International Journal of Fatigue 33 (2011) 492–499

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International Journal of Fatigue

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Technical note

Fatigue crack initiation at a notch

N. Ranganathan ⇑, H. Aldroe, F. Lacroix, F. Chalon, R. Leroy, A. TouguiUniversité François Rabelais de Tours, Laboratoire de Mécanique et Rhéologie, Polytech Tours, Département Mécanique et Conception de Systèmes, 7 Avenue MarcelDassault, 37200 Tours, France

a r t i c l e i n f o

Article history:Received 7 January 2010Received in revised form 1 September 2010Accepted 13 September 2010Available online 21 September 2010

Keywords:Fatigue crack initiationNotchLocal strainsModellingMarker bands

0142-1123/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijfatigue.2010.09.007

⇑ Corresponding author. Tel.: +33 2 47 64 36 17; faE-mail address: ranganathan@univ-tours.fr (N. Ran

a b s t r a c t

A short crack approach has been developed to determine the fatigue crack initiation life at a notch tip, inwhich the initiation life is assumed to correspond to the period during which an initial crack, equivalentto a microscopic defect, grows to attain a detectable size. This approach is compared to conventional localstrain approach and it is shown that the short crack approach gives acceptable predictions as compared toexperiments, in the 2024 T351 alloy. In the 7449 alloy, a different approach, using marker-band tech-nique has been applied to determine the short crack growth kinetics at a notch. The advantages and dis-advantages of the conventional strain based approach and the short crack model are discussed.

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1. Introduction

Determination of fatigue crack initiation life at a notch is veryimportant in mechanical, aeronautical and nuclear industries. Tothis end, classical methods are based on the determination of localstrain amplitudes in the vicinity of a notch using the Neuber’sempirical approach or the Glinka’s energy based model [1,2]. Thestrain amplitudes can then be used to determine an equivalentdamage based on the strain life relationship of the material deter-mined from conventional fatigue tests. A crack is supposed to ini-tiate at the notch tip when the damage sum reaches 1. Linear ornon linear damage laws can be used to this end. But one doesnot know the size of the initiated crack when the critical damagelevel is attained at the notch tip [3–5].

More recently, short crack propagation models have been pro-posed, that consider, usually, that a crack-like defect exists at thenotch tip right from the first cycle. Based on such hypothesis, thecrack initiation life represents, in fact the propagation life of theinitial crack as it grows under the fatigue loading to reach a size,detectable by current non-destructive inspection methods [6,7].One of the main questions regarding the short crack approach isthe assumption that there is no crack initiation period. Issuesregarding short crack propagation kinetics have been addressedin [7,8]. Another difficulty is that experimentally it is difficult tofollow short crack propagation. One can use metallurgical replicas

ll rights reserved.

x: +33 2 47 36 13 11.ganathan).

to follow surface crack growth [7,8]. In other approaches, the fati-gue crack growth kinetics at a notch is considered to be a two stageprocess. In the first stage the crack growth is governed by the notchplastic zone, and in the second stage by the crack tip plastic zone[9].

While there have been discussions on the validity of Neuber’s orGlinka’s approaches, the present study is focussed on determiningthe validity of the short crack approach based on experimentalobservations.

In this paper, we first study the short crack growth kinetics at anotch on notched specimens of the 2024 T351 aluminium alloy byconducting fatigue tests at positive load ratios on thin plates. Inthese tests short crack growth emanating from the notch was mon-itored by using metallurgical replicas. In this part of the study wecompare experimentally measured lives with predicted lives usinglocal strain based models and the short crack model, developed inour laboratory. This analysis shows that the short crack approachcan lead to good predictions as compared to strain based ap-proaches. However, these results also indicate at long lives theshort crack approach may lead to conservative life estimates,implying that crack initiation phase has to be considered.

To address this issue, specially designed tests are conducted inanother aluminium alloy, where fatigue crack initiation and crackgrowth kinetics at a notch are studied by using fatigue a marker-band technique. This technique permits to detect very low growthrates on the order of 10�9 m/cycle [10]. We show that the analysisof these results indicate that crack initiation period, under certaintest conditions, may not be negligible as compared the crackgrowth period. These aspects are then considered to propose

Table 1Paris law constants for the 2024 alloy.

