## 2 Determination of Local Stresses and Strains for the Notch Strain Approach Using FEA
### Cyclically Stabilised Uniaxial Material Behaviour  
For service life estimations carried out by means of the notch strain approach, the local stress-strain path is determined under consideration of elastic-plastic material behaviour [1; 2]. The uniaxial relationship between stress \(\sigma\) and strain \(\epsilon\) under cyclic loading is described by a material law. For the simplification of a cyclically stabilised deformation behaviour, the cyclic stress-strain curve can be described with the approach according to [PERSON] [44], independent from the prehistory of cyclic loading (see Equation (3)). \[\epsilon=\frac{\sigma}{\text{E}}+\left(\frac{\sigma}{\text{K}^{\prime}} \right)^{\frac{1}{N^{\prime}}} \tag{3}\]The approach of [PERSON] and [PERSON] is also used in the FKM guideline nonlinear [25]. [PERSON]'s modulus E is a fixed value in [25] depending on the material group involved--for example, E = 206 GPa for steel. The cyclic material parameters K\({}^{\prime}\) and n\({}^{\prime}\) can either be determined experimentally or estimated. In the FKM guideline nonlinear, the estimation method according to [PERSON] et al. [45; 46] is given for estimating the parameters of the cyclic stress-strain curve from the tensile strength R\({}_{\text{m}}\). For steel materials, this estimation is summarised in Equations (4) and (5):  
\[\text{n}^{\prime}=0.187 \tag{4}\]  
\[\text{K}^{\prime}= \frac{3.1148\text{ MPa }\cdot\left(\frac{\text{R}_{\text{m}}}{ \text{MPa}}\right)^{0.897}}{\left(\text{min}\bigg{[}0.338;\text{ }1033\cdot\left(\frac{\text{R}_{\text{m}}}{\text{MPa}}\right)^{-1.235}\bigg{]} \right)^{n^{\prime}}} \tag{5}\]  
Other approaches for describing the uniaxial cyclically stabilised material behaviour might also be utilized; e.g., a bilinear [2] or multilinear approach, or the modified Ramberg-Osgood approach according to [PERSON] and [PERSON] [47]. However, these approaches will not be considered in more detail in the following investigations, since the focus is on the FKM guideline nonlinear. The simplified assumption of cyclically stabilised material behaviour is often made within analytical fatigue assessments using strain-based concepts; e.g., [2; 48; 49; 50; 51]. This is mainly motivated by the fact that the number of material parameters required for the material model used can be kept within a reasonable range for a practical application. Throughout the literature, however, there are various complex approaches that promise to represent the physical cyclic material behaviour more accurately--see, e.g., [51; 52; 53]. Assuming a cyclically stabilised material behaviour, the deformation of the material during initial loading is described by Equation (3). The stress-strain path at load reversal is described according to [PERSON]'s behaviour [54], with the initial load curve doubled in stress and strain--see Equation (6) and Figure 1A. The doubled initial load curve is referred to as a hysteresis branch below. \[\Delta\varepsilon= \frac{\Delta\sigma}{\text{E}}+2\cdot\left(\frac{\Delta\sigma}{2 \cdot\text{K}^{\prime}}\right)^{\frac{1}{n^{\prime}}} \tag{6}\]  
In the local stress-strain diagram, a new coordinate system for the range of stress \(\Delta\sigma\) and strain \(\Delta\varepsilon\) is placed in each reversal point of the stress-strain curve and a new hysteresis  
Figure 1: Cyclically stabilised material behaviour: (**A**) [PERSON]’s behaviour; (**B**) local stress–strain path at constant amplitude loading. branch starts from there, which follows Equation (6). Its length depends on the level of stress. Figure 1B schematically shows the stress-strain path for constant amplitude loading. If the loading on a material shows variable amplitudes, the rules of material memory (memory rules) must also be taken into account; these describe at which points of the course the stress-strain path changes from an initial load curve to a hysteresis branch, and vice versa.