Ab Initio-Based Modelling of the Yield Strength in High-Manganese Steels

Abstract

An ab initio-based model for the strength increase by short-range ordering of C-Mn-Al clusters has been developed. The model is based on ab initio calculations of ordering energies. The impact of clusters on the yield strength of high-manganese austenitic steels (HMnS) is highly dependent on the configurational structure of the cells that carbon atoms will position themselves as interstitial atoms. The impact of the alloying elements C, Mn, and Al on the potential and actual increase in yield strength is analyzed. A model for the calculation of yield strengths of HMnS is derived that includes the impact of short-range ordering, grain size refinement, and solid solution strengthening. The model is in good agreement with experimental data and performs better than other models that do not include strengthening by short-range ordering.

1. Introduction

High Mn austenitic steels (HMnS) offer great combinations of hardness and ductility with a wide range of possible applications in various industries, such as the automotive industry [1,2,3]. Depending on the stacking fault energy, which is a result of alloying content, temperature, deformation grade, and other factors, HMnS exhibit a variety of different active deformation modes [4,5,6,7]. Those deformation modes include transformation induced plasticity (TRIP), twinning-induced plasticity (TWIP) or pronounced planar glide in combination with the formation of dislocation cells that is sometimes referred to as microband-induced plasticity (MBIP) [1,8,9,10]. The mode of deformation significantly influences the level of work hardening and plays a major role in determining the mechanical properties [7]. One of those material characteristics is the material’s yield strength (YS). In the past, various attempts have been made to model the YS of HMnS based on the chemical composition using linear regression [5,11,12,13]. However, these models often imply negative solid-solution strengthening by some of the alloying elements, like Mn, that are usually attributed to positive strengthening effects [2,14,15].

Interactions of carbon-metal pairs [16,17,18] and clusters are described by the concept of short-range ordering (SRO) phenomena [19] in HMnS and might offer an explanation for an additional increase in strength that might lead to a defective linear regression when using standard methods to calculate the impact of solid solution strengthening [14,20,21]. Short-range ordered clusters are a result of local differences in ordering energies that make particular unit cell configurations preferential to inhibit a carbon atom [22]. Ab initio calculations employing density-functional theory (DFT) have been shown to be suitable for calculations of ordering energies in HMnS for several chemical compositions. For example, thermodynamic predictions of κ-phase stability [23], interface energies [24], stacking fault energies [25], and hydrogen trap sites [26] showed good agreement with the experimental data.

Cell models [27] have been used to determine the location of carbon atoms in relation to the matrix or substitutional alloying atoms. Other studies show that the interaction of carbon atoms with each other is very repulsive and, thus, influences the diffusion behavior [4,25,28,29,30], which also points to a coordination of carbon atoms in octahedral sites dependent of surrounding solute atoms. Additionally, the position of C also influences the stacking fault energy of the material [31].

Short-range ordering has also been used to explain the pronounced planar glide that HMnS exhibit because of glide plane softening that occurs when short-range ordered C-Mn clusters are destroyed by dislocation stresses reorienting the carbon atom from its octahedral position to a tetrahedral position. This phenomenon also explains the serrated flow curves that can be found during tensile testing of HMnS, as the back stresses of dislocation pile-ups lead to an increase in strength. The reorientation of the carbon atom then leads to a temporary destruction of the short-range ordered cluster and as a result, a rapid drop in strength because of subsequent softening of this glide plane [32]. However, short-range ordering can be restored by short-range jumps of C back from the tetrahedral site to its initial octahedral site [1,2]. Those short-range jumps might be activated by temperature, but already occur at low temperatures where diffusion processes are usually negligible [1,32].

In this study, a basic cell model for the nearest-neighboring atoms of a central octahedral carbon atom is expanded to include the corner atoms of an elementary cell of an austenitic face-centered cubic (FCC) unit cell, which can be described as the second-nearest neighbors of the central octahedral site in an austenitic cell [27]. Results of ab initio calculations are employed to show that the position of Mn and Al being coordinated as a nearest neighbor or second-nearest neighbor to a central carbon atom is of major importance to the energy level of the system [33]. It is assumed that this also holds true in the case of a central interstitial nitrogen atom [2,33,34] but, in this work, the authors will focus solely on carbon. Based on the calculated ordering energies, the yield strength of HMnS is a result of a combination of short-range ordered clusters, regular solid-solution hardening, grain size effects, and the base strength of an austenitic lattice [14,35].

