Abstract
Cite this article as: I. Mejía, F. Reyes-Calderón and J.M. Cabrera, Modeling the hot flow behavior of a Fe-22Mn-0.41 C-1.6Al-1.4Si TWIP steel microalloyed with Ti, V and Nb, Materials Science & Engineering A, http://dx.doi.org/10.1016/j.msea. 2015.07.078 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Edificio "U", Ciudad Universitaria, 58066-Morelia, Michoacán, México. 
Abstract
The present research work analyses the influence of Ti, V and Nb microalloying elements on the hot flow behavior of a high-Mn TWIP (Twinning Induced Plasticity) steel. For this purpose, flow curves were obtained by uniaxial hot compression tests performed at four strain rates (10 -1 , 10 -2 , 10 -3 and 10 -4 s -1 ) and three temperatures (900, 1000 and 1100 ºC). The models of Estrin, Mecking and Bergstrom; Avrami and Tegart, and Sellars were applied to determine the hot working constants used to derive the constitutive equations describing the flow curves.
Abstract
The analysis of modeling parameters of the hot flow curves shows that Ti, V and Nb additions to TWIP steel generated slight increase in the peak stress ( p ), retardation of the dynamic recrystallization (DRX) onset, particularly at low temperature, and decrease in the activation energy required to recrystallization (Q t ). Likewise, the softening effect promoted by DRV and DRX was more evident at high temperatures and low strain rates. On the other hand, the resulting deformed microstructures, analyzed by the SEM-EBSD technique, showed that the most important refining effect on recrystallized austenitic grain was in the presence of V and
Abstract
Ti. The good agreement between the experimental and predicted hot flow curves demonstrated that the developed constitutive equations predict with reasonable accuracy the hot flow behavior of the studied TWIP steels.

 s  Saturation stress. 
Introduction
High-Mn TWIP steels provide a great potential to the manufacturing, building and automotive industries [1-6]. There are several studies [1-5] about the chemical compositions of TWIP steels and their influence on the deformation mechanisms, the mechanical properties, the stacking fault energy (SFE) and the phase transformations during cold deformation processes.
Introduction
However, there are few researches about the hot flow behavior [6] [7] [8] [9] [10] [11] [12] [13] [14] . These studies show, as expected, the influence of temperature (T), plastic deformation or strain () and strain rate ( ) on the hot flow behavior. Nevertheless, less effort has been addressed to perform a good control of these parameters to obtain an optimal microstructural conditions and therefore to promote their mechanical properties.
Introduction
Hot compression tests are usually carried out to simulate industrial processes such as forging, rolling, extrusion, etc. The obtained hot flow curves ( t - t ) show the typical plastic behavior of steels at high temperature, i.e., a single peak in the flow stress is noticed. The flow curves are furthermore characterized by three distinctive zones: i) A stage I that is related to the balance between the generated and the stored dislocations by strain hardening, together with those rearranged and annihilated by dynamic recovery (DRV) The appropriate design of the thermo-mechanical treatment is important in simulating hot forming processes, in order to control the grain size during the hot deformation. This is particularly true in TWIP steels [10] , where no phase transformations occurs during cooling.
Introduction
Additionally, the austenitic grains can be further refined by a fine dispersion of particles randomly distributed within the matrix. In this case, particles must be sufficiently stable to not dissolve or to not grow during the reheating and hot forming operations [24] . The addition of microalloying elements such as Nb and Ti can promote an extra strengthening via precipitation hardening or via grain refinement in high-Mn steels, and the imminent production of finer grain sizes at the end of the steel processing as previously was reported by relatively simple assuming that both U and Ω are independent of strain (ε). Assuming the classical relationship between the stress and dislocation density ():
Introduction
where  a is a material constant,   is the shear modulus and b the Burgers vector, then, Eq. (1) becomes in the following constitutive expression:
Introduction
being √ and √ , where ρ 0 is the initial dislocation density. Here is the saturation stress, i.e., the stress in the absence of DRX, equal to in the case of softening solely by DRV, and very close to peak stress ( ) in the presence of DRX. For the determination of the constitutive equations it is necessary to establish the influence of the strain rate ( ) and temperature (T) on  p , U and Ω.
