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The constitutive responses of Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy at high strain rates and elevated temperatures
Journal of Alloys and Compounds 647 (2015) 97e104
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The constitutive responses of Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy at high strain rates and elevated temperatures Jun Zhang a, Yang Wang a, *, Xiang Zan b, Yu Wang a a
CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, PR China b School of Materials Science and Engineering, Hefei University of Technology, Hefei 230009, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 May 2015 Accepted 16 May 2015 Available online 19 June 2015
The effects of strain rate and temperature on the uniaxial tension responses of Ti-6.6Al-3.3Mo-1.8Zr0.29Si alloy are systematically investigated over a wide range of strain rates, 0.001e1150 s1, and initial temperatures, 293e573 K. Dynamic tension and recovery tests are conducted using a splitHopkinson tension bar to obtain the adiabatic and isothermal stress-strain responses of the alloy at high strain rates. The experiments reveal that the tension behavior of the alloy is dependent on the strain rate and temperature. The value of initial yield stress increases with increasing strain rate and decreasing temperature, whereas the isothermal strain-hardening behavior changes little with strain rate and temperature over the full ranges explored. The adiabatic temperature rise is the main reason for the reduction of strain hardening rate during the high-rate deformation process. SEM observations indicate that the tension specimen is broken in a manner of ductile fracture. The phenomenologically based constitutive model, JohnsoneCook model, is suitably modified to describe the rate-temperature dependent deformation behavior of Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy. The model correlations are in good agreement with the experimental data within the investigated range of strain rates and temperatures. © 2015 Elsevier B.V. All rights reserved.
Keywords: Titanium alloy Uniaxial tension Rate-temperature sensitivity Adiabatic softening Constitutive model
1. Introduction
aþb titanium alloys are attractive structural materials due to their excellent combination of high strength, light weight, good formability and elevated corrosion resistance. With the wide engineering applications of these alloys involving high strain rates and elevated temperatures (such as hot-forming, high speed machining or foreign-object impact), it is critically important to have a fundamental understanding of the stress-strain behavior of the alloys over a wide range of strain rates and temperatures. Finite element analysis technique has been widely used as an effective numerical tool in forming process optimization and structural design. It is well recognized that the validity of the simulation results is greatly affected by the reliability and accuracy of prediction ability of the constitutive models. Therefore, it is crucial to propose valid formulations of rate and temperature dependent constitutive models in the numerical simulations. A considerable number of
* Corresponding author. E-mail address: [email protected] (Y. Wang). http://dx.doi.org/10.1016/j.jallcom.2015.05.131 0925-8388/© 2015 Elsevier B.V. All rights reserved.
experimental investigations and constitutive modeling studies have been performed on the stress-strain behavior for titanium alloys such as Tie6Ale4V which is one of the most extensively used two-phase titanium alloys [1e14]. Experimental investigation indicates that the mechanical responses of Tie6Ale4V are highly sensitive to the specified loading conditions such as strain rate and temperature. The yield strength of the alloy increases with increasing strain rate and exhibits more sensitivity to temperature than to the strain rate, while the strain hardening rate decreases with increasing strain rate. In addition, phenomenologically and physically based constitutive models have been used to characterize the rate and temperature dependent mechanical responses of titanium alloys at various strain rates and temperatures. Silva and Ramesh reviewed experimental results of high-strain-rate compressive deformation and failure modes on Tie6Ale4V and discussed the observations of adiabatic shear localization [1]. Macdougall and Harding performed tension and torsion tests on Tie6Ale4V alloy at high strain rates and used a Zerilli-Armstrong type constitutive relation to predict the observed mechanical responses in tensile impact tests [3]. Nemat-Nasser et al. investigated the compressive responses of Tie6Ale4V over a broad range of
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strain rates (103e7000 s1) and initial temperatures (77e1000 K) and proposed a physically based constitutive model in which the thermally activated mechanisms associated with the dislocation motion were taken into accounted [4]. Majorell et al. performed dynamic compressive tests on Tie6Ale4V in the temperature range of 77e1400 K and proposed a physically based constitutive model in which the flow stress contains a thermal and an athermal components [5,6]. Khan et al. [9] and Roy et al. [10] studied the compressive behaviors of Tie6Ale4V and Ti6Al4V-0.1B at high strain rates and elevated temperatures, respectively. Kotkunde et al. made a comparative study of constitutive modeling for Tie6Ale4V alloy using four models, namely the empirically based JohnsoneCook, Fields-Backofen and Khan-huang-Liang models and the physically based Mechanical Threshold Stress model [12]. Wang et al. [13] and Wu et al. [14] employed JohnsoneCook model to describe the flow responses of Tie6Ale4V at low strain rates and elevated temperatures and Tie6Ale2Sne2Zre3Moe1Cre2Nb at high strain rates and elevated temperatures. The split-Hopkinson pressure bar (SHPB) technique has been widely used to investigate the adiabatic compressive stress-strain behavior of metals at high strain rates [15]. It is well understood that the high strain-rate deformation process is adiabatic, where the deformation is instantaneous and the heat generated by the plastic work will be stored in the specimen. Thus there is a continuous temperature rise during the plastic deformation process and thermal work-softening effect coupled with the isothermal strain hardening effect should be taken into account in the modeling of stress-strain curves for materials whose flow stress is temperature dependent. Nemat-Nasser et al. proposed the recovery experimental technique on the SHPB to measure the isothermal compressive stress-strain responses of metals at high strain rates [16,17]. The advantage of this recovery technique is that it is possible to uncouple the thermal work-softening effect from the isothermal strain hardening effect on the plastic deformation under high strain-rate loading conditions. It can be found that most of the aforementioned studies have been mainly concentrated on the measurement and characterization of compression responses of titanium alloys. Few efforts have been reported on the tension stress-strain responses and constitutive modeling of titanium alloys at high strain rates and elevated temperatures, though it is found that a and aþb titanium alloys display the tension-compression asymmetry due to their lowsymmetry hcp systems [18,19]. The objective of this paper is to experimentally investigate the strain-rate and temperature sensitivity of the tension responses of Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy (referred as TC11 in China) over a wide range of strain rates (103e103 s1) and temperatures (293e573 K). The isothermal stress versus strain responses at high strain rates are measured to evaluate the strain hardening behavior of the alloy. The adiabatic and isothermal constitutive responses at various strain rates and temperatures are compared and a phenomenological constitutive model incorporating the thermal-mechanical coupling effect is proposed to characterize the measured tension responses of the alloy.
