Constitutive Analysis of the Deformation Behavior of Al-Mg-Si Alloy Under Various Forming Conditions Using Several Modeling Approaches

4.1. Shokry’s MJC-1 (S1-MJC)

The MJC model developed by Shokry [70] is modified in this study to improve its accuracy in predicting the deformation behavior of AA6082 at several values of

T

and

\dot{ε}

by incorporating a linear relationship that directly links

ε

with

T

and

\dot{ε}

. The S1-MJC model is formulated below:

$σ = (A + {B_{1} ε + B_{2} ε}^{2} + {B_{3} ε}^{3}) (1 + (C_{1} + C_{2} ε) \ln ε^{\cdot *}) (1 - {T^{*}}^{(m + m_{2} ε)})$

(29)

For obtaining the S1-MJC model’s constants,

T_{r}

and

{\dot{ε}}_{r}

are set as 400 °C and 0.001 s⁻¹. Therefore, Equation (29) simplifies to the following:

$σ = A + {B_{1} ε + B_{2} ε}^{2} + {B_{3} ε}^{3}$

(30)

Through regression analysis, $A$ was determined to be 58.071 MPa, $B_{1}$ was −11.065 MPa, $B_{2}$ was −22.881 MPa, and $B_{3}$ was 27.824 MPa.

After modifications and at

T

= 400 °C, Equation (29) is described below:

$\frac{σ}{A + {B_{1} ε + B_{2} ε}^{2} + {B_{3} ε}^{3}} - 1 = (C_{1} + C_{2} ε) \ln ε^{\cdot *}$

(31)

Using regression,

C_{1}

was determined to be 0.0616 and

C_{2}

was 0.0641. After applying the natural logarithm and simplifying, Equation (29) can be expressed in various

\dot{ε}

values as follows:

$\ln [1 - \frac{σ}{(A + {B_{1} ε + B_{2} ε}^{2} + {B_{3} ε}^{3}) (1 + (C_{1} + C_{2} ε) \ln ε^{\cdot *})}] = (m_{1} + m_{2} ε) T^{*}$

(32)

Using regression,

m_{1}

was determined to be 1.4884 and

m_{2}

was determined to be −0.3467. The S1-MJC model constants of the AA6082 alloy are written in Table 5.

Therefore, the S1-MJC model of AA6082 alloy is presented below:

$σ = (58.071 - 11.065 {ε - 22.881 ε}^{2} + {27.824 ε}^{3}) (1 + (0.0616 + 0.0641 ε) \ln ε^{\cdot *}) (1 - {T^{*}}^{(1.4884 - 0.3467 ε)})$

(33)

Figure 3 compares the experimental flow stresses of AA6082 and those determined via the S1-MJC model, with R = 0.983, AARE = 7.09%, and RMSE = 3.56 MPa, as listed in Table 6. As depicted, the results predicted by the S1-MJC model align closely with the experimental data, demonstrating a better fit than the MZA model; however, the S1-MJC model still does not outperform the accuracy achieved with CP modeling. This improved accuracy is because the S1-MJC model integrated both

\dot{ε}

and the softening effects and strain hardening in the model. It is well-established that dislocations are influenced by

\dot{ε}

and softening.

4.2. Shokry’s MJC-2 (S2-MJC)

The MJC model developed by Shokry et al. [71] is implemented in this study for determining the hot-flow behavior of AA6082 Al-Mg-Si over a wide range of

T

and

\dot{ε}

. Their model is represented below:

$σ = (\sum_{i = 0}^{3} A_{i} ε^{i}) (1 + (\sum_{i = 0}^{2} \sum_{j = 0}^{2} C_{i j} ε^{i} ε^{\cdot j}) \ln ε^{\cdot *}) \exp [(\sum_{i = 0}^{2} \sum_{j = 0}^{2} \sum_{k = 0}^{2} {m_{i j k} ε}^{i} ε^{\cdot j} T^{* k}) T^{*}]$

(34)

In this investigation, to determine the S2-MJC model’s constants,

{\dot{ε}}_{r}

was adjusted to 0.001 s^−1, and

T_{r}

was adjusted to 400 °C. Consequently, Equation (34) is simplified to the following:

$σ = \sum_{i = 0}^{3} A_{i} ε^{i}$

(35)

Expanding Equation (35) results in four parts including

ε

, each associated with four constants. Through regression, these constants were 58.071 MPa, −11.065 MPa, −22.881 MPa, and 27.824 MPa. Thus, at 400 °C, and after performing some adjustments, Equation (34) is presented below:

