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A postA post--processor for fatigueprocessor for fatigue--crack growth analysis crack growth analysis based on a finitebased on a finite--element stress field and its element stress field and its

application to components with random defects application to components with random defects

ETH Zürich, 25 Oktober 2007Gunnar Härkegård, NTNU, Trondheim

und Zentrum für Mechanik, ETH

Vortrag im Rahmen desKOLLOQUIUMS FUER TECHNISCHE WISSENSHAFTEN

und desSEMINARS IN MECHANIK

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AcknowledgementAcknowledgement

This presentation is largely based on work carried out at the Department of Engineering Design and Materials, NTNU, Trondheim, byArne Fjeldstad funded through the NorLight program and defending his PhD thesis 30/11, ‘Modelling of fatigue crack growth at notches and other stress raisers’Anders Wormsen funded by GE Energy and defending his PhD thesis 26/11, ‘A fatigue assessment methodology for notched components containing defects’

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ContentsContents

BackgroundCracks growing from a single defectGrowth of short fatigue cracksRandom-defect analysisConclusions, outlook

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BACKGROUNDBACKGROUND

5 Rolled material Cast material

Beachmarks of crack emanating from leading edge blade/ring transition (Huth)

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Beachmarks of crack emanating from blade/ring transition (Huth)

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Striations observed in an aluminium alloy Striations observed in an aluminium alloy (A. (A. JernbergJernberg, NTNU), NTNU)

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In situ observation of crack growth by In situ observation of crack growth by means of SEM in an aluminium alloy means of SEM in an aluminium alloy (M. (M. AnderssonAndersson, LTH, , LTH, dissdiss. 2005). 2005)

Thickness = 0.8 mm, crack depth > 2 mm

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‘‘A fatigue crack is growing from the very A fatigue crack is growing from the very first loading cycle.first loading cycle.’’Keith J. Miller, 1932Keith J. Miller, 1932--20062006

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DCPD measurement of naturally initiated DCPD measurement of naturally initiated fatigue crack at notch root: Test specimen fatigue crack at notch root: Test specimen

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DCPD measurement of naturally initiated DCPD measurement of naturally initiated fatigue crack at notch root: Test setupfatigue crack at notch root: Test setup

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Crack growth at the notch root from the first Crack growth at the notch root from the first cycle! (K. cycle! (K. StStäärkrk))

D1392 Risswachstum an Kerben St572S, R0.5, RTEndrisslänge Kerbe A 0.84mm, Kerbe B 0.78mm

02468

10121416

0 500 1000 1500 2000

Zyklenzahl N

Pote

ntia

länd

erun

g (%

)

POT1(% ) POT2(% )

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Growth of edge throughGrowth of edge through--cracks and cracks and semisemi--elliptical cracks in elliptical cracks in

inhomogeneous stress fieldsinhomogeneous stress fields

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Growth of (large) fatigue cracks Growth of (large) fatigue cracks

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Growth of a fatigue crack from Growth of a fatigue crack from aaii = 0.05 mm to= 0.05 mm toaaff = 5 mm in a = 5 mm in a homogeneoushomogeneous stress field stress field

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

N/N 0

a, m

m

Paris exponentm = 3

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SemiSemi--elliptic surface crack (throughelliptic surface crack (through--crack crack has has a/ca/c = 0)= 0)

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Surface crack in linearly decreasing stress Surface crack in linearly decreasing stress fieldfield

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Normalised fatigue life as a function of the Normalised fatigue life as a function of the relative stress gradientrelative stress gradient

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SemiSemi--elliptic crack at the root of a notchelliptic crack at the root of a notch

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Surface crack at the root of a notchSurface crack at the root of a notch

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Normalised fatigue life as a function of the Normalised fatigue life as a function of the curvature of the notch rootcurvature of the notch root

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MODELLING THE GROWTH OFMODELLING THE GROWTH OFA FATIGUE CRACK FROMA FATIGUE CRACK FROM

A SINGLE DEFECTA SINGLE DEFECT

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Different approaches to fatigue analysisDifferent approaches to fatigue analysis

Random DefectSingle DefectExplicit FCG analysisda/dn = f(Δσ, a; R),

Weakest LinkLocal StressImplicit FCG analysisS-N-curve (a > 1 mm),‘crack initiation’

