## 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  
To estimate the service life of a component according to the notch strain approach, it is necessary to convert the sequence of the external load (e.g., forces or moments) into a path of local elastic-plastic stresses and strains at the point of interest of the component. The dependency of local stresses and strains follows the principles given in Section 2.1 as long as the material is loaded in a uniaxial stress state. For multiaxial stress states, more or less complex hypothesis and material models need to be used. The FKM guideline nonlinear assumes that multiaxial proportional loading may also be described by the same principles as those for uniaxial loading (Section 2.1) if equivalent stresses according to the [PERSON] hypotheses are used [63; 64]. Since equivalent stresses are always positive, a sign needs to be added in this case [65]. The FKM guideline nonlinear suggests using the sign of hydrostatic stress \(\sigma_{\text{H}}\):  
\[\sigma_{\text{V}}=\frac{\text{sign}(\sigma_{\text{H}})}{\sqrt{2}}\sqrt{\left( \sigma_{\text{xx}}\text{--}\sigma_{\text{yy}}\right)^{2}\text{--}(\sigma_{ \text{xx}}\text{--}\sigma_{\text{zz}})^{2}\text{--}(\sigma_{\text{zz}}\text{-- }\sigma_{\text{yy}})^{2}\text{--}6(\tau_{\text{xy}}^{2}\text{--}\tau_{\text{yz }}^{2}\text{--}\tau_{\text{2x}}^{2})} \tag{7}\]  
\[\sigma_{\text{H}}=\frac{1}{3}\cdot\left(\sigma_{\text{xx}}\text{--}\sigma_{ \text{yy}}\text{--}\sigma_{\text{zz}}\right) \tag{8}\]  
For now, only components may be evaluated using the FKM guideline nonlinear [25] with uniaxial and multiaxial proportional loadings. This is due to the fact that neither the procedure described below of using a load-notch-strain curve to determine the stress-strain  
Figure 2: Memory rules. path cannot be applied to nonproportional loads, nor is a satisfactory consideration of the effects occurring under nonproportional loads, such as nonproportional hardening [66], or life extension or shortening, cf. e.g., [67], implemented within the FKM guideline nonlinear yet. For the application of a strain-based approach, besides the material behaviour, the dependency between the load (e.g., force, moment) and the local stress and strain needs to be known so that the local stress-strain path can be determined. Different procedures are available for determining the local stress-strain path:  
* Reversal point to reversal point with FEA: The most obvious option is to transfer the component into a finite element model, to deposit an elastic-plastic material behaviour according to Section 2.1, and apply the load-time history to the component in order to obtain the local stress-strain path. Since FEAs with elastic-plastic material behaviour--and especially with complex component geometries--still require large computing capacities and thus lead to high computing times, this procedure is only recommended for very short load-time histories. Section 5 illustrates this procedure with an example. * Reversal point to reversal point with notch root approximation: Significant time can be saved by using notch root approximations. These make it possible to estimate stresses and strains with elastic-plastic material behaviour from external loads or local elasticity-theoretical stress. The principle procedure is as follows [68]: the elastic-plastic stress-strain path is obtained by applying a notch root approximation to each reversal point of the load sequence or the elasticity-theoretical stress history, considering the memory rules [29; 62] and Masing's behaviour [54]. * Use of a load-notch-strain curve: With the previously described methods, a notch root approximation or an FEA must be carried out for each reversal point of the load-time history. This leads to a high calculation effort for long load-time histories, which can be shortened by using the so-called load-notch-strain curve. The load-notch-strain curve describes the relationship between the external load or local elasticity-theoretical stress and the local elastic-plastic strain.