Substructure evolution in two phases based constitutive model for hot deformation of TC18 in α + β phase region

Lin LI, Jixiong LIU,b, Ning DING, Mioqun LI,*

aSchool of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

bBAOTI Group Co., Ltd, Xi’an 721014, China

KEYWORDSConstitutive model;Dynamic recovery;Hot deformation;Substructure;TC18

AbstractThe α +β dual phase titanium alloys are key structural materials in aviation and aerospace industries,and the complicated flow behavior of these titanium alloys during hot deformation requires to establish a constitutive model incorporating physical mechanism for optimizing processing parameters and designing forming tools.This work aims to establish a constitutive model incorporating physical mechanism for hot deformation of TC18 in α+β phase region.Firstly,the flow behavior and microstructure evolution for hot deformation of TC18 in α+β phase region are characterized.The TC18 shows significant strain hardening rate and negative strain hardening exponent around and after peak flow stress, respectively.After peak flow stress, Dynamic Recovery (DRV)mechanism dominates the evolution of α and β phases according to the results of substructure evolution.Then, the internal state variables method is applied to establish a constitutive model incorporating physical mechanism for hot deformation of dual phase titanium alloys.The variation of dislocation density during the hot deformation of titanium alloys is modeled by considering the accumulation of dislocation due to the impediment to dislocation movement by substructure obstacles and the annihilation of dislocation due to the dynamic restoration effect.The interaction between dislocations,the subgrain boundaries and the grain/phase boundaries obstruct the dislocation movement in the α phase, and the first two obstructs the dislocation movement in the β phase during the hot deformation of TC18.The dislocation annihilation process in the α and β phases during the hot deformation of TC18 is dominated by DRV.Finally, the substructure evolution in the two phases based constitutive model for hot deformation of TC18 in α + β phase region is presented.This model is well applied to predict the flow stress and quantitively analyze the role of DRV effect in the evolution of α and β phases during the hot deformation of TC18.

1.Introduction

Titanium alloys have high strength,low density,excellent resistance against corrosion and high temperature performance,which are quite suitable for manufacturing the structural components in the aviation, aerospace, marine, and chemical industries.1–4Most of engineering titanium alloys consist of the Hexagonal Close-Packed (HCP) α and the Body-Centered Cubic (BCC) β phases, and the content and the microstructure including morphology, distribution and sizes of the α and β phases5significantly affected the mechanical property of dual phase titanium alloys.The microstructural features of the dual phase titanium alloys can be adjusted via thermomechanical processes.In practice, the hot deformation in the dual phase region (α + β deformation) was applied to titanium alloys so as to manufacture the component with balanced strength and ductility,and also to breakdown the lamellar α into equiaxed α grains,6–10and that in the single phase region (β deformation) was applied for the benefit of refining the β grain size.

The relationship between the processing parameters(deformation temperature, strain rate and strain) for the hot deformation and the deformation behavior, which was usually described as the constitutive relationship,11–16was quite important for manufacturing the titanium alloys components in industries.In general, the constitutive models during the hot deformation of titanium alloys can be divided into the following three types.Firstly, an empirical regression model was widely investigated, which was represented by the classical Arrhenius-type equation.17,18Especially, the straincompensated Arrhenius-type model simultaneously incorporating the effect of deformation parameters was used to well predict the flow stress of titanium alloys during the hot deformation.19–22The second was statistical modelling without understanding the physical background during the hot deformation, such as the Artificial Neural Network (ANN)model.23–25The third type was a constitutive model incorporating the physical deformation mechanism, in which the metallurgical activities such as Dynamic Recovery (DRV) and Dynamic Recrystallization (DRX) processes for the β deformation of Ti1726and Ti-10-2-327were taken into consideration.Thus, the constitutive model incorporating the physical deformation mechanism can describe the thermally activated mechanism in the hot deformation.

