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02 Low Cycle Fatigue for Base Material

Reference : Fundamentals of Metal Fatigue Analysis Ch. 2 Strain – Life

2017. 9

by Jang, Beom Seon


High Cycle Fatigue vs Low Cycle Fatigue Each failure occurs by apparently different

physical mechanisms

High cycle fatigue

Low cycle fatigue Significant plastic strain occurs during at least some

of the loading cycles.

Relatively short fatigue lives between 10~100,000 cycles

Ductility and resistance to plastic flow are important

Post welding treatment and high tensile material are not effective.

Engineering Structures are designed such that the nominal loads remain elastic.

However, stress concentrations often cause plastic strains to develop in the vicinity of notches.

Crack initiation life is estimated.

2.1 INTRODUCTION

2


Basic Definitions Engineering Stress and Strain

The true stress is defined as the ratio of the

applied load to the instantaneous cross

sectional area

The true strain is defined as the sum of all

the instantaneous engineering strains.

Incase of engineering strain.

2.2 MATERIAL BEHAVIOR - 2.2.1 Monotonic Stress-Strain Behavior

0A

PstressgengineerinS

00

0

l

l

l

llstraingengineerine

P = applied load

l0 = original length

d0 = original diameter

A0 = original area

l = instantaneous length

d = instantaneous diameter

Original and deformed configuration

A

Pstresstrue

00

lnl

l

dl ltrue strain

l l

3

0

0

0 0

engineeringstrainl

l

l ldle

l l


True and engineering stress-strain

True and Engineering Stress-Strain Relationship (valid up to necking)


AllA 00

0

0

l

l

A

A

A

A

l

l 0

0

lnln

0SAP A

P

A

AS 0

A

Ae 0ln1ln e

A

A10

)1( eS

lll 0The instantaneous length :

)1ln(1lnln00

0 el

l

l

ll

The true strain :

The volume remains constant up to necking

e 1lnComparison engineering and true

stress-strain

4




)1( eS

e 1ln(ε, σ) =(0.182, 456)(e,S)=(0.2,380)

Comparison engineering and true stress-strain

5


Stress-Strain relationship

Total true strain (εt) = Linear elastic strain (εe)+ plastic strain (εp)

For most metals a log-log plot of true stress versus true plastic strain

is modeled as a straight line.

K : strength coefficient, n : strain hardening exponent.

True fracture strength

Af : area at fracture, Pf : load at fracture.


True fracture ductility, true strain at final

fracture.

RA : Reduction in area

K can be defined in terms of σf and εf .

σf

εf

Elastic and plastic strain

pet

n

pK )( n

pK

1

)(

f

f

fA

P

0

00 ,1

1lnln

A

AARA

RAA

A f

f

f

n

ff K )( n

f

fK

6


Stress-Strain relationship

Plastic strain can be defined in terms of these quantities.

Total strain can be expresses as

Elastic strain

Total strain can be rewritten as


σf

εf

Elastic and plastic strain

n

pK

/1)(

n

f

fK

n

n

ff

p

/1)(

n

f

n

f /1)(

n

f

f

/1)(

pet

nt

KE

1

)(

Ee

7



Example of True Stress-Strain Curve

E = modulus of elasticity = 20600 MPa

n = cyclic strain hardening exponent =0.193

K = cyclic strength coefficient = 1210 MPa

8


pe

Engineering stress-strain curve (steel) Engineering stress-strain curve (aluminum)


Low Cycle Fatigue Calculation Procedure

2.1 INTRODUCTION

Strain-life curve

True Strain Amplitude

Low Cycle Fatigue

Stabilized Cyclic Strain-Stress Curve

True Strain-Stress Relations under inelastic

loading : Hysteresis Curve

1) Companion Sample 2) Incremental Step Test

Massing’s hypothesis

9



2.1 INTRODUCTION

Strain-life curve


Low Cycle Fatigue






10


Initial loading

2.2.2 Cyclic Stress-Strain Behavior – Hysteresis loop

Cyclic stress-strain curves are useful for assessing the

durability of structures and components subjected to repeated

loading.

Hysteresis loop : the response of a material subjected to

inelastic loading.

The area within the loop : plastic deformation work

done on the material

2.2 MATERIAL BEHAVIOR

Hysteresis Loop

Total strain range.

