, etc., where the strain level in these components arelow, now make extensive use of composites. However, for certifica-tion purposes, current designs are such that any delamination willnot grow. This approach places an artificial limit on the use of com-posite materials and the components that can be lightened throughtheir use. It also requires an accurate knowledge of the fatiguethreshold associated with the particular damage state.

In this context the boron epoxy doubler on the upper surface ofF-111 aircraft is a good example of an in-service delaminationproblem associated with a major load bearing component [1]. Inthis case the doublers were approximately 120 plies thick and tookapproximately 30% of the load in the critical section of the wing pi-vot fitting [2]. Even though the doublers passed cold proof load(CPLT) testing, small defects, which were thought to be less than0.1 mm in size and hence were undetectable, led to extensive

ll rights reserved.

. Jones).

delamination/disbonding in under 1000 flight hours [1]. At thisstage it should be stressed that the doublers met the RAAF’s initialdesign requirement, viz: to ensure that the doublers and the wingsdid not fail during CPLT, and that up till the retirement of the F-111C wings there was no further cracking in the critical locationpost doubler installation. Nevertheless, lessons learnt from thisprogram were the importance of designing to the fatigue thresh-old(s) associated with the inherent initial flaws and the associatedstress states and the need to be able to predict delaminationgrowth so as to establish the necessary inspection intervals. Thein-service delamination growth of small sub mm defects in com-posites and adhesively bonded joints is not unique to the F-111and was also observed on the boron epoxy repair to the CanadianCF-5 aircraft [3] as well as on the Canadian CF-5 and the F/A-18 fullscale (wing torsion box) fatigue tests [4].

The growth of such small defects under operational load spectraalso implies that the fatigue threshold was very low and as suchmirrors the findings presented in [5] where it was concluded thatunder operational loading the fatigue threshold for metallic struc-tures is very low and that crack growth can be assumed to com-mence on day one.

Given the failure during static testing of the Boeing 787 wing tofuselage joint and the in-service delaminations discussed abovethe need to be able to assess the fatigue performance of compositeswith small initial defects takes on an added significance. Indeed,the importance of working to the correct fatigue threshold, for agiven set of stresses and initial flaws, was highlighted by Schoenet al. [6], who stated:

‘‘During certification of the AIRBUS A320 vertical fin, nodelamination growth was detected during static loading. The

1344 R. Jones et al. / Composite Structures 94 (2012) 1343–1351

following fatigue loading of the same component had to beinterrupted due to large delamination growth. . . . This demon-strates the importance of using the threshold value instead ofthe static value for delamination growth in the design of com-posite structures.’’

Due to the low stress levels seen in the majority of compositestructural members there are only a few instances where therehas been in-service delamination growth or disbanding [1,3]. Thechallenge is to increase these working stress levels and yet stillbenefit from the advantages inherent in a no growth design. Thiscan only be achieved by knowing the precise boundary of the nogrowth design envelope and its relationship to the size and natureof the initial defect(s). Whereas for metals it is known [5] that forsmall defects subjected to operational flight load spectra the fati-gue threshold is very small and that fleet experience has shownthat delaminations can grow from small sub-mm initial defectsthere has, as yet, been no controlled study performed to establishthe behaviour of representative small (sub mm) defects in compos-ites. The in-service delaminations seen in the boron epoxy doubleron RAAF F-111 and Canadian CF-5 aircraft and the delaminationgrowth seen in various full scale fatigue tests also highlight theneed to develop a methodology that can be used to accuratelyand reliably predict the inspection intervals associated withdelamination growth arising from small naturally occurringdefects.

