Thermodynamic modeling of stacking fault energy in Fe–Mn–C austenitic steels

1 Introduction and background

Stacking fault energy (SFE) has been regarded as one of the most important thermodynamic parameters in the investigation of deformation mechanisms since the wide use of high alloyed austenitic steels. The strain-hardening behavior of face-centered cubic metals has also been found to be closely related to SFE. According to numerous studies reported, phase transformation-induced plasticity (TRIP) usually occurs at low SFE of less than 12 mJ·m⁻², while dislocation planar glide appears at high SFE above 80 mJ·m⁻² [1,2,3]. When the SFE values located within the range between 20 and 35 mJ·m⁻², an outstanding performance of twinning-induced plasticity (TWIP) in Fe–Mn–Si–Al and Fe–Mn–C–(–Al) austenitic steels was exhibited. [4,5]. The microbands and shear bands induced plasticity in the extremely high manganese and high aluminum steels could also be affected and described by SFE [6,7]. In our previous work, the TWIP effect and strain hardening behavior in Fe–Mn–C austenitic steels have been experimentally investigated, which confirms the importance of SFE in material design and deformation mechanism analysis [8,9,10,11].

Several different models for the SFE calculation of Fe–Mn–C austenite steels for automobiles with extremely high tensile strength and high ductility have been built up. Grässel et al. [3,4] employed a regular solution model in calculating the SFE in Fe–Mn–Si–Al TRIP or TWIP steels, and they pointed out that the critical value for the transition of TRIP to the TWIP effect was about 20 mJ·m⁻². Allain et al. [4,5] made a similar model to investigate the correlation between the calculated SFE and the plasticity mechanism in Fe–Mn–C steels, in which the effect of alloyed elements on SFE was theoretically revealed. A subregular solution model to predict SFE was developed by Saeed-Akbari et al. [12] for Fe–Mn–C(–Al) austenite steels, and the results calculated based on different thermodynamic parameters presented by Dew-Hughes and Kaufman [13,14] and Dinsdale [15] were schematically compared. Petrov [16], Petrov and Yakubtsov [17], and Yakubtsov et al. [18] studied the SFE of austenite stainless steels and high manganese steels with carbon and/or nitrogen alloyed by the Bragg-Williams model. Nonlinear relationships were found between SFE and interstitial concentration due to the Suzuki effect. This should be the most obvious difference between the models by Saeed-Akbari et al. [12] and Petrov [16]. Nakano and Jacques [19] built up a new model based on the two-sublattice method to investigate the effect of thermodynamic parameters on SFE calculations in the Fe–Mn and Fe–Mn–C systems. With the parameters taken from SGTE data [15] and Huang’s work [20], the phase transformation temperature and SFE prediction were roughly coincident with experiment values.

1.1 SFE by calculation and experiment

Figure 1 shows the calculated iso-SFE lines in the carbon/manganese (wt%) map at 300 K from different approaches [4,12,19], and straight lines proposed by Schumann [21] to describe the regimes of mechanical HCP and thermal HCP were also plotted. It can be seen the results based on different methods and parameters exhibit diverse characteristics. The SFE gradients in these three maps are found not very similar to one another within the range of 10–35 wt% manganese and 0–1.2 wt% carbon. At a given composition, the SFEs calculated by these three models are not quite equal and even in different ranges. The discrepancy between experiment results by X-ray diffraction, transmission electron microscopy, and embedded atom method and the thermodynamic model is still quite large [22,23,24,25,26,27], as shown in Table 1. It is difficult to distinguish the effective range for the application of these models. Further research, both on experiments and on thermodynamics, is always needed.

Figure 1

Calculated iso-SFE lines in the carbon/manganese (wt%) map at 300 K (unit: mJ·m⁻²): (a) by Allain et al. [4], (b) by Saeed-Akbari et al. [12], and (c) by Nakano and Jacques [19]. Straight lines proposed by Schumann to describe the regimes of mechanical HCP and thermal HCP were also plotted [21].

