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Microstructure analysis and yield strength simulation in high Co–Ni secondary hardening steel
Materials Science & Engineering A 669 (2016) 312–317
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Microstructure analysis and yield strength simulation in high Co–Ni secondary hardening steel Chenchong Wang a, Chi Zhang a,n, Zhigang Yang a, Jie Su b, Yuqing Weng a,b a b
Key Laboratory of Advanced Materials of Ministry of Education, School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Institute for Structural Materials, Central, Iron and Steel Research Institute, Beijing 100081, China
art ic l e i nf o
a b s t r a c t
Article history: Received 29 April 2016 Received in revised form 17 May 2016 Accepted 18 May 2016 Available online 18 May 2016
The microstructure of AerMet100 steel after aging was observed and analyzed. Based on the microstructure information of AerMet100 with different aging temperature, equivalent radius of M2C carbides was calculated and used in modified Wang's model to simulate the relation between yield strength and aging temperature. The simulation results showed that AerMet100 was mainly strengthened by M2C precipitate and solid solution. Solid solution strengthening decreased with the increasing of aging temperature because the growth of M2C and austenite gradually consumed Cr, Mo, Ni and C in the matrix. M2C strengthening was controlled by the opposite effect of growth rate and nucleation rate of M2C on strength. By adding the effect of austenite into the model and making simplification, the simulation results of the modified Wang's model were closer to the experiment results than the original Wang's model. & 2016 Elsevier B.V. All rights reserved.
Keywords: Microstructure Strength modeling Aging temperature Steel
1. Introduction With the development of materials design, computational materials engineering has been receiving great attention for a long time [1–3]. Recently, Materials Genome, a method of materials design from process and microstructure to properties, was studied by many experts in the field of steels [4–6]. Especially, the design of high Co–Ni secondary hardening steel was greatly concerned because of its good combination of strength and toughness [7]. For the composition and structure design, many previous studies reported and analyzed the process and microstructure of high Co–Ni secondary hardening steels in detail [8–10]. For the relation between microstructure and properties, several finite element method (FEM) and multi-scale simulation models were also reported to calculate the fracture toughness and hydrogen influenced critical stress intensity [11,12]. It was also generally accepted that precipitation strengthening was the main strengthening method for most high Co–Ni secondary hardening steels [13,14]. However, very limited studies reported the simulation or modeling process for the yield strength of these steels. Strengthening methods, especially precipitation strengthening, were studied in many different steels for a long period [15–18] and many classical models were built for a long time, as Orowan dislocation looping model [19] and Friedel's shear cutting model [20]. n
Corresponding author. E-mail address: [email protected] (C. Zhang).
http://dx.doi.org/10.1016/j.msea.2016.05.069 0921-5093/& 2016 Elsevier B.V. All rights reserved.
These models were modified by later studies in order to remove their limiting assumptions [21]. Apart from precipitation strengthening, several models were also built to simulate Peierls stress [22,23] and dislocation strengthening [24] in steels. Recently, Wang et al. [25] combined different strengthening models by superposition laws and simulated the yield strength of BA160 steel. However, the effect of reverted/retained austenite, which could be an important factor of the strength in some high Co–Ni secondary hardening steels (as AerMet100 and M54) [26], was not considered in this superposition model. In present study, the microstructure of AerMet100 steel was observed and analyzed. Wang's superposition model was modified by including the effect of reverted/retained austenite and making simplification. The microstructure information and equivalent radius of M2C carbides in AerMet 100 steel were used in the modified Wang's model. The yield strength of AerMet100 steel aging at 454, 468, 482, 510 and 538 °C for 5 h was simulated by modified Wang's model and the effect of aging temperature on yield strength was analyzed.
2. Materials and methods The concentrations of major elements of AerMet100 steel in this study was listed in Table 1. The specimens were austenitized at 890 °C for 1 h and quenched in oil to room temperature. Then immediately transferred to a cryogenic bath held at 73 °C for 2 h, and finally aged at 482 °C for 5 h as the accepted optimum heat treatment process.
C. Wang et al. / Materials Science & Engineering A 669 (2016) 312–317
313
AerMet100 was already analyzed in Ayer and Yoo's study [8,10,27], so it was not discussed in this work.
Table 1. Concentrations of major elements in AerMet100 steels. Element
C
Ni
Co
Mo
Cr
Fe
Content (wt%)
0.23
11.59
13.64
1.40
3.26
Bal.
