International Journal of Fatigue 31 (2009) 1771–1779

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International Journal of Fatigue

journal homepage: www.elsevier .com/locate / i j fa t igue

Invited Keynote Paper

Crack tip deformation fields and fatigue crack growth rates in Ti–6Al–4V q

Alexander M. Korsunsky a,*, Xu Song a, Jonathan Belnoue a, Terry Jun a, Felix Hofmann a,Paulo F.P. De Matos a, David Nowell a, Daniele Dini a,b, Olivier Aparicio-Blanco c, Michael J. Walsh d

a Department of Engineering Science and Rolls-Royce University Technology Centre (UTC) in Solid Mechanics, University of Oxford, Parks Road, Oxford OX1 3PJ, UKb Department of Mechanical Engineering, Imperial College, Exhibition Road, London SW7 2AZ, UKc ENSICAEN, 6 Boulevard Maréchal JUIN, 14050 CAEN Cedex, Franced Combustion Systems Engineering, Rolls-Royce plc, P.O. Box 31, Derby DE24 8BJ, UK

a r t i c l e i n f o

Article history:Received 2 October 2008Received in revised form 15 February 2009Accepted 24 February 2009Available online 14 March 2009

Keywords:FatigueEnergy dissipationCrack propagationStructural integritySynchrotron X-ray diffraction

0142-1123/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijfatigue.2009.02.043

q Paper presented at the International ConferenceStructural Materials, Hyannis, MA, USA, 15–19 Septem

* Corresponding author. Tel.: +44 1865 273043; faxE-mail address: alexander.korsunsky@eng.ox.ac.uk

a b s t r a c t

In this paper we present an overview of experimental and modelling studies of fatigue crack growth ratesin aerospace titanium alloy Ti–6Al–4V. We review work done on the subject since the 1980s to the pres-ent day, identifying test programmes and procedures and their results, as well as predictive approachesdeveloped over this period. We then present the results of some of our recent experiments and simula-tions. Fatigue crack growth rates (FCGRs) under constant applied load were evaluated as a function ofcrack length, and the effect of overload (retardation) was considered. Crack opening was measured duringcycling using digital image correlation, and residual stress intensity factor was determined using syn-chrotron X-ray diffraction mapping. Modelling techniques used for the prediction of FCGRs are thenreviewed, and an approach based on the analysis of energy dissipation at the crack tip is proposed.Finally, directions for further research are identified.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Reliable prediction for fatigue crack initiation and growth hasbecome a central theme in the design and lifing of aerospace com-ponents and assemblies, ever since the 1970s, when the so-calleddefect-tolerant approach to managing metal fatigue has foundincreasingly widespread use. The rationale of this approach is thatthe initiation and existence of defects and cracks in components isunavoidable and has to be accepted, e.g. due to the fact that theirsize often lies below the detection limit of the inspection tech-niques available. Therefore, regular inspections and health moni-toring on their own are insufficient to guarantee safe operation,and require being augmented by predictive defect growth model-ling that determines inspection frequency, so as to ensure that de-fects may not grow fast enough to reach the critical size in theinterval between inspections.

In this context the importance of developing and validating pre-dictive modelling capabilities for fatigue crack propagation be-comes self-evident. It is worth noting that even representingcorrectly the loading experienced by a cracked component in-ser-vice is itself often difficult, since loading histories are typicallycomplex and involve changing amplitudes, and sequences of over-

ll rights reserved.

on Fatigue and Damage ofber 2008.

: +44 1865 273010.(A.M. Korsunsky).

loads and underloads that result in fluctuations of fatigue crackgrowth rates (retardation and acceleration) that must be ac-counted for in crack growth calculations. Furthermore, alongsidethe major cycles associated with each flight (the low cycle, LCFcomponent), cycles of higher frequency and lower amplitude arealso present (the HCF component). Sensitivity to dwell atmaximum load is also often observed, even at relatively low tem-peratures. The influence of these time-dependent and history-dependent mechanisms on crack advance needs to be taken intoaccount within the framework of predictive modelling.

The first principal task here concerns the identification of crackgrowth correlation parameters. Paris et al. [1] postulated that thestress intensity factor range, DK, provides, in the first approxima-tion, a good correlator for the observed crack growth rates. Fur-thermore, he stated that the correlation between crack advanceper cycle, da/dN, and DK has the form of a simple power law. De-spite initially being received sceptically by many experts in thefield on the grounds of being too empirical, this relationship hasserved the engineering design community for many decades asthe basis for crack growth prediction.

