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ABSTRACT. The fatigue process in filletwelded joints is discussed and modeled.As a first approximation, a pure fracturemechanics model was employed to de-scribe the entire fatigue process. Themodel is calibrated to fit the crack growthmeasurements obtained from extensivetesting on fillet weld joints where cracksemanate from the weld toes. Emphasis islaid on the choice of growth parameters inconjunction with a fictitious initial cracksize distribution in order to obtain both re-liable crack growth histories and predic-tions of the entire fatigue life. The modelhas its shortcomings in describing thedamage evolution at low stress ranges dueto the presence of a significant crack initi-ation period in this stress regime. As an al-ternative to the fracture mechanics model,a two-phase model (TPM) for the fatigueprocess was developed and calibrated.The number of cycles to crack initiationwas modeled by a local strain approachusing the Coffin-Manson equation,whereas the propagation phase was mod-eled by fracture mechanics, adopting thesimple version of the Paris law. The notcheffect of the weld toe was treated by ex-treme value statistics for the weld toe ra-dius. To make the model fit all test data forcrack initiation and propagation, it is cru-cial to select a sufficiently low transitiondepth between the two phases. A transi-tion depth of 0.1 mm (0.004 in.) was se-lected. Furthermore, the material para-meters in the Coffin-Manson equationwere determined directly from the earlycracking in the weld toe for full-scale filletwelds and not from tests carried out withsmall-scale smooth specimens. This is es-sential if the model is to account for the ac-tual surface condition at the weld toe. TheS-N curves constructed by the presentphysical model were compared with the

median nonlinear S-N curve obtainedfrom the statistical random fatigue-limitmodel (RFLM) presented in Part 1 of thisinvestigation. The curves are almost iden-tical in the stress region where experimen-tal data exist and both models fit the ex-perimental data very well. However, theTPM does not predict any fatigue limit, incontrast to the RFLM. According to theTPM, the long fatigue life found at lowstress ranges is a result of a dominatinglong initiation period and is not a thresh-old phenomenon. The two models arecomplementary tools. The RFLM is apure statistical model, whereas the TPM isa semiempirical physical model. The TPMis capable of taking into account the effectof the global geometry of the joint, thelocal weld toe geometry, applied stressratio, and the residual stress condition.Both models can be used for fatigue lifepredictions, but only the physical TPM canbe used for planning of in-service inspec-tion strategies where damage evolution asa function of time is needed.

Introduction and Objectives

In this investigation, the fatigueprocess in fillet-welded joints wherecracks emanate from the weld toe wasstudied and modeled. In Part 1, a statisti-cal model based on a joint random fatiguelife and a random fatigue limit [random fa-tigue-limit model (RFLM)] was applied topredict the fatigue life of these types ofjoints. The S-N curve obtained from this

RFLM was nonlinear for a log-log scalebetween the stress range ∆S and numberof cycles N to failure. The model fits theassembled S-N data in the low stress re-gion far better than the traditional bilinearS-N curve found in rules and regulations.The fatigue behavior is obviously signifi-cantly more complex at low stress rangesthan the conventional curves are able todescribe. The RFLM-based nonlinearcurves give some new physical insight intothe fatigue process itself and this is pur-sued in this investigation by a semiempir-ical physical model. The model is a com-plementary tool to the statistical RFLMdeveloped in Part 1. It is important to con-trol the physical parameters that have aninfluence on the fatigue damage processas it evolves toward final failure. This isimportant when the following predictionsare required: • Predictions of fatigue life under condi-

tions for which experimental data donot exist

• Predictions of likely crack growth histo-ries leading to failure.The need for a more physical model for

carrying out the first type of predictions isobvious; fatigue life can only be predictedby a statistical model for joints pertainingto the populations and conditions forwhich the model was established. If basicfatigue properties such as joint geometryor loading modes are changed, it is only aphysical model that can predict the effectof such changes on the fatigue life. Hence,the physical model is an important tool insafe life analysis. The second type of pre-dictions are required if inspection plan-ning is to be carried out, i.e., a damage tol-erance approach. In this case it is anecessity to characterize the fatigueprocess itself, not only to determine thefinal fatigue life. This is essential if in-service inspections are to be planned; wemust know what crack sizes to look for atdifferent times before final failure. Thiswill make the scheduled inspection moreefficient and economical. Based on theseconsiderations, a physical model will be

Fatigue Behavior of Welded Joints Part 2:Physical Modeling of the Fatigue Process

Two complementary models were developed to predict fatigue behavior of joints infillet welds

BY P. DARCIS, T. LASSEN, AND N. RECHO

P. DARCIS and N. RECHO are with LERMES,Laboratoire d’Etudes et de Recherche enMEcanique de Structures, Université Blaise Pas-cal, Clermont II, France. T. LASSEN is withAgder University College, Faculty of Engineering,Grimstad, Norway.

KEYWORDS

Welded Steel JointsFatigue BehaviorInfluencing Physical ParametersCrack InitiationCrack PropagationLife PredictionsConstructed S-N Curves

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JANUARY 2006-s20

established to meet the following criteria: • The model should be corroborated by S-

N data for the joint in question whenthese are available. (Database 1 and 2in Part 1.)

