Ductile-to-Brittle Transition and Brittle Fracture Stress of Ultrafine-Grained Low-Carbon Steel

Ductile-to-brittle transition (DBT) temperature and brittle fracture stress, σF, are important toughness criteria for structural materials. In this paper, low-carbon steels with an ultrafine elongated grain (UFEG) structure (transverse grain size 1.2 μm) and with two ferrite (α)-pearlite structure with grain sizes 10 µm and 18 µm were prepared. The UFEG steel was fabricated using multipass warm biaxial rolling. The tensile tests with a cylindrical specimen and three-point bending tests with a single-edge-notched specimen were performed at −196 °C. The local stress near the notch was quantitatively calculated via finite element analysis (FEA). The σF for each sample was quantified based on the experimental results and FEA. The relationship between σF and dα in the wide range of 1.0 μm to 138 μm was plotted, including data from past literature. Finally, the conditions of grain size and temperature that cause DBT fracture in low-carbon steel were shown via the stress−d−1/2 map. The results quantitatively showed the superiority of α grain size for brittle fracture.


Introduction
Low-carbon steels are still the steel grades used in wide application due to their low price and good weldability, formability, and recyclability. They are normally composed of a ferrite(α)-pearlite microstructure through a thermomechanical control process [1,2], and they are changed to a α-cementite microstructure by refining α grains through a plastic deformation and heat treatment [3][4][5]. It is well known that yield strength, σ y , increases with a decrease in grain size, based on the Hall-Petch relationship [6][7][8][9]. Additionally, in uniaxial tensile tests, there are reports that the elongation is reduced by grain refinement, but the reduction in area is improved [10,11]. That is, ductility is not always deteriorated by grain refinement. With regard to toughness, which has a strong correlation with ductility, the ductile-to-brittle transition temperature (DBTT) obtained in the Charpy impact test is improved by refining the α grains [4,7,8]. This improvement in DBTT can be explained by the stress theory that the brittle fracture stress, σ F , increases significantly compared to the increase in σ y due to the refinement of α grains [4,12,13]. The improvement in σ F is one of the advantages of grain refinement. However, the σ F in ultrafine-grained steels has been little reported quantitatively, compared with the mechanical properties of σ y , ductility, DBTT, etc. This is attributed to the fact that steels with an ultrafine-grained (UFG) structure have a strong texture depending on process, so it is difficult to produce a bulk sample with an ultrafine-grained (UFG) structure and the local stress cannot be obtained directly from experiments alone, unlike the mechanical properties [14].
In the present study, an 800 MPa class low-carbon steel with an ultrafine elongated grain (UFEG) structure with an average transverse grain size of 1.2 µm was created by warm biaxial rolling process. For comparison, two low-carbon steels with a α-pearlite structure with grain sizes 10 µm and 18 µm were also prepared. The three-point bending tests were performed at −196 • C, and the σ F near the notch tip was quantitatively estimated based the data in the past literature [14][15][16] with α grain size in the range of 23 Finally, in low-carbon steel, the conditions of the α grain size and temperat the ductile-to-brittle transition fracture were clarified via the stress-(grain

Specimen Preparation
A low-carbon steel (0.15%C-0.3%Si-1.5%Mn) was used in this study prepared by vacuum melting and casting, homogenized at 1200 °C, hot-r square bar by a caliber-rolling simulator (Oono-roll Corporation, Tokyo, J then cut to 110 mm in length. The bar was hot-rolled to form a square ba sectional area of 2 cm 2 by the rolling simulator, then soaked to 900 °C for 1 air cooling (SM18 sample). The Vickers hardness was 146 ± 4HV0.1. In more finely grained structures, the hot-rolled 4 cm square bar was soaked h, and then it was caliber-rolled to form a square bar with a cross-sectiona followed by air cooling (SM10 sample). The hardness was 147 ± 9HV0.1 electron microscope (SEM), KEYWNCE VE-7800 (Keyence Corporation, Os ages of the microstructure for the SM18 and SM10 samples are shown in Fi samples have a typical α-pearlite structure. The average size of the α gra proximately 18 μm for the SM18 sample and approximately 10 μm for the Next, to obtain an UFG structure, the 4 cm square bar was austenitized at followed by ice-brine quenching. The UFG steel bar was produced via m rolling (WBR), as shown in Figure 2a, at a warm temperature of 550 °C. E mm square bar (WBR sample), shown in Figure 2b, was created. Detail of th was given previously [18,19]. The hardness was = 277 ± 7HV0.1. Figure  images of the microstructure. The average transverse size, dt, of the α grai [18], which was the same size from 1/2t to 1/8t.

