3.2.1. Johnson–Cook Model
The JC material model is a popular semi-empirical constitutive model used to describe the plastic behavior of materials under high strains, strain rates, and temperatures [
30]. Its simplicity, straightforward formulation, and ease of estimating material constants have made it used by researchers to forecast material flow behavior [
30]. The JC model can be represented as [
30,
31,
32,
33]:
where
A,
B,
n,
C,
T, and
m are the model coefficients [
27,
28,
29,
33]. Equation (
1) [
30] represents the elasto-plastic term, which shows the work hardening effect and viscosity term, which reveals strain-rate-strengthening effect and thermal softening term, which reveals the temperature effect that influences the material flow stress [
34,
35]. Here, the melting temperature (
), the reference temperature (
), and the reference strain rate (
) were assumed as 630 °C, 50 °C, and 0.005 s
, respectively. For example, for a 0° rolling direction (RD), the yield stress
A under the reference deformation conditions was determined as 186.163 MPa.
Determination of Material Constants: At reference conditions such as 50 °C and 0.005 s
, Equation (
1) can be altered into Equation (
2) [
10,
11,
12]:
Applying natural logarithms in Equation (
2) delivers Equation (
3) as shown below [
10,
11,
12]:
A correlation plot of
vs.
was outlined, as depicted in
Figure 5a. Thus,
B and
n were calculated as 536.359 MPa and 0.719, respectively.
Equation (
1) can be rearranged, when
T is
and expressed as Equation (
4) [
10,
11,
12].
By substituting, the material coefficients, like
A,
B, and
n, and corresponding stress values at reference conditions,
, were drawn, as represented in
Figure 5b. Then, the slope was obtained from the fitted curve as
C, and
C was estimated as 0.0049.
Similarly, Equation (
1) can be simplified, when
is
as [
36,
37,
38,
39]:
Equation (
6) can be received by applying the natural logarithm of Equation (
5) as [
36,
37,
38,
39]
By substituting, the material coefficients, like
A,
B, and
n, and corresponding stress values at reference conditions,
, were plotted, as shown in
Figure 6, and
m was obtained as 0.643. In conclusion, the JC flow stress model of AZ31B magnesium alloy material for a 0° rolling direction can be established as follows:
3.2.2. Modified Johnson–Cook Model
The modified JC model can also be used to describe the material deformation behavior of the AZ31B magnesium alloy, and it can be represented as [
18,
25]:
where
,
,
,
,
, and
are the model coefficients. Here,
and
are assumed as 50°, and 0.005 s
, respectively.
Determination of constants: For example, for a 0° RD, under the reference conditions 50 °C and 0.005 s
, Equation (
8) can be altered into Equation (
9) [
18,
25]:
As revealed in
Figure 7a, under the reference conditions, the relationship plot
and
was plotted using the second order polynomial equation, and the material parameters, such as
,
, and
, can be estimated as
,
, and
, respectively, from the coefficients of the fitted polynomial equation.
Under the reference temperature condition, Equation (
8) can be modified and rewritten as Equation (
10) [
18,
25]:
By substituting the estimated material constants, considering the stress values from the tested strain rate conditions at
, the relationship plot between the dimensionless strain rate,
and
, was plotted, as illustrated in
Figure 7b. Thus, the material coefficient,
, was computed as 0.0037.
For the tested conditions, Equation (
8) can be rearranged and written as Equation (
11) [
18,
25]:
Applying natural logarithm in Equation (
11) delivers Equation (
12) as follows:
Equation (
12) was simplified by adding a new coefficient, named
, which is equal to
, and
can be estimated from the correlation between
and
, as shown in
Figure 8a. Furthermore, in this study, we considered three strain rates, so three graphs were plotted and the introduced parameter,
, was derived from each plot, as depicted in
Figure 8b,c. The introduced new parameter,
, in Equation (
12) can be presented as Equation (
13) [
18,
25],
Eventually, as revealed in
Figure 8d, the model coefficients,
and
, are calculated as −0.00542 and 0.000659, respectively. Thus, the modified JC flow stress model of the AZ31B magnesium alloy material for a 0° rolling direction can be established as follows:
3.2.3. Modified Zerilli–Armstrong Model
The modified ZA model is also used to describe the material plastic deformation behavior of AZ31B magnesium alloy, and it can be represented as [
18,
25]:
where
,
,
n,
,
,
, and
are the model coefficients. Here,
and
are considered as 50 °C and 0.005 s
, respectively. For example, for 0° RD, the yield stress,
, under the reference deformation conditions was determined as 186.163 MPa.
