*4.1. Influence of Plastic Flow during Superplastic Deformation*

It is known that strain hardening and/or the strain-hardening rate can provide the plastic stability during superplastic tensile, defined by:

$$
\sigma = \mathbb{K}\_1 \dot{\varepsilon}^m \varepsilon^{n\_1},
\tag{4}
$$

where, *K*<sup>1</sup> is a constant, and *n*<sup>1</sup> is the coefficient of strain hardening. A logarithmic analysis of Equation (4) showed that *n*<sup>1</sup> *= dlnσ/dlnε* was calculated by taking the true stress and strain rates of the true strain, *ε* = 0.3–0.7, in the superplastic ductility, and the resulting *n*<sup>1</sup> values are shown in Table 1.


**Table 1.** Coefficients of strain hardening at different temperatures and strain rates.

There is no strain hardening in traditional superplastic materials [51]. In contrast, the 5A70 alloy showed extensive initial strain hardening, similar to other aluminum alloys with fine-grained structures [52]. In particular, the strain hardening at high temperatures and low strain rates is shown in Figure 4b, which significantly contributed to the elongation-to-failure of the superplastic tensile because the coefficient of strain rate sensitivity *m* was greater than 0.33. The increased dislocation density in the crystal lattice and the distortion of the grain structure were the main reasons for the strain hardening [53]. These *m* values were not sufficient to stop unstable plastic flow because of the pure strain rate hardening. However, the coefficient of strain hardening was relatively high (Table 1), and the strain hardening attributed to grain coarsening provided unstable plastic flow. Since the true strain rate declined with the accumulated strain, the *m* values provided a uniform deformation that

was visible within the gauge length up to the fracture during the strain-softening stage (Figure 3). The initial strain hardening in the superplastic flow stage provided stability of the plastic flow at the initial stage, with high *m* values.

The temperature-controlled shear modulus *<sup>G</sup>* (MPa) of pure aluminum, *<sup>G</sup>* = (3.022 × 104) − <sup>16</sup> *<sup>T</sup>*, was used [54]. Equation (1) with *p* = 2 and *n* = 2 is typically used to describe the superplastic flow of aluminum alloys with grains in the range of 1–10 μm [55]:

$$
\sigma = \sqrt{\frac{GkT}{AD\_{\mathcal{S}^b}}} \frac{d}{b^{3/2}} \dot{\varepsilon}^{1/2} \,, \tag{5}
$$

A plot of *<sup>σ</sup>* against . *ε* 1/2 using double linear scales was adopted to determine the threshold stress. Using the superplastic data gave the best linear fit for the assumed stress exponents for all the investigated temperatures. Therefore, all the values for the threshold stress were estimated by extrapolating the data to zero strain rates using rectilinear regression, as illustrated in Figure 9. The calculated threshold stresses were highly dependent on the deformation temperatures, as summarized in Table 2. The results demonstrated that the threshold stress was associated with a high density of dispersed particles, which impeded the movement of dislocations and grain boundaries during superplastic deformation. For the studied fine-grained 5A70 alloy, the *m* values tend to increase with increasing strain rate over temperatures ranging from 400 to 550 ◦C. Similarly, the decreased *m* value of the strain accumulation and the reduced superplastic flow stress led to the superplastic fracture.

**Figure 9.** Variation of the flow stress as a function of . *ε* 1/2 for the fine-grained 5A70 alloy subjected to RHT.

**Table 2.** Threshold stress of RHT 5A70 alloy at the studied temperatures.


The GBS was closely related to the microstructure recrystallization. In this case, the fine-grained structure and the aberration transformation of the lattice distortion produced by the dislocation slipping/climbing allowed the strain hardening rate to be eliminated during superplastic deformation. In addition, the full recrystallization of the deformed structure evidently does not attenuate the strain hardening. In contrast to the adopted true strain, the *m* values ranged from 0.56 to 0.38, as illustrated in Figure 5b. Nevertheless, the strain hardening occurred in the last stage of the formation, as shown in Figure 4b, and the strain hardening coefficients were 0.61 and 0.55, as shown in Table 1. Moreover, the strain hardening intensified during the superplastic deformation and corresponded to the coefficients

of strain hardening of *n*<sup>1</sup> = 0.79, 0.89 and 0.75. Meanwhile, it was not sufficient to compensate for extensive strain hardening at *m* ≤ 0.33, and the true stress under superplastic tensile began to show a general increase. However, the unstable superplastic flow occurred up to *n*<sup>1</sup> ≤ 0.50 without strain hardening, as shown in Figure 4a. The superplastic fracture for the fine-grained 5A70 alloy was accompanied by a sharp decline in the deformation and unsteady flow stresses for the strain hardening and strain rate sensitivity at high temperatures.

