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Modeling of strain induced transformation during hot deformation of an Mn–Al–C alloy
Journal Pre-proof Modeling of strain induced transformation during hot deformation of an Mn–Al–C alloy H. Dehghan, M. Rezayat, S.A. Seyyed Ebrahimi PII:
S0921-5093(20)30095-2
DOI:
https://doi.org/10.1016/j.msea.2020.139006
Reference:
MSA 139006
To appear in:
Materials Science & Engineering A
Received Date: 15 October 2019 Revised Date:
20 January 2020
Accepted Date: 23 January 2020
Please cite this article as: H. Dehghan, M. Rezayat, S.A.S. Ebrahimi, Modeling of strain induced transformation during hot deformation of an Mn–Al–C alloy, Materials Science & Engineering A (2020), doi: https://doi.org/10.1016/j.msea.2020.139006. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2020 Published by Elsevier B.V.
Credit Author Statement Hossein Dehghan: Investigation, Methodology, Writing - Reviewing and Editing. Mohammad Rezayat: Software, Reviewing and Editing. Seyyed Ali Seyyed Ebrahimi: Supervision and Editing.
Modeling of Strain Induced Transformation During Hot Deformation of an Mn-Al-C Alloy H. Dehghana, M. Rezayatb, S. A. Seyyed Ebrahimia* a
Advanced Magnetic Materials Research Center, School of Metallurgy and Materials, College of
Engineering, University of Tehran, Tehran 1439957131, Iran b
Faculty of Materials Engineering, Sahand University of Technology, P. O. Box 51335-1996, Tabriz,
Iran Corresponding author e-mail: [email protected]
Abstract High-temperature deformation of Mn51Al47C2 alloy was investigated through the hot compression tests at the temperature range of 750°C to 800°C and strain rate of 0.001 to 1 s-1, where due to the dynamical transformation of τ phase to ε phase, an excessive softening occurs in flow curves. The kinetics of dynamic phase transformation is studied by modified Avrami’s equation, and the activation energy of deformation was assessed by applying a standard approach on critical and steady-state stress. It was revealed that the deformation activation energy changed as the phase transformation occurred. Moreover, a physically realistic model is proposed to describe the hot deformation flow curve including straininduced phase transformation behavior of Mn-Al alloy. The comparison of predicted data with measured ones conveys that the developed formulation has a reasonable accuracy to model the flow stress.
Keywords:
Mn-Al-C
alloy;
Hot
deformation
transformation; Flow stress modeling
1
activation
energy;
Strain
induced
1. Introduction The Mn-Al permanent magnet alloys have been considered as attracting magnetic materials based on their potentially better magnetic properties than hard ferrites and also more affordable prices than rare earth magnets [1]. Several methods have been proposed to produce Mn-Al permanent magnet alloys [2-4]. Meanwhile, it is clear that the bulk form of permanent magnets is an industrial demand and therefore, techniques based on bulk deformation have been subjected to various researches [3,5]. Casting and hot deformation in a proper condition is one of these methods. Deformation at elevated temperature is faced with different phenomena such as dynamic recovery [6], recrystallization [7], aging [8], phase transformation [9] and defect propagation [10], i.e. strain localization, cracking, shear bands, etc, which affect the final product properties. Therefore, deep knowledge about the hot deformation behavior and high temperature flow stress is required for the proper design of thermomechanical processing to achieve the best properties. The constitutive analysis and the mathematically expression of stress in different conditions have been used to this purpose [11,12]. Based on the constitutive equation the flow stress is defined as a function of deformation parameters, i.e., strain, strain rate, and temperature. Constitutive models help to predict the flow stress of the material affected by different hardening and/or softening phenomena at various deformation conditions. While several constitutive equations have been developed for this purpose, most attention has been paid to contribute the effect of DRX and DRV as the main softening mechanism on flow stress [13-15], and seldom researches have been conducted to model the dynamic phase transformation softening. In the Mn-Al-C alloy it has been shown recently that the dynamic phase transformation is the main reason for an excessive flow softening [16]; however, there is not any information about constitutive analysis and modeling of this alloy. Therefore, the aim of the present work is to model the dynamic phase transformation kinetics and the hot-deformation behavior of the Mn-Al-C alloy where the strain induced phase transformation occurs.