To establish a constitutive model incorporating the physical deformation mechanism of dual phase titanium alloys during the hot deformation was of great challenge owing to the complicated flow behavior,28,29significant microstructure evolution30–34and deformation heterogeneity35–37of these titanium alloys.The Internal State Variables (ISVs) method was quite effective to involve the deformation mechanism within the constitutive model of titanium alloys during the hot deformation.Constitutive modelling for Ti-6Al-4V during the α + β and β deformation was obtained by using a twoparameter ISVs,in which the changes in the grain size,the dislocation density and the phase volume fraction were taken into consideration.38Babu and Lindgren39used the dislocation density and the vacancy concentration as the ISVs in the constitutive modelling during the α + β deformation of Ti-6Al-4V,in which the dynamic globularization induced flow softening effect was well described.Gao et al.40established a constitutive model incorporating the physical mechanism during the α + β deformation of TA15 by considering the microstructure and damage via the ISV method.To sum up,a constitutive model incorporating the physical mechanism via the ISV method can simultaneously describe the microstructure development and the deformation behavior during hot deformation.

TC18,a dual phase titanium alloy showing the high toughness and the excellent thermal-treatability, has been widely applied in manufacturing the load-bearing components of large aircrafts.41Lin et al.42established the straincompensated Arrhenius-type and the ANN constitutive models to describe the flow behavior during the α+β deformation of a cast TC18 billet,and both models showed high prediction accuracy.Chang and Zheng43incorporated the Zener-Holloman parameter into the Arrhenius-type constitutive equation and achieved high accuracy to describe the relationship between the flow stress and the processing parameters in the α + β deformation of TC18.It is noted that there existed the significant evolution of the α and β phases during the α + β deformation of TC18,44,45and thus the constitutive model incorporating physical mechanism during the α + β deformation of TC18 should simultaneously involve the deformation behavior of the α and β phases.However, such a constitutive model for the hot deformation of TC18 has not been achieved, and the accurate prediction of the flow behavior of TC18 with sound physical meaning is not reached.This study aimed to establish a constitutive model incorporating the physical deformation mechanism in the α + β deformation of TC18 via ISV approach.In particular, a general description of the change in dislocation density in the hot deformation of titanium alloys was established, in which the accumulation of dislocations was modeled by considering the effect of different microstructure and/or substructure obstacles and the annihilation of dislocations was modeled by considering DRV and DRX processes.Finally,the change in dislocation density was applied to the α and β phases in the α + β deformation of TC18 according to the experimental characterization of microstructure evolution.

2.Material and experimental procedures

The chemical composition of TC18 was 5.28wt%Al,5.05wt%V, 4.88wt% Mo, 1.09wt% Cr, 1.03wt% Fe, <0.01wt% Zr,<0.01wt% Si, 0.032wt% C, 0.001wt% H, 0.1wt% O,0.008 wt% N and Ti (bal.).The temperature for the α to β phase transformation of TC18 was 1138 K.There were the equiaxed primary α grains,the acicular secondary α grains and the retained β phase of the as-received hot forged TC18 bar with 400 mm in diameter (Fig.1(a), where T is the absolute deformation temperature, K), in which the average grain size of the equiaxed primary α grains is 3.8 μm.The cylindrical TC18 specimens with 12 mm in height and 8 mm in diameter manufactured via wire electrical discharge machining were subjected to the α + β deformation on a Thermecmastor-Z simulator.The deformation temperatures were 1063, 1083,1103 K, the strain rates were 0.01, 0.5, 1 s-1, and the height reductions were 10%,30%and 50%.As shown in schematic diagram of Fig.1(b), the TC18 specimens were isothermally treated at the deformation temperature for 5 min before loading in order to achieve uniform temperature distribution within the specimen.The temperature during hot deformation was measured via thermocouple welded in the central surface of specimen and the graphite powder layer was placed between the specimens and the anvils for the benefit of reducing die friction.Following compression,the specimens were air cooled to room temperature.It is mentioned that the air cooling is fast enough to avoid phase transformation in TC18 during cooling.For the microstructural examination,the as-compressed TC18 specimens were cut along the axial sections, and the central portions were mechanical polished and then electropolished by using a solution of 6 % HClO4, 64 % CH3OH and 30 %CH3(CH2)3OH.Evolutions of the Optical Microstructure(OM)and the orientation of TC18 during the hot deformation were examined by using the optical microscope (Leica DMI3000M) and a Scanning Electron Microscope (SEM,TESCAN MIRA3 XMU) equipped with an Electron Backscattered Diffraction Detector (EBSD, NordlysMax),respectively.

