Fatigue crack growth prediction models for metallic materials.pdf

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doi: 10.1111/j.1460-2695.2012.01721.x Fatigue crack growth prediction models for metallic materials Part I: Overview of prediction concepts T. MACHNIEWICZ Faculty of Mechanical Engineering and Robotics, AGH University of Science and Technology, Al. Mickiewicza 30, 30–059 Krak´ ow, Poland Received in final form 11 July 2012 ABSTRACT Deterministic models developed to predict fatigue crack growth in metallic materials are considered with special emphasis on approaches suitable for variable amplitude load histories. Part I gives a concise review of available models and their assessment based on reported in the literature comparisons between predicted and observed results. It is concluded that the so-called strip yield model based on the plasticity induce crack closure mechanism is a most versatile predictive tool convenient to use in the case of mode I crack growth under arbitrary variable amplitude loading. Part II of the paper is focused on the strip yield model and its predictive capabilities. Implementations of this type prediction approach reported in the literature are reviewed. It is shown that decisions regarding the constraint factor conception, a choice of the crack driving force parameter, the crack growth rate description and various numerical details can have a profound effect on the model results and the prediction quality. Keywords crack closure; fatigue crack growth; load interaction effect; prediction models; variable amplitude loading. NOMENCLATURE a = crack length a fict = fictitious crack length CA = constant amplitude CDF = crack driving force COD = crack opening displacement da/ dN = fatigue crack growth rate FE = finite element K = stress intensity factor (SIF) K eq = equivalent stress intensity factor K max = maximum stress intensity factor value K min = minimum stress intensity factor value K op = crack opening stress intensity factor K r = stress intensity factor due to residual stress K red = reducing stress intensity factor in Willenborg model OL = overload (subscript ‘OL’ is used for parameters associated with the overload occurrence) r p = plastic zone size r pOL = plastic zone size generated by overload R = stress ratio RMS = root mean square Correspondence: T. Machniewicz. E-mail: [email protected] c 2012 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 36, 293–307 293

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Page 1: Fatigue crack growth prediction models for metallic materials.pdf

doi: 10.1111/j.1460-2695.2012.01721.x

Fatigue crack growth prediction models for metallic materials

Part I: Overview of prediction concepts

T . MACHNIEWICZFaculty of Mechanical Engineering and Robotics, AGH University of Science and Technology, Al. Mickiewicza 30, 30–059 Krakow, Poland

Received in final form 11 July 2012

A B S T R A C T Deterministic models developed to predict fatigue crack growth in metallic materialsare considered with special emphasis on approaches suitable for variable amplitude loadhistories. Part I gives a concise review of available models and their assessment basedon reported in the literature comparisons between predicted and observed results. It isconcluded that the so-called strip yield model based on the plasticity induce crack closuremechanism is a most versatile predictive tool convenient to use in the case of mode I crackgrowth under arbitrary variable amplitude loading. Part II of the paper is focused on thestrip yield model and its predictive capabilities. Implementations of this type predictionapproach reported in the literature are reviewed. It is shown that decisions regardingthe constraint factor conception, a choice of the crack driving force parameter, the crackgrowth rate description and various numerical details can have a profound effect on themodel results and the prediction quality.

Keywords crack closure; fatigue crack growth; load interaction effect; predictionmodels; variable amplitude loading.

N O M E N C L A T U R E a = crack lengthafict = fictitious crack lengthCA = constant amplitude

CDF = crack driving forceCOD = crack opening displacement

da/dN = fatigue crack growth rateFE = finite element

K = stress intensity factor (SIF)K eq = equivalent stress intensity factor

Kmax = maximum stress intensity factor valueKmin = minimum stress intensity factor valueKop = crack opening stress intensity factorK r = stress intensity factor due to residual stress

K red = reducing stress intensity factor in Willenborg modelOL = overload (subscript ‘OL’ is used for parameters associated with the

overload occurrence)rp = plastic zone size

rpOL = plastic zone size generated by overloadR = stress ratio

RMS = root mean square

Correspondence: T. Machniewicz. E-mail: [email protected]

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294 T . MACHNIEWICZ

ROL = K OLmax/Kmax

RSO = overload ratio needed to crack arrestRu = SUL/SOL

Smax = maximum stressSmaxRMS = maximum root mean square stress

Smin = minimum stressSminRMS = minimum root mean square stress

Sop = crack opening stressSIF = stress intensity factorSY = strip yieldU = �K eff /�K

UL = underload (subscript ‘UL’ is used for parameters associated with theunderload occurrence)

VA = variable amplitude�a = crack growth increment�K = Kmax – Kmin

�K eff = Kmax – Kop

�K th = threshold �K�KT = intrinsic �K th

�SRMS = Smax RMS–Smin RMS

ρ∗ = crack tip radiusσ y = yield stress

φ, φR = retardation factors

I N T R O D U C T I O N

According to the damage tolerance philosophy, currentlyadopted by many industrial branches, flaws of varioustypes are unavoidable in a structure. In that case, de-termination of the service life of a component subjectto fatigue loading requires crack growth predictions. De-pending on whether the crack increment in a given loadcycle is uniquely determined from the input data or it isdeemed a random variable, probabilistic and determin-istic prediction models can be differentiated. The prob-abilistic approaches, which otherwise start from deter-ministic models, are amply considered elsewhere.1,2 Thispaper focuses on the deterministic concepts for crackgrowth predictions applicable to variable amplitude (VA)load sequences. As amply documented in the literature,3,4

load interaction effects, namely crack growth accelerationor retardation occur in crack growth under VA loadingconditions.