cooling down to the room temperature in the air. Metallographic examination of the undeformed specimen reveals that the microstructure of the alloy is duplex, containing a lamellar Widmanstatten structure dispersed between primary globular a grains, as shown in Fig. 1. 2.2. Experimental procedure Uniaxial tension testing was conducted on the alloy under quasi-static and dynamic loading conditions. Quasi-static tension tests were carried out on an MTS810 servo-hydraulic testing system at a rate of 0.001 s1 to obtain the isothermal stress-strain curves. Dynamic tension tests were performed using a split Hopkinson tension bar (SHTB) system at strain rates of 190e1150 s1 to determine the adiabatic stress-strain curves. Fig. 2 shows the sketch of the SHTB system which comprises an impact apparatus, a prefixed short metal bar, an incident bar and a transmitted bar [20]. The SHTB system was operated as follows. The tensile incident pulse was produced by the deformation of the prefixed metal bar due to the impact between the hammer and the impact block. The amplitude and duration of the incident pulse can be controlled by adjusting the length and diameter of the prefixed metal bar and the impact velocity of the hammer. The role of the prefixed metal bar (made of 2A12 aluminum alloy in China) is not only the pulse producer but also the low-pass mechanical filter due to its plastic deformation. The plate-shaped tension specimen was connected to the incident/transmitted bars using adhesive connection. Dynamic tension recovery tests were performed on the SHTB system also at strain rates of 190e500 s1 to determine the isothermal stressstrain responses of the alloy under high rate loading conditions. The tensile incident pulse with short duration was generated to make the specimen deform into the plastic deformation at a certain level of plastic strain. Then the deformed specimen was allowed to return to its initial testing temperature. Namely the specimen was subjected to a single loading and unloading at a high strain rate. By repeating the loading-unloading several times on the same specimen at the same strain rate, the dynamic tension recovery tests came into operations. The isothermal curves at high strain rates were obtained by connecting the points associating with the initial yielding upon each loading. The aforementioned quasi-static and high-rate tests in tension were also performed as a function of temperature. The investigated testing temperature range was 293e573 K. A rapid-contact localized heating technique was used to obtain the elevated temperatures up to 573 K. The detailed description of this heating technique and the calibration of the heating system can be found elsewhere [21].
2. Experimental details 2.1. Material The titanium alloy used in this investigation was obtained as commercial forged rods, 38 mm in diameter, purchased from BaoTi Group Co. Ltd. of China. The chemical composition (in wt%) of the alloy is 6.6Al, 3.3Mo, 1.8Zr, 0.29Si, 0.07Fe, 0.01C, 0.01N, 0.004H, 0.13O and balance Ti. The samples were prepared by annealing for 2 h at 1228 K, followed by aging for 6 h at 803 K and finished by
Fig. 1. Microstructure of the undeformed alloy.
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Fig. 2. Schematic diagram of the split-Hopkinson tension bar system.
2.3. Specimen preparation The dumb-bell shaped flat specimens were used in the uniaxial tension tests and were cut directly along the axis of the aging rod. The gage length, width and fillet radius of the dumbbell-shape plate specimen used in dynamic tension tests are 10 mm, 3.5 mm and 2 mm, respectively. The specimen thickness is 1.1 mm. The specimen geometry used in quasi-static tension tests is similar except that the gage length of 30 mm is long enough to avoid the end effects. The tension tests under each loading conditions were repeated at least three times and the good repeatability of the data was obtained. 3. Results and discussions 3.1. The tension stress-strain responses of Ti-6.6Al-3.3Mo-1.8Zr0.29Si alloy The engineering stress e engineering strain curves of Ti-6.6Al3.3Mo-1.8Zr-0.29Si alloy were obtained from the quasi-static and dynamic tests. According to the constant volume assumption, the true stress and true strain were converted from engineering values. Fig. 3 shows the true stress-strain responses of the alloy as a function of four strain rates at three initial temperatures. As presented in Fig. 3(a), the responses of the alloy at room temperature exhibit the obvious strain rate sensitivity and elasticeplastic deformation characteristics. Since no obvious yield point can be found in the stress-strain curve, the flow stress at 0.2% plastic strain is adopted as initial yield stress. There is a positive strain-rate sensitivity with respect to the initial yielding behavior. The values of initial yield stress at high strain rates increase significantly compared with quasi-static results, which presents a considerable strain-rate strengthening phenomenon. However, the plastic response at high strain rates is absolutely different from that under quasi-static loading conditions. The working hardening rate decreases as the strain rate increases and thermal softening behavior can be observed in the adiabatic experimental results at high strain rate tests, which are consistent with most of the FCC and BCC metals. Also, this observation is similar to other investigations on the rate-sensitive compression responses of aþb titanium alloys [7,9]. When the temperature increases, the strain rate also has a profound effect on the initial yielding and strain hardening behavior. Similar trend of strengthening in yield stress and reduction in strain hardening rate can be seen at high strain rates. The effects of strain rate and temperature on the initial yield stress are summarized in Fig. 4. It is obvious that the initial yield stress increases when strain rate increases and temperature decreases. Such strain rate and temperature sensitivity of Ti-6.6Al-3.3Mo-1.8Zr0.29Si alloy is comparable to that of Tie6Ale4V under compressive loading conditions [2,4,9,10]. To clearly understand the influence of strain rate on the tension strain hardening behavior of the alloy, the loading-unloadingreloading results at the rate of 500 s1 and temperature of 293 K are shown in Fig. 5. The dashed curves represent a series of multiloading-unloading stress-strain results of the same specimen. The
isothermal stress-strain relation at the rate of 500 s1 is obtained by connecting the points associating with initial yielding upon each loading. The results of the adiabatic tests performed at the rate of 500 s1 and the isothermal tests performed at the rate of 0.001 s1 are also presented in Fig. 5. It can be noticed an obvious difference exists between isothermal and adiabatic curves at the strain rate of 500 s1. The strain hardening rate of adiabatic curve is much lower than that of isothermal curve, which is solely due to the thermal softening caused by the adiabatic temperature rise. Namely, the adiabatic stress-strain response in the course of plastic deformation at a high strain rate is an essentially coupled result of strain-rate strengthening, strain hardening and adiabatic softening effects. By means of dynamic tension recovery technique on the SHTB, the adiabatic softening effect can be experimentally uncoupled from the strain-rate strengthening and strain hardening effects. Moreover, the strain hardening behavior in the isothermal responses at 500 s1 changes little compared to the isothermal result under quasi-static loading conditions, which indicates that the strain rate has totally different influence on the initial yielding and strain hardening behavior of the alloy. Such rate-independent behavior of isothermal strain hardening for Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy is comparable to the compressive thermomechanical responses of Ti6Al4V alloys and a tantalumetungsten alloy investigated by Nemat-Nasser et al. [4,17]. Fig. 6 summarizes the isothermal relations of the alloy at different strain rates and temperatures. The aforementioned rate-independent behavior of isothermal strain hardening is also found at the strain rate of 190 s1 and temperatures of 293 and 423 K. Comparing the quasi-static isothermal results (0.001 s1) obtained at 293, 423 and 573 K, the levels of isothermal strain hardening rate at different initial temperatures are approximately identical, showing that the isothermal strain hardening exhibits some degree of insensitivity on the initial temperature within the investigated temperature range. Such temperature-independent isothermal strain hardening behavior is also observed at the strain rate of 190 s1. In order to understand the failure mechanism of the alloy, the microstructure near the broken area of the specimen and the fracture surface were examined using the optical microscopy and the scanning electron microscopy, as shown in Fig. 7. Naked-eye observation indicates that the localization necking occurs during the course of plastic deformation. The metallographic examination of the necking zone indicates that the grains are stretched obviously along the loading direction and the crack propagates through both primary a and lamellar Widmanstatten grains. The SEM fractographs of the alloy display three kinds of dimple morphologies, namely round dimples, shallow dimples and parabolic dimples corresponding to fibrous zone, radiation area and shearing lip, as marked with A, B and C in Fig. 7. Fig. 8 displays the fractographs of the specimen tested at strain rates of 0.001 and 1150 s1 and at temperatures of 293 and 573 K. It can be seen that micrographs of the fracture surfaces for all loading cases are similar and exhibit the typical dimple morphology, which indicates that the tension specimen is broken in a manner of ductile fracture with good elongation. The size and depth of ductile dimples change little with the change of strain rates and temperatures, as seen in Fig. 8.
