## 4 Constitutive Model and Finite Element Implementations
### Comparison of Simulation and Experimental Results  
The three-dimensional geometry of the sample was built with Abaqus-CAE sketcher tool. The geometry of the sample consisted of a cylindrical solid with a reduced cross-section length in the middle, as given in Figure 15. There were different types of finite elements, and the element size also affected the precision and computing performance. It was important to balance both element type and element size in order to obtain results with a fairly small error and in a relatively short time. Within the framework of the finite element model, one end of the model was completely fixed (U1 = U2 = U3 = UR1 = UR2 = UR3 = 0). A reference point RP-1 was established at the center of the top surface of the model and coupled to the top surface, and the cyclic axial load was applied to the reference point RP-1. As depicted in Figure 15b, the Mises stress that emerged in the middle section of the specimen was presented. Consequently, the ratcheting strain located in the middle section of the specimen was singled out for the purpose of conducting a comparative analysis. In this paper, an axisymmetric 2D model was also used to investigate the effect of model space on accuracy. As shown in Figure 16a, symmetric constraints were applied along the left edge of the 2D model, i.e., U1 = UR2 = UR3 = 0. At the center of the top surface, a reference point RP-1 was created and coupled to the top surface for applying axial loads, and these constraints took into account the possible displacement limitations in that direction. As shown in Figure 16b, simulations of the uniaxial ratchet effect were carried out using both the 2D axisymmetric model and the 3D model. It was found that the ratchet strains predicted by both models were the same and the model space had no effect on the ratchet effect simulation results. \begin{table}
\begin{tabular}{c} \(E\) = 193 GPa,\(\
u\) = 0.3,\(\sigma_{0}\) = 100 MPa \\ \(r_{(1-6)}\)10, 65, 60, 39, 73, 162 \(\gamma_{(1-6)}\)4915, 2451, 1232, 612.5, 406.5, 32.125 \\ \(\mu\) = 0.6, \(Q\) = 59 MPa, \(b\) = 2.3, \(m\) = 2.5 \\ \end{tabular}
\end{table} 
Table 4: Material parameters. Figure 14: Uniaxial stretch curve. In order to demonstrate that the numerical results were not affected by spurious mesh correlations, Figure 17 showed the comparison of the ratcheting strains of models using different finite element types under loading conditions with an average stress of 300 MPa and a stress amplitude of 300 MPa. Figure 17a showed the simulation results of the finite element models using different grid cell shapes (hexahedral, tetrahedral, and wedge), and the gauge length in all simulation analyses was 0.25 mm. The total number of cells generated using hexahedral finite elements, tetrahedral finite elements, and wedge cell shapes was 11,780, 39,624, and 20,064, respectively. The results showed that the use of hexahedral finite elements was more suitable for further analysis. As shown in Figure 17b, the gauge lengths in this finite element model were simulated using three different sizes (0.1 mm, 0.25 mm, and 1 mm). It could be seen from the figure that the simulation results using the three sizes of hexahedral cell meshes were basically the same, and the use of a 0.25 mm gauge length was appropriate. Figure 18a,b compared the experimental and predicted results of the stress-strain hysteresis loops for the first 200 cycles under a stress amplitude of 300 MPa and a mean stress of 500 MPa, respectively. It could be seen that the shapes of the simulated hysteresis loop were in agreement with the experimental results. Figure 19 presents the evolution of the ratcheting strain. It can be observed that the predicted results of uniaxial ratcheting strain with ABAQUS were consistent with the experimental data. The relative errors between the simulated and experimental ratcheting strains during stress loading are given in Table 5.  
Figure 16: (**a**) Axisymmetric model; (**b**) comparison of the predictions of the two models with the experimental.