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. 2013 Mar 15;14(1):014207. doi: 10.1088/1468-6996/14/1/014207

A thermo-mechanical correlation with driving forces for hcp martensite and twin formations in the Fe–Mn–C system exhibiting multicomposition sets

Jinichiro Nakano 1,✉
PMCID: PMC5090573  PMID: 27877555

Abstract

The thermodynamic properties of the Fe–Mn–C system were investigated by using an analytical model constructed by a CALPHAD approach. The stacking fault energy (SFE) of the fcc structure with respect to the hcp phase was always constant at T0, independent of the composition and temperature when other related parameters were assumed to be constant. Experimental limits for the thermal hcp formation and the mechanical (deformation-induced) hcp formation were separated by the SFE at T0. The driving force for the fcc to hcp transition, defined as a dimensionless value –dGm/(RT), was determined in the presence of Fe-rich and Mn-rich composition sets in each phase. Carbon tended to partition to the Mn-rich phase rather than to the Fe-rich phase for the compositions studied. The results obtained revealed a thermo-mechanical correlation with empirical yield strength, maximum true stress and maximum true strain. The proportionality between thermodynamics and mechanical properties is discussed.

Keywords: 10.07

Keywords: stacking fault energy, high manganese steel, TWIP, TRIP, shape memory alloys

1. Introduction

Fe–Mn base austenitic alloys have been the subject of various studies because of their superior mechanical properties useful to many industrial applications [9, 21, 22, 27, 39, 40]. These alloys exhibit considerable strength and ductility depending on deformation modes that the alloys undergo. The high work hardening rate of these alloys is related to the formation of deformation twins in the fcc matrix as well as bct (α′) and hcp (ε) martensitic phases. It has been discussed that the stacking fault energy (SFE) may indicate the magnitude of dislocation dissociations into Shockley partials, affecting the subsequent work hardening rate. Alloys with low SFE tend to form mechanical (deformation-induced) hcp martensite and those with high SFE exhibit only dislocation glide; mechanical twining occurs for intermediate SFE values [3, 32]. A mechanical correlation with the thermodynamic property can be rationalized as a stacking fault comprises an embryo of hcp stacking or a twin. Correlations of the SFE values with deformation modes, i.e. the formation of hcp martensite, twin or only dislocation glide, have been extensively reported [2, 3, 14, 19, 32, 33]. The SFE values may not be the sole parameter responsible for such deformation behaviors [6].

In this work, an analytical model for the hcp phase in the Fe–Mn–C system constructed by a calculation of phase diagrams (CALPHAD) technique [26] was used to investigate correlations between mechanical and thermodynamic properties. The SFE of the fcc phase with respect to the hcp phase in the Fe–Mn–C system was evaluated with an emphasis on the alloys’ mechanical properties. Thermo-Calc software (version S) [4] was used for the calculations with the proposed thermodynamic parameters. Peculiarities associated with other related thermodynamic properties, such as T0 and driving force for the hcp transformation, were discussed along with the mechanical properties of alloys with different compositions.

2. Calculation method

2.1. Gibbs free energy of the hcp and fcc phases in the Fe–Mn system

Only the fcc and hcp phases were considered in this study. Thermodynamic parameters for the hcp phase were taken from [26] and those for the fcc phase were adopted from the work of Huang [18], where the stable phases in the Fe–Mn system are well assessed. The lattice stability values of the hcp and fcc phases of Fe and Mn are taken from Scientific Group Thermodata Europe (SGTE) [11]. Both phases are treated as solutions with random mixing in two sublattices, i.e. (Fe, Mn)1(Va)1 for fcc and (Fe, Mn)1(Va)0.5 for hcp, where Va is the vacancy. The molar Gibbs energy of phase ϕ (=fcc or hcp) is expressed by

2.1.

where x is the mole fraction of a component, 0Gx is the lattice stability of the phase of element x, R is the gas constant and T is the temperature. The excess term is described by the Redlich–Kister polynomial [31]

2.1.

The interaction parameter L in equation (2) is temperature dependent:

2.1.

where a, b, c, … are the parameters determined in the previous work [26]. The magnetic order–disorder transition modifies the Gibbs energy of a phase and thus the relative stability of the phases. By practically simplifying Inden's model [20], Hillert and Jarl [16] described the magnetic contribution to the Gibbs energy as

2.1.

where β is an average magnetic moment per atom in Bohr magnetons. The last term is a function of τ (=T/Tc), with Tc being the magnetic transition temperature. Details of the magnetic contribution to the Fe–Mn system are described elsewhere [26].

