Microstructural Analysis and Constitutive Modeling of Superplastic Deformation Behavior of Al-Mg-Zn-Cu-Zr-xNi Alloys with Different Ni Contents

Abstract

Superplastic forming is a process that enables the production of complex-shaped parts using metallic alloys. To design the optimal forming regimes and ensure the success of forming operations, it is essential to use mathematical models that accurately represent the superplastic deformation behavior. This paper is concerned with the study of the microstructure and superplastic deformation behavior, with the construction of a constitutive model, of Al-Mg-Zn-Cu-Zr aluminum alloys with varying Ni contents. The aluminum solid solution and coarse precipitates of the T(Mg₃₂(Al,Zn)₄₉ and Al₃Ni second phases were formed in the studied alloy and Cu dissolved in both second phases. The deformation behavior was investigated in the temperature range of 400–480 °C and the strain rate range of 10⁻³–10⁻¹ s⁻¹. Due to the fine Al₃Zr precipitates, the alloys exhibit a partially recrystallized grain structure before the onset of superplastic deformation. Coarse precipitates of the second phases facilitate dynamic recrystallization and enhance superplasticity at the strain rates and temperatures studied. The alloys with ~6–9% particles exhibit high-strain-rate superplasticity at temperatures of 440–480 °C and strain rates of 10⁻²–10⁻¹ s⁻¹. The presence of high fractions of ~9% Al₃(Ni,Cu) and ~3% T-phase precipitates provided high-strain-rate superplasticity with elongations of 700–800% at a low temperature of 400 °C. An Arrhenius-type constitutive model with good agreement between the predicted and experimental flow stresses was developed for the alloys with different Ni contents.

1. Introduction

Superplasticity is a property that has been extensively studied due to its applications in the aerospace and automotive industries [1,2,3,4] and other applications [5]. Superplastic forming (SPF) is based on this phenomenon, i.e., the possibility of ensuring a stable flow and large elongations without necking due to the high sensitivity of the flow stress to the strain rate for alloys with grains below 10 µm [6,7,8]. Conventional Al-based alloys exhibit superplasticity at strain rates of 10⁻⁴–10⁻³ s⁻¹ [9,10]. To increase the productivity and economic efficiency of SPF, the strain rates should be above 10⁻² s⁻¹ [11]. For higher strain rates, a finer grain structure is required [4,12,13,14,15]. Optimizing the microstructural heterogeneity through the presence of both coarse and fine particles helps to refine the grains [16]. The secondary-phase particles influence the dislocation density and their distribution during thermomechanical treatment, following the recrystallization kinetics and grain structure. Coarse hard particles (~1 µm) accelerate the formation of recrystallization nuclei [17,18,19,20,21], while fine particles inhibit grain growth [7,22,23,24,25,26,27,28]. Both modes help to refine the microstructure and may improve the superplasticity [16,29,30,31].

The development of alloys with superplastic behavior at increased strain rates and decreased temperatures is an actual task within the SPF of Al-based alloys. A group of high-strength AA7XXX-series alloys are utilized for SPF, but the complicated processing scheme, the high cost of fine-grained sheets, significant residual cavitation, and low forming rates make them commercially unattractive [32,33,34,35,36]. The development of new alloys based on this system with improved properties is a current issue. The deformation behavior is significantly influenced by the presence of secondary-phase particles, and coarse 1–2 µm particles can significantly enhance the superplasticity [16,29,30]. The evolution of the microstructure, including the parameters of the precipitates, the related grain size, and the thermal stability of the grain structure, significantly influences the superplastic deformation behavior. Al-Mg-Zn-Cu-Ni-Zr alloys exhibit promising superplastic behavior with elongations above 400% in a strain rate range of 10⁻³–10⁻¹ s⁻¹, thus showing a high-strain-rate superplasticity [16,37,38]. However, the effect of the volume fraction of coarse particles on the superplastic behavior has been insufficiently investigated, and this is one of the tasks of this study.