DK range (MPaffiffiffiffiffimp

C n

N. Ranganathan et al. / International Journal of Fatigue 33 (2011) 492–499 493

flimiting domains for the use of the short crack model. The advan-tages and the disadvantages of the local strain approach and theshort crack model are also briefly discussed.

1–2 1e-8 7.592–5 4.58e-8 1.975–10 1.67e-8 4

2. Theoretical basis

2.1. Classical local strain approach

In this local strains at the notch tip are obtained by using eitherthe Neuber approach or the Glinka model. The former is based onthe following hyperbolic relationship:

K2T ¼ Kr � Ke ð1aÞ

where, KT is the geometrical stress intensity factor, Ke is the localstrain concentration factor (the ratio of the local to remote strain)and Kr is the local stress concentration factor (the ratio of the localto remote stress)

In the case of fatigue loading, this equation can be rewritten as,

ðKT � Dr1Þ2

2E¼ Dr � De ð1bÞ

In this equation, �Dr1 represents the remotely applied stressand Dr and De, the stress and strain amplitudes at the notch tip.

In Glinka’s approach the local strains and stresses should repre-sent an energy equivalence as compared the remote loading condi-tions, leading to the following equation:

ðKT � Dr1Þ2

2E¼ Dr2

4Eþ Dr

n0 þ 1Dr2K 0

� � 1n0

ð2Þ

In this equation K0 and n’ correspond to the material’s cyclichardening law.

To determine the local strains, either Eq. (1) or (2) can then becoupled with a cyclic stress–strain response model (Ramberg–Os-good constitutive Eq. (3) lead to a system of two equations, andDr, the local stress range and De, the local strain range for each cy-cle which can be numerically determined.

De2¼ Dr

2Eþ Dr

2K 0

� � 1n0

ð3Þ

A comparison of both ways of evaluating local stress and strainhistories was done by Glinka et al. [2], and they give quite close re-sults, comparable with finite element method (FEM) estimation.

The local strain amplitude can then be used to determine thelife to failure by using the strain life relationship of the form

De2¼ rf 0

ENrb þ ef 0Nrc ð4Þ

where rf0, b, ef0 and c are materials constants determined constantamplitude fatigue tests and Nr the number of cycles to failure forsmooth specimens in the low cycle and high cycle life ranges sub-jected to fully reversed loading (Re = �1). The life to failure thus ob-tained corresponds in fact to fatigue crack initiation life Ni at anotch.

Alternatively, one can determine the damage corresponding toeach remote loading cycle i in a variable loading spectrum usingan elementary damage variable, Di.

Di ¼1Ni

ð5aÞ

Generally on can use linear summation models to determine tolife to crack initiation according to

XN

i¼1

Di ¼ D ð5bÞ

Here N represents the number of cycles in the remote loading spec-trum containing N cycles. A crack is supposed to exist when thedamage D attains the value of one. Thus, implicitly, damage in thisapproach corresponds to the pre-crack initiation phase. In a moresophisticated approach, non linear damage summation modelscan be used [5]. But such an approach require specially designedtests to indentify the model parameters [5]

In a software developed in our laboratory, the above mentionedlocal strain approaches have been incorporated along with the fol-lowing mean stress corrections (applicable if the local strains arepurely elastic): Morrow, Manson–Halford and Smith–Watson–Topper. The Manson–Coffin expression for the strain-life curve isavailable too. Details of these corrections are given in [8,11].

This strain based approach does give life to crack initiation at anotch, but it does not give the size of the initiated crack.

2.2. The short crack model

In the ‘short crack model’, the local stress amplitudes are usedto determine the fracture mechanics parameter DK, by taking intoaccount the crack length ‘‘a”. Analytical expressions are finite ele-ment estimations are available for common geometries such as athrough-thickness crack, a corner crack or an embedded crack inRefs. [6,11,12].

