materials Article
Constitutive Model Prediction and Flow Behavior Considering Strain Response in the Thermal Processing for the TA15 Titanium Alloy Jiang Li 1, * , Fuguo Li 1, * and Jun Cai 2 1 2
*
State Key Laboratory of Solidification Processing, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an 710072, China School of Metallurgical Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China;
[email protected] Correspondence:
[email protected] (J.L.);
[email protected] (F.L.); Tel.: +86-029-88492643 (J.L.)
Received: 3 September 2018; Accepted: 12 October 2018; Published: 15 October 2018
Abstract: To investigate the flow stress, microstructure, and usability of TA15 titanium alloy, isothermal compression was tested at 1073–1223 K and strain rates of 10, 1, 0.1, 0.01, and 0.001 s−1 , and strain of 0.9. The impact of strain and temperature on thermal deformation was investigated through the exponent-type Zener–Hollomon equation. Based on the influence of various material constants (including α, n, Q, and lnA) on the TA15 titanium alloy, the strain effect was included in the constitutive equation considering strain compensation, which is presented in this paper. The validity of the proposed constitutive equation was verified through the correlation coefficient (R) and the average absolute relative error (AARE), the values of which were 0.9929% and 6.85%, respectively. Research results demonstrated that the strain-based constitutive equation realizes consistency between the calculated flow stress and the measured stress of TA15 titanium alloy at high temperatures. Keywords: TA15 titanium alloy; constitutive equation; strain compensation; flow stress
1. Introduction TA15 alloy, a near-α titanium alloy, is extensively used in the aerospace industry due to its various advantages, such as weldability, mechanical capability at high temperatures, superior creep and erosion resistance, and large strength-to-weight ratio [1]. The mechanical properties and physical characteristics of TA15 titanium alloy have led to its wide application. The microstructure evolves in the process, and is influenced by process parameters: strain rate, strain, and temperature [2]. There is an interactive correlation between deformation behavior and microstructure evolution [3]. The required shape, properties, and microstructure can be obtained by optimizing the thermal deformation parameters in thermomechanical processing. Therefore, deformation and flow are being studied at elevated temperatures, which is conducive to investigating the thermal deformation ofTA15 titanium alloy. Scholars have investigated the deformation and microstructural evolution of TA15 titanium alloy [4–6]. Processing the components is difficult given the complex shape, hard-to-deform properties, and requirement of forming quality [7]. Gao et al. studied the microstructure evolution and flow behavior of near-α titanium alloy, whose microstructure is inhomogeneous in thermal deformation, finding that nonuniform microstructure before deformation was composed of β phase, lamellar α, and equiaxed α in the colony form. There was a correlation between the Burgers orientation and β phase in α colony [6]. According to Zhao et al., thermal flow stress is not significantly influenced
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by the initial β grain size [5]. By investigating the influence of inhomogeneous deformation on the microstructure of the rib-web part of TA15 alloy, Fan et al. found four microstructures in the forming process [8]. The superplasticity of TA15 alloy was enhanced using thermomechanical techniques, which could refine the grains of the alloy [9]. Thermal uniaxial tensile tests were conducted and finite element models were established to explore the thermal formability of TA15 titanium alloy [10]. Zhu et al. explored the influence of cooling speed on the major α phase microstructure evolution of TA15 titanium alloy. According to the research results, the size distribution and volume fraction of the major α phase are influenced by cooling speed [11]. The kinetics rate and dynamic globularization kinetics of TA15 are influenced by deformation parameters. The dynamic globularized fraction is in direct proportion to strain and temperature, but inversely proportional to strain rate [12]. A prediction model was established on the basis of an improved back propagation (BP) neural network, and it was used to explore the quantitative evolution of aspect ratio, grain size, and volume fraction of equiaxed α for TA 15 alloy [13]. As a foundation of engineering parts production, thermal deformation processes require not only microstructural and mechanical properties, but also dimensional accuracy [14]. As excellent mechanical properties require fine microstructural characteristics, it was necessary to explore the thermomechanical process affecting the microstructural characteristics. To this end, we used theconstitutive equation at various temperatures and strain rates. The Arrhenius model, as a phenomenological constitutive model, has generally been used to present the correlations among temperature, flow stress, and strain rate in a constitutive study, particularly at high temperatures [15]. The influence of strain has been proven to be necessary to verify the constitutive equation involving the strain effect, which predicted flow behaviors at elevated temperatures in pure titanium [16] and titanium alloy [15], steel [17–19], aluminum alloy [14,20,21], and magnesium alloy [22]. This research attempted to represent thermal deformation of TA15 titanium alloy by formulating a proper constitutive correlation. The thermal compression was tested at different temperatures and strain rates. The flow stress was further analyzed based on the test results. A comprehensive constitutive model involving temperature, strain rate, and flow stress was established. Finally, the reliability of the constitutive model was verified. 