Investigation on the Mechanism of Hot Deformation Behavior of Aluminum Single Crystals Based on Grain Orientation

Abstract

Aluminum alloys are widely used in transportation industries due to their excellent specific strength, stiffness, and formability. Modifying the texture of aluminum alloys can further enhance their mechanical properties. This study explores the hot deformation behavior of high-purity aluminum single crystals (ASCs) with Brass and Goss orientations. We examine the influence of crystal orientation on deformation mechanisms and establish hot processing maps to identify optimal conditions for microstructural evolution. The results highlight the distinct behaviors of Goss- and Brass-oriented ASCs, with Goss exhibiting greater dynamic recrystallization potential and Brass showing higher strain rate sensitivity at elevated temperatures.

1. Introduction

Aluminum alloys offer excellent specific strength and formability, making them widely used in transportation industries such as aviation, shipping, and automotive [1]. With the ongoing development of these industries and the growing demand for energy efficiency, lightweight vehicles in aerospace and automotive sectors are becoming a crucial trend, placing higher demands on the performance of aluminum and its alloys [2,3,4]. Researchers have shown that modifying the texture components of aluminum alloys can further enhance their mechanical properties [5]. In some applications, aluminum alloy components are required to exhibit either anisotropic or isotropic mechanical properties. For instance, 7XXX series alloys (e.g., 7050-T7451) are used in aerospace for strength uniformity [6], while AA6061 is optimized for automotive forming through texture control [7]. Hot simulation uniaxial compression tests are used to investigate the relationship between flow stress and strain, as well as to explore the effects of deformation conditions such as strain rate, temperature, deformation amount, and microstructure evolution. These tests help establish the influence of initial crystal orientation, temperature, and strain rate on flow stress during hot plastic deformation.

Numerous studies have explored the flow stress–strain relationship models of aluminum alloys. Zhang et al. [8] conducted hot compression tests on 2195 aluminum alloy at temperatures between 300 °C and 520 °C, strain rates of 0.01 s⁻¹ to 10 s⁻¹, and 60% compression, focusing on the evolution of the alloy during compression and its temperature variation. Ke et al. [9] investigated the hot deformation behavior and microstructure evolution of uniformly annealed AA7020 aluminum alloy under isothermal compression at 400 °C to 500 °C and strain rates from 0.001 s⁻¹ to 1 s⁻¹, establishing a sixth-order polynomial strain-compensated Arrhenius-type constitutive equation to describe the alloy’s hot deformation behavior.

Understanding the mechanical behavior and microstructural evolution of metal single crystals during plastic deformation is fundamental to comprehending the deformation behavior of polycrystalline materials and complex multiphase materials [10,11]. Metal single crystals are influenced by factors such as crystal orientation, strain rate, deformation temperature, and loading mode during plastic deformation. Huang et al. [12] studied the plastic deformation relationship between polycrystalline and single-crystalline pure aluminum, demonstrating that the behavior of grains in polycrystals can be compared to single-crystal behavior for accurately estimating stress–strain curves using quantitative texture analysis. Luo et al. [13] reported that typical ASCs undergo stage hardening, a yield phenomenon directly linked to crystal rotation under uniaxial strain. Strain rate and deformation temperature also play significant roles in the deformation behavior of metal single crystals. Li et al. [14] performed quasi-static and dynamic high-strain-rate compression tests on [0001]-oriented magnesium single crystals, revealing that different strain rates lead to distinct final microstructures in the specimens. Caillard et al. [15] suggested that the plastic deformation of ASCs during cold deformation is driven solely by the octahedral slip system {111}<110>, while during hot deformation, non-octahedral slip systems {110}<110>, {100}<110>, and {112}<110> also contribute to plastic deformation above 400 °C. Paul et al. [16] studied the relationship between activated slip systems and the formation of the first grain nucleus during recrystallization in Goss {110}<001> oriented ASCs, highlighting the crucial role of the {111} plane in the early stages of recrystallization. Miura et al. employed rapid X-ray Laue methods to investigate dynamic recrystallization in high-purity aluminum single crystals under high-temperature deformation [17]. Similarly, Khan et al. examined the strain rate effects on high-purity aluminum single crystals through both experimental and simulation approaches [18]. Despite valuable insights, isothermal compression studies on ASCs remain scarce. Thus, further research is needed to analyze the influence of crystal orientation on the plastic deformation behavior of ASCs under varying deformation conditions.

