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Introduction
Professor Darrell F. SocieUniversity of Illinois at Urbana-Champaign
© 2003 Darrell Socie, All Rights Reserved
Multiaxial Fatigue
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 14
Contact Information
Darrell SocieMechanical Engineering1206 West GreenUrbana, Illinois 61801
Office: 3015 Mechanical Engineering Laboratory
[email protected]: 217 333 7630Fax: 217 333 5634
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 2 of 14
When is Multiaxial Fatigue Important ?
Complex state of stressComplex out of phase loading
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 14
Uniaxial Stress
one principal stressone direction
X
Z
Y
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 14
Proportional Biaxial
principal stresses vary proportionally but do not rotate
X
Z
Yσ1 = ασ2 = βσ3
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 14
Nonproportional Multiaxial
Principal stresses may vary nonproportionallyand/or change direction
X
Z
Y
2
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 14
Crankshaft
Time
-2500
2500
90450
Mic
rost
rain
0
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 14
Shear and Normal Strains
-0.0025 0.0025
-0.005
0.005
-0.0025 0.0025
-0.005
0.005
0º 90ºγ γ
εε
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 14
Shear and Normal Strains
-0.0025 0.0025
-0.005
0.005
-0.0025 0.0025
-0.005
0.005
45º 135º
ε
γ
ε
γ
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 14
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 StrainTr
ansv
erse
Com
pres
sion
Stra
in
Thickness
100
x
z
y
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 10 of 14
Notch Stresses
29.375.1-0.0010.015021.873.0-0.0020.013014.170.6-0.0030.0115
063.5-0.0050.017σzσxεzεxt
x
z
y
State of Stress
Professor Darrell F. SocieUniversity of Illinois at Urbana-Champaign
© 2003 Darrell Socie, All Rights Reserved
Multiaxial Fatigue
3
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 12 of 14
Outline
State of StressStress-Strain RelationshipsFatigue MechanismsMultiaxial TestingStress Based Models Strain Based ModelsFracture Mechanics ModelsNonproportional LoadingStress Concentrations
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 13 of 14
State of Stress
Stress componentsCommon states of stressShear stresses
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 14 of 14
Stress Components
X
Z
Y
σz
σy
σx
τzyτzx
τxz
τxyτyx
τyz
Six stresses and six strains
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 15 of 14
Stresses Acting on a PlaneZ
Y
X
Y’
X’Z’
σx’
τx’z’
τx’y’
θ
φ
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 16 of 14
Principal Stresses
σ3 - σ2( σX + σY + σZ ) + σ(σXσY + σYσZσXσZ -τ2XY - τ2
YZ -τ2XZ )
- (σXσYσZ + 2τXYτYZτXZ - σXτ2YZ - σYτ2
ZX - σZτ2XY ) = 0
σ1
σ3
σ2
τ13
τ12τ23
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 17 of 14
Stress and Strain Distributions
80
90
100
-20 -10 0 10 20
θ
% o
f app
lied
stre
ss
Stresses are nearly the same over a 10° range of angles
4
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 18 of 14
Tension
τ
σσx
γ/2
εε1
σ1 σ2
σ3
σ2 = σ3 = 0
σ1
ε2 = ε3 = −νε1
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 19 of 14
Torsion
σ2 τ
στxyε1
ε3
ε2
γ/2
ε
σ1
σ3
X
Yσ2
σ1 = τxy
σ3
σ1
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 20 of 14
τ
σ
γ/2
ε
σ2 = σy
σ1 = σx
σ3
σ1 = σ2
σ3
ε1 = ε2
εν−ν
−=ε12
3
Biaxial Tension
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 21 of 14
σ1
σ2
σ3 σ3
σ2
σ1
Maximum shear stress Octahedral shear stress
Shear Stresses
231
13
σ−σ=τ ( ) ( ) ( )2
322
212
31oct 31
σ−σ+σ−σ+σ−σ=τ
oct23
τ=σMises: 1313oct 94.022
3τ=τ=τ
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 22 of 14
Shear Stress Influence
1
1.2
1.4
1.6
1.8
2
0 0.2 0.4 0.6 0.8 1
σσ1
στ
σ
1σ
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 23 of 14
State of Stress Summary
Stresses acting on a planePrincipal stressMaximum shear stressOctahedral shear stress
5
Stress Strain Relationships
Professor Darrell F. SocieUniversity of Illinois at Urbana-Champaign