2. Experiment As it was described in the previous publication [16], the Mn-Al-C alloy with nominal composition of Mn51Al47C2 was prepared via controlled atmosphere (vacuuming and then argon purging) induction melting process which is followed with two times re-melting, homogenizing at 1100 °C for 12 h. The composition of the alloy was measured by the 2
Inductively Coupled Plasma-Optical Emission Spectroscopy (ICP-OES, 730-ES, Varian, USA) and carbon content was measured using the LECO CS-244 carbon determinator (ASTM E1019). X-ray diffraction measurements were performed by Rigaku Ultima IV with Cu-kα (λ=1.54060 Å). The thermal response of the alloy during heat treatment was investigated with a NETZSCH STA 409 PC/PG differential scanning calorimeter. Cylindrical hot compression test samples, with 6 mm diameter and 9 mm height were prepared from the homogenized ingot (ASTM E209). The compression tests were performed at the temperatures of 750°C, 775°C and 800°C and strain rates of 10-3 to 1 s-1 where the microstructure at the beginning and the end of the deformation consisted of τ phase, and ε+γ phases, respectively [5,16]. Before compression test each sample was dwelled for 10 min at deformation temperature for temperature stabilizing. An INSTRON 8502 universal testing machine equipped with computer-controlled furnace was utilized to perform hot compression tests. The obtained flow curves were corrected for the effects of friction and deformation heat by well-accepted formulation [17].
3. Results and discussion 3.1. Experimental results Stress flow curves obtained from the compression test at various temperatures and strain rates are presented in Fig. 1. According to this figure, as deformation temperature is increasing, the flow stress is gradually decreasing. Moreover, as it can be seen from the figure, there is peak stress (σp) for each curve which is classically characterized by the features of DRX and/or dynamic phase transformation and/or flow localization [9,18,19]. In other words, during deformation, the flow stress raises to peak stress (σp) at the peak strain (εp) and decreases to the steady-state-stress (σss). According to this flow curves, it seems that by increasing the deformation temperature and reducing strain rate, the peak stress and the peak strain decrease. It is common in engineering metals that by increasing strain rate and/or reducing temperature, the peak in flow curves disappears and softening mechanism transfers from DRX to DRV. However, as is obvious in Fig. 1, the difference between peak stress and steady-state stress increases by deformation at lower temperatures with higher strain rate. It was proved in previous work that during deformation at temperatures 750°C to 800°C the τphase (tetragonal) dynamically transformed to ε-phase (hexagonal) and γ/γ2 (Al8Mn5) and results in flow softening [16]. Therefore, at the beginning of the deformation, τ phase was deformed and at the end of the deformation ε+γ phases were under deformation. According to the flow curves, the steady-state stress which is contributed to ε+γ phases is reduced at lower 3
strain rate and higher temperatures; however, it seems that its dependency on deformation condition is different from peak stress.
Fig. 1: flow curves obtained for different deformation conditions.
3.2. Constitutive analysis The constitutive analysis is usually used to develop the constitutive equation for predicting stress during the hot deformation. Besides, the deformation mechanism and activation energy can be obtained. In hot deformation studies, it is common to consider the effects of temperature and strain-rate on the deformation behavior by Zener–Hollomon parameter (Z) [20-22]. Moreover, the most widely applied method for prediction of flow stress is relating the Z parameter to the flow stress (σ), i.e.: Z = ε& exp( Q / RT ) = f (σ )
(1)
where f(σ) for low strain rate and high temperature deformations is in form of power law equation, for higher strain rates and low temperature is in form of exponential law equation and for a wide range of deformation conditions is conventionally in form of hyperbolic sine law equation (Eq.(2)). n Z = ε& exp(Q / RT ) = A[sinh(ασ )]
(2)
where A and α are material constants. Q is the activation energy of deformation and relates to the dominant deformation mechanism. In order to find the activation energy from presented routs, specific stress that represents the same deformation or softening mechanism for all 4
flow curves, such as steady-state, peak or critical stress for initiation of DRX, has been used. Generally, the critical stress (σc), the beginning of the softening mechanism, is the most widely accepted parameter used to determine Q [20]. By taking natural logarithm from mentioned equations, and then using partial differentiation operation on them, at constant deformation temperature and activation energy, the apparent materials constant can be obtained from the following equations: ∂ ln σ ∂ ln ε& α = × ∂ ln ε& T ∂σ T
(3)
∂ ln ε& n= ∂ ln[sinh(ασ )] T
(4)
For the determination of activation energy at certain strain and constant strain rate, partial differentiation of Eq. (2) and substituting Eq. (4) in it, yields the following equation: ∂ ln[sinh(ασ )] ∂ ln ε& Q = R × ∂ (1 / T ) ∂ ln[sinh(ασ )] T ε&
(5)
Using the critical stress obtained from flow curves and linear regression of ln sinh(ασ ) -1/T, the activation energy of deformation can be calculated for each strain rate. According to linear regressions represented in Fig. 2 the average value for α, n, and Q are 0.0055, 2.5 and 281 kJ/mole, respectively. The obtained activation energy is apparent and to find it more physically meaningful, some of the material’s information including elastic modulus and the self-diffusion coefficient at different temperatures are necessary. However, the obtained activation energy is more than the activation energy of statically transformation of τ to ε, i.e. 189 kJ/mole [23] and also activation energy of Mn diffusion in Al, 220 kJ/mole [24]. The value of n indicates the activation of dislocation based mechanisms during deformation [20]. Based on Eq.(2) by taking a linear regression from lnZ and lnsinh(ασ) plot, the value of A and n also can be obtained, i.e. 2.82×1012 and 2.43 respectively. Therefore, the relationship between critical stress and Z parameter can be derived as: Z = ε& exp(281000 / RT ) = 2.82 × 1012 [sinh(0.0055σ c )]
2.5
(6)
As it is shown in Fig. 3, the correlation coefficient of regression values is 0.98 which conveys that the prediction has acceptable accuracy. The same formulation was used to find a relationship between steady-state and saturation stress with the Z parameter. According to Fig. 3 the power exponent (slope of the fitted line) for steady-state stress is different from others which may be due to different phases undergoing deformation and consequently different active deformation mechanisms. Applying the same route for steady-state stress
5
instead of critical stress results in the activation energy of 197 kJ/mole with a power exponent of 3.2.