This paper examines the effects of crystal orientation on the hot deformation behavior of ASCs using isothermal uniaxial compression tests. The study reveals the microstructural evolution mechanisms of ASCs with different crystal orientations and establishes hot processing maps and rheological models for each orientation. These findings provide valuable insights for texture control and performance optimization of aluminum alloys.

2. Experimental Procedure

This study primarily conducts uniaxial compression tests on ASCs with different crystal orientations. Single-grain aluminum ingots (50–80 mm in size) were prepared by stress relief annealing. To obtain cylindrical ingots with large grain sizes (>50 mm) for sampling, we employed a re-casting and slow cooling method. The re-casting process was conducted in a crucible furnace (Model AB123, Thermo Fisher Scientific, Waltham, MA, USA), where the original ingot was melted at 700 °C for 2 h and then slowly cooled to 610 °C over 12 h, followed by furnace cooling to 100 °C. The recast cylindrical ingot was cut into disk-shaped specimens using a Computer Numerical Control (CNC) wire-cut EDM machine (Model XY987, Mitsubishi Electric, Tokyo, Japan). The ingot surfaces were etched with a 5% HF (Sigma-Aldrich, St. Louis, MO, USA) solution for 30 minutes to reveal clear grain boundaries. Thin slices (5 × 8 × 1 mm³) were cut from individual grains, polished, and electrolytically double-sprayed, as illustrated in Figure 1.

Electron backscatter diffraction (EBSD) testing was performed on the prepared slices using a ZEISS EVO MA10 scanning electron microscope (Carl Zeiss AG, Oberkochen, Germany). Prior to the uniaxial compression tests, we conducted EBSD analysis using HKL’s Channel 5 software (Version 5.10, Oxford Instruments, Abingdon, UK) to determine the initial crystal orientations of the single crystal, as shown in Figure 2. The EBSD results were analyzed to identify the required crystal orientations (Brass, Goss) by rotating the three Euler angles in Euler space. Isothermal uniaxial compression tests were carried out using a Gleeble-3150 thermal simulation testing machine (Dynamic Systems Inc., Poestenkill, NY, USA) at deformation temperatures of 25 °C, 100 °C, 200 °C, and 300 °C, and strain rates of 0.001 s⁻¹, 0.01 s⁻¹, 0.1 s⁻¹, 1 s⁻¹, and 10 s⁻¹, respectively. After the compression, the specimens were quenched in water to preserve their microstructure. Due to the friction during compression, we applied a correction formula to adjust the true stress data, based on the system’s recommendations and friction coefficient calculations.

3. Results

3.1. True Stress–Strain Curves and Micrographs

The flow stress curve for the Brass orientation is shown in Figure 3a. At 25 °C, under strain rates of 1 s⁻¹ and 10 s⁻¹, the flow stress levels of Brass are nearly identical, differing by only 1.6 MPa. At a strain rate of 10 s⁻¹, the flow stress reaches a peak of 89.3 MPa. When the strain rate decreases to 0.1 s⁻¹, the flow stress drops significantly by nearly 20.0 MPa, reaching 71.4 MPa. Further decreasing the strain rate to 0.01 s⁻¹ results in a flow stress of 44.1 MPa, and at 0.001 s⁻¹, the flow stress is 42.1 MPa, differing by just 2.0 MPa. At 100 °C, the flow stress curve for Brass exhibits similar behavior to that at 25 °C, with the flow stress at 100 °C/1 s⁻¹ (66.4 MPa) slightly lower than at 100 °C/10 s⁻¹ (77.5 MPa). These results indicate that the Brass orientation exhibits unique strain rate sensitivity at 25 °C and 100 °C, with similar flow stress levels under both high strain rates (1 s⁻¹ to 10 s⁻¹) and low strain rates (0.001 s⁻¹ to 0.01 s⁻¹), but a notable difference at 0.1 s⁻¹.

