## 3 Results and Discussion
### Chaboche's Coefficients  
The material coefficients for the Chaboche nonlinear kinematic hardening model were calibrated using experimental hysteresis loops obtained from strain-controlled fatigue tests on steel samples. To ensure that the proper parameters were determined, the simulated loops needed to closely agree with the experimental results in both shape and area. Smaller values of the coefficients \(C_{1-3}\) and \(\gamma_{1-3}^{{}^{\prime}}\) result in narrower loops, which indicates slower hardening and less ratcheting. Larger coefficient values resulted in wider loops and higher ratcheting strains. Figure 5 shows the comparison between simulated and experimental stress-strain loops at three different temperatures under a constant strain amplitude of %3\(\pm\) for SA333. The agreement between the loops confirms the adequacy of the selected coefficients \(C_{1-3}\) and \(\gamma_{1-3}^{{}^{\prime}}\) for the cyclic behavior of SA333. The coefficients \(C_{1}\) and \(C_{3}\) represent the plastic modulus at yielding and the slope of the linear part of the hysteresis loop, respectively. The backstress values during loading, unloading, and reloading are noted as \(\alpha_{01}\), \(\alpha_{02}\), and \(\alpha_{03}\). These are calculated from Equation (11) and then used as initial values (\(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\)) for the next loading cycle. The parameters \(\gamma_{1}^{{}^{\prime}}\) and \(\gamma_{3}^{{}^{\prime}}\) are obtained from uniaxial ratcheting data, where \(\gamma_{1}^{{}^{\prime}}\) helps to stabilize hardening in the early cycles, and \(\gamma_{3}^{{}^{\prime}}\) controls the steady rate of ratcheting strain. The constants \(C_{2}\) and \(\gamma_{2}^{{}^{\prime}}\) are selected through a trial-and-error process to satisfy Equation (12):  
\[\left(\frac{C_{1}}{\gamma_{1}^{{}^{\prime}}}\right)+\left(\frac{C_{2}}{\gamma_ {2}^{{}^{\prime}}}\right)+\sigma_{y}=\sigma-\frac{C_{3}}{2}\left[\Delta \varepsilon_{p}\right] \tag{12}\]  
where \(\sigma\) is the applied stress amplitude, and \(\Delta\varepsilon_{p}\) is the plastic strain range.  
The loop is widest at 298 K, indicating strong kinematic hardening and limited dislocation mobility due to lower thermal activation. At 573 K, the loop narrows significantly, suggesting that dynamic strain aging (DSA) begins to influence the material response. The DSA effect, attributed to the interaction between diffusing interstitial carbon atoms and dislocations, temporarily hinders dislocation motion. This results in increased cyclic hardening but reduced plastic strain accumulation. Interestingly, at 623 K, the loop becomes slightly broader again, which may be due to the partial relaxation of DSA effects and increased thermal softening at higher temperatures. The evolution of loop width across the temperatures reflects the interplay among dislocation mobility, backstress evolution, and diffusion mechanisms. The final set of Chaboche coefficients used for each case is provided in the figure captions and further summarized in Table 1.  
Figure 4: Kinematic hardening coefficients (**a**) \(C\), \(\gamma_{1}\), and (**b**) \(\gamma_{2}\) determined for SA333 [52] steel samples at various temperatures.