© 2003 Darrell Socie, All Rights Reserved
Multiaxial Fatigue
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 25 of 14
Outline
State of Stress Stress-Strain RelationshipsFatigue MechanismsMultiaxial TestingStress Based Models Strain Based ModelsFracture Mechanics ModelsNonproportional LoadingStress Concentrations
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 26 of 14
“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 - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 27 of 14
Elastic Stress Strain Relationships
( ) ( )
σσστττ
ν ν
ν ν νν ν νν ν ν
ν
ν
ν
εεεγγγ
x
y
z
xy
yz
xz
x
y
z
xy
yz
xz
E
=+ −
−−
−−
−
−
1 1 2
1 0 0 01 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 - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 28 of 14
σ2
von Misesyield surface
Tresca yieldsurface
Yield Surfaces
σ1
3τxy
σX
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 29 of 14
Equations
F
F
orF
ys
ys
ys
1 1 3
2 1 2
3 2 3
0
0
0
= − − =
= − − =
= − − =
σ σ σ
σ σ σ
σ σ σ
F x y x y xy ys= + − + − =σ σ σ σ τ σ2 2 2 23 0
Tresca
Mises
6
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 14
π - planeyield cylinder axis
Mises Yield Surfaces
32 σys
σ1
σ3
σ2
σ1
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 31 of 14
σ
ε
Initial yield surface
Yield surface after loading
Isotropic Hardening
σ
3τ
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 32 of 14
Cyclic Loading
σ
ε ε
σ
stab
ilize
d st
ress
rang
e
Strain Control Stress Control
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 14
Kinematic Hardening
σ
ε
σ
τ3
A
B
C
σB
σ = σA y
ασ − 2σB y
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 14
Ratcheting
γ
ε
Cyclic torsion with a mean tension stress
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 14
Cyclic Mean Stress Relaxation
ε
σ
Cyclic strain with mean stress
7
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 14
Cyclic Creep
ε
σ
Cyclic stress with a mean stress
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 14
Plasticity Models
σ
ε
a
b
c
f
d
e
σ
a
σ
c
σ
b
σ σσde
f
3τ3τ3τ
3τ 3τ 3τ
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 38 of 14
Why is modeling needed?
σ
A B C D
A
B
CD
ε
ε
time
ε*
ε*
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 14
Flow and Hardening Rules
σ
τ3φ
dσij
n
σij
αij
dαij
d d Fijp
ij
ε λ∂
∂σ=
F ijp= + =f g( ) ( )σ ε 0
Flow rule
Hardening rule
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 40 of 14
Multiaxial Example
-.006
0.003
εx
-0.003
0.006γxy
-500 500
σx
-250
250τxy
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 41 of 14
Multiaxial Example
-0.006 0.006γxy
-250
250τxy
-0.003 0.003εx
-500
500σx
8
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 42 of 14
Summary
Isotropic HardeningKinematic HardeningCyclic creep or ratchetingMean stress relaxationNonproportional hardening
Fatigue Mechanisms
Professor Darrell F. SocieUniversity of Illinois at Urbana-Champaign
© 2003 Darrell Socie, All Rights Reserved
Multiaxial Fatigue
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 44 of 14
Outline
State of Stress Stress-Strain RelationshipsFatigue MechanismsMultiaxial TestingStress Based Models Strain Based ModelsFracture Mechanics ModelsNonproportional LoadingStress Concentrations
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 45 of 14
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 - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 46 of 14
The Fatigue Process
Crack nucleationSmall crack growth in an elastic-plastic stress fieldMacroscopic crack growth in a nominally elastic stress fieldFinal fracture
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 47 of 14
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
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 48 of 14
Crack Nucleation
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 49 of 14
Slip Band in Copper
Polak, J. Cyclic Plasticity and Low Cycle Fatigue Life of Metals, Elsevier, 1991
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 50 of 14
Slip Band Formation
Loading Unloading
Extrusion