Fig. 2: Plots used for determining of α, n, and Q based on equations (2)-(4).
Fig. 3: Plot used to derive the representative constitutive equation based on the hyperbolic sine law for critical, saturation and steady-state stress. 3.3. Descriptions of strain softening The appearance of inflection points in the work hardening rate-stress plot (θ-σ) can be considered as an applicable method to find if phase transformation has occurred during deformation dynamically or not [21]. In other words, because the transformed phases are softer than transforming phase [16], and transformation mechanism is nucleation and growth, the work hardening-stress curve should have an inflection point, i.e.: ∂ ∂θ =0 ∂σ ∂σ ε = ε c
and θ = ∂σ ∂ε
=0
(7)
ε =ε p
6
Fig. 4a describes the flow softening based on strain induced transformation. Transformation starts from εc and results in an excessive reduction in strain hardening behavior of the material. Without softening, material consist of τ phase was supposed to be hardened to a maximum value called saturation stress, σsat, which obtained from the interaction of dislocation generation and annihilation mechanisms. This stress can be found as shown in Fig. 4b by extrapolating the slope at inflection point to the horizontal axis where θ=0. The steady-state stress, σss, in which hardening and softening mechanisms of fully transformed phases are in balance, also can be obtained by extrapolation of the slope at the end of θ-σ curve to the horizontal axis. The obtained stress is the steady state stress of fully transformed new phases, i.e. ε and γ. It can also be found from stress-strain curve where the flow stress become horizontal. As it can be identified from Fig. 1, some of flow curves are not horizontal in the strain of 0.7. in other words, the strain hardening at this strain is not zero. It means that the phase transformation has not been completed. This fact can be confirmed with the microstructural observation presented in Fig. 5. According to these microstructures, the microstructures of the samples deformed at the temperatures and the strain rates of 750°C and 0.1s−1 contains three phases while the microstructures of the samples deformed at 800°C with strain rate of 0.001s−1 shows two phases. The fraction of transformation can be calculated by assuming the applicability of the mixture role, i.e. σ = Xσss + (1-X)σrec [25]. In other words, the transformation fraction for each strain will be obtained by: X SIT =
∆σ σ rec − σ ss
(8)
Based on the Avrami kinetic equation [26], it has been shown that progress in dynamic recrystallization as a dynamic softening mechanism depends on strain according to the following equation [27,28]: N ε − ε c X = 1 − exp − K ε p
(9)
where K and N are constant which may depend on temperature or even strain rate. In this equation, εc is the critical strain required for starting DRX, respectively. As it was mentioned, strain induced transformation in this alloy resulted in the occurrence of dynamic softening. Hence one can consider the Eq. (9) for prediction the fraction of transformation (XSIT).
7
Fig. 4: (a) representation of flow curve obtained from deformation at 775°C with a strain rate of 0.1 /s associated with a fraction of strain induced phase transformation and (b) illustrating the corresponded work-hardening curve to show how to find the saturation stress, steady-state stress, and critical stress.
Fig. 5: Microstructures of the samples deformed up to the strain of 0.7 at the temperatures and the strain rates of (a) 750°C-0.1s−1, and (b) 800°C-0.001s−1. It should be noted that the effect of dynamic recrystallization of τ and/or ε+γ phases was neglected. As a matter of fact, the critical strain, where excessive transformation softening starts, was not high enough to get a significant fraction of DRX in the τ phase and therefore, it did not affect the softening. Moreover, according to Fig. 6, the temperature at which ε and γ phases are stable is more than 800˚C, while the deformation temperatures were lower. Besides, the newly transformed ε and γ grains did not deform heavily and so could not provide sufficient energy for the occurrence of DRX. Therefore, there is a limited probability 8
for the occurrence of DRX in these phases. Nonetheless, both dynamic phase transformation and dynamic recrystallization in transformed phases (if any) leads to the work softening during deformation. This softening can be modeled by the following approach which considers the apparent softening flow curves.