3.Experimental results and discussion

Fig.2 shows the flow stress–strain curves in the α + β deformation of TC18.At the beginning of the deformation,the flow stress of TC18 increases rapidly to the peak flow stress, which should be ascribed to the strain hardening effect.46The strain hardening rate (θ)46to characterize the strain hardening effect in the α + β deformation of TC18 is

where σ is the flow stress, MPa; ε is the strain.

Fig.2 Flow stress–strain curves in α+β deformation of TC18.

Fig.3 shows the strain hardening rate curves at the beginning of α+β deformation of TC18,i.e.,at the strains ranging from 0 to 0.03.The strain hardening rate is very high at the beginning of deformation,and then decreases gradually to zero when reaching the peak flow stress (indicated by the lines in Fig.3).With the further increment of strain,the strain hardening rate become negative and the flow stress decreases.After the peak flow stress in the α + β deformation of TC18, the strain hardening exponent (n) to further characterize the flow behavior is47

Fig.3 Strain hardening rate curves in α + β deformation of TC18 at strains ranging from 0 to 0.03.

where ˙ε is the strain rate, s-1.

Fig.4 Strain hardening exponents in α + β deformation of TC18 at strains above 0.1.

Fig.4 shows the strain hardening exponents in the α + β deformation of TC18.It is seen that the strain hardening exponent in the α + β deformation of TC18 is negative at the strains above 0.1.Besides, the strain hardening exponents at larger strains are lower than that at lower strains, suggesting that the restoration effect during the α + β deformation of TC18 becomes more and more evident as the strain increases.The flow stresses in the α + β deformation of TC18 at higher deformation temperatures are lower than that at lower deformation temperatures since the enhanced α to β phase transformation occurs and the thermally activated movement of dislocations is promoted at higher deformation temperatures.Besides, the flow stress in the α + β deformation of TC18 at a higher strain rate is larger than that at a lower strain rate,which is due to the larger impediment to deformation by the large quantities of dislocations at a higher strain rate.It is concluded that there exists the significant strain hardening effect at the beginning of the α + β deformation of TC18, and the restoration effect dominates the α + β deformation of TC18 at larger strains.Besides, the deformation temperature and the strain rate show significant effect on the flow behavior.Therefore, the microstructure in the α + β deformation of TC18 should be characterized so as to clarify the deformation mechanism.

Fig.5 shows the OM images in the α + β deformation of TC18, in which Δ is height reduction, and the acicular secondary α grains in the as-received TC18 dissolve generally into the β matrix.The Compression Axis(CA)is along the vertical direction in all micrographs.Table 1 summarizes the α grain size and the volume fraction of α phase in the α+β deformation of TC18 shown in Fig.5.The primary α grains along the CA rotate gradually to be perpendicular to the CA as the strain increases at T = 1063 K, ˙ε=1 s-1(Figs.5(a) and (b)).The fragmentation of long α grains into equiaxed α grains(shown by the arrow in Fig.5(b))as the strain increases results in the slight decrease in the α grain size (Table 1).Furthermore, there is the slight decrease in the volume fraction of α phase as the strain increases (Table 1), which could be ascribed to the enhanced deformation heat effect with the progression of deformation.48Besides,it is reported that the α to β dynamic phase transformation occurs in the hot deformation of titanium alloy,49which will contribute to the decrease in the volume fraction of α phase as the strain increases.Following compression at ˙ε = 1 s-1, Δ = 50 % (Figs.5(b)–(d) and Table 1), the α to β phase transformation is much more enhanced at T = 1103 K in comparison with that at T = 1063 K and T = 1083 K, which results in a decrease in the α grain size.Following compression at T = 1063 K,Δ = 50 % (Figs.5(b), (e), (f) and Table 1), the effect of the strain rate on the α grain size is negligible.The fragmentation of large α grains during the α+β deformation of TC18 will be enhanced at a high stress level,i.e.,during the deformation at a high strain rate.However, the static grain growth50at a low strain rate is more sufficient in comparison with that at a high strain rate.Moreover,the plastic strain induced dynamic grain growth and the dislocation density induced grain size variation51also affect the grain size.As a result, the strain rate affects the α grain size slightly under the combination effects of the above-mentioned factors.In addition, the strain rate shows negligible influence on the volume fraction of α phase in the α + β deformation of TC18 at T = 1063 K,Δ = 50 %, as shown in Table 1.