The development of alloys with the required properties can only be solved using experimental means, which significantly increases the time and labor costs associated with the process. To reduce the number of experiments, a computer modeling method was successfully implemented recently which makes it possible to predict the properties as a function of the alloy’s composition [10,39,40,41,42,43,44]. Mathematical models can be employed to predict the shape change and development of the forming modes for parts with complex shapes [45,46,47,48,49,50,51]. For successful simulation of the SPF process, it is important to accurately characterize the relationship between the flow behavior and the strain rate, strain, and temperature [50,51,52,53,54,55]. The modeling of the deformation behavior helps to reduce the time, effort, number of tests, material consumption, and thus the production costs and simplifies the superplastic forming process. Models based on the Zener–Hollomon parameter and Arrhenius-type equations, which used in this work, are widely used to describe the behavior of hot deformation of materials characterized by a steady flow, including superplastic deformation [56,57,58].

Thus, the aims of this work are (a) to study the effect of the Ni content on the microstructure and superplasticity of Al-Mg-Zn-Cu-Zr alloys in order to define the fraction of coarse particles required for a pronounced PSN effect and high-strain-rate superplasticity and (b) to use the experimental stress–strain data to construct an Arrhenius-type constitutive equation model that accounts for varying proportions of Al₃Ni particles.

2. Materials and Methods

The chemical compositions of the experimental alloys studied are listed in

Table 1. Chemical compositions of the studied aluminum-based alloys (nominal/EDS analysis) (wt.%).

Alloy	Mg	Zn	Ni	Zr	Cu	Al
1Ni	4/4.1	4/3.9	1/1.1	0.25/0.25	0.7/0.72	Bal.
2Ni	4/4.1	4/4.1	2/1.9	0.25/0.26	0.7/0.69	Bal.
3Ni	4/4.0	4/4.0	3/3.2	0.25/0.26	0.7/0.71	Bal.
4Ni	4/4.0	4/3.8	4/3.8	0.25/0.25	0.7/0.72	Bal.

. The concentrations of Mg, Zn, Cu, and Zr were similar in all of the alloys, whereas the Ni concentration varied from 1.1 to 3.8 wt.%. The alloys were processed in laboratory conditions using Al (99.99%), Zn (99.97%), Mg (99.97%), and Al-18 wt.% Ni, Al-53.5 wt.% Cu, and Al-5 wt.% Zr master alloys. The melt was prepared in a fireclay crucible using a Nabertherm S3 furnace (Nabertherm GmbH, Lilienthal, Germany) at a casting temperature of 760 ± 10 °C. Ingots with a size of 20 × 40 × 100 mm³ were cast into a water-cooled copper mold at a cooling rate of ~15 K/s.

Following solidification, the alloys were annealed in an electric furnaceNabertherm N30/65A (Nabertherm GmbH, Lilienthal, Germany) in two stages for homogenization of the microstructure. The first stage was set to 450 °C for 3 h to eliminate the non-equilibrium phases and to precipitate the L1₂ Al₃Zr phase, and the second stage was set to 500 °C for 3 h to finish the homogenization process and to spheroidize the Ni-rich particles. The alloys were hot-rolled at 420 ± 20 °C with a total reduction of 85% (from 20 to 3 mm) in ten passes, followed by cold rolling with a total reduction of 60% (from 3 to 1 mm) in six passes. The sheet processing scheme is presented in Figure 1.

The solidus (incipient melting point) temperatures of the alloys studied were determined via a differential thermal analysis (DTA) using a Setaram Labsys DTA/DSC 1600 calorimeter (KEP TECHNOLOGIES EMEA, Caluire, France) in an Ar atmosphere in their as-rolled states. The heating rate was 5 K/min.

X-ray diffraction (XRD) was carried out using a D8 Discover diffractometer (Bruker Corporation, Billerica, MA, USA) in a Cu-Kα radiation for 2θ angles of 20–90°.