Fig. 6: DSC analysis with a rate of 10 °C/min and the corresponded X-ray diffraction
The flow stress behavior prior to εc (τ phase) is characterized by σrec; afterward, the effect of the transformed phase should be considered which depends on transformation fraction. Again using mixture rule, the flow stress for a given strain can be obtained by Eq. (10). ε < εc σ σ (ε , ε&, T ) = rec σ rec − X SIT (σ rec − σ ss ) ε ≥ ε c
(10)
It has been stated that critical strain is a part of the peak strain [22,29]. Fig. 7(a) shows the relation, i.e. εc = 0.7 εp. According to this figure for each deformation condition, an individual peak strain is obtained.
In other words, peak and critical strains are the functions of
temperature and strain rate (Fig. 7(b)). Cabrera et al. [22], stated that critical strain depends on initial grain size and Zener–Hollomon (Z) parameter and Serajzadeh [28] proposed Eq. (11) to represent this relation. ε p = BZ b
(11)
Based on the fact that in the present study the initial condition for all samples was similar, the effect of grain size can be ignored and the mentioned equation will be applicable to predict the peak strain. The plot showed in Fig. 7(b) was utilized to find b and B in Eq. (11). Knowing the dependency of critical and peak strain to the Z parameter, the linear regression of lnln(1/(1-X)-ln((ε-εc)/εp) was used to find N and K for each flow curve (Fig. 8a). based on
9
the proposed approach, the average values of N and K are 1.086 and 0.37, respectively. This value of N indicates the diffusional based phase transformation. According to Fig. 8b, it seems that N and K are independent of Z.
Fig. 7: (a) the value of the critical strain with respect to peak stress and (b) plot used to find the variation of peak strain with the Z parameter.
Fig. 8: Plots used to obtain N and K.
3.4. Descriptions of strain Hardening Bergstrom [30] has suggested that the dislocation variation during dynamic recovery is affected with two mechanisms, dislocation generation (work hardening) and dislocation annihilation and immobilization (work softening), i.e. ∂ρ ∂ε = h − rρ . Which in this equation,
10
h describes the work hardening and r represents the dynamic recovery during deformation at a given temperature and strain rate. With the assumption of strain independent of h and r, and considering the initial and final condition, i.e. ε = 0, ρ = ρ 0 and ε → ∞ , ∂ρ ∂ε = 0 , the mentioned differential equation can be solved by using the relation between dislocation density and stress as below [29]: 2 2 2 σ rec = σ sat − (σ sat − σ o2 ) exp( − rε )
(12)
The yield stress, σo, of high temperature flow curve can be neglected in comparison with saturation stress, hence: σ rec = σ sat [1 − exp( −rε )]
1/ 2
(13)
The softening term in a certain strain rate and temperature can be found from linear regression of ln(1-(σrec/σrec)2 vs. ε for the early stage of deformation: ∂ ln(1 − (σ rec / σ sat ) 2 ) r = − ∂ε T ε&
(14)
By applying this method for each curve, the value of softening coefficient can be plot as a function of the Z parameter [31]. It has been reported that r relates to temperature and strain rate, or in other words, the Z parameter. The Eq.(15) has been applied for addressing the relation [32]. Based on data shown in Fig. 9, the material constant of the mentioned equation can be calculated. With accordance to this figure by increasing the Z parameter, i.e. reducing temperature and increasing strain rate, the softening coefficient decreased. r = CZ c
(15)
Fig. 9: Plot used to determine the softening coefficient as a function of the Z parameter.
3.5. Modeling results Considering the proposed equations mentioned in previous sections, the flow stress affected by strain induced transformation softening and strain hardening can be obtained as follow: 11
ε < εc σ σ (ε , ε&, T ) = rec σ rec − X SIT (σ rec − σ ss ) ε ≥ ε c σ rec = σ sat [1 − exp( −2014 Z −0.126 ε ) ]
1/ 2
1 / 2.31
σ sat
Z 1 = sinh −1 12 0.055 1.94 × 10
(16)
1 / 4.29
σ ss =
1 Z sinh −1 13 0.055 6.12 ×10
1.09 ε X SIT = 1 − exp − 0.37 − 0 . 7 0.068 0.0085Z
In order to verify the developed model, comparisons between the experimental and predicted results were carried out in Fig. 10(a-c). As it can be seen the predicted data in most of the deformation conditions have acceptable accuracy comparing with the measured data. This fact also can be quantitatively found from the value of R-squared for the plot of measured saturation, steady-state, peak and critical stresses with respect to predicted ones (Fig. 10(d)).
Fig. 10: the comparison between the (a-c) experimental and modeled flow carves and (d) saturation, steady-state, peak and critical stresses by the proposed equation.
12
4. Conclusion A physically-based model is proposed to describe the flow stress during high temperature deformation of Mn51Al47C2 alloy with considering the effect of strain induced phase transformation as the main work softening mechanism. The outputs of the proposed model are in good agreement with experimental results and the formulation seems to have suitable accuracy to predict the steady-state and peak stress. The kinetics of dynamic strain induced τ phase to ε phase transformation was investigated according to a modified Avrami’s equation. It was found that the activation energy of deformation is changed during transformation.