The morphology of the steel was observed using a S-4700 scanning electron microscopy (SEM). Thin foils for transmission electron microscopy (TEM) were cut and ground to the thickness of 50 mm. TEM samples were electropolished in a perchloric acidethanol solution at 30 to 40 °C. Also, Carbon replica was used to obtain the TEM results of carbides in the steel. After coating with carbon, replicas were extracted in 10 vol% nital. TEM samples were observed by JEOL JEM2011 (Japan Electron Optics Ltd., Tokyo) at 200 kV. The microstructure of AerMet100 was also analyzed by D/max 2500 V X-ray diffraction (XRD) and the samples were subjected to Cu-Ka radiation with a scanning speed of 2°/min.
3. Microstructure Fig. 1 showed the XRD and SEM results of martensite in AerMet100. Except for martensite as matrix, no peaks of austenite or carbides were found by XRD results. Similar findings were also reported in previous study about AerMet100 [10]. No austenite or carbides were found by XRD in previous study, because their size and content were too small (below the test limit of XRD). Austenite layers were observed at the interface of martensite laths by TEM (Fig. 2(a)). Their thickness was about 10 nm. These austenite layers were analyzed in detail by many previous studies and they probably had great effect on both fracture toughness and hydrogen embrittlement resistant property [11,12]. Needle-like carbides in fine scale were also observed in the martensite laths as Fig. 2(b). Fig. 2(c) showed the diffraction rings of these carbides. The calibration results of the diffraction rings showed that these carbides were M2C, which were important strengthening precipitate in many high Co–Ni secondary hardening steels. The length of M2C was about 10 nm, which was similar with the results reported in Ayer's study [10]. According to the XRD and TEM observation results, Fig. 3 showed the schematic microstructure of AerMet100. The Previous studies [10,27] showed that all the microstructure of AerMet100 aged at the range of 454–566 °C contained M2C, austenite layer and martensite. With the increase of aging temperature, Only the size and content of M2C and austenite layer increased. The relationship between aging temperature and microstructure in
4. Modeling of yield strength 4.1. Superposition of strengthening contributions According to the microstructure of AerMet100, the main phase was martensite as matrix. Also, austenite layers formed at the interface of martensite laths. The yield strength of steels with both martensite and austenite was studied in dual-phase steels [28,29]. It was accepted that the yield strength of steels with both martensite and austenite could be estimated by Eq. (1).
(
)
σy = f A σA + 1 − f A σM
(1)
where σy was the yield strength of the steel, fA was the phase fraction of austenite, σA and σM were the yield strength of austenite and martensite. The volume fraction of austenite ( fA ) in AerMet100 with different aging temperature was reported in Ayer's study [10]. Also, the effect of austenite layer on the fracture toughness of AerMet100 was analyzed in Wang's study [11], in which the yield strength of austenite layer (σA ) was set as 380 MPa. Strengthening constituents in martensite laths included HallPetch strengthening, dislocation strengthening, solid solution strengthening and precipitation strengthening. A superposition equation was built in Wang's study [25] to combine all these strengthening constituents as Eq.(2).
⎡ σM = M ⎢ τd + τssk + τpk ⎢⎣
(
1⎤ k
) ⎥⎥⎦ + σ
H− P
(2)
where M¼2.8 was the Taylor factor for bcc-metals [25], τd , τss and τp were the critical-resolved shear stress(CRSS) for dislocation strengthening, solid solution strengthening and precipitation strengthening respectively, k ¼1.8 is the superposition exponent for solid solution strengthening and Orowan precipitation strengthening [30], σH − P was the yield strength for Hall-Petch strengthening. Therefore, the final superposition equation for the yield strength of AerMet100 steel was deducted as Eq. (3). τd , τss , τp and σH − P in Eq. (3) could be simulated by different models.
⎧ ⎡ ⎪ M ⎢ τd + τssk + τpk σy = f A σA + 1 − f A ⎨ ⎪ ⎩ ⎢⎣
(
)
Fig. 1. XRD and SEM results of martensite: (a) XRD result; (b) SEM result.
(
1⎤ k
) ⎥⎥⎦ + σ
⎫ ⎪
H− P⎬ ⎪
⎭
(3)
314
C. Wang et al. / Materials Science & Engineering A 669 (2016) 312–317
Fig. 2. Austenite layer and M2C carbides: (a) Austenite layer at the interface of martensite lath; (b) M2C carbide; (c) the diffraction rings of M2C.
Fig. 4. Equivalent radius and volume fraction of M2C used in simulation.
the equivalent radius of M2C carbides. The volume fraction (φM C ), 2
radius (r) and length (l) of M2C carbides were studied by previous studies [8,10,25,27,31]. The calculated results of equivalent radius and the volume fraction of M2C used in the simulation were shown in Fig. 4. Based on the equivalent radius, the CRSS for precipitation strengthening in AerMet100 could be calculated by the modified Orowan bypass equation (Eq. (5)) [21]. Fig. 3. Schematic microstructure of AerMet100 after aging.