In spite of the conspicuous (and verifiable) success of Paris’ law,its limitations in application to ductile fracture can be readily iden-tified. This concerns, for example, the lack of dependence on theloading R-ratio and the lack of dependence on the loading history.Indeed, the key distinction between linear elastic and inelastic re-sponse lies in the fact that the structure in question does not returnto its original state following inelastic deformation. A plastically

1772 A.M. Korsunsky et al. / International Journal of Fatigue 31 (2009) 1771–1779

deformed structure retains a ‘‘memory” of prior deformation that islikely to affect its subsequent response, so that the same appliedDK will result in different fatigue crack growth rates (FCGRs). Thus,crack growth in ductile metals must be expected to show somedependence on the loading history.

Two specific consequences of material ductility are often iden-tified in the context of crack growth analysis. Firstly, in the partic-ular case of cracked structures the permanent set strain (residualinelastic strain) and permanent distortion that persist after unloadmanifest themselves in local stretching of material in the vicinityof the crack tip, and may result in plasticity-induced crack closure[2]. Secondly, the presence of (non-uniform) permanent inelasticstrains (eigenstrains) within the body leads to the creation of resid-ual stresses. These stresses are also concentrated near the crack tip.They modify the crack tip loading R-ratio, and may also change theeffective DK.

Although crack closure and crack tip residual stresses are themanifestation of the same physical phenomenon, a degree of con-troversy has persisted over a few years as to which approach pro-vides a better basis for crack growth prediction. Instead ofattempting to resolve this issue, one may try to identify somenon-controversial general principles that could help develop ra-tional correlations for FCGRs.

(i) Crack advance is controlled by the damage processes occur-ring within the highly localised fracture zone immediatelyahead of the crack tip. This zone is embedded within thecrack tip plastic zone, that in turn is surrounded by the zoneof domination of the elastic crack tip stress field solution (K-field). In the case of quasi-brittle fracture under monotonicloading one may argue that it is the K-field that determinescrack tip plasticity, that in turn controls the fracture process(the small scale yielding hypothesis, [3]). However, the sameargument is unacceptable in the case of fatigue crack propa-gation: it is manifest that prior history of crack growthexerts very strong influence over subsequent propagation.

(ii) Nevertheless, on the grounds of physical objectivity one mayargue the following. Elemental act of crack advance is deter-mined by the failure of the small material volume within theprocess zone [4]. Therefore, prediction of crack advanceshould be possible given sufficient insight into the deforma-tion behaviour of these small volumes at the crack tip,including residual stress and damage accumulation, i.e. his-tory effects.

(iii) Strictly speaking, distinction ought to be made betweenplasticity and damage: although both are dissipative mech-anisms, it is ultimately the damage component that deter-mines failure. However, in many metals the two may belinked, i.e. damage at every material point may be assumedto be a function of plastic strain (unless non-local effects aretaken into account, e.g. [5]). Therefore, FCGR should be cor-related with the plastic deformation processes at the cracktip.

The remaining task is then to seek a parameter or descriptor ofcrack tip plastic deformation that would provide the best correla-tion with experimental observations. For this purpose, Pommier[6] suggested the use of a single crack tip blunting parameter, q,and expressed all dissipative processes associated with the crackadvance as functions of either crack length, a, or crack tip bluntingparameter, q. Analysis based on the Clausius–Duhem inequality(second law of thermodynamics) leads to the inequality essentiallyof the form

dadt

P1G

dWpl

dt; ð1Þ

where dWpl is the energy dissipation due to plasticity, and G de-notes the strain energy release rate. On this basis a heuristic crackgrowth law is introduced in the form of an equality that links therate of crack advance, da/dt, with the rate of crack tip blunting,dq/dt (assuming the latter provides full characterisation of plastic-ity-induced energy dissipation). This is a significant deviation fromthe LEFM and Paris-type correlation for crack growth description,since the principal emphasis is being placed on the local crack tipdeformation parameters, and not on the description based on re-mote stress fields and the assumption of small scale yielding. Hav-ing thus introduced the crack growth law formulation andcalibrated it on the basis of FE simulations, Pommier and Risbet[7] then develop a constitutive crack growth law outside the FEframework, allowing predictions to be obtained without numeri-cally laborious simulation procedures. The cornerstone of this ap-proach is the following simple form of the correlation betweenthe crack growth rate and crack tip blunting rate:

dadt¼ a

2dqdt: ð2Þ

Note the use of equality in this equation, instead of the inequalityappearing in Eq. (1). The remaining calibration parameter is thenthe coefficient a that must be determined by comparison withexperimental results. This is discussed in more detail below.