Determination of constants: At
, Equation (
14) can be rearranged and represented as Equation (
15) [
18,
25]:
Then, by applying the natural logarithm in Equation (
15) [
25], Equation (
16) [
25] can be obtained as [
18,
25]:
By putting flow stress at
into Equation (
16),
and
can be computed from
vs.
, as illustrated in
Figure 9a. The steps were repeated for other strain values, and then, Equation (
19) [
25] was received by applying the natural logarithm in Equation (
17) as [
18,
25]:
At
, by adopting stress values from the entire temperature range and using the estimated values of
and
, the correlation plot of
vs.
was achieved, as shown in
Figure 9b. Thus, the model coefficients,
and
n, were determined as
and 0.6453, respectively, from the information of the fitted curve.
Similar to the coefficients
and
n, at
, by substituting estimated
into the discrete true strains, the coefficients
and
were computed as 0.0052 and 0.0052, respectively, from the linear model information of
vs.
, as depicted in
Figure 10a.
Applying the natural logarithm in Equation (
14) delivers Equation (
20) as follows [
25]:
For accounted strain rates with respect to one temperature, the relationship plot of
vs.
can be made, as illustrated in
Figure 10b. Then, the coefficient,
, was estimated from
Figure 10b at a specific temperature. For five temperatures, five different values of
were determined, and thereafter, the parameters
and
were computed as 0.0149 and 0.0004, respectively, from the fitted curve information, as depicted in
Figure 11. Thus, the modified ZA flow stress model of the AZ31B magnesium alloy material for 0° RD can be established as follows:
Using the estimated model parameters of the proposed constitutive models in
Table 1,
Table 2 and
Table 3, AZ31B magnesium alloy flow stress data under the considered deformation conditions for three rolling directions were calculated. In order to assess the accuracy of the proposed flow stress models, the AARE can be computed by comparing the test data with the predicted data using the following equation [
28,
40,
41,
42]:
where
,
, and
n are the experimental true stress, the predicted true stress, and the total number of true stress data, respectively. Subsequently, the prediction errors were estimated using Equation (
22) and plotted in
Figure 12. As shown in
Figure 12a, the original JC model could not sufficiently represent AZ31B magnesium alloy deformation flow behavior, as it shows higher prediction errors ranging from 11.19% to 15.57% across all rolling directions and deformation conditions. On the other hand,
Figure 12b,c demonstrate that the proposed modified JC and ZA models offer good prediction of flow stress values for the AZ31B magnesium alloy. Additionally, the prediction errors are estimated to be about 4.30% to 8.51% across all rolling directions and deformation conditions considering both the modified JC and ZA models. The prediction error comparison confirms that the prediction error is reduced by about 45.34% to 61.57% when compared against the minimum and maximum prediction error of the original JC model. To assess the predictive capability of the proposed flow stress models, a comparison plot was created depicting the predicted flow curves alongside the experimental data. This allows for a detailed discussion of the model’s accuracy with respect to each test condition.
According to Lin et al. [
43], the original JC model’s predictability is constrained to a specific
and
. This limitation arises from the model’s assumption on the coupled effects and independent phenomena. However, in practice, it is essential to account for the combined effects on the flow behavior of the AZ31B magnesium alloy [
18,
43,
44]. A comparison between the test and calculated data from the modified JC and ZA models under the tested conditions is outlined in
Figure 13 and
Figure 14.