τp = Y 4.2. Precipitation strengthening model
( ) ( )
ln 2Kb ⎛ 2wDR ⎞ ln⎜ ⎟ wLR ⎝ b ⎠ ln
2wDR b wLR b
(5)
Previous studies [8,27] reported that the length of M2C carbides formed in AerMet100 with 454 °C aging temperature was larger than the critical size for transition from the shearing to looping mechanism of precipitation strengthening. Therefore, Orowan dislocation looping model was used for the simulation of precipitation strengthening in AerMet100. However, M2C carbides in AerMet100 were cylindrical rather than spherical. In order to meet the assumption of Orowan dislocation looping model, the equivalent radius of M2C carbides in AerMet 100 steel was calculated based on constant volume law as Eq. (4).
where
4 3 πr 2l = πR 3
where Y ¼0.85 is a precipitate spatial-distribution parameter for Orowan dislocation looping [25]. wr = 0.82 and wq = 0.75 are the mean radius and the mean area of the particle intersection with
(4)
where r and l are the actual radius and length of M2C carbides, R is
wL =
πwq φM
− 2wr
2C
−1 ⎛ 1 1 ⎞ wD = ⎜ + ⎟ 2wr ⎠ ⎝ wL
K=
μ 4π
1 1−ν
(6)
(7)
(8)
C. Wang et al. / Materials Science & Engineering A 669 (2016) 312–317
Table 2. Solid solution strengthening coefficients, Δτ /Δc (MPa). Element
C
Ni
Co
Mo
Cr
kss, i
958.8
405.8
200.0
953.5
174.0
the glide plane [21], b is the Burgers vector for bcc iron, μ = 77 GPa and ν = 0.3 are the shear modulus and Poisson’s ratio. 4.3. Solid solution strengthening model The solid solution strengthening of martensite is expressed by Eq. (9) [25].
⎛ ⎞1/2 τss = ⎜⎜ ∑ kss2 , ici⎟⎟ ⎝ i ⎠
where kss, i is the strengthening coefficient of element i in martensite. The value of kss, i is listed in Table 2 [32,33]. ci is the atomic fraction of element i in martensite. The strengthening element in martensite included C, Co, Ni, Cr and Mo. Cr, Mo and C would be consumed by forming M2C carbides. Also, Ni would segregate in austenite layer. Therefore, the content of strengthening element in martensite should not be the same with the original composition of the steel. Because the mole fraction of M2C and austenite was much less than that of martensite, the atomic fraction of element i in martensite could be estimated by Eq. (10–12).
cj ≈ c0, j − 3X M2C ⋅xj
temperature and composition in martensite are shown in Fig. 5. According to the calculation results, it is clear that the content of Mo, Cr, Ni and C in martensite decreased with the increasing of aging temperature, because the growth of M2C and austenite would lead to more consumption of Mo, Cr, Ni and C. 4.4. Dislocation strengthening model For most high Co–Ni secondary hardening steels, the dislocation density ( ρ) remained a high value in different aging conditions, because Co would greatly reduce the recovery rate of the dislocation. According to Keh and Weissman's model [34], the relation between CRSS for dislocation strengthening and dislocation density could be expressed by Eq. (13).
τd ∝ (9)
(10)
ρ
(13)
According to Eq. (13), if the dislocation density of the steel remained the same value in different aging condition, the value of τd would also be constant. In Wang's model [25], τd was calculated in BA160(a secondary hardening steel). The results showed that the τd remained the same value (49 MPa) in different aging conditions. Since the dislocation strengthening contributed very little to the yield strength of high Co–Ni secondary hardening steels, τd = 49 MPa was used in this study in order to simplify the model. 4.5. Hall-Petch strengthening model As the last part of the yield strength simulation, Hall-Petch strengthening could be expressed by Eq. (14). 1
σH − P = σ0 + KH − Pd− 2 cNi ≈ c0, Ni − XA⋅xNi
(11)
cCo ≈ c0, Co
(12)
where cj is the atomic fraction of element j(j was Cr, Mo or C) in martensite, c0, j is the original atomic fraction of element j as the composition of the steel, X M2C and XA are the mole fraction of M2C and austenite, which could be estimated by the phase fraction reported in previous works [8,10,27]. And xj is the atomic fraction of element j in M2C ( xNi was for Ni in austenite), which was also reported in Ayer's work [10]. The calculation results of the relation between aging
Fig. 5. Calculation results of the relation between aging temperature and composition in martensite.
