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Workability characteristics and mechanical behavior modeling of severely deformed pure titanium at high temperatures
Materials and Design 53 (2014) 749–757
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Workability characteristics and mechanical behavior modeling of severely deformed pure titanium at high temperatures Seyed Vahid Sajadifar, Guney Guven Yapici ⇑ Mechanical Engineering Department, Ozyegin University, Istanbul, Turkey
a r t i c l e
i n f o
Article history: Received 12 April 2013 Accepted 17 July 2013 Available online 25 July 2013 Keywords: Titanium Equal channel angular extrusion/pressing Severe plastic deformation Modeling of hot deformation
a b s t r a c t In the present study, compression tests were performed at temperatures of 600–900 °C and at strain rates of 0.001–0.1 s1 to study the deformation and workability characteristics of commercially pure titanium after severe plastic deformation (SPD). It was found that the effects of temperature and strain rate are significant in dictating the steady state flow stress levels and the strain values corresponding to peak flow stress. The strain rate sensitivity (m) during hot compression of severely deformed Ti was shown to be strongly temperature dependent, where m increased with the increase in deformation temperature up to 800 °C. High temperature workability was analyzed based on the flow localization parameter (FLP). According to the FLP values, deformation at and below 700 °C is prone to flow localization. The flow behavior was predicted using Arrhenius type and dislocation density based models. The validities of the models were demonstrated with reasonable agreement in comparison to the experimental stress– strain responses. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Due to its high specific strength, pure titanium is an excellent candidate for engineering applications requiring light structural components. One of the available methods for enhancing strength as well as ductility is grain refinement via severe plastic deformation (SPD). SPD shows promise for generating an ultra-fine-grained (UFG) microstructure leading to enhanced physical and mechanical properties in titanium and its alloys [1–7]. Microstructure and mechanical properties of pure Ti processed by SPD and cold rolling were investigated [2–4]. It was seen that SPD followed by cold rolling at ambient and at cryogenic temperatures is a viable method for increasing the ultimate tensile strength levels with moderate ductility [2,3]. Yapici et al. [4] investigated the flow anisotropy of UFG Ti with a similar two step processing history. Wear properties after SPD was another topic of interest showing contradictory results [5,6]. While coarse grained (CG) pure Ti displayed extensive plastic deformation and wedge formation that generated large wear debris, wear of the UFG Ti was dominated by abrasive wear mechanisms and resulted in lower wear rates [6]. In contrast, a separate study demonstrated no significant improvement on the wear resistance of the UFG microstructure [5]. A common two phase Ti alloy (Ti64) was also subjected to SPD and the texture development and resulting anisotropic mechanical properties were examined [7]. Most of the aforementioned studies showed that ⇑ Corresponding author. Tel.: +90 216 564 9115; fax: +90 216 564 9057. E-mail address: [email protected] (G.G. Yapici). 0261-3069/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.matdes.2013.07.057
equal channel angular extrusion (ECAE/ECAP) is one of the most favorable SPD techniques for enhancing mechanical properties in engineering applications due to its capability of maintaining dimensions of the initial billet after multiple processing steps [1– 5]. The demand for obtaining necessary forces to be applied during warm and hot working (typically rolling or forging) of materials has driven studies focused on workability and modeling of respective flow stress curves. It is well known that study of flow stress behavior at elevated temperatures is crucial for the adjustment of warm and hot working process parameters leading to improved workability. These characteristics were previously investigated for pure Ti and other metallic materials [8–13]. Among them, few studies mentioned the workability of pure Ti at elevated temperatures, as observed in hot deformation tests [8,11–13]. These studies mainly focused on the microstructural evolution of coarse grained pure Ti during hot deformation along with the constitutive modeling of the mechanical behavior. However, they typically lacked discussion on the workability parameters which is important for better understanding the stress–strain response. Moreover, neither characterization nor modeling at elevated temperatures (up to 900 °C) has been a topic of study for Ti with ECAE induced UFG microstructure. The present investigation aims to fill this gap by (a) demonstrating the effect of ECAE processing on warm and hot flow characteristics and workability behavior of pure Ti at elevated temperature, (b) predict the flow stress–strain response with two modeling approaches.
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Various constitutive equations based on dislocation density evolution during hot forming have been developed from the experimentally measured data to describe the sensitivity of the flow stress to the strain [9,14]. Mathematical approaches utilize the Zener–Hollomon parameter in a hyperbolic sinusoidal equation form for polynomial fitting up to the power of four or five in some alloys [13,15]. Both methods are suitable tools to numerically analyze the hot deformation process as a function of applied strain, strain rate and deformation temperature. Here, the model is extended up to sixth polynomial power and predictions are compared with those obtained from microstructural parameters. The objective of this work is to provide an insight into the elevated temperature formability and workability characteristics of severely deformed pure Ti along with efforts to simulate the experimental flow response.
Hot compression tests were conducted under isothermal conditions at three different strain rates of 0.001, 0.01 and 0.1 s1 and at temperatures of 600, 700, 800 and 900 °C. The temperature range is in the warm to hot work region, i.e. 0.45Tm–0.6Tm. All specimens were heated up to the deformation temperature and then the samples were deformed in a single loading step. The hot compression tests were conducted inside a temperature-controlled furnace mounted on an Instron mechanical testing frame. Temperatures were maintained throughout the tests to ensure uniformly heated samples. Optical microscopy was used to monitor the changes in the microstructure. The specimens were prepared using standard polishing techniques and then etched with Kroll’s reagent. 3. Results and discussion 3.1. Microstructural evolution
2. Materials and experimental procedure The as-received commercial purity grade 2 Ti was received in bar form with the composition listed in Table 1. The bars were coated with a graphite base lubricant before extrusion and were heated in a furnace to the deformation temperature of 300 °C where they were held for 1 h before extrusion. Finally, they were transferred to the 25.4 mm 25.4 mm cross section, 90° angle ECAE die which was preheated to 300 °C. Extrusion took place at a rate of 1.27 mm/s. Eight ECAE passes were performed following route E accumulating a total strain of 9.24 in the as-processed material [3]. Route E was selected as the ECAE processing route. This hybrid route results in the largest fully worked region in a given billet and provides high volume fraction of high-angle grain boundaries and equiaxed grain morphology [16]. Route E consists of an alternating rotation of the billet by +180° and +90° around its long axis, between successive passes. Between each pass, the billet was heated in the furnace for 15 min at the deformation temperature. Following each extrusion pass, the billets were water quenched to maintain the microstructure achieved during ECAE. Lowest possible processing temperatures were crucial in preventing possible recrystallization and partly achieved using the sliding walls concept that helps reducing the die frictional effects [17]. Using this method, a number of difficult-to-work materials were processed without macroscopic localization [18,19]. For studying the hot deformation behavior of pure Ti after ECAE, compression experiments were performed at high temperatures according to ASTM: E209-00. The main benefit of compression testing is that the imposed stress state is close to that occuring in actual bulk deformation operations as in, extrusion, forging, rolling and others alike. The compression specimens were electro-discharge machined (EDM) in rectangular blocks shape, 4 mm 4 mm 8 mm, with compression axis parallel to the extrusion direction [20]. All samples were ground and polished to remove major scratches and eliminate the influence of residual layer from EDM. The reduction in height has been chosen as 60% (true strain: 0.9) at the end of the compression tests to avoid barreling. This strain level, which is required to capture the effect of both dynamic recovery and recrystallization on deformation behavior, has been applied in several researches [21,22]. In addition, lubrication with graphite was used during hot compression tests to decrease friction effects and minimize barreling [22].