Noroozi et al. [4] also seek to obviate the need to use finite ele-ment computations by employing simplified description of cracktip yielding using Ramberg–Osgood form of uniaxial inelasticstress–strain relationships. To obtain predictions of crack advancethey apply the Coffin–Manson strain-life relationship to elemen-tary material volumes ahead of the advancing crack front. In a sub-sequent paper [8] they also employ FE simulations to provide abetter account for the effect of deformation history.

The approach used in the present paper builds on several previ-ously published studies. It draws on the ideas of Glinka et al. [4]and Pommier and Risbet [7] to seek correlations between localdeformation fields and crack growth rates, and also on the use ofcriteria based on energy dissipation to predict crack initiation dem-onstrated, e.g. by Korsunsky et al. [9–11]. Detailed numerical sim-ulations of crack tip deformation fields are considered, andvalidation against experimental measurements is presented.

A series of experiments on crack growth rate determination wasconducted in the present study. Direct observation of surface crackopening history during fatigue cycling was used to determine clo-sure conditions. Synchrotron X-ray diffraction was used for thepurpose of mapping residual elastic strains around the crack tipin unloaded specimens. The measurements were interpreted bymatching the measured maps to LEFM solutions, allowing theresidual stress intensity factors to be determined. This provided alink between crack closure and medium range residual stressesin fatigue samples. Finally, numerical modelling of near-tip defor-mation fields was conducted in order to provide input for calibrat-ing FCGR models.

Titanium alloys form an important class of aerospace materialsoffering an excellent combination of specific strength and specificstiffness, and finding widespread use in the manufacture of fansand compressors for gas turbines of aero-engines. Titanium alloyspossess complex, processing-dependent microstructure, and maycontain different phases, typically a combination of hcp a-phaseand bcc b-phase, although the volume fraction of the latter is oftenrelatively small. The majority a-phase displays significant anisot-ropy at grain crystallite level, both in terms of elastic properties(stiffness) and plastic deformation (slip resistance on differentcrystallographic planes). This strong crystallographic dependencesometimes becomes manifest even at the macroscopic level, e.g.through the dependence of overall stiffness on texture, andthe dependence of macroscopic strength on the ‘‘weakest link”

Fig. 1. (a) Illustration of the experimental setup for the study of fatigue crackgrowth. Note the use of digital camera in combination with the Questar telescope inorder to obtain images of sample surface in the vicinity of the crack tip. (b)Dimensions of the compact tension (CT) specimen.

A.M. Korsunsky et al. / International Journal of Fatigue 31 (2009) 1771–1779 1773

represented by a ‘‘rogue grain” possessing particular orientationwith respect to its neighbour(s) and to the overall loading direction[17]. The combination of crystal plasticity modelling and experi-mental characterisation of grain orientation (e.g. by EBSD) andgrain-level strains (e.g. by diffraction) provides an efficient meansof developing improved insight into the deformation behaviour oftitanium alloys [18]. From the practical viewpoint the ultimatepurpose of these refined analyses remains the development of im-proved, more reliable predictive models that demonstrate goodagreement with experimental observations. In this context exten-sive experimental data, such as those contained in archival reports[15], continue to provide a valuable resource.

The experimental results of the present study were comparedwith FCGR data for Ti–6Al–4V obtained under test programme ondamage tolerant design (DTD) for lifing of airframe structures. Ini-tially introduced as a military specification in the early 1970s [12],in the early 1980s the formulation of systematic procedures forDTD was pursued by the Advisory Group for Aerospace Researchand Development (AGARD) headquartered in France. AGARDorganised a cooperative test programme between NATO countries[13] with the purposes of (i) promoting familiarity of laboratorieswith test techniques for DTD of engine disc materials, (ii) standar-dising test specimen geometries and test techniques, and (iii) cal-ibrating the results from participating laboratories through a roundrobin test programme [14]. A core part of the AGARD test pro-gramme focused on Ti–6Al–4V using detailed test procedures[14] (216 tests carried out by 12 different laboratories). The Ti–6Al–4V material for the round robin test programme was providedby Rolls-Royce from RB211 fan disc forgings. Good repeatability ofcrack growth testing was confirmed, with potential drop techniqueproving extremely accurate in measuring crack sizes. The predic-tive models compared in the Supplemental Programme [15] reliedon Elber’s plasticity-induced crack closure concept [2] and theeffective stress intensity factor range DKeff, except for the Rolls-Royce approach. The latter approach used the Walker model [16]in the form

dadN¼ CðDKð1� RÞmÞn; ð3Þ

where R denotes the ratio of minimum to maximum stress (stressintensity factor). In the present study only the experimental datafrom AGARD report were used.

