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MMJ1133 – FATIGUE AND FRACTURE MECHANICS

D – FATIGUE: STRAIN-LIFE APPROACH

FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM


Course Content:

A - INTRODUCTION

Mechanical failure modes; Review of load and stress analysis –

equilibrium equations, complex stresses, stress transformation,

Mohr’s circle, stress-strain relations, stress concentration; Fatigue

design methods; Design strategies; Design criteria.

B – MATERIALS ASPECTS OF FATIGUE AND FRACTURE

FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 2

Static fracture process; Fatigue fracture surfaces; Macroscopic features; Fracture mechanisms; Microscopic features.

C – FATIGUE: STRESS-LIFE APPROACH

Fatigue loading; Fatigue testing; S-N curve; Fatigue limit; Mean

stress effects; Factors affecting S-N behavior – microstructure, size

effect, surface finish, frequency.


D – FATIGUE: STRAIN-LIFE APPROACH

Stress-strain diagram; Strain-controlled test methods; Cyclic

stress-strain behavior; Strain-based approach to life estimation;

Strain-life fatigue properties; Mean stress effects; Effects of surface

finish.

E – LINEAR ELASTIC FRACTURE MECHANICS

Fundamentals of LEFM – loading modes, stress intensity factor, K;

Geometry correction factors; Superposition for Mode I; Crack-tip


Geometry correction factors; Superposition for Mode I; Crack-tip

plasticity; Fracture toughness, KIC ; Plane stress versus plane strain

fracture; Extension to elastic-plastic fracture.

F – FATIGUE CRACK PROPAGATION

Fatigue crack growth; Paris Law; da/dN-∆K; Crack growth test method; Threshold ∆Kth ; Mean stress effects; Crack growth life

integration.

.


Tegasan ( M

Pa)

0

200

400

600

Ujikaji A

Ujikaji B

Mechanical (tension) test

P


Terikan

0.0 0.1 0.2 0.3 0.4 0.50

4

P

lo+∆lloAo

• Effects of high straining rates

• Effects of test temperature

• Foil versus bulk specimens


Tensile testing machine

Load cell


Specimen

Load cell

Specimen grips

Crosshead

Data acquisition

system

Extensometer


Monotonic stress-strain behavior Engineering stress

oA

PS =

A

P=σ

( )oo

o

l

l

l

lle

∆=

−=

dl=ε

== ∫ldl

ε

Typical low carbon steel

True stress

Engineering strain

True or natural strain


l

dld =ε

== ∫

ol

l

l

dllnε

( )eS += 1σ

( )eA

Ao +== 1lnlnε

For the assumed constant

volume condition,

Then;

oo lAlA =


Monotonic stress-strain behavior

Bridgman correction for cylindrical

specimen of ductile material

Necking cause biaxial stress state at

the neck surface and triaxial stress

state at the neck interior.


+

+

=

R

D

D

R

A

P

f

f

f

41ln

41 min

min

σ


Plastic strains

Elastic region – Hooke’s law:

εσ E=

Tegasan ( M

Pa)

200

400

600

εσ log)1(loglog += E

Plastic region:

( )npK εσ =

εσ logloglog nK +=SS316 steel


Terikan

0.0 0.1 0.2 0.3 0.4 0.50

200

Ujikaji A

Ujikaji B

Engineering stress-strain diagram

True stress-strain diagram

MMJ1133 – FATIGUE AND FRACTURE MECHANICSSTRESS, σ (MPa)

400

500

600

700

1000

199.03.747 pεσ =

Plastic strains - Example

Non-linear /Power-law

SS316 steel ( )npK εσ =


PLASTIC STRAIN, εp

0.0 0.1 0.2 0.3 0.4 0.5 0.6

STRESS,

0

100

200

300

PLASTIC STRAIN, εp

0.01 0.1 1

STRESS, σ (MPa)

100

σ = Kεn

log K = 2.8735n = 0.1992

r2 = 0.9772


Loading-unloading behavior


n

peKE

1

+=+=σσ

εεεBauschinger

Effect


Strain-controlled tests

Stress response:

Cyclic hardening


Cyclic softening


Cyclic stress-strain behavior



Hysteresis Loops


∆σ - stress range

∆εe – elastic strain range

∆εp – elastic strain range

∆ε – total elastic strain range

ppeE

εσ

εεε ∆+∆

=∆+∆=∆


Stable cyclic stress-strain hysteresis loops

The type of behavior shown

by gross plastic deformation

is similar to that which

occurs locally at notches

and crack tips.