Mechanical Properties and Microstructure
Tensile tests at −196 °C were performed with a crosshead speed of 0.85 mm·min −1 using specimens with a round cross section of 6.0 mm and a gage length of 30 mm [20]. In order to obtain strength data on the anisotropy in the WBR sample for FEA, tensile tests of small plate specimens were also performed at −196 °C with a crosshead speed of 0.11 mm·min −1 . The specimens with a parallel length of 2.6 mm, a width of 2 mm, and a thickness of 1 mm were machined from the bars in the RD, in the normal direction (ND), and in the transverse direction (TD) A three-point bending test at a crosshead speed of 0.5 mm min −1 was performed at −196 °C with a single-edge-notched specimen, as shown in Figure  3a. To verify the effect of the root radius (ρ) in initial notch on the σF, the SM10 specimen with a notch of ρ = 0.25 mm was also prepared. For the WBR sample, in order to investigate the effect of notch orientation, two kinds of specimens, LD//ND (WBR(N)) and LD//TD (WBR(T)), were prepared, as shown in Figure 3b,c. The specimens after test were observed through a digital camera and a digital microscope, and the fracture surfaces were observed through SEM operated at 15kV. Electron backscattered diffraction (EBSD) in a Schottky-type SEM operated at 15kV was used for observing the microstructures (JEOL JSM-7001F, Tokyo, Japan). EBSD analysis was performed via a JEOL-7001F equipped with a TSL-OIM analytical system (TSL solutions, Sagamihara, Japan).

Mechanical Properties and Microstructure
Tensile tests at −196 • C were performed with a crosshead speed of 0.85 mm·min −1 using specimens with a round cross section of 6.0 mm and a gage length of 30 mm [20]. In order to obtain strength data on the anisotropy in the WBR sample for FEA, tensile tests of small plate specimens were also performed at −196 • C with a crosshead speed of 0.11 mm·min −1 . The specimens with a parallel length of 2.6 mm, a width of 2 mm, and a thickness of 1 mm were machined from the bars in the RD, in the normal direction (ND), and in the transverse direction (TD) A three-point bending test at a crosshead speed of 0.5 mm min −1 was performed at −196 • C with a single-edge-notched specimen, as shown in Figure 3a. To verify the effect of the root radius (ρ) in initial notch on the σ F , the SM10 specimen with a notch of ρ = 0.25 mm was also prepared. For the WBR sample, in order to investigate the effect of notch orientation, two kinds of specimens, LD//ND (WBR(N)) and LD//TD (WBR(T)), were prepared, as shown in Figure 3b,c. The specimens after test were observed through a digital camera and a digital microscope, and the fracture surfaces were observed through SEM operated at 15kV. Electron backscattered diffraction (EBSD) in a Schottky-type SEM operated at 15kV was used for observing the microstructures (JEOL JSM-7001F, Tokyo, Japan). EBSD analysis was performed via a JEOL-7001F equipped with a TSL-OIM analytical system (TSL solutions, Sagamihara, Japan).

Mechanical Properties and Microstructure
Tensile tests at −196 °C were performed with a crosshead speed of 0.85 mm·min −1 using specimens with a round cross section of 6.0 mm and a gage length of 30 mm [20]. In order to obtain strength data on the anisotropy in the WBR sample for FEA, tensile tests of small plate specimens were also performed at −196 °C with a crosshead speed of 0.11 mm·min −1 . The specimens with a parallel length of 2.6 mm, a width of 2 mm, and a thickness of 1 mm were machined from the bars in the RD, in the normal direction (ND), and in the transverse direction (TD) A three-point bending test at a crosshead speed of 0.5 mm min −1 was performed at −196 °C with a single-edge-notched specimen, as shown in Figure  3a. To verify the effect of the root radius (ρ) in initial notch on the σF, the SM10 specimen with a notch of ρ = 0.25 mm was also prepared. For the WBR sample, in order to investigate the effect of notch orientation, two kinds of specimens, LD//ND (WBR(N)) and LD//TD (WBR(T)), were prepared, as shown in Figure 3b,c. The specimens after test were observed through a digital camera and a digital microscope, and the fracture surfaces were observed through SEM operated at 15kV. Electron backscattered diffraction (EBSD) in a Schottky-type SEM operated at 15kV was used for observing the microstructures (JEOL JSM-7001F, Tokyo, Japan). EBSD analysis was performed via a JEOL-7001F equipped with a TSL-OIM analytical system (TSL solutions, Sagamihara, Japan).

