## 7 Results and Discussion  
### Force, Displacement, and Equivalent Strain Responses in Different Stress States  
In Figure 7, the global forces measured by the fracture tests using specimens of four different shapes are shown. In addition, the global forces and equivalent strains predicted by the FE analysis of the fracture tests are also shown. Strain localization prior to ductile fracture accompanied by necking was observed during the fracture tests. The black dashed line indicates the position where strain localization (hereafter referred to as SL) occurred. In each figure, the global forces (measured vs. predicted) are compared first, and variations in the equivalent strain are explained later. In Figure 7a, the predicted and measured forces are in good agreement when the NT10 specimen is used. Similar to the uniaxial tensile test with the standard (dog bone) tensile specimen (Figure 5b), the force gradually decreased after SL. This confirms that the combined model [15] used in the FE analysis is appropriate. A softening of the flow curve (from SL to the onset of fracture) was observed. This is attributed to the fact that even though \(\eta_{avg}\) is high (0.5349), the absolute value of \(\overline{\theta}_{avg}\) is relatively low (0.4551). Hence, the onset of fracture due to the shearing of the voids was delayed. Note that the smaller the Lode angle parameter, the faster is the occurrence of the fracture due to void shear. The calculated values of \(\eta_{avg}\), \(\overline{\theta}_{avg}\), and \(\overline{\varepsilon}_{f}\) are summarized in Table 3. The magnitude of \(\eta_{avg}\) is generally classified as follows: 0 or less, low; 0-1/3, moderate; 1/3-2/3, high; 2/3 or more, very high. However, there is no classification of the magnitude of the Lode angle parameter. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No. of Elements** & **2** & **4** & **6** & **8** & **10** \\ \hline Equivalent strain at fracture & 0.5962 & 0.5844 & 0.5786 & 0.5767 & 0.5770 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Variations in the equivalent strain at the fracture as number of elements in the thickness direction of the strip changes. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Specimen Type** & \(\eta_{avg}\) & \(\overline{\theta}_{avg}\) & \(\overline{\varepsilon}_{f}\) \\ \hline Standard (dog-bone) & 0.4424 & 0.7366 & 0.7893 \\ NT10 & 0.5349 & 0.4551 & 0.5416 \\ NT30 & 0.4916 & 0.6145 & 0.6755 \\ SNT & 0.8301 & 0.2235 & 0.2116 \\ IPS & 0.0118 & 0.0305 & 1.0993 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Loading parameters and equivalent strain at fracture for different types of specimens. Figure 7b shows that as the notch radius of the NT30 specimen is large, more deformation (longer cross head displacement) is required until the specimen reaches fracture after SL. The predictions and measurements are very similar, which means that the selection of the combined model is appropriate. As with the NT10 specimen, softening was observed for the NT30 specimen. Considering the product of \(\eta\) and \(\overline{\theta}\) values of two specimens (NT10 and NT30) as a measure trade-off, the reason for the softening is explained. In the SNT specimen (Figure 7c), unlike in the specimens with large notch radii such as NT10 and NT30, SL and fracture occur almost simultaneously. This is explained as follows: when \(\eta\) is very high, the strain to localization decreases because of the void growth [50,51]. However, if the absolute value of \(\overline{\theta}\) is small, fracture occurs more rapidly by shearing of the voids, and therefore, little softening is observed [52]. For the SNT specimen, \(\eta\) is as very high as 0.8301, but \(\overline{\theta}\) is as small as 0.2235. In Figure 7d, the IPS and SNT specimens are similar in that softening is hardly observed owing to rapid fracture. However, the IPS specimen can withstand approximately 12 times greater displacement than the SNT specimen until the onset of fracture.