At 200 °C and 300 °C, flow stress levels of Brass vary significantly with strain rate. At 200 °C/10 s⁻¹, the flow stress is 65.9 MPa, while at 300 °C/10 s⁻¹, it decreases to 47.6 MPa. At 200 °C/0.001 s⁻¹ and 300 °C/0.001 s⁻¹ to 0.01 s⁻¹, the flow stress reaches steady-state values of 19.6 MPa, 18.6 MPa, and 21.5 MPa, respectively. The flow stress curve for the Goss orientation is shown in Figure 3b. At 25 °C and 100 °C, the flow stress of Goss-oriented ASCs is relatively insensitive to strain rate. At 25 °C, the flow stress varies slightly from 53.0 MPa to 63.4 MPa as the strain rate increases from 0.001 s⁻¹ to 10 s⁻¹. Similarly, at 100 °C, the flow stress changes by only 2.0 to 3.0 MPa across the strain rate range. This suggests that Goss-oriented ASCs exhibit minimal strain rate sensitivity at 25 °C and 100 °C. At 200 °C, the flow stress increases linearly with strain rate, ranging from 24.7 MPa at 0.001 s⁻¹ to 49.8 MPa at 10 s⁻¹. At 300 °C/0.001 s⁻¹ to 0.01 s⁻¹, the flow stress curve reaches steady-state values of 20.7 MPa and 14.0 MPa. While the Brass orientation demonstrates more pronounced strain rate sensitivity across a broader temperature range, the Goss orientation remains relatively insensitive to strain rate at lower temperatures, with noticeable strain rate dependence emerging only at higher temperatures.

As shown in Figure 4 and

Table 1. Relationship between dynamic softening types and deformation conditions in Brass orientation.

	$\dot{\mathit{\epsilon }}$/s−1	0.001	0.01	0.1	1	10
T/°C
25		DRV	DRV	DRV	DRV	DRV + DDRX
100		DRV	DRV	DRV	DRV + DDRX	DRV + DDRX
200		DRV	DRV	DRV	DRV + DDRX + CDRX	DRV + DDRX + CDRX
300		DRV + DDRX	DRV + DDRX	DDRX + CDRX	DDRX + CDRX	DDRX + CDRX

, DRX behavior is observed in Brass-oriented ASCs at strain rates of 10 s⁻¹ and 1 s⁻¹ under deformation temperatures of 25 °C, 100 °C, and 200 °C. At 300 °C, DRX occurs across all strain rates from 0.001 s⁻¹ to 10 s⁻¹. Due to its common occurrence as a softening mechanism in aluminum alloys, DRX is categorized into discontinuous dynamic recrystallization (DDRX) and continuous dynamic recrystallization (CDRX). Under other deformation conditions, the deformed matrix of ASCs consists of dislocation configurations formed by DRV through dislocation motion. Typical dislocation patterns include deformation bands (Figure 4b5), dislocation micro-bands parallel to the activated slip plane (Figure 4b3), and intersecting groups of dislocations in a diamond shape (Figure 4b4,c4,c5).

As shown in Figure 5 and

Table 2. Relationship between dynamic softening types and deformation conditions in Goss orientation.

	$\dot{\mathit{\epsilon }}$/s−1	0.001	0.01	0.1	1	10
T/°C
25		DRV	DRV	DRV	DRV	DRV
100		DRV	DRV	DRV	DRV	DRV
200		DRV + DDRX	DRV + DDRX	DRV + DDRX	DRV + DDRX + CDRX	DRV + DDRX
300		DRV	DRV	DRV + DDRX	DRV + DDRX	DRV + DDRX

, Goss-oriented ASCs exhibit DRX under the condition of 200 °C/0.1 s⁻¹. Deformation bands parallel to the activated slip plane are observed in the deformed matrix of Goss-oriented ASCs under the conditions of 100 °C/(0.001 to 1 s⁻¹) and 200 °C/(0.001 to 0.01 s⁻¹). At 200 °C/(0.1 to 10 s⁻¹), elongated, strip-like DRX grains form along the direction parallel to the activated slip plane.

