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a

m

FL

FLP

UTSY

a FL=

m = 0

t ime

a FLP=

a

m FLP=

Plasticdeformations

FLP

a

mFLP

UTSY

FL

FLP

Loadedvolume

Surface roughness

Size of raw

material

Haigh diagram I

YY

Haigh diagramReduced Haigh diagram

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a

m

UTSY

P

a

mP

( , )K Kt m f a

Haigh diagram II

A

A

C

OB

SFaAA'

AP=

SFmOB'

OA=

SFam

OC'

OP=

m const=

a const=

K

K

f a

t m const

=

K q K

K K

f t

f t

= +

1 1( )

Service stress Safety factors

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Modern Fatigue Design

Background

Evolution in structural design due to increased computat ional power CA D /CA E - software

Need for new fatigue design methods that are va lid for a general type of loading easy to implement in a computer code

Several options, but no method with general validity

HCF: equivalent stress is defined and compared to afatigue limit (expressed in the equivalent stress)

LCF: ca lcula tion of damage connected to theconstitutive model of the materia l. Fat igue damageconnected to the plastic deformation

LEFM: effective stress intensity range

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Multiaxial high cycle fatigue initiation

Problem:

The H aigh diagra m is valid for

U niaxia l loading

O ne stress component

Solution:

Assume that, in the general case,

fatigue behaviour is influenced by

Applied shear stress amplitude

Hydrostatic stress

B ased on these assumptions, derive a fa tigue initiationcriterion that defines a limiting stress magnitude forwhich fatigue cracks will develop) for a general type ofloading.

Assumes undamaged material (continuum mechanics)

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Hydrostatic stress

The hydrostatic stress is the mean valueof normal stressesacting on the materialpoint (positive in tension)

A tensile (positive) hydrostatic stressopens up microscopic cracks (Stage IIcrack growth)

h = + +( )1

3x y z

The hydrostatic stress is a stress invariant

ij =

11 12 13

21 22 23

31 32 33

h = = + +( )1

3

1

311 22 33ii

regardless of coordinate system

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Shear stress measures

The shear stress initiates slip bands whichleads to microscopic cracks (stage I crackgrowth)

Since a static shear stress have noinfluence on the fatigue damage, theshear stress amplitude is employed

Two measures

Tresca shear stress

Tresca =1 32

von Mises stress

We need to define the amplitudes of these

vM = ( ) + ( ) + ( )1

21 2

22 3

23 1

2

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Equivalent Stress Measures

Uniaxial CaseOne stress component

Mid value and amplitude of this stresscomponent are taken to reflect thefatigue properties

The stresses during a load cycles are

defined by a service stress

Multiaxial CaseSix stress components (general case)

Hydrostatic stress and shear stressamplitude are taken to reflect thefatigue properties

The stresses during a load cycles are

defined by a closed curve

Y

FLP

m

Plastic

zone

UTSY

FL

Service

stress

a

FATIGUE

NO FATIGUE

FATIGUE

FATIGUE

NO

FATIGUE

Shear stressamplitude

NO

FATIGUE

Stressesduring oneload cycle

Plasticzone

Plasticzone

e3

c3

e3

e3

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Shear Stress Amplitude

General

I t ha s been found empirically tha t a superposed staticshear stressdoes not ha ve any influence on the fa tigue

initiation FL FLP= whereas FL FLP

In order to eliminate the influence of a superposed shear

stress, the shear stress amplitude is norma lly used inmultiaxia l H C F-criteria

This amplitude is the difference between the currentshear stressmagnitude and the mid value of the shear

stress for the current stress cycleFor the genera l case, this amplitude is rathercomplicated to compute (see Fat igue a Sur vey,Appendix I)

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Shear stress Uniaxial case

Mohrs stress circle for loa ding in a uniaxia l case

2 = 0

max

1 = max

x

2 = 0

max

1 = max

x

2 = 0

max

1 = max

x

time

Ma x normal and shear stress correspond to the samedirections throughout the loa d cycle

45 45 45

t i m e

maxt i m e

P

t i m e

maxmid

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The deviatoric stress tensor

The stress tensor

ij

xx xy xz

yz yy yz

zx zy zz

=

Split into volumetric and a deviatoricpart

ij

xx xy xz

yz yy yz

zx zy zz

xx xy xz

yz yy yz

zx zy zz

=

=

+

= +

h

h

h

h

hd

1 0 0

0 1 0

0 0 1

I

The volumetricpart conta ins the hydrostatic stress

The deviatoricpart reflects influence of shear stresses

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Mid value of the deviatoric stress tensor

