In this blog article I would like to show you a simple fatigue analysis. The analysis is done the way a fatigue specialist will do it and the way a finite element engineer is doing it often.

### The Analysis

We have 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. Transition from finite to infinite life is at 10^6 cycles.

Note that the S-N curves 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 level is S = 90 MPa, R = 0.

The fatigue specialist will make this calculation, using the S-N curve for Kt = 2.5:

N = 10E6 * (68/90)^7.1 = 1.37E5 cycles.

The FE engineer first models the strip and will find a peak stress of approximately 225 MPa. He or she might use the S-N curve for Kt = 1.0 and find this result:

N = 10E6 * (155/225)^8.1 = 7.52E2 cycles.

Here a complete different result is found.

### What Went Wrong?

The fatigue specialist merely used the data fit to find the life of the strip for which test data was available. So he or she must be right. The FE engineer translated the nominal stresses to local stresses and applied an S-N curve for Kt = 1.0 to it, assuming that this Kt = 1.0 S-N curve would be representative for local stresses in a notched geometry.

This assumption proves to be a fundamental mistake.

In the figure below the S-N curves for Kt = 1.0 and 2.5 are show, 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. Let’s assume that that will be done. In that case, the fatigue limit for the Kt = 1.0 curve is corrected for notch sensitivity and the curves for local stresses look like as shown below.

Correcting for notch sensitivity is an improvement. The gradients however are still very different. Now why would that be. By 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.

Most FE engineers use k = 5, this value is taken from all kind of guidelines. For our analysis in this article, the result would be, with 170 MPa being Sf corrected for notch sensitivity :

N = 10E6 * (170/225)^5 = 2.46E5 cycles.

Now this is much closer to the correct result. Note however that the FE engineer now predicts a factor 2 longer life. That is rather unconservative and a too large factor to be comfortable with.

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 absolutely too short-sighted.

### So why should you use Local Stresses?

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. FE 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 many FE engineers fall into 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.

### It could be worse

There are FE engineers arguing that, since there is some plasticity at the notch, strain-life approaches should be used. As explained in an earlier blog, such a situation differs completely from LCF (low cycle fatigue).

Strain-life curves have a plastic part and an elastic part. It is then argued that the elastic part could represent HCF (high cycle fatigue). In that case, local strains are taken from the FE model and the corresponding life is established. Since strain-life curves are determined for un-notched specimens, the same mistakes will be made as with local stresses.

There is another issue. The elastic part of strain-life curves intersects with N=1 at a fixed point, being the strain that is corresponding with the elastic part of the true stress at static failure. What has that to do with fatigue? We have seen earlier that gradients of S-N curves can differ, so assuming a fixed intersect at N=1 is not correct. Even for Kt = 1 curves, such an intersect will depend on aspects like surface conditions which will have no effect on static failure.

So be aware; using strain-life approaches are absolutely inappropriate for HCF analyses.

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