Numerical Procedure
A three-dimensional elastic-plastic FEA was performed using the FE-code ABAQUS/Standard (ver.6.5.4, Dassault systems, Tokyo, Japan). A 1/4 model was used by adopting the symmetry condition, as shown in Figure 4a. A 20-node quadratic element was used for the specimen, and the mesh in the specimen included 44,686 nodes and 52,110 elements. An 8-node linear element was used for the upper die and the lower die. The elements in the contact area between the specimen and the dies were made relatively fine, and the area near the initial notch was made finer, as shown in Figure 4b. A Young's modulus of 214 GPa and a Poisson ratio of 0.3 as the conventional material properties for low-carbon steel at −196 • C [21] were used in the FEA. The strain-stress curves shown in a later figure were used for the each specimen. The yield strength, σ y , at −196 • C was 789 MPa for the SM18 specimen, 913 MPa for the SM10 specimen, and 1256 MPa for the WBR specimen. Furthermore, in the WBR specimen, the yield condition of the anisotropy in which the y and z directions yield with a stress of 0.97 times the x direction was included in the model as the anisotropy of the yield strength. The anisotropic strength, YS ND /YS RD and YS TD /YS RD , was about 0.96 and 0.97, respectively. Here, YS RD is the 0.2% offset yield stress parallel to the RD, and the value was 1252 MPa. The YS RD = 1252 MPa was in agreement to the σ y = 1256 MPa. YS ND and YS TD denote the 0.2% offset yield stress parallel to the ND and TD, respectively. A three-dimensional elastic-plastic FEA was performed using the FE-code ABAQUS/Standard (ver.6.5.4, Dassault systems, Tokyo, Japan). A 1/4 model was used by adopting the symmetry condition, as shown in Figure 4a. A 20-node quadratic element was used for the specimen, and the mesh in the specimen included 44,686 nodes and 52,110 elements. An 8-node linear element was used for the upper die and the lower die. The elements in the contact area between the specimen and the dies were made relatively fine, and the area near the initial notch was made finer, as shown in Figure 4b. A Young's modulus of 214 GPa and a Poisson ratio of 0.3 as the conventional material properties for low-carbon steel at −196 °C [21] were used in the FEA. The strain-stress curves shown in a later figure were used for the each specimen. The yield strength, σy, at −196 °C was 789 MPa for the SM18 specimen, 913 MPa for the SM10 specimen, and 1256 MPa for the WBR specimen. Furthermore, in the WBR specimen, the yield condition of the anisotropy in which the y and z directions yield with a stress of 0.97 times the x direction was included in the model as the anisotropy of the yield strength. The anisotropic strength, YSND/YSRD and YSTD/YSRD, was about 0.96 and 0.97, respectively. Here, YSRD is the 0.2% offset yield stress parallel to the RD, and the value was 1252 MPa. The YSRD = 1252 MPa was in agreement to the σy = 1256 MPa. YSND and YSTD denote the 0.2% offset yield stress parallel to the ND and TD, respectively.  Figure 5a,b show the orientation maps along the RD for the SM18 sample and the SM10 sample. They are α structures without the strong texture observed in conventional heat-treated steel. Figure 5d,e show the orientation maps along the RD and ND for the WBR sample and the (001) pole figures corresponding to these maps. Here, the orientation maps on the cross-sectional plane parallel to the RD are given in previous paper [11]. The texture was dominated by {001}<100> cube orientations, which scarcely occurs in bodycentered cubic (bcc) metals. The {111}<110> was observed as a suborientation. The fraction of the cube texture was about 26.3% under a tolerance angle of 15°.  Figure 5a,b show the orientation maps along the RD for the SM18 sample and the SM10 sample. They are α structures without the strong texture observed in conventional heat-treated steel. Figure 5d,e show the orientation maps along the RD and ND for the WBR sample and the (001) pole figures corresponding to these maps. Here, the orientation maps on the cross-sectional plane parallel to the RD are given in previous paper [11].

Tensile Properties
The nominal stress−nominal strain curves at −196 °C are shown in Figure 6. The strength of the SM10 sample was greater than that of the SM18 sample, and the total elongation, TEL, was large. The strength of the WBR sample significantly increased compared to the other two samples, and the TEL decreased. However, the reduction in area, RA, did not decrease significantly regardless of the improvement in strength. We found that the strength-RA balance of UFEG steel is superior to that in α-pearlite steel [10,11]. All samples exhibited a yield-drop phenomenon followed by adequate elongation. Furthermore, the SM18 and SM10 samples had a typical curve that work hardening occurs after the Lüders elongation. Sharp yield-drop phenomena are commonly seen in ultrafine-grain materials and at low temperature [22,23].

