Change in Dislocation Characteristics with Cold Working in Ultralow-carbon Martensitic Steel

Abstract

In a previous study, the authors used X-ray analysis with the classical Williamson–Hall (CWH) method to suggest that charging a small amount of cold working markedly decreases the dislocation density of ultralow-carbon martensitic steel, although this heightens the 0.2% proof stress. However, this method does not consider the dislocation arrangement. In the present study, a modified Williamson–Hall/Warren–Averbach (MWH/WA) method was applied to ultralow-carbon martensitic steel (Fe–18%Ni alloy) in order to evaluate not only the dislocation density but also the dislocation arrangement. Their effects on the yielding behavior were examined. With the MWH/WA method, the dislocation density did not change up to 40% cold rolling. On the other hand, the dislocation arrangement parameter M was high (M > 1) in the as-quenched state and became smaller (M < 1) when a small plastic strain was charged. This means that the dislocation distribution is random in as-quenched martensite but changes the cell structure with cold working. Owing to such a dislocation arrangement, the CWH method tends to overestimate the dislocation density of as-quenched martensite compared to the MWH/WA method. Tensile testing revealed that the elastic limit was very low in as-quenched martensite and high in cold-rolled martensite. In the case of a tangled dislocation structure, a higher stress should be required because of the stable dislocation structure. On the other hand, the random dislocations introduced by martensitic transformation can easily move at a low stress level owing to their unstable distribution, which leads to the low elastic limit in as-quenched martensite.

1. Introduction

As-quenched martensitic steels are characterized by a low elastic limit, which is likely due to the existence of “mobile dislocations” that are introduced by the martensitic transformation because they can move at low levels of applied stress. Because the dislocation density is very high in as-quenched martensite, a slight movement of dislocations should produce a fairly large amount of plastic strain. In order to clarify the yielding mechanism of martensitic steels, the authors examined the change in dislocation density with tensile deformation in ultralow-carbon martensitic steel (Fe–18%Ni alloy).¹⁾ The results of the classical Williamson–Hall (CWH) method^2,3) proved that the dislocation density markedly decreases just after yielding and reaches a constant value. In contrast, the 0.2% proof stress is heightened by charging a small amount of cold working and then levels off at a constant value with increasing pre-strain. This result suggests that the yield stress of martensitic steels cannot be explained by only the dislocation density but depends on the dislocation arrangement: mobile or tangled.¹⁾ Harjo et al.⁴⁾ also reported that the dislocation arrangement for 0.22%C martensitic steel changes during tensile deformation. The stress required to move tangled dislocations corresponds to the substantial yield stress of martensitic steels; thus, the authors defined the density of tangled dislocations as the effective dislocation density.⁵⁾

The CWH method is a convenient process for estimating the dislocation density, but some problems remain with correcting the elastic anisotropy on each crystal plane. Recently, Ungár et al.⁶⁾ proposed the modified Williamson–Hall and modified Warren–Averbach (MWH/WA) method, which performs line profile analysis by considering the anisotropic crystal strain and size of the strain field of dislocation. It has been found that the CWH method tends to overestimate the dislocation density compared with line profile analysis.⁷⁾ This is because the CWH method converts all of the line profile broadening in the X-ray diffraction peaks to the dislocation density, even though other information may be contained in the line profile broadening. In this study, line profile analysis⁶⁾ was applied to evaluate the dislocation density and arrangement in ultralow-carbon martensitic steel (Fe–18%Ni alloy), and the change in yielding behavior during tensile testing was evaluated in terms of the change in dislocation arrangement.