$(\frac{σ}{\sum_{i = 0}^{3} A_{i} ε^{i}} - 1) / \ln ε^{\cdot *} = \sum_{i = 0}^{2} \sum_{j = 0}^{2} C_{i j} ε^{i} ε^{\cdot j}$

(36)

Expanding Equation (36) results in nine parts, including

\dot{ε}

and

ε

, each linked to a corresponding constant. Through regression, these constants were 0.0550, 0.0828, −0.0888, 0.1637, −1.2841, 1.2652, −0.1126, 1.0707, and −1.0288. After applying the natural logarithm and making adjustments, Equation (34) can be expressed in various

\dot{ε}

values as follows:

$\frac{\ln [\frac{σ}{(\sum_{i = 0}^{3} A_{i} ε^{i}) (1 + (\sum_{i = 0}^{2} \sum_{j = 0}^{2} C_{i j} ε^{i} ε^{\cdot j}) \ln ε^{\cdot *})}]}{T^{*}} = \sum_{i = 0}^{2} \sum_{j = 0}^{2} \sum_{k = 0}^{2} {m_{i j k} ε}^{i} ε^{\cdot j} T^{* k}$

(37)

Expanding the right side of Equation (37) results in 27 parts, including

T,

\dot{ε}

, and

ε

. Through regression, these constants were obtained to be −0.6668, 2.7593, −2.4307, −0.2755, −5.5232, 5.0762, −0.1888, 2.1973, −1.5661, −0.0198, −10.3673, 9.3805, −2.597, 56.627, −53.125, 2.2348, −35.0907, 32.2649, −1.6642, 16.4578, −14.7332, 3.5334, −73.087, 68.9637, −2.7157, 52.349, and −49.0335, as listed in Table 7.

To determine the constants for the SI-MJC and S2-MJC models, the corresponding equations are rearranged to isolate the constants on the right-hand side. Although the relationships between the predictors $ε$ , $ε^{\cdot}$ , and $T^{*}$ and the response (the equation’s output on the left-hand side) are nonlinear, the equations are linear with respect to the constants (coefficients). Therefore, a linear regression model using the least-squares method is employed to compute the constants for the three models in MATLAB.

Figure 4 shows comparisons between the experimental stresses of AA6082 and those determined via the S2-MJC model, with R = 0.99, AARE = 1.87%, and RMSE = 0.95 MPa. As depicted in Figure 4 and Table 8, the result predicted by the S2-MJC model closely aligns with the experimentation, showing a better fit than the previous models and close to the accuracy achieved with CP modeling. This improved accuracy is because the S2-MJC model integrated both

\dot{ε}

and

ε

on the one hand, and

T,

\dot{ε}

, and

ε

on the other hand, facilitated by an extensive set of constants that link the softening and dynamic recovery components.

The R values for the developed models are presented in Figure 5a. The CP and S2-MJC models achieve the highest R values close to 1, with values of 0.999 and 0.99, respectively. In contrast, R for the MZA is 0.951 and for S1-MJC is 0.983. Similarly, Figure 5b,c display these models’ AARE and RMSE values. The CP and S2-MJC models exhibit the best performance with the lowest values of AARE, which are 1.1% and 1.87%, respectively. The values of RMSE are 0.55 MPa and 0.95 MPa, respectively. Moreover, the MZA and S1-MJC models yield higher AARE values of 11.67% and 7.09% and RMSE values of 7.23 MPa and 3.56 MPa, respectively. The obtained values of R, AARE, and RMSE indicate that the MZA and S1-MJC models can be used to predict the flow behavior of the studied alloy but with limited accuracy. The MZA model accounts for the coupling effects between

ε

and

T

as well as between

\dot{ε}

and

T

. In contrast, the S1-MJC model considers the coupling effects between

ε

and

T

as well as between

ε

and

\dot{ε}

. Moreover, the CP and S2-MJC models incorporate the coupling effects between

T,

\dot{ε}

, and

ε

. Given the nonlinear nature of the flow behavior of the studied alloy, models that account for the more comprehensive coupling between

T,

\dot{ε}

, and

ε

are expected to provide more accurate predictions.

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Bandar Alzahrani www.mdpi.com

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Constitutive Analysis of the Deformation Behavior of Al-Mg-Si Alloy Under Various Forming Conditions Using Several Modeling Approaches

4.1. Shokry’s MJC-1 (S1-MJC)

4.2. Shokry’s MJC-2 (S2-MJC)

Greenberg

4.1. Shokry’s MJC-1 (S1-MJC)

4.2. Shokry’s MJC-2 (S2-MJC)

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