Fatigue initiation Low Cycle Fatigue FATIGUE - Chalmersanek/teaching/fatfract/98-11.pdf · Low Cycle Fatigue Stresses close to (or at) the yield limit Small stress increment large

Fatigue a survival kitSolid Mechanics Anders Ekberg

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FATIGUE

A Crash CourseCrack propagation

Retardation effects

Fatigue initiation

Limited High Cycle Fatigue life

Low Cycle Fatigue

Stress Concentrations in LCF

Stress cycle identification

Damage accumulationMultiaxial fatigue initiation

Crack closure

Crack growth arrestment

Crack growth in mixed mode loading


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Damage mechanisms

ab

cd

el

pl

YFL

pl

FL

Y

el

Plastic shakedown limit

Elastic shakedown limit

Yield limit

Fatigue limit


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Fatigue design methods

Parameters

Magnitude of loading

Complexity of loading

Damage of material

Environment

Possibility of experimentalvalidations

Experience

...Low StressMagnitudes

No Cracks Large Cracks

Multia

xial

Stres

s

High StressMagnitudes

Uniax

ial

Stre

ssA D

B CE

F G

H


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Philosophies in Design

Safe life means that the component is designed to last a previously determined life

Equivalent stress criterion (no cracks initiated) Fracture mechanics analysis (crack does not

propagate enough to cause failure in operational life)

Crcaks are not allowed to grow to failure.Also, (partial) failure of a part of the component through crack development must not endanger safety

Design steps A flaw size, that will lead to fracture, is estimated. A tolerable flaw size is defined An initial flaw size is defined The time to grow the crack from initial to

tolerable size is calculated Based on this, inspection schemes are defined

Fail Safe

Safe Life


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Fatigue

How?

Micro structural changes which cause nucleation of permanent damage. Persistent slip bands (PSB)

Creation of microscopic cracks

Growth and coalescence of microscopic flaws to form "dominant" cracks

Stable propagation of dominant macro crack(s)

Structural instability or complete failure

Where?

Initiation where s > 1

Normally weak spots

- inclusions- small cracks

stress concentrations- inclusions- corrosion pits

Fatigue will then grow (propagate) in the direction of max sInitiation propagation


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Fatigue crack initiation and propagation

Small cracks Shear driven Interact with microstructure Mostly analyzed by continuum

mechanics approaches

Large cracks Tension driven Fairly insensitive to

microstructure Mostly analyzed by fracture

mechanics models

Stage IITension driven crack

(propagation)

Stage IShear driven crack

(initiation)


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Influencing factors

Stress concentrations

Give rise to increased stress levels

Volumetric effects

Increased volume subjected to high load magnitude gives larger probability of a weak, highly stressed spot

Large raw material gives reduced fatigue resistance due to manufacturing quality

Environmental effects

Corrosion Corrosion pits which act as stress concentrators. Cracks will always form due to corrosion -> no fatigue limit

Heat will often decrease the fatigue strength and the fatigue limit may diminish at higher temperatures.


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a

m

FLFLP

UTSY

a FL=

m = 0

time

a FLP=a

m FLP=

Plasticdeformations

FLP

a

mFLP

UTSY

FL

FLP

Loaded volume

Surface roughness

Size of rawmaterial

Haigh diagram I

YY

Haigh diagram Reduced Haigh diagram


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a

mUTSY

P

a

mP

( , )K Kt m f a

Haigh diagram II

A

A

C

O B

SFaAA'AP

=

SFmOB'OA

=

SFamOC'OP

=

m const=

a const=

KK

f a

t mconst

=K q K

K Kf t

f t

= +

1 1( )

Service stress Safety factors


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N

10

FL

steel

aluminum

1 102 103 104 105 106 107

a

Given stress amplitude

Gives pertinent

fatigue life

Given service life

Gives allowable stress amplitude

No fatigue damage is induced, the component can sustain an infinite number of load cycles

The Whler diagram can be used to design for finite (and infinite) life

This can be done either for a given service loading or a given service life

This slope on the Whler curve can be described by the equation

am Nf = K

Whler (S-N) curve

1k

or be approximated as a straight line

Valid only for a certain R-ratio


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t

1

2

35

4

678

Stress cycle identification rainflow counting

1 passes an equally large maximum2 passes a larger minimum3 passes a larger maximum4 reaches the run of drop 25 reaches the run of drop 16 falls out7 falls out8 reaches the run of drop 61 and 6, 2 and 5, 3 and 4; 7 and 8 form closed loops (i.e. stress cycles)

Depict the loading sequence as a function of time. start with largest max or smallest min use straight lines between min and max

Let drops start from every max and min nd stop if: it starts from max and passes a larger or equal max it starts from min and passes a larger or equal min it reaches the run of another drop

Identify closed loops by joining drops


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Assume that, during the service life, we have 500 loadings of type 1 (defined by mid-value and magnitude), 1000 loadings of type 2 and 10000 loadings of type 3

The Palmgren Miner rule states that failure occurs when

DnNjj

Ji

ii

I= =

= =

1 1

1

where Dj is the damage of a load cycle, ni is the number of applied load cycles of type i, and Ni is the pertinent fatigue life

