Scripta Materialia 177 (2020) 176–180

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Scripta Materialia
journal homepage: www.elsevier.com/locate/scriptamat

Effect of dislocation distribution on the yield stress in ferritic steel

under identical dislocation density conditions
Yuki Tanaka a,∗, Takuro Masumura b,c, Toshihiro Tsuchiyama b,c,d, Setsuo Takaki b
a
School of Engineering Technical Division, Kyushu University, 744, Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan
b
Research Center for Steel, Kyushu University, 744, Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan
c
International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu University, 744, Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan
d
Department of Materials Science and Engineering, Kyushu University, 744, Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan

a r t i c l e i n f o a b s t r a c t

Article history: Yield stresses of ferritic iron pre-deformed at 298 K and 213 K were compared under an identical dis-
Received 1 August 2019 location density. The specimens pre-deformed at 213 K introduced uniformly distributed dislocations,
Revised 25 September 2019
while those at 298 K formed dislocation cell structure. After the pre-deformation, tensile tests and stress-
Accepted 12 October 2019
relaxation tests were performed at room temperature. The specimens with uniformly distributed disloca-
tions exhibited significantly lower yield stress and larger stress-relaxation than those with cell-structured
Keywords: dislocations even under almost same dislocation density. This result suggests that the uniformly dis-
Ferritic steels tributed dislocations contain a large amount of mobile dislocations which can easily move at a low stress
Yielding behavior compared with the cell-structured dislocations.
Dislocation distribution
© 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Dislocation density
Transmission electron backscatter
diffraction

It is generally accepted that the yield stress of work-hardened between cold-rolled and as-quenched martensitic steel is derived
metallic materials is linearly increased against the square root of from the difference in dislocation distribution: The dislocations in
dislocation density, which is called Bailey-Hirsh relation [1]. cold-rolled steel are tangled each other and form cell structures,
Regarding work-hardened iron, some researchers [2–4] inves- while those in as-quenched martensitic steel are relatively uni-
tigated the relation between dislocation density and tensile flow formly distributed without tangling.
stress, and authors have also reported the following equation of Since such a difference in dislocation distribution would in-
the relation between yield stress and dislocation density in cold- fluence the mobility of dislocations and also the yielding stress,
rolled industrial pure iron [5–7]. its effect should be also considered when the strength of work-
√ hardened steel is discussed. Therefore, we tried to control the dis-
σy [Pa] = 0.5 × 108 + 18 ρ (1)
location distribution by changing pre-deformation temperature in
We confirmed that this equation can be applied to up to order to clarify the effect of dislocation distribution on the yield
90% cold-rolled specimen with dislocation density of 2 × 1015 [/m2 ]. stress in ferritic steel. It is known that dislocation accumulation
However, the equation includes only the effect of dislocation den- behavior and development of dislocation substructure depend on
sity but does not consider the inhomogeneity of dislocation dis- the deformation temperature in ferritic steel [2,3,10,11]. For exam-
tribution caused by dislocation cell formation or localization of ple, Keh et al. reported that dislocation cell structure becomes hard
dislocations. As a typical example of varying yield stress due to to be formed in ferritic steel deformed at low temperature due to
dislocation distribution, martensitic steel is known to exhibit a the suppression of cross slip [2]. Therefore, it is expected that a
changeable yielding behavior depending on its dislocation distri- ferritic steel with uniformly distributed dislocation would be ob-
bution. The as-quenched martensitic steel yields continuously with tained by pre-tensile deformation at a low temperature.
a significantly lower elastic limit and proof stress compared with In this study, deformation temperature dependence of dislo-
those of cold-rolled steel containing the same dislocation density cation structures and its effect on the yield stress were investi-
[8,9]. It is explained that such a difference in yielding behavior gated by using the specimens pre-deformed at low temperature
and room temperature as comparison, and then the role of the dis-
location mobility (mobile or immobile) in the yielding stress and
∗
Corresponding author.
the yielding behavior was discussed.
E-mail address: tanaka.yuki.099@m.kyushu-u.ac.jp (Y. Tanaka).

https://doi.org/10.1016/j.scriptamat.2019.10.018
1359-6462/© 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
 Y. Tanaka, T. Masumura and T. Tsuchiyama et al. / Scripta Materialia 177 (2020) 176–180 177

2.0 㽢1014
ρ=1.5㽢1020ε2

Dislocation density, ρ /m-2

1.5
at 213K

1.0

at 298K

0.5

0
0 0.05 0.10 0.15 0.20 0.25 0.30

True strain, εt
Fig. 1. Change in dislocation density estimated by X-Ray diffractometry with true strain on specimens tensile deformed up to necking at 298 K or 213 K.

