Globularization and Recrystallization of a Ti-6Al-2Sn-4Zr-6Mo Alloy under Forging in the (α+β) Region: Experiment, Phenomenological Modeling and Machine Learning

Abstract

The Ti–6Al–2Sn–4Zr–6Mo (Ti-6246) alloy is an α + β titanium alloy with good combination of strength, toughness, high temperature creep property and crack growth resistance. This work examined the dynamic microstructural conversion behavior (specifically focusing on dynamic globularization of α phase) occurred under hot forging process (at temperatures ranging from 750°C to 1050°C) of the Ti-6246 alloy with various kinds of starting microstructure. Plastic flow behavior of the microstructure having the α-lamellae exhibited a peak stress followed by continuous flow softening due to frequent occurrence of lamellae kinking and globularization of the α phase. For the microstructural conversion of β phase, continuous dynamic recrystallization (CDRX) was dominantly occurred. And the enhanced CDRX is observed for the lamellar starting microstructure as compared to the bimodal and the equiaxed starting microstructures. On the globularization kinetics according to a phenomenological Avrami approach and a machine learning approach, when the processing conditions were categorized into 3 groups from a hierarchical clustering, the estimated globularization fraction from Avrami approach corresponded reasonably well with the experimental data only in a specific process condition group in which corresponded with a higher temperature and a higher dynamic globularization fraction. Thus, this work points out the possibility of a reliable prediction of dynamic globularization according to an optimal combination of the Avrami approach and the machine learning approach.

1. Introduction

In the thermomechanical processing of ingot -metallurgy α + β titanium alloy, primary hot working is applied for the conversion of ingot structure to fine equiaxed structure. In this regard, the main objectives are the homogenization of microstructure and the break-down of lamellar microstructure that involves with dynamic globularization behavior as microstructural conversion. The kinetics and mechanisms that control the globularization of the platelet, or Widmanstatten, α morphology have been investigated.^1,2,3,4) Semiatin et al. pointed out that the nucleation sites for globularization occurred at kinks in the lamellae as well as some of the prior-β grain boundaries.³⁾ This type of microstructural change involving a reduction in the aspect ratio of the plates can be interpreted in terms of geometric dynamic recrystallization process which is proposed by McQueen et al.^5,6) In general, globularization behavior exhibits in sigmoid way with an increasing of strain under a thermal activation process.^3,7,8,9) In this regard, Wang et al.⁸⁾ and the present author et al.⁹⁾ have pointed out that the globularization fraction as a function of strain could be reasonably expressed in terms of an Avrami type equation. The factor that dominates the dynamic globularization behavior is dependent on deformation condition. In deformation regime that the effect of thermal activation process is weak (in condition with a decreasing of test temperature and an increasing of strain rate), a kinking of α-lamellae accompanied by a shear localization is a dominant mode, resulting in a mechanical fragmentation into nearly equiaxed α grain. Herein, deformation behavior exhibits highly nonuniform. On the other hand, dynamic recrystallization (accompanied by frequent subgrain formation) is frequently occurred from the main driving force by intense shear localization within the α lamellae, and surface-tension driven penetration of the α phase by β phase is enhanced due to the enhanced atomic diffusion in the deformation regime in which the effect of thermal activation process is relatively strong (at a higher temperature). Thus, the dominating mode of globularization of α phase depends on the deformation condition.

The Ti-6246 alloy (Ti–6Al–2Sn–4Zr–6Mo (in wt.%)), developed by TIMET, is an α + β titanium alloy with good combination of strength, toughness, high temperature creep property and crack growth resistance.¹⁰⁾ This alloy is used in the compressor disks and fans of aero engines at 400 and 540°C for long-term and short-term load-carrying, respectively. Isothermal forging behavior and microstructural conversion on the Ti-6246 alloy have been examined.^11,12,13)

In the present study, the objective was to obtain a more quantitative understanding of the phenomenology and mechanism of flow behavior and microstructural conversion specifically focusing on dynamic globularization behavior on the forging process of the Ti-6246 alloy having various starting microstructures. For it, isothermal hot compression tests were carried out on the Ti-6246 alloy with three starting microstructures (a lamellar type, a bimodal type and an equiaxed type) followed by detailed microstructural analysis. Additionally, the occurrence of dynamic globularization in relation to deformation conditions (a strain, a strain rate and a temperature) was quantitatively summarized in terms of a phenomenological Avrami approach and a machine learning approach, followed by implementation into FEM code (DEFORM-3D, v.12.).