Data Availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0921-5093(20)30095-2
DOI:
https://doi.org/10.1016/j.msea.2020.139006
Reference:
MSA 139006
To appear in:
Materials Science & Engineering A
Received Date: 15 October 2019 Revised Date:
20 January 2020
Accepted Date: 23 January 2020
Please cite this article as: H. Dehghan, M. Rezayat, S.A.S. Ebrahimi, Modeling of strain induced transformation during hot deformation of an Mn–Al–C alloy, Materials Science & Engineering A (2020), doi: https://doi.org/10.1016/j.msea.2020.139006. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2020 Published by Elsevier B.V.
Credit Author Statement Hossein Dehghan: Investigation, Methodology, Writing - Reviewing and Editing. Mohammad Rezayat: Software, Reviewing and Editing. Seyyed Ali Seyyed Ebrahimi: Supervision and Editing.
Modeling of Strain Induced Transformation During Hot Deformation of an Mn-Al-C Alloy H. Dehghana, M. Rezayatb, S. A. Seyyed Ebrahimia* a
Advanced Magnetic Materials Research Center, School of Metallurgy and Materials, College of
Engineering, University of Tehran, Tehran 1439957131, Iran b
Faculty of Materials Engineering, Sahand University of Technology, P. O. Box 51335-1996, Tabriz,
Iran Corresponding author e-mail: [email protected]
Abstract High-temperature deformation of Mn51Al47C2 alloy was investigated through the hot compression tests at the temperature range of 750°C to 800°C and strain rate of 0.001 to 1 s-1, where due to the dynamical transformation of τ phase to ε phase, an excessive softening occurs in flow curves. The kinetics of dynamic phase transformation is studied by modified Avrami’s equation, and the activation energy of deformation was assessed by applying a standard approach on critical and steady-state stress. It was revealed that the deformation activation energy changed as the phase transformation occurred. Moreover, a physically realistic model is proposed to describe the hot deformation flow curve including straininduced phase transformation behavior of Mn-Al alloy. The comparison of predicted data with measured ones conveys that the developed formulation has a reasonable accuracy to model the flow stress.
Keywords:
Mn-Al-C
alloy;
Hot
deformation
transformation; Flow stress modeling
1
activation
energy;
Strain
induced
1. Introduction The Mn-Al permanent magnet alloys have been considered as attracting magnetic materials based on their potentially better magnetic properties than hard ferrites and also more affordable prices than rare earth magnets [1]. Several methods have been proposed to produce Mn-Al permanent magnet alloys [2-4]. Meanwhile, it is clear that the bulk form of permanent magnets is an industrial demand and therefore, techniques based on bulk deformation have been subjected to various researches [3,5]. Casting and hot deformation in a proper condition is one of these methods. Deformation at elevated temperature is faced with different phenomena such as dynamic recovery [6], recrystallization [7], aging [8], phase transformation [9] and defect propagation [10], i.e. strain localization, cracking, shear bands, etc, which affect the final product properties. Therefore, deep knowledge about the hot deformation behavior and high temperature flow stress is required for the proper design of thermomechanical processing to achieve the best properties. The constitutive analysis and the mathematically expression of stress in different conditions have been used to this purpose [11,12]. Based on the constitutive equation the flow stress is defined as a function of deformation parameters, i.e., strain, strain rate, and temperature. Constitutive models help to predict the flow stress of the material affected by different hardening and/or softening phenomena at various deformation conditions. While several constitutive equations have been developed for this purpose, most attention has been paid to contribute the effect of DRX and DRV as the main softening mechanism on flow stress [13-15], and seldom researches have been conducted to model the dynamic phase transformation softening. In the Mn-Al-C alloy it has been shown recently that the dynamic phase transformation is the main reason for an excessive flow softening [16]; however, there is not any information about constitutive analysis and modeling of this alloy. Therefore, the aim of the present work is to model the dynamic phase transformation kinetics and the hot-deformation behavior of the Mn-Al-C alloy where the strain induced phase transformation occurs.
2. Experiment As it was described in the previous publication [16], the Mn-Al-C alloy with nominal composition of Mn51Al47C2 was prepared via controlled atmosphere (vacuuming and then argon purging) induction melting process which is followed with two times re-melting, homogenizing at 1100 °C for 12 h. The composition of the alloy was measured by the 2
Inductively Coupled Plasma-Optical Emission Spectroscopy (ICP-OES, 730-ES, Varian, USA) and carbon content was measured using the LECO CS-244 carbon determinator (ASTM E1019). X-ray diffraction measurements were performed by Rigaku Ultima IV with Cu-kα (λ=1.54060 Å). The thermal response of the alloy during heat treatment was investigated with a NETZSCH STA 409 PC/PG differential scanning calorimeter. Cylindrical hot compression test samples, with 6 mm diameter and 9 mm height were prepared from the homogenized ingot (ASTM E209). The compression tests were performed at the temperatures of 750°C, 775°C and 800°C and strain rates of 10-3 to 1 s-1 where the microstructure at the beginning and the end of the deformation consisted of τ phase, and ε+γ phases, respectively [5,16]. Before compression test each sample was dwelled for 10 min at deformation temperature for temperature stabilizing. An INSTRON 8502 universal testing machine equipped with computer-controlled furnace was utilized to perform hot compression tests. The obtained flow curves were corrected for the effects of friction and deformation heat by well-accepted formulation [17].