Total strain amplitude

the elastic term may be replaced

pe

222

pe

222

p

E

11


2.2.2 Cyclic Stress-Strain Behavior – Baushinger effect


1. Tensional loading : past

the yield strength, σy, to some

value σmax

2. Compressive loading :

inelastic (plastic) strains

develop before –σy is reached.

σy

σmax

2σy

σy

σmax

σy

σmax

12


Constant strain amplitude

Stress response

Cyclic stress-strain

response

2.2.3 Transient Behavior : Cyclic Strain Hardening

The stress-strain response of metals is often drastically altered due

to repeated loading.

1. Cyclically harden : maximum stress increases with each cycle of strain.

→ requires more load to keep imposing the constant strain.


Cyclic Hardening

13


Constant strain

amplitude

Stress

response

Cyclic stress-

strain response

2.2.3 Transient Behavior : Cyclic Softening

2. Cyclically soften : maximum stress increases with each cycle of strain

→ requires less load to keep imposing the constant strain.

3. Be cyclically stable : requires the same load

4. Have mixed behavior(soften or harden depending on strain range)


Cyclic Softening

14


What is dislocation?

Dislocation : a crystallographic(결정학상의)

defect, or irregularity, within a crystal structure.

A crystalline material : consists of a regular array

of atoms, arranged into lattice planes.

An edge dislocation : a defect where an extra

half-plane of atoms is introduced mid way through

the crystal, distorting nearby planes of atoms.

A screw dislocation : Imagine cutting a crystal

along a plane and slipping one half across the

other.


Crystal lattice showing atoms and lattice planes

Edge dislocation

Screw dislocation

15

//upload.wikimedia.org/wikipedia/commons/a/aa/Dislocation_edge_b.jpg

//upload.wikimedia.org/wikipedia/commons/a/aa/Dislocation_edge_b.jpg

//upload.wikimedia.org/wikipedia/commons/9/99/Dislocation_edge_d2.svg

//upload.wikimedia.org/wikipedia/commons/9/99/Dislocation_edge_d2.svg

//upload.wikimedia.org/wikipedia/commons/5/52/Dislocation_screw_e.svg

//upload.wikimedia.org/wikipedia/commons/5/52/Dislocation_screw_e.svg


2.2.3 Transient Behavior : Cyclic Hardening and Softening

The reason of materials soften or harden

For soft material : initially the dislocation density is low. The density

rapidly increases due to cyclic plastic straining contributing to

significant cyclic strain hardening

For hard material : subsequent strain cycling causes a

rearrangement of dislocations which offers less resistance to

deformation and the material cyclically softens.

: the material will cyclically harden

: the material will cyclically soften


4.1y

ult

2.1y

ult

16

1.4, 0.2ult

y

n

Hard material

Soft material

1.2, 0.1ult

y

n


2.2.3 Transient Behavior : Cyclic Hardening and Softening

Between 1.2 and 1.4, small change in cyclic response.

Monotonic strain hardening exponent, n, can be used to predict the

material's cyclic behavior.

n> 0.20 the material will cyclically harden

n< 0.10 the material will cyclically soften

Cyclically stable condition reaches after 20~40% of the fatigue life.

Fatigue properties are usually specified at 50% of fatigue life when

the material response is stabilized.


17

n

pK )(



2.1 INTRODUCTION

Strain-life curve


Low Cycle Fatigue


using 1) Companion Sample 2) Incremental Step Test


loading : Hysteresis Curve Massing’s hypothesis

18


2.2.4 Cyclic Stress-Strain Curve Determination

1. Companion samples : A series of

samples are tested at various strain

levels and the stabilized hysteresis

loops are superimposed and the tips

of the loops are connected. Time

consuming.

2. Incremental step test : widely

accepted since quick and good results.

The response stabilizes after 3-4

blocks and fails after about 20 blocks.

The tips of the stabilized hysteresis

loops are connected → Cyclic Stress-

Strain Curve


Incremental step test

Companion samples

19



An example of Incremental step test


Strain HistoryHysteresis loop &

Cyclic Stress-Strain Curve

20



2.1 INTRODUCTION

Strain-life curve


Low Cycle Fatigue






21



After the incremental step test, if the specimen is pulled to failure,

the stress-strain curve will be nearly identical to the one obtained by

connecting the loop.

Massing’s hypothesis : the stabilized hysteresis loop may be

obtained by doubling the cyclic stress-strain curve.