Tensile Properties
The nominal stress−nominal strain curves at −196 • C are shown in Figure 6. The strength of the SM10 sample was greater than that of the SM18 sample, and the total elongation, TEL, was large. The strength of the WBR sample significantly increased compared to the other two samples, and the TEL decreased. However, the reduction in area, RA, did not decrease significantly regardless of the improvement in strength. We found that the strength-RA balance of UFEG steel is superior to that in α-pearlite steel [10,11]. All samples exhibited a yield-drop phenomenon followed by adequate elongation. Furthermore, the SM18 and SM10 samples had a typical curve that work hardening occurs after the Lüders elongation. Sharp yield-drop phenomena are commonly seen in ultrafine-grain materials and at low temperature [22,23].

Tensile Properties
The nominal stress−nominal strain curves at −196 °C are shown in Figure 6. The strength of the SM10 sample was greater than that of the SM18 sample, and the total elongation, TEL, was large. The strength of the WBR sample significantly increased compared to the other two samples, and the TEL decreased. However, the reduction in area, RA, did not decrease significantly regardless of the improvement in strength. We found that the strength-RA balance of UFEG steel is superior to that in α-pearlite steel [10,11]. All samples exhibited a yield-drop phenomenon followed by adequate elongation. Furthermore, the SM18 and SM10 samples had a typical curve that work hardening occurs after the Lüders elongation. Sharp yield-drop phenomena are commonly seen in ultrafine-grain materials and at low temperature [22,23].

Three-Point Bending Properties and Fracture Stress
Figure 7a,c shows the experimental result (solid line) and FE-result (broken line) of comparing the relations of bending load P and displacement u at −196 • C for the SM10 sample, as well as the appearance of the sample after the test. The specimen immediately fractured at displacement u = 0.327 mm at ρ = 0.13 mm and u = 0.396 mm at ρ = 0.25 mm. Although the FEA result at r = 0.25 mm shows a slight loss of linearity just before fracture, it can be seen that the P-u relations obtained via FEA are in good agreement with the experimental results regardless of ρ. The stresses, σ XX , σ YY , and σ ZZ , and the equivalent plastic strain, ε eq , obtained via FEA at the u = 0.327 mm and u = 0.396 mm, respectively, are shown in Figure 7b,d. Here, the FEA results show distributions of the stresses and strain on the central cross section (the plane of symmetry in the z direction in Figure 4a). From Figure 7b, it is predicted that the SM10 specimen at ρ = 0.13 mm fractured at peak maximum stress σ XX(max) , 2134 MPa (at this time, ε eq = 0.0044). This fracture stress σ F corresponds to 2.3 times the yield strength σ y of the SM10 sample. In the SM10 specimen at ρ = 0.25 mm, the σ XX(max) , i.e., σ F , was 2097 MPa (at this time, ε eq = 0.0038). The location of the peak maximum stress at ρ = 0.25 mm is farther from the notch tip than that at ρ = 0.13 mm due to the effect of the notch tip. The brittle fracture stress showed almost the same value regardless of ρ. The results for the SM18 sample are shown in Figure 8. The SM18 specimen at ρ = 0.13 mm brittlely fractured at u = 0.164 mm, and at that time the σ XX(max) was 1624 MPa (at this time, ε eq = 0.0034). This fracture stress σ F is much lower than that of the SM10 sample.
The P-u relations for the WBR samples and the appearance of the sample after the test are shown in Figure 9. Here, two results performed at each notch orientation of the WBR(N) specimen (LD//ND) and the WBR(T) specimen (LD//TD) are represented. The crack advanced vertically to the LD regardless of the notch orientation, i.e., the crack branched to the longitudinal direction of the specimen. The P sharply dropped after it attained a maximum load, P m ; beyond several load drops, P i , became almost constant, and decreased with several large load drops again thereafter. Finally, the all tests were terminated at u = 10 mm. A plateau region that appears after the P m results from the delamination caused by crack branching [4,11]. The P-u relations after the P m did not completely match in all specimens due to the difference of occurrence, propagation, and linkage of the microcracks on the delaminating cracks, but they showed almost the same behavior. The first load drop in the WBR(N)-1 specimen and the WBR(N)-2 specimen occurred at (u 1 , P 1 ) = (0.31 mm, 7.0 kN) and (0.28 mm, 6.3 kN), respectively. In the WBR(T)-1 specimen and the WBR(T)-2 specimen, it occurred at (u 1 , P 1 ) = (0.29 mm, 6.6 kN) and (0.29 mm, 6.4 kN), respectively. The results of comparing the P-u relations for the WBR(N)-1 specimen and the WBR(T)-1 specimen between the experiment (solid line) and the FEA (broken line) are shown in Figure 10a. Here, the FEA shows the results up to the displacement amount u 1 in which the P 1 occurred in the experiments, and it can be seen that the two are in good agreement for both specimens. Figure 10b shows the distributions of the σ XX , σ YY , σ ZZ and ε eq near the initial notch obtained via FEA at u 1 = 0.29 mm for the WBR(T)-1 specimen. Although the peak maximum stress, σ XX(max) , was 2662 MPa, the stresses to pay attention to would be σ YY(max) or σ ZZ(max) normal to the X direction, because the crack branched to the longitudinal direction of the specimen. Due to σ YY(max) < σ ZZ(max) , the σ ZZ(max) = 1627 MPa (at this time, ε eq = 0.0034) might be the fracture stress that caused the first microcrack. mm, the σXX(max), i.e., σF, was 2097 MPa (at this time, εeq = 0.0038). The location of the peak maximum stress at ρ = 0.25 mm is farther from the notch tip than that at ρ = 0.13 mm due to the effect of the notch tip. The brittle fracture stress showed almost the same value regardless of ρ. The results for the SM18 sample are shown in Figure 8. The SM18 specimen at ρ = 0.13 mm brittlely fractured at u = 0.164 mm, and at that time the σXX(max) was 1624 MPa (at this time, εeq = 0.0034). This fracture stress σF is much lower than that of the SM10 sample.  The P-u relations for the WBR samples and the appearance of the sample after the test are shown in Figure 9. Here, two results performed at each notch orientation of the WBR(N) specimen (LD//ND) and the WBR(T) specimen (LD//TD) are represented. The crack advanced vertically to the LD regardless of the notch orientation, i.e., the crack branched to the longitudinal direction of the specimen. The P sharply dropped after it attained a maximum load, Pm; beyond several load drops, Pi, became almost constant, and decreased with several large load drops again thereafter. Finally, the all tests were termi-