2. Materials and Methods

Many models for the calculation of the yield strength of austenitic steels have been published in the past. Those models generally employ two mechanisms to describe the yield strength: a strengthening by grain size reduction, the Hall-Petch effect, and the alloying elements’ contribution to the material’s strength [15,36,37]. Many of those models have used databases with significant amounts of different alloys to employ a linear regression in order to get parameters that determine the effect of alloying elements on the yield strength of the austenitic steel. Assuming that SRO in HMnS influences the yield strength ${\mathsf{\sigma }}_Y$ significantly, a term describing this effect has to be added to the equation [14], resulting in:

$${\mathsf{\sigma }}_Y{=\mathsf{\sigma }}_0{+\mathsf{\sigma }}_{SS}{+\mathsf{\sigma }}_{GS}{+\mathsf{\sigma }}_{SRO}$$

(1)

where ${\mathsf{\sigma }}_0$ is the base strength of a pure austenitic iron lattice and is assumed to be $90\mathrm{MPa}$ [14]. ${\mathsf{\sigma }}_{SS}$ is the contribution of alloying elements to the change in yield strength relative to a pure austenitic lattice. Since this paper focuses on HMnS with the main alloying elements C, Mn, and Al, those elements are assumed to be the main contributors to solid solution strengthening and the contributions of other alloying elements are neglected. It is assumed that this does not lead to a large mistake in calculating ${\mathsf{\sigma }}_{SS}$ because, apart from Mn and Al, no substitutional alloying element exceeds alloying contents of more than $1\mathrm{wt}.\%$.

${\mathsf{\sigma }}_{SRO}$ is the change in yield strength stemming from the existence of short-range ordered regions in the material. Based on the work of [14], this contribution to yield strength can be calculated as:

$${\mathsf{\sigma }}_{SRO}{=M\mathsf{\tau }}_{SRO}=M\left(\frac{E_{Order}}{b^3}\right)=M\left(\frac{E_{random}{-E}_{SRO}}{b^3}\right)$$

(2)

where $M=3.06$ is the Taylor-factor for fcc with $b=\frac{{\mathsf{\alpha }}_{\mathsf{\gamma }}}{\sqrt{2}}$ being the magnitude of the Burgers vector and the lattice parameter in austenite in Ȧ depending on the molar fractions $x_i$ of the alloying elements and temperature in K [38]:

$${\mathsf{\alpha }}_{\mathsf{\gamma }}=\left(\begin{array}{c}3.5780+{0.033x}_C+{0.00095x}_{Mn}-{0.0002x}_{Ni}\\ +{0.0006x}_{Cr}+{0.0056x}_{Al}+{0.0031x}_{Mo}+{0.0018x}_V\end{array}\right)\times \left(1+{2.065\times 10}^{-9}\times (\mathrm{T}-300)\right)$$

(3)

Looking at Equation (3), the temperature dependence of the lattice parameter is assumed to be linear.

${\mathsf{\sigma }}_{GS}$ describes the strengthening by grain refinement in accordance with the Hall-Petch effect and can be formulated as:

$${\mathsf{\sigma }}_{GS}{=k}_y{d}^{-1/2}$$

(4)

Even though $k_y$ depends on the cooling procedure and the carbon content [39], the value $k_y=11.3\mathrm{MPa}{\mathrm{mm}}^{1/2}$ seems to be a reasonable assumption for HMnS in general, based on the findings in the literature [1,2,14,39,40,41].

As can be seen in Figure 1, one unit cell consists of 14 atoms situated on the lattice positions of the FCC bracket and one central octahedral interstitial site, where a single carbon atom can be situated.

First-nearest neighbors (1NN) are the six direct neighbors of the carbon atom. In Figure 1, they are shown in green. Second-nearest neighbors (2NN) in this model are the grey atoms on the corners of the FCC lattice. Although third-nearest neighbors (3NN) can be included in first-principle calculations, a recent study has shown that neither 3NN Al atoms nor 3NN Mn atoms have a positive impact on the ordering energy [33]. To reduce the complexity of the calculations and limit the amount of possible cell configurations, the presence of 3NN alloying atoms is not accounted for in the analysis carried out in this paper. Instead, it is assumed that the material is the sum of FCC-unit cells that do not impact each other. This is obviously not true in reality, but for the purposes of this study, which is aimed at providing a basic model to connect ab initio calculations with experimental results, it is suitable, since it leads to reduced mathematical complexities. The term “cell configuration” is used to describe the combination of alloying atoms and matrix atoms in the cell. “Coordination sphere” is used throughout this paper to refer to the position of atoms relative to the central carbon atom.

