## 2 Determination of Local Stresses and Strains for the Notch Strain Approach Using FEA
### Simulation of Local Stress-Strain Paths and Load-Notch-Strain Curve
If the previously mentioned assumption of cyclically stabilised material behaviour is made, the relationship between an external load and the local elastic-plastic stresses and strains is always the same for a single load branch. The load-notch-strain curve can be regarded as a type of template. The load-notch-strain curve may therefore be determined before the simulation of the local stress-strain path itself using either a notch root approximation or FEA. For longer load-time histories with a large number of reversal points, this procedure is much less time-consuming than a point-by-point FEA or a point-by-point notch root approximation. The idea of using a load-notch-strain curve instead of a point-by-point calculation can most likely be traced to [PERSON] et al. [48], and was advanced by the work of [29]. In the following, the procedure for determining and using the load-notch-strain curve is described using the procedure of the FKM guideline nonlinear. For the application of the FKM guideline nonlinear, the load-time history is classified, which means that only a discretised load-notch-strain curve is required. The class width for the discretisation is determined by dividing the largest load L\({}_{\text{max}}\) by the number of classes. The FKM guideline nonlinear suggests using a total of 100 classes between L = 0 and L = L\({}_{\text{max}}\); whereas in Figure 3A, a class number of 4 is used for didactic reasons. In the classification, the load at each reversal point is shifted to the next largest class limit in terms of absolute value; see the red points in Figure 3A. In order to simplify the representation and make it independent of different units of load, as well as to correspond to the format of the FKM guideline nonlinear, the load-notch-strain curve is represented in this paper as the relationship between the linear-elastic stress and the notch strain. The same is true for the load-time history, which is also understood to be local linear-elastic stress. However, the load-notch-strain curve can also be used as a relation between external load and notch-strain. Subsequently, the load-notch-strain curve for the initial load is determined as discrete points for each class of linear-elastic stresses--i.e., the elastic-plastic stress and strain must be calculated for each class of linear-elastic stress. Figure 3B shows the qualitative calculation of the elastic-plastic strains and stresses with a notch root approximation. From the four elasticity theoretical stresses \(\sigma_{\text{el},1...4}\) at the class limits, which lie on a straight line with the slope of the elastic modulus E in the stress-strain diagram (Hooke's law), the elastic-plastic stresses, which must lie on the cyclic stress-strain curve of the material (shown in red), are inferred. The load-notch-strain curve is obtained by plotting the elasticity-theoretical stress amplitudes at the class limits \(\sigma_{\text{el},1...4}\) over the corresponding strains \(\epsilon_{\text{l},1...4}\) for the initial load. Due to [PERSON]'s behaviour, the local stress-strain path after load reversal does not follow the shape of the initial load curve (Equation (3)); instead, it follows the double initial load curve, the so-called hysteresis branch (Equation (6)). Therefore, in addition to the load-notch-strain curve for the initial load, the load-notch-strain curve for the hysteresis branch must also be determined. Within the load sequence, maximum stress ranges \(\Delta\sigma_{\text{el}}\), which correspond to twice the maximum elasticity-theoretical stress \(\sigma_{\text{el},\text{max}}\), can occur. The load-notch-strain curve for the hysteresis branch is therefore determined for elasticity-theoretical stresses of up to \(2\cdot\sigma_{\text{el},\text{max}}\), and is shown in green in Figure 3B. The local elastic-plastic stress-strain path can now be determined without much computational effort. In Figure 4A, the path is determined up to the first reversal point.