Table 1

SFEs measured by various methods in high manganese steel

No.	Chemical composition (wt%)	SFE (mJ·m−2)	Method	References
1	Fe–25Mn–0.15C–0.6Al	7.75	XRD	[22]
2	Fe–25Mn–0.15C–1.5Al	10.67	XRD	[22]
3	Fe–25Mn–0.15C–2.2Al	15.12	XRD	[22]
4	Fe–25Mn–0.15C–3.1Al	14.95	XRD	[22]
5	Fe–25Mn–0.15C–4.8Al	54.74	XRD	[22]
6	Fe–31Mn–0.17C	17.53	XRD	[23]
7	Fe–19Mn–5Cr–0.25C–1.2Al	21	TEM node	[24]
8	Fe–19Mn–5Cr–0.25C–2.5Al	31	TEM node	[24]
9	Fe–19Mn–5Cr–0.25C–3.3Al	39	TEM node	[24]
10	Fe–19Mn–5Cr-0.25C–4.0Al	47	TEM node	[24]
11	Fe–18Mn–0.6C–1.5Al	26.4	TEM node	[25]
12	Fe–18Mn–0.6C–1.5Al	30.0	TEM WBDF	[25]
13	Fe–18Mn–1.3C	15.0	TEM WBDF	[26]

2 Thermodynamic modeling description

As early proposed by Hirth [28] in 1970, the thermodynamic models of SFE in high alloyed steels were reported by Ferreira and Mullner [29], Olson and co-workers [30,31], and Lee and Choi [32]. The SFE can be calculated by the following equation:

where ρ is the molar surface density of {1 1 1} planes, Δ G γ → ε is the molar Gibbs free energy of the transformation of γ → ε , and σ is the interfacial energy of γ / ε .

The molar surface density ρ is defined by introducing the lattice parameter a of the investigated alloy:

where a is the lattice parameter and N is the Avogadro number.

The lattice parameter of austenite is composition and temperature dependent in the present model, and it can be determined by the following formulas [33,34]:

where X i is the molar fraction of element i and a 0 is the lattice parameter of this alloy at 300 K. β is the linear thermal expansion coefficient, and it is about 2.065 × 10⁻⁵ K⁻¹ for austenite steel [34]. T is the temperature in Kelvin.

The molar Gibbs free energy change Δ G γ → ε during γ → ε transformation is expressed by

where Δ G b γ → ε and Δ G s γ → ε are the change of Gibbs free energies in bulk and stacking fault regions, respectively. Δ G grain γ → ε is the change of Gibbs free energy related to the matrix grain size. Δ G strain γ → ε is the composition-dependent Gibbs free energy change from strain contribution, which is less than 40 J·mol⁻¹ in the present composition range [35].

For the current system, Δ G b γ → ε can be calculated by

where G m ε and G m γ are the molar free energies of HCP and FCC, respectively, which can be calculated by the two-sublattice model. Both phases were treated as random mixing solutions. Substitutional atoms (ferrite and manganese) were totally in the substitutional sublattice, while interstitial atoms (carbon) and vacancies were in the interstitial sublattice. The following formula was introduced in the present model:

with υ equal to 1 for FCC and 0.5 for HCP. Va stands for vacancies in the lattice.

The molar Gibbs energy of a phase φ (FCC or HCP) can be expressed by

where y i j is the site fraction of component i on sublattice j . Here, j is not necessary for clarity and is omitted. y Fe + y Mn = 1 and y C + y Va = 1 . G Fe : Va φ 0 and G Mn: Va φ 0 are the molar Gibbs energy of pure Fe and Mn with phase φ , while G Fe : C φ 0 and G Mn : C φ 0 are the molar Gibbs energy of Fe–C and Mn–C systems with all the octahedron sites occupied by C atoms. G m φ Ex is the excess term, G m φ Mag is the Gibbs free energy related to the magnetic phenomenon.

