Components which are subjected to loading which varies with time can fail
at stresses well below the material's ultimate strength.
This is known as fatigue failure
and it accounts for the vast majority of mechanical engineering
failures worldwide.
The bolts in an office chair,
the crank arm on your bicycle,
and pressurized oil pipelines, are just a few examples of components
which are subjected to time varying loads, and may be at risk of fatigue failure.
Fatigue failure occurs due to the formation and propagation of cracks.
It is a three-stage process.
The first stage is crack formation. This usually
occurs at free surfaces and at stress concentrations.
In stage 2 the crack grows in size,
and in stage 3,
after the crack has grown to a critical size,
fracture occurs.
So how can we figure out whether a component is likely to fail due to fatigue?
One common approach is to run fatigue tests by subjecting a component
or test piece to a large number of constant amplitude stress cycles, and
counting the number of cycles until it fractures.
If we repeat this test a large
number of times with different applied stress ranges, we can plot the results on
a graph, with the number of cycles to failure N on the horizontal axis and the
applied stress range S on the vertical axis.
Because the number of cycles to
failure can be very large, a log scale is usually used for the horizontal axis.
By fitting a curve to the data points we obtain what is known as an S-N curve.
The S-N curve allows you to calculate the number of cycles until a component is
likely to fail for a given stress range. For example if we have a stress range of
100 MPa or 15 ksi this S-N curve tells us that the number of cycles to
failure is 500,000. If we know that our component is subjected to one cycle per
minute we could predict that our component will fail due to fatigue after
approximately one year.
Fortunately we don't have to perform these
time-consuming fatigue tests ourselves. S-N curves for many different materials
are already published in different engineering codes.
For some materials, and in particular for ferrous materials,
is important to note that the S-N curve
at a very large number of cycles becomes a horizontal line.
This is known as the endurance limit.
Theoretically the component could be cycled at stress
ranges below this level forever, and it will never fail due to fatigue.
This makes the endurance limit an important fatigue design parameter.
It is common to differentiate between high cycle and low cycle fatigue.
High cycle fatigue occurs when the applied cyclical stresses are low and failure
occurs after a very large number of cycles, typically more than 10,000 cycles.
Because the stresses are low we are only dealing with elastic deformation.
Low cycle fatigue involves higher applied cyclical stresses and failure occurs
after fewer cycles. Because the stresses involved are above the material yield
stress both elastic and plastic deformation occur.
In these cases a strain based approach, using for example the Coffin-Manson relation,
is usually preferred to the S-N curve approach.
If we return to the data from our fatigue tests, we can see that there is a
large amount of variability in the data. This is typical for fatigue tests even
when identical test pieces are used.
If we use a best fit S-N curve, as we have
done here, there is a significant possibility that our component will fail
at a much smaller number of cycles than the curve predicts.
This test piece for example failed at
a much lower number of cycles than predicted by our S-N curve.
For this reason S-N curves published in engineering codes are normally shifted
downwards by a certain number of standard deviations to give a reduced
probability of failure.
Here, by shifting the mean curve down on the vertical axis
by two standard deviations, we have reduced the probability of failure from 50% to 1%.
Fatigue tests are usually run for the constant amplitude fully reversing
cycles you can see here. The same stress magnitude is applied in tension
and in compression.
Let's define a few terms - the stress range is defined as the
difference between the maximum and minimum stresses.
The stress amplitude is defined as half of the stress range.
The mean stress is the average of the maximum and minimum stresses.
In this case the mean stress is zero.
But this is only one very specific type of loading.
In some cases we might have a mean
stress which is not equal to zero, as shown here.
This mean stress will have an effect on the fatigue life.
A tensile mean stress will typically result in a shorter fatigue life.
One way to account for a tensile mean stress is to use S-N
curves derived for specific values of mean stress.
But these are often not available,
or would be time consuming to obtain.
Another approach is to use the Goodman diagram,
which adjusts the endurance limit to account
for a mean stress.
Let's see how it works.
On a Goodman diagram the mean stress is shown
on the horizontal axis and the stress amplitude is shown on the vertical axis.
A straight line is drawn between the endurance limit at a mean stress of 0
and the material ultimate tensile strength at a stress amplitude of 0.
If our cyclic loading conditions are located below the Goodman line,
our component will be safe from fatigue failure.
There are a few different variations of this diagram,
as you can see here.
This approach can only be used to
determine whether a component will have an infinite life. It doesn't allow us to
calculate a fatigue life.
In many real-world cases the applied loading is
likely to be far more complex than what we have considered so far.
We can use techniques like the Rainflow counting method to simplify a complex stress
spectrum into a number of simpler constant amplitude cycles
Miner's rule allows us to account for the cumulative damage caused by each of
these different constant amplitude stress ranges.
It calculates the damage fraction D as the sum of the fatigue damage contributions for each stress
range. The individual contributions are calculated by dividing the number of
cycles by the number of cycles to failure for that stress range.
The damage contributions from all stress ranges are then summed.
If the total summed damaged fraction is greater than one
fatigue failure is considered to have occurred.
In this example the damage fraction D sums to 0.94. This is less
than 1, and so fatigue failure has not occurred.
If the structure we are assessing contains an existing crack,
the S-N approach is not suitable for
determining the fatigue life.
If the dimensions of the crack are known, we can
instead determine the fatigue life using a Linear Elastic Fracture Mechanics approach.
This involves calculating a critical crack size which would result in fracture,
and using a crack growth law to calculate
the time required for the crack to grow to this critical size.
But that's enough about fatigue for now.
Stay tuned for more engineering videos!