Effect of Stress on Variant Selection in Lath Martensite in Low-carbon Steel

Abstract

The effect of stress on the variant selection in lath martensite in a low-carbon steel (Fe–0.18%C–0.89%Mn–2.88%Ni–1.51%Cr–0.40%Mo) was investigated using electron backscatter diffraction pattern (EBSP) analysis. The steel was continuously cooled from a fully austenitic temperature to room temperature under uniaxial compressive stress applied during the martensitic transformation. It was demonstrated that certain variants maintaining the Kurdjumov-Sachs (K-S) orientation relationship with the prior austenite were preferentially selected under the applied stress only in blocks larger than the average block size. Otherwise, no clear variant selection was found. The applied stress and the external work done during the martensitic transformation, which was evaluated from the transformation strain, showed that the variants with greater external work values were more likely to be selected. However, both the shift in the martensite start temperature and the selected variants indicate that only the invariant line transformation strain was effective for variant selection in lath martensite in the low-carbon steel, unlike in nickel steels where the lattice-invariant shear has been additionally included in the literature.

1. Introduction

Many crystallographic studies have been performed on the martensitic transformation. In low-carbon, low-alloy steels, the Kurdjumov-Sachs (K-S) orientation relationship, specified as (111)_γ//(011)_α′ and [ $\overline{1}01$ ]_γ//[ $\overline{1}\overline{1}1$ ]_α′, holds between the parent austenite phase (γ) and the martensite phase (α′). There are 24 combinations in the K-S relationship, which are called variants. The martensitic structure is a hierarchical structure composed of blocks of the same variant and then of packets, which are collections of blocks sharing the same {111}_γ plane. The specific hierarchical structure formed is known to be largely influenced by the variant selection.¹⁾ Moreover, it is known that the hierarchical structure of the martensite significantly influences the mechanical properties of steel. For example, the strength of lath martensite in low-carbon steel follows the Hall-Petch law when the packet size is taken as the effective grain size.²⁾ In addition, Wang et al. have reported that the toughness of lath martensite depends on the packet size.³⁾ Therefore, elucidation of the variant selection, which determines the base of the hierarchical structure, is a key to improving the properties of martensitic steel.

Based on this background, many studies have been conducted on variant selection during the martensitic transformation in recent years. Kitahara et al.⁴⁾ analyzed the crystallographic features of martensitic structures in low-carbon steels using the electron backscatter diffraction pattern (EBSP) technique and demonstrated that during the transformation from austenite to lath martensite, not all 24 possible variants necessarily appeared within a single prior austenite grain and that the number of variants appearing depended on the size of the packet. Morito et al.¹⁾ found that sub-blocks, defined as specific combinations of two variants misoriented by 10.53°, existed in lath martensite.

Studies on the strain-induced formation of martensite in high-alloy steels and nickel steels have shown that such variant selection in martensite is significantly influenced by the external stress applied during the martensitic transformation. For example, Gey et al.⁵⁾ assessed the crystallographic features of the residual austenite phase and the strain-induced martensite using the EBSP technique in a SUS304 stainless steel elongated by 10% strain at –60°C. In this study, variant selection in the strain-induced martensite was found to depend on crystal orientation of variant relative to the stress axis. Abreu et al.⁶⁾ also obtained a similar result for AISI 301LN stainless steel. In other words, mechanical driving forces generated by stress application are different for each variant because the martensitic transformation is a displacive transformation and the orientations of the transformation strain are different for each variant. Therefore, variants with larger driving force were considered to be formed preferentially. In both experiments, however, both rolling texture and dislocation occurred because of the application of stresses larger than the yield stress, and this may influence variant selection, as well. Furthermore, the influence of stress on variant selection in low-carbon, low-alloy steels, which have high martensite start temperatures M_s and undergo the martensitic transformation during a conventional cooling process, has not been clarified yet.

Therefore, this study aims to clarify the influence of stress on variant selection during the martensitic transformation of low-carbon, low-alloy steel. Stresses lower than the elastic limit were applied to during the martensitic transformation occurring as the steel cools from austenitic conditions, and the crystal orientations of the obtained martensitic structure were analyzed.