G m φ Ex can be expressed as follows:

The interaction parameters L Fe , Mn : C φ i , L Fe , Mn : Va φ i , L Fe : C , Va φ i and L Mn : C , Va φ i in equation (9) were all temperature dependent.

The Gibbs free energy related to the magnetic phenomenon was studied by Inden [36] and Hillert and Jarl [37] as follows:

where β φ is the magnetic moment per atom as Bohr magneton numbers, τ is ratio of temperature T , and magnetic transition temperature T c .

According to Gutierrez-Urrutia and Raabe [6], β φ in different phase can be calculated by

where X i is the molar fraction of element i .

The magnetic transition temperature T c can be calculated by the following equation:

where T c φ is the magnetic transition point of phase φ , T c , Fe : Va φ and T c , Mn : Va φ are the magnetic transition temperatures for pure Fe and Mn, T c , Fe : C φ and T c , Mn : C φ are the magnetic transition temperatures for the Fe–C and Mn–C binary system with the C atom occupying all the octahedron sites. T Fe , Mn : C φ , i , T Fe , Mn : Va φ , i , T Fe : C , Va φ , i , and T Mn : C , Va φ , i is the interaction parameters in Fe–Mn–C ternary systems.

The function g ( τ ) is expressed as follows [38]:

For τ ≤ 1 :

For τ > 1 :

As shown by Ishida [39], the Gibbs free energy change due to interstitial segregation could be divided into three parts:

where Δ G chm is the chemical free energy change related to the Suzuki effect, Δ G sur is the surface free energy change due to the concentration variation between the matrix and stacking faults, and Δ G els is the elastic free energy change due to the difference of segregated atoms’ sizes.

It is assumed that the chemical potential of FCC is equal to that of HCP when the equilibrium is reached. For the binary system with one substitutional element and one interstitial element, the following equation can be obtained:

with G γ and G ε being the free energy of FCC and HCP and X b and X s being the element concentration in bulk and stacking fault, respectively.

Petrov and Yakubtsov [17] employed Ishida’s assumption [39] that the systems were ideal solutions, and the following equation can be obtained for the ternary system:

where Δ G i γ → ε and Δ G j γ → ε are the free energy change of pure i and j . A reasonable simplification was introduced by Petrov and Yakubtsov [17], considering that thermodynamic parameters for interstitials are not very suitable for this γ → ε transformation:

where Λ is defined as the interaction energy of the interstitial element with stacking fault. Taking the lack of data in the Fe–Mn–C system, the parameter Λ can be simplified to the interaction effect of carbon to dislocation in FCC alloys [17].

Then, the chemical free energy change Δ G chm can be calculated by

The surface free energy change due to the segregated alloying elements in stacking faults was analyzed by Ericsson [40]:

where Λ is the interaction energy of the alloying element with stacking faults.

According to Suzuki’s study, the elastic free energy change Δ G els could be expressed by

where μ is the shear modulus, ν is the Poisson ratio, and V is the molar volume of the alloy. Actually, the elastic free energy is negligible in the present work because it shows little effect on SFE [39].

Nakatsu et al. [41] and Lee and Choi [32] suggested that the free energy change linked to the grain sizes could not be neglected particularly below 30 μm, and it can be calculated as follows:

where d is the austenite grain size in μm, and it is assumed to be 100 μm in average in this work.

3 Calculated SFE in the concerned systems

3.1 Fe–Mn binary system

For the Fe–Mn binary system, the site fraction y C = 0 and y Va = 1 . Thus, the calculation formula (8) can be expressed as follows:

(24) G m φ = y Fe G Fe : Va φ 0 + y Mn G Mn : Va φ 0 + RT [ y Fe ln y Fe + y Mn ln y Mn ] + G m φ Ex .

The excess molar Gibbs free energy can be shown by

(25) G m φ Ex = y Fe y Mn ∑ i = 0 z L Fe , Mn : Va φ i ( y Fe − y Mn ) i .