ProbabilisticDeterministicMaterial properties

Crack Growth

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Comprises all four approaches to fatigue designWritten in standard FORTRAN Can be operated under Windows and UNIX/LINUXCompatible with standard finite element

codes such as ABAQUS, ANSYS and I-DEAS

P•FAT – Probabilistic Fatigue Assessment Tool developed at

NTNU/IPM 2003-2007

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P•FAT: Single Defect

Simulates growth of fatigue crack from crackSimulates growth of fatigue crack from crack--like defectlike defectArbitrary component geometrySingle crack-like defect of elliptical shape

• Embedded crack• Surface crack• Corner crack

Short-crack modelMean-stress correction (residual stress)Adaptive step-size controlUpdates location of crack front relative to free surface

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Definition of crack plane and localDefinition of crack plane and localcoco--ordinate systemordinate system

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Crack configurations implemented in Crack configurations implemented in PP••FATFAT

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Computation of Computation of KKII(P) from stress field (FEA) (P) from stress field (FEA) and weight function (integration mesh)and weight function (integration mesh)

AyxgyxKA

z d)P,,(),()P(crack

I ′′⋅′′= ∫ ′σ

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WeightWeight functionsfunctions for for embeddedembedded crackscracks

( ) 3 2 21 2 3 4

2', '; ' 18 8 8 8

s s s s sg x y Pπ ρ ρ ρ ρ ρ

= − − − −

Ref: Wang et al., Engineering Fracture Mechanics 59(3):381-392, 1998

( ) ( )els Gauss Gauss

i

N

i j j1 1 1

, , ; 'N N

k i j

K g Pσ ξ η ξ η= = =

⎛ ⎞≈ ⎜ ⎟

⎝ ⎠∑ ∑ ∑

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KK estimationestimation accuracyaccuracy ((emdeddedemdedded crackscracks))

0

KFaσ π

=( ) 0''

iyya

σ σ ⎛ ⎞= ⎜ ⎟⎝ ⎠

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WeightWeight functionsfunctions for for semisemi--ellipticelliptic crackscracks

( ) ( )( )( )

AA

2 1 '', / , /''; /

2 ''

f y a c a Lg y a c

a yπ

+=

−( ) ( )( )C

C

2 1 '', / , /''; /

''

f y a c a Lg y a c

yπ

+=

Ref: Shen and Glinka, Theoretical and Applied Fracture Mechanics 15:247-255, 1991

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KK estimationestimation accuracyaccuracy ((semisemi--ellipticelliptic crackscracks))

0

KFaσ π

=( ) 0''

iyya

σ σ ⎛ ⎞= ⎜ ⎟⎝ ⎠

331

2

3

Smooth blockSmooth block--shaped specimen under axial shaped specimen under axial pushpush--pull loading, pull loading, RR = = --11

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FCG through crossFCG through cross--section of smooth section of smooth blockblock--shaped specimenshaped specimen

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Fatigue lives of cracked smooth specimens Fatigue lives of cracked smooth specimens based on Pbased on P••FAT and an eigenstrain methodFAT and an eigenstrain method

1027711419422

1588016164221

2018822901211

P•FATDai et al.Lciai

Cycles until failureInitial crack geometry (mm)

Dai et al. studied the growth of near surface cracks in a wide body by using a solution for the stress intensity factor based on eigen-strains.

Ref: Dai et al. Engineering Fracture Mechanics, 59(4):415-424, 1998

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1

2

3

Notched blockNotched block--shaped specimen under shaped specimen under axial pushaxial push--pull loading, pull loading, RR = = --11

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FCG through crossFCG through cross--section of notched section of notched blockblock--shaped specimenshaped specimen

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Fatigue lives of cracked notched wide plate Fatigue lives of cracked notched wide plate based on Pbased on P••FAT and an approximate FAT and an approximate

analytical methodanalytical method

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GROWTH OF SMALL FATIGUE CRACKSGROWTH OF SMALL FATIGUE CRACKS

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KitagawaKitagawa--Takahashi diagram and Takahashi diagram and El Haddad intrinsic crack length modelEl Haddad intrinsic crack length model

for steel (for steel (RRp0.2p0.2 = 200 = 200 –– 800 800 MPaMPa), Cu and Al), Cu and Al