• The model should predict a crack evo-lution that coincides with measuredcrack growth histories before failure.(Database 1 in Part 1.)With this background, we endeavored

to model the fatigue process in fillet weldjoints. The total fatigue life was consid-ered to be the sum of the cycles spent inthe crack initiation phase and the crackpropagation phase:

NT = Ni + NP (1)

For some time there has been a debateamong researchers and engineers as towhether or not the crack initiation periodis important. The traditional belief hasbeen that fatigue crack growth often startsfrom surface-breaking defects in the weldtoe region. The initial flaw has often been

assumed to have a depth greater than 0.1mm, sometimes 0.25 mm. In rules and reg-ulations, an initial crack depth even asdeep as 0.5 mm (0.02 in.) has been recom-mended (Ref. 1). These flaws act as directstarters for the fatigue crack propagationand the fatigue initiation period can be ne-glected. However, with advanced fatiguetesting where the entire crack depth his-tory, and not only the final fatigue life, ismonitored, it has become obvious that thissimple approach does not fit the facts. Sev-eral researchers have obtained experi-mental evidence that supports the exis-tence of a crack initiation period (Refs. 2,3). The conclusion is that rather thanspeaking of small microcracks or inclu-sions in the vicinity of the weld toe, it ismore correct to use the notion of unfavor-able surface condition, which gives arather short initiation period under accel-erated laboratory conditions. The earlydamage mechanism is a combination ofcrack nucleation and microcrack growth.In welded joints subjected to high stresses

in accelerated laboratory conditions (typ-ically stress range of 120–150 MPa(17.4–21.7 ksi)), the initiation period, de-fined as time to reach a crack depth of 0.1mm, is typically 30% of the entire fatiguelife (Ref. 3). This means that if the samejoint is subjected to stress levels that aretypical of service conditions (equivalentstress range 50–80 MPa (7.3–11.6 ksi), thecrack initiation will totally dominate thefatigue life. Due to this fact, a fracture me-chanics model (FMM) will not be able tomeet both of the model criteria listedabove. With this background, a TPM wasinvestigated and compared with the FMMas well as with the results obtained by theRFLM in Part 1 of this article. It wasLawrence et al. who first suggested theTPM and a good overview is given in Refs.4 and 5. As the method now stands, its ac-curacy depends greatly on the calibrationexperiments. Our objective here is to elab-orate and calibrate the model to fit thevarious test series that were presented inPart 1. The predictions made by the model

Fig. 1 — Joint configuration with crack shape parameters a and c.

Fig.3 — Experimental a-N curve (Database 1). Fig. 4 — Histogram for ln C (MPa, m).

Fig.2 — Growth parameters obtained from experiments together with the BS7910 scatter-band. A — Linear relationship; B — bilinear relationship.

A B

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will be compared with the predictionsmade by the RFLM in Part 1.

The Fracture MechanicsApproximation

Before developing the TPM we shallpresent a brief overview of the fracturemechanics model and point out its short-comings when it comes to describing theentire fatigue process for high-qualitywelds. It is assumed that the FMM canpredict the propagation rate of a semiel-liptical crack, as shown in Fig. 1, from itsinitial crack depth to final critical crackdepth. By integration of the Paris law thenumber of cycles to failure can be pre-sented in an S-N format:

where C and m are treated as material pa-rameters for a given mean stress and envi-ronmental condition. F(a) is a dimension-less geometry function accounting forloading mode, crack, and joint geometry.The equation is valid when the stress in-tensity factor range (SIFR) ∆K =∆S√πaF(a) is greater than the threshold

value ∆K0. Below this threshold value, it isassumed that the crack will not grow. Rec-ommendations are given in BS 7910 (Ref.6) for the growth rate parameters C and mand the threshold value ∆K0. The geome-try function F(a) is also given. Regardingthe parameters C and m, two alternativesare suggested for the relationship betweenthe growth rate da/dN and the SIFR for alog-log scale. The first alternative is basedon a single linear relationship, whereas thesecond alternative proposes a bilinear re-lationship — Fig. 2. The data points on thefigure are derived from growth measure-ments and are discussed below. The maindifference between the two models is thatthe bilinear one models the gradual de-crease in the growth rate for low values ofthe SIFR before the threshold value is fi-nally reached. As a first approximation,the FMM was used to fit all the measuredgrowth histories given in Fig. 3 (Database1). Each of the 34 curves was fitted by theFMM by adopting the slope m given forthe growth curves in Fig. 2. For each fit theparameters a0 and C were determined.The a-N curves obtained from the FMMwere quite close to the experimental onesin Fig. 3. This was true for both the singlelinear and bilinear relationships betweenthe growth rate and the SIFR. Detailsfrom this analysis are found in Ref. 7. Ahistogram for the natural logarithm of Cderived under the assumption of a single

linear relationship between the crackgrowth rate and the SIFR for a log-logscale is shown in Fig. 4. In rules and regu-lations, the distribution of ln(C) is as-sumed to be normally distributed (as thefatigue life is assumed lognormally dis-tributed). The present distribution is notobviously a normal distribution. The re-sults are also plotted in Fig. 2 and com-pared with BS 7910 recommendations. Ascan be seen, the results are in good agree-ment with the mean line given in BS 7910.This was not the case for the results de-rived for the bilinear relationship. Thegrowth rates are close to four times higherthan the mean values given in BS 7910 forthe lower line segment — Fig. 2. Thesehigh growth rates are probably due to thefact that the lower part of the bilinearcurve is derived from tests with relativelylong cracks (several millimeters) in com-pact tension specimens. The fatigueprocess in welded joints comprises crackgrowth of small surface-breaking ellipticalcracks with depths less than 0.1 mm. Thesecracks may grow considerably faster thanthe lower part of the bilinear growth curvein BS 7910 prescribes. Hence, one shouldtake care when using the bilinear relation-ship given in BS 7910 to calculate the earlyfatigue crack growth in welded joints.

