Weakest Link ZfM 071018

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Fatigue assessment based onFatigue assessment based onweakestweakest --link theorylink theory

ETH Zrich, 18 Oktober 2007Gunnar Hrkegrd, NTNU, Trondheim

und Zentrum fr Mechanik, ETH

Vortrag im Rahmen desKOLLOQUIUMS FUER TECHNISCHE WISSENSHAFTEN

und desSEMINARS IN MECHANIK


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NorwegischeNorwegische TechnischTechnisch --NaturwissenschaftlicheNaturwissenschaftliche

UniversitUniversit tt NTNU, TrondheimNTNU, Trondheim


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Over 150 years of fatigue failuresOver 150 years of fatigue failures

a nevera never --ending story?ending story?


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A nonA non --local fatigue assessment method based onlocal fatigue assessment method based onweakestweakest --link theory and statistics of extremeslink theory and statistics of extremes

and its application to componentand its application to component --like specimenslike specimens


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AcknowledgementAcknowledgementThis presentation is largely based on work carried

out at the Department of Engineering Design andMaterials, NTNU, byAnders Wormsen funded by GE Energy(defending his PhD thesis 26/11)Arne Fjeldstad funded through the NorLightprogram (defending his PhD thesis 30/11)

Thanks are also due to Bjrn Sjdin, SiemensTurbomachinery, Sweden, who provided the first

version of the weakest-link program


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ContentsContents

BackgroundLocal-stress approachWeakest-link approach

Elementary model of chainMaterial defects and extreme value distributionsWeibull distributed fatigue strength

Experimental investigationsConclusions, outlook


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BACKGROUNDBACKGROUND


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Traupel on methods for the assessment ofTraupel on methods for the assessment ofmechanical integritymechanical integrity

Der Kontrast zwischen der Geschlossen-heit und Strenge der Theorien, die uns

zur Berechnung von Spannungs- zustnden dienen, und der Lcken-

haftigkeit, scheinbaren Inkohrenz undUndurchsichtigkeit, die wir zu deren

Beurteilung heranziehen, ist fr jeden

wissenschaftlich denkenden Ingenieurtief unbefriedigend .Walter Traupel, Professor fr

Thermische Turbomaschinen an derETH 1954-1983


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What requirements should be fulfilledWhat requirements should be fulfilledby a modern fatigue life predictor?by a modern fatigue life predictor?

Should have a sound physical basisCracks are initiated at randomly distributed metallurgical defectsCracks grow from these defects

Should handle all geometries and loading casesin a consistent manner

Smooth and notched componentsSmall and large components

Push-pull, rotating bending, alternating torsionShould offer a consistent model to account forthe scatter in fatigue properties

Should be fully compatible with FEA


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Scatter in fatigue testingScatter in fatigue testing


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Different approaches to fatigue analysisDifferent approaches to fatigue analysis

Material properties

Crack Growth

Deterministic Probabilistic

Implicit FCG analysis S-N -curve ( a > 1 mm),crack initiation

Local Stress Weakest Link

Explicit FCG analysisda /d n = f( , a ; R ),

Single Defect Random Defect


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Comprises all four approaches to fatiguedesignWritten in standard FORTRANCan be operated under Windows and

UNIX/LINUXCompatible with standard finite element

codes such as ABAQUS, ANSYS and I-DEAS

P FAT Probabilistic FatigueAssessment Tool developed at

NTNU/IPM 2003-2007


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LOCALLOCAL --STRESS APPROACHSTRESS APPROACH


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Assumed equivalenceAssumed equivalencebetween a component and abetween a component and a

reference specimenreference specimen

( ) =t a m _

( ) =t a m _

( ) =t _ max m,max a,max


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FEA of Francis turbine runnerFEA of Francis turbine runner


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CastCast 13Cr4Ni steel fatigue test specimen13Cr4Ni steel fatigue test specimen


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Octant of fatigue test specimen modelledOctant of fatigue test specimen modelledwith twentywith twenty --node hex elementsnode hex elements

net net

y y

x

80

b =/2 40

l

=

/ 2

4 0

z

r =

5

R =

1 0 R

= 1 0

x

72.5 4 0

9 0

z

5

a) b)


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Local stress approachLocal stress approach

200

250

150

300

350

400

450

500

550

s a

2

N / m m

101010 10 10 10543 6 7 8

N


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LocalLocal --stress approachstress approach


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Why isWhy is local stresslocal stress unsatisfactory?unsatisfactory?

