Application of the theory

The method is based on local stresses. In the strain life approach, the real stresses are calculated from the pseudo stresses using for example a Neuber correction. The stress gradient however is calculated based on the pseudo stresses in Simcenter 3D.

During proportional loading the normalized stress gradient does not change with time for a fixed location of the FE structure. Thus, the correction factor does not change with time. In such cases, the correction factor can be used to modify the stress mode accordingly and then enter the fatigue life evaluation as usual. In the case of non-proportional loading, this is no longer true.

In practice, you have to apply the concept of stress gradients to non-proportional loading conditions. Here, the normalized stress gradient changes from time point to time point.

There are two alternatives for handling this problem.

Two approaches for handling nonproportional loading

• Correction based on stress modes:Calculate a correction factor for each applied load case (stress mode).Apply the correction to the stress modes.Superimpose the corrected stress modes.

• Correction based on superimposed stresses:Calculate the gradient of the full stress tensor for all modes.Superimpose the original stress modes and the stress tensor gradients.Use the gradient of the stress tensor to calculate the normalized gradient of the stress variable which is considered.Calculate the correction based on the superimposed stresses and normalized stress gradients.Apply the correction to the superimposed stress.

Comparison of the approaches

Both approaches are the same if applied to proportional loading. However approach a) is dependent on how the unit load cases are set up. In order to see this consider the following simple beam:

Tension in x-direction (load case 1)

The stress is given by:

It is constant with respect to y, thus the normalized gradient is 0.

For this load case, the correction factor f1= 1.

Bending in y-direction (load case 2)

The stress is given by:

At the surface of the beam (y=h/2) we have:

For this load case, we get a correction factor f2 calculated from the normalized gradient 2/h and the correction curve relating the gradient and the factor.

Mixed tension and bending (load case 3)

Here, the stress is the sum of the first two load cases.

At the surface (y=h/2), the normalized stress gradient is:

For this load case we get another correction factor f3 calculated from the normalized gradient and the correction curve.

Mixed load case build from the corrected load cases 1 and 2

If the mixed load case is calculated using approach a) as described above, then the factor between the stresses without correction and the stress with the correction based on the individual load cases, can easily be calculated:

This factor differs from f3. The size of the deviation depends on the parameters of the model and the correction curve.

Conclusion

This simple example shows that it is easy to get different results for loading situations which are essentially the same.

Assume that the loading of the beam is a proportional one given by the mixed case tension and bending. You can either set up an FE-analysis leading to two unit load cases for tension and bending separately or he can setup an FE-analysis where he gets only one load case for the mixed loading. At this point of the analysis both methods are perfectly justified. If you now want to use the stress gradient correction based on the approach a) he applies the same load time series to either both unit load cases for the separate case or to the combined load case. Also what he models with both approaches is the same. Nevertheless he gets different results depending on the way the FE-analysis was performed.

If the second approach b) is used, the results do not depend on how the FE analysis was setup. In both cases the same results (here the correction factor f3) are obtained.

Thus the second approach is implemented in Specialist Durability.

Numerical calculation of stress gradients

The calculation of the stress gradients is a non-trivial task. The stresses are known only at discrete locations (node of element, Gauss points, center stresses) and are discontinuous across elements. We need the gradient at the surface of the structure. When going into the inside of the structure, the stress value and the gradient may change rapidly. Calculating the gradient based on simply averaging the difference quotients will most probably underestimate the gradient. Thus a calculation of the gradient based on one element only seems to be good approximation to the problem.

If the stresses are available at the position node of element, they are averaged at the nodes to get a smoothed stress plot for the calculation of derivatives. If only center stresses are available they are mapped to position node of element and averaged at the nodes. If the stresses are at position node, they are used without modifications. The averaged nodal stresses are used for the gradient calculation only. The remaining process works on the element nodal stresses (if available) in analogy to the durability analysis without stress gradient effects.

Starting at a node of the surface, the normal at this point (with respect to the element under consideration) is calculated. Next, the element interpolation functions Ni (for Tetra4, Tetra8, Penta6, Penta15, Hexa8, Hexa20, Pyra5, Pyra13) are considered, such that

gives the value of the arbitrary quantity f for an arbitrary location in the element given by (r,s,t). From this formula, we are able to calculate the gradient in the local coordinate system of the element.

This gradient is taken at the position M of the node under consideration and projected onto the normal at M. Taking f to be the components of the nodal (averaged) stress tensor we get the required stress tensor gradient. This procedure is performed for all faces and all nodes at the surface giving a stress tensor gradient for each face and each node at the surface.

Parameters for the calculation of the correction factors

Once the stress gradients have been calculated, the correction factors need to be deduced from it. To this end the following parameters can be selected:

The stress variable to be used in the calculation of the normalized gradient . This is one of:

• von Mises

• Max principal

• Pressure

The relationship:

This is a simple list of curves to be implemented within Simcenter 3D and based on the FKM recommendations. The following curves can be chosen (from FKM paper referenced above):

• Stainless steel

• Steel

• Cast steel

• Ductile graphite iron

• Malleable cast iron

• Gray cast iron

• Forgeable aluminum alloy

• Aluminum cast material

The dependency is either calculated from the values a, b.

	a	b
Stainless steel	0.4	2400
Steel	0.5	2700
Cast steel	0.25	2000
Ductile graphite iron	0.05	3200
Malleable cast iron	-0.05	3200
Gray cast iron	-0.05	3200
Forgeable aluminum alloy	0.05	850
Aluminum cast material	-0.05	3200

For

Learn more

The fatigue notch factor

Examples for size effects

Stress gradients

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Macroscopic yielding

Neuber's approach to micro-yielding

Summary for size effects

Surface effects

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