2. Experimental Procedure

A Ti-6246 alloy with a chemical composition (in wt.%) of Ti-5.99Al-2.01Sn-3.97Zr-6.04Mo-(0.056O-0.0097N) was used in this work. Figure 1 shows the starting microstructures for forging test in this work. For preparation of these starting microstructures, the as received Ti-6246 bar with a diameter of 5 mm having the bimodal microstructure [as shown in Fig. 1(b)] was heat treated at 1000°C for 30 min followed by furnace cooling in order to form a lamellar (α+β) microstructure [as shown in Fig. 1(a)]. Additionally, the as received Ti-6246 alloy was heat treated at 880°C for 30 min followed by air cooling for 10 s and oil-quenching so as to have an equiaxed microstructure [as shown in Fig. 1(c)]. Thus, three types of starting microstructures (shown by SEM-BSE image), a lamellar (α+β) microstructure [as shown in Fig. 1(a)], a bimodal (α+β) microstructure [as shown in Fig. 1(b)] and an equiaxed microstructure [as shown in Fig. 1(c)] were prepared in this work. Total fraction of α phase in the lamellar microstructure [Fig. 1(a)] is 62.2%. The fraction of equiaxed α grain in the bimodal microstructure [Fig. 1(b)] and the equiaxed microstructure [Fig. 1(c)] are 35.2% and 36%, respectively. The grain size of coarse prior-β grain for all starting microstructures was measured to be approximately 1 mm, according to an optical microscopy analysis.

Fig. 1.

Starting microstructures (depicted by SEM-BSE images) of (a) lamellar (α+β), (b) bimodal (α+β), and (c) equiaxed (α+β) microstructures of the Ti-6246 alloy. In the SEM images, the white regions correspond to the β phase and the black regions correspond to the α phase.

An isothermal forging test was carried out using an AUTOGRAPH AG-X plus precision universal tester (SHIMADZU) with a 50 kN load cell. Forging temperatures were 750°C, 800°C, 850°C, 950°C and 1050°C, and deformation was conducted at constant strain rates of 10⁻³ s⁻¹, 10⁻² s⁻¹, 10⁻¹ s⁻¹ and 1 s⁻¹. The specimens (5 mm in diameter and 7.5 mm in height) were heated at approximately 0.3°C/s up to the testing temperature and maintained for 10 min for ensuring a uniform temperature distribution throughout the specimen, followed by forging at a height true strain of 0.75 (corresponding to a height reduction of 50%). After forging tests, the specimens were cooled in air for 10 s (to avoid a martensitic transformation), and then quenched in oil. The experimentally obtained stress- strain curves were corrected by the friction correction method and adiabatic correction method on the basis of reports by Li et al.^14,15) and Mataya and Sackschewsky.¹⁶⁾

Microstructures were identified with a field emission scanning electron microscope (FE-SEM) fitted with an electron back-scattering diffraction (EBSD) analyzer equipped with the HKL Channel 5 software. The OLYMPUS-Stream software was used on the SEM-backscattered electron (BSE) images or the EBSD image to evaluate the microstructural factors of grain size of the α and β phases and the average aspect ratio of α lamellae. In order to analyze dynamic globularization during hot deformation, globularization was deemed to have occurred when the aspect ratio of α lamellae exhibited an aspect ratio less than 4. The initial average aspect ratio of α lamellae in starting microstructures [Figs. 1(a) 1(b)] was more than 20. The microstructure was observed at five locations in a forged sample (as illustrated in Fig. 2) so as to identify the effect of different distributions of strain, strain rate and temperature on microstructural formation.

Fig. 2.

Schematic drawing of the forged specimen showing the locations (A–E) for microstructural observation.

The forging characteristics of the Ti-6246 alloy having a lamellar starting microstructure [Fig. 1(a)] were analyzed by FEM with a DEFORM-3D software (v.12) using a user-defined subroutine. The simulated condition is basically similar to the experimental forging condition. Herein, the corrected strain- stress flow behaviors obtained in this work was programed into the FEM code. For predicting the microstructural evolution specifically focusing on dynamic globularization, the constitutive equations on basis of a physical model of Avrami approach and a machine learning approach were established and implemented in the FEM code.

Here, dynamic globularization behavior was also modeled in terms of a machine learning approach. A machine learning is a programming technique used to automate the construction of analytical models and a process of examining, comparing and modeling data sets and their relationships. The machine learning approach has resulted in successful predictions in various fields. In this work, two machine learning algorithms of a hierarchical clustering and a neural network as implemented in the scikit-learn python module (v.0.21.3)¹⁷⁾ are made to conduct the regression and clustering of the experimental data. In section 3-4, the methods for its calculation will be explained more in detail.

3. Results and Discussion

3.1. Flow Behavior

Typical measured and corrected true stress – true strain (flow) curves of the Ti-6246 alloy with the three different microstructures [(a) the lamellar microstructure, (b) the bimodal microstructure, (c) the equiaxed microstructure] are summarized in Fig. 3. Here, testing conditions are the temperatures of 750°C, 850°C and 900°C and the strain rates of 1 and 10⁻³ s⁻¹. The corrected curves are shown by the squared plots.

Fig. 3.