2. Experimental

Table 1 shows the chemical composition of the steel (SNCM616) used in this study. SNCM616 is a low-alloy steel with a martensite start temperature M_s of 370°C and a martensite finish temperature M_f of 260°C. The samples were cylinders 8 mm in diameter and 12 mm in length that were heated as shown in Fig. 1 in a vacuum of 1.0 × 10^–1 Pa. First, the samples were heated to 1200°C at a heating rate of 10°C/s for austenitization and held for 300 s to allow the austenite grains to grow. The samples were then cooled at a rate of 1°C/s, and a uniaxial compressive stress of 70 MPa was applied to the samples from 400°C, just above M_s, to 200°C, just below M_f, during the cooling process. It should be noted that the steel does not yield under a compressive stress of 70 MPa, since the yield stress of the steel at 400°C is 105 MPa. Samples without applied stress were also prepared similarly. For microstructure observation, the samples were cut in half perpendicular to the stress axis. The cut surface of the samples was polished to a mirror finish, chemically etched with 2% nital solution, and finally observed with an optical microscope. For the EBSP analysis, the surface of the samples was polished to a mirror finish and electropolished with a mixed solution of chromic acid and phosphoric acid. The EBSP analysis was done using a field emission scanning electron microscope (FE-SEM) (JSM-7001FA, JEOL Ltd., Japan) with an accelerating voltage of 15 kV, an electron beam diameter of approximately 20 nm, and a step distance of 0.5 μm.

Table 1. Chemical composition of the steel used (weight%).

C	Si	Mn	Ni	Cr	Mo
0.18	0.21	0.89	2.88	1.51	0.4

Fig. 1.

Thermal history used in this study.

For the crystal orientation analysis, prior austenite grains oriented along specific directions were selected from the samples produced with and without the applied stress. For comparison purposes, the variants were divided into groups A, B, and C with the same Bain correspondence, as shown in Table 2. Variants belonging to the same Bain correspondence group have the same direction out of the three possible Bain deformation directions,⁷⁾ as indicated in Fig. 2. Therefore, as shown in the (001) pole figures representing the K-S orientation relationship in Fig. 3, variants belonging to the same Bain correspondence group were generated from the {001}_γ plane surrounding the prior austenite grains. It should be noted that a boundary with a misorientation angle higher than 5° was defined as a block boundary in this study, because the misorientation angles between all variants were higher than 5°, as shown in Table 2. It should be also noted that a rotation around the stress axis was performed as illustrated in Fig. 4 so that the grains could be evaluated independently of their deviation from the stress axis.

Table 2. The 24 crystallographic variants satisfying the K-S orientation relationship.

Variant	Plane parallel	Direction parallel	Angle from V1	Group
V1	(111)γ//(011)α′	[-1 0 1]γ // [-1-1 1]α′	–	A
V2		[-1 0 1]γ // [-1 1-1]α′	60.00	B
V3		[ 0 1-1]γ // [-1-1 1]α′	60.00	C
V4		[ 0 1-1]γ // [-1 1-1]α′	10.53	A
V5		[ 1-1 0]γ // [-1-1 1]α′	60.00	B
V6		[ 1-1 0]γ // [-1 1-1]α′	49.47	C
V7	(1-11)γ//(011)α′	[ 1 0-1]γ // [-1-1 1]α′	49.47	B
V8		[ 1 0-1]γ // [-1 1-1]α′	10.53	A
V9		[-1-1 0]γ // [-1-1 1]α′	50.51	C
V10		[-1-1 0]γ // [-1 1-1]α′	50.51	B
V11		[ 0 1 1]γ // [-1-1 1]α′	14.88	A
V12		[ 0 1 1]γ // [-1 1-1]α′	57.21	C
V13	(-111)γ//(011)α′	[ 0-1 1]γ // [-1-1 1]α′	14.88	A
V14		[ 0-1 1]γ // [-1 1-1]α′	50.51	C
V15		[-1 0-1]γ // [-1-1 1]α′	57.21	B
V16		[-1 0-1]γ // [-1 1-1]α′	20.61	A
V17		[ 1 1 0]γ // [-1-1 1]α′	51.73	C
V18		[ 1 1 0]γ // [-1 1-1]α′	47.11	B
V19	(11-1)γ//(011)α′	[-1 1 0]γ // [-1-1 1]α′	50.51	C
V20		[-1 1 0]γ // [-1 1-1]α′	57.21	B
V21		[ 0-1-1]γ // [-1-1 1]α′	20.61	A
V22		[ 0-1-1]γ // [-1 1-1]α′	47.11	C
V23		[ 1 0 1]γ // [-1-1 1]α′	57.21	B
V24		[ 1 0 1]γ // [-1 1-1]α′	21.06	A