Like other physical processes, fatigue damagerequires a finite volume to developThe maximum stress amplitude is unlikely to

coincide with the largest defectThe use of local stress overestimates thecrack growth rate, when the crack is growinginto a decreasing stress fieldIt does not offer a consistent model to accountfor the scatter in fatigue properties


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WEAKESTWEAKEST --LINK APPROACHLINK APPROACH


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Elementary weakestElementary weakest --link link model of a chainmodel of a chain


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s, link *

s, chain s, link *

Weibull distribution of link strength:

exp

Ditto of chain composed of links:

exp

b

b

n

P

n

P P n

=

= =


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Weibull distribut ion for the probability of surv ivalof a single link

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,5 1 1,5 2

P r o

b a

b i l i t y

o f s u r v

i v a

l

b = 3b = 10

b = 30b = 100


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Weibull distribution fo r the probability of surv ivalof a chain of n = 100 links

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

P r o

b a

b i l i t y

o f s u r v

i v a

l

b = 3b = 10

b = 30b = 100


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Material defects and extreme valueMaterial defects and extreme valuedistributionsdistributions

i k l b l i l i fF i k l b l i l i f


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Fatigue cracks at globular inclusions ofFatigue cracks at globular inclusions ofcalcium aluminate in a lowcalcium aluminate in a low --alloyalloy

carburising steel (carburising steel ( JuvonenJuvonen ))

InclusionInclusion cutcut byby freefree surfacesurface SubSub --surfacesurface inclusioninclusion

i k i l ldF i k i l ld


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Fatigue cracks at pores in steel weldFatigue cracks at pores in steel weld(Berge)(Berge)


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Fatigue cracks at pores in cast 13Cr4NiFatigue cracks at pores in cast 13Cr4Nisteel (steel ( HuthHuth ))

10 mm


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Probability of survival of a (homogeneouslystressed) volume element V >> V 0:Pr( A > a) = P s, V =

(1 - z 1 V )V / V exp(- z 1V )

By applying this formula to a volumeelement, V i , subjected to a nearly homo-geneous stress amplitude, a i , one obtains

P s, V i = exp(- z 1( a i ) V i )


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Methods based on the statistics of extremesMethods based on the statistics of extremes

for estimating the size of the maximum defectfor estimating the size of the maximum defectin a block of materialin a block of material

Block maximumdetermine the size of the largest defect in each ofk equally sized polished cross-sections

generalised extreme value distribution, GEVPeak over threshold

determine the size of all defects > a suitablychosen threshold in each of k equally sizedpolished cross-sectionsgeneralised Pareto distribution

GEV distribution


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The probability that a defect of size A max a max islocated within a volume element V is given by:

GEV distribution

= 0, > 0 and < 0 correspond to the Gumbel,Frchet and reversed Weibull or the Type I, II andIII extreme value distributions, respectively.


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Weibull distributed fatigue strength


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Under certain simplifying assumptions, i. a., 0, the no. of

defects per unit volume > a crit may be written as:

Further assuming > 0 (Frchet distribution) and (K-T)

yields a two-parameter Weibull distribution for theprobability of survival in terms of the stress amplitude:


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Piecewise powerPiecewise power law approximation oflaw approximation of


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Piecewise powerPiecewise power --law approximation oflaw approximation of

KitagawaKitagawa

--Takahashi diagramTakahashi diagram

survival

failure

-1 -1

3 2

W ib ll t lit d

a a,max =


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A homogeneously stressed reference specimen of volumeV 0 has the same probability of survival as an arbitrarycomponent, if the stress amplitude of the reference specimen

is chosen such that

Weibull stress amplitude

For b = , i.e., for a material without scatter,

a a,max =

a a,max =

and the weakest-link and local stress approaches becomeequivalent!

W ib ll t lit d td

a a,max =


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Based on a finite element stress analysis, the FEApost-processor PFAT computes the stress integral

Weibull stress amplitude, contd

as a sum of all element integrals. Higher orderintegration ( N Gauss 10) may be required:


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Experimental investigationsExperimental investigations


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(1) Weibull analysis of(1) Weibull analysis of S S -- N N data for adata for a

total of > 2,000 fatigue test specimenstotal of > 2,000 fatigue test specimensfrom 12 different alloysfrom 12 different alloys


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0

0

as, *

A0

s, *0

Weibull distribution of fatigue strength:

exp

Ditto of fatigue life:

expn

b

V

b

V

P

nP N

=

=

Probability distribution of WeibullProbability distribution of Weibull


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Probability distribution of Weibullobab ty d st but o o We buexponentexponent bb

Forged steels (6)Forged steels (6) Cast steels (4)Cast steels (4)


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(2) Fatigue testing and Weibull life(2) Fatigue testing and Weibull life

prediction of hydroprediction of hydro --turbine blade modelturbine blade model

C tC t 13C 4Ni l f i i (13C 4Ni t l f ti t t i ( H hH th ))


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CastCast 13Cr4Ni steel fatigue test specimen (13Cr4Ni steel fatigue test specimen ( HuthHuth ))

Smooth fatigue specimen for axial loadingSmooth fatigue specimen for axial loading


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Smooth fatigue specimen for axial loadingSmooth fatigue specimen for axial loading

((FraunhoferFraunhofer LBF, Darmstadt)LBF, Darmstadt)

LocalLocal --stress approachstress approach


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LocalLocal stress approachstress approach

WeakestWeakest --link approachlink approach


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Weakest link approachpp


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(3) Fatigue testing and Weibull life(3) Fatigue testing and Weibull life

prediction of notched rotating bendprediction of notched rotating bendspecimens from two lowspecimens from two low --alloy steelsalloy steels