Current approaches to predicting the growth of delaminationdamage are based on those used to predict crack growth in metals.The difference lies in the fact that whereas in metals the drivingparameter is based on the stress intensity factor range DK andKmax, in composites it is based on DG and/or Gmax [7–15] and theequation for the increment in the delamination size with cyclestakes the form:

da=dN ¼ C:ðDGÞm ð1Þ

or variants thereof. Unfortunately, when expressed in this formthe exponent m is usually very large, in [11] m has values greaterthan 10. As such [10] commented

‘‘For composites, the exponents for relating propagation rate tostrain energy release rate have been shown to be high, espe-cially in Mode I. With large exponents, small uncertainties inthe applied loads will lead to large uncertainties (at least oneorder of magnitude) in the predicted delamination growth rate.This makes the derived power law relationships unsuitable fordesign purposes. Hence, for composite materials more emphasismust be placed on the strain energy release rate threshold.Therefore, it is important to ensure that the threshold valueobtained corresponds to no delamination growth in thestructure.’’

As a result the common design philosophy is to adopt a nogrowth design [16,17]. This approach, which is partially drivenby the large value of the exponent in Eq. (1), places an artificial lim-it on the use of composite materials and the components that canbe lightened through their use. It also requires an accurate knowl-edge of the fatigue threshold(s) associated with the particulardamage state. However, it appears that even for large initial defectsthese thresholds can be a function of the test methodology [10,15].

In this context it should also be noted that it is well known that,for metals, whilst the fatigue threshold DKth is a function of the Rratio it is also a function of the size of the initial defect [18–20].Frost [18] was the first to reveal that DKth is also a function ofthe crack length and that DKth decreases as the crack length de-creases. This phenomenon was subsequently confirmed by Usamiand Shida [19] and Kitagawa and Takahashi [20]. Unfortunately,this effect has not been studied in any detail for composites.

As such given the current status for metals, as outlined above,we also need to address the questions:

� Is the fatigue threshold for defects in composite structures afunction of the test geometry?� Is the fatigue threshold for defects in composite structures a

function of the size of defect for small defects in compositestructures?� Does the fatigue threshold for small (sub mm) defects become

vanishingly small under operational load spectra?� Is the large value of the exponent m in Eq. (1) due to attempting

to force a power law relationship between da/dN and DG (orGmax).

Until these questions are answered there can be little confi-dence in any methodology that is used to assess whether or notthere will be damage growth in complex built structures underoperational loading. There can also be little confidence in anystatements on inspection intervals. This will continue to meanthe use of large safety factors and therefore excessively heavy com-posite structures, replacing/repairing structural components whendamage is found and structures that cannot effectively competewith metals at higher strains. It will also mean a heavy relianceon safety by inspection. Having identified a number of key ques-tions the present paper focuses on just one: Whether a variant ofthe Hartman–Schijve equation, with a � 2, can be used to repre-sent Modes I and II fatigue delamination growth in composites.

2. Fatigue crack growth

The mathematical equations that are commonly used to repre-sent delamination growth in composites are founded on those usedto model crack growth in metals. Thus let us begin with a brief re-view of the equations used to represent crack growth in metals. Itis commonly thought that the increment in the crack length per cy-cle, da/dN, can be related to DK and/or the maximum stress inten-sity factor Kmax. This approach was first suggested in 1961 by Paris,Gomez and Anderson [21] and also in [24], who related (da/dN) tothe maximum stress intensity factor Kmax. Paris and Erdogan [22]subsequently noted that the work of Liu [23] implied that crackgrowth was a function of the stress intensity factor range DK.Forman et al. [25] subsequently extended the Paris equation toaccount for Kmax approaching its fracture toughness Kc, viz:

da=dN ¼ CðDKÞm=ðð1� RÞKc � DKÞ ð2Þ

In 1970 Hartman and Schijve [26] suggested that da/dN shouldbe dependent on the amount by which DK exceeds the fatiguethreshold DKth, i.e. on DK � DKth. In this paper we build on theHartman–Schijve concept, i.e. that da/dN should be dependent onthe amount by which DK exceeds the fatigue threshold DKth ofthe material under the associated test environment, to investigatea variant of the original Hartman–Schijve equation, viz:

da=dN ¼ DðDkÞa ð3Þ

where the crack driving force, DK, is taken to be

Dk ¼ ðDK � DKthÞ=pð1� Kmax=AÞ ð4Þ

and where D and A are constants and DKth is the fatigue thresh-old. This particular variant of the Hartman–Schijve equation hasthe advantage that it allows the constant a to be determined di-rectly from the slope of the da/dN versus Dj curve.