Undeformedmaterial
Intrusion
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 51 of 14
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
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 52 of 14
Crack Initiation at Inclusions
Langford and Kusenberger, “Initiation of Fatigue Cracks in 4340 Steel”, Metallurgical Transactions, Vol 4, 1977, 553-559
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 53 of 14
Subsurface Crack Initiation
Y. Murakami, Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions, 2002
10
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 54 of 14
Fatigue Limit and Strength Correlation
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 55 of 14
Crack Nucleation Summary
Highly localized plastic deformationSurface phenomenaStochastic process
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 56 of 14
100 µm
bulksurface
10 µm
surface
20-25 austenitic steel in symmetrical push-pull fatigue (20°C, ∆εp/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
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 57 of 14
Stage I Stage II
loading direction
freesurface
Stage I and Stage II
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 58 of 14
Stage I Crack Growth
Single primary slip system
individual grain
near - tip plastic zone
S
SStage I crack is strongly affected by slip characteristics, microstructure dimensions, stress level, extent of near tip plasticity
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 59 of 14
Small Cracks at Notches
D acrack tip plastic zone
notch plastic zone
notch stress field
Crack growth controlled by the notch plastic strains
11
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 60 of 14
Small Crack Growth
1.0 mm
N = 900
Inconel 718∆ε = 0.02Nf = 936
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 61 of 14
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 - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 62 of 14
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
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 63 of 14
Strain-Life Data
Reversals, 2Nf
Stra
in A
mpl
itude
∆ε 2
10-5
10-4
0.01
0.1
1
100 101 102 103 104 105 106 107
10-3
10µm100 µm
1mmfracture Crack size
Most of the life is spent in microcrack growth in the plastic strain dominated region
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 64 of 14
Stage II Crack Growth
Locally, the crack grows in shear Macroscopically it grows in tension
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 65 of 14
Long Crack Growth
Plastic zone size is much larger than the material microstructure so that the microstructure does not play such an important role.
12
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 66 of 14
Material strength does not play a major role in fatigue crack growth
Crack Growth Rates of Metals
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 67 of 14
Maximum Load
monotonic plastic zone
σ
Stresses Around a Crack
σ
ε
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 68 of 14
Stresses Around a Crack (continued)
Minimum Load σ
ε
cyclic plastic zone
σ
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 69 of 14
Crack Closure
S = 250
b
S = 175
c
S = 0
a
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 70 of 14
Crack Opening LoadDamaging portion of loading history
Nondamaging portion of loading history
Opening load
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 71 of 14
Mode Iopening
Mode IIin-plane shear
Mode IIIout-of-plane shear
Mode I, Mode II, and Mode III
13
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 72 of 14
5 µmcrac
k gr
owth
dire
ctio
n
Mode I Growth
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 73 of 14
crack growth direction
10 µm
slip bandsshear stress
Mode II Growth
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 74 of 14
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
Frac
tion
N/N
f
100 µm crack
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 75 of 14
Fatigue Life, 2Nf
Dam
age
Frac
tion
N/N
ff