A Tescan-VEGA3 LMH scanning electron microscope (SEM, Tescan Brno s.r.o., Kohoutovice, Czech Republic) and a Zeiss Axiovert 200 light microscope (Carl Zeiss, Oberkochen, Germany) were used to analyze the microstructures. An energy-dispersive X-ray spectrometer (EDS) X—MAX80(Oxford Instruments plc, Abingdon, UK) was used to analyze the chemical composition and elemental distributions between the phases. To analyze the chemical compositions, ten images at a magnification of ×200 were studied and averaged. The standard deviation for the Mg, Zn, and Ni element content was ~0.1 wt%, while for Zr and Cu, it was below 0.03%. The samples were prepared through mechanical grinding on SiC papers of different dispersions and polishing with an OP-S suspension on a Struers LaboPol machine (Struers APS, Ballerup, Denmark). To analyze the grain structure of the alloys with polarized light, the samples were subjected to anode oxidation, which was carried out in a 10% water solution of hydrofluoric acid at a voltage of 20 V for a duration of 50 s. Lead was used as the cathode. The grain size, size, and volume fraction of the Al₃Ni particles were measured using the random secant method using at least 300 intersections. Error bars were calculated for a confidence probability of 0.95. The high-angle grain boundaries (HAGBs) and low-angle grain boundaries (LAGBs) were determined from the microstructure ascertained using an EBSD analysis (area—150 × 150 μm²; step—0.3 μm). The EBSD analysis was performed using an HKL Nordlys Max detector (Oxford Instruments plc, Abingdon, UK). The grain boundaries with misorientation angles of 2–14° were considered LAGBs, and those with a misorientation angle ≥ 15° were considered HAGBs.

The deformation behavior was characterized via uniaxial tensile tests using a Walter Bai LFM-100 machine (Walter + Bai AG, Löhningen, Switzerland) with the DionPro v.4.8 software (Walter + bai company, Löhningen, Switzerlan) to control the crosshead motion and to maintain a constant strain rate according to the ASTM E2448-11 standard [59]. The test machine was equipped with a force sensor with a force accuracy of 0.5% to 2 kN. Proportional samples with a gauge size of 6 × 1 mm² and a length of 14 mm were used. Samples were cut from cold-rolled sheets in the longitudinal direction (Figure 1). Tensile tests were performed in a temperature range of 400–480 °C (0.85–0.95 Tm) with constant strain rates in a range of 10⁻³ to 10⁻¹ s⁻¹. Three samples per test condition were tested.

3. Results and Discussion

3.1. Microstructure Analysis

According to the DTA (Figure 2a), the solidus temperatures (incipient melting points (T_i.m.)) were around ~520 °C and were insignificantly dependent on the Ni content. Two overlapping endothermic peaks were observed in the DTA curves for the alloys with 1.1–3.2Ni, and one peak was defined for the alloy of a near-eutectic composition, which matched the phase diagram (Figure 2b). The temperature range of 400–480 °C, corresponding to T_i.m of 0.85–0.95, was selected for studying the microstructure and the deformation behavior of the alloys.

According to the Thermo-Calc data (Figure 2b), the Al₃Ni and T(Al₂Zn₃Mg₃/Mg₃₂(Al,Zn)₄₉) [60] phases can precipitate in the alloys. These data are consistent with the X-ray phase analysis, which confirms the formation of an aluminum-based solid solution and both second phases (Figure 2c). According to SEM-EDS studies, after hot rolling and subsequent cold rolling and low-temperature annealing at 400 °C, the alloys contained uniformly distributed Ni,Cu-rich particles and a high fraction of Mg,Zn,Cu-rich micrometer and submicrometer particles in the (Al) matrix (Figure 3 and Figure 4). According to the EDS data, Cu was dissolved in the T phase (Figure 4), which agreed with [61,62]. The EDS data also confirmed an increased Cu content in the areas belonging to the Ni-rich Al₃Ni phase (the Cu and Ni elemental distribution maps in Figure 4b). The mean Cu content was ~4.7 ± 0.3 wt.% for the Ni-rich precipitates and ~0.7 ± 0.1 wt.% for the Al-based solid solution (Al). The T phase precipitated in the Al solid solution’s interior and at the Al₃Ni interphases (Figure 3 and Figure 4).

The particle sizes and volume fractions were analyzed in the as-rolled state and after annealing at the deformation temperatures (

Table 2. Particle volume fraction and sizes for the alloys studied.

