Microstructural characterization and strengthening mechanisms of a 15Cr-ODS steel produced by mechanical alloying and Spark Plasma Sintering

Fusion Engineering and Design 137 (2018) 71–78

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Microstructural characterization and strengthening mechanisms of a 15CrODS steel produced by mechanical alloying and Spark Plasma Sintering

T

⁎

Weijuan Lia, Haijian Xua,b,c,d,e, , Xiaochun Shac, Jingsong Mengc, Wenzhong Wangc, Chao Kangc, Ximin Zhanga, Zhaodong Wange a

School of Materials and Metallurgy, University of Science and Technology Liaoning, Anshan, 114051, PR China State Key Laboratory of Metal Material for Marine Equipment and Application, Ansteel Group Corporation, Anshan, 114009, PR China c Angang Steel Company Limited, Anshan, 114009, PR China d Material Science and Engineering, Faculty of Engineering, Hokkaido University, Sapporo, 060-8628, Japan e School of Material Science and Engineering, Northeastern University, Shenyang, 110819, PR China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Spark Plasma Sintering (SPS) Microstructure Small angle X-ray scattering (SAXS) Yield strength Ti and Zr additions

The microstructure of the oxide dispersion strengthened (ODS) Fe-15Cr ferritic steel with Ti and Zr additions was characterized using transmission electron microscopy (TEM), elemental mapping, synchrotron small angle X-ray scattering (SAXS) and electron backscatter diffraction (EBSD) techniques. The results showed that bimodal grain size distribution in the matrix was observed, which is attributed to the heterogeneous recrystallization process during the Spark Plasma Sintering (SPS). TEM and SAXS results showed that very high density nanoscale oxides are formed in 15Cr-ODS steel. Large concentration of nm-scale trigonal-phase Y4Zr3O12 oxides and some large oxides of monoclinic CrTi2O5 are observed in specimen by high resolution TEM (HRTEM) and Energy-dispersive X-ray spectroscopy (EDS) mapping. Finally, the yield stress was experimentally measured and quantitatively estimated. The findings indicated that theoretical calculation was in accordance with the experimental results, and strengthening contributions from solute atoms, grains boundaries, dislocations and oxides was presented.

1. Introduction Oxide dispersion strengthened (ODS) ferritic steels are amongst the most promising candidates for large scale structural materials in fast breeder reactors and blanket applications for fusion reactors. The steels that exhibit remarkable irradiation swelling resistance and excellent tensile, creep strength are being developed by powder metallurgy, wellknown to produce nanostructure and ultrafine grained materials [1–5]. The superior performance of ODS ferritic steels over their conventional heat resistance steels and austenitic steels is attributed to the high number density of nano-sized oxides within the matrix [6–9]. To do so, powder metallurgy is the main manufacture process of ODS steels [10]. Hot Isostatic Pressing (HIP) and/or Hot Extrusion (HE) are classic, almost unique, methods to consolidate the ODS steels for controlling the microstructures and mechanical properties of ODS steels [5,11]. However, abnormal grain growth and precipitation coarsening are often observed since these processes require exposure for several hours at high temperatures over 1273 K (1000 ℃) [12,13]. Compared to HIP and/or HE methods, Spark Plasma Sintering (SPS) is a novel

⁎

powder consolidation technology [14]. The advantages of SPS spend much less time in both heating and cooling during sintering process. SPS has been widely used because of its ability to heat up the materials very quickly, providing a powerful tool to retain the original nanograins. Recently SPS has been employed to prepare ODS steels. Bimodal microstructures were obtained in ODS steels after consolidation (SPS), which are different from the ODS steels microstructure produced by HIP and HE [15–18]. Grain structures composed of ultrafine grains with coarse grains improve ductility when compared to monomodal nanostructure materials [19–22]. Some groups have found that the YeO and Y-Ti-O nano-sized oxides were observed in Ti-free and Ti-containing ODS steels produced by SPS, respectively [23,24]. The advantage of those nanoscale oxides is described as follows: (1) to inhibit the migration of dislocations and grain boundaries to enhance high temperature strength; (2) to provide sinks for the irradiation induced defects; and (3) to supply a great number of nucleation sites for small helium bubbles and promote radiation-induced vacancy-interstitial recombination [25–27]. All of the above attribute to high microstructural stability in retardation of loss of toughness at lower irradiation

Corresponding author at: School of Materials and Metallurgy, University of Science and Technology Liaoning, Anshan, 114051, PR China. E-mail address: [email protected] (H. Xu).

https://doi.org/10.1016/j.fusengdes.2018.08.020 Received 3 June 2018; Received in revised form 27 August 2018; Accepted 30 August 2018 0920-3796/ © 2018 Elsevier B.V. All rights reserved.

