*3.3. Processing Maps*

The constitutive equation may be helpful in the interpretation of strain–stress curves. However, the processing map could be a straightforward method to describe the workability of the studied alloys. Although it is an explicit representation of the response of studied alloys, it has been widely applied in many process parameter selections.

According to Prasad and Srivatsana [27,28], the input energy causing deformation at a given temperature could be dissipated by heat or the so-called "conduction entropy" and microstructural changes induced by dislocation movement.

$$P = G + J = \int\_0^{\dot{\varepsilon}} \sigma d\dot{\varepsilon} + \int\_0^{\sigma} \dot{\varepsilon} d\sigma \tag{11}$$

where the first integral (G content) is dissipated energy as temperature arises, while the second integral (J co-content) is energy dissipated due to microstructural changes.

$$m = \frac{\partial f}{\partial G} = \frac{\dot{\varepsilon}\partial\sigma}{\sigma\partial\dot{\varepsilon}} = \frac{\partial\ln\sigma}{\partial\ln\dot{\varepsilon}}\tag{12}$$

The strain rate sensitivity (*m*) is given by Equation (9). This strain rate sensitivity value defines the relationship between ln . *ε* and ln *σ*. According to Prasad [4,5,8], the *m* value is generally between 0 and 1 for aluminium alloys. Equation (8) states that:

$$
\Delta \text{J/} \Delta P = \frac{m}{m+1} \tag{13}
$$

The efficiency of power dissipation (*η*) is, therefore, defined by:

$$\frac{\Delta f/\Delta P}{(\Delta f/\Delta P)\_{linear}} = \frac{2m}{m+1} = \eta \tag{14}$$

The denominator in Equation (11) indicates that the G-content is equal to J co-content in the ideal dissipation system. The efficiency of power dissipation (*η*) describes how close the current system is compared to the ideal dissipation system, since the J-content is more related to the microstructural changes. Therefore, the efficiency of power dissipation (*η*) essentially describes the microscopic deformation mechanism of the materials within the range of applied temperatures and strains. The efficiency of power dissipation (*η*) changes with temperature and strain rate to form a power dissipation map, representing the microstructure change in the studied materials. Since various failures (such as void formation and cracking propagation) or metallurgical changes (such as dynamic recovery, dynamic recrystallization, etc.) in the plastic deformation process dissipate input energy, with the help of microstructural characterization, the power dissipation diagram can be used to analyze different deformation mechanisms under other deformation conditions. It is necessary, first, to determine the processing instability zone of the studied alloys. According to Prasad [5,8,29], the instability criteria are given by:

$$\xi(\dot{\varepsilon}) = \frac{\partial \ln \left(\frac{m}{m+1}\right)}{\partial \ln \dot{\varepsilon}} + m < 0 \tag{15}$$

Figure 5 shows the constructed processing maps for the studied alloy with different Zn/Mg ratios at different strain conditions. The processing map is an instability diagram overlapped with an energy dissipation diagram at the current strain condition. As shown in Figure 5, the yellow shaded region represents the instability regions. When comparing with other publications, we have presented a series of processing maps that show a systematic change with different conditions. In general, the instability regions for studied alloys are within the lower-temperature and higher-strain-rate conditions. This is consistent with the idea that alloy AA7003 is easier to be deformed when compared to other high-strength 7xxx series alloys [5,9,26]. It is interesting that the studied alloys represent different workability with different Zn/Mg ratios, i.e., different rows in Figure 5. The alloys with a lower Zn/Mg ratio exhibit more significant instability regions. Therefore, it is concluded that alloy AA7003 with a higher Zn/Mg ratio could have better formability than lower Zn/Mg ratios. When comparing different strain conditions, i.e., different columns, it is clear that the higher-strain condition exhibits more significant instability regions. This also agrees with our shared knowledge that higher-strain conditions could lead to void formation and cracking propagation. The result also agrees with Figure 4 that the calculated activation energy value for the alloy with a Zn/Mg ratio of 10.8 is significantly lower than the other two alloys.

*Metals* **2022**, *12*, 1452

**Figure 5.** The processing maps for alloys with (**a**,**b**) Zn/Mg = 6.3, (**c**,**d**) Zn/Mg = 8.3, (**e**,**f**) Zn/Mg = 10.8, at strain of 0.6 (left column) and 0.8 (right column), respectively, while the shaded region represents the instability regions.

The contour lines in Figure 5 represent the efficiency of power dissipation (*η*) at the current strain condition. As discussed earlier, the maximum efficiency is 100%, while G-content is equal to J co-content in the ideal dissipation system. In the current dissipation system, the maximum efficiency (*η*) is about 50%, and the high efficiency of power dissipation (*η*) is always shown in the lower-right corner of each diagram. This indicates that the input energy is more likely to be dissipated at higher temperatures and low-strain-rate conditions without cracking or instability. Therefore, it is concluded that optimized deformation parameters for ternary alloy AA7003 are within a temperature range of 653–813 K and with strain rates lower than 0.3 S<sup>−</sup>1.

