The Constitutive Equation-Based Recrystallization Mechanism of Ti-6Al-4V Alloy during Superplastic Forming

Abstract

The present paper is concerned with the dynamic recrystallization of the Ti-6Al-4V alloy. Electron Backscatter Diffraction (EBSD) observations are performed after high-temperatures tensile tests, with the temperature ranging from 700 to ~950 °C, and the strain rates varying between 10⁻⁴ and 10⁻²/s. Based on the analysis of flow behavior, the dominant mechanism is identified, and a mechanism map is proposed. In particular, the conditions of 890 °C and strain rates ranging from 10⁻³ to ~10⁻²/s serve as the delineating boundary of dynamic recovery (DRV) and dynamic recrystallization (DRX). For superplastic deformation, the dominant softening mechanism is DRV. Consequently, the occurrence of continuous dynamic recrystallization (CDRX) can naturally be ascribed to the process of grain refinement. Then, a multi-scales physical-based constitutive model of CDRX is developed, demonstrating a good agreement is obtained between the experimental and calculated grain sizes, so the above model could be used to describe the grain growth for superplastic deformation. In conclusion, DRV and DRX in the superplastic forming of Ti-6Al-4V are studied in this study, the condition boundaries of their occurrence are distinguished, and a constitutive equation-based CDRX recrystallization mechanism is given, which might be employed in the fracture mechanism research.

1. Introduction

Owing to its exceptional combination of high specific strength, low density, and excellent corrosion resistance, among other advantages, the Ti-6Al-4V alloy finds extensive applications in aviation, medical, automotive, and various other industries [1,2,3]. Over the 80 years since its discovery, superplasticity refers to the performance of a material that can achieve large elongation without fracture in a specific temperature and strain rate range [4,5]. Superplastic forming (SPF) enables the processing of complex structures [6,7]. The current research focus lies on the shape and property control methods of SPF. The essence of this challenge is to enhance the formability of parts and understand the microstructure evolution during the forming process [8,9]. Hence, a well-defined superplastic deformation mechanism and its application in the SPF process are crucial to promoting the development of this technology.

Polycrystalline metals with finer grains are highly likely to have better superplasticity. However, the mechanism of balance of stable fine grain and grain boundary strain incompatibility has puzzled researchers for several decades [10,11,12]. The TA15 titanium alloy with an initial size of 3.2 μm was subjected to heat treatment by Zhao et al. [13], resulting in samples with grain sizes of 3.3, 5.7, 7.2, and 9.7 μm, respectively. Subsequent high-temperature tensile tests revealed that the elongation of the TA15 sample increased as the grain size decreased within the temperature range of 500~800 °C. Shaysultanov et al. [14] studied the mechanism of grain size evolution during superplastic deformation of the AlCrCuNiFeCo high-entropy alloy. The fracture strain at 900~1000 °C is much larger than at 800 °C, which can be attributed to the fact that finer grains can be obtained at higher temperatures. However, this conclusion does not take into account the possible increase in grain size at high temperatures. Peng et al. [15] systematically characterized the substructure of eutectic Au-Sn alloy, which consists of ζ-Au₅Sn and δ-AuSn phases, and achieved a tensile strain of 2430% at 473 K. The study demonstrated that node-like decomposition is the fundamental cause of fine-grain stabilization and superplasticity, ultimately leading to excellent superplasticity. This research advances the application of phase separation in engineering processes for superplastic forming technology. The evolution of grain size also affects the mechanical properties of materials, as demonstrated by the performance of structural parts [16]. In engineering production, a common method to obtain fine grains in materials is through refining grains via cooling deformation, hot deformation, and heat treatment. Another method involves regulating the microstructure of materials using in situ heating technology to enhance their mechanical properties [17,18]. Otto et al. [19] investigated the dependence of yield stress on deformation temperature with different grain sizes. It can be seen that the yield stress has strong temperature dependence; the grain size dependence of yield stress is also significant. At a temperature below 0.5 T_m, the yield strength of the material is significantly dependent on grain size. The influence of dislocation evolution on the yield strength of the material is greater at higher temperatures. Thus, the flow behavior and microstructure are more determined by microscopic variables such as dislocation density [20], etc.