2. Experimental Details TA15 titanium alloy, a near-α titanium alloy that has the chemical composition of Ti-6.5Al-2Zr-1Mo-1V (in wt%), was used in the research. The original specimen was sectioned from the bar axial, ground to 2000 grit with sand paper, and then polished to 0.5 µm. Finally, the specimen was etched to carry out optical microscopy (GX51F, OLYMPUS, Tokyo, Japan) observation, and Kroll’s agent (2% HF, 4% HNO3 , and 94% H2 O) was used to etch the specimens for 3–5 s. The β transus temperature of TA15 was measured as approximately 1258 K, and all specimens used in the experiments were annealed for homogenization by heating at 1073 K for 1 h and cooling in the furnace. The microstructure shown in Figure 1 primarily consisted of a number of coarse strip-shaped α phases; meanwhile, there were only a few thin strip-shaped α phases on the β matrix. The isothermal compression simulation test is the most commonly used method to investigate the deformation and microstructural evolution of materials at high temperatures. An isothermal compression simulation experiment was carried out on a Gleeble-3500 tester (DSI Corporation, New York, USA). Cylindrical φ10 mm × 15 mm specimens were used in this study, as shown in Figure 2. The upper and lower end faces were parallel to each other. The mechanical perpendicularity of the vertical face was maintained, and the two end surfaces were smooth to decrease the effects of transverse friction on deformation. The axis of the cylindrical specimen was the axial line of the bar billet. In the experiment, a radial sensor was used in the deformation of the specimen to collect and record the cross-section area and the collected signals to control the parameters of the tester. The specimen was compressed at a constant strain rate. The whole test was electrically heated,
TA15 titanium alloy, a near-α titanium alloy that has the chemical composition of Ti-6.5Al-2Zr1Mo-1V (in wt%), was used in the research. The original specimen was sectioned from the bar axial, ground to 2000 grit with sand paper, and then polished to 0.5 μm. Finally, the specimen was etched to carry out optical microscopy (GX51F, OLYMPUS, Tokyo, Japan) observation, and Kroll’s agent (2% HF, 4% HNO3, and 94% H2O) was used to etch the specimens for 3–5 s. Materials 2018, 11, 1985 3 of 15 The β transus temperature of TA15 was measured as approximately 1258 K, and all specimens used in the experiments were annealed for homogenization by heating at 1073 K for 1 h and cooling and thefurnace. temperature was obtained with a thermocouple. The temperature deviation was in in the The microstructure shown in Figure 1 primarily consisted of a number ofcontrolled coarse strip◦ C increments. ± 1 shaped α phases; meanwhile, there were only a few thin strip-shaped α phases on the β matrix.
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The isothermal compression simulation test is the most commonly used method to investigate the deformation and microstructural evolution of materials at high temperatures. An isothermal compression simulation experiment was carried out on a Gleeble-3500 tester (DSI Corporation, New York, USA). Cylindrical Ф10 mm × 15 mm specimens were used in this study, as shown in Figure 2. The upper and lower end faces were parallel to each other. The mechanical perpendicularity of the vertical face was maintained, and the two end surfaces were smooth to decrease the effects of transverse friction on deformation. The axis of the cylindrical specimen was the axial line of the bar billet. In the experiment, a radial sensor was used in the deformation of the specimen to collect and record the cross-section area and the collected signals to control the parameters of the tester. The specimen was compressed at a constant strain rate. The whole test was electrically heated, and the temperature was obtained thermocouple. temperature deviation was controlled in ±1 °C Figure 1. with Initial of The the specimens of TA15 alloy. Initialamicrostructure microstructure titanium alloy. increments.
Figure Figure2.2.The Theappearance appearanceofofthe thespecimen: specimen:(a) (a)before beforedeformation deformationand and(b) (b)after afterdeformation. deformation.
The isothermal simulation compression experiment temperature ranged from 1073 K to 1223 K in increments of 50 K. The strain rate varied from 0.001 s–1 to 10 s–1, and the corresponding reduction ranges were 10%, 20%, and 60%. A graphite flake cushion was used between the specimen and the pressure head to decrease transverse friction. We heated the specimen to deformation temperature
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The isothermal simulation compression experiment temperature ranged from 1073 K to 1223 K in increments of 50 K. The strain rate varied from 0.001 s−1 to 10 s−1 , and the corresponding reduction ranges were 10%, 20%, and 60%. A graphite flake cushion was used between the specimen and the pressure head to decrease transverse friction. We heated the specimen to deformation temperature at a heating rate of 10 ◦ C/s, which was maintained for 3 min to guarantee Materials 2018, 11, x FOR PEER REVIEW 4 of 15 temperature uniformity. Thermal deformation occurred at the isothermal constant strain rate. The stress-strain curves were automatically recorded as the isothermal compression at elevated specimens were quenched immediately in water after the compression test to maintain the temperature was tested. The specimens were quenched immediately in water after the compression organization. test to maintain the organization.
3. Results and Discussion 3. Results and Discussion 3.1. Experiment Results and and Flow Flow Stress Stress Behavior Behavior 3.1. Experiment Results Figure 33depicts depicts stress-strain curves of titanium TA15 titanium alloy indeformation various deformation Figure thethe stress-strain curves of TA15 alloy in various conditions. conditions. The flow stress rises substantially as the strain increases due to the work-hardening The flow stress rises substantially as the strain increases due to the work-hardening resulting from resulting from the increasingly higher dislocation density. Flow stress showed at nearlyand all the increasingly higher dislocation density. Flow stress showed peaks at nearly allpeaks temperatures temperatures and strain rates, and a phenomenon clear flow softening phenomenon occurred. Peakless strain strain rates, and a clear flow softening occurred. Peak strain was generally thanwas 0.1, generally less than 0.1, and it did not change obviously with thermal deformation conditions. and it did not change obviously with thermal deformation conditions.
