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Introduction
Professor Darrell F. SocieUniversity of Illinois at Urbana-Champaign
© 2003-2008 Darrell Socie, All Rights Reserved
Multiaxial Fatigue
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 202
Contact Information
Darrell SocieMechanical Science and Engineering1206 West GreenUrbana, Illinois 61801
Office: 3015 Mechanical Engineering Laboratory
dsocie@illinois.eduTel: 217 333 7630Fax: 217 333 5634
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 2 of 202
When is Multiaxial Fatigue Important ?
Complex state of stress
Complex out of phase loading
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 202
Uniaxial Stress
one principal stressone direction
X
Z
Y
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 202
Proportional Biaxial
principal stresses vary proportionally but do not rotate
X
Z
Y1 = 2 = 3
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 202
Nonproportional Multiaxial
Principal stresses may vary nonproportionally and/or change direction
X
Z
Y
2
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 202
Crankshaft
Time
-2500
2500
90
45
0M
icro
stra
in
0
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 202
Shear and Normal Strains
-0.0025 0.0025
-0.005
0.005
-0.0025 0.0025
-0.005
0.005
0º 90º
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 202
Shear and Normal Strains
-0.0025 0.0025
-0.005
0.005
-0.0025 0.0025
-0.005
0.005
45º 135º
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 202
3D stresses
0.004 0.008 0.0120
0.001
0.002
0.003
0.004
0.005
50 mm
30 mm
15 mm
7 mm
Longitudinal Tensile StrainT
rans
vers
e C
ompr
essi
on S
trai
n
Thickness
100
x
z
y
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 10 of 202
Notch Stresses
t x z x z
7 0.01 -0.005 63.5 0
15 0.01 -0.003 70.6 14.1
30 0.01 -0.002 73.0 21.8
50 0.01 -0.001 75.1 29.3
x
z
y
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 11 of 202
Outline
State of Stress
Stress-Strain Relationships
Fatigue Mechanisms
Multiaxial Testing
Stress Based Models
Strain Based Models
Fracture Mechanics Models
Nonproportional Loading
Stress Concentrations
3
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 12 of 202
Book
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 13 of 202
Outline
State of Stress
Stress-Strain Relationships
Fatigue Mechanisms
Multiaxial Testing
Stress Based Models
Strain Based Models
Fracture Mechanics Models
Nonproportional Loading
Stress Concentrations
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 14 of 202
State of Stress
Stress components
Common states of stress
Shear stresses
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 15 of 202
Stress Components
X
Z
Y
z
y
x
zyzx
xz
xyyx
yz
Six stresses and six strains
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 16 of 202
Stresses Acting on a PlaneZ
Y
X
Y’
X’Z’
x’
x’z’
x’y’
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 17 of 202
Principal Stresses
3 - 2( X + Y + Z ) + (XY + YZXZ -2XY - 2
YZ -2XZ )
- (XYZ + 2XYYZXZ - X2YZ - Y2
ZX - Z2XY ) = 0
1
3
2
13
1223
4
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Stress and Strain Distributions
80
90
100
-20 -10 0 10 20
% o
f app
lied
stre
ss
Stresses are nearly the same over a 10° range of angles
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Tension
x
1
1 2
3
2 = 3 = 0
1
231
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 20 of 202
Torsion
2
xy1
3
2
1
3
X
Y2
1 = xy
3
1
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 21 of 202
2 y
1 x
3
1 2
3
1 2
1
23
Biaxial Tension
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 22 of 202
1
2
3 3
2
1
Maximum shear stress Octahedral shear stress
Shear Stresses
231
13
232
221
231oct 3
1
oct2
3Mises: 1313oct 94.0
22
3
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Maximum and Octahedral Shear
0.2
0.4
0.6
0.8
1.0
-1 -0.5 0 0.5 1
18%
6%
torsion tension biaxial tension
Octahedral shear
Maximum shear
3/1
5
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 24 of 202
Shear Stress Influence
1
1.2
1.4
1.6
1.8
2
0 0.2 0.4 0.6 0.8 1
1
1
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 25 of 202
State of Stress Summary
Stresses acting on a plane
Principal stress
Maximum shear stress
Octahedral shear stress
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 26 of 202
Outline
State of Stress
Stress-Strain Relationships
Fatigue Mechanisms
Multiaxial Testing
Stress Based Models
Strain Based Models
Fracture Mechanics Models
Nonproportional Loading
Stress Concentrations
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 27 of 202
“About six months ago I wrote a paper, knowing that I should be very busy in the autumn and made a model to illustrate a point in it. But as I played with the model to learn how to use it, it grew too strong for me and took command and for the last six months I have been its obedient slave --- for the model explained the whole of my subject Fatigue.”