The dominant mechanism of superplastic deformation is grain boundary sliding, and grain structure evolution is very important for material deformation behavior. Dynamic recrystallization is an important grain growth mechanism, and studying its evolution process is conducive to clarifying the coordination mechanism of grain boundary slip stress concentration, such as the evolution of dislocation density during dynamic recrystallization. During high-temperature superplastic deformation, dynamic recovery (DRV) and dynamic recrystallization (DRX) occur simultaneously. Alabort et al. [21], based on the strain rate sensitivity factor, presented a mechanism map showing how changes in microstructure influence flow stress. When the strain rate is 10⁻⁴/s at 900 °C, the grain size still decreases in the DRV region, allowing for simultaneous DRX processes. This fact is valid due to the continuous dynamic recrystallization (CDRX) mechanism [22]. At other deformation temperatures and strain rates, discontinuous dynamic recrystallization (DDRX) is still dominant. At present, it is still controversial whether the DRX process is CDRX or DDRX. Matsumoto et al. [23,24] proposed that the grain growth of titanium alloy is computationally affected by such factors as grain size, phase composition, and grain growth process. Therefore, superplastic deformation grain growth should be a complicated evolution process, so it needs deeper research.

Based on these gaps, this paper investigates the dynamic recrystallization of the Ti-6Al-4V alloy during superplastic deformation. Firstly, the high-temperature tensile tests and microstructure observation are shown. Then, a grain growth model for DDRX is employed to distinguish the dynamic recrystallization model. Finally, a CDRX model is established to describe the grain refinement in superplastic deformation, followed by a discussion of the calculated results.

2. Experiment Methods

2.1. Materials

This research employed a 1 mm thick rolled Ti-6Al-4V alloy titanium sheet, and its chemical composition (wt%) is detailed in

Table 1. The chemical composition for Ti-6Al-4V alloy, wt%.

Chemical Composition	Ti	Al	V	Fe	H	O
wt%	Bal.	5.59	4.45	0.11	0.008	0.11

. The received material exhibited a mean grain size of 3.77 μm, with the refined grain contributing to enhanced superplastic and diffusion bonding (DB) properties.

2.2. High-Temperature Mechanical Testing

In this paper, high-temperature tensile tests were conducted using the Shimazu Electronic Universal Testing Machine (AG-Xplns100kN). The equipment is capable of performing tensile tests within a temperature range of room temperature to 1100 °C, with a maximum tensile stress capacity of 100 kN and a variable tensile rate between 0.001 and 800 mm/min. The test strain rates are controlled by a constant beam rate, ensuring that the initial strain rates at the beginning of the tensile test meet the specified requirements. During the tests, the equipment can collect the beam load-displacement data with a minimum sampling interval of 1 ms and a position detection resolution of 0.033 mm. It is noteworthy that, as no effective anti-oxidation treatment measures were employed in this experiment, the formation of an α-phase high-temperature oxide layer may contribute to a reduction in the failure strains of the samples [25].

2.3. Microstructure Observation by Electron Backscatter Diffraction

EBSD tests were performed on a SU3500 Tungsten Scanning Electron Microscope equipped with an EBSD probe. Prior to EBSD tests, the samples need polishing application for EBSD analysis. The obtained results were analyzed using Channel 5 software. Notably, the values of grain size can be used to provide the original parameters for quantitative analysis and characterize the multi-scale physical grain growth model. During the analysis, a misorientation threshold of 15° was set for the grain boundary, allowing for the classification of grain boundaries into high-angle grain boundary (HAGB, >15°) and low-angle grain boundary (LAGB, 2~15°).

3. Results

Flow Behavior for the Ti-6Al-4V Alloy

The flow stress curves obtained from the tests are shown in Figure 1. For the stress–strain curves at different temperatures, it becomes evident that the strain rates have a significant effect on the flow behavior. From the perspective of stress–strain trends, the flow stress curves collectively exhibit similar characteristics:

(i)

At high strain rates (10⁻³ to 10⁻²/s), the flow stress experiences a rapid increase at the onset of deformation, attributed to dislocation multiplication. Once the stress reaches its maximum, a balance is struck between dislocation multiplication and annihilation, leading to subsequent strain softening. At low strain rate (10⁻⁴/s), only strain hardening is observed in the results, with the stress stabilizing after reaching its peak.

(ii)

The flow stresses of the Ti-6Al-4V alloy display high sensitivity to strain rates. Higher strain rates correlate with elevated rheological stresses, a phenomenon arising from the intensified dislocation density and vigorous dislocation motion during deformation at high strain rates.

(iii)

For a given set of strain rates, higher temperatures consistently correspond to lower flow stresses. This correlation is attributed to the capacity of elevated temperatures to diminish interatomic bonding forces and increase the free energy of atoms, thereby promoting dislocation motion, and enhancing deformation.