Figure Figure 3. 3. True stress versus true strain curves of TA15 titanium titanium alloy alloy obtained obtained by isothermal isothermal hot compression compression tests tests at at temperatures temperatures of: of: (a) (a) 1073 1073 K, K, (b) (b) 1123 1123 K, K, (c) (c) 1173 1173 K, K, and and (d) (d) 1223 1223 K. K.
As can be seen from Figure 3, the stress-strain curves of TA15 titanium alloy were obtained by As can be seen from Figure 3, the stress-strain curves of TA15 titanium alloy were obtained by testing the isothermal compression at 1073 K to 1223 K in intervals of 50 K. The flow stress first peaked testing the isothermal compression at 1073 K to 1223 K in intervals of 50 K. The flow stress first peaked and then decreased as the strain increased. In addition, flow stress was highly sensitive to temperature. and then decreased as the strain increased. In addition, flow stress was highly sensitive to It first sharply decreased and then flattened out as the temperature rose. We ignored the deformation temperature. It first sharply decreased and then flattened out as the temperature rose. We ignored heating due to the low strain rates (0.001 s−1 and 0.01 s−1 ). At–11173 K and–11223 K, the flow stress curves the deformation heating due to the low strain rates (0.001 s and 0.01 s ). At 1173 K and 1223 K, the flow stress curves flattened out as the strain rate decreased. Formation at low strain rates can only enhance dynamic recovery (DRV), but can also inhibit dynamic recrystallization (DRX). Above the phase transition temperature, the β grain size was large as the temperature rose. The microstructure indicates that the DRV was dominant with little DRX during formation. At different experimental
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flattened out as the strain rate decreased. Formation at low strain rates can only enhance dynamic recovery (DRV), but can also inhibit dynamic recrystallization (DRX). Above the phase transition temperature, the β grain size was large as the temperature rose. The microstructure indicates that the DRV was dominant with little DRX during formation. At different experimental temperatures, the volume fraction and grain size of the major α phase were inversely proportional to temperature [4]. TA15 titanium alloy presented flow softening, which was closely related to the thermal processing Materials 2018, 11, x FOR PEER REVIEW 5 of 15 conditions and the initial microstructure, which is common in the metal deformation process. The dynamic was inhibited DRV. Theby flow stress tended be stable after process. The recrystallization dynamic recrystallization wasbyinhibited DRV. The flow to stress tended tosaturation. be stable A small amount of DRX occurred at higher strain. The β-transus temperature was 1263 K, and the DRV after saturation. A small amount of DRX occurred at higher strain. The β-transus temperature was was the main softening mechanism in the β-phase field because of the body-centered cubic crystal 1263 K, and the DRV was the main softening mechanism in the β-phase field because of the bodystructure cubic with rapid and highself-diffusion stacking faultand energy centered crystalself-diffusion structure with rapid high[23]. stacking fault energy [23]. The microstructure variations of the specimens at 1223 K are are shown The microstructure variations of the specimens at 1223 K shown in in Figure Figure 4, 4, and and the the −1 to 10 s−1 . Figure 4a indicates that equiaxial recrystal deformation strain rates increased from 0.001 s −1 –1 deformation strain rates increased from 0.001 s to 10 s . Figure 4a indicates that equiaxial recrystal −1 probably grains were grains were found, found, which which replaced replaced the the initial initial and andprimary primarystrip-shaped strip-shapedgrain grainof of0.001 0.001ss–1,, probably due to microstructure under thisthis deformation condition. As the due to the thedeformation deformationrecrystallization recrystallizationofofthe the microstructure under deformation condition. As deformation condition changed, the primary strip-shaped α phase volume fraction varied, and the the deformation condition changed, the primary strip-shaped α phase volume fraction varied, and β matrix volume fraction increased gradually with increasing the β matrix volume fraction increased gradually with increasingdeformation deformationtemperature. temperature. ItIt can can be be seen from the change of the microstructure evolution that the recrystallization volume fraction seen from the change of the microstructure evolution that the recrystallization volume fraction was was influenced by by the Considering the the trend trend in in the influenced the strain strain rate rate and and decreased decreased as as the the latter latter increased. increased. Considering the variation of the stress-strain curve under the corresponding conditions, the α phase and β variation of the stress-strain curve under the corresponding conditions, the α phase and β phase phase recrystallizations were were the the potential reason for The flow flow recrystallizations potential reason for flow flow softening, softening, as as shown shown in in Figure Figure 4a–e. 4a–e. The softening mechanism is more complicated when deformed in the two-phase zone for titanium alloy. softening mechanism is more complicated when deformed in the two-phase zone for titanium alloy. As As shown shown in in Figure Figure 4c,d, 4c,d, the the initial initial strip-shaped strip-shaped grain grain was was elongated elongated and and smooth smooth along along the the vertical vertical compression direction, which proved that no obvious DRX occurred in this process. Some squashed compression direction, which proved that no obvious DRX occurred in this process. Some squashed 1 were observed and a few β matrix grains were compressed and partial β grains β grains grains at at10 10ss–1−were β observed and a few β matrix grains were compressed and partial β grains were were equiaxial, proved the occurrence of recrystallization in β phase, as shown in Figure 4e. equiaxial, whichwhich proved the occurrence of recrystallization in β phase, as shown in Figure 4e. The The adiabatic was another important thesoftening, flow softening, especially at the adiabatic shearshear bandband was another important reasonreason for thefor flow especially at the higher −1 in this study; however, no obvious adiabatic shear band was found higher strain rate, such as 10 s strain rate, such as 10 s–1 in this study; however, no obvious adiabatic shear band was found (Figure (Figure 4e). 4e).