Actually, for Fe–Mn binary systems with no interstitial element alloyed, the present model based on the two-sublattice method can be simplified to a regular solution model. The thermodynamic parameters are presented in Table 2. In fact, most of these parameters are from SGTE data by Dinsdale [15], based on which Huang [20,42] and Djurovic et al. [43] calculated the experimentally verified Fe–C, Mn–C, and Fe–Mn–C phase diagrams.

Table 2

Thermodynamic parameters for the Fe–Mn–C system (SI units)

FCC : ( Fe , Mn ) 1 ( C , Va ) 1	Reference
G Fe : Va γ 0 = ‒ 236.7 + 132.416 T ‒ 24.6643 T ln T ‒ 0.00375752 T 2 ‒ 5.89269 × 10 ‒ 8 T 3 + 77358.5 T ‒ 1	[15]
G Mn : Va γ 0 = ‒ 3439.3 + 131.884 T ‒ 24.5177 T ln T ‒ 0.006 T 2 + 69600 T ‒ 1	[15]
G Fe : C γ 0 ‒ G Fe γ 0 ‒ G C γ 0 = 77 , 027 ‒ 15.877 T	[45]
G Mn : C γ 0 ‒ G Mn γ 0 ‒ G C γ 0 = 502 + 15.261 T	[42]
L Fe : C , Va γ 0 = ‒ 34 , 671	[45]
L Mn : C , Va γ 0 = ‒ 43 , 433	[42]
L Fe , Mn : C γ 0 = 20 , 082 ‒ 11.6312 T	[43]
L Fe , Mn : Va γ 0 = ‒ 7 , 762 + 3.865 T	[20]
L Fe , Mn : Va γ 1 = ‒ 259	[20]
T c , Fe : Va γ = 67	[45]
T c , Mn : Va γ = 540	[20]
T c , Fe , Mn : Va γ 0 = 760.67	[20]
T c , Fe , Mn : Va γ 1 = 689.33	[20]
HCP : ( Fe , Mn ) 1 ( C , Va ) 0.5
G Fe : Va ε 0 = ‒ 2480.08 + 136.725 T ‒ 24.6643 T ln T ‒ 0.00375752 T 2 ‒ 5.89269 × 10 ‒ 8 T 3 + 77358.5 T ‒ 1	[15]
G Mn : Va ε 0 = ‒ 4439.3 + 133.007 T ‒ 24.5177 T ln T ‒ 0.006 T 2 + 69600 T ‒ 1	[15]
G Fe : C ε 0 ‒ G Fe ε 0 ‒ 0.5 G C ε 0 = 52 , 905 ‒ 11.9075 T	[45]
G Mn : C ε 0 ‒ G Mn ε 0 ‒ 0.5 G C ε 0 = ‒ 16 , 992 + 4.319 T	[42]
L Fe , Mn : Va ε 0 = ‒ 69.41 + 2.836 T	[20]
L Fe , Mn : Va ε 1 = 273	[20]
L Fe , Mn : C ε 0 = 21 , 742 ‒ 50.2073 T	[43]
L Fe : C , Va ε 0 = ‒ 17 , 335	[45]
L Mn : C , Va ε 0 = ‒ 8 , 552 ‒ 9.367 T	[42]
T c , Mn : Va ε = 580	[20]

The change of Gibbs free energy as a function of manganese content predicted by the present model for the Fe–Mn binary system at 298 K is shown in Figure 2, in which the calculation results from previous studies are also plotted [19,20,32]. The Gibbs free energy change during γ → ε transformation decreases with manganese content increasing in the range between 0 and 12 wt%, followed by an increase with manganese increasing from 12 to 60 wt%. The variation trend of Gibbs free energy change in the present study is close to that predicted by Lee and Choi’s work [32] in 0–50 wt% Mn. For higher Mn concentration range, it is between the curves by Nakano and Jacques [19] and Huang [20]. The explanation should be the difference between the thermodynamic model and parameters. The measured data in Witusiewicz et al.’s work [44] are also presented in Figure 2. It is seen that the critical Mn contents for γ → ε phase transformation by experiment match the present data well, suggesting a good precision of this thermodynamic model.