4. Discussion

4.1. Mechanism Establishment Based on the DDRX Model

In the superplastic deformation process of the Ti-6Al-4V alloy, the microstructure evolution can be characterized by the following equation [26]. The change in grain size is intricately linked to both dislocation density and dynamic recrystallization. It is essential to emphasize that the dynamic recrystallization described by this model exclusively pertains to the DDRX process. In instances where dynamic recovery predominates as the primary dislocation annihilation mechanism, the calculated result for this internal variable is zero.

$$\left\{\begin{array}{c}\dot{d}={\Psi }_1{d}^{-{\Gamma }_1}+{\Psi }_2{\dot{\epsilon }}_p{d}^{-{\Gamma }_2}-{\Psi }_3{\dot{X}}^{{\Gamma }_4}{d}^{-{\Gamma }_3}\\ \dot{\rho }={\dot{\epsilon }}_p\left({\mathrm{\Pi }}_1\sqrt{\rho }-{\mathrm{\Pi }}_2\rho \right)-\frac{\dot{X}}{1-X}\rho \left({\epsilon }_p-{\epsilon }_c\right)\\ \dot{X}={\Theta }_0\theta \left(1-X\right){\left({\dot{\epsilon }}_p\right)}^{{\Theta }_1}{\left({\epsilon }_p-{\epsilon }_c\right)}^{{\Theta }_2}{M}_{b}P/d\end{array}\right.,$$

(1)

where $d$ is the grain size, $\dot{d}$ is the grain size evolution rate, ${\dot{\epsilon }}_p$ is the strain rate, $\dot{X}$ is the dynamic recrystallization volume fraction, $\rho$ is the dislocation density, $\dot{\rho }$ is the dislocation density evolution rate, ${\epsilon }_p$ is the plastic strain, ${\epsilon }_c$ is the critical strain of dynamic recrystallization, $\theta$ is the step function, $M_b$ is the grain boundary mobility, $P$ is the driving force per unit area, $P=\rho G{b}^2/2$, is the shear modulus, $b$ is the Burgers vector, $b=2.86\times {10}^{-10} \mathrm{m}$, ${\Psi }_{1~3}$, ${\Gamma }_{1~4}$, ${\mathrm{\Pi }}_{1~2}$, ${\Theta }_{0~2}$ are the constitutive parameters. The evolution of dislocation density and the volume fraction of DDRX, as calculated by Equation (1), is illustrated in Figure 2 and Figure 3, respectively. When subjected to deformation at low strain rates of 10⁻⁴/s, the dislocation density undergoes minimal change, and the calculated results indicate the absence of DDRX. The flow stress in Figure 1 corroborates this observation, as no stress softening is evident in the material at this strain rate. Consequently, only the DRV mechanism operates, and stresses attain equilibrium values when dynamic recovery and dislocation multiplication are in balance. In Figure 2a,b, a decrease in dislocation density is observed at 800 °C, indicative of dynamic softening. This observation aligns with the variations in the volume fraction of discontinuous dynamic recrystallization shown in Figure 3a,b, clearly indicating that dynamic recrystallization serves as the predominant dislocation annihilation mechanism within this temperature range. Similar outcomes are observed for the high strain rate of 10⁻²/s at 890 °C. However, at a strain rate of 10⁻³/s, a decrease in dislocation density is noted even in the absence of dynamic recrystallization, signifying the initiation of dominance by DRV under such deformation conditions. Therefore, based on the analysis of the multi-scale physical constitutive model’s calculation results, it can be inferred that 890 °C acts as the critical temperature between DRX and DRV as the dominant softening mechanism, particularly within the specified strain rate range of 10⁻³/s to 10⁻²/s. The dominance of DRV should be attributed to superplastic deformation, necessitating the application of the CDRX model to describe grain refinement.

4.2. Dynamic Mechanism of the Ti-6Al-4V Alloy during Superplastic Forming

Figure 4 illustrates the Inverse Pole Figure (IPF) maps of the high-temperature tensile specimen obtained through EBSD analysis. The temperatures and strain rates are 890 °C at 10⁻²/s, 890 °C at 10⁻³/s, and 920 °C at 10⁻⁴/s, as depicted in (a)~(c), respectively. Across all conditions, the grain morphologies consistently maintain an equiaxed form similar to the received material, indicating that grain boundary sliding predominantly governs the deformation mechanism. Recrystallized grain nucleation of DDRX is observable at 890 °C and 10⁻²/s, predominantly distributed around larger grains, as depicted in Figure 4a with the white box line. While subgrain boundaries are also observed, their contribution to generating recrystallized grains is minimal. In Figure 4b, at 890 °C and 10⁻³/s, a fully recrystallized microstructure is achieved. Due to elevated temperatures and deformation, an equilibrium between grain growth and recrystallization is established, resulting in grains exhibiting a comparatively uniform distribution of equiaxed morphology. Nucleation observations of DDRX are reduced, indicating that dynamic recrystallization predominantly occurs through CDRX. In Figure 4c, at 920 °C and 10⁻⁴/s, the process of grain refinement is primarily achieved through the transformation of low-angle grain boundaries into high-angle grain boundaries, representing a characteristic manifestation of the continuous dynamic recrystallization evolution.

