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)
© 2010 Grzegorz Glinka. All rights reserved. 2
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
© 2010 Grzegorz Glinka. All rights reserved. 3
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;
© 2010 Grzegorz Glinka. All rights reserved. 4
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
© 2010 Grzegorz Glinka. All rights reserved. 5
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.
© 2010 Grzegorz Glinka. All rights reserved. 6
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.
© 2010 Grzegorz Glinka. All rights reserved. 7
Information Path for Strength and Fatigue Life Analysis
ComponentGeometry
LoadingHistory
Stress-StrainAnalysis
Damage Analysis
Fatigue Life
MaterialProperties
© 2010 Grzegorz Glinka. All rights reserved. 8
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
© 2010 Grzegorz Glinka. All rights reserved. 9
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
© 2010 Grzegorz Glinka. All rights reserved. 10
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
© 2010 Grzegorz Glinka. All rights reserved. 11
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
© 2010 Grzegorz Glinka. All rights reserved. 12
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
© 2010 Grzegorz Glinka. All rights reserved. 13
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
© 2010 Grzegorz Glinka. All rights reserved. 14
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
© 2010 Grzegorz Glinka. All rights reserved. 15
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
© 2010 Grzegorz Glinka. All rights reserved. 16
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.
© 2010 Grzegorz Glinka. All rights reserved. 17
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
© 2010 Grzegorz Glinka. All rights reserved. 18
''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
© 2010 Grzegorz Glinka. All rights reserved. 19
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
© 2010 Grzegorz Glinka. All rights reserved. 20
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
© 2010 Grzegorz Glinka. All rights reserved. 21
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
© 2010 Grzegorz Glinka. All rights reserved. 22
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
© 2010 Grzegorz Glinka. All rights reserved. 23
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.
© 2010 Grzegorz Glinka. All rights reserved. 24
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
© 2010 Grzegorz Glinka. All rights reserved. 25
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
© 2010 Grzegorz Glinka. All rights reserved. 26
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
© 2010 Grzegorz Glinka. All rights reserved. 27
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
© 2010 Grzegorz Glinka. All rights reserved. 28
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
© 2010 Grzegorz Glinka. All rights reserved. 29
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
© 2010 Grzegorz Glinka. All rights reserved. 30
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
© 2010 Grzegorz Glinka. All rights reserved. 31
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
© 2010 Grzegorz Glinka. All rights reserved. 32
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
© 2010 Grzegorz Glinka. All rights reserved. 33
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
© 2010 Grzegorz Glinka. All rights reserved. 34
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
© 2010 Grzegorz Glinka. All rights reserved. 35
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
© 2010 Grzegorz Glinka. All rights reserved. 36
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
© 2010 Grzegorz Glinka. All rights reserved. 37
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
© 2010 Grzegorz Glinka. All rights reserved. 38
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
© 2010 Grzegorz Glinka. All rights reserved. 39
Strain-Stress Hysteresis Loops vs. “Rainflow Counted”Cycles
© 2010 Grzegorz Glinka. All rights reserved. 40
Stress history
Mathematical Description of Material Stress-Strain ResponseInduced by a Variable Amplitude Stress or Strain History
© 2010 Grzegorz Glinka. All rights reserved. 41
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
© 2010 Grzegorz Glinka. All rights reserved. 42
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
© 2010 Grzegorz Glinka. All rights reserved. 43
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
© 2010 Grzegorz Glinka. All rights reserved. 44
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
© 2010 Grzegorz Glinka. All rights reserved. 45
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.
© 2010 Grzegorz Glinka. All rights reserved. 46