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Understanding the factors controlling the hardness in martensitic steels
Scripta Materialia xxx (2015) xxx–xxx
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Understanding the factors controlling the hardness in martensitic steels E.I. Galindo-Nava ⇑, P.E.J. Rivera-Díaz-del-Castillo SKF University Technology Centre, Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Rd, Cambridge CB3 0FS, UK
a r t i c l e
i n f o
Article history: Received 22 July 2015 Accepted 10 August 2015 Available online xxxx Keywords: Martensitic steels Hardness Tempering Dislocation Modeling
a b s t r a c t A unified description for the hardness in martensitic steels for a wide range of carbon contents is presented. It is based on describing the strength contributions of lath and plate martensite, precipitates and retained austenite. Descriptions of the dislocation density in both martensitic structures are obtained in terms of carbon content and tempering conditions. It is shown that a peak in hardness usually observed for carbon contents ranging 0.6–1 wt% is a result of a compromise between the strength of martensite, and the increase in retained austenite. A parametric analysis is performed suggesting possible scenarios for hardness improvement. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Martensitic steels are amongst the strongest materials for structural applications. This is attributed to their complex microstructure, which is highly affected by chemical composition. Two distinct martensitic structures are identified in low-alloy steels depending on carbon content [1]: Laths form in the compositional range 0–0.6 wt%, whereas plates become the dominant structure above 1 wt%; mixed lath and plate structures are present in the 0.6–1 wt% C range, where the fraction of plate martensite increases with increasing carbon content. Lath martensite consists of fine units (100–300 nm thick) hierarchically arranged in substructures within the prior–austenite grains, namely packets and blocks of individual laths. These complex arrangements accommodate the crystallographic distortions during the transformation from austenite and ensuring that the net strain in the prior austenite grain is pure dilatation [2]. This structure is characterised by containing a high dislocation density and carbon redistribution at the lath boundaries. Conversely, plate martensite does not display an evident hierarchic structure [1,3], and plates appear as individual units of various sizes. These units are composed by a set of finely spaced transformation twins crossing throughout the plate (midribs) and dislocation arrays at the boundaries [4]. Additionally, Sherby et al. [5] have pointed out that the crystal structure transitions from being mainly cubic (BCC) to tetragonal (BCT) at the critical carbon content of x0C ¼ 0:6 wt%; plates start forming at this point. Additionally, the fraction of retained austenite increases significantly for carbon contents above this value, although thin films have also been found in steels with lower carbon content [6]. Fig. 1
⇑ Corresponding author.
shows a schematic representation of the martensitic structures for various carbon contents. It is well accepted that retained austenite aids in improving toughness and ductility, which content is highly affected by the processing conditions and carbon content. However, it also reduces the hardness or can lead to microstructural instabilities during tempering [7]. The variation in these microstructural features also reflects the wide spread in hardness of martensitic steels for various carbon contents [8]. For instance, the hardness increases monotonically in as–quenched conditions when increasing carbon content up to x0C . This increment is directly related to the increase in dislocation density, grain boundary area and precipitation nucleation. However, pronounced variations in the hardness have been observed for higher carbon contents [1]. This is mostly due to the increase in the fraction of retained austenite and its variation with the quenching conditions [5]. For instance, Litwinchuk et al. [9] observed a peak in the hardness at a carbon content of 1 wt% in as–quenched conditions, further decreasing with increasing carbon content. However, with proper quenching and tempering conditions it is possible to increase the hardness up to 1100 HV by reducing the fraction of retained austenite and promoting precipitation hardening [5]. These results illustrate the significant challenges in prescribing the hardness of martensitic steels in terms of their initial microstructure and chemical composition. The objective of this work is to postulate a model for describing the hardness evolution in martensitic Fe–C steels. This includes identifying how retained austenite affects the overall strength of the martensitic matrix and describing the strength of plate martensite stemming from dislocations at the plate interiors and midribs. Tempering conditions and chemical composition are included in
E-mail address: [email protected] (E.I. Galindo-Nava). http://dx.doi.org/10.1016/j.scriptamat.2015.08.010 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
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(a)
0 < C ≤ 0.6 wt%
(b)
0.6 < C ≤ 1 wt%
(c)
C > 1 wt%
Plates
Packets
γ laths
twins Blocks
Prior-austenite grain
Retained austenite
Carbon redistribution
Fig. 1. (a) Lath martensite with hierarchically arranged blocks and packets produced by carbon redistribution; (b) mixed structure of laths, packets and retained austenite; and (c) plate martensite containing plates with midribs (twins) and a considerable fraction of retained austenite.
the evolution of the dislocation density. A model describing the microstructure and strength evolution of lath martensite has been introduced in previous work [10]. This article further extends the theory to define a unified approach for steels with carbon content in the range 0–2 wt%, where both lath and plate martensite feature. It is shown how these microstructures affect the hardness and a parametric analysis is performed to suggest possible scenarios for hardness improvement. The yield strength of martensitic steels mainly stems from four contributions [1]: (1) solid solution rss , (2) lath/plate strength rMart , (3) precipitation hardening rp and (4) retained austenite. Most precipitation occurs where carbon partitions at dislocations, lath/plate boundaries or midribs [11]; this suggests that the strength of martensite structures and the precipitation hardening qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi contributions act in a combined form [12]: r2Mart þ r2p . Additionally, the strengthening effects of laths/plates and precipitates are effective only in the areas where martensite forms, i:e. in the volume fraction ð1 f c Þ, where f c is the volume fraction of retained austenite, and the strengthening contribution of the previous decreases according to ð1 f c Þ [1,8] (item 4). The effective strength of c is f c rc , where rc is the yield stress of austenite. However, in practice it is difficult to measure rc in martensitic steels. Nevertheless, rc is much lower than the strength of the matrix (f c rc rMart ) [1] and it can be assumed that f c rc 0. Since most of the measurements have been reported in terms of the Vickers hardness Hv and the previous strengthening contributions are defined in terms of the yield stress rY , the following equation for martensitic steels is employed to validate the strengthening mechanisms with experimental data [13]:
Hv ¼ 0:4ðrY þ 110Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:4 50 þ rss þ 1 f c r2Mart þ r2p þ 110 ;
ð1Þ