The statistics for the calculated initial

NC

da

S aF am

a

ac

=

( )

∫1(2)

0 ∆ π

Table 1 — Statistics for Local Weld Toe Geometry, Database 1

Weld toe angle θ (degrees) Weld toe radius ρ (mm)Mean Standard Deviation Mean Standard Deviation 58 9 1.6 0.7

Table 2 — Measured Number of Cycles (in1000) Spent in Various Phases. AcceleratedLaboratory Condition 150 MPa (Database 2)

Ni is defined as time to reach 0.1 mm Ni Np Nt Nt F-class140 330 468 513

Fig. 5 — Histogram for the initial crack depth. Fig. 6 — Schematic illustration of the local stress-strain hysteresis loopanalysis.

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JANUARY 200622-s

crack depths a0 are given in Fig. 5. Themean value is 0.015 mm (0.0006 in.) andthe upper bound is close to 0.03 mm(0.0012 in.). It should be emphasized thatthis initial crack depth distribution is apurely theoretical concept, i.e., it cannotbe proven that the crack depths are re-lated to initial flaws created by the weld-ing process. The derived mean values forthe a0, C, and m are substituted into Equa-tion 2 to calculate both crack evolutionand fatigue life at various constant ampli-tude stress levels. Both the linear and bi-linear relationship is used. The followingshortcomings were revealed: the predic-tions made by the linear relationship cor-respond to the predictions made by the S-N curve F-class above the fatigue limit,whereas the predictions made by the bi-linear curve do not. It is also to be notedthat both the F-class and the FMM fail topredict the long experimental lives givenby the data points in Database 2 at lowstress levels. In addition, the FMM fails topredict the fatigue limit. Based on thefracture mechanics threshold criterion,

∆S0 = √πa0F(a0) = ∆K0, the fatigue-limitis 95 MPa based on the initial cracks givenin Fig. 5. This is far too high comparedwith the limit of 56 MPa given by the F-class and 60 MPa obtained from theRFLM. The reason is the same as for thediscrepancy found between the experi-mental growth rates and the growth ratesgiven in the lower part of the bilinear re-lationship in BS 7910. Both the prescribeddrop in the growth rate and the final stopat low SIFR are valid for larger cracksonly. They are not applicable to shallow el-liptical cracks at the weld toe. In conclu-sion, the FMM should not be used tomodel the entire fatigue process in high-quality welds proven free from detectableinitial cracks. Although the model is capa-ble of describing the crack evolution atone given stress level, it will fail to predictthe change in slope of the S-N curve as thestress range decreases. Furthermore, thefatigue-limit will be overly optimistic. TheFMM should only be applied if cracks arefound and sized. In other cases a crack ini-tiation phase should be modeled before

the crack propagation phase is added on.This is shown in the next section.

Modeling the Fatigue CrackInitiation Period

Basic Concept and Equations for theLocal Stress-Strain Approach

The predictions for the number of cy-cles to crack initiation, Ni, are based on theCoffin-Manson equation with Morrow’smean stress correction (Refs. 4, 5):

Here ∆ε is the local strain range and σm isthe local mean stress at the weld toe. Theparameters b and c are the fatiguestrength and ductility exponents, and σ’fand ε’f are the fatigue strength and ductil-ity coefficients, respectively. The localstress and strain behavior is given by theRamberg-Osgood stabilized cyclic straincurve:

where K’ and n’ are the cyclic strength co-efficient and strain hardening exponent,respectively. Equation 4 is combined withthe Neuber rule as follows:

where ∆S is the nominal stress range, E is

∆ ∆∆

ε σ =( )K S

E

t

2

(5)

∆ ∆ ∆ε σ σ= +′

′

E K

n2

2(4)

1

∆εσ σ

ε2

2 2 (3)=−

( ) + ( )f m

i

b

f i

c

EN N

'

'

Fig. 7 — Definition of the toe notch geometry in one specimen by extremevalue statistics.

Fig. 8 — S-N curves constructed from the RFLM and TPM together with theF-class median curve and test data.

