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Factors Influencing Fatigue Mean Stress
Professor Darrell F. Socie Mechanical Science and Engineering
University of Illinois
© 2004-2013 Darrell Socie, All Rights Reserved
Fatigue and Fracture
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Factors Influencing Fatigue
Mean Stress Variable Amplitude Stress Concentrations Surface Finish
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Mean Stresses
mean stress
stress range
stre
ss
2SSS minmax
mean+
=
Smax
Smin
max
min
SSR =
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General Observations
Tensile mean stresses reduce the fatigue life or decrease the allowable stress range
Compressive mean stresses increase the fatigue life or increase the allowable stress range
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Mechanism
Fatigue damage is a shear process
∆S
Tensile mean stresses open microcracks and make sliding easier
2Smean
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Goodman 1890
Mechanics Applied to Engineering John Goodman, 1890
“.. whether the assumptions of the theory are justifiable or not …. We adopt it simply because it is the easiest to use, and for all practical purposes, represents Wöhlers data.
Sultimate = Smin + 2 ∆S
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Goodman Diagram
Mean stress
Alte
rnat
ing
stre
ss
Su 0
Se
107 cycles
105 cycles
−
∆
=∆
−= ultimate
mean
1R SS1
2S
2S
R = -1 R = 1
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Test Data ( 1941 )
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
1.0
1.2
0
Mean Stress Ultimate Strength
Alte
rnat
ing
Stre
ss
Fatig
ue L
imit
J.O. Smith, The Effect of Range of Stress on the Fatigue Strength of Metals, Engineering Experiment Station Bulletin 334, University of Illinois, 1941
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Compression
Se
Su 0 R = -1 R = 1
-Su R = - ∞
no influence
detrimental
beneficial
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Modified Goodman ( no yielding )
Se
Su 0 R = -1 R = 1
-Su R = - ∞
Sys
Sys
-Sys
ysmean SS2S
<+∆
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Mean Stress Influence on Life
0.1
1
10
100
1000
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 Mean Stress
Ultimate Strength
Rel
ativ
e Fa
tigue
Life
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Stress Concentrations
Plastic Zone
∆ε
∆ε The elastic material surrounding the plastic zone around a stress concentration forces the material to deform in strain control
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Mean Stresses at Notches
σ
ε
σ
ε
σ
ε
elastic plastic
Nominal Notch
Notch Nominal
Nominal mean stress is less than notch mean stress
Nominal mean stress is greater than notch mean stress
Nominal Notch
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Morrow Mean Stress Correction
cf
'f
bf
mean'f )N2()N2(
E2ε+
σ−σ=
ε∆
10-5
10-4
0.001
0.01
0.1
1
Reversals, 2Nf
Stra
in A
mpl
itude
100 101 102 103 104 105 106 107
Emeanσ
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Smith Watson Topper
σ
ε
∆ε
σmax cb
f'f
'f
b2f
2'f
max )N2()N2(E2
+εσ+σ
=ε∆
σ
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Mean Stress Relaxation
Stadnick and Morrow, “Techniques for Smooth Specimen Simulation of Fatigue Behavior of Notched Members” ASTM STP 515, 1972, 229-252
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Loading Histories
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Crack Growth Physics
σ
σ Maximum load
Minimum load
σ
ε
Mean stresses in plastic zone are small
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Mean Stress Effects
From: Dowling and Thangjitham, An Overview and Discussion of Basic Methodology for Fatigue, ASTM STP 1389,2000, 3-38
( )γ−∆
=R1KC
dNda m
0 < γ < 0.5
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Compression
Crack open
Crack closed
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Compressive Stresses
Compressive stresses are not very damaging in crack growth
Stre
ss Crack opening level
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Sources of Mean/Residual Stress
Loading History Fabrication Shot Peening Heat Treating
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Loading History
Tension overloads produce favorable compressive residual stress
Compressive overloads produce unfavorable tensile residual stress
σ
ε
σ
ε
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Cold Expansion
1965 Basic Cx process conceptualized (Boeing) The split sleeve is
slipped onto the mandrel, which is attached to the hydraulic puller unit.
The mandrel and sleeve are inserted into the hole with the nosecap held firmly against the workpiece.