H glHourglass h d hshaped push ll i (16+20)pull specimen (16+20)


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HourglassHourglass --shaped pushshaped push --pull specimen (16+20)pull specimen (16+20)

A

A

3 9

2 6

2 5

110

185

20

= 0.5, 1.5, 5K t = 3.3, 2.2, 1.4

20 6

110

B - B21

B

B 9

=

3 0

a)

b)

Notched rotating bend specimen (3+6)Notched rotating bend specimen (3+6)

Stress distribution in rotating bendStress distribution in rotating bend


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g

specimen withspecimen with K K tt = 1.4= 1.4

(Ave. Crit.: 75%)S, Mises

+3.010e-06+1.090e-01+2.180e-01+3.270e-01+4.360e-01+5.450e-01+6.540e-01+7.630e-01+8.720e-01+9.811e-01+1.090e+00+1.199e+00+1.308e+00

1

2

3

Weibull stress amplitude vs no of cyclesWeibull stress amplitude vs no of cycles


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Weibull stress amplitude vs. no. of cyclesWeibull stress amplitude vs. no. of cycles

49MnVS3 steel49MnVS3 steel 42CrMo4 steel42CrMo4 steel

Measured and predicted lives of Measured and predicted lives of


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notched rotating bend specimensnotched rotating bend specimens


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(4) Fatigue testing and Weibull(4) Fatigue testing and Weibullprediction of the fatigue limit forprediction of the fatigue limit for

30CrNiMo8 steel30CrNiMo8 steel

Smooth (8 shapes) and notched (18 shapes) specimensSmooth (8 shapes) and notched (18 shapes) specimensof 30CrNiMo8 steel for pushof 30CrNiMo8 steel for push --pull or rotating bendingpull or rotating bending


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of 30CrNiMo8 steel for pushof 30CrNiMo8 steel for push -pull or rotating bendingpull or rotating bending

(total no of specimens investigated > 700)(total no of specimens investigated > 700)

L

L

D

D

g

g

d D E

D E

t

a)

b)

x

y

z

x

y

z

Comparison between localComparison between local --stress and weakeststress and weakest --


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link predictions by means oflink predictions by means of

Error indicesError indices

PublicationsPublications


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PublicationsPublications

A. Wormsen, G. HrkegrdA statistical investigation of fatigue behaviour according toWeibulls weakest link theory, Proceedings of the 15th EuropeanConference on Fracture , Stockholm, Sweden, 2004.A. Wormsen, G. Hrkegrd, H.J. HuthProbabilistic fatigue assessment of a hydro-turbine blade model.9th International Fatigue Congress , Atlanta, Georgia, 2006.

A. Wormsen, G. HrkegrdWeibull fatigue analysis of notched components under constantand variable amplitude loading. 9th International FatigueCongress , Atlanta, Georgia, 2006.A. Wormsen, B. Sjdin, G. Hrkegrd, A. FjeldstadNon-local stress approach for fatigue assessment based onweakest-link theory and statistics of extremes. To be publishedin Fatigue & Fracture of Engineering Materials & Structures .

CONCLUSIONSCONCLUSIONS


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CONCLUSIONSCONCLUSIONS

A weakest-link module for fatigue life

prediction is available in PFATWeakest-link modelling is closely relatedto the statistical distribution of materialdefectsGood agreement with fatigue tests oncomponent-like specimensPredestined to become the new industrystandard
http://www.alstom.com/http://www.alcoa.com/global/en/home.asp


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OUTLOOKOUTLOOK
http://en.wikipedia.org/wiki/Image:Elkem_logo.pnghttp://en.wikipedia.org/wiki/Image:RR_logo.pnghttp://www.gjuteriforeningen.se/main.htmhttp://en.wikipedia.org/wiki/Image:VestasLogo.gifhttp://en.wikipedia.org/wiki/Image:StatoilLogo.gifhttp://en.wikipedia.org/wiki/Image:Volvo_logo.pnghttp://en.wikipedia.org/wiki/Image:Siemens_logo2.pnghttp://en.wikipedia.org/wiki/Image:Saab-logo.pnghttp://www.manbw.com/http://en.wikipedia.org/wiki/Image:BMW_Logo.svghttp://www.alstom.com/http://www.alcoa.com/global/en/home.asp


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OUTLOOK OUTLOOK

Database describing the relevant statisticaldistributions of fatigue strength and lifeVerification testing of real components

Random defect analysis: Statisticaldistribution of material defects, short fatiguecrack growth

These are objectives of a Norwegian-Danishprogram (3.5 MCHF, 2007-2010) on castcomponents for large wind turbines


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More about random defects and fatigue inMore about random defects and fatigue ina weeka week same time and place!same time and place!

A postA post --processor for fatigueprocessor for fatigue --crack crack --growthgrowthanalysis based on a finite element stressanalysis based on a finite element stress

field and its application to components withfield and its application to components withsmall, random defectssmall, random defects

Weakest Link ZfM 071018

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