Fig. 2.

(a) Conventional FCC unit cell. (b) Relation between FCC and BCT cells of austenite. (c) BCT cell of austenite. (d) Bain strain deforming the BCT austenite lattice into a BCC martensite lattice.⁷⁾

Fig. 3.

{001} pole figures showings the orientations of the 24 martensite variants that maintain the K-S orientation relationship transformed from a single-crystal austenite with (a) [001]_γ//ND, (b) [011]_γ//ND, and (c) [111]_γ//ND orientation.

Fig. 4.

Schematic model of rotation operation.

3. Results and Discussion

3.1. Variant Selection in Specific Grains

The sizes of the prior austenite grains observed ranged from 100 to 200 μm, and the prior austenite grains maintained the K-S orientation relationship regardless of whether stress was applied. Figure 5 shows examples of prior austenite grains observed in samples produced with and without applied stress. These prior austenite grains had good, symmetrical, specific crystal orientations relative to the stress axis (hereafter defined as ND axis), namely, [001]_γ//ND, [011]_γ//ND, and [111]_γ//ND. The blocks in the grains were divided into two types based on the block area: large blocks with areas larger than the 50 μm², the average of block areas, and small blocks with areas smaller than 50 μm². The pole figures of large blocks and of small blocks were made separately, and group A was colored with red, group B with green, and group C with blue. Figures 6(a) and 6(b) show the pole figures of the prior austenite grains shown in Fig. 5(a), Figs. 6(c) and 6(d) are for the grains shown in Fig. 5(b), Figs. 7(a) and 7(b) are for the grains shown in Fig. 5(c), Figs. 7(c) and 7(d) are for the grains shown in Fig. 5(d), Figs. 8(a) and 8(b) are for the grains shown in Fig. 5(e), and Figs. 8(c) and 8(d) are for the grains shown in Fig. 5(f). Two pole figures were made for the following reason. It has been reported that blocks formed in the early stages of the transformation tend to grow larger.⁸⁾ Therefore the pole figures of the large blocks can be less-influenced by the intricate internal stress resulting from the surrounding martensitic transformation on the variant selection and can clarify the variant selection behavior of the martensite formed in the early stages of the transformation.

Fig. 5.

Crystal orientation map of a prior austenite with a [001]_γ//ND orientation produced (a) without stress and (b) with stress, with a [011]_γ//ND orientation produced (c) without stress and (d) with stress, and with a [111]_γ//ND orientation produced (e) without stress and (f) with stress.

Fig. 6.

(001)_α′ pole figures of a prior austenite with a [001]_γ//ND orientation produced without stress for (a) only small blocks and (b) only large blocks and produced with stress for (c) only small blocks and (d) only large blocks.

Fig. 7.

(001)_α′ pole figures of a prior austenite with a [011]_γ//ND orientation produced without stress for (a) only small blocks and (b) only large blocks and produced with stress for (c) only small blocks and (d) only large blocks.

Fig. 8.

(001)_α′ pole figures of a prior austenite with a [111]_γ//ND orientation produced without stress for (a) only small blocks and (b) only large blocks and produced with stress for (c) only small blocks and (d) only large blocks.