Table 1 Chemical composition of pure Ti used in this study. Element
C
Fe
H
N
O
Ti
wt.%
0.008
0.041
0.002
0.006
0.15
Balance
The initial microstructure of as-received Ti is shown in Fig. 1a. It can be easily observed that the structure consists of coarse grains with an average grain size of 43 lm. The microstructure of the as-received Ti subjected to eight passes of ECAE is demonstrated in Fig. 1b. ECAE processing leads to a significant refinement in the microstructure and transforms the coarse grained as-received microstructure to one with ultra-fine grains after eight passes. In addition, micrographs depicting the microstructural evolution before and after hot compression testing are also demonstrated. The microstructure of UFG Ti followed by 3 min annealing at 600 °C and 900 °C can be seen in Fig. 1c and d, respectively. Annealing period is selected based on the time it takes to achieve uniform heating of the samples at the desired compression test temperature. After the 600 °C heat treatment, the microstructure consists of both slightly coarsened grains and regions with ultra-fine grains. In contrast, the 900 °C heat treatment resulted in severe coarsening of the structure with an average grain size of 25 lm. The existence of several recrystallized equiaxed grains can be observed in Fig. 1d. Considering the development of microstructure during hot compression, the micrographs after deformation at 600 °C and 900 °C can be seen in Fig. 1e and f, respectively. The sample subjected to compression at 600 °C exhibits a fine and homogeneous microstructure with slightly larger grains in comparison with those of the pre-compression sample. After compression at 900 °C, the microstructure contained a mixture of fine and coarse grains. The evolution of such a microstructure could be attributed to the deformation of the recrystallized grains that formed during the pre-test heat treatment and the formation dynamically recrystallized during the compression test. 3.2. Flow stress behavior Fig. 2 shows the true stress–true strain curves for the compression tests of severely deformed pure Ti at various temperatures for the strain rates of 0.001, 0.01 and 0.1 s1. It can be observed that the flow stress of pure Ti subjected to ECAE is sensitive to both temperature and strain rate. In the previous work on the high temperature flow behavior of pure Ti with 40 lm grain size, compressive responses were depicted up to 700 °C [13]. In comparison with the present results, it can be seen that ECAE could increase the high temperature strength of pure Ti up to as high as 700 °C. The rise of flow stress levels in UFG materials during hot deformation was also reported for an aluminum 6063 alloy at a temperature range of 300–450 °C [23]. At all deformation temperatures, the flow stress levels slightly increase to a peak and then decrease gradually to a steady state level. Regardless of the strain level (low, peak or high), the flow stress levels show similar dependence to deformation temperature
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Fig. 1. The microstructure of pure Ti (a) in as-received condition, (b) after 8 passes of ECAE processing, (c) after 8 passes of ECAE processing followed by 3 min of annealing at 600 °C, (d) after 8 passes of ECAE followed by 3 min of annealing at 900 °C, (e) after 8 passes of ECAE processing followed by compression at 600 °C with a strain rate of 0.1 s1, and (f) after 8 passes of ECAE processing followed by compression at 900 °C with a strain rate of 0.1 s1.
and strain rate. Fig. 3 exhibits this behavior for the case of peak strain, i.e. strain corresponding to peak flow stress. Such flow behavior is a typical characteristic of hot working that is accompanied by dynamic recrystallization softening. The work hardening at 600 °C is higher than those obtained at temperatures above it. In contrast, the flow stress levels exhibit slight transitional drops indicating thermal softening for deformation temperatures above 600 °C. The observed stress levels decrease with increasing temperature and decreasing strain rate because lower strain rates and higher temperatures provide longer time for energy accumulation and higher dislocation mobility around boundary regions. This evidently results in the nucleation and growth of dynamically recrystallized grains and dislocation annihilation.
Fig. 4 summarizes the values of peak strain as a function of deformation temperature and strain rate. It is worth noting that the effects of temperature and strain rate are significant on the peak strain level during hot deformation. For most materials both in pure and alloy form, the peak strain decreases with higher temperatures or lower strain rates [9,24]. A similar behavior was observed for the case of Ti after SPD. As can be seen in Fig. 4, the peak strain value at any given temperature increases with increasing strain rate, though the effect is mostly pronounced at the lowest testing temperature of 600 °C. Since dynamic recrystallization (DRX) typically initiates at a critical strain before the peak stress [25], it is accelerated by the rise in temperature and the drop in strain rate. Although, Ti and its alloys are generally considered as
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Fig. 3. The variation of peak stress as a function of strain rate at different deformation temperatures.
Fig. 4. The dependence of peak strain to (a) deformation temperature and (b) strain rate.
of DRX related phenomena is favored. With increasing deformation temperatures, the sensitivity of peak strain to strain rate diminishes. Still, both applied strain rate and deformation temperature are important parameters that dictate the amount of critical strain required for the onset of DRX in ECAE processed pure Ti. 3.3. Evolution of material properties Fig. 2. True stress–true strain response of severely deformed Ti with different strain rates at (a) 600 °C, (b) 700 °C, (c) 800 °C, and (d) 900 °C.
high stacking fault energy (SFE) materials that are expected to be softened mainly by dynamic recovery (DRV), DRX processes have also been found to take place depending on the deformation parameters [26–29]. In the present study, especially at temperatures above 600 °C and/or strain rates below 0.01 s1, occurrence
Strain rate sensitivity is a significant property, which can affect workability of materials and deserves investigation for the present work. Strain rate sensitivity, m, is related to the effect of strain rate on dislocation generation and propagation and as such is an index of workability. The following well-known relation was employed to calculate the strain rate sensitivity, m, for hot deformation.
m¼
@ lnðrÞ @ lnðe_ Þ e
ð1Þ
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Fig. 5. Strain dependence of strain rate sensitivity at various temperatures.