Figure 4. Cont.
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Figure 4. Cont.
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Figure 4. 4. Microstructure at Figure Microstructure of of TA15 TA15 titanium titanium alloy alloy specimens specimens compressed compressed to to strain strain 0.9 0.9 at at 1223 1223 K K at −1, (b) 0.01 s−1, (c) 0.1 s−1, (d) 1 s−1, and (e) 10 s−1. different strain rates: (a) 0.001 s different strain rates: (a) 0.001 s−1 , (b) 0.01 s−1 , (c) 0.1 s−1 , (d) 1 s−1 , and (e) 10 s−1 .
3.2. Constitutive Modeling 3.2.1. Constitutive Equation Derivation The flow stress of TA15 titanium alloy was affected by the strain rate and temperature in the evaluated temperature deformation. The stress-strain curve was obtained by testing the isothermal compression at different temperatures and strain rates, so that the constants of the presented constitutive equation could be obtained. The Arrhenius equation, which is a phenomenological model, was used to predict the constitutive equation [24], illustrate the correlation among temperature, flow stress and strain, and express the special parameter Z [25,26]: . Q (1) Z = ε exp RT . Q ε = AF (σ) exp − (2) RT where
n0 σ F (σ) = exp( βσ ) [sinh(ασ)]n
ασ < 0.8 ασ > 1.2 f or all σ
(3)
where T is absolute temperature in K, R is a gas constant, 8.3145 J/mol·K, Q is the deformation activation energy of thermal deformation in J/mol, ´E is strain rate in s−1 , and A, n’, β, α, and n are constants of materials, α = β/n’. 3.2.2. Material Constants Material constants in the constitutive equation were evaluated through the stress-strain curve based on experimental data. However, the strain effect was not considered in Equations (1) and (2). Previous research mainly studied the Arrhenius model, which accurately illustrates the correlation among flow stress, deformation temperature, and strain. The following process was evaluated at a strain of 0.2. The values of F(σ) were replaced in Equation (2). Correlation between low stress and high stress is as follows: 0 . ε = Bσn (for ασ < 0.8) (4)
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.
ε = C exp( βσ ) (for ασ > 1.2)
(5)
where B and C are material constants that are independent from the deformation temperature. The natural logarithms of both sides of Equations (4) and (5) were taken, and the equations are as follows: . 1 1 ln(σ) = 0 ln ε − 0 ln( B) (6) n n . 1 1 σ = ln ε − ln(C ) (7) β β The corresponding value under the strain of 0.2 was substituted into the above equations. The correlation plots of ln(σ)−ln(´E) and σ −ln(´E) were obviously approximated by parallel and straight lines, respectively. β and n’ were obtained from the line slopes in the ln(σ)−ln(´E) plot and σ −ln(´E) plot, as shown in the following equations and Figure 5. The averages of n’ and β were 5.6875 and 0.0384 MPa−1 , respectively. . ∂ ln ε n0 = (8) Materials 2018, 11, x FOR PEER REVIEW 8 of 15 ∂ ln σ T . ∂ ln ε as shown in the following equations and Figure (9) β = 5. The averages of n’ and β were 5.6875 and 0.0384 ∂σ T MPa–1, respectively.
Figure5.5. Evaluating Evaluatingthe thevalue valueof of(a) (a)n’ n’by byplotting plottinglnσ lnσversus versusln´ lnέ and(b) (b)ββby byplotting plottingσσversus versusln´ lnέ. Figure E and E.
Subsequently, Equation (2) was rewritten for all strain levels as follows:
∂ ln ε n′ = ∂ nlnexp σ T− Q ε = A[sinh(ασ )]
(8) (10)
.
RT
∂ ln ε The natural logarithms of both sides were above equation. The equation is as follows: β taken = for the (9) ∂σ T
.
ln ε Q ln A ln[sinh − (ασ)] =for all+strain levels Subsequently, Equation (2) was rewritten as follows: n nRT n n Q At a certain temperature, afterε Equation = A sinh(11) ασis differentiated: exp −
(
)
RT
(11)
(10)
1 ∂ ln[sinh(ασ)] = (12) The natural logarithms of both sides were n ∂taken ln ε for Tthe above equation. The equation is as follows: where the n value was taken based on the line slopes of ln[sinh(ασ)]−ln´E in Figure 6, which was 3.5987. ln ε of Figure Q 6.ln A And the value of lnA can be derived from(ασ the )intercept + − ln sinh = (11)
n
nRT
n
At a certain temperature, after Equation (11) is differentiated:
1 ∂ ln sinh (ασ ) = ∂ ln ε n T
(12)
where the n value was taken based on the line slopes of ln[sinh(ασ)]−lnέ in Figure 6, which was 3.5987.
