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Work hardening, dislocation structure, and load partitioning in
Title

lath-martensite determined by in situ neutron diffraction line
profile analysis

Author(s)Citation
Text Version

Harjo S., Kawasaki Takuro, Tomota Yo, Gong W., Aizawa Kazuya,
Tichy G., Shi Z., Ungar T.
Metallurgical and Materials Transactions A, 48(9), p.4080-4092
Author's Post-print

URL

https://jopss.jaea.go.jp/search/servlet/search?5058393

DOI

https://doi.org/10.1007/s11661-017-4172-0
This is a post-peer-review, pre-copyedit version of an article

Rightpublished in Metallurgical and Materials Transactions A.
The final authenticated version is available online at:
http://dx.doi.org/10.1007/s11661-017-4172-0.

Japan Atomic Energy Agency

1

TITLE:

2

Work Hardening, Dislocation Structure and Load Partitioning in Lath Martensite

3

Determined by In Situ Neutron Diffraction Line Profile Analysis

4
5

AUTHOR NAMES and AFFILIATIONS:

6

Stefanus Harjo1, Takuro Kawasaki1, Yo Tomota2, Wu Gong1, Kazuya Aizawa1, Geza

7

Tichy3, Zengmin Shi4 and Tamas Ungár3,5

8
9
10
11

1

J-PARC Center, Japan Atomic Energy Agency, 2-4 Shirane Shirakata, Tokai-mura,Naka-gun, Ibaraki, 319-1195, Japan
2

Graduate School of Science and Engineering, Ibaraki University, 4-12-1,

12Nakanarusawa, Hitachi, Ibaraki, 316-8511, Japan (Present address: Research Center for

13

Structure Materials, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba,

14

Ibaraki 305-0047, Japan)

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16
17
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20

3

Department of Materials Physics, Eötvös University, Budapest, PO Box 32, H-1518,

Hungary
4

College of Materials and Chemical Engineering, China Three Gorges University, 8Daxue Rd, Xiling, Yichang, Hubei, China
5

Materials Performance Centre, The University of Manchester, M13 9PL, Manchester,

UK

21
22

CORRESPONDING AUTHOR:

23

Stefanus Harjo

24

Neutron Science Section, J-PARC Center, Japan Atomic Energy Agency

25

2-4 Shirane Shirakata, Tokai-mura, Naka-gun, Ibaraki, 319-1195, Japan

26

Email: stefanus.harjo@j-parc.jp

27

Tel: +81 29 2843266

Fax: +81 29 2843370

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29

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1

ABSTRACT2

A lath martensite steel containing 0.22 mass% carbon was analyzed in situ during

3

tensile deformation by high-resolution time-of-flight neutron diffraction to clarify the4large work-hardening behavior at the beginning of plastic deformation. The diffraction

5

peaks in plastically deformed states exhibit asymmetries as the reflection of

6

redistributions of the stress and dislocation densities/arrangements in two lath packets:

7

soft packet, where the dislocation glides are favorable, and hard packet, where they are

8

unfavorable. The dislocation density was as high as 1015 m−2 in the as-heat-treated state.

9

During tensile straining, the load and dislocation density became different between the10

two lath packets. The dislocation character and arrangement varied in the hard packet

11

but hardly changed in the soft packet. In the hard packet, dislocations that were mainly

12

screw-type in the as-heat-treated state, became primarily edge-type and rearranged

13

towards a dipole character related to constructing cell walls. The hard packet played an

14

important role in the work hardening in martensite, which could be understood by

15

considering the increase in dislocation density along with the change in dislocation

16

arrangement.

17
18

KEYWORDS:

19

Lath martensite; dislocation; work hardening; neutron diffraction; electron microscopy

20
21

2

1

1. Introduction

2Lath martensite steel is widely used in high-strength structural materials. It is3

obtained by quenching to room temperature (RT) from a temperature at which the4austenitic phase is stable. The martensitic phase transformation produces a fine-grained

5

structure with an extremely high dislocation density (> 1015 m−2).[1] The microstructure

6of lath martensite typically comprises several packets with different crystallographic

7

orientations in a prior austenite grain, where the packets are formed by several

8

blocks.[2,3] The blocks are subdivided into sub-blocks with the same variant, and the9smallest constituents are plate-like crystals called laths with sizes of several tens to10several hundreds of nm.

11

The elastic limit of an as-quenched Fe-18Ni lath martensite steel is relatively low

12(300 MPa), and the tensile strength is 760 MPa at a nominal strain of approximately

13

1.5%.[4] This indicates a very high level of work hardening after yielding at the14

beginning of plastic deformation. Cold rolling was reported to increase the elastic limit

15

substantially, resulting in higher 0.2% proof stress with increasing equivalent plastic

16

strain.[4] To explain this deformation behavior, the changes in dislocation density (ρ) in17

the cold-rolled and tensile-deformed lath martensitic Fe–18Ni alloys were measured by

18

X-ray diffraction (XRD)[4] and neutron diffraction (ND)[5] based on the classical19

Williamson-Hall (W-H) plot[6]. The ρ values were found to decrease with plastic

20

deformation, as evidenced by the decrease in the slopes of the classical W-H plots with21

plastic deformation.

22
23
24

In general, the change in flow stress (Δσ) attributed to dislocations can be evaluatedusing Taylor’s equation[7]:Δσ = σ - σ0 = α μ MT b √ρ ,

[1]

25

where σ is the flow stress attributed to dislocations, σ0 is the sum of the friction stress of

26

dislocations and the stress attributable to the effect of solute element strengthening, α is27

a geometric coefficient between zero and unity, μ is the shear modulus, MT is the Taylor

28

factor, which accounts for the effect of texture, and b is the Burgers vector.

29

The value of α is usually assumed to be unchanged during deformation; hence, the30

increase in ∆σ is caused solely by the increase in ρ, unless the grain size is very small.