The values of m can be obtained from the slope of the lines plotted according to Eq. (1) at a constant strain. The variation of the strain rate sensitivity as a function of strain is plotted in Fig. 5 exhibiting strong dependence to deformation temperature. This phenomenon can be attributed to the increased activity of diffusion related mechanisms and dynamic recrystallization at higher temperatures resulting in higher strain rate sensitivity [30]. It is worth noting that strain rate sensitivity shows a considerable increase above 700 °C. The strain rate sensitivity values obtained at 800 °C are slightly greater than those obtained at 900 °C for the whole strain interval. It is expected that m values increase with the rise of deformation temperature due to thermally activated mechanisms. However, constancy of m can be attributed to the onset of DRX processes which may occur over a range of strain rates and temperatures [31,32]. In the present case, dominance of DRX around 800–900 °C might explain the similar strain rate sensitivity values at both temperatures. Considering the strain dependence of rate sensitivity, the m values at 600 °C and 700 °C remarkably increase and peak at a strain of 0.5 and 0.35, respectively. Then, the m values decrease with increasing strain and plateau at a constant value. Similar behavior is observed at a less pronounced degree for 800 °C, where the strain rate sensitivity peaks at a strain level of 0.25 and then drops gradually and remains almost constant for the rest of the strain range. At the highest deformation temperature, m is fairly constant and can be considered to be independent of true strain. It can be concluded that for severely deformed pure Ti, higher strain rate sensitivity levels are observed with increasing deformation temperature as expected. However, m is not a strong function of strain at or above 800 °C. The other crucial factor affecting the elevated temperature deformation behavior is the strain hardening rate, c, which is defined as:
c¼
1 dr r de e_
ð2Þ
where r is the flow stress, e_ is the strain rate and e is the true strain. The variation of strain hardening rate values with strain at different temperatures and strain rates is presented in Fig. 6. The strain hardening rates during hot deformation, as defined in Eq. (2), exhibit sharp decrease at low strain levels (below 0.2) while they stay almost constant at higher strains. Moreover, again at low strain levels, higher deformation temperatures result in lower strain hardening rates, indicating the dominance of dislocation rearrangement and recovery mechanisms. Since more recrystallization is expected at higher temperatures and lower strain rates, increased softening and thereby reduced flow stress levels after peak stress could be attributed to a higher volume fraction of recrystallized grains in the microstructure [33]. Irrespective of the deformation tempera-
Fig. 6. Strain hardening rate versus true strain plot demonstrating behavior at different temperatures and at a strain rate of (a) 0.1 s1 (b) 0.01 s1 and (c) 0.001 s1.
ture and strain rate, constant c was observed at above true strain levels of 0.2. In order to estimate the workability of metals and alloys under hot deformation conditions, flow localization parameter (FLP) has been proposed [34,35]. This factor can be represented as follows:
FLP ¼
c m
ð3Þ
Using Eq. (3), the average flow localization parameters for the whole strain range at different temperatures and strain rates, are calculated and summarized in Table 2. According to Table 2, FLP values increase with decreasing deformation temperature and increasing strain rate. It was stated that the materials with FLP P 5 are prone to flow localization [34,35]. Therefore, it is expected that UFG Ti is not expected to have favorable workability at temperatures below 800 °C due to flow localization. The values of FLP were highest and exceed 21 at 600 °C for all strain rates. The high magnitude of flow localization can be attributed to the lack of dynamic recrystallization, which might eventually result in failure [36]. In contrast, the values of FLP in all strain rates remain less than 2 at 800 and 900 °C. The effect of
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Table 2 Flow localization parameter for severely deformed Ti under different deformation temperatures and strain rates. Strain rate (s1)
600 °C
700 °C
800 °C
900 °C
0.1 0.01 0.001
22.45 21.02 22.13
7.35 7.15 7.29
1.91 0.80 1.56
0.29 0.35 0.13
strain rate on the FLP is less straightforward. Despite this, highest FLP values are observed for the highest deformation rates for all temperatures. It is clear that the ductile behavior during hot working of ECAE processed pure titanium is closely associated with deformation temperature and to a smaller extent with deformation rate. With reduced tendency to flow localization, deformation over 700 °C could be preferred for metal forming purposes in severely deformed Ti. 3.4. Prediction of flow stress by Arrhenius modeling Constitutive equations proposed by Zener and Hollomon have been used to describe the deformation behavior at high temperatures. In hot working processes, several constitutive equations have commonly been applied. These equations are based on the Zener–Holloman formulation and have been used in various hot deformation investigations and calculation details can be found elsewhere [13,15,37,38]. The flow stress can be written as a function of the Zener–Hollomon parameter, considering the definition of the hyperbolic law [13,14]:
8 #12 9 1=n " 1=n = 1 < Z Z r ¼ Ln þ þ1 ; A a : A Z ¼ e_ exp
Q RT
n
¼ A½sin hðarÞ
ð4Þ
ð5Þ
The values of material constants (A, a and n) and the apparent activation energy for hot deformation (Q) of the constitutive equations were computed for 0.05 incremental strain levels within the range of 0.05–0.9. For this purpose, the least squares method was utilized. The variations of these constants with true strain can be represented by polynomial functions of strain. For the sake of brevity, only the plot pertaining to Q is shown in Fig. 7. The activation energy ranges between 323 and 344 kJ/mol. Since Q represents the energy for deformation, higher Q would indicate the requirement for higher energy to activate relevant deformation mechanisms. The Q value for coarse grained pure Ti was reported as 218–
240 kJ/mol, which is considerably lower than that observed in the present work [13]. ECAE processing increased the amount of activation energy to sustain post-ECAE hot deformation. At the beginning of the deformation, the activation energy sharply decreases with increasing strain followed by a dip around 0.2 true strain and then resumes its increase. The polynomial fitting results of n, a, Q and ln(A) for the present material are shown below and summarized in Table 2.
a ¼ B0 þ B1 e þ B2 e2 þ B3 e3 þ B4 e4 þ B5 e5 þ B6 e6 Q ¼ C 0 þ C 1 e þ C 2 e2 þ C 3 e3 þ C 4 e4 þ C 5 e5 þ C 6 e6 ln A ¼ D0 þ D1 e þ D2 e2 þ D3 e3 þ D4 e4 þ D5 e5 þ D6 e6 n ¼ E0 þ E1 e þ E2 e2 þ E3 e3 þ E4 e4 þ E5 e5 þ E6 e6
ð6Þ
The constants, summarized in Table 3, are substituted into Eqs. (4)–(6), to predict the flow behavior of UFG pure Ti. The stress– strain curves predicted by the model at different temperatures and strain rates are illustrated in Fig. 8. The Arrhenius model employed herein provides decent description of the flow response in the examined range of strain rates and temperatures except 900 °C. It is worth noting that the typical peak stress behavior induced by the interplay of dynamic recovery and recrystallization can be successfully captured in the predicted stress–strain curves. Including the experiment at 900 °C, apart from a slight undershoot of the flow stress level at 0.01 s1 and lack of stress plateau at 0.1 s1, the current approach can be of significant use for modeling the high temperature flow behavior of severely deformed Ti where competition between work hardening and softening is effective. 3.5. Prediction of flow stress by dislocation density based modeling This section presents the dislocation density based approach used to describe the flow behavior in terms of work hardening (WH), dynamic recovery (DRV) and dynamic recrystallization (DRX) regions. This technique is based on Avrami’s formulation and has been utilized in various hot deformation studies and calculation details can be found elsewhere [9,39,40]. The fundamental flow stress relations used are:
r2 ¼ ½r2DRV þ ðr20 r2DRV ÞeXe ðe < eC Þ
r ¼ rDRV ðrp rDRX Þ 1 exp K
e eC ep
ð7Þ nd
ðe P eC Þ
ð8Þ
where eC is the critical strain for dynamic recrystallization and e is the total strain. r0, rDRV and rDRX are the yield stress, the steady state stress due to dynamic recovery and dynamic recrystallization, respectively. rp and ep are the peak stress, and the corresponding strain, respectively. The coefficient of dynamic recovery, X, and dynamic recrystallization parameters, K and nd, for can be calculated as a function of the Zener–Holloman parameter (Z) for different deformation conditions and the below relations can be obtained following the methodology outlined in [9,39]:
Table 3 Constants used for polynomial fitting in the determination of a, Q, ln(A), and n within the Arrhenius model utilized.
