the local stress-strain fatigue method · pdf filebecause the crack initiation period occupies...

The Local Stress-Strain Fatigue Method ( -N)

© 2010 Grzegorz Glinka. All rights reserved. 1

IntroductionAlthough most engineering structures and machine components are designed such that

the nominal stress remains elastic (Sn< ys) stress concentrations often causeplastic strains to develop in the vicinity of notches where the stress is elevated dueto the stress concentration effect. Due to the constraint imposed by the elasticallystresses material surrounding the notch-tip plastic zone deformation at the notchroot is considered strain controlled.

The basic assumption of the strain-life fatigue analysis approach is that the fatiguedamage accumulation and the fatigue life to crack initiation at the notch tip are thesame as in a smooth material specimen (see the Figure) if the stress-strain states inthe notch and in the specimen are the same. In other words:

The local strain approach relates deformation occurring in the immediate vicinity of astress concentration to the remote or local pseudo-elastic stresses and strains usingthe constitutive response determined from fatigue tests on simple laboratoryspecimens (i.e. the cyclic stress-strain curve and the strain-life curve.

From knowledge of the geometry and imposed loads on notched components, the localstress-strain histories at the tip of the notch must be determined (Neuber or ESEDmethod).

Fatigue damage must be calculated for each cycle of the local stress-strain history(hysteresis loops, linear damage summation)


a) Specimen

b) Notched component

yy

yyx

y

z

yy

''2 2

2bf

f f fN N cE

log

(/2

)

log (2Nf)

f

0

The Similitude Concept states that ifthe local notch-tip strain history in thenotch tip and the strain history in thetest specimen are the same, then thefatigue response in the notch tip regionand in the specimen will also be thesame and can be described by thematerial strain-life ( -N) curve.

yy

Fi

Plastic zone

f/E

The Basic Concept of the -N Method


a)

b)

a) smooth specimenrepresenting the state of affairsat he notch tip of a notchedbody; b) smooth specimenrepresenting the state of affairsat the weld toe in a weldment

peak

x

y

z

peak

x

peakSt

ress

y

S

S

peak

Stre

ss

x

y

M

M

The principal idea of the local stress strainapproach to fatigue life prediction method;


Fatigue damages:

1 21 2

3 4 53 4 5

1 1; ;

1 1 1; ; ;

D DN N

D D DN N N

Total damage: 1 2 3 4 5 ;D D D D D D Fatigue life: NT = N blck=1/D

2

: tK SNeuber

E

ija, ij

a

FiPlasticZone

s

'1

'

n

E K

0

Main Steps in the Strain-Life Fatigue Analysisof Notched Bodies

p e=

m

0

''2 2

2bf

f f fN N cE

se/E

log

(/2

)

log (2Nf)

sf/E

f

02Ne2N

S=

f(Fi)

t

1

2

3

4

5

6

7

8

1'

0

3

2,2'

4

5,5'7,7'

68

1,1'

0


The stepwise - N procedure for estimating fatigue life (canbe summarised as follows - see the Figure below).

• Analysis of external forces acting on the structure and the component in question (a),

• Analysis of internal loads in chosen cross section of a component (b),

• Selection of critical locations (stress concentration points) in the structure (c),

• Calculation of the elastic local stress, peak, at the critical point (usually the notch tip, d)

• Assembling of the local stress history in form of the form of peak and valley sequence (f),

• Determination of the elastic-plastic response at the critical location (h),

• Identification (extraction) of cycles represented by closed stress-strain hysteresis loops (h, i),

• Calculation of fatigue damage (k),

• Fatigue damage summation (Miner- Palmgren hypothesis, l),

• Determination of fatigue life (m) in terms of number of stress history repetitions, Nblck, (No. ofblocks) or the number of cycles to fatigue crack initiation, N.

The details concerning many other aspects of that methodology are discussed below.


Because the crack initiation period occupies major part of a fatigue life of a smoothspecimen the life of the specimen is assumed to be equal to the fatigue crack initiation life.Therefore, only the fatigue crack initiation life at the notch tip can be estimated from thefatigue data obtained form a smooth specimen subjected to the same stress-strain historyas that one occurring in the notch tip. The same history means the same magnitudes of allstress and strain components. If such conditions are satisfied the equality of one stress orstrain component in the notch and the smooth specimen assures that the othercomponents are the same as well. Therefore, it is possible to use in such a case only onestrain or one stress component as a parameter for fatigue damage calculation and fatiguelife estimation. It means that one component characterizes in those cases the entire stress-strain state.

