## 03 Aug 2021 Neuber’s Rule

When performing a FEA stress analysis, a linear-elastic (https://www.fea-solutions.co.uk/elasticity-and-plasticity/) material model is usually selected. In case the resulting peak stresses are found to be above the yield strength of the material, the underlying assumption that the material would behave in a linear-elastic manner has been violated, and hence the results are by definition invalid. They are still useful, though, because they show where the peak stresses occur, and how much they are above the Yield Strength.

For most applications, stresses above yield are not permitted. Hence, with the linear-elastic FEA results on hand, the design should be modified and re-analysed.

However, sometimes it is useful or required to know from linear-elastic FEA results what the equivalent elastic-plastic stresses are.

This is when Neuber’s rule is used, as follows:

1. For a given linear-elastic stress value, its corresponding point can be plotted on the line of Young’s Modulus (https://www.fea-solutions.co.uk/elastic-modulus/), which describes linear-elastic behaviour. This will give an area of elastic stress x elastic strain.

2. The corresponding point which has equal area to the elastic stress-strain value is found that lies on the Ramberg-Osgood Line, which is an approximation of the stress-strain curve in the plastic region.

This relationship is between the linear-elastic stress strain and plastic stress strain is the following:

_{
}This allows Neuber’s rule to predict non-linear behaviour from a linear-elastic model.

However, if stresses are expected to be above the yield strength of the material, an elastic-plastic analysis including non-linear material behaviour (https://www.fea-solutions.co.uk/non-linear-material/) should be used from the outset to represent plasticity, rather than performing a linear-elastic analysis and applying the Neuber’s rule.

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