## 2 Materials and Methods
Since the main assumption of the EMC is to have the same Young modulus (E) and K-based fracture toughness (\(\mathrm{K_{mat}}\)) for both the real ductile material and the equivalent (virtual) brittle material, Equation (6) may be rewritten as follows:  
\[\mathrm{SED_{EM}}=\frac{\sigma_{\mathrm{f}}^{*2}}{2\mathrm{E}} \tag{7}\]  
Now, given that the real material and the equivalent material develop the same SED at fracture, Equations (5) and (7) may be equated:  
\[\frac{\sigma_{\mathrm{Y}}^{2}}{2\mathrm{E}}+\frac{\mathrm{K}}{\mathrm{n}+1}[( \varepsilon_{\mathrm{u,True}})^{\mathrm{n}+1}-(0.002)^{\mathrm{n}+1}]=\frac{ \sigma_{\mathrm{f}}^{*2}}{2\mathrm{E}} \tag{8}\]  
Finally, Equation (9) is proposed by the EMC for calculating the \(\sigma_{\mathrm{f}}^{*}\):  
\[\sigma_{\mathrm{f}}^{*}=\sqrt{\sigma_{\mathrm{Y}}^{2}+\frac{2\mathrm{EK}}{ \mathrm{n}+1}[(\varepsilon_{\mathrm{u,True}})^{\mathrm{n}+1}-(0.002)^{\mathrm{ n}+1}]} \tag{9}\]  
where \(\varepsilon_{\mathrm{u,True}}\) (the true plastic strain at peak point) can be easily calculated from the corresponding engineering plastic strain (\(\varepsilon_{\mathrm{u}}\)), using the following expression:  
\[\mathrm{SED}_{\mathrm{tot.}}=\mathrm{SED}_{\mathrm{e}}+\mathrm{SED}_{\mathrm{P} }=\frac{1}{2}\sigma_{\mathrm{Y}}\cdot\varepsilon_{\mathrm{Y}}+\int\limits_{ \varepsilon_{\mathrm{P}}^{\mathrm{Y}}}^{\varepsilon_{\mathrm{P}}}\sigma\cdot \mathrm{d}\varepsilon_{\mathrm{P}} \tag{2}\]0  
With respect to the ASED approach [2; 3; 4; 5], it establishes that, under mode I loading, fracture takes place when the average value of the SED (\(\mathrm{W_{avg}}\)) over a certain control volume (defined by radius \(\mathrm{R_{0}}\)[5]; see Figure 3) reaches a critical value (\(\mathrm{W_{c}}\)):  
\[\mathrm{SED}_{\mathrm{tot.}}=\mathrm{SED}_{\mathrm{e}}+\mathrm{SED}_{\mathrm{P} }=\frac{1}{2}\sigma_{\mathrm{Y}}\cdot\varepsilon_{\mathrm{Y}}+\int\limits_{ \varepsilon_{\mathrm{P}}^{\mathrm{Y}}}^{\varepsilon_{\mathrm{P}}}\sigma\cdot \mathrm{d}\varepsilon_{\mathrm{P}} \tag{2}\]1  
where \(\mathrm{W_{c}}\) depends on the material and, in principle, can be estimated by using Equation (12):  
\[\mathrm{SED}_{\mathrm{tot.}}=\mathrm{SED}_{\mathrm{e}}+\mathrm{SED}_{\mathrm{P} }=\frac{1}{2}\sigma_{\mathrm{Y}}\cdot\varepsilon_{\mathrm{Y}}+\int\limits_{ \varepsilon_{\mathrm{P}}^{\mathrm{Y}}}^{\varepsilon_{\mathrm{P}}}\sigma\cdot \mathrm{d}\varepsilon_{\mathrm{P}} \tag{2}\]2  
with \(\sigma_{\mathrm{u}}\) being the ultimate tensile strength and E being the Young's modulus. Under plane strain conditions, \(\mathrm{R_{0}}\) may be estimated by using Equation (13) [2; 23]:  
\[\mathrm{SED}_{\mathrm{tot.}}=\mathrm{SED}_{\mathrm{e}}+\mathrm{SED}_{\mathrm{P} }=\frac{1}{2}\sigma_{\mathrm{Y}}\cdot\varepsilon_{\mathrm{Y}}+\int\limits_{ \varepsilon_{\mathrm{P}}^{\mathrm{Y}}}^{\varepsilon_{\mathrm{P}}}\sigma\cdot \mathrm{d}\varepsilon_{\mathrm{P}} \tag{2}\]3When plane stress conditions are dominant, R\({}_{0}\) follows Equation (14):  
\[\mathrm{SED}_{\mathrm{tot.}}=\mathrm{SED}_{\mathrm{e}}+\mathrm{SED}_{\mathrm{P} }=\frac{1}{2}\sigma_{\mathrm{Y}}\cdot\varepsilon_{\mathrm{Y}}+\int\limits_{ \varepsilon_{\mathrm{P}}^{\mathrm{Y}}}^{\varepsilon_{\mathrm{P}}}\sigma\cdot \mathrm{d}\varepsilon_{\mathrm{P}} \tag{2}\]4  
[PERSON] and [PERSON] [4] have derived a useful, straightforward analytical expression for the ASED referred to the control volume:  
\[\mathrm{SED}_{\mathrm{tot.}}=\mathrm{SED}_{\mathrm{e}}+\mathrm{SED}_{\mathrm{P} }=\frac{1}{2}\sigma_{\mathrm{Y}}\cdot\varepsilon_{\mathrm{Y}}+\int\limits_{ \varepsilon_{\mathrm{P}}^{\mathrm{Y}}}^{\varepsilon_{\mathrm{P}}}\sigma\cdot \mathrm{d}\varepsilon_{\mathrm{P}} \tag{2}\]5  
where F(2\(\alpha\)) is a function that depends on the notch opening angle (\(\alpha\)) whose values are gathered in [4]. Here, it is sufficient to say that for U-shaped notches (2\(\alpha\)=0\({}^{\circ}\)) F is equal to 0.785.