## 1 Introduction
\[R_{c}=\left(\frac{\sqrt{2\epsilon_{1}}\cdot\Delta K_{L,th}^{V}}{\Delta\sigma_{ 0}}\right)^{\frac{1}{1-\lambda_{1}}} \tag{2}\]  
The geometry parameter \(e_{1}\) depends on the notch opening angle \(2\alpha\)[28], \(\Delta K_{Lth}^{V}\) represents the long-crack growth threshold stress intensity factor range for V-notched components subjected to mode I loading [29] and \(\lambda_{1}\) depicts Williams' eigenvalue for mode I loading conditions [30]. [PERSON] et al. [31] derived a solution for the determination of \(R_{c}\) by evaluating the SED for a specimen featuring a straight edge (representing a notch opening angle of \(2\alpha=180^{\circ}\)) and a specimen featuring a crack (depicting a notch opening angle of \(2\alpha=0^{\circ}\)). As the critical value of the SED is independent of \(2\alpha\), equating the SED results of corresponding specimens yields Equation (3) for the evaluation of \(R_{c}\). Figure 1: Control volume \(\Omega\) of a blunt V-notch. \[R_{c}=\frac{(1+\
u)(5-8\
u)}{4\pi}\left(\frac{\Delta K_{th,lc}}{\Delta\sigma_{0}} \right)^{2} \tag{3}\]  
Comparing Equation (3) to Equation (1) and considering a [PERSON]'s ratio for metallic materials of \(\
u=0.3\), one obtains \(R_{c}=0.845\cdot a_{0}\). Dealing with blunt V-shaped notches, the linear-elastic strain energy density \(\overline{W}^{(e)}\) for mode I loading conditions averaged over the control volume \(\Omega\) can be evaluated by the analytical framework developed by [PERSON] and [PERSON] [27], given by Equation (4). \[\overline{W}^{(e)}=F(2\alpha)\cdot H\Big{(}2\alpha,\frac{R_{c}}{\rho}\Big{)} \cdot\frac{\sigma_{tip}^{2}}{E} \tag{4}\]  
Herein the parameter \(\sigma_{tip}\) represents the maximum stress at the notch root, which is evaluated utilizing linear-elastic finite element analysis. It should be noted, that \(\sigma_{tip}=\Delta\sigma_{tip}\) under tumescent loading at a stress ratio of \(R=0\). The [PERSON]'s modulus \(E\) is the result of tensile testing and the parameters \(F(2\alpha)\) and \(H(2\alpha,R_{c}/\rho)\) depict functions that depend on the notch geometry and control radius. Values for \(F\) and \(H\) are reported in the literature [27; 32; 33] for a variety of notch opening angles and control to notch root radius fractions. Concerning plain specimens subjected to uniaxial tensile loading, the evaluation of the linear-elastic strain energy density range takes on a simplified form given by Equation (5). \[\Delta\overline{W}^{(e)}=\int_{0}^{\Delta\varepsilon}\Delta\sigma d\varepsilon =\frac{\Delta\sigma^{2}}{2E} \tag{5}\]  
As the SED represents the area underneath the stress-strain curve [34], the linear-elastic definitions given by Equations (4) and (5) are strictly only valid for tumescent loading at a stress ratio of \(R=0\). In order to enhance the analytical framework to other stress ratios, [PERSON] et al. [35] introduced the mean stress correction coefficient \(c_{w}\), which relates the linear-elastic SED at a given stress ratio to the one obtained at \(R=0\). During analysis, Equations (4) and (5) are multiplied by \(c_{w}\), defined by Equation (6). The definition of the linear-elastic SED under consideration of \(R\) according to \(c_{w}\) is depicted in Figure 2. As the applied stress range \(\Delta\sigma\) is identical for the exemplified cases, the effect of \(R\) can be perceived clearly. \[c_{w}= \begin{cases}\frac{1+R^{2}}{(1-R)^{2}}&R<0\\ \frac{-R^{2}}{(1-R)^{2}}&0\leq R\leq 1\end{cases}  \tag{6}\]  
Based on the analytical framework reported in the literature, the linear-elastic SED concept depicts an engineering-feasible tool for the fatigue assessment of notched or notch-like components, such as welded structures. The methodology has been successfully applied to a variety of materials and boundary conditions, including elevated-temperature environments and multiaxial loading investigations [36; 37; 38; 39; 40; 41; 42; 43; 44]. Figure 2: Linear-elastic strain energy density under cyclic loading at different stress ratios.