4. Bypass mechanism and hardness

Precipitation-hardening behavior in medium carbon bainitic steels with added elements of Nb, Ti and V were precisely investigated. It was found that the amount of precipitation hardening became larger by the multiple element additions of Nb, Ti and V than the single addition of each element. Hardness changes due to change in dislocation bypass mechanism of particles. Even while hardness changes also depending on volume of the dispersed particles, the total amounts of particles contained in the present steels are assumed not largely different. The mechanisms of the change in the amount of precipitation hardening due to multiple element addition are precisely discussed below.

#### 4.1. Dominant dislocation-bypass mechanism

There are two well-known mechanisms of dislocation to pass through particle at ambient temperature, that is, the Orowan [9] and the cutting mechanisms [10], which control threshold stress for dislocation motion in the particle-bearing materials. Their activation and role during plastic deformation change sensitively depending on precipitate size. Takaki has estimated the critical particle diameter for the cutting mechanism to be 7–10 nm at maximum for the MC-type carbides such as NbC, TiC and VC, whereas the Orowan mechanism can activate at ranges of 7–10 nm and over [11]. The precipitation size observed in the present study is approximately 5–8 nm in average (Figure 3).

When a gliding dislocation is pinned by a dispersed particle, the increase in the tensile stress (Δσ) can be expressed by the following equation [12, 13]:

$$
\Delta \sigma = \beta \mathbf{G} \mathbf{b} / \lambda \cdot \sin \theta,\tag{3}
$$

the former one [15]. From the above arguments, the activated dislocation-bypass mechanism is concluded as a cutting mechanism irrespective of the present samples. The dominant dislocation bypass mechanism is closely related with the deformation stress [15]. However, it is revealed from the above argument that the bypass mechanism itself should not cause the change in hardness in the present samples. Nevertheless, according to the criteria proposed by Takaki [11], activation of the Orowan mechanism might be partially activated for such

Mechanisms of Significant Precipitation Hardening in a Medium Carbon Bainitic Steel by Complex Nanocarbides…

Gerold and Haberkorn have theoretically shown, considering the lattice misfit parameter at precipitate/matrix interface, that the increase in shear stress (Δτ0) due to dislocation detach-

> <sup>1</sup>:<sup>5</sup> fr b <sup>0</sup>:<sup>5</sup>

where k is a constant, f is the volume fraction of precipitates, r is the radius of the precipitate [10], ε is the lattice misfit parameter between the precipitate and matrix and ε is approximately

<sup>ε</sup> <sup>¼</sup> <sup>2</sup>

<sup>δ</sup> <sup>¼</sup> dp � dm dm

where dp and dm are the lattice constants of precipitate and matrix, respectively [10]. Eq. (4) indicates that the shear stress (Δτ0) increases with increasing misfit parameter, size and volume

The complex carbides of (Nb, V)C and (Ti, V)C are known to be continuous solid solution over the entire composition range [18]. The lattice constant dp of the complex carbide such as (Nb, V)C, (Ti, V)C and (Nb, Ti, V)C is proportional to the composition and estimated by the

where i denotes the alloying element (i.e., Nb, Ti and V) to compose carbide, and ni and di are the expediently estimated fraction of each carbide and the lattice constant, respectively. In the present study, dp and dm correspond to the lattice constants of {001}MC of carbide and of {011}<sup>α</sup>

Change in the lattice misfit parameter between the matrix and carbide could be evaluated using Eqs. (5)–(7). The calculated misfit parameter (ε) of (Nb, V)C and (Ti, V)C is displayed

, (4)

http://dx.doi.org/10.5772/intechopen.80273

31

<sup>3</sup> <sup>δ</sup>, (5)

, (6)

dp ¼ Σnidi, (7)

inhomogeneously and largely coarsen precipitates observed in Figure 3(a).

Δτ<sup>0</sup> ¼ kGj j ε

4.2. Origin of higher hardness due to complex carbides

ment from precipitate is expressed by

2/3 of the lattice misfit δ [16, 17]:

and δ is expressed by

fraction of precipitates.

following equation [19]:

of ferrite [8].

where G is the shear modulus of matrix, b is the Burgers vector, λ is the mean particle spacing, θ is the bowing angle between the dislocation and the straightline that connects two particles, andβis a constant composed of Taylor's factor and the coefficient of shear stress, for which the value is approximately 3.2 [14]. The values of λ in steels D and E were measured by TEM observation to be 20 nm and 10 nm, respectively. The values of G and b are 80 GPa and 0.25 nm [14], respectively. Assuming that the tensile stress is empirically evaluated by one third of the hardness [13], the increase in the tensile stress (Δσ) of steels D and E can be estimated as 120 and 270 MPa, respectively, from their increase in hardness (ΔHv) 40 and 90 by aging. Finally, theθ values in steels D and E can be calculated using Eq. (3) to be approximately 2.1 and 2.4�, respectively. These values are much smaller than those of particle/dislocation interaction by the Orowan mechanism, which shows a characteristic θ value of approximately 90� [12]. The Orowan mechanism should be, therefore, not activated as the dominant dislocation bypass mechanism in the present samples.