To examine the applicability of Eq. (3) let us first consider crackgrowth in a range of aerospace metals. To this end Fig. 1 presents aplot of da/dN against DK for the crack growth behaviour of theaerospace steel D6ac [27–29] for R ratios ranging from 0.1 to 0.9,

y = 2.31E-11 x2.66

R2 = 0.999

1.0E-11

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

0.1 1 10 100

K (MPa √metres)

da/dN(m

/cyc

le)

D6ac Kmax Ct3-5-lt

D6ac R = 0.9 Ct3-12-lt

D6ac R = 0.7 Ct3-25-lt

D6ac R = 0.8 Ct3-47-lt

D6ac R = 0.1 Ct3-46-lt

D6ac R = 0.3 Ct3-29-lt

D6ac R = 0.3 Ct3-10b-lt

D6ac R = 0.9 Ct3-27-lt

Δ

Fig. 1. Crack growth rate representation for D6ac steel.

y = 2.36E-10 x1.95

R2 = 0.988

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

0.1 1 10 100 1000

(ΔK ΔK- th)/(1-Kmax/Kcy)1/2 (MPa m)

da/d

N(m

/cyc

l e)

D6AC C3-5-ltD6ac Ct3-12-ltD6ac Ct3-27-ltD6ac Ct3-25-ltD6ac Ct3-47-ltD6ac Ct3-46-ltD6ac Ct3-29-ltD6ac Ct3-10b-lt4340 R = 0.14340 R = 0.710-Ni R = 0.110-Ni R = 0.8

√

Fig. 2. A Hartman–Schijve like representation of crack growth in D6ac, 4340 and10Ni–8Co–1Mo steels.

R. Jones et al. / Composite Structures 94 (2012) 1343–1351 1345

see Table 1. Here we see the traditional sigmoidal shape that iscommonly associated with the da/dN versus DK relationship. Thisdata, which shows little R ratio dependency, has a Paris (region)exponent of approximately 2.66 and a well defined fatigue thresh-old. Fig. 2 presents this data redrawn as per Eq. (3) together withthe data associated with the aerospace steels 4340 [27] and

Table 1Data and threshold values for the various tests.

Coupon Type oftest

Fatigue thresholdb DKIth

(MPap

m)A(MPa

pm)

D6acCt3-5-tl [27] Kmax = 20

LR2.0 220

Ct3-10b-lt [27] R = 0.3 Li 2.0 220Ct3-12-lt [27] R = 0.9 Li 2.0 220Ct3-25-lt [27] R = 0.7 Li 2.0 220Ct3-27-lt [27] R = 0.9 Li 2.0 220Ct3-29-lt [27] R = 0.3 Li 2.0 220Ct3-46-lt [27] R = 0.1 Li 2.0 220Ct3-47-lt [27] R = 0.8 Li 2.0 2204340 [27] R = 0.1 Li 7.5 1604340 [27] R = 0.7 Li 5.0 1604340 [36] R = 0.1 Li 1.5 1004340 [36] R = 0.7 Li 0.9 10010Ni-8Co-Imo

[30]R = 0.1 4.4 250

10Ni-8Co-Imo[30]

R = 0.8 1.8 250

Aluminium alloy7050-T7451

[33]R = 0.7 CA 0 32

7050-T7451[33]

R = 0.5 CA 0 32

7050-T7451[33]

R = 0.1 CA 0 32

7050-T7451[33]

R = -0.3 CA 0 32

7050-T7451[34] R = 0.7a 0.98 387050-T7451[35] R = 0.7 CA 1.4 327050-T7451[35] R = 0.5 CA 1.6 327050-T7451[35] R = 0.1 CA 2.5 32

LR = ASTM Load reducing test;Li = ASTM Load increasing test;Kmax = constant Kmax test;CA = constant amplitude loading, fixed R ratio;

a The nature of the test is unknown.b In this study the values of DKth were estimated from the da/dN versus DK data.