The configurations of the cells based on the alloying contents for this 14-cell model are calculated according to [27]:

$$n_{kl}=\frac{14!}{\left(14-k-l\right)!k!l!}{\mathsf{\theta }}_{Fe}^{(14-k-l)}{\mathsf{\theta }}_{Mn}^k{\mathsf{\theta }}_{Al}^l$$

(5)

where $n_{kl}$ is the fraction of all cells that have a configuration with k Mn atoms and l Al atoms with ${\mathsf{\theta }}_i$ being the molar fraction of Fe, Mn, and Al, denying the existence of other substitutional alloying elements and ignoring C, since carbon is an interstitial alloying element. Since diffusion rates of substitutional atoms in austenitic steels are very low, it is assumed that the distribution of substitutional elements according to Equation (5) is fixed and that the carbon atom is distributed in accordance with a minimum ordering energy. For clusters of nitrogen and chromium, this approach has been shown to be a realistic approximation, since the presence of nitrogen does not negatively influence the homogeneous distribution of chromium [34]. Cells of type $n_{kl}$ that contain a carbon atom are classified as $n_{Ckl}$. It should be noted that the assumption that substitutional atoms are distributed according to minimum ordering energies with respect to the presence of a central carbon atom is a very strong assumption, which is unlikely to be true in reality. However, without this assumption, the calculation of ordering energies $\Delta E_{Ckl}$ is not possible since ab initio results are presently not available for those unfavorable configurations.

The ordering energy $\Delta E_{Ckl}$ of a cell with k Mn atoms, l Al atoms and a central carbon atom is the combination of two energy contributions in comparison to a FCC-unit cell consisting only of Fe atoms and a central interstitial carbon atom. First, the energy required to bring a carbon atom in solution $\Delta E_{Sol}$ in a FCC-unit cell with respect to the presence of k Mn atoms and l Al atoms in the first two coordination spheres. The second factor is the change in energy level $\Delta E_{Lat}$ resulting from a lattice distortion due to the presence of Mn and Al:

$$\Delta E_{Ckl}=\Delta E_{Sol}+\Delta E_{Lat}$$

(6)

Ab initio calculations show that Al is preferentially situated in the second coordination sphere and Mn has a positive effect on the ordering energy level if it sits on a lattice position as a 1NN [22,25,33]. According to those studies, this means that the difference in ordering energy is so profound that the likelihood of a cell configuration that contains 1NN Al atoms instead of 2NN Al atoms or 2NN Mn atoms instead of 1NN Mn atoms is very low. Accordingly, for further calculations in this work and to reduce the model’s complexity, it is assumed that for any number of Mn atoms up to six, they will always be 1NN and for any number of Al atoms up to 8, they will always be 2NN.

Using the calculated ordering energies $\Delta E_{Ckl}$ from [33], the six cells with the lowest ordering energies are listed in

Table 1. Cell configurations of lowest ordering energy $\Delta E_{Ckl}.$

Cell Name ${\mathit{n}}_{\mathit{C}\mathit{k}\mathit{l}}$	k Mn Atoms	l Al Atoms	$\mathbf{\Delta }{\mathbf{E}}_{\mathit{C}\mathit{k}\mathit{l}}\left[\mathbf{eV}\right]$
$n_{C\mathit{08}}$	0	8	−1.2867
$n_{C\mathit{18}}$	1	8	−1.1524
$n_{C\mathit{28}}$	2	8	−1.0559
$n_{C\mathit{07}}$	0	7	−0.9304
$n_{C\mathit{68}}$	6	8	−0.9287
$n_{C\mathit{38}}$	3	8	−0.9233

and shown in Figure 2.

In Equation (2), $E_{random}$ describes the energy state of an unordered alloying system. In this case, a single carbon atom is situated in the central octahedral sites of a randomly chosen FCC-unit cell. This distribution is based on the assumption that there are no preferential octahedral sites, so neither the presence of Mn nor Al influences the position of carbon atoms. Thus, it can be assumed that, for the calculation of $E_{random}$, the likelihood of the presence of carbon is the same for each type of cell.

Using the random C distribution for $E_{random}$, $E_{random}$ is calculated according to:

$$E_{random}={\displaystyle \sum }_{k=0}^{14}{\displaystyle \sum }_{l=0}^{14}{n}_{Ckl,random}\Delta E_{Ckl},\mathrm{for}k+l\le 14\mathrm{and}k,l\ge 0$$

(7)

$$n_{Ckl,random}{=n}_{kl}{\times \mathsf{\theta }}_C$$

(8)

Since ${\displaystyle \sum }{n}_{kl}=1$, this method of determining $n_{Ckl,random}$ ensures that carbon is distributed homogeneously in the material, as well as ensuring that every available carbon atom is situated in an octahedral site of a cell.