State/Alloy	Volume Fraction (%)/Particle Size, d (μm)
	1Ni	2Ni	3Ni	4Ni
Cold rolling (CW)	9 ± 1/0.6 ± 0.1	13 ± 1/0.7 ± 0.1	15 ± 2/0.9 ± 0.1	16 ± 2/1.2 ± 0.1
Annealing at 400 °C	5 ± 1/0.6 ± 0.1	8 ± 1/0.6 ± 0.1	10 ± 1/0.9 ± 0.1	12 ± 2/1.2 ± 0.1
Annealing at 440 °C	2.3 ± 0.5/1.0 ± 0.1	6 ± 1/0.9 ± 0.1	7 ± 1/1.0 ± 0.1	9 ± 1/1.3 ± 0.1
Annealing at 480 °C	2.1 ± 0.2/0.9 ± 0.1	6 ± 1/0.9 ± 0.1	7 ± 1/1.0 ± 0.1	9 ± 1/1.3 ± 0.1

, Figure 3j–l). The volume fraction of the particles (for both phases) increased with an increasing Ni content and a decreasing annealing temperature (Table 2). The mean concentration of soluble Zn, Mg, and Cu elements was similar to their content in the alloys, and a small number of fine T-phase precipitates were defined at 440 °C. Precipitates of the T phase were dissolved at 480 °C, which agreed with the Thermo-Calc data (Figure 2b). As a result, the mean particle size increased and their fraction decreased, while the Cu content in the Al₃Ni particles remained at an enhanced level of about ~3%. The most reasonable explanation is that Cu atoms partially substitute Ni in the Al₃Ni phase. Therefore, the Al₃(Ni,Cu) and T(Al,Zn,Mg,Cu) second phases were observed for the alloys studied in the cold-worked state and at temperatures below 440 °C, and Al₃(Ni,Cu) was fined at the higher temperatures. At a temperature of 440 °C, the T-phase fraction was insignificant, at <1%.

Due to the T-phase precipitates, prior to the onset of superplastic deformation at 400 °C, the fraction of particles in the (Al) matrix was significantly higher than that at 440–480 °C. The particle size was ~1 μm for the alloys with 1.1–3.2%Ni and ~1.3 μm for the alloy with 3.8%Ni. The particles had a slightly elongated shape, with an aspect ratio of 0.75–0.82.

Zirconium-was homogeneously distributed in the (Al) solid solution (Figure 4). It is well known that Zr-containing nanosized (~10 nm) dispersoids of the Al₃Zr phase, precipitating during heat and thermomechanical processing, increase the recrystallization resistance and inhibit grain growth in Al-based alloys [17,63,64,65,66]. These precipitates are important to the grain refinement and superplasticity of the alloys studied.

Due to the Zr alloying and the Al₃Zr precipitates, the microstructure of the alloys before the onset of a superplastic flow at a low temperature of 400 °C (0.85 T_m) consisted of elongated non-recrystallized grains, and the width of grains varied from 0.5 to 12.5 μm (Figure 5a–d). Recrystallization occurred at 440 °C, while the structure of the alloys was incompletely recrystallized even at 480 °C (0.95 T_m) (Figure 5e–h).

According to the EBSD maps (Figure 6a), before the start of deformation at a temperature of 440 °C, the 1Ni alloy with ~2% coarse particles exhibited the largest recrystallized grains of 4.7 ± 0.4 μm and the smallest fraction of HAGBs of 49%. For the alloys with a higher Ni content and 6–9% of the coarse particles (Figure 6b–d), the recrystallized grain size was (2.2 ± 0.2)–(2.7 ± 0.3) μm, and the fraction of HAGBs increased to 64–69%. According to the EBSD data, the fraction of the recrystallized grains increased from 19% for 1Ni to 25% for 2Ni, 33% for 3Ni, and 40% for 4Ni alloys. With an increasing particle fraction from ~6 to ~9% (increasing Ni from 1.9 to 3.8%), the fraction of high-angle grain boundaries increased from 64 to 69%, while the mean grain size changed insignificantly considering the error bars. Thus, the main increase in the recrystallized grain fraction and the effect of grain refinement were defined with an increase in the particle fraction from ~2 to ~6%.