True stress- true strain curves (experimentally obtained curves and corrected curves) of the Ti-6246 alloy having (a) a lamellar, (b) a bimodal and (c) a equiaxed (α+β) starting microstructures tested at 750°C, 850°C, 900°C and strain rates of 10⁻³ s⁻¹ and 1 s⁻¹. The corrected curves are expressed by the squared plots. (Online version in color.)

The stresses after friction correction according to the reports^14,15) is calculated when the flow stress and friction factor, m_fric are substituted into the following equation,   

$$\sigma =\frac{{\sigma }_Z}{1+\left(A+B\epsilon +C{\epsilon }^2+D{\epsilon }^3\right)m_{fric}}$$(1)

where σ_Z is the true stress obtained experimentally; m_fric and ε are friction factor modified according to the reports,^14,15) and true strain, respectively.

The temperature rise due to adiabatic heating during deformation is usually calculated by the following equation,¹⁶⁾   

$$\mathrm{\Delta }\text{T}=\frac{{\eta }_e}{\rho c}{\int }_0^{\epsilon }\sigma d\epsilon =\frac{{\eta }_e}{\rho c}W$$(2)

where ΔT is the temperature rise due to the work done on sample, c is the heat capacity of material, ρ is density of sample, η_e is the heat efficiency and W is the power inputted into the sample. With respect to the heat efficiency η_e in Eq. (2), the η_e is generally taken for convenience to be a constant in the range of 0.90–0.96. Actually, the heat-dissipation time is closely related to strain rate: the longer the deformation time (for the case at lower strain rate), the lower the value of η_e is exhibited. It has been suggested that η_e varies linearly with $\text{log(}\dot{\epsilon }\text{)}$, so that η_e = 0 at $\dot{\epsilon }$ of 10⁻³ s^–1 and η_e < 0.95 at $\dot{\epsilon }$ equal to or greater than 1 s^–1.¹⁶⁾ Following this suggestion, Li et al. expressed the η_e as equation of η_e = (0.316)log$\dot{\epsilon }$+0.95.¹⁵⁾ According to it, the higher strain rate results in an increasing of the deformation heating temperature.

All of the curves exhibit a peak flow stress at a yielding point followed by moderate to extensive flow softening. The continuous flow softening is seen to be remarkable at a low testing temperature and a high strain rate. The flow softening observed in flow curve of the Ti-6246 alloy is deduced to be arisen from two possible sources of deformation heating and microstructure changes. The latter may include changes in phase morphology, dislocation substructure, and texture formation. With respect to the possible influence of deformation heating, we can observe that the stress values are quite dissimilar before and after correction, especially at a higher strain rete of 1 s⁻¹ and lower temperatures. This observed flow instability behavior is mainly due to the effect of adiabatic heating during the deformation at a higher strain rate, as described elsewhere.¹⁸⁾ At a high testing temperature and a low strain rate, there is no difference in the stress values before and after correction owing to a weakening of the influence of adiabatic heating. Additionally, the flow softening observed even after correction is mainly ascribed to the microstructure change of lamellae kinking and globularization behavior of the α phase. Indeed, we can note that the overall degree of flow softening is smaller for the equiaxed starting microstructure than the lamellar and bimodal starting microstructures. Among the three starting microstructures, flow softening behavior is the most remarkable for the lamellar starting microstructure, which is indicative of the frequent occurrence of lamellae kinking and globularization of the α phase. With an increase in testing temperature, flow behavior exhibits a steady state behavior, implying that either dynamic recovery (DRV) or continuous dynamic recrystallization (CDRX) is the dominant mode for microstructural conversion. During hot deformation of α and near-α Ti alloys at high temperatures, DRV and/or dynamic recrystallization (DRX) occur, resulting in microstructural conversion.¹⁸⁾ CDRX is well recognized to be frequently enhanced in hot deformation of (α+β) Ti alloys.^19,20,21) Here, CDRX is the behavior in which subgrains with low-angle boundaries are formed and subsequently evolve into grains with a high fraction of high-angle boundaries with increasing strain.

3.2. Kinetic Analysis of Hot Deformation

Constitutive equation for high temperature deformation according to the model proposed by Sellars and McTegart²²⁾ is expressed as following equation.   

$$\dot{\epsilon }=A_1{\left[\text{sinh}\left(\alpha \sigma \right)\right]}^n\text{exp}\left(-\frac{Q}{RT}\right)$$(3)