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Fig. 4. Effects of strain rate and temperature on the initial yield stress.
Fig. 3. True stress-strain curves obtained at strain rates of 0.001, 190, 500 and 1150 s1 and temperatures of (a) 293 K, (b) 423 K and (c) 573 K. Fig. 5. Comparison of isothermal and adiabatic responses at 293 K.
3.2. Constitutive modeling of tension behavior Johnson and Cook proposed an empirical constitutive model for metals subjected to large strain, high strain rates and various temperatures [22]. This constitutive model is one of the most well-
used phenemonologically based models in the applicable computer codes because of its simplicity and the availability of model parameters. The strain hardening, strain rate strengthening and thermal softening are taken into account in the model as a simple
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multiplication form. The flow stress, s, is expressed as a function of strain, ε, strain rate, ε_ , and temperature, T, as follows.
ε_ T Tr m 1 s ¼ ðA þ Bεn Þ 1 þ C ln ε_ 0 Tm Tr
Fig. 6. Comparison of isothermal responses at different strain rates and temperatures.
(1)
where A is the initial yield stress, B is the coefficient of strain hardening and n is the strain hardening exponent. ε_ 0 and C are the reference strain rate and the coefficient of strain rate strengthening, respectively. Tr and Tm are the reference temperature and melting temperature, and m is the thermal softening exponent. The expression in the first set of parentheses corresponds to the initial yielding and strain hardening behavior of materials and their strain-rate and temperature dependences are both represented by the second and third sets of parentheses, respectively. Khan and Liang found that the original JohnsoneCook model has inherent problems of modeling the dependence of strainhardening behavior of metals on strain-rate and temperature [23]. It was pointed out that the model was unable to accurately describe the decrease of work hardening with the increasing strain
Fig. 7. Metallographic examination and SEM observation for the tensile specimen loaded at the strain rate of 0.001 s1 and temperature of 423 K.
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Fig. 8. Fracture surface morphologies at (a) 0.001 s1, 293 K, (b) 1150 s1, 293 K, (c) 0.001 s1, 573 K and (d) 1150 s1, 573 K.
rate. They suggested that to better predict the work-hardening behavior, strain and strain rate must have some coupled effects on the description of work-hardening relation of the materials. From isothermal stress-strain curves of Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy shown in Fig. 4, it should be noted that the strain-rate and temperature effects on the initial yield stress and the isothermal strain hardening are not identical. Moreover, the adiabatic strain hardening is dependent on the temperature rise during the highrate deformation process. Due to such experimental observation, a modified JohnsoneCook model is proposed to improve the accuracy of the original model.
s ¼ A 1 þ C1 ln_ε* 1 T *m1 þ Bεn 1 þ C2 ln_ε* 1 T *m2 elDT
DT ¼
g rCp
(1) By taking the plastic strain as zero in Eqn. (2), the yield stress, syield, is expressed as a function of strain rate and temperature as follows.
(2)
syield ¼ A 1 þ C1 ln_ε* 1 T *m1
(3)
By comparing the experimental data at different strain rates and initial temperatures, as presented in Fig. 4, the values of three material constants, A, C1 and m1, are obtained.
Zε sdε
applied to adiabatic loading conditions corresponding to high strain-rate loadings. For isothermal loading conditions, the added term in Eqn. (2), elDT, needs to be omitted in the modified model. To determine the material constants for the modified model, isothermal and adiabatic experimental data at various strain rates and temperatures are first used to obtain an initial set of values for the material constants with the following steps.
0
where DT is the adiabatic transient temperature rise in the specimen during the high-rate loading process and l is the adiabatic softening coefficient. g is the conversion factor of plastic work to heat, r is the mass density and Cp is the specific heat capacity at constant pressure. ε_ * ¼ ε_ =_ε0 and T*¼(TTr)/(TmTr) correspond to the dimensionless strain rate and temperature, respectively. Compared with the original JohnsoneCook model, two modifications have been made in the present paper. Firstly, two pairs of model parameters, C1,m1 and C2,m2, are introduced to represent the strain-rate and temperature effects on the initial yield stress and strain hardening behavior, respectively. Secondly, a modified form of strain hardening incorporating the adiabatic softening is included in the second term of the model to describe the constitutive behavior of the alloy at high strain rates. Eight material constants, A, B, n, C1, C2, m1, m2 and l, are involved in the modified JohnsoneCook model. It should be emphasized that Eqn. (2) is
(4)
(2) By considering the isothermal constitutive relation at different strain rates and temperatures, Eqn. (2) will reduce to
sisothermal ¼ A 1 þ C1 ln_ε* 1 T *m1 þ Bεn 1 þ C2 ln_ε* 1 T *m2 (5) By fitting Eqn. (5) to the experimental results for isothermal stress-strain responses, as shown in Fig. 6, the values of four material constants, B, n, C2 and m2, are evaluated. (3) The final step is to determine the remaining material constant l. By comparing the adiabatic results with the isothermal results at high strain rates, as shown in Figs. 3 and 6, the adiabatic softening coefficient, l, can be derived as follows.
J. Zhang et al. / Journal of Alloys and Compounds 647 (2015) 97e104
Bεn 1 þ C2 ln_ε* 1 T *m1 1 ln l¼ DT sadiabatic sisothermal þ Bεn ð1 þ C2 ln_ε* Þð1 T *m1 Þ (6) where sisothermal and sadiabatic correspond to the isothermal and adiabatic flow stresses under high strain rate loading conditions, respectively. The adiabatic temperature rise, DT, is estimated using Eqn. (3). For the investigated Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy, its mass density is 4486 kg/m3 and its melting temperature is 1875 K. The conversion factor of plastic work into heat is assumed to be 0.9 during the course of high rate deformation. Since the specific heat capacity at constant pressure is temperature dependent, an approximation of this temperature dependence is given as
Cp ¼ 588:8 0:3419T þ 0:00065T 2
J=ðkg KÞ
(7)
The values of material constants estimated according to aforementioned steps are used as initial input into an optimization procedure based on the least square method. Here, the reference strain rate and temperature, ε_ 0 and Tr, are taken to be 0.001 s1 and 293 K, respectively. The final values of eight material constants for Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy are A ¼ 942.5 MPa, B ¼ 1420.6 MPa, n ¼ 0.721, C1 ¼ 0.034, C2 ¼ 0.001, m1 ¼ 0.726, m2 ¼ 27.1 and l ¼ 0.064 K1, respectively. The predicted yield stress from Eqn. (4) is shown in Fig. 4. The comparison of model correlations with the experimental stress-strain results is shown in Fig. 9. It can be seen that good agreement between the model results and experimental data is obtained, which indicates that the modified model is applicable to describe the rate and temperature
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dependent deformation behavior of Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy subjected to uniaxial tension loading over a wide range of strain rates and temperatures. Furthermore, the dynamic tension recovery tests can achieve the experimental uncoupling between the strain hardening behavior and adiabatic softening behavior of rate-temperature dependent metals, which provides an effective way to obtain a better set of material constants for the constitutive modeling. 4. Conclusion Uniaxial tension stress-strain behavior of Ti-6.6Al-3.3Mo-1.8Zr0.29Si alloy with duplex microstructure was investigated as a function of strain rate and temperature. The high strain-rate tension tests were performed using the split Hopkinson tension bar apparatus to obtain the adiabatic experimental results. Furthermore, the dynamic tensile recovery tests based on the Hopkinson bar technique were conducted to evaluate the isothermal stressstrain responses of the alloy under high strain-rate loading conditions. The experiment reveals that the tension responses of Ti6.6Al-3.3Mo-1.8Zr-0.29Si alloy are sensitive to strain rate and temperature. The values of initial yield stress decrease with increasing temperature and increase significantly with increasing strain rate, while the isothermal strain hardening behavior is essentially independent of strain rate and temperature. The reduced strain hardening behavior was observed at high strain rates due to adiabatically thermal softening. Microstructure observations of the fracture specimen indicate that the tension specimen is broken in a manner of ductile fracture. The phenomenologically-based JohnsoneCook model was modified to enable the different rate-temperature sensitivities of initial yielding
Fig. 9. Model correlations with the experimental data at temperatures of 293, 423 and 573 K and strain rates of (a) 0.001 s1, (b) 190 s1, (c) 500 s1 and (d) 1150 s1.