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 - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 76 of 14
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
Frac
tion
N/N
f
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 77 of 14
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
Frac
tion
N/N
f
14
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 78 of 14
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
Frac
tion
N/N
f
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 79 of 14
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
Frac
tion
N/N
f
Stress Based Models
Professor Darrell F. SocieUniversity of Illinois at Urbana-Champaign
© 2003 Darrell Socie, All Rights Reserved
Multiaxial Fatigue
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 81 of 14
Outline
State of Stress Stress-Strain RelationshipsFatigue MechanismsMultiaxial TestingStress Based Models Strain Based ModelsFracture Mechanics ModelsNonproportional LoadingStress Concentrations
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 82 of 14
Fatigue Mechanisms Summary
Fatigue cracks nucleate in shearFatigue cracks grow in either shear or tension depending on material and state of stress
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 83 of 14
Stress Based Models
SinesFindleyDang Van
15
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 84 of 14
1.0
00 0.5 1.0 1.5 2.0
Shear stressOctahedral stress
Principal stress
0.5
She
ar s
tress
in b
endi
ng
1/2
Bend
ing
fatig
ue li
mit
Shear stress in torsion
1/2 Bending fatigue limit
Bending Torsion Correlation
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 85 of 14
Test Results
Cyclic tension with static tensionCyclic torsion with static torsionCyclic tension with static torsionCyclic torsion with static tension
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 86 of 14
1.5
1.0
0.5
1.5-1.5 -1.0 1.00.5-0.5 0
Axi
al s
tress
Fatig
ue s
treng
th
Mean stressYield strength
Cyclic Tension with Static Tension
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 87 of 14
1.5
1.0
0.5
1.51.00.50
Shea
r Stre
ss A
mpl
itude
Shea
r Fat
igue
Stre
ngth
Maximum Shear StressShear Yield Strength
Cyclic Torsion with Static Torsion
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 88 of 14
1.5
1.0
0.5
3.00 2.01.0
Bend
ing
Stre
ssBe
ndin
g Fa
tigue
Stre
ngth
Static Torsion StressTorsion Yield Strength
Cyclic Tension with Static Torsion
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 89 of 14
1.5
1.0
0.5
1.5-1.5 -1.0 1.00.5-0.5 0
Tors
ion
shea
r stre
ss
Shea
r fat
igue
stre
ngth
Axial mean stressYield strength
Cyclic Torsion with Static Tension
16
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 90 of 14
Conclusions
Tension mean stress affects both tension and torsionTorsion mean stress does not affect tension or torsion
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 91 of 14
Sines
β=σα+τ∆ )3(2 h
oct
β=σ+σ+σα
+τ∆+τ∆+τ∆+σ∆−σ∆+σ∆−σ∆+σ∆−σ∆
)(
)(6)()()(61
meanz
meany
meanx
2yz
2xz
2xy
2zy
2zx
2yx
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 92 of 14
Findley
∆τ2
+
=k nσ
maxf
tension torsionMultiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 93 of 14
Bending Torsion Correlation
1.0
00 0.5 1.0 1.5 2.0
Shear stressOctahedral stress
Principal stress
0.5
She
ar s
tress
in b
endi
ng
1/2
Bend
ing
fatig
ue li
mit
Shear stress in torsion
1/2 Bending fatigue limit
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 94 of 14
Dang Van
τ σ( ) ( )t a t bh+ =
m
V(M)
Σij(M,t) Eij(M,t)
σij(m,t)εij(m,t)
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 95 of 14
Isotropic Hardening
σ
ε
stab
ilize
d st
ress
rang
e
Failure occurs when the stress range is not elastic
17
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 96 of 14
Oo
σ3
Co
Ro
ρ∗
CL
Oo
Loading path
Yield domain expands and
translates
a)
c) d)
b)σ3 σ3
σ3
σ1σ1
σ1σ1
σ2 σ2
σ2 σ2
Oo Oo
RL
OL
Multiaxial Kinematic and Isotropic
ρ* stabilized residual stressMultiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 97 of 14
τ
σh
τ
σh
Loading path
Failurepredicted
τ(t) + aσh(t) = b
