## 4 Discussion
### Single Loading Cases
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Deformation** & \multicolumn{1}{c}{**Shear**} & \multicolumn{1}{c}{**Compression**} & \multicolumn{1}{c}{**Bending**} \\ \hline
**F** & \(\begin{bmatrix}0.9664\\ -0.0203\end{bmatrix}\) & \(\begin{bmatrix}0.2038\\ 0.9948\end{bmatrix}\) & \(\begin{bmatrix}1.1936\\ 0.2360\end{bmatrix}\) & \(\begin{bmatrix}-0.2406\\ 0.7829\end{bmatrix}\) & \(\begin{bmatrix}0.9263&-1.662\\ 0.0458&1.0779\end{bmatrix}\) \\ \hline
**R** & \(\begin{bmatrix}0.9935\\ -0.1137\end{bmatrix}\) & \(\begin{bmatrix}0.1137\\ 0.9935\end{bmatrix}\) & \(\begin{bmatrix}0.9721\\ 0.2344&0.9721\end{bmatrix}\) & \(\begin{bmatrix}0.9945&-0.1052\\ 0.1052&0.9945\end{bmatrix}\) \\ \hline
**U** & \(\begin{bmatrix}0.9625\\ 0.0894\end{bmatrix}\) & \(\begin{bmatrix}0.0894\\ 1.0115\end{bmatrix}\) & \(\begin{bmatrix}1.2157\\ -0.0504&0.8175\end{bmatrix}\) & \(\begin{bmatrix}0.9260&-0.0519\\ -0.0519&1.0894\end{bmatrix}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Deformation gradient **F**, rigid body rotation **R**, and stretch tensor **U** of critical elements under single loading cases. Figure 5: Position of critical elements for (**a**) shear, (**b**) compression, and (**c**) bending loadings. For a more detailed analysis of the evolving inelastic deformation changes under single loading conditions, the deformation paths of the critical elements were traced and plotted on the stress triaxiality and equivalent plastic strain diagrams, as depicted in Figure 7. A red line represents the proposed fracture limit strain obtained in Figure 1b. Measurements were obtained until the fracture, or at the final deformation in the absence of a fracture, as listed in Table 6. In the case of the shear loading shown in Figure 7a, the elements at the BL and TR corners exhibited a positive range of triaxiality [2; 3.5] and eventually reached the fracture limit at strain values of 0.1 and 0.111, respectively. However, the critical elements for the TL and BR corners reached much higher strain levels without fracture because the TL and BR were in the negative triaxiality zone, where the fracture limit was much higher. From the perspective of the evolving deformation mode, the influence of the top and bottom was greater than that of the left and right. Both TL and TR had larger absolute values of triaxiality compared to BL and BR, and the absolute values of triaxiality for TL and TR indicated that the deformation path evolved increasingly in the direction where these values became larger on both sides. For compressive loading shown in Figure 7b, all the points were in almost the same deformation mode. In this case, the deformation mode hardly changed, and fracture did not occur because the vulnerable regions are in a negative triaxiality range where the fracture limit is very high. From a deformation point of view, compressive loading can be seen as creating a uniform and stable deformation across the entire structure. In the bending case shown in Figure 7c, fracture occurred at a lower strain of 0.108 in the TL corner where a positive triaxiality was applied. However, no fracture appeared in the opposite corner, even though the strain was twice as high as that in the TL corner. This can be attributed to the negative triaxiality value that can prevent fracture despite high plastic \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Loading Conditions** & **TL** & **TR** & **BL** & **BR** \\ \hline Shear (S) & 0.198 (O) & 0.100 (X) & 0.111 (X) & 0.423 (O) \\ \hline Compression (C) & 0.243 (O) & 0.246 (O) & 0.240 (O) & 0.241 (O) \\ \hline Bending (B) & 0.108 (X) & 0.274 (O) & 0.041 (O) & 0.011 (O) \\ \hline \hline \end{tabular} Note: (X) denotes fracture and (O) denotes safety.  
\end{table} 
Table 6: The largest equivalent plastic strain values for critical elements under the single loading cases. Figure 6: Normalized deformation of critical element by green strain tensor for (**a**) shear, (**b**) compression, and (**c**) bending.