where the term 50 MPa is the lattice friction stress [10]. rss is obtained with Fleischer’s equation estimating the increment in the critical resolved–shear stress due to the presence of substituP 1=2 tional solute atoms [14]: rss ¼ i ðb2i xi Þ , where xi is the atom fraction of alloying element i and bi is a constant accounting for the local modulus and lattice distortions of element i with respect to pure iron. This formulation has been successfully applied to martensitic steels [10]. The Orowan–Ashby equation dictates the increase in the applied stress for dislocations to bypass fine carbides r 1=2 [10]: rp ¼ 0:26 lrpb f p ln bp , where f p and r p are the volume fraction and mean radius of the carbide, respectively. rMart depends on the relative fraction of lath and plate martensite and it can be described by a mixture rule:
rMart ¼ rlath f lath þ rplate ð1 f lath Þ;
ð2Þ
where rlath and rplate are the strength of lath and plate martensite, respectively, and f lath is the volume fraction of lath martensite. The strength of lath martensite is controlled by the increase in grain boundary area and dislocation density [10]. The former is expressed in terms of a Hall–Petch equation for the block size dblock , as it is considered as the ‘‘effective” grain size [15,10]. The dislocation density in the laths has been obtained by equating the dislocation energy at the lath boundaries and the lattice strain energy produced by carbon redistribution. rlath equals to:
300
pffiffiffiffiffiffiffiffiffi
rlath ¼ pffiffiffiffiffiffiffiffiffiffiffi þ 0:25Mbl qlath qlath
dblock 3E 4e2 dCottrell ; ¼ 2 ð1 þ 2m Þl d2lath b
ð3Þ
where M ¼ 3 is the Taylor orientation factor, b = 0.286 nm is the magnitude of the Burgers vector, l = 80 GPa is the shear modulus, E = 211 GPa is the Young’s modulus, m = 0.3 is the Poisson ratio, e is the lattice strain produced by carbon redistribution, dlath is the lath boundary thickness, and dCottrell ¼ 7 nm is the thickness of a Cottrell atmosphere. Details on e estimation can be found in [10]. The block size is proportional to the prior–austenite grain size Dg according to: dblock ¼ 0:067Dg . dlath has been proposed to be arranged in such form that it ensures complete carbon segregation to the lath boundaries and it equals to [10]: 0 2=3 0 pffiffiffiffiffiffiffiffiffiffiffi 0 þ k0 xaC Ddiff t , where xaC is the carbon atom dlath ¼ dCottrell ðxaC Þ fraction in the matrix, k0 ¼ b=dCottrell is a constant accounting for a diffusion barrier for carbon atoms segregated into the Cottrell 2 atmospheres, and Ddiff ¼ 6:2 107 exp 80;000 m /s is the diffuRT sion constant of carbon in iron [8]. The first term in dlath represents 0 pffiffiffiffiffiffiffiffiffiffiffi the lath thickness for as–quenched conditions and xaC Ddiff t represents the mean carbon diffusion length during tempering. The strength of plate martensite is dictated by grain boundary strengthening, high dislocation density and transformation twins [1]. However, as opposed to the hierarchically arranged structure of lath martensite, the effect of these features in the overall strength of low–alloy steels is less understood. For instance, although the apparent twin density at the midribs increases with carbon content [4,16], it is not evident how plate refinement affects the strength. Nevertheless, it has been observed that the plate size decreases when decreasing the prior–austenite grain size, whilst preserving its morphology and aspect ratio [17,18]. This suggests that the grain boundary strengthening produced by the plate size is to some extent analogous to the Hall–Petch equation in lath martensite when considering the prior–austenite size: 600 p ffiffiffiffi; this value is obtained from [19]. Dg
The midribs mark the starting point of the transformation, where they grow by the formation of mechanical twins [4]. Carbon atoms redistribute mostly within the twins [20]; this mechanism has been observed directly by atom–probe
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experiments [21], and indirectly by measuring carbide precipitation mostly occurring at the midribs [22]. This implies that twinning is the dominant mechanism to handle the lattice distortions produced initially by carbon atoms. Similarly to the case of lath martensite, it is proposed that the strain energy of a twin Etwin equals the lattice strain energy Elattice . The former can be approximated by considering an equivalent dislocation density qt as
[10]: Etwin ¼ 12 lb qt ðdtwin wtwin Þ, where dtwin and wtwin are the length and thickness of a twin, respectively. The lattice strain energy is
Table 1 Chemical composition (in wt%) of the steels tested in this work. Steel
C
Si
Mn
Cr
Ni
Mo
Refs.
A B 100Cr6 C
0.12–0.98 0.09–1.91 0.97 0.7
0.28 0.3
– – 0.28 2.15
– – 1.38 0.54
– – 0.18 0.13
– – 0.06 0.44
[25] [9] [16] [26]
2
approximated by the Stibitz equation [23]: Elattice ¼
3Ee2t 2 b 2ð1þ2m2 Þ
where
et is the strain accommodated by a twin. f112g twins are frepffiffi quently found in plate martensite [3,1,24], leading to et ¼ 22. Combining these expressions, the dislocation density around the twin boundaries can be obtained. The length of the transformation twins in Fe–C martensite usually lies in the range 100–300 nm [16,20]. dtwin ¼ 200 nm is considered in these calculations. The increase in the twin density can be represented by decreasing the average twin thickness with carbon content. Following the same formalism for lath boundary refinement with carbon content, the twin thickness for as–quenched conditions should be such that carbon mostly redistributes to the twins in the midrib region; this l2
2=3
1=3
a C gives [10]: wAQ where lC ¼ b=ðxaC Þ is the twin ¼ 2 b ¼ 2bðxC Þ mean carbon spacing in the matrix and the factor 2 is a constant to adjust the model to experimental measurements. Stormvinter et al. [20] have estimated the twin thickness in Fe–1.6C to be 5– 10 nm, whereas the previous formula gives wtwin 4 nm; this is consistent with experiments. Additionally, dislocation recovery occurs during tempering as the lattice strains are released by carbon migration. Analogous to lath boundaries, this effect can be represented by adding an extra term accounting for the coarsening pffiffiffiffiffiffiffiffiffiffiffi a0 D t . produced by carbon diffusion: wtwin ¼ wAQ diff twin þ k0 xC As the plates grow there is build–up of elastic strain that should be relieved by dislocations outside the midribs [4]. Stormvinter et al. [20] have found that the width of smaller martensite units (containing only twins) are in the same order of magnitude as the midribs (twin) of larger martensite units [20]. This indicates that twins can only accommodate a fraction of the lattice distortion and the strain energy of the dislocations at the plate interiors is constrained to be lower or equal to the density at the midribs [24]. The lowest strain energy that twins can accommodate corresponds to the critical point where plate martensite is ignited (x0C ¼ 0:026 at.1). As a first approximation the density of the plate interiors is set equal to qt at x0C . Since wtwin is the parameter reflecting carbon effects, qplate is obtained by replacing wtwin in qt with 0 pffiffiffiffiffiffiffiffiffiffiffi 2=3 w0twin ¼ 2bðx0C Þ þ k0 xaC Ddiff t . Combining the previous results, this gives the strength of plate martensite to be equal to:
600 Dg
0
pffiffiffiffiffiffiffiffiffiffiffi
0
pffiffiffiffiffi
rplate ¼ pffiffiffiffiffiffi þ 0:25Mbl qplate þ 0:25Mbl qt qplate ¼ qt ¼
3E e2t ; 2 ð1 þ 2m Þl dtwin w0twin 3E e2t : ð1 þ 2m2 Þl dtwin wtwin
ð4Þ
During quenching the total carbon content in the steel xC (at.) partitions into the martensitic matrix (a0 ) and the retained austenite (c). Since carbon has a high solubility in austenite (up to 2.5 wt% 0 [8]), it can be assumed that: xcC xC ; this implies that xaC xC for as quenched conditions. Toji et al. [21] have found in a steel with 1.07 wt% of carbon that the carbon content in austenite is approximately the same to the nominal concentration and it is 1
This value is approximately 0.6 wt%.