The area fractions of the large blocks and small blocks in the prior austenite grains are summarized by group in Table 3. It demonstrates that for all orientation relationships, the area fractions of groups A, B, and C were equal for both large and small blocks produced without stress and for small blocks produced with stress. However, the area fraction of large blocks for the sample produced with applied stress was different, depending on the orientation relationship. For [001]_γ//ND, the area fraction of group A large blocks was remarkably large for the sample produced with applied stress. For [011]_γ//ND, on the other hand, the area fraction of group A large blocks was somewhat small for the sample produced with applied stress. For [111]_γ//ND, the area fractions of large blocks of all groups were almost equal at 1/3 for the sample produced with applied stress. Next, six prior austenite grains with the [001]_γ//ND orientation, [011]_γ//ND orientation, or [111]_γ//ND orientation were selected from each sample produced with and without applied stress, respectively. The pole figures obtained from each grain were superimposed, as shown in Figs. 9, 10, 11. It should be noted that a deviation of up to 15° from the ND axis was permitted; this deviation was corrected by performing a rotation for each γ grain. Figures 9, 10, 11 show that for all orientation relationships, the amounts of each of the three groups generated were almost equal in the large blocks in the sample produced without applied stress, in the small blocks in the sample produced without applied stress, and the in the small blocks in the sample produced with applied stress. Among the large blocks in the sample produced with applied stress, on the other hand, group A was formed dominantly for [001]_γ//ND, while fewer such blocks were formed for [011]_γ//ND. For [111]_γ//ND, each of the three groups accounted for approximately 1/3 of the blocks, even among the large blocks in the sample produced with applied stress.

Table 3. Area fraction of prior austenite (a) with [001]_γ//ND orientation, (b) with [011]_γ//ND orientation, and (c) with [111]_γ//ND orientation.

Fig. 9.

(001)_α′ pole figures of a prior austenite with a [001]_γ//ND orientation produced without stress for (a) only small blocks and (b) only large blocks and produced with stress for (c) only small blocks and (d) only large blocks.

Fig. 10.

(001)_α′ pole figures of a prior austenite with a [011]_γ//ND orientation produced without stress for (a) only small blocks and (b) only large blocks and produced with stress for (c) only small blocks and (d) only large blocks.

Fig. 11.

(001)_α′ pole figures of a prior austenite with a [111]_γ//ND orientation produced without stress for (a) only small blocks and (b) only large blocks and produced with stress for (c) only small blocks and (d) only large blocks.

This result indicates that stress application influenced the variant selection. This influence is evident only for relatively large blocks, while small blocks were not affected by stress application. The large blocks are thought to be generated in the early stages of the transformation, and thus are strongly affected by the external stress. On the other hand, since the α′ phase generated in the early stages forms a complex stress field and dislocations that lead to nucleation of martensite in the surrounding γ phase as the transformation proceeds, the influence of the external stress on the small blocks that form between the large blocks in the later stages of transformation is small.

Figure 12 shows the rates of transformation of the samples produced with and without applied stress, which were calculated from the measured thermal expansion. The rate of transformation X(T₁) at a given temperature T₁ was calculated using the following Eq. (1):⁹⁾   

$$X(T_1)=\frac{L(T_1)-L_{\gamma }(T_1)}{L_{\alpha }(T_1)-L_{\gamma }(T_1)}$$(1)

where L_γ(T₁) and L_α′(T₁) are the values of the thermal expansion at the temperature T₁ on lines extended in the higher or lower temperature direction, L_γ(T) and L_α′(T), respectively, and L(T₁) is the measured value of the thermal expansion at T₁. Figure 12 shows that stress application increased the transformation start temperature by 20°C but hardly changed the transformation finish temperature. This fact supports the conclusion that the large blocks generated in the early stages of the transformation are strongly affected by the external stress, while the small blocks generated in the later stages of the transformation are less influenced by the external stress.

Fig. 12.

Rate of transformation.

3.2. Relationship between the Stress Axis and Selected Variants

In the three orientations [001]_γ//ND, [011]_γ//ND, and [111]_γ//ND analyzed so far, it was observed that among the large blocks of the samples produced with applied stress, the Bain correspondence groups with small angles θ between ND and (001)_γ occupy a large area. To examine the relationship between the angle θ and the variant selection, 20 prior austenite grains were randomly selected from each of the samples produced with and without applied stress, and the area fraction of each of the Bain correspondence groups among large blocks was calculated. Figure 13 reveals that while the three groups were generated in almost equal proportions of 1/3 regardless of θ in the samples produced without applied stress, the Bain correspondence groups with small θ values tended to have larger areas in the samples produced with applied stress. In other words, in the samples produced with applied stress, the variants belonging to the Bain correspondence groups with small θ values are generated preferentially. Thus, it is suggested that because the angle θ between ND and (001)_γ corresponds to the angle between the stress direction and the compression direction of the Bain deformation, the relationship between the Bain deformation and the stress direction influences the variant selection.