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and strain hardening and the adiabatic softening at high strain rates to be accommodated. Dynamic tensile recovery experimental results were used to accurately determine the material constants in the modified constitutive model. Comparison between the experimental data and model results indicates that such model is capable of capturing the tensile thermomechanical responses of Ti-6.6Al3.3Mo-1.8Zr-0.29Si alloy within the investigated range of strain rates and temperatures.
[10]
[11]
[12]
Acknowledgments
[13]
The authors acknowledge the financial support by the National Natural Science Foundation of China under Contract No. 11172288.
[14]
References [1] M.G. Silva, K.T. Ramesh, The rate-dependent deformation and localization of fully dense and porous Ti-6Al-4V, Mater. Sci. Eng. A 232 (1997) 11e22. [2] W.S. Lee, C.F. Lin, Plastic deformation and fracture behaviour of Ti-6Al-4V alloy loaded with high strain rate under various temperatures, Mater. Sci. Eng. A 241 (1998) 241e248. [3] D.A.S. Macdougall, J. Harding, A constitutive relation and failure criterion for Ti6Al4V alloy at impact rates of strain, J. Mech. Phys. Solids 47 (1999) 1157e1185. [4] S. Nemat-Nasser, W.G. Guo, V.F. Nesterenko, S.S. Indrakanti, Y.B. Gu, Dynamic response of conventional and hot isostatically pressed Ti-6Al-4V alloys: experiments and modeling, Mech. Mater. 33 (2001) 425e439. [5] A. Majorell, S. Srivatsa, R.C. Picu, Mechanical behavior of Ti-6Al-4V at high and moderate temperatures-part I: experimental results, Mater. Sci. Eng. A 326 (2002) 297e305. [6] R.C. Picu, A. Majorell, Mechanical behavior of Ti-6Al-4V at high and moderate temperatures-part II: constitutive modeling, Mater. Sci. Eng. A 326 (2002) 306e316. [7] A.J.W. Johnson, C.W. Bull, K.S. Kumar, C.L. Briant, The influence of microstructure and strain rate on the compressive deformation behavior of Ti-6Al4V, Metallurgical Mater. Trans. A 34 (2003) 295e306. [8] D.G. Lee, S. Lee, C.S. Lee, S. Hur, Effects of microstructural factors on quasistatic and dynamic deformation behaviors of Ti-6Al-4V alloys with wid€tten structures, Metallurgical Mater. Trans. A 34 (2003) 2541e2548. mansta [9] A. Khan, R. Kazmi, B. Farrokh, M. Zupan, Effect of oxygen content and
[15] [16]
[17]
[18] [19]
[20] [21]
[22]
[23]
microstructure on the thermo-mechanical response of three Ti-6Al-4V alloys: experiments and modeling over a wide range of strain-rates and temperatures, Int. J. Plasticity 23 (2007) 1105e1125. S. Roy, S. Suwas, The influence of temperature and strain rate on the deformation response and microstructural evolution during hot compression of a titanium alloy Ti-6Al-4V-0.1B, J. Alloys Compd. 548 (2013) 110e125. C.H. Park, J.H. Kim, Y.T. Hyun, J.T. Yeom, N.S. Reddy, The origins of flow softening during high-temperature deformation of a Ti-6Al-4V alloy with a lamellar microstructure, J. Alloys Compd. 582 (2014) 126e129. N. Kotkunde, A.D. Deole, A.K. Gupta, S.K. Singh, Comparative study of constitutive modeling for Ti-6Al-4V alloy at low strain rates and elevated temperatures, Mater. Des. 55 (2014) 999e1005. F.Z. Wang, J. Zhao, N.B. Zhu, Z.L. Li, A comparative study on Johnson-Cook constitutive modeling for Ti-6Al-4V alloy using automated ball indentation (ABI) technique, J. Alloys Compd. 633 (2015) 220e228. H.B. Wu, S. To, Serrated chip formation and their adiabatic analysis by using the constitutive model of titanium alloy in high speed cutting, J. Alloys Compd. 629 (2015) 368e373. M.A. Meyers, Dynamic Behavior of Materials, John Wiley & Sons, New York, 1994. S. Nemat-Nasser, J.B. Isaacs, J.E. Starrett, Hopkinson techniques for dynamic recovery experiments, Proc. R. Soc. A: Math. Phys. Eng. Sci. 435 (1991) 371e391. S. Nemat-Nasser, J.B. Isaacs, Direct measurement of isothermal flow stress of metals at elevated temperatures and high strain rates with application to Ta and Ta-W alloys, Acta Mater. 45 (1997) 907e919. R.D. Luntz, R.M. Griffin, S.J. Green, S.C. Chou, High-strain-rate tests on titanium 6-6-2 utilizing a unique rate-testing machine, Exp. Mech. 15 (1975) 396e402. J.J. Fundenberger, M.J. Philippe, F. Wagner, C. Esling, Modelling and prediction of mechanical properties for materials with hexagonal symmetry, Acta Mater. 45 (1997) 4041e4055. Y.M. Xia, Y. Wang, Dynamic testing of materials with the rotating disk indirect bar-bar tensile impact apparatus, J. Test. Eval. 35 (2007) 31e35. W. Huang, X. Zan, X. Nie, M. Gong, Y. Wang, Y.M. Xia, Experimental study on the dynamic tensile behavior of a poly-crystal pure titanium at elevated temperatures, Mater. Sci. Eng. A 443 (2007) 33e41. G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures, in: Proceeding of the Seventh International Symposium on Ballistic, 1983, p. 541. The Hague, The Netherlands. A.S. Khan, R.Q. Liang, Behaviors of three bcc metal over a wide range of strain rates and temperatures: experiments and modeling, Int. J. Plasticity 15 (1999) 1089e1109.