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Figure 6. 6. Evaluating thethe value of of n by plotting ln[sinh(ασ)] versus ln´Elnέ. . Figure Evaluating value n by plotting ln[sinh(ασ)] versus
AtAs a specific straininrate, Equation (11) is strain differentiated: can be seen Figure 7, when the was 0.2, flow stress and deformation temperatures plotting (13),ln[sinh(ασ)] Figure 6. Evaluating the value of n by versusoflnέ. at a certain strain rate were substituted to Equation and the value Q was derived in the ∂ ln ε ∂ ln[sinh(ασ)] Q as =R (13) plotting slope of ln[sinh(ασ)] a function of 1/T (Q was in kJ/mol for. ln[sinh(ασ)]−1000/T). Average ∂ ln[sinh(ασ)] T ∂(1/T ) ε Asdifferent can be seen inrates Figure 7, 584.63 when the strain 0.2,offlow stress and deformation Q at strain was kJ, and thewas value Q was obtained, which was temperatures consistent with at the a As certain strain werekJ/mol to Equation (13), and of Qfor was the current reference (654 bythe Zhang [27] byvalue Luo [28] αtemperatures +derived β phase in region). can be seenrate in Figure 7,substituted when strain wasand 0.2,588.7 flow kJ/mol stressthe and deformation at a plotting slope of ln[sinh(ασ)] as a function of 1/T (Q was in kJ/mol for ln[sinh(ασ)]−1000/T). Average Whenstrain deformation activation energy Q in deformation approached diffusion activation certain rate were substituted to Equation (13), and the value of Q was derived in theenergy, plottingthe Q major at different strain rates was 584.63 and of Q was obtained, was activation consistent with material softening mechanism was dynamic recovery. When thewhich diffusion slope of ln[sinh(ασ)] as a function of kJ, 1/T (Q the wasvalue in kJ/mol for ln[sinh(ασ)] −1000/T). Averageenergy Q at the current reference (654 kJ/mol by Zhang [27] and 588.7 kJ/mol by Luo [28] for α + β phase region). was much than584.63 the deformation activation energy, the DRX been with the the main different strainlower rates was kJ, and the value of Q was obtained, whichmay was have consistent When deformation activation energy Qofin[27] deformation approached diffusion activation energy, the mechanism. According to the Briottet al., the deformation activation ofregion). titanium current reference (654 kJ/mol bystudy Zhang andet588.7 kJ/mol by Luo [28] for α +energy β phase major softening mechanism was dynamic recovery. thediffusion diffusion activation energy alloymaterial is high in theactivation two-phase region to the change in When temperature, whichactivation changes the phase When deformation energy Qdue in deformation approached energy, was much lower than the deformation activation energy, the DRX may have been the main fraction and ultimately affectswas the flow stress throughWhen the modeling analysis [29]. The value thevolume major material softening mechanism dynamic recovery. the diffusion activation energy mechanism. According to the study of Briottet et al., the deformation activation energy of titanium of lnA at a certain strain was 57.1869, determined from the intercept of Figure 7. The dislocation climb, was much lower than the deformation activation energy, the DRX may have been the main mechanism. alloy is high two-phase region due the change inreflected temperature, which changes theisphase recrystallization, and recovery were related to Q, whichactivation the characteristics of deformation According to in thethe study of Briottet et al., thetodeformation energy of titanium alloy high volume fraction and ultimately affects the flow stress through the modeling analysis [29]. The value in[30]. the two-phase region due to the change in temperature, which changes the phase volume fraction of lnA at a certain strain was 57.1869, the intercept Figure The dislocation climb, and ultimately affects the flow stress determined through the from modeling analysis of [29]. The 7. value of lnA at a certain recrystallization, and recovery were related to Q, which reflected the characteristics of deformation strain was 57.1869, determined from the intercept of Figure 7. The dislocation climb, recrystallization, [30].recovery were related to Q, which reflected the characteristics of deformation [30]. and
Figure 7. Evaluating the value of activation energy by plotting ln[sinh(ασ)] versus 1000/T.
Subsequently, similarthe processes followed the above-mentioned material constants Figure 7. Evaluating value of were activation energyso bythat plotting ln[sinh(ασ)] versus 1000/T. 7. Evaluating the valuestrains of activation energy plotting versus 1000/T. could beFigure obtained at deformation ranging fromby0.1 to 0.8 ln[sinh(ασ)] at 0.1 intervals. Subsequently, similar processes were followed so that the above-mentioned material constants could be obtained at deformation strains ranging from 0.1 to 0.8 at 0.1 intervals.
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Subsequently, similar processes were followed so that the above-mentioned material constants could be obtained at deformation strains ranging from 0.1 to 0.8 at 0.1 intervals. 3.2.3. Strain Effect
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Temperature and strain rate were the parameters of concern for elevated temperature 3.2.3. Strain Effect stress-strain curves. Strain cannot significantly influence flow stress. However, strain can affect Temperature andenergy strain rate were common the parameters of concernwas for elevated stressdeformation activation Q. This phenomenon studied temperature in Ti-6Al-4V titanium Strain cannot significantly influence flow stress. However, strain affect deformation alloystrain [15],curves. Ti-modified austenitic stainless steel [31], aluminum alloyscan[14,20,21], magnesium Q. This[17–19]. common phenomenon wascompression studied in Ti-6Al-4V titanium alloytemperatures, [15], Tialloyactivation [22,32], energy and steels In isothermal tests at elevated modified austenitic stainless steel [31], aluminum alloys [14,20,21], magnesium alloy [22,32], and strain generally influenced flow stress, which was particularly significant at high deformation steels [17–19]. In isothermal compression tests at elevated temperatures, strain generally influenced temperature for TA15 alloy, as can be seen from Figure 8. Therefore, the strain effects cannot be flow stress, which was particularly significant at high deformation temperature for TA15 alloy, as neglected. The results show that strain significantly influenced the materials constants α, β, n, lnA, can be seen from Figure 8. Therefore, the strain effects cannot be neglected. The results show that and Q in the range of strain. strain significantly influenced the materials constants α, β, n, lnA, and Q in the range of strain.
