iii fatigue models - illinois

III Fatigue Models

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

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Outline

nnSources of knowledgeSources of knowledgenn ModelingModelingnn Crack nucleationCrack nucleationnn Non propagating cracksNon propagating cracksnn Crack growth Crack growth

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The intrinsic fatigue properties of the component’s material are determined using small, smooth or cracked specimens.

Source of knowledge

AM 11/03 4This information can be used to predict, to compute the answer to…….

Smooth specimens

Crack growth specimen

Laboratory fatigue test specimens

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Outline

nn Sources of knowledgeSources of knowledge

nn ModelingModelingnnCrack nucleationCrack nucleationnn Non propagating cracksNon propagating cracksnn Crack growthCrack growth

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The three BIG fatigue questions

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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

n Simulate local cyclic stress-strain behavior at a notch root.

n May need to include cyclic hardening and softening behavior and mean stress relaxation 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 strains cause the accumulation of “fatigue damage.”

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 = ∆ε e + ∆εp =σ' fE

2Nf( )b + ε' f 2Nf( )c

MODEL: Strain-life curve

Fatigue test data from strain-controlled tests on smooth specimens.

Elastic component of strain, ? εe

Plastic component of strain, ? εp

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Transition Fatigue Life

106

105

104

103

.0001

.001

.01

.1


Reversals (2Nf)

σ'f b

c

ε' f

E

²e

Transition fatigue life, NtrElastic strains = plastic strains

2Ntr =ε f

' E

σ f'

1b−c

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10 710 610 510 41

10

100

Fatigue Life ( N, cycles)

Endurance Limit

2,000,000 cycles

N

ni

i

niN i

≥ 1i

∑ .0

Miner's Rule of Cumulative Fatigue Damage is the best and simplest tool available to estimate the likelihood of fatigue failure under variable load histories. Miner's rule hypothesizes that fatigue failure will occur when the sum of the cycle ratios (ni/Ni) is equal to or greater 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: Linear cumulative 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 stress correction

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 gage mounted in a high stress region of a component indicates a cyclic strain of 0.002. The number of reversals to nucleate a crack (2NI) is estimated to be 86,000 or 43,000 cycles.

∆ε t = ∆ε e + ∆εp =σ' fE

2Nf( )b + ε' f 2Nf( )c

106

105

104

103

.0001

.001

.01

.1


Reversals (2Nf)

σ'f b

c

ε' f

0.002

86,000 reversals

E

²e

MODEL USES

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Outline

nn Sources of knowledgeSources of knowledgenn ModellingModellingnn Crack nucleationCrack nucleation

nnNon propagating cracksNon propagating cracksnn Crack growthCrack growth

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Sharp notches may nucleate cracks but the remote stress may not be large enough to allow the crack to leave the notch stress field.

Non-propagating cracks

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Threshold Stress Intensity

? S

a

∆K = Y ∆S πa

The forces driving a crack forward are related to the stress intensity factor

At or below the threshold value of ? K, the crack doesn’t grow.

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MODEL: Non-propagating cracks

Crack length

Endurance limit for smooth specimens

Stre

ss ra

nge,

?S

Non-propagatingcracks

Failureath =

1π

∆Kth∆S

2

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? S

? K ? K

? P = ? S W

Groove Welded Butt Joint

Tensile-Shear Spot Weld

W? P

∆K =Y2

∆P / Bπa

+ ∆S πa

∆K = ∆S πa

a a

initial crack length

W˜ 8

P2Cycles, N P2Cycles, N

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MODEL USES

a

n Will the defect of length a grow under the applied stress range ?S?

n How sensitive should our NDT techniques be to ensure safety?

? S

ath =1π

∆Kth∆S

2

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Outline

nn Fatigue mechanismsFatigue mechanismsnn Sources of knowledgeSources of knowledgenn ModellingModellingnn Crack nucleationCrack nucleationnn Non propagating cracksNon propagating cracks

nnCrack growthCrack growth

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Measuring Crack Growth - Everything we know about crack growth begins here!

Cracks grow faster as their length increases.

How fast will it grow???

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Growth of fatigue cracks is correlated with the range in stress 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 growth rate (da/dN) is related to the crack tip stress field and is thus strongly correlated with the range of stress intensity factor: (? K=Y? Svpa).

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 carbon steels behave about the same irrespective of strength or grain size!

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

Crack growth rates of metals∆KE

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Forman’s equation which includes the effects of mean stress.

Walker’s equation which includes the effect of mean stress.

MODEL: Mean stresses (R ratio)

dadN

= C ∆K m

1− R( )Kc − ∆K

dadN

= C ∆K m

1− R( ) γ

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While some applications are actually constant amplitude, many or most applications involve variable amplitude loads (VAL) or variable load histories (VLH).

Aircraft load histories

PROBLEM: Variable load histories?

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FIX: Root mean cube method

∆K =∆Ki( )mni∑

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. Too frequent overloads very damaging.

Effects of Periodic Overloads

PROBLEM: Periodic overloads?

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FIX: MODEL for crack tip stress fields

ry = 1βπ

Kσ 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 growth retardation ceases when the plastic zone of ith cycle reaches the boundary of the prior overload plastic zone.

ß is 2 for plane stress and 6 for plain strain.

ry = 1βπ

Kσ y

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MODEL: Retarded crack growth

dadN

VA

= βdadN

CA

β =2 ry

a p − a i

k

ry =1

2 πK

σ ys

2

Wheeler model

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Retarded crack growth

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PROBLEM: Periodic overloads?

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Sequence Effects

Last one controls!

Compressive overloads accelerate cdrack growth by reducing roughness of fracture surfaces.

Compressive followed by tensile overloads…retardation!

Tensile followed by compressive, 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 of crack tip.

B. Dissolution plus H+ acceleration.

C. H+ acceleration

D. Corrosion products may retard crack 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

Stre

ss, S

Time, t

Smax

Sopen, Sclose

Rem

ote

Stre

ss, S

Time, t

Smax

Sopen, Sclose

Crack open

Crack closed

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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 history

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 =∆Keff

∆K=

Smax −Sopen

Smax −Smin=

11− R

1−Sopen

Smax

Initial crack length

A''A A'

A, A', A'' Crack tip positions

Plastic zones for crack 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 the effective stress intensity factor accounts for the effects of mean stress.

dadN

= C ∆K( )n

dadN

= C' ∆Keff( )n= C' U∆K( )n

MODEL USE: ?Keffective

iii fatigue models - illinois

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