Study of Thermal Compression Constitutive Relation for 5182-Sc-Zr Alloy Based on Arrhenius-Type and ANN Model

Abstract

Hot compression experiments were performed on alloy 5182 with small additions of Sc and Zr. The 5182 alloy containing Sc and Zr is critical for expanding the 5182 alloy’s range of applications, and a thorough understanding of its thermal processing behavior is of great importance to avoid processing defects. Alloy microstructure, including grain structures and Al₃(Sc_xZr_1−x) dispersoids were analyzed by EBSD and TEM. Stable flow stresses were observed below a strain rate of 1 s⁻¹ for the Sc-Zr containing alloy. The results of constitutive models, with and without strain−compensation, and artificial neural network (ANN) were used to compare to the experimental results. The Al₃(Sc_xZr_1−x) dispersoid data was introduced into the ANN model as a nonlinear influence factor. Addition of the Al₃(Sc_xZr_1−x) dispersoid information as input data improved the accuracy and practicality of the artificial neural network in predicting the deformation behavior of the alloy. The squared correlation coefficients of ANN prediction data reached 0.99.

1. Introduction

The 5xxx aluminum alloys such as 5182 and 5083 have been utilized successfully in applications ranging from beverage can ends to automobile panels, shipyard plates, rail cars, and armor materials [1]. When processed to the correct temper, they offer an attractive combination of high strength/weight ratio, weldability, formability, and corrosion resistance [2]. However, typical 5xxx alloys are limited in strength for some applications because of their lack of heat-treatability and low recrystallization temperatures. Alloying with rare earth elements has been explored as a means for further increasing the strength and recrystallization resistance of 5xxx commercial alloys [3,4,5,6]. Trace additions of rare earth elements such as Sc and Er increase the strength of cast and rolled aluminum alloys considerably [7,8,9,10]. The addition of Zr (which has a low diffusion coefficient) and special homogenization processes have been the subject of a great deal of research. The Al₃(Sc_xZr_1−x) dispersoids exhibit a Sc-core Zr-shell structure, where the shell rich in Zr improves the thermal stability of the dispersoid particles [11,12]. Most research has focused on model alloys with Sc-Zr additions, rather than commercial alloys such as 5182. The present study was looking into investigating the effects of Sc and Zr additions on the hot deformation behavior of 5182 alloy. It has also been demonstrated that the addition of Sc and Zr causes a reduction in thermal deformation stability of aluminum alloys [13]. We try to consider the influence of the entire production process, homogenization, and post-deformation annealing process on the hot working process.

The thermal and deformation parameters during wrought processes such as hot rolling produce different microstructures in 5xxx alloys, which ultimately determine the mechanical properties of the final product [14]. The flow stress curve can reflect not only the energy consumed to change the shape, but also phenomena such as dynamic recovery (DRV) and intermittent recrystallization during the high temperature deformation process. It is critical to establish the link between the flow stress curve and the product’s microstructure evolution during hot deformation.

A functional model relating the flow stresses to deformation parameters (temperature, strain rate, and true strain) is a desirable tool to predict the behavior at high temperatures [15,16]. In this study, true thermal compression data were used to develop an Arrhenius constitutive model. However, the intrinsic workability of a material can also be described by a process diagram model, which includes kinetic, atomic, and dynamic material models (DMM) [17,18]. The DMM employed in this study consists of a power dissipation diagram with an instability diagram superimposed on it, a commonly used processing diagram that can visually identify the thermal processing safety zone and instability zone [19,20,21].

