There is a difference between the calculation of the fatigue life with a nominal stresses and stress concentration factor, and when using FEA. This article explains an important point to be considered when using the S-N curve with the local stresses from an FEA study.

The Analysis

Consider the case of a strip with a hole, with stress concentration factor Kt = 2.5, made of an arbitrary material. From test data we have obtained S-N curves for R=0 for this material for both Kt = 1.0 and Kt = 2.5 (same material, same geometry, same surface condition and same environment as the strip to be analysed), see figure below.

For Kt = 1.0, the fatigue limit is Sf = 155 MPa and the exponent in the Basquin equation (S^k * N = c) is k = 19.3. For Kt = 2.5, the fatigue limit is Sf = 68 MPa and the exponent is k = 7.1. The transition from finite to infinite life is at 10^6 cycles.

Note that the S-N curves in the figure are quite flat, that is because of the stress ratio R = 0. With increasing mean stress and stress ratio, the upper asymptote decreases faster than the fatigue limit and the S-N curve becomes flatter.

The applied stress amplitude is S = 90 MPa, R = 0.

The using the analytical approach this calculation, using the S-N curve for Kt = 2.5 will be made:

N = 1x10^6 * (68/90)^7.1 = 1.37*10^5 cycles.

In the FEA approach a peak stress of approximately 225 MPa will be found. The temptation here is to use the S-N curve for Kt = 1.0, as the peak stress has been found directly, which would give this result:

N = 1x10^6 * (155/225)^19.3 = 7.52*10^2 cycles.

It is thus seen that the two methods are not the same.

What Went Wrong?

In the analytical approach we merely used the data fit to find the life of the strip for which test data was available, so this must be correct. In the approach with FEA it had been incorrectly assumed that the S-N curve for Kt = 1.0, which represents an unnotched geometry, is also representative for local stresses in a notched geometry.

Let's look into the reasoning further. In the figure below the S-N curves for Kt = 1.0 and 2.5 are shown, but now for local stresses. For Kt = 1.0, nothing will change, for Kt = 2.5 the curve will shift upwards (all stresses are multiplied with the Kt value).

As you can see, these curves are not the same, far from it. Both the fatigue limit and the gradient are different. The difference in fatigue limit is explained by notch sensitivity. The larger the Kt and the more ductile the material is, the larger the difference in fatigue limit will be. There are methods to take the notch sensitivity into account, so the fatigue limit for the Kt = 1.0 curve is corrected for notch sensitivity. The curves for local stresses are consequently as shown in the next figure.

Correcting for notch sensitivity is an improvement. The gradients however are still very different. Why is this?

In the analysis using local stresses only the Kt (i.e, the ratio between the local and nominal stresses) is taken into account. But that is not the only parameter that plays a role. Also the stress gradient at the notch is very important. For Kt = 1.0 (un-notched specimen) the stress gradient is flat, For notched geometries, the gradient increases with the Kt.

The gradient of the S-N curve also depends on the applied stress ratio, with increasing R, the curve becomes flatter.

Typically in an FEA study k = 5 of would be used for the gradient, which is taken from many general guidelines. For our analysis in this article, the result would be, with 170 MPa being Sf corrected for notch sensitivity :

N = 1x10^6 * (170/225)^5 = 2.46x10^5 cycles.

Now this is much closer to the correct result, but still predicts a factor 2 longer life which is not conservative.

The table below gives an indication of Basquin exponents k depending on Kt and stress ratio. The exponent k ranges from 3.6 to over 15. So always using k = 5 is clearly too short-sighted.

So why should you use Local Stresses and FEA?

The analysis above was for a real simple geometry for which the stress concentration factor could be established very easily and accurately. For complex structures that is often much more difficult, if not impossible. This is where a FE analysis becomes useful. FEA is a very powerful and accurate tool to determine stress levels in complex geometries. And accurate stresses is what we need for accurate fatigue analyses.

The pitfall that arises often in FEA studies is the misconception that stresses from FE, a Kt = 1 fatigue limit and a Basquin exponent of k = 5 will lead to an accurate fatigue analysis. Only the input (i.e, stresses) for the analysis might be accurate that way, the subsequent step certainly is not.

For more details on the theory of fatigue check out the learning paths Essentials of Metal Fatigue or Comprehensive Guide to Fatigue.

This blog was curated by Johannes Homan from Fatec Engineering.