# In or Wöhler curve, which is a plot

In this chapter basic fatigue
theory, stress-life (S-N) curve, Strain life (E-N) curve is provided. In
addition, rainflow counting algorithm is discussed, which is used to simplify
the load history cycles. Following that, review of S-N analysis in the time domain
is briefly discussed.

2.1.
Fundamentals of fatigue analysis

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In a car body most of the
mechanical components which satisfy the mechanical strength criteria are
subjected to cyclic or fluctuating loads because of the different loading
conditions the car body has to encounter during its lifespan. The result of
such loading induces fluctuating or non-proportional cyclic stresses which
mostly leads to failure by fatigue phenomenon. The damage which occurs because
of this fatigue process is cumulative and cannot be recovered back.

Stress life (SN) and strain life (EN) are used to anticipate the overall
service life of the desired component. The choice of using from these two
methods solely depend upon the application.

Stress life curve

The first method which was effectively developed and used to predict
fatigue damage was the SN curve method.

The basis of this method is the stress life or
Wöhler curve, which is a plot between the alternating stress versus the number
of cycles to failure on a log-log graph. The alternating stress is usually
expressed in terms of stress amplitude.

Figure 2.1 describes the S-N curve on a log-log
plot. In this approach all the strains are treated as elastic. N (cycles to
failure) is always plotted on the x-axis.
Stress(S) is plotted on the y-axis, giving the relationship as follows:-

=

(2.1)

Where b is the inverse
slope of the line and is specified as the Basquin exponent, a is related to the
intercept on the y axis of the curve

SN curve is limited to high cycle fatigue. The
calculated elastic stress range is used with the SN curve to effectively
predict the damage inculcated by a specific stress range. The total damage is
calculated by collectively accumulating the damage throughout the operational
history.

Strain life curve

This method for life estimation is applicable to both low-cycle as well
as high cycle fatigue. The main difference of this method when compared with SN
is that in this type the elastic-plastic strains are used to predict damage.                              The basic steps
which are involved in the strain life (EN) approach are as follows:

In the test for fatigue the strain range is controlled and the resulting stabilized
stress range  together
with the cycles to failure is recorded. In this strain life technique, the
cycles to failure is converted to reversals to failure. Each cycle has a total
of two reversals with a symbol 2 being
used.

The total strain is obtained by addition of both the elastic strain (and plastic strain (to formulate a relationship between the applied
strain and the fatigue life.

Then:-

=  +

(2.2)

(2.3)

Where, Fatigue ductility
coefficient

:  Fatigue strength
coefficient

c     :  Fatigue ductility exponent

b     :  Fatigue strength exponent

2:  Number of half cycles,
reversals, to failure

A detailed derivation of Equation (2.3) can be found in BISHOP ET AL.
(2000).

Mean stress effects

Most of the High and low cycle fatigue data is generated by using a
stress cycle which is fully reversed therefore implying that the mean stresses
are zero. However, in actual scenario the loading application has some non-zero
mean stress values on which the alternating stress is superimposed as described
in Figure

Terminology for alternating stress or definition
for cyclic stressing

The mean stress ( can be expressed in terms of maximum stress (and minimum stress

(2.4)

The Cyclic stress amplitude ( is defined as

(2.5)

The stress range) is expressed as

(2.6)

The ratios associated with the fatigue data are
defined as 4

Stress ratio, R =   and Amplitude ratio, A =   =

Mean stress effects
are performed independently in stress life and strain life fatigue 5 .A Haigh
diagram 6 is used in order to present the results from a fatigue test using a
non-zero mean stress. Mean stress is plotted along the X-axis and alternating
stress along Y-axis. Many empirical relations have been used that effectively
relate mean stress to alternating stress.

Example of a Haigh diagram,

The following relations are available in the
stress-life module

Goodman:  +      = 1

Gerber:           + = 1

Sorderberg:    +      = 1

Morrow:           +      = 1

Where is the endurance limit, is the ultimate strength, is the yield strength and is the fatigue strength.

Comparison of these above mentioned relations is
depicted in Fig. … below

2.1
Neuber’s rule

The elastic–plastic strains with the help of a nonlinear FEA can be
easily estimated or can be calculated from a linear–elastic FEA using an
elastic–plastic correction method, such as Neuber’s rule.

It states,
that the product of the nominal stress (S) with the nominal strain (e) is proportional
to the product of local elastic plastic stresses () and strains).

A detailed
description of this rule can be found in RALPH IVAN ET AL. (2001)

Neuber’s Hypothesis states:

(2.4)

Where, is the strain concentration factor

And,   is the stress concentration factor

Before yielding point is achieved, both are equal to

Rearranging equation (2.4) give
the following,

(2.5)

2.2
Rainflow counting

It is a method that is widely used in fatigue durability analysis to
count number of cycles of stress of varying amplitudes by using a fluctuating
stress history. A detailed information on this method can be found in 7. 8
Counting is carried out by looking into the stress strain behavior of the
material.

Procedure
for rainflow counting

A sample stress time history is shown in the figure below, to which the
method of rainflow counting is performed. The basic procedure to obtain the
cycle count is thoroughly explained in the following steps.

·
Peaks and troughs are extracted from the time signal and the points
between adjacent peaks and troughs are discarded.

·
Starting point for a rainflow counting method is either at a peak or a
trough.

·
If the rainflow counting starts at a peak, it is then compared with the
adjacent peak. If the adjacent peak is larger or equal to the starting peak
then the flow stops.

·
If the adjacent peak is smaller than that of the starting peak the flow
continues and then compares with the subsequent peaks in the signal.

·
Similarly, if the rainflow counting starts at a trough, it is stopped by
a trough which is smaller than the starting point.

·
If a new path follow and intercept the flow from an earlier drawn path,
the new path is then stopped.

Eventually, all the segments of the signal will be counted as cycles.

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