When modeling the thermal processing of materials, it is necessary to consider that nonlinear variables have an impact on flow stresses [22,23]. The current constitutive model is a mathematical representation of the link between experimental thermal compression and flow stresses. In the field of materials, promising machine learning algorithms can immediately acquire non-linear correlations between input parameters and output data without the requirement for assumptions or knowledge of the data’s mathematical or physical properties [24]. Because predicting thermal deformation flow stresses is a regression problem, machine learning algorithms that tackle classification problems, such as decision trees and vector machines, are not suitable for this work. Regression-solving artificial neural networks provide the advantage of introducing nonlinear elements and boosting model accuracy [25,26]. According to Ashtiani’s research, a neural network utilizing a feed-forward error back-propagation algorithm modifies the network’s hidden layer weights and thresholds using gradient descent to reduce the error between the target and predicted outputs [27]. In Sheikh−Ahmad’ study, a novel artificial neural network (ANN) construction method (training and verification) was successfully applied to the sparse high strain-rate regime [28]. The predicted values of the ANN model are more consistent with the original experimental data in the study on thermal deformation behavior of 5083 and 5754 alloys [29]. In previous hot compression studies, the artificial neural network models for alloys containing Sc and Zr were constructed on the basis of processing conditions, as in the constitutive models. The Al₃(Sc_xZr_1−x) dispersoids have a significant impact on the recrystallization behavior and strength of Sc and Zr reinforced aluminum alloys, and these effects should be factored into the neural network design [22]. In this study, Al₃(Sc_xZr_1−x) dispersoids information is introduced to establish an accurate flow stress prediction model using artificial neural networks.

2. Experimental Materials and Methods

2.1. Materials

The materials employed in this study were produced from a base 5182 alloy composition (alloy A). Alloy B contained 0.05% Sc and 0.1% Zr and alloy C contained 0.1% Sc and 0.1% Zr. The chemical compositions of the three alloys are shown in

Table 1. Chemical composition of as-cast samples (wt.%).

Materials	Mg	Mn	Ti	Fe	Si	Zr	Sc
A alloy	4.53	0.22	0.005	0.175	0.068	-	-
B alloy	4.50	0.20	0.010	0.190	0.070	0.1	0.05
C Alloy	4.44	0.22	0.005	0.168	0.067	0.1	0.1

.

2.2. Hot Compression Experiment and Microstructure Observation

The ingots were homogenized in two stages (heated at 275 and 440 °C for 20 h and 12 h, respectively) before being cut into Φ10 mm × 15 mm cylinders. Gleeble-3500 was subjected to isothermal compression tests. All cylinders were heated up at 10 °C/s and held for 3 min before compression. The hot compression temperatures were 300 °C~ 500 °C (Temperature interval of 50 °C) with strain rates of 0.01, 0.1, 1, and 10 s⁻¹. Figure 1 shows parts of the samples before and after compression, including the C alloy in the hot compressed state of 300 °C—0.01, 1, and 10 s⁻¹. No macroscopic cracks were found in the appearance of all the samples after compression.

Electron backscatter diffraction (EBSD) data were collected by a FEI Tescan at 20 kV with a step of 0.4 µm. Transmission electron microscopy (TEM) samples with a diameter of Φ3 mm were prepared by twin-jet polishing and then examined by a FEI Talos F200S electron microscope operating at 200 kV. For dark field observation, [110] crystal orientation of the Al matrix was selected.

3. Results and Discussion

3.1. True Stress−Strain Curves

Figure 2, Figure 3 and Figure 4 show results for the hot compression tests, expressed as true stress−strain curves for each of the three alloys. It can be seen from the curves that the flow stress falls as the temperature rises and increases when the strain rate rises. During the hot compression deformation process, the material undergoes two opposing changes: hardening and softening. The softening effect is produced by recovery and/or recrystallization and the hardening effect is caused by dislocation and subgrain hardening. Theoretically, a faster deformation rate at the same temperature would result in a greater work-hardening effect on the material. However, at a strain rate of 10 s⁻¹, the flow stress of the samples reached a peak and began to decrease. When compressed at 300 and 350 °C, the curves for the 10 s⁻¹ strain rate continued to decrease before unloading and eventually fell below the flow stress observed at a 1 s⁻¹ strain rate.