Part I of this paper gives a concise review of availablecrack growth prediction models that account for the loadinteraction phenomena. These concepts are classified ac-cording to their physical basis and their predictive capabil-ities are assessed based on reported comparisons betweenpredicted and observed results. It is concluded that theso-called strip yield (SY) model of crack closure is a mostversatile predictive tool convenient to use in the case ofmode I crack growth under arbitrary VA load histories.

Part II of this paper is focused on the SY model. Thoughin models of this category the major components remainessentially the same, specific SY model implementationsdiffer in many respects, such as the conception of con-straint factors, crack driving force (CDF) parameters, thetype of crack growth input data, and various numerical de-tails. All above aspects can essentially affect SY model re-sults and the corresponding decisions and computationalchoices as well as their consequences for the predictionresults are addressed in the paper.

C L A S S I F I C A T I O N O F F A T I G U E C R A C KG R O W T H P R E D I C T I O N M O D E L S

Deterministic models for fatigue crack growth predictioncan be classified as proposed in Fig. 1. It is seen that withmost concepts crack growth analysis for a given load his-tory is carried out cycle-by-cycle, which implies summingup fatigue crack growth increments (�ai) associated withconsecutive load cycles. The crack length (ai) at a certaincycle number is therefore computed as

ai = ai−1 + �ai . (1)

The crack growth increment in a load cycle, i.e. the fa-tigue crack growth rate (da/dN ) in that cycle, is computedutilizing a crack growth equation. The best known one,

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FATIGUE CRACK GROWTH PREDICT ION MODELS FOR METALL IC MATERIALS 295

Fig. 1 Classification of deterministic models for fatigue crack growth predictions.

according to Paris, relates da/dN to the stress intensityfactor (SIF) range defined as �K = Kmax – Kmin, whereKmax and Kmin are the maximum and minimum SIF valuesin a load cycle. The Paris equation reads

dadN

= C�K m, (2)

where m and C are the material and environment depen-dent parameters, the C-value being, in addition, affectedby the stress ratio (R). Modifications of Eq. (1) have beenproposed in order to account for the effect of R and tocover crack growth in the near threshold regime. A re-view of various concepts can be found in the literature.5,6

Alternative to �K fracture mechanics parameters, for ex-ample the crack opening displacement range (�CTOD)or the Rice integral range (�J), are sometimes employedto represent the CDF.

Elber7 observed under constant amplitude (CA) pulsat-ing tension that the fatigue crack was closed above theminimum load of the cycle. He postulated that the crackcould not grow when the crack surfaces were in contactand proposed that da/dN be related to the effective rangeof SIF rather than to the total range. That effective rangeis defined as

�Keff = Kmax − Kop, (3)

where Kop is the SIF level at which the crack becomesfully open during uploading. The Kop/Kmax-value in agiven cycle depends on the material and the precedingload history.

A CDF parameter employed in a crack growth equationand, hence, the computed �ai depends on the currentcrack size in cycle i. For that reason, consecutive �ai

increments are determined for consecutive upward stressranges in a load history, which implies a simple range

counting method. Essentially, the rainflow count, widelyapplied in fatigue crack initiation analyses, can only beused in the case of the equivalent SIF (K eq) approach (cf.Fig. 1). As well known, a unique feature of the rainflowtechniques is that the largest load range, i.e. the variationbetween the highest and the lowest minimum in a load-time trace, is always counted. If, however, for a lengthyload history considerable crack growth occurs betweenthe lowest minimum and the highest maximum, it maybe physically questionable to couple these two remoteevents into one cycle. Applicability of the rainflow countto fatigue crack growth is addressed by Skorupa3 in thecontext of the so-called incremental crack growth law byde Koning.8

E Q U I V A L E N T S I F M O D E L S

A basic idea behind the K eq approach, illustrated inFig. 2, is replacing a VA sequence with a CA sequencedefined by an equivalent stress range �Seq correspondingto a SIF range of �K eq such that an average da/dN un-der the VA sequence is the same as under the equivalentCA loading. One of the earliest implementations of thisconcept is the so-called Root Mean Square (RMS) modelproposed by Barsom9 and subsequently modified by oth-ers.10 The value of �K eq is assumed to correspond to anequivalent stress range defined as �SRMS = Smax RMS –Smin RMS with

Smax RMS =√√√√1

m

m∑i=1

(Smax i )2; Smin RMS =√√√√1

m

m∑i=1

(Smin i )2,

(4)

where Smax i and Smin i are the consecutive extremes ofthe actual VA load history comprising m cycles. Other

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296 T . MACHNIEWICZ

Fig. 2 Equivalent SIF concept.

Fig. 3 Post-OL plastic zones considered in Wheeler’s model.

proposals for adopting K eq can also be found in the liter-ature, as reviewed by Skorupa3 and Sander.11

L I N E A R ( N O N - I N T E R A C T I O N ) M O D E L

The linear model ignores a dependency of the crackgrowth increment in a given cycle on the preceding loadhistory. Hence, this approach is conceptually parallel tofatigue life estimates employing Miner’s rule. For someVA load histories, applying the linear model requires thatda/dN values over a wide range of R-ratios and in thethreshold regime be available. In general, a correlationbetween the linear model predictions and experimentalresults can only be obtained when overall load interactioneffects are negligible. This may occur when the retarda-tion and acceleration phenomena cancel out.