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independent of the martensite volume fraction. During tempering, carbon atoms diffuse from the matrix to form precipitates with fraction f p . If cementite is assumed to form, this gives the effective carbon content to be: xaC ¼ xC ð1 0:25f p Þ [10]. The model is tested against experimental measurements obtained from the literature in a number of steels. Table 1 shows their chemical composition and (commercial) denomination; substitutional elements with less than 0.05 wt% were neglected. The hardness is obtained using Eqs. (1)–(4). The input parameters of the model are the chemical composition, Dg ; f c ; f p ; rp and tempering conditions. Details on how estimating rss can be found in [10]. A MATLAB script with the model solution is included as Supplementary material. Speich [27] has identified experimentally the variation of f lath for as–quenched conditions and carbon content in the range 0–1.3 wt%. This parameter is converted to an analytical expression to capture its carbon variation and simplify calculations: f lath = 1.2 2.344C + 8.2C2 13.13C3 + 6.034C4 for C P 0:2 wt% and f lath ¼ 1 if C < 1 wt%. Fig. 2(a) shows the model predictions and experimental measurements on the Vickers hardness (HV30) for as–quenched conditions (f p ¼ 0) in Steels A2 and B. In this case, Dg ¼ 45 lm, f p ¼ 0 [25]; Fig. 3(b) shows the experimental estimations of f c and f lath 0
from [25,27]. f c ¼ 0:031 0:165C þ 0:3C 2 for C P 0:35 wt% was fitted to simplify calculations for Steel A. f c in Steel B was not reported in [9] and the same fraction was assumed than Steel A. Additionally, the model predictions are shown when f c ¼ 0 to illustrate the relative variation in the hardness when modifying the austenitisation temperature. The model shows very good agreement for steel A and for steel B with carbon content up to 0.4 wt%. However, higher hardness are measured in B for C P 0:4 wt%; this can be due to the variation in the fraction of retained austenite that can increase the hardness up to 200 HV (dotted line); nevertheless, the model reproduces well the peak in hardness at C 1 wt% and its subsequent decrease when increasing carbon content. This peak results from the compensation between the strength of the matrix (dotted line) and the increase in the retained austenite fraction. These results show that hardness can be improved if f c is reduced. Fig. 2(b) also shows the evolution of the dislocation density in the laths, plates and twins. Higher lath dislocation density is predicted; this is due to the fact that dislocations are confined to smaller volumes since lath boundaries are finer than plates [1]. qplate remains constant, whereas qt increases with carbon content due to twin boundary refinement, confirming that midribs (twins) are the main contributors to the strength of the plates. This behaviour has been observed in FeNi plate martensite [24], where similar dislocation density was measured at the midribs and plate interiors. Fig. 3(a) shows the model predictions and experimental measurements on the hardness (HV30) in 100Cr6 as a function of the tempering time at 160 C. Fig. 3(b) shows the measured values of f c ; f p and rp 3; the prior–austenite grain size was assumed to equal
2 HV20 were measured in this case, and a factor of ð20=30Þ1=2 was added to the experiments to transform them to HV30. 3 In the case of f p , 4% fraction of primary cementite particles at the prior–austenite grain boundaries was measured, but not considered in the calculations since they do not contribute to the overall hardness [1].
P.E.J. Rivera-Díaz-del-Castillo, Scripta Mater. (2015), http://dx.doi.org/10.1016/j.
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(b)
800
100
Volume fraction (%)
900
HV30
700 600 500 400
ρlath
Exp - Speich flath - fit
80
60
15
10 40
-2
f - fit
20
As-quenched
100 0.0
0.2
0.4
0.6
16
10
ρt
300 200
ρplate
Dislocation density (m )
Exp - Grange et al. Exp - Litwinchuk et al. Mod Mod - No fγ
(a)
Exp - Speich Exp - Grange et al.
0.8
1.0
1.2
1.4
1.6
1.8
0 0.0
2.0
C (wt%)
14
0.2
0.4
0.6
0.8
1.0
1.2
1.4
C (wt%)
1.6
10 2.0
1.8
Fig. 2. (a) Carbon effects on hardness in Steels A and B and (b) the values of f lath and f c . Dislocation densities of lath and plate martensite as function of carbon content.
(a) 950
(b)
900
800 750 700
f
12
fp
12 8 9 6 6
4
3
2 0
600 0
1
2
(c)
3
t (h)
40
Exp 1 Exp 2 Mod 1 Mod 2
800
700
2
1.4
550
1.2
f - Exp 2 15
4
300
250
500
30
20
3
t (h)
(d)
f (%)
f - Exp 1
1
35
25
600
0 0
4
900
HV30
18 15
0.97 C(wt%) o 160 C tempering
650
rp
10
C (wt%)
HV30
850
14
rp (nm)
Volume fraction (%)
Exp Mod
750
1.0
250
600 450
0.8 750 0.6
500
350
700 650
10 400
300 0
100
200
o
300
400
0.4
5
0.2
0 500
0.0
T ( C)
500
0
100
200
300
o
400
400
500
600
700
T ( C)
Fig. 3. Hardness of (a) 100Cr6 and (b) its respective microstructure for different tempering times. Hardness of (c) Steel C and (d) hardness in the matrix for various temperatures and carbon contents.