The structure of presentation in this paper is as follows. Theexperimental programme for crack growth characterisation is de-scribed, together with the results relating to the evidence of crackclosure obtained by digital image correlation techniques. The re-sults for fatigue crack growth rates (FCGRs) obtained in this studyare compared with the AGARD data. The entire body of data is thenanalysed based on the Glinka et al. [4] approach, and a particularmeans of describing the transition between elasticity-dominatedand plasticity-dominated responses is presented. Finally, finite ele-ment analysis of crack tip deformation fields in titanium alloys ispresented. The results are used to predict FCGRs obtained in ourexperiments using the Pommier and Risbet [7] approach, and alsousing the newly proposed method based on the analysis of energydissipation in the vicinity of the crack tip. A discussion of the re-sults concludes the presentation.

2. Experimental investigation of crack growth and closure in Ti–6Al–4V

An illustration of the experimental setup used for the study offatigue crack growth rates in the titanium alloy under investigationis presented in Fig. 1a. Compact tension specimens (Fig. 1b) weremachined from a 5 mm-thick rolled plate of Ti–6Al–4V alloy so

that the crack propagation direction was perpendicular to the lon-gitudinal (rolling) direction within the plate. Two holes separatedby 33 mm (inter-centre distance) were used to grip the CT sampleusing pins. The servo-hydraulic loading device was equipped witha 15 kN load cell and driven by a programmable controller allow-ing flexible variation of amplitude and frequency of the load. Forthe majority of tests the load ratio of R = 0.1 was used.

The key component of the acquisition system was the digitalcamera attached to Questar long range telescope. The camera al-lowed continuous filming at 30 frames per second and 640 � 420pixel matrix. Since the field of view of the telescope could be ad-justed to approximately 0.65 mm, the pixel size for this imagingarrangement was close to 1 lm, and accuracy close to or betterthan 0.1 lm can be expected. In practice, however, the attainablespatial resolution depends strongly of the quality of surface prep-aration, lighting and contrast, etc.

The procedure for experimental data collection was as follows.The sample was mounted in the grips and the optical systemaligned by transverse translation of the telescope on a table with

da/dN = 4.60E-12 ΔK3.58E+00

1.E-07

1.E-06

1.E-0500101

ΔK, MPa √mda

/dN

(m/c

ycle

) CT3CT4CT2CT1RRCT24RRCT23CT5CT6Power (CT4)

Fig. 3. Paris crack growth rate diagram for Ti–6Al–4V incorporating results for thesamples used in the present study designated CT1 and CT2, and the data fromAGARD report for samples RRCT23 and RRCT24. Note the retardation observed dueto overload in specimen CT1. Power law dependence of crack growth rate on theapplied stress intensity factor range is indicated by the fit line (specimen CT4).

1774 A.M. Korsunsky et al. / International Journal of Fatigue 31 (2009) 1771–1779

digital position readout. Collocating the cross-hairs with the notchroot and zeroing the readout provided a convenient means of pre-cision monitoring of the crack length subsequently. The samplewas pre-cracked and then subjected to the desired load history(predominantly constant amplitude loading) at the frequency ofa few Hz. The crack growth experiment was periodically inter-rupted, the telescope position adjusted and the precise cracklength noted. The cycling frequency was then reduced to 0.25 Hz,and a video of the near crack tip deformation recorded. In the caseof overload analysis, the setup allows video record to be collectednot only in the overload cycle, but also in the preceding and subse-quent normal load cycles, if required.

Fig. 2 illustrates the crack growth curves obtained for two sam-ples designated OXCT1 and OXCT2. Note the retardation in one ofthe samples due to the application of a 100% overload cycle. Thiseffect is discussed in somewhat greater detail below in the contextof crack closure.

Fig. 3 shows the results for the samples series CT1–CT6 plottedtogether with archival data for two samples from AGARD report[15,19], illustrating satisfactory agreement obtained between thetwo sets of data on the Paris fatigue crack growth diagram. Apower law relationship in the form

dadN¼ CDKm ð4Þ

provides a satisfactory description of the dependence of the fatiguecrack growth rate (FCGR) on the applied stress intensity factorrange DK, illustrated in Fig. 3 for sample CT4.