P L A S T I C Z O N E M O D E L S

Earliest load interaction models were proposed by Wil-lenborg et al.12 and Wheeler13 in order to reproduce theeffect of a single overload (OL) on fatigue crack growthrates. In both models it is assumed that da/dN becomesreduced as long as the current plastic zone of size rp isembedded within the OL plastic zone of size rpOL gen-erated by the OL cycle at a crack length of aOL, Fig. 3.Subsequent modifications of the Willenborg and Wheelermodels aimed at accounting for acceleration and reducedretardation due to underloads (ULs).

Wheeler model

According to Wheeler’s model,13 after an OL the fatiguecrack growth rate computed for �K corresponding to thecurrent cycle is multiplied by a retardation factor φR givenby

φR =

⎧⎪⎨⎪⎩(

rp

aOL + rpOL − a

)w

for a + rp < aOL + rpOL

1 for a + rp ≥ aOL + rpOL

,

(5)

where w is a parameter that must be derived empiricallyfor a given material and class of loading spectra.

Equation (5) implies that after an OL (i.e. for a > aOL)φR adopts values below unity until a crack length at whichthe current plastic zone touches the boundary of the OL-induced plastic zone. Gray and Gallagher14 redefined theφR coefficient relating it to Kmax/K ∗

max, where K ∗max is the

hypothetic Kmax value that would make the plastic zonefor the current crack length reach the boundary of the OLplastic zone:

φR =

⎧⎪⎪⎨⎪⎪⎩

(Kmax

K ∗max

) 2wπ

for Kmax < K ∗max

1 for Kmax ≥ K ∗max.

(6)

In addition, the w exponent was expressed by Gray andGallagher as:14

w = m2

((log

�K�Kth

)/log RSO

)(7)

where m is the exponent in the Paris equation, �Kth de-notes the threshold SIF range and RSO is a value of theOL ratio ROL = K OL

max/Kmax needed to arrest the crack.Another attempt to improve the Wheeler model perfor-

mance was by Huang et al.15 who addressed the influenceof an UL immediately following an OL in a load history.

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FATIGUE CRACK GROWTH PREDICT ION MODELS FOR METALL IC MATERIALS 297

The resulting modified expression for φR was in the form:

φR =

⎧⎪⎨⎪⎩(

rp

aOL + rpOL − a − r�

)w

for a + rp < aOL + rpOL − r�

1 for a + rp ≥ aOL + rpOL − r�

,

(8)

where r� is the distance between the boundary of the OLplastic zone and of the reversed plastic zone generatedafter the UL.

The original model of Wheeler is included in the com-mercially available AFGROW16 software package.

Willenborg model

Willenborg et al.12 assume that due to residual com-pressive stresses within the OL plastic zone the extremeK-values of a current load cycle (Kmax and Kmin) are re-duced by an amount of Kred, which yields a reduced crackgrowth rate due to the corrected, lower R-ratio. In ad-dition, in the case of Kmin < Kred only a positive portionof the K range is assumed to control the crack growthrate. The Kred-value corresponds to such an increase inthe current Kmax that would cause the current plastic zoneto touch the OL generated plastic zone boundary. Gal-lagher17 proposed a following equation for Kred

Kred = K OLmax

(1 − a − aOL

rpOL

) 12

− Kmax, (9)

where K OLmax is the maximum SIF corresponding to the OL

cycle.Against experimental observations indicating that the

RSO-value (cf. Eq. 7) is material and loading parametersdependent, Eq. (9) implies crack arrest if the OL ratioROL = K OL

max/Kmax ≥ 2 because in that case K red = Kmax

immediately after the OL application (a = aOL). To avoidthe corresponding unconservative predictions Gallagherand Hughes18 proposed that K red be multiplied by a factordefined as

φ =1 − �Kth

�K(RSO − 1)

. (10)

The Willenborg model modified according to Gallagherand Hughes is often referred to as the generalized Wil-lenborg model. Another improvement of the Willenborgmodel proposed by Brussat19 and referred to as the modi-fied generalized Willenborg model aimed at addressing areduction in crack growth retardation due to an UL. Tothis end, a dependency of the φ factor (Eq. 10) on the ULand OL levels was introduced according to the following

relationship:

φ =

⎧⎪⎨⎪⎩

2.523φ0

1 + 3.5(0.25 − RU)0.6 for RU < 0.25

1 for RU ≥ 0.25, (11)

where RU = SUL/SOL and φ0 is a material constant de-fined as the φ-value for RU = 0.

Another attempt to account in the Willenborg modelfor acceleration and reduced retardation due to ULs wasby Chang and Engle.20 The latter version is referred to asthe Walker–Chang–Willenborg model.

Three different versions of the Willenborg model,all referred to earlier in this section, namely the gen-eralized model, the modified generalized model andthe Walker–Chang–Willenborg model are available inthe NASGRO19 software. The generalized Willenborgmodel is also included in the AFGROW16 software. An-other interaction model available in the latter softwarepackage and closely related to the plastic zone mod-els is a Hsu model, which is an empirically based ap-proach that borrows and builds on the concepts offeredby the Wheeler and Willenborg models, but also accountsfor Elber’s crack closure mechanism. The Hsu modelis unable to predict crack growth under compression–compression cycles.