Contrary to the homogeneous distribution of carbon for $E_{random}$, the distribution of C for the calculation of $E_{SRO}$ is based on the calculated ordering energies [33]. Since the differences in ordering energy between different cell configurations are significant, as is shown in Table 1, it is assumed that carbon is always situated in the most favorable cell that has not already been assigned a carbon atom. Assuming that at higher temperatures, for example during recrystallization annealing, the diffusion of carbon as an interstitial alloying element is possible, carbon will diffuse to positions with minimal ordering energy in order to minimize the system’s Gibbs energy. This means that the cells selected for saturation with carbon are the cells with the lowest $\Delta E_{Ckl}$ until $n_{Ckl,SRO}{=\mathsf{\theta }}_C$.

The calculation of ordering energies in the model presented within this work was performed via ab initio simulations based on quantum-mechanical structure optimizations employing density-functional theory (DFT) within the Vienna ab initio Simulation Package (VASP) [33]. The quantum-mechanical structure optimizations are performed based on calculations of lowest-energy states of the electron structure and thus, are conducted at 0 K. Results from ab initio approaches employing DFT have offered good results that are in line with experimental evidence. For example, Mössbauer spectroscopy [17], atom-probe tomography (APT) [42], in situ synchrotron X-ray diffraction measurements [23], and correlative TEM/APT approaches [43] have shown that results from ab initio simulations employing DFT are applicable to systems containing nano-sized κ-carbides that are usually formed during isothermal annealing at temperatures of 600 °C and above. Generally speaking, the effect of temperature changes on the lattice parameter and on intrinsic material properties that derive from electron effects like electrical conductivity is often approximated to be linear [44]. The serrated flow curves of many high-manganese TWIP steels usually occur at temperatures of up to about 200 °C. At higher temperatures, this dynamic strain aging-like effect is either very small or is not detected at all [2,7].

In order to model the temperature dependence of ${\mathsf{\sigma }}_{SRO}$, the ab initio results are fitted using a linear approximation of a decrease in ${\mathsf{\sigma }}_{SRO}$ between 0 K and 500 K. For this purpose, it is assumed that the fact that no serrated flow curve is expected at 500 K can be interpreted as a result of no significant strength increase by short-range ordered clusters.

Based on those assumptions and equations, ${\mathsf{\sigma }}_{Y,calc}$ in MPa was calculated using Equations (2), (3), and (7) for 13 different HMnS with known average grain sizes $d_m$ in μm, listed in

Table 2. Chemical composition of HMnS used in this study in wt. %; grain size d_m in μm; experimentally determined and calculated yield strength σ_Y,exp and σ_Y,calc in MPa, experimental data of alloys marked with * are taken from [14].

Alloy	C	Mn	Al	${\mathit{d}}_{\mathit{m}}$	${\mathsf{\sigma }}_{\mathit{Y},\mathit{e}\mathit{x}\mathit{p}}$	${\mathsf{\sigma }}_{\mathit{Y},\mathit{calc}}$
X60MnAl17-1	0.56	17.00	1.35	8	328	356
X50MnAl15-1 *	0.49	15.40	1.30	27	294	281
X30MnAl22-1	0.33	22.46	1.21	18	277	275
X30MnAl17-1	0.32	16.80	1.47	17	255	276
X70Mn24	0.71	23.50	0.01	16	334	334
X60Mn23	0.57	23.21	0.01	10	328	334
X50Mn30 *	0.50	30.00	0.00	30	276	273
X50Mn22	0.54	21.95	0.00	14	328	311
X50Mn18 *	0.53	17.90	0.01	8	356	341
X30Mn28	0.28	28.00	0.00	9	291	294
X30Mn23	0.32	22.79	0.01	10	284	291
X30Mn22 *	0.31	22.28	0.00	3	381	384
X30Mn13 *	0.30	12.74	0.01	10	266	274

. The alloying contents are given in $\mathrm{wt}.\%$.

Experimental yield strength data was taken from the literature [5,14]. Additionally, quasistatic tensile tests have been carried out using a Zwick Z250 tensile testing machine of the AllroundLine by Zwick Roell (Ulm, Germany).

${\mathsf{\sigma }}_{SS}$ has been calculated using a linear approximation to explain the differences between experimental and calculated values for the yield strength using the following equation that is a direct result of Equation (1):

$${\mathsf{\sigma }}_{SS,exp}{=\mathsf{\sigma }}_{Y,exp}{-\mathsf{\sigma }}_{\mathsf{\gamma }-Fe}{-\mathsf{\sigma }}_{GS}{-\mathsf{\sigma }}_{SRO}$$

(9)

This way, ${\mathsf{\sigma }}_{SS,exp}$ was determined for all steels in Table 2. A regression analysis was performed with respect to the alloying contents of C, Mn, and Al.