The material used in this study was a commercial interstitial- and {220} [13]. It is known that dislocation density ρ is propor-
free steel (IF steel), in which the effect of solute carbon and ni- tional to the square of ε [14]. In this study, the value of ρ was
trogen on the strength should be negligibly small. The as-received estimated by the following equation [15] which was established
material was firstly cold-rolled at a reduction of 90%, and then an- against the dislocation density measured by TEM.
nealed at 1073 K for 1.0ks to obtain a recrystallized ferritic single  
ρ /m2 = 1.5 × 10 ε 2
20

structure with a grain size of around 50 μm. The annealed spec- (3)
imens were machined to tensile test pieces with a gauge size of X-ray diffractometry was performed in 3–5 specimens and the
60l × 12.5w × 1t mm3 , and then tensile-deformed by varied degrees mean value of ε was applied for the estimation of dislocation den-
of strain at room-temperature (298 K) or a low temperature (213 K) sity.
(pre-deformation) to give different dislocation distribution. After Fig. 1 shows the relation between dislocation density and true
that, the pre-deformed specimens were again tensile-tested at an strain obtained by tensile tests (pre-deformation) at different tem-
initial strain rate of 8.3 × 10−4 s−1 to fracture at room temperature. perature of 298 K and 213 K. It is found that the dislocation density
Microstructures of specimens were observed by optical mi- is linearly increased with increasing the true strain at both temper-
croscopy, transmission electron microscopy (TEM: JEM-2010 atures, but its value in the 213 K pre-deformed specimens is always
developed by JEOL Ltd.) and transmission EBSD (t-EBSD) method higher at any strain than that in the 298 K ones. This suggests that
[12]. t-EBSD was conducted using field emission scanning electron dislocation accumulation takes place more effectively in the 213 K
microscopes with EBSD system (FE-SEM: SU6600 developed by pre-deformed specimens; in other words, the dynamic recovery
Hitachi High-Technologies). Working distance and step size for during pre-deformation occurred more frequently at a higher tem-
t-EBSD were set as 6 mm and 0.025 μm, respectively. The captured perature. In order to clarify the testing temperature dependence
EBSD patterns were analyzed by a software, OIM analysis ver. 7.1.0 of dislocation distribution, the 298 K and the 213 K pre-deformed
developed by TSL solutions. EBSD-KAM was determined by first specimens with the same dislocation density were prepared based
nearest-neighbor points in hexagonal grid under a condition that on the result of Fig. 1. The deformation degrees to equalize the
the misorientation angle greater than 5° is ignored. dislocation density are 9% and 20% at 298 K, and 5% and 18% at
X-ray diffractometry was conducted using Cu-Kα on the speci- 213 K; the dislocation density of lower strained specimens and
mens with electrical polishing. The authors confirmed that the sur- higher strained specimens are estimated to be 6.0 × 1013 [/m2 ] and
face layer damaged by sand paper grinding has been completely 1.3 × 1014 [/m2 ], respectively.
removed by the electrical polishing. The diffraction peaks were Fig. 2 shows TEM images observed from 111 direction (a-d)
treated with the computer program PDXL2 to remove the back- and KAM maps obtained by t-EBSD method at the same regions
ground noise and the effect of Kα 2. Effect of instrumental func- as TEM observation (e–f). The graphs under the KAM maps are
tion was corrected using well annealed 0.006%C steel on the basis line profiles for the misorientation along the arrow in the fig-
of Gaussian function fitting. From the diffraction angle θ [rad] and ures, and the led lines and black lines denote the misorientation
the full width at half maximum β m[rad], the micro-strain ε of cold between adjacent measurement point, and the misorientation
rolled specimens was estimated by the following equation, between measurement point and origin point of the graph, re-
spectively. A significant difference was found in the dislocation
cos θ 0·9 sin θ
β = + 2ε (2) distribution by comparing each lower strained specimens (a, b):
λ D λ In the 298 K pre-deformed specimen, the dislocations are tangled
where λ and D denote the wave length of X-ray (0.154 nm) and the and a typical dislocation cell structure is formed, while in the
crystallite size, respectively. 213 K pre-deformed specimen, dislocations are separately and
In order to avoid the effect of elastic anisotropy, the value of uniformly dispersed though they seem to be on some particular
ε was determined using the three plots corresponding {110}, {211} slip planes. In the higher strained specimens, dislocation cell
178 Y. Tanaka, T. Masumura and T. Tsuchiyama et al. / Scripta Materialia 177 (2020) 176–180