20 lm and f lath 0 (see Fig. 2(b)). The model shows good agreement with the experiments with a maximum variation of 50 HV30. Maximum hardness is observed when tempering for 0.25 h, as very fine precipitates are present, although f c is still high (11%). For this condition the matrix holds the maximum contribution (592 HV30)4, followed by precipitation hardening (207 HV30), and 4 The hardness of the matrix, precipitates and solid solution are obtained with the rp rMart expressions [10]: HMart ¼ 0:4ð1 f c ÞrMart pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Hp ¼ 0:4ð1 f c Þrp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and r2Mart þr2p r2Mart þr2p Hss ¼ 0:4rss , respectively.
very low solid solution strengthening (47 HV30). After 4 h of tempering, the retained austenite fraction decrease compensates the softening in the matrix from recovery (629 HV30), and the increase in precipitate size decreases its strengthening contribution (53 HV30). Higher carbon content in the martensite aids in accelerating precipitation. Fig. 3(c) shows the hardness predictions in Steel C with two austenitisation temperatures and the experimental values of f c [26]; this provides two ranges of retained austenite for the same steel. Dg ¼ 10 lm, f p ¼ 5 % and r ¼ 10 nm were assumed for all temperatures since these parameters were not characterised.
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The model shows very good agreement with experiments confirming that it can capture correctly the effects of f c during tempering. Fig. 3 (d) shows a parametric analysis on carbon and tempering temperature effects in the hardness of the matrix HMart ¼ 0:4ð1 f c ÞrMart (no precipitation strengthening is included in this term) tempered for 1 h; Dg ¼ 10 lm was considered and f c values were taken from Fig. 2(b). The model predicts that 700 HV30 in the matrix can be obtained in a high carbon steel (C = 1 wt%) tempered at 150 C than in a medium carbon steel with 0.6 C (wt%) tempered at 220 C. These results illustrate that inappropriate tempering treatments may lead to overaging conditions where the strength in the matrix can drop substantially. It is interesting noting that the temperature window of constant hardness in the matrix in the lower temperature regime decreases with carbon content; this is due to the increase in the precipitation fraction. Additionally, these results can also help to understand how segregation affects the hardness in the matrix during steels’ processing. A unified description for the hardness in martensitic steels for wide carbon contents has been presented. This is based on describing the strength contributions of the dislocation density in lath and plate martensite, precipitates and retained austenite. The strength of the matrix is the main contributor to the overall hardness, followed by precipitation, which effects increase with reducing the austenite volume fraction. Additionally, it was shown that the peak in hardness usually observed in steels with carbon content 0.6– 1 wt% results from the compensation between the strength of the matrix (lath/plate or both) and the increase in the retained austenite fraction. This effect can be prevented by modifying the austenitisation temperature, where the hardness can increase up to 900– 1000 HV30 for steels with carbon content higher than 0.6 wt%.
Acknowledgement This research was supported by the grant EP/L014742/1 from the UK Engineering and Physical Sciences Research Council (EPSRC).
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5
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.scriptamat.2015. 08.010.These data include MOL files and InChiKeys of the most important compounds described in this article. References [1] G. Krauss, Steels: Processing, Structure and Performance, ASM Int, Ohio, 2005. [2] C.C. Kinney, K.R. Pytlewski, A.G. Khachaturyan, J.W. Morris Jr., Acta Mater. 69 (2014) 372–385. [3] A. Stormvinter, G. Miyamoto, T. Furuhara, P. Hedström, A. Borgenstam, Acta Mater. 60 (2012) 7265–7274. [4] A. Shibata, T. Murakami, S. Morito, T. Furuhara, T. Maki, Mater. Trans. 49 (2008) 1242–1248. [5] O.D. Sherby, J. Wadsworth, D.R. Lesuer, C.K. Syn, Mater. Trans. 49 (2008) 2016– 2027. [6] D.H. Sherman, S.M. Cross, S. Kim, F. Grandjean, G.J. Long, M.K. Miller, Metall. Mater. Trans. A 38 (2007) 1698–1711. [7] H. Nakagawa, T. Miyazaki, J. Mater. Sci. 34 (1999) 3901–3908. [8] H.K.D.H. Bhadeshia, R.W.K. Honeycombe, Steels: Microstructure and Properties, B-H, Oxford, 2006. [9] A. Litwinchuk, F.X. Kayser, H.H. Baker, J. Mater. Sci. 11 (1976) 1200–1206. [10] E.I. Galindo-Nava, P.E.J. Rivera-Díaz-del-Castillo, Acta Mater. 98 (2015) 81–93. [11] Z. Hou, P. Hedström, Y. Xu, W. di, J. Odqvist, ISIJ Int. 54 (2014) 2649–2656. [12] P.E.J. Rivera-Díaz-del-Castillo, K. Hayashi, E.I. Galindo-Nava, Mater. Sci. Tech. 29 (2013) 1206–1211. [13] E.J. Pavlina, C.J. Van Tyne, J. Mater. Eng. Perform. 17 (2008) 888–983. [14] R.L. Fleischer, Acta Metall. 11 (1963) 203–209. [15] S. Morito, H. Yoshida, T. Maki, X. Huang, Mater. Sci. Eng. A 438 (2006) 237–240. [16] A.T.W. Barrow, J.H. Kang, P.E.J. Rivera-Díaz-del-Castillo, Acta Mater. 60 (2012) 2805–2815. [17] P. Vivesvaran, Metall. Mater. Trans A 27 (1996) 973–980. [18] A. Shibata, H. Jafarian, N. Tsuji, Mater. Trans. 53 (2012) 81–86. [19] S. Takaki, Mater. Sci. Forum 706 (2012) 181–185. [20] A. Stomvinter, P. Hedström, A. Borgenstam, J. Mater. Sci. Technol. 29 (2013) 373–379. [21] Y. Toji, G. Miyamoto, D. Raabe, Acta Mater. 86 (2015) 137–147. [22] J.R. Yang, H.Y. Lee, H.W. Yen, H.T. Chang, Solid State Phenom. 172 (2011) 67– 72. [23] G.R. Stibitz, Phys. Rev. 49 (1936) 859. [24] A. Shibata, S. Morito, T. Furuhara, T. Maki, Acta Mater. (2009) 483–492. [25] R.A. Grange, C.R. Hribal, L.F. Porter, Metall. Trans. A (1977) 1775–1785. [26] A. Kokosza, J. Pacyna, Arch. Mater. Sci. Eng. 31 (2008) 87–90. [27] G.R. Speich, W.C. Leslie, Metall. Trans. 3 (1972) 1043–1054.