Fig. 13.

Relationship between area fraction and angle between ND and (001)_γ for samples produced (a) without stress and (b) with stress.

3.3. Driving Force of the Transformation and Work Done by the External Force

According to Abreu et al.,⁶⁾ the driving force of the transformation (ΔG) can be expressed as the sum of the chemical driving force (ΔG_chem) and the mechanical driving force (ΔG_mech), and the mechanical driving force corresponds to work done by an external force, that is, the external work (U). These are represented by the following Eqs. (2) and (3):   

$$\mathrm{\Delta }G=\mathrm{\Delta }{G}_{chem}+\mathrm{\Delta }{G}_{mech}$$(2)

  

$$\mathrm{\Delta }{G}_{mech}=U=\sigma :\mathit{\epsilon }$$(3)

where σ is the tensor of the applied stress (70 MPa uniaxial compressive stress) and ε is the strain tensor affecting the driving force of the transformation. As a function of the deformation gradient F, ε can be calculated by the following Eq. (4):   

$$\mathit{\epsilon }=(F^{t}F-I)/2$$(4)

There are two possible equations for the deformation gradient F, as follows:   

$$F^1=RB{S}_2{S}_1$$(5)

  

$$F^2=RB$$(6)

F¹ is the deformation gradient according to the double lattice-invariant shear formulation derived by Kelly.¹⁰⁾S₁ and S₂ correspond to lattice-invariant shear, B corresponds to the Bain deformation, and R corresponds to a rigid body rotation. The S₁ and S₂ values used here are the same as those used in the analysis performed by Morito et al.¹⁾F² is the deformation gradient in the case where only RB, the invariant line strain, is assumed to affect the variant selection.⁷⁾ The difference between the deformation values obtained using F¹ and F² is illustrated in Fig. 14. In general, the total strain obtained using F¹ is smaller than that obtained using F² because of the lattice-invariant shear. The parameters used for this calculation are listed in Table 4. When F¹ is used, a plane close to (5, 7, 5)_γ becomes the habit plane, and S₁, S₂, and R are selected to satisfy the K-S orientation relationship. Because the habit plane (0.49714, 0.71113, 0.49714)_γ corresponds to an invariant plane, Eq. (3) can be expressed as the sum of the product between the shear strain on the habit plane and the stress and the product between the normal strain on the habit plane and the stress. When F² is used, on the other hand, because R and B are the same as those in F¹ and only the lattice-invariant shear is removed, F² also satisfies the K-S orientation relationship, but the habit plane does not exist. Therefore, in this case, the external work U can be given by the scalar product of the strain tensor and the stress tensor, as expressed in Eq. (3).

Fig. 14.

Schematic illustrations of the mechanisms (a) F¹ and (b) F².

Table 4. Results of calculation for F¹ and F².

Input
Lattice parameter:
Austenite:	a0 = 0.36313 nm
Martensite:	a = 0.28974 nm
Lattice invariant shear
S1:	(101)[-101]γ (112)[-1-11]α′
S2:	(100)[01-1]γ (110)[-11-1]α′
Shape strain of S2:	0.09122
Output
Shape strain of S1:	0.26488
Habit plane:	(0.49714, 0.71113, 0.49714)γ
Shape strain direction:	[–0.20113, 0.70712, –0.67789]γ
Shape strain matrix(F1):	$\left(\begin{array}{l}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{0.97578}\hspace{1em} -\text{0.03465}\hspace{1em} -\text{0.02422}\\ \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{0.08515}\hspace{1em}\hspace{1em} \text{1.12181}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} } \text{0.08515}\\ -\text{0.08164}\hspace{1em} -\text{0.11677}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} } \mathrm{\hspace{0.17em} }\text{0.91837}\end{array}\right)$
Shape strain matrix(F2):	$\left(\begin{array}{l}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} } 1.11744\hspace{1em} -\text{0.03465}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} } \text{0.10822}\\ \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{0.01828}\hspace{1em}\hspace{1em} \text{1.12181}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} } \text{0.08515}\\ -\text{0.15586}\hspace{1em} -\text{0.11677}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} } \mathrm{\hspace{0.17em} }\text{0.78593}\end{array}\right)$
Orientation relationship (F1 and F2):	(111)γ – (011)α′: 0.23 degree
	[-101]γ – [-1-11]α′: 3.41 degree