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Generalisation of El HaddadGeneralisation of El Haddad’’s model s model for the fatigue limitfor the fatigue limit

‘‘IntrinsicIntrinsic’’ crack length:crack length:

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ModifiedModified KK--T diagram T diagram

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KK--T T ‘‘Fatigue Assessment DiagramFatigue Assessment Diagram’’

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El HaddadEl Haddad’’s model in terms of s model in terms of ΔΔKKand and ΔΔσσ::

2

A

2th

eff 1 ⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎟⎠⎞

⎜⎝⎛

ΔΔ

+Δ=Δσσ

KKKK

Stress intensity range corrected Stress intensity range corrected with respect to (short) crack length:with respect to (short) crack length:

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‘‘EquivalentEquivalent’’ stress of a crack in an stress of a crack in an inhomogeneous stress fieldinhomogeneous stress field

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ddaa/d/dnn vs. vs. ΔΔKK for short cracks in AA6082for short cracks in AA6082--T6 T6 with the crack depth with the crack depth aa as a parameteras a parameter

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1 10

ΔK , MPa m1/2

da/d

n, m

/cyc

le

a = 0.01 mma = 0.02 mma = 0.05 mma = 0.1 mma = 0.2 mma = 0.5 mma = 1 mma = 10 mm

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ddaa/d/dnn vs. vs. ΔΔKK for short cracks in AA6082for short cracks in AA6082--T6 T6 with the stress range with the stress range ΔΔσσ as a parameteras a parameter

1.00E-11

1.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

0.1 1 10

ΔK , MPa m1/2

da/d

n, m

/cyc

le

Δσ = 0Δσ = 40Δσ = 64Δσ = 80Δσ = 100Δσ = 125Δσ = 160Δσ = 200

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ddaa/d/dnn in AA6082in AA6082--T6 vs. (a) T6 vs. (a) ΔΔKK and (b) and (b) ΔΔKKeqeq

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ddaa/d/dnn in A508 steel vs. (a) in A508 steel vs. (a) ΔΔKK and (b) and (b) ΔΔKKeqeq

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Prediction of selfPrediction of self--arresting cracksarresting cracksat notches at notches –– Reanalysis of FrostReanalysis of Frost’’s s

classical fatigue testsclassical fatigue tests

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Notched fatigue test specimens forNotched fatigue test specimens for(a) push(a) push--pull and (b) rotating bendingpull and (b) rotating bending

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Stress intensity range corrected with Stress intensity range corrected with respect to (short) crack lengthrespect to (short) crack length

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Stress intensity range corrected with Stress intensity range corrected with respect to (short) crack lengthrespect to (short) crack length

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Application of PApplication of P••FAT to the analysis FAT to the analysis of fatigueof fatigue--crack growth from a small crack growth from a small

weld defectweld defect

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TT--joint made by welding together two joint made by welding together two AA6082AA6082--T6 RHS profilesT6 RHS profiles

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WWööhler diagram for Thler diagram for T--joint under 4PB: Measured joint under 4PB: Measured and predicted lives incl. the influence of (simulated) and predicted lives incl. the influence of (simulated) residual stresses from welding residual stresses from welding

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MODELLING THE GROWTH OFMODELLING THE GROWTH OFFATIGUE CRACKS FROMFATIGUE CRACKS FROM

RANDOM DEFECTSRANDOM DEFECTS

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Different approaches to fatigue analysisDifferent approaches to fatigue analysis

Random DefectSingle DefectExplicit FCG analysisda/dn = f(Δσ, a; R)

Weakest LinkLocal StressImplicit FCG analysisS-N-curve (a > 1 mm)

ProbabilisticDeterministicMaterial properties

Crack Growth

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P•FAT: Random Defect

For every finite element of a component, random For every finite element of a component, random defects are generated based on the underlying defects are generated based on the underlying statistical distributions (number , location, size).statistical distributions (number , location, size).The defects are The defects are ‘‘rankedranked’’ according to their effective according to their effective stress intensity ranges.stress intensity ranges.This procedure is repeated for a large number of This procedure is repeated for a large number of nominally identical components.nominally identical components.For each component, the number of cycles to failure is For each component, the number of cycles to failure is determined by means of the determined by means of the ‘‘Single defectSingle defect’’ option. option. This is a This is a ‘‘Monte CarloMonte Carlo’’ simulation of the life simulation of the life distribution of the component. distribution of the component.