  

$$\dot{\epsilon }=A_2{\sigma }^n\text{exp}\left(-\frac{Q}{RT}\right)\hspace{1em}\left[\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{a}\mathrm{\hspace{0.17em} } \mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}\mathrm{\hspace{0.17em} } \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{\hspace{0.17em} } \mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n}\right]$$(4)

where A₁ and A₂ are the material constants, R is the universal gas constant (8.314 J/(mol K)), n=(1/m) is the stress exponent, m is the strain rate sensitivity defined by δ(logσ)/δ(log$\dot{\epsilon }$), and T is the absolute temperature. Since hot deformation of Ti-6246 alloy in this work is carried out in a lower temperature region of the (α+β) region, the Eq. (3) is used for identifying the deformation mode and the apparent activation energy Q for hot deformation. The parameters in Eq. (3) are estimated by nonlinear regression analysis using the corrected flow curves. The estimated parameters and Q are summarized in Table 1. The estimated Q values are 319.3, 370.8 and 452.6 kJ/mol for the lamellar, the bimodal and the equiaxed starting microstructures, respectively. The Q values for the lamellar and the bimodal starting microstructures are similar to the Q values (ranging from 300 to 455 kJ/mol) reported previously for hot deformation of various Ti alloys.^{11,20,23,24,25,26,27)} Compared to the reported Q value (455 kJ/mol) for the Ti-6Al-4V alloy having the lamellar microstructure,²⁶⁾ the Q value (319.3 kJ/mol) of the lamellar starting microstructure in the present Ti-6246 alloy exhibits relatively a lower value. This result implies that a thermal activation process associated with the enhanced diffusion for α-globularization is frequently activated in the Ti-6246 alloy with a lamellar microstructure. Additionally, a higher Q values more than 450 kJ/mol is observed for the equiaxed microstructure. As described below showing the deformed microstructures, the α-precipitation from supersaturated β phase occurs dynamically under hot deformation for the equiaxed starting microstructure, thus affecting an apparent increase in the Q value.

Table 1. The estimated parameters of constitutive Eq. (3) determined by nonlinear regression analysis using the corrected flow curves.

	lamellar M	Bimodal M	Equiaxed M
A1	3.84×1012	1.37×1016	1.20×1020
α	0.008962	0.0062	0.005637
n	2.45	3.09	3.61
Q (kJ/mol)	319.3	370.8	452.6

Deformation in (α+β) region, M: Microstructure

With respect to the stress exponent n, the n value close to 4 is typical for dislocation mobility induced viscos flow (associated with the power-law creep mechanism) in many pure metal and solution-hardened alloys including α-Ti and β-Ti^28,29,30) and (α+β) Ti-6Al-4V alloy.³⁰⁾ Additionally, the n value close to 3 corresponds to the operation of viscos drag dislocation motion induced by solute atoms. From Table 1, the high n value of 3.61 (which is close to 4) is observed for the equiaxed starting microstructure, which is indicative of activation of dislocation mobility induced viscos flow. On the other hand, the low n value of 2.45 (which is close to 3) is observed for the lamellar starting microstructure. It indeed points out that the enhanced diffusion for α-globularization is frequently occurred for the lamellar starting microstructure.

Hereafter, the Zener-Hollomon (Z) parameter given by   

$$Z=\dot{\epsilon }\text{exp}\left(\frac{Q}{RT}\right)$$(5)

(where R is gas constant, T is absolute temperature and Q is the activation energy obtained from Table 1), indicating the temperature compensated strain rate parameter is used for kinetic analyzing. Figure 4 summarizes the relationship of flow stress (at a true plastic strain of 0.6) and Z parameter in Log-Log scale for all starting microstructures. Linear relationships are clearly seen for all starting microstructures, indicating that thermally activated process is indeed dominant in hot deformation of all starting microstructures.

Fig. 4.

Relationship between Z parameter and the flow stress value (at a strain of 0.6) of the forged Ti-6246 alloy with a lamellar, a bimodal and an equiaxed (α+β) starting microstructures.

3.3. Deformed Microstructures

Figure 5 shows the deformed microstructures compressed at temperatures of 750°C, 850°C, at strain rates of 10⁻³ s⁻¹ and 1 s⁻¹ and at a height true strain of 0.75 of the Ti-6246 alloys having various starting microstructures. On testing at 750°C for the lamellar and bimodal starting microstructures, a lamellae kinking phenomenon in which the α phase exhibits a bending resulting in a kinked morphology, could be observed. The decrease in the α fraction, coarsening of the α grain size, and the frequent activation of dynamic globularization of the α phase are found to be dominant with increasing forging temperatures up to 850°C and decreasing strain rate up to 10⁻³ s⁻¹.

Fig. 5.

Deformed microstructures (i.e., SEM-BSE images) at the central location of the specimens [(a-1–4) lamellar. (b-1–4) bimodal and (c-1, 2) equiaxed starting microstructures]. Forged temperatures are (a-1, 3) (b-1, 3) (c-1) 750°C and (a-2, 4) (b-2, 4) (c-2) 850°C, and strain rates are (a-1, 2) (b-1, 2) (c-1, 2) 10⁻³ s⁻¹ and (a-3, 4) (b-3, 4) 1 s⁻¹, respectively.