315
(14)
where KH − P = 0.2 MPa m1/2 is the Hall-Petch constant for martensite packets, d = 2.7 μm was the packet size of martensite [25]. Based on the Peierls stress model [22,23], σ0 could be expressed as Eq.(15).
σ0 = M
⎡ 2π ds ⎤ μ ⎥ exp⎢ − ⎢ 1−ν ⎣ b( 1 − ν ) ⎥⎦
(15)
where ds is the spacing of the slip planes in bcc iron.
5. Results and discussion Compared with the original Wang's model [25], the modified Wang's model used in this work added the calculation of equivalent radius for the M2C carbides into the model in order to meet the assumption of Orowan dislocation looping model. Also, in the original Wang's model, the dislocation density and composition of martensite should be tested by experiment as the parameters used in simulation. However, the accurate value of these parameters was difficult to measure by experiment. These parameters, which were not critically important for the simulation, would greatly increase the complexity of the model. In the modified Wang's model, The CRSS for dislocation strengthening and composition of martensite was estimated by several equations, so that experiment results of dislocation density and composition of martensite were not needed in the simulation and the model was simplified. In addition, the effect of austenite on the yield strength was added into the model in order to make the microstructure information more accurate. Table 3 summarized the simulation results of strengthening fraction for all the strengthening constituents in different aging temperature. It can be concluded that precipitation strengthening (about 50%) and solid solution strengthening (about 22%) were the
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C. Wang et al. / Materials Science & Engineering A 669 (2016) 312–317
Table 3. Simulation results of strengthening fraction for all the strengthening constituents in AerMet100 after aging. Strengthening constituents
Hall–Petch strengthening
Dislocation Strengthening
Solid-solution strengthening
Precipitation strengthening
Austenite layer
Strengthening fraction (%)
16.8–21.7
7.1–9.3
21.0–22.5
45.2–54.6
0.1–1.5
Fig. 6. Simulation results of CRSS for solid solution and M2C strengthening.
increasing of aging temperature because of the coarsening of M2C. The relation between yield strength and aging temperature was shown in Fig. 7. When the aging temperature was in the range of 454–482 °C, the simulation results were higher than experiment results. In Ayer's work [10], small amount of cementite was also observed at the grain boundary of AerMet100 when the aging temperature was in the range of 454–482 °C. It was because the aging temperature was not high enough for the complete dissolution of cementite. And the cementite at the grain boundary would probably lead to the decrease of strength. Therefore, the experiment results of yield strength was lower than the ideal value when the aging temperature was in the range of 454–482 °C. The curve of simulation-to-experiment ratio showed that the difference between simulation and experiment results was less than 10%. Also, compared with original Wang's model, the simulation results obtained by the modified Wang's model were closer to the experiment results. It means that the modified Wang's model used in this paper was more suitable for AerMet100 than the original one.
6. Conclusion
Fig. 7. Simulation results of the relation between yield strength and aging temperature.
main strengthening methods for AerMet100 steel. This conclusion was also accepted by most high Co–Ni secondary hardening steels [13,14]. Fig. 6 showed the simulation results of CRSS for solid solution and M2C strengthening. With the growth of M2C and austenite, element Cr, Mo, Ni and C in the matrix would be gradually consumed. So, the CRSS for solid solution decreased with the increasing of aging temperature. For M2C strengthening, the size and volume fraction of M2C carbides had opposite effect on strength. According to Orowan dislocation looping model, the increasing of volume fraction was beneficial to the strength, but the increasing of radius was not. So, the strength was controlled by both growth rate and nucleation rate of M2C. When the aging temperature increased from 454 to 468 °C, the growth of M2C carbides was not significant. So, the increasing of nucleation rate occupied the leading position and led to the increasing of CRSS. When the aging temperature was higher than 468 °C, the growth rate of M2C carbides would increase significantly with the increasing of aging temperature. So, CRSS for M2C strengthening decreased with the
(1) As the observation results, the microstrucure of AerMet100 steels aged at 482 °C for 5 h included martensite as matrix, needle-like M2C carbides inside the martensite laths and fine austenite layers at the interface of martensite laths. (2) According to the simulation results by modified Wang's model, precipitation and solid solution strengthening were the main strengthening methods for AerMet100 steel after aging. (3) As the simulation results, CRSS for solid solution decreased with the increasing of aging temperature because the growth of M2C and austenite would gradually consume Cr, Mo, Ni and C in the matrix. CRSS for M2C strengthening in AerMet100 was controlled by the opposite effect of growth rate and nucleation rate of M2C on strength. (4) The difference between simulation and experiment results was less than 10%. Compared with original Wang's model, the simulation results obtained by the modified Wang's model were closer to the experiment results.