Damage accumulation Palmgren - Miners Rule

1 2 3

11

12

13

14

15

16 N71

a

DnNjj

Ji

ii

I= = + +

= 12 1 22

2 32

3 12

, , , , , , c

EQS ad

ad

S h,mid eS= + >32 ij ij

c, ,

Crossland criterion

EQC a a a a a a C h,max eC= ( ) + ( ) + ( ) + >12 1 22

2 32

3 12

, , , , , , c

EQS ad

ad

C h,max eC= + >32 ij ij

c, ,

Dang Van criterion

EQDV1,a 3,a

DV h,max eDV2=

+ >c

Valid only for proportional loading

(in-phase and fixed principal directions)


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Equivalent stress criteria components

Shear stress measures

The shear stress initiates microscopic cracks (stage I crack growth) A static shear stress have no influence onfatigue damage the shear stress amplitude is employed

Hydrostatic stressMean value of normal stresses that opensup cracks (Stage II crack growth)

h = = + +( )13

13 11 22 33ii

regardless of coordinate system(stress invariant)


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The deviatoric stress tensor

The stress tensor can be split into deviatoric and volumetric part

ij

xx xy xz

yz yy yz

zx zy zz

xx xy xz

yz yy yz

zx zy zz

ij ij kk

=

=

+

= + = +

h

h

h

h

dh

d

1 0 0

0 1 0

0 0 113

s I

The volumetric part contains the hydrostatic stressThe deviatoric part reflects influence of shear stresses

Midvalue:

ij

xx xy xz

yx yy yz

zx zy zz

,md

md

dm

dm

dm

dm

dm

dm

dm

dm

dm

= =

s (proportional loading)

Amplitude: ij ij ijt t,ad d

,md( ) = ( ) (or s s sa

d dmdt t( ) = ( ) )


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x

x

x

x

Low Cycle Fatigue

Stresses close to (or at) the yield limitSmall stress increment large strain increment. Best resolution if strains are employed in fatigue model

Induced fatigue damage due to global plasticity

Loading above yield limit, (LCF) gives

With stress concentration factor

K

max

and strain concentrationfactor

K

max

we get K K

K

Kt

K

K

K > K

YtK


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Stress concentrations in LCF Neubers rule

At a stress concentration,Neubers rulegives the ralation between stress and strain as

max maxf2

max,a max,af2

,a

=

=

KEK

E

2

2

This equation has two unknown

Stress and strain must also fulfil constitutiverelationship (for cyclic loading)

2 equations and 2 unknown

max

max

Constitutive relation

Neuber hyperbola


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LCF Design Rules

According to Morrow, the relationship between strain amplitude, a , and pertinent number of load cycles to failure, Nf can be written as

a f f f f= ( ) + ( )

EN Nb c2 2

or, with a static mean stress 0

a f m f f f= ( ) ( ) + ( )

EN Nb c2 2

According to Coffin Manson, the relationship can be simplified as

a = 1.75UTS

ENf

0.12 + 0.5D0.6Nf 0.6

Morrow

Morrow with mean stress correction

Coffin Manson


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Fatigue crack growth

In experiments, crack propagation has been measured as a function of the stress intensity factor

I II III

logdadN

log KKth KC

There exists a threshold value of K below which fatigue cracks will not propagate

At the other extreme, Kmax will approach the fracture toughness KC, and the material will fail

Linear relationship between log d d

aN( ) and K in region II

d da N depends also on crack size. This is not shown in the plot


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Paris law

Paris law can be written asdd

aN

C K m= (C and m are material parameters)

1. Find stress intensity factor for the current geometry

2. Find crack length corresponding to K Kmax = C3. (Check if LEFM is OK)4. Integrate Paris law5. Solve for the number of stress cycles corresponding to failure

Paris law does not account for mean stress effects (described by the R-ratio) history effects (introduced by H)

and is only valid for uniaxial loading long cracks LEFM-conditions


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Variable amplitude loading

A (tensile) overload will introduce (compressive) residual stresses

These residual stresses will influence K and thus the rate of crack propagation

The Wheeler model is used to defines reduction of the crack growth rate due to overload

The reduction factor is defined as

R c0d

= +

a d

Reduced crack growth rate is then calculated as

dd

ddR

RaN

aN

=

crack

d0

adc


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Crack closure

Normally cracks only grow when they are open

The Elber accounts for crack closure, also for tensile loads by defining an effective stress intensity range

Paris lawK K K

K K

= [ ]max min

min minmax ,0

Elber correction for crack closure att K=KopK K Keff op max

Modified Paris lawd

d effa

N C Km=

Empirical relation

K R K

R R R Rop =

= + +

( )

( ) . . . max , where

0 25 0 5 0 25 1 12


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Crack closure

Kmax

Keff

KopKmin

K

Crack closure

Using Elber correction in Paris law is non-conservative (predicts a longer fatigue life)compared to standardParis law

The only difference when using Elber correction is in a new, higher Kmin

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1-10

-8

-6

-4

-2

0

2

4

6

8

10

RK

op a

nd K

min

KopKmin

Smallest magnitude of Kmin in Paris law

KParisKElber

KParis


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Crack arrest at different scales

A The load magnitude is below the fatigue limit we will not initiate any (macroscopic cracks)

B The applied load gives a stress intensity below the fatigue threshold stress intensity macroscopic cracks will not continue to grow

A

B

Fatiguefailure

loga

KI,th = U a

No fatigue failure

No propagation

log

log e

Documents

Fatigue initiation Low Cycle Fatigue FATIGUE - Chalmersanek/teaching/fatfract/98-11.pdf · Low Cycle Fatigue Stresses close to (or at) the yield limit Small stress increment large