CAE-Companion-2018-2019

Theory WISSEN CAE

Comparison of Notch Stresses from Elastic Plastic FEA and Neuber Approximation

Background For durability analysis, the Neuber approximation is a well- known method to estimate the local elastic-plastic stresses σ plast in notches based on linear-elastic FEA stress results σ elast . With the Young’s Modulus Ε , and the non-linear strain σ plast it can be expressed as

The following FEAanalyseswereperformedon a fine-grain steel: „ „ Linear-elastic material behavior and subsequent Neuber correction of the elastic stresses „ „ Cyclic-plastic material curve and nominal stress-strain definition with non-linear geometry „ „ Cyclic-plastic material curve and true stress-strain definition with non-linear geometry As mentioned above the Neuber rule applies to regions with limited local plastic zones. The different regions (fig. 1) for the degree of plastification are as follows: „ „ In region 1 (0 - 10 kN) local plastification restricted to the notch prevails. „ „ In region 2 (10 - 20 kN) the entire cross-section begins to plastify. „ „ In region 3 (above 20 kN) the entire cross-section plastifies. For our example the Neuber rule can be applied up to around 20 kN, because the initial cross-section plastifies only very little up to this limit. Due to the loss of stiffness there is a large increase in local stresses at a load of 40 kN. Fig. 2 shows a comparison of the stress amplitudes for the non-linear stresses and the Neuber corrected linear stresses. From this diagram it can be seen that for this material the correlation is very good up to a load of about 20 kN.

Using the Ramberg-Osgood equation for the description of the cyclic stress strain curve,

the non-linear stresses σ plast can be computed as the intersection with the Neuber hyperbola ( K´ and n´ denoting the cyclic hardening coefficient and the cyclic hardening exponent respectively). The following case study illustrates the applicability of the Neuber method, since it is recommended for local plastifica- A notched tensile specimen of steel with notch radius 0.7 mm (stress concentration factor α = 2.15), was analyzed with an alternating load of up to 40 kN (stress ratio R = -1). In fig. 1 the stress amplitude at the base of the notch can be seen for the v. Mises and max. principal normal stress. From the diagram it can be seen that initial notch plastification occurs at approx. 8 kN. tion only. Example

Figure 2: Comparison of stress amplitudes for elastic-plastic stresses and Neuber corrected linear stresses Conclusion Although a multi axial stress state already exists in the notch, the Neuber method provides useful results up to approxi- mately 20 kN (~ 0.6 % plastic strain) for the local stress in the notch. If higher strain or stress multiaxiality are anticipated deviations may increase and a non-linear material law should then generally be employed. As long as the load level is smaller than approximately three times the yield level a lot of CPU time can be safed in durability analysis. Of course this result depends on the stress concentration and on the plastic material behavior.

Figure 1: Notch stress development in the notch base with elastic-plastic material behavior

CAEWissen by courtesy of Engineering Center Steyr, Austria, Author: Gerhard Spindelberger. For more information see www.femfat.com

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