5. Constitutive Equation Establishment for CDRX

One of the objectives of this paper is to study the CDRX during superplastic forming. With the discussion on the experimental data and constitutive model calculated results, it is necessary to describe the flow behavior and microstructure evolution with a CDRX constitutive model. The physics-based model with internal state variables (ISVs), dislocation density, grain size, etc., is considered to illustrate the deformation mechanism of material. The following assumptions are made during the modeling process: (i) It is assumed that the recrystallization mechanism remains singular under the condition of material deformation, and (ii) the recrystallization mechanism of the material does not undergo changes during deformation.

5.1. Flow Stress Model

Flow stress $\sigma$ of superplastic forming is the macroscopic characterization of material properties, and shear stress $\tau$ is the microscope description. Taylor factor $M$ ($M=3.06$ for Ti-6Al-4V alloy) is related to $\sigma$ and $\tau$, and the formulation is expressed as [27]

$$\sigma =M\tau .$$

(2)

Shear stress is used to overcome the resistance of a material to deformation. Luo et al. [28] established a constitutive model by treating $\tau$ as ${\tau }_{fd}$, ${\tau }_{ta}$, and ${\tau }_{GB}$, which represent the stress related to the forest dislocation, thermal activation, and grain boundary in high-temperature deformation, respectively. The mathematical expression is as follows:

$$\tau ={\tau }_{fd}+{\tau }_{ta}+{\tau }_{GB}.$$

(3)

To take into account the effect of voids in grain rearrangement and grain rotation (GR) for titanium in SPF, Equation (3) is modified as [26]

$$\tau ={K(\tau }_{fd}+{\tau }_{ta}+{\tau }_{GB}+{\tau }_{GR}),$$

(4)

where $K$ is the parameter related to the voids and cavitation, and the stress ${\tau }_{GR}$ is related to the effect of GR. The expression of $K$ is given by [29]

$$K={(1-n_1{ϵ}^{n_2})}^{n_3},$$

(5)

where $n_1$, $n_2$, and $n_3$ are material constants. The evolution of voids during superplastic forming could be represented by the volume fraction of cavitation, which is expressed as

$$\dot{ϵ}={\dot{ϵ}}_{nucleation}+{\dot{ϵ}}_{growth},$$

(6)

where ${\dot{ϵ}}_{nucleation}$, ${\dot{ϵ}}_{growth}$ is the change rate of $ϵ$ due to the nucleation and the growth of cavitation, which can be represented by

$$\left\{\begin{array}{c}{\dot{ϵ}}_{nucleation}=\mu \sigma {\epsilon }^{\lambda }{\dot{\epsilon }}_p^{1.4}/\left(1-ϵ\right)\\ {\dot{ϵ}}_{growth}=\eta \left(1-ϵ\right){\dot{\epsilon }}_p^{1.6}\end{array}\right.,$$

(7)

with material constants $\mu$, and $\lambda$, and where $\eta$.${\tau }_{fd}$ can be written as [30]

$${\tau }_{fd}=\alpha Gb{\rho }^{0.5},$$

(8)

where $\alpha$ is the proportionality factor between shear stress and forest dislocation density; $G=49.02-5281/\left(exp\frac{181}{T}-1\right)$[31]. Shear stress ${\tau }_{ta}$ can be expressed by [32]

$${\tau }_{ta}={\tau }_0{\left[1-{\left(\frac{RT}{\Delta G}ln\frac{{\dot{\epsilon }}_r}{{\dot{\epsilon }}_p}\right)}^{1/q}\right]}^{1/p},$$

(9)

where ${\tau }_0$ is shear stress ${\tau }_{ta}$ at absolute zero; $R$ is gas constants and $R=8.314 \mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}$; $T$ is the temperature in Kelvin; $\Delta G$ is the active energy for material, which can be used to overcome the obstacles in deformation; ${\dot{\epsilon }}_r$ is the reference strain rate, and ${\dot{\epsilon }}_p$ is the strain rate in deformation; $p$ and $q$ are material constants. The effect of the grain boundary is given by

$${\tau }_{GB}=k{d}^{-0.5}/M,$$

(10)

where $k$ is the Hall–Petch coefficient, $k=12.7$; $d$ is the grain size for the Ti-6Al-4V alloy. The shear stress which is related to grain rotation is expressed as [26]

$${\tau }_{GR}={\beta }_1{{\epsilon }_p}^{{\gamma }_1}{d}^{{\gamma }_2}+{\beta }_2\left(lg{\epsilon }_p\right)d^{{\gamma }_3},$$

(11)

where ${\beta }_1$, ${\beta }_2$, ${\gamma }_1$,${\gamma }_2$, and ${\gamma }_3$ are material constants. The first term on the right side of Equation (11) is promotion grain rotation, and the second term is resistance.

