## 3 Results
### Predictions of Fracture Load Using EMC-SED  
The average strain energy density from the five tested tensile specimens is \(W_{c,EMC}=2.357\pm 0.15\) MJ/m\({}^{3}\). The equivalent fracture stress \(\sigma_{f}^{*}\) can be calculated using  
\[\sigma_{f}^{*}=\sqrt{(W_{c,EMC})(2E)} \tag{7}\]  
which gives \(\sigma_{f}^{*}=103.39\) MPa. The average strain energy density of \(2.357\) MJ/m\({}^{3}\) and the specimen with \(\rho~{}=1.5\) mm and \(d_{p}=6.0\) mm was then used to determine the critical radius of the control volume \(R_{c}\). This was achieved by applying the average fracture force  
Figure 11: Fracture behavior of specimen with \(\rho=3.5\) mm and \(d_{p}=4.0\) mm at different times. (a) t = 1 s, (b) t = 50 s, (c) t = 75 s, (d) t = 89 s. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\rho\) (mm) & \(d_{p}\) (mm) & \(F_{1}\)(N) & \(F_{2}\) (N) & \(F_{3}\) (N) & \(F_{arg}\) (N) & St. Dev (N) \\ \hline
1.5 & 6.0 & 6870.6 & 6954.4 & 6832.4 & 6885.8 & 50.94 \\
2.0 & 6.0 & 6914.0 & 6741.0 & 6456.5 & 6703.8 & 188.63 \\
2.5 & 6.0 & 6966.0 & 7063.0 & 7026.0 & 7018.3 & 39.97 \\
3.0 & 6.0 & 6996.3 & 6988.1 & 6410.8 & 6798.4 & 274.12 \\
3.5 & 6.0 & 6614.9 & 6415.0 & 6882.0 & 6637.3 & 191.33 \\
4.0 & 6.0 & 6510.0 & 6495.0 & 6341.5 & 6448.8 & 76.14 \\
4.5 & 6.0 & 6583.0 & 6600.0 & 6316.0 & 6499.7 & 130.06 \\
5.0 & 6.0 & 6566.0 & 6444.0 & 6271.0 & 6427.0 & 121.03 \\
3.5 & 4.0 & 7045.0 & 7115.0 & 7003.0 & 7054.3 & 46.20 \\
3.5 & 5.5 & 6554.0 & 6616.0 & 6577.0 & 6582.3 & 25.59 \\
3.5 & 7.0 & 6582.0 & 6669.0 & 6522.0 & 6591.0 & 60.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental results of U-notched specimens. Figure 10: Fracture behavior of specimen with \(\rho=5.0\) mm and \(d_{p}=6.0\) mm at different times. (a) t = 1 s, (b) t = 46 s, (c) t = 67 s, (d) t = 90 s, (e) t = 108 s, (f) t =111 s.  
6885.8 N obtained from the experiment as listed in Table 3 as a load in the finite element model. As mentioned earlier, the material is modeled as linear-elastic. Therefore, the average modulus of elasticity listed in Table 1, \(E=2267.6\) MPa, and a Poisson's ratio of 0.38 were used in the material mode. Knowing that the value \(r_{0}\) for the U-notch geometry is \(\rho/2\), the radius \(R_{c}+r_{0}\) shown in Figure 3 is changed incrementally and the corresponding strain energy and volume enclosed by the produced crescent shapes are evaluated using ANSYS for each increment. By dividing the strain energy over the volume, the strain energy density is obtained for each increment. Because the fracture load is applied, the value of \(R_{c}+r_{0}\) that produces a strain energy density equal or close to \(W_{c,EMC}=2.357\) MJ/m\({}^{3}\) must corresponds to the onset of failure for the notch specimen. The process for determining the value of the critical radius is shown in Figure 12 and a value of \(R_{c}=2.115\) mm was obtained. As stated by the strain energy density criterion, the critical radius of the control volume is a material property [23]. Therefore, the size of the crescent for other U-notched specimens is only controlled by \(r_{0}=\rho/2\). Comparison between the experimental and the estimated fracture load is shown in Table 4. It can be seen that the estimation for all U-notched specimens with depth \(d_{p}=6.0\) mm are nearly within 4.4% of that obtained from the experiment. However, the accuracy of the estimation drops for the specimens with \(d_{p}=4.0\) mm and \(d_{p}=7\) mm reaching to 25.5% and 14.1% that of the experimental values, respectively. A synthesis of the fracture data for all U-notched specimen in terms of the square root of the strain energy density evaluated within the control volume, \(\overline{W}\), over the critical strain energy density, \(W_{c,EMC}\) with respect to the notch radius is reported in Figure 13.  
Figure 12: The control volume for specimen with \(\rho=1.5\) mm and \(d_{p}=6.0\) mm: (**a**) elastic strain energy density; (**b**) determination of the critical radius \(R_{c}\). Figure 13: Synthesis of fracture data in terms of average strain energy density.