3 multiaxial fatigue seminar - university of illinois3 multiaxial fatigue ©2001-2005 darrell socie,...
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Professor Darrell F. SocieDepartment of Mechanical and Industrial Engineering
University of Illinois at Urbana-Champaign
© 2001-2005 Darrell Socie, All Rights Reserved
Multiaxial Fatigue
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Contact Information
Darrell SocieMechanical Engineering1206 West GreenUrbana, Illinois 61801USA
[email protected]: 217 333 7630Fax: 217 333 5634
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Outline
n State of Stress ( Chapter 1 )
n Fatigue Mechanisms ( Chapter 3 )
n Stress Based Models ( Chapter 5 )
n Strain Based Models ( Chapter 6 )
n Fracture Mechanics Models ( Chapter 7 )
nNonproportional Loading ( Chapter 8 )
nNotches ( Chapter 9 )
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State of Stress
n Stress componentsnCommon states of stressn Shear stresses
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Stress Components
X
Z
Y
σz
σy
σx
τzyτzx
τxz
τxyτyx
τyz
Six stresses and six strains
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Stresses Acting on a PlaneZ
Y
X
Y’
X’Z’
σx’
τx’z’
τx’y’
θ
φ
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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
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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
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Tension
τ
σσx
γ/2
εε1
σ1 σ2
σ3
σ2 = σ3 = 0
σ1
ε2 = ε3 = −νε1
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Torsion
σ2 τ
στxyε1
ε3
ε2
γ/2
ε
σ1
σ3
X
Yσ2
σ1 = τxy
σ3
σ1
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τ
σ
γ /2
ε
σ2 = σy
σ1 = σx
σ3
σ1 = σ2
σ3
ε1 = ε2
εν−ν−=ε
12
3
Biaxial Tension
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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
oct2
3τ=σMises: 1313oct 94.0
223
τ=τ=τ
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State of Stress Summary
n Stresses acting on a planen Principal stressnMaximum shear stressnOctahedral shear stress
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Fatigue Mechanisms
nCrack nucleationn Fracture modesnCrack growthn State of stress effects
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Crack Nucleation
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Slip Bands
Loading Unloading
Extrusion
Undeformedmaterial
Intrusion
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Slip Bands
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Mode Iopening
Mode IIin-plane shear
Mode IIIout-of-plane shear
Mode I, Mode II, and Mode III
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Stage I Stage II
loading direction
freesurface
Stage I and Stage II
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Case A and Case B
Growth along the surface Growth into the surface
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5 µmcrac
k gr
owth
dire
ctio
n
Mode I Growth
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crack growth direction
10 µm
slip bandsshear stress
Mode II Growth
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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
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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
Frac
tion
N/N
f
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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
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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
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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
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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
Frac
tion
N/N
f
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Fatigue Mechanisms Summary
n Fatigue cracks nucleate in shearn Fatigue cracks grow in either shear or tension
depending on material and state of stress
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Stress Based Models
n Sinesn FindleynDang 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
fatig
ue li
mit
Shear stress in torsion
1/2 Bending fatigue limit
Bending Torsion Correlation
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Test Results
nCyclic tension with static tensionnCyclic torsion with static torsionnCyclic tension with static torsionnCyclic 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
sF
atig
ue s
tren
gth
Mean stressYield strength
Cyclic Tension with Static Tension
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1.5
1.0
0.5
1.51.00.50
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
engt
h
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
Tor
sion
she
ar s
tres
s
She
ar fa
tigue
str
engt
h
Axial mean stress
Yield strength
Cyclic Torsion with Static Tension
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Conclusions
n Tension mean stress affects both tension and torsionn Torsion mean stress does not affect tension
or torsion
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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
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Findley
∆τ2
+
=k nσ
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
fatig
ue li
mit
Shear stress in torsion
1/2 Bending fatigue limit
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Dang Van
τ σ( ) ( )t a t bh+ =
m
V(M)
Σij(M,t) E ij(M,t)
σij(m,t)εij(m,t)
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τ
σh
τ
σh
Loading path
Failurepredicted
τ(t) + aσh(t) = b
Dang Van ( continued )