31

Therefore, the decrease in ρ for lath martensitic Fe–18Ni alloy, as reported in

32

References 4 and 5, is puzzling. The results of ρ reported in References 4 and 5 remain

33

questionable despite the fact that the large ρ value invoked by martensitic

31transformation can decrease slightly as a result of plastic deformation, as reported in

2

Reference 8. Hutchingson et al.[9] carried out similar experiments but interpreted the3slopes of the classical W-H plots to indicate residual intragranular shear stresses

4

generated during martensitic transformation. They claimed that the residual

5

intragranular shear stresses were reduced in magnitude by plastic deformation,6subsequently controlling the stress-strain behavior. However, their interpretation is7

questionable when considering the diffraction profile analysis presented in this paper.

8

In situ ND is a powerful tool for clarifying phenomena in various engineering

9

applications.[10-17] We have reported in situ high-resolution ND experiments of a lath10

martensite steel containing 0.22 mass% carbon during tensile deformation.[17] We found

11

that the initial homogeneous lath structure was disrupted by plastic tensile deformation,

12

producing a composite on the length scale of martensite lath packets. The diffraction

13

profiles of plastically strained martensite steel revealed characteristically asymmetric as14

observed in materials with heterogeneous dislocation structures[18,19]. The diffraction

15

patterns were evaluated by the convolutional multiple whole profile (CMWP) procedure

16

based on physically modeled profile functions for dislocations, crystallite size, and17

planar defects.[20,21] The lath packets oriented favorably for dislocation glide became

18

soft (soft-packet orientation components, SO), and those unfavorably for dislocation

19

glide became hard (hard-packet orientation components, HO), causing dislocation

20

density to become smaller and larger compared to the initial average dislocation density,

21

respectively. The decomposition into SO and HO was accompanied by load

22

redistribution and the formation of long-range internal stress between the two lath

23

packets.

24

In the present work, which is the second part of Reference 17, the evolution of25

dislocation properties and lattice strain during tensile deformation is discussed in terms

26of the composite behavior of the lath-packet structure. The average dislocation densities

27

provided by neutron line profile analysis are compared with scanning transmission

28

electron microscopy (STEM) observations. The changes in dislocation character and29

dislocation arrangement during tensile deformation in the two types of lath packets are30

discussed in relation to work hardening. The work-hardening mechanism of the lath31martensite is further discussed by correlating the dislocation structure with the flow

32

stress in Taylor’s equation.

33

4

1
2

2. Experimental
The sample used in this study was a lath martensite steel with the chemical

3composition of Fe–0.22C–0.87Si–1.64Mn–0.024Ti–0.0015B–0.0025N (mass%).[22]4Specimens were prepared from a 20-mm-thick plate that was austenitized at 1173 K

5(900°C) for 3.6 ks, quenched, and then tempered at 453–473 K (180°C–200°C) for6

approximately 10.8 ks. The average packet and block sizes were 20 and 4 μm,

7respectively. A rod-shaped specimen with a diameter of 5 mm and a length of 15 mm

8

was prepared for in situ ND experiments during tensile testing using TAKUMI[23], a9high-resolution time-of-flight (TOF) neutron diffractometer for engineering materials

10

sciences at the Materials and Life Science Experimental Facility of the Japan Proton

11

Accelerator Research Complex.

12

Tensile deformation for in situ ND was performed in a stepwise manner with load

13

control in the elastic region, whereas in a continuous manner in the plastic region. The

14crosshead speed was constant (the strain rate was 10−5 s−1) in the plastic region. The

15strain was monitored by a strain gauge glued to the specimen. The deformation in the

16plastic region was increased stepwise to arbitrary strains followed by unloading. The

17

ND data were collected continuously using an event-recording mode during tensile18

deformation. Further details regarding the ND conditions are given in our previous

19

paper[17]. The diffraction patterns related to the step load-holding states, plastic

20

deformations, and unloaded states after plastic deformation were then extracted

21

according to the macroscopic stress and strain data. The macroscopic stress and strain

22

values relevant to the diffraction patterns were averaged over the interval times for data

23

extraction. Figure 1 shows the macroscopic stress–strain curve of the specimen. The

24

elastic limit was approximately 350 MPa; therefore, the rate of work hardening was

25

extremely high. In the macroscopic stress–strain curve obtained from continuous

26

loading under the same strain rate until fracture, a very high tensile strength of27

approximately 1.65 GPa and a uniform strain of approximately 6.1% were confirmed.

28

Data analyses for evaluating the lattice constant, phase fraction, and lattice strain

29

were performed using Z-Rietveld software[24], while dislocations were analyzed using

30

the CMWP procedure. The diffraction profiles of LaB6 powder measured under the31same conditions as the in situ ND measurements were used to determine the32

instrumental peak profiles for the dislocation analyses. Figure 2 shows the observed and33

Rietveld-calculated or CMWP-fitted ND patterns before tensile deformation. During the5

1

Rietveld or CMWP fitting, the second phase of γ was also analyzed to exclude its

2

influence on the results of the main phase of martensite. The data analyses using

3

Z-Rietveld were conducted for all diffraction patterns, whereas the dislocation analyses

4using the CMWP procedure were performed only on the diffraction profiles collected

5from the unloaded states after plastic deformation.

6

7
8

Figure 1 Macroscopic stress–strain curve of the lath martensite steel in this study.

9

10
11

Figure 2 The observed (black circles) and Rietveld-fitted [green line in (a)] or

12

CMWP-fitted [red line in (b)] ND profiles before tensile deformation. K = 1 / d, where d

13

is the lattice spacing. The blue line is the residual between the fitted and observed

14

profiles. The embedded figure in (a) or (b) shows the enlarged profile with log scale on15

the vertical axis for the high-index peak range. M and A indicate martensite and retained

16

austenite, respectively.

6

1(color for online only)

2
3

STEM observations were performed using an electron microscope (Tecnai G2F20)

4with bright field (BF) and annular dark field (ADF) modes operated at 200 kV. The

5thickness of the observation area in the STEM foil was estimated using electron

6

energy-loss spectroscopy[25], and the ρ value was determined using the linear

7

cross-sectioning method.