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temperatures and to potential degradation of creep strength at higher temperatures. Some research groups have also reported the benefit of Ti addition in refining oxide particles (Y2O3) sizes, and increasing their number densities, through changing their chemical composition [28,29]. According to the first principle calculation results, the binding energy of Y-Zr-O phase is higher than that of Y-Ti-O phase in Fe matrix, which means that the Y-Zr-O phase is easier to form and more stable than Y-Ti-O phase [30]. Some researchers have found that the addition of element Zr can refine the oxides due to smaller oxide formation energy [31]. For this, some researchers have suggested utilizing of Zr rather than Ti in Al-ODS steel which promoted formation of finer Y-ZrO particles compared to coarse Y-Al-O particles [32]. So it was expected that the addition of Zr elements in ODS steels without Al addition would refine the oxide particles and increase the number density of oxide particles by forming Y-Zr-O phase instead of Y-Ti-O and/or Y-O phase, which can improve the high temperature strength with a good corrosion resistance ability. In the present research, we aimed at investigating the 15Cr-ODS steel with Ti and Zr additions fabricated by MA and SPS. The grain morphologies and the nano-sized oxides were observed by means of EBSD, SAXS, TEM and HRTEM. The yield strengths of 15Cr-ODS steel were measured at room temperature, 550 ℃ and 650 ℃. The strengthening mechanisms were estimated based on the comparison between the theoretical calculation and experimental measurements.

Bcosθ/λ = 0.9D+2εsinθ/λ

Where B is the integral widths that were attained by eliminating the instrumental broadening, θ is the diffraction angle, λ is the wavelength of X-ray (1.541 Å), D is the grain size, and ε is the strain in matrix. Hence, the dislocation density ρ was derived as [35]: ρ=14.4ε2/b2

A 15Cr-ODS steel with nominal chemical composition of Fe-15Cr2W-0.3Ti- 0.3Zr-0.3Y2O3 (wt. %) was manufactured by using highpurity elemental powders and nanoscale Y2O3 powders. The mixed powders were mechanically alloyed (MA) in a FRITSCH Pulverisette 5 planetary mill for 50 h with a rotation speed of 260 rpm and ball-topowder weight ratio of 10:1 under high-purity Ar atmosphere at room temperature. The stainless steel ball was applied as milling media. Its actual composition after MA was measured by inert gas pulse infrared thermal conductive method (ASTME1019-2003) and is listed in Table 1. The MA powders were then consolidated by SPS to form a dense cylindrical pellet of 30 mm diameter and 6 mm height. SPS device was a Sumitomo graphite SPS-1050 sintering system (Japan). Sintering cycles were performed under 50 MPa average pressure, with a heating rate of 400 K min−1 up to the holding temperature for a soaking time of 5 min. The soaking temperature was chosen at 950 ℃. The cooling was ensured by direct contact with water-cooled punches, which induced a cooling rate of 200 K min−1. Relative density of SPS compacts was measured using the Archimedes' principle where the mass of the sample emerged and immersed in water is measured using a very precise weighing scale, and the density of ferritic steel (7.82 g/cm3) was chosen as the theoretical density. The relative density of 15Cr-ODS steel fabricated by SPS is reaches to 97.7%. X-ray diffraction (XRD) was carried out on the specimen to analyze the crystal structure, using a Rigaku Smartlab, using Cu Kα radiation, at a voltage of 40 kV, a current of 40 mA and a speed of 2 (°)/min. The diffraction angle (2θ) range of all the measurement was restricted to 15–120°. The lattice parameter a was calculated by the Cohen-Wagner extrapolation plot (ahkl vs. cos2θ/sinθ) [33], the lattice strain (ε) of specimen was determined by means of Williamson-Hall equation from XRD patterns according to Scherrer formula after subtracting the widths due to instrumental broadening and strain effects using the equation [34]:

q=

W

Ti

Zr

Y

O

N

C

Bal.