Figure 8. The variation withstrain strainof of the the material (a)(a) α, (b) n, (c) (d) lnA. Figure 8. The variation with materialconstants: constants: α, (b) n, Q, (c)and Q, and (d) lnA.
strain compensation was so that flow stress of TA15 alloy accurately The The strain compensation wasconsidered considered so the that the flow stress of could TA15bealloy could be predicted. The model between strain and material constants was established using the polynomial accurately predicted. The model between strain and material constants was established using the function. The fitted polynomial ranged from two to eight. The optimal fitting model of fifth-order polynomial function. The fitted polynomial ranged from two to eight. The optimal fitting model of polynomial was selected. The material constants were evaluated at strains from 0.1 to 0.8. Equation fifth-order polynomial selected. Theequation. materialGeneralization constants were at strains from (14) is the fifth-orderwas polynomial fitting and evaluated representation would be lost0.1 to 0.8. Equation is the fifth-order polynomial fitting equation. Generalization andpolynomial representation due to the (14) overfitting of higher-order (more than five) polynomials. Table 1 shows the would be lost due to theconstants overfitting ofQ, higher-order (more than five) polynomials. Table 1 shows the fitting of the material α, n, and lnA of TA15 alloy.
polynomial fitting of the material constants α, n, Q, and lnA of TA15 alloy.
αα = = CC0 ++CC1εε ++CC2εε22 + C3ε 33 + C4ε 44 + C5ε 55 0 2 + C3 ε + C4 ε + C5 ε 1 2 3 4 5 =D D0 + +D nn= D11εε + +DD22εε2 ++DD3ε3 ε3 ++DD4ε4 ε4 ++DD5ε5 ε5 2
3
4
5
Q ==EE0++EEε1 ε++EEε2 ε2 ++EE3εε3 + +EE4εε 4 + E5εε 5 Q +E 0 1 2 3 4 5
ln A = F0 + F1 ε + F2 ε22 + F3 ε33 + F4 ε44 + F5 ε55
ln A = F0 + F1ε + F2ε + F3ε + F4ε + F5ε
(14)
(14)
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Table 1. Polynomial fitting results of material constants (α, n, Q, and lnA). Table 1. Polynomial fitting results of material constants (α, n, Q, and lnA). α Coefficient n Coefficient Q Coefficient lnA Coefficient
C0=0.00684 D0=4.29525 n Coefficient C1=0.00248 D1=–3.41358 C0 = 0.00684 4.29525 0 2==–4.96388 C2=0.00883 DD C1 = 0.00248 D1 = −3.41358 3=–0.01072 C2 =C0.00883 D2D=3=29.91754 −4.96388 C40.01072 =–0.00216 D3D=4=–35.6602 C3 = − 29.91754 C4 = −C0.00216 D4D=5=13.30015 −35.6602 5=0.00544 α Coefficient
C5 = 0.00544
D5 = 13.30015
E0=567.4279 Q Coefficient E1=742.9928 = 567.4279 EE20=–5372.16 E1 = 742.9928 EE3=12837.02 2 = −5372.16 EE43=–13555.2 = 12837.02 = −13555.2 EE54=5382.073
F0=55.57609 lnA Coefficient F1=74.06573 0 = 55.57609 F2F=–541.173 F1 = 74.06573 F3F=1300.602 2 = −541.173 F4F=–1381.13 3 = 1300.602 = −1381.13 FF54=551.388
E5 = 5382.073
F5 = 551.388
The constants of materials were calculated by predicting the flow stress through hyperbolic sine function. The constitutive equation the Zener-Holloman parameter and flowthrough stress was written as The constants of materials wereofcalculated by predicting the flow stress hyperbolic follows considering Equations (1) andof (10): sine function. The constitutive equation the Zener-Holloman parameter and flow stress was written as follows considering Equations (1) and (10): 1/2
1 Z " Z σ = + 1#1/2 ln 1/n + 2/n 1 α Z A Z σ = ln + A + 1 α A A 1/ n
2/ n
(15) (15)
Validation Constitutive Modeling 4.4.Validation ofof Constitutive Modeling 4.1. 4.1.Results ResultsofofConstitutive ConstitutiveModeling Modeling AsAsshown shownininFigure Figure9,9,the theexperimental experimentaldata dataand andpredicted predicteddata datawere werecompared comparedtotoverify verifythe the constitutive modeling result considering the strain compensation. The experimental data were constitutive modeling result considering the strain compensation. The experimental data were consistent in most mostdeformation deformationconditions. conditions. In the processing condition consistentwith withthe the predicted predicted data in In the processing condition (1073 –1; Figure (1073 K, when s−1 ; Figure 9a), there variation between the predicted experimental 9a), there was was variation between the predicted datadata andand the the experimental data K, when 0.1 s0.1 data due to the high nonlinearity of the flow behavior at high temperatures, which was affected by due to the high nonlinearity of the flow behavior at high temperatures, which was affected by many many factors, limiting accuracy applicablerange rangeand andthe theflow flow stress. The model factors, limiting the the accuracy of of thethe applicable modelfitting fittingofofthe the materials constants and some deviations may introduce variation into thethe modeling. materials constants and some deviations may introduce variation into modeling.
