6. Fatigue strength
Fatigue fracture
Under a constant cyclic loading or irregular cyclic loading, What is fatigue fracture?
Component fractures
◎ Stress level
Under low stress level below yield stress, fatigue fracture happens
Static fracture
13%
Corrosion, Rupture
3%
Delayed fracture
Stress corrosion 5%
Thermal fatigueCorrosion fatigue
Fretting fatigue
11% Fatigue60%
Low cycle fatigue
8%
◎ Fracture cause
80-90% of fracture is by fatigue
◎ How to fracture
Under cyclic loading, componentsuddenly fractures
Fatigue and fatigue fracture surface
(1)Origin …
(2)Crack growth …
(3)Macroscopic …
(4)Microscopic …
(5)Other …
Surface of componentStress concentration (Notch, Key, Inclusion)
After crack initiation, along max. stress plane
Smooth surface, macroscopically few plasticdeformation
Beach mark (under irregular cyclic stress)
Striation
When crack grows, cross section decreases
Ductile fracture
Rough surface
Fatigue fracture
Fatigue fracture factor
time
Stre
ss
Tens
ion(
+)
Com
pres
sion
(-)
Main factor
(1) Max. tensile stress
(2) Cyclic stress
(3) Number of cyclic stress
Large
・ Stress concentration
・ Environment
・ Residual stress
・ Metallurgical
・ Combined stress
・ Over loading
◎ other causes
Cyclic stress I
(b)Partial alter
σm< σa‐1 < R < 0
(c)Pulsating
R = 0σm= σa
(d)Partial Pulsating
σm> σa
R > 0
σmin
σm
σmax
σa
σm : Mean stress
σa : Stress amplitude
R : Stress ratio
2minmax σσ
σ+
=m
2minmax σσ
σ−
=a
max
min
σ
σ=R
(a)Alternating
σm= 0R = ‐1
Tens
ion(
+)
Com
pres
sion
(-)
Stre
ss
Time
Definition of cyclic stress
Cyclic stress test
Alternating
σm= 0R = ‐1
Tens
ion(
+)
Com
pres
sion
(-)
Stre
ss
Time
Rotating fatigue tester
A point on surface of sample
Upper
Side
Lower
Side
Upper
Compression
Tension
0
0
Compression
1cy
cle
Rotating fatigue tester
Specimen
N1
σ1 P
What does P mean
σa=σ1 cyclically applies
Specimen fails N = N1
SーN curve I
Aluminum alloy FailureNon failure
Mildsteel
105 106 107 1080
100
200
300
Cyclic number to failure N
Stre
ss a
mpl
itude
σa
[M
Pa]
S-N curves of almimum alloy and mild steel
Rotaing bendingσm=0 (R=‐1)
Fundamental diagram to evaluate fatigue property
Cyclic stress(stress amplitude σa)- Cyclic number to failure
Nominal stress。 Fatigue lifelog Nf
SーN curve II
Aluminumalloy
Mild steel
Cyclic number to failure N(S-N curves aluminum alloy and mild steel)
FailureNon failure
105 106 107 1080
100
200
300
Stre
ss a
plitu
deσ
a[
MPa
]
Rotating bendingσm=0 (R=‐1)
Fatig
ue li
mit
Fatigue limit … Clear knee point (Mild steel, Titanium, Carbon steel)⇒ Over 107 cycle, fatigue life = ∞
(Super long life region, σa decreases)
Mild steel
Fatigue limit = 162MPa
Fatig
ue st
reng
th a
t107
107cycles fatiguestrength …
Not clear knee point(Non iron metals)⇒ Fatigue life is not ∞
at N=107cycles , fatigue strength
Aluminum alloy
107fatigue strength= 135MPa