The interpretation of crack tip video recordings was carried outas follows. The video was processed so as to extract a sequence ofbitmap images from the video files. Designating one of the imagesas reference, digital image correlation [20] can be used to extractdisplacements and ultimately strain maps as a function of point

CT1 and CT2 (Pmax=7.5kN,Pmin=0.75kN,Pol=15kN)

10

15

20

25

30

35

0 5000 10000 15000 20000 25000 30000N (cycles)

a (m

m)

Fig. 2. Example of crack growth diagrams as a function of cycles from initiation, fortwo specimens, one subjected to constant remote applied load (triangular markers),one showing retardation following a 100% overload (circle markers).

in the loading cycle. In the present study the point-wise methodof De Matos [21] was used only to monitor crack opening by track-ing the distance between two points selected to lie on two oppositecrack faces some way behind the crack tip.

Fig. 4 illustrates the crack opening behaviour during cycling ex-tracted from video recordings using digital image correlation. Thetwo sets of data correspond to cycles designated 27,210 (duringconstant applied load amplitude) and 75,453 (during the retarda-tion stage of crack growth due to the application of a 100% over-load). The evidence of crack closure is apparent at minimumapplied load during the cycle, particularly during the retardationstage in the crack growth experiment.

3. Evaluation of residual stress intensity factors

X-ray diffraction provides an extremely useful tool for spatiallyand directionally resolved determination of residual strains incrystals and polycrystals. Synchrotron-based X-ray instrumentsare particularly attractive for the purpose of bulk residual stressanalysis, due to the availability of high flux, high energy X-raysensuring deep penetration, and spatial and angular resolution ofthe measurements [22].

Spatially resolved measurements of strains in the crack openingdirection in unloaded samples CT1 and CT6 were conducted intransmission mode on the ID15 beamline at the European Synchro-tron Radiation Facility (ESRF, Grenoble, France). The use of energy-dispersive detector allowed the collection of entire diffractionprofiles that were interpreted in terms of average macroscopicstrain using Rietveld refinement [23]. The strain map in the vicinityof the crack tip in sample CT6 is illustrated in Fig. 5a. It is apparentthat, even in the absence of external loading, the strain distributionindicates intensification in the vicinity of the crack tip, confirmingthat the ‘‘wedging” effect of crack closure gives rise to a residualstress intensity factor.

0

2

4

6

8

10

12

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2Cycle fraction

Ope

ning

(mic

rom

etre

s)

7545327210

Fig. 4. Crack opening diagram indicating the separation between points lying on opposite shores of the crack behind the crack tip during cycling. Cycle 75,453 corresponds tothe retardation stage following the application of 100% overload.

Fig. 5. (a) Experimentally measured map of residual strain in the crack opening direction in specimen CT6 immediately after a 100% overload. (b) The best match LEFMsimulated strain map in the crack opening direction, corresponding to the residual stress intensity factor KI = 17 MPa

pm.

A.M. Korsunsky et al. / International Journal of Fatigue 31 (2009) 1771–1779 1775

In order to achieve quantitative interpretation of this experi-mental result, we first consider the LEFM expression for strainaround the tip of a plane strain crack, given by:

eðr; #Þ ¼ ð1þ mÞE

KIffiffiffiffiffiffiffiffiffi2prp cosð#=2Þ½1� 2mþ sinð#=2Þ sinð3#=2Þ�: ð5Þ

Let us denote the experimentally measured strain at position (xj,yj)by ej, and use ej to denote the corresponding value calculated usingEq. (5).

The only adjustable parameter in Eq. (5), the stress intensityfactor KI, can be readily found as follows. We form a sum of leastsquares measure of misfit between model and experiment,

J ¼XN

j¼1

ðej � ejÞ2; ð6Þ

and seek the best match value of KI [24] as

K�I ¼ arg minK I

J: ð7Þ

The application of this method to the experimental data for speci-men CT6 (Fig. 5a) yields the stress intensity factor value ofK�I ¼ 17 MPa

ffiffiffiffiffimp

. The significance of this value is clearly appreci-ated by comparison with the maximum value of KI applied duringconstant load crack growth experiment of 38.3 MPa

pm. Thus, it is

confirmed that a 100% overload ðKOLI ¼ 76:6 MPa

ffiffiffiffiffimpÞ causes signif-

icant crack closure and the associated residual KI of about 22% of themaximum overload SIF value.