Figure Comparison between betweenthe theexperimental experimental predicted the temperature: Figure9.9. Comparison andand predicted flowflow stressstress at theat temperature: (a) 1073 (a)K,1073 K, (b) 1123 K, (c) 1173 K, and (d) 1223 K. (b) 1123 K, (c) 1173 K, and (d) 1223 K.
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Figure 10 depicts the correlation of experimental data of flow stress and the predicted data in the constitutive 4.2. Constitutive equation Modeling incorporating Verification the effect of strain over the entire strain range, strain rate, and temperature. Standard statistical parameters, like R and AARE, can quantify the predictability of the Figure 10 depicts the correlation of experimental data of flow stress and the predicted data in constitutive equation: the constitutive equation incorporating the effect of strain over the entire strain range, strain rate, N and temperature. Standard statistical parameters, can quantify the predictability of Eilike − ER and Pi −AARE, P i =1 R= the constitutive equation: (16) 2 N 2 N N E − E P − P E − E P − P ∑ i i1 i i= (16) R = q i =1 i=1i 2 N 2 N ∑i=1 Ei − E ∑i=1 Pi − P 1 N E −P AARE = 1 N i =E1 −i P i ×100 (17) AARE = N ∑ i Eii × 100 (17) N i =1 Ei
( (
)( ) ) ( )
_
_
where N is the data number used in the research, P and E are average values of P and E, P is flow where N is the data number used in the research, P and E are average values of P and E, P is flow stress stress predicted from the constitutive equation involving the strain compensation, and E is the predicted from the constitutive equation involving the strain compensation, and E is the measured measured flow stress. flow stress.
Figure 10. predicted data andand experimental datadata over over the entire range range of strain, Figure 10. The Thecorrelation correlationbetween between predicted data experimental the entire of strain rate, and temperature. strain, strain rate, and temperature.
As aa common statistical parameter, parameter, R R provides about the the relationship relationship As common statistical provides strength strength information information about between the the calculated calculated value value and and the the observed observed value value [21]. [21]. Since Since there there is is aa tendency tendency for for the the equation equation between to deviate toward lower or higher values, a higher R value does not represent better performance. to deviate toward lower or higher values, a higher R value does not represent better performance. AARE was was also by comparing the relative As aa result, it is AARE also computed computed by comparing the relative error. error. As result, it is an an unbiased unbiased statistical statistical parameter used to determine predictability. The values of AARE and R are 6.85% and and 0.9929, 0.9929, parameter used to determine predictability. The values of AARE and R are 6.85% respectively, shown in Figure 10. This indicates that the proposed deformation constitutive equation respectively, shown in Figure 10. This indicates that the proposed deformation constitutive equation involving strain strain compensation compensation can involving can estimate estimate flow flow stress stress for for TA15 TA15 titanium titanium alloy alloy accurately. accurately. 4.3. Prediction of Constitutive Model 4.3. Prediction of Constitutive Model Flow stress is complex and associated with complicated metallurgical phenomena, across which Flow stress is complex and associated with complicated metallurgical phenomena, across which hot deformation changes considerably. However, the developed model predicted flow stress in a wide hot deformation changes considerably. However, the developed model predicted flow stress in a range of strains, strain rates, and temperatures. This can be considered as the major potential of the wide range of strains, strain rates, and temperatures. This can be considered as the major potential of developed model compared to the analytical, phenomenological, and traditional empirical model the developed model compared to the analytical, phenomenological, and traditional empirical model that is applied in a specific processing domain [33]. Validating the accuracy of the prediction is good that is applied in a specific processing domain [33]. Validating the accuracy of the prediction is good not only for the experimental data but also for the extensional conditions. Because of that higher not only for the experimental data but also for the extensional conditions. Because of that higher temperature values can introduce the β phase transition in the material, the temperature selected for temperature values can introduce the β phase transition in the material, the temperature selected for the specimens prediction accuracy is lower, at 1023 K. The phase change in the microstructure easily the specimens prediction accuracy is lower, at 1023 K. The phase change in the microstructure easily produces different stress-strain results, so lower temperatures ensure that the microstructure of the produces different stress-strain results, so lower temperatures ensure that the microstructure of the materials is filled with a single α phase. A similar study found that the simulation error of extreme materials is filled with a single α phase. A similar study found that the simulation error of extreme points can be decreased by increasing the temperature and strain rate ranges for the near-α titanium alloy [34].
