Professor Darrell F. Socie Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign © 2001-2011 Darrell Socie, All Rights Reserved Multiaxial Fatigue
Professor Darrell F. Socie Department of Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
© 2001-2011 Darrell Socie, All Rights Reserved
Multiaxial Fatigue
Multiaxial Fatigue © 2001-2011 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 125
Contact Information
Darrell Socie Mechanical Science and Engineering 1206 West Green Urbana, Illinois 61801 USA [email protected] Tel: 217 333 7630 Fax: 217 333 5634
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Outline
State of Stress ( Chapter 1 )
Fatigue Mechanisms ( Chapter 3 )
Stress Based Models ( Chapter 5 )
Strain Based Models ( Chapter 6 )
Fracture Mechanics Models ( Chapter 7 )
Nonproportional Loading ( Chapter 8 )
Notches ( Chapter 9 )
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State of Stress
Stress components Common states of stress 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 Plane Z
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
τ12 τ23
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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
−=ε12
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
oct23
τ=σMises: 1313oct 94.022
3τ=τ=τ
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State of Stress Summary
Stresses acting on a plane Principal stress Maximum shear stress Octahedral shear stress
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Fatigue Mechanisms
Crack nucleation Fracture modes Crack growth State of stress effects
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Crack Nucleation
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Slip Bands
Loading Unloading
Extrusion
Undeformed material
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
free surface
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 bands shear stress
Mode II Growth
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1.0
0.2
0
0.4
0.8
0.6
1 10 10 2 10 3 10 4 10 5 10 6 10 7
Nucleation
Tension Shear
304 Stainless Steel - Torsion
Fatigue Life, 2N f
Dam
age
Frac
tion
N/N
f
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304 Stainless Steel - Tension
1 10 10 2 10 3 10 4 10 5 10 6 10 7
1.0
0.2
0
0.4
0.8
0.6 Nucleation
Tension
Fatigue Life, 2N f
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 10 2 10 3 10 4 10 5 10 6 10 7
Inconel 718 - Torsion
Fatigue Life, 2N f
Dam
age
Frac
tion
N/N
f
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Inconel 718 - Tension
Shear
Tension
Nucleation
1 10 10 2 10 3 10 4 10 5 10 6 10 7
1.0
0.2
0
0.4
0.8
0.6
Fatigue Life, 2N f
Dam
age
Frac
tion
N/N
f
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Fatigue Life, 2N f
Dam
age
Frac
tion
N/N
f f
Nucleation
Shear
Tension
1 10 10 2 10 3 10 4 10 5 10 6 10 7
1.0
0.2
0
0.4
0.8
0.6
1045 Steel - Torsion
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1045 Steel - Tension
Nucleation Shear
Tension
1.0
0.2
0
0.4
0.8
0.6
1 10 10 2 10 3 10 4 10 5 10 6 10 7
Fatigue Life, 2N f
Dam
age
Frac
tion
N/N
f
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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
0 0 0.5 1.0 1.5 2.0
Shear stress Octahedral stress
Principal stress
0.5
She
ar s
tress
in b
endi
ng
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
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 1.0 0.5 -0.5 0
Axi
al s
tress
Fa
tigue
stre
ngth
Mean stress Yield strength
Cyclic Tension with Static Tension
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1.5
1.0
0.5
1.5 1.0 0.5 0
She
ar S
tress
Am
plitu
de
She
ar F
atig
ue S
treng
th
Maximum Shear Stress Shear Yield Strength
Cyclic Torsion with Static Torsion
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1.5
1.0
0.5
3.0 0 2.0 1.0
Ben
ding
Stre
ss
Ben
ding
Fat
igue
Stre
ngth
Static Torsion Stress Torsion Yield Strength
Cyclic Tension with Static Torsion
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1.5
1.0
0.5
1.5 -1.5 -1.0 1.0 0.5 -0.5 0
Tors
ion
shea
r stre
ss
She
ar fa
tigue
stre
ngth
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
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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
0 0 0.5 1.0 1.5 2.0
Shear stress Octahedral stress
Principal stress
0.5
She
ar s
tress
in b
endi
ng
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)
Σ ij (M,t) E ij (M,t)
σ ij (m,t) ε ij (m,t)
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τ
σh
τ
σh
Loading path
Failure predicted
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
Plastic Work Brown and Miller Fatemi and Socie Smith Watson and Topper Liu
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10 10 2 10 3 10 4 10 5 0.001
0.01
0.1
Cycles to failure
Pla
stic
oct
ahed
ral
shea
r stra
in ra
nge 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 Torsion
Axial 0
90
180
135
45
30
T
A T
T T
T T
T
A
A
A A
A
Plastic Work
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0.005 0.010.0
102
2 x102
5 x102
103
2 x103Fa
tigue
Life
, Cyc
les
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
', '( ) ( )
22
2 2+ =−
+S AE
N B Nnf n mean
fb
f fc
σ σε
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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 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
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(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
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Cyclic Torsion
Cyclic Shear Strain Cyclic Tensile Strain
Shear Damage Tensile Damage
Cyclic Torsion
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Cyclic Torsion Static 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 Damage Shear Damage Cyclic Torsion Static Compression
Cyclic Torsion with Compression
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Cyclic Torsion Static Compression
Hoop Tension
Cyclic Shear Strain Cyclic Tensile Strain
Tensile Damage Shear 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
All critical plane models correctly predict these results
Hydrostatic stress models can not predict these results
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0.006 Axial strain
-0.003
0.003 S
hear
stra
in
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
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
τ
σ
σ 2
σ 1 = τ xy
σ 3
σ 1
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(
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
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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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σ
