Home | Fracture Control Program - Modelling2014/07/03  · C. H+ acceleration D. Corrosion products...

Fred LawrenceUniversity of Illinois at Urbana

flawrenc@uiuc.edu ! [217] 333-6928

Modelling

2

Outline

"" Fatigue mechanismsFatigue mechanisms"" Sources of knowledgeSources of knowledge"" ModellingModelling

"" Crack nucleationCrack nucleation"" Non propagating cracksNon propagating cracks"" Crack growthCrack growth

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Stages in the development of a fatiguecrack

3. Intrusionsand extrusions(surfaceroughening)

2. Persistentslip bands

4. Stage I fatiguecrack growth

5. Stage II fatiguecrack growth

1. Cyclic slip

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First cycle Many cycles later - cyclic hardening

Cyclic slip

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Cyclic stress-strain behavior

•Development of cell structures (hardening)•Increase in stress amplitude (under strain control)•Break down of cell structure to form PSBs•Localization of slip in PSBs

PSB

cyclic hardening cyclic softening

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Slip Band Formation

Cyclically hardened material

Extrusion

Cyclically hardened material

Intrusion

Cyclically hardened material

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Intrusions and extrusions

Intrusions andextrusions on thesurface of a Nispecimen

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Crack initiation

Fatigue crack initiation at an inclusionCyclic slip steps (PSB)Fatigue crack initiation at a PSB

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Cyclic Plastic Zone

Shear band model

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Stage I crack growth

Single primary slip system

individual grain

near - tip plastic zone

S

S

Stage I crack growth (rc ≤ d)is strongly affected by slipcharacteristics, microstructuredimensions, stress level, extentof near tip plasticity

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Stage II crack growth

Stage II crack growth (rc >> d)

Fatigue crack growing inPlexiglas

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Outline

"" Fatigue mechanismsFatigue mechanisms"" Sources of knowledgeSources of knowledge"" ModellingModelling

"" Crack nucleationCrack nucleation"" Non propagating cracksNon propagating cracks"" Crack growthCrack growth

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The intrinsic fatigueproperties of thecomponent’s material aredetermined using small,smooth or crackedspecimens.

Source of knowldege

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This information can be used to predict, to compute the answer to…….

Smoothspecimens

Crack growthspecimen

Laboratory fatigue test specimens

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1. Will a crack nucleate? 2. Will it grow? 3. How fast will it grow?

The three BIG fatigue questions

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Outline

"" Fatigue mechanismsFatigue mechanisms"" Sources of knowledgeSources of knowledge"" ModellingModelling

"" Crack nucleationCrack nucleation"" Non propagating cracksNon propagating cracks"" Crack growthCrack growth

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1. Will a crack nucleate? 2. Will it grow? 3. How fast will it grow?

Crack nucleation

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Cyclic behavior of materials

Bauschinger effect:monotonic and cyclic stress-strain different!

monotonic

cyclic

∆ε

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Cyclic Deformation

Hysteresis loops∆ε

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MODEL: Cyclic stress-strain curve

ε

σ Cyclic stress-strain curve

Hysteresis loops for different levels of applied strain

∆ε

ε = σE

+ σK '

1/n '

∆ε = ∆σE

+2∆σ2K '

1/n '

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MODEL: Cyclic stress-strain curve

ε = σE

+ σK '

1/n '

ε = σE

+ σK

1/n

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MODEL USES

" Simulate localcyclic stress-strainbehavior at a notchroot.

" May need toinclude cyclichardening andsoftening behaviorand mean stressrelaxation effects.

Kt

S(t)

Notched component

σ, ε

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Planar and wavy slip materials

Wavy slip materials Planar slip materials

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Concept of fatigue damage

Plastic strainscause theaccumulationof “fatiguedamage.”

1954 - Coffin and Manson

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106

105

104

103

.0001

.001

.01

.1

total strain ampltudeelastic strain ampltudeplastic strain ampltude

Reversals (2Nf)

σ'f b

c

ε' f

E

²e

∆ε t = ∆εe + ∆εp =σ' f

E2Nf( )b + ε' f 2Nf( )c

MODEL: Strain-life curve

Fatigue test datafrom strain-controlled tests onsmooth specimens.

Elastic component ofstrain, ∆εe

Plastic component ofstrain, ∆εp

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Transition Fatigue Life

106

105

104

103

.0001

.001

.01

.1

total strain ampltudeelastic strain ampltudeplastic strain ampltude

Reversals (2Nf)

σ'f b

c

ε' f

E

²e

Transition fatigue life, Ntr

Elastic strains = plastic strains

2Ntr =ε f

' E

σ f'

1

b−c

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10 710 610 510 41

10

100

Fatigue Life ( N, cycles)

EnduranceLimit

2,000,000 cycles

N

ni

i

ni

Ni≥ 1

i∑ .0

Miner's Rule ofCumulative FatigueDamage is the bestand simplest toolavailable to estimatethe likelihood offatigue failure undervariable loadhistories. Miner's rulehypothesizes thatfatigue failure willoccur when the sumof the cycle ratios(ni/Ni) is equal to orgreater than 1

ni = Number of cycles experienced at a given stress or load level.Ni = Expected fatigue life at that stress or load level.

