Low Fatigue - Introduction

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Low Cycle Fatigue (LCF)Solid Mechanics Anders Ekberg
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Low Cycle Fatigue Ð Introduction
Finite or Infinite Fatigue Life?
HCF
•Both infinite or finite fatigue life is possible and 
can be analyzed 
LCF
•Only finite fatigue life is possible and should be 
analyzed using LCF-criteria
Stress or Strain?
HCF
•Elastic material
Small strain increment -> large stress increment
LCF
• Stresses close to (or at) the yield limit
Small stress increment -> large strain increment. 
Best “resolution” if strains are employed in 
fatigue model.
σ
ε
∆σ
∆ε
x
x
x
x∆σ
∆ε
Low Cycle Fatigue (LCF)Solid Mechanics Anders Ekberg
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LCF Ð Introduction contÕd
Damage Mechanisms
Induced fatigue damage will be due to 
• global plasticity
• local plasticity (same as in HCF)
For high load amplitudes (and/or high maximum 
magnitudes of loading) global plasticity will be the 
dominating cause
For low load magnitudes, the model should tend to 
similar results as for HCF criteria (i.e. the Wöhler 
curve)
Note that the static load carrying capacity must not be 
exceeded. This must be checked separately
Low Cycle Fatigue (LCF)Solid Mechanics Anders Ekberg
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Morrow Design Rule
According to Morrow, the relationship between 
strain amplitude, εa , and pertinent fatigue life, Nf 
can be written as
ε ε ε
σ
εa a
el
a
pl f
f f f= + =
′ ( ) + ′ ( )
E
N Nb c2 2
Nf is the number of load cycles to failure
logεalog ′εf
log ′σf
E
εa
el
εa
pl
εa = εa
el + εa
pl
0 2 4 6 log 2Nf( )
σ
ε
x
ε el ε pl
Low Cycle Fatigue (LCF)Solid Mechanics Anders Ekberg
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Mean Stress Effects
According to Morrow, the effect of a mean stress, 
sm, can be taken into account by redefining the 
criterion as
ε ε ε
σ σ
εa a
el
a
pl f m
f f f= + =
′ −( ) ( ) + ′ ( )
E
N Nb c2 2
logεa
log 2Nf( )0 2 4 6
s m < 0
(compressive)
s m > 0
(tensile)
Low Cycle Fatigue (LCF)Solid Mechanics Anders Ekberg
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Coffin Ð Manson Design Rule
For the elastic part, the relationship between strain amplitude 
and fatigue life can be approximated by
ε
σ
a
el UTS
f=
−1 75 0 12. .
E
N
The fatigue life in the plastic part can be approximated by
εa
pl
f=
−0 5 0 6 0 6. . .D N
where D is the ductility, defined as
D A
A
≡ ≈ln 0
fra
fraε
This yields the Coffin – Manson relationship
εa =1.75
σUTS
E
Nf−0.12 + 0.5D0.6Nf−0.6
Low Cycle Fatigue (LCF)Solid Mechanics Anders Ekberg
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LCF Design Ð some notes
Comparison of the criteria
The Morrow criterion includes five material parameters
E b c, , , ,′ ′σ εf f
The Coffin – Manson criterion includes three material 
parameters
E D, ,σUTS
It is also possible to express the criteria using the strain 
range, ∆ε (and not the strain amplitude εa) in the 
Coffin-Manson criterion (see Eq. 5h
Approximations
ε ε
ε ε
a f
a f
= ⇒ = ⇒ =
= ⇒ = ⇒ =
1 2 10
0 5 1 10
3
4
% %
. % %
∆
∆
N
N
Low Cycle Fatigue (LCF)Solid Mechanics Anders Ekberg
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Stress and strain concentrations
Stress concentration
The stress concentration factor ahead of a 
notch is defined as
Kσ
σ
σ
≡
∞
max
In a similar manner, the strain 
concentration factor ahead of a
notch is defined as
Kε
ε
ε
≡
∞
max
If we load above the yield limit, which is 
the case for LCF conditions, we get 
∆ ∆σ ε ε σ≠ ≠ and K K
K
Kt
σ
∞
σY
Kt
K
s
K
e
Kε > Kσ
Low Cycle Fatigue (LCF)Solid Mechanics Anders Ekberg
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NeuberÕs rule
K K Kε σ⋅ = f
2 where K q Kf t= + −1 1( )
which yields
σ ε σ ε σ εσ εmax max f
2
⋅ = ⋅ =
∞ ∞ ∞ ∞
K K K
Assuming elastic conditions far from the 
notch, we get the Neuber hyperbola
σ ε
σ
max max
f
2
⋅ =
∞
K
E
2
The equation of the Neuber hyperbola has 
two unknown
But, the stress must also fulfil constitutive 
relationship between stress and strain
Thus, two equations and two unknown
σ
εεmax
σmax
Constitutive 
relation
Neuber 
hyperbola