The influence of temperature and grain size on the diffusion coefficient during the superplastic deformation process in the typical superplastic flow theory of fine-grained alloys was proven through experiments. The effect of the diffusion process was related to the effective diffusion coefficient *Deff*, which included the lattice diffusion coefficient, *DL*, and the grain boundary diffusion coefficient, *Dgb*, whose effective diffusion coefficient is defined as follows [56]:

$$D\_{eff} = D\_L + \varkappa \frac{\pi w}{d} D\_{\S^{\flat \mu}} \tag{6}$$

where *<sup>x</sup>* is a constant equal to 1.7 × <sup>10</sup>−<sup>2</sup> and *<sup>w</sup>* is the grain boundary width (*w =* <sup>2</sup>*<sup>b</sup>* and *<sup>b</sup>* = 2.863 × <sup>10</sup>−<sup>10</sup> m) [30]. Equation (6) indicates that the two diffusion paths were independent, and both *Dgb* and *DL* contribute simultaneously to the superplastic deformation. As is well-known, the *DL* of pure aluminum is *DL* (m2/s) = 1.86 × <sup>10</sup>−<sup>4</sup> exp(−143400/RT), and the *Dgb* of pure aluminum is *Dgb* (m2/s) = 10−<sup>4</sup> exp(−84000/RT) [41]. In this study, at 400 and 550 ◦C, using the Equation: *ϕ* = *x*(*πw*/*d*)(*Dgb*/*DL*), *ϕ* = 0.022–0.085 < 1 was obtained. Therefore, the dominant diffusion mechanism at the temperature range from 400 to 550 ◦C was lattice diffusion.

## *4.2. Effect of Temperature on the Grain Growth and Superplastic Behavior*

Equation (1) shows that with a constant strain rate, the superplastic elongation temperature is inversely proportional to the *n*-th power of the true strain. That is, the higher the temperature, the longer the true strain of the superplastic *δ* value. Meanwhile, the true stress evidently declines. Figure <sup>10</sup> shows the EBSD analyses of the 5A70 alloy deformed at different temperatures for . *<sup>ε</sup>*= 1 × <sup>10</sup>−<sup>3</sup> <sup>s</sup>−<sup>1</sup> after the superplastic fracture. The color of each grain was coded by its crystal orientation based on the [001] inverse pole figure in Figure 10a. The aggregation of a large number of ultrafine grains was near the small cavities in the tensile specimen. In addition, the cavity interlinkage and coalescence were precisely identified via the supporting microscopy evidence. Nevertheless, new ultrafine grains occurred in the limited region and were generated near the initial deformed grains, indicating that dynamic recrystallization occurred. Compared with the microstructures of the sample before deformation (Figure 2b), it can be inferred from Figure 10a–d that the grains were gradually elongated in the tensile direction as the temperature increased.

Figure 10a–d show that the microstructure mainly consists of grain sizes that were larger than 10 μm. Dynamic recrystallization occurred during superplastic deformation, which generated the recrystallized grains. At 400 ◦C and 1 × 103 <sup>s</sup><sup>−</sup>1, dynamic recrystallization occurred without obvious grain growth, giving a grain size of 9.60 μm. The grain boundary character distribution data and average grain sizes of the samples in Figure 10 are listed in Table 3. Compared with Figure 2b, it can be observed that the recrystallized grains gradually coarsen when increasing the deformation temperatures or increasing the deformation degree. However, the final grain structure had an average recrystallized grain size of less than 15 μm for 400–500 ◦C in this work, revealing that the 5A70 alloy had a strong ability to inhibit the grain growth during superplastic deformation. The proportion of the LAGBs fraction at 400, 450, and 500 ◦C gradually increased from 11.4 to 16.4 and 22.8%, but decreased to 18.5% at 550 ◦C. In contrast, the corresponding grain boundary angle decreased initially and then increased, which was due to the distorted grain structure at 550 ◦C. These fractions indicated that most boundaries were at high-angles, which can indirectly prove the occurrence of grain boundary or interphase boundary sliding. Nevertheless, the abnormal grain growth led to an increased LAGBs and grain misorientation angle, as shown in Figure 10d,h. The grain boundary sliding along the [110] crystal direction and the orientation grain boundary angle account for more than 50% during the superplastic deformation. Sakai et al. [57] found that the accompanying grain refinement and grain boundary rotation from the large flow softening took place due to the operation of GBS at low strain rates. However, it is known that the glide plane of the face-centered cubic structure in the 5A70 alloy is {111}, and the grain boundary slip direction is <110> [58]. This demonstrates that the sliding mainly took place in the slip direction to balance the grain-to-grain deformation and the reaction stress during superplastic tensile. It is noted that there was positive evidence for the grain rotation and grain boundary sliding for the 5A70 alloy during superplastic deformation.