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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
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Strain Based Models
n Plastic Workn Brown and Millern Fatemi and Socien Smith Watson and Toppern Liu
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10 10 2 10 3 10 4 10 50.001
0.01
0.1
Cycles to failure
Pla
stic
oct
ahed
ral
shea
r st
rain
ran
ge Torsion
Tension
Octahedral Shear Strain
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1
10
100
102 103 104
A
Fatigue Life, Nf
Pla
stic
Wor
k pe
r Cyc
le, M
J/m
3
T TorsionAxial0o
90o
180 o
135 o
45o
30o
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
Fatig
ue L
ife, C
ycle
s
Normal Strain Amplitude, ∆εn
∆γ = 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 )
( )∆ ∆γ ∆ε$maxγ α α α= + S n
1
∆γ∆εmax
', '( ) ( )
2
22 2+ =
−+S A
EN B Nn
f n meanf
bf f
cσ σ
ε
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Fatemi and Socie
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C F G
H I J
γ / 3
ε
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
Leng
th, m
m
Crack Length Observations
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Fatemi and Socie
cof
'f
bof
'f
y
max,n )N2()N2(G
k12
γ+τ
=
σσ
+γ∆
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Smith Watson Topper
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SWT
cbf
'f
'f
b2f
2'f1
n )N2()N2(E2
+εσ+σ
=ε∆
σ
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Liu
∆WI = (∆σn ∆εn)max + (∆τ ∆γ)
b2f
2'fcb
f'f
'fI )N2(
E4
)N2(4Wσ
+εσ=∆ +
∆WII = (∆σn ∆εn ) + (∆τ ∆γ)max
bo2f
2'fcobo
f'f
'fII )N2(
G4
)N2(4Wτ
+γτ=∆ +
Virtual strain energy for both mode I and mode II cracking
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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 ∆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
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Conclusions
n All critical plane models correctly predict these resultsnHydrostatic stress models can not predict
these results
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0.006Axial strain
-0.003
0.003
She
ar s
trai
n
Loading History
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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
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Strain Based Models Summary
n Two separate models are needed, one for tensile growth and one for shear growthnCyclic plasticity governs stress and strain
rangesnMean stress effects are a result of crack
closure on the critical plane
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Separate Tensile and Shear Models
τ
σ
σ2
σ1 = τxy
σ3
σ1
Inconel 1045 steel stainless steel
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Cyclic Plasticity
∆ε∆γ∆εp
∆γp
∆ε∆σ∆γ∆τ∆εp∆σ∆γp∆τ
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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(
E
2)S7.03.1(S
2ε++
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
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Fracture Mechanics Models
nMode I growthn TorsionnMode II growthnMode III growth
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Mode I, Mode II, and Mode IIIMode Iopening
Mode IIin-plane shear
Mode IIIout-of-plane shear
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σ
σ
σ σ
Mode I
Mode II
Mode I and Mode II Surface Cracks
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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
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Mode II
Mode III
Surface Cracks in Torsion
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Transverse Longitudinal Spiral
Failure Modes in Torsion
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1500
1000
Yie
ld S
tren
gth,
MP
a
40
30
50
60
Har
dnes
s R
c200 300 400 500
Shear Stress Amplitude, MPa
No Cracks
SpiralCracks
Transverse Cracks
Longi-tudinal
Fracture Mechanism Map
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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
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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
79
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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()(K
5.02
I2
IIeq π
ε∆+γ∆
ν+=ε∆
( ) ak1GFKys
max,neq π
σσ
+γ∆=ε∆
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Fracture Surfaces
Bending Torsion
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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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Fracture Mechanics Models Summary
nMultiaxial loading has little effect in Mode InCrack closure makes Mode II and Mode III
calculations difficult
83
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Nonproportional Loading
n In and Out-of-phase loadingnNonproportional cyclic hardeningn Variable amplitude
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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
85
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θε1
γ xy
22τxy
σx
εx
∆σx
∆εx
∆γxy
2∆2τxy
In-Phase and Out-of-Phase
86
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εx εx
εx εx
γxy/2 γxy/2
γxy/2γxy/2 cross
diamondout-of-phase
square
Loading Histories
87
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in-phase
out-of-phase
diamond
square
cross
Loading Histories
88
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Findley Model Results
∆τ/2 MPa σn,maxMPa ∆τ/2 + 0.3 σ
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
εx
γxy/2cross
εx
γxy/2diamondout-of-phase
εx
γxy/2square
in-phase
εx
γxy/2
89
