## 4 Discussion  
The results presented in Section 3 clearly show that the FFM is able to characterize the fracture process of single crystal silicon nanoscale samples. The main advantage of the approach is that, similarly to the TCD [30], it can easily handle notched and cracked geometries. Difficulties in realizing accurate cracks, when dealing with ideal brittle fracture, are indeed well-known and could result in apparent scale dependent fracture toughness. Figure 3 clearly shows that also for a very small notch radius, the apparent generalized fracture toughness deviates from the sharp case. This means that small notch radii cannot be assumed or simplified as sharp. This mistake is commonly made when the FIB processing system (or chemical etching) is used to directly introduced a crack-like notch, based on the assumption that for a small \(\rho/a\) ratio the geometry could be treated as a crack. Unfortunately, \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\omega\) (deg) & \(\rho/l_{ch}\) & \(K_{Ic}^{V}\) & \(K_{Ic}^{V,\rho}\) **FFM** & \(K_{Ic}^{V,\rho}\) **exp.** \\ \hline
33 & 1.97 & 1.04 & 1.50 & 1.43 \\
48 & 2.67 & 1.08 & 1.70 & 1.70 \\
59 & 3.90 & 1.15 & 2.05 & 2.04 \\
68 & 1.22 & 1.25 & 1.56 & 1.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between experimental (denoted by _exp._) and FFM apparent generalized fracture toughness (MPa m\({}^{1-\lambda}\)) for the considered nano-cantilevers. For the sake of clarity, dimensionless values of the notch root radius are reported as well, where \(l_{ch}=(K_{Ic}/\sigma_{u})^{2}\approx 5.18\) nm. Figure 5: Comparison between experimental (denoted by _exp._) and Finite Fracture Mechanics (FFM) apparent generalized fracture toughness for the considered nano-cantilevers. The ratio is plotted against the dimensionless notch root radius. this assumption does not hold at a very small scale for ideal brittle materials such as the single crystal silicon, that would require, ideally, an atomically sharp crack (ideal Griffith's crack) [22] to determine the proper constant and scale independent fracture toughness. The FFM approach proposed here, instead, provides directly the generalized apparent fracture toughness and gives the possibility to handle different notch radii and opening angles. This conclusion is supported by Figure 5, where the failure assessment is reliable for all the geometries, with negligible differences. FFM is added therefore to the list of methods that have been proven to be successful when applied at small scale, such as the TCD [30] and the strain energy density [17]. Indeed, these methods share several similarities. Strain energy density and FFM have the same proportionality between the critical notch stress intensity factor and the two fundamental properties i.e., the fracture toughness and the ultimate tensile strength. The difference is that in the FFM the relation depends on notch opening angle only, while in the strain energy density it depends on Poisson's ratio as well [51]. TCD and FFM, instead, both rely on the definition of a critical length scale parameter. While according to the TCD the critical distance is a material constant, the FFM critical crack advancement \(l_{c}\) depends both on the notch radius and amplitude [33]. Figure 4 shows that when the notch radius tends to large values, i.e., smooth samples, all the curves converge towards the same constant value. In this case, the energy requirement provides a constant crack advancement in agreement with the quantized fracture mechanics (QFM) [52], while the stress condition reduces to \(\sigma=\sigma_{u}\) and determines the final failure. When the notch radius tends to zero, the geometry reverts to the sharp V-notch case: In the crack case (\(\omega=0^{\circ}\)), the energy balance reverts to \(K_{I}=K_{lc}\) and governs the failure mechanism, while the stress requirement leads to the constant critical crack advancement related to the TCD line method [36].