14.8

1.9

0.29

0.27

0.25

0.21

0.03

0.08

4π sin θ λ

(3)

Where λ is the X-ray wavelength, 2θ is the scattering angle. The IRENA package developed by Argonne National Laboratory was used to fit the SAXS curves [39]. The fitting equation used in IRENA package is given below [40]:

I (q) ≈ G exp(

−q2Rg2 3

) + B[

erf (qRg / 6 )) 3 q

]p

(4)

Where G is a constant defined by the specifics of composition and concentration of the oxides. For dilute oxides, G=Npnp2, where Np is the number of oxides in the scattering volume and np is the number of excess electrons in an oxide compared to Fe matrix. Thus, G=Np(ρeVp)2, where Vp is the volume of an oxide and ρe is the electron-density difference between the oxide and Fe matrix. Rg is radius of gyration. B=Np2ρπe2Sp, where Sp is the oxide surface area. The number density distribution N(r) of oxides with radius r is assumed to have a log normal distribution, which is defined as follows:

N(r ) =

Table 1 The actual composition of ODS steel (wt. %). Cr

(2)

Where b is the Burgers vector (0.248 nm) [36]. FIB lift-out method was employed to prepare transparent electron microscopy (TEM) specimens by using JEOL JIB-4601 F FIB system. Low-energy Ga beam (5 kV and 1 kV) was used to remove the higher energy ion damage. Finally, those micro-samples were electro-polished using flash electropolishing at 20 V in a solution of 80%C2H5OH+20% H2SO4 at about 0 ℃ for 0.02 s. The microstructure was characterized using a JEOL JEM-2100 TEM with an acceleration voltage of 200 kV. To determine the number density of oxide particles, the local TEM foil thicknesses were measured by convergent beam electron diffraction method (CBED) 34 [34,37]. High angle annual dark field (HAADF) and scanning transmission electron microscopy (STEM) images were taken in an aberration corrected FEI Titan3 G2 60–300 microscope operated at 300 kV. EDS characterization was done in STEM mode at 300 keV to obtain elemental mapping. The EBSD samples with dimension 3 × 5×5 mm3 for grain morphologies and size analyzing were mechanical polished and then electro-polished in order to remove the deformation layer introduced during mechanical milling. The grain morphologies and size were analyzed by EBSD, which EBSD experiment was carried out on JEOL JSM6500 F Scanning Electron Microscopy (SEM) in step length of 50 nm in Hokkaido University. Synchrotron radiation SAXS experiments were performed in transmission mode using two systems at the National Institute for Materials Science (NIMS). To obtain a reasonable transmission rate, the samples were mechanically thinning to less than 30 μm A two-dimensional imaging detector was used to collect the scattering patterns. The distance between the samples and the detector was 2 m. After correction (using cowhells as standard sample), the two-dimensional scattering rings were transformed into one-dimensional scattering curves by using Fit 2D software [38]. The scattering vector q is defined as:

2. Experimental

Fe

(1)

1 1 [ln(r / R 0)2] exp{ − } 2σ 2 2π rσ

(5)

Where R0 is the radius at the peak position of number density, and the standard deviation σ is the width of log normal distribution of N(r). The average diameter dave and standard deviation σ in this study were obtained from IRENA package directly. The tensile tests were carried out using Shimadzu AG-Xplus electro72

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Fig. 1. (a) XRD pattern of the specimen, (b) plot of lattice parameter a versus cos2θ/sinθ, and (c) plot of Williamson-Hall equation.

mechanical machine at the constant strain rate of 2 × 10−3 mms-1 at room temperature. Specimen size is 13 mm in gauge length, 3 mm in width and 1 mm in thickness. Tension tests were performed three times at each temperature. 3. Result and discussion 3.1. Microstructure characterization 3.1.1. Dislocation density Fig. 1 shows XRD pattern and corresponding extrapolation plots of the specimen. The diffraction peaks were indexed in Fig. 1a, and the lattice parameter a was also estimated as 0.28743 nm in Fig. 1b. Besides, the micro-strain (ε) was decided based on the plot as shown in Fig. 1c. According to Eq. (3), the dislocation density was approximately determined as 3.7 × 1014 m2, which was in the same order as ferritic ODS steel [41] and much higher than that of conventional steels, indicating recovery slightly occurred during the final annealing presumably attributing to inhibition of nano-sized oxides. 3.1.2. Grains Fig. 2 shows an EBSD image taken from the specimen fabricated by SPS, with grain colours related to grain orientation expressed in the inset standard triangle. Grain boundaries having a misorientation > 15 ℃ were identified as “high angle boundaries’’, and those with misorientation of 2–15 ℃ were identified as “sub-grain boundaries’’. The orientation imaging map indicated that grain orientation was essentially random, and most of the ferritic grains are smaller than 500 nm and some grains are larger than 1 μm, the average grain size calculated