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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)
90
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600
-600
300
-300
-0.003 -0.0060.003 0.006
Axial Shear
In-Phase
91
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Axial
600
-600
-0.003 0.003
Shear
-300
300
0.006-0.006
90° Out-of-Phase
92
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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
93
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0 1 2 3 4
5 6 7 8 9
1011 12 13
Loading Histories
94
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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
She
ar S
tres
s (
MP
a )
Stress-Strain Response
95
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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
She
ar S
tress
( M
Pa
)
Axial Stress ( MPa )
96
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1000
200102 103 104
Equ
ival
ent S
tres
s, M
Pa
2000
Fatigue Life, Nf
Maximum Stress
Nonproportional hardening results in lower fatigue lives
All tests have the same strain ranges
97
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Case A Case B Case C Case D
σx σx σx σx
σy σy σy σy
Nonproportional Example
98
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Case A Case B Case C Case D
τxy τxy τxy τxy
Shear Stresses
99
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-0.003 0.003
-0.006
0.006γ xy
-300 300
σx
-150
150
εx
τ xy
Simple Variable Amplitude History
100
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-0.003 0.003εx
-300
300σ x
-0.005 0.005γxy
-150
150τ xy
Stress-Strain on 0° Plane
101
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-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
102
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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
103
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time
-0.005
0.005
She
ar s
trai
n, γ
Shear Strain History on Critical Plane
104
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Fatigue Calculations
Load or strain history
Cyclic plasticity model
Stress and strain tensor
Search for critical plane
105
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Nonproportional Loading Summary
nNonproportional cyclic hardening increases stress levelsnCritical plane models are used to assess
fatigue damage
106
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Notches
n Stress and strain concentrationsnNonproportional loading and stressingn Fatigue notch factorsnCracks at notches
107
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τθzσθ
σz
MTMX
MY
P
Notched Shaft Loading
108
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0
1
2
3
4
5
6
0.025 0.050 0.075 0.100 0.125Notch Root Radius, ρ/d
Str
ess
Con
cent
ratio
n F
acto
r
Tor
sion
Ben
ding
2.201.201.04
D/d
Dd
ρ
Stress Concentration Factors
109
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λσ
λσ
σ σθ
a
r
σr
τrθ
σr
σθ
σθ
τrθ
Hole in a Plate
110
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-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
111
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0
0.5
1.0
1.5
1 2 3 4 5ra
τrθ
σ
Shear Stresses during Torsion
112
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Torsion Experiments
113
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Multiaxial Loading
nUniaxial loading that produces multiaxial stresses at notchesnMultiaxial loading that produces uniaxial
stresses at notchesnMultiaxial loading that produces multiaxial
stresses at notches
114
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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
115
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MX
MY
1 2 3 4
MX
MY
Applied Bending Moments
116
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A
B
C
D
A
C
A’
C’
BD
B’ D’
Location
MX
MY
Bending Moments on the Shaft
117
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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
118
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t1 t2 t3 t4
MX
MT
t1
σz
σ1σT
σT
t2
σz
σ1
σT
σT
t3
σ1 = σz
t4
σT
σ1 = σT
Torsion Loading
Out-of-phase shear loading is needed to produce nonproportional stressing
MX
MT
119
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Kt = 3 Kt = 4
Plate and Shell Structures
120
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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 BendingKt Torsion
Kf Torsion
Kf Bending
Dd
ρ
ρ
d
= 2.2Dd
Fatigue Notch Factors
121
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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
122
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Fracture Surfaces in Torsion
Circumferencial Notch
Shoulder Fillet
123
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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
∆= aFK I πσ∆=
124
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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
125
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Notches Summary
nUniaxial loading can produce multiaxial stresses at notchesnMultiaxial loading can produce uniaxial
stresses at notchesnMultiaxial stresses are not very important in
thin plate and shell structuresnMultiaxial stresses are not very important in
crack growth
126
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Final Summary
n Fatigue is a planar process involving the growth of cracks on many size scalesnCritical plane models provide reasonable
estimates of fatigue damage
127
University of Illinois at Urbana-Champaign
Multiaxial Fatigue