Figure 2

Gibbs free energy change during γ → ε transformation against manganese content at 298 K in the Fe–Mn system.

Figure 3 shows the relationship of calculated transformation temperature T ₀ with manganese content in the Fe–Mn binary system. T ₀ is defined as the temperature at which the Gibbs free energy of FCC is equal to that of HCP. As manganese content increases from 0 to 35 wt%, the transformation temperature decreases gradually. It can be seen the predicted T ₀ values followed well with the experimental results from previous studies [46,47,48,49].

Figure 3

Comparison of calculated and experiment transformation temperature T ₀ during γ → ε .

The SFE variation against manganese content in the Fe–Mn system at 298 K is shown in Figure 4. The calculated results by Huang [20], Nakano and Jacques [19], Lee and Choi [32], and Saeed-Akbari et al. [12] and experiment values by Schumann [50], Petrov et al. [51], Volosevich et al. [52] are also presented. SFEs decrease as manganese content increases from 0 to 10 wt% and turn to increase as manganese content increases further. Nakano and Jacques [19] predicted the minimum SFE of about 12 mJ·m⁻² appeared at about 5 wt% Mn, while Lee and Choi [32] and Saeed-Akbari et al. [12] found the extreme point of the SFE line was at 13 wt% Mn. Additionally, even the results proposed by Huang [20] and Allain et al. [4] were observed to be monotonic increasing. Schumann [50] presented a sharp SFE curve with the trough point located at 13 wt% Mn. Volosevich et al. [52] experimentally estimated that the minimum SFE of 15 mJ·m⁻² corresponds to a Mn content of 22 wt%, and Petrov et al. [51] measured the SFE of Fe–Mn alloys and indicated that the minimum of empirical SFE shift between 20 and 25 wt% Mn related to the specimens’ purity. In the recent work on SFE in high manganese TWIP steels by Pierce et al. [53], the original experimental SFEs on the Fe–Mn system by Schumann [50] and Volosevich et al. [52] were proved to be much higher than the recent calculation and experiment results. The variation trend and the location of the present curve are quite close to that of the values proposed by Lee and Choi [32], Saeed-Akbari et al. [12], and Schumann [50], especially when the Mn concentration is in 10–30 wt%.

Figure 4

Comparison of calculated and experiment SFE in the Fe–Mn system against Mn content.

Interfacial energy is an important parameter in the thermodynamic model of SFE during phase transformation in the Fe–Mn–C system [4,12,19]. However, no exact value from theory or experiment was proposed in previous studies except inverse calculation [35,53]. Lee and Choi [32] and Allain et al. [4] assumed it to be around 5–15 mJ·m⁻², Saeed-Akbari et al. [12] investigated the effect of interfacial energy on SFE within the range from 5 to 20 mJ·m⁻², but it was neither composition nor temperature dependent. Nakano and Jacques [19] determined the interfacial energy to be 16 mJ·m⁻² to fit the SFE to experimental data in the Fe–Mn binary system. Cotes et al. [35] calculated the interfacial energy as functions of temperature and composition based on experimental results by the reverse method. In the latest experimental and theoretical research by Pierce et al. [53], a Gibbs energy change-dependent interfacial energy was proposed for the Fe–Mn–Al–Si system with a mathematical fitting method. In the present investigation, the interfacial energy during FCC to HCP transformation was determined by the approach of Pierce et al. [53] to fit the pure ferrite SFE to 13 mJ·m⁻²:

(26) σ = 5.418 × 10 − 6 ⋅ Δ G γ → ε ⋅ Δ G γ → ε + 29.021 ,

where σ is the interfacial energy of FCC to HCP in mJ·m⁻² and Δ G γ → ε is the Gibbs free energy change during the transformation. The lowest interfacial energy is located where the Gibbs free energy change approaches zero according to the above parabolic relationship, which should be in good agreement with the physical transformation process. Considering the variable Δ G γ → ε in formula (26) is temperature and composition-dependent, the interfacial energy calculated is also determined by the system constituent and environment indirectly. Consequently, the method can be extended to predict the SFE of the Fe–Mn–C ternary system without a significant loss of accuracy.