2.2. Stacking fault energy

Following the model of Olson and Cohen [28], the SFE is calculated as

2.2.

where n is the number of planes (assumed to be 2), ρ is the atomic density of the {111} planes, Inline graphic is the Gibbs free energy difference between the hcp and fcc phases, ΔGstrain is the strain energy and σ is the energy at the hcp/fcc interface. The atomic density ρ is of the order of 10−5 m2 [3, 24]. According to Cotes et al [10], ΔGstrain exhibits minimal composition dependence for 0–28 wt% Mn and it was therefore assumed to be 37 J mol−1 in this study; σ = 0.016 J m−2 was assumed following previous work [26].

3. Results

3.1. Evolution of stacking fault energy

Figure 1 shows a variation of SFE with respect to Mn and C contents along with experimental data [29, 41, 42]. The SFE values decrease at small Mn contents and increase with higher amounts of Mn, indicating the presence of an SFE minimum in the system. A carbon addition moves the minimum to the Mn-richer side, which modifies a trend in the SFE evolution with the same amount of Mn in the alloy. Even in trace amounts, carbon exhibits a substantial effect on increasing the SFE where larger effects are expected for lower Mn contents. For example, an addition of 2 wt% C to Fe–35Mn raises the SFE by only 23 mJ m−2, whereas the same amount of carbon increases the SFE by 59 mJ m−2 in the Mn-free Fe–C system. With the carbon addition alone, there are already three potential composition sets to be considered in each phase. Evolutions of the SFE in higher-order systems with additions of Al, Cr, Si, etc may not follow a constant trend like that seen in the Fe–Mn–C system due to multidimensional miscibility gaps. Modeling of SFE in such higher-order systems is under way.

Figure 1.

Figure 1.

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Calculated SFE versus Mn content in Fe–Mn–C alloys at 298 K. Experimental SFE values [29, 41, 42] are also shown for different C contents.

3.2. Effect of grain sizes and magnetic transition on SFE

The SFE is generally measured by taking a separation of partial dislocation nodes, which is inversely proportional to the SFE [35]. Small impurities may affect SFE to some extent [30]. If one assumes that carbon and other elements or impurities are pinned at grain boundaries during grain growth, grain sizes of austenite may influence the SFE as the local lattice concentration is also affected. Inline graphic of the bulk fcc material is also influenced by the grain size. If Inline graphic of austenite decreases with increasing austenite grain size, the Gibbs free energy of the fcc phase with respect to the hcp phase increases, consequently reducing the SFE. However, such a grain boundary effect must be very subtle, especially for decent grain sizes.

The magnetic transition temperature for the fcc solution in Fe–Mn is composition dependent [3, 8, 18, 23, 45], and can be expressed as

3.2.

where Tcϕ is the magnetic transition temperature of the phase ϕ in a Fe–Mn solution, TcFeϕ is the magnetic transition point of the phase ϕ in pure Fe and exTcFe, Mnϕ is the excess temperature term from the interaction between Fe and Mn. Further details can be found in [26]. The calculated Inline graphic values are shown in figure 2. Mn increases the critical temperature for magnetic transition, while C has an opposite effect. The C effect on Inline graphic is relatively small compared to that on T0. Below T0, hcp martensite can transform from the fcc matrix on cooling, which would not occur above T0 where formation of hcp martensite is only possible by mechanical deformation. Figure 2 indicates separations between magnetically ordered or disordered phases, above T0 or below it. The addition of carbon effectively suppresses the thermal formation of hcp martensite while it stabilizes the paramagnetic fcc phase. The heat evolved from the application of external stress (e.g. rolling) moves the system into a different regime in figure 2. For example, an Fe–25Mn alloy is located in the ‘antiferromagnetic fcc + thermal hcp’ region at room temperature, but it moves into a region where thermal hcp is suppressed and the mechanical hcp formation is possible. Furthermore, the fcc phase is magnetically disordered in this region, increasing the relative stability of the hcp phase.

Figure 2.

Figure 2.

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Effect of carbon on the Néel temperature Inline graphic of the fcc phase along with the T0 in Fe–Mn–C alloys.

3.3. SFE along T0 temperature

Figure 3 shows iso-SFE contours versus Mn and C concentrations in the Fe–Mn–C system. The SFE along T0 never crosses the iso-SFE contours and is always constant. When the Gibbs free energy of the hcp phase is equal to that of the fcc phase, Olson and Cohen's equation suggests a constant SFE by assuming other parameters to be constant. The constant SFE along T0, determined as 32.7 mJ m−2, is independent of the composition and temperature studied as long as changes in the atomic density, strain energy and surface energy are negligible.

Figure 3.

Figure 3.

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Iso-SFE contours in Fe–Mn–C alloys at 298 K showing constant SFE along T0, which separates the limits of mechanical and thermal hcp formations. These experimental limits were reported by Schuman [36].