However, if the stress-strain state in the notch tip and in the specimen are not the samecalculations based on only one stress or strain component might be inaccurate.

Therefore, it seems important to review the elastic plastic stress-strain behavior ofmaterials and their mathematical models used in fatigue applications. It is also important toknow the modifications, which should be applied before the uni-axial strain-life ( N)properties can be used if the stress-strain state in the notch tip is not the same as that onein the material specimen used for obtaining relevant material properties.


Information Path for Strength and Fatigue Life Analysis

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Fatigue Life

MaterialProperties


0 0 00 0

0 0 0ij yy

0 00 0

0 0

xx

ij yy

zz

Stress and strain state in specimens usedfor determination of material properties

yy

yy

yy

xx

xx zz yy

33

6-8mm

yy

xy

zyy

yy

Smooth Laboratory Specimens Used for the Determinationof the Curve under Monotonic and Cyclic Loading


Determination of the Stabilized Cyclic Stress-Strain Curve

Incremental Step Test Program

0

0.01

0.02

- 0.01

- 0.0220 cycles

Stra

in,

Time, t

Time, t

Multiple Step Test Program

Stra

in,

0

Stabilized cyclicstress-strain curve

0


Mathematical Expressions Describing the Stress-Strain Curve andthe Shape of the Hysteresis Lop

1'

'2 2 2 2 2npe

E K

1'

'

ne p E K

1'

'

n

E K

1'

'2 2 2n

E K

0

Equation of the cyclic stress-strain curve Equation of the hysteresis loop branch

1'

'2 2 2n

E K

0

E

ep


The Massing Hypothesis

Massing’s hypothesis states that the stabilised hysteresis loop branch may beobtained by doubling the basic material stress-strain curve.

-cyclic stress – strain curve (amplitudes)

- doubled stress – strain curve (ranges)

or

11/

'

n

a aa E K

11/

'2 2 2

n

E K

11/

'2

2

n

E K


Monotonic and cyclic stress-strain curves for variousmetallic materials

50

100

Stre

ss,

[ksi

]

0 0.01 0.02

Strain,

cyclic

monotonic

2024-T4

0.01 0.02

50

100

Stre

ss,

[ksi

]

0

Strain,

cyclic

monotonic

7075-T6

0.01 0.02

50

100

Stre

ss,

[ksi

]

0

Strain,

cyclic

monotonic

Man-Ten Steel

0.01 0.02

50

100

Stre

ss,

[ksi

]

0

150

Strain,

cyclic

monotonic

SAE 4340350 BHn

0.01 0.02

50

100

Stre

ss,

[ksi

]

0

150

Strain,

Ti - 811

cyclic

monotonic

0.01 0.02

50

100

Stre

ss,

[ksi

]0

150

Strain,

cyclic

monotonic

Waspalloy A


The stress-strain response of metals is often drastically altered due torepeated loading. The material may:

– Cyclically harden– Cyclically soften– Be cyclically stable– Have mixed behaviour (soften or harden depending on )

• The reason materials soften or harden appears to be related to thenature and stability of the dislocation substructure of the material.

• For a soft material, initially the dislocation density is low. The densityrapidly increases due to cyclic plastic straining contributing tosignificant cyclic strain hardening.

• For a hard material subsequent strain cycling causes a rearrangementof dislocations, which offers less resistance to deformation and thematerial cyclically softens

If the material will cyclically harden

If the material will cyclically soften

1.4ult

y

1.2ult

y


1. Smooth laboratory specimens are used for thedetermination of the and - N curves.

2. The data points are obtained at half life ofeach specimen to assure that the material isstabilized.

3. 80% -95% of the specimen life spent to createa crack up to 0.5 -1 mm deep.diameter: 6 - 8 mm 6-8mm

Determination of the fatigue strain-life curve


Determination of the Fatigue Strain-Life Curve

' 22

cppa f fN

Number of cycles, Nf

10-4

10 102 103 105 106104

10-2

10-3

Plas

ticst

rain

ampl

itude

ap

f’=1.38

Plastic

0.704RQC-100 Steeluts=758 MPa

Number of cycles, Nf

'

22

bfeea fN

E

Elastic a/E

f’=938 MPa

0.0648

RQC-100 Steeluts=758 MPa

10-4

10 102 103 105 106

10-2

104

10-3

Elas

ticst

rain

ampl

itude

ae

0

E

ep


Fatigue Strain – Life PropertiesIn 1910, Basquin observed that stress-life (S-N) data could be plotted linearly on a log-log scale.

