This article explains how a “Neuber Stress” is calculated from fatigue test data obtained from un-notched axially-loaded specimen.

In the presence of a geometric stress concentration feature on a part, the fatigue life of that part is often under-predicted as a result of over-prediction of the effect of stress concentration. Actually, the stress around these zones will be reduced by plastic redistribution of stresses by the material. The Neuber Stress correction factor is used to reduce the apparent high stresses caused near geometric stress concentration zones. Traditionally, when a part with a geometric stress concentration is modelled with a linear elastic material model, the geometric stress concentration factor K_{t}, is used to evaluate stresses near the stress concentrations, resulting in, high stress values. Being a conservative design strategy, this leads to a potentially heavier component. The linear elastic material model does not account for material plasticity near the stress concentration zones. A K_{t} value of 3 or higher would be associated with traditionally found stress concentration zones, resulting in a three-fold higher stress value. Conventionally, the stresses around stress concentration zones are calculated by multiplying the nominal stress S_{n} by ‘k_{t}’, as shown in Figure 1. In FEA packages, if the same linear elastic material models are used for stress analyses, a maximum stress result of a higher magnitude will be observed, as shown in Figure 2.

Figure 1^{[1]}: Conventional method of calculating stresses around stress concentration zones.

Figure 2^{[1]}: FEA analysis calculating stress at a change in geometry using an elastic model (transition radius)

So, how is the actual stress around the stress raisers attained?

In Figure 3, the linear elastic model predicts a stress **S _{LE}** and a corresponding strain value of

**e**. To find the corrected stress and strain values, a curve of constant total energy (usually a hyperbola) is used, and is intersected with the elasto-plastic stress strain curve of the material. The point where this intersection occurs, gives the corrected stress and strain values, S

_{LE}_{c}and e

_{c }respectively.

Figure 3^{[2]}: Stress correction considering the material plasticity.

The corrected strain value, e_{c}, is higher than the modelled strain, e** _{LE}**, which occurs as a result of the plastic redistribution of material around the stress concentration zones.

For factoring the correction in, the total energy in both cases is equated and the corrected stress and strain values are determined i.e., **S _{LE }*** e

_{LE }=

**S*** e

_{c }_{c }.

For static analyses, a spreadsheet of Neuber (corrected) stress values can be found via this link: **Static Analysis Neuber Corrected Stress Values**.

In case of a fatigue analysis, the procedure remains fairly similar. Here, the material behavior is divided into two parts: the material follows a cyclic stress-strain curve for initial loading, and it follows a hysteresis stress-strain behavior for subsequent loading reversals. This is illustrated in Figure 4.

Figure 4^{[1]}: Stress correction in material fatigue cases

Note: The graph in Figure 4 uses ‘σ’ and ‘ϵ’ for **S _{LE }**and

**e**, respectively, as discussed in this blog.

_{LE }A Neuber stress plot, is essentially a plot of energy vs fatigue life. The “energy” here, is calculated as √(E ∗ Δe ∗ Δ**S**), E being the Modulus of Elasticity of the material. This quantity, having the units of stress, represents the energy at a fatigue hot-spot, such as the region at the root of a stress concentration, a notch for example, from where a fatigue crack can initiate.

The **S _{c }*** e

_{c }=

**S*** e

_{LE }_{LE }equation can be written as:

**S _{c }*** e

_{c }* E=

**S*** e

_{LE }_{LE }* E

**S _{c }*** e

_{c }* E =

**S***

_{LE }**S**

_{LE}√(**S _{c }*** e

_{c * }E) =

**S**= Neuber stress = S.

_{LE }After running fully axial (reversed tension/compression) fatigue tests, the constant amplitude stress/strain data will be generated. The elastic modulus, and the stress and strain ranges are taken from the stabilized stress-strain plots, and the Neuber Stress is computed to create a fatigue life plot, as shown in Figure 5.

Figure 5^{[1]}: Axial test specimen, stabilized stress-strain plots.

The Neuber stress, √(E ∗ Δe ∗ Δ**S**) is then plotted along Y-axis to generate a Neuber Stress Plot, and ultimately the fatigue life of a specimen under a given Neuber Stress value is estimated. The material plasticity correction automatically gets built into the graph. An example of such a graph is shown in Figure 6.

Figure 6^{[1]}: An example of a Neuber Stress Plot

The advantage of a Neuber stress plot over a stress-life or strain-life plot is that by integrating three variables – stress, strain and the elastic modulus of a material, it accounts for the plastic redistribution of stresses around the stress concentration zones and models it more accurately.

Neuber stress plots can be useful to fatigue design engineers, and part 2 of this blog post will delve more into this.

REFERENCES:

- Al Conle’s study “Plasticity Corrections for Elastic Analysis Results: Neuber Method –http://fde.uwaterloo.ca/Fde/Notches.new/neuberStress4AISIpt1.pdf
- Neuber Stress explanation from Abbott Aerospace – https://www.abbottaerospace.com/neuber-method-for-reducing-elastic-stress-values