When the puller is activated, the mandrel is drawn through the sleeve radially expanding the hole.
Courtesy of Fatigue Technology Inc.
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Theory of Cold Expansion
Courtesy of Fatigue Technology Inc.
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Fatigue Life Improvement
Courtesy of Fatigue Technology Inc.
100
200
300
0 108 107 106 105 104 103
Nom
inal
Stre
ss
Fatigue Life
Cold Expanded
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Shot Peening
-600
-400
-200
0
200
0 0.2 0.4 0.6 0.8 Depth (mm)
Res
idua
l Stre
ss (M
Pa)
Residual stress in a shot peened leaf spring
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Shot Peening Results
www.metalimprovement.com
500
1000
1500
500 0
0 1000 1500 2000 2500 Fatig
ue s
treng
th a
t 2x1
06 c
ycle
s, M
Pa
Ultimate Tensile Strength, MPa
2SS u
FL =
Smooth
Notched
Shot Peened Smooth & Notched
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Heat Treating
200
600
400
0 0 5 10 15
Brin
ell H
ardn
ess
Depth, mm 0 5 10 15
-1500
-1000
-500
0
500
1000
Depth, mm R
esid
ual S
tress
, MP
a
Radial
Circumferential
Axial
50 mm diameter induction hardened 1045 steel shaft
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Things Worth Remembering
Local mean stress rather than the nominal mean stress governs the fatigue life
Mean stress has the greatest effect on crack nucleation
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Factors Influencing Fatigue
Mean Stress Variable Amplitude Stress Concentrations Surface Finish
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Variable Amplitude Loading
How to you identify cycles ?
How do you assess fatigue damage for a cycle ?
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Rainflow Cycle Counting
What could be more basic than learning to count correctly?
Matsuishi and Endo (1968) Fatigue of Metals Subjected to Varying Stress – Fatigue Lives Under Random Loading, Proceedings of the Kyushu District Meeting, JSME, 37-40
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Rainflow strain
A
F
D
B
E
I, A H G
C
AD
BC
CB
DA
Counts 1/2 cycles
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Rainflow and Hysteresis A
C B
D E F G
H I
ε
σ
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Cumulative Damage
…. ….
…. ….
High - Low
Low - High nH nL
nL nH
∆SL
∆SH
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Linear Damage
∆SH
∆SL
Nf H Nf L
Lf
L
Hf
H
F Nn
Nn
NnDamage +==∑
Miner’s Rule:
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Nonlinear Damage
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
0 0
Cycle ratio,
Dam
age
fract
ion
n Nf
0040.=ε∆
0200.=ε∆
ΣD = 0.7
ΣD = 1.3
ΣD ~ 1
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Periodic Overload Results
10 100 103 104 105 106 107 108 109
10-3
10-4
0.01
0.1
Stra
in A
mpl
itude
Fatigue Life
….
….
The fatigue limit is reduced by a factor of 3 when a few large cycles are applied
Bonnen and Topper, “The Effects of Periodic Overloads on Biaxial Fatigue of Normalized SAE 1045 Steel” ASTM STP 1387, 2000, 213-231
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Fatigue Damage Calculations
100
1000
10000
Stre
ss A
mpl
itude
, MP
a
100
Cycles 101 102 103 104 105 106 107
( )bf'f NS
2S
=∆
1
10
b1
'f
f S2SN
∆=
10SDamage ∆∝
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Crack Growth Data
10-12
10-11
10-10
10-9
10-8
10-7
10-6
1 10 100
Cra
ck G
row
th R
ate,
m/c
ycle
mMPa,K∆
mKCdNda
∆=
∆KTH
Kc
1
3
