Three Stages of Fatigue Failure Identifying Fatigue · PDF fileFracture ¾sudden, brittle-like failure ... crack initiation), for low-cycle fatigue Strain-Life stress-based, for high-cycle

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Over the Next Several DaysWhat is Fatigue? Types of Fatigue LoadingEmpirical DataEstimating Endurance/Fatigue StrengthStrategies for Analysis

Uniaxial Fully ReversedUniaxial FluctuatingMultiaxialCrack Growth

Some HistoryRail car axles The all-important microcrackRole of stress concentrations

Comet airplanes

Three Stages of Fatigue FailureCrack InitiationCrack Propagation

oscillating stress… crack grows, stops growing, grows, stops growing… with crack growth due to tensile stresses

Fracturesudden, brittle-like failure

Identifying Fatigue Fractures

beachmarks

Three Approaches

best model of crack propagation, for low-cycle fatigue

LEFM (Fracture Mechanics)

useful when yielding begins (i.e., during crack initiation), for low-cycle fatigueStrain-Life

stress-based, for high-cycle fatigue, aims to prevent crack initiationStress-Life

Low vs. High Cycle>103 cycles, high cycle fatigue

<103 cycles, low cycle fatigue

car crank shaft –

manufacturing equipment @ 100 rpm –

ships, planes, vehicle chassis

~2.5 E8 Rev/105 miles

1.25 E8 Rev/year

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Types of Fatigue Loading

minmax σσσ −=∆ stress range

2σσ ∆

=aalternating component

2minmax σσσ +

=mmean

component max

minσσ

=Rstress ratio

m

aAσσ

=amplitude ratio

Fully Reversed Repeated Fluctuating

UpdateWhat is fatigue? Types of Fatigue LoadingEmpirical DataEstimating Endurance/Fatigue StrengthStrategies for Analysis


Testing Fatigue PropertiesRotating Beam – most data is from this type

Axiallower or higher? Why?

CantileverTorsion

Fully Reversed Empirical Data

Wrought Steel

An S-N Curve (Stress-Life)

Fully Reversed Empirical Data

Aluminum

Endurance Limit

A stress level below which a material can be cycled infinitely without failure

Many materials have an endurance limit:low-strength carbon and alloy steels, some stainless steels, irons, molybdenum alloys, titanium alloys, and some polymers

Many other materials DO NOT have an endurance limit:aluminum, magnesium, copper, nickel alloys, some stainless steels, high-strength carbon and alloy steels

eS ′

for these, we use a FATIGUE STRENGTH defined for a certain number of cycles (5E8 is typical)fS ′

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minmax σσσ −=∆

2σσ ∆

=a

2minmax σσσ +

=m

m

aAσσ

=

max

minσσ

=R

stress range

alternating component

mean component

stress ratio

amplitude ratio


Getting Fatigue Data1) Test a prototype2) Test the exact material used3) Published fatigue data4) Use static data to estimate

Estimating Se´ From Static Datasee page 324 in your book…

ksi 40for ksi 19ksi 40for 4.0ksi 60for ksi 24ksi 60for 4.0ksi 200for 100ksi200for 5.0

85@

85@

≥≅≤≅≤≅≤≅≥≅≤≅

′

′

′

′

′

′

utf

ututf

ute

utute

ute

utute

SSSSSSSSSSSksiSSSS

E

E

steels

irons

aluminums

BUT, these are all for highly polished, circular rotating beams of a certain size

Correction Factors

freliabtempsurfsizeloadf

ereliabtempsurfsizeloade

SCCCCCS

SCCCCCS

′

′

=

=

pages 326+ in your book

Constructing Estimated S-N Curves

Sm=0.9Sut for bendingSm=0.75Sut for axial loading

The material strength at 103 cycles, Sm:

The line from Sm to Se or Sf, Sn=aNb

or logSn=loga + blogN

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Fatigue Stress Concentration

Kf = 1+q(Kt-1)

q = notch sensitivityfunction of material, Sut, Neuber constant, anotch radius, r

ra

q+

=1

1



0=mσ 0≠mσ

Uniaxial

Multiaxial



2σσ ∆

=a

2minmax σσσ +

=m

m

aAσσ

=

max

minσσ

=R

stress range


mean component

stress ratio

amplitude ratio


Uniaxial, Fully Reversed StrategyLoading & Stress Half

N (umber of cycles) Fluctuating Load (Fa)

Tentative DesignTentative Material

Kt σa (nominal)

σ1, σ2, σ3 (principal)

σ´ (von Mises)

Kf

σa

Uniaxial, Fully Reversed StrategyFatigue Half

Se´ or Sf´

Se or Sf

CloadCsurfCsizeCtempCreliab

Estimated S-N Curve


Tentative DesignTentative Material Tentative DesignTentative Material


σ´ (von Mises)


σ´ (von Mises)

