Microstructure Evolution and Constitutive Modelling of Deformation Behavior for Al-Mg-Si-Cu-Sc-Zr Alloy Processed with Isothermal Multidirectional Forging

Abstract

This research is devoted to the microstructure evolution and deformation behavior of the Al-1.2Mg-0.7Si-1.0Cu-0.1Sc-0.2Zr alloy during the isothermal multidirectional forging (MDF) in a large cumulative strain and temperature range. The structure investigation of the studied alloy revealed several phases precipitated during solidification, among which θ(Al₂Cu), Q(Al₅Cu₂Mg₈Si₆), Mg₂Si, Sc-bearing W(AlScCu) and V(AlSi₂Sc₂) phases were observed. The MDF at 150–350 °C and a maximum cumulative strain of 14.4 significantly refined grain structure providing a mean grain size of 1.2–2.1 µm. The L1₂ structured Al₃(Sc,Zr) dispersoids with a mean size of 10 ± 1 nm were formed during two-step homogenization annealing. Due to Zener pinning of the nanoscale dispersoids and fine-grained structure, the alloy exhibited near-superplastic behavior in a temperature range of 460–500 °C and strain rate range of 2 × 10⁻³–1 × 10⁻² s⁻¹ with the maximum elongation to failure of ~300%. After a strengthening heat treatment, the forged alloy exhibited the yield strength of 326 ± 5 MPa, ultimate tensile strength of 366 ± 5 MPa, and elongation of 10 ± 3%. The hot deformation behavior was described using the Arrhenius type model. The developed model demonstrated high predictability accuracy with a maximum average absolute relative error of 6.6%.

1. Introduction

The fine-grained Al-based alloys attract regular interest due to their combination of properties, among which are increased plasticity, formability, and strength [1]. Grain refinement in Al-based alloys is currently the main focus of multiple research studies that reflect relevant industrial needs [2]. The popular Al-based heat-treatable 6xxx-type Al-Mg-Si-based alloys exhibit good corrosion resistance and outstanding strength-to-weight ratio [3,4]. Moreover, the Al-Mg-Si-based alloys demonstrate good formability at low temperatures [5]. However, at elevated temperatures, owing to the low alloyed solid solution, the 6xxx-type alloys are prone to a grain growth that complicates the formation of a fine-grained structure [6]. Thus, the intense dynamic grain growth of these alloys leads to poor superplasticity [7,8]. The grain growth leads to structure inhomogeneity that provides inhomogeneous flow and results in a decrease in uniform elongation.

A fine-grained structure is commonly achieved through a severe plastic deformation (SPD) method that has been proven to be universal and effective for various metals and alloys [9]. The SPD is realized by various techniques that are united by the ultra-high plastic strain accumulated during material processing. Popular SPD techniques, such as the equal channel angular pressing (ECAP) [10], friction stir processing (FSP) [11] and high pression torsion (HPT) [12], have demonstrated high efficiency in grain refinement and mechanical properties improvement. However, the high energy consumption of the previously mentioned SPD approaches limits the sample size and industrial application. A promising SPD technique that allows for processing bulk samples while maintaining their geometry is isothermal multidirectional forging (MDF) [13,14,15,16,17]. MDF has been effectively used for achieving outstanding grain-refinement in the Al-based alloys [18,19], by which a grain size of 0.5–2 µm was achieved in 5xxx type alloys [20,21]. A similar grain size was observed in 6xxx-type alloys after multidirectional forging at cryogen temperatures [22,23].

Although the MDF treatment effectively refines the grains, the thermal stability of the refined grain structure requires special approach. To suppress grain growth, Al alloys are usually doped with small additions of transition metals (TM) and rare-earth elements (RE) [22,24,25], including Sc and Zr which have proven their significant effectiveness [26,27]. These metals exhibit low equilibrium solubility in aluminum. During rapid solidification, the solubility of Sc and Zr can be increased to allow the formation of a supersaturated solid solution. Subsequent heat treatment provides the decomposition of supersaturated solid solution with a precipitation of fine nanoscale dispersoids of Al₃Sc or Al₃Zr phases with an L1₂ crystal structure type [28,29,30]. The co-adding of Sc and Zr results in a formation of a complex Al₃(Sc,Zr) phase with core-shell structure [31,32] that is distinguished by an increased potential resistance to coarsening at elevated temperatures [33]. The nanoscale dispersoids provide for the Zener pinning effect that suppress static and dynamic grain growth [34]. The positive effect of Sc and Zr co-addition on grain stability at elevated temperatures has been multiply reviewed for thermomechanical-treated Al alloys [35,36]. As a result, the improved superplastic behavior due to the Zener pinning effect has been previously demonstrated in 2xxx- [37], 5xxx- [38,39,40,41], 6xxx- [42,43], and 7xxx-type [44,45] Al alloys. The effectiveness of microalloying directly depends on the particle parameters. Following the Smith–Zener Equation (1), the pinning pressure increases with increasing particle volume fraction and decreasing particle radius [46].

$$P_{SZ}=\frac{3f\gamma }{2r},$$

(1)

where f is the volume fraction, r is the radius of particles, and γ is the grain boundary energy.

The parameters of heat and thermomechanical treatment play a principal role in dispersoid precipitation. Different researchers have observed that the high number density and nanoscale size of dispersoid in Sc- and Zr-bearing Al alloys can be achieved through a low temperature annealing [30,47] or two-step annealing with a low temperature first stage at 300–400 °C [48,49]. The first annealing step intensifies the nucleation of the dispersoids, whereas a high temperature second step accelerates the growth of them. Thus, multiple studies confirmed the effectiveness of the SPD method and microalloying for grain refinement in Al-based alloys. However, the effect of the MDF technique on the grain refinement and structural stability of low alloyed Al-Mg-Si-based alloys is still poorly studied.

In this context, the current study had two aims: (1) to achieve a refined grain structure of an Al-Mg-Si-Cu-based alloy through two simultaneous approaches, an MDF and an effective SPD approach and microalloying with Sc and Zr; and (2) to improve the hot deformation behavior of the studied alloy by suppressing the dynamic grain growth through activation of the Zener pinning effect. In this work, the structural evolution and near superplastic behavior of the Al-Mg-Si-Cu-Sc-Zr alloy processed by isothermal MDF in the wide temperature and cumulative strain ranges were investigated. In addition, the hot deformation behavior of the Al-Mg-Si-Cu-Sc-Zr alloy was constitutively modelled.

2. Materials and Methods

The alloy with the following composition: Al-1.2Mg-0.7Si-1.0Cu-0.1Sc-0.2Zr wt.%, was prepared using a graphite–fireclay crucible in an inductive furnace (Interselt, Saint Petersburg, Russia). When the temperature reached 780 °C, the casting was carried out in a copper water-cooling mold with a cooling rate of ~15 K/s. The temperature of the melt was measured using a chromel-alumel thermocouple. To prepare the melt, the following components were co-added: 99.99Al, 99.95Mg, Al-12Si, Al-53.5Cu, and Al-5Zr, Al-2Sc. The master alloys were placed in the molten Al in decreasing order of their solidus temperatures.

The obtained ingots were annealed in two steps at 350 °C for 8 h and subsequently at 480 °C for 3 h. The annealing was carried out using a Nabertherm N30/65A air-forced convection furnace (Nabertherm GmbH, Lilienthal, Germany).

The multidirectional forging (MDF) was carried out using a hydraulic press machine (Nordberg N3650E, Changshu Tongrun Auto Acessory Co., Ltd., Changshu, China) providing a force of 50 tons. The scheme of the MDF process is demonstrated in Figure 1. The forging process included several circles. Each circle involved three passes in mutually perpendicular directions. During the MDF, the deformation occurred in two directions. The sample shape remained unchanged at the end of each pass. After each pass, the samples were rotated through 90° to maintain multiaxial deformation. One pass provided a strain of 0.8 and the cumulative strain for a full circle was 2.4. The initial crosshead speed for each pass was 5 mm/min. The samples exhibited parallelepipedal shape with a size l in one dimension and a similar size a in two other dimensions (Figure 1). The true stress-true strain curves and structure evolution during forging were studied using small samples with l = 16 mm and a = 8 mm. The superplastic behavior was analyzed using bulk samples with l = 56 mm and a = 28 mm. The MDF procedure is described in detail in ref. [50]. The maximum number of MDF circles for the studied alloy was 6 (18 passes) that provided for a maximum cumulative strain of 14.4.

The microstructure evolution was analyzed using a Tescan-VEGA3 LMH scanning electron microscope (Tescan Brno s.r.o., Kohoutovice, Czech Republic). The elemental distribution maps were obtained using an energy dispersive spectroscopy (EDS) detector (Oxford Instruments plc, Abingdon, UK). An electron backscatter diffraction (EBSD) maps were obtained using an HKL NordlysMax detector (Oxford Instruments plc, Abingdon, UK). The accelerating voltage and operating distance were 20 kV and 10–15 mm, respectively. The samples’ SEM were preliminarily prepared by grinding and polishing in a Struers LaboPol machine (Struers APS, Ballerup, Denmark).

The analysis of nanoscale dispersoids was performed using a JEOL JEM transmission electron microscope (JEOL, Tokyo, Japan). Disc-shaped specimens with a diameter of 3 mm and a thickness of 0.18–0.22 were examined. The specimens were preliminarily electrochemically polished at 20 ± 1 °C and 19 ± 2 V in a 30% HNO₃ in methanol electrolyte using a Struers TenuPol-5 twinjet polishing machine (Struers APS, Ballerup, Denmark).

The EBSD data were obtained with a scanning step of 0.15 µm in the area of 250 × 250 μm, and the maps were analyzed using Aztec (EBSD) software (v.2.0, Oxford Instruments plc, Abingdon, UK). The high-angle grain boundaries (HAGBs) were identified as boundaries between the grains with a misorientation angle higher than 15°. The low-angle grain boundaries (LAGBs) were identified as boundaries between the grains with a misorientation angle of 2–15°, respectively.

The deformation behavior of the alloy was analyzed during a constant strain rate tensile test at 440–520 °C and 2 × 10⁻³–1 × 10⁻² s⁻¹ strain rate. The tensile test was carried out using a Walter-Bai LFM 100 universal testing machine (Walter + Bai AG, Löhningen, Switzerland). The samples for the tensile test had a gauge part size of 4 × 6 × 1 mm. The samples were cut from the as-forged specimens using a Charmilles Robofil 190 wire-electro discharge machine (Agie Charmilles, Geneva, Switzerland).

The room temperature mechanical properties were determined using a tensile test in the Zwick Z250 universal testing machine (Zwick Roell Group, Ulm, Germany). The gauge part of samples was 18 × 6 × 1 mm in size. Three samples were used per state.

3. Results

3.1. Microstructure in As-Cast and Homogenized States

Figure 2a demonstrates the BSE images of the studied alloy in the as-cast state. Several phases distinguished by contrast were revealed in the images. The aluminum solid solution (Al) was the predominant microstructural feature. The phases with dark and bright contrast were found on the periphery of dendrite cells. The bright areas predominantly corresponded to the Cu-rich phases (CuAl₂ and Q(Al₅Cu₂Mg₈Si₆)) while the dark areas correspond to the Mg₂Si phase. The volume fraction of Cu-rich phases and Mg₂Si phase were 1.8 ± 0.2% and 2.1 ± 0.1%, respectively. The two-step homogenization annealing provided for near complete dissolution of the Cu-rich phases (Figure 2b). The Mg₂Si particles were partially fragmented and spherodized. After homogenization, the volume fraction of the Mg₂Si phase was 2.0 ± 0.1% that was similar to as-cast state. The small fraction of particles enriched with Sc were also defined after annealing (Figure 2b). The size of particles was varied from 0.2 to 2 µm.

The TEM studies of the as-homogenized alloy revealed nanoscale dispersoids (Figure 2a,b). Typical for the L1₂-Al₃(Sc,Zr) phase superlattice reflections were observed in the selected area electron diffraction (SAED) (Figure 2d insert). The ordered position of the superlattice reflections against the reflections of Al confirm at least partial coherency of the dispersoids and the matrix. The mean dispersoid size was 10 ± 1 nm.

3.2. Deformation Behavior during MDF

The MDF was performed at 150, 250, and 350 °C. The true stress vs. true strain dependencies were corrected considering the increase in the strain rate during compression [51] (Figure 3). Relatively stable flow stage occurred in a strain interval of 0.1–0.4. Further deformation to a strain higher than ~0.4 led to intense strengthening, which was attributed to the increased friction between the die and the matrix due to the increase in processed surface during the pass.

At the steady stage of deformation, the stress-strain behavior depended on the strain and temperature. An insignificant softening was observed at 150 °C (Figure 3a). At 250 °C, the softening was observed at a cumulative strain below 7 (Figure 3b). At a higher cumulative strain, the flow was accompanied by dynamic strengthening effect. At 350 °C, dynamic softening occurred only during the first pass and subsequent passes were accompanied with strengthening (Figure 3c). The temperature increase provided for a stress decrease at the stable stage. The flow stress also decreased with cumulative strain. A significant decrease of flow stress was observed during the first 11 passes at 150 °C and 9 passes at 250–350 °C. During a subsequent increase in cumulative strain, the flow stress changed insignificantly.

3.3. The Microstructure of Alloy after MDF

The microstructure was analyzed after three and six circles of multidirectional forging (Figure 4). During the forging process, the Mg₂Si particles were fragmentized. The particle parameters are presented in

Table 1. Parameters of the Mg₂Si particles after MDF.

Particle Parameter	Number of Circles	Temperature of MDF
		150 °C	250 °C	350 °C
Mean particle size, µm	3	1.3 ± 0.1	1.3 ± 0.2	1.5 ± 0.2
	6	1.3 ± 0.1	1.4 ± 0.1	1.5 ± 0.2
Aspect ratio	3	0.62 ± 0.04	0.68 ± 0.03	0.65 ± 0.04
	6	0.63 ± 0.03	0.67 ± 0.03	0.66 ± 0.03
Mean interparticle distance, µm	3	53 ± 11	47 ± 10	53 ± 8
	6	49 ± 12	44 ± 8	43 ± 8
Standard deviation of interparticle distance	3	42.9	36.8	34.9
	6	49.2	35.1	36.7

. The particle mean size and aspect ratio were insignificantly influenced by temperature and cumulative strain. However, the decrease in standard deviation of interparticle distance confirmed the increasing particle distribution homogeneity with increasing MDF temperature. The bright Sc-bearing particles which were observed after homogenization were also found for the sample subjected to MDF. The analysis of the EDS maps revealed the particles (marked as type-A) with bright contrast enriched with Cu, Sc, Si, and a minor amount of Zr (Figure 5a–c). However, the localization of Cu was not found in the particles marked as type-B. Higher magnification BSE images and EDS analysis also revealed fine precipitates enriched with Cu in the samples as-forged at 250 °C and 350 °C. The precipitates size increased with an increase in forging temperature and cumulative strain.

The EBSD IPF mapping of the studied alloy was carried out after six circles of forging (Figure 6). The grain and subgrain size increased with increasing temperature. The subgrain/grain sizes were 0.9 ± 0.1/1.2 ± 0.1 µm, 1.7 ± 0.1/2.0 ± 0.1 µm, and 1.8 ± 0.1/2.1 ± 0.1 µm for 150, 250, and 350 °C, respectively. The grain boundary misorientation histograms revealed that the value of LAGBs decreased with increasing temperature from 150 to 250 °C. The further increase in temperature to 350 °C insignificantly changed the fraction of LAGBs and HAGBs.

3.4. The Hot Deformation Behavior of Alloy after MDF

The hot deformation behavior was studied for the bulk sample subjected to six circles of MDF at 350 °C. The true stress vs. true strain curves for a constant strain rate tensile test in a temperature range of 460–500 °C and a strain rate of 2 × 10⁻³–1 × 10⁻² s⁻¹ are shown in Figure 7. The flow stress increased with the strain and decreased with the deformation temperature. The elongation decreased with increasing deformation temperature (

Table 2. Elongation to failure (%) after tensile tests at various temperatures and strain rates.

Strain Rate (s−1)	Temperature
	460 °C	480 °C	500 °C
1 × 10−2	310 ± 20	130 ± 10	110 ± 20
5 × 10−3	290 ± 10	210 ± 10	120 ± 20
2 × 10−3	310 ± 10	240 ± 20	200 ± 20

). The maximum values of elongations to failure within the range of 290–310% were observed at 460 °C under studied strain rate conditions.

3.5. Room Temperature Mechanical Properties

The mechanical characteristics of the alloy studied at room temperature were examined after six MDF circles at 350 °C. The samples were preliminarily solid solution treated (SST) at 520 °C for 30 min and artificially aged at a temperature of 180 °C for 8 h. The values of yield strength (YS), ultimate tensile strength (UTS) and elongation to failure (El) were 326 ± 5 MPa, 366 ± 5 MPa, and 10 ± 3%, respectively.

3.6. Arrhenius Constitutive Model

The stress–strain curves depicted in Figure 7 were utilized in the development of an Arrhenius constitutive model for the purpose of predicting and modeling the flow behavior of the alloy under investigation. The characterization of the deformation response of this particular alloy, taking into account the effects of strain rate and temperature, can be effectively clarified by the utilization of a Zener–Hollomon parameter (Equation (2)) and an exponent-type formula (Equation (3)) [52,53].

$$Z={\dot{\epsilon }}^{.}{exp}^{\frac{Q}{RT}}$$

(2)

$$\dot{{\epsilon }^{.}}=Af\left(\sigma \right){exp}^{\left(-\frac{Q}{RT}\right)}$$

(3)

In the given Equation (2), A represents a material constant, $\dot{\epsilon }$ denotes the strain rate measured in s⁻¹, T signifies the temperature measured in Kelvin, Q represents the effective activation energy of deformation measured in kJ/mol, R is the gas constant equal to 8.314 J/(molK), and $f\left(\sigma \right)$ is a function related to the flow stress (Equations (4)–(6)):

$$\left(\sigma \right)=\{\begin{array}{}\mathrm{(4)}& {\sigma }^{n_1}, for \alpha \sigma <0.8\mathrm{(5)}& {exp}^{\left(\beta \sigma \right)}, for \alpha \sigma >1.2 \mathrm{(6)}& {\left[\sinh \left(\alpha \sigma \right)\right]}^{n_2}, for various \sigma \end{array}$$

The constants of the material are denoted as β, $n_1$, n₂ $\mathrm{a}\mathrm{n}\mathrm{d} \alpha$, where α is defined as the ratio of β to $n_1$. While Equations (4) and (5) are suitable for low stress conditions (ασ < 0.8) and high stress conditions (ασ > 1.2), respectively, the hyperbolic sine law (Equation (6)) can be regarded as universally applicable and can be employed across a broad range of stress levels. The Equations (4)–(6) can be expressed in the form provided below (Equations (7) and (8)), taking into account the Zener–Holomon parameter:

$$\dot{\epsilon }=\{\begin{array}{}\mathrm{(7)}& A_1{\sigma }^{n_1}{exp}^{\left(-\frac{Q_1}{RT}\right)}-Power law\hfill \mathrm{(8)}& A_2{e}^{\left(\beta \sigma \right)}{exp}^{\left(-\frac{Q_2}{RT}\right)}-Exponential law\hfill \mathrm{(9)}& A_3{\left[\sinh \left(\alpha \sigma \right)\right]}^{n_2}{exp}^{\left(-\frac{Q_3}{RT}\right)}-Hyperbolic sine law\hfill \end{array}$$

3.6.1. Calculation of the Arrhenius Constitutive Model Constants

The material constants were determined by utilizing the true stress and true strain data. The calculating approach was demonstrated using a true strain of 0.4 (50%) as a pilot route. Equations (10) and (11) were derived through the natural logarithm of Equations (7) and (8), respectively.

$$\ln \dot{\epsilon }=\ln A_1+n_1\ln \sigma -\frac{Q_1}{RT}\Rightarrow n_1={\left[\frac{\partial \ln \dot{\epsilon }}{\partial \ln \sigma }\right]}_T$$

(10)

$$\ln \dot{\epsilon }=\ln A_2+\beta \sigma -\frac{Q_2}{RT}\Rightarrow \beta ={\left[\frac{\partial \ln \dot{\epsilon }}{\partial \sigma }\right]}_T$$

(11)

By applying a logarithm to Equation (9), Equation (12) can be derived, which was subsequently utilized for the computation of Q₃ and n₂. Equations (13) and (14) were derived by differentiating Equation (12).

$$\ln \dot{\epsilon }=\ln A_3+n_2\ln \sinh \left(\alpha \sigma \right)-\frac{Q_3}{RT}$$

(12)

$$n_2={\left[\frac{\partial \ln \dot{\epsilon }}{\partial \ln \left[\sinh \left(\alpha \sigma \right)\right]}\right]}_T$$

(13)

$$Q_3=R\times {\left[\frac{\partial \ln \dot{\epsilon }}{\partial \ln \left[\sinh \left(\alpha \sigma \right)\right]}\right]}_T\times {\left[\frac{\partial \ln \left[\sinh \left(\alpha \sigma \right)\right]}{\partial \left(\frac{1}{T}\right)}\right]}_{\dot{\epsilon }}$$

(14)

The determined constants corresponding to a strain level of 0.4 (50%) are compiled in

Table 3. The computed values of A₁, A₂, A₃, n₁, n₂, β, $Q_{\mathrm{1,2},3}$ and α constants at strain level of 0.4 (50%).

ln(A1)	n1	Q1 [KJ/mol]	ln(A2)	β [MPa−1]	Q2 [KJ/mol]	α	ln(A3)	n2	Q3 [KJ/mol]
23	3.5	51 ± 5	17	0.22	50 ± 5	0.062	14	2.67	51 ± 6

.

The pattern between $T,\dot{\epsilon },\mathrm{a}\mathrm{n}\mathrm{d}\sigma$ was expressed as follows (Equation (15)):

$$\sigma =\frac{1}{\alpha }\ln \left\{{\left(\frac{z}{A_3}\right)}^{\frac{1}{n_2}}+{\left[{\left(\frac{z}{A_3}\right)}^{\frac{2}{n_2}}+1\right]}^{\frac{1}{2}}\right\}$$

(15)

where $Z=\dot{\epsilon }{exp}^{\frac{Q_3}{RT}}$(Equation (2)).

3.6.2. Compensation of Strain Effect

The magnitude of strain has a noticeable impact on the material constants. Hence, it is imperative to take into account the influence of strain while figuring out flow stress. The material constants were obtained for flow stress values in strain ranging from 0.05 to 0.6. The constants (α, n₂, $Q_3$ and ln(A₃)) strain dependances are illustrated in Figure 8. The values of the constants (α, n₂, $Q_3$ and ln(A₃)) were obtained through the utilization of polynomial functions of strain, as described in Equation (16). The coefficients of Equation (16) are listed in

Table 4. Coefficients in Equation (20).

Parameter	Y0	B1	B2	B3	Adj. R-Square (Third Order)	Adj. R-Square (Fifth Order)
$\alpha$	0.161	−0.729	1.792	−1.478	0.96	0.99
$n_2$	2.176	−2.210	16.755	−20.177	0.98	0.98
${\mathrm{l}\mathrm{n}(A}_3)$	22.178	−64.456	190.923	−200.369	0.97	0.99
$Q$	103.819	−394.307	1143.484	−1192.050	0.98	0.99

. The dependances of the constants on strain were fitted using third-order polynomials fit based on the Adj. R-Square values (Table 4); there were no significant differences in the Adj. R-Square values when using higher order polynomials, such as fifth-order polynomials (Table 4). The number of constants when using the fifth-order polynomials is more than that of third-order polynomials fitting. Thus, the third-order polynomials exhibited the required accuracy with a lower number of constants than in fifth-order polynomials.

$$\left\{\begin{array}{c}\alpha =Y_0+B_1{\epsilon }^1+B_2{\epsilon }^2+B_3{\epsilon }^3\\ n_2=Y_0+B_1{\epsilon }^1+B_2{\epsilon }^2+B_3{\epsilon }^3\\ A_3=Y_0+B_1{\epsilon }^1+B_2{\epsilon }^2+B_3{\epsilon }^3\\ Q_3=Y_0+B_1{\epsilon }^1+B_2{\epsilon }^2+B_3{\epsilon }^3\end{array}\right.$$

(16)

In order to assess the appropriateness of the acquired constitutive model, the correlation coefficient (R), average absolute relative error (AARE), and root mean square error (RMSE) were calculated using Equations (17)–(19) and accounted for 0.93, 6.6%, and 1.15, respectively. The comparison between the experimental and predicted values of flow stress is illustrated in Figure 9.

The patterns collected from the data demonstrate a satisfactory level of consistency between the experimental results and support the model’s high level of prediction (Figure 9d).

$$R=\frac{\sum _{i=1}^N\left(E_i-\overline{E}\right)\left(P_i-\overline{P}\right)}{\sqrt{{\sum }_{i=1}^N{\left(E_i-\overline{E}\right)}^2{\sum }_{i=1}^N{\left(P_i-\overline{P}\right)}^2}}$$

(17)

$$AARE=\frac{1}{N}\sum _{i=1}^N\left|\frac{E_i-P_i}{E_i}\right|$$

(18)

$$RMSE=\sqrt{\frac{1}{N}\sum _{i=1}^N{\left(E_i-P_i\right)}^2}$$

(19)

4. Discussion

The analysis of the as-cast structure of the studied alloy revealed Mg₂Si and Cu-enriched Al₂Cu and Q phases. The non-equilibrium Cu-rich phases dissolved during homogenization annealing whereas the Mg₂Si particles were fragmented and spherodized. The spherodization and fragmentation of phases of eutectic origin during annealing positively influenced the structural homogeneity and particles distribution after thermomechanical treatment [54,55]. The homogenization annealing of the alloy studied also provided for the formation of the nanoscale L1₂ phase dispersoids, which were homogeneously distributed in the Al-based matrix. The dispersoids provided a strong Zener pinning effect during MDF [56,57,58,59] and hot deformation inhibiting the static and dynamic grain growth. It should be noted that the Si atoms can also dissolve in the Al₃(Sc,Zr) phase, increasing its thermal stability due to the formation of an (Al,Si)₃(Sc,Zr) phase [60,61]. In the present study, dispersoids exhibited a size of ~10 nm after a second high temperature homogenization step that can be a result of partial Al substitution by Si atoms, which increase the thermal stability of dispersoids. This is in agreement with our earlier studies; dispersoids with similar parameters were found after two-step homogenization in 6xxx-type alloys [42,62].

The increase in cumulative strain and MDF temperature resulted in an increase in particle distribution homogeneity. The homogeneously distributed microscale particles can contribute to grain refinement during thermomechanical treatment and improve deformation behavior due to the particles’ stimulated nucleation effect (PSN) [46,51,63,64,65]. During MDF, the Mg₂Si phase was noticeably fragmentized, the result of both the temperature and strain effect. The Mg₂Si particle size after MDF was in the range of 1.3–1.5 µm. Despite a small volume fraction, these particles can provide for the PSN effect and additionally refine grain in the alloys studied. The positive effect of Mg₂Si particles with similar parameters on grain refinement was previously discussed in refs. [7,42].

The Sc-rich type-A and type-B phases observed in the as-forged sample exhibited a solidification origin. It was suggested that the Cu-bearing type-A phase corresponded to the W(AlCuSc) phase [66,67]. The Si atoms may dissolve in W-phase by analogy with the (Al,Si)₃(Sc,Zr) phase. The Cu-free type-B phase should correspond to the V(AlSc₂Si₂) phase discussed in refs. [60,66]. The Cu-enriched microscale particles with bright contrast observed in the as-forged at 250–350 °C samples were formed due to the ageing that occurred during MDF. These particles corresponded to the Q-(AlMgSiCu)-phase [68,69] or/and the Al₂Cu phase.

The MDF significantly refined grain structure in the alloy studied. The grain-refinement effect accompanied by a decrease in stress value with increasing cumulative strain reflects recrystallization duringac MDF. The decrease in temperature of MDF to 150 °C resulted in a decrease in grain/subgrain size. A high-volume fraction of LAGBs was observed after forging at 150 °C. This phenomenon suggests the continuous dynamic recrystallization [46] as the dominant recrystallization mechanism occurred during MDF of the alloy studied [70,71]. The continuous dynamic recrystallization is the result of LAGBs transforming into HAGBs induced by the strain [46]. A similar process can be observed during the heating of a pre-strained alloy [71,72,73]. We suggest that the dynamic recrystallization is the dominant process at 150 °C, whereas at higher temperatures of 250–350 °C, post-dynamic and static recrystallization can also occur. The static recrystallization may occur when the sample is heated to a treatment temperature before starting the forging. The increase in grain size with MDF temperature can be explained using a Zener factor [46]. The L1₂ –phase dispersoids prevent the intense grain coarsening due to the Zener pinning effect. The dispersoids exhibited a mean size of ~10 nm. Taking into account that all atoms of Sc and Zr precipitated from the solid solution and formed an Al₃(Sc,Zr) phase, the dispersoid volume fraction f is approximately ~0.005. The Zener factor allows for calculating the particles’ contribution to the grain growth suppression [46,74]:

$$Z_H=\frac{3fd}{4r}$$

(20)

where d is the mean grain size. The grain size after MDF at 150 °C was ~1.2 µm and at 250–350 °C it was ~2.0–2.1 µm. The corresponding Zener factor value was equal to 0.9 at 150 °C and ~1.5 at 250–350 °C. The Z = 1 is a threshold value above which the static grain growth is suppressed [74,75]. The Zener parameter value Z > 1 after MDF at both 250 and 350 °C. The threshold grain size value above which the grain growth is considered suppressed is ~1.3 µm. While the Zener factor does not consider the strain-induced grain growth, an insignificant difference in the grain size after MDF at 250 and 350 °C is most obviously associated with the Zener effect.

The fine-grained structure of the alloy after MDF and the Zener pinning effect provided a near superplastic behavior in a temperature range of 460–500 °C. The dynamic strengthening observed in the stress–strain curves is associated with dynamic grain growth that occurs at elevated temperatures [6]. This explains a decrease of elongation with increasing temperature during hot deformation. A hyperbolic Arrhenius constitutive model [76,77,78] is successfully constructed to estimate the flow stress behavior of this alloy in the studied temperature and strain rate range. The developed model demonstrated high predictability accuracy with the maximum error AARE of 6.6%.

The as-forged alloy demonstrated improved mechanical properties. The same level of mechanical characteristics was observed in the alloy with similar composition but increased Sc after thermomechanical treatment included hot and cold rolling [43]. The as-rolled and further annealed sheets with Sc exhibited a non-recrystallized structure that contributes significantly to strength due to dislocation strengthening mechanism [79,80]. Moreover, Sc and Zr provide for a high precipitation contribution to strength (Orowan strengthening) due to L1₂ dispersoids [81]. In the present study, the alloy contains low Sc of 0.1% and exhibits a recrystallized structure, whereas the strength was maintained at a similar level Sthat was achieved through both grain refinement Hall-Petch mechanism and Zener pinning effect. A near-superplastic behavior and high strength observed in the bulk samples offers additional opportunities for industrial application of the studied alloy and MDF technique.

The Arrhenius type constitutive model based on the experimental stress–strain results was constructed to model the hot flow behavior of this alloy. The constants of the Arrhenius type constitutive model were illustrated on the dependance of the deformation level. The stress exponent n is commonly employed in order to determine the primary mechanism governing the hot deformation process. Hence, the diverse methods of superplastic deformation can be correlated with independent values of n. Based on the findings of previous studies [82,83], it has been observed that the grain boundary sliding and the dislocation glide mechanism can result in values of n that are approximately two and three, respectively. On the other hand, processes associated with n values ranging from four to six are attributed to dislocation climb. According to the data presented in Figure 8b, the n values exhibit a range between 2.10 and 2.75 in the studied range of the strain and within the studied strain rates and temperatures, and this revealed that the grain boundary sliding and dislocation glide mechanisms controlled the hot deformation. The phenomenon of grain boundary sliding is observed to occur simultaneously with continuous dynamic recrystallization in both deformed region and substructured volume as the applied strain increases. The observed decrease in the n value at higher strains could potentially be attributed to the development of a recrystallized grain structure dominated in all samples. Furthermore, it is likely that the process of active grain boundary sliding is facilitated by dislocation mechanisms.

The material constant α, as shown in Figure 8d, exhibits a consistent decrease as the strain increases, and the same characteristic is observed for the activation energy Q. The specific deformation mechanisms and related mechanisms were also proposed by the effective activation energy of deformation [84]. The Q value exhibited an impressive decrease, reaching a strain value of 0.6 (Figure 8a). The observed reduction in Q as strain increases is consistent with the findings reported in previous studies [82,83,85,86]. The observed decrease in the Q value provides clear evidence of a reduction in the stored energy within the material as a result of strain [84,87].

5. Conclusions

The microstructure evolution and deformation behavior of the Al-1.2Mg-0.7Si-1.0Cu-0.1Sc-0.2Zr alloy after the homogenization annealing and multidirectional forging (MDF) to a cumulative strain of ∑e = 14.4 in a temperature range of 150–350 °C were studied. The main conclusions are drawn as follows:

• Two Sc-bearing phases were formed in the studied alloy during solidification: the Cu-rich W(AlCuSc)-phase with dissolved Si atoms and the Cu-free V(AlSc₂Si₂) phase were suggested. A minor amount of particles of these phases with a size in a range of 0.2–2 µm were also observed after homogenization annealing and MDF. The dispersoids of the L1₂-Al₃(Sc,Zr) phase with a mean size of 10 ± 1 nm were precipitated during homogenization annealing.
• Due to the Zener pinning effect of nanoscale L1₂ phase dispersoids and a large cumulative strain of 14.4, the fine-grained structure with a mean grain size of 1.2–2.1 µm was formed after MDF.
• Fine-grained as-forged at 350 °C alloy exhibited the maximum elongation to failure of ~300% at 460 °C in a strain-rate range of 2 × 10⁻³–1 × 10⁻² s⁻¹; at room temperature, the alloy exhibited yield strength of 326 ± 5 MPa, ultimate tensile strength of 366 ± 5 MPa, and elongation at fracture of 10 ± 3%.
• The hot deformation behavior was described using the Arrhenius type constitutive model. The developed model demonstrated a high predictability accuracy with a maximum average absolute relative error of 6.6%.