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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
The constitutive responses of Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy at high strain rates and elevated temperatures Jun Zhang a, Yang Wang a, *, Xiang Zan b, Yu Wang a a
CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, PR China b School of Materials Science and Engineering, Hefei University of Technology, Hefei 230009, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 May 2015 Accepted 16 May 2015 Available online 19 June 2015
The effects of strain rate and temperature on the uniaxial tension responses of Ti-6.6Al-3.3Mo-1.8Zr0.29Si alloy are systematically investigated over a wide range of strain rates, 0.001e1150 s1, and initial temperatures, 293e573 K. Dynamic tension and recovery tests are conducted using a splitHopkinson tension bar to obtain the adiabatic and isothermal stress-strain responses of the alloy at high strain rates. The experiments reveal that the tension behavior of the alloy is dependent on the strain rate and temperature. The value of initial yield stress increases with increasing strain rate and decreasing temperature, whereas the isothermal strain-hardening behavior changes little with strain rate and temperature over the full ranges explored. The adiabatic temperature rise is the main reason for the reduction of strain hardening rate during the high-rate deformation process. SEM observations indicate that the tension specimen is broken in a manner of ductile fracture. The phenomenologically based constitutive model, JohnsoneCook model, is suitably modified to describe the rate-temperature dependent deformation behavior of Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy. The model correlations are in good agreement with the experimental data within the investigated range of strain rates and temperatures. © 2015 Elsevier B.V. All rights reserved.
Keywords: Titanium alloy Uniaxial tension Rate-temperature sensitivity Adiabatic softening Constitutive model
1. Introduction
aþb titanium alloys are attractive structural materials due to their excellent combination of high strength, light weight, good formability and elevated corrosion resistance. With the wide engineering applications of these alloys involving high strain rates and elevated temperatures (such as hot-forming, high speed machining or foreign-object impact), it is critically important to have a fundamental understanding of the stress-strain behavior of the alloys over a wide range of strain rates and temperatures. Finite element analysis technique has been widely used as an effective numerical tool in forming process optimization and structural design. It is well recognized that the validity of the simulation results is greatly affected by the reliability and accuracy of prediction ability of the constitutive models. Therefore, it is crucial to propose valid formulations of rate and temperature dependent constitutive models in the numerical simulations. A considerable number of
* Corresponding author. E-mail address: [email protected] (Y. Wang). http://dx.doi.org/10.1016/j.jallcom.2015.05.131 0925-8388/© 2015 Elsevier B.V. All rights reserved.
experimental investigations and constitutive modeling studies have been performed on the stress-strain behavior for titanium alloys such as Tie6Ale4V which is one of the most extensively used two-phase titanium alloys [1e14]. Experimental investigation indicates that the mechanical responses of Tie6Ale4V are highly sensitive to the specified loading conditions such as strain rate and temperature. The yield strength of the alloy increases with increasing strain rate and exhibits more sensitivity to temperature than to the strain rate, while the strain hardening rate decreases with increasing strain rate. In addition, phenomenologically and physically based constitutive models have been used to characterize the rate and temperature dependent mechanical responses of titanium alloys at various strain rates and temperatures. Silva and Ramesh reviewed experimental results of high-strain-rate compressive deformation and failure modes on Tie6Ale4V and discussed the observations of adiabatic shear localization [1]. Macdougall and Harding performed tension and torsion tests on Tie6Ale4V alloy at high strain rates and used a Zerilli-Armstrong type constitutive relation to predict the observed mechanical responses in tensile impact tests [3]. Nemat-Nasser et al. investigated the compressive responses of Tie6Ale4V over a broad range of
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strain rates (103e7000 s1) and initial temperatures (77e1000 K) and proposed a physically based constitutive model in which the thermally activated mechanisms associated with the dislocation motion were taken into accounted [4]. Majorell et al. performed dynamic compressive tests on Tie6Ale4V in the temperature range of 77e1400 K and proposed a physically based constitutive model in which the flow stress contains a thermal and an athermal components [5,6]. Khan et al. [9] and Roy et al. [10] studied the compressive behaviors of Tie6Ale4V and Ti6Al4V-0.1B at high strain rates and elevated temperatures, respectively. Kotkunde et al. made a comparative study of constitutive modeling for Tie6Ale4V alloy using four models, namely the empirically based JohnsoneCook, Fields-Backofen and Khan-huang-Liang models and the physically based Mechanical Threshold Stress model [12]. Wang et al. [13] and Wu et al. [14] employed JohnsoneCook model to describe the flow responses of Tie6Ale4V at low strain rates and elevated temperatures and Tie6Ale2Sne2Zre3Moe1Cre2Nb at high strain rates and elevated temperatures. The split-Hopkinson pressure bar (SHPB) technique has been widely used to investigate the adiabatic compressive stress-strain behavior of metals at high strain rates [15]. It is well understood that the high strain-rate deformation process is adiabatic, where the deformation is instantaneous and the heat generated by the plastic work will be stored in the specimen. Thus there is a continuous temperature rise during the plastic deformation process and thermal work-softening effect coupled with the isothermal strain hardening effect should be taken into account in the modeling of stress-strain curves for materials whose flow stress is temperature dependent. Nemat-Nasser et al. proposed the recovery experimental technique on the SHPB to measure the isothermal compressive stress-strain responses of metals at high strain rates [16,17]. The advantage of this recovery technique is that it is possible to uncouple the thermal work-softening effect from the isothermal strain hardening effect on the plastic deformation under high strain-rate loading conditions. It can be found that most of the aforementioned studies have been mainly concentrated on the measurement and characterization of compression responses of titanium alloys. Few efforts have been reported on the tension stress-strain responses and constitutive modeling of titanium alloys at high strain rates and elevated temperatures, though it is found that a and aþb titanium alloys display the tension-compression asymmetry due to their lowsymmetry hcp systems [18,19]. The objective of this paper is to experimentally investigate the strain-rate and temperature sensitivity of the tension responses of Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy (referred as TC11 in China) over a wide range of strain rates (103e103 s1) and temperatures (293e573 K). The isothermal stress versus strain responses at high strain rates are measured to evaluate the strain hardening behavior of the alloy. The adiabatic and isothermal constitutive responses at various strain rates and temperatures are compared and a phenomenological constitutive model incorporating the thermal-mechanical coupling effect is proposed to characterize the measured tension responses of the alloy.