3.2 Fe–Mn–C ternary system

All the thermodynamic parameters involved in calculating the SFE in the Fe–Mn–C system are from Table 1. The interaction energy of the interstitial element with stacking faults Λ in this Fe–Mn–C system can be simplified as that of the alloying element with dislocations after Yakubtsov et al. [18]. The same method was employed in the present work due to the lack of experimental parameters. Gavrilyuk and Duz [54] conducted internal friction tests to determine the interaction of carbon atoms with dislocations in FCC steels. Composition-dependent interaction energy in the range of 0.14–0.40 wt% carbon was obtained. Petrov [55] measured the carbon concentration of Fe–22Mn–0.6C austenite steel in the bulk and stacking faults by EDS, respectively. With the data from Gavrilyuk and Duz [54] and Petrov [55], a square root formula was proposed as follows:

(27) Λ = a x C ,

where Λ is the interaction energy and x C is carbon concentration in the bulk in weight percent. a is the fitting coefficient and equal to 4,550.

The change of Gibbs free energy Δ G γ → ε during γ → ε transformation was expressed as equation (5), in which Δ G strain γ → ε was usually very small compared to others, and Δ G grain γ → ε can be neglected when the grain size was set to 100 μm. Thus, the change of Gibbs free energy Δ G γ → ε can be determined by the other two terms, Δ G b γ → ε and Δ G s γ → ε .

Figure 5 shows bulk Gibbs free energy change Δ G b γ → ε , stacking fault Gibbs energy change Δ G s γ → ε , magnetic Gibbs free energy change Δ G mag γ → ε , and total Gibbs free energy change Δ G γ → ε as functions of carbon and manganese concentration in the Fe–Mn–C system at 298 K. It can be seen that the bulk Gibbs free energy change Δ G b γ → ε takes dominant effect on the total Gibbs energy change during phase transformation, followed by the stacking fault Gibbs energy change Δ G s γ → ε related to the Suzuki effect. The bulk Gibbs free energy change Δ G b γ → ε increases almost linearly with carbon concentration increasing but varies non-monotonously with manganese concentration increasing. The highest value was observed to locate where carbon content is 1.2 wt% and manganese content is 30 wt%. The stacking fault Gibbs energy change Δ G s γ → ε decreases nonlinearly with carbon concentration increasing, but shows no obvious relationship with manganese concentration. The magnetic Gibbs free energy change Δ G mag γ → ε increases significantly as manganese concentration increases but decreases gradually with carbon concentration increasing. Consequently, the highest point for Δ G mag γ → ε is 30 wt% manganese and zero carbon. The total Gibbs free energy change Δ G γ → ε varies non-monotonously with manganese concentration increasing, and increases nonlinearly as carbon concentration increases.

Figure 5

Gibbs free energy change in the Fe–Mn–C system at 298 K: (a) Δ G b γ → ε , (b) Δ G s γ → ε , (c) Δ G m a g γ → ε , and (d) Δ G γ → ε .

Figure 6 shows the variation of carbon segregated concentration (equation (19)) in stacking fault regions and the resulting SFE_seg against its concentration in the bulk. It can be seen the carbon segregated concentration increases nonlinearly with bulk concentration increasing, leading to a decrease of SFE_seg in this composition range. These are similar to the variation characteristics of nitrogen concentration in Fe–18Cr–10Ni–xN stainless steels investigated by Yakubtsov et al. [18].

Figure 6

Carbon segregation against its concentration in the bulk, together with the SFE variation in the Fe–22Mn–xC system.