The main reason for the above softening is the adiabatic temperature rising during the deformation process. At $\dot{\epsilon }$ ≥ 1 s⁻¹, the heat generated by deformation cannot be transferred quickly enough, causing the sample temperature to rise and promoting softening of the samples [30,31]. To investigate the effect of adiabatic heat on flow stress, the following equation can be used.

$${\sigma }_{c\left(T_{,}\epsilon ,\dot{\epsilon }\right)}={\sigma }_{o\left(T,\epsilon ,\dot{\epsilon }\right)}-{\frac{\partial {\sigma }_o\left(T\right)}{\partial T}|}_{\epsilon ,\dot{\epsilon }}\mathsf{\Delta }T$$

(1)

σ_o is the original flow stress, σ_c is the corrected flow stress, T, ε, and $\dot{\epsilon }$ represent temperature, strain, and strain rate, respectively. ∆T is the actual elevated temperature. The corrected flow stress obtained by Equation (1) is shown in Figure 5. From the corrected data, it is found that significant hardening of the materials occurred at 10 s⁻¹ (strain rate).

3.2. Processing Maps and Microstructural Evolution

Thermal processing maps can be important tools for understanding the deformation mechanisms during hot working, such as dynamic recovery, adiabatic shear, localized flow, stored energy, and subsequent recrystallization behavior [21]. These maps can be used effectively to avoid dangerous processing zones prone to fracture and defect generation, and to identify temperature/strain conditions for controlling the microstructure [16]. In this study, the machining map applicable to the new type 5182 alloy was determined based on thermal simulation compression experimental results and dynamic material model (DMM) theory, combined with the Murty–Rao (M-R) destabilization criterion [17]. The thermal processing map contains a contour map of the material power dissipation factor and a destabilization factor map. The approach for calculating both factors is as follows.

3.2.1. Establishment of Dynamic Material Model (DMM)

The DMM model combines the external power and the energy consumed by the material deformation [32,33,34]. The external power is the input energy (P). The model uses input energy (P) to represent the external power. The energy consumed by the deformation is divided into two parts, one is the dissipation energy G (mainly the heat and crystal defect energy generated during deformation) and another one is the dissipation coefficient J (the energy consumed by microstructure evolution). The model is shown in the following Equation (2) [34]:

$$P=\mathsf{\sigma }\dot{\epsilon =G+J={{\displaystyle \int }}_0^{\dot{\epsilon }}\sigma d\dot{\epsilon }+}{{\displaystyle \int }}_0^{\sigma }\dot{\epsilon }d\sigma$$

(2)

The distribution of the input energy P between the dissipation energy G and the dissipation coefficient J is represented by the following strain rate sensitivity index m [35]:

$$\mathrm{m}={\left(\frac{\partial J}{\partial G}\right)}_{T,\epsilon }={\left(\frac{\partial \ln \sigma }{\partial \ln \dot{\epsilon }}\right)}_{T,\epsilon }$$

(3)

Assuming m = 1 and that the material is in an ideal linear dissipative state, the dissipative coefficients used for microstructure evolution can be expressed in terms of the power dissipation factor η.

$$\mathsf{\eta }=\frac{J}{J_{max}}=\frac{J}{P/2}=2\left(1-\frac{G}{P}\right)=2\left[1-\frac{1}{\sigma \dot{\epsilon }}{{\displaystyle \int }}_0^{\dot{\epsilon }}\sigma d\dot{\epsilon }\right]$$

(4)

The power dissipation factor η is of thermodynamic significance and represents the relative entropy production related to the microstructure evolution of materials [36,37]. By calculating the dissipation factors corresponding to different deformation parameters, the power dissipation factor contour map can be drawn.

3.2.2. The Processing Map Established Based on M-R Instability Criterion

According to Murty’ study [32,33,34,38], the strain rate sensitivity index m varies with the strain rate $\dot{\epsilon }$ as the material undergoes plastic deformation, and the Murty–Rao destabilization criterion applicable to any ε-σ relationship is derived, written as:

$$\xi \left(\dot{\epsilon }\right)=2m-\eta <0$$

(5)

ξ is the instability factor of the material under certain deformation conditions. The final thermal processing maps of the three materials are obtained according to the equations, as shown in Figure 6.

