Fatigue i

fatiguefatigue

L. VerganiL. Vergani

L. Vergani L. Vergani -- approccio Sapproccio S--NN

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•From latin fatigare.

• Components of machines, vehicles and structures are frequently subjected to repeated loads (cyclic loads) and the resulting cyclic stresses can lead to microscopic damage. This damage can accumulate until it develops into a crack that leads the failure of the component, even at stresses well below the ultimate strength of materials. This process of accumulating damage is the FATIGUE.

•Mechanical failure due to:

55% high cycle fatigue (HCF)

10% low cycle fatigue (LCF)

15% fatigue (RCF, CF, creep-fatigue,…)

20% static loading

10% other kind of failure

Fatigue of Materials


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Mechanical failures due to fatigue have been studied for more than 150 years. One early study was in 1828 by Albert in Germany. Fatigue was studied in the mid-1800s by several researchers in response to failures of components as railway axles, shaft, gears….. The fatigue failures are frequent also at present…

Fatigue of materials


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The damage is characterized by three steps:

•Nucleation of the crack (from the surfaces or internal from existing defects)

•Propagation (short crack, long crack)

•Final failure

Fatigue damage MECHANISMS


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•The nucleation of a micro-crack is due to the plastic strain and the persistent slip band (PSB).

Nucleation of a micro-crack


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•In the most strained zones the material fails and micro-cracks nucleate. In the first stage these micro-cracks propagate in the direction of the maximum tangential stress (Stage I). They can be inter-granular or trans-granular.

Nucleation of a micro-crack

Da Suresh - “Fatigue of Materials”


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•During the Stage I the propagation of the micro-cracks is influenced by the microstructure of the material.

•When the dimensions of the micro-crack are increasing the friction between the crack faces is increasing too. The propagation of the cracks continues in a plane perpendicular to the applied load (Stage II), until the sudden failure.

Propagation of a micro-crack

Da Suresh - “Fatigue of Materials”


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•The fatigue propagation zone is fairly flat and marked by the beach marks.

•When the crack has reached a sufficient size a final failure occurs.

Propagation of a micro-crack

The final failure can be ductile (involving large deformation) or brittle (involving little deformation) depending on the material.


9Description of a cyclic loading

Variable loading

off-shore structure

airplane

Da Broek - “The Practical Use of Fracture Mechanics”


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With the aim to evaluate the effect of the fatigue the cyclic loading could be schematized as:

Alternating stress

Mean stress

Amplitude ratio

2minmax σσσ −

=a

2minmax σσσ +

=med

max

min

σσ

=R

Description of a cyclic loading


11Fatigue design

A D

B C

No cracks Large cracks

Low stress amplutude

High stress amplutude

uni-axial loading

A. HCF (fatigue strength, Haigh diagram, Wöhler curves).

B. LCF (Coffin-Manson curves)

C. Elasto-plastic fracture mechanics (EPFM)

D. Linear-elastic farcture mechanics (LEFM).


12Fatigue design

AD

B C

F

E

G

H

No cracks Large cracks

Low stress amplitude

High stress amplitude

Uniaxial Stress

Multiaxial Stress

Amplitude of the loading cycles

Complexity applied loadings

Geometry and dimensions

Acceptability of the

damage

Environment

Experience


13Fatigue characterization of materials

Wöhler curves (S-N curves)

R=costant (very often =-1)

The diagram of curves S-N is a log-log diagram

UTS


14S-N curves

We can enter in these curves by considering the life or by considering the stress amplitude

σFA

UTS


15Fatigue test machines

Rotating bending test machine scheme

da Davoli, Vergani, Beretta, Guagliano, Baragetti “Costruzione di macchine 1” McGraw-Hill


16Macchine di prova

Rotating bending test machines


17Fatigue test machines

Axial loading test machine

da Davoli, Vergani, Beretta, Guagliano, Baragetti “Costruzione di macchine 1”McGraw-Hill

L. Vergani L. Vergani –– approccio Sapproccio S--NN

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POLITECNICO DI MILANO

Fatigue characterization of materials

The tests to evaluate the fatigue strength of materials are carried out by using standard specimens

(norma ISO 1143)

d = 10 mm

Ra= 0.3μm

Kt = 1

σFAf/Rm= 0.4 - 0.6

σFAa/Rm= 0.3 - 0.45

τFAt/Rm= 0.23 - 0.33



If multiple fatigue tests are run at one stress level, there is always considerable statistical scatter in the fatigue life. If the statistical scatter in cycles failure is considered a distribution as in figure is obtained.

If the logarithm of Nf is considered as the variable a symmetrical distribution is obtained: standard Gaussian (normal) (equivalent to lognormal distribution of Nf) statistical use is resonable.


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Statistical analysis of fatigue data permits the average fatigue curves t.o be established along with additional S-N curves for various probabilities of failure


Fatigue characterization of materials

The experimental data are treated by the statistical approach: STAIR CASE.This approach allows to determine the fatigue strength characterized by the 50% of probability of failure.A large number of specimens to be experimentally tested is required.The number of specimens has to be odd in order to have a different number of failure and survivors.Before starting the tests the maximum number of loading cycles and the value of Δσ are chosen.

