## 3 Results and Discussion
### Finite Element Method  
To simulate the ratcheting behavior of the SA508 and SA333 steel samples, finite element models were constructed using ANSYS 2022 R [55]. As shown in Figure 6, two different element types were examined to evaluate mesh sensitivity in the analysis. Each sample has a different geometry because the data were obtained from different experimental studies, and the specimen designs reflect the original configurations used in those sources [51; 52]. The geometries correspond to standard axisymmetric tensile specimens, and both nonlinear brick and tetrahedral solid elements were implemented. The solid elements were generated using the sweep mesh technique, which involves creating a mapped mesh on a source face and extruding it through the volume to form structured meshing. The mesh structures were designed with brick and tetrahedron elements containing \begin{table}
\begin{tabular}{l l l l} \hline
**Material** & **Temperature** & **Models** & **Parameters and Coefficients** \\ \hline \multirow{8}{*}{SA508} & \multirow{3}{*}{298 K} & Isotropic & \(Q\) (MPa) = 70, \(b\) = 20 \\ \cline{3-4}  & & A-V & \(C\) (GPa) = 50, \(\gamma_{1}\) = 200, \(\gamma_{2}\) = 10 \\ \cline{3-4}  & & Chaboche & \(C_{1-3}\) (GPa) = 7, 2.8, 0.8, \(\gamma_{1-3}^{{}^{\prime}}\) = 100, 25, 30 \\ \cline{3-4}  & & Isotropic & \(Q\) (MPa) = 53, \(b\) = 15 \\ \cline{3-4}  & & A-V & \(C\) (GPa) = 60, \(\gamma_{1}\) = 225, \(\gamma_{2}\) = 5 \\ \cline{3-4}  & & Chaboche & \(C_{1-3}\) (GPa) = 10, 2.2, 0.474, \(\gamma_{1-3}^{{}^{\prime}}\) = 100, 25, 30 \\ \cline{3-4}  & & Isotropic & \(Q\) (MPa) = 41, \(b\) = 12.5 \\ \cline{3-4}  & & A-V & \(C\) (GPa) = 38, \(\gamma_{1}\) = 245, \(\gamma_{2}\) = 9 \\ \cline{3-4}  & & Chaboche & \(C_{1-3}\) (GPa) = 33, 5.8, 0.1, \(\gamma_{1-3}^{{}^{\prime}}\) = 250, 25, 30 \\ \hline \multirow{8}{*}{SA333} & \multirow{3}{*}{298 K} & Isotropic & \(Q\) (MPa) = 60, \(b\) = 20 \\ \cline{3-4}  & & A-V & \(C\) (GPa) = 60, \(\gamma_{1}\) = 200, \(\gamma_{2}\) = 7 \\ \cline{3-4}  & & Chaboche & \(C_{1-3}\) (GPa) = 1123, 50.5, 5.9, \(\gamma_{1-3}^{{}^{\prime}}\) = 280,750, 950, 9 \\ \cline{3-4}  & & Isotropic & \(Q\) (MPa) = 50, \(b\) = 014 \\ \cline{3-4}  & & A-V & \(C\) (GPa) = 50, \(\gamma_{1}\) = 230, \(\gamma_{2}\) = 11 \\ \cline{3-4}  & & Chaboche & \(C_{1-3}\) (GPa) = 1450, 45, 3.2, \(\gamma_{1-3}^{{}^{\prime}}\) = 275,000, 950, 9 \\ \cline{3-4}  & & Isotropic & \(Q\) (MPa) = 45, \(b\) = 11 \\ \cline{3-4}  & & A-V & \(C\) (GPa) = 45, \(\gamma_{1}\) = 245, \(\gamma_{2}\) = 8 \\ \cline{3-4}  & & Chaboche & \(C_{1-3}\) (GPa) = 650, 97.5, 0.8, \(\gamma_{1-3}^{{}^{\prime}}\) = 300,500, 950, 9 \\ \hline \end{tabular}
\end{table} 
Table 1: Coefficients of hardening rules to assess ratcheting. Figure 5: Chaboche’s coefficients \(C_{1-3}\) and \(\gamma_{1-3}^{{}^{\prime}}\) for SA333 steel samples tested at (a) 298 K, (b) 573 K, and (c) 623 K [54]. 20 and 10 nodes, respectively. The details of the element type, shape, and nodal features employed in this study are provided in Table 2. The models were constrained at one end to restrict motion in all directions, and cyclic boundary conditions were applied at the opposite end to replicate consistent sinusoidal loading patterns. To ensure mesh-independent results in simulation of the ratcheting response, convergence studies were carried out for the SA508 and SA333 steel samples using different element types. Figure 7 illustrates the displacement values recorded for varying number of elements. Figure 7a,b display the convergence results for SA508 samples meshed with nonlinear brick and tetrahedron elements, respectively. In both cases, the displacement response stabilized with increasing number of elements, indicating acceptable convergence. Specifically, the displacement variation became negligible beyond approximately 800 elements for the brick type and 1200 elements for the tetrahedral mesh. Similarly, Figure 7c,d present the convergence for SA333 samples, showing that the response remained unchanged when more than 660 and 925 elements were employed, respectively. These