Fig.5 OM images in α + β deformation of TC18.

Table 1 α grain size and volume fraction of α phase in α + β deformation of TC18.

Fig.6 EBSD orientation distribution (//CA) of α phase.

Fig.7 Grain boundary misorientation distribution in α phase.

Fig.8 Grain boundary misorientation distribution in β phase.

Fig.6 shows the α phase orientation distribution (//CA)during the α + β deformation of TC18, in which TD and RD represent two mutually perpendicular radial directions of the cylindrical specimen, the β phase is represented in band contrast,CA is along the vertical direction in all micrographs.The High-Angle Grain Boundaries(HABs,>15°)in Fig.6 are shown by the black lines and the Low-Angle Grain Boundaries(LABs, 2°–15°) are shown by the white lines.Figs.7 and 8 show the grain boundary misorientation distribution for the α and β phases in Fig.6, respectively.Following compression at T=1063 K, ˙ε=1 s-1,there are mainly the HABs in the primary α grains of TC18 at Δ=10%(Figs.6(a)and 7(a)),and more LABs occur in the primary α grains of TC18 at Δ=50%(shown by the arrows in Figs.6(b)and 7(b)).Moreover, there are mainly discontinuous LABs in the β phase at T = 1063 K, ˙ε=1 s-1, Δ = 10 % (Fig.6(a)) while a subgrain structure occurs at T = 1063 K, ˙ε=1 s-1, Δ = 50 % (Fig.6(b)).The fractions of LABs in the β phase in the α+β deformation of TC18 at T=1063 K, ˙ε=1 s-1are quite high(Figs.8(a) and (b)).The evolution of LABs in the α and β phases of TC18 at T = 1063 K, ˙ε=1 s-1indicates that the microstructure evolution is induced by DRV, in which the accumulated dislocations with the increasing of strain are rearranged into LABs.52Following compression at T = 1103 K, ˙ε=1 s-1,Δ = 50 % (Fig.6(c)), the primary α grains of TC18 generally show similar orientation within the grains, and the fraction of LABs for the α phase (Fig.7(c)) is much lower than that at 1063 K(Fig.7(b)).The promoted α to β phase transformation of TC18 at 1103 K in comparison with that at 1063 K will largely consume the dislocations, contributing to the lower fraction of LABs for the α phase.The β phase of TC18 at T=1103 K, ˙ε=1 s-1,Δ=50%(Fig.6(c))is generally characterized by the subgrains.The fraction of LABs for the β phase at T=1103 K, ˙ε=1 s-1,Δ=50%(Fig.8(c))is similar with that at T = 1063 K, ˙ε=1 s-1, Δ = 50 % (Fig.8(b)),which is due to that the evolution of β phase is dominated by the significant DRV effect, as reported in the hot deformation of titanium alloys.35,52,53Following compression at T = 1063 K, ˙ε=0.01 s-1, Δ = 50 % (Fig.6(d)), the fraction of LABs for the α phase of TC18(Fig.7(d))is lower than that at T=1063 K, ˙ε=1 s-1,Δ=50%(Fig.7(b))since the large quantity of dislocations occur at a higher strain rate.Even though the subgrain in the β phase of TC18 at T = 1063 K,˙ε=0.01 s-1, Δ = 50 % (Fig.6(d)) is not as evident as that at T = 1063 K, ˙ε=1 s-1, Δ = 50 % (Fig.6(b)), the fraction of LABs in the β phase at T = 1063 K, ˙ε=0.01 s-1,Δ = 50 % (Fig.8(d)) is still very high due to the significant DRV effect in the evolution of the β phase.