In the computer programme developed in our laboratory, ana-lytical expressions from [6,8,11,12] are incorporated to determinethe local stress intensity factor amplitude DK depending on thecrack geometry.

It has been shown, for a given crack length,

Kthrough thickness > Kembedded > Kcorner crack

This aspect has to be born in mind, as it is quite difficult to pre-dict the kind of crack configuration, for a given loading conditionsand the material studied. The initial crack size is the typical micro-structural feature size, such as an inclusion or a cavity. In fact, inthe short crack model, unlike the local strain approach, there isno ‘initiation phase’ as such. Once the crack driving force parame-ter is evaluated, the crack advance per cycle has to be computed.The crack growth law used represents the behaviour of long cracksat high R ratios, using Paris’ law (Eq. (6)). In the case of the 2024T351 alloy, three different domains were considered with differentslopes and constants c in different ranges of DK–Table 1 [7,8].

dadNðmm=cycleÞ ¼ cDKn ð6Þ

The crack initiation life can now be obtained by a linear summa-tion of crack advance between the initial and final crack lengths, ai

and af.

NiðpredÞ ¼Z af

ai

dacDKn ð7Þ

The main differences between the local strain based approachand the short crack approach are given in Fig. 1.

3. Brief experimental details

The present study was conducted on two aluminium alloys, the2024 alloy in the T351 condition and the 7449 alloy in the T7951

Fig. 1. Flow-chart for the short crack and local strain based approaches.

Table 2Composition (in wt.%) of the studied alloys (Al not indicated).

Alloy Zn Mg Cu Cr Fe Si Mn Ti

2024 0.04 1.5 4.46 0.01 0.22 0.1 0.66 0.027449 7.5–8.7 1.8–2.7 1.4–2.1 – 0.15 max 0.12 max 0.2 max 0.25 (Ti + Zr)

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condition. The nominal composition and full mechanical propertiesare given in Tables 2 and 3 respectively. Fig. 2 shows the staticstress–strain behaviour of the studied materials.

Fig. 2 compares the static behaviour of the two studied material.In the 2024 T351 alloy, the experiments were carried out using

the following specimen geometry, Fig. 3.Variable amplitude tests were carried out using a portion of the

TWIST spectrum, called TWIST extract. A sample of the load spec-trum is given in Fig. 4.

The short crack growth was monitored using optical measure-ment on the polished surface of the specimen, and replicas weremade at regular intervals, with a view to detect cracks as early aspossible (the minimal detectable size was close to 10 lm). These

Table 3Nominal mechanical properties.

Alloy Young’smodulus (GPa)

Yield strength(MPa)

Ultimatetensilestrength (MPa)

Elongation(%)

2024 T351 73 300 500 167449 T7951 71.5 598 623 8

Fig. 2. Static stress–strain diagrams.

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replicas were then observed with an optical microscope at a mag-nification of (1000�).

For constant amplitude loading, the stress level was held at themaximum stress value during the replicating process, whereas forthe variable amplitude loading, the stress was held ‘‘as is” in orderto avoid overloads.

Fig. 3. Notched specimen for tests in the 2024 T351 alloy, KT = 3.14(thickness = 1.2 mm).

Fig. 4. Typical flight from TWIST extract (23 flights): normalized stress versusnumber of cycles.

Fig. 5. Notched specimens for tests in the 7449 T7951 alloy, KT = 3.41(thickness = 5 mm).

3.1. Marker band tests

These tests were carried out on the Al alloy 7449 Al alloy in theT7951 temper.

Constant amplitude fatigue crack propagation tests from nearthreshold to moderate growth rates were carried out using10 mm thick CT50 specimens. In these tests crack growth ratewas monitored on the polished size of the specimen using an opti-cal microscope (X25).

Fig. 6. Marker band cycles.

Table 4Comparison between predicted crack initiation lives (corner and through-thicknesscrack configurations) and measured ones, constant amplitude loading.