The above discussion of crack closure and residual stress effectson the deformation fields in the vicinity of crack tips can be con-cluded as follows. The two effects are intimately connected, andboth exert an influence over the effective crack growth conditions.The analysis of these effects is of great importance in the context ofestablishing the correct correlation between the applied loads, andthe ‘applied’ stress intensity factors computed on the basis of LEFM(long and medium range fields). However, on the grounds of objec-tivity of the crack growth process it is possible to argue that it isthe local deformation conditions at the crack tip (short rangefields) that determine the rate of crack advance. Therefore, oncethe effective stress intensity factor range and the plastic deforma-tion history throughout the crack growth are fully established, thisinformation should be sufficient for the purposes of building pre-dictive models for crack advance. In the sequel we therefore con-centrate only on this aspect of analysis, and no longer concernourselves with the consideration of crack closure.

4. Crack growth rate prediction

Having obtained experimentally information about fatiguecrack growth rate (FCGR) as a function of the applied stress

1776 A.M. Korsunsky et al. / International Journal of Fatigue 31 (2009) 1771–1779

intensity factor range DK under constant load amplitude (i.e.slowly increasing SIF range), the correlation may be established be-tween FCGR and physical parameters that characterise the localcrack tip deformation fields. Once such relationship is found, onewould seek to eliminate the global, remote parameter DK thatserves as an intermediate means of correlation; and to hypothesizea direct relationship between FCGR and some aspect of crack tipdeformation.

The implementation of this approach involves several distinctsteps.

(1) The calibration of FCGR against applied stress intensity fac-tor range DK.

(2) The calibration of ‘‘constant amplitude” crack tip deforma-tion field parameter against applied stress intensity factorrange DK.

(3) Cross-calibration of FCGR against crack tip deformation fieldparameter by elimination of DK.

The first step above has already been accomplished by experi-mental tests described in the previous section. This section is de-voted to steps (2) and (3).

In what follows, two candidate parameters that capture thecrack tip deformation field are considered, namely (i) the cracktip blunting parameter, q, as utilised by Pommier and Risbet [7]in their studies, and (ii) the crack tip energy dissipation parameter(per cycle). In principle these parameters may be measureableexperimentally. For example, the digital image correlation (DIC)technique hold out the promise of providing extremely high spatialresolution (sub-micron) characterisation of crack tip deformationfields, including crack tip strains and crack opening profiles. Inpractice, however, these objectives remain elusive, with publishedresults pertaining primarily to the extraction of global parameters,such as stress intensity factors [22]. As mentioned previously, ourown interpretation of digital images allowed extraction of such rel-evant local parameters as crack opening displacement, but theaccuracy attained at present was deemed insufficient to quantifystrain distributions or crack blunting. Similarly, although experi-mental measurements of spatially and temporally resolved tem-perature changes due to energy dissipation have been reported[25], reliable conversion of these results into dissipated energyper cycle remains problematic.

Taking account of the above considerations, the approachadopted here relied on numerical modelling of crack tip deforma-tion fields using the finite element method. The description ofhardening behaviour of titanium alloy Ti–6Al–4V was calibratedby comparison with the results for unnotched dogbone tension–compression specimens. Plane strain model was used, and the re-

Fig. 6. Illustration of the crack opening profile obtained by load application via a slparticularly severe deformation at the crack tip.

gion of crack propagation was represented by a particularly finemesh with element size of 10 lm. Mesh refinement studies werecarried out, and confirmed that, while further reduction of the ele-ment size led to a considerable increase in the time for computa-tion, no significant difference was observed in terms of theoverall response, such as the crack opening profile. Crack growthwas simulated by progressive release of nodes ahead of the cracktip after one or two complete loading cycles, in order to ensuresome degree of fatigue cycle saturation. The manner of applying re-mote loading that induces crack tip deformation was found to beparticularly significant. For example, if large DK were applied atthe start of crack growth, an extensive region of plastic deforma-tion (and associated residual stress) was induced ahead of thecrack tip that exerted strong influence on subsequent deformation.In order to overcome this problem, the remotely applied load wasapplied via a slow ramp, allowing the system a better means ofapproaching a quasi steady state situation. Fig. 6 illustrates thesmooth crack opening profile obtained in this way, together withthe contours of von Mises stress. Numerical results of this kind ad-mit interpretation in terms of crack tip deformation parameters.