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(a) Potentially life(a) Potentially life--controlling defects in controlling defects in a single componenta single component

(b) A priori vs. a (b) A priori vs. a posteriori posteriori ‘‘lifetime lifetime

rankingranking’’

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LifeLife--controlling defects of 500 nominally controlling defects of 500 nominally equal specimensequal specimens

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3D distribution of potentially critical defects3D distribution of potentially critical defects

0

100

200

300

400

500

−50

0

50

−100

−50

0

50

100

amin

= 10 μm, amax

= 600 μm, amean

= 40 μm

Truncated exponential distribution

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Fatigue lives of 1000 nominally equal Fatigue lives of 1000 nominally equal specimens specimens –– Monte Carlo simulationMonte Carlo simulation

1 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

Specimen number

N/N

mea

n

amin

= 10 μm, amax

= 600 μm, amean

= 40 μm

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Specimens considered to illustrate a Monte Carlo lifeSpecimens considered to illustrate a Monte Carlo life--time simulation based on FCG from random defectstime simulation based on FCG from random defects

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Probability of failure under homogeneous stress Probability of failure under homogeneous stress based on the Kbased on the K--T diagram and the size T diagram and the size

distribution of defects distribution of defects

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Monte Carlo simulation of the size of the lifeMonte Carlo simulation of the size of the life--controlling defect assuming the defect size to be controlling defect assuming the defect size to be

GumbelGumbel distributeddistributed

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Monte Carlo simulation of the smooth and Monte Carlo simulation of the smooth and notched fatigue limits assuming the defect size to notched fatigue limits assuming the defect size to

be be GumbelGumbel distributeddistributed

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The smooth and notched fatigue limits are well The smooth and notched fatigue limits are well described by the threedescribed by the three--parameter Weibull parameter Weibull distribution:distribution:

This observation connects the This observation connects the randomrandom--defect defect simulation with the simulation with the weakestweakest--linklink approach.approach.

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Welded pipe subjected to axial tensionWelded pipe subjected to axial tension

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FE mesh of welded pipe (60,000 elements)FE mesh of welded pipe (60,000 elements)

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FCG through pipe weldFCG through pipe weld

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PublicationsPublications

A. Fjeldstad, G. Härkegård, A. WormsenThe influence of a stress gradient on the growth of a fatigue crack. 9th International Fatigue Congress, Atlanta, Georgia, 2006.A. Wormsen, A. Fjeldstad, G. HärkegårdA post-processor for fatigue crack growth analysis based on a finite element stress field. To be published in Computer Methods in Applied Mechanics and Engineering.A. Fjeldstad, A. Wormsen, G. HärkegårdSimulation of fatigue crack growth in components with random defects. To be published in Engineering Fracture Mechanics.A. Fjeldstad, A. Wormsen, G. HärkegårdA reanalysis of Frost’s classical fatigue tests on self-arresting cracks at notches. Dept. of Engineering Design and Materials, NTNU, Trondheim, 2007.

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ConclusionsConclusions

Fatigue is basically caused by crack growth from stochastically distributed material defectsProbabilistic fatigue assessment may be carried out by means of the analysis of fatigue crack growth from random defectsA tool for such assessment, P•FAT, post-processing stresses from a FEA, is being developed at NTNUP•FAT offers a physically sound and robust method for the fatigue assessment of materials and structures

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OUTLOOKOUTLOOK

Physically sound and robust models (and data!) for the growth of short fatigue cracksDitto for the initiation of cracks at defects that are not crack-like, e.g., poresStatistical distribution of material defectsVerification testing of real componentsThese are objectives of a Norwegian-Danish program (3.5 MCHF, 2007-2010) on cast components for large wind turbines

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Seminar presentationsSeminar presentations

This presentation and that from 18.10.2007, ‘A non-local fatigue assessment method based on weakest-link theory and statistics of extremesand its application to component-like specimens’,can both be found as PDFs by means of the linkhttp://www.zfm.ethz.ch/e/v/sem/.