stress concentration - efatigue concentration © 2002-2014 darrell socie, all rights reserved 2 of...
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Professor Darrell F. Socie
© 2004-2014 Darrell Socie, All Rights Reserved
Stress Concentration
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Outline
1. Stress Concentration 2. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and Kf
6. Small Crack Growth 7. Small Notches
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Stress Concentration Factor
Applied stress Local stress
ρ+σ=σ
a21appliedlocal
Inglis Solution 1910
2a
ρ
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Circular Notch
λσ
λσ
σ σ θ
a
r σr
τrθ
σθ
σr
σθ
τrθ
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Stresses
θ
−
+
λ−+
−
λ+=
σσ 2cos431
211
21 242
ar
ar
arr
θ
+
λ−−
+
λ+=
σσθ 2cos31
211
21 42
ar
ar
θ
+
−
λ−−=
στ θ 2sin231
21 24
ar
arr
Independent of size, dependant only on r/a
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Stress Distribution For tension λ = 0 and θ = 90
0
1
2
3
0 1 2 3 4
stre
ss
σσr
σσθ
λσ
λσ
σ σ θ a
r
ra
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Stress Ratio Effects
-4
-3
-2
-1
0
1
2
3
4
30 60 90 120 150 180 Angle
σ θ σ
λ = 1
λ = 0
λ = -1
λσ
λσ
σ σ
stresses around the circumference of a hole
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Elliptical Notches
abaKT
221 =ρ
ρ+=
2a
2b
Sharp Notch: high KT high gradient
Blunt Notch: low KT low gradient
stre
ss
KT
KT
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Stress Concentration in a Bar
1
2
3
4
5
r
Axial, σz
Tangential, σθ
Radial, σr ρ
σ σo
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Bridgeman Analysis (1943)
r
σz
ρ a
σr
r
σz
ρ a
σr
Elastic stress distribution Plastic stress distribution
σys σys
ttancons2
rz =σ−σ
=τ
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Stresses
ρ
−ρ++σ=σ
a2ra2aln1
22
oz
∫ σπ=a
0zz drr2P
ρ
+
ρ
+σπ=2a1ln
a21aP flow
2max
Pmax = Anet σflow CF
CF constraint factor
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Constraint Factors
0 11 1.212 1.384 1.648 1.7320 2.63
2.96∞
CF a /ρ
Pmax = Anet σflow CF
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Effect of Constraint
Stre
ss, σ
z
Strain, εz
Increasing constraint
Higher strength and lower ductility
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Outline
1. Stress Concentration 2. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and Kf
6. Small Crack Growth 7. Small Notches
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Notch Rules Neuber
Glinka
Seeger
n1
22
t KEeSK
σ
σ+σ
=εσ=
n1
22
t Kn11
EdeSK
σ
σ+
+σ
=εσ= ∫
SKK*SKSK
KEeSK
p
t
y
tLimitp
n1
2**2
p
=σ
=
σ
σ+σ
=εσ=
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Define Kσ and Kε after Yielding
S , e
σ , ε
Define: nominal stress, S nominal strain, e
notch stress, σ notch strain, ε
Stress concentration
Strain concentration S
K σ=σ
eK ε
=ε
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Kσ and Kε
Stre
ss (M
Pa)
Strain
KtS
Kte
σ ε
SK σ
=σ
eK ε
=ε
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Stress and Strain Concentration
Kt
Nominal Stress
Stre
ss/S
train
Con
cent
ratio
n
1K →σ
2tKK →ε
First yielding
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Neuber’s Rule St
ress
(MPa
)
Strain
KtS
Kte
σ
ε εσ=eKSK tt
Stress calculated with elastic assumptions
Actual stress
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Neuber’s Rule for Fatigue
2222eKSK tt ε∆σ∆
=∆∆
Stress and strain amplitudes
Elastic nominal stress
E2S
2e ∆
=∆
Substitute and rearrange
22E
2SKt
ε∆σ∆=
∆
The product of stress times strain controls fatigue life
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SN Materials Data
10 1 10 102 103 106 107 105 104
Fatigue Life, Reversals
100
1000
10000
Stre
ss A
mpl
itude
, MP
a
93 steels
17 aluminums
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εN Materials Data
93 steels
17 aluminums
10-4
10-2
10-3
0.1
10
1
Stra
in A
mpl
iture
1 10 102 103 106 107 105 104
Fatigue Life, Reversals
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1 106 107 105 10 102 103 104
Fatigue Life, Reversals
1
10
102
105
104
103
2 2
E K
t ε ∆
σ ∆ or
2 S ∆
2 2 E
ε ∆ σ ∆
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Glinka’s Rule St
ress
(MPa
)
Strain
KtS
Kte
σ
ε
Add in the ”missing” strain energy
εσ= ∫ d2eKSK tt
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Outline
1. Stress Concentration 2. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and Kf
6. Small Crack Growth 7. Small Notches
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A Dilemma
Stress analysis and stress concentration factors are independent of size and are related only to the ratio of the geometric dimensions to the loads
Fatigue is a size dependant phenomena
How do you put the two together ?