In-phaseij ij ija c f t = + ( )

aij and cij are consta nts

f t( ) is a common t ime dependent functionFixed principal directionsE very component of corresponds to a fixed directionthroughout the loading

ij

xx xy xz

yx yy yz

zx zy zz

t

t t t

t t t

t t t

t

t

t

a

a

d

d d d

d d d

d d d

d

d

d

d

( ) =

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

=

( )

( )

( )

=

1

2

3

11

22

0 0

0 0

0 0

0 0

0dd

d

d

d

d

0

0 0

0 0

0 0

0 033

11

22

33a

c

c

c

f t

+

( )

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Movie 1 Click me!
http://www.am.chalmers.se/~anek/teaching/fatfract/Movie1.movhttp://www.am.chalmers.se/~anek/teaching/fatfract/Movie1.movhttp://www.am.chalmers.se/~anek/teaching/fatfract/Movie1.mov

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Mid value of the deviatoric stress tensor III

Limitations In-phase loading

In in-phase loading, thestresscomponents have their

max- and min-magnitudes atthe same instant in time

In out-of-phase loa ding, max-and min magnitudes occur a t

different instants of time fordifferent stress components

time

stress

time

stress

The case of out-of-phase loading is much more difficult toana lyse, for insta nce due to difficulties in

D efining a stress cycle D efining a mid value of the shear stress

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Mid value of the deviatoric stress tensor IV

Limitations Fixed principal directions

Rotating principal directions

" "

ij,pd

d

d

d

=

1

1

3

0 0

0 0

0 0

corresponds to a rotating coordinate system

Instead w e have to look a t the full devia toric stress tensor a ndfind its mid va lue

ij

xx xy xz

yz yy yz

zx zy zz

,md

md

h

h

h m

= =

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

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Amplitude of the deviatoric stress tensor

The mid value of the deviatoric stress tensor is found as

ij,md

md

dm

dm

dm

= =

1

1

3

0 0

0 0

0 0

(proportional loading)

( m denotes mid-value of component during stress cycle)

or as

ij

xx xy xz

yz yy yz

zx zy zz

,md

md

h

h

h m

= =

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

(general)

the amplitude of the deviatoric stress tensor is defined as

ij ij ijt t,ad d

,md( ) = ( ) (or a

d dmd

t t( ) = ( ) )

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Amplitude of the deviatoric stress tensor II

For in-phase loading with fixed principal directions (proportionalloading), we can express the amplitude of the Tresca a nd von Misesstress using the amplitude of the devia toric stress tensor

Tresca,a 1,a

d

3,a

d

2( ) ( ) ( )t t t=

where ( 1,ad 1d 1,md( ) ( )t t= etc)

vM,a a a a a a a( ) ( ) ( ) ( ) ( ) ( ) ( ), , , , , ,t t t t t t t = ( ) + ( ) + ( )1

21 2

22 3

23 1

2

(it can be shown tha t using a or ad

gives the same results)

The ma x values are given a s

Tresca,a1,a

d

3,a

d

2= where (

1,ad 1,max

d

1,min

d

2= )

vM,a a a a a a a= ( ) + ( ) + ( )1

21 2

22 3

23 1

2, , , , , ,

Amplitude of the deviatoric stress tensor II

For in-phase loading with fixed principal directions (proportionalloading), we can express the amplitude of the Tresca a nd von Misesstress using the amplitude of the devia toric stress tensor

Tresca,a 1,a

d

3,a

d

2( ) ( ) ( )t t t=

where ( 1,ad 1d 1,md( ) ( )t t= etc)

vM,a a a a a a a( ) ( ) ( ) ( ) ( ) ( ) ( ), , , , , ,t t t t t t t = ( ) + ( ) + ( )1

21 2

22 3

23 1

2

(it can be shown tha t using a or ad

gives the same results)

The ma x values are given a s

Tresca,a1,a

d

3,a

d

2= where (

1,ad 1,max

d

1,min

d

2= )

vM,a a a a a a a= ( ) + ( ) + ( )1

21 2

22 3

23 1

2, , , , , ,

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Equivalent stress criteria

Sines criterion

EQS ad

ad

ad

ad

ad

ad

S h,mid eS= ( ) + ( ) + ( ) + >1

21 2

2

2 3

2

3 1

2

, , , , , , c

Crossland criterion

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

2

1 2

2

2 3

2

3 1

2

, , , , , , c

Dang van criterion

EQDV

1,a 3,a

DV h,max eDV2=

+ >c

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Concluding remarks

Fatigue analysis

C alcula te the state of stress

Apply the equivalent stress criterion, fatigue if

eq e>

In the case of no f at igue, ca lculate safety coefficient a s

SF=

e

EQ

Pros Cons

Suitable for computer ana lysis C orrosion correction etc.

G eneral state of stress La ck of empirical knowledge

Id entify critical parts of component Separa tes betw een fa tigue /no fa tigue

Have a physical basis

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Lunch

Documents

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