Acknowledgements This work was financially supported by National Basic Research Programs of China (Nos. 2015GB118001 and 2015CB654802). Greatly acknowledged the financial support provided by the National Natural Science Foundation of China (Grant No. 51471094).
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Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Microstructure analysis and yield strength simulation in high Co–Ni secondary hardening steel Chenchong Wang a, Chi Zhang a,n, Zhigang Yang a, Jie Su b, Yuqing Weng a,b a b
Key Laboratory of Advanced Materials of Ministry of Education, School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Institute for Structural Materials, Central, Iron and Steel Research Institute, Beijing 100081, China
art ic l e i nf o
a b s t r a c t
Article history: Received 29 April 2016 Received in revised form 17 May 2016 Accepted 18 May 2016 Available online 18 May 2016
The microstructure of AerMet100 steel after aging was observed and analyzed. Based on the microstructure information of AerMet100 with different aging temperature, equivalent radius of M2C carbides was calculated and used in modified Wang's model to simulate the relation between yield strength and aging temperature. The simulation results showed that AerMet100 was mainly strengthened by M2C precipitate and solid solution. Solid solution strengthening decreased with the increasing of aging temperature because the growth of M2C and austenite gradually consumed Cr, Mo, Ni and C in the matrix. M2C strengthening was controlled by the opposite effect of growth rate and nucleation rate of M2C on strength. By adding the effect of austenite into the model and making simplification, the simulation results of the modified Wang's model were closer to the experiment results than the original Wang's model. & 2016 Elsevier B.V. All rights reserved.
Keywords: Microstructure Strength modeling Aging temperature Steel
1. Introduction With the development of materials design, computational materials engineering has been receiving great attention for a long time [1–3]. Recently, Materials Genome, a method of materials design from process and microstructure to properties, was studied by many experts in the field of steels [4–6]. Especially, the design of high Co–Ni secondary hardening steel was greatly concerned because of its good combination of strength and toughness [7]. For the composition and structure design, many previous studies reported and analyzed the process and microstructure of high Co–Ni secondary hardening steels in detail [8–10]. For the relation between microstructure and properties, several finite element method (FEM) and multi-scale simulation models were also reported to calculate the fracture toughness and hydrogen influenced critical stress intensity [11,12]. It was also generally accepted that precipitation strengthening was the main strengthening method for most high Co–Ni secondary hardening steels [13,14]. However, very limited studies reported the simulation or modeling process for the yield strength of these steels. Strengthening methods, especially precipitation strengthening, were studied in many different steels for a long period [15–18] and many classical models were built for a long time, as Orowan dislocation looping model [19] and Friedel's shear cutting model [20]. n
Corresponding author. E-mail address: [email protected] (C. Zhang).
http://dx.doi.org/10.1016/j.msea.2016.05.069 0921-5093/& 2016 Elsevier B.V. All rights reserved.
These models were modified by later studies in order to remove their limiting assumptions [21]. Apart from precipitation strengthening, several models were also built to simulate Peierls stress [22,23] and dislocation strengthening [24] in steels. Recently, Wang et al. [25] combined different strengthening models by superposition laws and simulated the yield strength of BA160 steel. However, the effect of reverted/retained austenite, which could be an important factor of the strength in some high Co–Ni secondary hardening steels (as AerMet100 and M54) [26], was not considered in this superposition model. In present study, the microstructure of AerMet100 steel was observed and analyzed. Wang's superposition model was modified by including the effect of reverted/retained austenite and making simplification. The microstructure information and equivalent radius of M2C carbides in AerMet 100 steel were used in the modified Wang's model. The yield strength of AerMet100 steel aging at 454, 468, 482, 510 and 538 °C for 5 h was simulated by modified Wang's model and the effect of aging temperature on yield strength was analyzed.
2. Materials and methods The concentrations of major elements of AerMet100 steel in this study was listed in Table 1. The specimens were austenitized at 890 °C for 1 h and quenched in oil to room temperature. Then immediately transferred to a cryogenic bath held at 73 °C for 2 h, and finally aged at 482 °C for 5 h as the accepted optimum heat treatment process.
C. Wang et al. / Materials Science & Engineering A 669 (2016) 312–317
313
AerMet100 was already analyzed in Ayer and Yoo's study [8,10,27], so it was not discussed in this work.
Table 1. Concentrations of major elements in AerMet100 steels. Element
C
Ni
Co
Mo
Cr
Fe
Content (wt%)
0.23
11.59
13.64
1.40
3.26
Bal.