σ
σ σ
Mode I
Mode II
Mode I and Mode II Surface Cracks
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λ = -1 λ = 0 λ = 1
λ = -1 λ = 0 λ = 1
∆σ = 193 MPa ∆σ = 386 MPa 10-3
10-4
10-5
10-6
da/d
N m
m/c
ycle
10 100 200 20 50 10 100 200 20 50 ∆K, 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
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Transverse Longitudinal Spiral
Failure Modes in Torsion
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1500
1000
Yiel
d S
treng
th, M
Pa
40
30
50
60
Har
dnes
s R
c
200 300 400 500 Shear Stress Amplitude, MPa
No Cracks
Spiral Cracks
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 10 100 50 20 ∆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 = 0 Mode II R = -1
7075 T6 Aluminum
Mode I R = -1 Mode II R = -1
SNCM Steel
Mode I and Mode II Growth
∆KI , ∆KII MPa
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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()(K5.0
2I
2IIeq 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 1.0 Crack 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
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Fracture Mechanics Models Summary
Multiaxial loading has little effect in Mode I Crack closure makes Mode II and Mode III
calculations difficult
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Nonproportional Loading
In and Out-of-phase loading Nonproportional cyclic hardening Variable amplitude
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εx γxy
εy
t
t
t
t εx = 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 ν 1 +
∆γ
γ ν 1 +
In and Out-of-Phase Loading
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θ ε 1
γ xy
2 2 τ xy
σ x
ε x
∆σ x
∆ε x
∆γ xy
2 ∆2τ xy
In-Phase and Out-of-Phase
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εx εx
εx εx
γxy/2 γxy/2
γxy/2γxy/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
∆τ/2 MPa σ n,max MPa ∆τ/2 + 0.3 σ
n,max N/N ip
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
ε x
γ xy /2 cross
ε x
γ xy /2 diamond out-of-phase
ε x
γ xy /2 square
in-phase
ε x
γ xy /2
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Nonproportional Hardening
t
t
t
t εx = εosin(ωt)
γxy = (1+ν)εosin(ωt)
In-phase
Out-of-phase
εx
εx
γxy
γxy
εx = εocos(ωt)
γxy = (1+ν)εosin(ωt)
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600
-600
300
-300
-0.003 -0.006 0.003 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 = 38,500 Nf = 310,000
Nf = 3,500 Nf = 40,000
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0 1 2 3 4
5 6 7 8 9
10 11 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
Shea
r Stre
ss (
MPa
)Stress-Strain Response
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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
Shea
r Stre
ss (
MPa
)
Axial Stress ( MPa )
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1000
200 10 2 10 3 10 4
Equ
ival
ent S
tress
, MP
a
2000
Fatigue Life, N f
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 σ x σ x σ x σ x
σ y σ y σ y σ y
Nonproportional Example
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Case A Case B Case C Case D
τxy τxy τxy τxy
Shear Stresses
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-0.003 0.003
-0.006
0.006
-300 300
σx
-150
150
εx
Simple Variable Amplitude History
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-0.003 0.003 εx
-300
300
-0.005 0.005 γxy
-150
150
τ xy
Stress-Strain on 0° Plane
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-0.003 0.003 ε30
-300
300
-0.003 0.003
-300
300 30º plane 60º plane
ε60
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
300 150º plane 120º plane
ε120 ε150
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time
-0.005
0.005
She
ar s
train
, γ
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
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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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Notches
Stress and strain concentrations Nonproportional loading and stressing Fatigue notch factors Cracks at notches
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τ θ z σ θ
σ z
M T M X
M Y
P
Notched Shaft Loading
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0
1
2
3
4
5
6
0.025 0.050 0.075 0.100 0.125 Notch Root Radius, ρ/d
Stre
ss C
once
ntra
tion
Fact
or
Tors
ion
Ben
ding
2.20 1.20 1.04
D/d
D d
ρ
Stress Concentration Factors
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λσ
λσ
σ σ θ
a
r
σr
τrθ
σr
σθ
σθ
τrθ
Hole in a Plate
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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
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0
0.5
1.0
1.5
1 2 3 4 5 r a
τrθ σ
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
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0.004 0.008 0.012 0
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
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M X
M Y
1 2 3 4
M X
M Y
Applied Bending Moments
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A
B
C
D
A
C
A’
C’
B D
B’ D’
Location
MX
MY
Bending Moments on the Shaft
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Bending Moments
∆ M A B C D 2.82 1 1 2.00 3 2 1.41 2 1 1.00 2 0.71 2
∆ ∆M M= ∑ 55
A B C D ∆ M 2.49 2.85 2.31 2.84
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t 1 t 2 t 3 t 4
M X
M T
t 1
σ z
σ 1 σ T
σ T
t 2
σ z
σ 1
σ T
σ T
t 3
σ 1 = σ z
t 4
σ T
σ 1 = σ T
Torsion Loading
Out-of-phase shear loading is needed to produce nonproportional stressing
MX
MT
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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 0.125 Notch root radius,
Stre
ss c
once
ntra
tion
fact
or
Kt Bending Kt Torsion
Kf Torsion
Kf Bending
D d
ρ
ρ d
= 2.2 D d
Fatigue Notch Factors
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1.0
1.5
2.0
2.5
1.0 1.5 2.0 2.5 Experimental Kf
Cal
cula
ted
Kf conservative
non-conservative
Fatigue Notch Factors ( continued )
bending torsion
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 3.00 0.00
0.25
0.50
0.75
1.00
1.25
1.50
F
λ = −1
λ = 0
λ = 1 σ
λσ
σ
λσ
R
a
a R
Stress Intensity Factors
( )meqKCdNda
∆= aFKI πσ∆=
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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
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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
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Final Summary
Fatigue is a planar process involving the growth of cracks on many size scales
Critical plane models provide reasonable estimates of fatigue damage