Dang Van ( continued )
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 98 of 14
Stress Based Models Summary
Sines:
Findley:
Dang Van: τ σ( ) ( )t a t bh+ =
β=σα+τ∆ )3(2 h
oct
∆τ2
+
=k nσ
maxf
Strain Based Models
Professor Darrell F. SocieUniversity of Illinois at Urbana-Champaign
© 2003 Darrell Socie, All Rights Reserved
Multiaxial Fatigue
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 100 of 14
Outline
State of Stress Stress-Strain RelationshipsFatigue MechanismsMultiaxial TestingStress Based Models Strain Based ModelsFracture Mechanics ModelsNonproportional LoadingStress Concentrations
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 101 of 14
Strain Based Models
Plastic WorkBrown and MillerFatemi and SocieSmith Watson and TopperLiu
18
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10 10 2 103 104 1050.001
0.01
0.1
Cycles to failure
Plas
tic o
ctah
edra
lsh
ear s
train
rang
e Torsion
Tension
Octahedral Shear Strain
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 103 of 14
1
10
100
102 103 104
A
Fatigue Life, Nf
Plas
tic W
ork
per C
ycle
, MJ/
m3
T TorsionAxial0o
90o
180o
135o
45o
30o
T
A T
TT
TT
T
A
A
A A
A
Plastic Work
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 104 of 14
0.005 0.010.0
102
2 x102
5 x102
103
2 x103
Fatig
ue L
ife, C
ycle
s
Normal Strain Amplitude, ∆εn
∆γ = 0.03
Brown and Miller
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 105 of 14
Case A and B
Growth along the surface Growth into the surface
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 106 of 14
Uniaxial
Equibiaxial
Brown and Miller ( continued )
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Brown and Miller ( continued )
( )∆ ∆γ ∆ε$ maxγ α α α= + S n
1
∆γ∆εmax
', '( ) ( )
22
2 2+ =−
+S AE
N B Nnf n mean
fb
f fc
σ σε
19
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 108 of 14
Fatemi and Socie
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 109 of 14
C F G
H I J
γ / 3
ε
Loading Histories
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 110 of 14
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 - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 111 of 14
Fatemi and Socie
cof
'f
bof
'f
y
max,n )N2()N2(G
k12
γ+τ
=
σσ
+γ∆
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 112 of 14
Smith Watson Topper
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 113 of 14
SWT
cbf
'f
'f
b2f
2'f1
n )N2()N2(E2
+εσ+σ
=ε∆
σ
20
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 114 of 14
Liu
∆WI = (∆σn ∆εn)max + (∆τ ∆γ)
b2f
2'fcb
f'f
'fI )N2(
E4)N2(4W σ
+εσ=∆ +
∆WII = (∆σn ∆εn ) + (∆τ ∆γ)max
bo2f
2'fcobo
f'f
'fII )N2(
G4)N2(4W τ
+γτ=∆ +
Virtual strain energy for both mode I and mode II cracking
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 115 of 14
Cyclic Torsion
Cyclic Shear Strain Cyclic Tensile Strain
Shear Damage Tensile Damage
Cyclic Torsion
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 116 of 14
Cyclic TorsionStatic Tension
Cyclic Shear Strain Cyclic Tensile Strain
Shear Damage Tensile Damage
Cyclic Torsion with Static Tension
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 117 of 14
Cyclic Shear Strain Cyclic Tensile Strain
Tensile DamageShear DamageCyclic TorsionStatic Compression
Cyclic Torsion with Compression
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 118 of 14
Cyclic TorsionStatic Compression
Hoop Tension
Cyclic Shear Strain Cyclic Tensile Strain
Tensile DamageShear Damage
Cyclic Torsion with Tension and Compression
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 119 of 14
Test Results
Load Case ∆γ/2 σhoop MPa σaxial 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
21
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 120 of 14
Conclusions
All critical plane models correctly predict these resultsHydrostatic stress models can not predict these results
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 121 of 14
0.006Axial strain
-0.003
0.003
Shea
r stra
in
Loading History
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 122 of 14
Model Comparison Summary of calculated fatigue lives
