## 3 Possibilities for the Efficient Design of FEA for the Calculation of the Load-Notch-Strain Curve
### Reduction in the Number of Load Steps to Be Simulated
#### 3.4.2 Regression with Power Function  
As an alternative to the interpolation with a polygonal chain, the load-notch-strain curve can be described by a power function. The power function is chosen based on the material law according to [PERSON] and [PERSON] [44]; see Equation (3). [PERSON] [15] also used this approach, and was thus able to describe load-notch-strain curves from FEA with a functional relationship in a good approximation. As described in Section 3.2, only the determination of the load-notch-strain curve for the hysteresis branch is necessary. This is described with Equation (16) following Equation (6), with which the range of the strain \(\Delta\epsilon_{\mathrm{P}}\) (index P for power function) is calculated:  
\[\Delta\epsilon_{\mathrm{P}}(\Delta\sigma_{\mathrm{el}})=\frac{\Delta\sigma_{ \mathrm{el}}}{\mathrm{E}}+2\left(\frac{\Delta\sigma_{\mathrm{el}}}{2\cdot \mathrm{c}}\right)^{\frac{1}{d}} \tag{16}\]  
The aim is now to determine the parameters \(\mathrm{c}\) and \(\mathrm{d}\). For this purpose, the front part--Hooke's law described by the modulus of elasticity E--is subtracted from the stress range \(\Delta\epsilon_{\mathrm{P}}\). The remaining power approach represents a linear course by double logarithmic scaling. A comparison with the results from the FEA in Figure 8 obviously does not show a linear course. This can be explained by the approximation to the net section plasticity [28,75].  
A linear regression through all the simulated load steps results in the red curve, which shows deviations from the simulated ones in all areas. In particular, the load steps for small loads \(\Delta\sigma_{\mathrm{el}}\) deviate strongly from a linear course; thus, it makes sense to exclude them for a regression. If only load steps above a proposed limit value of \(\log(\Delta\epsilon_{\mathrm{FEA}}-\Delta\sigma_{\mathrm{el,FEA}}/\mathrm{E}) \geq-4\) are considered in the regression, the load steps of the medium and high loads can be well approximated (see the black curve). The large deviation found for small loads only appears very large due to the logarithmic scaling applied.  
Figure 7: Interpolation of the load–notch-strain curve by polygonal chain with \(\mathrm{n_{H}}=2\) load steps.  
Figure 9 shows the load-notch-strain curves with both regressions in a linear representation. In the range of low loads, there is no apparent difference between the load-notch-strain curves; in the range of high loads, the load-notch-strain curve with consideration of the limit value of \(\log(\Delta\varepsilon_{\text{FEA}}\text{-}\Delta\sigma_{\text{el,FEA}}/\text {E})\geq-4\) reproduces the results of the FEA significantly better. Therefore, this limit value was used in the following. More details on the regression and the determination of the parameters c and d are given in Appendix A.  
The stress ranges at the class limits for the load-notch-strain curve of the hysteresis branch are then determined using Equation (16). For the load-notch-strain curve for the initial loading, proceed as in Section 3.2 or use Equation (17):  
Figure 8: Regression of the load–notch-strain curve with the power approach.  
Figure 9: Load–notch-strain curve with the power approach.  
\[\epsilon_{\rm P}(\sigma_{\rm el})=\frac{\sigma_{\rm el}}{\rm E}+\left(\frac{\sigma_{ \rm el}}{\rm c}\right)^{\frac{1}{2}} \tag{17}\]  
The advantage of this approach is that the load-notch-strain curve is not described by a sectionwise-defined function. This makes it possible, for example, to estimate load-notch-strain curves from the existing FEA results when the load requirements change, even if the new maximum load \(\rm L_{\rm max}\) is slightly higher than the maximum simulated load. This is not readily possible with sectionwise-defined functions. The disadvantage of this is that the load-notch-strain curve does not necessarily run through the FEA results due to the linear regression.