### LCF-analysis

```Low Cycle Fatigue (LCF) analysis
(last updated 2011-10-05)
Kjell Simonsson
1
Aim
fatigue life calculations are generally strain-based.
The aim of this presentation is to give a short introduction to strain-based
Low Cycle Fatigue (LCF) analysis.
Kjell Simonsson
2
Basic observations; Wöhler-diagrams/SN-curves
Basquin’s relation
Experiments made on smooth test specimens result in stress-life curves (so
called SN-curves) of the type shown below (such diagrams are also
referred to as Wöhler-diagrams)
 max   min


a
log  a
2
 fl Fatigue limit (or Endurance limit),
“utmattningsgräns” in Swedish
log 
'
f
log  fl
Nf
No. of cycles to failure
2 N f No. of load reversals to failure
log 2 N f
In the region of long lives, the curve may often be described by
Basquin’s relation
log  a  log  'f  b log2 N f   log  a  log  'f 2 N f    a   'f 2 N f 
b
Kjell Simonsson
3
b
Wöhler-diagrams/SN-curves; cont.
Basquin’s relation; cont.
Basquin’s relation may be rewritten in terms of elastic strains, or
total strains, since the stress is low for long lives
 a   'f 2 N f b 
log  a
a 
log
 'f
E
 'f
E
2 N f b
or
c.f.
log  a  log
log 2 N f
 'f
E
 b log2 N f 
Now, it is important to note that even if the experimental curve (blue above)
was simply scaled by the Young’s modulus for long lives, this is not the case
for short lives, since the presence of plasticity will make the
stress-total strain relation non-linear.
Kjell Simonsson
4
Wöhler-diagrams/SN-curves; cont.
The Coffin-Manson relation
In the strain representation introduced on the previous slide, it has been
found that the SN-curve for short lives also can be described by a linear
relation in the log-log diagram
log  a
log 
'
f

'
f
log
c.f.
log  a  log  'f  c log2 N f 
or
E
 a   'f 2 N f c
log 2 N f
The relations above are different versions of the so called CoffinManson relation.
Kjell Simonsson
5
Wöhler-diagrams/SN-curves; cont.
Putting it all together; Morrow’s relation
From a mathematical point of view, it is clear that due to the log-log scale
• the contribution to the strain amplitude from Basquin’s relation
is negligible compared to the contribution from Coffin and Manson’s
relation for short lives
• the contribution to the strain amplitude from Coffin and Manson’s relation
is negligible compared to the contribution from Basquin’s
relation for long lives
Furthermore, since both relations lie below the experimental one in the
“overlapping” region, a direct addition of the two relations has a potential to
reasonably well describe the behavior for all fatigue lives. The so obtained
model, found below, is referred to as Morrow’s relation.
Kjell Simonsson
6
Wöhler-diagrams/SN-curves; cont.
Morrow’s relation; cont.
We thus have
log  a
log  'f
log

'
f
a 
 'f
E
2 N f b   'f 2 N f c
E
log 2 N f
Often the two terms in the Morrow relation are identified as the elastic and
plastic part of the total strain, but this is only completely true for long lives
(with no plastic strains) and approximately true for very short lives (with
very small elastic strains compared to the plastic ones). In the
“overlapping” region this interpretation is not applicable!
7
Kjell Simonsson
Wöhler-diagrams/SN-curves; cont.
Morrow’s relation; cont.
Strength coefficient
The following labeling is common
Ductility coefficient
log  a
log  'f
log

a 
'
f
E
 'f
E
2 N f b   'f 2 N f c
Strength exponent
log 2 N f
Kjell Simonsson
8
Ductility exponent
Wöhler-diagrams/SN-curves; cont.
Morrow’s relation; cont.
In order to account for mean stress effects, the Morrow relation is modified
as exaggeratedly illustrated below for a positive mean stress, where the
black curves represents the mean stress corrected relations.
log  a
log  'f
log
 'f
a 
 'f   m
E
2 N f b   'f 2 N f c
E
log 2 N f
Kjell Simonsson
9
An example
Let us assume that a material is subjected to repeated strain sequences for
which we, by a cycle count method (e.g. the Rain Flow Count method), have
found the following inherent strain cycles
• 1 cycle with
 a  0.00275
• 2 cycles with
 a  0.00125
• 2 cycles with
 a  0.0015
The question is now how many cycles the material can withstand if the
influence of the mean stress can be neglected, that the Palmgren-Miner
linear damage accumulation hypothesis is applicable and if the material
data is given by
E  200GPa ,  'f  900 MPa ,  'f  0.26 , b  0.095 , c  0.47
What we need to do, before being able to consider the damage calculation
for one load sequence, is to calculate the number of cycles to failure for
each type of load cycle. This is accomplished by using the Morrow equation.