Stabilized Cyclic Stress-Strain Curve

Doubling Hysteresis Loop

22


Cyclic true stress versus plastic strain

Log-log plot of the completely reversed stabilized cyclic true stress

versus true plastic strain

Where, = cyclically stable stress amplitude

ε = cyclically stable plastic strain amplitude

K′ = cyclic strength coefficient

n′ = cyclic strain hardening exponent (0.10 ~0.25, average 0.15)

2.3 STRESS-PLASTIC STRAIN POWER LAW RELATION

Log-log plot of true cyclic stress versus true

cyclic plastic strain

Total strain is

'( )n

pK 1/ '( ) n

pK

1/ '( ) n

E K

23


Total strain is

An arbitrary point P1(σ1,ε1)

on Cyclic Stress-Strain Curve,

From Massing’s hypothesis, P1 can be located on hysteresis curve ,

P′1(∆σ1, ∆ε1) .

Hysteresis loop by Massing’s hypothesis


Cyclic stress-strain Curve

11 1 '

1 ( ) n

E K

1111 2,2 '1

')

2(2 n

KE

'

1

')

2(

22n

KE

Hysteresis Curve

1/ '( ) n

E K

24

This formula represents

this curve not Hysterias

curve itself. Using this,

we can calculate ∆σ for

given ∆ε


Example of Experiment data

Example of actual Monotonic and Cyclic Stress Strain

Curve.

2.5 DETERMINATION OF FATIGUE PROPERTIES

25


Initial application of strain follows stress- strain curve

All successive strains follows hysteresiss curve.

Example 2.1

Q : Consider a test specimen with the following material properties :

E = modulus of elasticity = 30 X 103 ksi

n′ = cyclic strain hardening exponent =0.202

K′ = cyclic strength coefficient = 174.6 ksi

Fully reversed cyclic strain with a strain range, ∆ε, of 0.04. Determine the stress-strain response of the material.


11 1

1 ( ) n

E K

202.0

11

3

1 )4.176

(1030

02.0ksiksi

ksi1.771

1

2( )2

n

E K

ksi2.154

ksi1.7712

ksi02.012

26

①

②

-0.01 ②’

If ① → ②’

144.4ksi

2 1 67.3ksi 2 1 0.01ksi

-0.01

-67.3



2.1 INTRODUCTION

Strain-life curve


Low Cycle Fatigue






27


Strain Life Curve

Stress life (S-N) data on a log-log scale.

Plastic strain –life (ε1-N) data on log-log coordinates by Coffin and Manson

Total strain and the elastic term

2.4 STRAIN-LIFE CURVE

(2 )2

p c

f fN

∆εp /2 = plastic strain amplitude

2Nf = reversals to failure

ε′f = cyclic strength coefficient (≈true facture ductility, εf )

c = fatigue ductility exponent (-0.5~-0.7)

222

pe

(2 )2

b

f fN

∆ /2= true stress amplitude

2Nf = reversals to failure (1 rev =1/2 cycle)

′f = fatigue strength coefficient

b = fatigue strength exponent

E

e

22

(2 ) (2 )2

f b c

f f fN NE

elastic plastic

Strain-life curve

28

2Nf

2Nt


Strain Life Curve


maN

2 fN N

One cycleOne reversal

( ) ( )2

b b

f

EN

(2 )2

f b

fNE

( )2

f bNE

When N=12

f

E

(2 )2

b

f fN

( )2

b

f N

1( ) ( ) ( )

2

b b b

f

N c

Low cycle S-N curve High cycle S-N curve

Low cycle S-N curve in strain-life relationship

29

2Nf

2Nt


Strain Life Curve

Transition fatigue life, 2Nt

Short lives : more plastic strain, wider loop.

Long lives : less plastic loop, narrower loop.

As the ultimate strength increases, the transition life decreases and elastic

strains dominate for a greater portion of the life range.


Relationship between transition life and hardness for steels

22

pe

1/( )2 ( )

f b c

t

f

EN

(2 ) (2 )f b c

f f f f tN N at N NE

Shape of the hysteresis curve in relation to the strain-life curve

30

As the ultimate strength increases


Brinell Hardness number

A measure of the hardness of a material obtained by pressing a hard

steel ball into its surface.

the ratio of the load on the ball in kilograms to the area of the

depression made by the ball in square millimeters

Endurance limit (Fatigue Limit) of S-N Curve of base material is

related to hardness

Se(ksi) ≈ 0.25 x BHN for BHN <400

100 ksi (=689MPa) for BHN > 400

Reference

)(

2

22

iDDDD

FBHN

31


Strain-Life Curve

Definition of failure

Separation of specimen : common for uniaxial loading

Development of given crack length (often 1.0mm)

Loss of specified load carrying capability (often 10 or 50% load

drop)

→ Not a large difference in life between these criteria

Factor of 2

Strain-life approach measure life in terms of reversals (2N), the

stress-life method Cycles (N)

Strain-life approach uses both strain range (∆ε) and amplitude

(εa).

hysteresis curve can be modeled as twice the Cyclic σ-ε curve

versus


32


Methods to determine Properties

The strain-life equation requires four empirical constants ( ).

