# Fatigue crack growth prediction models for metallic

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### Transcript of Fatigue crack growth prediction models for metallic

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, 30059 Krakow, Poland

Received in final form 11 July 2012

ABSTRACT 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.

NOMENCLATURE a = crack lengthafict = fictitious crack lengthCA = constant amplitude

CDF = crack driving forceCOD = crack opening displacement

da/dN = fatigue crack growth rateFE = finite elementK = stress intensity factor (SIF)

K eq = equivalent stress intensity factorKmax = 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: machniew@agh.edu.pl

c 2012 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 36, 293307 293

294 T . MACHNIEWICZ

ROL = KOLmax/KmaxRSO = 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 amplitudea = crack growth incrementK = Kmax Kmin

K eff = Kmax KopK th = threshold KKT = intrinsic K th

SRMS = Smax RMSSmin RMS = crack tip radius y = yield stress

, R = retardation factors

I N T RODUC T I ON

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 available

crack 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 SYmodel. 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 LAS S I F I CA T I ON OF FA T I GU E CRACKGROWTH PRED I C T I ON MODE 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 = ai1 + 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,

c 2012 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 36, 293307

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 andminimum SIF valuesin a load cycle. The Paris equation reads

dadN

= CKm, (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 equation

and, hence, the computed ai depends on the currentcrack size in cycle i. For that reason, consecutive aiincrements 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

EQU I VA L EN T S I F MODE 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

SmaxRMS =1

m

mi=1

(Smax i )2; SminRMS =1

m

mi=1

(Smin i )2,

(4)

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

c 2012 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 36, 293307

296 T . MACHNIEWICZ

Fig. 2 Equivalent SIF concept.

Fig. 3 Post-OL plastic zones considered in Wheelers model.

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

L I N EAR (NON - I N T ERAC T I ON ) MODE 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 Miners 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 ob

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