Table 3 — Cyclic Mechanical Properties and Parameters in the Coffin Manson EquationCalibrated for Time to Reach 0.1 mm, HB = 202

Parameter, Symbol (units) Value

Cyclic yield stress, S’y (MPa) 424 (61 ksi)Ultimate strength, Su (MPa) 697 (101 ksi)Young modulus, E (GPa) 206 (30000 ksi)Fatigue strength exponent, b –0.089Fatigue strength coefficient, σ’f (MPa) 1032 (150 ksi)Fatigue ductility exponent c –0.6Fatigue ductility coefficient ε’f 0.81Cyclic strength coefficient, K’ (MPa) 1064 (154 ksi)Strain hardening exponent, n’ 0.148

Number of cyclesToe radius [mm]

Young’s modulus, and Kt is the stress con-centration factor at the welded toe. Equa-tion 5 is sometimes modified by introduc-ing the fatigue notch factor Kf instead ofKt . It is argued that the fatigue notch fac-tor better quantifies the severity of a dis-continuity in the fatigue life calculation.Yung and Lawrence concluded (Ref. 4)that Peterson’s equation correctly interre-lates the fatigue notch factor to elasticstress concentration factors for weldedjoints:

Here ρ is the weld toe root radius and apis Peterson’s material parameter. The lat-ter may be approximated by the expres-sion 1.087 × 105 Su–2 (in mm, N/mm2)where Su is the tensile strength of the steel.The following expression for Kt is used(Ref. 8):

where θ is the weld toe angle (radians) andT is the plate thickness — Fig. 1.

The definition of the local stress-strainvariation is illustrated in Fig. 6. The nom-inal stress S (left) and the local stress σ(right) are shown for the first reversal(0-1) and the stabilized hysteresis loop(1-2-3). The local ∆σ and ∆ε and the meanstress, σm, corresponding to the cyclicloading, are determined. The effect ofcyclic hardening or cyclic softening is neglected.

In the case of an elastic notch root con-dition at the weld toe, Equation 3 reducesto the Basquin equation, neglecting thesecond term on the right-hand side. Thisequation can be rearranged as follows:

From this equation one may construct anS-N curve with slope b.

Definition of the Initiation Phase andDetermination of Parameters

When applying the Coffin-Manson ap-proach to predict time to crack initiation

at the weld toe, threequestions arise. Thefirst is what stressconcentration factorKt one shouldchoose to character-ize the notch effectfrom the highly vari-able local toe geom-etry. The next is thedefinition of time tocrack initiation, i.e.,what crack depth isreached at the end ofthe phase beforepropagation is con-sidered to take over.This depth will be re-ferred to as the tran-sition depth. If toolarge a depth is cho-sen, the initiationphase will not obeyEquation 3 due tothe substantialamount of crackgrowth involved. Several definitions arepossible for the transition depth and this isprobably one of the reasons why so fewtwo-phase models have been applied inpractice. The last question is how to de-termine the cyclic mechanical parametersand the parameters in the Coffin-Mansonequation such that they are valid for theweld toe condition. Usually these parame-ters are determined from tests with small-scale smooth specimens. However, in afull-scale weld there may be size effectsand surely surface finish effects involved.Tests with smooth specimens may not berepresentative for the rough surface con-ditions found in the vicinity of the weldtoe. It is our hypothesis that the actual pa-rameters should be determined directlyfrom full-scale tests with welded jointswhere the early cracking is measured.Database 1 gives us this possibility. Basedon the discussion above we will emphasizethree topics:• Local toe geometry and stress concen-

tration factor• The choice of the transition depth• The determination of the parameters in

Coffin-Manson law.

Local Toe Geometry and StressConcentration Factor

The statistics for the local geometry forthe specimens tested in series 1 (Database1) are given in Table 1. If we substitute themean values in Table 1 into Equation 7 thiswill give Kt = 2.9. This value is corrobo-rated by a refined finite element analysis.However, it is highly likely that the cracksinitiate at a more unfavorable geometry. Itis above all the variability in the toe radius

that yields Kt values in the range from 2.9to 7. To circumvent this problem of vari-ability, Lawrence et al. suggested using thefatigue notch factor Kf instead of Kt. Fur-thermore, a worst-case notch was definedby setting the toe radius equal to the Pe-terson constant in Equation 6. In our casethis will approximately give ρ = ap = 0.3mm, which corresponds to Kf = 3.1. Theproblem with the Kf concept is that thephysical interpretation is not obvious. Ithas been claimed (Ref. 9) that the need fora definition of a fatigue notch factor is dueto the fact that the crack initiation life inreality includes an appreciable amount ofcrack growth. If the initiation phase werelimited to pure nucleation of a micro-crack, the stress concentration factorcould have been used directly in the cal-culations. Furthermore, the smallest ra-dius in a joint is a random variable and itsmean value will be a function of the lengthof the welded joint. This is pursued below.Let us set the toe angle constant at themean value and try to determine the mostlikely smallest toe radius found within onetest specimen. The statistics in Table 1were derived from 300 measurementstaken with 5.5-mm spacing along a weldedjoint with a total length 1650 mm. It ap-peared that two neighboring radii with aspacing of 5.5 mm could be very different,whereas at a closer distance there is a cor-relation. Based on this observation we as-sume that at a 5.5-mm spacing the mea-sured radii will be independent. Hence, awelded joint with length W mm contains k= W/5.5 totally independent radii. The ex-tremum distribution for the smallest valuewill then read:1 – Fρmin(ρ) = (1 – Fρ(ρ))k (9)

NK S

K S

if

f m

b

f m

f

b

b

=′ −( )

=′ −( )

( )−

−

1

2 2

1

2

2(8)

1

1

1

∆

∆

σ σ

σ σ

/

/

/

KT

t = + ( )

1 0.5121 (7)0.572

0.469

θρ

KK

af

t

p

= +−( )

+

11

1

(6)

ρ

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Fig. 9 — S-N curves constructed from the RFLM and TPM in the low-stressregion.