3.2. Superplastic Deformation Behavior

The stress–strain data, the strain rate sensitivity m values, and the elongation to failure in a temperature range of 400–480 °C with strain rates in a range of 10⁻³ s⁻¹ to 10⁻¹ s⁻¹ are presented in Figure 7 and Figure 8. A higher sensitivity to the strain rate provided better superplasticity with a more uniform flow. The m values above 0.3 provided a superplastic flow for the fine-grained alloys, with m = 0.5 indicating the dominance of grain boundary sliding; that is the deformation mechanism provided the most uniform flow. Meanwhile, for the alloys with a non-recrystallized structure, the m values were smaller at low strain due to the low fraction of HAGBs, and m increased with increasing strain due to the dynamic recrystallization effect and the increase in the fraction of HAGBs having the capability to cause grain boundary sliding. The alloys exhibited superplasticity (elongations > 400% and a strain rate sensitivity coefficient m > 0.3 at a steady flow stage) in a wide range of strain rates, including a high strain rate range (Figure 8). At a low temperature of 400 °C (Figure 7a and Figure 8a,d), the alloy with 1.1%Ni demonstrated a weak superplasticity, with m < 0.3 and elongations of 200–300% in the studied strain rate range resulting in the inhomogeneous and coarse grain structure. The combination of a low temperature and low number of coarse particles resulted in weak recrystallization kinetics, with large grains and a weak superplastic response.

On increasing the Ni content to 1.9–3.8%, the m values reached 0.4–0.8, and the elongation increased above 400% (Figure 8d–f). The high-Ni alloys exhibited good superplasticity and elongations > 400% in a wide temperature–strain rate range: at a low temperature of 400 °C and an elevated temperature of 480 °C for strain rates below 5 × 10⁻² s⁻¹ and at a temperature of 440 °C up to a strain rate of 10⁻¹ s⁻¹. Low temperatures weaken the recrystallization kinetics, but high temperatures promote grain growth, and both are detrimental to superplasticity.

A high fraction of ~12% of the secondary phases (T and Al₃(Ni,Cu)) at 400 °C provided high m values and elongations at high strain rates for the alloy with 3.8% Ni. A strain rate sensitivity above 0.4 and elongations of 740–780% were observed for this alloy at the low temperature of 400 °C and a high strain rate of up to 5 × 10⁻² s⁻¹ (Figure 7a).

The deformation behavior and elongations depend on the precipitate fraction and the related dynamic recrystallization and grain growth kinetics. At small strains of <0.2–0.4, strain hardening was observed for all of the alloys (Figure 7). It is likely that the deformation at this stage was controlled by dislocation slip and that the dislocation density increased due to a high fraction of non-recrystallized grains before the start of the deformation. At the steady flow stage of the superplastic flow, strain softening was revealed for the alloys with 1.1–3.2% Ni. It is reasonable to suggest that dynamic recrystallization occurred during straining, the fraction of high-angle grain boundaries increased, which facilitated grain boundary sliding, and finally, this provided high elongation [37,67,68,69,70,71,72]. Different behavior with strain hardening at the steady deformation stage was observed for the alloy with 3.8%Ni and a high number and density of coarse particles. Due to their high fraction, coarse particles facilitated recrystallization during heating and the initial stage of deformation, and the grain growth of the recrystallized grains was pronounced in the steady state. Meanwhile, the high proportion of particles facilitating dynamic recrystallization in this alloy provided improved superplastic properties at the low deformation temperature of 400 °C compared to the alloys with lower Ni contents.

Thus, due to the PSN effect, coarse particles facilitated recrystallization and increased the fraction of high-angle grain boundaries, which significantly improved the superplastic properties of the studied alloys. Fine precipitates of the Al₃Zr phase inhibited grain growth and critically impacted the good superplasticity of the high-Ni alloys, while for the alloys with a small fraction of coarse particles, their effect was insufficient to provide good superplastic properties. The maximum elongation of ~950% was observed at 1 × 10⁻² s⁻¹ and 440 °C for the alloy with ~3%Ni and ~8% coarse particles (Figure 6e). Thus, this combination of a tensile regime and the precipitate parameters provided a good balance between recrystallization and grain growth.