SーN curve Ⅲ
(Extremely Low Cycle Fatigue)
(Low Cycle Fatigue)
(High Cycle Fatigue)
Hysteresis loop σa ; High level
Plastic deformation
Elastic regioσa ; Elastic stress
σa
Cyclic number to failureNf
101 102 103 104 105 106 107
PーSーN curve
N1
P=0.01
N0
P = 0.5(S‐N curve)
P=0.50
50%
σ1
Cyclic number to failure log N
Stre
ss a
mpl
itude
σa
N2
P-S-N curve
P=0.90P=0.99
10% P=0.10
Fatigueprobability
Unevenness of fatigue life
Material, geometry, stress ratio, stress amplitude are constant
Life differs 10 times
N
PN
PN ; Fracture probability at N
Area ; Fracture probability till N
Fatigue limit and mechanical prpertiesⅠMechanical properties
・ Yield strength σS
・ Tensile strength σB
・ Brinnel HB
・ Vickers HV
Fatigue limit σw0
Vickers hardness
SampleDiamond probe Probe
d1
d2
21
22cos2 oaddd
WaSurfaceare
PLHV °==
][ 2
21
/ 854.1 mmkgfdd
W=
(Rotating bending )
◎ Aluminum alloy …
◎ Cupper alloy … Bw σσ 25.00 ≅Bw σσ 33.00 ≅
◎ Iron and steel … Bw σσ 5.00 ≅ ( ) Vw H1.06.10 ±≅σ,
Fatigue limit and mechanical propertiesⅡFatigue limits and static strength of iron and steel
Ratio of tensile
strength (/σB)
Ratio of Vickers (/HV)
Alternating torsion 0.32 0.10
Alternating tension
compression 0.43 0.15
Pulsating tension 0.33 0.11
Plane bending 0.52 0.18
Fatigue limits and static strength of iron and steel
Low cycle fatigue
(Extremely Low Cycle Fatigue)
(Low Cycle Fatigue) Hysteresis loopσa ; High level
(Plastic deformation)
σa
Cyclic number to failureNf
101 102 103 104 105 106 107
Under high temperature,Machine
Cyclic thermal strain
・ Nuclear vessel・ Steam turbine
Short fatigue life
Hysteresis loopⅠ
σa
BHigh loading
AYielding
C
Yielding
(Bauschinger effect )Δσ
εa
σa
ΔεCompression
εpa
σm
εm
D
E
Δεp
Tensile strain
εea
εa
Unloading
Compressionεpa
0
Stre
ssσ
Strain ε
Hysteresis loop
Hysteresis loopⅡ
Δσ
εa
εpa
εa
σm
σa
Δε
εea
εm
D
E
0
Stre
ssσ
Strainε
A
σa
B
εpa
C
Δεp
Hysteresis loop
εpa ; Plastic strain
Δεt ; Total strain range
εea ; Elastic strain
εm ; Mean strain
Δεp ; Plastic strain range
Δεe ; Elastic strain range
ppet EΔε
ΔσΔεΔεΔε +=+=
Area of hysteresis loop = Plastic work/ volume
Low cycle fatigue ⇒ Transfer to heat ⇒ Low speed
Hysteresis loop Ⅲ
Static stress-strain curve
Cyclic stress-strain curve'
2'
2
n
K ⎟⎠⎞
⎜⎝⎛=ΔεΔσ
Δσ ; Cyclic Stress
K’ ;
n’ ; Cyclic hardening index
( General n’≒ 0.15)
Stress range changes with increasing N
・ Annealed steel Δσ increase・ Cold rolling steel Δσdecrease
Till 50% of lifeShape of hysteresis loop saturates
Stre
ss
Δσ
Strain
Δε
Cyclic stress-strain curve
Strain range and fatigue life
103 104102
101
100
10510-3
10-1
10-2
100 101
10-1
Cyclic number to failure Nf
Plas
tic st
rain
ran
geΔε
p
Manson-Coffin law
41=fN
fε2
15.0≈b
CN bfp =Δε
Manson-Coffin relation
Relation between Δεpand Nf of low cycle fatigue
CN bfp =Δε …(式 6.15)
b,C ; Constant(For many materials,b≒0.5)