KfKf

σaσa

Kt σa (nominal)Kt σa (nominal)Kt σa (nominal)

Uniaxial Fully Reversed Strategy

Se´ or Sf´

Se or SfSe or Sf



Estimated S-N CurveEstimated S-N Curve

σ ′= n

fSN

Nf = fatigue safety factor; Sn = Fatigue strength at n cycles;σ ´= largest von Mises alternating stress

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0=mσ 0≠mσ

Uniaxial

Multiaxial



2σσ ∆

=a

2minmax σσσ +

=m

m

aAσσ

=

max

minσσ

=R

stress range


mean component

stress ratio

amplitude ratio


Does Mean Stress Matter? The Data

Transform S-N --> σa-σm Fluctuating Stress Failure Plotσa

σm

Sy

Se or Sf

SutSy

Failure

Safety

constructed for a given number of cycles N

GerberSoderberg

Yield

modified-Goodman

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Definitions Factors of Safety Four cases

1) σa constant, σm varies2) σa varies, σm constant3) σa and σm increase at constant ratio4) σa and σm increase independentlyIf you know how the stress can vary, only use one of four casesIf stress can vary in any manner, Case 4 should be used (the most conservative)

Four Cases (Use Ruler!)“Augmented” Modified-Goodman Plot

von Mises calculated for σa and for σm separately

σa

σm

Se or Sf

SutSy

Sy

Syc

1=′

+′

y

a

y

mSSσσ

1=′

+′

f

a

ut

mSSσσ1=

′+

′−

yc

a

yc

mSSσσ

fa S=′σ

Uniaxial Fluctuating StrategyStress & Loading


Tentative DesignTentative Material

Kf

Kt

Kfm

σ1a, σ2a, σ3a; σ1m, σ2m, σ3m (principal)

σá, σ´m (von Mises)

σa

σm (nom) σa (nom)

σm

Uniaxial Fluctuating Strategy Fatigue Aspects

Se´ or Sf´

Se or Sf


Modified-Goodman Diagram

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Uniaxial Fluctuating StrategyN (umber of cycles) Fluctuating Load (Fa)

Tentative DesignTentative Material Tentative DesignTentative Material

σ1a, σ2a, σ3a; σ1m, σ2m, σ3m (principal)

σá, σ´m (von Mises)

Kf

σa

Kt σm (nom) σa (nom)

Kfm

σm

Se´ or Sf´

Se or SfSe or Sf



Modified-Goodman DiagramModified-Goodman Diagram

Nf

StrategyFind σá and σ´m with appropriate stress concentration factorsFind Se

Plot modified-Goodman diagramFind factor of safety



0=mσ 0≠mσ

Uniaxial

Multiaxial



2σσ ∆

=a

2minmax σσσ +

=m

m

aAσσ

=

max

minσσ

=R

stress range


mean component

stress ratio

amplitude ratio


Multiaxial Fatiguesimple multiaxial stress

periodic, synchronous, in-phasecomplex multiaxial stress

everything elseassuming synchronicity and being in-phase is usually conservative

Fully Reversed MultiaxialFind von Mises equivalent stress for alternating component

Cload implications

( ) ( ) ( )2

213

232

221 aaaaa a

aσσσσσσ

σ−+−+−

=′

a

yf

SN

σ ′=

8

Fluctuating MultiaxialSines MethodVon Mises Method

( ) ( ) ( )2

213

232

221 aaaaa a

aσσσσσσ

σ−+−+−

=′

( ) ( ) ( )2

213

232

221 mmmmm m

mσσσσσσ

σ−+−+−

=′

Modified-Goodman Diagram



0=mσ 0≠mσ

Uniaxial

Multiaxial

Crack Growth Experiments

dNdaln

( )K∆ln

n

Time

Stre

ss Measure a vs. cycles

( ) aK πσβ ∆=∆Calculate:

Then plot

( )nKAdNda

∆=Which Follows

Crack Growth Analysis

aK πβσ=

Calculate:

cc aK πβσ=22

2

σπβc

cKa =∴

dNdaln

( )K∆ln

n

( )nKAdNda

∆=From

( )( )( )daK

AKAdadN n

n−∆=

∆=

1

( )( ) ( )( )

daaA

daKA

N cc a

a

nnna

af ∫∫⎟⎠⎞

⎜⎝⎛ −

−−

∆=∆=00

211 πσβ

( )( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +−∆

⎟⎠⎞

⎜⎝⎛ +−−

−=

=

=

12

)1(

12

1

0

nn

N anA

caa

aa

f πσβ

( )( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛ −∆

−−−−

=)

21(

0)

21()1(

211 nn

cn

N aanA

f πσβ

Overall Strategy

Documents

Three Stages of Fatigue Failure Identifying Fatigue · PDF fileFracture ¾sudden, brittle-like failure ... crack initiation), for low-cycle fatigue Strain-Life stress-based, for high-cycle