' (2 )2

bf fN

where: 2/ - true stress amplitude; fN2 - reversals to failure (1 rev = ½ cycle);'f - fatigue strength coefficient, b - fatigue strength exponent (Basquin’s exponent)

Parameters f’ and b are fatigue properties of the material. The fatigue strength coefficient, f

’, isapproximately equal to the true fracture strength at fracture f. The fatigue strength exponent, b,varies in the range of 0.05 and –0.12.

Manson and Coffin, working independently (1950), found that plastic strain-life data ( p-N) could belinearized in log-log co-ordinates.

' (2 )2

p cf fN

where: 2p - plastic strain amplitude; 2Nf - reversals to failure; f

’ - fatigue ductility coefficient c - fatigue ductility exponent

Parameters f’ and c are fatigue properties of the material. The fatigue ductility coefficient, f

’, isapproximately equal to true fracture ductility (true strain at fracture), f

’. The fatigue ductility exponent,c, varies in the range of –0.5 and –0.7.


2 22

b cfa f f fN N

E

''

Fatigue Strain-Life CurveSt

rain

ampl

itude

,a

=/2

Number of cycles to failure, Nf

Elastic a/E

f’=938 MPa

0.0648

RQC-100 Steeluts=758 MPa

Plastic

0.704

f’=1.38

Total

102 103 104 105 1061010-4

10-4

10-4

Stra

inam

plitu

de,

a=

/2(lo

gsc

ale)

Number of reversals to failure, 2Nf

2Nt1 (log scale)

f’/E

f’

Total

Elastic

c

b


''2 22

b cf N Nf f fE

'1

'n

E K

The Strain-life and the Cyclic Stress-Strain Curve Obtained fromSmooth Cylindrical Specimens Tested Under Strain Control

(Uni-axial Stress State)

Strain - Life Curve

log

(/2

)

log (2Nf)

f/E

f

0 2Ne

e/E

c

b

0

Stress -Strain Curve

E


CYCLIC PROPERTIES

K - cyclic strength coefficientn - cyclic strain hardening exponent

ys - cyclic yield strengthE - modulus of elasticity

n

KE

/1

FATIGUE PROPERTIES

f - fatigue ductility coefficientc - fatigue ductility exponent

f - fatigue strength coefficientb - fatigue strength exponent

''(2 ) (2 )

2f b c

f f fN NE

The Ramberg-Osgood curve

The Manson-Coffin curve


The Mean Stress Effect

m,1 < 0;

m,2 = 0;

m,3 > 0;

Number of reversals, log(2Nf)100 102 104 106

f’

Stra

inam

plitu

de,l

og(

/2)

m,1

m,2

m,3max,2

min,

2

max,

1

min,1

max,3

min,3

Time

Stre

ss

m,3> 0

m,1< 0m,2 = 0

Stre

ss

Strain


Morrow'

'2 22

b cff

mf fN N

E

Manson-Halford/' '

''2 2

2

c bb cf f

f fm

f

mfN N

E

Smith-Watson-Topper (SWT)

'max

2'2 '2 2

2b b cf

f f f fN NE

Mean Stress Effect Correction Models


0

20

40

60

80

100

120

140

-350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350

Mean Stress m (ksi)

Stre

ssAm

plitu

de

Morrow

SWT

SAE 8620 Alloy Steela - m diagrams for N=106 cycles

a(k

si)

Comparison of Constant Life a - m Curves According to Morrow’sand the SWT Mean Stress Correction Model;

SAE 8620 Alloy Steel; at Nf =106 cycles


Limitations and Physical Interpretation of MeanStress Correction Models

Morrow’s model• The predictions made with Morrow’s mean stress correction model

are consistent with the observations that mean stress effects aresignificant at low values of plastic strain, where the elastic straindominates. The correction also reflects the trend that meanstresses have little effect at shorter lives, where plastic strains arelarge.

• However Morrow,s mean stress model incorrectly predicts that theratio of elastic to plastic strain range is dependent on mean stress.This is clearly not true, because the shape of the stress-strainhysteresis loop does not depend on the mean stress.

• Although Morrow’s mean stress correction model violets theconstitutive relationship, it generally correctly predicts mean stresseffects.


Limitations and Physical Interpretation of MeanStress Correction Models

Manson and Halford model• Manson and Halford modified both the elastic and plastic terms of the

strain-life equation to maintain the independence of the elastic-plastic strainratio from mean stress.