◎ Nf =1/4 cycle , Δεp=2εf
⎟⎠⎞
⎜⎝⎛
−=⎟
⎟⎠
⎞⎜⎜⎝
⎛=
φε
100100lnln 0
ff A
A◎
A0 ; Cross section beforeA ; Cross section afterφ ; Reduction of areaεf ; Failure ductile
Microscopic fracture appearanceSu
rfac
e
Cyclic stress
Surf
ace
Crack initiation, First stage of crack growth
(Ⅰ)First stage of crack
growth
Enlargement
・ Aluminum alloy…Crack initiation continuously relates to growth
・ Steel, Titanium …Crack size is similar to grain size
Extrusion
Intrusion
Slip band
Microscopic fracture appearanceⅡ
Stage IIa of crack growth process
(Ⅱa)Stage II crack growth
(Ⅰ)
Direction of crack growth
Surf
ace
Cyclic stress
試験片表面
Small crack ⇒ Grow in grain(along slip plane)
Stress concentration gives riseto damage at crack tip
Continuous
Crack growth rate
dNdahrateCrackgrowt =
(a ; crack length、N ; Cyclic number
Crack tip
Intergranular ⇒ DelayGranular ⇒ High
Microscopic fracture appearanceⅢStage IIb of crack growth process
Crack growth direction
Surf
ace
Cyclic stress
試験片表面
(Ⅱa)(Ⅰ) (Ⅱb)Stage II of crack
growth
Microscopic structure effect
Mechanics factor(Stress intensity factor)
transfer
(Striation)
Pure Titanium
cyclemdNda /10分の数μ=
Striation spacing⇒ crack growth rate
Microscopic fracture appearanceⅣStage IIc of crack growth process
(Ⅱa)(Ⅰ) (Ⅱb)
Crack growth direction
Surf
ace
Cyclic stress
(Ⅱc)Stage II of crack growth
Striation
High crack growth rate(High strength steel
⇒Cleavage, intergranularcracking )
Final fracture
Ductile fracture
Crack growth lawⅠLinear fracture mechanics
Small yielding condition ⇒Application to fatigue crack
At crack tip, the same fracture happens
σ
ρ=0
Plastic zone
(b)Same Plastic elastic stress
σ
a1a2
KⅠ1 KⅠ2=
(a)Same elastic stress field
For different crack length, the same stress intensity factor
Elastic stress and elastic plastic stress becomes the same
Crack growth properies
FaKKK ⋅=−= πΔσΔ minmax
Stress intensity factor range ΔK
FaK ⋅= πσmaxmax FaK ⋅= πσminmin,
Crack growth rateⅡ
Δσ
2
Δσ
1
Stre
ss r
ange
Time t(a)
き裂長さ a1
き裂長さ a2 For long a ,
S. I. F. KDriving force Large
(b)
S.I.F
.ran
geΔ
K
Time t
Stress intensity factor range,ΔK, change
aK ∝Δ
Stress ratio = 0
Δσ ; Cyclic stress,a ;
ΔK
1=Δ
K2
Δσ1 >Δσ2
2211 aa ΔσΔσ =
21 KK ΔΔ =⇒
Crack growth rate Ⅲ(Paris law)
・ Resistance of crack growth・ Fatigue life estimation
Threshold
ΔKth ; Threshold S.I.F.
ΔK decrease ⇒ da/dN → 0Lower limit of crack growth
Final failure
Before fractureΔK
(R ; stress ratio,Kfc ; Fatigue fracture toughness)
( )RKK fc −= 1Δ
Stress intensity factor log(ΔK)
Cra
ck g
row
th r
ate
log(
da/d
N)
m
1
Steady growth Paris law
( )mKCdNda
Δ=
C, m ; material constantFor many materilas, m = 2~7
Notch effectⅠ(Notch)
・ Stress concentration at notch root・ Fast crack growth
◎ Notched component⇒ few data of fatigue
Cross section suddenly changes
Hole、 Screw、Key、Defect etc.