The regression analysis lead to the following equation to calculate ${\mathsf{\sigma }}_{SS}$ with respect to the alloying contents ${\mathrm{\chi }}_i$ in weight percent:

$${\mathsf{\sigma }}_{SS}=84.55{\mathrm{\chi }}_C+1.35{\mathrm{\chi }}_{\mathrm{Mn}}+{0.44\mathrm{\chi }}_{\mathrm{Al}}$$

(10)

Overall, the lattice distortion effect of carbon is by far the strongest contribution to solid solution hardening. This is in accordance with calculations of the misfit of C in an octahedral site using the crystallographic correlation $\frac{{\mathrm{r}}_{\mathrm{OL}}}{{\mathrm{r}}_{\mathrm{Fe}}}=0.41$. Comparing the size of the octahedral site to the size of a carbon atom shows that a carbon atom is approximately 8% bigger than the octahedral site, leading to a lattice distortion. Using the lattice parameter calculated according to Equation (3), this size difference can be calculated to approximately 0.03 nm. Comparing this value with the lattice distortion effects by Al and Mn in an austenitic steel according to [45], which are $0.000065\frac{\mathrm{nm}}{\mathrm{wt}.\%\mathrm{Mn}}$ and $0.00095\frac{\mathrm{nm}}{\mathrm{wt}.\%\mathrm{Al}}$ respectively, the difference in contributions between C and Mn and Al in Equation (10) appear to be reasonable.

3. Results and Discussion

Figure 3 shows that the calculations for the yield strength ${\mathsf{\sigma }}_{Y,calc}$ using the proposed model show good agreement with the experimental data.

The calculations used in this work lead to differences between ${\mathsf{\sigma }}_{Y,calc}$ and ${\mathsf{\sigma }}_{Y,exp}$ that are smaller than 10%. Those small differences between ${\mathsf{\sigma }}_{Y,calc}$ and ${\mathsf{\sigma }}_{Y,exp}$ might be a result of a negligence of other alloying elements that participate in solid solution hardening, minor inaccuracies of the measurement of the alloying contents, or $d_m$.

Since ${\mathsf{\sigma }}_{SRO}$ can also be understood as a deviation from linear solid solution hardening behavior, a close relationship between ${\mathsf{\sigma }}_{SS}$ and ${\mathsf{\sigma }}_{SRO}$ seems logical. Higher alloying contents of C, Mn, and Al lead to a higher solid solution strengthening, but also to more carbon atoms integrated in Mn-C, Al-C, and Mn-Al-C clusters, as well as a higher amount of cells with more Al atoms in the second coordination sphere or Mn atoms in the first coordination sphere according to Equation (5).

The individual contributions of solid solution hardening ${\mathsf{\sigma }}_{SS}$ and ${\mathsf{\sigma }}_{SRO}$ are illustrated in Figure 4.

The strengthening effect of short-range ordered clusters ${\mathsf{\sigma }}_{SRO}$ is dependent on the chemical composition. For the steel compositions analyzed within the scope of this work, ${\mathsf{\sigma }}_{SRO}$ at room temperature is approximately in the range of 20 MPa for X30Mn28, containing 0.28 wt. % C and 28.00 wt. % Mn, and 65 MPa for X60MnAl17-1, containing 0.56 wt. % C, 17.00 wt. % Mn, and 1.35 wt. % Al.

Based on Figure 4, one could think, that the connection between ${\mathsf{\sigma }}_{SS}$ and ${\mathsf{\sigma }}_{SRO}$ is approximately linear. However, it needs to be kept in mind that the calculation of ${\mathsf{\sigma }}_{SS}$ is based on linear approximations as well. Additionally, it should be pointed out that although the contribution of ${\mathsf{\sigma }}_{SS}$ to ${\mathsf{\sigma }}_Y$ is always greater than that of ${\mathsf{\sigma }}_{SRO}$, the proportion between ${\mathsf{\sigma }}_{SS}$ and ${\mathsf{\sigma }}_{SRO}$ does not seem to follow a linear trend based on chemical composition. This is illustrated by the line in Figure 4 that represents a relation of $\frac{{\mathsf{\sigma }}_{SS}}{{\mathsf{\sigma }}_{SRO}}=1$. To further point out the differences in the contributions to ${\mathsf{\sigma }}_Y$, the individual contributions of ${\mathsf{\sigma }}_{\mathrm{GS}}$, ${\mathsf{\sigma }}_{SS}$, and ${\mathsf{\sigma }}_{SRO}$ are portrayed in Figure 5.

