## 3 Results  
The analytical frame developed in Section 2.1 is fully applicable provided the low ratios \(a/W\) and \(\rho/a\) presented in the previous section (Table 2). By recalling the apparent fracture toughness of a sharp V-notch defined according to [35]  
\[K_{lc}^{V}=\lambda^{\lambda}\left[\frac{(2\pi)^{2\lambda-1}}{\beta^{2}/2} \right]^{1-\lambda}\frac{K_{lc}^{2(1-\lambda)}}{\sigma_{u}^{1-2\lambda}}, \tag{11}\]  
the dimensionless apparent generalized fracture toughness is plotted in Figure 3 versus the dimensionless notch root radius \(\rho/l_{ch}\), where \(l_{ch}=(K_{lc}/\sigma_{u})^{2}\approx 5.18\) nm is the so called \"Irwin's length\" (see Section 2.2 for the mechanical properties). In addition to the geometries considered in the present work, Figure 3 also includes the results for the opening angle \(\omega=0^{\circ}\) and \(\omega=150^{\circ}\), as references. When \(\rho\to 0\), the fracture toughness ratio correctly tends to 1 since \(K_{lc}^{V,\rho}\) is approaching the value of the sharp (\(\rho=0\)) V-notch \(K_{lc}^{V}\). Given a fixed value of the normalized notch root radius, the ratio increases as the the opening angle \(\omega\) decreases, i.e., the lower the notch amplitude, the higher the influence of the radius. Figure 3: Dimensionless apparent generalized fracture toughness versus dimensionless notch root radius for the opening angles of the nano-cantilever considered in Section 2.2. For the sake of clarity, two additional opening angles are added as references, i.e., \(\omega=0^{\circ}\) and \(\omega=150^{\circ}\). Irwin’s length \(l_{ch}=(K_{lc}/\sigma_{u})^{2}\approx 5.18\) nm. The dimensionless crack extension is plotted in Figure 4. As \(\rho\to 0\), the geometry reverts to the sharp V-notch case, with the curves approaching the values provided by the function [35]:  
\[l_{c}^{V}=\frac{2}{\lambda\beta^{2}(2\pi)^{2(1-\lambda)}}l_{ch}, \tag{12}\]  
which leads to a decreasing crack length for higher notch amplitudes. As \(\rho\) increases the element becomes smoother, and each function converges to the asymptotic value \(l_{c}=2/\pi\left(\frac{K_{lc}}{\Gamma 12\sigma_{u}}\right)^{2}\). The most significant deviations from the sharp case are again generally observed as the notch amplitude decreases. Semi-analytical values of crack advancements are also reported in Table 3, showing that for the current geometries all the crack advances were within 10%. Dimensionless values of the notch root radius are reported as well. Finally, the theoretical FFM estimations based on Figures 3 and 4 are compared with experimental values. In order to obtain the experimental apparent fracture toughness, the stress at the notch root \(\sigma_{max}=\sigma_{y}(0)\) at incipient failure must be evaluated through FEM analysis. Indeed, numerical simulations were already carried out in [30] to study the local stress state and are here exploited, for the sake of simplicity. The experimental values of \(K_{lc}^{V,\rho}\) can be then easily obtained from Equation (13):  
Figure 4: Critical crack advancement versus notch root radius normalized by Irwin’s length, \(l_{ch}=(K_{lc}/\sigma_{u})^{2}\approx 5.18\) nm. \begin{table}
\begin{tabular}{c c c c c} \hline \(\omega\) (**deg**) & \(\rho l_{ch}\) & \(l_{c}^{V}\) (**nm**) & \(l_{c}\) (**nm**) & \(l_{c}l_{c}^{V}\) \\ \hline
33 & 1.97 & 3.27 & 2.64 & 0.806 \\
48 & 2.67 & 3.25 & 2.62 & 0.807 \\
59 & 3.90 & 3.23 & 2.60 & 0.805 \\
68 & 1.22 & 3.17 & 2.83 & 0.894 \\ \hline \end{tabular}
\end{table} 
Table 3: Critical crack advancement for the geometries under consideration. The reference crack advancement related to a sharp notch provided by Equation (12) is also reported. Irwin’s length is equal to \(l_{ch}=(K_{lc}/\sigma_{u})^{2}\approx 5.18\) nm.