(Notch)Origin of crack
Fracture
◎ How to evaluate stress concentration⇒ FEM
Decrease of fatigue strength
凹凸
Non propagating crack
Notch effectⅡ(Fatigue limit of notched material)Fa
tigue
lim
itσ
w
Stress concentration Kt
Fatigue limit of notched material
ρ
Rotating bendA
CFatigue strength σw1
σw0
① Fatigue strengthσw1
For Smooth specimen,Limit stress not to initiate crack
B D
Crack strength σw2
Branch ρ=ρ0
② Crack strength σw2
Fracture stress to occur non-propagatingcrack
Fatigue limit (Two types)
Fatigue limit of notched material
Crack initiates, but not fracture
Notch effectⅢ(Fatigue notch factor Kf)
ρ
Rotating bending
Non-propagting crack
Fatig
ue li
mit
σw
Stress concentration Kt
Fatigue limit of notched specimen
A
CFatigue strength σw1
σw0
B D
Crack strength σw2
Branchρ=ρ0
Branch point B
Material constantDependence on ρ0
① Fatigue strength σw1
ρ>ρ0 ; No non-propagating crack② Crack strength σw2
ρ<ρ0 ; Non-propagating crack
How much is Fatigue limit decreasedby the notch
Fatigue limit of smooth specimen σw0
1
01
w
wfK
σ
σ=
2
02
w
wfK
σ
σ=,
Fatigue notch factor Kf
Fatigue limit and stress concentration
Kt=Kf
Stress concentration Kt
1 2 3 40
0.5
1.0
Fatig
ue li
mit
σw
1/ σ
w0
, σw
2/ σ
w0 S30C
ρ
Rotating bending(d=5mm)
td
t = 0.5mm
Notch depth t = 0.1mm
Different notch depth⇒ Kt of branch point differs
⇒ Kt of branch point differsDifferent diameter
Branch point0
1
w
w
σ
σ
0
2
w
w
σ
σ
・ Fatigue strength σw1
・ Crack strength σw2
0
1
w
w
σ
σ
0
2
w
w
σ
σ
Kt / Kf1
Kt / Kf2
1/ρand Kt/KfⅠ
0 2010
1.0
2.0
0
1/ρ [mm-1]
Kt /
Kf1
, Kt /
Kf2
Relation between Kt / Kf1, Kt / Kf2 and 1/ρ
S30C
Notch depth t = 0.1mmt = 0.5mm
ρ0≒0.5mm
2
「If Elastic Max. stress and notch radii is the same, Fatigue limit is the same. 」
Fatigue limit and non-propagating crackⅠ
σmin=-σa
σmac=+σa
l
Plastic zone
Plastic zone
x Micro-non-propagating crack
δ
(δ: Crack opening displacement)
No opening at crack tip ⇒ Non-propagation
Non-propagation of micro-crack
・ δ ; very small at crack tip
・ like closing
① Size of non-propagating crack
② Size of inclusion and defectEffect on fatigue limit
Fatigue limit of steel
After initiated crack grows,Limited stress which non-propagates
(Threshold stress which crack does not grow.)
Fatigue limit and non-propagating crackⅡ
Fatigue limit of smooth and notched specimen of steelNo notched specimen
Non-propagation
Notch sensitivity
Kt / Kf1
Kt / Kf2
0 2010
1.0
2.0
0
1/ρ [mm-1]
Kt /
Kf1
, Kt /
Kf2