In Figure 6, besides the processing maps of the alloys showing the safe working zone, there is a corresponding microstructure. The figures show that a large number of adiabatic shear bands unfavorable to material deformation appear in the hot-working instability zone, that is, the high strain rate deformation (10 s⁻¹). Recrystallized grains appear in the alloy microstructure with a low strain rate (≤1 s⁻¹) and there is no flow instability. The processing condition reflected by the processing map is consistent with the actual microstructure.

Table 2. Recrystallization degree of 3 alloys after thermal deformation (%).

Materials	Recrystallization Degree of 300 °C, 10 s−1	Recrystallization Degree of 500 °C, 0.01 s−1
A alloy	17.3	35.9
B alloy	15.6	16.9
C alloy	14.1	16.2

shows the recrystallization degree of some samples after thermal deformation.

The aim of studying thermal processing of materials is not only to obtain materials of a certain size and shape, but more importantly to study the evolution of the microstructure and properties during deformation to regulate and improve the properties of the material. The instability during hot compression is a process that evolves with time. Figure 7 compares the effect of Sc and Zr additions on the destabilization zone during evolution. As can be seen in Figure 7, the instability of all three materials occurs between 450 and 500 °C at a strain of 0.3, and the area of the instability zone is larger in the C alloy than in other two alloys. As the strain increases, the area of the destabilization zone increases, and when the strain reaches 1.0 (close to the final strain), the C alloy has the largest destabilization region among the three alloys. Stable and continuous thermal deformation of the material necessitates an equilibrium condition of work hardening and reversion recrystallization softening throughout the deformation process. The presence of Al₃(Sc_xZr_1−x) dispersoids prevents alloys B and C from dynamically recrystallizing, which violates the dynamic equilibrium condition of deformation and causes the alloy to destabilize more readily during the hot working process. When combined with the thermal processing map (shown in Figure 6), it is clear that the destabilizing phenomena occur in all three alloys at strain rates larger than 1 s⁻¹ in the temperature range of 300 °C~500 °C, and deformation is safe at strain rates less than or equal to 1 s⁻¹. After Mg₂Si dispersoids strengthening, the Al−3%Mg₂Si alloy instability zone is large in range including 300 °C~340 °C, 0.01 s⁻¹~5 s⁻¹ and 360 °C~460 °C, 0.01 s⁻¹~0.1 s⁻¹ in the study of Hu et al., and the instability zone to be avoided during hot working is more problematic [39]. In addition to the Al₃Zr dispersoids, there is also Al₂Cu₂Mn₃ phase that exists in the Al−Cu−Mg−Mn−Zr alloy, and the strengthening phase causes the alloy to distort at low strain rates of 0.001 s⁻¹~0.01 s⁻¹ with the possibility of plastic instability, affecting subsequent machining performance [40]. Similarly, Yang’s research discovered that after 0.12% Zr microalloying, the Al−Cu−Li alloy is destabilized at low strain rates [41]. In contrast to the previous work, the instability zone during thermal processing of the 5182 alloy microalloyed by Sc and Zr remained in the region of strain rate larger than 1 s⁻¹, with a wider window of conditions within which it could be thermally processed safely. The findings show that the Al₃(Sc_xZr_1−x)-reinforced 5182 alloy retains good hot workability and that processing at a strain rate of 1 s⁻¹ does not produce flaws that could cause problems during subsequent processing.

3.3. Establishment of Constitutive Models and Artificial Neural Network

The constitutive equation is used to quantitatively describe the dynamic response of flow stress to deformation parameters (temperature, strain rate, strain, etc.) or microstructure parameters (elements in solution, particles, grain size, texture, etc.) during hot deformation.