Fig. 2. TEM images and transmission EBSD maps showing difference in dislocation distribution in tensile deformed IF-steel. (a) 0.09 strained at 298 K, (b) 0.05 strained at
213 K, (c, e) 0.2 strained at 298 K and (d, f) 0.18 strained at 213 K.

structure was confirmed in each specimen (c, d). However, a cell wall, while in the 213 K pre-deformed specimen, the small
remarkable difference became clear in KAM maps between the KAM value dispersed uniformly and a gradual change of crystal
298 K and 213 K pre-deformed specimens (e, f); In the 298 K orientation were confirmed though they tend to slightly localize
pre-deformed specimen, the localization of higher KAM value and at corresponding the dislocation cell wall. Ikeda et al. investi-
the large misorientation more than 1° in the misorientation profile gated the deformation temperature dependence of dislocation
were confirmed, which corresponds to the position of dislocation structure by TEM observation from various directions for single
 Y. Tanaka, T. Masumura and T. Tsuchiyama et al. / Scripta Materialia 177 (2020) 176–180 179

400 120
(a) ρ=6.0㽢1013
Low strained specimens High strained specimens 0.09 strained at 298K
ρ=6.0㽢1013 ρ=1.3㽢1014 100

Nominal stress, σn / MPa

True stress, σt / MPa

300 at 298K at 298K 0.05 strained at 213K

at 213K
80
Magnified figure
at 213K
200
60
0.09 strained at 298K

40
100 0.05 strained at 213K

20
0.1
0
True strain, εt 0
0 50 100 150 200 250 300
Testing time, t / s
Dislocation density, ρ/m-2
1x1013 5x1013 1x1014 3x1014 Fig. 4. Relaxation behavior of IF-steels with same dislocation density and different
0.4 dislocation distribution.
(b)
σ[Pa]=0.5+1.8㽢10-8√ρ
0.2% proof stress, σ0.2 / GPa