When the carbides were examined precisely by TEM, a typical contrast owing to coherent strain as well as Moiré fringe was observed (Figure 3). This indicates the presence of misfit strain due to interface coherency between carbides and matrix. Miura et al. have experimentally shown that dislocation passes through incoherent particles by the Orowan mechanism and coherent particles by cutting mechanism, and the latter deformation stress is higher than the former one [15]. From the above arguments, the activated dislocation-bypass mechanism is concluded as a cutting mechanism irrespective of the present samples. The dominant dislocation bypass mechanism is closely related with the deformation stress [15]. However, it is revealed from the above argument that the bypass mechanism itself should not cause the change in hardness in the present samples. Nevertheless, according to the criteria proposed by Takaki [11], activation of the Orowan mechanism might be partially activated for such inhomogeneously and largely coarsen precipitates observed in Figure 3(a).

#### 4.2. Origin of higher hardness due to complex carbides

Gerold and Haberkorn have theoretically shown, considering the lattice misfit parameter at precipitate/matrix interface, that the increase in shear stress (Δτ0) due to dislocation detachment from precipitate is expressed by

$$
\Delta \tau\_0 = \mathbf{kG} |\varepsilon| ^{1.5} \left(\frac{\text{fr}}{\text{b}}\right)^{0.5} \text{ }\tag{4}
$$

where k is a constant, f is the volume fraction of precipitates, r is the radius of the precipitate [10], ε is the lattice misfit parameter between the precipitate and matrix and ε is approximately 2/3 of the lattice misfit δ [16, 17]:

$$
\varepsilon = \frac{2}{3} \text{ \(\delta\)}\tag{5}
$$

and δ is expressed by

4. Bypass mechanism and hardness

30 New Uses of Micro and Nanomaterials

addition are precisely discussed below.

average (Figure 3).

4.1. Dominant dislocation-bypass mechanism

(Δσ) can be expressed by the following equation [12, 13]:

Precipitation-hardening behavior in medium carbon bainitic steels with added elements of Nb, Ti and V were precisely investigated. It was found that the amount of precipitation hardening became larger by the multiple element additions of Nb, Ti and V than the single addition of each element. Hardness changes due to change in dislocation bypass mechanism of particles. Even while hardness changes also depending on volume of the dispersed particles, the total amounts of particles contained in the present steels are assumed not largely different. The mechanisms of the change in the amount of precipitation hardening due to multiple element

There are two well-known mechanisms of dislocation to pass through particle at ambient temperature, that is, the Orowan [9] and the cutting mechanisms [10], which control threshold stress for dislocation motion in the particle-bearing materials. Their activation and role during plastic deformation change sensitively depending on precipitate size. Takaki has estimated the critical particle diameter for the cutting mechanism to be 7–10 nm at maximum for the MC-type carbides such as NbC, TiC and VC, whereas the Orowan mechanism can activate at ranges of 7–10 nm and over [11]. The precipitation size observed in the present study is approximately 5–8 nm in

When a gliding dislocation is pinned by a dispersed particle, the increase in the tensile stress

where G is the shear modulus of matrix, b is the Burgers vector, λ is the mean particle spacing, θ is the bowing angle between the dislocation and the straightline that connects two particles, andβis a constant composed of Taylor's factor and the coefficient of shear stress, for which the value is approximately 3.2 [14]. The values of λ in steels D and E were measured by TEM observation to be 20 nm and 10 nm, respectively. The values of G and b are 80 GPa and 0.25 nm [14], respectively. Assuming that the tensile stress is empirically evaluated by one third of the hardness [13], the increase in the tensile stress (Δσ) of steels D and E can be estimated as 120 and 270 MPa, respectively, from their increase in hardness (ΔHv) 40 and 90 by aging. Finally, theθ values in steels D and E can be calculated using Eq. (3) to be approximately 2.1 and 2.4�, respectively. These values are much smaller than those of particle/dislocation interaction by the Orowan mechanism, which shows a characteristic θ value of approximately 90� [12]. The Orowan mechanism should be, therefore, not activated as the dominant dislocation bypass mechanism in the present samples.