C R A C K C L O S U R E M O D E L S

Because Elber’s discovery of plasticity induced crack clo-sure an opinion prevails that fatigue crack growth be-haviour of metals is to a large extent controlled by thismechanism. Especially the influence of the stress ratio,stress level and thickness as well as load interaction phe-nomena occurring under VA loading are attributed tocrack closure. Consequently, a number of crack growthprediction models utilizing �Keff (cf. Eq. 3) as the CDFparameter have been developed. These type approachescan be classified as semi-empirical models and SY models.With the semi-empirical prediction concepts, the crackopening stress (Sop) corresponding to the Kop level of SIFis adjusted in every cycle according to specified rules em-ploying material and loading dependent parameters. Onthe contrary, the SY models involve a determination ofSop based on the computed distribution of local plasticstretches behind the crack tip. Though none of the crackclosure models model imposes any limitations or require-ments with respect to the load sequence or material, allhave been developed especially for applications to flight-simulation loading on aircraft alloys.

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298 T . MACHNIEWICZ

Semi-empirical crack closure models

ONERA model

According to the ONERA model proposed by Baudinand Robert,21 Kop in cycle i depends on equivalent Klevels, Kmin,eq and Kmax,eq, that are load history dependentand must be adjusted in every cycle. Thus, Kmax,eq,i is aminimum Kmax,i-value required to induce in the followingcycle a primarya plane stress plastic zone, while a change inKmin,eq between two successive cycles is related to a changein the applied loading between the previous (i–1) andcurrent (i) cycle according to the following, empiricallyderived relationship:

�Kmin,eq,i = Kmin,eq,i − Kmin,eq,(i−1)

= (Kmin,eq,i − Kmin ,eq,(i−1))(

Kmax,i

Kmax,eq,i

)2+t/2

,

(12)

where t is the material thickness in mm.The resulting crack opening level in cycle i is expressed

as

Kop,i = Kmax,eq,i (αONR · f1(Req,i ) + (1 − αONR) f2(Req,i )),(13)

where Req,i denotes an equivalent stress ratio (Req,i =Kmin,eq,i

Kmax,eq,i), whilst f 1 and f 2 are empirical functions which

must be determined experimentally for a given material.The αONR parameter can adopt values between zero andunity, to make the Kop fall between two extreme cases,namely a CA loading (αONR = 0) and a single OL (αONR

= 1).

PREFFAS model

With the PREFFAS model, first proposed by Aliagaet al.22 Kop in a given cycle is defined as the maximumvalue of crack opening levels Kop,i,j computed for thatcycle considering the previous cycles j (1 ≤ j ≤ i):

Kop,i = max(Kop,i, j )

= max(Kmax, j − U(Kmax, j − Kmin,Low)). (14a)

Here, Kmin,Low denotes the lowest Kmin-value for the con-sidered cycles and the crack closure ratio U is expressedas

U = A + B · R (14b)

where R = Kmin,Low/Kmax,j, whereas A and B are the ma-terial and thickness dependent, empirically determinedconstants.

a A primary plastic zone is generated in material that has not been plasticallydeformed before.

CORPUS model

The CORPUS model developed by de Koning and co-workers23,24 aimed at covering the effect of periodic OLson crack growth. Contrary to the ONERA and PREFFASmodels, material memory rules to account for load inter-action effects adopted in CORPUS stem directly fromconsidering plastic deformations in the crack wake, whichis convincing from the plasticity induced crack closurepoint of view. However, in contrast to the SY model ad-dressed in the next section, some a priori assumptionsabout the shape of the crack plastic wake are adopted. Asillustrated in Fig. 4, the fatigue crack surfaces are coveredwith ‘humps’, each hump being associated with a plasticzone generated at a previous maximum stress. A hump cre-ated during an upward load range can be partially flattenedby subsequent downward ranges. Therefore, if an OL ap-plied after smaller amplitude cycles generates a prominenthump associated with elevating the crack opening level,this effect can be partly annihilated by an UL, which re-sults in a reduction in Sop.

The Sop-value is identified as the applied stress level atwhich a last contact between the humps is lost (Fig. 4):

Sop = max(Sop,n), (15)

where Sop,n corresponds to a hump created in cycle n, at acrack length of an, and associated with a plastic zone sizeof rp,n. Sop,n depends on the maximum stress in cycle n(Smax,n) and a minimum stress occurring after the subse-quent loading history (Smin,n) according to the followingrule:

Sop,n ={

0 for a > an + rp,n

g(Smax,n − Smin,n) · h for an ≤ a ≤ an + rp,n

,

(16)

where g is the material and Smin,n/Smax,n ratio dependentfunction, whilst the correction function h accounts for theSmax,n/σy ratio, σ y being the material yield stress.Plastic zones of various sizes can be generated during

crack growth under VA loading. The corresponding in-formation must be stored as long as they can still affectthe crack opening level in subsequent cycles. It is assumedthat the influence of a previous cycle i on Sop persists un-til, in a current cycle j, aj +rp,j > ai +rp,i. In the case ofperiodic OLs, each OL causes an increase in Sop until alimiting value given by

Supperboundop = Sop,n + mst,n

(Smax,n − Sop,n

)(17)

is attained, where the ms t,n parameter depends on a ra-tio of the crack growth increment between the OLs tothe plastic zone size generated by an OL prior to thatincrement (�a/rp). After an OL the mst,n-value initially

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FATIGUE CRACK GROWTH PREDICT ION MODELS FOR METALL IC MATERIALS 299

Fig. 4 Crack opening behaviour according to the CORPUS model.