3.3.1. Determination of Material Constants in Arrhenius Constitutive Equation

A typical Arrhenius constitutive equation can be expressed in the following three equations [42,43]:

$$\dot{\epsilon }=A{\sigma }^{n_1}exp\left[-Q/\left(RT\right)\right] \alpha \sigma <0.8$$

(6)

$$\dot{\epsilon }=Aexp\left(\beta \sigma \right)exp\left[-Q/\left(RT\right)\right] \alpha \sigma >1.2$$

(7)

$$\dot{\epsilon }=A{\left[\sinh \left(\alpha \sigma \right)\right]}^{n}exp\left[-Q/\left(RT\right)\right] All condition$$

(8)

where $\dot{\epsilon }$ is the strain rate (s⁻¹); σ is the flow stress (MPa); Q represents the deformation activation energy (J/mol); R is the Planck constant, 8.314 J/(mol·K); T is the absolute temperature, K; A, α, β, n₁, and n are material constants, where α = β/n₁.

The thermoplastic deformation of the material is a thermally activated process, and the effect of the deformation constant temperature T and strain rate $\dot{\epsilon }$ on the flow stress σ can be described by the Zener−Hollomon parameter (Z parameter) [44]:

$$Z=\dot{\epsilon }exp\left[Q⁄\left(RT\right)\right]$$

(9)

An alternative expression for flow stress is obtained by combining Equation (9) (Z−parameter equation) and Equation (8) (hyperbolic sine function) [42]:

$$\sigma =\left(1/\alpha \right)\ln \left\{{\left(Z/A\right)}^{1/n}+{\left[{\left(Z/A\right)}^{2/n}+1\right]}^n\right\}$$

(10)

Before establishing the constitutive equations, it is first assumed that the value of activation energy Q is independent of the deformation temperature or is constant over a certain temperature range. Taking the logarithm of Equations (6)–(8) yields Equations (11)–(13) [45].

$$n_1={\left[\frac{\partial \ln \dot{\epsilon }}{\partial \ln \sigma }\right]}_T$$

(11)

$$\mathsf{\beta }={\left[\frac{\partial \ln \dot{\epsilon }}{\partial \sigma }\right]}_T$$

(12)

$$n={\left\{\frac{\partial \ln \dot{\epsilon }}{\partial \ln \left[\sinh \left(\alpha \sigma \right)\right]}\right\}}_T$$

(13)

The lnσ−ln$\dot{\epsilon }$, σ−ln$\dot{\epsilon }$ and ln[sinh(ασ)]−ln$\dot{\epsilon }$ linear plots were created using a least square method to obtain n₁, β, and n. In addition, α was obtained by α = β/n₁.

Transforming Equation (8) to obtain Equation (14):

$$\ln \left[\sinh \left(\alpha \sigma \right)\right]=\frac{Q}{nR}\times \frac{1}{T}+\frac{\ln \dot{\epsilon }-\ln A}{n}$$

(14)

From Equation (14), it can be seen that the relationship of 1/T−ln[sinh(ασ)] is linear at the same strain rate. The value of Q can be obtained by Equation (15):

$$\frac{Q}{nR}={\left\{\frac{\partial \ln \left[\sinh \left(\alpha \sigma \right)\right]}{\partial 1/\mathrm{T}}\right\}}_{\dot{\epsilon }}$$

(15)

Substituting the data into the above equation obtains the average activation energy, and the Z value under different deformation situations was determined by Equation (9). Substituting Equation (8) into Equation (9) and taking logarithms for both sides of the equation:

$$\ln Z=\ln A+n\ln \left[\sinh \left(\alpha \sigma \right)\right]$$

(16)

A linear fit of ln[sinh(ασ)]−lnZ by least squares is obtained with a linear slope of n and an intercept of lnA. The data corresponding to a strain of 1.0 for the three alloys were substituted into the above equation to obtain the corresponding constitutive equation at a strain of 1.0, as shown in

Table 3. Coefficients in constitutive equations.

Materials	α	n	Q (KJ/mol)	lnA
A alloy	0.009946	6.38	195.37	32.89
B alloy	0.009745	6.61	193.70	32.51
C alloy	0.009739	5.95	196.04	33.09

.