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Understanding the factors controlling the hardness in martensitic steels E.I. Galindo-Nava ⇑, P.E.J. Rivera-Díaz-del-Castillo SKF University Technology Centre, Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Rd, Cambridge CB3 0FS, UK
a r t i c l e
i n f o
Article history: Received 22 July 2015 Accepted 10 August 2015 Available online xxxx Keywords: Martensitic steels Hardness Tempering Dislocation Modeling
a b s t r a c t A unified description for the hardness in martensitic steels for a wide range of carbon contents is presented. It is based on describing the strength contributions of lath and plate martensite, precipitates and retained austenite. Descriptions of the dislocation density in both martensitic structures are obtained in terms of carbon content and tempering conditions. It is shown that a peak in hardness usually observed for carbon contents ranging 0.6–1 wt% is a result of a compromise between the strength of martensite, and the increase in retained austenite. A parametric analysis is performed suggesting possible scenarios for hardness improvement. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Martensitic steels are amongst the strongest materials for structural applications. This is attributed to their complex microstructure, which is highly affected by chemical composition. Two distinct martensitic structures are identified in low-alloy steels depending on carbon content [1]: Laths form in the compositional range 0–0.6 wt%, whereas plates become the dominant structure above 1 wt%; mixed lath and plate structures are present in the 0.6–1 wt% C range, where the fraction of plate martensite increases with increasing carbon content. Lath martensite consists of fine units (100–300 nm thick) hierarchically arranged in substructures within the prior–austenite grains, namely packets and blocks of individual laths. These complex arrangements accommodate the crystallographic distortions during the transformation from austenite and ensuring that the net strain in the prior austenite grain is pure dilatation [2]. This structure is characterised by containing a high dislocation density and carbon redistribution at the lath boundaries. Conversely, plate martensite does not display an evident hierarchic structure [1,3], and plates appear as individual units of various sizes. These units are composed by a set of finely spaced transformation twins crossing throughout the plate (midribs) and dislocation arrays at the boundaries [4]. Additionally, Sherby et al. [5] have pointed out that the crystal structure transitions from being mainly cubic (BCC) to tetragonal (BCT) at the critical carbon content of x0C ¼ 0:6 wt%; plates start forming at this point. Additionally, the fraction of retained austenite increases significantly for carbon contents above this value, although thin films have also been found in steels with lower carbon content [6]. Fig. 1
⇑ Corresponding author.
shows a schematic representation of the martensitic structures for various carbon contents. It is well accepted that retained austenite aids in improving toughness and ductility, which content is highly affected by the processing conditions and carbon content. However, it also reduces the hardness or can lead to microstructural instabilities during tempering [7]. The variation in these microstructural features also reflects the wide spread in hardness of martensitic steels for various carbon contents [8]. For instance, the hardness increases monotonically in as–quenched conditions when increasing carbon content up to x0C . This increment is directly related to the increase in dislocation density, grain boundary area and precipitation nucleation. However, pronounced variations in the hardness have been observed for higher carbon contents [1]. This is mostly due to the increase in the fraction of retained austenite and its variation with the quenching conditions [5]. For instance, Litwinchuk et al. [9] observed a peak in the hardness at a carbon content of 1 wt% in as–quenched conditions, further decreasing with increasing carbon content. However, with proper quenching and tempering conditions it is possible to increase the hardness up to 1100 HV by reducing the fraction of retained austenite and promoting precipitation hardening [5]. These results illustrate the significant challenges in prescribing the hardness of martensitic steels in terms of their initial microstructure and chemical composition. The objective of this work is to postulate a model for describing the hardness evolution in martensitic Fe–C steels. This includes identifying how retained austenite affects the overall strength of the martensitic matrix and describing the strength of plate martensite stemming from dislocations at the plate interiors and midribs. Tempering conditions and chemical composition are included in
E-mail address: [email protected] (E.I. Galindo-Nava). http://dx.doi.org/10.1016/j.scriptamat.2015.08.010 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
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(a)
0 < C ≤ 0.6 wt%
(b)
0.6 < C ≤ 1 wt%
(c)
C > 1 wt%
Plates
Packets
γ laths
twins Blocks
Prior-austenite grain
Retained austenite
Carbon redistribution
Fig. 1. (a) Lath martensite with hierarchically arranged blocks and packets produced by carbon redistribution; (b) mixed structure of laths, packets and retained austenite; and (c) plate martensite containing plates with midribs (twins) and a considerable fraction of retained austenite.
the evolution of the dislocation density. A model describing the microstructure and strength evolution of lath martensite has been introduced in previous work [10]. This article further extends the theory to define a unified approach for steels with carbon content in the range 0–2 wt%, where both lath and plate martensite feature. It is shown how these microstructures affect the hardness and a parametric analysis is performed to suggest possible scenarios for hardness improvement. The yield strength of martensitic steels mainly stems from four contributions [1]: (1) solid solution rss , (2) lath/plate strength rMart , (3) precipitation hardening rp and (4) retained austenite. Most precipitation occurs where carbon partitions at dislocations, lath/plate boundaries or midribs [11]; this suggests that the strength of martensite structures and the precipitation hardening qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi contributions act in a combined form [12]: r2Mart þ r2p . Additionally, the strengthening effects of laths/plates and precipitates are effective only in the areas where martensite forms, i:e. in the volume fraction ð1 f c Þ, where f c is the volume fraction of retained austenite, and the strengthening contribution of the previous decreases according to ð1 f c Þ [1,8] (item 4). The effective strength of c is f c rc , where rc is the yield stress of austenite. However, in practice it is difficult to measure rc in martensitic steels. Nevertheless, rc is much lower than the strength of the matrix (f c rc rMart ) [1] and it can be assumed that f c rc 0. Since most of the measurements have been reported in terms of the Vickers hardness Hv and the previous strengthening contributions are defined in terms of the yield stress rY , the following equation for martensitic steels is employed to validate the strengthening mechanisms with experimental data [13]:
Hv ¼ 0:4ðrY þ 110Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:4 50 þ rss þ 1 f c r2Mart þ r2p þ 110 ;
ð1Þ