The calculated T0 values are plotted together with experimental Ms (hcp martensite start) and As (austenite start) temperatures in figure 4 [1, 5, 9, 12, 13, 15, 21, 32, 37, 38, 39, 40]. Thermodynamic parameters to calculate T0 were adopted from [26] where the parameters were optimized over T0 temperatures determined by averaging the experimental Ms and As temperatures. Note that some data points in figure 4 were reported independently rather than as Ms–As pairs. Because Ms is always located below T0, Ms is always located at lower Mn than the Mn concentration at T0. If one assumes that Ms is located about 50 °C below T0, then the Mn concentration gap between Ms and T0 varies depending on the Mn content in the Fe–Mn system. T0 drops more rapidly below Inline graphic as the fcc phase becomes more stable due to the magnetic transition from paramagnetism to antiferromagnetism. The difference between the Ms concentration and the T0 concentration is approximately constant above Inline graphic. As the alloy's Mn concentration approaches T0, this difference becomes smaller and, therefore, Ms becomes closer to T0 below Inline graphic. Inline graphic in Fe–Mn is located at approximately 15 wt% Mn at 298 K and only a few per cent increase is predicted with the addition of 1.2 wt% C. A part of the trend can be seen in an empirical limit for the thermal hcp formation from the fcc phase reported by Schumann [36]. The T0 SFE seems to separate the thermal limit and the mechanical limit. As external stress is applied and/or heat is evolved from the stress, the T0 SFE shifts to higher concentrations due to the higher molar volume of the fcc structure. This shift increases the driving force to facilitate the hcp martensite or twin formation even at temperatures above T0. For C-rich Fe–Mn–C alloys, on the other hand, the T0 SFE moves to the lower Mn concentration if they are located on the Mn-poor side of the SFE minimum.

Figure 4.

Figure 4.

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Calculated T0 and experimental Ms and As temperatures.

3.4. Driving force for the transformation to the hcp phase

The driving force for the diffusionless transformation, determined at Ms and As, is not constant in the Fe–Mn system [26]. The driving force for the martensitic transition at Ms decreases with decreasing transformation temperature while that for the reverse transformation at As increases with decreasing transition temperature. Less undercooling may be required for an hcp martensitic transition from the fcc phase at high Mn content, while the hcp to fcc transition shows the opposite trend. Ms temperatures are thus located closer to T0 temperatures at Mn contents greater than the Inline graphic intersection, whereas As temperatures are further away from T0 temperature at higher Mn, depending on the locations relative to the Mn concentration where a magnetic transition occurs. As a result, the driving force for the fcc to hcp transition tends to increase with temperature (i.e. lower Mn), while that for the reverse transition tends to decrease with temperature (lower Mn). A sufficient amount of the positive driving force is required to trigger the formation of hcp martensite, which means the chemical composition of the alloy at Ms temperatures must be located sufficiently below T0. The diffusionless formation of hcp martensite may still be mechanically induced for alloys at temperatures above T0 if enough stress is applied to increase the relative difference in the Gibbs free energy, thus shifting the T0 up. If the Mn content for the shifted T0 is greater than that in the alloy, a positive driving force can be obtained.

An alloy composition should be intact during the martensitic transformation. The driving force for the martensitic transition is, therefore, calculated by taking the difference between the Gibbs energies of the two phases at the alloy composition. The driving force determined this way exhibits no clear correlation with mechanical properties of interest. Twinning induced plasticity (TWIP) steels, on the other hand, are strengthened by the formation of deformation twins in the fcc matrix. If the twins are assumed to be embryos of the hcp phase, then their formation is related to the driving force for the hcp transition. Recent studies related the serrations on a stress–strain curve for TWIP steels to the Portevin–Le Chatelier effect promoted by diffusing solutes interacting with dislocations [7, 44]. The fcc alloy is not thermodynamically stable if a composition separation is expected at equilibrium. The alloy is, however, stable if no such separation is possible to further minimize the energy. The difference between the lowest energy for the fcc phase and the corresponding energy value for the hcp phase determines the globally stable driving force for the hcp transformation. In this work, the driving force was determined by taking the lowest energies of the two phases at the alloy composition, which lie on the equilibrium planes drawn through all the composition sets if available. Note that this driving force should not be confused with that for a complete precipitation, which is determined by the difference between the energy of the matrix phase and that of the equilibrium tangent as each phase partitions to a new equilibrium composition. The present driving force is the energy difference between the two phases taken at the most stable stages independent of each other, which is determined at the alloy composition. In the present study, phase separations are hypothetically assumed to estimate the lowest possible energy of each phase.