3. Results and discussion 3.1. Experimental results Stress flow curves obtained from the compression test at various temperatures and strain rates are presented in Fig. 1. According to this figure, as deformation temperature is increasing, the flow stress is gradually decreasing. Moreover, as it can be seen from the figure, there is peak stress (σp) for each curve which is classically characterized by the features of DRX and/or dynamic phase transformation and/or flow localization [9,18,19]. In other words, during deformation, the flow stress raises to peak stress (σp) at the peak strain (εp) and decreases to the steady-state-stress (σss). According to this flow curves, it seems that by increasing the deformation temperature and reducing strain rate, the peak stress and the peak strain decrease. It is common in engineering metals that by increasing strain rate and/or reducing temperature, the peak in flow curves disappears and softening mechanism transfers from DRX to DRV. However, as is obvious in Fig. 1, the difference between peak stress and steady-state stress increases by deformation at lower temperatures with higher strain rate. It was proved in previous work that during deformation at temperatures 750°C to 800°C the τphase (tetragonal) dynamically transformed to ε-phase (hexagonal) and γ/γ2 (Al8Mn5) and results in flow softening [16]. Therefore, at the beginning of the deformation, τ phase was deformed and at the end of the deformation ε+γ phases were under deformation. According to the flow curves, the steady-state stress which is contributed to ε+γ phases is reduced at lower 3
strain rate and higher temperatures; however, it seems that its dependency on deformation condition is different from peak stress.
Fig. 1: flow curves obtained for different deformation conditions.
3.2. Constitutive analysis The constitutive analysis is usually used to develop the constitutive equation for predicting stress during the hot deformation. Besides, the deformation mechanism and activation energy can be obtained. In hot deformation studies, it is common to consider the effects of temperature and strain-rate on the deformation behavior by Zener–Hollomon parameter (Z) [20-22]. Moreover, the most widely applied method for prediction of flow stress is relating the Z parameter to the flow stress (σ), i.e.: Z = ε& exp( Q / RT ) = f (σ )
(1)
where f(σ) for low strain rate and high temperature deformations is in form of power law equation, for higher strain rates and low temperature is in form of exponential law equation and for a wide range of deformation conditions is conventionally in form of hyperbolic sine law equation (Eq.(2)). n Z = ε& exp(Q / RT ) = A[sinh(ασ )]
(2)
where A and α are material constants. Q is the activation energy of deformation and relates to the dominant deformation mechanism. In order to find the activation energy from presented routs, specific stress that represents the same deformation or softening mechanism for all 4
flow curves, such as steady-state, peak or critical stress for initiation of DRX, has been used. Generally, the critical stress (σc), the beginning of the softening mechanism, is the most widely accepted parameter used to determine Q [20]. By taking natural logarithm from mentioned equations, and then using partial differentiation operation on them, at constant deformation temperature and activation energy, the apparent materials constant can be obtained from the following equations: ∂ ln σ ∂ ln ε& α = × ∂ ln ε& T ∂σ T
(3)
∂ ln ε& n= ∂ ln[sinh(ασ )] T
(4)
For the determination of activation energy at certain strain and constant strain rate, partial differentiation of Eq. (2) and substituting Eq. (4) in it, yields the following equation: ∂ ln[sinh(ασ )] ∂ ln ε& Q = R × ∂ (1 / T ) ∂ ln[sinh(ασ )] T ε&
(5)
Using the critical stress obtained from flow curves and linear regression of ln sinh(ασ ) -1/T, the activation energy of deformation can be calculated for each strain rate. According to linear regressions represented in Fig. 2 the average value for α, n, and Q are 0.0055, 2.5 and 281 kJ/mole, respectively. The obtained activation energy is apparent and to find it more physically meaningful, some of the material’s information including elastic modulus and the self-diffusion coefficient at different temperatures are necessary. However, the obtained activation energy is more than the activation energy of statically transformation of τ to ε, i.e. 189 kJ/mole [23] and also activation energy of Mn diffusion in Al, 220 kJ/mole [24]. The value of n indicates the activation of dislocation based mechanisms during deformation [20]. Based on Eq.(2) by taking a linear regression from lnZ and lnsinh(ασ) plot, the value of A and n also can be obtained, i.e. 2.82×1012 and 2.43 respectively. Therefore, the relationship between critical stress and Z parameter can be derived as: Z = ε& exp(281000 / RT ) = 2.82 × 1012 [sinh(0.0055σ c )]
2.5
(6)
As it is shown in Fig. 3, the correlation coefficient of regression values is 0.98 which conveys that the prediction has acceptable accuracy. The same formulation was used to find a relationship between steady-state and saturation stress with the Z parameter. According to Fig. 3 the power exponent (slope of the fitted line) for steady-state stress is different from others which may be due to different phases undergoing deformation and consequently different active deformation mechanisms. Applying the same route for steady-state stress
5
instead of critical stress results in the activation energy of 197 kJ/mole with a power exponent of 3.2.
