## 3 Results and Discussion
### Backstress Increments and Yield Surface Evolution  
In the deviatoric stress space, the backstress increments represent the magnitude and direction of the yield surface translation over the loading process beyond the elastic limit. According to Chaboche postulation, the backstress increment was achieved through integration of backstress increments, \(d\overline{\alpha}=\sum_{i=1}^{M}d\overline{\alpha}_{i}\). Figure 7 represents the yield surface translation for the AM aluminum sample for three different element types. The initial yield surface of the Al 4043 sample is depicted by a solid line. This figure also presents the stress-strain curve of additively manufactured Al 4043 through using right vertical and upper horizontal axes. When the applied load increases beyond the elastic limit, the yield surface translates (dashed-line surfaces) into the deviatoric stress space, intercepting the stress-strain curve. Beyond the elastic limit, the initial yield surface of Al 4043 with the origin of \(O\) was shifted to center \(O^{*}\). The translated yield surface is shown with dashed lines intercepting the stress-strain curve at point \(P^{*}\). The evolution of the yield surface by means of different element types resulted in the same direction of backstress increment \(d\overline{\alpha}\) with various magnitudes as shown in Figure 7. The lower amount of backstress increment resulted in a smaller vector of \(\overline{OO_{1}^{*}}\) achieved through the second-order brick elements, while the linear brick elements possessed a larger translation from \(\overline{OO_{3}^{*}}\). By surpassing the  
Figure 4: Mesh structures of the Al 4043 sample with (**a**) brick solid elements and (**b**) tetrahedron solid elements.  
yield limit, the origin of the yield surface with the second-order tetrahedron element was shifted to \(O_{2}^{*}\) and was placed between the translated centers for two other element types.  
Figure 5: Convergence trials for three different element types of (**a**) the linear brick, (**b**) the quadratic tetrahedron, and (**c**) the quadratic brick.  
Figure 6: Distribution of total strain on the entire sample undergoing loading cycles.  
Figure 7: Initial and translated yield surfaces over the loading process for three different element types of linear break, quadratic tetrahedron, and quadratic brick.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Element Type** & **Ansys Software Code** & **Integration Points** & **Pictorial [33]** \\ \hline
8-Node Brick & SOLID185 & \(2\times 2\times 2\) & \\ \hline
10-Node Tetrahedron & SOLID187 & 4 & \\ \hline
20-Node Brick & SOLD186 & \(2\times 2\times 2\) & \\ \hline \end{tabular}
\end{table}
Table 2: Different mesh structures information [33].  
The mean values of backstress increments determined from the peak and valley of each loading cycle were plotted for different element types at various applied stress levels in Figure 8. The yield surface translation was governed by the Chaboche hardening rule through the backstress increments as the number of cycles progressed. Figure 8 shows that as the number of cycles increases, the average backstress moderately declines and results in a steady-state condition at longer cycles. The average backstress through using the second-order brick elements possessed a lower amount as compared to other element types. The highest backstress increment was associated with the first-order brick elements, which manifested the larger translation of the yield surface in Figure 7. The backstress curve of the quadratic tetrahedron element collapsed between the two other element curves consistent with the vector \(\overline{OO_{2}^{*}}\) and the corresponding translated yield surface.