where the term 50 MPa is the lattice friction stress [10]. rss is obtained with Fleischer’s equation estimating the increment in the critical resolved–shear stress due to the presence of substituP 1=2 tional solute atoms [14]: rss ¼ i ðb2i xi Þ , where xi is the atom fraction of alloying element i and bi is a constant accounting for the local modulus and lattice distortions of element i with respect to pure iron. This formulation has been successfully applied to martensitic steels [10]. The Orowan–Ashby equation dictates the increase in the applied stress for dislocations to bypass fine carbides r 1=2 [10]: rp ¼ 0:26 lrpb f p ln bp , where f p and r p are the volume fraction and mean radius of the carbide, respectively. rMart depends on the relative fraction of lath and plate martensite and it can be described by a mixture rule:
rMart ¼ rlath f lath þ rplate ð1 f lath Þ;
ð2Þ
where rlath and rplate are the strength of lath and plate martensite, respectively, and f lath is the volume fraction of lath martensite. The strength of lath martensite is controlled by the increase in grain boundary area and dislocation density [10]. The former is expressed in terms of a Hall–Petch equation for the block size dblock , as it is considered as the ‘‘effective” grain size [15,10]. The dislocation density in the laths has been obtained by equating the dislocation energy at the lath boundaries and the lattice strain energy produced by carbon redistribution. rlath equals to:
300
pffiffiffiffiffiffiffiffiffi
rlath ¼ pffiffiffiffiffiffiffiffiffiffiffi þ 0:25Mbl qlath qlath
dblock 3E 4e2 dCottrell ; ¼ 2 ð1 þ 2m Þl d2lath b
ð3Þ
where M ¼ 3 is the Taylor orientation factor, b = 0.286 nm is the magnitude of the Burgers vector, l = 80 GPa is the shear modulus, E = 211 GPa is the Young’s modulus, m = 0.3 is the Poisson ratio, e is the lattice strain produced by carbon redistribution, dlath is the lath boundary thickness, and dCottrell ¼ 7 nm is the thickness of a Cottrell atmosphere. Details on e estimation can be found in [10]. The block size is proportional to the prior–austenite grain size Dg according to: dblock ¼ 0:067Dg . dlath has been proposed to be arranged in such form that it ensures complete carbon segregation to the lath boundaries and it equals to [10]: 0 2=3 0 pffiffiffiffiffiffiffiffiffiffiffi 0 þ k0 xaC Ddiff t , where xaC is the carbon atom dlath ¼ dCottrell ðxaC Þ fraction in the matrix, k0 ¼ b=dCottrell is a constant accounting for a diffusion barrier for carbon atoms segregated into the Cottrell 2 atmospheres, and Ddiff ¼ 6:2 107 exp 80;000 m /s is the diffuRT sion constant of carbon in iron [8]. The first term in dlath represents 0 pffiffiffiffiffiffiffiffiffiffiffi the lath thickness for as–quenched conditions and xaC Ddiff t represents the mean carbon diffusion length during tempering. The strength of plate martensite is dictated by grain boundary strengthening, high dislocation density and transformation twins [1]. However, as opposed to the hierarchically arranged structure of lath martensite, the effect of these features in the overall strength of low–alloy steels is less understood. For instance, although the apparent twin density at the midribs increases with carbon content [4,16], it is not evident how plate refinement affects the strength. Nevertheless, it has been observed that the plate size decreases when decreasing the prior–austenite grain size, whilst preserving its morphology and aspect ratio [17,18]. This suggests that the grain boundary strengthening produced by the plate size is to some extent analogous to the Hall–Petch equation in lath martensite when considering the prior–austenite size: 600 p ffiffiffiffi; this value is obtained from [19]. Dg
The midribs mark the starting point of the transformation, where they grow by the formation of mechanical twins [4]. Carbon atoms redistribute mostly within the twins [20]; this mechanism has been observed directly by atom–probe
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experiments [21], and indirectly by measuring carbide precipitation mostly occurring at the midribs [22]. This implies that twinning is the dominant mechanism to handle the lattice distortions produced initially by carbon atoms. Similarly to the case of lath martensite, it is proposed that the strain energy of a twin Etwin equals the lattice strain energy Elattice . The former can be approximated by considering an equivalent dislocation density qt as
[10]: Etwin ¼ 12 lb qt ðdtwin wtwin Þ, where dtwin and wtwin are the length and thickness of a twin, respectively. The lattice strain energy is
Table 1 Chemical composition (in wt%) of the steels tested in this work. Steel
C
Si
Mn
Cr
Ni
Mo
Refs.
A B 100Cr6 C
0.12–0.98 0.09–1.91 0.97 0.7
0.28 0.3
– – 0.28 2.15
– – 1.38 0.54
– – 0.18 0.13
– – 0.06 0.44
[25] [9] [16] [26]
2
approximated by the Stibitz equation [23]: Elattice ¼
3Ee2t 2 b 2ð1þ2m2 Þ
where
et is the strain accommodated by a twin. f112g twins are frepffiffi quently found in plate martensite [3,1,24], leading to et ¼ 22. Combining these expressions, the dislocation density around the twin boundaries can be obtained. The length of the transformation twins in Fe–C martensite usually lies in the range 100–300 nm [16,20]. dtwin ¼ 200 nm is considered in these calculations. The increase in the twin density can be represented by decreasing the average twin thickness with carbon content. Following the same formalism for lath boundary refinement with carbon content, the twin thickness for as–quenched conditions should be such that carbon mostly redistributes to the twins in the midrib region; this l2
2=3
1=3
a C gives [10]: wAQ where lC ¼ b=ðxaC Þ is the twin ¼ 2 b ¼ 2bðxC Þ mean carbon spacing in the matrix and the factor 2 is a constant to adjust the model to experimental measurements. Stormvinter et al. [20] have estimated the twin thickness in Fe–1.6C to be 5– 10 nm, whereas the previous formula gives wtwin 4 nm; this is consistent with experiments. Additionally, dislocation recovery occurs during tempering as the lattice strains are released by carbon migration. Analogous to lath boundaries, this effect can be represented by adding an extra term accounting for the coarsening pffiffiffiffiffiffiffiffiffiffiffi a0 D t . produced by carbon diffusion: wtwin ¼ wAQ diff twin þ k0 xC As the plates grow there is build–up of elastic strain that should be relieved by dislocations outside the midribs [4]. Stormvinter et al. [20] have found that the width of smaller martensite units (containing only twins) are in the same order of magnitude as the midribs (twin) of larger martensite units [20]. This indicates that twins can only accommodate a fraction of the lattice distortion and the strain energy of the dislocations at the plate interiors is constrained to be lower or equal to the density at the midribs [24]. The lowest strain energy that twins can accommodate corresponds to the critical point where plate martensite is ignited (x0C ¼ 0:026 at.1). As a first approximation the density of the plate interiors is set equal to qt at x0C . Since wtwin is the parameter reflecting carbon effects, qplate is obtained by replacing wtwin in qt with 0 pffiffiffiffiffiffiffiffiffiffiffi 2=3 w0twin ¼ 2bðx0C Þ þ k0 xaC Ddiff t . Combining the previous results, this gives the strength of plate martensite to be equal to:
600 Dg
0
pffiffiffiffiffiffiffiffiffiffiffi
0
pffiffiffiffiffi
rplate ¼ pffiffiffiffiffiffi þ 0:25Mbl qplate þ 0:25Mbl qt qplate ¼ qt ¼
3E e2t ; 2 ð1 þ 2m Þl dtwin w0twin 3E e2t : ð1 þ 2m2 Þl dtwin wtwin
ð4Þ
During quenching the total carbon content in the steel xC (at.) partitions into the martensitic matrix (a0 ) and the retained austenite (c). Since carbon has a high solubility in austenite (up to 2.5 wt% 0 [8]), it can be assumed that: xcC xC ; this implies that xaC xC for as quenched conditions. Toji et al. [21] have found in a steel with 1.07 wt% of carbon that the carbon content in austenite is approximately the same to the nominal concentration and it is 1
This value is approximately 0.6 wt%.
