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 fatigue

Corrosion fatigue

Fretting fatigue

11% Fatigue 60%

Low cycle fatigue

8%

◎ Fracture cause

80-90% of fracture is by fatigue

◎ How to fracture

Under cyclic loading, component

suddenly fractures

Fatigue and fatigue fracture surface

（１）Origin …

（２）Crack growth …

（３）Macroscopic …

（４）Microscopic …

（５）Other …

Surface of component

Stress concentration （Notch, Key, Inclusion）

After crack initiation, along max. stress plane

Smooth surface, macroscopically few plastic

deformation

Beach mark （under irregular cyclic stress）

Striation

When crack grows, cross section decreases

Ductile fracture

Rough surface

Fatigue fracture

Fatigue fracture factor

time

Str

ess

Ten

sio

n（＋）

C

om

pre

ssio

n（－）

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 = 0

σm= σa

（d）Partial Pulsating

σm> σa

R > 0

σmin

σm

σmax

σa

σm ： Mean stress

σa ： Stress amplitude

R ： Stress ratio

2

minmax σσσ

m

2

minmax σσσ

a

max

min

σ

σR

（a）Alternating

σm= 0

R = ‐1

Ten

sion（＋

） C

om

pre

ssio

n（－

）

Str

ess

Time

Definition of cyclic stress

Cyclic stress test

Alternating

σm= 0

R = ‐1

Ten

sion（＋

） C

om

pre

ssio

n（－）

Str

ess

Time

Rotating fatigue tester

A point on surface of sample

Upper

Sid

e

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 Failure

Non failure

Mild

steel

105 106 107 108 0

100

200

300

Cyclic number to failure N

Str

ess

am

pli

tud

e σ

a

［

MP

a］

S-N curves of almimum alloy and mild steel

Rotaing bending

σm＝0 （R=‐1）

Fundamental diagram to evaluate fatigue property

Cyclic stress（stress amplitude σa）- Cyclic number to failure

Nominal stress。 Fatigue life

log Nf

SｰN curve II

Aluminum

alloy

Mild

steel

Cyclic number to failure N

（S-N curves aluminum alloy and mild steel）

Failure

Non failure

105 106 107 108 0

100

200

300

Str

ess

ap

litu

de

σa

［

MP

a］

Rotating bending

σm＝0 （R=‐1）

Fa

tig

ue

lim

it

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

Fa

tig

ue

stre

ng

th a

t10

7

107cycles fatigue

strength …

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 regio

σa ; Elastic stress

σ

a

Cyclic number to failure

Nf

101 10

2 10

3 10

4 10

5 10

6 10

7

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

Str

ess

am

pli

tud

e σ

a

N2

P-S-N curve

P=0.90

P=0.99

10% P=0.10

Fatigue

probability

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 prpertiesⅠ

Mechanical properties

・ Yield strength σS

・ Tensile strength σB

・ Brinnel HB

・ Vickers HV

Fatigue limit σw0

Vickers hardness

Sample

Diamond probe Probe

d1

d2

21

22cos2 oad

dd

W

aSurfaceare

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 propertiesⅡ

Fatigue 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 loop

σa ; High level

（Plastic deformation）

σ

a

Cyclic number to failure

Nf

101 10

2 10

3 10

4 10

5 10

6 10

7

Under high temperature,

Machine

Cyclic thermal strain

・ Nuclear vessel

・ Steam turbine

Short fatigue life

Hysteresis loopⅠ

σa

B

High loading

A Yielding

C

Yielding

（Bauschinger effect ）

Δσ

εa

σa

Δε Compression

εpa

σm

εm

D

E

Δεp

Tensile strain

εea

εa

Unloading

Compression εpa

0

Str

ess

σ

Strain ε

Hysteresis loop

Hysteresis loopⅡ

Δσ

εa

εpa

εa

σm

σa

Δε

εea

εm

D

E

0

Str

ess

σ

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

ppetE

ΔεΔσ

ΔεΔεΔε

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 life

Shape of hysteresis loop saturates

Str

ess

Δσ

Strain

Δε

Cyclic stress-strain curve

Strain range and fatigue life

103 104 102

101

100

105

10-3

10-1

10-2

100 101

10-1

Cyclic number to failure Nf

Pla

stic

str

ain

ran

ge

Δε p

Manson-Coffin law

41fN

fε2

1

5.0b

CNb

fp Δε

Manson-Coffin relation

Relation between Δεpand Nf of low cycle fatigue

CNb

fp Δε …（式 6.15）

b，C ； Constant

（For many materials,b≒0.5）

◎ Nf =1/4 cycle , Δεp＝2εf

φε

100

100lnln 0

f

fA

A◎

A0 ； Cross section before

A ； Cross section after

φ ； Reduction of area

εf ； Failure ductile

Microscopic fracture appearance S

urf

ace

Cyclic stress

Su

rface

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 appearance Ⅱ

Stage IIa of crack growth process

（Ⅱa）

Stage II crack growth

（Ⅰ）

Direction of crack growth

Su

rface

Cyclic stress

試験片表面

Small crack ⇒ Grow in grain

（along slip plane）

Stress concentration gives rise

to damage at crack tip

Continuous

Crack growth rate

dN

dahrateCrackgrowt

（a ； crack length、N ； Cyclic number）

Crack tip

Intergranular ⇒ Delay

Granular ⇒ High

Microscopic fracture appearance Ⅲ

Stage IIb of crack growth process

Crack growth direction

Su

rfa

ce

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