Max. remote stress (MPa) R Ni (cycles) prediction Ni (cycles)measured(M)

Ratio

CC/M TT/MCornercrack(CC)

Through-thickness(TT)

160 0.1 32,230 20,160 36,000 0.90 0.56160 0.1 34,810 21,360 32,900 1.06 0.65160 0.1 34,960 21,420 27,000 1.29 0.79140 0.1 57,580 31,830 40,000 1.44 0.80140 0.1 57,580 31,830 58,000 0.99 0.55120 0.1 119,650 57,110 82,000 1.46 0.70120 0.1 119,650 57,110 189,000 0.63 0.30180 0.1 24,400 15,770 29,190 0.84 0.54180 0.1 24,400 15,770 29,000 0.84 0.54180 0.3 58,030 32,030 37,000 1.57 0.87180 0.3 58,030 32,030 40,000 1.45 0.80154 0.3 120,370 57,430 63,710 1.89 0.90216 0.5 119,560 57,070 49,000 2.44 1.16252 0.5 57,730 31,890 29,000 1.99 1.10

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The crack initiation tests on a notched specimen was then car-ried out, using a single edge notched specimen, KT = 3.41 [10]. Twotests at a load ratio of 0.1, were carried out with nominal remotestress amplitudes of 120 and 140 MPa respectively. The specimengeometry is given in Fig. 5. The specimen thickness was 5 mm.

The marker-band technique used here consists of adding peri-odically low amplitude, high R ratio cycles with a small sequenceof overload–underload cycle. The loading pattern used is given inFig. 6. This type of marker-band cycles were found to give detect-able traces (under a scanning electron microscope) even at verylow growth rates [10,13].

Tests with marker bands were initially carried out during con-stant amplitude tests on CT specimens to verify that marker bandsare observed in all the crack growth rage studied. This kind of mar-ker bands have been previously used to study crack growth kinet-ics under flight simulation loading [13] Marker bands observationswere made under JEOL KV4860 scanning electron microscope.

4. Experimental results and analysis

4.1. 2024 T351 alloy

Fig. 7 shows the evolution of crack length for three tests carriedout at a load ratio of 0.1 and at a nominal stress amplitude of160 MPa.

In this figure, experimental measurements are compared withpredictions using Eqs. (6) and (7), considering through thicknessor edge crack geometries. It can be seen that the crack geometrycan be either of the through thickness type or a corner crack andthat the crack grows faster in the through thickness configuration.

The results for constant amplitude tests are compiled in Table 4.In this table the predicted lives correspond to the summation of Eq.(7) with ai = 10 lm and af = 300 lm.

Comparing predicted and measured lives the ratios reported inthe above table lead to a range of 1.34 ± 0.51 for the corner crackconfiguration while the corresponding range is within 0.73 ± 0.23for the through thickness configuration Fig. 8.

This indicates that the through-thickness crack configurationcan be an acceptable compromise for prediction of crack initiationlives using the short crack approach for this alloy and for the spec-imen geometry. The predictions are always conservative for thethrough thickness configuration except for two tests at R = 0.5.However, it can also be seen that at long lives the predictionscan be over conservative – but in these cases (at a stress amplitudeof 120 MPa and R = 0.1), slightly better estimations are obtainedusing the corner crack configuration.

Fig. 7. Measured crack length evolutions for three tests as compared to estimatedones for two crack configurations.

A similar analysis was carried out for the variable amplitudetests using the TWIST extract spectrum. It was observed that forthe three tests studied here, the crack geometry observed wasthe through-thickness crack. The life range covered was relativelyshort from 2500 to 4600 flights and the predictions using thethrough thickness configurations were within 10% of the measuredlives (slightly non conservative).

A further analysis was carried out for a test programme carriedout the working group No. 4 of the Société Française de métallurgieet matériaux (SF2M) [14]. In this study fatigue crack initiation atnotch was studied on the same alloy using a theoretical aeronauti-cal wing spectrum covering a large life range. The KT value of thestudied geometry is 2.75 [14].