The experimental parameters were substituted into Equation (10) to obtain the predicted flow stresses, and the comparison of the predicted and experimental values is shown in Figure 8. To check the accuracy of the Arrhenius−type constitutive equation established above, the squared correlation coefficient (R²) and the average absolute relative error (AARE) were used to statistically test the calculated and experimental values for a strain rate of 1.0 conditions. R² is a quantity to assess the degree of linear correlation between the experimental and calculated values, and AARE is an unbiased statistical parameter to determine the accuracy of the constitutive equation [22,46,47]. The accuracy of the constitutive equation is excellent when the correlation coefficient is close to 1 and AARE is close to 0. R and AARE are calculated by Equations (17) and (18), respectively [47,48].

$$\mathrm{R}=\frac{{\sum }_{i=1}^N\left(X_i-X_a\right)\left(Y_i-Y_a\right)}{\sqrt{{\sum }_{i=1}^N{\left(X_i-X_a\right)}^2}\sqrt{{\sum }_{i=1}^N{\left(Y_i-Y_a\right)}^2}}$$

(17)

$$\mathrm{AARE}=\frac{1}{N}{{\displaystyle \sum }}_{i=1}^N\left|\frac{Y_i-X_i}{X_i}\right|\ast 100\%$$

(18)

Xi and Yi are the experimental and calculated values of the flow stresses. Xa and Ya are the average experimental flow stress and calculated flow stress, respectively. N is the number of data in the studied dataset.

3.3.2. Strain Compensated Constitutive Equations

The presently established constitutive equations only consider the effects of temperature and strain rate without introducing the effect of total strain, for which the strain-compensated constitutive equations should be established [45,49]. The strain−compensated constitutive equation is a ninth-order polynomial function of strain and contains four material parameters (α, n, Q, lnA) as follows [22,45,50,51]:

$$\alpha =B_0+B_1\epsilon +B_2{\epsilon }^2+B_3{\epsilon }^3+B_4{\epsilon }^4+B_5{\epsilon }^5+B_6{\epsilon }^6+B_7{\epsilon }^7+B_8{\epsilon }^8+B_9{\epsilon }^9$$

(19)

$$n=C_0+C_1\epsilon +C_2{\epsilon }^2+C_3{\epsilon }^3+C_4{\epsilon }^4+C_5{\epsilon }^5+C_6{\epsilon }^6+C_7{\epsilon }^7+C_8{\epsilon }^8+C_9{\epsilon }^9$$

(20)

$$Q=D_0+D_1\epsilon +D_2{\epsilon }^2+D_3{\epsilon }^3+D_4{\epsilon }^4+D_5{\epsilon }^5+D_6{\epsilon }^6+D_7{\epsilon }^7+D_8{\epsilon }^8+D_9{\epsilon }^9$$

(21)

$$\ln A=E_0+E_1\epsilon +E_2{\epsilon }^2+E_3{\epsilon }^3+E_4{\epsilon }^4+E_5{\epsilon }^5+E_6{\epsilon }^6+E_7{\epsilon }^7+E_8{\epsilon }^8+E_9{\epsilon }^9$$

(22)

The flow stresses corresponding to 0.1 to 1.0 strains on the hot compression strain stress curve are extracted at 0.1 intervals, and the material parameters corresponding to different strains are obtained and then polynomially fitted to obtain the coefficients in Equation (19) to Equation (22).

Table 4. The fixed coefficients of three alloys with strain rate of 1.0.

Materials	α	n	Q (KJ/mol)	lnA
A alloy	0.00996	6.38	198.81	33.02
B alloy	0.00985	6.64	196.57	32.97
C alloy	0.00983	5.78	194.32	36.08

shows the fixed coefficients of three alloys with a strain rate of 1.0.

Substitution of Equation (19) to Equation (22) into Equation (10) yields the constitutive equations considering the strain effect [52]:

$$\sigma =\frac{1}{\alpha }\ln {\left\{{\left(\frac{\dot{\epsilon }exp\left(Q/RT\right)}{A}\right)}^{\frac{1}{n}}+{\left[{\left(\frac{\dot{\epsilon }exp\left(Q/RT\right)}{A}\right)}^{\frac{2}{n}}+1\right]}^{\frac{1}{2}}\right\}}_{\epsilon }$$

(23)

Figure 9 shows the comparison between the predicted and experimental values of the constitutive equations. The correlation coefficient between experimental values and predicted values commonly used to judge model accuracy is shown in Figure 9. R² is improved slightly compared with values before modification.