Introduction
The stage II is controlled by the DRX process. As any process of solid state transformation that takes place by nucleation and growth, DRX has a kinetic that can be modeled by the
Introduction
where X is the dynamically recrystallized volume fraction in a given period of time (t), and n A and k are the constants related to the nucleation and growth rate mechanism, respectively. If the peak strain (ε p ), the strain rate ( ) and the progressing strain (ε) are known, then:
Introduction
The time for 100% DRX may stabilize at low slope, which would increase the error while calculating the constant k. Hence, it is customary to define the recrystallization kinetics in terms of the time for 50% of recrystallization (t 50% ). Since exp(-0.693) ≈ 0.5, the relationship between t 50% and the constant k can be expressed as:
Introduction
It has been experimentally verified [42, 43] that t 50% depends on the deformation parameters, such as temperature (T), strain rate ( ) and initial grain size (d 0 ) as follows:
Introduction
where k t , m t and n t are material constants, Q rec is an activation energy (driving force) for DRX and R is the universal gas constant. Combining the Eqs. (4) and (6), the recrystallized fraction can be expressed as [23]:
Introduction
Finally, it is assumed that X is proportional to the stress softening:
Introduction
Therefore, the flow curve under DRX conditions can be expressed as:
Introduction
where  ss is the steady state stress attained at large strains as already pointed out. This equation is only valid for ε>ε p . In general, the use of Eqs. (3), (7) and (10) in conjunction with some additional kinetic equations that will be shown in the next paragraph will allow modeling the experimental hot flow curves, whether or not DRX is occurring.
Introduction
Finally, the stage III is completed with the determination of the kinetic equation for the steadystate stress, which can be also be applied to the peak stress. For this purpose, the hyperbolic sine law is traditionally applied [22, 28, 44]:
Introduction
where A is a material constant, n is the creep exponent and  is the inverse of the stress behavior. The unified expression is defined as follows:
Introduction
In this equation, the temperature dependent stress is normalized by the temperature dependent Young's modulus ( p (T)/E(T)) and also the strain rate is normalized by the -Fe lattice selfdiffusion coefficient ( (T)). In this case, the diffusion coefficient and Young's modulus as a function of the temperature [45] are defined as:
Introduction
and
Introduction
where  Cylindrical samples (7 mm in diameter x 11 mm in length) were machined using electric discharging machining (EDM). Uniaxial hot compression tests were carried out in an Instron machine coupled with a radiant cylindrical furnace and using inert atmosphere of Ar at three temperatures (900, 1000 and 1100 ºC) and four constant strain rates ( 
Modeling the hardening and recovery stages
As previously mentioned, Eq. (2) allows the calculation of U and  parameters of the stage I.
Modeling the hardening and recovery stages
Assuming  0 = 0 and using a least square method to fit experimental data to Eq. (2), the hardening (U) and softening () terms were derived and plotted as a function of the ZenerHollomon parameter (Z), i.e., the strain rate is compensated by temperature, which in turn is defined as follows:
Modeling the hardening and recovery stages
where Q HW is the activation energy for hot working, which were previously calculated [48] for the present TWIP steels (Q NM = 366, Q Nb = 434, Q TW V = 446 and Q TW Ti = 425 kJ/mol),
Modeling the hardening and recovery stages
T is the deformation temperature, ̇ is the strain rate and R is the universal gas constant (8.314 J/mol·K). As shown in Fig. 3 , a potential relationship between both terms and Z can be easily noticed, responding to the following mathematical relationships:
Modeling the hardening and recovery stages
Experimental values of  and U were also fitted by the least-square method to Eqs. (16) and (17) . Then, the modeling constant values (K Ω , m Ω and K U , m U ) were obtained. A similar procedure was used to calculate U (Fig. 3(b) ). In this case, the highest m U exponent is observed in the TW-Nb steel, followed by the TW-V and TW-Ti steel, respectively. The highest hardening term is observed in the TW-V steel, followed by the TW-Ti and TW-Nb, irrespective of the Z value considered, although at the highest Z values all three steel seems to offer similar hardening values, i.e., the chemical composition does not affect the hardening term at the highest strain rate and the lowest temperature.
Modeling the dynamic recrystallization (DRX) stage
The  p values were obtained directly from the experimental data, which correspond to the maximum stress value for each flow curve. These values are commonly associated, for simplification purposes, to the DRX onset. On the other hand,  p depends on the deformation temperature, the strain rate and the initial grain size as a power function of the form: (18) where K ε , n ε and m ε are material constants, d 0 is the initial grain size and Z is the ZenerHollomon parameter. If d 0 remains constant, as in this case of microalloyed TWIP steels, Eq.