Relation between Kt / Kf1, Kt / Kf2 と 1/ and ρ
S30C
Notch depth t = 0.1mmt = 0.5mm
ρ0≒0.5mm
2
Increase of notch sensitivity
Kt = Kf(Max. notch sensitivity)
11
−
−=
t
f
KK
η (0<η<1)
Notch sensitivity factor
Kt = 1.67 and ρ=1mm
η= 0.28
η= 0.69
η= 0.88
Pure Ti
S10C
Al alloy
Insensibility
Sensitivity
Size effectρ1
Rotating bending
ρ2
Rotating bending
② Large Surface area (statistical factor)① Stress gradient
Two main factors
Similar size of specimens
⇒ 1/ρ increases
⇒ For the same Kt , Kf1 and Kf2 decrease
Big Size ⇒Strength decreaseFor the same materials,
Size effect
⇒21
, f
t
f
t
KK
KK
increases ⇒ Decrease of fatigue strength
⇒ Probability of existing microcrack increases
Dangerous cross section increases
2
02
1
01 ,
w
wf
w
wf KK
σ
σ
σ
σ==⇒ Σw1 andσw2 increase
Area FECD ;Possible area of Safety use
Mean stress effectⅠ
45°
AlternatingPush-pull
45°
PulsatingPush-pull
0
Stre
ss a
mpl
itude
Mean stress
Diagram of fatigue limit
Diagram of fatigue limit
σT
σw
0
A
B
Effect of mean stress on fatigue limit
A ; Smooth specimen σw0
B ; True facture stress σT
σm
σa
GC
E
-σS
σS
σS
DF
H
Large plastic deformation happens
σS ; Yield stress
Surface effectⅠ (Effect of surface roughness)
0.1 1 10 100
1.0
0.5
Surface roughness Hmax’ [μm]
Fatig
ue st
reng
thm
= σ
w’/σ
w
Annealed steel
SteelTi alloyAl alloy
Bending fatigue
Effect of surface roughness
Decrease of fatigue limitLarge Surfaceroughness
凹凸
Surface effect
Estimation of fatigue lifeⅠ (Low cycle fatigue)
102 103 105104
10-2
10-3
10-1
Stra
in r
ange
Δε
p, Δε
t
Number of cycle to failure Nf
Strain range – fatigue life curve
Manson-coffin law
CN bfp =⋅Δε
Steel
fC ε=55.0≈b 、
ffp N εΔε =⋅ 55.0
Plastic strain range-life curve
Total strain range-life curve
( ) 03.100251.0 53.0 =− ft NΔε
Important total strain Range-fatigue life curve
Practical aspect
Low cycle fatigue
Estimation of fatigue lifeⅡ (Crack growth life)
Crack growth life
Paris law
( )mKCdNda
Δ=
C, m ; Material constantFor many materials,m = 2~7
( ) ( ) ( )∫∫⋅
=⋅
= c
i
c
i
a
a mm
a
a mC daaFCFaC
daN 11
πΔσπΔσ
( ) ( ){ }( ) ( )m
mc
mi
FCm
aa
⋅−
−=
−−−−
πΔσ2
2 2222
⇒
ai ; Initial crack lengthac ; Critical crack length
( )∫ ∫== C c
i
N a
a mC daKC
dNN0
1Δ
Integration
Linear cumulative damage lawⅠ
Under Fluctuating stress, estimation of fatigue life①
Time
Stre
ss
σ1σ2
(a)
Time
Stre
ss
σ1σ2
(b)
2 step 2 stage fluctuating stress
For Stressσ1 , Fatigue life Nf = N1
For Stressσ2 , Fatigue life Nf = N2
Stress range changes during cycle