3.3.3. Establishment and Prediction of Artificial Neural Network

Flow stress is affected by many nonlinear factors during thermal processing. The constitutive modeling approach, which has been in use for many years, is mature, but it has a number of drawbacks, including a high computing effort, poor forecast accuracy, and a lack of coverage of nonlinear components. In order to improve upon the traditional constitutive approach, this research incorporated an artificial neural network (ANN) model to process the thermal processing data. The ANN builds the prediction model by training data and previous studies have proved that feedforward neural network models can predict the flow stress of metal well [53,54,55,56]. Normally, compression temperature, strain rate, and true strain are set as input data and flow stress is set as output data. The neural network is implemented using MATLAB software.

The data were preprocessed before building the neural network model and substituted into the data set to train the neural network. The model was revised to incorporate the influence of Al₃(Sc_xZr_1−x) dispersoids, including size and volume fraction as input data to better compare with the results predicted by the present constitutive equations.

Figure 10 shows a series of dark field images of the Al₃(Sc_xZr_1−x) corresponding to 300, 350, 400, 450, and 500 °C thermal compression temperatures and 0.01 s⁻¹ strain rate for the B and C alloys. The average diameter and volume fraction (f_v) of the Al₃(Sc_xZr_1−x) dispersoids observed at different hot deformation temperatures were obtained by counting the dispersoids in multiple dark field images and are included in

Table 5. Characteristics of Al₃(Sc_xZr_1−x) dispersoids.

Characteristics of Al3(ScxZr1−x)	5182-0.05Sc-0.1Zr Alloy					5182-0.1Sc-0.1Zr Alloy
	300 °C	350 °C	400 °C	450 °C	500 °C	300 °C	350 °C	400 °C	450 °C	500 °C
Diameter (nm)	9.73	10.08	11.22	11.93	13.40	7.65	7.96	9.36	12.75	15.13
fv (10−3)	1.0	1.6	1.0	2.5	2.7	1.0	1.2	1.2	4.4	5.2

. Previous research has discovered that Al₃(Sc_xZr_1−x) has a considerable impact on the mechanical properties of aluminum alloys, as measured by Orowan strengthening [12,57,58,59]. For hot compression experiments, there are many nonlinear factors, so the method of training the neural network using information for Al₃(Sc_xZr_1−x) as the input data is simple and can make the model more accurate. The size and volume fraction of the dispersoids, as shown in Figure 10, were measured and reported in Table 5.

The artificial neural network was performed in MATLAB. Artificial neural networks are divided into three layers: an input layer, a hidden layer, and an output layer. In the heat deformation model, the input layer contains thermal compression temperature (T, °C), strain rate ($\dot{\epsilon }$, s⁻¹), strain (ε, %), dispersoid size (D, mm), and dispersoids volume fraction (f_v, %) and the output data are the corresponding flow stresses (σ, MPa). The middle hidden layer modifies the network weights and thresholds with the goal of minimizing the mean square error, and higher accuracy for the neural network model is obtained by training with existing data. In order to improve the training speed and accuracy of the ANN, the domain scope of input data set and output data set need to be unified. The experimental temperature (T, °C) changes to 1/T(1/K), $\dot{\epsilon }$(s⁻¹) can be rewritten as ln($\dot{\epsilon }$), and the flow stress σ changes to σ/σ_max.

The widely used feedforward backpropagation neural network model was used in this study. During the model training process, the backpropagation error modifies the corresponding weights and thresholds of the hidden layer neurons [55]. Figure 11 shows a schematic diagram for the ANN. Figure 12 shows the comparison of experimental and ANN predicted values.