4.Constitutive model

4.1.Dislocation density

The plastic deformation of metallic materials is attributed to the dislocation motion,39,54in which the mutual interactions between dislocations and other obstacles such as grain boundaries hinder the motion of dislocations.Therefore, an increase in stress is needed for an increase in strain, which is usually defined as the strain hardening effect.On the other hand, the occurrence of dynamic restoration process such as DRV and DRX30,31,35,52,53in the hot deformation of metallic materials counteracts the strain hardening effect via annihilation or decomposition of dislocations.Therefore, the flow stress in the hot deformation of metallic materials can reveal the average behavior of the dislocations,and the dislocation density is widely chosen as the ISV for constitutive modeling of metallic materials.Besides,the evolution of dislocations in the metallic materials is quite important since it will significantly affect the mechanical performance of materials.For instance, the stored dislocations in the materials after hot deformation are a combined result of the strain hardening process and the dynamic restoration process,which directly affect the strength of material.Su et al.55shows that the annihilation of dislocations could provide more spaces for the storage of dislocation and thus enhances the ductility of Ti-6Al-4V alloy.As a result,clarifying the change in the dislocation density in the hot deformation of metallic materials is quite important for tailoring the mechanical property of metallic materials via thermomechanical processes.

In terms of the Kocks-Mecking theory,56the variation of dislocation density (q) with shear strain (dq/dγ) for the hot deformation of metals is the combination result from the accumulation of dislocations ((dq/dγ)+) and annihilation of dislocations ((dq/dγ)–), which is expressed as

where n′is the number of obstacles;Liis the mean free path for a specific obstacle, μm.

Generally,there exist the following three kinds of substructure obstacles that decide the mean free path for the dislocation movement of titanium alloys during the hot deformation.Firstly, the interaction between the dislocations during the random trapping of mobile dislocations will lead to the dislocation accumulation.In this case, the distance between the dislocations decides the dislocation accumulation and the corresponding mean free path (Ls) is54,57,58

As seen from Eqs.(6) and (8), the distance between the dislocations and the subgrain boundaries have an analogous effect on the mean free path for the dislocation movement during the hot deformation of titanium alloys.Therefore,the influence of the interaction between dislocations and subgrain boundaries on the mean free path can be expressed in

where k1is the material constant.

Thirdly,the grain/phase boundaries obstruct the movement of dislocations and the corresponding mean free path (Lg)depends on the grain size (dg)60as

where C is the proportionality constant; X is the DRX fraction; ˙X is the DRX rate, s-1.

Therefore, by considering the dislocation accumulation process resulting from the impediment to dislocation movement by the substructure obstacles (the interaction between the dislocations, the subgrain and grain/phase boundaries)and the dislocation annihilation process due to DRV and DRX processes during the hot deformation, the dislocation density variation is given in

where M is the Tayor factor.

4.2.Grain size

The variation of grain size for the hot deformation of titanium alloys consists of the grain boundary migration induced static grain growth, the plastic strain induced dynamic grain growth and the dislocation density induced grain size variation.38,51

The static grain growth rate (˙dstatic) is highly related to the deformation temperature as given in

where δ is the thickness of grain boundary, m; D0is the selfdiffusion constant,m2?s-1;Qgbis the grain boundary diffusion activation energy,kJ?mol-1;k is the Boltzman’s constant,k=1.38 × 10-23J?K-1.

The static grain growth rate (˙dstatic) is rewritten as Eq.(16)by substituting Eq.(15) into Eq.(14):

4.3.Constitutive law

The total shear stress τ is the sum of a thermally activated component τ* and an athermal component τμin terms of the Taylor’s assumption, which is shown in61

where τ* corresponds to the stress needed for the dislocations to travel across the lattice and pass through the short-range obstacles such as impurities and solute atoms, MPa; τμresults from the long-range strain hardening effect induced by the substructure obstacles (the interaction between the dislocations, the subgrain and grain/phase boundaries) and larger microstructural pinning features such as the second phase particles, MPa.