(elastic constraint

surrounding a local plastic


surrounding a local plastic

zone)


Cyclic Stress-Strain Curves Difficult to predict fatigue strength of

a material from values of monotonic

yield and ultimate strength


Ausformed H-11

steel, 660 Bhn

SAE 4142 steel,

400 Bhn


Cyclic Stress-Strain Behavior

n

p

a K

′

∆′=

2

εσ

∆∆∆ εεε


50

ksi

0.01

in./in. naa

n

pea

KE

KE

′

′

′

+=

′

∆+

∆=

∆+

∆=

∆=

1

1

22

222

σσ

σσ

εεεε


Strain-Life Approach The strain-life approach or local strain approach

is able to account directly for the plastic strains

often present at stress concentration.

To relate life to nucleation of small macrocrack

(initiation life) for notched part

TO

Life of small unnotched specimen

Cycled to the same strain as the material at

notch root.Elastic zone

σ


notch root.

Inconsistent definition of Low Cycle Fatigue life:

-Life to a small detectable crack

- life to a certain percentage (50%)

decrease in tensile load

- life to a certain decrease in the ratio of

unloading to loading moduli

- life to fracture

Elastic zone

Notch plastic

zone


Strain-Life Curves

( ) ( )cff

b

f

f

pea

NNE

22

222

εσ

εεε

ε

′+′

=

∆+

∆==

∆

For elastic behavior:

( )bN2σσσ

′==∆


( )ffa N2

2σσ ′==

(Basquin’s equation)

For elastic behavior:

( )cff

pN2

2ε

ε′=

∆

(Manson-Coffin relationship)


exponentstrengthFatigueb

exponentductilityFatiguec

tcoefficienstrengthFatigueσ

tcoefficienductilityFatigueε

F

F

−

−

−′

−′Strain-Life Curves

Transition life, 2Nt occurs when:

∆∆ εε


SAE 1020 steel

(similar to BS 070M20)

( ) ( )

cb

f

f

t

c

tf

b

t

f

pe

EN

NNE

−

′

′=

′=′

∆=

∆

1

2

22

22

σ

ε

εσ

εε

Nt


Cyclic Properties of Some High Strength Steels



Method of Universal Slopes

(To simplify the job of estimating fatigue life. or failure)

( ) ( )cF

bF NNE

222

εσε

′+′

=∆ Difficult to evaluate total strain at localized

strain concentration region

Simplification

(Based on fitting of data from different metals including steels, Al and Ti alloys)


6.0

12.0

5.3

+=∆NEN

S fUε

ε

(Based on fitting of data from different metals including steels, Al and Ti alloys)

Ref: S.S. Manson, “Fatigue: A Complex Subject –

Some Simple Approximation,” Exp. Mech. Vol. 5,

No. 7, July 1965, pp. 163.

( ) ( ) ( ) 56.0155.009.0

832.0

20196.02623.02

−− +

=∆

fffU NNE

Sε

ε

Ref: U. Muralidharan and S.S. Manson, “Modified

Universal Slopes Equation for Estimation of Fatigue

Characteristics,” Trans. ASME, J.Eng. Mater. Tech.,

vol. 110, 1988, pp. 55.


For Design Purpose

To determine σa when 2N is specified

naa

′

′+=

∆1

σσε


KE

′

+=2

( ) ( )cF

bF NNE

222

εσε

′+′

=∆

( )nF

FK ′′

′=′εσ

Use

Iterate these

equations to solve

for σa


Mean Stress Effects


In LCF , εa ↑ , σm relaxation↑ , σm effect ↓


Mean Stress Effects

Morrow’s mean stress method

( ) ( )cff

b

f

mf

a NNE

222

εσσ

εε

′+−′

==∆


( ) ( )fffa NN

E22

2εε +==

( ) ( ) ( ) cb

fff

b

ffa NENE+′′+′= 22

22

max εσσεσ

Smith, Watson and Topper (SWT parameter)

( ) ( )cf

b

c

f

mf

f

b

f

mf

a NNE

222

′

−′′+

−′==

∆σ

σσε

σσε

ε


Surface Finish Effects

Fatigue cracks nucleate early in

LCF due to large plastic

deformation, thus little influence

of surface finish at short life.

For long life, the effect is


For long life, the effect is

handled by modifying the slope

b to b’ .

For steels with Sf at 106 cycles:

b’ = b + 0.159 log ks

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