Fig. 2: Plots used for determining of α, n, and Q based on equations (2)-(4).
Fig. 3: Plot used to derive the representative constitutive equation based on the hyperbolic sine law for critical, saturation and steady-state stress. 3.3. Descriptions of strain softening The appearance of inflection points in the work hardening rate-stress plot (θ-σ) can be considered as an applicable method to find if phase transformation has occurred during deformation dynamically or not [21]. In other words, because the transformed phases are softer than transforming phase [16], and transformation mechanism is nucleation and growth, the work hardening-stress curve should have an inflection point, i.e.: ∂ ∂θ =0 ∂σ ∂σ ε = ε c
and θ = ∂σ ∂ε
=0
(7)
ε =ε p
6
Fig. 4a describes the flow softening based on strain induced transformation. Transformation starts from εc and results in an excessive reduction in strain hardening behavior of the material. Without softening, material consist of τ phase was supposed to be hardened to a maximum value called saturation stress, σsat, which obtained from the interaction of dislocation generation and annihilation mechanisms. This stress can be found as shown in Fig. 4b by extrapolating the slope at inflection point to the horizontal axis where θ=0. The steady-state stress, σss, in which hardening and softening mechanisms of fully transformed phases are in balance, also can be obtained by extrapolation of the slope at the end of θ-σ curve to the horizontal axis. The obtained stress is the steady state stress of fully transformed new phases, i.e. ε and γ. It can also be found from stress-strain curve where the flow stress become horizontal. As it can be identified from Fig. 1, some of flow curves are not horizontal in the strain of 0.7. in other words, the strain hardening at this strain is not zero. It means that the phase transformation has not been completed. This fact can be confirmed with the microstructural observation presented in Fig. 5. According to these microstructures, the microstructures of the samples deformed at the temperatures and the strain rates of 750°C and 0.1s−1 contains three phases while the microstructures of the samples deformed at 800°C with strain rate of 0.001s−1 shows two phases. The fraction of transformation can be calculated by assuming the applicability of the mixture role, i.e. σ = Xσss + (1-X)σrec [25]. In other words, the transformation fraction for each strain will be obtained by: X SIT =
∆σ σ rec − σ ss
(8)
Based on the Avrami kinetic equation [26], it has been shown that progress in dynamic recrystallization as a dynamic softening mechanism depends on strain according to the following equation [27,28]: N ε − ε c X = 1 − exp − K ε p
(9)
where K and N are constant which may depend on temperature or even strain rate. In this equation, εc is the critical strain required for starting DRX, respectively. As it was mentioned, strain induced transformation in this alloy resulted in the occurrence of dynamic softening. Hence one can consider the Eq. (9) for prediction the fraction of transformation (XSIT).
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Fig. 4: (a) representation of flow curve obtained from deformation at 775°C with a strain rate of 0.1 /s associated with a fraction of strain induced phase transformation and (b) illustrating the corresponded work-hardening curve to show how to find the saturation stress, steady-state stress, and critical stress.
Fig. 5: Microstructures of the samples deformed up to the strain of 0.7 at the temperatures and the strain rates of (a) 750°C-0.1s−1, and (b) 800°C-0.001s−1. It should be noted that the effect of dynamic recrystallization of τ and/or ε+γ phases was neglected. As a matter of fact, the critical strain, where excessive transformation softening starts, was not high enough to get a significant fraction of DRX in the τ phase and therefore, it did not affect the softening. Moreover, according to Fig. 6, the temperature at which ε and γ phases are stable is more than 800˚C, while the deformation temperatures were lower. Besides, the newly transformed ε and γ grains did not deform heavily and so could not provide sufficient energy for the occurrence of DRX. Therefore, there is a limited probability 8
for the occurrence of DRX in these phases. Nonetheless, both dynamic phase transformation and dynamic recrystallization in transformed phases (if any) leads to the work softening during deformation. This softening can be modeled by the following approach which considers the apparent softening flow curves.