With respect to the globularization kinetic of Ti alloy, for example, Semiatin et al. presented that the deformed microstructures of the Ti-6Al-4V alloy having a lamellar (α+β) microstructure comprised bent and kinked lamellae and an absence of globularization at low strains, on the other hand, almost fully globularized microstructures were exhibited at large strains.³⁾ Here, Semiatin et al. pointed out globularization initiated at strains of the order of 1.0 and the globularization process was complicated by strains of ≈ 2.5 for the deformation at a strain rate of 10⁻³ s⁻¹.³⁾ The nucleation sites for globularization occurs at kinks in the lamellae, and this type of microstructural conversion involving a reduction in the aspect ratio of the lamellae is interpreted in terms of geometric dynamic recrystallization.^5,6) There are two main mechanisms recognized for globularization process. One is the boundary splitting process that shear localization within the lamellae serves to fragment the lamellae into nearly equiaxed grain.³¹⁾ And the other is the termination migration process in which the driving force is caused by curvature differences between the lamellae and the flat lamellae interfaces, and the transfer of atoms from the edge surface of the lamellae terminations to the flat surface results in globularization.³²⁾ For denoting the globularization behavior in numerical modeling, Wang et al.⁸⁾ and the present author⁹⁾ adopted an Avrami type equation, resulting in a good correlation with the experimental results. So, this work also adopted an Avrami approach for expressing the dynamic globularization behavior. The estimated activation energy for dynamic globularization in the case of the lamellar starting microstructure according to an Avrami approach (as pointing out in the following section) is about 86.8 kJ/mol, which is close to the activation energy of grain boundary diffusion of Ti element in α phase of 97 kJ/mol.³³⁾ This estimated energy interestingly points out that dynamic globularization process of the α phase in the Ti-6246 alloy with a lamellar starting microstructure proceeded under enhanced diffusion of atoms from the boundaries split by shear localization by deformation. Dynamic globularization behavior observed in this work will be stated in detail as following section.

On the forged microstructures for the equiaxed starting microstructure [Figs. 5(c-1) 5(c-2)], formation of fine α precipitates from supersaturated β phase is observed in forging at 750°C and 10⁻³ s⁻¹. As stated in section 3-2, such the dynamic α precipitation is deemed to result in a high apparent activation energy and a low stress exponent value estimated in the constitutive model of Eq. (3).

Regarding the microstructural conversion of the β phase, Fig. 6 shows the EBSD images showing the orientations of the β phase of the forged Ti-6246 alloy tested at (a) (c) (e) 750°C, (b) (d) (f) 850°C and strain rates of (a) (b) (e) (f) 10⁻³ s⁻¹ and (c) (d) 1 s⁻¹. Herein, starting microstructures are the lamellar starting microstructure [(a)–(d)] and the bimodal starting microstructure [(e) (c)], respectively. The orientation color shown in Fig. 6 indicates the orientation normal to the cross section of forged specimen. In addition, the microstructures shown in Fig. 6 are observed in the central region of the forged specimen. From Fig. 6, the fine equiaxed grained β phase can be observed in all deformed microstructures. The fraction of high angle boundary (HAGB) at the β/β interface tends to exhibit a higher fraction with an increasing of forging temperature, and exhibits a highest angle of 72.9% tested at 850°C and 10⁻³ s⁻¹ for the lamellar starting microstructure. This result points out that CDRX of the β phase accompanied by evolution into grains with a high fraction of high angle boundary is enhanced under hot forging. Here, we can note that CDRX is enhanced more frequently in the lamellar starting microstructure [as shown in Fig. 6(b)] than the bimodal starting microstructure [as shown in Fig. 6(f)], being attributable to a large amount of strains (dislocations) locally accumulated in the lamellar starting microstructure having a higher contacting area of the α/β interface.

Fig. 6.

Deformed microstructures (i.e., EBSD-orientation images focusing on β phase) at the central location of the specimens [(a–d) lamellar starting microstructure and (e) (f) bimodal starting microstructure]. Forged temperatures are (a) (c) (e) 750°C and (b) (d) (f) 850°C, and strain rates are (a) (b) (e) (f) 10⁻³ s⁻¹ and (c) (d) 1 s⁻¹, respectively. (Online version in color.)

Figures 7(a) 7(b) show the dependence of the Z parameter (in Log scale) on (a) a fraction of dynamic globularization of the α phase and (b) an average width of the α lamellae (for the secondary α phase in the case of the bimodal starting microstructure). Figure 7 shows the results measured at the central location of the forged specimens with (a) the lamellar starting microstructure and (b) the lamellar and bimodal starting microstructures. Here, globularization was deemed to have occurred when the α lamellae exhibited an aspect ratio less than 4 (the initial average aspect ratio was more than 20) in order to analyze dynamic globularization behavior quantitatively. From Fig. 7(a), a linear relationship can be clearly seen, indicating that globularization process itself is depending on the thermally activated process associated with the enhanced diffusion process. From Fig. 7(b), it is interestingly noted that there are two linear relationships in the lower log₁₀ (Z) region and higher log₁₀ (Z) region. In the lower log₁₀ (Z) region having a higher slope value, the α width of lamellae strongly depends on Z parameter, indicating that grain sizes itself are dependent on the thermally activated process up to forging condition of the Z parameter about 15. Thus, it can be found that microstructural conversion mechanism controlling the grain size becomes different at and from the Z parameter of approximately 15. Additionally, we can observe that the dependence of the Z parameter on the α width is remarkable for the lamellar starting microstructure as compared to the bimodal microstructure. In this regard, the remarkable dependence is deduced to relate to the enhanced CDRX of the β phase occurred in the lamellar starting microstructure as abovementioned.