WELDING JOURNAL 23 -s

Number of cycles

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JANUARY 200624 -s

where Fρ(ρ) is the cumulative distributionfunction (CDF) of an arbitrary radius asgiven in Table 1, whereas Fρmin(ρ) is theCDF for the smallest value over the lengthW. The mean and peak values for thissmallest radius can be derived from thecorresponding probability density func-tion (PDF). The peak value is by defini-tion the most likely value for the smallestradius within the length W. If we assumethat ρ is Weibull distributed (Ref. 10) andW is set to 1650 mm (k = 300), then ourapproach will give a peak value for thesmallest radius close to 0.1 mm. The small-est radii found in various test series are ac-tually close to 0.1 mm (Ref. 10). Hence,these results support our approach. If weuse the same approach within the width ofone test specimen (W = 60 mm, k = 60/5.5= 10), we get ρ = 0.42 mm. The PDFs areshown in Fig. 7. As can be seen, the meanvalue and the peak value are not very dif-ferent for the relatively narrow symmetri-cal extreme value distribution. The result-ing figures will give a stress concentrationfactor of 4.5 that will be used in our calcu-lations.

Transition Depth

In earlier work, the transition depthhas often arbitrarily been set to 0.25 mm(0.01 in.). However, the time to developsuch a deep crack will include a largeamount of crack growth. This has in manycases resulted in models that give reason-ably good predictions at short fatigue lives(less than 106 cycles), but an overpredic-tion of the fatigue lives for low stress lev-els. The reason is that the initiation part atlow stress levels will have a life curve witha slope close to the parameter b ≅ –1/10(see Equation 8). This is only true for pureinitiation, whereas crack growth will havea slope according to the Paris law, 1/m ≅–1/3 relative to the applied stress range.Hence, a phase that contains both nucle-ation and growth will have a slope between–1/3 and –1/10. If a slope of –1/10 is as-sumed for such a mixed process, it willoverestimate the fatigue life significantlyat low stresses. In more recent work,Lawrence et al. (Ref. 5) suggested that thetransition depth should be between 0.05and 0.1 mm. In the present work, we in-vestigated the results obtained by settingthe transition depth equal to the lower andupper bound of this given range. The fol-lowing arguments support a transitiondepth of 0.1 mm (0.004 in.):• To apply the facture mechanics model at

crack depths smaller than 0.1 mm maybe dubious because one approaches thegrain size of the steel (typically 0.01mm).

• In laboratory tests it is hardly possible tomeasure any crack less than 0.1 mm with

sufficient accuracy without using de-structive methods.

• Cracks with a depth below 0.1 mm arenot of interest in in-service inspection asno common nondestructive examina-tion method can detect such smallcracks. Hence, for inspection planningwe do not need the notion of a cracksmaller than 0.1 mm.Our arguments are partly theoretical,

partly practical. The actual number of cyclesto reach 0.1 mm is given in Table 2 for Data-base 1. In contrast to the fictitious initialcrack depths obtained from the FMManalysis in the previous section, this is ameasurable quantity. The fatigue initiationlife is close to 30% of the entire fatigue life.The entire fatigue life is 10% shorter thanthe predictions from the F-class curve. Thearguments for a transition depth of 0.05 mmare less obvious. It is to be noted that eventhis shallow depth is well above the initialcrack distribution obtained from the frac-ture mechanics model (see Fig. 5). Theupper bound was found to be 0.03 mm forthese cracks. Hence, the transition depth of0.05 mm is about the smallest transitiondepth possible based on what may be inter-preted as crack size of possible initial flaws.The time to arrive at 0.05-mm crack depthwas not measurable for the tests carried outfor Database 1. The number of cycles toreach this depth can be found by backcalcu-lation from the first measurable crack size(0.1 mm) by applying the fracture mechan-ics model. In this way the number of cyclesto reach 0.05 mm was determined to 90,000cycles, i.e., 20% of total fatigue life.

Cyclic Mechanical Properties andParameters in Coffin-Manson Equation

It is our hypothesis that parameters de-termined from small-scale smooth speci-mens are not directly applicable for weld toeconditions. Hence, we will determine theseparameters directly from the time to earlycracking as given Table 2. We now seek theparameters in Equations 3 and 4 that corre-spond to the initiation time given in Table 2.The calibration is carried out by assuming adependency between the various parame-ters with the Brinell hardness (HB) as thekey parameter. The following equationswere applied (Ref. 11):

where Su and Sy are the tensile stress andyield stress respectively, and Sy ,is the