MODEL: Linearcumulative damage

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Mean stress effects

log 2Nf

log∆ε

/2

Compressive mean stressZero mean stressTensile mean stress

Good!

Bad!

Effects the growth of (small) fatigue cracks.

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MODEL: Morrow’s mean stresscorrection

log 2Nf

log∆ε

/2

Zero mean stress

Tensile mean stress

σf’/E

σm/E

∆ε2

=σ f

' −σ m

E2N f( )b

+ε f' 2N f( )c

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0.0001

0.001

0.01

0.1

1E+02 1E+03 1E+04 1E+05 1E+06

Cycles

Morrow -200 MPa

SWT -200 MPa

Morrow +200 MPa

SWT +200MPa

σmax∆ε2

=σ f

' 2

E2N f( )2b

+σ f' ε f

' 2N f( )b +c

MODEL: Mean stress effects

∆ε2

=σ f

' −σ m

E2N f( )b

+ε f' 2N f( )cMorrow

Smith-Watson-Topper

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A strain gagemounted in a highstress region of acomponentindicates a cyclicstrain of 0.002. Thenumber of reversalsto nucleate a crack(2NI) is estimated tobe 86,000 or 43,000cycles.

∆ε t = ∆εe + ∆εp =σ' f

E2Nf( )b + ε' f 2Nf( )c

106

105

104

103

.0001

.001

.01

.1

total strain ampltudeelastic strain ampltudeplastic strain ampltude

Reversals (2Nf)

σ'f b

c

ε' f

0.002

86,000 reversals

E

²e

MODEL USES

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Outline

"" Fatigue mechanismsFatigue mechanisms"" Sources of knowledgeSources of knowledge"" ModellingModelling

"" Crack nucleationCrack nucleation"" Non propagating cracksNon propagating cracks"" Crack growthCrack growth

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1. Will a crack nucleate? 2. Will it grow? 3. How fast will it grow?

The three BIG fatigue questions

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Sharp notchesmay nucleatecracks but theremote stressmay not belarge enoughto allow thecrack to leavethe notchstress field.

Non-propagating cracks

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Threshold Stress Intensity

∆S

a

∆K = Y ∆S πa

The forces driving a crack forward arerelated to the stress intensity factor

At or below thethreshold valueof ∆K, thecrack doesn’tgrow.

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MODEL: Non-propagating cracks

Crack length

Endurance limit for smooth specimens

Stre

ssra

nge,∆S

Non-propagatingcracks

Failureath =

1π

∆Kth

∆S

2

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∆S

∆K ∆K

∆P = ∆S W

GrooveWelded ButtJoint

Tensile-ShearSpot Weld

W

∆P

∆K =Y2

∆P / Bπa

+ ∆S πa

∆K = ∆S πa

a a

initial crack length

W≈ ∞

P2Cycles, N P2Cycles, N

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MODEL USES

a

" Will the defect oflength a growunder the appliedstress range ∆S?

" How sensitiveshould our NDTtechniques be toensure safety?

∆S

ath =1π

∆Kth

∆S

2

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Outline

"" Fatigue mechanismsFatigue mechanisms"" Sources of knowledgeSources of knowledge"" ModellingModelling

"" Crack nucleationCrack nucleation"" Non propagating cracksNon propagating cracks"" Crack growthCrack growth

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1. Will a crack nucleate? 2. Will it grow? 3. How fast will it grow?

The three BIG fatigue questions

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Measuring Crack Growth - Everything weknow about crack growth begins here!

Cracks growfaster as theirlengthincreases.

How fast will it grow???

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Growth of fatigue cracks iscorrelated with the range instress intensity factor: ∆K.

MODEL: Paris Power Law

dadN

= C ∆K( )n

∆K = Y ∆S πa

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Fatigue fracture surface

Scanning electronmicroscope image -striations clearly visible

Schematic drawing ofa fatigue fracturesurface

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Crack growthrate (da/dN) isrelated to thecrack tip stressfield and is thusstronglycorrelated withthe range ofstress intensityfactor:(∆K=Y∆S√πa).

dadN

= C ∆K( )n

Paris power law

MODEL: Crack growth rate versus ∆K

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Crack Growth Data

All ferritic-pearlitic,that is, plain carbonsteels behave aboutthe sameirrespective ofstrength or grainsize!

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The fatigue crack growth rates for Al and Ti are much more rapid than steel fora given ∆K. However, when normalized by Young’s Modulus all metals exhibitabout the same behavior.