Delaminating Crack
As seen in Figures 7 and 8, in the SM10 specimen and the SM18 specimen, the crack advanced directly across the central portion of the specimen and the σF was represented by a maximum peak stress, σXX(max), parallel to the longitudinal direction. In contrast, in the WBR specimen with the UFEG structure, the crack advanced vertically to the LD. The σF was represented by a maximum peak stress, σZZ(max), normal to the longitudinal direction. The initial microcrack is considered to have occurred when the first load drop appeared during the bending test. In order to verify this, the cross section of the specimen after the bending test was observed. Figure 11 shows a cross-sectional image of the WBR(T)-1 specimen after the test. Several branching cracks parallel to the RD can be observed. The branching crack near the notch tip is the first branching crack that occurred during the bending test, and it occurred approximately 0.14 mm from the notch tip. This

Delaminating Crack
As seen in Figures 7 and 8, in the SM10 specimen and the SM18 specimen, the crack advanced directly across the central portion of the specimen and the σF was represented by a maximum peak stress, σXX(max), parallel to the longitudinal direction. In contrast, in the WBR specimen with the UFEG structure, the crack advanced vertically to the LD. The σF was represented by a maximum peak stress, σZZ(max), normal to the longitudinal direction. The initial microcrack is considered to have occurred when the first load drop appeared during the bending test. In order to verify this, the cross section of the specimen after the bending test was observed. Figure 11 shows a cross-sectional image of the WBR(T)-1 specimen after the test. Several branching cracks parallel to the RD can be observed. The branching crack near the notch tip is the first branching crack that occurred during the bending test, and it occurred approximately 0.14 mm from the notch tip. This