Modeling the dynamic recrystallization (DRX) stage
(18) can be simplified to:
Modeling the dynamic recrystallization (DRX) stage
The experimental relationships of  p  and p respect the Zener-Hollomon parameter (Z) are shown in Fig. 4 . As can be seen, DRX is a thermally activated phenomenon and the amount of stress and deformation necessaries to the DRX onset are proportional to Z values [49].
Modeling the dynamic recrystallization (DRX) stage
Microalloying elements in TWIP steels produced a slight increase on  p values, as can be seen in Fig. 4(a) . It is evident that  p values increase either at high strain rates or when test temperature is low. This in turn means that the higher the Z values the higher the stress required to the DRX onset. Then, it was possible to determine the influence of microalloying elements on the  p necessary to the DRX onset. Fig. 4(a) shows how at high temperatures or low strain rates there is a slight deviation on the  p values, and the lowest value is for TW-NM. On the other hand, the highest  p value at high Z value is for TW-Nb, although the deviation respect to other TW steels is minimal. As a result of this analysis, the slope of the predicted values represented as the exponent (m p ) is close to 0.15 for all analyzed TWIP steels. Such value is a little bit lower than the typical stress exponent (0.2) which is an indication of some extra-hardening promoted by precipitated particles. When the deformation mechanism is only governed by the glide and climb of dislocations such exponent should be close to 0.2. A diminution of this value typically indicates some precipitation hardening, which in the present case is at the most, very minor. It is worth noting that particles are even present in the TW-NM steel in the form of aluminum nitrides. The most important effects on  p and  p are shown in TW-Ti and TW-V steels, particularly at low Z values. In Fig. 4(b) it is possible to appreciate how the V and Nb presence reduces the  p values. Therefore, the DRX onset is slightly accelerated in TW-V steel.
Dynamic recrystallization (DRX) kinetics
The Avrami relationship showed in Eq. (4) The results in Fig. 5(a) show that, the DRX kinetics retardation is due to the microalloying presence such as Ti, but it produces an acceleration of the DRX onset. Microalloying elements delay the DRX, therefore the recrystallization process is delayed too. Traditionally, the Avrami exponent (n A ) is determined by using a linear data fitting and measuring the slope of the double logarithmic plot ln(ln(1/(1-X))) vs. ln((ε-ε p )/ ), as can be seen from Fig. 5(a) . The Fig. 5(b) . These analyses also corroborate the grain refinement produced by single peak flow curves obtained to the four TWIP steels of this work. The low dependence of n A on temperature and strain rate showed in Fig. 5(b) can be expressed by the following fit:
Dynamic recrystallization (DRX) kinetics
As already mentioned, the recrystallized volume fraction (X) is assumed proportional to the softening observed in the flow curve. This in turn is considered to calculate the time necessary to produce 50% of recrystallized volume (t 50% ). If the initial grain size (d 0 ) is kept constant as assumed in the present research work, Eq. (7) can be simplified as follows:
Dynamic recrystallization (DRX) kinetics
The determined t 50% values were fitted using Eq. (21) , and applying the least squares method the parameters k t50% , m t50% and Q t were obtained, which are listed in Table 2 . Then, the time necessary for 50% recrystallization for analyzed TWIP steels are: t 50%(TW-NM) = 11.09 s, t 50%(TW-Nb) = 2.7 s, t 50%(TW-V) = 3,067.27 s, and t 50%(TW-Ti) = 2,274.81 s. As can be seen from 
According to Steinmetz et al. [59] temperature clearly delays the onset of twinning by
simultaneously decreasing the flow stress, increasing the rate of dislocation annihilation through climb and increasing the SFE, which in turn increases the critical stress needed to initiate deformation twinning. They also determined that at elevated temperatures the effective shear stress never reaches the critical twinning stress. This change influences directly the microstructure, texture, work hardening and mechanical properties [60] . On the other hand, it is remarkably that k t50% values are considerably higher for microalloyed TWIP steels than in TW-NM steel. This indicates that DRX kinetics is faster in TW-NM steel. Additionally, the activation energy (Q t ) required to start the DRX is higher in TW-NM than in microalloyed TWIP steels. Again, there are no previous reference values of k t50% and Q t for TWIP steels, but compared with boron microalloyed and medium carbon microalloyed steels previously mentioned, the calculated values present are lower.
Modeling the peak stress ( p ) and steady-state stress ( ss )
The final modeling stage is completed with the determination of kinetic equations for the peak stress ( p ) and the steady-state stress ( ss ), Eqs. (11) to (14) . In these equations, the constant Fig. 6(a) .