10Ni–8Co–1Mo [30]. (Here we have plotted da/dN (m/cycle)against the quantity Dj so that a relationship of the form y = x2

corresponds to Eq. (3) with a = 2.) Details of the nature of thesetests and the values of A and DKth used are given in Table 1.

Although [31] was the first to show that the growth of small submm cracks could be related to da/dN to (DK � DKth)2 [32] was thefirst to reveal that when formulated in this fashion, i.e. by relatingda/dN to (DK – DKth)2, the same relationship held for both long andsmall cracks. To illustrate how this approach unifies small and longcrack growth we examined the crack growth data presented in [33]for small, i.e. approximately 0.007 mm deep, initial cracks in 7050-T7451 aluminium alloy. To this end Fig. 3 presents a plot of the da/dN versus DK relationship presented in [33] for the R = �0.3, 0.1,

da/dN = 1.15E-09 ΔK2.03

R2 = 0.922

da/dN = 7.18E-10 K2.54

R2 = 0.964

da/dN = 2.89E-11 K3.65

R2 = 0.987da/dN = 5.27E-11 Δ

Δ

Δ

K3.12

R2 = 0.992

0.1 1 10 100

K MPa m

da/dN(m/cycle)

NASA Kmax DSTO CT R = 0.1DSTO CT R = 0.1 NASA R = 0.7NASA R = 0.1 DSTO CT R = 0.5Short crack Test 2 R = 0.1 Short crack Test 2 R = 0.7DSTO MT R = 0.2 Short crack Test 3 R = 0.1Short crack Test 3 R = 0.7

short crack effect ?10-11

10-10

10-9

10-8

10-7

10-6

10-5

da/dN = 1.34E-09 K2.06

R2 = 0.961Δ

Fig. 3. Comparison of the various da/dN versus DK test data for 7050-T7451, from[33].

Fig. 4. Long and small cracks in 7050-T7451.

Fig. 5. Long and small cracks in 4340 steel.

1 It would appear that Andersons et al. were unaware of either [26], [32], or [49].

1346 R. Jones et al. / Composite Structures 94 (2012) 1343–1351

0.5 and 0.7 short crack tests discussed in [33] together with thelong crack growth data presented in [34] and [35]. In this figurewe clearly see the so called short crack anomaly, first documentedin [36] whereby for a given DK the crack growth rate is signifi-cantly greater than that seen in tests with long cracks.

Fig. 4 presents this data redrawn as per Eq. (3) together with aplot of Eq. (3) with a = 2 and a value of D = 0.7 � 10�9. Here the val-ues of A and DKth used when plotting both the long and short crackdata are given in Table 1.

From Fig. 4 we see that when plotted in this fashion the long[34,35] and small crack data [33] essentially coincide. We alsosee that, in this instance, Eq. (3) with the value of the constant D(=0.7 � 10�9) and with a = 2 also provides a reasonably good repre-sentation of both the small and long crack data. As such the socalled short crack anomaly, which is evident in [33] and Fig. 3when da/dN is plotted against DK, vanishes when crack growth isplotted as per Eq. (3).

To further illustrate this Fig. 5 presents a plot of the short crackdata presented in [37] for short cracks in 4340 steel, in this in-stance the initial flaws had crack lengths that varied from 0.1 to0.13 mm, together with the 4340 long crack data presented in

[27] and shown in Fig. 2. Here we see that when plotted in thisfashion the long [27] and short crack data [37] (again) essentiallycoincide. We also see that, in this instance, Eq. (3) with the valueof the constant D (=2.3 � 10�10) and a = 2 also provides a reason-ably good representation of both the small and long crack data.