Kjell Simonsson
10
An example; cont.
For the first load cycle we have
9E8 
0.095
0.47

0.00275  
 0.262 N f 1 
2 N f 1 
 2 E11 
This is a non-linear equation for Nf1, which needs to be solved numerically.
In many pocket-calculators there are features for doing this. However, since
one notes that the Right Hand Side (RHS) of the above relation is a
decreasing function of Nf1, we may find a solution by a simple trial and error
N=10 000
Kjell Simonsson
=>
RHS=?
11
An example; cont.
For the first load cycle we have
9E8 
0.095
0.47

0.00275  
 0.262 N f 1 
2 N f 1 
 2 E11 
Nf1=10 000
=>
RHS=4.23E-3 larger than the LHS, must
increase Nf1 !
Nf1=100 000
=>
RHS=?
Kjell Simonsson
12
An example; cont.
For the first load cycle we have
9E8 
0.095
0.47

0.00275  
 0.262 N f 1 
2 N f 1 
 2 E11 
Nf1=10 000
=>
RHS=4.23E-3
larger than the LHS, must
increase Nf1 !
Nf1=100 000
=>
RHS=2.25E-3
smaller than the LHS, must
decrease Nf1 !
Nf1=50 000
=>
RHS=?
Kjell Simonsson
13
An example; cont.
For the first load cycle we have
9E8 
0.095
0.47

0.00275  
 0.262 N f 1 
2 N f 1 
 2 E11 
Nf1=10 000
=>
RHS=4.23E-3
larger than the LHS, must
increase Nf1 !
Nf1=100 000
=>
RHS=2.25E-3
smaller than the LHS, must
decrease Nf1 !
Nf1=50 000
=>
RHS=2.67E-3
smaller than the LHS, must
decrease Nf1 !
Nf1=40 000
=>
RHS=?
Kjell Simonsson
14
An example; cont.
For the first load cycle we have
9E8 
0.095
0.47

0.00275  
 0.262 N f 1 
2 N f 1 
 2 E11 
Nf1=10 000
=>
RHS=4.23E-3
larger than the LHS, must
increase Nf1 !
Nf1=100 000
=>
RHS=2.25E-3
smaller than the LHS, must
decrease Nf1 !
Nf1=50 000
=>
RHS=2.67E-3
smaller than the LHS, must
decrease Nf1 !
Nf1=40 000
=>
RHS=2.83E-3
larger than the LHS, must
increase Nf1 !
Nf1=45 000
=>
RHS=?
Kjell Simonsson
15
An example; cont.
For the first load cycle we have
9E8 
0.095
0.47

0.00275  
 0.262 N f 1 
2 N f 1 
 2 E11 
Nf1=10 000
=>
RHS=4.23E-3
larger than the LHS, must
increase Nf1 !
Nf1=100 000
=>
RHS=2.25E-3
smaller than the LHS, must
decrease Nf1 !
Nf1=50 000
=>
RHS=2.67E-3
smaller than the LHS, must
decrease Nf1 !
Nf1=40 000
=>
RHS=2.83E-3
larger than the LHS, must
increase Nf1 !
Nf1=45 000
=>
RHS=2.75E-3
= LHS !!!!
Kjell Simonsson
16
An example; cont.
For the first load cycle we have
9E8 
0.095
0.47

0.00275  
 0.262 N f 1 
2 N f 1 
 2 E11 
By doing the same type of calculations for the other two load cycle types, we
get in total
Nf1=4.5E4 , Nf2=2.2E6 , Nf3=7.2E5
Palmgren-Miner then gives the following damage for 1 sequence
D fs 
1
2
2
1



4.5E 4 2.2 E 6 7.2 E 5 38500
Thus, the expected number of load sequences to failure (when D=1) is
N f D fs  1  N f 
Kjell Simonsson
1
 38 500
D fs
17
Wöhler-diagrams/SN-curves; cont.
It is to be noted that there are other ways to describe the SN-curve and its
mean stress sensitivity. However, these are outside the scope of this
presentation.
Kjell Simonsson
18
Topics still not discussed
Topics for the next lecture
On the next lecture we are to look more closely into the description of the
cyclic plastic behavior of metallic materials, and to introduce approximate
methods for estimating the cyclic plastic flow at stress concentrations (in an
otherwise elastic structure/body)
Kjell Simonsson
19
```