Figure 7 shows the calculated carbon- and manganese-dependent iso-SFE map in the Fe–Mn–C ternary system at 298 K, together with the straight lines criterion for the formation of mechanical or thermal HCP proposed by Schumann [21] and the calculation and experiment values in previous study [3,53,56]. The iso-SFE lines calculated by this model follow Schumann’s lines in large range, which is in good agreement with an activation of the mechanical HCP martensite transformation controlled by SFE values about 10–20 mJ·m⁻² with carbon and manganese ranges between 0–0.9 and 10–30 wt%, respectively. Comparing Figure 7 to Figure 1, the SFE gradient against manganese concentration predicted by the present model was similar to that in Allain et al. [4] and Saeed-Akbari et al. [12] work. The SFE gradient against carbon concentration, however, was observed to be different from each other. The SFE calculated by the new sublattice model as the function of carbon concentration was just between the results of Allain et al. [4] and Saeed-Akbari et al. [12] model. It should be noticed that the SFE of a given composition in the Fe–Mn–C system predicted by Nakano et al. research was much higher than that of other approaches, including the present. The lack of consideration for the Suzuki effect should be the main explanation except for the thermodynamic parameters difference. The widely accepted SFE for the TWIP effect in high manganese steels during γ → ε transformation at 298 K is within 18–20 mJ·m⁻², which was proposed by Frommeyer and Bruex [2], Grassel et al. [3], Allain et al. [4], and Rémy et al. [57] based on the theoretical and experimental investigation. Actually, this is also in line with the present prediction in the composition range of 18–22 wt% manganese and 0.5–0.7 wt% carbon. In our previous study [58,59,60], the deformation behavior of Fe–Mn–C austenite steel was investigated, and the strengthening mechanism was confirmed by XRD and microstructure observation, as shown in Table 3. The Fe–20Mn–0.4C steel is with the TRIP + TWIP effect, while others are with the TWIP effect. The data are also marked in Figure 7, and it is shown that the results are coincident with the criterion by Frommeyer and Bruex [2] and Grassel et al. [3].

Figure 7

Calculated iso-SFE lines in the Fe–Mn–C system at 298 K. Straight lines proposed by Schumann were also plotted [21].

Table 3

Deformation mechanism for different Fe–Mn–C steels

	Mn	C	SFE (mJ·m−2)	Mechanism
1	19.74	0.46	14.0	TRIP + TWIP
2	20.08	0.61	20.4	TWIP
3	22.28	0.42	19.4	TWIP
4	22.18	0.52	23.0	TWIP
5	22.07	0.60	26.3	TWIP
6	22.03	0.73	27.4	TWIP
7	23.77	0.60	33.5	TWIP

Saeed-Akbari et al. [12] made a comparison of the iso-SFE maps calculated on the basis of Dew-Hughes and Kaufman [13,14] and Dinsdale [15] at 300 K. Large negative SFE values were found in the range of 10–35 wt% manganese and 0–1.2 wt% carbon when SGTE data [15] were selected. However, in this work, the composition and temperature-dependent SFE calculated based on the SGTE database [15] were found to be coincident with previous modeling and experimental results. It could be inferred that both the thermodynamic calculation method, such as the regular solution model [3,5,6], subregular solution model [12], Bragg-Williams model [16,17,18], and two-sublattice model [19], and the combined parameters [13,14,15] are of influential importance for the modeling accuracy when predicting SFE.