Jenkin 1922
“Fatigue in Metals,” The Engineer, Dec. 8, 1922
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 28 of 202
Elastic Stress Strain Relationships
x
y
z
xy
yz
xz
x
y
z
xy
yz
xz
E
1 1 2
1 0 0 0
1 0 0 0
1 0 0 0
0 0 01 2
20 0
0 0 0 01 2
20
0 0 0 0 01 2
2
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 29 of 202
von Misesyield surface
Tresca yieldsurface
Yield Surfaces
3xy
X
6
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 202
Equations
F
F
or
F
ys
ys
ys
1 1 3
2 1 2
3 2 3
0
0
0
F x y x y xy ys 2 2 2 23 0
Tresca
Mises
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 31 of 202
- planeyield cylinder axis
Mises Yield Surfaces
32 ys
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Initial yield surface
Yield surface after loading
Isotropic Hardening
3
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Cyclic Loading
stab
ilize
d st
ress
ran
ge
Strain Control Stress Control
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 202
Kinematic Hardening
3
A
B
C
B
A y
B y
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 202
Ratcheting
Cyclic torsion with a mean tension stress
7
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Cyclic Mean Stress Relaxation
Cyclic strain with mean stress
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Cyclic Creep
Cyclic stress with a mean stress
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Plasticity Models
a
b
c
f
d
e
a
c
b
d
e
f
333
3 3 3
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 202
Why is modeling needed?
A B C D
A
B
CD
time
*
*
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 40 of 202
Flow and Hardening Rules
3
dij
n
ij
ij
dij
d dF
ijp
ij
F ijp f g( ) ( ) 0
Flow rule
Hardening rule
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 41 of 202
Multiaxial Example
-.006
0.003
x
-0.003
0.006xy
-500 500
x
-250
250xy
8
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 42 of 202
Multiaxial Example
-0.006 0.006xy
-250
250xy
-0.003 0.003x
-500
500x
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Summary
Isotropic Hardening
Kinematic Hardening
Cyclic creep or ratcheting
Mean stress relaxation
Nonproportional hardening
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 44 of 202
Outline
State of Stress
Stress-Strain Relationships
Fatigue Mechanisms
Multiaxial Testing
Stress Based Models
Strain Based Models
Fracture Mechanics Models
Nonproportional Loading
Stress Concentrations
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 45 of 202
10-10 10-8 10-6 10-4 10-2 100 102
Specimens StructuresAtoms Dislocations Crystals
Size Scale for Studying Fatigue
Understand the physics on this scale
Model the physics on this scale
Use the models on this scale
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 46 of 202
The Fatigue Process
Crack nucleation
Small crack growth in an elastic-plastic stress field
Macroscopic crack growth in a nominally elastic stress field
Final fracture
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1903 - Ewing and Humfrey
Cyclic deformation leads to the development of slip bands and fatigue cracks
N = 1,000 N = 2,000
N = 10,000 N = 40,000 Nf = 170,000Ewing, J.A. and Humfrey, J.C. “The fracture of metals under repeated alterations of stress”, Philosophical Transactions of the Royal Society, Vol. A200, 1903, 241-250
9
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Crack Nucleation
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Slip Band in Copper
Polak, J. Cyclic Plasticity and Low Cycle Fatigue Life of Metals, Elsevier, 1991
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Slip Band Formation
Loading Unloading
Extrusion
Undeformedmaterial
Intrusion
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Slip Bands
Ma, B-T and Laird C. “Overview of fatigue behavior in copper sinle crystals –II Population, size, distribution and growthKinetics of stage I cracks for tests at constant strain amplitude”, Acta Metallurgica, Vol 37, 1989, 337-348
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2124-T4 Cracking in Slip Bands
N = 60
N = 2000N = 1200
N = 300N = 240
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2219-T851 Cracked Particle
10mJames & Morris, ASTM STP 811 Fatigue Mechanisms: Advances in Quantitative Measurement of Physical
Damage, pp. 46-70, 1983.
10
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Initiation in Ti
Slip bands Hard alpha inclusion
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Crack Initiation at Inclusions
Langford and Kusenberger, “Initiation of Fatigue Cracks in 4340 Steel”, Metallurgical Transactions, Vol 4, 1977, 553-559
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 56 of 202
Subsurface Crack Initiation
Y. Murakami, Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions, 2002
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Fatigue Limit and Strength Correlation
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Crack Nucleation Summary
Highly localized plastic deformation
Surface phenomena
Stochastic process
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100 µm
bulksurface
10 µm
surface
20-25 austenitic steel in symmetrical push-pull fatigue (20°C, p/2= 0.4%) : short cracks on the surface and in the bulk
Surface Damage
From Jacques Stolarz, Ecole Nationale Superieure des MinesPresented at LCF 5 in Berlin, 2003
11
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Stage I Stage II
loading direction
freesurface
Stage I and Stage II
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Stage I Crack Growth
Single primary slip system
individual grain
near - tip plastic zone
S
S
Stage I crack is strongly affected by slip characteristics, microstructure dimensions, stress level, extent of near tip plasticity
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Small Cracks at Notches
D acrack tip plastic zone
notch plastic zone
notch stress field
Crack growth controlled by the notch plastic strains
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Small Crack Growth
1.0 mm
N = 900
Inconel 718 = 0.02Nf = 936
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 64 of 202
0
0.5
1
1.5
2
2.5
0 2000 4000 6000 8000 10000 12000 14000
J-603
F-495 H-491
I-471 C-399
G-304
Cycles
Cra
ck L
engt
h, m
m
Crack Length Observations
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 65 of 202
Crack - Microstructure Interactions
Akiniwa, Y., Tanaka, K., and Matsui, E.,”Statistical Characteristics of Propagation of Small Fatigue Cracks in Smooth Specimens of Aluminum Alloy 2024-T3, Materials Science and Engineering, Vol. A104, 1988, 105-115
10-6
10-7
0 0.005 0.01 0.015 0.02 0.0250.03 0.025 0.02 0.015 0.01 0.005
A B CD
F
E
Crack Length, mm
da/dN, mm/cycle
12
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Strain-Life Data
Reversals, 2Nf
Str
ain
Am
plitu
de 2
10-5
10-4
0.01
0.1
1
100 101 102 103 104 105 106 107
10-3
10m100 m
1mmfracture Crack size
Most of the life is spent in microcrack growth in the plastic strain dominated region
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Stage II Crack Growth
Locally, the crack grows in shear Macroscopically it grows in tension
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Long Crack Growth
Plastic zone size is much larger than the material microstructure so that the microstructure does not play such an important role.
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Material strength does not play a major role in fatigue crack growth
Crack Growth Rates of Metals
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Maximum Load
monotonic plastic zone
Stresses Around a Crack
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Stresses Around a Crack (continued)
Minimum Load
cyclic plastic zone
13
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Crack Closure
S = 250
b
S = 175
c
S = 0
a
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Crack Opening LoadDamaging portion of loading history
Nondamaging portion of loading history
Opening load
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Mode Iopening
Mode IIin-plane shear
Mode IIIout-of-plane shear
Mode I, Mode II, and Mode III
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 75 of 202
5 mcrac
k g
row
th d
ire
ctio
n
Mode I Growth
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crack growth direction
10 m
slip bandsshear stress
Mode II Growth
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1045 Steel - Tension
NucleationShear
Tension
1.0
0.2
0
0.4
0.8
0.6
1 10 102 103 104 105 106 107
Fatigue Life, 2Nf
Dam
age
Fra
ctio
n N
/Nf
100 m crack
14
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Fatigue Life, 2Nf
Dam
age
Fra
ctio
n N
/Nf
f
Nucleation
Shear
Tension
1 10 102 103 104 105 106 107
1.0
0.2
0
0.4
0.8
0.6
1045 Steel - Torsion
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 79 of 202
1.0
0.2
0
0.4
0.8
0.6
1 10 102 103 104 105 106 107
Nucleation
TensionShear
304 Stainless Steel - Torsion
Fatigue Life, 2Nf
Dam
age
Fra
ctio
n N
/Nf
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304 Stainless Steel - Tension
1 10 102 103 104 105 106 107
1.0
0.2
0
0.4
0.8
0.6 Nucleation
Tension
Fatigue Life, 2Nf
Dam
age
Fra
ctio
n N
/Nf
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 81 of 202
Nucleation
Tension
Shear
1.0
0.2
0
0.4
0.8
0.6
1 10 102 103 104 105 106 107
Inconel 718 - Torsion
Fatigue Life, 2Nf
Dam
age
Fra
ctio
n N
/Nf
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 82 of 202
Inconel 718 - Tension
Shear
Tension
Nucleation
1 10 102 103 104 105 106 107
1.0
0.2
0
0.4
0.8
0.6
Fatigue Life, 2Nf
Dam
age
Fra
ctio
n N
/Nf
Fatigue Made Easy
Similitude
Professor Darrell F. SocieMechanical Science and Engineering
University of Illinois
© 2002-2011 Darrell Socie, All Rights Reserved
15
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 84 of 202
Fatigue Analysis
MaterialData
ComponentGeometry
ServiceLoading
AnalysisFatigue
Life Estimate
?
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The Similitude Concept
Why Fatigue Modeling Works !
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 86 of 202
What is the Similitude Concept
The “Similitude Concept” allows engineers torelate the behavior of small-scale cyclicmaterial test specimens, defined undercarefully controlled conditions, to the likelyperformance of real structures subjected tovariable amplitude fatigue loads under eithersimulated or actual service conditions.
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 87 of 202
Fatigue Analysis Techniques
Stress - Life
BS 7608, Eurocode 3
Strain - Life
Crack Growth
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 88 of 202
Life Estimation
MethodStress-LifeBS 7608
Strain-LifeCrack Growth
PhysicsCrack Nucleation
Crack GrowthMicrocrack GrowthMacrocrack Growth
Size0.01 mm
1 - 10 mm0.1 - 1 mm
> 1mm
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 89 of 202
Stress-Life Fatigue Modeling
P
FixedEnd
The Similitude Concept states that if theinstantaneous loads applied to the ‘test’structure (wing spar, say) and the testspecimen are the same, then the responsein each case will also be the same and canbe described by the material’s S-N curve.Due account can also be made for stressconcentrations, variable amplitude loadingetc.
100
1000
10000
Str
ess
Am
plitu
de, M
Pa
100
Cycles101 102 103 104 105 106 107
16
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Fatigue Analysis: Stress-Life
MaterialData
ComponentGeometry
ServiceLoading
AnalysisFatigue
Life Estimate
SN curveKa, Ks, …
Kf
S , Sm
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 91 of 202
Stress-Life
Major Assumptions:Most of the life is consumed nucleating cracks
Elastic deformation
Nominal stresses and material strength control fatigue life
Accurate determination of Kf for each geometry and material
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Stress-Life
Advantages:Changes in material and geometry can easily be
evaluated
Large empirical database for steel with standard notch shapes
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Stress-Life
Limitations:Does not account for notch root plasticity
Mean stress effects are often in error
Requires empirical Kf for good results
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 94 of 202
BS 7608 Fatigue Modeling
The Similitude Concept states that if theinstantaneous loads applied to the ‘test’structure (welded beam on a bulldozer, say)and the test specimen (standard fillet weld)are the same, then the response in eachcase will also be the same and can bedescribed by one of the standard BS 7608Weld Classification S-N curves.
10
100
1000
105
Cycles
Str
ess
Ran
ge, M
Pa
108107106
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Weld Classifications
D E
F2 G
17
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Fatigue Analysis: BS 7608
MaterialData
ComponentGeometry
ServiceLoading
AnalysisFatigue
Life Estimate
Weld SN curve
Class
S
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BS 7608
Major Assumptions:Crack growth dominates fatigue life
Complex weld geometries can be described by a standard classification
Results independent of material and mean stress for structural steels
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BS 7608
Advantages:Manufacturing effects are directly included
Large empirical database exists
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BS 7608
Limitations:Difficult to determine nominal stress and weld
class for complex shapes
No benefit for improving manufacturing process
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Strain-Life Fatigue Modeling
The Similitude Concept states that if theinstantaneous strains applied to the ‘test’structure (vehicle suspension, say) and thetest specimen are the same, then theresponse in each case will also be the sameand can be described by the material’s e-Ncurve. Due account can also be made forstress concentrations, variable amplitudeloading etc.
10-5
10-4
0.001
0.01
0.1
1
Reversals, 2Nf
Str
ain
Am
plit
ude
100 101 102 103 104 105 106 107
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Fatigue Analysis: Strain-Life
MaterialData
ComponentGeometry
ServiceLoading
AnalysisFatigue
Life Estimate
N curve curve
Kf
S , Sm
18
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Strain-Life
Major Assumptions: Local stresses and strains control fatigue
behavior
Plasticity around stress concentrations
Accurate determination of Kf
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Strain-Life
Advantages:Plasticity effects
Mean stress effects
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Strain-Life
Limitations:Requires empirical Kf
Long life situations where surface finish and processing variables are important
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Crack Growth Fatigue Modeling
The Similitude Conceptstates that if the stressintensity (K) at the tip of acrack in the ‘test’ structure(welded connection on an oilplatform leg, say) and thetest specimen are the same,then the crack growthresponse in each case willalso be the same and can bedescribed by the Parisrelationship. Account canalso be made for localchemical environment, ifnecessary.
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
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Fatigue Analysis: Crack Growth
MaterialData
ComponentGeometry
ServiceLoading
AnalysisFatigue
Life Estimate
da/dN curve
K
S , Sm
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Crack Growth
Major Assumptions:Nominal stress and crack size control fatigue life
Accurate determination of initial crack size
19
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Crack Growth
Advantage:Only method to directly deal with cracks
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Crack Growth
Limitations:Complex sequence effects
Accurate determination of initial crack size
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Choose the Right Model
Similitude Failure mechanism
Size scale
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Design Philosophy
Safe Life
Damage Tolerant
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Safe Life
0
100
200
300
400
500
104 105 106 107 108 109
Str
ess
Am
plit
ude
, MP
a
Fatigue Life
99 90 11050
Percent Survival
0
100
200
300
400
500
104 105 106 107 108 109
Str
ess
Am
plit
ude
, MP
a
Fatigue Life
99 90 1105099 90 11050
Percent Survival
Choose an appropriate risk and replace critical partsafter some specified interval
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Damage Tolerant
Inspect for cracks larger than a1 and repair
Cycles
Cra
ck s
ize
a1
a2
Safe Operating Life
Inspection
20
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Inspection
A Boeing 777 costs $250,000,000
A new car costs $25,000
For every $1 spent inspecting and maintaining a B 777 you can spend only 0.01¢ on a car
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Things Worth Remembering
Questions to askWill a crack nucleate ?
Will a crack grow ?
How fast will it grow ?
Similitude Failure mechanism
Size Scale
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Outline
State of Stress
Stress-Strain Relationships
Fatigue Mechanisms
Multiaxial Testing
Stress Based Models
Strain Based Models
Fracture Mechanics Models
Nonproportional Loading
Stress Concentrations
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Fatigue Mechanisms Summary
Fatigue cracks nucleate in shear
Fatigue cracks grow in either shear or tension depending on material and state of stress
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Stress Based Models
Sines
Findley
Dang Van
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1.0
00 0.5 1.0 1.5 2.0
Shear stressOctahedral stress
Principal stress
0.5
She
ar s
tres
s in
ben
ding
1/2
Ben
ding
fat
igue
lim
it
Shear stress in torsion
1/2 Bending fatigue limit
Bending Torsion Correlation
21
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Test Results
Cyclic tension with static tension
Cyclic torsion with static torsion
Cyclic tension with static torsion
Cyclic torsion with static tension
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1.5
1.0
0.5
1.5-1.5 -1.0 1.00.5-0.5 0
Axi
al s
tres
s
Fat
igue
str
eng
th
Mean stressYield strength
Cyclic Tension with Static Tension
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1.5
1.0
0.5
1.51.00.5
0
She
ar S
tres
s A
mpl
itude
She
ar F
atig
ue S
tren
gth
Maximum Shear StressShear Yield Strength
Cyclic Torsion with Static Torsion
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1.5
1.0
0.5
3.00 2.01.0
Ben
ding
Str
ess
Ben
ding
Fat
igue
Str
eng
th
Static Torsion StressTorsion Yield Strength
Cyclic Tension with Static Torsion
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1.5
1.0
0.5
1.5-1.5 -1.0 1.00.5-0.5 0
Tors
ion
shea
r st
ress
She
ar f
atig
ue s
tren
gth
Axial mean stress
Yield strength
Cyclic Torsion with Static Tension
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Conclusions
Tension mean stress affects both tension and torsion
Torsion mean stress does not affect tension or torsion
22
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Sines
)3(2 h
oct
)(
)(6)()()(6
1
meanz
meany
meanx
2yz
2xz
2xy
2zy
2zx
2yx
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Findley
2
k n
maxf
tension torsion
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Bending Torsion Correlation
1.0
00 0.5 1.0 1.5 2.0
Shear stressOctahedral stress
Principal stress
0.5
She
ar s
tres
s in
ben
ding
1/2
Ben
ding
fat
igue
lim
it
Shear stress in torsion
1/2 Bending fatigue limit
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Dang Van
( ) ( )t a t bh
m
V(M)
ij(M,t) Eij(M,t)
ij(m,t)ij(m,t)
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Isotropic Hardening
stab
ilize
d st
ress
ran
ge
Failure occurs when the stress range is not elastic
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Oo
3
Co
Ro
CL
Oo
Loading path
Yield domain expands and
translates
a)
c) d)
b)3 3
3
11
11
2 2
2 2
Oo Oo
RL
OL
Multiaxial Kinematic and Isotropic
* stabilized residual stress
23
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h
h
Loading path
Failurepredicted
taht= b
Dang Van ( continued )
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Stress Based Models Summary
Sines:
Findley:
Dang Van: ( ) ( )t a t bh
)3(2 h
oct
2
k n
maxf
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Model Comparison R = -1
Stress ratio,
Rel
ativ
e F
atig
ue L
ife
0.001
0.01
0.1
1
10
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Torsion Tension Biaxial Tension
Goodman
Findley
Sines
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Outline
State of Stress
Stress-Strain Relationships
Fatigue Mechanisms
Multiaxial Testing
Stress Based Models
Strain Based Models
Fracture Mechanics Models
Nonproportional Loading
Stress Concentrations
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Strain Based Models
Plastic Work
Brown and Miller
Fatemi and Socie
Smith Watson and Topper
Liu
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10 10 2 103 104 105
0.001
0.01
0.1
Cycles to failure
Pla
stic
oct
ahed
ral
shea
r st
rain
ran
ge Torsion
Tension
Octahedral Shear Strain
24
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1
10
100
102 103 104
A
Fatigue Life, Nf
Pla
stic
Wor
k pe
r C
ycle
, M
J/m
3
T TorsionAxial0
90
180
135
45
30
T
A T
TT
TT
T
A
A
A A
A
Plastic Work
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0.005 0.010.0
102
2 x102
5 x102
103
2 x103
Fa
tigu
e L
ife,
Cyc
les
Normal Strain Amplitude, n
= 0.03
Brown and Miller
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Case A and B
Growth along the surface Growth into the surface
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Uniaxial
Equibiaxial
Brown and Miller ( continued )
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Brown and Miller ( continued )
max S n
1
max
', '( ) ( )
2
22 2
S A
EN B Nn
f n meanf
bf f
c
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Fatemi and Socie
25
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C F G
H I J
/ 3
Loading Histories
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0
0.5
1
1.5
2
2.5
0 2000 4000 6000 8000 10000 12000 14000
J-603
F-495 H-491
I-471 C-399
G-304
Cycles
Cra
ck L
engt
h, m
m
Crack Length Observations
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Fatemi and Socie
cof
'f
bof
'f
y
max,n )N2()N2(G
k12
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Smith Watson Topper
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SWT
cbf
'f
'f
b2f
2'f1
n )N2()N2(E2
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Liu
WI = (n nmax + (
b2f
2'fcb
f'f
'fI )N2(
E
4)N2(4W
WII = n n ) + (max
bo2f
2'fcobo
f'f
'fII )N2(
G
4)N2(4W
Virtual strain energy for both mode I and mode II cracking
26
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Cyclic Torsion
Cyclic Shear Strain Cyclic Tensile Strain
Shear Damage Tensile Damage
Cyclic Torsion
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Cyclic TorsionStatic Tension
Cyclic Shear Strain Cyclic Tensile Strain
Shear Damage Tensile Damage
Cyclic Torsion with Static Tension
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Cyclic Shear Strain Cyclic Tensile Strain
Tensile DamageShear DamageCyclic TorsionStatic Compression
Cyclic Torsion with Compression
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Cyclic TorsionStatic Compression
Hoop Tension
Cyclic Shear Strain Cyclic Tensile Strain
Tensile DamageShear Damage
Cyclic Torsion with Tension and Compression
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Test Results
Load Case /2 hoop MPa axial MPa Nf
Torsion 0.0054 0 0 45,200 with tension 0.0054 0 450 10,300
with compression 0.0054 0 -500 50,000 with tension and
compression 0.0054 450 -500 11,200
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Conclusions
All critical plane models correctly predict these results
Hydrostatic stress models can not predict these results
27
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0.006Axial strain
-0.003
0.003
Sh
ear
stra
in
Loading History
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Model Comparison
Summary of calculated fatigue lives
Model Equation LifeEpsilon 6.5 14,060Garud 6.7 5,210Ellyin 6.17 4,450
Brown-Miller 6.22 3,980SWT 6.24 9,930Liu I 6.41 4,280Liu II 6.42 5,420Chu 6.37 3,040
Gamma 26,775Fatemi-Socie 6.23 10,350
Glinka 6.39 33,220
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Strain Based Models Summary
Two separate models are needed, one for tensile growth and one for shear growth
Cyclic plasticity governs stress and strain ranges
Mean stress effects are a result of crack closure on the critical plane
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Separate Tensile and Shear Models
2
1 =
xy
3
1
Inconel 1045 steel stainless steel
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Cyclic Plasticity
p
p
pp
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Mean Stresses
eqf mean
fb
f fc
EN N
''( ) ( )2 2
R1
2)()(W maxnnI
cf
'f
bf
n'f
nmax )N2()S5.05.1()N2(
E
2)S7.03.1(S
2
cof
'f
bof
'f
y
max,n )N2()N2(G
k12
cbf
'f
'f
b2f
2'f1
n )N2()N2(E2
28
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Outline
State of Stress
Stress-Strain Relationships
Fatigue Mechanisms
Multiaxial Testing
Stress Based Models
Strain Based Models
Fracture Mechanics Models
Nonproportional Loading
Stress Concentrations
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Fracture Mechanics Models
Mode I growth
Torsion
Mode II growth
Mode III growth
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Mode I, Mode II, and Mode III
Mode Iopening
Mode IIin-plane shear
Mode IIIout-of-plane shear
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Mode I
Mode II
Mode I and Mode II Surface Cracks
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= -1 = 0 = 1
= -1 = 0 = 1
= 193 MPa = 386 MPa10-3
10-4
10-5
10-6
da/d
N m
m/c
ycle
10 100 20020 50 10 100 20020 50K, MPa m
10-3
10-4
10-5
10-6
da/d
N m
m/c
ycle
Biaxial Mode I Growth
K, MPa m
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Mode II
Mode III
Surface Cracks in Torsion
29
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Transverse Longitudinal Spiral
Failure Modes in Torsion
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1500
1000
Yie
ld S
tren
gth
, M
Pa
40
30
50
60
Har
dnes
s R
c
200 300 400 500
Shear Stress Amplitude, MPa
No Cracks
SpiralCracks
Transverse Cracks
Longi-tudinal
Fracture Mechanism Map
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 170 of 202
10-6
10-5
10-4
10-3
da/d
N,
mm
/cyc
le
5 10 1005020KI , KIII MPa m
KI
KIII
Mode I and Mode III Growth
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 171 of 202
1 10 100
10-7
10-6
10-5
10-4
10-3
10-2
da/d
N,
mm
/cyc
le
m
Mode I R = 0Mode II R = -1
7075 T6 Aluminum
Mode I R = -1Mode II R = -1
SNCM Steel
Mode I and Mode II Growth
KI , KII MPa
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 172 of 202
Fracture Mechanics Models
meqKCdN
da
25.04III
4II
4Ieq )1(K8K8KK
5.02III
2II
2Ieq K)1(KKK
5.02IIIII
2Ieq KKKKK
a)EF())1(2
EF()(K
5.0
2I
2IIeq
ak1GFKys
max,neq
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Fracture Surfaces
Bending Torsion
30
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10-9
10-8
10-7
10-6
10-5
10-4
0.2 0.4 0.6 0.8 1.0Crack length, mm
Cra
ck g
row
th r
ate,
mm
/cyc
le K = 11.0
K = 14.9
K = 10.0
K = 12.0
K = 8.2
Mode III Growth
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0 5 10 15 20 25
5
10
15
20
25
tensile growth
shear growthno growth
KI
K
II
K
K
Growth of Inclined Cracks
From: Otsuka et.al. Engineering Fracture Mechanics, Vol 7, 1975
1010 Steel
Cracks grow in either tension or shear
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 176 of 202
Otsuka
sinK
2
3
2cosK
2cosK II
2I
1cos3K
2sinK
2cos
2
1K III
Tensile growth:
Shear growth:
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 177 of 202
Strain Energy Density
2III33
2II22III12
2I11 KaKaKKa2KaS
S
at o 0
2
2 0S
at o
S a K K a K K K K
a K K a K K
o Imean
I o IImean
I Imean
II
o IImean
II o IIImean
III
2 11 12
22 33
[
]
( ) ( ) ( )
( ) ( )
Strain energy density at the crack tip:
Necessary and sufficient conditions for crack growth:
Cyclic strain energy density:
Sih, G.C and Barthelemy, B.M. “Mixed Mode Fatigue Crack Growth Predictions” Engineering Fracture Mechanics, Vol. 13, 1980
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Fracture Mechanics Models Summary
Multiaxial loading has little effect in Mode I
Crack closure makes Mode II and Mode IIIcalculations difficult
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Outline
State of Stress
Stress-Strain Relationships
Fatigue Mechanisms
Multiaxial Testing
Stress Based Models
Strain Based Models
Fracture Mechanics Models
Nonproportional Loading
Stress Concentrations
31
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Nonproportional Loading
In and Out-of-phase loading
Nonproportional cyclic hardening
Variable amplitude
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 181 of 202
x
xy
y
t
t
t
tx osint
xy osint
In-phase
Out-of-phase
x
x
xy
xy
x ocost
xy osint
In and Out-of-Phase Loading
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xy
22xy
x
x
x
x
xy
2xy
In-Phase and Out-of-Phase
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x x
x x
xy/2 xy/2
xy/2xy/2 cross
diamondout-of-phase
square
Loading Histories
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in-phase
out-of-phase
diamond
square
cross
Loading Histories
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Findley Model Results
MPa n,maxMPa
n,maxN/Nip
in-phase 353 250 428 1.0
90 out-of-phase 250 500 400 2.0
diamond 250 500 400 2.0
square 353 603 534 0.11
cross - tension cycle 250 250 325 16
cross - torsion cycle 250 0 250 216
x
xy/2cross
x
xy/2diamondout-of-phase
x
xy/2square
in-phase
x
xy/2
32
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Nonproportional Hardening
t
t
t
tx osint
xy osint
In-phase
Out-of-phase
x
x
xy
xy
x ocost
xy osint
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600
-600
300
-300
-0.003 -0.0060.003 0.006
Axial Shear
In-Phase
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Axial
600
-600
-0.003 0.003
Shear
-300
300
0.006-0.006
90° Out-of-Phase
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600
-600
-0.004 0.004
Proportional
-600
600
-0.004 0.004
Out-of-phase
Critical Plane
Nf = 38,500Nf = 310,000
Nf = 3,500Nf = 40,000
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0 1 2 3 4
5 6 7 8 9
1011 12 13
Loading Histories
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-300
-150
0
150
300
-600 -300 0 300 600-300
-150
0
150
300
-600 -300 0 300 600-300
-150
0
150
300
-600 -300 0 300 600
-300
-150
0
150
300
-600 -300 0 300 600-300
-150
0
150
300
-600 -300 0 300 600-300
-150
0
150
300
-600 -300 0 300 600
Case 3Case 2Case 1
Case 4 Case 5 Case 6
Sh
ear
Str
ess
( M
Pa
)
Stress-Strain Response
33
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Stress-Strain Response (continued)
-300
-150
0
150
300
-600 -300 0 300 600-300
-150
0
150
300
-600 -300 0 300 600-300
-150
0
150
300
-600 -300 0 300 600
-300
-150
0
150
300
-600 -300 0 300 600
-300
-150
0
150
300
-600 -300 0 300 600-300
-150
0
150
300
-600 -300 0 300 600
Case 7 Case 9 Case 10
Case 13Case 12Case 11
Sh
ear
Str
ess
( M
Pa
)
Axial Stress ( MPa )
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 193 of 202
1000
200102 103 104
Equ
ival
ent
Str
ess,
MP
a
2000
Fatigue Life, Nf
Maximum Stress
Nonproportional hardening results in lower fatigue lives
All tests have the same strain ranges
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Case A Case B Case C Case D
x x x x
y y y y
Nonproportional Example
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Shear Stresses on Planes
= /2
= /2
= 0
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Shear Stresses on Planes
= /2
= /2
= 0
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Combined
= /2
= /2
34
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Case A Case B Case C Case D
xy xy xy xy
Shear Stresses
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Variable Amplitude
Time
-2500
2500
90
45
0
Mic
rost
rain
0
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Cycle Counting and Damage?
-0.0025 0.0025
-0.005
0.005
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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, AH
G
C
AD
BC
CB
DA
Counts 1/2 cycles
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Rainflow and Hysteresis
A
CB
D EF G
HI
35
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Principal Stress Directions
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Signed Equivalent Stress
2 2 2
eq 1 2 1 3 2 3 max
1sgn
2
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Torsion Stresses
eq
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Torsion Stresses
45
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Real Data
-60
-40
-20
0
20
40
60
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Signed Equivalent Stress
-100
-80
-60
-40
-20
0
20
40
60
80
100
36
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-0.003 0.003
-0.006
0.006
-300 300
x
-150
150
x
Simple Variable Amplitude History
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-0.003 0.003x
-300
300
-0.005 0.005xy
-150
150
xy
Stress-Strain on 0° Plane
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 212 of 202
-0.003 0.00330
-300
300
-0.003 0.003
-300
30030º plane 60º plane
60
Stress-Strain on 30° and 60° Planes
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Stress-Strain on 120° and 150° Planes
-0.003 0.003
-300
300
-0.003 0.003
-300
300150º plane120º plane
120 150
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time
-0.005
0.005
She
ar s
trai
n,
Shear Strain History on Critical Plane
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Fatigue Calculations
Load or strain history
Cyclic plasticity model
Stress and strain tensor
Search for critical plane
37
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Analysis model Single event16 input channels2240 elements
An Example
From Khosrovaneh, Pattu and Schnaidt “Discussion of Fatigue Analysis Techniques for Automotive Applications”Presented at SAE 2004.