These results are now analyzed using the conventional ap-proach and the short crack model. For the conventional approach,Smith–Watson–Topper model (SWT) and Morrow’s correctionwere used [11]. It can be seen here that the short crack model pre-dictions are quite comparable with those obtained from the SWTmodel. The Morrow correction seems to give slightly non conserva-tive predictions. For the lowest stress level tested, experimentalscatter data is available as the experiments were repeated. Thescatter in prediction was considered for the short crack model for

Fig. 8. Comparison of measured and predicted lives: local strain and short crackapproaches – aircraft wing spectrum.

Fig. 9. Marker bands (shown by arrows) at low DK values, CT specimen.

Fig. 10b. First two marker bands (shown by arrows).

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clarity’s sake. Scatter in prediction for this model is basically gov-erned by the initial flaw size (assumed to between 5 and 20 lm)and the slope of Paris law .

4.2. 7449 T7951 alloy

In this alloy the studies are intended to determine the crackgrowth kinetics at a notch using marker-band cycles. The kind ofmarker bands (Fig. 6) were determined by a careful study of differ-ent configurations possible. It was first verified from constantamplitude tests on CT specimens that Marker bands are visiblefrom very low growth rates near threshold conditions and thatthe Marker bands do not induce any load interaction effects [9].

Fig. 9 shows marker bands from SEM observations, for nearthreshold conditions. Three marker bands are clearly visible. Thecrack growth direction is from right to left. The marker bands markthe crack front and it can be seen that the crack front geometry isirregular and the distance between the bands are not constant.Typically five measurements are made and we get an average valueof 30 lm. The interval between successive marker bands applica-tion is 10,000 cycles. Thus the average growth rate, obtained frommarker band spacing is 30 � 10–6/10,000, i.e. an average growthrate of 3 � 10�9 m/cycle, which is quite comparable to the macro-scopic growth rate measured by optical methods on the specimensurface.

Figs. 10a and 10b show marker bands on a notch fatigue test.

Fig. 10a. Overall view crack at a notch.

In Fig. 10a one can see a general view of the fracture surface forthe test conducted at a stress amplitude of 120 MPa. The first twomarker bands are shown by arrows in Fig. 10b. The distance be-tween the bands is again not uniform and one can estimate anaverage growth rate of 8 � 10�10 m/cycle. In this case, the crackstarted spreading out in the radial direction after this stage.

A second test was carried out at higher stress amplitude of140 Mpa. In this case also marker bands were clearly detectableright from the initial defect, an example is shown in Fig. 11. Thespacing between the bands is again irregular indicating thatgrowth rates are non uniform along the crack front. From thesemeasurements an average spacing between the bands was deter-mined from three locations. This average distance was convertedto equivalent growth rates as the number of cycles between eachmarker band load application is 10,000 cycles. These results are gi-ven in Fig. 12 for both the tests, in which the estimated growthrates are given with respect to the square root of distance fromthe notch (this representation is chosen to give a fracture mechan-ics based analysis as the stress intensity factor is related to thesquare root of crack length). It can be seen here that the shortestcrack lengths, the crack growth rate can be about 10 times higherat 140 MPa than at the lower stress. The differences between thetwo tests are negligible at the end of the tests (crack length closeto failure).

These curves can then be numerically integrated to get thenumber of measured crack growth cycles Np. Then knowing the

Fig. 11. Marker bands for second test.

Fig. 12. Comparison crack growth rates measured from marker band spacing forboth the tests.

Table 5Crack initiation and propagation lives from marker band analysis.

Stress amplitude (Mpa) Cycles (Nf) Cycles (Np) Cycles (Ni)

120 3.48E5 2.65E5 8.3E4140 1.93E5 1.87E5 6E3

Table 6Comparison of both approaches.

Local strain approach Short crack model

Estimation of notch tip strainamplitude – Neuber or Glinkamodels

Estimation of notch tip strainamplitude – Neuber or Glinka models

Strain life relationship Appropriate Paris lawDamage summation Determination of crack advanceLife to crack initiation when damage

equals oneLife corresponds to a detectable cracksize

Crack size unknown K calibration needed for new crackconfigurationsCrack geometry unknown

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number of cycles to failure, Nf. Then the number of crack initiationcycles can be estimated by the following equation:

Ni ¼ Nf � Np ð8Þ

These results are given in Table 5 for the two tests.This analysis indicates that for the test at 120 MPa the crack ini-

tiation period seems to be significant (30% of total life) while forthe test at 140 MPa, the initiation period is negligible (3% of totallife).