The morphology of the steel was observed using a S-4700 scanning electron microscopy (SEM). Thin foils for transmission electron microscopy (TEM) were cut and ground to the thickness of 50 mm. TEM samples were electropolished in a perchloric acidethanol solution at 30 to 40 °C. Also, Carbon replica was used to obtain the TEM results of carbides in the steel. After coating with carbon, replicas were extracted in 10 vol% nital. TEM samples were observed by JEOL JEM2011 (Japan Electron Optics Ltd., Tokyo) at 200 kV. The microstructure of AerMet100 was also analyzed by D/max 2500 V X-ray diffraction (XRD) and the samples were subjected to Cu-Ka radiation with a scanning speed of 2°/min.
3. Microstructure Fig. 1 showed the XRD and SEM results of martensite in AerMet100. Except for martensite as matrix, no peaks of austenite or carbides were found by XRD results. Similar findings were also reported in previous study about AerMet100 [10]. No austenite or carbides were found by XRD in previous study, because their size and content were too small (below the test limit of XRD). Austenite layers were observed at the interface of martensite laths by TEM (Fig. 2(a)). Their thickness was about 10 nm. These austenite layers were analyzed in detail by many previous studies and they probably had great effect on both fracture toughness and hydrogen embrittlement resistant property [11,12]. Needle-like carbides in fine scale were also observed in the martensite laths as Fig. 2(b). Fig. 2(c) showed the diffraction rings of these carbides. The calibration results of the diffraction rings showed that these carbides were M2C, which were important strengthening precipitate in many high Co–Ni secondary hardening steels. The length of M2C was about 10 nm, which was similar with the results reported in Ayer's study [10]. According to the XRD and TEM observation results, Fig. 3 showed the schematic microstructure of AerMet100. The Previous studies [10,27] showed that all the microstructure of AerMet100 aged at the range of 454–566 °C contained M2C, austenite layer and martensite. With the increase of aging temperature, Only the size and content of M2C and austenite layer increased. The relationship between aging temperature and microstructure in
4. Modeling of yield strength 4.1. Superposition of strengthening contributions According to the microstructure of AerMet100, the main phase was martensite as matrix. Also, austenite layers formed at the interface of martensite laths. The yield strength of steels with both martensite and austenite was studied in dual-phase steels [28,29]. It was accepted that the yield strength of steels with both martensite and austenite could be estimated by Eq. (1).
(
)
σy = f A σA + 1 − f A σM
(1)
where σy was the yield strength of the steel, fA was the phase fraction of austenite, σA and σM were the yield strength of austenite and martensite. The volume fraction of austenite ( fA ) in AerMet100 with different aging temperature was reported in Ayer's study [10]. Also, the effect of austenite layer on the fracture toughness of AerMet100 was analyzed in Wang's study [11], in which the yield strength of austenite layer (σA ) was set as 380 MPa. Strengthening constituents in martensite laths included HallPetch strengthening, dislocation strengthening, solid solution strengthening and precipitation strengthening. A superposition equation was built in Wang's study [25] to combine all these strengthening constituents as Eq.(2).
⎡ σM = M ⎢ τd + τssk + τpk ⎢⎣
(
1⎤ k
) ⎥⎥⎦ + σ
H− P
(2)
where M¼2.8 was the Taylor factor for bcc-metals [25], τd , τss and τp were the critical-resolved shear stress(CRSS) for dislocation strengthening, solid solution strengthening and precipitation strengthening respectively, k ¼1.8 is the superposition exponent for solid solution strengthening and Orowan precipitation strengthening [30], σH − P was the yield strength for Hall-Petch strengthening. Therefore, the final superposition equation for the yield strength of AerMet100 steel was deducted as Eq. (3). τd , τss , τp and σH − P in Eq. (3) could be simulated by different models.
⎧ ⎡ ⎪ M ⎢ τd + τssk + τpk σy = f A σA + 1 − f A ⎨ ⎪ ⎩ ⎢⎣
(
)
Fig. 1. XRD and SEM results of martensite: (a) XRD result; (b) SEM result.
(
1⎤ k
) ⎥⎥⎦ + σ
⎫ ⎪
H− P⎬ ⎪
⎭
(3)
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Fig. 2. Austenite layer and M2C carbides: (a) Austenite layer at the interface of martensite lath; (b) M2C carbide; (c) the diffraction rings of M2C.
Fig. 4. Equivalent radius and volume fraction of M2C used in simulation.
the equivalent radius of M2C carbides. The volume fraction (φM C ), 2
radius (r) and length (l) of M2C carbides were studied by previous studies [8,10,25,27,31]. The calculated results of equivalent radius and the volume fraction of M2C used in the simulation were shown in Fig. 4. Based on the equivalent radius, the CRSS for precipitation strengthening in AerMet100 could be calculated by the modified Orowan bypass equation (Eq. (5)) [21]. Fig. 3. Schematic microstructure of AerMet100 after aging.