The thermally activated component τ* is highly related to the deformation temperature and the strain rate:56

where τ0is the thermal activated stress at 0 K,MPa;ΔG is the deformation activation energy, kJ?mol-1; ˙ε0is the reference strain rate representing the dislocation vibrational frequency arrested at an obstacle,63s-1; p and q are the material constants.

Since the dislocation accumulation associated with the substructure obstacles is presented in Section 4.1, the athermal activated stress τμresponsible for the strain hardening effect can be expressed by a one-parameter internal variable, i.e.,the total dislocation density q, is

where a is an empirical constant usually in the range of 0.2–0.5;G is the temperature dependent shear modulus, MPa.

For titanium alloy, G is expressed as61

Then, the shear stress τ can be converted into the normal stress by using a Taylor factor in

4.4.Constitutive model for hot deformation of TC18

4.4.1.Model description

According to the microstructural characterization in Section 3,the β phase during the α+β deformation of TC18 is generally characterized by the subgrains, while the high-angle grain boundaries occur in the α phase.Thus,the interaction between the dislocations and the subgrain boundaries obstruct the dislocation movement in the β phase during the α + β deformation of TC18,and the interaction between the dislocations,the subgrain boundaries and the grain/phase boundaries obstruct the dislocation movement in the α phase.In addition, DRV plays a significant role in the α and β phases evolution during the α + β deformation of TC18.However, DRX does not occur in the α and β phases during the α + β deformation of TC18.Consequently, the role of DRV in the dislocation annihilation in the α and β phases during the α + β deformation of TC18 must be considered.According to the analysis in Section 4.1, the dislocation density rate in the α phase during the α + β deformation of TC18 is where Tβis the absolute temperature for the α to β phase transformation, Tβ= 1138 K.

Therefore, the substructure evolution in two phases based constitutive model for the α + β deformation of TC18 by incorporating the physically deformation mechanisms is

4.4.2.Identification of material constants

Tables 2 and 3 shows the experimental α grain size and flow stress in the α + β deformation of TC18 (teacher’s sample data) used to identify the material constants, respectively.The genetic algorithm-based objective optimization method is applied to identify the material constants by minimizing the objective function.The squares of the difference between the experimental and the calculated α grain size and the flow stress are used to define the objective functions as

4.4.3.Validation of model

Tables 2 and 3 show the experimental α grain size and flow stress during the α + β deformation of TC18 used to verify the present model, i.e., testing sample data, respectively.Fig.9 shows the experimental and calculated α grain size in the α + β deformation of TC18.The average relative errorbetween the experimental grain size and the calculated of the α phase (AARE(dα)) is

Table 3 Flow stress in α + β deformation of TC18 to identify material constants (teacher’s samples) and verify model (testing samples).

Fig.9 Comparison between experimental and calculated α grain size in α + β deformation of TC18.

According to Eq.(35),AARE(dα)is calculated to be 7.4%,suggesting that the calculated α grain size in the α + β deformation of TC18 coincides well with the experimental.

Fig.10 shows the experimental flow stress and the calculated in the α + β deformation of TC18.The average relative error between the experimental and the calculated flow stress((AARE(σ))) is

According to Eq.(36), AARE(σ)is calculated to be 3.2%.Therefore, the flow stress during the α + β deformation of TC18 by applying the present model is accurately predicted.Fig.11 shows the variation of the calculated flow stress with strain in the α + β deformation of TC18.As seen from Fig.11, the calculated flow stress in the α + β deformation of TC18 decreases as the strain increases at different deformation temperatures and strain rates, which coincides well with that of the experiment.

Fig.10 Comparison between experimental and calculated flow stress in α + β deformation of TC18.

Fig.11 Comparison between variation of calculated flow stress with strain and experimental flow stress–strain curves in α + β deformation of TC18.