2. Experimental Procedures

2.1. Material and Heat Treatment

An ultralow-carbon martensitic steel (Fe–18%Ni–0.002%C alloy) was used in this study. Table 1 presents the chemical composition of the steel. The amounts of carbon and nitrogen were sufficiently low that the effect of strain aging could be neglected. An ingot of 30 kg was produced by vacuum melting and hot-forged into 35 mm-thick steel plates at 1473 K. The steel plate was cold-rolled for a 50% reduction in thickness and then subjected to a solution treatment at 1173 K for 1.8 ks in the austenite single-phase region followed by water quenching. After the surface of the steel plate was ground, the as-quenched steel was cold-rolled for a thickness reduction of 40%. In order to avoid the effect of texture, the amount of cold working was limited to a 40% thickness reduction. The Ms temperature was measured with a thermal dilatation tester. The microstructure of the specimens was observed with an optical microscope (OM), scanning electron microscope equipped with an electron back scattering diffraction system (EBSD-SEM), and transmission electron microscope (TEM). Tensile tests were performed with an Instron type testing machine (49 kN) at the crosshead speed of 1 mm/s for JIS-13B test pieces with gauge dimensions of 50 mm in length, 12.5 mm in width, and 1 mm in thickness.

Table 1. Chemical composition of specimens used in this study (mass%).

	C	Si	Mn	P	S	Ni	Fe	Ms(K)
Fe-18Ni	0.0017	0.009	0.43	0.0019	0.0013	17.9	Bal.	560

Specimens with dimensions of 10 mm^l × 10 mm^W × 1 mm^t were prepared for X-ray diffraction (XRD) analysis. The surfaces of the specimens were flattened by grinding with sandpaper and then electrolytically polished to eliminate the surface layer strained by sandpaper grinding. The XRD measurements for the line profile analysis⁶⁾ were performed with Bragg–Brentano (2θ–θ) geometry using an X-ray diffractometer (RINT2100, Rigaku Co. Lid.) equipped with a Cu–Kα radiation source (40 kV, 40 mA). The diffraction lines were recorded for 2θ = 40°–145° in increments of 0.01° to cover the diffractions of (110), (200), (211), (310), and (222). The observed diffraction peaks were separated to the two peaks diffracted by Cu-Kα₁ and Cu-Kα₂ radiation.⁸⁾ The peaks corresponding to Cu-Kα₁ radiation were fitted to the peaks formed by the Voigt function and used for the analysis. The instrumental line broadening was evaluated by using the standard reference material (annealed pure iron), and the corresponding instrumental width was subtracted from each peak width in the measured specimens.

2.2. Method of X-ray Diffraction Line Profiles Analysis

Williamson et al. reported that the XRD peak width is related to the crystallite size and lattice strain through the following CWH equation:²⁾   

$$\mathrm{\Delta }\text{K}=\frac{0.9}{D}+\epsilon K\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\left(\mathrm{\Delta }K=\beta \frac{cos\theta }{\lambda },K=\frac{2sin\theta }{\lambda }\right)$$(1)

where β, θ, λ, ε, and D are the full-widths at half-maximum (FWHMs) of the XRD line profile, diffraction angle, X-ray wavelength, lattice strain, and crystallite size, respectively. The dislocation density ρ is given by Eq. (2) under the assumption that the lattice strain is produced by dislocations only:³⁾   

$$\rho =B{\left(\frac{\epsilon }{b}\right)}^2$$(2)

where b is the magnitude of the Burgers vector (0.25 nm for bcc-Fe) and B is an unknown conversion coefficient between ε and ρ. However, the elastic constant is different in each crystal plane, so Eq. (1) is not a monotonic function against sin θ. In order to correct the elastic anisotropy, Ungár et al.⁶⁾ proposed the following modified Williamson–Hall (MWH) equation that considers the contrast factor C:   

$$\mathrm{\Delta }\text{K}=\frac{0.9}{D}+{\left(\frac{\pi A^2{b}^2}{2}\right)}^{\frac{1}{2}}{\rho }^{\frac{1}{2}}\left(K{\overline{C}}^{\frac{1}{2}}\right)+O\left(K^2\overline{C}\right)$$(3)

where A depends on the effective outer cutoff radius of dislocations; R_e. O represents higher-order terms in $K{\overline{C}}^{\frac{1}{2}}$ . Here, C is the average contrast factor of dislocations and depends on the relative orientations between the Burgers vector, line vector of a dislocation, and the diffraction vector as follows:⁹⁾   