τp = Y 4.2. Precipitation strengthening model
( ) ( )
ln 2Kb ⎛ 2wDR ⎞ ln⎜ ⎟ wLR ⎝ b ⎠ ln
2wDR b wLR b
(5)
Previous studies [8,27] reported that the length of M2C carbides formed in AerMet100 with 454 °C aging temperature was larger than the critical size for transition from the shearing to looping mechanism of precipitation strengthening. Therefore, Orowan dislocation looping model was used for the simulation of precipitation strengthening in AerMet100. However, M2C carbides in AerMet100 were cylindrical rather than spherical. In order to meet the assumption of Orowan dislocation looping model, the equivalent radius of M2C carbides in AerMet 100 steel was calculated based on constant volume law as Eq. (4).
where
4 3 πr 2l = πR 3
where Y ¼0.85 is a precipitate spatial-distribution parameter for Orowan dislocation looping [25]. wr = 0.82 and wq = 0.75 are the mean radius and the mean area of the particle intersection with
(4)
where r and l are the actual radius and length of M2C carbides, R is
wL =
πwq φM
− 2wr
2C
−1 ⎛ 1 1 ⎞ wD = ⎜ + ⎟ 2wr ⎠ ⎝ wL
K=
μ 4π
1 1−ν
(6)
(7)
(8)
C. Wang et al. / Materials Science & Engineering A 669 (2016) 312–317
Table 2. Solid solution strengthening coefficients, Δτ /Δc (MPa). Element
C
Ni
Co
Mo
Cr
kss, i
958.8
405.8
200.0
953.5
174.0
the glide plane [21], b is the Burgers vector for bcc iron, μ = 77 GPa and ν = 0.3 are the shear modulus and Poisson’s ratio. 4.3. Solid solution strengthening model The solid solution strengthening of martensite is expressed by Eq. (9) [25].
⎛ ⎞1/2 τss = ⎜⎜ ∑ kss2 , ici⎟⎟ ⎝ i ⎠
where kss, i is the strengthening coefficient of element i in martensite. The value of kss, i is listed in Table 2 [32,33]. ci is the atomic fraction of element i in martensite. The strengthening element in martensite included C, Co, Ni, Cr and Mo. Cr, Mo and C would be consumed by forming M2C carbides. Also, Ni would segregate in austenite layer. Therefore, the content of strengthening element in martensite should not be the same with the original composition of the steel. Because the mole fraction of M2C and austenite was much less than that of martensite, the atomic fraction of element i in martensite could be estimated by Eq. (10–12).
cj ≈ c0, j − 3X M2C ⋅xj
temperature and composition in martensite are shown in Fig. 5. According to the calculation results, it is clear that the content of Mo, Cr, Ni and C in martensite decreased with the increasing of aging temperature, because the growth of M2C and austenite would lead to more consumption of Mo, Cr, Ni and C. 4.4. Dislocation strengthening model For most high Co–Ni secondary hardening steels, the dislocation density ( ρ) remained a high value in different aging conditions, because Co would greatly reduce the recovery rate of the dislocation. According to Keh and Weissman's model [34], the relation between CRSS for dislocation strengthening and dislocation density could be expressed by Eq. (13).
τd ∝ (9)
(10)
ρ
(13)
According to Eq. (13), if the dislocation density of the steel remained the same value in different aging condition, the value of τd would also be constant. In Wang's model [25], τd was calculated in BA160(a secondary hardening steel). The results showed that the τd remained the same value (49 MPa) in different aging conditions. Since the dislocation strengthening contributed very little to the yield strength of high Co–Ni secondary hardening steels, τd = 49 MPa was used in this study in order to simplify the model. 4.5. Hall-Petch strengthening model As the last part of the yield strength simulation, Hall-Petch strengthening could be expressed by Eq. (14). 1
σH − P = σ0 + KH − Pd− 2 cNi ≈ c0, Ni − XA⋅xNi
(11)
cCo ≈ c0, Co
(12)
where cj is the atomic fraction of element j(j was Cr, Mo or C) in martensite, c0, j is the original atomic fraction of element j as the composition of the steel, X M2C and XA are the mole fraction of M2C and austenite, which could be estimated by the phase fraction reported in previous works [8,10,27]. And xj is the atomic fraction of element j in M2C ( xNi was for Ni in austenite), which was also reported in Ayer's work [10]. The calculation results of the relation between aging
Fig. 5. Calculation results of the relation between aging temperature and composition in martensite.
315
(14)
where KH − P = 0.2 MPa m1/2 is the Hall-Petch constant for martensite packets, d = 2.7 μm was the packet size of martensite [25]. Based on the Peierls stress model [22,23], σ0 could be expressed as Eq.(15).