(D;cumulative damage)
12
2
1
1 =+=Nn
NnD … (
Fatigue damage
After σ1 cycles n1(n1<N1),σ2 cycles n2
Miner law)
Linear cumulative damage lawⅡ
Under Fluctuating stress, estimation of fatigue life②
12
2
1
1 =+=Nn
NnD
After σ1 cycles n1(n1<N1),σ2 cycles n2
Miner law
(When D =1 , fatigue fracture)
D<1 D>1
Time
Stre
ss
σ1σ2
(a)
2 step 2 stage fluctuating stress
(a) Cyclic stress
Actually,
High ⇒ LowTime
Stre
ss
σ1σ2
(b)
(b)Cyclic stress
Low ⇒ High
(For some case, D=0.1~20 ⇒ must modify)
Linear cumulative damage lawⅢ
Stre
ss a
mpl
itude
σa
Number of cycles N
Linear cumulative damage law
σ1
σ2
σ3
N1
σW
n3n1 n2 N2
Miner lawΣ(ni/Ni)=1
N3*
N3=∞
Modified miner law
平均応力・残留応力の影響Ⅱ(疲労限度線図②)
平均応力の影響
残留応力の影響
・ 圧縮残留応力 ⇔ 圧縮の平均応力が作用する
・ 引張り残留応力 ⇔ 引張りの平均応力が作用するに対応する
0
疲労限度σ
a
平均応力 ; σm
疲労限度
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
n
B
mwa
σ
σσσ 10
σB ; 引張り強さ
σm ; 平均応力
σw0 ; 平滑材の疲労限度
σw
0
σB
修正グッドマン線図
n = 1 … 直線
ゲルバー線
n = 2 … 放物線
σS
ゾーダーベルク線
n = 1 … σSに置き換えた
(教科書 P184 図 6.16(b))
切欠材の疲労限度の推定
切欠材の疲労限度の推定
縦軸 ;f
w
K0σ
横軸 ; 引張り強さ σB
σw0 ; 平滑材の疲労限度
Kf ; 切欠係数
σw
0 / K
f
切欠材の疲労限度として評価
0
疲労限度σ
a平均応力 ; σm
σw
0
σB
修正グッドマン線図
ゲルバー線
σS
ゾーダーベルク線
図.疲労限度線図
(教科書 P184 図 6.16(b))
組み合わせ応力下の疲労強度Ⅰ
yM
Mx
z
ab c
d
y
M
M
T
Tx
z
ab c
d
yTx
z
ab c
d
σ σ
y
き裂発生領域
x
45°
(a)曲げ (c)ねじり
y
き裂発生領域
xτ
τ
τ
τ
図.曲げ、ねじり組み合わせ応力における最大せん断応力
2maxσ
τ =
σ σ
y
き裂発生領域
xτ
τ
τ
τ
θ
(b)曲げ+ねじり
22
max 2τ
στ +⎟
⎠⎞
⎜⎝⎛= ττ =max
(教科書 P186 図 6.17)
組み合わせ応力下の疲労強度Ⅱ
疲労き裂 ⇒ 主として、せん断応力の繰り返しで発生
12
2
max
2
max
=⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛τ
τ
τ
σ⇒
12
0
2
0
=⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛
ww τ
τ
σ
σ⇒2
0max
wστ =
0max wστ =
(曲げのとき)
(ねじりのとき)
近似
( ) 20
22 2 wστσ =+⇒ … (式 6.27)
00
2 ww τ
σ= と仮定する
22
max 2τ
στ +⎟
⎠⎞
⎜⎝⎛=
y
M
M
T
Tx
z
ab c
d
(注意)
① 曲げとねじりの応力サイクルの位相差がある時、ない時に比べてτmaxが小さい。
② 切欠材では、曲げとねじりで Kt およびρ0 が異なる。
表面効果と疲労限度Ⅲ (環境の影響)
① 塩水などによる腐食
②高温環境
⇒ 腐食ピットの発生
応力腐食割れ
⇒ 疲労限度 低下
⇒ 疲労限度 低下
⇒ 表面層の酸化、軟化
表面効果と疲労限度Ⅱ (表面処理の影響)
①鋼材の焼きなまし、焼きならし、脱炭
②浸炭、窒化
⑤ メッキ
⑥ コーティング、イオン注入
⇒ C含有量の減少により軟化
⇒ 疲労き裂が発生しやすい
引張り残留応力の発生
⇒ 疲労限度 低下
⇒ 硬度が増す
圧縮残留応力の発生
⇒ 疲労限度 上昇
③表面圧延
④ ショットピーニング⇒ 加工硬化、圧縮残留応力の発生
⇒ 疲労限度 上昇
⇒ メッキ層中の微細な割れ
⇒ 疲労限度 低下
⇒ 強化層の形成 (研究中)
⇒ 疲労限度 上昇
1/ρとKt/Kfによる整理Ⅱ表.材料の引張り強さ、疲労強度、ρ0と近似式の係数
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+=
ρρρ
1111 3
32
2
31
11
AAAKK
f
t …(式 6.20) (ρ>ρ0)
A , B ; 係数
ρ ; 切欠半径⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+=
ρρρ
1111 3
32
2
31
12
BBBKK
f
t …(式 6.21) (ρ<ρ0)
(教科書 P181 表 6.2)