Fig. 5 Schematic of discretized plastic strip in the strip yield model.

increases but it immediately drops to zero when the crackgrows beyond the last OL plastic zone (�a/rp > 1).

Strip yield model

The SY model is based on the Dugdale theory of cracktip plasticity modified to account for the crack closurephenomenon. A conception of this type prediction modelwas conceived by Dill and Saff,25 but the first SY modelsuitable for application in the case of arbitrary VA loadingsequences was developed by Newman.26 Newman was thefirst to formulate the model flow diagram, to develop anSop computation procedure and to propose a means ofaccounting for the 3D stress conditions at the crack tip.

As well known, according to the Dugdale theory the plas-tic zone ahead of the crack tip is modelled as an infinitelythin strip located in the net section ahead of the crack tip.The strip length, i.e. the plastic zone size (rp), is equal tothe distance between the real crack tip and the fictitiouscrack tip, Fig. 5. To make the fictitious COD for a ≤ x ≤afict equal to the plastic deformation of the strip material,the fictitious crack surface in this region is loaded witha compressive yield stress (σ y). The concept is valid for

stationary crack. In order to adapt the Dugdale model tothe case of a growing fatigue crack, the strip material isdisconnected over a distance corresponding to the crackgrowth increment. Consequently, a strip of plastically de-formed material is building up in the crack wake. In thatcase, the displacement compatibility between the plasticstrip and the surrounding elastic material requires apply-ing stresses also on some segments in the crack wake (0 ≤x < a) where the plastic elongation of the strip, L(x), ex-ceeds the fictitious crack COD, V (x). Stresses applied inthe crack wake to make L(x) = V (x) are referred to as thecontact stresses. For computational purposes the problemis discretized by dividing the strip into bar elements thatare intact ahead of the crack tip and broken behind it,as shown in Fig. 5. Stresses and lengths of the strip ele-ments are computed for a maximum and minimum loadof a given fatigue cycle by considering displacement com-patibility conditions along the fictitious crack surface. Asolution for stresses and lengths of the strip elements be-ing known, the Sop stress is calculated, usually using oneof the methods proposed by Newman,26,27 and the �K eff

parameter can be determined. Finally, the crack growthincrement incurred in that cycle is determined from theda/dN versus �K eff material input data, in line with El-ber’s concept. The SY model is considered in more detailin Part II of this paper.

R E S I D U A L S T R E S S M O D E L S

Disparities between observed crack growth and that pre-dicted from crack closure measurements as well as a fail-ure of crack closure prediction models to reflect someexperimentally observed crack growth phenomenaprompted a development of alternative to crack closurebased prediction approaches. Several researches recog-nize great significance of residual stresses ahead of thecrack tip for the fatigue crack growth behaviour, at thesame time questioning the influence of crack closure oreven neglecting the occurrence of this phenomenon.28–30

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300 T . MACHNIEWICZ

Fig. 6 Stress distributions considered in the UniGrow model at: (a) Smax; (b) Smin.

UniGrow model

Noroozi et al.30 attribute load interaction effects in crackgrowth, i.e. retardation or acceleration, to residual stressesahead of the crack tip. They assume that the residual stressdistribution in that region depends on the current cracklength and the load history. In their crack growth predic-tion model termed UniGrow, da/dN is related to a SIFrange defined as �K – K r, where K r is the SIF value asso-ciated with the residual stress distribution. An importantparameter in the model is the crack tip radius ρ∗, Fig. 6,assumed to be material dependent. Because ρ∗ is of a fi-nite value, crack growth is predicted using a methodologyapplicable to notched members. In UniGrow, a numberof cycles to grow the crack ρ∗ increment is computed bythe local strain approach according to the Manson–Coffinequation. The local strain amplitude at the crack tip is es-timated from Neuber’s rule utilizing a linear solution byCreager and Paris31 for the stress distribution ahead of ablunted crack tip computed at the Smax and Smin appliedstresses, Fig. 6, and assuming the material constitutive re-sponse according to the Ramberg–Osgood relationship.Because the phenomenon of crack closure is not mod-elled, the distribution of residual stresses in the crack wakeat Smin (contact stresses in terms of crack closure) is arbi-trarily assumed to be a mirror image of the compressivestress distribution ahead of the crack tip according to theCreager and Paris solution.31

Lang’s model

A crack growth prediction model developed by Lang28,29

was intended for arbitrary irregular loading histories. TheCDF parameter was defined as

�Keff,L = Kmax − KPR − �KT, (18)

where �KT is the intrinsic threshold and KPR is the K-value (above Kmin) at which ’the stresses ahead of the cracktip switch from compression to tension’ on uploading.

The above interpretation of KPR implies that the crackgrowth rate in a given cycle is controlled by compressiveresidual stresses generated during the preceding unload-ing. According to Eq. (18), the onset of crack growth(identified with da/dN of 1 × 10−7 mm/cycle) occurswhen, after exceeding the KPR level, the �KT thresh-old is overcome. Lang does not negate the occurrenceof crack closure, but considers it contribution negligible.Measurements of both KPR and �KT required to cali-brate the model for a given material are extremely diffi-cult and lengthy, as coupled with detecting crack growthrates in the order of 10−7 mm/cycle. The KPR evolutionin the Lang model is govern by a number of empiricalequations. Rules for their application are formulated forspecific events in the load history, for example after a stepchange in the SIF, after and OL, in the course of a blockof OLs, etc. Time-consuming experiments that would berequired to derive the governing equations for a new ma-terial practically exclude applicability of Lang’s model byindependent users.