With the addition of carbon to the Fe–Mn system, there are potentially three composition sets to consider for each phase. In the Fe–Mn–C system with up to 35 wt% Mn and 1.2 wt% C, the hcp and fcc phases both separate into an Fe-rich phase and a Mn-rich phase. It must be emphasized that, for these phases, the two composition sets occurred only in the presence of carbon in the system. No phase separation was noted in the Fe–Mn binary system. Carbon tended to partition to the Mn-rich phase rather than to the Fe-rich phase across the compositions studied. The present driving force is determined as the dimensionless Gibbs energy difference (-dGm/(RT)), which is plotted against Mn and C contents in figure 5. –dGm/(RT) is negative for Mn-rich systems, and rapidly turns positive with increasing carbon concentration. For example, the Fe–35Mn alloy (SFE = 45 mJ m−2) falls into a negative -dGm/(RT) region, but the Fe–30Mn–0.5C alloy falls into a positive -dGm/(RT) region, although it has the same SFE. If a sufficiently positive driving force is needed to form deformation-induced hcp martensite or twin, Mn and C contents can be adjusted accordingly at production. This result may explain previous reports [6, 17, 43] where deformation twins were either present or absent in alloys having similar SFE values. Bouaziz et al [6] compared Fe–30Mn and Fe–22Mn–0.6C alloys with similar SFE values where, after deformation, extensive mechanical twinning was observed in the Fe–22Mn–0.6C alloy, whereas no twins were found in the Fe–30Mn alloy (figure 6). The confusion with the conventional SFE concept may be attributed to the -dGm/(RT) value, which is negative for the Fe–30Mn alloy (-0.1) and positive to a large extent for the Fe–22Mn–0.6C alloy (+0.4).

Figure 5.

Figure 5.

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− dGm/(RT) versus C content at 298 K in Fe–Mn–C alloys. Pairs of different alloys of the same SFE may have positive (Fe–30Mn–0.5C) or negative (Fe–35Mn) −dGm/(RT) values.

Figure 6.

Figure 6.

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Transmission electron microscopy (TEM) images showing distinct deformation structures: (a) dislocation cell structure developed in Fe–30Mn and (b) deformation twinning observed in Fe–22Mn–0.6C [6].

Figures 7–9 show the empirical maximum true stress, maximum true strain and yield strength [6, 19, 25, 34] as a function of -dGm/(RT). The maximum stress and strain here correspond to sample failure in tensile tests. The Mn and C contents in collected data varied from 12 to 30 wt% and from 0 to 1.2 wt%, respectively. A positive correlation exists between these values for the given experimental data. This proportionality between mechanical and thermodynamic properties may be related to the increased thermodynamic stability of the hcp relative to fcc phase corresponding to -dGm/(RT), which contributed to the formation of deformation-induced twins upon introduction of sufficient stress. A general increase in strain with carbon content implies a correlation of the alloy strengthening with twinning. The increasing trends seem to be minimized toward points where -dGm/(RT) = 0 (dashed curves). It is, however, not certain from the available data whether the trend is continuous into the negative -dGm/(RT) region (dotted curves). It should be noted that Fe–Mn alloys with low Mn (<15 wt.%) have low SFE but high -dGm/(RT) values. Undercooling below the phase boundary into the ‘fcc + hcp region’ (or even T0) becomes sufficiently large, and the thermal hcp phase may already be activated on cooling, provided that the Mn content is not low enough for bct martensite to be a dominant phase. As a result, the proportionality found in these figures is expected to rapidly fall for low-Mn Fe–Mn alloys without the addition of carbon.

Figure 7.

Figure 7.

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Experimental maximum true stress versus -dGm/(RT). A dotted curve shows a trend across all the compositions, while a dashed curve indicates a trend over alloys with positive -dGm/(RT).

Figure 9.

Figure 9.

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Experimental yield strength versus -dGm/(RT). A dotted curve shows a trend across all the compositions, while a dashed curve indicates a trend over alloys with positive -dGm/(RT).

Figure 8.

Figure 8.

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Experimental maximum true strain versus -dGm/(RT). A dotted curve shows a trend across all the compositions, while a dashed curve indicates a trend over alloys with positive -dGm/(RT).

4. Conclusions

SFE in the Fe–Mn–C system was calculated using the model developed by a CALPHAD approach. SFE is always constant (32.7 mJ m−2) at T0, independent of composition and temperature, provided that the other parameters are intact. The experimental limits for the thermal hcp formation and the mechanical hcp formation were separated by the SFE determined at T0. The driving force for transition to the hcp phase was determined as -dGm/(RT), the system value with consideration of multidimensional miscibility gaps. The proportionality of experimental maximum true stress, maximum true strain and yield strength to -dGm/(RT) implied the need for positive -dGm/(RT) to promote the formation of mechanical twins.

Acknowledgments

This technical effort was performed in support of the National Energy Technology Laboratory's ongoing research into the development of coal gasification under RES contract no. DE-FE0004000. Valuable discussions with Mr James Bennett and Dr Kyei-Sing Kwong of NETL are gratefully acknowledged.

Disclaimer. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

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