cyclic yield stress. The correlation be-tween the various parameters, using theHB as a master variable, is not exact. InRef. 12, it was found that the relationshipmight lead to significant overestimation ofthe initiation life. Furthermore, Dowling(Ref. 9) suggested that the surface condi-tion at the weld toe would primarily alterthe fatigue strength exponent b. However,in the present work we accept the rela-tionships given in Equation 10 but not theabsolute values. An absolute value issought such that the time to reach a giventransition crack depth coincides with whatactually has been measured on the weldedjoints in Database 1. The solution gives aHB close to 202 for the cycles to reach acrack depth of 0.1 mm. This HB is veryclose to the value actually measured in theHAZ of the weld toe. The values mea-sured for the base metal was HB = 145and value of the HAZ was 213. Hence, oursolution is only 5% less than the highestvalue measured at the potential cracklocus. If we had applied HB = 213 directly,the solution would give the time to reacha crack depth of 0.3 mm. Hence, a signifi-cant amount of crack propagation wouldhave been included in the initiation phase.When using the search scheme for deter-mining the parameters with a transitiondepth of 0.05 mm, we obtained a HB of180, i.e., still within the range measured onthe specimens, but 15% less than the valueat the HAZ.

The parameters corresponding to HB= 180 (a = 0.05 mm) and HB = 202 (a =0.1 mm) were used in Equations 3 and 4 todefine the first part of the TPM. The prop-agation phase (Equation 2) was subse-quently added to calculate the entire fa-tigue life. The model predicts exactly themean value for the fatigue lives of Data-base 1 at a stress range of 150 MPa (21.7ksi). When the stress range was decreasedfrom 150 MPa (21.7 ksi) to below 100 MPa(14.5 ksi), the model based on a transitiondepth of 0.1 mm predicted somewhatlonger lives than the median line obtainedfrom the RFLM, which is representativefor Database 2 in this stress region. Themodel based on a transition depth of 0.05mm predicted results somewhat shorterlives than figures obtained from theRFLM in this region. At a stress range of80 MPa, the model with a = 0.1 mm pre-dicts 28% longer life than the median lineof the RFLM, whereas the model witha=0.05 mm predicts 20% shorter life thanthe same median line. These results arediscussed more in detail in the next sec-tion. The results indicate that the intervalfor a transition depth between 0.05 and 0.1mm as proposed by Lawrence et al. (Ref.5) is a reasonable choice. Furthermore,any transition depth in this narrow bandwill predict fatigue life well within the scat-ter band of Database 2. Based on the ar-

S HB MPa nb

c

S S MPa K S MPa

bS

S MPa

c cK

u

y u y

n

uf u

ff

l n

= ′ =

′ = ′ = ( )= +

′ = +

= < < ′ =′

′

− ′

′

3.45

0.608

0.1667 2.1917

0.95 370 (10)

0.7

0 002.

– log

– –0.5

/

σ

εσ

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guments given in the beginning of this sec-tion, we have selected a transition depth of0.1 mm in what follows. The correspond-ing parameters in the Coffin-Mansonequation are given in Table 3. As we al-ready have shown, the solution given inTable 3 is not unique. Other solutionswithout total dependency between the pa-rameters are possible. However, these so-lutions will not be far from the one givenin Table 3. Hence, we regard the solutionrepresentative of prediction of time toreach a depth of 0.1 mm in welds madefrom C-Mn steel with a yield stress closeto 345 MPa (50 ksi).

Constructing the S-N Curve fromthe Two-Phase Model

One of our main goals is to construct S-N curves from the TPM that are consistentwith the RFLM curves obtained in Part 1of this investigation. As pointed out in Part1, the RFLM curve fits the data points farbetter than the F-class curve at low stressranges. Although the TPM is semiempiri-cal, it has a more physical-theoretical basisthan the RFLM, which is based on purelystatistical methods. The TPM is capable ofpredicting the influence of, for instance,local weld toe geometry, stress ratio, andstress relieving. Thus, high-quality jointswill have long fatigue lives, whereas poorquality will be penalized. The TPM modelcan also be applied to calculate fatiguelives at low stress levels where experimen-tal data do not exist and where it is dubi-ous to extrapolate the statistical RFLM.

Let us begin by demonstrating that themodel can predict fatigue lives that are ingood agreement with the S-N curve ob-tained from the RFLM under appropriateassumptions of the quality of the joint. Forjoints that are stress relieved (Database 1),the TPM will predict fatigue lives as givenin Table 4 at various stress levels. As can beseen from the table, the time to crack ini-tiation at a test stress range of up to 150MPa (21.7 ksi) is 30% of the entire fatiguelife, whereas it is 88% of the fatigue life ata stress range of up to 80 MPa (11.5 ksi).These results pinpoint the importance ofthe crack initiation life at low stress ranges,i.e., in the stress region where servicestresses usually occur. The table also liststhe fatigue lives predicted by the RFLMand the F-class. At stress ranges below 100MPa (14.5 ksi), the TPM predicts some-what longer lives than the S-N curve basedon the RFLM and significantly longer livesthan the F-class S-N curve. As can be seenfrom Table 4, the total TPM fatigue life isclose to 1.6 times longer than the life ob-tained from the RFLM and 5.5 timeslonger than prediction made by the F-classat 80 MPa (11.5 ksi).