Delaminating Crack
As seen in Figures 7 and 8, in the SM10 specimen and the SM18 specimen, the crack advanced directly across the central portion of the specimen and the σ F was represented by a maximum peak stress, σ XX(max) , parallel to the longitudinal direction. In contrast, in the WBR specimen with the UFEG structure, the crack advanced vertically to the LD. The σ F was represented by a maximum peak stress, σ ZZ(max) , normal to the longitudinal direction. The initial microcrack is considered to have occurred when the first load drop appeared during the bending test. In order to verify this, the cross section of the specimen after the bending test was observed. Figure 11 shows a cross-sectional image of the WBR(T)-1 specimen after the test. Several branching cracks parallel to the RD can be observed. The branching crack near the notch tip is the first branching crack that occurred during the bending test, and it occurred approximately 0.14 mm from the notch tip. This position corresponds to the position of the maximum peak stress, σ ZZ(max) , shown in Figure 10b. The mechanism of the first branching crack formation was shown in Figure 12. In the UFEG structures with a strong cube texture, there are {100} cleavage planes along the RD and LD, as illustrated in Figure 12c. The crack//RD is mainly situated in the ferrite matrix and its grain boundaries. The bcc iron cleaves on {100} planes and the coherence length on {100} corresponds to the cleavage crack length. The coherence length on {100} for crack//RD in the elongated grains with a cube texture is much longer than that for crack⊥RD in these grains. This structure produces a condition in which the main crack can run along the longitudinal direction. The stresses near the notch tip have a relation of σ XX >> σ ZZ > σ YY , as shown in Figure 10b. Moreover, the fracture stress has a relation of σ F⊥RD << σ F//RD from σ F ∝ d eff−0.5 on the basis of the Griffith theory, where d eff denotes an effective grain size for cleavage fracture. Since the brittle fracture occurs when the stress σ ij exceeds the fracture stress, a delaminating crack related to a brittle fracture induced by σ ZZ , σ YY > σ F⊥RD occurred before σ XX > σ F//RD or a ductile fracture. As a result, the first crack parallel to the RD near the initial notch occurred as shown in Figure 11, and the first load drop, P 1 , appeared (Figure 10a). position corresponds to the position of the maximum peak stress, σZZ(max), shown in Figure  10b. The mechanism of the first branching crack formation was shown in Figure 12. In the UFEG structures with a strong cube texture, there are {100} cleavage planes along the RD and LD, as illustrated in Figure 12c. The crack//RD is mainly situated in the ferrite matrix and its grain boundaries. The bcc iron cleaves on {100} planes and the coherence length on {100} corresponds to the cleavage crack length. The coherence length on {100} for crack//RD in the elongated grains with a cube texture is much longer than that for crack⊥RD in these grains. This structure produces a condition in which the main crack can run along the longitudinal direction. The stresses near the notch tip have a relation of σXX >> σZZ > σYY, as shown in Figure 10b. Moreover, the fracture stress has a relation of σF⊥RD << σF//RD from σF ∝ deff−0.5 on the basis of the Griffith theory, where deff denotes an effective grain size for cleavage fracture. Since the brittle fracture occurs when the stress σ ij exceeds the fracture stress, a delaminating crack related to a brittle fracture induced by σZZ, σYY > σ F⊥ RD occurred before σXX > σF//RD or a ductile fracture. As a result, the first crack parallel to the RD near the initial notch occurred as shown in Figure 11, and the first load drop, P1, appeared (Figure 10a).    Figure  10b. The mechanism of the first branching crack formation was shown in Figure 12. In the UFEG structures with a strong cube texture, there are {100} cleavage planes along the RD and LD, as illustrated in Figure 12c. The crack//RD is mainly situated in the ferrite matrix and its grain boundaries. The bcc iron cleaves on {100} planes and the coherence length on {100} corresponds to the cleavage crack length. The coherence length on {100} for crack//RD in the elongated grains with a cube texture is much longer than that for crack⊥RD in these grains. This structure produces a condition in which the main crack can run along the longitudinal direction. The stresses near the notch tip have a relation of σXX >> σZZ > σYY, as shown in Figure 10b. Moreover, the fracture stress has a relation of σF⊥RD << σF//RD from σF ∝ deff−0.5 on the basis of the Griffith theory, where deff denotes an effective grain size for cleavage fracture. Since the brittle fracture occurs when the stress σ ij exceeds the fracture stress, a delaminating crack related to a brittle fracture induced by σZZ, σYY > σ F⊥ RD occurred before σXX > σF//RD or a ductile fracture. As a result, the first crack parallel to the RD near the initial notch occurred as shown in Figure 11, and the first load drop, P1, appeared (Figure 10a).

Brittle Fracture Stress vs. Grain Size
From the Griffith equation, the σ F is represented by Here, E denotes Young's modulus, γ s is the surface energy of the fracture, and ν is Poisson's ratio [24]. Actually, a shape factor related to the method of fracture test must be taken into account in Equation (1). Under this three-point bending test, the σ F is represented by the following [16,25]: Generally, the d eff is the α grain size and the σ F is determined through experiments and FEA. In the case of the WBR samples, the d eff (in this case, d effL ) for the fracture stress σ F⊥RD , as shown in Figure 12c, corresponds to the average grain size, d L , in a longitudinal direction. It is difficult to accurately measure the size of d L from EBSP maps and the SEM images of the delaminating fracture surface, but the d effL can be calculated from the results of the equiaxed grain sample and σ F⊥RD . Assuming the γ s , E, and ν in the SM18 sample and the WBR sample are the same, the effective grain size, d effL(WBR) , of the WBR sample is expressed by The σ F//RD(WBR) is 6284 MPa using d efft(WBR) = 1.2 µm. That is, UFG steel with a grain size of 1.2 µm is a very high fracture stress of about 6.3 GP. Figure 13 shows the fracture stresses plotted as a function of the inverse square root of d eff , together thewith the data shown in past literature [14][15][16]. The effective surface energy is estimated to be 158 J m −2 based on this linear relation and Equation (2). Here, E = 214 GPa and ν = 0.30 were used in this relation [21]. This value is included in the ranges of 90-190 J m −2 for C-Mn steels with α-pearlite microstructure [26].