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Crack Growth Data
10-12 10-11 10-10 10-9 10-8 10-7 10-6
Crack Growth Rate, m/cycle
1
10
100
m
MP
a , K
∆
3SDamage ∆∝
( )2/m1SC
aaN2m
m
2/m1i
2/m1f
f
−π∆
−=
−−
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Multiple Choice
Which cycles do the most fatigue damage ? (a) a few large cycles (b) a moderate number of intermediate cycles (c) a large number of small cycles
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Fatigue Data
1
10
100
Am
plitu
de
100
Cycles 101 102 103 104 105 106 107
1000
n = 10
n = 5
n = 3 nSDamage ∆∝
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Loading History
Time (Secs)
50 100 150 200 250 300
750
500
250
0
-250
-500
-750
Stra
in G
age
(ust
rain
)
Bracket.sif-Strain_b43
0
1500
-750
750
0
750 C
ount
s
rf_000.sif-Strain_b43
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Slope = 3
0
1500
-750
750
3.15
% d
amag
e
Damage
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Slope = 5
0
1500
-750
750
5.14
% d
amag
e
Damage
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Slope = 10
0
1500
-750
750
20.78
% d
amag
e
Damage
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Mechanisms and Slopes
100
103
104
Stre
ss A
mpl
itude
, MPa
100
Cycles 101 102 103 104 105 106 107
1 10
10-12 10-11 10-10 10-9 10-8 10-7 10-6
Crack Growth Rate, m/cycle
1
10
100
m
MPa
,
K ∆
1
3
Crack Nucleation
Crack Growth 10
100
103 104 105 106 107 108
Total Fatigue Life, Cycles
1
5 1
Equ
ival
ent L
oad,
kN
Structures
A combination of nucleation and growth
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Equivalent Load
n
N
1i
ni
N
SS
∑=
∆=∆
Equivalent constant amplitude loading
Typically n ranges from 4 to 6 for structures
N cycles at an amplitude of does as much damage as the entire loading history
S∆
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SAE Keyhole Specimen
Suspension
Transmission
Bracket
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SAE Keyhole Test Data ManTen RQC 100
10
100
103 104 105 106 107 108
Total Fatigue Life, Cycles
1
Suspension Transmission
Bracket
Constant Amplitude
5 1
Equi
vale
nt L
oad,
kN
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Things Worth Remembering
Rainflow counting is employed to identify cycles
The slope of the fatigue curve ( damage mechanism) has a large influence on how much damage is caused by smaller cycles
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Factors Influencing Fatigue
Mean Stress Variable Amplitude Stress Concentrations Surface Finish
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Stress Concentration Factor
Applied stress Local stress
ρ+σ=σ
a21appliedlocal
Inglis Solution 1910
2a ρ
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Kσ and Kε
Stre
ss (M
Pa)
Strain
KtS
Kte
σ ε
SK σ
=σ
eK ε
=ε
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Neuber’s Rule St
ress
(MPa
)
Strain
KtS
Kte
σ
ε εσ=eKSK tt
Stress calculated with elastic assumptions
Actual stress
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Neuber’s Rule for Fatigue
2222eKSK tt ε∆σ∆
=∆∆
Stress and strain amplitudes
Elastic nominal stress
E2S
2e ∆
=∆
Substitute and rearrange
22E
2SKt
ε∆σ∆=
∆
The product of stress times strain controls fatigue life
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A Dilemma
Stress analysis and stress concentration factors are independent of size and are related only to the ratio of the geometric dimensions to the loads
Fatigue is a size dependent phenomenon
How do you put the two together ?