3.3. The Constitutive Model

The stress–strain data were used to construct a constitutive model for predicting the superplastic behavior. The effect of temperature and strain rate on hot deformation behavior can be represented using the Zener–Hollomon parameter (Z) in an exponent-type equation [73,74] as follows:

$$Z=\dot{\epsilon }·exp\left({\displaystyle \frac{Q}{RT}}\right)$$

(1)

$$\dot{\epsilon }=\left\{\begin{array}{c}{A}_1{\sigma }^{n_1}(exp\left(-{\displaystyle \frac{Q_1}{RT}}\right))-Power law \\ A_2\left(exp\left(\beta \sigma \right)\right)\left(exp\left(-{\displaystyle \frac{Q_1}{RT}}\right)\right)-Exponential law\\ A_3{\left[\sinh \left(\alpha \sigma \right)\right]}^{n_2}\left(exp\left(-{\displaystyle \frac{Q_1}{RT}}\right)\right)-Hyperbolic sine law\end{array}\right.$$

(2)

where

σ true stress, MPa;

$\dot{\epsilon }$ true strain rate, s⁻¹;

R universal gas constant with a value of 8.314 J·mol⁻¹·K⁻¹;

T absolute temperature, K;

Q activation energy, J/mole.

A, n₂, β, and α = β/n₁ are the constants specific to the material.

The hyperbolic sine law is used to analyze a wide range of stresses, so in this study, it was used to develop the constitutive model for accurately simulating and predicting the superplastic flow characteristics of these alloys.

For the investigated alloys, the true flow stress (σ) was a function of the deformation temperature (T), true strain rate ($\dot{\epsilon }$), true strain (ε), and weight percent of Ni. Therefore, the relationship was as follows in Equation (2).

$$\sigma =f(T, \dot{\epsilon }, \epsilon , wt.\%Ni)$$

(3)

The material constants and parameters of the equation were determined logarithmically using both parts of Equation (2) in the following, Equations (4)–(6).

$$\ln \dot{\epsilon }=\ln A_1+n_1\ln \sigma -{\displaystyle \frac{Q_1}{RT}}$$

(4)

$$\ln \dot{\epsilon }=\ln A_2+\beta \sigma -{\displaystyle \frac{Q_2}{RT}}$$

(5)

$$\ln \dot{\epsilon }=\ln A_3+n_2\ln \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\alpha \sigma )-{\displaystyle \frac{Q_3}{RT}}$$

(6)

Through partial differentiation of Equations (4)–(6), we obtain Equations (7) and (8):

$$n_1={\left[{\displaystyle \frac{\partial \ln \dot{\epsilon }}{\partial \ln \sigma }}\right]}_T; \beta ={\left[{\displaystyle \frac{\partial \ln \dot{\epsilon }}{\partial \sigma }}\right]}_T; n_2={\left[{\displaystyle \frac{\partial \ln \dot{\epsilon }}{\partial \ln \left[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\alpha \sigma \right)\right]}}\right]}_T;$$

(7)

$$Q_3={R·{\left[{\displaystyle \frac{\partial \ln \dot{\epsilon }}{\partial \ln \left[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\alpha \sigma \right)\right]}}\right]}_T·\left[{\displaystyle \frac{\partial \ln \left[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\alpha \sigma \right)\right]}{\partial \left(\frac{1}{T}\right)}}\right]}_{\dot{\epsilon }}$$

(8)

According to Equation (3), the parameters of the constants α, Q₃, A₃, and n₂ were dependent on the deformation temperature (T), true strain rate ($\dot{\epsilon }$), true strain (ε), and weight percent of Ni. For guidance, the parameters in Equations (7) and (8) were calculated at a constant strain level of 0.69 (100% deformation) and are listed in

Table 3. Calculated values of the material constants/parameters at a deformation level of 0.69 for the alloys with different Ni contents.

Alloy	1Ni	2Ni	3Ni	4Ni
$\mathbf{l}\mathbf{n}\left({\mathit{A}}_1\right)$	14.9	21.4	16.9	7.4
${\mathit{n}}_1$	1.8	2.6	2.5	2.3
${\mathit{Q}}_1[{\displaystyle \frac{\mathbf{k}\mathbf{J}}{\mathbf{m}\mathbf{o}\mathbf{l}}}]$	148 ± 6	200 ± 11	167 ± 6	111 ± 8
$\mathbf{l}\mathbf{n}\left({\mathit{A}}_2\right)$	19.7	17.3	12.9	5.5
m	0.71	0.49	0.51	0.57
β [MPa−1]	0.08	0.11	0.12	0.14
${\mathit{Q}}_2[{\displaystyle \frac{\mathbf{k}\mathbf{J}}{\mathbf{m}\mathbf{o}\mathbf{l}}}]$	157 ± 6	144 ± 9	119 ± 6	90 ± 8
α	0.04	0.04	0.05	0.06
$\mathbf{l}\mathbf{n}\left({\mathit{A}}_3\right)$	22.5	27.7	22.3	12
${\mathit{n}}_2$	1.4	2.0	1.9	1.7
${\mathit{Q}}_3[{\displaystyle \frac{\mathbf{k}\mathbf{J}}{\mathbf{m}\mathbf{o}\mathbf{l}}}]$	162 ± 7	193 ± 15	159 ± 7	107 ± 6

. The dependencies of α, Q₃, ln(A₃), and n₂ on the strain and wt.%Ni of the alloys studied are shown in Figure 9. The detailed method for the determination of these parameters is described in [75,76].

The effective activation energy Q decreased as the fraction of precipitates increased. The values changed from ~150–160 kJ/mol to ~90–100 kJ/mol. This can be explained by the PSN effect of the coarse particles facilitating dynamic recrystallization. A higher fraction of HAGBs provided a lower effective activation energy due to the involvement of the grain boundary sliding deformation mechanism, the importance of which increased with an increasing Ni content.

To model the flow behavior of the investigated alloys, the effect of strain on the material parameters in Table 3 should be considered. The dependence of the hyperbolic sine law equation constants ($Q_3$, $n_2$, $A_3$, α) on the strain is represented using Equation (9).

$$\left\{\begin{array}{c}{Q}_3=C_0+C_1{\epsilon }^1+C_2{\epsilon }^2+C_3{\epsilon }^3+C_4{\epsilon }^4+C_5{\epsilon }^5\\ n_2=D_0+D_1{\epsilon }^1+D_2{\epsilon }^2+D_3{\epsilon }^3+D_4{\epsilon }^4+D_5{\epsilon }^5\\ A_3=E_0+E_1{\epsilon }^1+E_2{\epsilon }^2+E_3{\epsilon }^3+E_4{\epsilon }^4+E_5{\epsilon }^5\\ \alpha =F_0+F_1{\epsilon }^1+F_2{\epsilon }^2+F_3{\epsilon }^3+F_4{\epsilon }^4+F_5{\epsilon }^5\end{array}\right.$$

(9)

where C₀–C₅, D₀–D₅, E₀–E₅, and F₀–F₅, are the coefficients; ε is the true strain.

A fifth-degree polynomial provides a small error (R = 0.98–1.0) compared to a polynomial of the fourth degree or lower. A higher-order (i.e., >5) polynomial would over-fit the calculated data, thus losing its ability to truly represent and generalize them.

Using the expression of the hyperbolic sine law (Equation (2)), the constitutive model relating the true flow stress and the Zener–Hollomon parameter can be written as follows:

$$\sigma ={\displaystyle \frac{1}{\alpha }}\mathrm{l}\mathrm{n}\left\{{({\displaystyle \frac{\dot{\epsilon }·exp(\frac{Q_3}{RT})}{A_3}})}^{{\displaystyle \frac{1}{n_2}}}+{\left[({{\displaystyle \frac{\dot{\epsilon }·exp(\frac{Q_3}{RT})}{A_3}})}^{{\displaystyle \frac{2}{n_2}}}+1\right]}^{{\displaystyle \frac{1}{2}}}\right\}$$

(10)

Figure 10 shows a comparison of the experimental and modeled strain curves for the alloys at the three temperatures of 400, 440, and 480 °C. It can be seen from the graphs that as the strain temperature increases, the strain curves, both the experimental and modeled curves, shift toward lower stress values. This is explained by the fact that with increasing temperature, the mobility of the atoms in the crystal lattice increases, which promotes easy sliding and reorientation of the grains, thus facilitating the deformation process. As a result, less stress is required to achieve the same strain.

