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MMJ1133 – FATIGUE AND FRACTURE MECHANICS
D – FATIGUE: STRAIN-LIFE APPROACH
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
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.
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
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
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 3
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.
.
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Tegasan ( M
Pa)
0
200
400
600
Ujikaji A
Ujikaji B
Mechanical (tension) test
P
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM
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
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Tensile testing machine
Load cell
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 5
Specimen
Load cell
Specimen grips
Crosshead
Data acquisition
system
Extensometer
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
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
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 6
l
dld =ε
== ∫
ol
l
l
dllnε
( )eS += 1σ
( )eA
Ao +== 1lnlnε
For the assumed constant
volume condition,
Then;
oo lAlA =
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
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.
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 7
+
+
=
R
D
D
R
A
P
f
f
f
41ln
41 min
min
σ
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
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
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 8
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 =
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 9
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
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Loading-unloading behavior
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 10
n
peKE
1
+=+=σσ
εεεBauschinger
Effect
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Strain-controlled tests
Stress response:
Cyclic hardening
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 11
Cyclic softening
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Cyclic stress-strain behavior
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 12
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Hysteresis Loops
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 13
∆σ - stress range
∆εe – elastic strain range
∆εp – elastic strain range
∆ε – total elastic strain range
ppeE
εσ
εεε ∆+∆
=∆+∆=∆
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
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
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 14
surrounding a local plastic
zone)
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Cyclic Stress-Strain Curves Difficult to predict fatigue strength of
a material from values of monotonic
yield and ultimate strength
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 15
Ausformed H-11
steel, 660 Bhn
SAE 4142 steel,
400 Bhn
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Cyclic Stress-Strain Behavior
n
p
a K
′
∆′=
2
εσ
∆∆∆ εεε
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 16
50
ksi
0.01
in./in. naa
n
pea
KE
KE
′
′
′
+=
′
∆+
∆=
∆+
∆=
∆=
1
1
22
222
σσ
σσ
εεεε
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
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
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM
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
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Strain-Life Curves
( ) ( )cff
b
f
f
pea
NNE
22
222
εσ
εεε
ε
′+′
=
∆+
∆==
∆
For elastic behavior:
( )bN2σσσ
′==∆
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM
( )ffa N2
2σσ ′==
(Basquin’s equation)
For elastic behavior:
( )cff
pN2
2ε
ε′=
∆
(Manson-Coffin relationship)
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
exponentstrengthFatigueb
exponentductilityFatiguec
tcoefficienstrengthFatigueσ
tcoefficienductilityFatigueε
F
F
−
−
−′
−′Strain-Life Curves
Transition life, 2Nt occurs when:
∆∆ εε
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 19
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
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Cyclic Properties of Some High Strength Steels
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 20
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
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)
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 22
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.
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
For Design Purpose
To determine σa when 2N is specified
naa
′
′+=
∆1
σσε
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 23
KE
′
+=2
( ) ( )cF
bF NNE
222
εσε
′+′
=∆
( )nF
FK ′′
′=′εσ
Use
Iterate these
equations to solve
for σa
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Mean Stress Effects
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 24
In LCF , εa ↑ , σm relaxation↑ , σm effect ↓
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Mean Stress Effects
Morrow’s mean stress method
( ) ( )cff
b
f
mf
a NNE
222
εσσ
εε
′+−′
==∆
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 25
( ) ( )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
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
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
FATIGUE: STRAIN-LIFE APPROACH M.N.Tamin, CSMLab, UTM 26
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