• This equation tends to predict too much mean stress effect at short lives orwhere plastic strains dominate. At high plastic strains, mean stressrelaxation occurs.

Smith, Watson, and Topper (SWT) model• Since SWT parameter is in the general form of

it becomes undefined when max is negative ( max < 0). The physicalinterpretation of this approach assumes that no fatigue damage occurswhen the maximum stress is compressive.

max 0ff N


Stress States in a Notched Body

22

22

2233

33

11

C

A, B

D

Stresses in axisymmetric notchedbody under axial loading

Stresses in axisymmetric notched bodyunder axial, bending and torsion loading

22

22

2233

33

11

C

A, B

12

23

D

n = F/Anet peak = max = Kt n

1

peak

n

22

11

33

3

2

A B

C

M

F

D 0

T

F

T M


ee

E

e

e

E

Fi

ije, ij

e

( )a

a afE

a

a

E

Non-linear elastic-plastic body

Linear elastic body

FiPlasticZone

ija, ij

a

The localized plasticity phenomenon


22 22 22 2222 22

2

2222 22

e ea a

a

n a

a

t a

a

Kor

E

fE

22e

22a

n

2

1

2L

P

P

0 0

E

0

0

0?

forEfor

1'

'

n

KE

The Neuber Rule and material stress-strain curve


Neuber’s Rule The ESED Method

22e 22a

22e

22a

0

B

A

22e 22a

22e

22a

0

B

A A

22 22

2

22 22e e a atnK

E

22

22 2222

2

22

02 2

a

ae

n te

aKd

E22

22 22

aa af

E22

22 22

aa af

E


Neuber’s Rule and theRamberg-Osgood curve

22e 22a

22e

22a

0

B

A

22e 22a

22e

22a

0

B

A A

22 22

2

22 22e e a atnK

E22 22 2 22 2

22

2 2

1

2 2 2 1

a ae et

an

nKE E n K

22 222

1

2

a aa

n

E K22 22

2

1

2

a aa

n

E K

The ESED method and theRamberg-Osgood curve


Graphical solution to the Neuber rule and the equationof the Stress-Strain curve

Cyclic stress-strain curve22

a22

a f( 22a )

Stre

ss

Strain

22a

2

e e22 22 2 2

n a2 2

t aKE

Neuber’s rule

22a0

Elastic behavior


P

timeO

P2

P1

Notched Body

2

1

2L

P

P

S S

a

e

Material Response at the Notch TipDue to Loading and UnloadingReversals of Load P

Load History Stress-Strain Response at the Notch Tip

0

Cyclic stress-straincurve

Not

chtip

stre

ss,

a

Notch tip strain, a

Hysteresis stress-strain curve

Neuber’s hyperpola,

2

maxax

mm

ax2a etK S

m2

ax

2a atK S

P1

P1


22

1 1t nK

E1

10

01, 03

02

2

23

3

22

2 2t nK

E

n

y

dn

0

T

peak

Stre

ss

- curve

'1

'

n

E K

'12

1 11 1 1 '

n

E K'

122 2

2 2 2 '22

n

E K

Simulation of Stress-Strain Response at the NotchTip (Neuber’s Rule) Induced by Cyclic Loading

time

Nom

inal

stre

ss,

n

n1

0

Nominal stress history

n3

n2 n4


Simulation of Stress-Strain Response at the NotchTip (ESED Method) Induced by Cyclic Loading

1 '1

12

1 11 1 ' '

0 2 1n

dE n K

1 '12

2 2 22 2 ' '

0

22 1 2

nd

E n K

n

y

dn

0

T

peak

Stre

ss

- curve

'1

'

n

E K

time

Nom

inal

stre

ss,

n

n1

0

Nominal stress history

n3

n2 n4

1

10

01, 03

022

23

3

1

1 10

12

2ntK

dE

22

2 20

2

2ntK

dE


1'

max maxmax '

1'

'

min max

min max

22

a a na

a a na

a a a

a a a

E K

E K

Cyclic loading and cyclic stress-strain responsesmooth component, non-linear elastic-plastic stress-strain curve

amax

0t

a

amin

0

0

a

amax, a

max

1'

max maxmax '

a a na

E K

1'

'22

a a na

E Ka

amin ,

amin


2 2,max

max max

11''

max maxmax ''

min max min max

22

;

t n t n a aa a

a aa a nnaa

a a a a a a

K KEE

E KE K

Cyclic loading and cyclic stress-strain responsenotched component, non-linear elastic-plastic stress-strain curve