The Optical Microscopy (OM) results indicate significant differences in the microstructural morphology of Brass and Goss oriented ASCs under different conditions, particularly at 25 °C/0.001 s⁻¹, 25 °C/10 s⁻¹, 300 °C/0.001 s⁻¹, and 300 °C/10 s⁻¹. Therefore, specimens deformed under these conditions were selected for EBSD characterization.

Figure 6 shows the Inverse Pole Figure (IPF) maps of Brass and Goss oriented ASCs under the conditions of 25 °C/0.001 s⁻¹ and 300 °C/0.001 s⁻¹. After compression, the deformed matrix of Brass-oriented ASCs transforms to an S orientation, while the matrix of Goss-oriented ASCs remains unchanged. At 25 °C/0.001 s⁻¹, the deformation bands in Brass-oriented ASCs align along the (11-1) slip system, while Goss-oriented ASCs form two sets of deformation bands along the (-111) and (11-1) slip systems, as shown in Figure 6a,c. At 300 °C/0.001 s⁻¹, both Brass and Goss oriented ASCs exhibit two sets of deformation bands along the (11-1) and (-111) slip systems, as shown in Figure 6b,d. At lower temperatures, due to lower activation energy during deformation, cross-slip is difficult to initiate, impeding dislocation movement and leading to a discontinuous distribution of small-angle grain boundaries in both Brass and Goss oriented ASCs. In contrast, at higher temperatures, the higher activation energy promotes dislocation movement, facilitating cross-slip along the (11-1) and (-111) slip systems and forming a continuous distribution of small-angle grain boundaries. Additionally, the two sets of deformation bands interlock, with the dislocation bands gradually subdividing into multiple dislocation cells, each approximately 20–30 μm in size.

Figure 7 and Figure 8 show the Kernel Average Misorientation (KAM) maps and local orientation difference distributions of Brass and Goss oriented ASCs at a strain rate of 0.001 s⁻¹. As seen in Figure 7, under the condition of 25 °C/0.001 s⁻¹, the local orientation difference for both Brass and Goss oriented ASCs predominantly ranges from 1° to 2°, with some areas showing differences between 3° and 4°. These larger orientation differences correspond to the deformation bands. Moreover, for Goss-oriented ASCs, the deformation bands along the (-111) slip system exhibit larger local orientation differences than those along the (11-1) slip system, suggesting that the activation of the (-111) slip system is dominant. After deformation at 300 °C/0.001 s⁻¹, the dislocation density within the deformation bands of Brass and Cube oriented ASCs is very low, with dislocations concentrated at the boundaries, forming geometrically necessary dislocation boundaries. The higher temperature facilitates dislocation motion and cross-slip, leading to dislocation annihilation and rearrangement at small-angle grain boundaries, resulting in a decrease in dislocation density with temperature [19]. The GND (Geometrically Necessary Dislocation) density was calculated from the KAM map using the method described by Ma et al. [20]. Consequently, the ρ^GND (density of geometrically necessary dislocations) value for Brass-oriented ASCs decreases from 0.476 × 10¹⁴ m⁻² to 0.274 × 10¹⁴ m⁻² (Figure 6a,b), while for Goss-oriented ASCs, it decreases from 0.388 × 10¹⁴ m⁻² to 0.226 × 10¹⁴ m⁻² (Figure 8c,d).

Figure 9 shows the IPF maps of Brass and Goss oriented ASCs under the conditions of 25 °C/10 s⁻¹ and 300 °C/10 s⁻¹. As seen in Figure 9a, at 25 °C/10 s⁻¹, full DRX occurred in Brass-oriented ASCs, with high-angle grain boundaries (HAGBs) dominating, and only a few low-angle grain boundaries (LAGBs) present. A large number of elongated DRX grains are parallel to the activated slip system (-111), while a smaller number are parallel to the (11-1) slip system. The pole figure in Figure 9a shows that after deformation at 25 °C/10 s⁻¹, the microstructure of Brass-oriented ASCs consists of four orientations: JT, G1, G2, and G3. The G1 texture is very similar to the matrix (JT) orientation, both being S-orientation variants, while the G2 texture corresponds to R-Cube orientation. In Figure 9b, after deformation at 300 °C/10 s⁻¹, a large number of equiaxed DRX grains were formed in the deformed matrix of Brass-oriented ASCs. The grain sizes vary significantly, with large grains reaching up to 900 μm and small grains around 20 μm. Most of the grain boundaries are LAGBs, with some being HAGBs. The pole figure in Figure 9b shows that at 300 °C/10 s⁻¹, the microstructure of Brass-oriented ASCs consists of three orientations: JT, G4, and G5, with G5 being a variant of the S orientation similar to the matrix (JT). Figure 9c,d show that at both 25 °C/10 s⁻¹ and 300 °C/10 s⁻¹, Goss-oriented ASCs exhibit only slight DRX, with DRX grains aligned along the (11-1) slip system. Additionally, at 300 °C/10 s⁻¹, large deformation bands formed in Goss-oriented ASCs, with numerous deformation microbands distributed along the (111) slip system.