89

3. Results and discussion

10

3.1 Crystal structure and phase fraction

11

The crystal structure used in the Rietveld analysis [Figure 2(a)] for martensite was

12

BCC. The crystal structures of lath martensite steels with carbon contents below 0.6

13mass% were reported to be BCC at RT.[26] Although martensite in a Fe–30Ni–0.2C alloy14

was reported to have a BCT structure with a c/a ratio of approximately 1.02[27], the15sample used in this study was Ni-free, and the martensite peaks in Figure 2(a) were16

perfectly fitted using the TAKUMI instrumental profile shape function with a BCC

17

structure. A random texture was found in the as-heat-treated state (before tensile18

deformation) from the ratio of hkl peak integrated intensity. A weak α-fiber texture was19developed after 4.7% tensile deformation.

20

Retained austenite (γ) was confirmed in the specimen, as shown in Figure 2(a), and21

its fraction before tensile deformation was refined to be approximately 3.7%. The lattice

22

constants of martensite and γ were determined to be 0.28646(0) nm and 0.35912(3) nm,

23

respectively. Figure 3 shows the fractions of γ measured in the unloaded states after

24

plastic tensile deformation. The γ phase still existed after 4.7% tensile deformation, but

25

its fraction decreased to approximately 2.2%. A small amount of γ might transform to26martensite during plastic tensile deformation. The existence of γ was difficult to confirm

27

in the microscopy images, likely because of its very small size and/or martensitic

28

transformation during specimen preparation.

29

7

1
2

Figure 3 Fractions of retained austenite measured after plastic tensile deformation in the3

unloaded states.

45

3.2 Strain anisotropy and elastic anisotropy

6

The hkl-dependent Young’s modulus (Ehkl) values for martensite obtained from the7lattice strain results are summarized in Table 1. The Young’s modulus values in a cubic

8

crystal must follow the following linear relation[28]:91/Ehkl = B + FH2,

10where B and F are constants, and H2 is the fourth order invariant of hkl, H2 = (h2k2 +11h2l2 + k2l2) / (h2 + k2 + l2)2. The inverses of the measured Ehkl values are plotted versus

12

H2 in Figure 4, indicating that Eq. [2] was fulfilled perfectly within the experimental

13

errors with B = 0.0059 and F = −0.0062. B and F are related to the elastic constants (c11,14c12, and c44) as follows [28]:

15

=

[2]

and

=

−

.

[3]

16

B and F are clearly insufficient to provide three elastic constants without any further

17

information. Fortunately, we know that the c44/c12 ratio for metals is usually between 0.5

18

and 0.7.[29] Taking c44/c12 = 0.6, using Eq. [3], the values of B and F provide the elastic

19

constants for the martensite investigated here:20

c11 = 283(5) GPa, c12 = 161(4) GPa, and c44 = 97(4) GPa.

21

[4]

With these elastic constants, the elastic anisotropy (A) of our martensite material was

22

determined to be 1.59. The A value of α-Fe is 2.4.[30] The A value of a martensite steel

23

investigated in Reference 30 was 1.01. However, the compositions of the martensite

24

investigated here and that reported in Reference 30 are different. The composition of the8

1

present martensite steel is Fe–0.22C–0.87Si–1.64Mn–0.024Ti (mass%), whereas the2composition of the steel reported in Reference 30 is Fe–0.52C–0.22Si–1.0Mn–0.3Al

3(mass%). The A value of 1.59 is between the values of α-Fe and the martensite steel in4

Reference 30. This indicates that the elastic anisotropy is rather sensitive to the5composition and probably the exact quenching conditions of martensitic steel.

6
7

5

-1

1/Ehkl (10 GPa )

6

-3

4
3
2
1
0
0.00

0.10

0.20

8

0.30

2

7

H

Figure 4 Measured 1/Ehkl values versus H2.

910Strain anisotropy line broadening means that the full width at half maximum

11(FWHM) values of the diffraction peaks are not a monotonic function of diffraction

12order.[31] Figure 5(a) shows the FWHM values of martensite steel before deformation

13

and with 0.6%, 3%, and 4.7% tensile deformation versus K = 1/d, where d is the lattice14

spacing. The FWHM values were evaluated by a Gaussian function from the physical15

profiles of the diffraction peaks that are free from instrumental effects, as provided by

16

the CMWP procedure. The increase in FWHM with K indicates substantial microstrain17

caused by the large dislocation density. The apparent scatter of the FWHM values

18

around the global ascending trend is typical for strain anisotropy. Strain anisotropy can

19

be rectified by accounting for the hkl-dependent dislocation contrast C(hkl).[31] In

20polycrystalline cubic materials, C(hkl) can be averaged over the permutations of hkl and22

written as [32]
̅= ̅
1–

23

where

24

depends on the dislocation character (e.g., screw- or edge-type) and the elastic

25

anisotropy of the material. In References 31 and 33, the apparently irregular behavior of

21

̅

,

[5]

is the average contrast for h00-type reflections, and q is a parameter that92

the FWHM values in the conventional W-H plot was rectified when K was replaced byK√ ̅ in the modified W-H plot. The irregular behavior of the FWHM values in Figure

35(a) was rectified when q was 1.7, as shown in Figure 5(b). According to the theoretical

4computation for BCC with a slip system of <111> {110}, A = 1.6, and c44/c12 = 0.6, q

5values of 0.2 and 2.5 correspond to edge-type and screw-type dislocations,6

respectively.[33] Therefore, the q value of 1.7 in Figure 5(b) indicates that the7dislocations have a mixed edge and screw character with screw type being dominant.

8

Figure 5(b) shows that the FWHM values follow a perfect straight line, confirming the9evaluation of the elastic constants and the q value of 1.7. According to a TEM study[34],10

a dislocated martensite structure consists of two kinds of dislocations: straight11

screw-type dislocations induced by lattice invariant shear and tangled dislocations

12

generated in the austenite matrix to relax the internal stress caused by transformation

13

strain. The tangled dislocations are inherited in martensite. This TEM work supports the14

obtained q value along with the mixture of screw- and edge-type dislocations in

15

as-quenched martensite.

1

16

The slopes of the straight lines in Figure 5(b) decrease slightly with increasing

17

macroscopic strain. It is important to note here that the profile does contain the width

18

part and the tail part. The tail is however ignored in the FWHM value. The decrease in

19

FWHM was also accompanied by changes in peak shape from Gaussian to Lorentzian.

20

This peak shape change might be associated with the change in dislocation arrangement.

21

The dislocation densities, characteristics, and arrangements evaluated by analyzing the22whole profile using the CMWP procedure will be discussed in detail in the next

23

sections.

24
25

10

1
2

Figure 5 (a) FWHM values of the physical profiles free from instrumental effects (as3provided by CMWP analysis) versus K=1/d for martensite steel before deformation and

45

at after 0.6%, 3%, and 4.7% tensile deformation. (b) The same FWHM values as in (a)versus K√ ̅ in the modified W-H plot with q = 1.7.

6

(color for online only)

7
8

3.3 Dislocation densities based on CMWP analysis assuming symmetrical peak profiles

9and STEM observation

10

In this section, we first explain the results of the CMWP analysis under the11

assumption that a symmetrical peak profile was maintained throughout tensile12

deformation, although we reported that the symmetrical diffraction profiles before

13

tensile deformation became asymmetric as a result of plastic strain.[17] This analysis was

14

performed to obtain average dislocation densities and compare them with the15dislocation densities based on STEM observations and the CMWP analysis considering

16

peak asymmetry (described later).

17

The average values of ρ (ρave) in the axial direction are summarized in Figure 6. The

18

parameters are labeled as averages here to express the results from all packets regardless

19of the presence of SO and HO. The value of ρave before tensile deformation were20

already high (approximately 4.0 × 1015 m−2). This value is consistent with that reported

21

for a lath martensite steel with a similar carbon content (0.18 mass%) determined using

22

TEM[35]. This high value is attributed to martensitic transformation, which is difficult to23

achieve by plastic tensile deformation. The value of ρave changed slightly with24

increasing macroscopic strain, although an increase in flow stress was observed. These25

ρave values lie on the same experimental curves as those obtained in cold-rolled lath11

1martensite steel plates when they were replotted as a function of the equivalent plastic

2

strain.

3

4
5

Figure 6 Dislocation densities obtained from CMWP fitting assuming a symmetrical6

peak profile in the axial direction.

7
8

TEM observations were used to confirm the change in dislocation density, although

9

the CMWP fitting of TOF ND profiles was already demonstrated to be reliable[36].

10

Figure 7(a) shows the STEM-BF and STEM-ADF images obtained from a specimen

11

before tensile deformation, and Figure 7(b) shows the images after 4.7% tensile12

deformation. The dislocation densities were determined using five ADF images; three

13

images with the incident beam parallel to <111> and two images with the incident beam

14

parallel to <001> (all a/3 <111>-type dislocations were visible under these incident

15

beam conditions). The ρave value before tensile deformation was determined to be16

between 8.79 × 1014 and 1.48 × 1015 m−2 (average = 1.17 × 1015 m−2), which was quite

17

close to the TEM-based value reported by Morito et al.[35] for a lath martensite steel

18

with a similar carbon concentration (average = 1.11 × 1015 m−2 in an Fe–0.18C steel).

19

Meanwhile, the ρave value after 4.7% tensile deformation was determined to be between

20

9.05 × 1014 and 1.45 × 1015 m−2 (average = 1.18 × 1015 m−2), indicating no significant

21

difference between the two conditions. These values are smaller than those determined

22

by the CMWP method using the ND profiles presented in Figure 6. The dislocation

23

densities determined by TEM are lower than those determined by diffraction methods in24

many cases. In our case, this is because the present TEM observations mainly counted

25

dislocations located inside of lathes, whereas the CMWP method evaluated all12

1

dislocations, including those at the sub-boundaries. Huang et al.[37] reported that the2

total dislocation density in lath martensite of an interstitial free steel containing Mn and3

B is the sum of the dislocations in sub-block boundaries (2 × 1014 m−2), in lath4boundaries (3 × 1014 m−2; they are called dislocation boundaries in Reference 37), and5in the volume between boundaries (3 × 1014 m−2). They evaluated dislocation

6

boundaries using the misorientation angle of the sub-block or lath boundary and the7boundary area per unit area of sub-block or lath. Because the steel used in the present

8

study contained 0.22 mass% carbon, the dislocation boundaries must be higher than9

those reported by Huang et al.[37]. Hence, the total dislocation density can be roughly

10

estimated to be three times higher than that inside of laths. In conclusion, the results

11

confirm that the change in ρave during tensile deformation was small and did not exhibit

12

a decreasing trend. The decreasing ρ value with deformation progress determined using

13

the classical W-H plot based on peak width reported in References 4 and 5 might be14erroneous because the entire peak shape (including the tail part) was not taken into

15

account in the analysis.

16

17
18

Figure 7 STEM images (a) before tensile deformation and (b) after 4.7% tensile19

deformation. The incident beam was parallel to the <001> orientation.

20

13

1

3.4 Dislocation density and dislocation character obtained by CMWP analysis with

2

dual-packet contribution

3

As described in our previous paper[17], the diffraction profiles of plastically strained

4

martensite steel revealed characteristically asymmetric. We have proposed a fitting

5

procedure to analyze the ND patterns in the unloaded states after plastic tensile6

deformation using a dual-packet contribution composed of two BCC structures in the

7

CMWP analyses. The details are described in Reference 17. This fitting procedure was

8

supported by a crystallographic relationship in low carbon martensite [i.e., the prior9

austenite (111) plane is parallel to the martensite (110) plane, and the habit plane of lath10

martensite is nearly (110)].[2,3] For example, the orientation difference in the diffracted11(110) plane with respect to the lath boundary [another (110)] is either 60° or 90°, and in12

the diffracted (200) plane, 45° or 90°. However, these analyses could not be performed

13

for the ND patterns taken during loading because the statistical accuracy of the data was

14

insufficient. The fraction of HO (fHO) was found to be approximately 50% and was

15

unchanged during tensile deformation.

16

Figure 8(a) shows the dislocation densities in the packet components (ρHO for HO

17and ρSO for SO) obtained from the CMWP fitting assuming dual-packet contribution.

18

The ρHO value increased with increasing macroscopic strain up to the order of 1016 m−2,19whereas the ρSO value decreased rapidly at the beginning of deformation to on the order20of 1014 m−2 and then hardly changed. Further details regarding ρHO and ρSO are reported

21

in our previous paper[17]. The total average dislocation density (ρt) calculated from the22

ρHO and ρSO values as the weighted average according ρt = fHO ρHO + (1 – fHO) ρSO

23

showed a similar tendency as the ρave value shown in Figure 6 but with slightly larger

24

values. It is important to note here that the ρave values in Figure 6 were obtained by the25

CMWP procedure assuming a symmetrical profile, whereas ρHO and ρSO were provided

26

by allowing the existence of two different packet populations. Using this procedure, the27

asymmetries in the peak profiles were correctly taken into account, and the obtained28

results are considered to be physically correct.

29

Figure 8(b) shows the values of q for HO and SO (qHO and qSO, respectively). The q

30

value obtained before tensile deformation was approximately 1.7, indicating that before

31

tensile deformation, the dislocations were of mixed edge- and screw-type with a larger

32

proportion of screw-type dislocations. Screw-type dislocations are mainly found in

33

BCC polycrystalline materials.[33,38] The qSO values were almost unchanged with14

1

deformation from the state before tensile deformation, indicating that dislocations with2

screw character were dominant in the SO. In contrast, the qHO value decreased largely at3the beginning of tensile deformation to be approximately 0.6, indicating that the4

proportion of edge dislocations increased in the HO. These results support the5

simulation results reported in our previous paper (Table 1 in Reference 17). Screw6

dislocations can move in any direction and therefore are annihilated relatively easily,

7even when they are far apart from each other.[39] Edge dislocations must either glide on

8

slip planes or climb to be annihilated and therefore are only annihilated within short9

distances.[39]

10

11
12

Figure 8 (a) Dislocation density and (b) parameter depending on the dislocation

13

character (q) in the HO or SO obtained from CMWP fitting assuming multi-packet

14

contribution in the axial direction.

15(color for online only)

16
17

The relatively unchanged q value of 1.7 and the decreasing dislocation density

18

during deformation in the SO are consistent with the results of the modified W-H plot,15

1

as described in Section 3.2, in which good linearity was maintained with q = 1.7, and2

the slopes decreased slightly with increasing macroscopic strain. Therefore, the FWHM

3values of the profiles are mainly of the profile parts of the SO. As shown in our previous

4

paper [Figures 5(c) and 5(d) in Reference 17], the total physical diffraction profiles in5

the plastically tensile-deformed martensite consisted of two peaks. The peak with larger

6

intensity and smaller FWHM corresponded to the SO, whereas the other peak with7

smaller intensity and larger FWHM corresponded to the HO. The FWHM values shown8

in Figure 5 clearly correspond to the peaks with larger intensity, for which the FWHM

9values decreased slightly with strain.

1011

3.5 Dislocation arrangement and crystallite size based on CMWP analysis

12

The parameter M, which is the product of the effective cutoff radius of dislocation

13(Re) and the square root of ρ (M = Re√ρ), indicates the dislocation arrangement.[20] A

14small or large value of M indicates that the dipole character and the screening of the15

displacement field of dislocations are strong or weak, respectively.

16

Figure 9 shows the average values of M (Mave) obtained from CMWP fitting

17

assuming a symmetrical peak profile and M value corresponding to the HO (MHO)18

obtained from the CMWP fitting assuming multi-packet contribution. The values of19

Mave and MHO were large before tensile deformation. They decreased rapidly at the20

beginning of deformation and then gradually decreased with the progress of tensile21

deformation, finally becoming less than 1.0. Meanwhile, the values of M for SO (MSO)

22

remained large during tensile deformation. The large values of MSO suggest that it has

23

little effect on dislocation density, which can be attributed to the balanced competition

24of dislocation generation and annihilation, resulting in small work softening. The values

25of Mave were consistent with those of MHO within the analytical error. Therefore, the26

profile shapes corresponding to Re or M can be concluded to mainly be the profile parts

27of the HO. The decrease in MHO indicates that the dislocations in the HO rearranged

28

towards a configuration with a stronger dipole character of dislocation. A similar

29

tendency for M with respect to the reduction in thickness was also observed by XRD in30

a carbon-free Fe–18Ni alloy after cold rolling[40]. These results suggest that the31interactions between dislocations and solute carbon atoms do not affect the32

re-arrangement of dislocations during RT deformation.

33

16

1
2

Figure 9 Average arrangement parameter M obtained from CMWP fitting assuming a

3

symmetrical peak profile and parameter M in the HO obtained from CMWP fitting

4

assuming multi-packet contribution in the axial direction.

5
6

Figure 10 shows the area-weighted average crystallite size, which is relevant to the7subgrain size in the HO in the present case. The subgrain size decreased with increasing

8

macroscopic strain. TEM studies indicated that the lath martensite structure changes to a9

deformation cell structure with plastic deformation.[4,40-42] The lath boundaries became

10

difficult to be distinguish and changed to cell structures with dense dislocation walls

11

after cold rolling. These findings indicate that the dislocation cell boundaries increased,

12

while the subgrain size decreased. Therefore, the results in Figure 10 are in good13

agreement with these previous TEM works. The decreasing trend in the subgrain size in

14

the HO (Figure 10) is similar to the decreasing trend in MHO shown in Figure 9.

15

Therefore, the decrease in MHO indicates that two effects (i.e., increasing dipole

16character of the dislocation structure and decreasing subgrain size related to the17formation of dislocation cells) acted simultaneously. Decreasing trends in both M and18

crystallite size were also observed by Stráská et al.[43] in a magnesium alloy processed19

by high-pressure torsion.

20

17

Area-weighted average crystallite size (nm)

300

Axial
HO

250

200

150

100

50
0

1
2

1

2
3
Macroscopic true strain (%)

4

Figure 10 Area-weighted average crystallite size (subgrain size) in the HO.

34

3.6 Lattice strain

5First, all ND patterns were fitted using Z-Rietveld assuming a symmetrical peak6

profile to determine the average lattice constants and peak positions. The lattice strain

7

can be evaluated from the peak shift according to the following equation:8

εhkl = (dhkl – d0hkl) / d0hkl,[6]9

where ε, d, and d0 are the lattice strain, measured lattice spacing, and reference lattice10

spacing, respectively. The lattice spacing determined before tensile deformation was

11

used as d0. Figure 11 shows the lattice strains in the axial direction measured for12martensite and γ. In Figure 11(a), all martensite-hkl lattice strain responses to the13

macroscopic stress deviated from linearity to have smaller rates of increase. In contrast,

14

the γ <311> lattice strains had larger values than the martensite lattice strains at the15

related macroscopic stresses. Note that the <311> lattice strain represents the bulky

16

elastic strain for FCC polycrystalline materials.[10,15] In Figure 11(b), the average

17

residual lattice strain in the unloaded state after plastic tensile deformation for

18

martensite that was averaged over <hkl> decreased and became compressive with19

increasing macroscopic strain, whereas that for γ increased in the opposite tensile

20

direction. These results indicate that γ plays the role of the hard phase in the material

21

used in this study. Similar behaviors have been observed in transformation-induced

22

plasticity-aided multiphase steels[12,14]. In these steels, retained austenites show higher

23

flow stress than the ferrite–bainite matrix because of carbon enrichment. This effect was24

not observed in the lath martensite steel used in this study because carbon enrichment

25

was minor. Similar behavior was observed in Fe–Cu alloy[16], in which tiny copper

18

1

precipitates behaved as the hard phase despite the low flow stress at the elasto-plastic

2

deformation in copper polycrystalline aggregates[13]. Extremely small austenite particles

3

embedded in the strong martensite matrix have been speculated to exhibit high flow

4

resistance similar to the tiny Cu particles in iron. However, the martensite lattice strains5

are still maintained in the increasing tendency with increasing macroscopic stress,

6

indicating work hardening.

7

8
9

Figure 11 (a) Lattice strains measured during tensile deformation and (b) residual lattice10strains measured in unloaded states after plastic tensile deformation in the axial

11

direction. M and A indicate martensite and retained austenite, respectively.

12(color for online only)

13
14

Next, the ND patterns of the unloaded states after plastic tensile deformation were15

analyzed to determine the peak positions of the SO and HO based on a dual-packet

16

contribution composed of two BCC structures in both the Z-Rietveld and CMWP

17

analyses. Figure 12 shows the fits obtained using Z-Rietveld. The fit was improved by18

using two sub-peaks corresponding to contributions from SO and HO. The SO sub-peak

19

had a higher intensity and smaller FWHM value, while the HO sub-peak had a lower

20

intensity and larger FWHM value.

21

19

1
2

Figure 12 Martensite-200 diffraction profiles in the 4.2%-deformed state in the axial

3

direction. (a) Measured and Z-Rietveld-calculated profiles assuming a symmetrical4peak-profile. (b) Measured and Z-Rietveld-calculated profiles assuming a dual-packet

5

contribution composed of two BCC structures. The sub-profiles in (b) correspond to SO

6

and HO. The peak positions of the calculated profiles are shown with vertical bars. M

7indicates martensite.

8(color for online only)

9
10

Residual strains operating in the two components of lath martensite, SO and HO,

11were computed using a composite model assuming zero stress balance. In fact, the12balances of residual strains in the SO and the HO are the average residual lattice strains13

for martensite shown in Figure 11(b) because of the presence of γ. Figure 13 shows the14

residual component strains in the SO and the HO measured in the unloaded states after

15

plastic tensile deformation in the axial direction. The results obtained from both the16

Rietveld and CMWP analyses were in good agreement within the analytical error. The

20

1

residual component strains in the SO were compressive, whereas those in the HO were2

tensile, and their absolute values became larger with increasing macroscopic strain. This3

indicates that work softening occurs in the SO as opposed to work hardening in the HO.

4

The increases in the residual component strain values in the SO and HO became small

5

at macroscopic strain values above approximately 2.5%, and the increase in flow stress6(Figure 1) was also small. The difference in the residual component strain at the largest

7

macroscopic true strain was approximately 0.29% (570 MPa).

8

Average residual component strain

0.002

0.001
Axial
HO
SO
Hollow: Rietveld, Filled: CMWP

0.000

-0.001-0.002
0

9

1

2

3

4

Macroscopic true strain (%)

10

Figure 13 Residual component strain as a function of macroscopic strain in the HO and11

SO analyzed using the Rietveld and CMWP methods.

12(color for online only)

13
14

Figure 14 shows the lattice strain distribution among γ, SO, and HO, which was

15evaluated as follows. The lattice strain responses to macroscopic stress in Figure 11(a)16were averaged and smooth-interpolated to determine the phase strain and phase stress of17martensite. The stress balances of residual component stresses in the SO and the HO

18

were considered to be the martensite phase stresses for the related macroscopic stresses

19

by assuming that the Young’s moduli of SO and HO were identical, and that no20

stress-relaxation occurred during unloading. The lattice strain distribution reflects the21

partitioning of load among γ, SO, and HO. The lattice strain of γ showed the largest

22

value during macroscopic plastic tensile deformation; however, its contribution to the23

entire flow stress was less than 6% because of its small volume fraction. Therefore, the24

HO is considered to play the most important role in work hardening in this specimen

25

during tensile deformation.

21

1

2
3

Figure 14 (a) Lattice strain distribution during tensile deformation estimated from the4lattice strains in Figure 11(a) and the residual component strains in Figure 13. M and A

5

indicate martensite and retained austenite, respectively. (b) The relevant macroscopic

6

stress–strain data.

7(color for online only)

89
10

3.7 The α coefficient in Taylor’s equation
Since the average dislocation densities in the present lath martensite steel were found11

to hardly change during plastic tensile deformation, the observed large work hardening

12

was hypothesized to be related to an increase in the α coefficient in Taylor’s equation.

13

The α coefficients for HO (αHO) and for SO (αSO) for this specimen can be estimated14from the macroscopic stress−strain curve and the values of ρHO and ρSO based on a

15

composite model using the following equation:16

Δσ = σ - σ0 = μ MT b (fHO αHO √ρHO + (1 – fHO) αSO √ρSO) .

[7]

17

The values of σ0, μ, MT, and b used in the calculations were 350 MPa, 77.3 GPa, 2.8,

18

and 0.248 nm, respectively. The αSO value at the beginning of deformation was

19determined to be approximately 0.18 and was fixed during further tensile deformation20

because of the work softening in the SO.

21

Figure 15 shows the calculated αHO values. The value of αHO clearly increased

22

rapidly at the beginning of plastic deformation and then gradually varied with the23

progress of tensile deformation. The αHO value saturated at approximately 0.4, which is24

the value frequently used for metallic materials[44]. However, although the values of α

25vary widely[45,46], α is considered to be constant during deformation in many studies[4,45–

22

1

47]

. The α coefficient is determined from the angle between adjacent dislocation

2

segments at a point where the dislocation breaks free from an obstacle.[48] In an in situ

3

ND study during the tensile loading of a stainless steel, the α coefficients were found to4

differ depending on the individual <hkl> grain families.[36] The α coefficient was large

5

in <hkl> grain families with larger Schmid factors, in which dislocations were arranged

6

in longitudinal bands frequently divided by sub-boundaries, and low in the other7

families with smaller Schmid factors, in which the cell structure was evolved.[36]

8

9
10

Figure15 Values of α calculated from the dislocation densities according to Taylor’s

11equation (Eq. [7]) and its relationships with the change in flow stress caused by

12

dislocations and the parameter M determined from the stress–strain curve for the HO.

13(color for online only)

14
15

The values of Δσ and MHO are superimposed in Figure 15. Note that the vertical axis

16

depicting MHO in Figure 15 is in reverse order. Thus, in Figure 15, a rapid increase in Δσ

17value is proportional to a rapid decrease in MHO, which is related to an increase in αHO.

18

Schafler et al.[49] also reported that M can be linked to α in Taylor’s equation of flow

19

stress, although their results did not indicate a direct relationship. The change in α with

20changes in dislocation arrangement during plastic tensile deformation was recently

21

discussed in detail by Mughrabi[50]. According to Mughrabi’s composite model, α is

22

proportional to the square root of the cell wall volume fraction, where an increase in cell

23

wall volume fraction increases α. Hence, the decrease in MHO with increasing plastic

24

deformation suggests that the dislocations are rearranged, becoming dipole character25

related to constructing cell walls, and αHO increases as a result.

23

1
2

4. Conclusions

3

In situ ND was performed during the tensile deformation of a lath martensite steel

4containing 0.22 mass% carbon using a high-resolution TOF neutron diffractometer. The

5

sample showed extremely large work hardening at the beginning of plastic deformation.

6

The results are summarized as follows.

7(1) The dislocation density of the lath martensite in the as-heat-treated state was on the8order of 1015 m−2. The average dislocation density obtained from CMWP analysis

9changed little during tensile deformation, in good agreement with the STEM

10

observations of microstructure.

11(2) The diffraction peaks in the plastically deformed states were asymmetric, reflecting

12

the partitioning of load and different dislocation densities/arrangements in the two

13lath packets: SO, where dislocation glides are favorable, and HO, where they are

14

unfavorable. During tensile straining, the dislocation density increased in the HO

15

accompanied by an increase in load sharing, indicating work hardening. In contrast,16

the dislocation density decreased in the SO, indicating work softening. The

17

dislocation character and arrangement varied in the HO but hardly changed in the18

SO. In the HO, the dislocations in the as-heat-treated state, which were mainly

19

screw-type, became primarily edge-type and rearranged towards a dipole character

20

related to constructing cell walls.

2122(3) The HO played an important role in work hardening in the lath martensite steel
during tensile deformation.

23(4) The extremely large work hardening could not be sufficiently accounted for by the24

increase in dislocation density; it was also necessary to consider the change in25

dislocation arrangement. Dislocation arrangement could be accounted for through26

the α coefficient in Taylor’s equation, which could be estimated from the variation

27

in M determined by CMWP analysis.

28
29

ACKNOWLEDGMENTS30

The authors acknowledge Prof. Shigeo Sato of Ibaraki University for the valuable31

discussion. ND experiments were performed at BL19 in the Materials and Life Science32

Facility of J-PARC under proposals 2014I0019 and 2014P0102. Partial financial

33

support was provided by the Japan Society for the Promotion of Science Grant-in-Aid24

1

for Scientific Research (grant numbers 26289264, 15H05767) and Education

2

Commission of Hubei Province of China (grant number B20161203).

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20

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21
22
23

Table 1 The values of hkl-dependent Young’s modulus (Ehkl).
hkl

110

200

211

220

310

222

Ehkl

233(1) 167(2) 233(2) 229(3) 183(3) 250(8)

24
25
26

FIGURE CAPTIONS:

27

Figure 1 Macroscopic stress–strain curve of the lath martensite steel in this study.

28
29

Figure 2 The observed (black circles) and Rietveld-fitted [green line in (a)] or

30

CMWP-fitted [red line in (b)] ND profiles before tensile deformation. K = 1 / d, where d

31

is the lattice spacing. The blue line is the residual between the fitted and observed27

1

profiles. The embedded figure in (a) or (b) shows the enlarged profile with log scale on2

the vertical axis for the high-index peak range. M and A indicate martensite and retained3

austenite, respectively.

4
5

Figure 3 Fractions of retained austenite measured after plastic tensile deformation in the6unloaded states.

7
8

Figure 4 Measured 1/Ehkl values versus H2.

910

Figure 5 (a) FWHM values of the physical profiles free from instrumental effects (as11provided by CMWP analysis) versus K=1/d for martensite steel before deformation and12

at after 0.6%, 3%, and 4.7% tensile deformation. (b) The same FWHM values as in (a)versus K√ ̅ in the modified W-H plot with q = 1.7.