Fig. 7. Relationship between predicted values of activation energy and true strain for severely deformed pure Ti.
a
Q
ln(A)
n
B0 = 0.017 B1 = 0.001 B2 = 0.020 B3 = 0.025 B4 = 0.147 B5 = 0.153 B6 = 0.047
C0 = 378145.520 C1 = 946360.310 C2 = 6168607.150 C3 = 19347678.650 C4 = 32002373.400 C5 = 26650354.280 C6 = 8776537.410
D0 = 33.547 D1 = 7.914 D2 = 20.509 D3 = 29.633 D4 = 91.292 D5 = 33.085 D6 = 21.472
E0 = 5.428 E1 = 21.653 E2 = 99.354 E3 = 268.380 E4 = 417.190 E5 = 337.330 E6 = 108.920
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Fig. 9. Comparison between the dislocation density based model predictions and the experimental results for the flow response of severely deformed Ti at (a) 600 °C, (b) 700 °C, (c) 800 °C, and (d) 900 °C.
Fig. 8. Comparison between the Arrhenius model predictions and the experimental results for the flow response of severely deformed Ti at (a) 600 °C, (b) 700 °C, (c) 800 °C, and (d) 900 °C.
X ¼ 96:47 Z 0:042 K ¼ 1:03 103 Z 0:214
actual softening behavior. Since this model uses the Zener–Hollomon parameter with different deformation conditions, the flow stress behavior can be predicted in the entire domain where the apparent activation energy for hot working is similar. 3.6. Evaluation of the flow models
ð9Þ
ð10Þ
The flow behavior can be predicted by applying the above relations to Eqs. (7) and (8) as demonstrated in Fig. 9. This model is accurate in predicting the mechanical behavior at all deformation temperatures except 600 °C. At 600 °C and above 0.001 s1, driving force for dynamic recrystallization may not be enough for initiation and thus the model may lose its accuracy in determining the
In order to verify the accuracy of the applied constitutive models, the deviations between the predicted stress (rp) and experimental stress (rE) values were obtained as:
rp rE 100 Eð%Þ ¼
rE
ð11Þ
Fig. 10a demonstrates the mean error values for the first modeling approach based on Arrhenius formulation. The mean error plot reveals that simulations exhibit a better agreement at rela-
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Comparing both models, it can be seen that the error levels for the Arrhenius model are larger than the dislocation density based prediction for most of the loading conditions. Due to the underlying assumptions, for temperatures above 600 °C, the dislocation density based model should be preferred; whereas below 600 °C, the mathematical model based on the Arrhenius formulation may lead to better predictions.
4. Conclusion The deformation and workability characteristics of severely deformed pure Ti were studied at various deformation conditions in the warm to hot working regime. Mechanical behavior was modeled by two distinct approaches. The following conclusions can be drawn:
Fig. 10. Mean error levels in the prediction of flow stress behavior by (a) Arrhenius formulation based modeling and (b) dislocation density based modeling.
tively lower temperatures. With increasing deformation temperature, a steady increase was observed in the error level. It was seen that the highest error level is less than 7% in all cases except at a temperature of 900 °C and strain rates of 0.01 and 0.001 s1. Iterations in this work revealed that polynomial fitting up to a minimum of sixth power was required to achieve the current error levels. The major reason for higher discrepancy in the highest temperature can be attributed to the relatively lower stress levels that magnify the error. Moreover, this model does not account for softening mechanisms such as recovery or recrystallization. Furthermore, performance of the present model was compared with a similar Arrhenius type polynomial model up to fourth power for annealed pure Ti in a separate study [13]. In line with the current results, error levels increased with higher forming temperatures for annealed pure Ti as well. However, the error values shown here are noticeably less and hence improve our confidence in the proposed polynomial model for accurate flow behavior estimation of ECAE processed pure titanium in the warm to hot working regime. For the second model, the mean error levels for different deformation conditions are presented in Fig. 10b revealing better agreement at relatively higher temperatures. It was seen that except at a temperature of 600 °C and a strain rate of 0.1 s1, the highest error level is less than 5%. The main reason for higher discrepancy in the lowest temperature can be attributed to the assumptions of the present model. Typically, the employed model is reliable for modeling single peak dynamic recrystallization flow stress curves. At the lowest temperature, the deformation mechanism is expected to be dominated by recovery mechanisms rather than recrystallization and thus the imposed softening could lead to flawed predictions. Moreover, it is observed that error levels decrease by reduction of the strain rate. Lower strain rates should provide the necessary time for dynamic recrystallization to initiate. With mostly favorable predictions, this model is acceptable for determining the elevated temperature compressive response of UFG Ti with active recrystallization mechanisms.
(i) Before and during elevated temperature testing, grain growth occurred in the UFG microstructure obtained after ECAE. Post-ECAE compression up to 900 °C led to a mixture of fine and coarse grains. (ii) The flow response of ECAE processed pure Ti at elevated temperatures depends on both deformation temperature and strain rate. Higher flow stress levels are demonstrated with the increase in strain rate and the decrease in deformation temperature. (iii) The high temperature strain rate sensitivity of severely deformed pure Ti was seen to be strongly temperature dependent. It was seen to sharply increase with the rise of deformation temperature up to 800 °C. In contrast, strain rate sensitivity is fairly independent of strain, especially above 800 °C. (iv) The tendency for localized flow was probed by determining the flow localization parameter for all testing conditions. Accordingly, the workability at elevated temperatures is satisfactory at 800 °C and above. From a metal forming perspective in the high temperature regime, deformation over 700 °C could be preferred due to comparably lower amount of flow localization. (v) Both modeling approaches demonstrated stress–strain curves with reasonable agreement in the examined strain rate and temperature range. In this range, the error levels on stress predictions remained less than 10%. For temperatures above 600 °C, the dislocation density based model should be preferred; whereas below 600 °C, the mathematical model based on the Arrhenius formulation.