0

0

a

a

1'

max maxmax '

a a na

E K

1'

'22

a a na

E Kamin,

amin

amax,

amax

a

n

FF

n,min

0 t

n

n,max

n


Stress-strain response at the notch tip

Smax

Smin

SSm

Sa

Nominal stress S

S

0

Elastic-perfectly plastic material

Material curve

ys

E

0

Not

chtip

stre

ss

max

min

ysmax

min

max

min

ys ys

No yielding Yielding on the 1st cycle Reversed yieldingys

00 0

Notched component

P

P

Kt

S

2ysmaxt

yst

For K andK

SS 2

ysmaxt

yst

For K andK

SS

2 yst SFor K


The ‘erratic’ relationship between the nominal mean stress Smand the local (at the notch tip) mean stress m

m=0

ys

ys

0 t

Local notch tip stress histories

m

ys

a

0 t

2ysmaxt

yst

For K andK

SS

m

0

a

t

2ysmaxt

yst

For K andK

SS

max

2 yst SK

max

a

m

min

t

t

t

t t

max

a

m

max a

KKK

SKS

2K

SS

S

max

max

a

m

am

in

ys

ys

ys t

a

a

t

t

KK2K

SSS

0

max

a

m

min

ys

ys

ys

max

min min

min

max

Sm

S

0 t

Smax

Nominal stress history S

Sa S


Material Stress-Strain ResponseDue to Variable AmplitudeCyclic Loading

Stre

ss,

Resultingstress history

b)

Strain,

Applied strainhistory

Resultingstress-strain

path

1

2’

3

4

5

6 7

1’

8

7

2


Strain-Stress Hysteresis Loops vs. “Rainflow Counted”Cycles


Stress history

Mathematical Description of Material Stress-Strain ResponseInduced by a Variable Amplitude Stress or Strain History


The linear hypothesis of Fatigue Damage accumulation (the Miner rule)

11

1 ;f

DN

2 51 3 4

1 11 1 1 1 1ff f f

RfN NN N

LND

22

1 ;f

DN 3

31 ;

f

DN

44

1 ;f

DN 5

5

1 ;f

DN

53 41 2

5

532 41

1

1 1 1 1 1 ;

1 !!f f ff f

ii

D D D

N

D

N

DD

N NN

D

if D Failure

2Nf2

a

2Nf1 2Nf3

a

Stra

inam

plitu

de,

a

2Nf4 2Nf5 Reversals

a

a

a

''2 22

b cfa f f fN NE

''

, 2 22 fi

i

b cf

a i fi f fi NN NE


a

a

0 m

A

B

C, E

D

a

a0

m

A

B

C, E

D

t

S

0

A

C

D

E

B

S, a

200 MPa

a

0.005

S

0

A

C

D

E

B

t

The Loading Sequence Effect

2

1

2L

P

P

S S

a

e


Modeling the residual stress effect2

t r N N22 22

K SE

- Neuber’srule

r

e

2

22 22t rK S

E

2

22 22tK SE

t rK SE

tK SE

e

222

22 2202

E

t r E EK Sd

E- ESED method

r

E

e

e

2

2t rK S

E

2

2tK SE

tK SE

t rK SE


Nom

inal

stre

ssS

Time

B

AC

Residual Stress Effect onthe Stress-Strain Responseat the Notch Tip

x

peak

Stre

ss

y

S

S r > 0!

0

Bmax

m

min

r A

Cr= 0!

m

C

A

B

r> 0!Case 1

r A

0

B

min

max

m

Case 2

m

B

r= 0!

A

r< 0!

r< 0!

C

C


Summary of the Local Strain-Life ( -N) Approach

Advantages:• The method takes into account the actual stress-strain response of the material due

to cyclic loading.• Plastic strain, and the mechanism that leads to crack initiation, is accurately

modeled.• This method can model the effect of the residual mean stresses resulting from the

sequence effect in load histories and the manufacturing residual stresses. Thisallows for more accurate damage accumulation under variable amplitude cyclicloading.

• The -N method can be more easily extrapolated to situations involving complicatedgeometries.

• This method can be used in high temperature applications where fatigue-creepinteraction is critical.

• In situations where it is important, this method can incorporate transient materialbehavior.

• This method can be used for both low cycle (high strains) and high cycle fatigue (lowstrains)

• There is only one essential empirical element in the method, i.e. the correction forthe mean stress effect.


the local stress-strain fatigue method · pdf filebecause the crack initiation period occupies...

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