According to Figure 5, the contribution of ${\mathsf{\sigma }}_{SRO}$ to ${\mathsf{\sigma }}_Y$ is significant. The proportional contribution of ${\mathsf{\sigma }}_{SRO}$ to ${\mathsf{\sigma }}_Y$ is in the range of 7.1% for X30Mn28 up to 19.1% for X70Mn24. Since a grain size effect on ${\mathsf{\sigma }}_{SRO}$ is not assumed in this work, the impact of ${\mathsf{\sigma }}_{SRO}$ for a given steel composition with larger grain size is higher than for the same steel composition with finer grains. Correspondingly, the proportional contribution of ${\mathsf{\sigma }}_{SRO}$ that is portrayed in Figure 5 is higher for compositions with larger grains and thus, with smaller grain size strengthening effects ${\mathsf{\sigma }}_{GS}$.

The comparison of ${\mathsf{\sigma }}_{SS}$ and ${\mathsf{\sigma }}_{SRO}$ for the steels X30Mn13 and X30Mn22 could lead to the conclusion that an increase in Mn content does not significantly influence ${\mathsf{\sigma }}_{SRO}$. On the other hand, by comparing X30Mn22 with X30Mn28, it does appear that Mn actually reduces ${\mathsf{\sigma }}_{SRO}$. Accordingly, based on these calculations, no clear tendency of the impact of a single alloying element on ${\mathsf{\sigma }}_{SRO}$ can be found.

For a more detailed analysis, the impact of C and Mn on ${\mathsf{\sigma }}_{SRO}$ is portrayed in Figure 6 and Figure 7 with regards to the development of ${\mathsf{\sigma }}_{SRO}$, ${\mathsf{\sigma }}_{SS}$, $E_{random}$, and $E_{SRO}$ for steels with and without Al.

Figure 6 analyzes a steel with a constant Mn content of 22 wt. % with increasing C content from 0.0 wt. % up to 0.8 wt. %. Additionally, seven steels that were studied in this work were selected to illustrate the impact of changes in Mn content and Al content.

According to Figure 6a, ${\mathsf{\sigma }}_{SRO}$ and ${\mathsf{\sigma }}_{SS}$ increase approximately linearly with C content. This can be explained by an increasing number of cells with carbon atoms. In theory, this increase in ${\mathsf{\sigma }}_{SRO}$ will only occur as long as there are cells with a preferential structure available, which leads to a decrease in ordering energy $E_{SRO}$, as can be seen in Figure 6b. If no preferential cells were available, C atoms would have to be situated in non-preferential cells with ${\Delta E}_{Ckl}>0\mathrm{eV}$. To a small degree, this effect is included in the calculation of $E_{random}$ because C is assumed to be homogeneously distributed among all cells, including those with positive and, thus, unfavorable values of ${\Delta E}_{Ckl}$. This explains the lower rate of decrease of $E_{random}$ in Figure 6b compared to $E_{SRO}$. However, ${\mathsf{\sigma }}_{SRO}$ will always remain positive because, according to Equation (2), as long as $E_{random}$ is bigger than $E_{SRO}$, ${\mathsf{\sigma }}_{SRO}$ will be positive. This premise is always fulfilled by using the assumption in this work, that carbon will be situated in the cells of lowest possible ordering energy first and that the distribution of alloying elements for the calculation of $E_{random}$ and $E_{SRO}$ is the same.

Looking at Figure 6a, the steels with Al, X30MnAl22-1 and X30MnAl17-1, show positive deviations for the calculated values of ${\mathsf{\sigma }}_{SRO}$ and X30MnAl17-1 shows a negative deviation for ${\mathsf{\sigma }}_{SS}$. In the case of solid solution strengthening for X30MnAl17-1, this can be explained by the low Mn content of 16.80 wt. % compared to the 22.00 wt. % used as a basis of calculation of ${\mathsf{\sigma }}_{SS}$ in Figure 6. Accordingly, X70Mn24 and X60Mn23 show slightly higher values for than ${\mathsf{\sigma }}_{SS}$ it would be the case for 22.00 wt. % Mn.