cyclemdN

da/10分の数μ

Striation spacing

⇒ crack growth rate

Microscopic fracture appearance Ⅳ

Stage IIc of crack growth process

（Ⅱa） （Ⅰ） （Ⅱb）

Crack growth direction

Su

rfa

ce

Cyclic stress

（Ⅱc）

Stage II of crack growth

Striation

High crack growth rate

（High strength steel

⇒Cleavage, intergranular

cracking ）

Final fracture

Ductile

fracture

Crack growth lawⅠ

Linear 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

a1 a2

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 πσmaxmaxFaK πσminmin

，

Crack growth rateⅡ

Δσ

2

Δσ

1

Str

ess

ran

ge

Time t （a）

き裂長さ a1

き裂長さ a2 For long a ,

S. I. F. K

Driving force Large

（b）

S.I

.F.r

an

ge

ΔK

Time t

Stress intensity factor range,ΔK, change

aK Δ

Stress ratio ＝ 0

Δσ ; Cyclic stress，a ;

ΔK

1=

ΔK

2

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 → 0

Lower 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

lo

g（

da/d

N）

m

1

Steady growth Paris law

mKCdN

daΔ

C, m ; material constant

For 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） F

ati

gu

e li

mit

σw

Stress concentration Kt

Fatigue limit of notched material

ρ

Rotating bend A

C Fatigue 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-propagating

crack

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

Fati

gu

e li

mit

σw

Stress concentration Kt

Fatigue limit of notched specimen

A

C Fatigue strength σw1

σw0

B D

Crack

strength σw2

Branchρ=ρ0

Branch point B

Material constant

Dependence on ρ0

① Fatigue strength σw1

ρ＞ρ0 ; No non-propagating crack

② Crack strength σw2

ρ＜ρ0 ; Non-propagating crack

How much is Fatigue limit decreased

by 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 4

0

0.5

1.0

Fati

gu

e li

mit

σ

w1/

σw

0 ,

σw

2/

σw

0

S30C

ρ

Rotating bending（d=5mm）

t d

t = 0.5mm

Notch depth t = 0.1mm

Different notch depth

⇒ Kt of branch point differs

⇒ Kt of branch point differs

Different diameter

Branch point 0

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 20 10

1.0

2.0

0

1/ρ ［mm-1］

Kt /

Kf1

, K

t /

Kf2

Relation between Kt / Kf1 , Kt / Kf2 and 1/ρ

S30C

Notch depth t = 0.1mm

t = 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 defect Effect 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 steel

No notched specimen

Non-

propagation

Notch sensitivity

Kt / Kf1

Kt / Kf2

0 20 10

1.0

2.0

0

1/ρ ［mm-1］

Kt /

Kf1

, K

t /

Kf2

Relation between Kt / Kf1 , Kt / Kf2 と 1/ and ρ

S30C

Notch depth t = 0.1mm

t = 0.5mm

ρ0≒0.5mm

2

Increase of notch sensitivity

Kt = Kf（Max. notch sensitivity）

1

1

t

f

K

Kη （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 effect ρ1

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 decrease

For the same materials,

Size effect

⇒ 21

, f

t

f

t

K

K

K

Kincreases

⇒ 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°

Alternating

Push-pull

45°

Pulsating

Push-pull

0

Str

ess

am

pli

tud

e

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

G C

E

－σS

σS

σS

D F

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］

Fati

gu

e st

ren

gth

m =

σw

’/σ

w

Annealed steel

Steel Ti alloy Al alloy

Bending fatigue

Effect of surface roughness

Decrease of fatigue limit Large Surface

roughness

凹凸

Surface effect

Estimation of fatigue lifeⅠ （Low cycle fatigue）

102 103 105 104

10-2

10-3

10-1

Str

ain

ran

ge

Δ

ε p, Δ

ε t

Number of cycle to failure Nf

Strain range – fatigue life curve

Manson-coffin law

CNb

fp Δε

Steel

fC ε55.0b 、

ffp N εΔε 55.0

Plastic strain range-life curve

Total strain range-life curve

03.100251.053.0 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

mKCdN

daΔ

C, m ; Material constant

For many materials,m = 2～7

c

i

c

i

a

a mm

a

a mC daaFCFaC

daN

11

πΔσπΔσ

m

m

c

m

i

FCm

aa

πΔσ2

22222

⇒

ai ； Initial crack length

ac ； Critical crack length

C c

i

N a

a mC daKC

dNN0

1

Δ

Integration

Linear cumulative damage lawⅠ

Under Fluctuating stress, estimation of fatigue life①

Time

Str

ess

σ1 σ2

（a）

Time

Str

ess

σ1 σ2

（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 N

n

N

nD … （

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 N

n

N

nD

After σ1 cycles n1（n1＜N1）,

σ2 cycles n2

Miner law

（When D =1 , fatigue fracture）

D＜1 D＞1

Time

Str

ess

σ1 σ2

（a）

2 step 2 stage fluctuating stress

（a） Cyclic stress

Actually,

High ⇒ Low

Time

Str

ess

σ1 σ2

（b）

（b）Cyclic stress

Low ⇒ High

（For some case, D＝0.1～20 ⇒ must modify）

Linear cumulative damage lawⅢ

Str

ess

am

pli

tud

e

σa

Number of cycles N

Linear cumulative damage law

σ1

σ2

σ3

N1

σW

n3 n1 n2 N2

Miner law

Σ（ni/Ni）=1

N3*

N3=∞

Modified miner law