AFGROW closure model

The crack growth description with the so-called closuremodel available in the AFGROW16 software is appar-ently in accordance with the crack closure concept, i.e.da/dN is related to �Keff defined by Eq. (3). However,the determination of Sop stems from premises very differ-ent than in the case of crack closure based models as thecrack closure phenomenon is attributed to compressiveresidual stresses in the reversed plastic zone ahead of thecrack tip rather than to the contact stresses behind thecrack tip. For that reason the AFGROW closure model

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FATIGUE CRACK GROWTH PREDICT ION MODELS FOR METALL IC MATERIALS 301

can be counted among the residual stress concepts. TheSop stress is defined as a stress level at which the com-pressive stresses are overcome by applied tensile loading.Rules and constraints governing variations in Sop underVA loading stem mainly from the available experimentalevidence on crack growth behaviour for aircraft AL alloys.Though, as said earlier, several other interaction predic-tion approaches are available in AFGROW, the closuremodel is commonly referred to in the literature as theAFGROW model.

F E M O D E L S

FE analyses provide a complete stress tensor, strain ten-sor and displacement vector at any point of the model atany applied load level. Hence, any CDF parameter linkedto the stress/strain/displacement behaviour of a crackedbody can be determined. Through-the-thickness, straightfront cracks are usually analysed using 2D FE modelsassuming plane stress state or plane strain state condi-tions,32–34 whilst a 3D modelling finds application in thecase of curvilinear 2D or 3D cracks35,36 or when the effectof thickness on local stresses in the crack vicinity is inves-tigated.37 Crack growth is usually modelled by releasingsuccessive nodes along the crack line. This implies thatvalues of successive crack growth increments are definedby the mesh size, i.e. spacings between the nodes on thecrack path. In order to avoid unduly large crack growthincrements the mesh spacing along the crack line mustbe sufficiently refined. Instructive data on the mesh suffi-ciency to produce mesh independent results in the case of2D FE models can be found in studies by McClung andSehitoglu32 and Park et al.33

The literature evidence indicates an extremely strongdependency of the FE crack growth analysis results on anumber of computational choices, like the mesh size,33

the material constitutive equation34 and, in the case of2D analyses, plain stress versus plain strain assumption.For example, Sander et al.38 reported a quarter of a �Krange difference between the plane strain and plane stressKop-values.

For broken elements behind the crack tip FE modellingshould assure the capability of carrying compressive (con-tact) stresses and, at the same time, exclude their abilityto carry tensile stresses. To this end, a specific FE codeinfrastructure, if available (e.g. the DEBOND option inABAQUS), or user-introduced techniques can be utilized.For example, Pommier34 applied spring elements to linkcoincident nodes on the crack path. A crack growth incre-ment was modelled by changing the stiffness of the springfor a node just ahead of the crack tip from symmetric intension and compression to zero in tension, and a verylarge value in compression.

A stable material response after a crack advance can onlybe achieved when several loading cycles (between 2 and5) are applied on the model. Hence, assigning to a givencrack length a proper cycle number, i.e. crack growth pre-dictions, can only be done in a post-processing stage, uti-lizing available from fatigue tests material data on da/dNversus a chosen CDF parameter. However, such a pro-cedure is not applicable for VA loading sequences. Inthat case, crack length increments must correspond tothe consecutive loading cycles. FE codes oriented towardscrack growth predictions for VA loading referred to as thesmoothed FEM (S-FEM) have been developed by Kikuchiet al.36 and Kamaya et al.39 With the S-FEM, a hierarchi-cal modelling is applied, i.e. the total model area consistsof a global part and local parts, each one discretized in-dependently. Due to this concept a change of the crackgeometry (e.g. crack length) is only associated with a re-discretization of a small local region, while the meshingof the rest of the model area remains undisturbed.

D I S C U S S I O N A N D C O N C L U S I O N S

Reported in the literature prediction results from modelsreviewed in this paper suggest that the adequacy of a givenconcept is strongly related to the type of material andloading. This implies that a ‘universal’ model, that is theone adequate for an arbitrary loading once calibrated fora given material, is still lacking. Efforts into developingsuch a prediction approach can either involve a totally newconception or an improvement of one of already availablemodels. In the latter case it is obvious that a selected modelshould be characterized by possibly few constraints on itsapplicability and a possibly small amount of empiricism.

The linear model which, by definition, does not accountfor load interaction effects is evidently off a list and canonly serve as a tool to quantify the magnitude of load in-teraction phenomena for a given VA load sequence. Fig-ure 7 gives fatigue crack growth rates experimentally ob-served (open symbols) and predicted by the linear model(closed symbols) for M(T) specimens from the D16 Alalloy (Russian equivalent of 2024-T3) subject to the mini-FALSTAFF flight-simulation load spectrum. Three char-acteristic stress levels (Smax) are considered. The predictedda/dN-values, which are two to three times higher thanthe measured data, indicate that significant crack growthretardation has occurred under this load sequence.

When the equivalent SIF concept is used, criteria formaking the best choice for K eq are difficult to be rational-ized as the approach has no physical basis. According toSander and Richard,40 the K eq model can yield acceptablepredictions only when the VA load sequence representsa stochastic uniform distribution. However, Schijve’s re-sults41 suggest that it is a necessary but not a sufficient

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Fig. 7 Comparison of fatigue crack growth rates observed in the D16 Al alloy under miniFALSTAFF sequences and predictions from thenon-interaction model.

criterion for the adequacy of this type approaches. Theavailable literature evidence reviewed by Skorupa3 sug-gests that it depends on both the type of VA loadinghistory and material which K eq parameter will be mostadequate.