The above model of flow stress includes ISVs$—\rho$ and $d$, which are used to describe the evolution of dislocation density and grain size related to the superplastic forming process. Therefore, it is necessary to model ISVs based on temperatures, strain rates, and strains.

5.2. Grain Size Model

Titanium alloy is a typical two-phase alloy, but the experiments show that the grain sizes of the α and β phases are on the same order of magnitude at 900 °C [33]. Therefore, the effect of different phases is not considered in the modeling of grain size, that is, the changes in two-phase content ratio with temperature are ignored.

The evolution of grain size is competitively affected by static grain growth ${\dot{d}}_{static}$, dynamic grain growth ${\dot{d}}_{dynamic}$ and continuous dynamic recrystallization ${\dot{d}}_{CDRX}$; thus,

$$\dot{d}={\dot{d}}_{static}+{\dot{d}}_{dynamic}+{\dot{d}}_{CDRX}.$$

(12)

Static grain growth can be considered as grain growth caused by atom diffusion under high temperatures [34], which is realized by grain boundary migration. Static grain growth is represented by

$${\dot{d}}_{static}={\beta }_3{d}^{{-\gamma }_4},$$

(13)

where ${\beta }_3$ and ${\gamma }_4$ are material constants. The second term on the right side of Equation (12) is dynamic grain growth, which is caused by the increase in plastic strain. The formula can be characterized as

$${\dot{d}}_{dynamic}={\beta }_4{\dot{\epsilon }}_p{d}^{{-\gamma }_5},$$

(14)

where ${\beta }_4$ and ${\gamma }_5$ are material constants. For continuous dynamic recrystallization, the evolutions of LAGBs in sub-grains are related to GNDs. Therefore, the evolution of CDRX should be related to the changes in dislocation densities. The change rate of grain size can be described by [35]

$${\dot{d}}_{CDRX}=-{\beta }_5{\dot{\rho }}^{{\gamma }_7}{d}^{{-\gamma }_6},$$

(15)

where ${\beta }_5$, ${\gamma }_6$ and ${\gamma }_7$ are material constants.

5.3. Dislocation Density Model

At present, although different measurement methods and theories of dislocation density have been reported [36,37], it is still difficult to obtain dislocation density values accurately. During the SPF process, the evolution of dislocation density is simultaneously controlled by the increment due to work hardening ${\dot{\overline{\rho }}}_{hardening}$ and a decrease in dynamic softening ${\dot{\overline{\rho }}}_{softening}$, so

$$\dot{\overline{\rho }}={\dot{\overline{\rho }}}_{hardening}+{\dot{\overline{\rho }}}_{softening},$$

(16)

where $\overline{\rho }$ is normalized dislocation density, $\overline{\rho }=(\rho -{\rho }_i)/{\rho }_s$, ${\rho }_i$ is the initial dislocation density, and ${\rho }_s$ is saturated dislocation [38,39]. For annealed alloys, the dislocation before deformation is 10¹⁰ m⁻², and ${\rho }_s$ is 2~3 orders of magnitude larger than ${\rho }_i$. The constitutive model describing the evolution of dislocation density caused by work hardening is as follows:

$${\dot{\overline{\rho }}}_{hardening}={\beta }_6{\left({SH}_0/SH\right)}^{{\gamma }_8}\left(1-\overline{\rho }\right){\dot{\epsilon }}_p,$$

(17)

where ${\beta }_6$ and ${\gamma }_8$ are material constants. ${SH}_0$ is the annealed HAGBs area for the initial Ti-6Al-4V alloy, and it can be obtained by 2/$d_0$ with initial grain size $d_0$. $SH$ represents the area of HAGBs. Term ${\left({SH}_0/SH\right)}^{{\gamma }_8}$ represents the influence of grain size on dislocation density; dislocation density is negatively related to grain size, which is consistent with the relationship between dislocation density and the dynamic recrystallization process [40]. Term $\left(1-\overline{\rho }\right){\dot{\epsilon }}_p$ is dislocation density evolution due to deformation and dynamic recovery; for that, the increase in deformation degree contributes to dislocation multiplication, but the increase rate of dislocation density decreases when dislocation annihilation occurs [41]:

$${\dot{\overline{\rho }}}_{softening}=-{\alpha }_1{\overline{\rho }}^{{\gamma }_9}-{\alpha }_{2}S{\overline{\rho }}^{{\gamma }_{10}},$$