Brittle Fracture Stress vs. Grain Size
From the Griffith equation, the σF is represented by Here, E denotes Young's modulus, γs is the surface energy of the fracture, and ν is Poisson's ratio [24]. Actually, a shape factor related to the method of fracture test must be taken into account in Equation (1). Under this three-point bending test, the σF is represented by the following [16,25]: Generally, the deff is the α grain size and the σF is determined through experiments and FEA. In the case of the WBR samples, the deff (in this case, deffL) for the fracture stress σF ⊥ RD, as shown in Figure 12c, corresponds to the average grain size, dL, in a longitudinal direction. It is difficult to accurately measure the size of dL from EBSP maps and the SEM images of the delaminating fracture surface, but the deffL can be calculated from the results of the equiaxed grain sample and σF ⊥ RD. Assuming the γs, E, and ν in the SM18 sample and the WBR sample are the same, the effective grain size, deffL(WBR), of the WBR sample is expressed by Using deff(SM18) = 18.0 μm, σF(SM18) = 1624 MPa, and σF ⊥ RD(WBR) = 1627 MPa, the deffL(WBR) is estimated to be 17.9 μm. The fracture stress, σF//RD(WBR), parallel to the RD is described by The σF // RD(WBR) is 6284 MPa using defft(WBR) = 1.2 μm. That is, UFG steel with a grain size of 1.2 μm is a very high fracture stress of about 6.3 GP. Figure 13 shows the fracture stresses plotted as a function of the inverse square root of deff, together thewith the data shown in past literature [14][15][16]. The effective surface energy is estimated to be 158 J m −2 based on this linear relation and Equation (2). Here, E = 214 GPa and ν = 0.30 were used in this relation [21]. This value is included in the ranges of 90-190 J m −2 for C-Mn steels with α-pearlite microstructure [26]. Figure 13. Fracture stress vs. effective grain size. Figure 13. Fracture stress vs. effective grain size.