When comparing these figures we must

bear in mind that the figures derived fromthe TPM correspond to the test series inDatabase 1, i.e., stress relieved (SR) andwith an applied stress ratio of R = 0.3. Thisstress relieving has a strong bearing on thetime-to-crack initiation through the Mor-row mean stress effect at long lives (Equa-tion 3). The RFLM S-N curve is domi-nated by Database 2 in the low stressregion. These tests are carried out on non-load-carrying fillet weld joints with thick-nesses in the range of 16 to 38 mm. Thespecimens are all in the as-welded (A-W)condition and with a positive stress ratio,i.e., there may be large residual stressespresent in the specimens. Furthermore,the vast majority of tests used to deter-mine the F-class curve are in A-W condi-tions and often tested at a stress ratio closeto R = 0.1.

Thus, our next step is to simulate theseconditions for the initiation part of theTPM by setting the residual stress equal tothe actual material yield stress, i.e., 400MPa (58 ksi). The results are given inTable 5. As can be seen, the TPM resultsare now almost identical to the nonlinearS-N curve obtained from the RFLM, butthe model still predicts a fatigue life 2.5times longer than the F-class at 80 MPa(11.5 ksi). These results are illustrated inFig. 8 where the F-class and RFLM-basedS-N curves are drawn together with theTPM S-N curve and the test results. As canbe seen, we have basically two types ofcurves. The F-class curve is bilinear,whereas the RFLM and TPM curves arecontinuously changing slope. All three S-N curves coincide at high stress levels.Hardly any discrepancy in fatigue life (lessthan 10%) is found above a stress rangelevel of 120 MPa (17.4 ksi). When the

stresses are lowered to under 100 MPa,the RFLM curve and the TPM curve stillcoincide, but they predict 2–9 times longerlives than the F-class curve as long as thestress range is above the F-class fatigue-limit of 56 MPa (8.1 ksi). It is our judgmentthat the F-class curve is too conservative inthe stress region under consideration, aswas already discussed in Part 1. This is dueto the fact that it is a straight line andbased on test results that have the centerof gravity for the stress ranges between120 and 150 MPa (17.4 and 21.7 ksi).Hence, the curve fails to take into accountthe increasing fatigue life due to the im-portance of an initiation phase below 100MPa. The experimental results plotted inthis stress region corroborate the predic-tions made by the TPM. It has been shownhow the TPM is capable of correctly tak-ing into account the effect of residualstresses and loading ratio. This is shown inmore detail in Fig. 9. The life curve ob-tained for the A-W condition coincideswith the median curve (i.e., the RFLMcurve) for Database 2.

Finally, it should be noted that al-though the statistically based RFLM andthe physically based TPM give the samelife predictions at almost any stress rangelevel, there is one fundamental differencebetween them. The RFLM does prescribea fatigue limit, whereas the TPM does not.This is illustrated in Fig. 9 where the focusis on the lower stress region. The slope ofthe S-N curve derived from the TPM willnot be smaller than b = –1/10 and willnever become horizontal as is the casewith the RFLM. This gives a discrepancybetween the curves at very long lives(longer than 108 cycles) such that theRFLM is more optimistic. The TPM pre-

Table 5 — Results Derived from the TPM at Various Stress Ranges. As Welded (AW), R = 0.1

Stress Range Ni (cycles) Np (cycles) Nt (cycles) Ni/Nt %(MPa) TPM TPM TPM TPM

150 1.3 × 105 3.3 × 105 4.6 × 105 28120 4.3 × 105 6.4 × 105 1.1 × 106 40100 1.3 × 106 1.1 × 106 2.4 × 106 5480 6.6 × 106 2.2 × 106 8.8 × 106 7560 8.1 × 107 5.1 × 106 8.5 × 107 94

Table 4 — Results Derived from the TPM at Various Stress Ranges. Stress Relieved (SR), R = 0.3

Stress Range Ni (cycles) Np (cycles) Nt (Cycles) Ni/Nt % Nt Nt(MPa) TPM TPM TPM TPM RFLM F-class

150 1.4 × 105 3.3 × 105 4.7 × 105 30 4.6 × 105 5.1 × 105

120 5.6 × 105 6.5 × 105 1.2 × 106 47 1.1 × 106 1.0 × 106

100 2.1 × 106 1.1 × 106 3.2 × 106 66 2.5 × 106 1.7 × 106

80 1.6 × 107 2.2 × 106 1.8 × 107 88 11.0 × 106 3.4 × 106

60 3.7 × 108 5.1 × 106 2.9 × 108 99 ∞ 8.0 × 106

WELDING RESEARCH

JANUARY 2006-s26

dicts that any joint will eventually fail if thenumber of cycles is high enough. It is infact possible to build a fatigue limit intothe TPM by assuming that after crack ini-tiation has taken place the crack may stopto grow, due to the fact that it has a SIFRbelow the threshold value. However, nodata are available to corroborate such be-havior for shallow surface-breakingcracks.

Conclusions

Our study was primarily carried out fornon-load-carrying fillet weld joints madeof C-Mn steel with nominal yield stressclose to 345 MPa (50 ksi). A pure fracturemechanics model (FMM) and a two-phasemodel (TPM) were used to predict the fa-tigue life. The initiation life was modeledby the Coffin-Manson equation, whereasthe crack propagation was based on thesimple version of the Paris law. The mod-els were validated and calibrated with theuse of large databases. The criteria for ac-ceptance of the models were that theyshould predict both damage evolution andfinal fatigue life at any stress level. TheFMM failed to fulfill these criteria,whereas the TPM gave an excellent fit toboth measured crack growth histories andexperimental fatigue lives. The followingconclusions were drawn.