(18)

where ${\gamma }_9$ and ${\gamma }_{10}$ are the material constants, and ${\alpha }_1$ and ${\alpha }_2$ are modeled as

$$\left\{\begin{array}{c}{\alpha }_1=c_{{\alpha }_1}exp\left(-\frac{Q_a}{RT}\right)\\ {\alpha }_2=\frac{c_{{\alpha }_2}}{T}exp\left(-K_{g}T-\frac{Q_b}{RT}\right)\end{array}\right.,$$

(19)

where $c_{{\alpha }_1}$, $c_{{\alpha }_2}$ are material constants, $K_{\mathrm{g}}=5.4\times {10}^{-4}/\mathrm{K}$ for the Ti-6Al-4V alloy during superplastic forming, and $\mathrm{K}$ is the temperature unit of Kelvin [42]; $Q_a$ is the thermal activation energy, and $Q_b$ is the activation energy of the grain boundary self-diffusion coefficient. Terms $-{\alpha }_1{\overline{\rho }}^{c_1}$ and $-{\alpha }_{2}S{\overline{\rho }}^{{\gamma }_9}$ represent the effects of dynamic recovery and dynamic recrystallization on dislocation density, respectively.

5.4. LAGB and HAGB Area Model

A kinematic model for the area of LAGBs, which accounts for the effect of accumulation of dislocation, migration of LAGBs, sub-grains rotation, and the swept LAGBs by migration of HAGBs, is established by

$$\dot{SL}={\dot{SL}}_{dis}+{\dot{SL}}_{sub}+{\dot{SL}}_{LABs}+{\dot{SL}}_{HABs}.$$

(20)

During superplastic deformation, LAGBs can hinder dislocation motion, resulting in the accumulation of dislocation and the formation of new LAGB areas. The evolution of LAGB area caused by the accumulation of dislocation is modeled as [40]

$${\dot{SL}}_{dis}={\alpha }_4{\overline{\rho }}^{{\gamma }_{11}}{\dot{\epsilon }}_p,$$

(21)

where ${\alpha }_4$ is the material parameter, ${\alpha }_4=c_{{\alpha }_4}exp\left(-\frac{Q_{\alpha }}{RT}\right)$. In CDRX, sub-grain rotation occurs, which increases the misorientation of LAGBs, and some LAGBs are transitioned into HAGBs. Thus, the decrease in the LAGBs area can be described as [43]

$${\dot{SL}}_{sub}=-{\beta }_{7}SL{\dot{\overline{\theta }}}_{sub},$$

(22)

where $c_2$ is a parameter associated with the reduction in LAGBs due to sub-grain rotation, ${\beta }_7={\theta }_c{c}_{07}$, and ${\theta }_c$ is the critical misorientation value (${\theta }_c=15°$). ${\overline{\theta }}_{sub}$ is the normalized misorientation angle, ${\overline{\theta }}_{sub}={\theta }_{sub}/{\theta }_c$. The third term on the right side of Equation (21) represents the migration of LAGBs, as follows [40]:

$${\dot{SL}}_{LABs}={-\alpha }_3{SL}^2{\overline{\theta }}_{sub}^{{\gamma }_{\theta }}\left(1-ln\left({\overline{\theta }}_{sub}\right)\right){\overline{\rho }}^{{\gamma }_{12}},$$

(23)

where ${\gamma }_{\theta }$, ${\gamma }_{12}$ are the material constants; ${\alpha }_3=\frac{c_{{\alpha }_3}}{T}exp\left(-K_{g}T-\frac{Q_b}{RT}\right)$. Then, the migration of HAGBs may sweep some LAGBs, which can also reduce the area of LAGBs [44],

$${\dot{SL}}_{HABs}=-{\alpha }_{5}SL{\overline{\rho }}^{{\gamma }_{13}}SH,$$

(24)

where ${\gamma }_{13}$ is the material constant; ${\alpha }_5$ is a material parameter related to thermal activation, ${\alpha }_5=C_{{\alpha }_5}exp\left(-Q_a/\left(RT\right)\right)$.