When the carbides were examined precisely by TEM, a typical contrast owing to coherent strain as well as Moiré fringe was observed (Figure 3). This indicates the presence of misfit strain due to interface coherency between carbides and matrix. Miura et al. have experimentally shown that dislocation passes through incoherent particles by the Orowan mechanism and coherent particles by cutting mechanism, and the latter deformation stress is higher than

Δσ ¼ βGb=λ � sinθ, (3)

$$\delta = \frac{\mathbf{d}\_{\rm p} - \mathbf{d}\_{\rm m}}{\mathbf{d}\_{\rm m}},\tag{6}$$

where dp and dm are the lattice constants of precipitate and matrix, respectively [10]. Eq. (4) indicates that the shear stress (Δτ0) increases with increasing misfit parameter, size and volume fraction of precipitates.

The complex carbides of (Nb, V)C and (Ti, V)C are known to be continuous solid solution over the entire composition range [18]. The lattice constant dp of the complex carbide such as (Nb, V)C, (Ti, V)C and (Nb, Ti, V)C is proportional to the composition and estimated by the following equation [19]:

$$\mathbf{d}\_{\mathbf{P}} = \Sigma \mathbf{n}\_{i} \mathbf{d}\_{i} \tag{7}$$

where i denotes the alloying element (i.e., Nb, Ti and V) to compose carbide, and ni and di are the expediently estimated fraction of each carbide and the lattice constant, respectively. In the present study, dp and dm correspond to the lattice constants of {001}MC of carbide and of {011}<sup>α</sup> of ferrite [8].

Change in the lattice misfit parameter between the matrix and carbide could be evaluated using Eqs. (5)–(7). The calculated misfit parameter (ε) of (Nb, V)C and (Ti, V)C is displayed

Figure 4. Calculated change in the lattice misfit parameter ε between matrix and complex precipitates of (Nb, V)C and (Ti, V)C depending on chemical composition.

in Figure 4. It is clear that the misfit parameter (ε) of the complex carbides (Nb, V)C and (Ti, V) C is larger than that of VC though smaller than those of NbC and TiC. Misfit parameter of (Nb, Ti, V)C complex carbide must be also linearly changed sensitively depending on the chemical composition.

In the same way, the lattice constant of (Nb, Ti, V)C complex carbide as well as the other carbides was determined. For this purpose, for an example of steel E, the chemical composition of the complex carbide at 873 K (i.e., aging temperature) was calculated to be (Nb0.03, Ti0.12, V0.85)C in atomic fraction by means of the "Thermo-Calc" software using the chemical composition in Table 1. Figure 5 summarizes the calculated lattice misfit parameters (ε) of the carbides precipitated in the tested samples. It is evident that the misfit parameters (ε) of (Nb, Ti, V)C is greater than that of VC and smaller than those of NbC and TiC.

Miyamoto et al. have suggested that precipitation hardening (ΔHv) due to carbides can be converted into increase in shear stress (Δτ0) by using the Taylor's factor of 2.08 in nontextured bcc metals by the following equation [20]:

$$
\Delta \pi\_0 = \mathbf{3} \cdot \Delta \mathbf{H} \mathbf{v} / 2.08,\tag{8}
$$

to TiN precipitation could be reasonably understood by the quantitative estimation based on

Figure 5. Comparison of the calculated lattice misfit parameter ε between matrix and precipitates of NbC, TiC, VC and

Mechanisms of Significant Precipitation Hardening in a Medium Carbon Bainitic Steel by Complex Nanocarbides…

The amount of precipitation hardening (ΔHv) of the tested samples can be estimated from

The constant k in Eqs. (4) and (9) is reported to be approximately 1–3 [22, 23]. The precipitation hardening (ΔHv) by addition of 0.5 mass% Mo was quite small (Figure 2). Furthermore, Mo carbides were not detected by TEM observation in all the samples. From these reasons, the effect of Mo carbides on precipitation hardening can be excluded from the estimations. Therefore, carbides that mainly contribute to precipitation hardening in steels B, C, D and E are