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Similitude
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Fatigue of Notches
From Dowling, Mechanical Behavior of Materials, 1999
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Notch Size
Kt Kf Kt
Kf
Large Notch Small Notch
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Microstructure Size
Kt Kf
Kt Kf
Low Strength High Strength
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Stress Gradient
Kt Kf
Kt
Kf
Low Kt High Kt
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Kt vs Kf
0
2
4
6
8
10
0 2 4 6 8 10 Kt
Kf
Kf = Kt
Experiments
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Peterson’s Equation
ρα
+
−+=
1
1K1K tf
mmMPa2070025.08.1
u
σ
=α
0
1
2
3
4
10-4 10-3 10-2 0.1 1 10 102 103
Kf
ρα
No effect when ρ << α
Full effect when ρ >> α
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Pererson’s Constant
1000 500 2000 1500
0.1
0.03
0.7
0.3
Ultimate Strength, MPa
α, m
m
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Static Strength
hole Kt = 2.5
slot Kt = 5
diamond Kt = 20
edge Kt = 20
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1018 Steel Test Data
0
20
40
60
80
100
0 2 4 6 8 10
load
, kN
displacement, mm
edge diamond slot hole
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Notched SN Curve
100
1000
10000
100
Cycles
Stre
ss A
mpl
itude
, MPa
101 102 103 104 105 106 107
Smooth specimen data
Notched specimen Kf
Stress concentrations are not very important at short lives
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Outline
1. Stress Concentration 2. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and Kf
6. Small Crack Growth 7. Small Notches
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Smith - Miller
aK
D13.0l
πσ=
ρ>
lD69.71K
D13.0l
πσ
ρ+=
ρ<
Long crack
Short crack
Kfatigue
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Cracks at Notches
D
a a D + a
KtS S S
a << D a >> D aSKK t π= ( )aDSK +π=
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Stress Intensity Factors
0 0.1 0.2 0.3 0.4 0.5 0.6
1.0
0
3.0
2.0
Da
DSK
aSKK t π=
( )aDSK +π=
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Cracks at Holes
10 1 1 12
Once a crack reaches 10% of the hole radius, it behaves as if the hole was part of the crack
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Outline
1. Stress Concentration 2. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and Kf
6. Small Crack Growth 7. Small Notches
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Crack Growth Data
10-12
10-11
10-10
10-9
10-8
10-7
10-6
1 10 100
Cra
ck G
row
th R
ate,
m/c
ycle
mMPa,K∆
mKCdNda
∆=
∆KTH
Kc
a212.1KTH ππ
σ∆>∆
Nonpropagating cracks
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Frost Data nucleation fracture
Rotating bending Notched bar
Notched plate
1 3 5 7 9 11 13 15
1.0
0.8
0.6
0.4
0.2
0
Kt
S nom
inal
S fat
igue
lim
it
tK1 nonpropagating cracks
Frost, “A Relation Between the Critical Alternating Propagation Stress and Crack Length for Mild Steel” Proceedings of the Institute for Mechanical Engineers, Vol. 173, No. 35, 1959, 811-836
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Significance
DKS th
flπ
∆=
For Kt > 4, the notch acts like a crack with a depth D
Kt does not play a role for sharp notches !
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Specimens with Similar Geometry
Kt = 2.4 Kt = 10.7
Ultimate Strength 780 MPa Yield Strength 660 MPa
25 25
2.5 2.5
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Test Results
1
100
1000
Nom
inal
Stre
ss A
mpl
itude
1 10 100 103 104 105 106 107
Total Fatigue Life, Cycles
Strength Limited
Crack Growth Dominated Fatigue Strength Dominated
Threshold Stress Intensity Dominated
Kt = 2.4 Kt = 10.7
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Outline
1. Stress Concentration 2. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and Kf
6. Small Crack Growth 7. Small Notches
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Small Crack Growth
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Threshold
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Normalized Thresholds
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Growth from Notches
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Cracks at Notches
D a crack tip plastic zone
notch plastic zone
notch stress field
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Crack Growth
∆K or a
Cra
ck g
row
th ra
te
Long cracks Crack tip plasticity controls
Short cracks Notch plasticity controls
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Closure Observations
a b
d c c 1st subcycle
d 500th subcycle
σ
ε
a
b
1026 steel
∆ε1/2 = 0.005
∆ε2/2 = 0.001
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Closure Correlation
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Outline
1. Stress Concentration 2. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and Kf
6. Small Crack Growth 7. Small Notches
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area
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Small Notches
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Threshold Stress Intensity
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Hardness Corelation
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Flaw Sensitivity
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Fatigue Limit
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Stress Concentration