Model Equation Life Epsilon 6.5 14,060 Garud 6.7 5,210 Ellyin 6.17 4,450
Brown-Miller 6.22 3,980 SWT 6.24 9,930 Liu I 6.41 4,280 Liu II 6.42 5,420 Chu 6.37 3,040
Gamma 26,775 Fatemi-Socie 6.23 10,350
Glinka 6.39 33,220
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 123 of 14
Strain Based Models Summary
Two separate models are needed, one for tensile growth and one for shear growthCyclic plasticity governs stress and strain rangesMean stress effects are a result of crack closure on the critical plane
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 124 of 14
Separate Tensile and Shear Models
τ
σ
σ2
σ1 = τxy
σ3
σ1
Inconel 1045 steel stainless steel
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 125 of 14
Cyclic Plasticity
∆ε∆γ∆εp
∆γp
∆ε∆σ∆γ∆τ∆εp∆σ∆γp∆τ
22
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 126 of 14
Mean Stresses
∆ε eqf mean
fb
f fc
EN N=
−+
σ σε
''( ) ( )2 2
[ ]
−
γ∆τ∆+ε∆σ∆=∆R1
2)()(W maxnnI
cf
'f
bf
n'f
nmax )N2()S5.05.1()N2(
E2
)S7.03.1(S2
ε++σ−σ
+=ε∆+γ∆
cof
'f
bof
'f
y
max,n )N2()N2(G
k12
γ+τ
=
σσ
+γ∆
cbf
'f
'f
b2f
2'f1
n )N2()N2(E2
+εσ+σ
=ε∆
σ
Fracture Mechanics Models
Professor Darrell F. SocieUniversity of Illinois at Urbana-Champaign
© 2003 Darrell Socie, All Rights Reserved
Multiaxial Fatigue
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 128 of 14
Outline
State of Stress Stress-Strain RelationshipsFatigue MechanismsMultiaxial TestingStress Based Models Strain Based ModelsFracture Mechanics ModelsNonproportional LoadingStress Concentrations
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 129 of 14
Fracture Mechanics Models
Mode I growthTorsionMode II growthMode III growth
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Mode I, Mode II, and Mode IIIMode Iopening
Mode IIin-plane shear
Mode IIIout-of-plane shear
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 131 of 14
σ
σ
σ σ
Mode I
Mode II
Mode I and Mode II Surface Cracks
23
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 132 of 14
λ = -1λ = 0λ = 1
λ = -1λ = 0λ = 1
∆σ = 193 MPa ∆σ = 386 MPa10-3
10-4
10-5
10-6
da/d
Nm
m/c
ycle
10 100 20020 50 10 100 20020 50∆K, MPa m
10-3
10-4
10-5
10-6
da/d
Nm
m/c
ycle
Biaxial Mode I Growth
∆K, MPa m
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 133 of 14
Mode II
Mode III
Surface Cracks in Torsion
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Transverse Longitudinal Spiral
Failure Modes in Torsion
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 135 of 14
1500
1000
Yiel
d St
reng
th,M
Pa
40
30
50
60
Har
dnes
sR
c
200 300 400 500Shear Stress Amplitude, MPa
No Cracks
SpiralCracks
Transverse Cracks
Longi-tudinal
Fracture Mechanism Map
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 136 of 14
10-6
10-5
10-4
10-3
da/d
N,
mm
/cyc
le
5 10 1005020∆KI , ∆KIII MPa m
∆KI
∆KIII
Mode I and Mode III Growth
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 137 of 14
1 10 100
10-7
10-6
10-5
10-4
10-3
10-2
da/d
N, m
m/c
ycle
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
24
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 138 of 14
Fracture Mechanics Models
( )meqKCdNda
∆=
[ ] 25.04III
4II
4Ieq )1(K8K8KK ν−∆+∆+∆=∆
[ ] 5.02III
2II
2Ieq K)1(KKK ∆ν++∆+∆=∆
[ ] 5.02IIIII
2Ieq KKKKK ∆+∆∆+∆=∆
a)EF())1(2
EF()(K5.0
2I
2IIeq π
ε∆+γ∆
ν+=ε∆
( ) ak1GFKys
max,neq π
σσ
+γ∆=ε∆
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 139 of 14
Fracture Surfaces
Bending Torsion
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 140 of 14
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 ra
te, m
m/c
ycle ∆K = 11.0
∆K = 14.9
∆K = 10.0
∆K = 12.0
∆K = 8.2
Mode III Growth
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 141 of 14
Fracture Mechanics Models Summary
Multiaxial loading has little effect in Mode ICrack closure makes Mode II and Mode IIIcalculations difficult
Nonproportional Loading
Professor Darrell F. SocieUniversity of Illinois at Urbana-Champaign
© 2003 Darrell Socie, All Rights Reserved
Multiaxial Fatigue
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 143 of 14