Modeling the peak stress ( p ) and steady-state stress ( ss )
The dependence of  ss on strain rate and temperature, as well as the corresponding values of A and  are shown in Fig. 6(b) . The creep exponent was fixed at n = 5 in the four analyzed TWIP steels, i.e., it was assumed that the deformation mechanism was controlled by the glide and climb of dislocations. When the ( / ) 1/5 vs. ( p /E(T)) plot is presented in Fig. 6(a) , it is possible to appreciate a good alignment and fitting using the hyperbolic sine equation . It only depends on the nature of the material considered. In this case, A is an apparent value because the internal microstructure of analyzed TWIP steels was not considered. In the case of the A p constant obtained in relation with  p , the highest value is for TW-Nb steel while the lowest value is for TW-V steel ( Fig. 6(a) ). In reference to the A ss constant related to  ss , the highest value is for TW-Nb steel while the lowest value is for TW-Ti steel (Fig. 6(b) ). In contrast,  p and  ss values of TW-Nb steel are the lowest of the studied TWIP steels for both calculations ( p and  ss ). The good agreement between experimental and estimated data for  p and  ss support the hypothesis [23] that there is no significant precipitation during hot compression and the proper use of the stress exponent (n = 5).
Validation of the model and constitutive equations
The constitutive equations to the hot flow behavior of the studied TWIP steels were solved by substitution of the calculated parameters in the above procedure, which are shown in Table 3 .
Validation of the model and constitutive equations
Those parameters were evaluated at the strain range ( t = 0 to 0.8) used in the experimental tests, in order to be compared with the predicted behavior, as shown in Figs. 7 (a-l) . It is possible to see that the model has a good agreement to the experimental data. However, there are some deviations in tests performed at high temperature and high strain rate (1100 ºC and 10 -1 s -1 ), as shown in Figs. 7(f, i, and l) ). These deviations are associated to several experimental causes, e.g., experimental errors because of the high strain rate test and the slow machine response. Additionally, it is important to take into account that theoretical curves consider an isothermal test, but experimental test are realized with adiabatic heating, which produces a higher heating than the expected one.
Validation of the model and constitutive equations
It is important to mention that previous research works have widely studied the Nb effect, in order to analyze its influence on processing parameters as shown in Table 3 . Cuddy [61, 62] determined that in microalloyed Nb steel, the presence of Nb carbide and carbonitride postponed coarsening to higher temperatures, the austenite recrystallization and growth presumably by particle pinning of grain boundaries. But, an abrupt abnormal growth occurred between 1100 to 1150°C producing a duplex austenite structure. Nevertheless, at certain compositional Nb variations there is no effect on precipitation, boundary pinning and grain [67] analyzed the Nb effect on SRX of TWIP steels. They concluded that SRX in high-Mn steels is markedly slower than in low-Mn carbon steels, and Nb presence significantly slows down the rate of SRX in both steels. Thus, the Nb effect is less pronounced when high Mn levels are present. Despite these results, in a previous study of microalloyed Nb TWIP steels
Validation of the model and constitutive equations
Hamada et al. [68] determined that the lower the Nb alloying presence as precipitated particles, the better the hot ductility presented. And the most important variables to control de hot ductility of TWIP steels are the grain size and strain hardening. Consequently, for microalloyed high-Mn TWIP steels of this research work the chemical composition and the interaction of alloying elements with temperature and strain rate, are the main factors controlling the operative deformation mechanisms.
Microstructural evolution.
The microstructural evolution of the studied microalloyed TWIP steels was analyzed by using the generated orientation imaging microscopy (OIM) maps. The high angle boundaries (misorientations>15º) and low angle boundaries (misorientations<15º) are shown as black and white lines, respectively (Figs. 8 and 9) . Additionally, the different coincidence site lattice (CSL) relationships are marked with different colors. 3 twin (misorientation of 60º about a <111> axis), 9 and 27 boundaries are shown as red, blue and yellow lines, respectively. The quantitative method performed to determine the recrystallized grains was the grain average misorientation (GAM), which analyzes the average misorientation between all neighboring pairs of points in a grain [69, 70] . In the present research work, the GAM by consideration of a threshold value of 1.55º was used to distinguish between recrystallized and deformed grains.
Microstructural evolution.