cooling down to the room temperature in the air. Metallographic examination of the undeformed specimen reveals that the microstructure of the alloy is duplex, containing a lamellar Widmanstatten structure dispersed between primary globular a grains, as shown in Fig. 1. 2.2. Experimental procedure Uniaxial tension testing was conducted on the alloy under quasi-static and dynamic loading conditions. Quasi-static tension tests were carried out on an MTS810 servo-hydraulic testing system at a rate of 0.001 s1 to obtain the isothermal stress-strain curves. Dynamic tension tests were performed using a split Hopkinson tension bar (SHTB) system at strain rates of 190e1150 s1 to determine the adiabatic stress-strain curves. Fig. 2 shows the sketch of the SHTB system which comprises an impact apparatus, a prefixed short metal bar, an incident bar and a transmitted bar [20]. The SHTB system was operated as follows. The tensile incident pulse was produced by the deformation of the prefixed metal bar due to the impact between the hammer and the impact block. The amplitude and duration of the incident pulse can be controlled by adjusting the length and diameter of the prefixed metal bar and the impact velocity of the hammer. The role of the prefixed metal bar (made of 2A12 aluminum alloy in China) is not only the pulse producer but also the low-pass mechanical filter due to its plastic deformation. The plate-shaped tension specimen was connected to the incident/transmitted bars using adhesive connection. Dynamic tension recovery tests were performed on the SHTB system also at strain rates of 190e500 s1 to determine the isothermal stressstrain responses of the alloy under high rate loading conditions. The tensile incident pulse with short duration was generated to make the specimen deform into the plastic deformation at a certain level of plastic strain. Then the deformed specimen was allowed to return to its initial testing temperature. Namely the specimen was subjected to a single loading and unloading at a high strain rate. By repeating the loading-unloading several times on the same specimen at the same strain rate, the dynamic tension recovery tests came into operations. The isothermal curves at high strain rates were obtained by connecting the points associating with the initial yielding upon each loading. The aforementioned quasi-static and high-rate tests in tension were also performed as a function of temperature. The investigated testing temperature range was 293e573 K. A rapid-contact localized heating technique was used to obtain the elevated temperatures up to 573 K. The detailed description of this heating technique and the calibration of the heating system can be found elsewhere [21].
2. Experimental details 2.1. Material The titanium alloy used in this investigation was obtained as commercial forged rods, 38 mm in diameter, purchased from BaoTi Group Co. Ltd. of China. The chemical composition (in wt%) of the alloy is 6.6Al, 3.3Mo, 1.8Zr, 0.29Si, 0.07Fe, 0.01C, 0.01N, 0.004H, 0.13O and balance Ti. The samples were prepared by annealing for 2 h at 1228 K, followed by aging for 6 h at 803 K and finished by
Fig. 1. Microstructure of the undeformed alloy.
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Fig. 2. Schematic diagram of the split-Hopkinson tension bar system.
2.3. Specimen preparation The dumb-bell shaped flat specimens were used in the uniaxial tension tests and were cut directly along the axis of the aging rod. The gage length, width and fillet radius of the dumbbell-shape plate specimen used in dynamic tension tests are 10 mm, 3.5 mm and 2 mm, respectively. The specimen thickness is 1.1 mm. The specimen geometry used in quasi-static tension tests is similar except that the gage length of 30 mm is long enough to avoid the end effects. The tension tests under each loading conditions were repeated at least three times and the good repeatability of the data was obtained. 3. Results and discussions 3.1. The tension stress-strain responses of Ti-6.6Al-3.3Mo-1.8Zr0.29Si alloy The engineering stress e engineering strain curves of Ti-6.6Al3.3Mo-1.8Zr-0.29Si alloy were obtained from the quasi-static and dynamic tests. According to the constant volume assumption, the true stress and true strain were converted from engineering values. Fig. 3 shows the true stress-strain responses of the alloy as a function of four strain rates at three initial temperatures. As presented in Fig. 3(a), the responses of the alloy at room temperature exhibit the obvious strain rate sensitivity and elasticeplastic deformation characteristics. Since no obvious yield point can be found in the stress-strain curve, the flow stress at 0.2% plastic strain is adopted as initial yield stress. There is a positive strain-rate sensitivity with respect to the initial yielding behavior. The values of initial yield stress at high strain rates increase significantly compared with quasi-static results, which presents a considerable strain-rate strengthening phenomenon. However, the plastic response at high strain rates is absolutely different from that under quasi-static loading conditions. The working hardening rate decreases as the strain rate increases and thermal softening behavior can be observed in the adiabatic experimental results at high strain rate tests, which are consistent with most of the FCC and BCC metals. Also, this observation is similar to other investigations on the rate-sensitive compression responses of aþb titanium alloys [7,9]. When the temperature increases, the strain rate also has a profound effect on the initial yielding and strain hardening behavior. Similar trend of strengthening in yield stress and reduction in strain hardening rate can be seen at high strain rates. The effects of strain rate and temperature on the initial yield stress are summarized in Fig. 4. It is obvious that the initial yield stress increases when strain rate increases and temperature decreases. Such strain rate and temperature sensitivity of Ti-6.6Al-3.3Mo-1.8Zr0.29Si alloy is comparable to that of Tie6Ale4V under compressive loading conditions [2,4,9,10]. To clearly understand the influence of strain rate on the tension strain hardening behavior of the alloy, the loading-unloadingreloading results at the rate of 500 s1 and temperature of 293 K are shown in Fig. 5. The dashed curves represent a series of multiloading-unloading stress-strain results of the same specimen. The
isothermal stress-strain relation at the rate of 500 s1 is obtained by connecting the points associating with initial yielding upon each loading. The results of the adiabatic tests performed at the rate of 500 s1 and the isothermal tests performed at the rate of 0.001 s1 are also presented in Fig. 5. It can be noticed an obvious difference exists between isothermal and adiabatic curves at the strain rate of 500 s1. The strain hardening rate of adiabatic curve is much lower than that of isothermal curve, which is solely due to the thermal softening caused by the adiabatic temperature rise. Namely, the adiabatic stress-strain response in the course of plastic deformation at a high strain rate is an essentially coupled result of strain-rate strengthening, strain hardening and adiabatic softening effects. By means of dynamic tension recovery technique on the SHTB, the adiabatic softening effect can be experimentally uncoupled from the strain-rate strengthening and strain hardening effects. Moreover, the strain hardening behavior in the isothermal responses at 500 s1 changes little compared to the isothermal result under quasi-static loading conditions, which indicates that the strain rate has totally different influence on the initial yielding and strain hardening behavior of the alloy. Such rate-independent behavior of isothermal strain hardening for Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy is comparable to the compressive thermomechanical responses of Ti6Al4V alloys and a tantalumetungsten alloy investigated by Nemat-Nasser et al. [4,17]. Fig. 6 summarizes the isothermal relations of the alloy at different strain rates and temperatures. The aforementioned rate-independent behavior of isothermal strain hardening is also found at the strain rate of 190 s1 and temperatures of 293 and 423 K. Comparing the quasi-static isothermal results (0.001 s1) obtained at 293, 423 and 573 K, the levels of isothermal strain hardening rate at different initial temperatures are approximately identical, showing that the isothermal strain hardening exhibits some degree of insensitivity on the initial temperature within the investigated temperature range. Such temperature-independent isothermal strain hardening behavior is also observed at the strain rate of 190 s1. In order to understand the failure mechanism of the alloy, the microstructure near the broken area of the specimen and the fracture surface were examined using the optical microscopy and the scanning electron microscopy, as shown in Fig. 7. Naked-eye observation indicates that the localization necking occurs during the course of plastic deformation. The metallographic examination of the necking zone indicates that the grains are stretched obviously along the loading direction and the crack propagates through both primary a and lamellar Widmanstatten grains. The SEM fractographs of the alloy display three kinds of dimple morphologies, namely round dimples, shallow dimples and parabolic dimples corresponding to fibrous zone, radiation area and shearing lip, as marked with A, B and C in Fig. 7. Fig. 8 displays the fractographs of the specimen tested at strain rates of 0.001 and 1150 s1 and at temperatures of 293 and 573 K. It can be seen that micrographs of the fracture surfaces for all loading cases are similar and exhibit the typical dimple morphology, which indicates that the tension specimen is broken in a manner of ductile fracture with good elongation. The size and depth of ductile dimples change little with the change of strain rates and temperatures, as seen in Fig. 8.