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points can2018, be decreased by increasing the temperature and strain rate ranges for the near-α titanium Materials 11, 1985 13 of 15 points alloy [34]. can be decreased by increasing the temperature and strain rate ranges for the near-α titanium alloy [34]. to predict the constitutive model of TA15 titanium alloy, the constitutive model is Therefore, Therefore,to topredict predictthe theconstitutive constitutive model of ofTA15 TA15 titanium alloy,the constitutive model to predict alloy, constitutive further Therefore, extended flow stress at a model temperature of titanium 1023 K, which isthe not included model in the isis further extended to predict flow stress at a temperature of 1023 K, which is not included the further extended to predict flow stress at a temperature of 1023 K, which is not included in previous compression tests. Experimental and predicted flow stresses at a temperature of 1023inKthe previous compression tests. Experimental and predicted flow stresses at a temperature of 1023 previous compression tests. and stresses temperature ofvalue 1023 KK were compared, and results are Experimental shown in Figure 11.predicted The sameflow change trendat inathe stress-strain werecompared, compared, andresults results areshown shown Figure11. 11.good Thesame same changetrend trend thestress-strain stress-strain value and are ininFigure The change the is were found in the experimental result, and prediction is at a temperature ofinin1023 K, as shownvalue in is found in the experimental result, and prediction is good at a temperature of 1023 K, as shown is found in the experimental result, and prediction is good at a temperature of 1023 K, as shown Figure 11b. Similarly, the quantization method with AARE and R, as mentioned above, was used to inin Figurethe 11b.prediction Similarly,accuracy. method with as above, was Figure 11b. Similarly, the quantization method with AARE AARE and R, asmentioned mentioned above, wasused usedto evaluate The values of AARE and Rand areR, 8.19% and 0.9881, respectively, evaluate the prediction accuracy. The values of AARE and R are 0.9881, respectively, to evaluate the prediction accuracy. The values of AARE and R 8.19% and 0.9881, respectively, shown in Figure 12. This indicates that the proposed constitutive equation considering strain effect shown inFigure Figure12. 12. Thisindicates indicates thatthe theproposed proposed constitutive equation consideringstrain straineffect effect This that constitutive equation considering hasshown good in prediction accuracy for the extended data range for TA15 titanium alloy. hasgood goodprediction predictionaccuracy accuracyfor forthe theextended extendeddata datarange rangefor forTA15 TA15titanium titaniumalloy. alloy. has
Figure 11. The comparison between the prediction and experimental results at temperature of 1023 Figure 11. between the prediction and experimental results at temperature of 1023 K: Figure 11.The Thecomparison comparison between the prediction and experimental results at temperature of 1023 K: (a) experimental results and (b) prediction results of different strain rates. (a) results and and (b) prediction results of different strain rates.rates. K:experimental (a) experimental results (b) prediction results of different strain
Figure correlation between predicted and experimental results at temperature 1023 Figure 12.12. TheThe correlation between predicted and experimental results at temperature of of 1023 K. K. Figure 12. The correlation between predicted and experimental results at temperature of 1023 K.
5. Conclusions 5. Conclusions 5. Conclusions In this investigation, flow stress at various strain rates and deformation temperatures was In this investigation, flow stress at various strain rates and deformation temperatures was predicted by investigation, establishing a flow strain-dependent constitutive equation model for TA15 titanium alloy. In by this stress at various strainequation rates and deformation predicted establishing a strain-dependent constitutive model for TA15 temperatures titanium alloy.was The following conclusions were drawn: establishing a strain-dependent constitutive equation model for TA15 titanium alloy. Thepredicted followingby conclusions were drawn: The following conclusions were drawn: (1) TA15 titanium alloy stress was sensitive to deformation temperature and strain rate; the value (1) TA15 titanium alloy stress was sensitive to deformation temperature and strain rate ; the value increased with increasing strain rate andtodecreased withtemperature increased deformation temperature. (1)increased TA15 titanium alloy stress wasrate sensitive deformation and straintemperature. rate ; the value with increasing strain and decreased with increased deformation The experimental result curves exhibited thedecreased typical flow behavior, anddeformation a constitutivetemperature. model was increased with increasing strain rate and with increased The experimental result curves exhibited the typical flow behavior, and a constitutive model introduced based on the Arrhenius equation; experimental exhibited the typical flow behavior, and a constitutive model wasThe introduced basedresult on thecurves Arrhenius equation; was introduced based Arrhenius (2) Material constants α, n,on Q,the and lnA wereequation; significantly influenced by the strain effect for TA15 titanium alloy, and the relationships between strain and material constants could be described using fifth-order polynomials with a good fitting correlation;
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Flow stress at various conditions of deformation was accurately predicted through the constitutive equation incorporating strain compensation. R and AARE were used to quantify the predictability of the constitutive equation, with values of 6.85% and 0.9929, respectively. The values were 8.19% and 0.9881, respectively, in the prediction using the extended stress-strain data for 1023 K, which proved the high accuracy of the constitutive equation compensated by the strain effect for TA15 titanium alloy.
Author Contributions: J.L. proposed the conceptualization and wrote the manuscript; J.L. and F.L. designed and performed the models and methods of analysis; J.C. designed and performed the isothermal compression simulation tests and other experiments; J.L. and F.L. discussed and analysed the results of the experiments, and J.L. was in charge of the response of the review comments. Funding: This research was funded by the National Natural Science Foundation of China with Grant No. 51605387 and No. 51275414, the Province Natural Science Foundation of Shaanxi with Grant No. 2015JM5204, the Fundamental Research Funds for the Central Universities with Grant No. 3102015BJ (II) ZS007. Acknowledgments: The authors thank Xiyuan Yao for her valuable help with the microstructure experiments and part of the pictures processing for the result. Conflicts of Interest: The authors declare no conflict of interest.
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