**Figure 10.** *Cont.*

**Figure 10.** Influence of the temperature on the grain growth in the deformation process at 400 ◦C with . *<sup>ε</sup>* = 1 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup>−1, where the grain and orientation angle are shown in (**a**,**e**), as well as 450 ◦C in (**b**,**f**), 500 ◦C in (**c**,**g**), and 550 ◦C in (**d**,**h**).

The grain growth of the fine-grained structure was based on the initial grain size during the fully dynamic recrystallization, which includes the static growth of the grain over a certain period of time, and the grain size increased under superplastic tensile. Therefore, the total grain growth during superplasticity is defined as *dD* = (*∂Dt*/*∂t*)*dt* + (*∂Dε*/*∂ε*)*dε*. Sato et al. [59] proposed the deformation induced grain growth (DIGG) model as: ln(*D*/*D*0) = *αε*. Based on the grain growth model introduced by Cao [60], the superplastic deformation grain growth model at a constant rate is shown as follows [61]:

$$d = \left[ d\_0^{\;q} \exp(aq\varepsilon) + \left( K/(aq\dot{\varepsilon}\_0) \right) \exp\varepsilon (\exp(aq\varepsilon) - 1) \right]^{1/q},\tag{7}$$

where, *d* is the grain size at the tensile time *t*, *d0* is the initial grain size, *α* is the grain growth exponent, *<sup>q</sup>* is the growth exponent, *<sup>ε</sup>* is the true strain, *<sup>K</sup>* is the grain growth rate constant, and . *ε*<sup>0</sup> is the initial strain rate. Then, the relation: *φ* = *wDgb*/(*D*0*DL*) is used, where *w* is the grain boundary width. Malopheyev et al. [62] obtained the value for the diffusion coefficient: at 400–550 ◦C, the calculated available *φ* = 0.02–0.13 < 1. Therefore, the lattice diffusion growth dominated the grain growth mechanism, where *q* =3[63]. The growth rate factor, *K,* was obtained with the temperature changes and strain rates shown in Table 3.


**Table 3.** Analysis of the grain growth for the superplastic fracture surfaces.

For temperatures of 400–500 ◦C, growth of the lattice diffusion promoted the grain structure distortion energy for the grain growth during superplastic deformation. Namely, grain growth along the lattice diffusion of the polymerization growth factors was *α* = 0.18–0.34, and the grain growth rate was *<sup>K</sup>* = 2.32 × <sup>10</sup><sup>−</sup>21–3.42 × <sup>10</sup>−<sup>21</sup> <sup>m</sup>3/s. This is because the dynamic recrystallization of the deformed structure is a function of the pinning effect of the precipitated particles. Moreover, the grain growth occurred towards the tensile direction as the tensile deformation was accumulated. At 550 ◦C, the lattice diffusion of the polymerization growth factors was *α* = 0.61 and 0.66. The abnormal grain growth was due to the weakening of the pinning effect from the dissolution of the precipitated particles, which were unable to effectively inhibit the grain growth during dynamic recrystallization [64]. In addition, the deformed distortion structure had a pronounced orientation along the tensile direction. Therefore, the grain growth exponent, *α*, indicates the influence of the dynamic grain growth for the 5A70 alloy superplasticity. Furthermore, the superplastic tensile of the 5A70 alloy exhibited a strong temperature dependence. This clearly clarified that the strain hardening was due to the distortion of the grain structure during dynamic recrystallization.

The lattice diffusion dominated the mechanism of the GBS-induced grain to rotate the slip surface under an applied stress and balance the stress tensor to maintain the superplastic flow. At *T* = 500 ◦C, the decreased precipitated particles played a significant role in promoting the recrystallized grain growth, which resulted in abnormal grain growth.
