## 3 Results and Discussion
### The Finite Element Method  
Through the use of Ansys 2022 R [31], a finite element model was developed to numerically assess the ratcheting response of AM Al 4043 samples. Figure 4 represents the flat shape geometry of aluminum samples with thickness of a 1 mm. Test sample movements in x, y, and z directions were limited by fixing one end of the sample. Sinusoidal loading cycles were applied with frequency of 0.01 mm/s from another specimen end. Through the developed FE model, different element types were examined as shown in Figure 4. The solid elements in this figure were created through using the sweep mesh method. Using the sweep method, a source surface was meshed, and it was then swept through a body to generate volume meshing. Three meshing structures of the linear brick elements with 8 nodes, the quadratic tetrahedron with 10 nodes, and the brick elements with 20 nodes were designed. Figure 4 presents mesh structures for the brick solid and the tetrahedron solid elements.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Material** & \(\sigma_{m}\)\(\pm\)\(\sigma_{a}\) (MPa) & **Chaboche Coefficients** \\ \hline \multirow{4}{*}{Al 4043} & \(75\pm 55\) & C\({}_{1}\) (GPa) = 280 & \(\gamma^{\prime}_{1}\) = 4500 \\ \cline{2-4}  & \(70\pm 55\) & C\({}_{2}\) (GPa) = 16 & \(\gamma^{\prime}_{2}\) = 800 \\ \cline{2-4}  & \(65\pm 65\) & C\({}_{3}\) (GPa) = 5 & \(\gamma^{\prime}_{3}\) = 0 \\ \cline{2-4}  & \(75\pm 60\) & & \\ \hline \end{tabular}
\end{table}
Table 1: Coefficients of the Chaboche hardening rule.  
Figure 3: Chaboche coefficients \(C_{1-3}\) and \(\gamma^{\prime}_{1-3}\) for Al 4043 alloy.  
The convergence of displacement for each element type was studied to determine the independent nature of the result from the number of elements. Figure 5a presents the convergence curve for the linear brick element. As the number of elements increased, there was no substantial change in the result and the convergence was achieved at 119 elements. The quadratic tetrahedron element in Figure 5b shows a constant displacement when the number of taken elements exceeded 190. In this figure, as the number of elements was increased from 192 elements to 500 elements, the difference in displacement was found to be lower than 0.14%. The trend of convergence the quadratic brick elements in Figure 5c revealed that there was no significant change in the result above 120 elements and the model converged at this point. A sufficient number of elements led to an accurate estimation of the maximum and minimum strain values at the mid-section (gauge length) of samples.  
Figure 6 exhibits the contour plot presentation of total strain for the AM Al 4043 sample under cyclic loading of \(75\pm 60\) MPa. The maximum total strain of 0.040996 was achieved at the specimen gauge length with the smallest area. The minimum strain of 0.000314 was obtained in the locations farther than critical area. In a distance far enough from this section, the contour lines of the total strain stay constant. As the elements corners are nodal locations, the Gaussian integration points are positioned within the element. Contour plots are generated from FE results by extrapolating stress and strain values from Gaussian integration points to nodal locations. This extrapolation process is performed using the shape functions. Table 2 presents the details of element type, shape, and nodal features employed in this study.