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5 broken specimens (3 σ2 and 2 σ3) 6 run out specimens (2 σ2, 3 σ1 and 1 σο)

2523 32 σσσσ Δ

−+

=FA

Less frequent event

failure


23S-N Curves


The S-N curves can be schematized

σFA

UTSUTS

Log

Logm

FAσ3

7

1010

=

m

KNma =σ

103 107


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From the specimen to the component

•Surface finish effect

•Dimension effect

•Notch effect

With the aim to consider these effects the follwing parameters are defined:


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•Surface finish effect: the coefficient b3 is defined equal to the ratio between the fatigue strength obtained by specimens with different surface finish and the fatigue strength obtained by standard specimens (roughness Ra=0.3 μm)

b3 pattern versus ultimate strength of materials(1- lucidato; 2-rettificato fine; 3-rettificato; 4,5-tornito)


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•Dimensional effect: the coefficient b2 is defined equal to the ratio between the fatigue strength obtained by specimens with generic dimensions and the fatiguestrength obtained by standard specimens (d=10mm).

b2 pattern versus the dimensions


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•Notch effect: the fatigue notch coefficient Kf is defined equal to the ratio between the fatigue strength obtained by standard specimens and the fatigue strength obtained by notched specimens

•Kf depend on Kt by the notch sensivity q.

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−

−=

t

f

KK

q Bending and axial fatigue

Torsion fatigue


29Notch sensitivity

Peterson rule: Neuber rule: q

r

=+

1

1 ρ

R [MPa]m

ρ

0.2

0.4

0.6

0.8

800 1200 1600

raq

+=

1

1

a=0.0634 (tempered and quenched steel)

a=0.254 (annealed steel)

a=0.634 (aluminium alloys)


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From specimen to the component

•Kf=1+q(Kt-1)

r M Mf1 f2

A B

rA> rB

At the same σmax :

From experimental tests: KfA=2,1 e KfB=3,6 If the notch radius is larger (KtA=2,5) the value of Kf decreases of 16%, on the contrary if the notch radius is lower (KtB=5), the value decreases of 28%.

KtA=2,5 KtB=5

2 tBfA fB fB

tA

KM M M

K= =

KfB> KfA The gradient effect is secondary


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•The fatigue limit of the component becames:

f

faFAfaFA K

bb 32),(),('

σσ =

The S-N curve of the component

σa

σFA

σ’FA


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Mean stress effect

Haigh diagram:

σ’FA

Rm


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Mean stress effect:

Semplified diagram:

σ’FA

Rm

σsn

σsn σsnRc

The yielding limit is considered


34Mean stress effect:


35Compression mean stress effect

The surface tretament are applied to improve the fatigue behavior of mechanical componentThermo-chemical treatment (carburizing and nitriding) Mechanical treatment (shot peening, cold rolling)

+-


36Compressione mean stress effect

σ

σ

σ

a

FAf

medRc

0

PPlim

σ

σa

medresσ

P'lim

P'

Rm

A

OPOPlim

1 =η

''lim

2 APAP

=η

Without residual stresses:

With residual stresses:


37Haigh diagram: torsion


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Finite life estimationIf the service required life is lower than the life corresponding to the fatigue limit (N=106÷107):

A semplified Wöhler diagram is constructed

(σ’F) value is determined

Log N

max

FAf

σ

σ

1 2 3 4 5 6 7

Rm

σF

σFAf

σF

Ν=105

σ’F> σ’FAf


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The shaft is loaded by a varaible bending momentN=105

b2 b3 Kf and matreial characteristics are known

r

Mf

M

fM

D d

′ = ⋅⋅

σ σFAf FAfff

b bK2 3

M f = Mo sinωt


40Finite life

Wöhler diagram

Log N

max

FAf

σ

σ

1 2 3 4 5 6 7

Rm

Fσ


41example

dL1

L2

F0 F0

Steel: 39NiCrMo3 (Rm=900MPa, Rsn=700MPa). L1=40mm L2=60mm d=15mm B=5mm F0=10.000N

F= F0(1+sinωt)

Νf>107

Failure?


42Example

σmed=-50MPaσa=60MPa

b2, b3, Kf, σFA

t5060

σ

stress:


43Example

σa>0 σm<0

σ

σ

σ

a

FAf

medRs

Rs

Rs mR

P

Plim

60

-50

Haigh diagram

σ σFA FAf

b bK

' = 2 3

limOPOP

η =


44example


45example

σ

σ

σ

a

FAf

medRc O

Plim

Rm

σrot


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Example

L’omologazione del manubrio illustrato in figura richiede che esso superi senza rompersi una prova di fatica dalla durata di 5x105 cicli con una carico applicato F1, alternato intorno al valor nullo, di 1 kN.

Si stimi se, assegnati il materiale e le dimensioni geometriche, il manubrio supererà la prova.

DatiMat.: 39NiCrMo3 (Rm=900 MPa, Rp0.2=650 MPa)

L=400 mm

a= 200 mm

Sez. A-A (circolare cava)

De= 25 mm

Di=20 mm

Kt=1.7

Α

Α

−F1

F1F1

−F1 A-AL

a

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