Acknowledgements The authors would like to acknowledge the support from the Marie Curie Career Integration Grant, Project No: 304150, within the EU-FP7 for conducting this research study. References [1] Stolyarov VV, Zhu YT, Alexandrov IV, Lowe TC, Valiev RZ. Influence of ECAP routes on the microstructure and properties of pure Ti. Mater Sci Eng A 2001;299:59–67. [2] Fan Z, Jiang H, Sun X, Song J, Zhang X, Xie C. Microstructures and mechanical deformation behaviors of ultrafine-grained commercial pure (grade 3) Ti processed by two-step severe plastic deformation. Mater Sci Eng A 2009;527:45–51. [3] Stolyarov VV, Zhu YT, Alexandrov IV, Lowe TC, Valiev RZ. Grain refinement and properties of pure Ti processed by warm ECAP and cold rolling. Mater Sci Eng A 2003;343:43–50. [4] Yapici GG, Karaman I, Maier HJ. Mechanical flow anisotropy in severely deformed pure titanium. Mater Sci Eng A 2006;434:294–302.
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Materials and Design journal homepage: www.elsevier.com/locate/matdes
Workability characteristics and mechanical behavior modeling of severely deformed pure titanium at high temperatures Seyed Vahid Sajadifar, Guney Guven Yapici ⇑ Mechanical Engineering Department, Ozyegin University, Istanbul, Turkey
a r t i c l e
i n f o
Article history: Received 12 April 2013 Accepted 17 July 2013 Available online 25 July 2013 Keywords: Titanium Equal channel angular extrusion/pressing Severe plastic deformation Modeling of hot deformation
a b s t r a c t In the present study, compression tests were performed at temperatures of 600–900 °C and at strain rates of 0.001–0.1 s1 to study the deformation and workability characteristics of commercially pure titanium after severe plastic deformation (SPD). It was found that the effects of temperature and strain rate are significant in dictating the steady state flow stress levels and the strain values corresponding to peak flow stress. The strain rate sensitivity (m) during hot compression of severely deformed Ti was shown to be strongly temperature dependent, where m increased with the increase in deformation temperature up to 800 °C. High temperature workability was analyzed based on the flow localization parameter (FLP). According to the FLP values, deformation at and below 700 °C is prone to flow localization. The flow behavior was predicted using Arrhenius type and dislocation density based models. The validities of the models were demonstrated with reasonable agreement in comparison to the experimental stress– strain responses. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Due to its high specific strength, pure titanium is an excellent candidate for engineering applications requiring light structural components. One of the available methods for enhancing strength as well as ductility is grain refinement via severe plastic deformation (SPD). SPD shows promise for generating an ultra-fine-grained (UFG) microstructure leading to enhanced physical and mechanical properties in titanium and its alloys [1–7]. Microstructure and mechanical properties of pure Ti processed by SPD and cold rolling were investigated [2–4]. It was seen that SPD followed by cold rolling at ambient and at cryogenic temperatures is a viable method for increasing the ultimate tensile strength levels with moderate ductility [2,3]. Yapici et al. [4] investigated the flow anisotropy of UFG Ti with a similar two step processing history. Wear properties after SPD was another topic of interest showing contradictory results [5,6]. While coarse grained (CG) pure Ti displayed extensive plastic deformation and wedge formation that generated large wear debris, wear of the UFG Ti was dominated by abrasive wear mechanisms and resulted in lower wear rates [6]. In contrast, a separate study demonstrated no significant improvement on the wear resistance of the UFG microstructure [5]. A common two phase Ti alloy (Ti64) was also subjected to SPD and the texture development and resulting anisotropic mechanical properties were examined [7]. Most of the aforementioned studies showed that ⇑ Corresponding author. Tel.: +90 216 564 9115; fax: +90 216 564 9057. E-mail address: [email protected] (G.G. Yapici). 0261-3069/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.matdes.2013.07.057
equal channel angular extrusion (ECAE/ECAP) is one of the most favorable SPD techniques for enhancing mechanical properties in engineering applications due to its capability of maintaining dimensions of the initial billet after multiple processing steps [1– 5]. The demand for obtaining necessary forces to be applied during warm and hot working (typically rolling or forging) of materials has driven studies focused on workability and modeling of respective flow stress curves. It is well known that study of flow stress behavior at elevated temperatures is crucial for the adjustment of warm and hot working process parameters leading to improved workability. These characteristics were previously investigated for pure Ti and other metallic materials [8–13]. Among them, few studies mentioned the workability of pure Ti at elevated temperatures, as observed in hot deformation tests [8,11–13]. These studies mainly focused on the microstructural evolution of coarse grained pure Ti during hot deformation along with the constitutive modeling of the mechanical behavior. However, they typically lacked discussion on the workability parameters which is important for better understanding the stress–strain response. Moreover, neither characterization nor modeling at elevated temperatures (up to 900 °C) has been a topic of study for Ti with ECAE induced UFG microstructure. The present investigation aims to fill this gap by (a) demonstrating the effect of ECAE processing on warm and hot flow characteristics and workability behavior of pure Ti at elevated temperature, (b) predict the flow stress–strain response with two modeling approaches.
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Various constitutive equations based on dislocation density evolution during hot forming have been developed from the experimentally measured data to describe the sensitivity of the flow stress to the strain [9,14]. Mathematical approaches utilize the Zener–Hollomon parameter in a hyperbolic sinusoidal equation form for polynomial fitting up to the power of four or five in some alloys [13,15]. Both methods are suitable tools to numerically analyze the hot deformation process as a function of applied strain, strain rate and deformation temperature. Here, the model is extended up to sixth polynomial power and predictions are compared with those obtained from microstructural parameters. The objective of this work is to provide an insight into the elevated temperature formability and workability characteristics of severely deformed pure Ti along with efforts to simulate the experimental flow response.
Hot compression tests were conducted under isothermal conditions at three different strain rates of 0.001, 0.01 and 0.1 s1 and at temperatures of 600, 700, 800 and 900 °C. The temperature range is in the warm to hot work region, i.e. 0.45Tm–0.6Tm. All specimens were heated up to the deformation temperature and then the samples were deformed in a single loading step. The hot compression tests were conducted inside a temperature-controlled furnace mounted on an Instron mechanical testing frame. Temperatures were maintained throughout the tests to ensure uniformly heated samples. Optical microscopy was used to monitor the changes in the microstructure. The specimens were prepared using standard polishing techniques and then etched with Kroll’s reagent. 3. Results and discussion 3.1. Microstructural evolution
2. Materials and experimental procedure The as-received commercial purity grade 2 Ti was received in bar form with the composition listed in Table 1. The bars were coated with a graphite base lubricant before extrusion and were heated in a furnace to the deformation temperature of 300 °C where they were held for 1 h before extrusion. Finally, they were transferred to the 25.4 mm 25.4 mm cross section, 90° angle ECAE die which was preheated to 300 °C. Extrusion took place at a rate of 1.27 mm/s. Eight ECAE passes were performed following route E accumulating a total strain of 9.24 in the as-processed material [3]. Route E was selected as the ECAE processing route. This hybrid route results in the largest fully worked region in a given billet and provides high volume fraction of high-angle grain boundaries and equiaxed grain morphology [16]. Route E consists of an alternating rotation of the billet by +180° and +90° around its long axis, between successive passes. Between each pass, the billet was heated in the furnace for 15 min at the deformation temperature. Following each extrusion pass, the billets were water quenched to maintain the microstructure achieved during ECAE. Lowest possible processing temperatures were crucial in preventing possible recrystallization and partly achieved using the sliding walls concept that helps reducing the die frictional effects [17]. Using this method, a number of difficult-to-work materials were processed without macroscopic localization [18,19]. For studying the hot deformation behavior of pure Ti after ECAE, compression experiments were performed at high temperatures according to ASTM: E209-00. The main benefit of compression testing is that the imposed stress state is close to that occuring in actual bulk deformation operations as in, extrusion, forging, rolling and others alike. The compression specimens were electro-discharge machined (EDM) in rectangular blocks shape, 4 mm 4 mm 8 mm, with compression axis parallel to the extrusion direction [20]. All samples were ground and polished to remove major scratches and eliminate the influence of residual layer from EDM. The reduction in height has been chosen as 60% (true strain: 0.9) at the end of the compression tests to avoid barreling. This strain level, which is required to capture the effect of both dynamic recovery and recrystallization on deformation behavior, has been applied in several researches [21,22]. In addition, lubrication with graphite was used during hot compression tests to decrease friction effects and minimize barreling [22].