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Fatigue of Notches
From Dowling, Mechanical Behavior of Materials, 1999
104 105 106 107 108
100
400
300
200
Fatigue Life
Nom
inal
Stre
ss, M
Pa
d d
Kt = 3.1
Kt = 3.1
Kf = 2.2
104 105 106 107 108
100
400
300
200
Fatigue Life
dd dd
Kt = 3.1
Kt = 3.1
Kf = 2.2
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Notch Size
Kt Kf Kt
Kf
Large Notch Small Notch
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Microstructure Size
Kt Kf
Kt Kf
Low Strength High Strength
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Stress Gradient
Kt Kf
Kt
Kf
Low Kt High Kt
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Kt vs Kf
0
2
4
6
8
10
0 2 4 6 8 10 Kt
Kf
Kf = Kt
Experiments
Effe
ctiv
e st
ress
con
cent
ratio
n
Calculated stress concentration
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Peterson’s Equation
ρα
+
−+=
1
1K1K tf
mmMPa2070025.08.1
u
σ
=α
0
1
2
3
4
10-4 10-3 10-2 0.1 1 10 102 103
Kf
ρα
No effect when ρ << α
Full effect when ρ >> α
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Peterson’s Constant
1000 500 2000 1500
0.1
0.03
0.7
0.3
Ultimate Strength, MPa
α, m
m
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Static Strength
hole Kt = 2.5
slot Kt = 5
diamond Kt = 20
edge Kt = 20
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1018 Steel Test Data
0
20
40
60
80
100
0 2 4 6 8 10
load
, kN
displacement, mm
edge diamond slot hole
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Notched SN Curve
100
1000
10000
100
Cycles
Stre
ss A
mpl
itude
, MPa
101 102 103 104 105 106 107
Smooth specimen data
Notched specimen Kf
Stress concentrations are not very important at short lives
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Fatigue of Notches
From Dowling, Mechanical Behavior of Materials, 1999
104 105 106 107 108
100
400
300
200
Fatigue Life
Nom
inal
Stre
ss, M
Pa
d d
Kt = 3.1
Kt = 3.1
Kf = 2.2
104 105 106 107 108
100
400
300
200
Fatigue Life
dd dd
Kt = 3.1
Kt = 3.1
Kf = 2.2
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Crack Growth Data
10-12
10-11
10-10
10-9
10-8
10-7
10-6
1 10 100
Cra
ck G
row
th R
ate,
m/c
ycle
mMPa,K∆
mKCdNda
∆=
∆KTH
Kc
a212.1KTH ππ
σ∆>∆
Nonpropagating cracks
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Frost Data nucleation fracture
Rotating bending Notched bar
Notched plate
1 3 5 7 9 11 13 15
1.0
0.8
0.6
0.4
0.2
0
Kt
S nom
inal
S fat
igue
lim
it
tK1 nonpropagating cracks
Frost, “A Relation Between the Critical Alternating Propagation Stress and Crack Length for Mild Steel” Proceedings of the Institute for Mechanical Engineers, Vol. 173, No. 35, 1959, 811-836
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Significance
DKS th
flπ
∆=
For Kt > 4, the notch acts like a crack with a depth D
Kt does not play a role for sharp notches !
A stress concentration behaves like a crack once a stress concentration becomes large (Kt > 4)
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Cracks at Notches
D
a a D + a
KtS S S
a << D a >> D
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Stress Intensity Factors
0 0.1 0.2 0.3 0.4 0.5 0.6
1.0
0
3.0
2.0
Da
DSK
aSKK t π=
( )aDSK +π=
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Cracks at Holes
20 1 1 22
Once a crack reaches 10% of the hole radius, it behaves as if the hole was part of the crack
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Specimens with Similar Geometry
Kt = 2.4 Kt = 10.7
Ultimate Strength 780 MPa Yield Strength 660 MPa
25 25
2.5 2.5
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Test Results
1
100
1000
Nom
inal
Stre
ss A
mpl
itude
1 10 100 103 104 105 106 107
Total Fatigue Life, Cycles
Strength Limited
Crack Growth Dominated Fatigue Strength Dominated
Threshold Stress Intensity Dominated
Kt = 2.4 Kt = 10.7
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Things Worth Remembering
Fatigue may be thought of as a failure of the average stress concept, consequently, fatigue usually begins at stress concentrators which are most frequently located on the surface
The severity of a stress concentrator in fatigue is size dependent
Small stress concentrators are more effective in high strength materials
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Factors Influencing Fatigue
Mean Stress Variable Amplitude Stress Concentrations Surface Finish
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Modern View of the Fatigue Limit The fatigue limit is the stress where a crack may nucleate but will not grow through the first microstructural barrier such as the grain size, pearlite colony size, prior austenite grain size, eutectic cell size or precipitate spacing.