2.1. The fatigue threshold DKth

Before discussing the applicability of Eq. (3) to composites weneed to discuss the dependency of the fatigue threshold DKth onspecimen geometry as well as on the test method. Although inthe development of the ASTM test standards [30] it was found thatthe value of DKth was specimen geometry independent subsequenttests [38–43] have shown that this is not always the case. It hasalso been suggested [44–46] that the ASTM load reducing testmethodology can produce erroneous values for DKth, and thereforeerroneous Region I data. However, this conclusion was questionedin [47]. In [39] it was reported that for high R ratio tests the mea-sured value of DKth appears to be very much less geometry and testmethodology dependent. Consequently, in those few cases whereconstant amplitude, Kmax (Gmax), or load increasing test data wasnot available we analysed high R ratio load reducing tests, i.e.R = 0.5 or greater. Similarly materials that show a pronounced‘Marci effect’ [48], i.e. where the fatigue threshold vanishes at highvalues of Kmax (Gmax), were also excluded from this study.

Having briefly discussed the origin of fatigue crack growth lawsas applied to metals, and how crack growth can often be repre-sented by a variant of the Hartman–Schijve equation with an expo-nent (a) that is approximately equal to 2, as first suggested in [49],let us now address delamination growth in composites.

3. A Hartman–Schijve variant for composites

In the previous sections we have seen that the growth of smallsub mm defects can play a central role in determining the fatiguelife of aircraft structures and that the crack growth rates associatedwith such small defects do not generally correspond to the Parislike representations determined from ASTM standard tests on largedefects. In this context we have also seen that if we express da/dNin terms of the amount by which DK exceeds its threshold ratherthan by DK alone then, for metals, this anomalous behaviour canvanish. It has also been shown that if the delamination growth rateis expressed using Paris like formulations we obtain relatively largeexponents and that the logical conclusion of such a formulation isthat designs should be kept beneath the fatigue limit. A problemwith this approach is that it is known that the fatigue limit associ-ated with naturally occurring defects under operational loading isvery much lower than that determined using ASTM standard testsand that experience has shown that small sub mm defects cangrow in service event though the structures have passed proof loadtests. It may therefore be advantageous to seek to represent delam-ination growth using equations that are based on the Hartman–Schijve concept rather than on Paris like equations.

Andersons et al. [50] were the first to derive an equation thatwas similar to the Hartman and Schijve equation1 for representingdelamination growth in composites, viz:

da=dN ¼ C½ðDK I � DK IthÞ=ð1� K I max=K IcyÞ�b ð5Þ

Here [50] explained that K is proportional top

G and that thefactor of proportionality is a function of the elastic moduli, seeSih, Paris and Irwin [51]. In Eq. (5) the terms b and C are constants,Kmax is the maximum value of K seen in the cycle and KIcy is theapparent (Mode I) cyclic toughness.

Fig. 6. The Paris representation of Mode I delamination growth in severalcomposites, from [50].

Fig. 8. The Hartman–Schijve representation for Mode II delamination in severalcomposites, adapted from [50].

R. Jones et al. / Composite Structures 94 (2012) 1343–1351 1347

Inspecting Eqs. (5) and (3) it would appear that when Kmax

(Gmax) is much less than the apparent cyclic toughness Kcy (Gcy),i.e. in Region I and the lower section of Region II, then Eq. (5)and the Eq. (3) are (essentially) operationally the same. Unfortu-nately, although [50] showed that delamination damage in severalcomposites conformed to Eq. (5) they did not give the associatedvalues of b.