Table 1 Chemical composition of pure Ti used in this study. Element
C
Fe
H
N
O
Ti
wt.%
0.008
0.041
0.002
0.006
0.15
Balance
The initial microstructure of as-received Ti is shown in Fig. 1a. It can be easily observed that the structure consists of coarse grains with an average grain size of 43 lm. The microstructure of the as-received Ti subjected to eight passes of ECAE is demonstrated in Fig. 1b. ECAE processing leads to a significant refinement in the microstructure and transforms the coarse grained as-received microstructure to one with ultra-fine grains after eight passes. In addition, micrographs depicting the microstructural evolution before and after hot compression testing are also demonstrated. The microstructure of UFG Ti followed by 3 min annealing at 600 °C and 900 °C can be seen in Fig. 1c and d, respectively. Annealing period is selected based on the time it takes to achieve uniform heating of the samples at the desired compression test temperature. After the 600 °C heat treatment, the microstructure consists of both slightly coarsened grains and regions with ultra-fine grains. In contrast, the 900 °C heat treatment resulted in severe coarsening of the structure with an average grain size of 25 lm. The existence of several recrystallized equiaxed grains can be observed in Fig. 1d. Considering the development of microstructure during hot compression, the micrographs after deformation at 600 °C and 900 °C can be seen in Fig. 1e and f, respectively. The sample subjected to compression at 600 °C exhibits a fine and homogeneous microstructure with slightly larger grains in comparison with those of the pre-compression sample. After compression at 900 °C, the microstructure contained a mixture of fine and coarse grains. The evolution of such a microstructure could be attributed to the deformation of the recrystallized grains that formed during the pre-test heat treatment and the formation dynamically recrystallized during the compression test. 3.2. Flow stress behavior Fig. 2 shows the true stress–true strain curves for the compression tests of severely deformed pure Ti at various temperatures for the strain rates of 0.001, 0.01 and 0.1 s1. It can be observed that the flow stress of pure Ti subjected to ECAE is sensitive to both temperature and strain rate. In the previous work on the high temperature flow behavior of pure Ti with 40 lm grain size, compressive responses were depicted up to 700 °C [13]. In comparison with the present results, it can be seen that ECAE could increase the high temperature strength of pure Ti up to as high as 700 °C. The rise of flow stress levels in UFG materials during hot deformation was also reported for an aluminum 6063 alloy at a temperature range of 300–450 °C [23]. At all deformation temperatures, the flow stress levels slightly increase to a peak and then decrease gradually to a steady state level. Regardless of the strain level (low, peak or high), the flow stress levels show similar dependence to deformation temperature
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Fig. 1. The microstructure of pure Ti (a) in as-received condition, (b) after 8 passes of ECAE processing, (c) after 8 passes of ECAE processing followed by 3 min of annealing at 600 °C, (d) after 8 passes of ECAE followed by 3 min of annealing at 900 °C, (e) after 8 passes of ECAE processing followed by compression at 600 °C with a strain rate of 0.1 s1, and (f) after 8 passes of ECAE processing followed by compression at 900 °C with a strain rate of 0.1 s1.
and strain rate. Fig. 3 exhibits this behavior for the case of peak strain, i.e. strain corresponding to peak flow stress. Such flow behavior is a typical characteristic of hot working that is accompanied by dynamic recrystallization softening. The work hardening at 600 °C is higher than those obtained at temperatures above it. In contrast, the flow stress levels exhibit slight transitional drops indicating thermal softening for deformation temperatures above 600 °C. The observed stress levels decrease with increasing temperature and decreasing strain rate because lower strain rates and higher temperatures provide longer time for energy accumulation and higher dislocation mobility around boundary regions. This evidently results in the nucleation and growth of dynamically recrystallized grains and dislocation annihilation.
Fig. 4 summarizes the values of peak strain as a function of deformation temperature and strain rate. It is worth noting that the effects of temperature and strain rate are significant on the peak strain level during hot deformation. For most materials both in pure and alloy form, the peak strain decreases with higher temperatures or lower strain rates [9,24]. A similar behavior was observed for the case of Ti after SPD. As can be seen in Fig. 4, the peak strain value at any given temperature increases with increasing strain rate, though the effect is mostly pronounced at the lowest testing temperature of 600 °C. Since dynamic recrystallization (DRX) typically initiates at a critical strain before the peak stress [25], it is accelerated by the rise in temperature and the drop in strain rate. Although, Ti and its alloys are generally considered as
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Fig. 3. The variation of peak stress as a function of strain rate at different deformation temperatures.
Fig. 4. The dependence of peak strain to (a) deformation temperature and (b) strain rate.
of DRX related phenomena is favored. With increasing deformation temperatures, the sensitivity of peak strain to strain rate diminishes. Still, both applied strain rate and deformation temperature are important parameters that dictate the amount of critical strain required for the onset of DRX in ECAE processed pure Ti. 3.3. Evolution of material properties Fig. 2. True stress–true strain response of severely deformed Ti with different strain rates at (a) 600 °C, (b) 700 °C, (c) 800 °C, and (d) 900 °C.
high stacking fault energy (SFE) materials that are expected to be softened mainly by dynamic recovery (DRV), DRX processes have also been found to take place depending on the deformation parameters [26–29]. In the present study, especially at temperatures above 600 °C and/or strain rates below 0.01 s1, occurrence
Strain rate sensitivity is a significant property, which can affect workability of materials and deserves investigation for the present work. Strain rate sensitivity, m, is related to the effect of strain rate on dislocation generation and propagation and as such is an index of workability. The following well-known relation was employed to calculate the strain rate sensitivity, m, for hot deformation.
m¼
@ lnðrÞ @ lnðe_ Þ e
ð1Þ
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Fig. 5. Strain dependence of strain rate sensitivity at various temperatures.
