*3.3. Cavity Growth*

3.3.1. Cavity Growth Controlled by Diffusion and Plastic

The mechanism of cavity growth in the superplastic tensile state is divided into two types: (1) the growth mechanism of stress promoting a spherical cavity growing by diffusion along the grain boundary; and (2) the cavity growth is controlled by the plastic flow of the surrounding material [27,32–35].

At 500 ◦C and 1 × <sup>10</sup>−<sup>3</sup> <sup>s</sup><sup>−</sup>1, the parallel sections of the superplastic tensile specimens were tested by the SEM with an aborted strain of *ε* = 0.65, 1.40 and 2.65; the corresponding results are presented in Figure 4. There are a large number of finely-dispersed second phase particles in the 2 mm thick deformed sheet, and the black points/areas show the cavity distribution during the superplastic tensile deformation. It is clear that the number and density of the cavities increased immediately, accompanied by the constant accumulation of the deformation in Figure 4a,c,e. In addition, the corresponding magnification is shown in Figure 4b,d,f.

**Figure 4.** The scanning electron microscopy results in the superplastic tensile stages and magnified regions at 500 ◦C and 1 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup><sup>−</sup>1: *<sup>ε</sup>* = 0.65 (**a**,**b**), *<sup>ε</sup>* = 1.40 (**c**,**d**) and *<sup>ε</sup>* = 2.65 (**e**,**f**).

(**e**) (**f**)

Figure 4b,d shows the cavity growth with a strain increase from *ε* = 0.65 to 1.40, and the cavities evidently grew and interlinked at *ε* = 2.65 in Figure 4e,f. The initial nucleation stage of the cavity mostly exhibited an O-shape due to the low surface energy of the spherical cavity, which remained relatively stable and easy to nucleate. O-shaped cavities were found at every stage, indicating that new cavity nucleation continued to occur during the process of strain accumulation in the superplastic tensile stage, as shown in Figure 4b,d,f.

At 500 ◦C and 1 × <sup>10</sup>−<sup>3</sup> <sup>s</sup><sup>−</sup>1, it is suggested that the applied stress reached the maximum value at *ε* = 0.65, as shown in Figure 1b. This is because the dislocations continuously accumulated in the second phase particles and the grain boundaries during deformation, as shown in Figure 2, which increased the strength of the 5A70 alloy and provided the driving force for cavity nucleation. Meanwhile, Mg atoms precipitated from the Al-matrix to form second phase particles. The pinning effect of the precipitated phase was encouraged to enhance the cavity nucleation and the chemical potential between the forceful grain boundary atoms and the free surface of the cavity [34,35]. However, the cavity growth rate under diffusion control was obtained as follows:

$$\frac{dr}{d\varepsilon} = \mathfrak{a}\_0 (\frac{2\Omega \delta D\_{\rm gb}}{r^2 kT}) [\frac{\sigma - (2\gamma/r)}{\dot{\varepsilon}}] \,\tag{3}$$

where Ω is the atomic volume, *δ* is the grain boundary width, *δ* = 2*b*. *Dgb* is the coefficient for grain boundary diffusion, *k* is Boltzmann's constant, *T* is the absolute temperature, *σ* is the applied stress, *γ* is the surface energy, *<sup>r</sup>* is the cavity radius, *<sup>α</sup>*<sup>0</sup> is the cavity spacing, . *ε* is the strain rate and *α<sup>0</sup>* is defined as *α*<sup>0</sup> = 1/{4 ln(*a*/2*r*) − [1 − (2*r*/*a*) 2 ][3 − (2*r*/*a*0) 2 ]}. The cavity size parameter was *α<sup>0</sup> =* 0.12 as the initial stage of cavity diffusion complied with *a*/2*r* ≥ 10.

The cavities grew due to plastic deformation in the surrounding crystalline lattice. The plasticcontrolled growth mechanism is given by [33–35]:

$$\frac{dr}{d\varepsilon} = r - \frac{3\gamma}{2\sigma'}\tag{4}$$

where the superplastic tensile of 5A70 aluminum alloy was tested at 400–500 ◦C, and the critical cavity radius, *rc*, denoting the transition from conventional diffusion growth for plastic-controlled growth was obtained from Formulas (3) and (4), so that [27,33]:

$$r\_c = \left(\frac{2\Omega\delta D\_{\rm gb}}{kT}\right)^{\frac{1}{\theta}} \left(\frac{\sigma}{\dot{\varepsilon}}\right)^{\frac{1}{\theta}},\tag{5}$$

Formula (5) demonstrates that when the temperature and strain rate are constant, the lower the applied stress in the superplastic tensile deformation, and the smaller the critical cavity radius, *rc*, exhibited. Figure 4 shows the tension strain *ε* = 0.65, 1.40 and 2.65, presenting the results of the SEM analysis as pointed in the Figure 1b. As noted above, the accumulation of tensile deformation causes a decrease in the critical nucleation radius, moreover, the cavity density increases immediately. This is a complex function of cavity nucleation related to the testing temperature, strain rate, microstructure size and phase-particle shape and size. Generally, larger particles and larger interfacial defects cause a more rapid separation of the Al-matrix and second phase particles with an increase in tensile deformation, while smaller particles have a smaller restricted plastic zone, and greater accumulated strain is needed to produce complete separation for continuous cavity nucleation [36,37]. Taking the data in Table 2 into Formulas (3) and (4), the cavity growth rate, *dr/dε*, can be drawn as shown in Figure 5.

**Figure 5.** Schematic illustration of diffusion growth and plastic-controlled growth, showing the critical radius *rc*.