## **4. Discussions**

As discussed earlier, the J co-content is energy dissipated due to microstructural changes. The alloy with a Zn/Mg ratio of 8.3 was chosen for typical microstructure characterization. The electron backscatter diffraction (EBSD) technique was selected to analyze deformed microstructures, recrystallized microstructures, and substructures quantitively. The specific microstructural characterization is shown in Figure 6. When the grain's misorientation angle (θc) exceeds 15◦, it is classified as a deformed microstructure. Grains consisting of subgrains whose internal misorientation is below 15◦, but whose misorientation from subgrain to subgrain is above 2◦, are classified as substructures. All the remaining grains are classified as recrystallized. In Figure 6, the colored maps present different microstructures, i.e., blue stands for recrystallized structures, yellow stands for substructures, and red stands for deformed structures. It is shown in Figure 6a that the primary remaining microstructures are deformed structures that coexist with a small number of substructures and few recrystallized structures. When deformed at a higher temperature, i.e., 733 K in Figure 6b, it is shown that more substructures and more recrystallized structures were found. It is shown in Figure 5b that the primary microstructures are substructures. Additionally, the recrystallized structures are significantly increased. Figure 6c shows that when deformed at 813 K, the recrystallized structures become the dominated microstructures.

The statistical analysis of EBSD mappings at a different temperature and at a strain rate of 0.1 s−<sup>1</sup> is shown in Figure 7. It is demonstrated in Figure 7a that the area fraction of recrystallized grains significantly increased with deformation temperatures, while the frequency of deformed microstructures declined dramatically. This indicates that recrystallization occurs rapidly for the alloy with Zn/Mg = 6.3. The frequency of recrystallized structures showed a moderate decrease for the alloy with Zn/Mg = 8.3. However, it is shown in Figure 7b that the substructures arose significantly at different temperature conditions. For the alloy that contained the highest Zn/Mg ratio (as shown in Figure 7c), the substructures exhibited a remarkable frequency at a temperature of 733 K and 813 K. It is also shown that the amount of recrystallized structures increased slightly with temperature. It also should be noticed that the deformed microstructures decreased with rising temperatures within all three alloys. In general, the EBSD results show that the alloy with Zn/Mg = 10.8 exhibited a significantly small amount of recrystallization, while the other two alloys exhibited a moderate amount of recrystallization. This could be an indication that the dynamic recrystallization is the main factor to dissipate the deformation energy. It is also shown that the alloy with Zn/Mg = 10.8 retained a large number of substructures. This could also be an indication that this alloy can be further deformed without cracking.

**Figure 6.** The typical EBSD mappings for the alloy with Zn/Mg = 6.3 deformed at (**a**) 653 K, (**b**) 733 K and (**c**) 813 K at a strain rate of 0.1 S<sup>−</sup>1, with the different colors representing different microstructures, i.e., yellow stand for substructures, blue stand for recrystallized structures and red stand for deformed structures; the bottom right image shows the compressed samples before and after deformation.

Many reports focus on the effects of dispersoid particles [30,31]. It is proven that these particles effectively inhibit the dislocation movement and grain boundary movements [32–35]. In the current research work, ~0.2% Zr was added to the studied alloys. The typical particles are known to be Al3Zr dispersoids. It is shown in Figure 8a that there are a significant number of spherical Al3Zr particles; they have an L12 crystalline structure (as shown by the selected area diffraction pattern (SADP)) and are confirmed to be coherent with the Al matrix. The SADP also indicates that these particles exhibit a simple cubic/cubic orientation relationship with the Al matrix, where the diffraction pattern from Al3Zr particles is located at {100} planes.

It is shown in Figure 8a that the size of Al3Zr dispersoids is within the range of 20–50 nm. In fact, due to the same homogenization treatment and Zr addition, all three studied alloys have a similar size distribution of Al3Zr dispersoids. However, it should be noted that these dispersoids are heterogeneously distributed within grains. Some regions (as shown in Figure 8b) have very-low-number density while others have relatively highnumber density (as shown in Figure 8a). Interestingly, these particles constantly interact with dislocations or grain boundaries. It is shown in Figure 8b that the dislocations bypass a single spherical Al3Zr dispersoid by bowing around. According to classical deformation theory, an Orowan loop is left afterward. The classical deformation theory considers the shear stress or line tension caused by dispersoids themselves. However, it should be noted that the studied alloys exhibit different thermal–mechanical behaviors, given they contain a similar amount of Zr addition. Therefore, it is concluded that the main alloying content, such as Zn/Mg ratios, could also affect the dynamic recrystallization process. In the current study, about the reaction with recrystallizations, three scenarios can be discerned: the low-Zn/Mg-ratio alloy being deformed with difficulty can cause recrystallization quickly, the medium-Zn/Mg-ratio alloy has a moderate trend of recrystallization, and the high-Zn/Mgratio alloy being deformed easily can lead to very low recrystallization. The underlying mechanism is not yet fully understood. More detailed work will be conducted in the future.

**Figure 7.** Statistical analysis of different area fractions of different microstructures at various deformation temperatures for (**a**) alloy with Zn/Mg = 6.3, (**b**) alloy with Zn/Mg = 8.3, and (**c**) alloy with Zn/Mg = 10.8.

**Figure 8.** TEM observation of the alloy with Zn/Mg = 6.3 deformed at 733 K at a strain rate of 0.1 S<sup>−</sup>1, (**a**) Distribution of Al3Zr dispersoids within grains (an alloy with Zn/Mg = 6.3), viewing from {100}Al, (**b**) an Al3Zr dispersoid tangled with dislocation movement during deformation.