The pole figures in Figure 9c,d indicate that at 25 °C/10 s⁻¹, the microstructure of Goss-oriented ASCs contains four orientations: JT, G6, G7, and G8. The deformed matrix (JT) remains Goss-oriented, G6 corresponds to Cube orientation, G7 is a variant of the S orientation, and G8 is similar to Cube orientation. At 300 °C/10 s⁻¹, the microstructure of Goss-oriented ASCs only shows recrystallization of Cube-oriented grains.

Figure 10 and Figure 11 show the KAM maps and local orientation difference distributions of Brass- and Goss-oriented ASCs at a strain rate of 10 s⁻¹. As seen in Figure 10, at both 25 °C/10 s⁻¹ and 300 °C/10 s⁻¹, Brass-oriented ASCs exhibit substantial DRX, although a small portion of the deformed matrix still contains high-density dislocation regions. At 25 °C/10 s⁻¹, the deformed matrix of Brass-oriented ASCs contains only a few LAGBs (Figure 8a), while at 300 °C/10 s⁻¹, a significant number of LAGBs are present (Figure 10b), indicating a much higher dislocation density at 300 °C/10 s⁻¹ than at 25 °C/10 s⁻¹. The elevated temperature activates dislocation movement, promoting the orientation transformation of the matrix. Consequently, the ρ^GND value of Brass-oriented ASCs increases from 0.117 × 10¹⁴ m⁻² at 25 °C to 0.134 × 10¹⁴ m⁻² at 300 °C (Figure 11a,b). At 25 °C/10 s⁻¹, Cube-oriented ASCs exhibit deformation bands with high dislocation density, as shown in Figure 11c.

At 300 °C/10 s⁻¹, although a small number of elongated DRX grains form in Cube-oriented ASCs, these grains have extremely low dislocation density, while the deformed matrix forms deformation bands with very high dislocation density, as seen in Figure 9d. As a result, the ρ^GND value of Cube-oriented ASCs increases from 0.226 × 10¹⁴ m⁻² to 0.429 × 10¹⁴ m⁻² as the deformation temperature increases (Figure 11c,d).

3.2. Establishment of Arrhenius Constitutive Equation for Strain Compensatideveloping Hot Studying Processes on of Aluminum Single Crystal Strip

Studying the changes in rheological stress of alloys during high-temperature deformation and establishing corresponding rheological stress models is crucial for providing key parameters and forming the basis for understanding the material’s behavior under extreme conditions and optimizing processing techniques.

The rheological stress equation can be expressed as follows [21]:

$$\sigma =f\left(\epsilon ,\dot{\epsilon },T,C,S\right)$$

(1)

In the formula, $\sigma$ is the rheological stress, $\epsilon$ is the strain amount, $\dot{\epsilon }$ is the strain rate, $T$ is the deformation temperature, $C$ is the material composition, and $S$ denotes the microstructure of the material.