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independent of the martensite volume fraction. During tempering, carbon atoms diffuse from the matrix to form precipitates with fraction f p . If cementite is assumed to form, this gives the effective carbon content to be: xaC ¼ xC ð1 0:25f p Þ [10]. The model is tested against experimental measurements obtained from the literature in a number of steels. Table 1 shows their chemical composition and (commercial) denomination; substitutional elements with less than 0.05 wt% were neglected. The hardness is obtained using Eqs. (1)–(4). The input parameters of the model are the chemical composition, Dg ; f c ; f p ; rp and tempering conditions. Details on how estimating rss can be found in [10]. A MATLAB script with the model solution is included as Supplementary material. Speich [27] has identified experimentally the variation of f lath for as–quenched conditions and carbon content in the range 0–1.3 wt%. This parameter is converted to an analytical expression to capture its carbon variation and simplify calculations: f lath = 1.2 2.344C + 8.2C2 13.13C3 + 6.034C4 for C P 0:2 wt% and f lath ¼ 1 if C < 1 wt%. Fig. 2(a) shows the model predictions and experimental measurements on the Vickers hardness (HV30) for as–quenched conditions (f p ¼ 0) in Steels A2 and B. In this case, Dg ¼ 45 lm, f p ¼ 0 [25]; Fig. 3(b) shows the experimental estimations of f c and f lath 0
from [25,27]. f c ¼ 0:031 0:165C þ 0:3C 2 for C P 0:35 wt% was fitted to simplify calculations for Steel A. f c in Steel B was not reported in [9] and the same fraction was assumed than Steel A. Additionally, the model predictions are shown when f c ¼ 0 to illustrate the relative variation in the hardness when modifying the austenitisation temperature. The model shows very good agreement for steel A and for steel B with carbon content up to 0.4 wt%. However, higher hardness are measured in B for C P 0:4 wt%; this can be due to the variation in the fraction of retained austenite that can increase the hardness up to 200 HV (dotted line); nevertheless, the model reproduces well the peak in hardness at C 1 wt% and its subsequent decrease when increasing carbon content. This peak results from the compensation between the strength of the matrix (dotted line) and the increase in the retained austenite fraction. These results show that hardness can be improved if f c is reduced. Fig. 2(b) also shows the evolution of the dislocation density in the laths, plates and twins. Higher lath dislocation density is predicted; this is due to the fact that dislocations are confined to smaller volumes since lath boundaries are finer than plates [1]. qplate remains constant, whereas qt increases with carbon content due to twin boundary refinement, confirming that midribs (twins) are the main contributors to the strength of the plates. This behaviour has been observed in FeNi plate martensite [24], where similar dislocation density was measured at the midribs and plate interiors. Fig. 3(a) shows the model predictions and experimental measurements on the hardness (HV30) in 100Cr6 as a function of the tempering time at 160 C. Fig. 3(b) shows the measured values of f c ; f p and rp 3; the prior–austenite grain size was assumed to equal
2 HV20 were measured in this case, and a factor of ð20=30Þ1=2 was added to the experiments to transform them to HV30. 3 In the case of f p , 4% fraction of primary cementite particles at the prior–austenite grain boundaries was measured, but not considered in the calculations since they do not contribute to the overall hardness [1].
P.E.J. Rivera-Díaz-del-Castillo, Scripta Mater. (2015), http://dx.doi.org/10.1016/j.
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(b)
800
100
Volume fraction (%)
900
HV30
700 600 500 400
ρlath
Exp - Speich flath - fit
80
60
15
10 40
-2
f - fit
20
As-quenched
100 0.0
0.2
0.4
0.6
16
10
ρt
300 200
ρplate
Dislocation density (m )
Exp - Grange et al. Exp - Litwinchuk et al. Mod Mod - No fγ
(a)
Exp - Speich Exp - Grange et al.
0.8
1.0
1.2
1.4
1.6
1.8
0 0.0
2.0
C (wt%)
14
0.2
0.4
0.6
0.8
1.0
1.2
1.4
C (wt%)
1.6
10 2.0
1.8
Fig. 2. (a) Carbon effects on hardness in Steels A and B and (b) the values of f lath and f c . Dislocation densities of lath and plate martensite as function of carbon content.
(a) 950
(b)
900
800 750 700
f
12
fp
12 8 9 6 6
4
3
2 0
600 0
1
2
(c)
3
t (h)
40
Exp 1 Exp 2 Mod 1 Mod 2
800
700
2
1.4
550
1.2
f - Exp 2 15
4
300
250
500
30
20
3
t (h)
(d)
f (%)
f - Exp 1
1
35
25
600
0 0
4
900
HV30
18 15
0.97 C(wt%) o 160 C tempering
650
rp
10
C (wt%)
HV30
850
14
rp (nm)
Volume fraction (%)
Exp Mod
750
1.0
250
600 450
0.8 750 0.6
500
350
700 650
10 400
300 0
100
200
o
300
400
0.4
5
0.2
0 500
0.0
T ( C)
500
0
100
200
300
o
400
400
500
600
700
T ( C)
Fig. 3. Hardness of (a) 100Cr6 and (b) its respective microstructure for different tempering times. Hardness of (c) Steel C and (d) hardness in the matrix for various temperatures and carbon contents.