To investigate the relationship between variant selection and the external work U, the relationship between U and the angle θ which is the angle between ND and (001)_γ was calculated for both formulations F¹ and F² (Fig. 15). To be specific, the U value of each variant was calculated for each θ, and the maximum, average, and minimum values obtained for each Bain correspondence group were calculated. Figure 15 shows that the Bain correspondence groups with smaller θ values have larger U values regardless of whether F¹ or F² is used. On the other hand, the actual measured data shown in Fig. 13(b) indicate that the Bain correspondence groups with smaller θ values have larger areas. This suggests that variants with greater external work values U are preferentially generated during the martensitic transformation under stress. It should be noted that it is difficult to determine from only Fig. 15 which of F¹ or F² affects the variant selection.

Fig. 15.

Relationship between U and the angle between ND and (001)_γ.

Therefore, to compare F¹ and F², the U value of each variant was calculated for each orientation of the γ phase. Figure 16 shows the (001)_α′ pole figures for each γ orientation, on which the magnitude of U is represented by circles with sizes proportional to its magnitude. The open circles represent variants with negative U values. Comparison of this figure with Figs. 9 10, 11(d) demonstrates that groups with greater U values were generated preferentially in all cases, regardless of whether F¹ or F² was used. In addition, to determine which variant was preferred within the same group, the variant pair marked with black arrows in Fig. 16 was investigated in detail. This variant pair was chosen because magnitude correlations of the U values of the two variants show large contrast between F¹ and F². When F² is used, there is no difference in U of the variant pair because the members of the variant pair belong to the same Bain correspondence group. On the other hand, when F¹ is used, there is a large difference in U of the variant pair because the lattice-invariant shear does a large amount of work. In this calculation, there is a difference of 59 J/mol between the U values of the members of this variant pair. In other words, if F¹ is applied, there should be a large difference between the generated amounts of each variant in the pair, whereas if F² applied, there should be almost no difference between the generated amounts of each variant in the pair. The actual measurement demonstrated that there is no difference between the generated amounts of each variant in the pair, as shown in Fig. 10(d). Therefore, it must be F², that is, invariant line strain must affect the variant selection.

Fig. 16.

Calculated (001)_α′ pole figures of a prior austenite.

To verify the obtained result, the maximum values of U for F¹ and F² were calculated and examined thermodynamically. There was a difference between the maximum values of U for F¹ and F²: the maximum value of U for F² was 92 J/mol and that for F¹ 51 J/mol. Equation (2) suggests that as the maximum value of U becomes larger, a smaller chemical driving force (ΔG_chem) is required for the transformation, causing the transformation start temperature to increase. The actual measurement, shown in Fig. 12, demonstrated that the transformation start temperature increased by 20°C because of the stress application. The energy corresponding to this 20°C increment was provided by the applied stress. Therefore, a calculation of ΔG_chem can verify which deformation gradient, F¹ or F², affected the variant selection.