The relationship between rheological stress and the coupling effect of deformation temperature and strain rate, as proposed by Zener and Hollomon [22] in 1944 and confirmed through high-speed tensile experiments on steel, can be expressed by a parameter:

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

(2)

This parameter, known as the Zener-Hollomon parameter ($Z$), represents the deformation rate factor for temperature compensation. In 1966, Sellars and Tegart [23] proposed a hyperbolic sine-corrected Arrhenius relationship, incorporating the activation energy ($Q$) of thermal deformation and the deformation temperature ($T$), based on their analysis of thermal processing data for various metal materials.

$$\dot{\epsilon }=f\left(\sigma \right)\mathit{exp}\left(-Q/RT\right)$$

(3)

The Arrhenius relationship for hyperbolic sine correction has three forms:

$$\Sigma =\left\{\begin{array}{c}{A}_1{\sigma }^{n_1}\mathit{exp}\left(-Q/RT\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha \sigma <0.8\hfill \\ A_2\mathit{exp}\left(\beta \sigma \right)\mathit{exp}\left(-Q/RT\right)\hspace{1em}\hspace{1em}\hspace{1em}\alpha \sigma >1.2\hfill \\ A_3{\left[\mathit{sinh}\left(\alpha \sigma \right)\right]}^n\mathit{exp}\left(-Q/RT\right)fullstressrange\hfill \end{array}\right.$$

(4)

In the above formula, $\alpha$, $\beta$, $n_1$, $n$, $A_1$, $A_2$, and $A_3$ are material constants, with $\alpha$ = $\beta$/$n_1$.

Using the flow stress and true strain data obtained from compression tests under varying processing conditions, such as deformation temperature and strain rate, the material constants of the constitutive equation were calculated. Since the plastic deformation behavior of metals typically reaches a steady-state flow, strain’s influence on stress is generally not considered in the constitutive equation, focusing instead on steady-state or peak stress [22,24]. Although strain significantly impacts material constants and follows a polynomial relationship, the true stress–strain curves of ASCs in all orientations in this study do not exhibit peak stress. Therefore, the Arrhenius equation with strain compensation was employed for data fitting. The parameters $A_1$, $A_2$, and $A_3$, $\alpha$, and $\beta$ were fitted under different strain values, and the corresponding fitted data are shown in the following Figure 12.

The material constants $\alpha$, $n$, $Q$, and lnA of the constitutive equation were calculated under different deformation conditions at intervals of 0.05 within the true strain range of 0.05 to 0.5. The calculated values of each material constant were plotted with true strain as the x-axis and then fitted using a five-term formula, as shown in the equation. The fitting results, which show good consistency with the experimental data, are presented in Figure 13. The parameters used to obtain the material constants α, n, Q, and lnA are summarized in

Table 3. Constant Fitting Values of Brass Orientation in Arrhenius Equation.

$\mathit{\alpha }$		$\mathit{n}$		$\mathit{Q}$		$\mathit{I}\mathit{n}\mathit{A}$
B1	−13.09271	C1	−1370.27179	D1	−1786.53483	E1	−795.08152
B2	21.82372	C2	1939.16177	D2	2118.98826	E2	1032.28878
B3	−14.32482	C3	−926.19797	D3	−641.1203	E3	−426.90976
B4	4.70243	C4	147.24373	D4	−29.31507	E4	57.44896
B5	−0.81209	C5	6.22845	D5	22.04151	E5	−2.34777
B6	0.08988	C6	3.60271	D6	46.23156	E6	7.4944

.

$$\left\{\begin{array}{c}\alpha =B_1{\epsilon }^5+B_2{\epsilon }^4+B_3{\epsilon }^3+B_4{\epsilon }^2+B_5\epsilon +B_6\\ n=C_1{\epsilon }^5+C_2{\epsilon }^4+C_3{\epsilon }^3+C_4{\epsilon }^2+C_5\epsilon +C_6\\ Q=D_1{\epsilon }^5+D_2{\epsilon }^4+D_3{\epsilon }^3+D_4{\epsilon }^2+D_5\epsilon +D_6\\ InA=E_1{\epsilon }^5+E_2{\epsilon }^4+E_3{\epsilon }^3+E_4{\epsilon }^2+E_5\epsilon +E_6\end{array}\right.$$

(5)

Similarly, the constants of the Arrhenius equation for the Goss orientation at different strains were calculated and fitted using a five-term equation, with the results presented in Figure 14 and Figure 15. The parameters used to obtain the material constants α, n, Q, and lnA are summarized in

Table 4. Constant fitting values of Goss orientation in Arrhenius equation.

