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