Fig.12 shows the λ values for the α and β phases in the α + β deformation of TC18, in which the negative value of λ presents that DRV effect shows the opposite effect to the change in the dislocation density in comparison with that of the strain hardening effect.At the strain rates of 1 s-1and 0.5 s-1,the DRV effect plays a more and more important role in the change in dislocation density both for the α(Figs.12(a)–(c)) and β (Figs.12(d)–(f)) phases as the strain increases in comparison with that of the strain hardening effect.In particular,the DRV effect on the change in the dislocation density is more evident at lower strain rates,especially at a strain rate of 0.01 s-1that the DRV effect on the change in the dislocation density is quite significant from the beginning of deformation.Therefore, Fig.12 quantitatively shows the role of the DRV effect in the evolution for the α and β phases in the α + β deformation of TC18, and the DRV effect is more evident at lower strain rates as reported in the hot deformation of titanium alloys.35,52,53

Fig.12 λ values for α and β phases in α + β deformation of TC18.

Fig.13 Structure illustration of ANN model and comparison between experimental and calculated flow stress in α+β deformation of TC18.

For comparison,the Backpropagation(BP)ANN model is applied to predict the flow stress in the α + β deformation of TC18 because the backpropagation ANN model is quite effective to predict a non-linear relationship.42Generally,there are an input layer, an output layer and several hidden layers in ANN structure.In the present ANN model, the parameters in the input layer are the deformation temperature (T, K),strain rate (˙ε, s-1) and strain (ε), the parameter in the output layer is the flow stress (σ, MPa), and the activation function in the hidden layer is a sigmoid function.There are seven neutrons in the hidden layer in the present ANN model determined by using trial–error method.Then, the present ANN structure is 3-7-1, as shown in Fig.13(a).The experimental data at 1063 K and 1103 K and different strain rates is used as the training data and that at 1083 K and different strain rates is used as the testing data.Fig.13(b) shows the experimental and calculated flow stress in the α + β deformation of TC18 via backpropagation ANN model, and the average relative error between the experimental and the calculated flow stress (AARE(σ)) for the testing data by using Eq.(36) is 3.3 %.As a result, it is seen that the substructure evolution in two phases based constitutive model during the α+β deformation of TC18 shows the same prediction accuracy as that by using ANN model.However, the backpropagation ANN model does not describe the physical mechanisms during the hot deformation of metallic materials.Instead, the substructure evolution in two phases based constitutive model lays foundation for studying the flow behavior during the α + β deformation of TC18, which is quite important for manufacturing the TC18 components with high properties.

5.Conclusions

The substructure evolution in the two phases based constitutive model for the hot deformation of TC18 in the α+β phase region is represented by taking the dislocation density and grain size as ISVs according to the experimental characterization of the flow behavior and the microstructure evolution.The following results are obtained.

(1) The significant strain hardening rate at the beginning of the α + β deformation of TC18 leads to the rapid increase in the flow stress to the peak flow stress.The negative strain hardening exponent in the α + β deformation of TC18 after the peak flow stress is due to the dominant role of DRV effect in the evolution of α and β phases.

(2) The variation of the dislocation density of titanium alloys during the hot deformation is modeled by considering the accumulation of dislocation due to the impediment to dislocation movement by the substructure obstacles and the annihilation of dislocation due to the dynamic restoration effect.

(3) Apply the substructure evolution in the two phases based constitutive model to the α + β deformation of TC18,the interactions between the dislocations,the subgrain and grain/phase boundaries decide the dislocation accumulation in the α phase and the first two decides the dislocation accumulation in the β phase.The model for dislocation annihilation in the α and β phases is represented by considering the DRV effect.

(4) The substructure evolution in the two phases based constitutive model is well applied to predict the flow stress in the α + β deformation of TC18, which shows the same accuracy with that via backpropagation ANN model.In addition, the substructure evolution in the two phases based constitutive model quantitatively predict the role of the DRV effect in the evolution of α and β phases of TC18.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China (Nos.51474375 and 51275416) and the China Postdoctoral Science Foundation (No.2018M633571).