This relation, i.e. Eq. (5), was shown [50] to hold for both ModesI and II growth, i.e. KI (GI) and KII (GII), in a range of unidirectional

y = 7.62E-08 x2.14

R2 = 0.981

y = 1.37E-08 x2.20

R2 = 0.981

y = 4.93E-09 x2.01

R2 = 0.830

y = 5.56E-07 x2.09

R2 = 0.982

1.0E-12

1.0E-11

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

0.001 0.01 0.1 1 10

(Δ ΔKI- KIth)/(1-KImax/KIc)1/2 (MPa √m)

da/dN(m

/cyc

le)

T300-2500 Mode I R = 0.2T300-2500 Mode I R = 0.5T300-2500 Mode I R = 0.7AS4 Peek R = 0.2AS4 Peek R = 0.5T800-#3631 R = 0.2 T800-#3631 R = 0.5T300-#3631 R = 0.2T300-#3631 R = 0.5

Fig. 7. The Hartman–Schijve representation of Mode I delamination growth inseveral composites, adapted from [50].

laminates provided that the corresponding values of DKth and Kc

were used. With this in mind equations for Modes I and II delam-ination growth can be formulated, viz:

da=dN ¼ DððDpGI � Dp

GIthÞ=pð1�pGI max=

pGIcyÞÞb ð6Þ

da=dN ¼ DððDpGII � Dp

GIIthÞ=pð1�pGII max =

pGIIcyÞÞb ð7Þ

where the terms Dp

GIth andp

GIcy are perhaps best viewed asparameters that are used to ensure that the entire range of data fitsthe equations A Paris like representation of the Mode I delaminationgrowth data presented in [50] is presented in Fig. 6. Here we see, asremarked above, that the Paris exponent m is very large. Fig. 7 plotsda/dN against ðDK I � DK IthÞ=

pð1� K I max=K IcyÞ for the Mode I datapresented in [50] whilst Fig. 8 plots da/dN against ðDK II � DK IIthÞ=pð1� K II max=K IIcyÞ for the Mode II data presented in [50]. In bothcases, i.e. for both Modes I and II, we see an exponent b that isapproximately equal to 2, allowing for experimental error, so thatthese data sets can be reasonably well represented by Eq. (3) witha, or the value of b in Eq. (6) and (7), approximately equal to 2.As such with this formulation, unlike the ‘‘Paris like’’ formulations,the exponent now has a value that is similar to that seen for metals.This suggests that when formulated as per Eq. (3), or alternativelyas per Eqs. (6) and (7), a traditional damage tolerance approachmay be possible. This approach also reinforces the importance ofan accurate knowledge of the thresholds GIcy and GIIcy. The values

Table 2Mode I delamination fracture toughness, threshold and R ratio.

Material KIc (MPap

m) DKIth (MPap

m) R ratio

T300/2500 1.57 0.640 0.2T300/2500 1.57 0.518 0.5T300/2500 1.57 0.385 0.7T800/#3631 1.28 0.716 0.25T800/#3631 1.28 0.530 0.5T300/#3631 1.28 0.710 0.5T300/#3631 1.28 0.502 0.5AS4/PEEK 4.2 1.250 0.2AS4/PEEK 4.2 1.000 0.5

Table 3Mode II delamination fracture toughness, threshold and R ratio.

Material KIIc (MPap

m) DKIIth (MPap

m) R ratio

T800H/#3631 4.5 0.95 0.2T800H/#3631 4.5 1.15 0.5T800/3900 8.5 2.10 0.1T800/3900 8.5 2.12 0.5AS4/PEEK 11.0 2.02 0.5AS4/PEEK 11.0 2.09 0.5

1348 R. Jones et al. / Composite Structures 94 (2012) 1343–1351

of the constants DKIth and KIc and DKIIth and KIIc used in these plotsand the particular fibre matrix configurations studied are given inTables 2 and 3 respectively.

Fig. 9. A Paris like law representation of the ESIS TC4 round robin data on IM7/977-2 obtained at Empa [12].

y = 1.16E-09 x2.13

R2 = 0.923

y = 9.90E-10 x2.03

R2 = 0.811

y = 1.67E-09 x2.14

R2 = 0.927

1.0E-11

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

0.1 1 10 100 1000

da/dN(m

/cyc

le)

( √GI - ΔΔ √GIth)/√(1-√GImax/√GIcy) √(J/m2)

DCB1DCB2DCB3DCB4DCB5

Fig. 10. The Hartman–Schijve representation of the ESIS TC4 round robin data onIM7/977-2 obtained at Empa [12].