Outline
State of Stress Stress-Strain RelationshipsFatigue MechanismsMultiaxial TestingStress Based Models Strain Based ModelsFracture Mechanics ModelsNonproportional LoadingStress Concentrations
25
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 144 of 14
Nonproportional Loading
In and Out-of-phase loadingNonproportional cyclic hardeningVariable amplitude
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 145 of 14
εx
γxy
εy
t
t
t
tεx = εosin(ωt)
γxy = (1+ν)εosin(ωt)
In-phase
Out-of-phase
εx
εx
γxy
γxy
εx = εocos(ωt)
γxy = (1+ν)εosin(ωt)ε
∆γ
ε
γ
ν1+
∆γ
γ
ν1+
In and Out-of-Phase Loading
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 146 of 14
θε1
γ xy
22τxy
σx
εx
∆σx
∆εx
∆γxy
2∆2τxy
In-Phase and Out-of-Phase
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 147 of 14
εx εx
εx εx
γxy/2 γxy/2
γxy/2γxy/2 cross
diamondout-of-phase
square
Loading Histories
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 148 of 14
in-phase
out-of-phase
diamond
square
cross
Loading Histories
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 149 of 14
Findley Model Results
∆τ/2 MPa σn,maxMPa ∆τ/2 + 0.3 σn,max N/Nipin-phase 353 250 428 1.090° out-of-phase 250 500 400 2.0
diamond 250 500 400 2.0square 353 603 534 0.11cross - tension cycle 250 250 325 16cross - torsion cycle 250 0 250 216
εx
γxy/2cross
εx
γxy/2diamondout-of-phase
εx
γxy/2square
in-phase
εx
γxy/2
26
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 150 of 14
Nonproportional Hardening
t
t
t
tεx = εosin(ωt)
γxy = (1+ν)εosin(ωt)
In-phase
Out-of-phase
εx
εx
γxy
γxy
εx = εocos(ωt)
γxy = (1+ν)εosin(ωt)
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 151 of 14
600
-600
300
-300
-0.003 -0.0060.003 0.006
Axial Shear
In-Phase
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 152 of 14
Axial600
-600
-0.003 0.003
Shear
-300
300
0.006-0.006
90° Out-of-Phase
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 153 of 14
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
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 154 of 14
0 1 2 3 4
5 6 7 8 9
1011 12 13
Loading Histories
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 155 of 14
-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
Shea
r Stre
ss (
MPa
)
Stress-Strain Response
27
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 156 of 14
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
Shea
r Stre
ss (
MPa
)
Axial Stress ( MPa )
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 157 of 14
1000
200102 103 104
Equ
ival
ent S
tress
,MPa
2000
Fatigue Life, Nf
Maximum Stress
Nonproportional hardening results in lower fatigue lives
All tests have the same strain ranges
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 158 of 14
Case A Case B Case C Case Dσx σx σx σx
σy σy σy σy
Nonproportional Example
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 159 of 14
Case A Case B Case C Case D
τxy τxy τxy τxy
Shear Stresses
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 160 of 14
-0.003 0.003
-0.006
0.006γ xy
-300 300
σx
-150
150
εx
τ xy
Simple Variable Amplitude History
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 161 of 14
-0.003 0.003εx
-300
300σ x
-0.005 0.005γxy
-150
150τ xy
Stress-Strain on 0° Plane
28
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 162 of 14
-0.003 0.003ε30
-300
300
-0.003 0.003
-300
30030º plane 60º plane σ 60
σ 30
ε60
Stress-Strain on 30° and 60° Planes
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 163 of 14
Stress-Strain on 120° and 150° Planes
-0.003 0.003
-300
300
σ 120
-0.003 0.003
-300
300150º plane120º plane
σ 150
ε120 ε150
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 164 of 14
time
-0.005
0.005
Shea
r stra
in, γ
Shear Strain History on Critical Plane
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 165 of 14
Fatigue Calculations
Load or strain history
Cyclic plasticity model
Stress and strain tensor
Search for critical plane
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 166 of 14
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.