The fatigue behavior of fillet weldjoints is far more complex at typical in-service stresses than fracture mechanicscan describe. This is due to the fact thatthe crack initiation phase dominates thefatigue life at these low stresses.

A TPM is capable of modeling thedamage evolution from the initial state tothe final fracture provided that the modelis accurately calibrated for this purpose.The notch factor at the weld toe is basedon extreme value statistics for the toegeometry, and the transition crack depthbetween the initiation phase and the prop-agation phase is set to 0.1 mm. The para-meters in the Coffin-Manson equationwere determined directly from earlycracking in full-scale welded joints.

As the TPM has a semiempirical phys-ical basis, the determining factors such asresidual stresses, global and local jointgeometry, and loading mode are readilyaccounted for.

The S-N curves constructed from themodel are nonlinear for a log-log scale andcoincide with the curves obtained from thestatistical RFLM in Part 1 of this investi-gation. Both models fit experimental datafar better than the conventional bilinear S-N curves.

There is a fundamental difference be-tween S-N curves obtained from theRFLM and the TPM in the way that thelatter curves do not predict any fatigue

limit. At stress ranges below 70 MPa theRFLM curve will appear flat, whereas theTPM curve will continue to fall with asmall slope close to the parameter b in theCoffin-Manson law. At present there areno data to corroborate either one of thesecurves, but the authors tend to have moreconfidence in the prediction made by thephysical-based TPM than the predictionsbased on the statistical RFLM when ex-trapolated outside the range of the data.

The first practical consequence of thepresent two-phase model is that it predictslonger lives at low stress ranges than theconventional S-N curves in rules and reg-ulations. With the application of the TPM-constructed S-N curve in the lower stressregion it is possible to reduce dimensionsby 30–40% and still achieve the same fa-tigue life as for the F-class S-N curve.

The second practical consequence isthat in-service inspection strategy may beoptimized. This is due to the fact that thecrack path leading to final fracture is quitedifferent from the path calculated by apure fracture mechanics model. The two-phase model with its long initiation phasewill give a more hidden path for crack evo-lution. Hence, an inspection program withincreased inspection frequency at the endof service life can be proven to be favorable.

Suggestions for Future Work

In this work, the TPM has been used toconstruct median S-N curves only. Futurework should focus on constructing quan-tile curves for design purposes as well.This can be done by Monte Carlo simula-tion treating the main determining factorsas random variables. The resulting curvesshould be compatible with the quantilecurves obtained from the RFLM. Further-more, as the TPM has no fatigue limit, itwill be interesting to analyze how themodel responds to variable loading byusing a damage accumulation law. Finally,the practical consequences of the model interms of joint dimensions and scheduledinspection programs should be studied inmore detail. Other types of joints such asbutt joints are also of great interest in thisregard.

References

1. American Bureau of Shipping. 2003.Guide for the Fatigue Assessment of OffshoreStructures. ABS, April.

2. Verreman, Y., and Nie, B. 1996. Early de-velopment of fatigue cracking at manual filletwelds. Fatigue & Fracture of Engineering Materi-als and Structures 19 (6): 669–681.

3. Lassen, T. 1990. The effect of the weldingprocess on the fatigue crack growth in weldedjoints. Welding Journal 96(2): 75-s to 85-s.

4. Yung, J. Y., and Lawrence, F. V. 1985. An-alytical and graphical aids for the fatigue design

of weldments. Fatigue Fract. Engn. Mater. Struct.8: 223–241.

5. Lawrence, F. V., Dimitrakis, S. D., andMunse, W. H. 1996. Factors influencing weld-ment fatigue. Fatigue and Fracture, ASM Hand-book, Vol. 19, pp. 274–286. Materials Park,Ohio: ASM International.

6. BS 7910, Guidance on Methods for As-sessing the Acceptability of Flaws in FusionWelded Structures. 2000. London, U.K.: BritishStandards Institution (BSI).

7. Darcis, P. et al. 2004. A fracture mechan-ics approach for the crack growth in weldedjoints with reference to BS 7910. European Con-ference on Fracture (ECF 15), Stockholm, Swe-den, August 11–13, 2005.

8. Niu, X., and Glinka, G. 1987. The weldprofile effect on the stress intensity factors inweldments. Int. J. of Fracture 35: 3–20.

9. Dowling, E. 1996. Estimating fatigue life.Fatigue and Fracture, ASM Handbook,Vol. 19,pp. 250–262. Materials Park, Ohio: ASM Inter-national.

10. Engesvik, K., and Lassen, T. 1988. Theeffect of weld toe geometry on fatigue life. The7th OMAE Conference, Houston, Tex., pp.441–445.

11. Testin, R. A., Yung, J. Y., Lawrence, F.V., and Rice, R. C. 1987. Predicting the fatigueresistance of steel weldments. Welding Journal66: 93-s to 98-s.

12. Tricoteaux, A., Fardoun, F., Degallaix,S., and Sauvage, F. 1995. Fatigue crack initiationlife prediction in high strength structural steelwelded joints. Fatigue Fract. Engng. Mater.Struct. 18(2): 189–200.

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