Slip Bands Crack
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Intrinsic Flaws
Little effect of surface pit because it is smaller than the grain size
Large effect of defect because it is larger than the grain size
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Surface Finish Influence
Method Stress-Life Strain-Life
Crack Growth
Physics Crack Nucleation
Microcrack Growth Macrocrack Growth
Size 0.01 mm
0.1 - 1 mm > 1mm
Influence of Surface Finish
Strong Moderate
None
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Sources of Surface Effects Machining
Cutting Grinding
Corrosion General Pitting
Processing Cutting/Shearing Casting Forging Plating
Foreign Object Damage Nicks Scratches
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Machining
σ1
Cracks start in machining marks not in the direction of the maximum principal stress
100 µm
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Casting
100 µm
Surface flaw in gray cast iron
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Nodular Iron Surface
Flake graphite formed on the surface of a nodular iron casting
Starkey and Irving, “A Comparison of the Fatigue Strength of Machined and As-cast Surfaces of SG Iron” International Journal of Fatigue, July, 1982, 129-136
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Test Data 10-1
10-2
10-3
10-4
102 103 104 105 106 107 108 10 1
Stra
in A
mpl
itude
Fatigue Life, Reversals
Cast Surface
Machined Surface
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Surface Reduction Factors
400 600 800 1000 1200 1400 1600
Ultimate Strength, MPa
0.2
0.4
0.6
0.8
1.0
0
Surfa
ce F
acto
r Polished
Ground
Forged
Hot Rolled
Machined
5.0
2.5
1.7
1.2
1.0
Fatig
ue N
otch
Fac
tor
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Noll and Lipson 1945
0
200
400
600
800
1000
400 600 800 1000 1200 1400 1600 1800 200 0 2000
Ultimate Strength, MPa
Fatig
ue L
imit,
MPa
Ground
Machined
Hot Rolled Forged
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Hiam and Pietrowski 1978
Driven for 1 or 2 years in Southern Ontario before making specimens to evaluate corrosion effects
Hiam and Pietrowski, “The Influence of Forming and Corrosion on the Fatigue Behavior of Automotive Steels”, SAE Paper 780040, 1978
Strain controlled fatigue testing
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Kf for pitting
Hot RolledSurface
CorrodedSurface
950X 1.12 1.49
0.06% C HSLA 1.18 1.65
0.18% C HSLA 1.90
Surface finish factor predicts Kf = 1.6 for a Hot Rolled Surface
from Hiam and Pietrowski
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Pit Depth Effects on Life 106
105
104
0.1 0.3 0.2 0 Pit Depth, mm
Fatig
ue L
ife, C
ycle
s
from Hiam and Pietrowski
950X Steel
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Fatigue Notch Factor for Pits
1.0
0.1 0.3 0.2 0 Pit Depth, mm
1.1
1.2
1.3
1.4
1.5
Kf
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Spring Failures
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Microscopic Examination
Corrosion Pits
Corrosion Pits
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Chrome Plating
105 107 106 108 104
Stre
ss A
mpl
itiud
e, M
Pa
500
200
400
300
Life, Cycles
Almen, “Fatigue Loss and Gain by Electroplating” , Product Engineering, Vol. 22, No. 5, 1951, 109-116
Chrome plated
Shot peened and chrome plated
Base steel
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Hard Chrome Plating
coating
Metals Handbook, Volume 9, Fractography and Atlas of Fractographs
In addition to cracks, coatings frequently have high tensile residual stresses
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Galvanized Steel
3
0.5 0
1
2
1
225 MPa
305 MPa
240 MPa
Life Fraction
Cra
ck D
ensi
ty m
m-1
Vogt, Boussac, Foct, “Prediction of Fatigue Resistance of a Hot-dip Galvanized Steel” Fatigue and Fracture of Engineering Materials and Structures, Vol. 23, No. 1, 2001,33-40
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Fatigue Limit for Galvanized Steel
100 10 100 1000
200
300
400
Fatig
ue L
imit
Coating Thickness, t µm
Uncoated Fatigue Limit
tKS TH
FLπ
∆=
Coatings can be modeled with a crack equal to the coating thickness
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Anodized Aluminum
Rateick et. al. “Relationshipp of Microstructure to Fatigue Strength Loss in Anodized Aluminum-Copper Alloys” Aeromet 2004, June 2004
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Pitting at Cu Rich Constituent
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Foreign Object Damage
http://www.eng.ox.ac.uk/~ftgwww/frontpage/fod2.html
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Foreign Object Damage
http://www.eng.ox.ac.uk/~ftgwww/frontpage/fod2.html
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Upper Control Arm
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Things Worth Remembering
Fatigue crack nucleation is a surface phenomena and everything about the surface affects the fatigue life
Most of the design rules are conservative having been developed for materials of the 1950’s