Figure 6 shows the relationship between the measured ΔHv values and the parameter G|ε|1.5 (fr/b)0.5 of the present steels. Calculated results using Eq. (9) and constant k of 1 and 3 are also shown by dotted lines. For the calculation of G|ε|1.5 (f r/b)0.5, the diameter of carbides (2r) was assumed to be 5 nm for simplicity, and the volume fraction of the precipitates (f) was estimated by the "Thermo-Calc" calculation using the chemical compositions in Table 1. It is evident in Figure 6 that the results derived from experimental data of steels D and E appear between the lines with constant k of 1 and 3. The above results indicate that the values of ΔHv calculated from Eq. (9) almost coincide with the present experimental data. That is, quantitative estimation of the amount of precipitation hardening (ΔHv) in the medium carbon bainitic steels with Nb, Ti and V additions can be reasonably understood by the carbide/matrix lattice misfit parameter and cutting

<sup>1</sup>:<sup>5</sup> fr b <sup>0</sup>:<sup>5</sup>

, (9)

http://dx.doi.org/10.5772/intechopen.80273

33

ΔHv ¼ 0:69 � kGj j ε

the cutting mechanism.

NbC, TiC, VC and (Nb, Ti, V)C, respectively.

Eqs. (4) and (7):

(Nb, Ti, V)C.

Equation (8) indicates that the value of Δτ<sup>0</sup> is proportional to precipitation hardening (ΔHv). Actually, Ronay has investigated precipitation-hardening behavior in low carbon steels and shown a linear relationship between shear stress (Δτ0) and hardness (ΔHv) due to TiN precipitation [21]. She also reported that the increase in the hardness and, therefore, shear stress due

Mechanisms of Significant Precipitation Hardening in a Medium Carbon Bainitic Steel by Complex Nanocarbides… http://dx.doi.org/10.5772/intechopen.80273 33

Figure 5. Comparison of the calculated lattice misfit parameter ε between matrix and precipitates of NbC, TiC, VC and (Nb, Ti, V)C.

to TiN precipitation could be reasonably understood by the quantitative estimation based on the cutting mechanism.

The amount of precipitation hardening (ΔHv) of the tested samples can be estimated from Eqs. (4) and (7):

in Figure 4. It is clear that the misfit parameter (ε) of the complex carbides (Nb, V)C and (Ti, V) C is larger than that of VC though smaller than those of NbC and TiC. Misfit parameter of (Nb, Ti, V)C complex carbide must be also linearly changed sensitively depending on the chemical

Figure 4. Calculated change in the lattice misfit parameter ε between matrix and complex precipitates of (Nb, V)C and

In the same way, the lattice constant of (Nb, Ti, V)C complex carbide as well as the other carbides was determined. For this purpose, for an example of steel E, the chemical composition of the complex carbide at 873 K (i.e., aging temperature) was calculated to be (Nb0.03, Ti0.12, V0.85)C in atomic fraction by means of the "Thermo-Calc" software using the chemical composition in Table 1. Figure 5 summarizes the calculated lattice misfit parameters (ε) of the carbides precipitated in the tested samples. It is evident that the misfit parameters (ε) of (Nb,

Miyamoto et al. have suggested that precipitation hardening (ΔHv) due to carbides can be converted into increase in shear stress (Δτ0) by using the Taylor's factor of 2.08 in nontextured

Equation (8) indicates that the value of Δτ<sup>0</sup> is proportional to precipitation hardening (ΔHv). Actually, Ronay has investigated precipitation-hardening behavior in low carbon steels and shown a linear relationship between shear stress (Δτ0) and hardness (ΔHv) due to TiN precipitation [21]. She also reported that the increase in the hardness and, therefore, shear stress due

Δτ<sup>0</sup> ¼ 3 � ΔHv=2:08, (8)

Ti, V)C is greater than that of VC and smaller than those of NbC and TiC.

bcc metals by the following equation [20]:

(Ti, V)C depending on chemical composition.

32 New Uses of Micro and Nanomaterials

composition.

$$
\Delta \mathbf{H} \mathbf{v} = 0.69 \cdot \mathbf{k} \mathbf{G} |\varepsilon|^{1.5} \left(\frac{\mathbf{f} \mathbf{r}}{\mathbf{b}}\right)^{0.5} \tag{9}
$$

The constant k in Eqs. (4) and (9) is reported to be approximately 1–3 [22, 23]. The precipitation hardening (ΔHv) by addition of 0.5 mass% Mo was quite small (Figure 2). Furthermore, Mo carbides were not detected by TEM observation in all the samples. From these reasons, the effect of Mo carbides on precipitation hardening can be excluded from the estimations. Therefore, carbides that mainly contribute to precipitation hardening in steels B, C, D and E are NbC, TiC, VC and (Nb, Ti, V)C, respectively.