Crack growth rates of metals∆K

E

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Forman’s equation which includes theeffects of mean stress.

Walker’s equation which includes theeffect of mean stress.

MODEL: Mean stresses (R ratio)

da

dN= C ∆K m

1−R( )Kc − ∆K

da

dN= C ∆K m

1− R( )γ

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While some applications areactually constant amplitude,many or most applicationsinvolve variable amplitudeloads (VAL) or variable loadhistories (VLH).

Aircraft loadhistories

PROBLEM: Variable load histories?

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FIX: Root mean cube method

∆K =∆Ki( )m

ni∑ni∑

m

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Overloads retard crack growth, under-loads accelerate crack growth.

PROBLEM: Overloads, under-loads?

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Infrequent overloads help, but more frequent may be even better. Toofrequent overloads very damaging.

Effects of Periodic Overloads

PROBLEM: Periodic overloads?

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FIX: MODEL for crack tip stress fields

ry = 1βπ

K

σ y

ß is 2 for (a) plane stress and 6 for (b)

Monotonic plastic zone sizeCyclic plastic zone size

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MODEL: Overload plastic zone size

Crack growthretardation ceaseswhen the plasticzone of ith cyclereaches theboundary of theprior overloadplastic zone.

ß is 2 for planestress and 6 forplain strain.

ry = 1βπ

K

σ y

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MODEL: Retarded crack growth

da

dN

VA

= β da

dN

CA

β =2 ry

a p − ai

k

ry = 12 π

K

σ ys

2

Wheeler model

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Infrequent overloads help, but more frequent may be even better. Toofrequent overloads very damaging.

Effects of Periodic Overloads

Retarded crack growth

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Infrequent overloads help, but more frequent may be even better. Toofrequent overloads very damaging.

Effects of Periodic Overloads

PROBLEM: Periodic overloads?

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Sequence Effects

Last one controls!

Compressive overloadsaccelerate cdrack growth byreducing roughness of fracturesurfaces.

Compressive followed by tensileoverloads…retardation!

Tensile followed bycompressive, little retardation

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Overloads retard crack growth, under-loads accelerate crack growth.

Sequence effects

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PROBLEMS: Small Crack Growth?

Small cracks don’t behave like long cracks! Why?

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A. Dissolution ofcrack tip.

B. Dissolution plusH+ acceleration.

C. H+ acceleration

D. Corrosionproducts may retardcrack growth at low∆K.

B

C D

A

PROBLEMS: Environment?

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FIX: Crack closure

Plastic wake New plastic deformation

S = Smax

S = 0

S

S

Rem

ote

Str

ess,

S

Time, t

Smax

Sopen, Sclose

Rem

ote

Str

ess,

STime, t

Smax

Sopen, Sclose

Crackopen

Crackclosed

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CAUSE: Plastic Wake

Short cracks can’t do this so they behave differently!

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“Effective” part of load history

Damaging portion of loading history

Nondamaging portion of loading hist

Opening load

S

S

S

S

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MODEL: Crack closure

dadn

= C ∆K( )m K max( )p ExtrinsicIntrinsic

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Other sources of crack closure

SopenSmax

(b)

Smax Sopen

(a)

Roughness inducedcrack closure (RICC)

Oxide inducedcrack closure (OICC)

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MODEL: Crack closure

U =∆K eff

∆K=

Smax − Sopen

Smax − Smin=

11− R

1−Sopen

Smax

Initial crack length

A''A A'

A, A', A'' Crack tip positions

Plastic zones forcrack positions A...A”

Plastic wake

∆Keff = U ∆K

Plasticity induced crack closure (PICC)

U ≈ 0.5 + 0.4R (sometimes)

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The use of theeffective stressintensity factoraccounts for theeffects of meanstress.

dadN

= C ∆K( )n

dadN

= C' ∆Keff( )n = C' U∆K( )n

MODEL USE: ∆Keffective

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²K( )a

a

U( )a 1

0 a

a

²K ( )aeff

a

da

dN

a i af ²K ( )aeff

dadN

a

Bilinear Paris Law

a th

1. Initiation?

2. Growth?

3. Life?

²K th

NF =dadN

a i

a f∫−1

da

Use of ∆Keff

Crack closure at anotch model.Information aboutU(a) becomes the“sticking point.”

MODEL USE

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PROBLEM: Fully plastic conditions?

∆ ∆ ∆σ∆σ ∆ε

K J E aa E

n

p2 22

1 65 0

1= = +

+.

.'

( )∆ ∆εK E Y a a=

Correlate crack growth rate with strain intensity?

Correlate crack growth rate with values of the J integral?

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PROBLEM: Fully plastic conditions?

COD

Correlate crack growth rate with values of the cracktip opening displacement (COD)?