The increase of ${\mathsf{\sigma }}_{SRO}$ can be explained by the impact of the introduction of new cell configurations containing Al atoms. According to Table 1, $n_{C\mathit{08}}$ is the lowest possible energy configuration. However, since the Al content of the steels in this study is very low, only very few cells with the configuration $n_{C\mathit{08}}$ are available. The majority of Al atoms is in cells also containing Mn atoms. According to [33], the introduction of Al atoms as a 2NN into cells with Mn atoms as 1NN does not necessarily lead to a lower ordering energy, but is rather dependent of the number of 1NN Mn atoms. The molar fraction of Mn in a steel containing 22 wt. % Mn obviously depends on the wt. % of the other alloying elements. For the X30MnAl22-1, the molar fraction of Mn is approximately $0.23$. According to Equation (5), the most common cell configuration is expected to be $n_{\mathit{30}}$. The ordering energy of the cell configuration $n_{C\mathit{30}}$ is $\Delta E_{C\mathit{30}}=-0.0501\mathrm{eV}$. This means that this cell configuration is preferential to a pure iron FCC-unit cell. Increasing the number of Al atoms in this cell configuration to 1 and 2 respectively results in $\Delta E_{C\mathit{31}}=-0.0082\mathrm{eV}$ and $\Delta E_{C\mathit{32}}=0.0338\mathrm{eV}$. Both cell configurations are not preferential with respect to $n_{C\mathit{30}}$ which according to previous assumptions means that the cells $n_{C\mathit{31}}$ and $n_{C\mathit{32}}$ will not contain a carbon atom as long as there are other more preferential cell configurations like $n_{\mathit{30}}$ that do not already contain a carbon atom. In turn, $E_{SRO}$ will not be negatively influenced by cell configurations like $n_{C\mathit{31}}$ and $n_{C\mathit{32}}$. However, $E_{random}$ will include the negative impact of unfavorable cell configurations which will lead to an increase of $E_{random}$ and a total increase of ${\mathsf{\sigma }}_{SRO}$.

Overall, it can be said that the positive impact of Al on ${\mathsf{\sigma }}_{SRO}$ is a result of an introduction of favorable cell configurations containing Al atoms that inhibit a carbon atom and the avoidance of cell configurations of unfavorable cell configurations.

Figure 7 analyzes a steel with a constant C content of 0.3 wt. % with increasing Mn content from 10 wt. % up to 30 wt. %. Seven steels with varying Al and C content were added to the diagram.

Contrary to the impact of C that is documented in Figure 6, a local maximum of ${\mathsf{\sigma }}_{SRO}$ at approximately 16 wt. % Mn is observed. Looking at Figure 7b, this can be explained by the stagnation of $E_{SRO}$ for Mn contents of 16 wt. % and higher. At the same time, $E_{random}$ is decreasing, leading to a smaller difference between $E_{random}$ and $E_{SRO}$ and accordingly to a lower ${\mathsf{\sigma }}_{SRO}$.

The stagnation of $E_{SRO}$ can be explained by considering the method of calculation for ${\mathsf{\sigma }}_{SRO}$. For Al-free steels, the energetically optimal cell is $n_{\mathit{60}}$. As long as ${\mathsf{\theta }}_C<n_{C\mathit{60}}$, an increase in the amount of C atoms will decrease $E_{SRO}$. This is due to the underlying assumption that C atoms will be situated in preferential cells of lowest possible ordering energy. At ${\mathsf{\theta }}_C{=n}_{C\mathit{60}}$, all C atoms are situated in cells of optimal configuration and thus, $E_{SRO}$ is minimized. Due to the way $E_{random}$ is calculated in Equation (7), $E_{random}$ continues to decrease at ${\mathsf{\theta }}_C>n_{C\mathit{60}}$. As a result, ${\mathsf{\sigma }}_{SRO}$ decreases.

In accordance with Figure 6a, Figure 7a shows that additions of Al and C increase ${\mathsf{\sigma }}_{SRO}$ significantly.

The contribution of solid solution hardening ${\mathsf{\sigma }}_{SS}$ increases linearly with the Mn and C content, as it is to be expected considering Equation (10).

To conclude, unlike for the addition of C, Mn additions do not always increase ${\mathsf{\sigma }}_{SRO}$. This is a result of a stagnation of $E_{SRO}$ because of a saturation of preferential cells with carbon. In theory, the same effect should be noticeable for C and Al, as well. However, because of the comparably low alloying contents of those elements, it is very unlikely that a point of saturation will be reached in common alloys.

This decrease in ${\mathsf{\sigma }}_{SRO}$ can be understood as the impact of the deviation from linear solid solution hardening. If the ratio of Mn atoms to Fe atoms becomes increasingly larger and the amount of cells with randomly-distributed Mn increases, the impact of an actual ordering of Mn decreases since fewer cells are brought into short-range ordering to minimize the ordering energy.