$$\overline{C}=C_{h00}\left(1-{\text{qH}}^2\right),\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{\text{H}}^2=\frac{h^2{k}^2+k^2{l}^2+l^2{h}^2}{{\left(h^2+k^2+l^2\right)}^2}$$(4)

where h, k, and l are the Miller indices of each peak. The value of the elastic compliance C_h00 is determined from the three elastic constants in the cubic system: c₁₁, c₁₂, and c₄₄. The parameter q indicates the characteristic of dislocations existing in a specimen: the fraction of screw and edge dislocations. In order to estimate the dislocation density and effective outer cutoff radius of a dislocation, a further analyses of the line profiles by using the modified Warren–Averbach (MWA) equation⁶⁾ is required:   

$$lnA\left(L\right)\cong ln{A}^s\left(L\right)-Y\left(L\right)\left(K^2\overline{C}\right)+O^{\prime }{\left(K^2\overline{C}\right)}^2$$(5)

  

$$Y\left(L\right)=\rho \frac{\pi b^2}{2}{\text{L}}^{2}ln\frac{R_e}{L}$$(6)

where L, A(L), A^S(L) and O′ are the Fourier variable, the real part of the Fourier coefficient, the size Fourier coefficient, and a coefficient for the higher-order term K²C. From Eq. (5), fitting ln (A(L)) against K²C by using various L values gives Y(L) as the coefficient of the first-order term in Eq. (5). Then, Eq. (6) leads to the following equation:   

$$\frac{Y\left(L\right)}{L^2}=\frac{\pi b^2\rho }{2}ln{R}_e-\frac{\pi b^2\rho }{2}lnL$$(7)

The dislocation density and outer cutoff radius of dislocations are obtained by fitting the plot with a linear line. In addition, R_e measured by MWA is approximated within a small range of the Fourier variable. Thus, the correct function in a wide range is given by R_e′, as reported by Wilkins.¹⁰⁾ In this study, R_e′ was calculated from the function R_e = exp(2) × R_e′.¹¹⁾

3. Results and Discussion

3.1. Analysis of Characteristics of Dislocations in Ultralow-carbon Martensitic Steel

Figure 1 shows the orientation imaging maps and inverse pole figures of the materials used in this investigation. The as-quenched specimen had a lath martensitic structure without retained austenite. The colored structure corresponds to the packet composed of several blocks. With cold rolling, such a microstructure is elongated toward the rolling direction. However, no large change was found in the macroscopic microstructure. In addition, the inverse pole figures revealed that crystal orientation of cold-rolled martensite is randomly distributed. Figure 2 displays normalized X-ray line profiles of the (310) diffraction for as-quenched and cold-rolled specimens. The as-quenched specimen had the broadest line profile, but the peak width decreased once with a charge of 5% cold rolling and then broadened again with a charge of 40% cold rolling. In addition, the X-ray peak ratio was confirmed to not greatly change up to a thickness reduction of 40%. Figure 3 shows the changes in the FWHM for each diffraction peak with cold rolling. At all diffraction peaks, the FWHM decreased once with a charge of 5% cold rolling and then stayed constant or slightly increased as the amount of cold working was increased. This result suggests that some changes occur in the dislocation arrangement during cold working. In order to investigate the characteristics of dislocations and the change in dislocation density, line profile analysis was performed with the MWH/WA method as follows.

Fig. 1.

Orientation image maps and inverse pole figures of as-quenched ultralow-carbon martensitic steel: (a), (d) as-quenched, (b), (e) 20% cold-rolled, (c), (f) 40% cold-rolled.

Fig. 2.

Changes in XRD line profiles with cold rolling in ultralow-carbon martensitic steel.

Fig. 3.

Changes in FWHM with cold rolling in ultralow-carbon martensitic steel.