$\mathit{\alpha }$		$\mathit{n}$		$\mathit{Q}$		$\mathit{I}\mathit{n}\mathit{A}$
B1	−16.07102	C1	43,540.41231	D1	87,930.87979	E1	30,287.39282
B2	25.08813	C2	−54,528.1365	D2	−104,011.02941	E2	−36,454.24904
B3	−15.20873	C3	24,566.73601	D3	42,320.00341	E3	15,218.55438
B4	4.61821	C4	−4921.57689	D4	−6890.39042	E4	−2589.37529
B5	−0.7858	C5	466.51424	D5	542.0963	E5	210.62506
B6	0.095	C6	1.57131	D6	55.31925	E6	11.47075

.

Using the formula, the rheological stress of ASCs under various deformation conditions can be predicted, and a plastic deformation constitutive model for ASCs, incorporating strain compensation and represented by the Zener-Hollomon parameter, can be established:

The rheological stress equation can be expressed as follows:

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

(6)

At a true strain of 0.51, the constitutive equations corresponding to the orientation of Brass and Goss ASCs are:

$$In\dot{\epsilon }=7.11-36374.05/RT+6.13ln\left[sinh\right(0.023216\sigma \left)\right]$$

$$In\dot{\epsilon }=39.37-140378.08/RT+18.13ln\left[sinh\right(0.021332\sigma \left)\right]$$

3.3. Verification of the Constitutive Equation for Aluminum Single Crystals

The constitutive equation does not fully account for the changes in microstructure and composition of ASCs during isothermal uniaxial compression, leading to some errors in predicting the rheological stress curve [25,26,27]. To quantitatively assess the accuracy of the established constitutive model, the correlation coefficient (R) and average relative error (AARE) were used to evaluate the prediction performance of the Arrhenius constitutive equation with strain compensation. The calculation formulas for R and AARE are as follows:

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

(7)

$$AARE={\displaystyle \frac{1}{N}}\sum _{i=1}^N\left|{\displaystyle \frac{E_i-P_i}{E_i}}\right|\times 100\%$$

(8)

In the formula, E represents the experimental value, P is the predicted value, and N is the total number of data points. Under deformation conditions with a temperature range of 25 °C to 300 °C and strain rates between 0.001 s⁻¹ and 10 s⁻¹, the correlation coefficient (R) and average relative error (AARE) for the Brass orientation between experimental and predicted rheological stress are 0.9936 and 6.428%, respectively. This indicates that the Arrhenius constitutive model with strain compensation provides a relatively accurate prediction for the Brass orientation. For the Goss orientation, R and AARE are 0.9907 and 7.074%, respectively, as shown in Figure 16b. The graph clearly shows a good correlation between the predicted and experimental values.

3.4. Establishment of Aluminum Single Crystal Processing Diagram

Based on ASCs thermal simulation experiments and Prasad’s hot studying diagram theory, this study investigates the thermal deformation behavior of ASCs with different crystal orientations by analyzing their microstructural evolution during deformation. The stable plastic forming conditions for ASCs with various orientations were determined, providing theoretical guidance for optimizing the hot studying process parameters of high-purity aluminum [28].

• Analysis of the Instability Diagram of Aluminum Single Crystal

Figure 17 presents the contour map of the instability factor (Zeta, ε) for Brass and Goss-oriented ASCs at a true strain of 0.51. The Zeta values along the contour lines represent the risk of flow instability during plastic deformation. As shown, significant differences in Zeta values and their distributions exist between the two orientations. For the Brass orientation, the instability factor Zeta (ε) is less than 0 (instability zone) primarily at strain rates greater than 0. The minimum Zeta value of −24.6 occurs at 1 s⁻¹, and the maximum value is 17.4. Notably, when the deformation temperature is between 100 °C and 150 °C, the strain rate ranges from 1 s⁻¹ to 10 s⁻¹, indicating relative stability.

