## 2 Materials and Methods  
### Fundamentals of FFM Approach for Blunt V-Notches  
The FFM approach [35] assumes that a crack advances of a finite length \(l\) when two criteria, one based on stress and the other on the energy balance, are full-filled simultaneously. The stress criterion requires that the average stress \(\sigma_{y}(x)\) upon the crack advance \(l\) is higher than the material tensile strength \(\sigma_{u}\):  
\[\int_{0}^{l}\sigma_{y}(x)\mathrm{d}x\geq\sigma_{u}l, \tag{1}\]  
where \((x,y)\) is the Cartesian coordinate system centered at the notch root (Figure 1). The energy criterion ensures that the energy available for a crack increment \(l\) is higher than the energy necessary to create the new fracture surface. [PERSON]'s relationship allows expressing the requirement in the form  
\[\int_{0}^{l}K_{l}^{2}(c)\mathrm{d}c\geq K_{lc}^{2}l, \tag{2}\]  
\(K_{l}(c)\) and \(K_{lc}\) being the stress intensity factor (SIF) related to a crack of length \(c\) stemming from the notch root (Figure 1) and the fracture toughness, respectively. At incipient failure, Equations (1) and (2) coalesce into a system of two equations in two unknowns: The critical crack advancement \(l_{c}\) and the failure load. Focusing on blunt V-notched geometries [37] and assuming the notch tip radius \(\rho\) sufficiently small with respect to the notch depth \(a\) (Figure 2), the stress field along the notch bisector can be expressed as [49]:  
\[\sigma_{y}(x)=\frac{K_{l}^{V,\rho}}{[2\pi(x+r_{0})]^{1-\lambda}}\left[1+ \left(\frac{r_{0}}{x+r_{0}}\right)^{\lambda-\mu}\eta_{\theta}(0)\right], \tag{3}\]  
where \(K_{l}^{V,\rho}\) is the apparent generalized (or notch) SIF, and  
\[r_{0}=\frac{\pi-\omega}{2\pi-\omega}\rho. \tag{4}\]  
The parameters \(\lambda\), \(\mu\) and \(\eta_{\theta}(0)\) depend on the notch amplitude \(\omega\), and their values are reported in Table 1. Furthermore, supposing a crack of length \(c\) sufficiently small with respect to the notch depth \(a\) (Figures 1 and 2), the following relationship was proposed for the SIF \(K_{l}\)[50]:  
Figure 1: Blunt V-notch geometry: \(c\) represent the length of a crack stemming from the notch tip. \[K_{I}(c)=\frac{\beta K_{I}^{V,\rho}c^{\lambda-0.5}}{\left[1+\left[\frac{r_{0}}{c} \left(\frac{\beta}{\alpha}\right)^{\frac{1}{1-\lambda}}\right]^{m}\right]^{\frac {1-\lambda}{m}}}, \tag{5}\]  
where  
\[\alpha=1.12\sqrt{\pi}\frac{[1+\eta_{\theta}(0)]}{(2\pi)^{1-\lambda}}, \tag{6}\]  
and \(\beta\),\(m\) are provided in Table 1.  
Within brittle structural behavior, it is legitimate to suppose that failure takes place as soon as the apparent generalized SIF reaches its critical value \(K_{I}^{V,\rho}=K_{Ic}^{V,\rho}\), usually termed as apparent generalized fracture toughness. As expected, FFM can be implemented by inserting Equations (3) and (5) into Equations (1) and (2), respectively, and integrating. Simple analytical manipulations lead to the following two coupled equations:  
\[\frac{K_{Ic}^{V,\rho}}{\sigma_{u}r_{0}^{1-\lambda}}=f(\bar{l}_{c}), \tag{7}\]  
and  
\[\int_{0}^{l}K_{l}^{2}(c)\mathrm{d}c\geq K_{lc}^{2}l, \tag{2}\]0  
where the functions \(f,h\) rising from the integration procedure are [37]:  
\[\int_{0}^{l}K_{l}^{2}(c)\mathrm{d}c\geq K_{lc}^{2}l, \tag{2}\]1 \[\int_{0}^{l}K_{l}^{2}(c)\mathrm{d}c\geq K_{lc}^{2}l, \tag{2}\]2 (10) \[\int_{0}^{l}K_{l}^{2}(c)\mathrm{d}c\geq K_{lc}^{2}l, \tag{2}\]3  
\(\bar{l}_{c}=l_{c}/r_{0}\) and \(\bar{c}=c/r_{0}\). The system of Equations (7) and (8) is solved numerically through a simple MATLAB(r) code: For a given structure, after extracting \(l_{c}\) from Equation (8), the apparent generalized fracture toughness can be estimated through Equation (7). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\omega\) (**deg**) & \(m\) & \(\lambda\) & \(\mu\) & \(\eta_{\theta}\)**(0)** & \(\beta\) \\ \hline
0 & 1.82 & 0.5000 & \(-0.5000\) & 1.000 & 1.000 \\
33 & 1.45 & 0.5021 & \(-0.4515\) & 1.035 & 1.005 \\
48 & 1.38 & 0.5040 & \(-0.4285\) & 1.007 & 1.010 \\
59 & 1