Fatigue of materials

Lec.1 High-cycle fatigue. Design with respect to fatigue limit.Wöhler diagram. Haigh diagram and reduction of the fatigue limit.

Lec.2 Design with respect to fatigue life, damage accumulationPalmgren-Miner hypothesis.

Lec.3 Fatigue from a material point of view.Guest lecturer. Johan Moverare

Lec.4 Fatigue from a structural mechanics point of view.Guest lecturer. Hans Ansell (Saab)

Fatigue of materialsIntroduction: Teaspoon example

Bend a teaspoonrepeatidly back and forth.After a while the spoon breaks with a snap.

Materials subjected to a time varying loading may fractureeven if the loading level is so low that stresses in the material always are below the yield limit of the material

Fatigue failure

Fig. Meso scale

The progression of a fatigue failure may be divided into 3 stages

1) Crack initiation phase2) Crackgrowth (crack propagation due to cyclic loading)3) Fracture (final rupture when the crack is long enough)

When fatigue life is to be determined the number of loading cycles to the final rupture has to be counted.

(no separaration between the different stages above is needed)

When designing for material fatigue, a calculated stress or strain in the material has to be compared with an experimentally determined fatigue strength of the material.

There is a high degree of uncertances

• The exact loading of a structure is perhaps not known• Hard to determine stresses due to local stress-concentrations or other geometrical imperfections

• The fatigue life will differ between identical tests

Material fatigue is sorted into

• High-Cycle Fatigue HCF (low stresses compared to the yield limit, large number of loading cycles to failure)

• Low-Cycle Fatigue LCF (high stresses, low number of loading cycles to failure, usually strain based fatigue life prediction)

High-Cycle Fatigue (HCF)

Definitions:

cycles loading ofNumber 2πωt

τtN ==

ωt σσσ(t) am sin+=

aσmσ

t

σ

ammax σσσ +=

ammin σσσ −=

)σ(σ21σ minmaxm +=

Mean stress:

)σ(σ21σ minmaxa −=

Stress amplitude :

Fatigue data are often given for two types of loadings

aσt

σ

mσt

aσσ

1) Alternating: ( )0am σσ 0σ == ,

Notation: 0σ σ ±=

Notation: 00 σ σ σ ±=

2) Pulsating: )σσ(σ 0am ==

B A σa += Nlog•

• ⇔= σa KNm

) (1σ a NloglogKm

log −=

aσ

Nlog0 1 2 3 4 5 6 7 8 9

0σm =

Fatigue life calculations (Wöhler diagram)

Nσ

N = number of loading cycles to failure

SN-curve (Wöhler diagram) for steel

Fatigue life calculations (Wöhler diagram)

SN-curve (Wöhler diagram) for steel

limit fatigueσσ FLu ==

aσ

Nlog0 1 2 3 4 5 6 7 8 9

uσ

0σm =

• The fatigue limit σu

Fatigue life calculations (Wöhler diagram)

SN-curves (Wöhler diagram) for different values of the mean stress σm

aσ

Nlog0 1 2 3 4 5 6 7 8 9

0σm =

Nσ

0σm1 >

m1m2 σσ >

• Effect of mean stress σm on fatigue life and fatigue limit σu

uσlimit fatigueσσ FLu ==

Fatigue life calculations (Wöhler diagram)

limit fatigueσσ FLu ==

aσ

Nlog0 1 2 3 4 5 6 7 8 9

uσ

• Statistical scatter

SN-curves is usually plotted for 50 % failure probability

Material data for fatigue limits are usually given fortension/compression, bending and torsion for alternating and pulsating loading.

Fatigue limits (notations)

Load

Tension/compression

Bending

Torsion

Alternating Pulsating

uσ±

ubσ±

uvτ±

upup σσ ±

ubpubp σσ ±

uvpuvp ττ ±

The Haigh diagram

Alternating: uσ±

Pulsating: upup σσ ±uam σσ , 0σ ==⇔

upaupm σσ , σσ ==⇔

aσ

Yσ

Yσ

uσ + +upσ

upσ mσUσ

Area where no fatigue failure occurs

Shows the fatigue limit as a function of the mean stress plotted in the σa - σm plane.Used to estimate the safety against fatigue failure

1) Insert the data for fatigue limits in the diagram

Uσ = ultimate strength

Yσ = yield limit

2) Insert the ultimate strength and the yield limit

To construct the diagram:

The Haigh diagram - Reduction of the fatigue limit

Factor

κ

λ

δ

Reduction of the fatigue limit due to:

Surface roughness

Size of raw material

Size of loaded volume ) τ,(σ uvub

Reduced diagram

uσ δ λκ

upσ δ λκ +

aσ

Yσ

Yσ

uσ + +upσ

upσmσ

Uσ

+

Reduction of the fatigue limit due to surface roughness (κ )

Reduction of the fatigue limit due to size of raw material (λ)

Reduction of the fatigue limit due to size of loaded volume (δ)

Inserting the service stress )σ,(σ: amP

P+

Reduced Haigh diagram

uσ δ λκ

upσ δ λκ +

aσ

Yσ

Yσ

uσ + +upσ

upσmσ

Uσ

+

limits fatigue reducing factors δ λ, κ,limits fatigue σ σ

strength ultimate σlimit yield σstressmean σamplitude stress σ

upuU

Y

ma

==

===

=

,

• Stress concentration factor Kt

In order to account for stress concentrations due to geometrical imperfections such as holes, fillets and notches in a loaded volume the mean stress is multiplied with the corresponding stress concentration factor Kt.

nommm σσ tK=

The stress amplitude is multiplied with the fatigue notch factor Kf at stress concentrations.

nomaa σσ fK=

• Fatigue notch factor Kf

nommm σσ tK=Stress concentration factor Kt

Stress concentration factor Ktnommm σσ tK=

Fatigue notch factor Kfnomaa σσ fK=

Safety against fatigue failure, safety factors

aσ

P

Reduced Haigh diagramYσ

Yσ

uσ upσ

upσmσ

Uσ

+

+ ++

+

O A

A’

B’

C’

Safety factors

constantσ when ma ==APAA'SF

constantσ when am ==OAOB'SF

Inserting the service stress P : )σ ,σ()σ ,(σ noma

nommam ft KK=

constantσσ when

m

aam ==

OPOC'SF