Fig. 2. EBSD orientation map of 15Cr-ODS steel produced by SPS. 73

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Fig. 3. HAADF image and EDS element mapping of nano-sized oxides.

from Fig. 2 is 0.51 μm by counting more than 2000 grains in the EBSD image taken in different regions of the specimen. Similar bimodal grain size distributions have been reported in other ODS alloys produced by SPS [21]. This heterogeneous microstructure is usually ascribed to: (1) the formation of the larger ferrite grains by recovery and grain growth in regions with a lower density of nano-sized oxides due to inhomogeneities in the starting Y distribution [42,43]; and/or (2) incomplete recrystallization and abnormal grain growth in which a few large grains break away from pinning nano-sized oxides and rapidly “consume’’ smaller, higher energy neighbours [44]. This heterogeneous microstructure will improve plasticity compared with other ODS steels in similar compositions fabricated by mechanical alloying (MA) and hot isostatic pressing (HIP) [7].

nm-scale oxides in 15Cr-ODS steel. Qualitatively, the areas of Zr, Y and O enrichment coincided with Fe and Cr depletion, and supported identification of the nano-sized oxides as being of the Y-Zr-O type. Fig. 4 shows HAADF micrograph and EDS element mapping of large oxides in 15Cr-ODS steel. Qualitatively, the areas of Cr Ti and O enrichment coincided with Fe depletion, and supported identification of the oxides as being of the Cr-Ti-O type. The contrast in STEM-HAADF images is highly sensitive to local variations in the atomic number (Zcontrast images), which is a more suitable approach for detailed imaging of nano-sized, disperse oxides [45]. Fig. 5 shows the bright-field TEM image of dislocations and nanooxides in the matrix (Red and yellow triangles represent dislocations and nano-oxides, respectively), and size distribution of the nano-oxides. A frequency plot of the nano-oxides diameter was determined by measuring 1402 particles from TEM images obtained from different regions of several specimens, and is shown in Fig. 5b. Average diameter and number density of the oxides were determined as 3.7 nm and 1.1 × 1023 m−3, respectively. The volume fraction of nano-sized oxides was, then, estimated as ∼0.35%, which are consistent with those in similar MA-SPS alloys [46], although somewhat at the smaller diameter

3.1.3. Chemical composition, size distribution, number density and crystallographic structures of oxides Two kinds of oxides with different size ranges are found, seen in Figs. 3 and 4. High-density nm-scale (< 10 nm) oxides are uniformly distributed in the specimen, and a few large oxides with size larger than 50 nm. Fig. 3 shows HAADF micrograph and EDS element mapping of

Fig. 4. HAADF image and EDS element mapping of large oxides. 74

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Fig. 5. (a) The bright-field TEM image of dislocations and nano-oxides, and (b) size distribution of nano-oxides from TEM.

Fig. 6. (a) HRTEM image of Y4Zr3O12 oxides, (b) FFT diagram of the micrograph in (a), (c) FFT filtered image derived from the image in (a), (d) simulated SAD pattern from [-2 1 0] Y4Zr3O12.

axis which agrees well with the FFT image from Y4Zr3O12 in Fig. 6b. No Y-O and/or Y-Ti-O oxides were identified. Y4Zr3O12 formation reveals the occurrence of internal oxidation reactions: 2Y2O3+3ZrO2= Y4Zr3O12 during consolidation due to the stronger interaction between Y and Zr [30]. Uchida et al. [47] have found that including Zr oxides had a smaller particle size due to the stronger interaction between Y, Zr and O and their cluster compared to Y, Ti and O. As a result, Y-Zr-O nano-oxides

and higher fraction end of those reported. HRTEM imaging was performed to identify the crystal structure of the Y-Zr-O enriched nano-sized oxides. Fig. 6 shows an HRTEM lattice image of an oxide particle (diameter: 4.2 nm), their FFT image, FFT filtered image and stimulated diffraction pattern from the [-2 1 0] zone axis. Two measured atomic plane distances are 3.038 Å and 3.008 Å with an angle of 109°. This result is in good agreement with the trigonal δ-phase Y4Zr3O12, which was indexed with [-2 1 0] zone axis. Fig. 6d shows that stimulated diffraction pattern (SAD) from the [-2 1 0] zone 75

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Fig. 7. (a) SAXS curve of 15Cr-ODS steel, and (b) Size distribution of nano-oxides from SAXS.