Figure 6 shows the relationship between the measured ΔHv values and the parameter G|ε|1.5 (fr/b)0.5 of the present steels. Calculated results using Eq. (9) and constant k of 1 and 3 are also shown by dotted lines. For the calculation of G|ε|1.5 (f r/b)0.5, the diameter of carbides (2r) was assumed to be 5 nm for simplicity, and the volume fraction of the precipitates (f) was estimated by the "Thermo-Calc" calculation using the chemical compositions in Table 1. It is evident in Figure 6 that the results derived from experimental data of steels D and E appear between the lines with constant k of 1 and 3. The above results indicate that the values of ΔHv calculated from Eq. (9) almost coincide with the present experimental data. That is, quantitative estimation of the amount of precipitation hardening (ΔHv) in the medium carbon bainitic steels with Nb, Ti and V additions can be reasonably understood by the carbide/matrix lattice misfit parameter and cutting

2. The amount of precipitation hardening in medium carbon bainitic steel with multiple element additions of Nb, Ti and V was larger than those in steels with single additions of

Mechanisms of Significant Precipitation Hardening in a Medium Carbon Bainitic Steel by Complex Nanocarbides…

http://dx.doi.org/10.5772/intechopen.80273

35

3. The dominant dislocation-bypass mechanism at room temperature of VC and (Nb, Ti, V)C carbides with a diameter approximately of 5–8 nm is the cutting mechanism, whereas that

4. The difference in the amount of precipitation hardening could be reasonably understood

5. Cutting mechanism would not be dominant when the precipitate/matrix misfit parameter

One of the authors, H.M., acknowledges the support by Japan Science and Technology Agency (JST) under Industry-Academia Collaborative R&D Program "Heterogeneous Structure Control: Towards Innovative Development of Metallic Structural Materials," and the authors deeply

1 Kimitsu R&D Lab., Nippon Steel and Sumitomo Metal Corporation, Kimitsu, Chiba, Japan

2 Pipe and Tube Research Lab., Steel Research Laboratories, Nippon Steel and Sumitomo

3 Department of Mechanical Engineering, Toyohashi University of Technology, Toyohashi,

[1] Funakawa Y, Shiozaki T, Tomita K, Yamamoto T, Maeda E. Development of high strength hot-rolled sheet steel consisting of ferrite and nanometer-sized carbides. ISIJ International.

[2] Miyata K, Sawaragi Y. Effect of Mo and W on the phase stability of precipitates in low Cr heat resistant steels. ISIJ International. 2001;41:281-289. DOI: https://doi.org/10.2355/

2004;44:1945-1951. DOI: https://doi.org/10.2355/isijinternational.44.1945

\*

these elements.

Acknowledgements

appreciate this support.

Author details

Makoto Okonogi<sup>1</sup>

Aichi, Japan

References

isijinternational.41.281

of NbC and TiC carbides appears different.

by the lattice misfit parameter between carbide and matrix.

and size of carbides become large enough due to loss of coherency.

, Takuya Hara<sup>2</sup> and Hiromi Miura3

\*Address all correspondence to: miura@me.tut.ac.jp

Metal Corporation, Amagasaki, Hyogo, Japan

Figure 6. Relationship between the measured precipitation hardening (ΔHv) and estimated parameter G|ε|1.5 (fr/b)0.5 using experimental data of the present steels, which are exhibited by symbols. For comparison, the relationship derived from Eq. (9) is also shown by dotted lines with k constants of 1 and 3.

mechanism for dislocation to pass through carbides. On the other hand, in steels B and C, the measured ΔHv values were far from the calculated data. This could be because of the misfit dislocations that were produced around the carbides to reduce the misfit strain between carbides and matrix, since the misfit parameter (ε) of NbC and TiC is very large. Hence, precipitation hardening in steels B and C would not be understood simply by the cutting mechanism only presumably because of the loss of coherency between the coarsen carbides and matrix. Reduction of ΔHv, therefore, took place because of a loss of coherency at the carbide/matrix interface [15]. Loss of coherency at the interface is well known to appear by coarsening [24, 25] and large lattice misfit strain [26]. Hence, complicated dislocation-bypass mechanisms could be activated depending on the size of precipitates in steels B and C.

#### 5. Conclusions

The precipitation hardening due to complex carbides in medium carbon bainitic steels was investigated. The yielded results are as follows:

1. Small amount of additional elements can cause drastic change in the structures and chemical compositions of nanocarbides dispersed in steels, which induce conversion in dislocation bypass mechanisms to improve mechanical properties.