Fig. 6: DSC analysis with a rate of 10 °C/min and the corresponded X-ray diffraction
The flow stress behavior prior to εc (τ phase) is characterized by σrec; afterward, the effect of the transformed phase should be considered which depends on transformation fraction. Again using mixture rule, the flow stress for a given strain can be obtained by Eq. (10). ε < εc σ σ (ε , ε&, T ) = rec σ rec − X SIT (σ rec − σ ss ) ε ≥ ε c
(10)
It has been stated that critical strain is a part of the peak strain [22,29]. Fig. 7(a) shows the relation, i.e. εc = 0.7 εp. According to this figure for each deformation condition, an individual peak strain is obtained.
In other words, peak and critical strains are the functions of
temperature and strain rate (Fig. 7(b)). Cabrera et al. [22], stated that critical strain depends on initial grain size and Zener–Hollomon (Z) parameter and Serajzadeh [28] proposed Eq. (11) to represent this relation. ε p = BZ b
(11)
Based on the fact that in the present study the initial condition for all samples was similar, the effect of grain size can be ignored and the mentioned equation will be applicable to predict the peak strain. The plot showed in Fig. 7(b) was utilized to find b and B in Eq. (11). Knowing the dependency of critical and peak strain to the Z parameter, the linear regression of lnln(1/(1-X)-ln((ε-εc)/εp) was used to find N and K for each flow curve (Fig. 8a). based on
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the proposed approach, the average values of N and K are 1.086 and 0.37, respectively. This value of N indicates the diffusional based phase transformation. According to Fig. 8b, it seems that N and K are independent of Z.
Fig. 7: (a) the value of the critical strain with respect to peak stress and (b) plot used to find the variation of peak strain with the Z parameter.
Fig. 8: Plots used to obtain N and K.
3.4. Descriptions of strain Hardening Bergstrom [30] has suggested that the dislocation variation during dynamic recovery is affected with two mechanisms, dislocation generation (work hardening) and dislocation annihilation and immobilization (work softening), i.e. ∂ρ ∂ε = h − rρ . Which in this equation,
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h describes the work hardening and r represents the dynamic recovery during deformation at a given temperature and strain rate. With the assumption of strain independent of h and r, and considering the initial and final condition, i.e. ε = 0, ρ = ρ 0 and ε → ∞ , ∂ρ ∂ε = 0 , the mentioned differential equation can be solved by using the relation between dislocation density and stress as below [29]: 2 2 2 σ rec = σ sat − (σ sat − σ o2 ) exp( − rε )
(12)
The yield stress, σo, of high temperature flow curve can be neglected in comparison with saturation stress, hence: σ rec = σ sat [1 − exp( −rε )]
1/ 2
(13)
The softening term in a certain strain rate and temperature can be found from linear regression of ln(1-(σrec/σrec)2 vs. ε for the early stage of deformation: ∂ ln(1 − (σ rec / σ sat ) 2 ) r = − ∂ε T ε&
(14)
By applying this method for each curve, the value of softening coefficient can be plot as a function of the Z parameter [31]. It has been reported that r relates to temperature and strain rate, or in other words, the Z parameter. The Eq.(15) has been applied for addressing the relation [32]. Based on data shown in Fig. 9, the material constant of the mentioned equation can be calculated. With accordance to this figure by increasing the Z parameter, i.e. reducing temperature and increasing strain rate, the softening coefficient decreased. r = CZ c
(15)
Fig. 9: Plot used to determine the softening coefficient as a function of the Z parameter.
3.5. Modeling results Considering the proposed equations mentioned in previous sections, the flow stress affected by strain induced transformation softening and strain hardening can be obtained as follow: 11
ε < εc σ σ (ε , ε&, T ) = rec σ rec − X SIT (σ rec − σ ss ) ε ≥ ε c σ rec = σ sat [1 − exp( −2014 Z −0.126 ε ) ]
1/ 2
1 / 2.31
σ sat
Z 1 = sinh −1 12 0.055 1.94 × 10
(16)
1 / 4.29
σ ss =
1 Z sinh −1 13 0.055 6.12 ×10
1.09 ε X SIT = 1 − exp − 0.37 − 0 . 7 0.068 0.0085Z
In order to verify the developed model, comparisons between the experimental and predicted results were carried out in Fig. 10(a-c). As it can be seen the predicted data in most of the deformation conditions have acceptable accuracy comparing with the measured data. This fact also can be quantitatively found from the value of R-squared for the plot of measured saturation, steady-state, peak and critical stresses with respect to predicted ones (Fig. 10(d)).
Fig. 10: the comparison between the (a-c) experimental and modeled flow carves and (d) saturation, steady-state, peak and critical stresses by the proposed equation.
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4. Conclusion A physically-based model is proposed to describe the flow stress during high temperature deformation of Mn51Al47C2 alloy with considering the effect of strain induced phase transformation as the main work softening mechanism. The outputs of the proposed model are in good agreement with experimental results and the formulation seems to have suitable accuracy to predict the steady-state and peak stress. The kinetics of dynamic strain induced τ phase to ε phase transformation was investigated according to a modified Avrami’s equation. It was found that the activation energy of deformation is changed during transformation.
Data Availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: