Work Hardening in Ferritic Steel Containing Ultra-fine Carbides

Abstract

Work hardening of ferritic steels containing fine carbides varied from 3 nm to 15 nm was investigated and compared to Ashby’s model, which is well known as a work hardening theory of metals containing hard particles. A specific work hardening behavior was observed in the steels strengthened by the nanometer-sized carbides; work hardening proceeded in two stages within a few plastic strains. In the former step, the matrix deformed without the geometrically necessary dislocation since a misfit strain between the carbides and matrix is close to the Burgers vector. So the Ashby’s model cannot explain this phenomenon. Yet in the later stage, the amount of work hardening was close to predicted value based on the Ashby’s model. The plastic strain at which the later stage started decreased with the increase in the diameter of carbides since the geometrically necessary dislocation is easier to be generated by the larger carbides. A new model which can be applied to steels containing the nanometer-sized carbides by focusing generating dislocation into the matrix around carbides was established.

1. Introduction

Shape of the stress-strain curve of the conventional precipitation hardening steel sheets by the tensile test generally depicts parabolic.¹⁾ Ashby’s work hardening model^2,3,4) has been used to explain the parabolic curve of the steels; a major factor determining the work hardening is the geometrically necessary dislocation (GN-dislocation) generated at interface boundaries between a hard particle and matrix.

In the case of steels strengthened by nanometer-sized carbides, their size is close to the Burgers vector. The work hardening of the steels can differ from the predicted one by Ashby’s model since the GN-dislocation seems to be difficult to be generated by the fine carbides.

The purpose of this study was to investigate a work hardening behavior of steels containing carbides with several diameters from 3 nm to 15 nm. Furthermore a new work hardening model that can apply to the steel containing nanometer-sized carbides was proposed.

2. Experimental Procedure

Low carbon steels containing of 0.02mass%C-0.09mass%Ti and 0.04mass%C-0.16mass%Ti were prepared in order to investigate a change of work hardening by the volume fraction of the carbides. The atomic concentration ratio of C and Ti were designed to be 1.0 to avoid an effect of large cementites on work hardening.^5,6) Manganese in the amount of 1.3 mass% was also added to obtain fine carbides by lowering γ/α transformation temperature^7,8) in the hot-rolling process as described bellow.

The vacuum-smelted and cast to 25 kg ingots were rough-rolled to 33 mm thickness bars and slabs in 70 mm width and 50 mm length were cut out from the bars for hot-rolling. The slabs were reheated at 1523 K for 3.6 ks and hot-rolled to 4 mm thickness by 5 passes; finishing temperature was controlled around 1223 K. They were rapid-cooled at a cooling rate of 30 K/s to 923 K, 973 K, or 1023 K in an alumina fluid furnace at a room temperature and then aged for 600 s in an electric furnace set up at 923 K, 973 K or 1023 K, followed by air-cooling. One of the hot-rolled steel sheets that were kept at 973 K for 600 s was wrapped in a stainless foil to retard decarburization and then reheated at 973 K for 86.4 ks to coarsen the carbides. In the following, sample A is 0.09mass%Ti added steel held at 923 K for 600 s after hot-rolling; samples B and C are 0.16mass%Ti added steel held at 923 K and 1023 K for 600 s, respectively, sample D is 0.16mass%Ti added steel held 973 K for 600 s and reheated at 973 K for 86.4 ks.

Samples for the tensile test of which the tensile direction was parallel to the rolling direction were cut from the hot-rolled sheets, and grinded to 2 mm in thickness in order to remove surface scale. The samples with 50 mm in gauge length and 12.5 mm in gauge width (JIS13B type specimen) were prepared, and then the tensile tests were carried out at a strain rate of 3.3×10⁻⁴ s⁻¹. The yield strength was defined as 0.2% proof stress since all the samples showed the continuous yielding.

The dislocation density in the hot-rolled sheets was evaluated using a X-ray diffractratometry⁹⁾ with a Co radiation source and was calculated by the half peak widths of bcc iron peaks (110), (211) and (220).

Samples for microstructure observation were etched by nital in a cross section along the rolling direction and were observed using a scanning electron microscope (SEM). Tens of horizontal and vertical lines were drawn on the SEM photographs; each line was 35 μm in real length, and the points at the intersection of these lines and ferrite grain boundaries were counted. The mean lineal intercept length was calculated on the average of each intercept length, and the mean ferrite grain diameter prescribed by ASTM¹⁰⁾ was calculated by multiplying the mean lineal intercept length by 1.13.

Thin foils were prepared from the hot-rolled steel by the twin-jet method and the carbides were observed by a transmission electron microscope (TEM) at 200 kV. Elements in the carbides were analyzed by energy dispersive X-ray spectrometry attached to TEM (TEM-EDS). The diameter of each carbide was measured from the TEM photographs using an image analysis software (Image Pro, Nippon Roper). The areas of carbides were measured and converted to the equivalent circle diameter. The mean diameter of carbides was then obtained by the average of them with more than 100 carbides.

Precipitates in the steels were extracted electrolytically with 10% acetyle acetone-tetramethyl ammonium chloride-methanol (10%AA) and filtered with a mesh with the 0.2 μm pore size to perform quantitative analysis of Fe and Ti concentrations by inductively coupled plasma-mass spectrometry (ICP-MS). The amount of Ti in carbides was determined by a following way. At first, solid solution Ti content was measured; Ti concentration in the electrolytic solution dissolving the matrix was detected to be solid-solute Ti. Secondary, the amounts of Ti as TiS and TiN were calculated from the amounts of S and N assuming that their compositions are stoicheiometric, respectively. The amount of Ti as TiC was calculated by subtracting the solid solution Ti content in matrix, Ti as TiS and Ti as TiN from the total Ti amount. The volume fraction of carbides (TiC) was calculated with 4.93 g/cm³ of the TiC density.¹¹⁾ Meanwhile, Fe content as cementite was measured from the filtered residue by ICP-MS assuming that the cementite composition is Fe₃C.

3. Results

3.1. Microstructures

Figure 1 shows the SEM micrographs of the samples A, B, C and D. All the microstructures were fully ferritic, and any bainite and martensite were not observed. Since the very small amounts of Fe as Fe₃C were in the range between 0.003 mass% and 0.006 mass%, the influence of large cementites on the work hardening was ignored in this study. The diameters of the ferrite grains were within a narrow range of 4–7 μm, therefore the influence of the change in the ferrite grain size³⁾ was also ignored.

Fig. 1.

Scanning electron micrographs of samples; (a) sample A, (b) sample B, (c) sample C and (d) sample D.

Figure 2 shows the TEM bright field images of the samples. Carbides are recognized to be black contrast indicated by the arrows on the photos. The ultra-fine carbides, of which the mean diameter was about 3 nm, were observed in the samples A and B, whereas the mean carbide diameters in the samples C and D were 8 nm and 15 nm, respectively.

Fig. 2.

Transmission electron micrographs of samples; (a) sample A, (b) sample B, (c) sample C and (d) sample D.

The volume fractions of TiC in the samples A, B and C were 0.12 vol.%, 0.23 vol.% and 0.29 vol.%, respectively. The volume fraction of TiC in the sample D was as large as that in the sample C.

Table 1 lists the mean diameter of the ferrite grains and carbides, and the volume fraction of the carbides.

Table 1. Diameters of ferrite grains and carbides and volume fraction of carbides in samples A, B, C and D.

Sample	Thermal history	Diameter of ferrite grain	Diameter of carbide		Volume fraction of carbide
			Mean	Standard deviation
		μm	nm		vol.%
A	923 K for 600 s	5	3.2	0.9	0.12
B		4	3.0	0.7	0.23
C	1023 K for 600 s	6	7.7	2.4	0.29
D	973 K for 86.4 ks	7	14.5	6.1	0.29

3.2. Mechanical Properties

Table 2 shows mechanical properties of the samples. Yield ratio (YR) values were calculated by dividing the yield strength by the tensile strength. The work hardening coefficients (n-value)¹⁾ were calculated at the true strains of 0.01 and 0.03. The yield strength of the sample A was lower than that of the sample B, but the YR and n-values of the samples A and B were at the same level. The yield strengths and the YR values decreased with increasing the diameter of the carbides while the n-values increased.

Table 2. Mechanical properties of samples A, B, C and D.

Sample	YS	TS	YR	El	n-value
	MPa	MPa	–	%	–
A	537	617	0.87	22	0.08
B	708	795	0.89	18	0.07
C	423	563	0.75	22	0.14
D	333	454	0.73	28	0.14

In order to confirm whether the steel containing the fine carbides is subject to Orowan mechanism¹²⁾ or cutting mechanism,^13,14) the precipitation hardening was estimated by assuming that the yield strength obeys a superposition law shown in Eqs. (1)¹⁵⁾ and (2).¹⁶⁾   

$$\text{YS(MPa)}={\sigma }_0+\mathrm{\Delta }{\sigma }_{\mathrm{s}\mathrm{o}\mathrm{l}}+\mathrm{\Delta }{\sigma }_{\mathrm{g}\mathrm{r}\mathrm{a}}+\mathrm{\Delta }{\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}}+\mathrm{\Delta }{\sigma }_{\mathrm{p}\mathrm{r}\mathrm{e}}$$(1)

  

$$\text{YS(MPa)}={\sigma }_0+\mathrm{\Delta }{\sigma }_{\mathrm{s}\mathrm{o}\mathrm{l}}+\mathrm{\Delta }{\sigma }_{\mathrm{g}\mathrm{r}\mathrm{a}}+{\left(\mathrm{\Delta }{\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}}^2+\mathrm{\Delta }{\sigma }_{\mathrm{p}\mathrm{r}\mathrm{e}}^2\right)}^{1/2}$$(2)

Here, σ₀ is the lattice friction (=53.9 MPa),¹⁷⁾ the amounts of hardening by each mechanism were calculated: Δσ_sol is the solid-solution hardening, Δσ_gra is the grain refinement hardening, Δσ_dis is the dislocation hardening and Δσ_pre is the precipitation hardening. The Δσ_sol and Δσ_gra were calculated by Eqs. (3) and (4),¹⁸⁾ respectively.   

$$\mathrm{\Delta }{\sigma }_{\mathrm{s}\mathrm{o}\mathrm{l}}=32.34\text{[\%Mn]}+\text{83}\text{.16[\%Si]}+354.2\text{[\%N]}$$(3)

  

$$\mathrm{\Delta }{\sigma }_{\mathrm{g}\mathrm{r}\mathrm{a}}=14.92d^{-\text{1/2}}$$(4)

Here, [%Mn], [%Si] and [%N] are the mass concentration of manganese, silicon and nitrogen in solution, respectively. [%Mn] and [%Si] were 1.3% and 0.01%, respectively, in all the samples. [%N], however, was zero since all nitrogen is consumed by forming large TiN. The d (mm) is the mean diameter of the ferrite grains. The Δσ_pre was obtained by Eq. (5)¹⁹⁾ which is based on the Orowan mechanism.   

$$\mathrm{\Delta }{\sigma }_{\mathrm{p}\mathrm{r}\mathrm{e}}=\frac{5\mathrm{\hspace{0.17em} }900}{d_{\mathrm{p}}}{f}^{1/2}ln\frac{d_{\mathrm{p}}}{b}$$(5)

Here, d_p (nm) and f are the mean diameter and the volume fraction of the carbides, respectively, b is the Burgers vector (=0.25 nm). The Δσ_dis is evaluated by Eq. (6)²⁰⁾ based on Bailey-Hirsh equation.²¹⁾   

$$\mathrm{\Delta }{\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}}=0.5Gb{\rho }^{1/2}$$(6)

Here, ρ is the dislocation density, G is the rigidity modulus. G was about 80 GPa measured by the resonance method. The dislocation density of the sample D was assumed to be identical to reported value,¹⁸⁾ of which the steel was held at 923 K for 48 hrs. Table 3 shows the calculation results of the amounts of each hardening and the comparison with the experimental yield strength. Δσ_gra and Δσ_pre were calculated using the values in Table 1. The calculated yield strengths obtained by Eq. (2) showed a good agreement with the experimental results. By contrast, the linear approximation, Eq. (1), was not appropriate. The Δσ_dis and Δσ_pre cannot be summable linearly when both spacings of dislocations and carbides are small.^19,25) It is important to note that the calculation result by Eq. (2) suggests the Δσ_pre can be obtained by Orowan mechanism even in the samples A and B. The yield strength could not be explained by the cutting mechanism at least since the calculated values by the cutting mechanism were bigger than the experimental values.

Table 3. Comparison of YS between calculated and measured value.

Sample	Solid-Solution hardening	Grain refinement hardening	Precipitation hardening		Dislocation density	Dislocation hardening	Calculated YS		Measured YS
			Average	Range			Eq. (1)	Eq. (2)
	MPa	MPa	MPa		x 1014 m−2	MPa	MPa	MPa	MPa
A	43	209	163	140–198	4.6	213	683	575	537
B	43	228	235	207–274	8.9	297	856	703	708
C	43	193	142	117–184	0.8	90	521	457	423
D	43	184	89	68–133	0.5	72	442	395	333

Funakawa et al.¹⁷⁾ had indicated the steel containing 3 nm of carbides that consisted of Ti and Mo was strengthened by the Ashby-Orowan mechanism. The result reported by Kamikawa et al.¹⁵⁾ also showed steels containing carbides of which one third was 3 nm or less and more than 3 nm of carbides were both strengthened by the Ashby-Orowan mechanism. In the case of 1.6 nm or more of the carbide, at least, the strength of the samples in this study will be applied to the Ashby-Orowan mechanism.

The initial dislocation densities of the samples in this study were different from each others, as shown in Table 3. The effect of the initial dislocation density on work hardening was anticipated to be small if interparticle spacing (λ₁) is smaller than mean spacing of dislocations (λ₂). The λ₁ and λ₂ can be calculated by Eqs. (7)²²⁾ and (8),²³⁾ respectively. The Eq. (11) was assumed to be random arrays of precipitates.   

$${\lambda }_1=\left[1.25\sqrt{\frac{\pi }{6f}}-\frac{\pi }{4}\right]d_{\mathrm{p}}$$(11)

  

$${\lambda }_2={\left(\frac{3}{\rho }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$(7)

A relationship between λ₁ and λ₂ is displayed in Fig. 3. An interaction between dislocation and carbide is considered to influence the work hardening behavior rather than an interaction between dislocations since λ₁ was almost identical with λ₂. So the pinning effect by carbide will be bigger than the interaction between dislocations. Hence, the effect of the initial dislocation density on the work hardening was not taken into account in this study.

Fig. 3.

Relationship between interparticle spacing (λ₁) and mean spacing of dislocations (λ₂) of samples in this study.

Figure 4 shows the true stress-true strain curves of the samples. The values of dσ/dε of the samples A and D were the lowest at the true strain of 0.05, whereas that of sample C was the highest; the magnitudes of dσ/dε were not dominated by the diameter of the carbides above the true strain of 0.05. Kamikawa et al.^15,24) reported that nanometer-sized precipitates help to enhance dynamic recovery. However the dσ/dε did not always decrease with decreasing the diameter of the carbides in Fig. 4. The ferrite grain size of this study was much smaller than that of the reports. This may lead the difference of the magnitude of dσ/dε.

Fig. 4.

True stress (σ) – true strain (ε) curve and work hardening ratio (dσ/dε) of samples A (d_p=3 nm, f=0.12 vol.%), B (d_p=3 nm, f=0.23 vol.%), C (d_p=8 nm, f=0.29 vol.%) and D (d_p=15 nm, f=0.29 vol.%). (Online version in color.)

3.3. Work Hardening Behavior in the Early Stage of Deformation

Ashby expressed the amount of work hardening of metals containing hard particles which cannot be deformed or fractured with deforming the matrix as Eq. (8).²⁾   

$$\tau -{\tau }_0=KG\sqrt{\frac{bf{\gamma }_{\mathrm{p}}}{d_{\mathrm{p}}}}$$(8)

Here, τ is the shear stress, τ₀ is the yield shear stress, γ_p is the plastic shear strain, K is a material constant. Figure 5 shows the incremental stresses at the true strains of 0.01, 0.025 and 0.05 and the parameter ( $\sqrt{bf\epsilon /d_p}$ ) calculated from the diameter and the volume fraction of the carbides. A result of Fe–Ta alloy containing more than 125 nm of precipitates²⁵⁾ is also shown in Fig. 5. The true stress (σ) and the true strain (ε) were calculated by the relationships of τ=σ/2 and γ=2ε, respectively. All lines were drawn through the original point. The slope of the sample A was as large as that of the sample B, whereas the slope increased as the diameter of the carbides increased.

Fig. 5.

Comparison of stress increment between this study results(■: sample A, ○: sample B, ▲: sample C and ▽: sample D) and Fe–Ta alloys²⁸⁾ containing coarse precipitates over 125 nm in diameter at 0.01, 0.025 and 0.05 of true plastic strains.

The slopes in Fig. 5 represent the material constant (K) in Eq. (8). By contrast, the slopes of each sample depicted the different magnitude. Figure 5 infers that the Ashby’s model cannot explain the work hardening of steels containing the ultra-fine carbides.

The results of a rearranged relationship between the square root of the plastic shear strain ( ${\gamma }_{\mathrm{p}}^{1/2}$ ) and the incremental shear stress (τ−τ₀) are shown in Fig. 6 to analyze a role of precipitation parameter ( $\sqrt{bf\gamma /d_{\mathrm{p}}}$ ). Strain regions in which the incremental stress (τ−τ₀) linearly increased with ${\gamma }_{\mathrm{p}}^{1/2}$ were observed in all the samples. A specific work hardening, however, was appeared in the samples A, B and C containing 8 nm or less of carbides; it can be seen that the slopes were changed at a plastic strain on the incremental shear stress – the square root of the plastic shear strain curves. The plastic shear strain at the change of the slope was defined to be γ_p′ in the following. The slopes before and after changing were distinguished as “first slope” and “second slope”, respectively, and tangents of them were defined as K₁′ and K₂′. The γ_p′ became smaller with decreasing the volume fraction or increasing the diameter of carbides. The K′ consists of the material constant (K), the rigidity modulus (G), and the diameter and the volume fraction of carbide, as shown in Eq. (9).   

$$K^{\prime }=KG\sqrt{\frac{bf}{d_{\mathrm{p}}}}$$(9)

Fig. 6.

Relationship between incremental shear stress from yield point as a function of square root of plastic shear strain and first and second slope of samples within 0.03 of plastic shear strain; (a) sample A, (b) sample B, (c) sample C and (d) sample D.

So the K can be calculated by Eq. (10).   

$$K=\frac{K^{\prime }}{G}\sqrt{\frac{d_{\mathrm{p}}}{bf}}$$(10)

Table 4 shows the calculation results for the K₁′, K₂′, K₁, K₂ and γ_p′. The result for the sample D was only the K₂′ and K₂ since the change of slope did not appear. The K₁ of the samples A, B and C were smaller than the K₂. The K₁ and K₂ of the sample A were almost the same as those of the sample B. The K₁ of the samples A and B were much smaller than that of the sample C and a reported value,²⁶⁾ which was from 0.35 to 0.52 in bcc metals. The K₂ of samples C and D were consistent with the reported value.

Table 4. Tangents of first slopes(K₁′ and K₁) and second slopes (K₂′ and K₂) in samples A, B, C and D.

sample	Diameter of carbide	Volume fraction of carbide	K1′	K1	γp′	K2′	K2
	nm	vol.%	MPa	–	–	MPa	–
A	3	0.12	153	0.19	0.009	180	0.23
B	3	0.23	194	0.17	0.010	261	0.24
C	8	0.29	264	0.35	0.004	333	0.43
D	15	0.29	–	–	–	302	0.54

The Ashby’s model can explain the work hardening of the sample D. Still, the K₁ of the samples A and B were much smaller than the predicted value by the Ashby’s model. It cannot also explain the change of the slopes observed in the samples A, B and C. Therefore, the Ashby’s model cannot apply to the steels with the nanometer-sized carbides while it can explain the work hardening of the steel with 15 nm or more of carbides.

4. Discussion

4.1. Work Hardening Mechanism of Steels Strengthened by Nanometer-sized Carbides

A new work hardening model of the steels strengthened by the ultra-fine carbides within a few plastic strains was considered. At first, the rapid increase of the flow stresses just after yielding in Fig. 6 were focused in order to discuss the specific work hardening that the slope changed at a plastic strain. The smallest square root of the plastic shear strain when τ−τ₀ increased linearly with ${\gamma }_{\mathrm{p}}^{1/2}$ was defined to be ${\gamma }_{\mathrm{s}}^{1/2}$ . In the sample D, the ${\gamma }_{\mathrm{s}}^{1/2}$ was zero since the stress increased linearly from the yield point. The ${\gamma }_{\mathrm{s}}^{1/2}$ of the samples A, B and C were 2.8×10⁻², 4.3×10⁻² and 2.3×10⁻², respectively. These ${\gamma }_{\mathrm{s}}^{1/2}$ values and the amount of work hardening from yield point to ${\gamma }_{\mathrm{s}}^{1/2}$ increased with increasing the volume fraction or decreasing the diameter of the carbides. The result was seemed to be related to the λ₁. The relationship between the λ₁ and the ${\gamma }_{\mathrm{s}}^{1/2}$ is presented in Fig. 7. The ${\gamma }_{\mathrm{s}}^{1/2}$ decreased linearly with increasing the λ₁. The rapid increase in the stress just after yielding was a specific phenomenon which can appear in the steel with the small λ₁ less than about 250 nm.

Fig. 7.

The square root of plastic shear strain starting the linear relationship between τ−τ₀ as a function of mean particle spacing; samples A: d_p=3 nm, f=0.12 vol.%, B: d_p=3 nm, f=0.23 vol.%, C: d_p=8 nm, f=0.29 vol.% and D: d_p=15 nm, f=0.29 vol.%.

The apparent λ₁ is expected to decrease by forming the Orowan loops on an activated slip plane. Therefore subsequent dislocations is anticipated to be more difficult to transit a pair of particles where the Orowan loop is already formed. Fisher et al.²⁷⁾ suggested that the Orowan loops exert back stress on the activated slip plane, and the back stress causes the work hardening. On the other hand, Ashby indicated that a precipitate should be fractured by the shear stress generated by the Orowan loops within small strain.²⁾ The Fisher’s model cannot be applied by itself in this study because the sharp increase of the stress just after yield point did not keep and this implied the work hardening mechanism changed. The change in the incremental stress in Fig. 6 cannot be also explained if the carbides are fractured by the loops. Hence, the work hardening from yield point to ${\gamma }_{\mathrm{s}}^{1/2}$ cannot be explainable on only an activated slip plane. The sharp increase in the work hardening right after yield point should be subject to the Fisher’s model. However, a pair of particles where a few dislocations have already bypassed will prevent subsequent dislocations from going through due to the back stress by the Orowan loops. Ultimately, it will be harder for them to bypass in the pair of particle having Orowan loops. As a result, the subsequent dislocation will glide on another slip plane when the λ₁ is small, and the increase of ${\gamma }_{\mathrm{s}}^{1/2}$ should be attributed to the number of the activated slip plane.

Next, the Ashby’s model was reviewed in order to make the reason why it cannot apply to the steel with the ultra-fine carbides clear. The GN-dislocation density generated at interfaces between hard precipitates and matrix was taken into account in Eq. (8). In addition, the amount of the misfit strain between the precipitate and matrix depends on the size of precipitate. Figure 8 illustrates the decrease in the number of the prismatic loops with the carbides diameter schematically. The misfit strain must be smaller with decreasing the size of precipitate, and be less than Burgers vector within a few plastic strains in the case of the ultra-fine carbides. Hence, the specific phenomenon that the plastic deformation progresses without generating GN-dislocation will be occurred under the first slope in Fig. 6.

Fig. 8.

Schematic illustration of deformed matrix around hard particle and expected generation behavior of the misfit dislocations.

The incremental stress from yield point to ${\gamma }_{\mathrm{s}}^{1/2}$ was mainly attributed to the back stress by the Orowan loops on a slip plane and the number of the activated slip plane. In order to consider the effect of the carbide size on the work hardening, the influence of the Orowan loop and the change of the number of the activated slip planes on work hardening from yield point to ${\gamma }_{\mathrm{s}}^{1/2}$ should be avoided since it depended on the λ₁. Therefore Fig. 9 shows the rearranged result of Fig. 6 that the incremental stresses at ${\gamma }_{\mathrm{s}}^{1/2}$ are all shifted to the origin point so as to eliminate the rapid increase of flow stress from yield point to ${\gamma }_{\mathrm{s}}^{1/2}$ . The plastic strain range showing the first slopes of the sample A was as large as that of the sample B; that of the sample C was smaller than those of the samples A and B. This inferred that the γ_p′ depends on the carbides size, and the generation of GN-dislocation seemed to dominate the γ_p′.

Fig. 9.

Incremental shear stress as a function of the square root of shear strain as a beginning point of ${\gamma }_{\mathrm{s}}^{1/2}$ to eliminate the effect of work hardening just after yield point; samples A: d_p=3 nm, f=0.12 vol.%, B: d_p=3 nm, f=0.23 vol.%, C: d_p=8 nm, f=0.29 vol.% and D: d_p=15 nm, f=0.29 vol.%. Solid and broken lines represent first slope and second slope, respectively.

4.2. Generation of GN-dislocation

The γ_p′ became smaller as generating the GN-dislocation easily since the misfit strain between the carbide and matrix increased with increasing the carbide size. In the Ashby’s model, the γ_p′ was considered to be explained by Eq. (12)²⁾ expressing the generation of GN-dislocations.   

$$n=\frac{{\gamma }_{\mathrm{p}}{d}_{\mathrm{p}}}{4b}$$(12)

Here, n is the number of GN-dislocations. Assuming one GN-dislocation generates at the γ_p′, those of steels containing 3 nm, 8 nm and 15 nm of carbides were calculated to be 0.33, 0.12 and 0.07, respectively. The observed γ_p′ decreased with increasing the diameter of carbides, but the calculated values were much bigger than the experimental results. Equation (12) is based on the strain applied to the tensile specimen. However, a strain gradient around carbide seems to be bigger than it. When the effect of the strain gradient expresses by using a coefficient, C, Eq. (12) can be converted into Eq. (13).   

$$n=\frac{C{\gamma }_{\mathrm{p}}{d}_{\mathrm{p}}}{4b}$$(13)

The subtracted γ_p′ values from γ_s of samples A, B and C in Table 4 are 8.0×10⁻³, 8.0×10⁻³ and 3.4×10⁻³, respectively. Substituting them into Eq. (13), the calculated C values of the samples A, B and C were 41, 41 and 36, respectively. These values resulted in constant, so it is necessary to take the strain gradient into account in considering the generation of GN-dislocation.

5. Conclusion

The work hardening behavior of the ferritic steels containing the carbides in the diameter of 3 nm, 8 nm and 15 nm were investigated. The following results were obtained.

(1) The Ashby’s model could explain the amount of work hardening in the steels containing 15 nm of the carbides. However, it cannot apply to the steels containing 3 nm and 8 nm of carbides.

(2) The rapid increase in the stress right after yield point was observed in the steel with the small interparticle spacing and the amount of work hardening increased with decreasing interpaticle spacing due to the increase of the back stress by Orowan loops and the change of the number of activated slip planes.

(3) The specific work hardening behavior that the incremental stress suddenly changed with the square root of the plastic strain in the steels containing 3 nm and 8 nm of carbides was observed. This was attributed to the change of generation behavior of GN-dislocation since the misfit strain between carbide and matrix is close to Burgers vector.

(4) The strain gradient around carbides should be considered to express the generation of GN-dislocation since the stress concentration was obtained to be about 40 times compared to the nominal stress.

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