Figure 4 plots the (a) CWH and (b) MWH results for as-quenched and cold-rolled martensite. The CWH plots show irregular changes due to the elastic anisotropy of each crystal plane. On the other hand, the MWH plots follow smooth curves that fit Eq. (3). The fitting curves slightly differed among the specimens. The average contrast factor C is a function of q, so the fitting curves can change depending on q. In other words, q is optimized by selecting the best fit to the experimental data. Figure 5 shows q obtained by the best fit as a function of the thickness reduction by cold rolling. The value of q depends on the fraction of screw and edge components in dislocations. The value of q in bcc-Fe is 1.2 for the full edge component and 2.8 for the full screw component.¹⁰⁾ Because q of as-quenched martensite was estimated to be approximately 2.6, the fraction of screw dislocations should be about 88% in as-quenched martensite. Moritani et al.¹²⁾ used high-resolution TEM to report that screw dislocations exist at the interface between martensite and retained austenite. This suggests that screw dislocations are mainly introduced during martensitic transformation. As the amount of cold rolling is increased, q gradually decreases. This result proves that the fraction of edge dislocations increases with cold working. The authors observed the change in the martensite lath with TEM and found that the lath structure is gradually broken as the amount of cold working is increased.¹³⁾ Thus, increasing the edge component seems to be related with such microstructural changes in lath martensite.

Fig. 4.

(a) Classical WH plots and (b) modified WH plots in ultralow-carbon martensitic steel.

Fig. 5.

Change in q with cold rolling in ultralow-carbon martensitic steel.

On the other hand, Fig. 6 shows the changes in dislocation density that were obtained with the MWH/WA and CWH methods.¹⁾ There were two large differences: [1] an overall overestimation by the CWH method and [2] a drastic decrease in dislocation density by charging 5% cold rolling. The overestimation [1] was probably due to an incorrect conversion coefficient B,¹⁴⁾ which is caused by experimental or fitting errors. However, the much larger difference [2] cannot be explained solely by the overestimation. That is, some information other than dislocation density should be included in the FWHM values. In the CWH method, the dislocation density is estimated from the lattice strain ε under the assumption that the effective outer cutoff radius of the dislocation is constant, but it may change depending on the dislocation arrangement. In the MWH/WA method, the dislocation arrangement parameter M can be estimated separately from the dislocation density.

Fig. 6.

Change in dislocation density with cold rolling in ultralow-carbon martensitic steel.

Figure 7 shows the changes in the (a) effective outer cutoff radius of dislocation R_e′ and (b) dislocation arrangement parameter M that were measured with the MWH/WA method. Here, M is a dimensionless parameter expressed as a function of ρ and R_e′ (M = R_e′√ρ)¹⁰⁾ and indicates an interactive effect between dislocations. In the case of randomly distributed dislocations, large values (M > 1) should be obtained because the screening of the displacement field of dislocations is weak. However, if the dislocations have formed cell structure or dislocation dipole, M becomes smaller than unity (M < 1) because of the strong interaction of dislocations.^10,15,16) As shown in Fig. 7(a), R_e′ of as-quenched martensite is as large as 55 nm; it decreases drastically with the charging of small plastic strain. M also changes just like R_e′ against the applied plastic strain because the dislocation density is almost the same regardless of the amount of charged plastic strain. These results indicate that the distribution of dislocations is random in as-quenched martensite but changes to the cell structure with the charging of a small amount of plastic strain, as schematically displayed in Fig. 7(b). In other words, the dislocation arrangement is so unstable in the as-quenched state that it can easily be changed to a stable one by charging a small amount of plastic strain.

Fig. 7.

Change in (a) effective outer cutoff radius and (b) dislocation arrangement parameter with cold rolling in ultralow-carbon martensitic steel.

Figure 8 represents the TEM images of the (a) as-quenched specimen and (b) 20% cold-rolled one as observed from the rolling direction. The as-quenched specimen has a typical lath martensitic structure, and the dislocation arrangement is random, as shown in the microstructure in Fig. 8(a). On the other hand, a typical dislocation cell structure was observed in the cold-rolled martensite, as shown in Fig. 8(b). Note that the dislocation arrangement of cold-worked martensite is characterized by a tangled dislocation structure. As a result, the CWH method was found to overestimate the dislocation density, as shown in Fig. 6, due to the effect of the dislocation arrangement. However, the CWH method can be used to reasonably evaluate the dislocation density for cold-worked martensite.