Grains with GAM less than 1.55º were considered as recrystallized grains. The considered GAM criteria was used successfully for austenitic stainless steels by Mirzadeh et al. (Figs. 8 (a, b, and c) ), and tested at 900 ºC and 10 -1 s -1 (Figs. 8 (d, e, and f) ). Figs. 8 (a, and d) show the microstructure maps with band contrast (BC), which in turn allows to visualize clearly the present microstructure, as well as the grain boundaries (GB) (thick black lines) and sub-grains (thin white lines). The results indicate that the higher the test temperature the higher the heterogeneous grain size (Fig. 8(a) ). This is also an evidence of dynamically recrystallized grains. The resulting grains are larger because of the test duration. The TW-NM steel tested at lower temperature ( Fig. 8(d) ) shows heterogeneous grain size and the largest grains are oriented toward the flow direction. The largest grains are sub-structured and surrounded by the classical necklace effect (finer recrystallized grains). deformed and the 58% is sub-structured, as shown in Fig. 8(c) . For the TW-NM steel at 900
Microstructural evolution.
°C the necklace grains are composed by 37% of recrystallized grains, the sub-structured grains are the 9% fraction and the fraction of deformed grains is the 54% located around the largest deformed grains. Fig. 9 shows the EBSD maps of deformed microstructures of TW-Ti steel. These grains are more deformed and elongated trough the flow direction than those presented to the TW-NM steel. The grain coarsening showed in Fig. 9(a) is due to the permanence at high temperature.
Microstructural evolution.
There is a few percent of annealing twins (CSL 3), which are located randomly into the finest grains, as shown in Fig. 9(b) . The largest sub-structured grains represent the highest volume fraction (51%), the recrystallized grains represent the 29% of volume fraction and the 21% remaining corresponds to deformed grains.
Microstructural evolution.
At high strain rate (10 -1 s -1 ) and low deformation temperature (900 ºC), the microstructure of TW-Ti steel displays deformed elongated grains surrounded by finer grains (Figs. 9 (d, e, and   f) ), which is an indicative that these test conditions were insufficient to complete the DRX.
Microstructural evolution.
The CSL 3, 9 and 27 are isolated into the microstructure (Figs. 9 (b, and e) ). The TW-Ti steel microstructure is more refined than the TW-NM steel at the same hot deformation condition. The recrystallized volume fraction (blue) is about 16%, the sub-structured grains (yellow) represent the 9% of the volume fraction and the deformed grains volume (red) is the 75%, as is shown in Fig. 9(f) . 
Microstructural evolution.
where K D is the Derby's coefficient, whose values are between 1 and 10; n D is the Derby's exponent, which is commonly n D =2/3;   is the shear modulus and b is the Burger's vector. (22), a large recrystallized grain size indicates a low  ss value. The TW-NM steel exhibited the largest recrystallized grain size at high temperature and low strain rate, as shown in Fig. 10(b) . On the other hand, it is evident the grain size growth inhibition produced by microalloying elements such as V and Ti on hot deformation, which is showed by the low slopes of Figs. 10(a, and b) .
Conclusions
1. The hot compression tests carried out to the studied TWIP steels allowed to determine the most important flow characteristics and the DRX onset, promoted by single peak curves.
Conclusions
2. Addition of Ti, V and Nb microalloying elements to TWIP steel produced a slight increase of peak stress ( p ).
Conclusions
3. The DRX onset for the studied TWIP steels was delayed in tests carried out at 900 ºC, accompanied by an increase of peak strain ( p ). The softening effect promoted by DRV and DRX was more evident at high temperatures (1000 and 1100 ºC) and low strain rate (10 -4 s -1 ), particularly in the TW-NM and TW-Nb steels.
Conclusions
4. A decrease on the activation energy required to recrystallization (Q t ) due to the microalloying elements addition was determined in the following order: i) TW-NM steel,
Conclusions
ii) TW-Ti steel, iii) TW-V steel and iv) TW-Nb steel. Likewise, the necessary time for 50% of recrystallization (t 50% ) was determined in the following order: i) TW-V, ii) TW-Ti and iii) TW-NM and iv) TW-Nb.
Conclusions
5. The good agreement between the experimental and predicted hot flow curves demonstrates that the constitutive equations predict with reasonable accuracy the hot flow behavior of the present TWIP steels. The short deviations presented at high temperature and high strain rates are attributable to experimental error conditions.
Conclusions
6. The addition of V and Ti to TWIP steel was more effective in the austenite grain growth control compared to that obtained in TW-NM and TW-Nb steels during hot deformation. ( ss /E(T)) of TW-NM steel and the determined curves by hyperbolic sine fitting. ( ss /E(T)) of TW-NM steel and the determined curves by hyperbolic sine fitting. 

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