Fig. 7.

(a) Relationship between Z parameter and dynamic globularization fraction of α phase in the case of the lamellar (α+β) starting microstructure. (b) Relationship between Z parameter and width of α lamellae in the deformed microstructure (circle plots: the lamellar starting microstructure, square plots: the bimodal starting microstructure).

3.4. Modeling of the Dynamic Globularization Behavior: Physical Approach of JMAK Model and Data Analytics of Machine Learning

Dynamic globularization behavior is summarized from a quantitative analysis of microstructures at 5 locations (as shown in Fig. 2) in each forged specimen with the data of effective strain, effective strain rate and temperature estimated by FEM code of a DEFORM-3D software. After that, microstructural modeling of the dynamic globularization of the α phase is established followed by implementation into the FEM code. On the dynamic globularization behavior, it is reported that the globularization fraction increases with an increasing strain in a sigmoid way for the Ti-6Al-4V alloy^3,34) and the Ti-17 alloy.^8,9) The dynamic globularization behavior can be expressed similar to the DRX mode, therefore, a modified Johnson-Mehl-Avrami-Kolmogorov (JMAK) type equation is adopted for denoting the dynamic globularization behavior.^7,8,35) The dynamic globularization fraction (f_DG) of the α phase is defined as follows:   

$$f_{DG}=1-\text{exp}\left[-k\times {\left(\epsilon -{\epsilon }_{\text{c}}\right)}^n\right]$$(6)

  

$${\epsilon }_{\text{P}}=\left(5.5\times {10}^{-3}\right)\cdot Z^{0.0778}$$(7)

  

$${\epsilon }_{\text{c}}=0.8\cdot {\epsilon }_P$$(8)

  

$$k=k_0{\dot{\epsilon }}^{m1}\text{exp}\left(-\frac{Q_1}{RT}\right)$$(9)

where f_DG is the dynamic gloularization fraction, k and n are material constants, ε is the true strain, ε_c is the critical strain for the onset of dynamic globularization, and ε_P is the true strain at the peak stress around yielding point in the flow curve that is experimentally identified. The Z is the Zener-Hollomon parameter as expressed in Eq. (5). The parameter k in Eq. (6) corresponds to the reaction rate, and expressed in Arrhenius equation as Eq. (9). The material constants in the above equations were determined on the basis of the minimization of the sum of errors between the experimental and the calculated data according to the non-linear regression analysis. The constants k₀, m₁, Q₁ and n₁ were determined to be 496.8, 0.241, 86.8 (kJ/mol) and 2, respectively.

This work also models the dynamic globularization behavior according to a machine learning approach. Among the machine learning algorithms, the neural network (NN) is an information treatment system with the characteristics of adaptive learning, which is suitable for treating non-linear phenomena and complex relationships.³⁶⁾ The NN is essentially an operation linking input to output data, by using a particular set of non-linear basis functions. The advantage of NN is their ability to learn or adapt to changing conditions. The architecture of the NN refers to the number of the layers in the NN and the number of the neurons in each layer. This work set the one layer and 4 neurons for NN calculation. Figure 8(a) illustrates the schematic structure of NN in this work: 1 hidden layer and 4 neurons including 1 bias. Herein, a sigmoid function [1/(1+e^−x)] is applied as the activation function, and the input and output variables (X) are normalized to be X_i as following Eq. (10):   

$$X_i=0.1+0.8\times \left(\frac{X-X_{min}}{X_{max}-X_{min}}\right)$$(10)

Thus, the dynamic globularization fraction (as output variable) of the forged specimen is calculated through the NN model with input variables of process conditions of temperature, strain rate, and strain.

Fig. 8.

(a) Architecture of the neural network regression process in this work, (b) Correlation between the experimental result and the predicted results (JMAK result and NN result) of the dynamic globularization fraction for all process conditions in this work (in the case of the lamellar starting microstructure).

Figure 8(b) summarizes the predicted results of JMAK model (from Eq. (6)) and NN model with the experimental results (for all testing conditions in this work). Coefficient of determination, R² is also indicated. Relatively, a higher R² is observed for the NN model than the JMAK model. However, a strong scattering is observed in the plots obtained by linear relation for both modeling approaches. This result implies that dynamic globularization behavior is not explained primary in relation to the JMAK model and NN model for all testing conditions in this work. Therefore, process conditions for forging in this work were hereafter categorized into several groups on the basis of the machine learning with an unsupervised learning method.