For the migration of LAGBs and sub-grain rotation process, the reduction in the LAGBs area could lead to the increase in the HAGBs area, so it can be given by

$$\dot{SH}={\beta }_{7}SL{\dot{\overline{\theta }}}_{sub}+{\alpha }_3{SL}^2{\overline{\theta }}_{sub}^{{\gamma }_{\theta }}\left(1-ln\left({\overline{\theta }}_{sub}\right)\right){\overline{\rho }}^{{\gamma }_{12}}.$$

(25)

When the migration of HAGBs occurs, the new HAGBs may annihilate the original HAGBs, and the evolution of the area of HAGBs caused by the migration of HAGBs is established [40]:

$$\dot{SH}={-\alpha }_2{SH}^2{\overline{\rho }}^{{\gamma }_{12}}.$$

(26)

Combining Equations (25) and (26), a constitutive model for the HAGBs area can be written as

$$\dot{SH}={\beta }_{7}SL{\dot{\overline{\theta }}}_{sub}+{\alpha }_3{SL}^2{\overline{\theta }}_{sub}^{{\gamma }_{\theta }}\left(1-ln\left({\overline{\theta }}_{sub}\right)\right){\overline{\rho }}^{{\gamma }_{12}}{-\alpha }_2{SH}^2{\overline{\rho }}^{{\gamma }_{12}}.$$

(27)

In addition, the evolution of ${\overline{\theta }}_{sub}$ is given by Li et al. [40]:

$${\dot{\overline{\theta }}}_{sub}={\alpha }_6{\dot{\epsilon }}_p{\overline{\rho }}^{{\gamma }_{14}}{SL}^2{\overline{\theta }}_{sub}\left({\theta }_r{\overline{\theta }}_{sub}-1\right)ln\left(n{\overline{\theta }}_{sub}\right),$$

(28)

where ${\gamma }_{14}$ is the material constant;${ \theta }_r={\theta }_c/{\theta }_{sat}$, ${\theta }_{sat}$ is the steady value of the average misorientation for sub-grains, and ${\theta }_{sat}=4~8°$; $n={\theta }_c/{\theta }_m$, ${\theta }_m$ is the misorientation for grain boundaries with maximum energy, ${\theta }_m=20~25°$; ${\alpha }_6$ is a material parameter related to thermal activation, ${\alpha }_6=C_{{\alpha }_6}exp\left(-Q_a/\left(RT\right)\right)$.

In summary, the CDRX constitutive model for the Ti-6Al-4V alloy during superplastic forming can be established as follows:

$$\left\{\begin{array}{c}\sigma ={M{\left(1-n_1{ϵ}^{n_2}\right)}^{n_3}(\tau }_{fd}+{\tau }_{ta}+{\tau }_{GB}+{\tau }_{GR})\\ {\tau }_{fd}=\alpha Gb{\rho }^{0.5}\\ {\tau }_{ta}={\tau }_0{\left[1-{\left(\frac{RT}{\Delta G}ln\frac{{\dot{\epsilon }}_r}{{\dot{\epsilon }}_p}\right)}^{1/q}\right]}^{1/p}\\ {\tau }_{GB}=k{d}^{-0.5}/M\\ {\tau }_{GR}={\beta }_1{{\epsilon }_p}^{{\gamma }_1}{d}^{{\gamma }_2}+{\beta }_2\left(lg{\epsilon }_p\right)d^{{\gamma }_3}\\ \dot{d}={\beta }_3{d}^{{-\gamma }_4}+{\beta }_4{\dot{\epsilon }}_p{d}^{{-\gamma }_5}-{\beta }_5{\dot{\rho }}^{{\gamma }_7}{d}^{{-\gamma }_6}\\ \dot{\overline{\rho }}={\beta }_6{\left({SH}_0/SH\right)}^{{\gamma }_8}\left(1-\overline{\rho }\right){\dot{\epsilon }}_p-{\alpha }_1{\overline{\rho }}^{{\gamma }_9}-{\alpha }_{2}S{\overline{\rho }}^{{\gamma }_{10}}\\ \dot{SL}={\alpha }_4{\overline{\rho }}^{{\gamma }_{11}}{\dot{\epsilon }}_p-{\beta }_{7}SL{\dot{\overline{\theta }}}_{sub}{-\alpha }_3{SL}^2{\overline{\theta }}_{sub}^{{\gamma }_{\theta }}\left(1-ln\left({\overline{\theta }}_{sub}\right)\right){\overline{\rho }}^{{\gamma }_{12}}-{\alpha }_{5}SL{\overline{\rho }}^{{\gamma }_{13}}SH\\ \dot{SH}={\beta }_{7}SL{\dot{\overline{\theta }}}_{sub}+{\alpha }_3{SL}^2{\overline{\theta }}_{sub}^{{\gamma }_{\theta }}\left(1-ln\left({\overline{\theta }}_{sub}\right)\right){\overline{\rho }}^{{\gamma }_{12}}{-\alpha }_2{SH}^2{\overline{\rho }}^{{\gamma }_{12}}\\ {\dot{\overline{\theta }}}_{sub}={\alpha }_6{\dot{\epsilon }}_p{\overline{\rho }}^{{\gamma }_{14}}{SL}^2{\overline{\theta }}_{sub}\left({\theta }_r{\overline{\theta }}_{sub}-1\right)ln\left(n{\overline{\theta }}_{sub}\right)\end{array}\right.$$

(29)

5.5. Determination of Material Constants

There is an abundance of parameters and material constants in the established constitutive model. Some parameters can be obtained from the published literature, as listed in

Table 2. Parameters from the literature in the established constitutive model.