0.3 Fig. 3(b). The Bailey-Hirsh relation reported in commercial pure

iron mentioned in the introduction of this paper (Eq. (1)) is also
drown in this figure (dotted line). The yield stress of the 298 K
pre-deformed specimens agrees with the previous results. How-
0.2
ever, a remarkable deviation is found in the 213 K pre-deformed
specimens in which the dislocations are uniformly distributed, sug-
gesting that tangled dislocations tend to enhance the dislocation
0.1 strengthening; however, the uniformly dispersed dislocations are
at 298K less effective for strengthening than tangled dislocations. Here, it
at 213K should be noted that the 213 K pre-deformed specimens exhibit a
㽢108 continuous type (round-house type) yielding and an unclear yield
0 point, suggesting this specimen contains a significant amount of
0 0.05 0.1 0.15 0.20
mobile dislocations as in the case of as-quenched martensite [8,9].
(DIslocation density, ρ)1/2 / m-1 When the significant amount of mobile dislocation is contained
Fig. 3. (a) True stress-strain carves and (b) Bailey–Hirsh relationship in IF-steels
within the metals, a large plastic strain is generated according to
with a different dislocation distribution, representing the effect of dislocation dis- a number of mobile dislocation because it can move even at a low
tribution on the yield stress. stress under yield point. As a result, the mobile dislocation move-
ment causes a continuous type yielding and unclear yield point in
tensile testing. In general, the plastic deformation derived from the
crystal irons pre-deformed at 296 K and 195 K, and consequently mobile dislocation movement will be detected clearly by stress-
proved that a dislocation tangling behavior depends on deforma- relaxation testing. In the stress-relaxation testing, a crosshead is
tion temperature; in the ambient temperature deformation, the stopped when a specified stress is applied to the specimen, and
dislocations are complicatedly and three-dimensionally tangled, then, the change in load (or stress) with the passage of time is
while in the low temperature deformation, they are simply tan- measured without crosshead shift. If the significant amount of mo-
gled two-dimensionally on the specific slip plane ((110) plane) bile dislocation is contained in the 213 K pre-deformed specimens,
[11]. Similarly, it could be accepted that the 298 K pre-deformed the load detected by the load cell should gradually decrease during
specimen exhibits the local distribution of higher KAM value due the stress-relaxation testing owing to the plastic strain generated
to complicatedly and locally tangled dislocations, while the 213 K by the movement of mobile dislocations within the specimen.
pre-deformed specimen exhibits the uniformly distributed KAM The stress-relaxation testing was carried out at ambient tem-
value because of the less dislocation tangling. perature (298 K) at applied stress less than the yield point (approx-
In order to clarify the effect of dislocation distribution on the imately 110 MPa) for the low strained specimens shown in Fig. 2(a,
yield stress, tensile testing was performed at ambient tempera- b) which have the different dislocation distribution and the same
ture (298 K) for the specimens with same dislocation density and dislocation density. The relation between stress and testing time
different dislocation distribution as shown in Fig. 2. The stress- is represented in Fig. 4. The stresses of both specimens decreased
strain curves of low strained (ρ =6.0 × 1013 m−2 ) and high strained just after onset of the stress-relaxation testing but the amount of
(ρ =1.3 × 1014 m−2 ) specimens pre-deformed at different tempera- the stress reduction is markedly different; by 9 MPa in 298 K pre-
ture were shown in Fig. 3(a). Even though dislocation density is deformed specimen and by 16.4 MPa in 213 K specimen. As for the
identical, the yield stress of these specimens are absolutely dif- high strained specimens, the same tendency was obtained. This re-
ferent, and the yielding of 213 K pre-deformed specimens takes sult suggests that a more mobile dislocation is contained in the
place at quite a lower stress in comparison with that of 298 K pre- 213 K pre-deformed specimens in comparison with the 298 K pre-
deformed specimens. deformed specimens.
The yield stresses of the specimens pre-deformed at 298 K and The density of mobile dislocation contained in the specimen
213 K are plotted against the square root of dislocation density in is quantitatively estimated as follows. Under relaxation conditions,
180 Y. Tanaka, T. Masumura and T. Tsuchiyama et al. / Scripta Materialia 177 (2020) 176–180

since total strain in the whole testing equipment system ε t is con- between the specimens pre-deformed at different temperature
stant, the following equation stands up, reveals that there is a significant difference more than two digits
in mobile dislocation density. This indicates that introduction
εt = εe + ε p = 0 (4)
of mobile dislocation are promoted below room temperature, in
thus, other words, most of the dislocations introduced by deformation
at room temperature are tangled and immobilized due to the
εe = −ε p (5)
interaction between dislocations relating to the cross slipping.
where, ε e and ε p denote the elastic strain of the specimen and The obtained results are summarized as follows;
holding tool, and the plastic strain of specimen, respectively. Re- The specimens with uniformly distributed dislocations (213 K
garding from Hooke’s law: σ =Eε e , the following equation is estab- pre-deformed), which is containing a significant amount of mo-
lished, bile dislocations, exhibits a markedly lower yield stress and a
larger stress-relaxation in comparison with that with dislocation
σ = E εe = −E ε p (6)
cell structure (298 K pre-deformed) even under the same disloca-
where, E denotes the elastic modulus of the testing equipment sys- tion density. This results suggest that an introduction of disloca-
tem. In general, the following equation is known as the relation tions does not necessarily strengthen the worked ferritic steel, and
between the total length of moving dislocation (that is mobile dis- sometimes, causes lower yield stress of the steel when a significant
location density) ρ m and the plastic strain ε p . amount of mobile dislocations is contained.
This research did not receive any specific grant from funding
ε p = ρm bx/2 (7)
agencies in the public, commercial, or not-for-profit sectors.
where, b and x are magnitude of Burgers vector of dislocation and
the mean distance of moved dislocations. From Eqs. (6) and (7),
Supplementary material
the mobile dislocation density ρ m can be calculated.
ρm = −2σ /Ebx (8) Supplementary material associated with this article can be
found, in the online version, at doi:10.1016/j.scriptamat.2019.10.
This equation suggests that ρ m proportional to the stress reduc-
018.
tion during stress-relaxation testing –σ and inversely proportional
to x. Since it is difficult to estimate theoretically the distance of all
of the moved dislocations, it is roughly estimated by the following References
methods based on the experimentally results shown in Fig. 2. The
[1] J.E. Bailey, P.B. Hirsch, Philos. Mag. 5 (1960) 485.
mean distance of mobile dislocation in the 213 K pre-deformed
[2] A.S. Keh, S. Weissmann, in: Electron Microscopy and Strength of Crystals, In-
specimens, whose dislocations are uniformly dispersed, were cor- terscience Publishers, 1963, p. 231.
responding to a half of mean spacing of uniformly distributed dis- [3] D.J. Dingley, D. MacLean, Acta Metall. 15 (1967) 885.
location λ(=(3/ρ )1/2 ). While, in the 298 K pre-deformed specimens, [4] J.T. Evans, R. Rawlings, Mater. Sci. Eng. 4 (1969) 297.
[5] K. Nakashima, M. Suzuki, Y. Futamura, T. Tsuchiyama, S. Takaki, Mater. Sci. Fo-
since dislocation cannot pass over complicatedly tangled disloca- rum 503-504 (2006) p. 627.
tions, the mean distance of dislocation was estimated as half of [6] S. Takaki, M. Natori, K. Nakashima, N. Nakada and T. Tsuchiyama: J. Jpn. Soc.
mean spacing of tangling (cell size) measured by intercept method. Heat Treat., 50 (2010), p. 173.
[7] Y. Tanaka, S. Takaki, T. Tsuchiyama, R. Uemori, ISIJ Int. 58 (2018) 1927.
The values calculated by Eq. (8) were 7.3 × 1011 [/m] in 298 K [8] H. Muir, B.L. Averbach, O. Miyagawa, Trans. Am. Soc. Met. 47 (1955) 380.
pre-deformed specimen and 1.4 × 1013 [/m] 213 K pre-deformed [9] K. Nakashima, Y. Fujimura, H. Matsubayashi, T. Tsuchiyama and S. Takaki:
specimen for lower strained specimens, and 7.5 × 1011 [/m] in 298 K Tetsu-to-Hagané, 93 (2007), 459.
[10] Y. Lan, H.J. Klaar, W. Dahl, Metall. Trans. A 23A (1992) 537.
pre-deformed specimen and 1.6 × 1013 [/m] 213 K pre-deformed [11] S. Ikeda, J. Phys. Soc. Japan 27 (1969) 1564.
specimen for higher strained specimens. It should be noted, [12] R.R. Keller and R.H. Geiss: J. Microsc., 245 (2012), 245.
however, that these values are the density of mobile dislocation [13] G.K. Williamson and W.H. Hall: Acta Metall., 1 (1953), 22.
[14] G.K. Williamson and R.E. Smallman: Philos. Mag., 8 (1956), 34.
which can move at a stress of approximately 110 MPa but not the
[15] D. Akama, T. Tsuchiyama, S. Takaki, J. Soc. Mater. Sci. Jpn. 66 (2017) 522.
density of all of mobile dislocation within specimen. Comparison