The crack tip blunting parameter q was estimated by Pommierand Risbet [7] by means of subtracting the elastic component ofdeformation from the elasto-plastic solution, and integration(averaging) of the remaining discontinuity over the segment ofthe crack lying between a/20 and a/10 of the crack length fromthe tip. An alternative approach adopted in our study involved asimple procedure for decomposition of crack opening displace-ment into the elastic and plastic parts, as follows. The crack open-ing as a function of distance r from the crack tip can beapproximately written as

DuðrÞ ¼ CKffiffiffirpþ q ð8Þ

where K denotes the applied stress intensity factor, C is a constantknown from linear elastic fracture mechanics, and q representsplasticity-induced crack blunting that in the first approximation isassumed to be constant. On the basis of Eq. (8) a simple procedurefor the determination of q can be proposed: if crack opening Du isplotted as a function of

ffiffiffirp

, then within the range of applicability of(8) its variation must be represented by a straight line with theintercept given by q. Fig. 7 provides an example of such plot. It isapparent that the plot deviates from linearity at larger distancesfrom the crack tip, due to the effects of sample geometry. For thisreason linear fit is used on the few points close to the tip, wherethe relationship appears linear, and insensitive to the selection ofpoints for the fit.

Dissipated energy computation was carried out for the entireelement lying immediately ahead of the crack tip. Although somedegree of mesh size dependence can be expected for such a

owly increasing ramp. The contours indicate von Mises stress distribution. Note

0

0.002

0.004

0.006

0.008

0.01

0.1 0.2 0.3 0.4 0.5 0.6

y = 0.00039687 + 0.013009x R= 0.99831

CO

D (

mm

)

r0.5 (mm0.5)

Fig. 7. Example of the plot of crack opening displacement vs. square root ofdistance from the crack tip. Note the deviation from the linear dependence of Eq. (7)at larger distances. Crack blunting parameter q is determined by fitting a straightline to the left-most part of the plot.

A.M. Korsunsky et al. / International Journal of Fatigue 31 (2009) 1771–1779 1777

calculation, it should also be noted that element-based, rather thanpoint-wise calculation, provides a certain degree of non-locality forthis evaluation. At any rate, the key aspect required of the ap-proach being developed here is internal consistency, and that issatisfied by the above choice.

The dependence of the crack tip blunting parameter increment,dq, on the applied stress intensity factor, DK, is illustrated in Fig. 8.The power law fit to this relationship is indicated by the straightline in bi-logarithmic coordinates. It is interesting to note thatthe power law exponent of 3.33 is very close to the value of 3.58obtained from the analysis of Paris diagram of FCGR vs. applied

10-7

10-6

10-5

0.0001

0.001

011 100

y = 5.8166e-10 * x^(3.3255) R= 0.99828

Δ K

δρ (mm)

Fig. 8. The dependence of crack blunting parameter increment (per cycle) on theapplied stress intensity factor, DK, that is described well by a power lawrelationship.

DK. Thus, the Pommier and Risbet [7] hypothesis of the simple lin-ear relationship between crack tip blunting and crack growth rateis borne out in the present study. The only remaining adjustableparameter is coefficient a in Eq. (2): once the value of this param-eter is found, crack growth rate predictions can be made on the ba-sis of crack tip blunting rate (see below).

The evolution of computed plastic energy dissipation (per cycle)with the applied stress intensity factor range is illustrated in Fig. 9.It is immediately apparent that this relationship is also well de-scribed by a power law function. However, the power law expo-nent (slope in bi-logarithmic axes) is different from that of theParis law for FCGR vs. stress intensity factor range. It thereforeseems logical, in this case, to postulate the following form of crackgrowth rate law:

dadN¼ b

dWpl

dN

� �q

ð9Þ

where the coefficient b and exponent q have been introduced. Theimplication of this form of relationship is that a fixed proportionof dissipated plastic energy is converted into damage that assistscrack propagation [26].

The above analyses lead to Eqs. (2) and (9) as the basis for theprediction of crack advance. It is difficult to argue in favour of pre-ferring one form over the other, since each appears to offer certainadvantages. For example, the linear nature of the relationship in (2)for the crack tip blunting parameter increment may appear moreconvenient for fitting analysis. On the other hand, the energy-based form of (9) may be more amenable to the incorporation ofmulti-axial loading and multi-mode crack propagation regimes inthe consideration. In this study the two forms of FCGR predictiveequations have been used to compute crack growth diagrams forthe samples studied experimentally.

