Weakest-link

analysis

2

200

400

600

800

4 5 6 7

log N

Smax, MPa

failures

runouts

T =1

T =10

What is the fatigue limit ?

Weakest-link

analysis

3

Defects and fatigue strength

Murakami’s experiments on specimens containing

microholes

25 µµµµm 50 µµµµm

Axial loading (bending) torsional loading (biaxial)

Fatigue limit is the ‘non-propagation’ condition

for small cracks emanating from the defects

Weakest-link

analysis

4

a/a0

Murakami’s idea: •defects can be treated as

short cracks;

threshold ∆K e ∆slim depend on the crack dimension

Short cracks and defects

Weakest-link

analysis

5

( ) 2/1

65.0 AreaK I ⋅⋅∆⋅=∆ πσMurakami’s

equation:

√ √ √ √area model (Murakami & Endo)

( ) ( ) ( ) 410226.03/13 2/5.0120103.3

−⋅+− −⋅⋅+⋅⋅=∆ Hv

th RareaHvK

Thresholds:

( ) ( ) 40.226 10

1/ 6

1200.5 / 2

Hv

w

HvC R

areaσ

−+ ⋅+= ⋅ ⋅ −

Fatigue limit:

C = 1.56 internal def.

C = 1.43 surface def.

Weakest-link

analysis

6

the fatigue is controlled by the extreme values of the

population of defects not by the average dimension

10 100 1000100

200

300

smooth specimens

interpolation

Stress amplitude, MPa

Hole diameter 2R, µµµµm

small holes bending

survival

failure

analysis of extremes based on extreme value sampling

Extreme defects

Weakest-link

analysis

7

•Methods based on Statistics of Extremes

•new technical recommendations (ESIS e ASTM)

Extreme defects

Weakest-link

analysis

8

Components with defects

Weakest-link model

Application to 3 strips of

‘super-clean’ steels

Comparison with fatigue

experiments

?

Weakest-link

analysis

9

Extremes

Let us consider m defects with distribution function F

the distribution

function of the

maximum defect:

( ) ( )max

m

F a F a=

m→∞

( ) ( )max , , exp expa

F a G aλλ δ

δ − ≅ = − −

Weakest-link

analysis

10

Extremes ⇒⇒⇒⇒ Weakest-link

m defectsLog S

Log a

Slim

alim

Smoothspecimen

Short cracks Long cracks

( ){ }lim limPr ( )R a a F a= ≤ =m

totR R=

Weakest-link

[ ] ( ){ }lim max lim( ) Prm

totR F a a a= = ≤

Approaches of WL and Extremes are coincident

Weakest-link

analysis

11

Weakest-link model

If we imagine the component divided into n domains:

Ri

tot iR R=∏ Weakest-link

( )lim,i V iR G a= ( ) 0

0

V

VV VG G=

Extremes

Weakest-link

analysis

12

Application

3 series of thin strips of super-clean steels (SANDVIK)

Material Thickness [ ]mm HV [ ]2/mmkgf Rm [ ]MPa

strip A 0.305 539 1705

strip B 0.305 556 1744

strip C 0.381 581 1649

inclusions at fracture origin

Weakest-link

analysis

13

Research of extreme defects

Polished sections

Distribution of maximum defects on So = 400 mm2

Weakest-link

analysis

14

FE analysis of fatigue specimens

• Calculation of failure probability as a function of S;

• determination of fatigue limit distribution function.

Weakest-link

analysis

15

Results

good fatigue strength

predictions

Weakest-link

analysis

16

Comparison among strips

strip B has the

best hardness why low fatigue strength ?

extreme defects ?

Weakest-link

analysis

17

Comparison among strips

Maximum stress limit

Dangerous defects

Strip A Strip B Strip C

7 11 11 Critical threshold [ ]mµ0.606 1.361 0.722 Critical density /

2400 mm

‘density’ of detrimental defects

Weakest-link

analysis

18

Conclusions

• fatigue limit in presence of defects can be estimated

from the Kitagawa diagram of the material under

examination

• a Weakest-link model has been proposed in

combination with ‘statistics of extremes’ for estimating

fatigue strength in mechanical components

• application shows that while for material qualification

the maximum defect is a sufficient information, the

calculation of the failure probability for a component

need also information about defect density

a weakest-link analysis for fatigue strength of components

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