4 SFE variation against temperature and grain size

The dependency of SFEs on temperature and grain size in the Fe–Mn–C system is of significant influence in investigating the deformation mechanism. Based on the thermodynamic model developed in Part 2, the carbon- and manganese-dependent SFE maps at different temperatures are presented in Figure 8. As temperature increases from 300 to 550 K, the SFE of the concerning system increases obviously. A similar variation of SFE in this composition range was demonstrated by Allain et al. [4] and Saeed-Akbari et al. [12]. For a given Fe–22Mn–0.6C austenite steel, the SFE increases from 26 to 70 mJ·m⁻² in the above temperature increasing range. This matches the calculated result by Allain et al. [4]. In a new study by Bleck [61], the SFE for Fe–22Mn–0.6 C steel at 300 K is about 24 mJ·m⁻², and it increases to 65–70 mJ·m⁻² at 550 K, following the prediction by the present model. Mosecker et al. [62] investigated the different deformation mechanisms in austenite steels and revealed that SFE increases monotonically with temperature increasing, totally in line with the data in this work.

Figure 8

Carbon and manganese-dependent SFE maps at different temperatures: (a) 300 K, (b) 350 K, (c) 400 K, (d) 450 K, (e) 500 K, and (f) 550 K.

Figure 9 shows the carbon- and manganese-dependent SFE maps with different grain sizes. A limited effect of grain size on SFE has been observed. As the grain size in Fe–Mn–C alloys increases from 1 to 70 μm, the SFE gradually decreases. However, the SFE sensitivity to grain size obviously drops when the average diameter exceeds 70 μm. Saeed-Akbari et al. [12] illustrated that the SFE decreased monotonously with grain size increasing between 5 and 50 μm. This is largely in line with the present results. However, for the Fe–Mn–C steels with more coarse-grained structure, the negligible influence of grain size on SFE has been found. For a given Fe–22Mn–0.6C steel, the SFE is about 37 mJ·m⁻² for the matrix with a grain size about 1 μm, while it is around 24 mJ·m⁻² for the coarse grain sized 70 μm. Lee et al. [63] measured the SFE in Fe–18Mn–0.6C steel with grain sizes of about 10 and 3 μm, and the results were 14 and 27 mJ·m⁻², respectively. Considering the experimental error, the smallest and largest difference is about 10 and 20 mJ·m⁻², largely consistent with the present results. An interesting phenomenon should be noticed that the twinning behavior can refine the original austenite grain. It means the grain size decreases gradually during deformation when the TWIP effect occurs, and the SFE increases accordingly, suppressing the further activation of twins. It is found that the critical stress for twins increases as grain size decreases [64]. The variation in SFE is also an explanation for the ceases or saturation of twins in the austenite steel.

Figure 9

Carbon and manganese-dependent SFE maps at 300 K with different grain sizes: (a) 1 μm, (b) 10 μm, (c) 30 μm, (d) 50 μm, (e) 100 μm, and (f) 200 μm.

5 Conclusions

The SFE models of Fe–Mn and Fe–Mn–C systems were thermodynamically analyzed in this work. The validation of model prediction by previous experimental results was also carried out. The main conclusions can be drawn as follows:

1. A new thermodynamic model to predict the SFE of Fe–Mn and Fe–Mn–C austenite steels based on the two-sublattice method has been developed. The calculated Gibbs free energy and transformation temperature during γ → ε , and SFE as functions of temperature and composition are coincident with the modeling and experimental values in previous studies. The interfacial energy during γ → ε transformation was determined by a parabolic function of Gibbs free energy change, which is indirectly temperature and composition dependent.

2. The interstitial atoms segregation and the resulting SFE_seg variation in Fe–Mn–C steels have been revealed. The carbon segregated concentration in the stacking faults is agreed well with the measured value. The Suzuki effect induced SFE_seg decreases slightly with carbon concentration increasing, while the total SFE increases in the concerning composition range. The SFE in the Fe–Mn–C ternary system was found to be significantly dependent on the bulk Gibbs free energy change, followed by that of the stacking faults regions.

3. The carbon- and manganese-dependent SFE maps have been plotted under different temperatures and grain sizes. For the steels with carbon and manganese range between 0–1.2 and 10–30 wt%, respectively, the SFE increases with temperature increasing. The SFE of the Fe–Mn–C system decreases as grain size increases from 1 to 70 μm. However, for the alloys with a more coarse structure, the sensitivity of SFE to grain size was not obvious.