Comparison of the experimental and modeled curves shows that at temperatures of 400 °C and 440 °C, slight differences are observed between the two. However, at 480 °C, the experimental and modeled curves show almost complete agreement, indicating the high adequacy of the model used and its ability to reliably describe the behavior of the alloys studied at this deformation temperature. Thus, the model allows for better predictions only for a limited processing range, where specific deformation mechanisms are operating. It is reasonable to assume that a higher accuracy was observed for the alloys with a stable grain structure and where dynamic recrystallization was less pronounced. To improve the predictability, the evolution of the microstructural parameters should be included in the model.

For evaluation of the accuracy, the predictability of the hyperbolic sine constitutive model considering strain compensation is quantified in terms of the correlation coefficient (R) (Equation (11)), the average absolute relative error (AARE) (Equation (12)), and the root mean square error (RMSE) (Equation (13)).

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

(11)

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

(12)

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

(13)

where E_i is the experimental finding, and P_i is the predicted value.

Figure 11 shows the correlation between the experimental flow stress data and the flow stress data predicted by the constructed model. Based on Figure 9, it can be argued that the assessment of the constructed model demonstrates a strong correlation between the experimental and predicted data, except in the instability regimes. Despite this divergence, the R, AARE (%), and RMSE are found to be 0.94, 10%, and 3.8, respectively, which reflects the good prediction capabilities of the strain-compensated constitutive model.

4. Conclusions

The effect of the Ni content on the phase composition and microstructure and superplastic deformation behavior was studied for thermomechanically treated Al-Mg-Zn-Cu-Zr alloys in the temperature range of 400–480 °C at constant strain rates in the range of 10⁻³–10⁻¹ s⁻¹. The experimental data were used to construct a strain-compensated constitutive model for predicting the superplastic deformation behavior of the investigated alloys. The main conclusions are as follows.

• According to the microstructural and XRD data, the alloys contained coarse particles of the Al₃Ni and T(Mg₃₂(Al,Zn)₄₉ phase in which Cu was dissolved. The T-phase fraction was ~3% for the low temperature of 400 °C and >1% at 440 °C, and it was completely dissolved at 480 °C. The particle size varied in the range of 0.6–1.3 µm, and the total fraction increased from 2 to 16% with an increasing Ni content and decreasing temperature.
• Prior to the onset of superplastic flow, the grain structure of the alloys was non-recrystallized at 400 °C and partially recrystallized at 440–480 °C. Increasing the fraction of micrometer-sized particles increased the proportions of high-angle grain boundaries and recrystallized grains and decreased the mean grain size. Significant changes were defined when the total fraction of particles increased from ~2.5 to ~6%.
• The alloys showed excellent superplasticity in a temperature range of 400–480 °C and a wide strain rate range of 10⁻³ to 10⁻² s⁻¹ with elongation above 400% and a strain rate sensitivity coefficient m above 0.3. The alloys with 1.9–3.2% Ni and 6–9% coarse particles were superplastic up to 10⁻¹ s⁻¹ at 440–480 °C with elongation up to 950%. A high fraction of second T-phase and Al₃Ni-phase particles of ~12% with a mean size of 0.6 µm provided good superplasticity, with an elongation of ~700–800% at a low temperature of 400 °C and a high strain rate of (0.5–1) × 10⁻² s⁻¹.
• The effect of strain on the flow stress was integrated into the material constants, specifically A, n, α, and Q. A fifth-order polynomial was identified as the optimal representation of the influence of strain on the material constants, exhibiting excellent correlation and generalization. The strain-compensated constitutive model demonstrated accurate prediction of the flow behavior of the investigated alloys. The effective activation energy (Q) of superplastic deformation decreased from ~150–160 kJ/mol to ~90–100 kJ/mol with an increase in the fraction of Al₃Ni particles, which facilitated dynamic recrystallization, high-angle grain boundary formation, and the grain boundary sliding deformation mechanism.