1058 ± 12 MPa, 11.2% ± 0.8, 553 ± 11 MPa, 18.5% ± 1.1 and 441 ± 8 MPa, 23.6% ± 1.3, respectively at room temperature (RT), 550 ℃ and 650 ℃. The total elongation of the specimen is lower 11.8% than other ODS steels in similar composition [32] at testing temperature of RT. Based on the microstructure characterization, it can be concluded that the plasticity decrease of the specimen is owing to the coarsening of the dispersed Cr-Ti-O oxides which are distributed in the grain boundary. The cracks initiate from these brittle oxides and grow as a result of cyclic plastic deformation under the conditions of service. Based on the microstructural characteristics of this material, the strength can be contributed by solid-solution strengthening (σss), grain boundary strengthening (σg), dislocation strengthening (σdis) and dispersion strengthening from nanoscale oxides (σp). The strain-stress curve of this steel tested at RT is shown in Fig. 9. It is widely accepted that the yield stress (YS) of the ODS alloys at RT can be estimated by a sum of strengthening contributions as follows [52,53]:

can hinder the grain growth. Otherwise Darling et al. [48] noticed that Zr in solute state can also prevent grain growth that Zr solutes in Fe will produce high elastic misfit strain energy and then lower the grainboundary free energy in nanocrystalline materials. So a constituent of Zr not only contribute to form oxides are dispersed inside the grain and along the grain boundary but also may be in solute state to hinder the grain growth. According to the TEM results (Fig. 5), the diameter of nano-sized oxides is smaller than 10 nm. For SAXS, the relationship between scattering vector q and oxides diameter d can be described as d = 2π/q [49]. Thus, the scattering vector range from 0.5 nm−1 to 1.5 nm−1 is analyzed for fitting the diameter distribution of nano-sized oxides, and the fitting process is exhibited in Fig. 7a. The diameter distribution of nano-sized oxides in 15Cr-ODS steel shows in Fig. 7b, the average diameter and number density of nano-sized oxides are 3.2 nm and 1.9 × 1023/m3, respectively. Both average size and size distribution of nano-sized particles obtained from SAXS show good agreement with TEM data. Whereas, the average size obtained from SAXS is smaller than that from TEM. The statistics difference between SAS and TEM data probably is due to the resolution restriction of TEM and/or the larger analyzed volume of SAXS [50,51]. Besides nano-sized particles, some large particles with size of 50–100 nm also are found in the matrix shown in Figs. 4 and 8. The EDS results of these particles indicate Cr, Ti, O elements are enriched in these particles. The selected area electron diffraction (SAED) pattern (inset into Fig. 8) confirms that the large particle is monoclinic CrTi2O5, which the zone axis direction is [1–32].

σy = σ0 + σss + σg +

2 σdis + σp2

(6)

Where σ0 is the Peierls-Nabarro friction stress in pure iron, it is needed for a dislocation to move into an atomic plane [54]. Based on Alinger's thesis [55], Hin et al. chose a value of 125 MPa in a simple strengthening model [56]. Solid solution σss is classically used for ODS alloy with the following form [57,58]:

σss =

∑ ki XiZ i

(7)

Where Xi is the atomic fraction and ki a hardening constant of the element i, Z = 3/4 For substitutional solid solution (Cr, W),

3.2. Analysis of strengthening mechanisms The yield strengths and total elongation of 15Cr-ODS steel are

Fig. 9. The strain-stress curve of the 15Cr-ODS steel tested at different testing temperatures.

Fig. 8. TEM bright field image and SAED pattern (inset) of CrTi2O5. 76

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was less relevant to predict the yield strength. The quadratic model takes into account the high probability of interacting mechanisms that are active in ODS alloys. Fig. 10 presents a comparison between the experiment data and calculation results of room temperature yield strength, the calculation results are very closely to the data measured in experiment which suggest that strengthening contributors from solid-solution atoms and grain boundaries are linear additive, while the dislocation strengthening and oxides strengthening were estimated by root mean square summation. The comparison results also manifest the Orowan strength strengthening has the highest contribution among the proposed mechanisms caused by nano-sized oxides in ODS alloy. High number density of nanoscale Y-rich oxides formed during SPS in the ODS alloys can suppress effectively dislocation movement, as well as grain boundaries migration. Some other performances, such as the irradiation resistance behavior, will be conducted in the next step work. Fig. 10. Comparison between the experimental data and the calculation value of yield strength at RT. Strengthening components from friction stress, solidsolution strengthening, grain boundary strengthening, dislocation strengthening, and dispersion strengthening were denoted by σ0, σss, σg, σdis, and σp, respectively.