Overall, the change of $E_{Order}$ with varying contents of alloying elements can be understood as the effectiveness of an alloy addition of those elements with respect to the increase in strength by short-range ordering. Since it is assumed that ${\mathsf{\sigma }}_{\mathrm{SS}}$ follows a linear trend and ${\mathsf{\sigma }}_{\mathrm{GS}}$ is fixed, $\left|\frac{{\mathsf{\delta }E}_{Order}}{{\mathsf{\delta }\mathrm{\chi }}_i}\right|$ is an indicator for the effectiveness of an addition of an alloying element. A higher $\left|\frac{{\mathsf{\delta }E}_{Order}}{{\mathsf{\delta }\mathrm{\chi }}_i}\right|$ means a greater efficiency in employing an addition of the alloying element i. Looking at Figure 6 and Figure 7, C and Al both appear to be more efficient than Mn in increasing the yield strength by short-range ordering.

In order to compare the accuracy of the model used in this work to calculate ${\mathsf{\sigma }}_Y$ with other descriptive models, ${\mathsf{\sigma }}_{Y,model}$ according to models by Pickering [15], Lorenz [15], Bouaziz [11], De Cooman [2], and Choi [46] has been calculated. The equations used for ${\mathsf{\sigma }}_{Y,model}$ are listed in

Table 3. Solid solution hardening models in the literature.

Model	${\mathsf{\sigma }}_{\mathbf{Y},\mathbf{model}}$
Pickering	68 + 493N + 354C + 3.7Cr + 14Mo + 20Si + 0.22d−1/2
Lorenz	40 + 450N + 525C + 2Mn + 8.4Cr + 22Mo + 5Cu + 0.77d−1/2
Bouaziz	228 + 137C − 2Mn + σGS
De Cooman	189 + 413/√(d[μm])
Choi	97 + 279C − 1.5Mn + 49.6Si + 20.5Al + σGS

. The coefficients used for the alloying contents are in $\frac{\mathrm{MPa}}{\mathrm{wt}.\%}$, coefficients for grain size influence are in MPam^0.5 unless stated otherwise. It needs to be pointed out that although all of the models are for austenitic steels, only the models of Bouaziz, De Cooman, and Choi are specifically for HMnS. Additionally, some of the models include coefficients for the Hall-Petch effect while others neglect them. In order to be able to compare the models, the grain size effect has been calculated according to Equation (4) and added to the model calculations of Choi and Bouaziz.

Figure 8 compares ${\mathsf{\sigma }}_{Y,model}$ with ${\mathsf{\sigma }}_{Y,calc}$ and ${\mathsf{\sigma }}_{Y,exp}$.

Due to the difference in the steel concepts those models were developed for, significant deviations from ${\mathsf{\sigma }}_{Y,exp}$ can be seen. Those deviations are a result of a focus on other alloying elements. For example, Pickering does not consider the impact of Mn on yield strength. Nevertheless, the models of Choi [46] and De Cooman [2] offer reasonable results, but are not as close to the experimental data as the model developed in this work. Additionally, the model by De Cooman does not take compositional variances into account, while the model of Choi considers the impact of Mn on yield strength to be negative.

The models of Pickering and Lorenz are not applicable to HMnS based on the results of the calculations. The model by Bouaziz tends to overestimate the yield strength.

4. Conclusions

A basic ab initio-based model for the calculation of ${\mathsf{\sigma }}_Y$ has been developed. The model shows good agreement with the experimental values. Discrepancies between experimental ${\mathsf{\sigma }}_{Y,exp}$ and calculation results ${\mathsf{\sigma }}_{Y,calc}$ are smaller than 10%. While many complexity-reducing assumptions have been made that lead to inaccuracies when calculating ${\mathsf{\sigma }}_{SRO}$, this model portrays that the general concept of combining ab initio simulations with experimental data can lead to feasible results on a macroscopic scale. This is also shown by the calculated positive impact of Mn and Al on yield strength by solid solution strengthening in a reasonable range.

The impact of the alloying content of C, Mn, and Al on the strengthening by short-range ordering was analyzed. For the given steel compositions with fully austenitic microstructures in this work, the values for ${\mathsf{\sigma }}_{SRO}$ are as high as approximately 65 MPa. This significant strengthening leads to a proportional contribution to ${\mathsf{\sigma }}_Y$ of nearly 20%.

While an increase in C and Al increases ${\mathsf{\sigma }}_{SRO}$ as long as a sufficient number of atoms of substitutional alloying elements are available to form preferential cell structures, a limiting factor for the effectiveness of strength increases by Mn was found.

The impact of short-range ordered clusters during deformation needs to be evaluated. A higher amount of more stable, energetically favorable clusters might not only lead to an increase in ${\mathsf{\sigma }}_{SRO}$ and ${\mathsf{\sigma }}_Y$, it might also impact the beginning of serrations in the flow curve, as well as their amplitude if the serrations are resulting from short-distance carbon atom jumps associated with the cutting of clusters.