In order to divide the process conditions into each category, a hierarchical clustering based on the Ward’ method³⁷⁾ was carried out for the dataset of process parameters of temperature, effective strain rate and effective strain which were estimated from FEM analysis. A clustering algorithm is recognized as one of unsupervised learning methods and classifies data points into different data aggregates (clusters). The Ward’s method merges the clusters to minimize the difference between the variance of the cluster after merging and the sum of variance of clusters before merging. Figure 9 summarizes the result of hierarchical clustering of 60 dataset of process conditions into 3 clusters (groups) shown by a dendrogram. From Fig. 9, as a tendency of characteristics of process conditions and the resultant dynamic globularization behavior for each group, the Gr.1 is the region with a higher temperature and a higher dynamic globularization fraction. The Gr.2 corresponds to the region with a lower temperature and in the middle region of dynamic globularization fraction, and the Gr.3 does the region with a lower temperature and a quite lower dynamic globularization fraction. Thus, the hierarchical clustering based on the Ward’s method could categorize into three groups of processing conditions. Figure 10 shows the predicted results (JMAK result and NN result) with experimental dynamic globularization fraction of each group of (a) Gr.1, (b) Gr.2 and (c) Gr.3. Coefficient of determination, R² is also indicated in Fig. 10. Good correlations between the predicted and experimental results are seen for the NN results in all process groups. Here, the R² of each group is noted to be higher than the R² obtained from the all data of Fig. 8(b), indicating that the combination with a hierarchical clustering in a machine learning approach successfully enables the reliable prediction of dynamic globularization. Furthermore, this result implies that dominant globularization mode of the present Ti-6246 alloy is different depending on process conditions. On the JMAK result from Fig. 10, we can observe a high R² value in Gr.1, while there is no correlation between the predicted and experimental results in Gr.2 and Gr. 3, pointing out that dynamic globularization dominated by a thermal activation process (expressed by the JMAK model of Eq. (6)) can be applied only in process conditions of Gr. 1. From this NN analysis, Figure 11 summarizes the sensitivity [expressed by fraction (in%)] showing the importance which affects the dynamic globularization behavior. This approach to assessing the predictor variables’ importance is to compute a sensitivity for each variable. The fraction of sensitivity that affects the dynamic globularization is derived from a measure how much the predicted value’s error increases when the variables are excluded from the model one at a time. Herein, instead of actually excluding variables, they are fixed at a constant value of an average value. So, the determined sensitivity (shown by fraction) indicates the importance affecting the behavior. From Fig. 11, we can observe the different result in variables’ importance among Gr.1–3; a strain rate and a strain strongly affect the dynamic globularization behavior in Gr. 2, while a temperature and a strain rate do it in Gr. 3. In contrast, a strain is found to affect the dynamic globularization behavior the most in Gr. 1 in which JMAK approach can be adopted. As we pointed out from Fig. 7(a), dynamic globularization behavior is reasonably explained in relation to the Z parameter (expressed by temperature and strain rate as Eq. (5)), being indicative of the occurrence under thermal activation process. Here, the obtained data of Fig. 7(a) are the result of the microstructure observed at central region of forged specimen, so the amount of effective strain accumulated under forging is almost similar among these microstructures. Thus, the strain rate and temperature indeed affect the dynamic globularization behavior. Furthermore, according to the sensitivity fraction obtained in Gr.1 (Fig. 11), it reveals that the amount of strain has the much strongest influence on dynamic globularization fraction among all process conditions. This is in good agreement with the experimental result by Shell et al.³⁸⁾ pointing out that large strains are required to initiate or complete the globularization owing to the difficulties of the development of the interfaces as the source of breakup and the large penetration depth causing the breakup within the lamellae. In this way, the dominating factors on dynamic globularization (affected by a temperature, a strain rate and a strain) can be quantitatively clarified thanks to the combination with the machine learning approach.

Fig. 9.

The dendrogram of process conditions analyzed by a hierarchical clustering (by Ward method). Herein, process conditions are classified into three groups of Gr. 1, Gr. 2 and Gr. 3.

Fig. 10.

Correlation between the experimental result and the predicted results (JMAK result and NN result) of the dynamic globularization fraction for the process condition of (a) Gr. 1, (b) Gr. 2 and (c) Gr. 3 (in the case of the lamellar starting microstructure).

Fig. 11.

Sensitivity (expressed in%) of the predictor variables’ importance on the dynamic globularization according to NN analysis for each group.