13
14
15

Figure 6 Dislocation densities obtained from CMWP fitting assuming a symmetrical

16

peak profile in the axial direction.

17
18

Figure 7 STEM images (a) before tensile deformation and (b) after 4.7% tensile19

deformation. The incident beam was parallel to the <001> orientation.

20
21

Figure 8 (a) Dislocation density and (b) parameter depending on the dislocation

22

character (q) in the HO or SO obtained from CMWP fitting assuming multi-packet

23

contribution in the axial direction.

24
25

Figure 9 Average arrangement parameter M obtained from CMWP fitting assuming a

26

symmetrical peak profile and parameter M in the HO obtained from CMWP fitting

27

assuming multi-packet contribution in the axial direction.

28
29

Figure 10 Area-weighted average crystallite size (subgrain size) in the HO.

30
31

Figure 11 (a) Lattice strains measured during tensile deformation and (b) residual lattice32

strains measured in unloaded states after plastic tensile deformation in the axial

33

direction. M and A indicate martensite and retained austenite, respectively.

28

1
2

Figure 12 Martensite-200 diffraction profiles in the 4.2%-deformed state in the axial

3

direction. (a) Measured and Z-Rietveld-calculated profiles assuming a symmetrical4peak-profile. (b) Measured and Z-Rietveld-calculated profiles assuming a dual-packet

5

contribution composed of two BCC structures. The sub-profiles in (b) correspond to SO

6

and HO. The peak positions of the calculated profiles are shown with vertical bars. M

7indicates martensite.

8
9

Figure 13 Residual component strain as a function of macroscopic strain in the HO and10

SO analyzed using the Rietveld and CMWP methods.

1112

Figure 14 (a) Lattice strain distribution during tensile deformation estimated from the13

lattice strains in Figure 11(a) and the residual component strains in Figure 13. M and A

14

indicate martensite and retained austenite, respectively. (b) The relevant macroscopic

15

stress–strain data.

16
17

Figure15 Values of α calculated from the dislocation densities according to Taylor’s

18

equation (Eq. [7]) and its relationships with the change in flow stress caused by

19

dislocations and the parameter M determined from the stress–strain curve for the HO.

20

29

50

0.06
0.08
0.10

100

0.12

0.10

0.14

d (nm)
0.16
0.18

A 111

0.09

A 200

M 200

0.08

A 220

M 211

0.07

A 311

A 311

A 222

M 220

M 310

M 222

M 321

(a)
Rietveld

M 220

A 420

M 400

150

M 310

M 420 M 332
M 411
M 400
M 321
M 222

Intensity (a.u.)
200
2

M 110

4

10

8

6

4

2

0.11

0

0.20
0.22

4

50

6
8
10

K (1/nm)
12
14

M 332

13
M 400

A 422

A 420

10

M 420

12

M 411

11
M 321

10

M 400

9

M 222

100
A 222

4

M 310

6

A 311

8

M 211

150

M 220

M 200

M 110

M 321

M 222

M 310

M 220

2

A 311

A 220

0
A 200

A 111

200
4

(b)
CMWP

2

14

16

0.06

Retained austenite fraction

0.05
0.04
0.03
0.02
0.01
0.00
0

1

2
3
Macroscopic true strain (%)

4

7

5

-1

1/Ehkl (10 GPa )

6

-3

4
3
2
1
0
0.00

0.10

0.20
2

H

0.30

0.20

FWHM (1/nm)

0.15

(a)
Before deformation
0.6% deformed
3.0% deformed
4.7% deformed

0.10

0.05

0.00
6

8

10

12

K (1/nm)

14

16

(b)
Before deformation
0.6% deformed
3.0% deformed
4.7% deformed

q = 1.7

2

3

4

5
1/2

KC

(1/nm)

6

7

10

m )

7

Dislocation density (10

-2

8

15

9

ave

Axial

6
5
4
3
2
1
0
0

1

2

3

Macroscopic true strain (%)

4

-2

Dislocation density (10 m )

15

14

12

10

8

6

4

2

0

(a)
Axial

2.5

(b)

Axial

q value

2.0

1.5

1.0

0.5

0.0
0

1

2
3
Macroscopic true strain (%)

4

10
Axial
9
Mave
MHO

8

Parameter M

7
6
5
4
3
2
1
0
0

1

2
3
Macroscopic true strain (%)

4

Area-weighted average crystallite size (nm)

300
Axial
HO
250

200

150

100

50
0

1

2
3
Macroscopic true strain (%)

4

1600

(a)
Lattice strain

Macroscopic true stress (MPa)

1400
1200
1000
800
600

Axial
M-110
M-200
M-211
M-310
A-311

400
200
0
0.000

0.005

0.010
Lattice strain

0.015

0.008(b)

0.006
Residual lattice strain

Axial

Residual lattice strain in unloaded state
after plastic deformation
M-average
A-311

0.004

0.002

0.000

-0.00201

23Macroscopic true strain (%)

4

30
25

Intensity (a.u.)

20
15
10
5
0

(a)
measured
Z-Rietveld
calculated
residual

4.2 % deformed
M-200
Axial

30

(b)

25

Intensity (a.u.)20

measured
Z-Rietveld
calculated
SO sub-peakHO sub peakresidual

4.2 % deformed
M-200
Axial

15
10
5
0
0.140

0.142

0.144
d (nm)

0.146

Average residual component strain

0.002

0.001
Axial
HO
SO
Hollow: Rietveld, Filled: CMWP

0.000

-0.001-0.00201

23Macroscopic true strain (%)

4

1600

(a)

Macroscopic true stress (MPa)

1400
1200
1000
800
600
400
M-HO
M-SO
A-phase

200
0
0.000

0.002

0.004 0.006 0.008
Lattice strain

0.010

0.012

(b)

0
1
2
3
4
5
Macroscopic true strain (%)

1400

0.3

800

value (MPa)

1000

2

3

0.2
Coefficient HO∆ valueParameter MHO

0.1
0

1

2
3
Macroscopic true strain (%)

4

600

400

4

5

Parameter M

1

1200

∆

coefficient0.4