20 lm and f lath 0 (see Fig. 2(b)). The model shows good agreement with the experiments with a maximum variation of 50 HV30. Maximum hardness is observed when tempering for 0.25 h, as very fine precipitates are present, although f c is still high (11%). For this condition the matrix holds the maximum contribution (592 HV30)4, followed by precipitation hardening (207 HV30), and 4 The hardness of the matrix, precipitates and solid solution are obtained with the rp rMart expressions [10]: HMart ¼ 0:4ð1 f c ÞrMart pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Hp ¼ 0:4ð1 f c Þrp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and r2Mart þr2p r2Mart þr2p Hss ¼ 0:4rss , respectively.
very low solid solution strengthening (47 HV30). After 4 h of tempering, the retained austenite fraction decrease compensates the softening in the matrix from recovery (629 HV30), and the increase in precipitate size decreases its strengthening contribution (53 HV30). Higher carbon content in the martensite aids in accelerating precipitation. Fig. 3(c) shows the hardness predictions in Steel C with two austenitisation temperatures and the experimental values of f c [26]; this provides two ranges of retained austenite for the same steel. Dg ¼ 10 lm, f p ¼ 5 % and r ¼ 10 nm were assumed for all temperatures since these parameters were not characterised.
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The model shows very good agreement with experiments confirming that it can capture correctly the effects of f c during tempering. Fig. 3 (d) shows a parametric analysis on carbon and tempering temperature effects in the hardness of the matrix HMart ¼ 0:4ð1 f c ÞrMart (no precipitation strengthening is included in this term) tempered for 1 h; Dg ¼ 10 lm was considered and f c values were taken from Fig. 2(b). The model predicts that 700 HV30 in the matrix can be obtained in a high carbon steel (C = 1 wt%) tempered at 150 C than in a medium carbon steel with 0.6 C (wt%) tempered at 220 C. These results illustrate that inappropriate tempering treatments may lead to overaging conditions where the strength in the matrix can drop substantially. It is interesting noting that the temperature window of constant hardness in the matrix in the lower temperature regime decreases with carbon content; this is due to the increase in the precipitation fraction. Additionally, these results can also help to understand how segregation affects the hardness in the matrix during steels’ processing. A unified description for the hardness in martensitic steels for wide carbon contents has been presented. This is based on describing the strength contributions of the dislocation density in lath and plate martensite, precipitates and retained austenite. The strength of the matrix is the main contributor to the overall hardness, followed by precipitation, which effects increase with reducing the austenite volume fraction. Additionally, it was shown that the peak in hardness usually observed in steels with carbon content 0.6– 1 wt% results from the compensation between the strength of the matrix (lath/plate or both) and the increase in the retained austenite fraction. This effect can be prevented by modifying the austenitisation temperature, where the hardness can increase up to 900– 1000 HV30 for steels with carbon content higher than 0.6 wt%.
Acknowledgement This research was supported by the grant EP/L014742/1 from the UK Engineering and Physical Sciences Research Council (EPSRC).
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5
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.scriptamat.2015. 08.010.These data include MOL files and InChiKeys of the most important compounds described in this article. References [1] G. Krauss, Steels: Processing, Structure and Performance, ASM Int, Ohio, 2005. [2] C.C. Kinney, K.R. Pytlewski, A.G. Khachaturyan, J.W. Morris Jr., Acta Mater. 69 (2014) 372–385. [3] A. Stormvinter, G. Miyamoto, T. Furuhara, P. Hedström, A. Borgenstam, Acta Mater. 60 (2012) 7265–7274. [4] A. Shibata, T. Murakami, S. Morito, T. Furuhara, T. Maki, Mater. Trans. 49 (2008) 1242–1248. [5] O.D. Sherby, J. Wadsworth, D.R. Lesuer, C.K. Syn, Mater. Trans. 49 (2008) 2016– 2027. [6] D.H. Sherman, S.M. Cross, S. Kim, F. Grandjean, G.J. Long, M.K. Miller, Metall. Mater. Trans. A 38 (2007) 1698–1711. [7] H. Nakagawa, T. Miyazaki, J. Mater. Sci. 34 (1999) 3901–3908. [8] H.K.D.H. Bhadeshia, R.W.K. Honeycombe, Steels: Microstructure and Properties, B-H, Oxford, 2006. [9] A. Litwinchuk, F.X. Kayser, H.H. Baker, J. Mater. Sci. 11 (1976) 1200–1206. [10] E.I. Galindo-Nava, P.E.J. Rivera-Díaz-del-Castillo, Acta Mater. 98 (2015) 81–93. [11] Z. Hou, P. Hedström, Y. Xu, W. di, J. Odqvist, ISIJ Int. 54 (2014) 2649–2656. [12] P.E.J. Rivera-Díaz-del-Castillo, K. Hayashi, E.I. Galindo-Nava, Mater. Sci. Tech. 29 (2013) 1206–1211. [13] E.J. Pavlina, C.J. Van Tyne, J. Mater. Eng. Perform. 17 (2008) 888–983. [14] R.L. Fleischer, Acta Metall. 11 (1963) 203–209. [15] S. Morito, H. Yoshida, T. Maki, X. Huang, Mater. Sci. Eng. A 438 (2006) 237–240. [16] A.T.W. Barrow, J.H. Kang, P.E.J. Rivera-Díaz-del-Castillo, Acta Mater. 60 (2012) 2805–2815. [17] P. Vivesvaran, Metall. Mater. Trans A 27 (1996) 973–980. [18] A. Shibata, H. Jafarian, N. Tsuji, Mater. Trans. 53 (2012) 81–86. [19] S. Takaki, Mater. Sci. Forum 706 (2012) 181–185. [20] A. Stomvinter, P. Hedström, A. Borgenstam, J. Mater. Sci. Technol. 29 (2013) 373–379. [21] Y. Toji, G. Miyamoto, D. Raabe, Acta Mater. 86 (2015) 137–147. [22] J.R. Yang, H.Y. Lee, H.W. Yen, H.T. Chang, Solid State Phenom. 172 (2011) 67– 72. [23] G.R. Stibitz, Phys. Rev. 49 (1936) 859. [24] A. Shibata, S. Morito, T. Furuhara, T. Maki, Acta Mater. (2009) 483–492. [25] R.A. Grange, C.R. Hribal, L.F. Porter, Metall. Trans. A (1977) 1775–1785. [26] A. Kokosza, J. Pacyna, Arch. Mater. Sci. Eng. 31 (2008) 87–90. [27] G.R. Speich, W.C. Leslie, Metall. Trans. 3 (1972) 1043–1054.
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