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Low Cycle Fatigue (LCF) analysis; cont.
(last updated 2011-10-11)
Kjell Simonsson
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Aim
For high loadings/short lives (with respect to the number of load cycles),
fatigue life calculations are generally strain-based.
Since global plastic flow generally is not acceptable, we will focus
attention on localized plastic flow at local stress raisers/stress
concentrations. Having found the total cyclic strain range occurring there,
we may then go on and calculate the expected life by e.g. Morrow’s
equation.
However, before that, as a requisite, we need to take a look at the typical
cyclic plastic behavior of metals.
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Plastic behavior under monotonic loading
The Ramberg-Osgood relation
The plastic behavior of metals can for monotonic loadings often be
described by the Ramberg-Osgood relation


E

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
 

 
E K
1/ n
,n  1
Plastic behavior under cyclic loading
The cyclic stress-strain curve
When a material is subjected to cyclic loading (inducing plastic flow) its
behavior typically gradually change for a number of loading cycles and then
stabilize.
As an example, for a strain controlled test (Rε=-1) of a cyclically softening
material we then schematically get (with the rate of softening strongly
exaggerated and with some appropriate units and numbers on the axes)


t
t
 min
R 
 1
 max
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Plastic behavior under cyclic loading; cont.
The cyclic stress-strain curve; cont.
After making a number of cyclic test, where one for each test finds the
stabilized conditions, the so called cyclic stress-strain curve can be set up
as schematically illustrated below, where it is to be noted that it does NOT
have anything to do with the monotonic loading curve discussed previously,
and that it generally does not look exactly as it.

E

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Plastic behavior under cyclic loading; cont.
The cyclic stress-strain curve; cont.
The cyclic stress strain curve can generally be described by a RambergOsgood type of relation (with DIFFERENT parameters compared to the
case of monotonic loading)


E

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
 

 
E  K' 
1 / n'
, n'  1
Plastic behavior under cyclic loading; cont.
The Masing behaviour
At stabilised cyclic conditions we typically find the behavior illustrated
below, where the inner curve is the cyclic stress-strain curve, and where the
outer curve describes a closed hysteresis loop, given by the relation below


E

1 / n'

  
2


 
 
E  K' 

 


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E


2
K
'


1 / n'
Plastic behavior under cyclic loading; cont.
The Masing behaviour; cont.
The so called Masing behaviour, described by the formulas on the previous
slide, can be used to construct closed hysteresis loops of any type and
location, as illustrated below.










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Plastic behavior under cyclic loading; cont.
A piece of reality
At the Divisions of Solid Mechanics and Engineering Materials some
work has been performed on modelling the cyclic plastic behaviour of
the nickel-base superalloy IN718. The work has been reported in the
following journal article:
Gustafsson D., Moverare J.J., Simonsson K. and Sjöström S. (2011),
Modelling of the constitutive behaviour of Inconel 718 at intermediate
temperatures, Journal of Engineering for Gas Turbines and Power,
133, Issue 9
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Plastic behavior under cyclic loading; cont.
A piece of reality; cont.
Material response 1,6% Strain range, 400°C, R=0
• As can be seen in the curves from the material testing, the hardening
curve of the first loading is lower than that of the following cycling
• This is due to a large
softening of the material,
possibly caused by
the formation of planar
slip bands
• As can be seen, limited
mean stress relaxation
occurs
• By combining the Ohno-Wang kinematic hardening model with an
isotropic softening description, a simple model capable of describing
the observed behavior was found
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Plastic behavior under cyclic loading; cont.
A piece of reality; cont.
Results 1,6% strain range (good fit with few material parameters)
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Plastic behavior under cyclic loading; cont.
A piece of reality; cont.
Results 1,0% strain range (with model optimized for the 1,6% case)
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Plasticity at notches
Stress- and strain concentration factors
At plastic flow, the stiffness of the material decreases with respect to the
elastic behavior. By this it follows, that we in the case of plastic flow at a
notch have
K   K
Schematically, we have the following situation, where Kt is the “ordinary”
stress (and strain) concentration factor for elastic conditions.
K elastic plastic
range
Kt
range
K
K
 nom
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Plasticity at notches; cont.
Neuber’s rule
Neuber suggested that
K K  Kt2
Leading to
K  K  nom  K t  nom K t nom
nom


  
 max,pl
 max,pl
 max,el
 max,el
or
1
E
2
2
 max, pl max, pl  E max,


el
max,el
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Plasticity at notches; cont.
Neuber’s rule; cont.
For a specified nominal ( and thus specified maximum elastic) stress or
strain, the Neuber rule represents a hyperbola, where the form of it is
governed by the RHS of the relation.
 max, pl
 max, pl
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Plasticity at notches; cont.
Neuber’s rule; cont.
However, since the material for cyclic conditions also is to obey the cyclic
stress-strain behavior, the actual stress and strain state at the notch will be
given by the intersection of these curves.
 max, pl
Actual stress and strain state
 max, pl
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Plasticity at notches; cont.
Neuber’s rule; cont.
Note that
• the Neuber rule provides a simple way of finding an approximation of the
stress and strain state at a notch in which the material flows plastically,
without actually carrying out any elasto-plastic calculation
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Topics still not discussed
Topics for the next lecture
On the next lecture we are to look at how fatigue analysis can be carried
out in an FE-context.
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