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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
40
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)?