These can be obtained from fatigue data.

Although these, relationships may be useful, Kʹ and nʹ are usually obtained

from a curve fit of the cyclic stress –strain data using


( )

f

n

f

K

bn

c

(2 )2

b

f fN

(2 )2

p c

f fN

/c

'

f

p

fε

ΔεN

1

22

2 2

b / c

p

f

f

Δε

ε

n

pK )(

( )

b / c

p n

f p'

f

εK

ε

2

2

1( ) ( )

b / c

b / c n

f p p

f

Kε

n

pK )(

, , ,f fb c

33


Methods to determine Properties

Approximate methods

Fatigue strength coefficient ′f

(corrected for necking)

(steels with hardness below 500BHN)

Fatigue strength exponent, b : -0.05~-0.12 for most metals, average of -

0.085

Fatigue ductility coefficient, ε′f :

RA : the reduction in area

Fatigue ductility exponent c : not well defined, -0.5~-0.7


f f

ksi50 uf S

f f RA

f

1

1lnwhere

34


Example 2.2


Q : From the monotonic and cyclic strain-life data for smooth steel specimens.

Determine the cyclic stress-strain and strain-life constants

Monotonic data Sy = 158 ksi, E=2.84 X103 ksi

Su = 168 ksi, f =228 ksi

%RA = 52 εf =0.734

E

ep

22222

35


Example 2.2


E

ep

22222

0

20

40

60

80

100

120

140

160

180

0 0.01 0.02 0.03 0.04 0.05

Δσ/2

Δε/22

p

E

e

22

36


Example 2.2

Fatigue strength coefficients ′f and b by fitting a power law relationship

between ∆/2 and 2N f

Fatigue ductility coefficients ε′f and c by fitting a power law relationship

between ∆ εp/2 and 2N f


(2 )2

b

f fN

(2 )2

p c

f fN

Strain-life Curve

(2 )2

f b

fNE E

f

E

37


Example 2.2

The cyclic strength coefficient, K′ and the cyclic strain hardening

exponent, n′.

1) by fitting a power law relationship to stress amplitude ∆/2 versus

plastic strain amplitude ∆ εp/2. → Preferred

2) From the relationship

From strain-life data v.s. from approximations


( )n

pK 094.0',216' nksiK

( )

f

n

f

K

bn

c 227 , 0.104K ksi n

38

f f

f f

b : average of -0.085

c : -0.5~-0.7


Mean stress effect

Mean strain is negligible but mean stress has a significant effect on

the fatigue life.

At longer lives, mean compressive stress effect is valid.

At high strain amplitudes (0.5% to 1% or above), mean stress tends

toward zero.

2.6 MEAN STRESS EFFECTS

Effect of mean stress on strain-life curve

Mean stress relaxation

39


Modification to strain-life equation I

Morrow suggested.

The strain-life equation,


0(2 )

2 2

f befN

E E

0(2 ) (2 )

2

f b c

f f fN NE

Morrow’s mean stress correction

Independence of elastic/plastic strain ratio from mean stress

Same ratio of elastic to plastic strain, but, vastly different mean

stress

Ratio of elastic to plastic strain is dependant on mean

stress?

40

02( )2 , 0

b

f

f

e

N here bE

“Fatigue life decreases as mean tensile stress 0 increase.”


Smith, Watson, and Topper (SWT)’s modification. For completely

reversed loading

Multiplying the strain-life equation by this term, results in

The term σmax

Modification to strain-life equation I

Manson and Halford’s modification

Too much mean stress effect at short

lives.


0 0(2 ) ( ) (2 )

2

cf fb cbf f f

f

N NE

2

2

max

( )(2 ) (2 )

2

f b b c

f f f fN NE

max (2 )2

b

f fN

0max2

fNmax

It becomes undefined when σmax is negative.

No fatigue damage occurs when σmax <0 ?

41

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