Parameter	Significance	Value	Origin
$M$	Taylor factor	3.06	Ref. [46]
$b$	The magnitude of the Burgers vector	$2.86\times {10}^{-10}\mathrm{m}$	-
$R$	Gas constant	$8.314 \mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}$	-
$k$	Hall–Petch coefficient	$12.7$	Ref. [47]
$K_g$	-	$5.4\times {10}^{-4}{\mathrm{K}}^{-1}$	Ref. [42]
${\theta }_c$	Critical misorientation value	$15°$	Ref. [48]
${\gamma }_{\theta }$	-	$6.18$	-

. Other material constants are optimized by the multi-objective genetic algorithm (GA), which is illustrated in ref. [45]. The optimization results of material constants in this step are listed in

Table 3. Parameters optimized for the grain size constitutive model.

${\beta }_3$	${\beta }_4$	${\beta }_5$	${\beta }_6$	${\beta }_7$	$C_{{\alpha }_1}$	$C_{{\alpha }_2}$	$C_{{\alpha }_3}$
5.97	6.17	5.29	2.17	0.05	0.09	2,568,716	152,549
$C_{{\alpha }_4}$	$C_{{\alpha }_5}$	$C_{{\alpha }_6}$	${\theta }_r$	${\gamma }_5$	${\gamma }_6$	${\gamma }_8$	${\gamma }_{10}$
12.69	19,953	2,729,480	1.77	7.83	1.41	0.013	3.87
${\gamma }_{11}$	${\gamma }_{12}$	${\gamma }_{13}$	${\gamma }_{14}$	$n$	$Q_b$	${\gamma }_{\theta }$
0.036	0.52	5.96	0.20	0.67	9625.58	4.20

.

Comparisons between the experimental data and predicted results of grain sizes are presented in Figure 5. The horizontal coordinate of the scatter is the experimental result, and the vertical coordinate is the calculated result. The dashed line is a 45° line, indicating that the horizontal and vertical coordinates are equal, so the closer the scatter point to the dashed line, the more accurate the calculation result. The comparisons between the experimental data and calculated results show good agreement. Thus, the CDRX constitutive model established in this paper can describe the microstructure evolution for the Ti-6Al-4V alloy during superplastic forming.

5.6. Areas of the HAGBs and LAGBs

Figure 6 describes the evolution of the areas of HAGBs, LAGBs, and total grain boundaries (GBs). As is shown in Figure 6a, for $\dot{\epsilon }={10}^{-2}/\mathrm{s}$, the ratio of the HAGB area (SH) to the GB area (S) increases with the deformation process. The present paper shows that the HAGB area is more and more abundant than the LAGB area (SL), which is due to the large number of fine grains caused by CDRX, and sub-grain boundaries evolve into grain boundaries. In Figure 6b, this phenomenon is further demonstrated. The area of HAGBs increases with strain; however, opposite trends are observed in the area of LAGBs. In other words, the conversion from LAGBs to HAGBs is further verified. On the contrary, for the lower strain rates (10⁻³ and 10⁻⁴/s), only small parts of LAGBs are converted into HAGBs, which demonstrates that CDRX does not play a dominant role. Thus, stable grain sizes are also the reason why the superplasticity of the Ti-6Al-4V alloy is good at this temperature.

In total, the DRV and DRX softening mechanism are distinguished by EBSD and constitutive modeling, then the CDRX recrystallization grain size model is put up. Distinct recrystallization modes coexist with varied coordination modes of grain boundary slip stress concentration. Building upon the insights gained from this study, it is anticipated that subsequent endeavors will witness advancements in the analysis of material fracture behavior and the elucidation of mixed recrystallization mechanisms.

6. Conclusions

(1)

Based on EBSD test observations and constitutive equation calculation results, this paper investigated the DRV and DRX of the Ti-6Al-4V alloy during superplastic forming;

(2)

It is found that 890 °C, 10⁻³/s~10⁻²/s are the critical values between DRX and DRV as the dominant softening mechanism. For the DRX case, recrystallized grain nucleation of DDRX is observable. Nevertheless, the evolution of grain boundary misorientation during the CDRX plays a more important role.

(3)

A constitutive model for the Ti-6Al-4V alloy during superplastic forming can be established, and the CDRX mechanism is emphasized. A good comparison can be obtained between the calculated results and experimental data on grain size. In addition, the conversion from LAGBs to HAGBs is also verified based on the established model.