Fig. 8.

TEM images of (a) as-quenched and (b) 20% cold-rolled ultralow-carbon martensitic steel observed parallel to the rolling direction.

3.2. Effect of Dislocations Characteristics on the Yielding of As-quenched Martensite

In order to clarify the effect of dislocation characteristics on the yielding behavior, tensile testing was carried out for as-quenched and 5% cold-rolled specimens. Figure 9 shows (a) the true stress–strain curves for both specimens and (b) the difference in plastic strain between them. Note that the yielding behavior was completely different in both specimens, even though they had almost the same dislocation density (2.1–2.2 × 10¹⁵ m⁻²). The elastic limit was very low in the as-quenched specimen, while it was sufficiently high in the 5% cold-rolled specimen. The yielding behavior corresponds to the results in the authors’ previous papers,^1,5) and the reason for the low elastic limit was been reported to be the mobile dislocations, which are unstably distributed and can move at low levels of stress in martensitic steel. In terms of the dislocation arrangements in Figs. 7(b) and 8, the randomly distributed dislocations are mobile dislocations in as-quenched martensite, but the tangled dislocation structure has a stable structure against deformation in cold-rolled martensite. Figure 8(b) reveals that a fairly large amount of plastic strain was produced in the as-quenched specimen as the applied stress was increased, and a plastic strain of 5.6 × 10⁻³ was introduced by the movement of mobile dislocations. In order to clarify the influence of mobile dislocations on the yielding behavior, the plastic stain was estimated as follows with regard to the movement of mobile dislocations.

Fig. 9.

(a) True stress–strain curves and (b) difference in true strain calculated from (a) for ultralow-carbon martensitic steel.

The plastic strain ε is known to be a function of the moving distance x:   

$$\epsilon ={\rho }_{m}bx/2$$(8)

where b is the Burgers vector of dislocation and ρ_m is the total length of dislocations that have moved in a unit volume. The above equation can be rewritten as follows to estimate ρ_m:   

$${\rho }_m=2\epsilon /bx$$(9)

By substituting the plastic strain ε = 5.6 × 10⁻³ into the above equation, ρ_m is given as a function of x. Figure 10 presents the calculated result. Based on the results in Fig. 6, the dislocation density in as-quenched martensite was estimated to be 2.1 × 10¹⁵ m⁻². When all dislocations have moved to produce the plastic strain of 5.6×10⁻³, they are estimated to have moved by only 21 nm. On the other hand, the mean spacing of dislocations was evaluated to be 38 nm (√(3/ρ) ≈ 38 nm) for as-quenched martensite. This result indicates that a plastic strain is easily produced by the slight movement of mobile dislocations, which results in the low elastic limit of martensitic steels. In order to obtain the substantial yield stress of martensitic steels, a slight deformation should be charged to change the dislocation arrangement from an unstable random distribution to a stable tangled distribution.

Fig. 10.

Relation between the density and distance of moving dislocations.

4. Conclusion

The MWH/WA method was used to investigate changes to the dislocation arrangement for as-quenched and cold-rolled specimens of ultralow-carbon lath martensite (Fe–18%Ni alloy). The results were as follows:

(1) As-quenched martensite is characterized by a random and unstable distribution of mobile dislocations, which causes the CWH method to tend to overestimate the dislocation density compared with the MWH/WA method.

(2) Because the dislocation arrangement is so unstable in as-quenched martensite, it can be changed from a random distribution to a tangled structure by charging a slight deformation. The low yield stress of as-quenched martensite was confirmed to be due to the plastic strain produced by a change in the dislocation arrangement.

Acknowledgments

The authors thank Prof. Shigeo Sato of Ibaraki University for his advice on the X-ray line profile analysis. This study was supported by a 25th ISIJ Research Promotion Grant from the Iron and Steel Institute of Japan.

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