The values of m can be obtained from the slope of the lines plotted according to Eq. (1) at a constant strain. The variation of the strain rate sensitivity as a function of strain is plotted in Fig. 5 exhibiting strong dependence to deformation temperature. This phenomenon can be attributed to the increased activity of diffusion related mechanisms and dynamic recrystallization at higher temperatures resulting in higher strain rate sensitivity [30]. It is worth noting that strain rate sensitivity shows a considerable increase above 700 °C. The strain rate sensitivity values obtained at 800 °C are slightly greater than those obtained at 900 °C for the whole strain interval. It is expected that m values increase with the rise of deformation temperature due to thermally activated mechanisms. However, constancy of m can be attributed to the onset of DRX processes which may occur over a range of strain rates and temperatures [31,32]. In the present case, dominance of DRX around 800–900 °C might explain the similar strain rate sensitivity values at both temperatures. Considering the strain dependence of rate sensitivity, the m values at 600 °C and 700 °C remarkably increase and peak at a strain of 0.5 and 0.35, respectively. Then, the m values decrease with increasing strain and plateau at a constant value. Similar behavior is observed at a less pronounced degree for 800 °C, where the strain rate sensitivity peaks at a strain level of 0.25 and then drops gradually and remains almost constant for the rest of the strain range. At the highest deformation temperature, m is fairly constant and can be considered to be independent of true strain. It can be concluded that for severely deformed pure Ti, higher strain rate sensitivity levels are observed with increasing deformation temperature as expected. However, m is not a strong function of strain at or above 800 °C. The other crucial factor affecting the elevated temperature deformation behavior is the strain hardening rate, c, which is defined as:
c¼
1 dr r de e_
ð2Þ
where r is the flow stress, e_ is the strain rate and e is the true strain. The variation of strain hardening rate values with strain at different temperatures and strain rates is presented in Fig. 6. The strain hardening rates during hot deformation, as defined in Eq. (2), exhibit sharp decrease at low strain levels (below 0.2) while they stay almost constant at higher strains. Moreover, again at low strain levels, higher deformation temperatures result in lower strain hardening rates, indicating the dominance of dislocation rearrangement and recovery mechanisms. Since more recrystallization is expected at higher temperatures and lower strain rates, increased softening and thereby reduced flow stress levels after peak stress could be attributed to a higher volume fraction of recrystallized grains in the microstructure [33]. Irrespective of the deformation tempera-
Fig. 6. Strain hardening rate versus true strain plot demonstrating behavior at different temperatures and at a strain rate of (a) 0.1 s1 (b) 0.01 s1 and (c) 0.001 s1.
ture and strain rate, constant c was observed at above true strain levels of 0.2. In order to estimate the workability of metals and alloys under hot deformation conditions, flow localization parameter (FLP) has been proposed [34,35]. This factor can be represented as follows:
FLP ¼
c m
ð3Þ
Using Eq. (3), the average flow localization parameters for the whole strain range at different temperatures and strain rates, are calculated and summarized in Table 2. According to Table 2, FLP values increase with decreasing deformation temperature and increasing strain rate. It was stated that the materials with FLP P 5 are prone to flow localization [34,35]. Therefore, it is expected that UFG Ti is not expected to have favorable workability at temperatures below 800 °C due to flow localization. The values of FLP were highest and exceed 21 at 600 °C for all strain rates. The high magnitude of flow localization can be attributed to the lack of dynamic recrystallization, which might eventually result in failure [36]. In contrast, the values of FLP in all strain rates remain less than 2 at 800 and 900 °C. The effect of
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Table 2 Flow localization parameter for severely deformed Ti under different deformation temperatures and strain rates. Strain rate (s1)
600 °C
700 °C
800 °C
900 °C
0.1 0.01 0.001
22.45 21.02 22.13
7.35 7.15 7.29
1.91 0.80 1.56
0.29 0.35 0.13
strain rate on the FLP is less straightforward. Despite this, highest FLP values are observed for the highest deformation rates for all temperatures. It is clear that the ductile behavior during hot working of ECAE processed pure titanium is closely associated with deformation temperature and to a smaller extent with deformation rate. With reduced tendency to flow localization, deformation over 700 °C could be preferred for metal forming purposes in severely deformed Ti. 3.4. Prediction of flow stress by Arrhenius modeling Constitutive equations proposed by Zener and Hollomon have been used to describe the deformation behavior at high temperatures. In hot working processes, several constitutive equations have commonly been applied. These equations are based on the Zener–Holloman formulation and have been used in various hot deformation investigations and calculation details can be found elsewhere [13,15,37,38]. The flow stress can be written as a function of the Zener–Hollomon parameter, considering the definition of the hyperbolic law [13,14]:
8 #12 9 1=n " 1=n = 1 < Z Z r ¼ Ln þ þ1 ; A a : A Z ¼ e_ exp
Q RT
n
¼ A½sin hðarÞ
ð4Þ
ð5Þ
The values of material constants (A, a and n) and the apparent activation energy for hot deformation (Q) of the constitutive equations were computed for 0.05 incremental strain levels within the range of 0.05–0.9. For this purpose, the least squares method was utilized. The variations of these constants with true strain can be represented by polynomial functions of strain. For the sake of brevity, only the plot pertaining to Q is shown in Fig. 7. The activation energy ranges between 323 and 344 kJ/mol. Since Q represents the energy for deformation, higher Q would indicate the requirement for higher energy to activate relevant deformation mechanisms. The Q value for coarse grained pure Ti was reported as 218–
240 kJ/mol, which is considerably lower than that observed in the present work [13]. ECAE processing increased the amount of activation energy to sustain post-ECAE hot deformation. At the beginning of the deformation, the activation energy sharply decreases with increasing strain followed by a dip around 0.2 true strain and then resumes its increase. The polynomial fitting results of n, a, Q and ln(A) for the present material are shown below and summarized in Table 2.
a ¼ B0 þ B1 e þ B2 e2 þ B3 e3 þ B4 e4 þ B5 e5 þ B6 e6 Q ¼ C 0 þ C 1 e þ C 2 e2 þ C 3 e3 þ C 4 e4 þ C 5 e5 þ C 6 e6 ln A ¼ D0 þ D1 e þ D2 e2 þ D3 e3 þ D4 e4 þ D5 e5 þ D6 e6 n ¼ E0 þ E1 e þ E2 e2 þ E3 e3 þ E4 e4 þ E5 e5 þ E6 e6
ð6Þ
The constants, summarized in Table 3, are substituted into Eqs. (4)–(6), to predict the flow behavior of UFG pure Ti. The stress– strain curves predicted by the model at different temperatures and strain rates are illustrated in Fig. 8. The Arrhenius model employed herein provides decent description of the flow response in the examined range of strain rates and temperatures except 900 °C. It is worth noting that the typical peak stress behavior induced by the interplay of dynamic recovery and recrystallization can be successfully captured in the predicted stress–strain curves. Including the experiment at 900 °C, apart from a slight undershoot of the flow stress level at 0.01 s1 and lack of stress plateau at 0.1 s1, the current approach can be of significant use for modeling the high temperature flow behavior of severely deformed Ti where competition between work hardening and softening is effective. 3.5. Prediction of flow stress by dislocation density based modeling This section presents the dislocation density based approach used to describe the flow behavior in terms of work hardening (WH), dynamic recovery (DRV) and dynamic recrystallization (DRX) regions. This technique is based on Avrami’s formulation and has been utilized in various hot deformation studies and calculation details can be found elsewhere [9,39,40]. The fundamental flow stress relations used are:
r2 ¼ ½r2DRV þ ðr20 r2DRV ÞeXe ðe < eC Þ
r ¼ rDRV ðrp rDRX Þ 1 exp K
e eC ep
ð7Þ nd
ðe P eC Þ
ð8Þ
where eC is the critical strain for dynamic recrystallization and e is the total strain. r0, rDRV and rDRX are the yield stress, the steady state stress due to dynamic recovery and dynamic recrystallization, respectively. rp and ep are the peak stress, and the corresponding strain, respectively. The coefficient of dynamic recovery, X, and dynamic recrystallization parameters, K and nd, for can be calculated as a function of the Zener–Holloman parameter (Z) for different deformation conditions and the below relations can be obtained following the methodology outlined in [9,39]:
Table 3 Constants used for polynomial fitting in the determination of a, Q, ln(A), and n within the Arrhenius model utilized.