The Wheeler model is one of the most empirical loadinteraction models and its results are strongly dependenton the w parameter that must be determined for a givenmaterial from tests under a spectrum and stress level ‘sim-ilar’ to the actual case. As such, it can be considered arelative approach, which assumes that similar load histo-ries applied on the same detail produce similar interactioneffects in fatigue crack growth. In fact, the literature evi-dence shows that a satisfactory prediction accuracy fromthe Wheeler model can only be obtained if the consideredspectrum is ‘similar’ to that previously used to determinethe w-value.3,42 The weak point of the model is that cri-teria for the spectrum similarity are not precisely defined.

It should be emphasized that both the original and thegeneralized Willenborg model will never yield a life pre-diction that is less than the non-interaction prediction.Hence, satisfactory results can be obtained for spectradominated by OLs, and therefore inducing crack growthretardation. As an example may serve crack growth pre-dictions from the generalized Willenborg model reportedrecently by Ghidini and Donne43 for Al alloy specimens.For spectra with a large percentage of compression loads,

the modified generalized Willenborg model referred toabove as well as other concepts proposed to account foracceleration and reduced retardation due to ULs (e.g. theWalker–Chang–Willenborg model20) show significantlyimproved predictive accuracy compared to the originalconcepts. This is, however, achieved at the cost of in-troducing various empirical parameters, which must bedetermined for a given material from tests under somesimple VA sequences. Note that the original Willenborgmodel does not incorporate empirical parameters besidesthe yield stress.

When applying the plastic zone models to predict OLeffects on crack growth it cannot be overlooked that nei-ther the Wheeler nor the Willenborg model in both theoriginal and modified form offer a means to predict thedelayed crack growth retardation, as well as the retarda-tion zone exceeding the OL plastic zone. The latter phe-nomenon is well documented in the literature, includingresults by the present author and co-workers.44 A sat-isfactory correspondence between predictions from theplastic zone models and experiment could be obtainedeither for some specific load histories, as in a work bySander and Richard,45 who applied the modified Wheelermodel, or when additional empirical material and loadingdependent parameters were introduced to reflect the de-layed retardation phenomenon in the modified Weelermodel, as demonstrated in a study by Yuen and Taheri.42

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Fig. 8 Fatigue crack growth rates observed in the 350WT steel and predicted using the original and modified Wheeler model42: (a) singleOL; (b) sequence with multiple OLs.

According to these authors, in order to more accuratelypredict the OL effect, namely the transient crack growthacceleration immediately after the OL, the subsequentdelayed retardation, and the interaction between multipleOLs, three additional material/loading dependent param-eters referred to as the delay parameter, OL interactionparameter and the accelerated SIF range were required inthe Wheeler model, in addition to the retardation factorφR (cf. Eq. 5). Only the model modified in the above wayallowed for a fairly adequate prediction of each stage ofthe post-OL crack growth and load interaction effects fora sequence with periodically repeated OLs, as it is seen inFig. 8a and b respectively.

Padmadinata46 evaluated performance of the three semi-empirical crack closure models addressed in “Crack clo-sure models” section by comparing their predictions withcrack growth observed in specimens from two aircraft Alalloys (Al 2024-T3 and 7075-T6) under VA simplified andstandard flight simulation loading. Specifically, the mod-els’ capability of reproducing various experimental trendsdue to the change of some loading parameters, as for ex-ample the ground stress level or gust spectrum severity, ordue to the change of sheet thickness was vetted. Thoughfor the flight simulation histories a systematic effect ofthe ground stress level was exhibited in the fatigue tests,the PREFFAS model did not predict any influence of theseverity of the ground load because all negative loads wereclipped to zero. As pointed out by Schijve,47 another lim-itation of the PREFFAS model is ignoring the effect ofcrack advance on Kop (cf. Eq. 14), which is acceptable onlywhen the �a increment is much smaller than the plasticzone size associated with the peak Kmax. Consequently,the PREFFAS model is not adequate in cases when sig-nificant �a can occur, i.e. at higher crack growth ratesand/or for long, periodically applied OL blocks. Con-

trary to the PREFFAS model results, predictions of theCORPUS and ONERA models reported by Padmadinatadid indicate a significant reduction in crack growth livewith a more severe ground stress level, but the COR-PUS model missed this trend when the gust load in flightsimulation loading was severe, i.e. if the maximum com-pressive gust load (occurring once in 2500 flights) was be-low the ground load level, and when OL–UL or OL–ULcombinations occurred in simplified load sequences. Pad-madinata46 introduced a modification in the CORPUSmodel to remove this deficiency. Figure 9 presents com-parisons between crack growth lives observed in fatiguetests on the 2024-T3 Al alloy under simplified flight sim-ulation loading sequences of two types and the lives ob-tained from the semi-empirical crack closure models, thelinear model and a SY model from the NASGRO19 soft-ware (variable constraint-loss version, see Part II of thispaper). The results are presented in terms of the pre-dicted to experimentally observed life ratio NPRED/NEXP.It is seen that of the semi-empirical models the modifiedCORPUS correlates best the test results. Note also thatfor some specific combinations of the load sequence andloading parameters, predictions from the semi-empiricalmodels (mainly PREFFAS and ONERA) are worse thanaccording to the linear model, which implies an overes-timation of load interaction. Interestingly, the results inFig. 9 reveal in several cases a somewhat inferior accuracyof the SY model compared to the semi-empirical models.It should be however, emphasized, that in contrast to thesemi-empirical models, which have been tuned to describecrack growth under the specific load histories consideredin Fig. 9 (flight simulation), no constraints of this typeare imposed on crack closure variations in the SY model.Also, it is worth noting that except for sequence no. 7,the SY model predictions are on the safe side, though the