**Table 2.** Constants used for the calculation of the cavity growth rate (*dr*/*dε*).

**\*** The material constants used for 5A70 alloy: *<sup>δ</sup>* = 5.72 × <sup>10</sup>−<sup>10</sup> m, *<sup>k</sup>* = 1.38 × <sup>10</sup>−<sup>23</sup> <sup>J</sup>·K−1, *Dgb* = 3.3 × <sup>10</sup>−<sup>14</sup> m3·s−1, *<sup>γ</sup>* = 0.64 J·m−<sup>2</sup> and *<sup>R</sup>* = 8.314 J·mol−1·K−1.

Figure 5 shows the relationship between the cavity growth rate, *dr/dε*, and the cavity radius, *r*, indicating the mechanism of cavity growth during the superplastic tensile state of FG 5A70 alloy. The figure illustrates that when the cavity radius *r* < *rc* = 1.52 μm, the cavity growth controlled by diffusion dominates the cavity growth mechanism. By contrast, the cavity growth is controlled by the plastic flow, exhibiting a power-law growth rate that is larger than the other type of growth rate. In addition, during plastic-controlled cavity growth the cavity volume fraction should increase exponentially with strain, and *η* is the growth rate parameter for the cavity volume [38,39].

$$\eta = \frac{3}{2} \frac{m+1}{m} \text{sinh} \frac{2(2-m)}{3(2+m)} \tag{6}$$

The coefficient of strain rate sensitivity, *m*, value continuously decreases along with the accumulation of tensile deformation. However, the cavity growth rate parameter increases as the strain increases simultaneously. Figure 6 shows the true strain range from *ε* = 0.1 to 1.6, while the growth rate parameters increased correspondingly, from *η* = 1.21 to 3.22. Therefore, the diffusion-controlled cavity growth with a small growth rate parameter occurred in the initial stage of superplastic tensile deformation. As the plastic-controlled growth dominated the cavity growth, the cavity growth rate parameter and the cavity growth rate increased simultaneously. Meanwhile, it was clearly found that plastic-controlled growth of the cavities occurred mainly due to cavity interlinkage and coalescence.

### 3.3.2. Effect of Superplastic Diffusion on Cavity Growth

Along with superplastic tensile deformation, the Mg atoms continuously separated from the distorted Al-matrix lattice into the generated vacancies. When the cavity size of the diffusion mechanism was larger than the grain size, the newly formed vacancies diffused into the neighboring cavity along the multiple grain boundaries under stress. Meanwhile, superplastic diffusion growth of the cavity nucleation region appeared, and then the cavities begin to interlink during the growth process. Superplastic diffusion growth was the dominant mechanism during this stage, when the voids vacated into the adjacent cavities, subject to diffusion-controlled growth.

Chokshi et al. [35] noted that the growth of cavities due to superplastic diffusion occurred while the cavity was large enough to intersect with multiple grain boundaries. This required a cavity size larger than half of the grain size, *d/2*, therefore, the grain size of cavity growth under superplastic diffusion was considered between *d/2* and *d*.

Figure 6 shows comprehensively the cavity growth parameter, *η*, cavity area fraction, *Sc*, and the grain diameter radius, *d*, with the cavity equivalent radius, *r*. Meanwhile, the corresponding *Sc* was presumed to be 0.24, 0.77 and 3.96%, as shown in Figure 4a,c,e.

**Figure 6.** The cavity growth parameter, *η*, cavity area fraction, *Sc*, cavity equivalent radius, *r*, and dynamic recrystallization grain size, *d*, at different superplastic tensile stages at 500 ◦C and <sup>1</sup> <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup><sup>−</sup>1.

Region A in Figure 6 shows an increase in the applied stress due to increased dislocation and the continuous nucleating of the cavity (Figure 4) at the tensile strain stage ε = 0–0.65, corresponding to cavity equivalent radius *r* = 9.76 μm and grain size *d* = 12.83 μm. The dominant mechanism of cavity growth transformed from diffusion growth to superplastic diffusion growth, judging by the inequality *rc* ≤ *d/2* ≤ *r* ≤ *d*. In region B, with a tensile strain of *ε* = 0.65–1.40, the applied stress continued to decline due to the cavity nucleation and cavity growth. Superplastic diffusion growth took effect because the cavity radius complied with *d*/2 ≤ r ≈ *d*, while the cavity equivalent radius was *r* = 13.90 μm and the grain size was *d* = 13.96 μm at tensile strain *ε* = 1.40. Figure 6 shows the cavity growth parameter, *η*, corresponding to the coefficients of strain rate sensitivity, *m*, which are distributed in the region A and region B. Therefore, superplastic diffusion growth dominated the cavity growth of tensile strain *ε* = 0.46–1.40. Ultimately, the plastic-controlled growth dominated the cavity growth, interlinkage and coalescence, when the superplastic tensile strain was *ε* > 1.40, as shown in region C and D.

Figure 6 demonstrates that dynamic recrystallization occurred during superplastic tensile deformation. Due to the pinning effect of a large number of dispersed Mg-rich phase particles, the grain size was 8.6 μm after rolling and 17.67 μm after superplastic fracture in the ND plane. The cavity size increased continuously related to the superplastic diffusion growth. Previous analysis in Figure 4b,d indicated that the cavity nucleation and growth was not interlinked extensively. However, the cavities interlinked and coalesced rapidly due to the cavity equivalent radius, *r*, being larger than the grain size, *d*, which coincided with the previous analysis in Figure 4f.