For the Goss orientation, three unstable regions are identified: (1) strain rates above 0.1 s⁻¹ below 50 °C; (2) strain rates less than 0.1 s⁻¹ between 75 °C and 155 °C; and (3) strain rates greater than 0.1 s⁻¹ between 160 °C and 260 °C. The minimum Zeta value for Goss orientation is −25.2, and the maximum value is 21.8. Comparing the two orientations, the Brass orientation is more prone to instability at higher strain rates, with a larger distribution of unstable regions and an absolute Zeta value of 24.6. Conversely, the Goss orientation may become unstable under various strain rate conditions, with an absolute Zeta value reaching 25.2.

• Analysis of the Power Dissipation and Thermal Processing Diagrams of Aluminum Single Crystal

Figure 18 illustrates that at a strain rate of 0, Brass-oriented ASCs are prone to rheological instability under strain rates of 1 s⁻¹ to 10 s⁻¹. In the unstable state, slight wrinkles appear on the ND/RD surface of Brass-oriented ASCs, as shown in Figure 18b,c. However, within the temperature range of 100 °C to 150 °C and under strain rates of 1 s⁻¹ to 10 s⁻¹, Brass-oriented ASCs can be processed stably, with a power dissipation rate of 10% to 14%, indicating low energy dissipation. At temperatures between 150 °C and 300 °C and strain rates of 0.001 s⁻¹ to 1 s⁻¹, ASCs exhibit stable rheological properties and power dissipation rates ranging from 29.0% to 33.9%, making them most suitable for processing with Brass orientation.

Under deformation temperatures ranging from 25 °C to 300 °C and strain rates between 0.001 s⁻¹ and 10 s⁻¹, Goss-oriented ASCs exhibit three unstable regions, namely type II, III, and IV. In the unstable zones, Goss-oriented ASCs develop folds on their ND/RD surfaces, and the bulging of the ND/TD surfaces is uneven, as shown in Figure 18b,c. In contrast, Goss-oriented ASCs undergoing stable deformation transition from a rectangular prism to a parallelepiped shape, with relatively uniform deformation, as seen in Figure 18d,e.

Figure 19 shows that the peak power dissipation rate of ASCs varies with crystal orientation under different deformation conditions at a true strain of 0.51. The highest power dissipation rates for Brass and Goss orientations are 33.9% and 27.7%, respectively. This suggests that the dynamic recrystallization (DRX) behavior in Brass-oriented ASCs leads to higher energy consumption for microstructural evolution, while Goss-oriented ASCs consume less energy.

4. Conclusions

In this study, a constitutive model for the flow stress behavior of ASCs was developed based on uniaxial compression tests conducted at deformation temperatures ranging from 25 °C to 300 °C and strain rates from 0.001 s⁻¹ to 10 s⁻¹. Hot processing maps for ASCs with different crystal orientations were established, and the flow instability and stability regions during isothermal compression deformation were analyzed. The stable hot processing temperature and strain rate ranges for each orientation were determined. The key findings are as follows:

• After isothermal uniaxial compression, the flow stress curves of ASCs exhibit characteristics similar to DRV behavior. At a true strain of 0.51, strain hardening dominates, causing the true stress to increase with strain. Most flow stress curves do not show a clear peak stress, and the primary softening mechanism is DRV.
• The flow stress of ASCs increases with decreasing deformation temperature and increasing strain rate, indicating positive strain rate sensitivity in the 25 °C to 300 °C range. The order of power dissipation rates for different orientations corresponds with the ease of dynamic recrystallization (DRX): Goss > Brass.
• At a true strain of 0.51, flow instability is likely to occur in ASCs within the medium temperature-low strain rate (100–200 °C, 0.001–0.01 s⁻¹) or high temperature-high strain rate (200–300 °C, 1–10 s⁻¹) ranges, leading to defects such as macroscopic folding bands. At high temperature-low strain rate (200–300 °C, 0.001–0.01 s⁻¹), ASCs experience relatively stable rheological flow, and energy dissipation efficiency is highest. This indicates that high deformation temperatures and low strain rates can be employed to control microstructural evolution during plastic deformation of ASCs.
• For high-temperature applications where stability and controlled microstructural evolution are critical, the Brass orientation may be the more suitable choice, particularly at temperatures above 200 °C. Conversely, for applications where stability at lower strain rates is important, the Goss orientation would be more beneficial.