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Topics in Ship Structures
02 Low Cycle Fatigue for Base Material
Reference : Fundamentals of Metal Fatigue Analysis Ch. 2 Strain – Life
2017. 9
by Jang, Beom Seon
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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
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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
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True and engineering stress-strain
True and Engineering Stress-Strain Relationship (valid up to necking)
2.2 MATERIAL BEHAVIOR - 2.2.1 Monotonic Stress-Strain Behavior
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
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True and engineering stress-strain
2.2 MATERIAL BEHAVIOR - 2.2.1 Monotonic Stress-Strain Behavior
)1( eS
e 1ln(ε, σ) =(0.182, 456)(e,S)=(0.2,380)
Comparison engineering and true stress-strain
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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.
2.2 MATERIAL BEHAVIOR - 2.2.1 Monotonic Stress-Strain Behavior
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
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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
2.2 MATERIAL BEHAVIOR - 2.2.1 Monotonic Stress-Strain Behavior
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
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True and engineering stress-strain
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
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2.2 MATERIAL BEHAVIOR - 2.2.1 Monotonic Stress-Strain Behavior
pe
Engineering stress-strain curve (steel) Engineering stress-strain curve (aluminum)
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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
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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
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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
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2.2.2 Cyclic Stress-Strain Behavior – Baushinger effect
2.2 MATERIAL BEHAVIOR
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
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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.
2.2 MATERIAL BEHAVIOR
Cyclic Hardening
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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)
2.2 MATERIAL BEHAVIOR
Cyclic Softening
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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.
2.2 MATERIAL BEHAVIOR
Crystal lattice showing atoms and lattice planes
Edge dislocation
Screw dislocation
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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
2.2 MATERIAL BEHAVIOR
4.1y
ult
2.1y
ult
16
1.4, 0.2ult
y
n
Hard material
Soft material
1.2, 0.1ult
y
n
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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.
2.2 MATERIAL BEHAVIOR
17
n
pK )(
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Low Cycle Fatigue Calculation Procedure
2.1 INTRODUCTION
Strain-life curve
True Strain Amplitude
Low Cycle Fatigue
Stabilized Cyclic Strain-Stress Curve
using 1) Companion Sample 2) Incremental Step Test
True Strain-Stress Relations under inelastic
loading : Hysteresis Curve Massing’s hypothesis
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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
2.2 MATERIAL BEHAVIOR
Incremental step test
Companion samples
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2.2.4 Cyclic Stress-Strain Curve Determination
An example of Incremental step test
2.2 MATERIAL BEHAVIOR
Strain HistoryHysteresis loop &
Cyclic Stress-Strain Curve
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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
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2.2.4 Cyclic Stress-Strain Curve Determination
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.
2.2 MATERIAL BEHAVIOR
Stabilized Cyclic Stress-Strain Curve
Doubling Hysteresis Loop
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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
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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
2.3 STRESS-PLASTIC STRAIN POWER LAW RELATION
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
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Example of Experiment data
Example of actual Monotonic and Cyclic Stress Strain
Curve.
2.5 DETERMINATION OF FATIGUE PROPERTIES
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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.
2.3 STRESS-PLASTIC STRAIN POWER LAW RELATION
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
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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
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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
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2Nf
2Nt
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Strain Life Curve
2.4 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
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2Nf
2Nt
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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.
2.4 STRAIN-LIFE CURVE
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
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As the ultimate strength increases
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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
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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
2.4 STRAIN-LIFE CURVE
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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
2.5 DETERMINATION OF FATIGUE PROPERTIES
( )
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
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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
2.5 DETERMINATION OF FATIGUE PROPERTIES
f f
ksi50 uf S
f f RA
f
1
1lnwhere
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Example 2.2
2.5 DETERMINATION OF FATIGUE PROPERTIES
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
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Example 2.2
2.5 DETERMINATION OF FATIGUE PROPERTIES
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
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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
2.5 DETERMINATION OF FATIGUE PROPERTIES
(2 )2
b
f fN
(2 )2
p c
f fN
Strain-life Curve
(2 )2
f b
fNE E
f
E
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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
2.5 DETERMINATION OF FATIGUE PROPERTIES
( )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
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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
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Modification to strain-life equation I
Morrow suggested.
The strain-life equation,
2.6 MEAN STRESS EFFECTS
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.”
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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.
2.6 MEAN STRESS EFFECTS
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 ?
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