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Fig. 4. Effects of strain rate and temperature on the initial yield stress.
Fig. 3. True stress-strain curves obtained at strain rates of 0.001, 190, 500 and 1150 s1 and temperatures of (a) 293 K, (b) 423 K and (c) 573 K. Fig. 5. Comparison of isothermal and adiabatic responses at 293 K.
3.2. Constitutive modeling of tension behavior Johnson and Cook proposed an empirical constitutive model for metals subjected to large strain, high strain rates and various temperatures [22]. This constitutive model is one of the most well-
used phenemonologically based models in the applicable computer codes because of its simplicity and the availability of model parameters. The strain hardening, strain rate strengthening and thermal softening are taken into account in the model as a simple
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multiplication form. The flow stress, s, is expressed as a function of strain, ε, strain rate, ε_ , and temperature, T, as follows.
ε_ T Tr m 1 s ¼ ðA þ Bεn Þ 1 þ C ln ε_ 0 Tm Tr
Fig. 6. Comparison of isothermal responses at different strain rates and temperatures.
(1)
where A is the initial yield stress, B is the coefficient of strain hardening and n is the strain hardening exponent. ε_ 0 and C are the reference strain rate and the coefficient of strain rate strengthening, respectively. Tr and Tm are the reference temperature and melting temperature, and m is the thermal softening exponent. The expression in the first set of parentheses corresponds to the initial yielding and strain hardening behavior of materials and their strain-rate and temperature dependences are both represented by the second and third sets of parentheses, respectively. Khan and Liang found that the original JohnsoneCook model has inherent problems of modeling the dependence of strainhardening behavior of metals on strain-rate and temperature [23]. It was pointed out that the model was unable to accurately describe the decrease of work hardening with the increasing strain
Fig. 7. Metallographic examination and SEM observation for the tensile specimen loaded at the strain rate of 0.001 s1 and temperature of 423 K.
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Fig. 8. Fracture surface morphologies at (a) 0.001 s1, 293 K, (b) 1150 s1, 293 K, (c) 0.001 s1, 573 K and (d) 1150 s1, 573 K.
rate. They suggested that to better predict the work-hardening behavior, strain and strain rate must have some coupled effects on the description of work-hardening relation of the materials. From isothermal stress-strain curves of Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy shown in Fig. 4, it should be noted that the strain-rate and temperature effects on the initial yield stress and the isothermal strain hardening are not identical. Moreover, the adiabatic strain hardening is dependent on the temperature rise during the highrate deformation process. Due to such experimental observation, a modified JohnsoneCook model is proposed to improve the accuracy of the original model.
s ¼ A 1 þ C1 ln_ε* 1 T *m1 þ Bεn 1 þ C2 ln_ε* 1 T *m2 elDT
DT ¼
g rCp
(1) By taking the plastic strain as zero in Eqn. (2), the yield stress, syield, is expressed as a function of strain rate and temperature as follows.
(2)
syield ¼ A 1 þ C1 ln_ε* 1 T *m1
(3)
By comparing the experimental data at different strain rates and initial temperatures, as presented in Fig. 4, the values of three material constants, A, C1 and m1, are obtained.
Zε sdε
applied to adiabatic loading conditions corresponding to high strain-rate loadings. For isothermal loading conditions, the added term in Eqn. (2), elDT, needs to be omitted in the modified model. To determine the material constants for the modified model, isothermal and adiabatic experimental data at various strain rates and temperatures are first used to obtain an initial set of values for the material constants with the following steps.
0
where DT is the adiabatic transient temperature rise in the specimen during the high-rate loading process and l is the adiabatic softening coefficient. g is the conversion factor of plastic work to heat, r is the mass density and Cp is the specific heat capacity at constant pressure. ε_ * ¼ ε_ =_ε0 and T*¼(TTr)/(TmTr) correspond to the dimensionless strain rate and temperature, respectively. Compared with the original JohnsoneCook model, two modifications have been made in the present paper. Firstly, two pairs of model parameters, C1,m1 and C2,m2, are introduced to represent the strain-rate and temperature effects on the initial yield stress and strain hardening behavior, respectively. Secondly, a modified form of strain hardening incorporating the adiabatic softening is included in the second term of the model to describe the constitutive behavior of the alloy at high strain rates. Eight material constants, A, B, n, C1, C2, m1, m2 and l, are involved in the modified JohnsoneCook model. It should be emphasized that Eqn. (2) is
(4)
(2) By considering the isothermal constitutive relation at different strain rates and temperatures, Eqn. (2) will reduce to
sisothermal ¼ A 1 þ C1 ln_ε* 1 T *m1 þ Bεn 1 þ C2 ln_ε* 1 T *m2 (5) By fitting Eqn. (5) to the experimental results for isothermal stress-strain responses, as shown in Fig. 6, the values of four material constants, B, n, C2 and m2, are evaluated. (3) The final step is to determine the remaining material constant l. By comparing the adiabatic results with the isothermal results at high strain rates, as shown in Figs. 3 and 6, the adiabatic softening coefficient, l, can be derived as follows.
J. Zhang et al. / Journal of Alloys and Compounds 647 (2015) 97e104
Bεn 1 þ C2 ln_ε* 1 T *m1 1 ln l¼ DT sadiabatic sisothermal þ Bεn ð1 þ C2 ln_ε* Þð1 T *m1 Þ (6) where sisothermal and sadiabatic correspond to the isothermal and adiabatic flow stresses under high strain rate loading conditions, respectively. The adiabatic temperature rise, DT, is estimated using Eqn. (3). For the investigated Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy, its mass density is 4486 kg/m3 and its melting temperature is 1875 K. The conversion factor of plastic work into heat is assumed to be 0.9 during the course of high rate deformation. Since the specific heat capacity at constant pressure is temperature dependent, an approximation of this temperature dependence is given as
Cp ¼ 588:8 0:3419T þ 0:00065T 2
J=ðkg KÞ
(7)
The values of material constants estimated according to aforementioned steps are used as initial input into an optimization procedure based on the least square method. Here, the reference strain rate and temperature, ε_ 0 and Tr, are taken to be 0.001 s1 and 293 K, respectively. The final values of eight material constants for Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy are A ¼ 942.5 MPa, B ¼ 1420.6 MPa, n ¼ 0.721, C1 ¼ 0.034, C2 ¼ 0.001, m1 ¼ 0.726, m2 ¼ 27.1 and l ¼ 0.064 K1, respectively. The predicted yield stress from Eqn. (4) is shown in Fig. 4. The comparison of model correlations with the experimental stress-strain results is shown in Fig. 9. It can be seen that good agreement between the model results and experimental data is obtained, which indicates that the modified model is applicable to describe the rate and temperature
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dependent deformation behavior of Ti-6.6Al-3.3Mo-1.8Zr-0.29Si alloy subjected to uniaxial tension loading over a wide range of strain rates and temperatures. Furthermore, the dynamic tension recovery tests can achieve the experimental uncoupling between the strain hardening behavior and adiabatic softening behavior of rate-temperature dependent metals, which provides an effective way to obtain a better set of material constants for the constitutive modeling. 4. Conclusion Uniaxial tension stress-strain behavior of Ti-6.6Al-3.3Mo-1.8Zr0.29Si alloy with duplex microstructure was investigated as a function of strain rate and temperature. The high strain-rate tension tests were performed using the split Hopkinson tension bar apparatus to obtain the adiabatic experimental results. Furthermore, the dynamic tensile recovery tests based on the Hopkinson bar technique were conducted to evaluate the isothermal stressstrain responses of the alloy under high strain-rate loading conditions. The experiment reveals that the tension responses of Ti6.6Al-3.3Mo-1.8Zr-0.29Si alloy are sensitive to strain rate and temperature. The values of initial yield stress decrease with increasing temperature and increase significantly with increasing strain rate, while the isothermal strain hardening behavior is essentially independent of strain rate and temperature. The reduced strain hardening behavior was observed at high strain rates due to adiabatically thermal softening. Microstructure observations of the fracture specimen indicate that the tension specimen is broken in a manner of ductile fracture. The phenomenologically-based JohnsoneCook model was modified to enable the different rate-temperature sensitivities of initial yielding
Fig. 9. Model correlations with the experimental data at temperatures of 293, 423 and 573 K and strain rates of (a) 0.001 s1, (b) 190 s1, (c) 500 s1 and (d) 1150 s1.