3.3.4. Comparison between Constitutive Equation and Neural Network

The flow behavior of 5182 alloy and Sc-Zr microalloyed 5182 alloy was investigated by three models in the temperature range of 300–500 °C and strain rate range of 0.01–10 s⁻¹. Corresponding predicted flow stress was obtained for all three models. Figure 13 shows the stress−strain curves for alloy C after thermal compression at 350 and 450 °C, as well as the predicted values. The predicted values of the constitutive equation and strain-compensated constitutive equation, which exhibit considerable inaccuracies in comparison to the flow stress curves, are shown in Figure 13a−d. The comparison of the ANN predictions with the flow stress curves is shown in Figure 13e,f, and it can be seen that the predictions coincide with the curves. Performances of constitutive equations, strain−compensated constitutive equations, and ANN in temperature range of 300–500 °C, strain rate range of 0.01–10 s⁻¹ are demonstrated in

Table 6. Performances of three models.

Materials	Performances	Constitutive Model	Strain-Compensated Model	ANN
A alloy	R2	0.93	0.94	0.99
	AARE%	8.76	8.65	0.28
B alloy	R2	0.93	0.94	0.99
	AARE%	8.61	8.34	0.24
C alloy	R2	0.93	0.98	0.99
	AARE%	9.39	9.15	0.58

using the mentioned squared correlation coefficient (R²) and average absolute relative error (AARE). According to R² and AARE, it can be seen that both constitutive equations were able to predict the flow stresses accurately. The neural networks have the highest accurate predicted values of these models, with R² of 0.99 and AARE all less than 1% for the three metal ANN prediction values. In this study, the ANN model was constructed using the deformation temperature, strain rate, and strain as input data, and information (the size and volume fraction) of the Al₃(Sc_xZr_1−x) dispersoids were also introduced, resulting in a more accurate model. From this study, it can be seen that the ANN model can find patterns from more relevant parameters to obtain more accurate predictions by leaving aside the empirical mathematical equations compared to the constitutive model.

3.3.5. Cross−Validation of ANN

The ANN model, as can be seen in the model accuracy comparison above, has exceptionally high accuracy. High model accuracy in ANN prediction models only relates to the single input dataset and does not mean that the model is capable of generalization [28,60]. As a result, the cross−validation approach was utilized to test the ANN model’s generalization capabilities. The data set in this study was divided into 6 equal−sized mutually exclusive subsets, of which 5 subsets were used to train the network and 1 subset was used to test the trained network at each time, and the mean of the 6 test results can be determined after 6 training sessions.

Figure 14 shows the results of the cross-validation ANN generalization capability. Figure 14a compares the training set data to the predicted data from six times of validation. The projected data versus the test set is depicted in Figure 14b. In both the training and test sets, good linearity between predicted and experimental data is maintained. The R² and AARE between the predicted data and the experimental data obtained by the ANN model are 0.98 and 3.25%, indicating that the ANN established in this study has a high generalization capability and is suitable for predicting the heat deformation flow stress of the 5182-Sc-Zr alloy.

4. Conclusions

In this paper, thermal deformation behaviors of 5182 alloy and the new type 5182 alloy with the addition of Sc−Zr were investigated at deformation temperatures of 300 °C~500 °C and strain rates of 0.01 s⁻¹~10 s⁻¹. The constitutive equations and ANN models were established based on the hot compression stress–strain curves, and the effects of trace additions of Sc and Zr on the microstructure were also investigated. Based on the above studies, the following conclusions can be drawn.

• The addition of Sc and Zr affects recrystallization degree of 5182 alloys after hot deformation significantly. The recrystallization degree of alloy 5182 was twice as high as that of 5182−0.05Sc−0.1Zr alloy and 5182−0.1Sc−0.1Zr alloy at the hot compression condition of 500 °C−0.01 s⁻¹.
• Al₃(Sc_xZr_1−x)−reinforced 5182 alloys can be safely hot worked at a strain rate of 1 s⁻¹ without unfavorable defects, indicating that the 5182 alloy has maintained good hot workability after strengthening.
• Comparing the R² and AARE of the constitutive equation, strain compensated constitutive equation, and ANN model, it can be found that the ANN model with the introduction of Al₃(Sc_xZr_1−x) dispersoids information leaving aside the empirical mathematical equations, has more accurate prediction of the flow stress.