4. Conclusion A 15Cr-ODS steel with a nominal composition of Fe-15Cr-2W-0.3Ti0.3Zr- 0.3Y2O3 was produced by MA and SPS. EBSD, SAXS and HRTEM techniques were employed to characterize the grain morphologies and nano-sized oxides in the SPS ODS steel. Furthermore, the yield strength was experimentally examined and quantitatively calculated based on various strengthening mechanisms, respectively. The following conclusions are obtained:

kCr = 9.95 MPa/at%3/4 and kW = 75.79 MPa/at%3/4, respectively. Z = 1/2 for interstitial element (C, N) is more suitable in bcc alloys, kC=kN = 1722.5 MPa/at%1/2. It is well known that the grain boundary strengthening is very important in polycrystalline materials. The grain boundary strengthening can be expressed by the Hall-Petch relationship as equation [59]: σg=khpd−1/2

(1) A bimodal grains are formed in 15Cr-ODS steel, which is attributed to the heterogeneous temperature field during SPS. (2) During hot processing, high-density nano-sized Y4Zr3O12 oxides are the main precipitates in the steel matrix, with slightly monoclinic CrTi2O5 oxides occasionally detected. The size distribution of oxides obtained from SAXS and TEM shows a good agreement. (3) A yield strength theoretical calculation based on the contribution of solute elements, grain size, dislocations and oxides allowed an accurate prediction of yield stress for the specimen.

(8)

Where khp is the Hall-Petch coefficient, which Kim et al. [60] demonstrated that the value of khp is ∼210 MPa μm1/2 in ferritic ODS alloys, and d is average grain size, which was determined as ∼0.51 μm in the specimen by EBSD analysis. The dislocation strengthening (σdis) should be taken into consideration since it is strongly related to strength in ODS alloy. This contribution is given by [36,61]:

σdis = MαGb ρ

Acknowledgments

(9)

This research is supported by the National Natural Science Foundation of China (51274121), the Provincial Natural Fund Project of Liaoning Province, China (201602396), the Science and Technology Project of the Education Department of Liaoning Province, China (2016TSZD03), United Fund between State Key Laboratory of Metal Material for Marine Equipment and Application, Ansteel Group Corporation and University of Science and Technology Liaoning (301002515, 301002520). The TEM analyses were carried out at the Joint-use Facilities: Laboratory of Nano-Micro Material Analysis, Hokkaido University. Finally, Haijian Xu thanks the inimitable care and support of Xiaoting Wang. I love you. Will you spend the rest of your life with me?

Where M is the Taylor factor (3.0), G is the shear modulus (85.3 GPa) [62], α is a numerical constant which equals to 0.38 in ferrite, ρ is the dislocation density. The dislocation density in ODS alloy can be evaluated by the following equation [63]:

ρdis = 2 3

(ξ 2)1/2 d×b

(10)

Where ε is lattice strain calculated by XRD profile and Eq. (1), d is the average grain size and b is the Burgers vector (0.248 nm). According to the calculation, the dislocation density of the specimen was 3.7 × 1014/ m2. Finally, the strengthening of enriched nano-sized oxides due to Orowan mechanism can be written as Eqs. (11) and (12) [36,64,65]

0.81MGb ln(2rs / r0) σp = 2π (1 − ν )1/2 λ − 2rs 1/2

λ=2

2 ⎡⎛ π ⎞ r ⎜ ⎟ 3 ⎢ ⎝ 4f ⎠ ⎣

⎤ − 1⎥ ⎦

References [1] [2] [3] [4] [5] [6] [7] [8]

(11)

(12)

Where λ is the mean planar center to the center distance between nanosized oxide particles, ν is the Poisson ratio (0.334) [66], r0 is the inner cut-off radius of a dislocation core, which is assumed to be the magnitude of the Burgers vector, rs = 0.816×r, r is the average planar oxide particles radius of the cross-section of a particle of radius, and f is the volume fraction of nano-oxides, which are determined by TEM and SAXS analysis. Also, Kim et al. showed that a linear superposition model

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