5. Discussion

The study conducted on the 2024 T351 alloy shows that theshort crack approach to fatigue crack initiation at a notch can givevery good results for constant amplitude as well as variable ampli-tude tests.

It should be borne in mind that these tests correspond to thinsheets and where plane stress conditions prevail.

In a previous study [15] it has been shown that a short crackgrowth model need to very good predictions for crack initiationlives in the aluminium alloys 2024 T3 and the 7075 T6 and the4340 steel. In this study, crack closure was taken into accountand the initial crack sizes assumed were between 6 and 20 lm. Dif-ferent crack configurations were considered, similar to the ones as-sumed in the current study. It was also shown that in the case ofthicker plates, one of the crack configuration possible is the semicircular embedded crack. We have found a similar result in the7449 alloy, confirming the presence of semi circular embeddedcracks (Fig. 10a). It should be remarked that for this alloy the spec-imen thickness was 5 mm.

In a larger perspective, a comparison can be made between theshort crack and the classical local strain approach, as in Table 6 –see also Fig. 1.

In the analysis given in [4], it has been proposed that the Neuberapproach gives acceptable results under plane stress conditionswhile the Energy based model seems to give better results underplane strain conditions. In the present study, the energy modelcoupled with Smith–Watson–Topper correction for mean stressgave results closest to experimental lives.

The main advantages of the short crack model is that strain liferelationship is not needed and one can use an appropriate Paris’law to determine the crack growth life. The main disadvantagesare the fact the crack configuration is unknown, but it appearsfor thin sheets a through-thickness crack growth can give accept-able results, for thin sheets.

The tests in the 7449 alloy using marker bands technique showthat crack initiation period cannot be neglected for long lives. Withrespect to the short crack model, this would indicate that at longlives the short crack model estimations can be conservative if theinitiation period is not taken into account, a result reported above.In another study made on cast iron, it was shown that the crack ini-tiation life in the vicinity of an inclusion (graphite) corresponds tothe period to create a stable crack configuration starting from theinitiation site, even though this period involves unevenly localizedcrack propagation [16].

Further tests are being carried out to ascertain the life belowwhich the crack initiation period can be neglected, to define thevalidity range of the short crack approach. Current results indicatethat this limiting life could depend upon the material tested.

Another aspect not treated in this study, is the possible exis-tence of a short crack effect close to a notch. Limited results froma previous study show that this effect is not significant in theexperimental conditions studied in the 2024 alloy [7,8]. In a previ-ous study it was shown that short crack effect, if any is confined toa zone within 50 lm near the notch and corresponds to the zonewhere crack closure effects develop [17].

The present study shows that the short crack model gives verygood results regarding the fatigue life to a given crack length, oncethe crack configuration is known. Crack initiation phase, if any, isnot significant in the experimental conditions studies here.

6. Conclusions

The study of short crack growth kinetics at a notch, in thinplates of aluminium alloy 2014 T351 show that two crack geome-tries are possible – corner crack or a through-thickness crack.

The short crack approach can give very good estimations of lifeto attain a detectable crack size at a notch, provide the crack geom-etry is known.

The predicted lives are within 30% of the measured lives exceptfor a few test conditions.

Conservative life estimations, especially at long lives, indicatethat under such conditions, the crack initiation period needs tobe evaluated.

N. Ranganathan et al. / International Journal of Fatigue 33 (2011) 492–499 499

In the 7449 alloy the marker-band technique used permit theevaluation crack growth rates at a notch from very low growthrates.

Subtraction of the measured crack growth life from the total lifegives an estimation of crack initiation life.

Crack initiation life cannot be neglected at long lives.

Acknowledgements

The aluminium alloy used in this study was furnished by Pechi-ney, France. Some of the tests were carried out by Mr Felix Mathieuand Mr Kunal Sharma during their training period in the LMRlaboratories.

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