Kaufman et al. proposed the following equation of ΔG_chem for pure iron from 200 K to 900 K:¹¹⁾   

$$\begin{array}{l}\mathrm{\Delta }{G}_{chem}^{Fe}=(1\mathrm{\hspace{0.17em} }202-2.63\times {10}^{-3}{T}^2\\ \hspace{1em}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} } +1.54\times {10}^{-6}{T}^3)\times 4.2\hspace{1em}\hspace{1em}[J/mol]\end{array}$$(7)

where T is the temperature. Zener proposed the following equation for alloy steel:¹²⁾   

$$\mathrm{\Delta }{G}_{chem}=(1-x)\mathrm{\Delta }{G}_{chem}^{Fe}+x\mathrm{\Delta }{H}_A$$(8)

In this equation, x is the mole fraction of an alloying element A and ΔH_A is the difference between the heats of mixing of the element A into α′ and into γ. Note that this equation does not depend on temperature. Wang et al. reported¹³⁾ the following equation as the temperature-dependent term of the driving forces of the transformation, $-\mathrm{\Delta }{G}_r^{*}$ , for Fe–(0.2–0.5)C–(0.5–2.0)Mn–(0.5–2.0)Si–(0.5–2.0)Cr–(0.1–0.7)Mo.   

$$-\mathrm{\Delta }{G}_r^{*}=3\mathrm{\hspace{0.17em} }247(J/mol)-4.8446(J/mol\cdot °C)\cdot Ms$$(9)

Estimation of the difference of the driving force from Eqs. (7), (8), and (9) at T = 663 K (390°C) and T = 643 K (370°C) resulted in external work values U corresponding to the stress application of 1.1×10² J/mol and 97 J/mol, respectively. These values are close to 92 J/mol, the maximum value of U for F²; the differences from this maximum are about 20 J/mol and 5 J/mol, respectively. Even if the larger difference 20 J/mol is converted into temperature using Eqs. (7) and (8), it is only 3°C. Because a temperature gradient occurs within the samples during cooling, this value of 3°C could be within an error. Accordingly, the transformation start temperature results also support that the invariant line strain described by F² affects variant selection. In addition, it is found that the variants with greater external work values U were more likely to be selected.

However, in studies about the changes in M_s in strain-induced martensite in high-nickel steel, Ueda et al. and Patel et al. have accurately predicted rises in M_s by using the work done by the external forces calculated from the shape strain induced by the transformation, including not only invariant line strain but also lattice-invariant shear.^14,15) The different results obtained here are considered to result from the difference between the mechanisms of transformation in high-nickel steel and in the low-carbon, low-alloy steel used in this study. The martensitic structure of high-nickel steel includes both lenticular martensite and plate martensite and contains many lattice defects caused by internal twin crystals. In addition, the habit plane in this structure is close to (2, 5, 9)_γ or (3, 10, 15)_γ, and the lattice correspondence relationship and habit planes can be explained by assuming only one lattice-invariant shear.^16,17) On the other hand, the martensitic structure of low-carbon, low-alloy steel is lath martensite and contains many lattice defects caused by dislocations. The habit plane is (1, 1, 1)_γ or (5, 5, 7)_γ, and the lattice correspondence relationship and habit planes cannot be explained unless at least two lattice-invariant deformations are assumed.^10,18,19,20) Therefore, the difference between these habit planes must not be a mere quantitative difference, but a more inherent one. In other words, this study and the studies performed by Ueda et al. and Patel et al. do not conflict, since these studies analyzed the mechanisms of different martensitic transformations. It should be noted that the changes in habit planes and the changes in the stress response due to changes in composition are not explained here, and further elucidation remains a challenge.

4. Conclusions

In this study, the effect of stress on variant selection during the martensitic transformation was investigated by applying stress to a low-carbon steel in the temperature range of the martensitic transformation and by analyzing the crystal orientation of the obtained martensitic structure. The following findings were obtained:

(1) Stress application tended to affect the variant selection only in large blocks that form in early stages of the transformation. In small blocks, on the other hand, no remarkable effect of the stress was found.

(2) When stress was applied, Bain correspondence groups with small angles between the stress axis and (001)_γ tended to be generated preferentially. This suggests that in the martensitic transformation under stress, variants with greater external work values U were formed preferentially.

(3) When it was assumed that the invariant line strain affected the external work U, the calculated variant selection in the transformation was found to be in good agreement with the experimental results.

(4) The mechanical driving force calculated assuming the invariant line strain agreed closely with the chemical driving force decrease as a result of increasing transformation start temperatures.

The above conclusions suggest that in the martensitic transformation of low-carbon, low-alloy steel under stress, only the invariant line strain among all the transformation strain components influences the variant selection.

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