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Biaxial and Uniaxial Solution10-2
1 10 100 1000 10000
Element rank based on critical plane damage
Da
ma
ge
Uniaxial solutionSigned principal stress
Critical plane solution
10-3
10-4
10-5
10-6
10-7
10-8
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Nonproportional Loading Summary
Nonproportional cyclic hardening increases stress levels
Critical plane models are used to assess fatigue damage
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Outline
State of Stress
Stress-Strain Relationships
Fatigue Mechanisms
Multiaxial Testing
Stress Based Models
Strain Based Models
Fracture Mechanics Models
Nonproportional Loading
Stress Concentrations
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 220 of 202
Notches
Stress and strain concentrations
Nonproportional loading and stressing
Fatigue notch factors
Cracks at notches
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 221 of 202
z
z
MTMX
MY
P
Notched Shaft Loading
38
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0
1
2
3
4
5
6
0.025 0.050 0.075 0.100 0.125Notch Root Radius, /d
Str
ess
Con
cent
ratio
n F
acto
r
Tors
ion
Ben
ding
2.201.201.04
D/d
Dd
Stress Concentration Factors
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 223 of 202
a
r
r
r
r
r
Hole in a Plate
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 224 of 202
-4
-3
-2
-1
0
1
2
3
4
30 60 90 120 150 180
Angle
= 1
= 0
= -1
Stresses at the Hole
Stress concentration factor depends on type of loadingMultiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 225 of 202
0
0.5
1.0
1.5
1 2 3 4 5ra
r
Shear Stresses during Torsion
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Torsion Experiments
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Multiaxial Loading
Uniaxial loading that produces multiaxial stresses at notches
Multiaxial loading that produces uniaxial stresses at notches
Multiaxial loading that produces multiaxial stresses at notches
39
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0.004 0.008 0.0120
0.001
0.002
0.003
0.004
0.005
50 mm
30 mm
15 mm
7 mm
Longitudinal Tensile Strain
Tra
nsve
rse
Com
pres
sion
Str
ain
Thickness
100
x
z
y
Thickness Effects
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Multiaxial Loading
Uniaxial loading that produces multiaxial stresses at notches
Multiaxial loading that produces uniaxial stresses at notches
Multiaxial loading that produces multiaxial stresses at notches
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MX
MY
1 2 3 4
MX
MY
Applied Bending Moments
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 231 of 202
A
B
C
D
A
C
A’
C’
BD
B’ D’
Location
MX
MY
Bending Moments on the Shaft
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Bending Moments
M A B C D2.82 1 12.00 3 21.41 2 11.00 20.71 2
M M 55
A B C D
M 2.49 2.85 2.31 2.84
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Combined Loading
=
40
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Maximum Tensile Stress Location
32
Kt = 31 =
Kt = 3.411 = 1.72
Kt = 41 =
45
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 235 of 202
Kt = 3 Kt = 4
In and Out of Phase Loading
In-phase Out-of-phase
Damage location changes with load phasing
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 236 of 202
Multiaxial Loading
Uniaxial loading that produces multiaxial stresses at notches
Multiaxial loading that produces uniaxial stresses at notches
Multiaxial loading that produces multiaxial stresses at notches
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 237 of 202
t1 t2 t3 t4
MX
MT
t1
z
1T
T
t2
z
1
T
T
t3
z
t4
T
T
Torsion Loading
Out-of-phase shear loading is needed to produce nonproportional stressing
MX
MT
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 238 of 202
0
1
2
3
4
5
6
0.025 0.050 0.075 0.100 0.125
Notch root radius,
Str
ess
conc
entr
atio
n fa
ctor
Kt Bending
Kt Torsion
Kf Torsion
Kf Bending
Dd
d
= 2.2Dd
Fatigue Notch Factors
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 239 of 202
1.0
1.5
2.0
2.5
1.0 1.5 2.0 2.5Experimental Kf
Cal
cula
ted
Kf conservative
non-conservative
Fatigue Notch Factors ( continued )
bendingtorsion
ra
1
1K1K T
f
Peterson’s Equation
41
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Fracture Surfaces in Torsion
Circumferencial Notch
Shoulder Fillet
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Neuber’s Rule
Str
ess
(MP
a)
Strain
KtS
Kte
eKSK tt
Stress calculated with elastic assumptions
Actual stress
ES2e
For cyclic loading
eS ee
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 242 of 202
Multiaxial Neuber’s Rule
ES2e
Define Neuber’s rule in equivalent variables
Stress strain curve
'n
1
'KE
Constitutive equation
xy
y
x
xy
y
x
f(E,K’,n’)
Five equations and six unknowns
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 243 of 202
Ignore Plasticity Theory
11
e2
e
2 e
e
11
e3
e
3 e
e
11
e2
e
2 S
S
11
e3
e
3 S
S
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 244 of 202
Hoffman and Seeger
1e
2e
1
2
S
S
1e
2e
1
2
e
e
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 245 of 202
Glinka
Strain energy density
Str
ess
(MP
a)
Strain
ee
eS
ije
ije
ije
ije
ijij
ijij
eS
eS
42
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Koettgen-Barkey-Socie
xy
y
x
xye
ye
xe
T
S
S
fo(Eo,Ko,no)
E
K’ n’
Material Yield SurfaceStructural Yield Surface
Ko no
Eo
xy
y
x
xy
y
x
f(E,K’,n’)
Se
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 247 of 202
1.00 1.50 2.00 2.50 3.000.00
0.25
0.50
0.75
1.00
1.25
1.50
F
R
a
aR
Stress Intensity Factors
meqKCdN
da aFKI
Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 248 of 202
0
20
40
60
80
100
0 2 x 10 6
Cra
ck L
eng
th,
mm
4 x 10 6 6 x 10 6 8 x 10 6
= -1 = 0 = 1
Cycles
Crack Growth From a Hole
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Notches Summary
Uniaxial loading can produce multiaxial stresses at notches
Multiaxial loading can produce uniaxial stresses at notches
Multiaxial stresses are not very important in thin plate and shell structures
Multiaxial stresses are not very important in crack growth
Multiaxial Fatigue
Recommended