###### Abstract
In the case of notch stress intensity factor (N-SIF)-based design, the fatigue limit is determined by the stress field distribution of V-notches. The stress field singularity at sharp notches \(r\to 0\) can be described using [PERSON]'s eigenvalues \(\lambda_{1},\lambda_{2}\)[39] for varying opening angles. If a polar coordinate system originating from the notch tip is aligned to the notch bisector, the N-SIF value quantifies the asymptotic stress distribution in the close neighborhood, as exemplified for Mode I in Equation (1). The numerical N-SIF assessment of sharp V-notches is evaluated accordingly in this study. \[K_{I}^{V}=\sqrt{2\ \pi}\underset{r\to 0}{\text{lim}}\Big{[}(\sigma_{\theta})_{ \theta=0}r^{1-\lambda_{1}}\Big{]} \tag{1}\]  
A link between the exact value of the notch stress intensity factor \(K_{I}^{V}\) of V-notches with varying opening angles and maximum elastic peak stress \(\sigma_{peak}\) at the notch tip has been presented in [40], as well as analytically proven. A non-dimensional ratio \(K_{FE,I}^{*}\) (Equation (2)) exists between the notch stress intensity factor \(K_{I}^{V}\), the peak stress \(\sigma_{peak}\), and the edge length of the linear quadrilateral finite element \(d\) at the notch tip, powered by \(1-\lambda_{1}\) as the notch angle contribution. The non-dimensional ratio is illustrated for Mode I, evaluated as \(K_{FE,I}^{*}=1.38\). \[K_{FE,I}^{*}=\frac{K_{I}^{V}}{\sigma_{peak}\ d^{1-\lambda_{1}}} \tag{2}\]  
It should be noted that the peak stress method includes the rapid fatigue assessment of the N-SIF for varying V-notch opening angles between \(0^{\circ}\leq 2\alpha\leq 135^{\circ}\) and linear element types [41]. As the finite element simulation requires only the linear-elastic material properties, the non-dimensional ratio \(K_{FE,I}^{*}\) is solver-independent [42]. For multi-axial stress conditions, an equivalent peak stress can be calculated if synchronous stress contributions are given [43]. In the case of non-proportional loads, a local biaxiality ratio selects the proper peak stress method-based design scatter band [44]. To apply the N-SIF approach to blunt cracks, the stress field is calculated based on the sharp notch stress field by adding a supplementary term [45; 46] and linked to the maximum principal stress at the notch tip. Evaluating the blunt V-notch stress intensity factor away from the notch tip, as generalized N-SIF, has been recommended in [47] and, accordingly, is applied in this study. The evaluated \(K_{p,I}^{V}\) is averaged across a distance of approximately \(r=r_{0}\to\ r_{0}+0.4\rho\) from the notch tip. The parameters \(\mu_{1}\) and \(\widetilde{\omega}_{1}\) depend on the notch opening angle \(2\alpha\), whereas \(r_{0}\) depends also on the notch radius \(\rho\). Mode I loading is detailed in Equation (3), for reference. \[K_{\rho,I}^{V}=\sqrt{2\ \pi}r^{1-\lambda_{1}}\frac{\left(\sigma_{\theta}\right)_{ \theta=0}}{1+\widetilde{\omega}_{1}\left(\frac{r}{r_{0}}\right)^{\mu_{1}- \lambda_{1}}} \tag{3}\]  
Elastic strain energy density (ESED) fatigue design is based on the distribution of the stress and strain fields away from the notch tip within a certain control volume. An energy-based fatigue assessment for sharp V-notches under fatigue loading has been presented in [48]. Under the plane strain condition, Equation (4) depicts the dependence of the critical radius \(R_{c}\)--and, therefore, the control volume \(\Omega\)--on the material threshold of the stress intensity factor range \(\Delta K_{th}\) for long cracks and the endurance stress range \(\Delta\sigma_{0}\). \[R_{c}=\frac{(1+\
u)(5-8\
u)}{4\pi}\left(\frac{\Delta K_{th}}{\Delta\sigma_{0} }\right)^{2} \tag{4}\]  
The El Haddad-Smith-Topper parameter \(a_{0}\)[32] is linked to the critical control radius \(R_{c}\) by Poisson's ratio \(\
u\) and the appropriate stress state. For \(\
u=0.3\), this equates to \(R_{c}=0.845a_{0}\) for plane strain and to \(R_{c}=1.025a_{0}\) for plane stress condition.