The results are illustrated in Fig. 10 for both crack tip bluntingand energy dissipation approaches. Note that the orientation ofthe chart reflects the nature of the prediction procedure utilisedhere: the crack length is used as the input for the computation ofcrack blunting and plastic energy dissipation, and hence the crack

0.001

0.01

20 30 40 50

y = 4.0617e-06 * x^(1.9897) R= 0.9976

ΔWp (μJ)

ΔK (MPa m1/2)

Fig. 9. The dependence of crack tip plastic energy dissipation (per cycle) on theapplied stress intensity factor, DK, that is described well by a power lawrelationship.

0

5000

10000

15000

20000

25000

30000

35000

0 5 10 15 20 25Crack length (mm)

No.

cyc

les

en.dissip.model(CT2)test(CT2)en.dissip. model(CT4)test(CT4)tip rad.model(CT2)tip rad.model (CT4)

Fig. 10. Prediction of crack growth rates for two samples from the series studiedexperimentally. Note that crack length is used as the argument, while the numberof cycles to attain this crack length is the objective of the computation. The legendcontains the reference to two samples considered (CT2 and CT4), the experimentalresults (‘‘test”), tip radius model (‘‘tip.rad”) and energy dissipation model (‘‘en.dis-sip. model”).

1778 A.M. Korsunsky et al. / International Journal of Fatigue 31 (2009) 1771–1779

growth rate. It is apparent that predictions of similar quality aregenerated by the two approaches.

It is important to evaluate the relationship in which the pres-ently proposed approach stands to earlier energy-based analysesof crack initiation and propagation. While the idea itself is clearlynot novel (e.g. [26] and references therein), previously it has beenapplied to macroscopic measures, i.e. attempt was made to connectthe loading conditions directly with the crack propagation rate. Ithas become progressively more and more apparent that this meth-odology is not capable of providing satisfactory prediction of FCGR,particularly in the presence of complex loading history. In contrast,the presently proposed consideration of local energy dissipationprocesses at the crack tip is likely to capture the key connection be-tween deformation and damage, and thus to furnish more reliablepredictive tools.

5. Discussion and conclusions

The present paper presented an overview of a range of experi-mental and modelling studies of fatigue crack growth in aerospacetitanium alloy Ti–6Al–4V.

Laboratory tests of crack growth rates in compact tension (CT)specimens were conducted, and the relationship between appliedstress intensity factor range and FCGR was found. For purposes ofvalidation, the results were compared with the historic data fromthe AGARD round robin tests.

The application of overload resulted in a clearly manifestedretardation of FCGR that resulted in a transient effect that never-theless persisted for many cycles. During this retardation stagecrack closure could be observed at sample surface by direct opticalphotography and digital image correlation. Furthermore, the pres-ence of residual strain intensification within the bulk of the sampledue to the ‘‘wedging” effect of crack closure could also be identifiedby means of high energy synchrotron X-ray diffraction. The col-lected strain maps were interpreted by matching to LEFM predic-tions, and it was demonstrated that the magnitude of residualSIF could be determined.

The key conclusion drawn from these analyses concerned thefact that crack closure and medium/long range residual stressesare only significant in terms of the link that they provide betweenthe external loading conditions, and the local deformation condi-tions at the crack tip. It is precisely these local conditions that ulti-mately control crack growth, and therefore FCGR.

Therefore, the focus of modelling efforts can be placed on theconsideration of local deformation fields, and establishing theparameters that would provide the best correlation with FCGR.Two approaches were considered in the present study: the ap-proach based on the consideration of crack tip blunting due toPommier and Risbet [7], and the presently proposed approachbased on the analysis of local energy dissipation in the immediatevicinity of the crack tip. The two approaches achieve similar qual-ity of FCGR prediction under constant amplitude loading condi-tions considered here.

Aspects of fatigue crack growth rate prediction addressed in thepresent study provide firmer basis for introducing a rational sepa-ration between macroscopic remote loading parameters (such asapplied SIF ranges) and the essentially local phenomenon of fatiguecrack advance.

Only partial validation of the proposed relationships has beenprovided. Further study of the effect of complex loading historieson the crack propagation rates is requires. In particular, the influ-ence of overloads and underloads must be considered, as must alsobe the various implementations of fatigue crack growth experi-ments, e.g. involving reducing and increasing applied stress inten-sity factor range DK while maintaining the maximum SIF Kmax

unchanged; the effects of spectrum loading, etc.

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