Numerous isothermal experiments were conducted over a practical range of deformation temperatures (650–950

${}^{\circ}$C) and strain rates (0.05–1.0 s

${}^{-1}$) to develop the JC material model to predict the flow stress data of AISI-1045 medium carbon steel. In addition, the experimental data obtained from the quasi-static strain rate tensile tests at room temperature were employed for the evaluation of damage model parameters. To verify the model adequacies and predictability, the proposed constitutive model predictions were compared with the experimental observations and were also incorporated into the numerical simulations for inverse calibrations.

Figure 11 depicts the comparison of experimental stress-strain flow curves with the predicted flow curves by using the proposed JC model, whereas the model parameters were estimated from the optimization method. The data plotted in

Figure 11 and the numerical data outlined in

Table 5 clearly display that the presented optimized JC model is in good agreement with the experimental observations at higher temperatures for all strain rates, and on the contrary, the model cannot offer a better prediction of flow stress at the deformation temperature, 650

${}^{\circ}$C, for all tested strain rates. Thus, from the prediction error comparison, the flow stress data obtained in the optimized JC model were found to be more consistent with the experimental data than the conventional JC model.

To perform the model evaluation, standard statistical measurements such as coefficient of determination (

${R}^{2}$) and an average absolute relative error (AARE) were adopted to quantify the proposed JC model predictability at discrete strains with an interval of 0.025 for all strain rates and temperatures. The

${R}^{2}$ provides information about the prediction strength of the linear relationship between the experimental observations and the predicted values, whereas AARE was estimated through a term-by-term comparison of the relative error. To perform this quantification, the following expressions were employed [

28,

29]:

where

${\sigma}_{\mathrm{exp}}$,

${\sigma}_{\mathrm{pred}}$,

${\overline{\sigma}}_{\mathrm{exp}}$ are the experimental flow stress, the predicted flow stress, and the mean flow stress, respectively, and

n is the total number of data points.

In this research, each test condition was examined by estimating the values of

${R}^{2}$ and AARE for each case rather than the traditional method, in which the entire data set was used to compute the statistical parameters as mentioned in

Table 5. In this way, the prediction strength of the proposed JC model can be discussed in detail in terms of each and every test condition. The predicted flow curves and the graphical validation of the optimized JC model are shown in

Figure 11 for all strain rates and deformation temperatures.

Figure 11a depicts the comparison plot between predicted flow curves and experimental data and it shows that the developed model overpredicts the flow stress data. The numerical values

${R}^{2}$ and AARE were found to be 0.0012, lacking the prediction of the linear relationship as illustrated in

Figure 12a, and 40.9341, respectively. From these numbers, it is clear that the optimized JC model cannot predict the material behavior at a deformation temperature of 923 K for all strain rate conditions. However, somehow, as shown in

Figure 12a, there is some abnormal behavior in the distribution of data points. To verify this phenomenon, the residual plot, decomposed into three parts—low, exact, and high predictions—is plotted in

Figure 12d. From

Figure 12d, it is clearly shown that the proposed model mostly under predicts the flow stress, as the most of the data points were distributed linearly, somehow in exponential form, above the exact prediction line. This phenomenon explains that by adding some noise function into a negative linear or exponential form to the original flow stress model, this prediction error can be avoided. Likewise,

Figure 11b shows that at a deformation temperature of 1123 K for all strain rates, it is evident that most of the predicted flow stress data are close to the experimental observations, whereas

Figure 12b exhibits a good correlation between actual and predicted data. In addition, the computed corresponding values of the statistical parameters,

${R}^{2}$, 0.8679, and AARE, 5.9313%, show that the proposed JC model has considerable potential to predict the flow stress under the tested conditions. Furthermore,

Figure 11c and

Figure 12c illustrate the good correlation between experimental and predicted data under the tested conditions.

${R}^{2}$ and AARE were found to be 0.8419 and 5.9689%, respectively. Besides, it is noted that the prediction error minimization considering the material parameters,

c and

m using the optimization procedure led to significant improvement in the JC model prediction. For all test conditions, the overall AARE reduced from 18.12% to 17.61%. The differences between the models may be small but this small error can cause the false estimation of flow stress which leads to the inaccurate prediction of material behavior.

Overall, it was observed that the optimized JC model for AISI-1045 medium carbon steel can be used for flow stress prediction at high temperatures over the entire tested range of strain rates. Even though the overall flow stress prediction was good, in a few cases, for example, at deformation temperature 1123 K and at a strain rate of 0.05 s${}^{-1}$, deviation was found to occur. The reason for the deviation is mainly because of the softening behavior and the drop in flow stress that happens in the early stages at the following deformation temperatures: 923 K (at 0.05 s${}^{-1}$∼1.0 s${}^{-1}$), 1123 K (at 0.05 s${}^{-1}$) and 1223 K (at 0.05 s${}^{-1}$). The decreased flow stress values led to the improper estimation of the model parameters, because the JC model is just a phenomenological model that does not consider any of the material physical aspects. In addition, sometimes numerical numbers such as ${R}^{2}$ can lead to error, even though the model is adequate numerically. So, in order to remove this misinterpretation, graphical validation is necessary, and if both numerical and graphical outputs are admissible, then the developed flow stress model is good to use for future the calculations.