Next, the behavior models of dynamic globularization on the basis of the JMAK approach and the machine learning (established as abovementioned) has been implemented into the FEM software of DEFORM-3D (v.12) through a user-subroutine followed by simulation of isothermal forging. Here, the incremental form of the equation was developed as expressed in Eqs. (11) and (12) for the calculation of JMAK approach of Eq. (6). Thanks to this incremental form, it enables to accumulate small changes of temperatures and strain rate at each step in FEM analysis.   

$$f_{DG}^{n+1}=f_{DG}^n+d{\left(f_{DG}\right)}^n$$(11)

  

$$\text{d}{\left(f_{DG}\right)}^n=\frac{\partial f}{\partial \epsilon }\text{d}\epsilon +\frac{\partial f}{\partial {\epsilon }_c}\text{d}{\epsilon }_c+\frac{\partial f}{\partial k}\text{d}k$$(12)

For FEM calculation combined with the machine learning approach, the obtained results of clustering and NN algorithms were implemented into FEM code followed by calculation. Figure 12 shows the distribution of dynamic globularization fraction of α phase across the cross section of the forged specimen tested at (a) (b) 850°C and 10⁻² s⁻¹ and (c) (d) 800°C and 10⁻³ s⁻¹ (at a height strain of 50%) calculated from (a) (c) the JMAK approach and (b) (d) the machine learning approach (with clustering and NN algorithms). We can observe from the heterogeneous distribution in Fig. 12 that dynamic globularization is frequently occurred especially at central region. This is mainly due to that strains are locally accumulated in the central region of a forged specimen. Comparing the distribution of the JMAK result to the machine learning result, a similar distributed behavior is apparently observed. In more detail, for the machine learning result, we can observe a wider region for testing at 850°C and 10⁻² s⁻¹ that dynamic globularization is activated. Additionally, we can observe a more local accumulation that dynamic globularization is enhanced in testing at 800°C and 10⁻³ s⁻¹. As abovementioned, there is a good correlation between the experimental result and the machine learning result. Hence, a further direction of this study for more precise prediction will be to reconsider and brash up of a physical model of JMAK approach in optimal combination with a machine learning approach followed by contribution to understanding of physical phenomena more deeply.

Fig. 12.

Distributions of the dynamic globularization fraction from the FEM analysis of (a) (c) JMAK approach and (b) (d) machine learning approach (clustering and neural network) at testing conditions of (a) (b) 850°C and 10⁻² s⁻¹ and (c) (d) 800°C and 10⁻³ s⁻¹. (Online version in color.)

4. Summary

This work experimentally analyzed the flow behaviors and the microstructural conversion mechanism during the hot forging process of the Ti-6246 alloy with various starting microstructures of a lamellar, a bimodal and an equiaxed microstructures. Additionally, dynamic globularization behavior was modeled in terms of a phenomenological Avrami approach and data analytics of a machine learning, followed by clarification of the globularization kinetics in relation to deformation conditions.

(1) Plastic flow behavior tested in an (α+β) temperature region exhibit an extensive flow softening for all starting microstructures. The continuous flow softening is remarkable with a decreasing of testing temperature and an increasing of strain rate. Among the all starting microstructures, the overall degree of flow softening is the most remarkable for the lamellar starting microstructure due to the frequent occurrence of lamellae kinking and globularization of the α phase. The estimated activation energies for hot deformation in an (α+β) temperature region are 319.3, 370.8 and 452.6 kJ/mol for the lamellar, the bimodal and the equiaxed starting microstructures, respectively.

(2) On the microstructural conversion of the α-lamellae in the lamellar and bimodal starting microstructures, lamellae kinking is dominantly occurred and dynamic globularization is enhanced with increasing testing temperature and decreasing strain rate. Herein, the globularization behavior is dominated by thermal activation process. On the other hand, CDRX is dominantly occurred for the β phase under deformation, and the enhanced CDRX is observed for the lamellar starting microstructure compared to the bimodal starting microstructure, which is attributable to a large amount of strains accumulated locally in the lamellar starting microstructure due to a higher contacting area of α/β interface.

(3) On the dynamic globularization behavior of the Ti-6246 alloy with a lamellar starting microstructure, a phenomenological model of JMAK and a neural network approach don’t correspond with the experimental globularization values if all testing conditions in the present study are taken into consideration and calculation. On the other hand, when the processing conditions are categorized (into 3 groups of Gr. 1–3) according to a hierarchical clustering, the estimated results by JMAK approach correspond reasonably well with the experimental data in process conditions of Gr. 1 (corresponding to a region with a higher temperature and a higher dynamic globularization fraction). Additionally, the sensitivity analysis from NN analysis can quantitatively estimate the variable’s importance and it reveals that a strain is judged to strongly affect the dynamic globularization behavior in process conditions of Gr. 1. Thus, this work points out the possibility of a reliable prediction of dynamic globularization according to an optimal combination of the JMAK model and the machine learning approach.

Acknowledgement

This work was financially supported in part by the Amada Foundation (AF-2018026) and a Grant-in-Aid for Scientific Research from the Light Metals Educational Fundation, Inc., Japan.

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