σ0 = M
⎡ 2π ds ⎤ μ ⎥ exp⎢ − ⎢ 1−ν ⎣ b( 1 − ν ) ⎥⎦
(15)
where ds is the spacing of the slip planes in bcc iron.
5. Results and discussion Compared with the original Wang's model [25], the modified Wang's model used in this work added the calculation of equivalent radius for the M2C carbides into the model in order to meet the assumption of Orowan dislocation looping model. Also, in the original Wang's model, the dislocation density and composition of martensite should be tested by experiment as the parameters used in simulation. However, the accurate value of these parameters was difficult to measure by experiment. These parameters, which were not critically important for the simulation, would greatly increase the complexity of the model. In the modified Wang's model, The CRSS for dislocation strengthening and composition of martensite was estimated by several equations, so that experiment results of dislocation density and composition of martensite were not needed in the simulation and the model was simplified. In addition, the effect of austenite on the yield strength was added into the model in order to make the microstructure information more accurate. Table 3 summarized the simulation results of strengthening fraction for all the strengthening constituents in different aging temperature. It can be concluded that precipitation strengthening (about 50%) and solid solution strengthening (about 22%) were the
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Table 3. Simulation results of strengthening fraction for all the strengthening constituents in AerMet100 after aging. Strengthening constituents
Hall–Petch strengthening
Dislocation Strengthening
Solid-solution strengthening
Precipitation strengthening
Austenite layer
Strengthening fraction (%)
16.8–21.7
7.1–9.3
21.0–22.5
45.2–54.6
0.1–1.5
Fig. 6. Simulation results of CRSS for solid solution and M2C strengthening.
increasing of aging temperature because of the coarsening of M2C. The relation between yield strength and aging temperature was shown in Fig. 7. When the aging temperature was in the range of 454–482 °C, the simulation results were higher than experiment results. In Ayer's work [10], small amount of cementite was also observed at the grain boundary of AerMet100 when the aging temperature was in the range of 454–482 °C. It was because the aging temperature was not high enough for the complete dissolution of cementite. And the cementite at the grain boundary would probably lead to the decrease of strength. Therefore, the experiment results of yield strength was lower than the ideal value when the aging temperature was in the range of 454–482 °C. The curve of simulation-to-experiment ratio showed that the difference between simulation and experiment results was less than 10%. Also, compared with original Wang's model, the simulation results obtained by the modified Wang's model were closer to the experiment results. It means that the modified Wang's model used in this paper was more suitable for AerMet100 than the original one.
6. Conclusion
Fig. 7. Simulation results of the relation between yield strength and aging temperature.
main strengthening methods for AerMet100 steel. This conclusion was also accepted by most high Co–Ni secondary hardening steels [13,14]. Fig. 6 showed the simulation results of CRSS for solid solution and M2C strengthening. With the growth of M2C and austenite, element Cr, Mo, Ni and C in the matrix would be gradually consumed. So, the CRSS for solid solution decreased with the increasing of aging temperature. For M2C strengthening, the size and volume fraction of M2C carbides had opposite effect on strength. According to Orowan dislocation looping model, the increasing of volume fraction was beneficial to the strength, but the increasing of radius was not. So, the strength was controlled by both growth rate and nucleation rate of M2C. When the aging temperature increased from 454 to 468 °C, the growth of M2C carbides was not significant. So, the increasing of nucleation rate occupied the leading position and led to the increasing of CRSS. When the aging temperature was higher than 468 °C, the growth rate of M2C carbides would increase significantly with the increasing of aging temperature. So, CRSS for M2C strengthening decreased with the
(1) As the observation results, the microstrucure of AerMet100 steels aged at 482 °C for 5 h included martensite as matrix, needle-like M2C carbides inside the martensite laths and fine austenite layers at the interface of martensite laths. (2) According to the simulation results by modified Wang's model, precipitation and solid solution strengthening were the main strengthening methods for AerMet100 steel after aging. (3) As the simulation results, CRSS for solid solution decreased with the increasing of aging temperature because the growth of M2C and austenite would gradually consume Cr, Mo, Ni and C in the matrix. CRSS for M2C strengthening in AerMet100 was controlled by the opposite effect of growth rate and nucleation rate of M2C on strength. (4) The difference between simulation and experiment results was less than 10%. Compared with original Wang's model, the simulation results obtained by the modified Wang's model were closer to the experiment results.
Acknowledgements This work was financially supported by National Basic Research Programs of China (Nos. 2015GB118001 and 2015CB654802). Greatly acknowledged the financial support provided by the National Natural Science Foundation of China (Grant No. 51471094).
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