To further evaluate the applicability of Eq. (6) to representdelamination growth let us consider the Mode I delamination dataobtained by Brunner [12] as part of the Technical Committee onFracture of Polymers, Composites and Adhesives of the EuropeanStructural Integrity Society (ESIS) activities with regard to fatiguedelamination propagation discussed in [12]. The specimens weremade from unidirectional carbon–fibre epoxy (IM7/977-2). The re-sults of five different R = 0.1 DCB tests, labelled DCB1 to DCB5 areshown in Fig. 9 where we plot da/dN versus Gmax. (Note the scatterin the data at low values of da/dN.) This data is re-drawn in Fig. 10where we plot da/dN against (D

pGI � D

pGIth)/

p(1-p

GImax/p

GIcy)with the values of D

pGIth and GIcy used being listed in Table 4. Note

that in this instance the scatter associated with the low delamina-tion growth rates, i.e. da/dN < 10�9 m/cycle, appears greater inFig. 10 than in Fig. 9.

The results of ESIS tests on unidirectional AS4-PEEK laminates,labelled 3205-7 to 3205-10 are shown in Figs. 11 and 12. The spec-imens have been made from unidirectional laminates and had beenstored at ambient conditions for an extended period of time. Inboth cases we again see an exponent of approximately 2 so thatthe data can be reasonably well represented by Eq. (6) with bapproximately equal to 2. As such we again see that with this for-mulation the delamination growth law resembles that seen forcrack growth in metals.

Let us next consider the Mode II delamination data presented byMatsubara, Ono and Tanaka [52] for a unidirectional T-glass fibrein a 180 �C cure Toray #3651 epoxy resin tested at a range of R

Table 4Mode I delamination fracture toughness and threshold.

Specimen GIc (J/m2) Dp

Gth (J/m2)

Carbon fibre epoxy specimen numberDCB1 134 8.18DCB2 170 9.27DCB3 220 9.04DCB4 180 8.08DCB5 190 9.31

AS4-PEEK specimen number3205-7 585 12.243205-8 880 19.873205-9 1432 20.983205-10 1200 18.71

Fig. 11. The Paris representation of delamination growth in AS4-PEEK (ESIS TC4round robin data obtained at EMPA).

Fig. 12. The Hartman–Schijve representation of delamination growth in AS4-PEEK(ESIS TC4 round robin data obtained at EMPA).

Fig. 13. Paris representation of Mode II delamination growth in Tglass-#3651, from[52].

Fig. 14. Hartman Schijve representation of Mode II delamination growth in Tglass-#3651.

Table 5Mode II delamination fracture toughness and threshold.

GIc (J/m2) Dp

Gth (p

J/m2) R

Tglass#3651 1750* 9.4 �0.5Tglass#3651 1750* 13.6 0.1Tglass#3651 1750a 14.2 0.3

a From [52].

R. Jones et al. / Composite Structures 94 (2012) 1343–1351 1349

ratio’s. The experimental data is reproduced in Figs. 13 and 14 andthe constants used are given in Table 5. Fig. 14 again reveals anexponent of approximately 2 so that the Mode II delaminationgrowth rate can be reasonably well represented by Eq. (7) with bapproximately equal to 2.

3.1. Discussion

The examples discussed above have shown that, although thereare a few instances where the scatter in the data is moderatelylarge, this formulation appears to hold for a number of problemsassociated with Modes I and II delamination growth in compositesand in each case the exponent b is approximately 2 which is con-sistent with the value found for the growth of both long and shortcracks in metals. Indeed, as shown in Section 2, the formulation isconsistent with the behaviour of a range of aerospace metals. Forthose problems that can be modelled using crack closure based ap-proaches it can also be recast in terms of crack closure concepts

and as such may be considered to be a variant of the McEvily crackgrowth law [53–55], see Appendix A.