Condition for Brittle Fracture
The σ y increases with k × d α −1/2 under the Hall-Petch relationship. In the previous study [11], the coefficient k was 0.6 MPa·m −1/2 regardless of temperature in low-carbon steel, and this is consistent with the result reported by other researchers [8,23]. The slope, 6.8, in the σ F − d eff −1/2 relation shown in Figure 13 is 10 times larger than the k. This indicates that the DBTT is significantly improved by grain refinement. To quantitatively verify this, the conditions of brittle fracture in low-carbon steel under the present bending test and Charpy impact test were examined through the results of Charpy test obtained in previous papers [11,20] and the results of the present study for the SM18 sample. Figure 14a shows variations in the Charpy absorbed impact energy, vE (Figure 9 in [11]), and fracture energy, J (Figure 8b in [20]), with the temperature for the SM18 sample. Here, the vE curve was obtained from the Charpy impact tests with full-size V-notched specimens under a 500 J capacity [11], and the J denotes the area under the P − u curve, as shown in Figure 8a. The sample exhibits a typical energy-transition curve, in which the vE and J decrease with decreasing temperature. In the three-point bending test, the J starts to increase from below about −140 • C. That is, the sample transfers from brittle fracture to ductile fracture at that point. The P − u relation at −150 • C and the stress distributions near the initial tip at u = 0.289 mm, in which the sample exhibited catastrophic fracture, are shown in Figure 14b. These results were obtained using the same approach as for the result at −196 • C (Figure 8a). The σ F at −150 • C was 1674 MPa, and this value is almost in agreement with the σ F at −196 • C, shown in Figure 8b. The σ F corresponds to about 3 times the σ y (=560 MPa) at −150 • C. On the basis of the Yoffee diagram [12], the DBTT occurs when the maximum tensile stress near the notch/crack tip, σ 1 , exceeds the σ F . The σ 1 has to be calculated via FEA. Since the σ 1 is about 3 times the σ y in the present bending test, a relation among the σ F , the σ 1 and temperature is shown in Figure 14c. The σ F is independent of temperature. Hence, the σ 1 that increases with decreasing temperature exceeds the σ F at about −140 • C. This indicates that the sample exhibits complete brittle fracture below −140 • C, complete ductile fracture above −30 • C, and ductile-to-brittle transition fracture (DBTF) between those temperatures. In contrast, in the Charpy impact test, although it is difficult to precisely calculate the σ 1 near the notch during the impact test due to high-speed deformation and heat generation, the σ 1 increases as the strain rate increases. According to the FEA results of Tanguy et al. [27] and Takashima and Minami [28], the σ 1 is about 3.5-4 times the σ y . Since the σ F is independent of the strain rate, complete brittle fracture can be predicted to appear even at temperatures above −140 • C obtained in the three-point bending test. The vE curve in Figure 14a,c proves this fact. The sample shows complete brittle fracture at below −90 • C, indicated by a black arrow. The σ y at −90 • C is about 430 MPa, and the σ F = 1674 MPa corresponds to about 3.9 times the σ y . This is consistent with the FEA results of σ 1 ≈ (3.5-4) σ y [27,28].
From Figure 13 and the Hall-Petch relationship, we can quantitatively show the superiority of the α grain size for brittle fracture. Figure 15 shows a comparison of the grain-size dependence of maximum tensile stress and brittle fracture stress. This result indicates that a low-carbon steel with a α grain size of 6.3 µm or more exhibits complete brittle fracture at −196 • C, and one with a α grain size of 4.3 µm or less exhibits complete ductile fracture even at −196 • C. When the ρ of the initial notch or ligament length in the test specimen is small or when a fracture test is conducted under a high strain rate, these grains shift toward a finer size. In the Charpy impact test, since the maximum tensile stress near the V-notch in a specimen is assumed to be 3.9 times the σ y , it is predicted via the stress-d α −1/2 map shown in Figure 15 that a low-carbon steel with a α grain size of 3.0 µm or more exhibits complete brittle fracture at −196 • C, and a steel with a α grain size of 2.5 µm or less exhibits complete ductile fracture. In the general fracture test, the maximum tensile stress near the crack or notch tip is of the order of (2-4) σ y [14,[27][28][29][30]. If a ferrite grain can be ultrarefined to 1 µm, the material would exhibit complete ductile fracture even under liquid nitrogen temperature. Through the maps shown in Figures 14c and 15, we can estimate the grain size and temperature at which ductile-to-brittle transition fracture occurs. From Figure 13 and the Hall-Petch relationship, we can quantitatively show the superiority of the α grain size for brittle fracture. Figure 15 shows a comparison of the grainsize dependence of maximum tensile stress and brittle fracture stress. This result indicates that a low-carbon steel with a α grain size of 6.3 μm or more exhibits complete brittle fracture at −196 °C, and one with a α grain size of 4.3 μm or less exhibits complete ductile fracture even at −196 °C. When the ρ of the initial notch or ligament length in the test specimen is small or when a fracture test is conducted under a high strain rate, these grains shift toward a finer size. In the Charpy impact test, since the maximum tensile stress near the V-notch in a specimen is assumed to be 3.9 times the σy, it is predicted via the stressdα −1/2 map shown in Figure 15 that a low-carbon steel with a α grain size of 3.0 μm or more exhibits complete brittle fracture at −196 °C, and a steel with a α grain size of 2.5 μm or less exhibits complete ductile fracture. In the general fracture test, the maximum tensile stress near the crack or notch tip is of the order of (2-4) σy [14,[27][28][29][30]. If a ferrite grain can be ultrarefined to 1 μm, the material would exhibit complete ductile fracture even under liquid nitrogen temperature. Through the maps shown in Figures 14c and 15, we can estimate the grain size and temperature at which ductile-to-brittle transition fracture occurs.

Conclusions
The UFEG low-carbon steel with atransverse grain size of 1.2 μm dominated by a cube texture was created via multipass warm biaxial rolling. Two low-carbon steels with Figure 15. Conditions for brittle fracture in low-carbon steel on the basis of a comparison of the grain-size dependence of maximum tensile stress and brittle fracture stress.

Conclusions
The UFEG low-carbon steel with atransverse grain size of 1.2 µm dominated by a cube texture was created via multipass warm biaxial rolling. Two low-carbon steels with a ferrite (α)-pearlite structure with grain size 10 µm and 18 µm were also prepared via conventional rolling and heat treatment. These samples were studied from the viewpoint of the toughness criterion, ductile-to-brittle transition fracture, and brittle fracture stress through mechanical tests and FEA. The main results are summarized as follows.
(1) The brittle fracture stress (σ F ) was estimated through the three-point bending test and FEA. The σ F for low-carbon steels with a grain size of 18 µm, 10 µm, and 1.2 µm was about 1674 MPa, 2134 MPa, and 6284 MPa, respectively.
(2) The σ F and ferrite grain size (d α ) had a relation of σ F = 6.8 d α −1/2 . The slope, 6.8, is 10 times larger than in the Hall-Petch relationship. This indicates that grain refinement significantly improves the DBTT.
(3) The conditions of α grain size and temperature that cause a ductile-to-brittle fracture in low-carbon steel were clarified. We can understand them via the stress −d α −1/2 map. The superiority of the α grain size for brittle fracture was quantitatively shown.