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 167 of 14
Biaxial and Uniaxial Solution10-2
1 10 100 1000 10000
Element rank based on critical plane damage
Dam
age
Uniaxial solutionSigned principal stress
Critical plane solution
10-3
10-4
10-5
10-6
10-7
10-8
29
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 168 of 14
Nonproportional Loading Summary
Nonproportional cyclic hardening increases stress levelsCritical plane models are used to assess fatigue damage
Stress Concentrations
Professor Darrell F. SocieUniversity of Illinois at Urbana-Champaign
© 2003 Darrell Socie, All Rights Reserved
Multiaxial Fatigue
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 170 of 14
Outline
State of Stress Stress-Strain RelationshipsFatigue MechanismsMultiaxial TestingStress Based Models Strain Based ModelsFracture Mechanics ModelsNonproportional LoadingStress Concentrations
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 171 of 14
Notches
Stress and strain concentrationsNonproportional loading and stressingFatigue notch factorsCracks at notches
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 172 of 14
τθzσθ
σz
MT MX
MY
P
Notched Shaft Loading
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 173 of 14
0
1
2
3
4
5
6
0.025 0.050 0.075 0.100 0.125Notch Root Radius, ρ/d
Stre
ss C
once
ntra
tion
Fact
or
Tors
ion
Bend
ing
2.201.201.04
D/d
Dd
ρ
Stress Concentration Factors
30
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 174 of 14
λσ
λσ
σ σθ
a
r
σr
τrθ
σr
σθ
σθ
τrθ
Hole in a Plate
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 175 of 14
-4
-3
-2
-1
0
1
2
3
4
30 60 90 120 150 180
Angle
σθ
σ
λ = 1
λ = 0
λ = -1
Stresses at the Hole
Stress concentration factor depends on type of loading
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 176 of 14
0
0.5
1.0
1.5
1 2 3 4 5ra
τrθσ
Shear Stresses during Torsion
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Torsion Experiments
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Multiaxial Loading
Uniaxial loading that produces multiaxial stresses at notchesMultiaxial loading that produces uniaxial stresses at notchesMultiaxial loading that produces multiaxial stresses at notches
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 179 of 14
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
Tran
sver
se C
ompr
essi
on S
train
Thickness
100
x
z
y
Thickness Effects
31
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 180 of 14
MX
MY
1 2 3 4
MX
MY
Applied Bending Moments
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 181 of 14
A
B
C
D
A
C
A’
C’
BD
B’ D’
Location
MX
MY
Bending Moments on the Shaft
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 182 of 14
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
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 183 of 14
t1 t2 t3 t4
MX
MT
t1σz
σ1σT
σT
t2σz
σ1
σT
σT
t3σ1 = σz
t4
σT
σ1 = σT
Torsion Loading
Out-of-phase shear loading is needed to produce nonproportional stressing
MX
MT
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 184 of 14
Kt = 3 Kt = 4
Plate and Shell Structures
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0
1
2
3
4
5
6
0.025 0.050 0.075 0.100 0.125Notch root radius,
Stre
ss c
once
ntra
tion
fact
or
Kt BendingKt Torsion
Kf Torsion
Kf Bending
Dd
ρ
ρ
d
= 2.2Dd
Fatigue Notch Factors
32
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 186 of 14
1.0
1.5
2.0
2.5
1.0 1.5 2.0 2.5Experimental Kf
Cal
cula
ted
K f
conservative
non-conservative
Fatigue Notch Factors ( continued )
bendingtorsion
ra1
1K1K Tf
+
−+=
Peterson’s Equation
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Fracture Surfaces in Torsion
Circumferencial Notch
Shoulder Fillet
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1.00 1.50 2.00 2.50 3.000.00
0.25
0.50
0.75
1.00
1.25
1.50
F
λ = −1
λ = 0
λ = 1σ
λσ
σ
λσ
R
a
aR
Stress Intensity Factors
( )meqKCdNda
∆= aFKI πσ∆=
Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 189 of 14
0
20
40
60
80
100
0 2 x 10 6
Cra
ck L
engt
h, m
m
4 x 10 6 6 x 10 6 8 x 10 6
λ = -1 λ = 0 λ = 1
Cycles
Crack Growth From a Hole
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Notches Summary
Uniaxial loading can produce multiaxial stresses at notchesMultiaxial loading can produce uniaxial stresses at notchesMultiaxial stresses are not very important in thin plate and shell structuresMultiaxial stresses are not very important in crack growth
Multiaxial Fatigue