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and strain hardening and the adiabatic softening at high strain rates to be accommodated. Dynamic tensile recovery experimental results were used to accurately determine the material constants in the modified constitutive model. Comparison between the experimental data and model results indicates that such model is capable of capturing the tensile thermomechanical responses of Ti-6.6Al3.3Mo-1.8Zr-0.29Si alloy within the investigated range of strain rates and temperatures.
[10]
[11]
[12]
Acknowledgments
[13]
The authors acknowledge the financial support by the National Natural Science Foundation of China under Contract No. 11172288.
[14]
References [1] M.G. Silva, K.T. Ramesh, The rate-dependent deformation and localization of fully dense and porous Ti-6Al-4V, Mater. Sci. Eng. A 232 (1997) 11e22. [2] W.S. Lee, C.F. Lin, Plastic deformation and fracture behaviour of Ti-6Al-4V alloy loaded with high strain rate under various temperatures, Mater. Sci. Eng. A 241 (1998) 241e248. [3] D.A.S. Macdougall, J. Harding, A constitutive relation and failure criterion for Ti6Al4V alloy at impact rates of strain, J. Mech. Phys. Solids 47 (1999) 1157e1185. [4] S. Nemat-Nasser, W.G. Guo, V.F. Nesterenko, S.S. Indrakanti, Y.B. Gu, Dynamic response of conventional and hot isostatically pressed Ti-6Al-4V alloys: experiments and modeling, Mech. Mater. 33 (2001) 425e439. [5] A. Majorell, S. Srivatsa, R.C. Picu, Mechanical behavior of Ti-6Al-4V at high and moderate temperatures-part I: experimental results, Mater. Sci. Eng. A 326 (2002) 297e305. [6] R.C. Picu, A. Majorell, Mechanical behavior of Ti-6Al-4V at high and moderate temperatures-part II: constitutive modeling, Mater. Sci. Eng. A 326 (2002) 306e316. [7] A.J.W. Johnson, C.W. Bull, K.S. Kumar, C.L. Briant, The influence of microstructure and strain rate on the compressive deformation behavior of Ti-6Al4V, Metallurgical Mater. Trans. A 34 (2003) 295e306. [8] D.G. Lee, S. Lee, C.S. Lee, S. Hur, Effects of microstructural factors on quasistatic and dynamic deformation behaviors of Ti-6Al-4V alloys with wid€tten structures, Metallurgical Mater. Trans. A 34 (2003) 2541e2548. mansta [9] A. Khan, R. Kazmi, B. Farrokh, M. Zupan, Effect of oxygen content and
[15] [16]
[17]
[18] [19]
[20] [21]
[22]
[23]
microstructure on the thermo-mechanical response of three Ti-6Al-4V alloys: experiments and modeling over a wide range of strain-rates and temperatures, Int. J. Plasticity 23 (2007) 1105e1125. S. Roy, S. Suwas, The influence of temperature and strain rate on the deformation response and microstructural evolution during hot compression of a titanium alloy Ti-6Al-4V-0.1B, J. Alloys Compd. 548 (2013) 110e125. C.H. Park, J.H. Kim, Y.T. Hyun, J.T. Yeom, N.S. Reddy, The origins of flow softening during high-temperature deformation of a Ti-6Al-4V alloy with a lamellar microstructure, J. Alloys Compd. 582 (2014) 126e129. N. Kotkunde, A.D. Deole, A.K. Gupta, S.K. Singh, Comparative study of constitutive modeling for Ti-6Al-4V alloy at low strain rates and elevated temperatures, Mater. Des. 55 (2014) 999e1005. F.Z. Wang, J. Zhao, N.B. Zhu, Z.L. Li, A comparative study on Johnson-Cook constitutive modeling for Ti-6Al-4V alloy using automated ball indentation (ABI) technique, J. Alloys Compd. 633 (2015) 220e228. H.B. Wu, S. To, Serrated chip formation and their adiabatic analysis by using the constitutive model of titanium alloy in high speed cutting, J. Alloys Compd. 629 (2015) 368e373. M.A. Meyers, Dynamic Behavior of Materials, John Wiley & Sons, New York, 1994. S. Nemat-Nasser, J.B. Isaacs, J.E. Starrett, Hopkinson techniques for dynamic recovery experiments, Proc. R. Soc. A: Math. Phys. Eng. Sci. 435 (1991) 371e391. S. Nemat-Nasser, J.B. Isaacs, Direct measurement of isothermal flow stress of metals at elevated temperatures and high strain rates with application to Ta and Ta-W alloys, Acta Mater. 45 (1997) 907e919. R.D. Luntz, R.M. Griffin, S.J. Green, S.C. Chou, High-strain-rate tests on titanium 6-6-2 utilizing a unique rate-testing machine, Exp. Mech. 15 (1975) 396e402. J.J. Fundenberger, M.J. Philippe, F. Wagner, C. Esling, Modelling and prediction of mechanical properties for materials with hexagonal symmetry, Acta Mater. 45 (1997) 4041e4055. Y.M. Xia, Y. Wang, Dynamic testing of materials with the rotating disk indirect bar-bar tensile impact apparatus, J. Test. Eval. 35 (2007) 31e35. W. Huang, X. Zan, X. Nie, M. Gong, Y. Wang, Y.M. Xia, Experimental study on the dynamic tensile behavior of a poly-crystal pure titanium at elevated temperatures, Mater. Sci. Eng. A 443 (2007) 33e41. G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures, in: Proceeding of the Seventh International Symposium on Ballistic, 1983, p. 541. The Hague, The Netherlands. A.S. Khan, R.Q. Liang, Behaviors of three bcc metal over a wide range of strain rates and temperatures: experiments and modeling, Int. J. Plasticity 15 (1999) 1089e1109.