One major difference between the present formulation and themore traditional Paris based laws lies in its potential to representthe growth of small naturally occurring defects. The design philos-ophy currently used for composite structures is to ensure that theenergy release rate associated with any defect lies beneath thethreshold determined from tests on specimens containing large de-fect. (This is the so called ‘‘no growth’’ design philosophy.) Unfor-tunately it is known that, for metals, predictions based on Parislike crack growth equations that use data obtained from tests onlarge cracks produce highly unconservative estimates for the fati-gue life of structures containing small naturally occurring defects.This phenomena is aptly illustrated in [57] which studied thegrowth of a small 0.003 mm initial defect in a 7050-T7451 alumin-ium bulkhead of an F/A-18 aircraft subjected to a representativefighter load spectra [57]. In this context the data presented in Sec-tion 2 for 7050-T7451 aluminium alloy has shown that the growthof both long and naturally occurring short cracks conform to theHartman–Schijve equation. This suggests that the present formula-tion, with the constants determined as above from the long delam-ination tests but with the threshold set to zero, may enable aconservative estimate of the time required for a defect to growfrom a small naturally occurring defect to an detectable size tobe computed. If this hypothesis can be substantiated then the cur-rent no growth philosophy could be extended to a statement suchas:

Within the lifetime of the aircraft there should be no growth toa size greater than that detectable using existing NDI tools.

4. Conclusion

The through life management of aircraft structural componentsrequires a means for assessing the effect of small naturally occur-ring defects on the structural integrity of the aircraft. In this con-text it has long been known that the da/dN versus DK

1350 R. Jones et al. / Composite Structures 94 (2012) 1343–1351

relationship and the associated fatigue thresholds obtained usingASTM standard tests are not readily related to the behaviour ofsuch small naturally occurring defects. As a result alternative nonParis based growth equations capable of representing both longand small crack (delamination) growth are required. This paperpresents one such possible formulation. To this end the experimen-tal data presented in this paper reveals that Modes I and II delam-ination growth can often be reasonably well represented by avariant of the Hartman–Schijve equation. We also find that whenexpressed in this form the exponent of this variant is approxi-mately 2 and as such is considerably lower than the exponent in‘‘Paris like’’ power law representations. As such we see that, inthese examples, the present variant of the Hartman–Schijve repre-sentation of delamination growth in composites is similar to thatseen for crack growth in metals. This suggests that the present for-mulation may be useful for the damage tolerant assessment ofsmall naturally occurring defects in composite structures.

Acknowledgements

The authors would like to acknowledge material supply for se-lected Mode I fatigue test data reported on IM7/977-2 (prepregfrom Cytec Inc., specimens prepared by Dr. D.D.R. Cartié at Cran-field University, UK) and for AS4/PEEK (from Montan University,Leoben, Austria).

Appendix A. A crack closure variant

The examples given above reveal that there are numerousmaterials which exhibit a da/dN versus D

pG relationship that is

a function of the R ratio. For those problems that can also be mod-elled using crack closure based approaches let us follow the ap-proach first suggested by Elber [56] and define an effective stressintensity factor range D

pGeff ð¼

pGmax �

pGopÞwhere Gop is the va-

lue of G at which the delamination first opens, and a function U(R),which is a function of the R ratio:

Dp

Geff ¼ UðRÞDpG ðA:1Þ

such that the various R ratio dependent da/dN versus Dp

G curvesreduce to a single da/dN versus D

pGeff curve which is (essentially)

coincident with the high R ratio da/dN versus Dp

G curve(s) for thematerial, see [56]. In such cases adopting the Hartman–Schijve ap-proach that da/dN is a function of the quantity (D

pG � D

pGth),

leads to an alternative relationship, viz:

da=dN ¼ HððDpGeff � Dp

Geff ;thÞ=pð1� Gmax=AÞÞb ðA:2Þ

It is interesting to compare this equation with that proposed in[53,54], viz:

da=dN ¼ FðDKeff � DKeff ;thÞ2 ðA:3Þ

where F is a constant which has been shown to hold for both shortand long cracks.

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