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Fig. 9 Predicted-to-observed life ratios according to different models for simplified flight simulation loading sequences on the 2024-T3 Alalloy (stress levels in MPa).

amount on conservatism involved in the predictions canbe overly excessive, as for sequences 6, 8, 10 and 12.

For semi-elliptical surface cracks in plane specimensof Al alloys under simplified (programmed) and stan-dard flight simulation loading Ichsan48 reported a fairlygood correlation between CORPUS model results andobserved crack growth, including crack front shape de-velopment predictions. However, it cannot be overlookedthat the evaluation of semi-empirical model prediction ca-pabilities in the above cited works46–48 is limited to crackgrowth under specific aircraft loading spectra and typicalaircraft materials. For other types of loading and othermaterials much less acceptable results were obtained. Forexample, Skorupa et al.49 noted unsatisfactory CORPUSmodel predictions on crack growth in plane specimensof a low alloy steel under single and periodically appliedOLs.

The UniGrow model addressed in “Residual stress mod-els” section involves a determination of the local stresses(σ min,net, Fig. 6) assuming a blunted crack tip of the sameradius ρ∗ at both Smax and Smin. This implies negationof the crack tip sharpening at Smin, which may be con-sidered a disputable concept. An early validation of theUniGrow model produced by Noroozi et al.30,50 was con-fined to CA loading, while predictions on the post-OLcrack growth response were not satisfactory, Fig. 10a. Ina more recent work, Mikheevskiy and Glinka51 demon-strated a very good correlation between predictions ofthe improved UniGrow model and experimental resultsfor various materials and various types of VA loading, asexemplified in Fig. 10b. Because detailed information onadjusting the K-factor during the VA load sequence is

lacking in this work, it is difficult to assess how generalare the concepts involved.

As already elucidated in “FE models” section, FE crackgrowth models can hardly be applied to real predictions.Rather, they are a useful tool to study the effect of somespecific loading events on the local stress/strain field.A better understanding of the crack growth mechanismgained in this way can then be utilized in crack growthprediction approaches.

A unique feature of the SY model is that for each load-ing cycle the Sop stress, and hence �K eff , is determinedin a consistent way, based on the computed solution forplastic strip element stresses and lengths at consecutiveextremes of the load history (cf. “Strip Yield model” sec-tion). This is in contrast to all other prediction modelsaddressed in this paper, which require adjusting the CDFparameter according to some empirically based rules re-lated to specific occurrences in a load sequence. Becausea theoretical foundation behind the SY model makes itcapable of reflecting any trend observed in crack growth,no limitations on the type of VA loading are imposed,as in the case of other models. For the above reasons,the SY model can be currently considered a most ver-satile and robust crack growth prediction concept. Re-ported SY model analyses of crack growth indicate thatit can describe a number of crack growth phenomenathat cannot be reproduced by any other prediction model,such as for example delayed retardation after an OL, ef-fect of periodically applied OLs, or the OL-influencedzone larger than the OL plastic zone.26,52 At the sametime, the literature evidence on a quantitative correlationbetween SY model predictions and observed crack growth

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Fig. 10 Comparison of fatigue crack growth data observed and predicted by using the UniGrow model: (a) 7075-T6 Al alloy, single OL;50

(b) 350WT steel, sequence with two OLs.51

is inconsistent in that it reveals both a very good,53 anda poor agreement,11 even for load histories of the sametype. A reason for contradictory data on the model predic-tive capabilities is differences between specific SY modelimplementations. Possible choices and decisions when de-veloping a SY model and their far-reaching consequencefor the model results are considered in detail in Part II ofthis paper.

When in a prediction model a range of SIF is employedas the CDF parameter, the model applicability to ’real-work’ problems is conditioned by the availability of anappropriate K-solution. A number of K-solutions whichcan be useful for practical problems are available in NAS-GRO19 and AFGROW.16 When analyses with predictionmodels incorporated in these software packages are run,also user-introduced correction factors can be applied ontop of the K-solutions already available in the programme.Alternatively, it is possible to use an external K-solver thatcommunicates with AFGROW, though this approach canbe very time-consuming. Crack growth predictions forinteracting multiple cracks growing simultaneously in astructure are currently not possible using the NASGROand AFGROW models. This issue is of practical impor-tance in the case of multi-site damage in joints with me-chanical fasteners, and the corresponding K-solutions andcrack growth prediction approaches are amply covered inthe recent book by Skorupa A. and Skorupa M.54

In the case of the SY model a solution for the crack sur-face displacements is needed, in addition to the solutionfor K . The approach allowing the treatment of any ge-ometry of interest using the SY model is covered in PartII of this paper.

Acknowledgement

This paper owes much to the support and helpful com-ments of Professor Małgorzata Skorupa, AGH Universityof Science and Technology.

The financial support from the governmental researchfunds within the years 2009–2012 is acknowledged.

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