Fig. 7. Relationship between predicted values of activation energy and true strain for severely deformed pure Ti.
a
Q
ln(A)
n
B0 = 0.017 B1 = 0.001 B2 = 0.020 B3 = 0.025 B4 = 0.147 B5 = 0.153 B6 = 0.047
C0 = 378145.520 C1 = 946360.310 C2 = 6168607.150 C3 = 19347678.650 C4 = 32002373.400 C5 = 26650354.280 C6 = 8776537.410
D0 = 33.547 D1 = 7.914 D2 = 20.509 D3 = 29.633 D4 = 91.292 D5 = 33.085 D6 = 21.472
E0 = 5.428 E1 = 21.653 E2 = 99.354 E3 = 268.380 E4 = 417.190 E5 = 337.330 E6 = 108.920
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Fig. 9. Comparison between the dislocation density based model predictions and the experimental results for the flow response of severely deformed Ti at (a) 600 °C, (b) 700 °C, (c) 800 °C, and (d) 900 °C.
Fig. 8. Comparison between the Arrhenius model predictions and the experimental results for the flow response of severely deformed Ti at (a) 600 °C, (b) 700 °C, (c) 800 °C, and (d) 900 °C.
X ¼ 96:47 Z 0:042 K ¼ 1:03 103 Z 0:214
actual softening behavior. Since this model uses the Zener–Hollomon parameter with different deformation conditions, the flow stress behavior can be predicted in the entire domain where the apparent activation energy for hot working is similar. 3.6. Evaluation of the flow models
ð9Þ
ð10Þ
The flow behavior can be predicted by applying the above relations to Eqs. (7) and (8) as demonstrated in Fig. 9. This model is accurate in predicting the mechanical behavior at all deformation temperatures except 600 °C. At 600 °C and above 0.001 s1, driving force for dynamic recrystallization may not be enough for initiation and thus the model may lose its accuracy in determining the
In order to verify the accuracy of the applied constitutive models, the deviations between the predicted stress (rp) and experimental stress (rE) values were obtained as:
rp rE 100 Eð%Þ ¼
rE
ð11Þ
Fig. 10a demonstrates the mean error values for the first modeling approach based on Arrhenius formulation. The mean error plot reveals that simulations exhibit a better agreement at rela-
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Comparing both models, it can be seen that the error levels for the Arrhenius model are larger than the dislocation density based prediction for most of the loading conditions. Due to the underlying assumptions, for temperatures above 600 °C, the dislocation density based model should be preferred; whereas below 600 °C, the mathematical model based on the Arrhenius formulation may lead to better predictions.
4. Conclusion The deformation and workability characteristics of severely deformed pure Ti were studied at various deformation conditions in the warm to hot working regime. Mechanical behavior was modeled by two distinct approaches. The following conclusions can be drawn:
Fig. 10. Mean error levels in the prediction of flow stress behavior by (a) Arrhenius formulation based modeling and (b) dislocation density based modeling.
tively lower temperatures. With increasing deformation temperature, a steady increase was observed in the error level. It was seen that the highest error level is less than 7% in all cases except at a temperature of 900 °C and strain rates of 0.01 and 0.001 s1. Iterations in this work revealed that polynomial fitting up to a minimum of sixth power was required to achieve the current error levels. The major reason for higher discrepancy in the highest temperature can be attributed to the relatively lower stress levels that magnify the error. Moreover, this model does not account for softening mechanisms such as recovery or recrystallization. Furthermore, performance of the present model was compared with a similar Arrhenius type polynomial model up to fourth power for annealed pure Ti in a separate study [13]. In line with the current results, error levels increased with higher forming temperatures for annealed pure Ti as well. However, the error values shown here are noticeably less and hence improve our confidence in the proposed polynomial model for accurate flow behavior estimation of ECAE processed pure titanium in the warm to hot working regime. For the second model, the mean error levels for different deformation conditions are presented in Fig. 10b revealing better agreement at relatively higher temperatures. It was seen that except at a temperature of 600 °C and a strain rate of 0.1 s1, the highest error level is less than 5%. The main reason for higher discrepancy in the lowest temperature can be attributed to the assumptions of the present model. Typically, the employed model is reliable for modeling single peak dynamic recrystallization flow stress curves. At the lowest temperature, the deformation mechanism is expected to be dominated by recovery mechanisms rather than recrystallization and thus the imposed softening could lead to flawed predictions. Moreover, it is observed that error levels decrease by reduction of the strain rate. Lower strain rates should provide the necessary time for dynamic recrystallization to initiate. With mostly favorable predictions, this model is acceptable for determining the elevated temperature compressive response of UFG Ti with active recrystallization mechanisms.
(i) Before and during elevated temperature testing, grain growth occurred in the UFG microstructure obtained after ECAE. Post-ECAE compression up to 900 °C led to a mixture of fine and coarse grains. (ii) The flow response of ECAE processed pure Ti at elevated temperatures depends on both deformation temperature and strain rate. Higher flow stress levels are demonstrated with the increase in strain rate and the decrease in deformation temperature. (iii) The high temperature strain rate sensitivity of severely deformed pure Ti was seen to be strongly temperature dependent. It was seen to sharply increase with the rise of deformation temperature up to 800 °C. In contrast, strain rate sensitivity is fairly independent of strain, especially above 800 °C. (iv) The tendency for localized flow was probed by determining the flow localization parameter for all testing conditions. Accordingly, the workability at elevated temperatures is satisfactory at 800 °C and above. From a metal forming perspective in the high temperature regime, deformation over 700 °C could be preferred due to comparably lower amount of flow localization. (v) Both modeling approaches demonstrated stress–strain curves with reasonable agreement in the examined strain rate and temperature range. In this range, the error levels on stress predictions remained less than 10%. For temperatures above 600 °C, the dislocation density based model should be preferred; whereas below 600 °C, the mathematical model based on the Arrhenius formulation.
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