Figure 13 and
Figure 14 show good agreement at high temperatures, indicating that both the modified JC and modified ZA models accurately predict AZ31B magnesium alloy flow stress values. These models are suitable for analyzing hot deformation behavior in sheet-metal-forming processes. The modified ZA model demonstrates good accuracy in forecasting deformation behavior at elevated temperatures compared to the modified JC model, as revealed in
Figure 13 and
Figure 14. In detail, as depicted in
Figure 13, the recognized modified JC model showed good prediction against the experimental observations under the reference conditions (50 to 250 °C and 0.005 to 0.0167 s
). Moreover, at 200 °C for the tested strain rates across all the rolling directions, the model provided better predictions against the test data. Similar observations were also made for 150 °C and 250 °C temperatures for the considered strain rates; however, there were some noticeable deviations in the predicted data.
Compared with the other test conditions,
Figure 13 reveals that the established modified JC model could not significantly forecast the material deformation behavior at 100 °C for the entirety of the test conditions. Subsequently, as shown in
Figure 14a–f, the calculated flow stress data from the modified ZA model falls close to the test observations at 100 °C to 250 °C at the tested strain rates for 0° and 45° RDs; however, for 90° RD, the proposed MZA model could represent the material flow behavior at only 200 °C and 250 °C, including reported strain rates, as depicted in
Figure 14. Furthermore, under 50 °C and 100 °C test conditions, high prediction deviations were observed, as shown in
Figure 14. In addition, for three rolling directions, the prediction errors were observed to be higher than the other test conditions. Thus, the modified ZA model outperforms the original JC and modified JC models in accurately representing AZ31B magnesium alloy deformation behavior across the considered processing conditions, as outlined in
Figure 14. The improved performance of the modified ZA model can be attributed to the combined effects of deformation temperature and strain rate on the flow stress.
The proposed constitutive equation predictability can be further verified through employing statistical metrics, like
and RMSE, as follows [
28,
40,
41,
42]:
The
measures the linear relationship strength, while RMSE provides information on the comparison of relative errors term by term.
Figure 15a reveals that the calculated flow stress data deviate from the best-fit line, with an estimated
value of 0.888. This suggests that the original JC model fails to accurately capture the material’s deformation behavior. Furthermore,
Figure 15d also demonstrates that the residual distribution is not random, indicating that the model lacks predictability due to missing terms in the constitutive equation. Moreover,
Figure 16a shows that the original JC model considerably underestimates the flow stress, which results in high prediction error. In contrast,
Figure 15b illustrates that the modified JC model yields predicted data that closely align with the best-fit line, with a correlation coefficient of 0.962, which is significantly higher than the original JC model. This signifies an improved correlation between the predicted and test data. However, despite the improvement in prediction,
Figure 15e reveals that the residual distribution is still non-random, indicating that the modified JC model also overlooks certain terms in the constitutive equation, resulting in remaining prediction errors.
Additionally,
Figure 16b displays that the modified JC model showed the prediction error reduction with high range of under predictions (within ±30%) of the flow stress. Similarly, as shown in
Figure 15c, the predicted data mostly fall near the best-fit line for the modified ZA model, as well with an
of 0.954, which is significantly higher than the original JC model. This also explains that the modified ZA model has better predictability than the original JC model. However,
Figure 15f demonstrates that the residual distribution is still not random and explains that the modified ZA model also misses some terms in the constitutive equation, which might be the reason for the remaining prediction error. Additionally,
Figure 16c displays that the modified ZA model showed a prediction error reduction somehow balanced in the range of under and over predictions (within ±20%). The predictability of the original JC, modified JC, and modified ZA models is summarized in
Table 4 and
Table 5. Based on the discussion above, it is evident that the proposed modified JC and ZA models provide good predictions for the flow stress values of the AZ31B magnesium alloy. These models are suitable and reliable for analyzing the AZ31B magnesium alloy hot deformation process.