## 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
Since this is the first load on the material, the load-notch-strain curve for the initial load is used. The amplitude of the first flank corresponds to the stress of the fourth class \(\sigma_{\text{el}}=\sigma_{\text{el},4}\). The corresponding elastic-plastic strain \(\epsilon_{1}\) is determined with the load-notch-strain curve and corresponds to the strain \(\epsilon_{1,4}\). The local stress-strain curve begins at the origin and continues to the strain \(\epsilon=\epsilon_{1}\). The corresponding stress \(\sigma\) and the course of the \(\sigma\)-\(\epsilon\) path are determined with the material law--see Equation (3). Figure 3: Load–notch-strain curve: (**A**) classification of the load-time history and transfer in elasticity-theoretical stresses; (**B**) determination of the load–notch-strain curve for the initial load and the hysteresis branch. Figure 4B shows the calculation up to the next reversal point. Due to [PERSON]'s behaviour at load reversal, the load-notch strain curve for the hysteresis branch is used. The stress range of the flank corresponds to the eighth class's stress \(\Delta\sigma_{\text{el}}=\sigma_{\text{el},8}\). The associated elastic-plastic strain \(\Delta\varepsilon_{2}\) is determined with the load-notch-strain curve. The local stress-strain curve is continued with a hysteresis branch (Equation (6)) with a stress range of \(\Delta\varepsilon_{2}\). The determination of the local curve for the third flank is conducted in Figure 4C analogous to Figure 4B. The simulation of the local stress-strain path is carried out in connection with the Rainflow-HCM algorithm [62], whereby the material memory is also taken into account when using a load-notch-strain curve, cf. [29]. Subsequently, the local stress-strain path is examined for closed hystereses, and these are evaluated with a load parameter in order to estimate the service life of the component. Since the focus of this contribution is the determination of the load-notch-strain curve, the service life estimation is not discussed in detail. The FKM guideline nonlinear uses the notch root approximation according to [PERSON] [27] extended by [PERSON] and [PERSON] [28] or the one according to [PERSON] and [PERSON] [30; 31; 32]  
Figure 4: Generating the local stress–strain path using a load–notch-strain curve. in order to convert linear-elastic stresses to elastic-plastic strains (for the calculation of load-notch-strain curve). Both notch root approximations depend on the knowledge of the limit load factor \(\mathrm{K_{p}}\) to account for the global deformation behaviour occurring due to net section plasticity [28]. The limit load factor describes how much the elastic limit load can be increased before plastic collapse occurs when the plastic limit load is reached. The yield initiation load is the load that leads to the yield point being reached in the case of elastic-ideal-plastic material behaviour. The plastic limit load is the load that occurs when the load is increased above the elastic limit load until the entire component cross-section is plastified and plastic collapse occurs. \(\mathrm{K_{p}}\) may be determined either analytically for simple geometries or by using FEA with elastic-ideal-plastic material behaviour for complex geometries [28]. While using FEA for the determination, the load for the plastic collapse of the component in question needs to be approximated. This means that the load is increased until no more equilibrium is found for the finite element problem. The calculation times up to this point can become quite large. This is especially noteworthy since the quantity \(\mathrm{K_{p}}\) is only used to correct the approximation methods of the previously mentioned notch approximations. As an alternative to the notch approximations in connection with the \(\mathrm{K_{p}}\) variable, which is difficult to determine, FEA with elastic-plastic material behaviour can also be used directly, this time using the material behaviour according to Equation (3). This procedure is described below.