## 5 Results and Discussion
### Cyclic Elasto-Plastic Response  
The elasto-plastic cyclic response was compared with the experimental results obtained in the cyclic tests. A simulation for each strain amplitude level applied in the experiments was performed up to the number of failure cycles. The kinematic model was compared with the cyclic Ramberg-Osgood model and the experimental results. Then, the isotropic model was combined with the kinematic model, and the numerical results were compared with the experimental results. Figure 12 compares the hysteresis loops for stabilized cyclic conditions obtained from the numerical and experimental models. Additionally, Figure 12 also presents a comparison of the Ramberg-Osgood curve with the maximum stress levels observed using the Chaboche kinematic hardening numerical model and considering the cyclic kinematic hardening properties in Table 7. In general, the maximum stress obtained from the Chaboche model showed a slight deviation in relation to the Ramberg-Osgood curve. This deviation tended to be higher as the strain amplitude was decreased. Additionally, from the comparison between the numerical and experimental outcomes, differences in the plastic strain energies were observed. Regarding the transient behavior, this was analyzed considering all the numerical hysteresis cycles, and the experimental and stabilized cycles, as shown in Figure 13. The cyclic behavior shown in Figure 13 was obtained by considering the cyclic hardening model combined with the properties shown in Table 7. The three levels of the strain amplitude shown were 0.5% (small), 0.9% (medium), and 1.5% (large). Comparing the strain tests with 0.5 and 1.5%, it was observed that the numerical model was able to predict the maximum and minimum stress levels for both the first hysteresis loop and for the stabilized hysteresis loop. However, for an applied strain amplitude of 0.9%, a deviation was verified for both the first and for the stabilized hysteresis loop. This deviation is also observed in Figure 12,  
Figure 11: Evolution of the decomposition components of the back stress, total back stress, and the sum between the total back stress and size of the elasticity limit surface up to 30% plastic strain. with the maximum stress of the hysteresis loop being lower than predicted by the Ramberg-Osgood curve. Since the isotropic hardening parameters were average values obtained for the test strain ranges, the deviation observed in Figure 13 for \(\varepsilon_{a}=0.9\%\) was to be expected. With respect to the area developed by the plastic work, the Chaboche hardening model developed a greater dissipative energy, being more conservative in the fatigue analysis. Another way of analyzing the evolution of isotropic hardening is the evaluation of the evolution of the maximum stress in each cycle, as represented in Figure 14. Figure 14 shows, in illustrative terms, a comparison o the numerical model results obtained using the combined Chaboche model. Experimentally, the softening gradient was greater up to approximately 5% of the total life. For normalized lifetimes greater than 10%, there was an almost linear softening, up to a normalized lifetime of 50%. Comparing the experimental data with the numerical results, it can be observed that, only as the strain amplitude increased did the lifetime associated with the stabilized behavior also tend to be higher. In the case of 0.5%, stabilization occurred to 20%, while for 0.9% stabilization occurred to approximately 50%. However, for a 1.5% applied strain amplitude, stabilization would occur for values greater than 50%. Figure 12: Comparison of the numerical results given by Chaboche kinematic hardening model, experimental data, and cyclic Ramberg-Osgood model for steady-state cyclic conditions under various strain amplitudes. Figure 13: Comparison of the numerical results given by the combined hardening model and experimental data for the first hysteresis loop and the steady-state cyclic conditions.