# Understanding Fatigue Failure and S-N Curves

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.

Understanding Plane Stress

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.

Understanding Young's Modulus

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.

Understanding Bernoulli's Equation

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!