2024 Volume 65 Issue 9 Pages 1170-1177
In ultra low-carbon Fe-18%Ni alloy with a lath martensitic structure, the effects of microstructure and dislocation density on the yielding and deformation behavior were investigated. The microstructure of the matrix became finer with decreasing austenite grain size but no difference was found in the yielding behavior up to 2% deformation. Dislocation density was constant independent of the microstructure and was estimated to be about 1.5 × 1015/m2. The dislocations that had been introduced by martensitic transformation formed a tangled dislocation cell structure within a lath and the amount of dislocation strengthening was determined by a critical stress that was required to make the tangled dislocations bow out. As a result, it was confirmed that the 2% proof stress of the steel used can be evaluated by adding the amount of dislocation strengthening to the friction stress.
This Paper was Originally Published in Japanese in J. Soc. Mater. Sci., Japan 73 (2024) 306–313. In section 4.3, a minor correction is added.
Fig. 11 Changes of the amount of dislocation strengthening Δσ as a function of the bowing angle θ at the given dislocation density.
The most distinctive characteristic of martensitic steels containing carbon is that high strength can be obtained using solid solution strengthening by carbon. The solubility limit of carbon in ferrite (bcc iron) is extremely low, making it difficult to use solid solution strengthening by carbon, but austenite (fcc iron: γ), which exists at high temperatures, can dissolve up to 2.1% (mass%) carbon. Therefore, a diffusionless transformation from fcc to bcc makes it possible to force the solid solution of carbon in fcc into bcc, and the supersaturated solid solution of carbon can significantly increase the strength (frictional force) of the microstructure. Moreover, dislocation strengthening can be used effectively in steels with a carbon content of 0.4% or less because high-density dislocations are introduced to offset the strain caused by changes in the crystal structure (supplementary deformation). Martensite that has undergone supplementary deformation by the introduction of dislocations is called lath martensite. In lath martensite, austenite grains are hierarchically divided by substructures such as packets, blocks, and laths to reduce anisotropy of transformation strain [1, 2]. The yield stress of martensitic steel is generally estimated by adding frictional force including the effect of solid solution strengthening by carbon, crystal grain refinement strengthening by the substructures, dislocation strengthening due to high-density dislocations, and strain aging due to auto-tempering [3, 4] during quenching [5]. However, martensitic steels containing carbon greatly differ in the degree of auto-tempering depending on the cooling speed [4]; therefore, the strengthening mechanism of martensite in the quenched state (as-quenched martensite) is expected to be very complex. When discussing strengthening mechanisms, it is essential to evaluate dislocation density; many theories and methods using X-ray and line profile analysis with synchrotron radiation have been proposed [6–8]. However, the X-ray diffraction method has the problem that it cannot accurately determine dislocation density due to the broadening of the full width at half maximum caused by the bct structure [9]. Ideally, pure iron, which contains no carbon, should be used to investigate the basic properties of lath martensitic steels, but in reality, it is extremely difficult to form a lath martensitic structure with pure iron. Therefore, when investigating the basic properties of lath martensitic steels, Fe-18%Ni alloys are often used, which makes it possible to easily form a lath martensitic structure without needing to add carbon [10–15]. Three types of strengthening are considered to have an effect on the yield stress of the alloy: frictional force including the contribution of solid solution strengthening by Ni, strengthening by crystal grain refinement due to the substructure, and dislocation strengthening. Past reports suggest that the microstructure has almost no effect on the tensile deformation behavior of Fe-18%Ni alloy [13, 14]. However, no past reports have comprehensively shown the effects of microstructure, including dislocation density and dislocation properties. Therefore, this study investigated the effect of austenite grain size on dislocation density and dislocation properties by conducting dislocation analysis using the modified Williamson–Hall (mWH) method [16] after preparing specimens with different austenite grain sizes and microstructures under different austenitization conditions. Furthermore, the mechanism of dislocation strengthening was quantitatively evaluated using the results of dislocation analysis.
The chemical composition of the Fe-18%Ni alloy used in this study was 17.88%Ni, 0.38%Mn, Si < 0.01%, C < 0.001%, P < 0.005%, and S < 0.001%. After vacuum-melting the alloy, we cast it in a mold to reduce solidification segregation and prepared a 20 kg ingot. We then hot-rolled it at 1200°C to a thickness of 12 mm to destroy the cast structure, to be used as the specimen. The specimen was homogenized by annealing at 1200°C for 5 hours. Its surface was ground to remove the oxide film and cold-rolled to 90% (thickness: approximately 1 mm). Elemental analysis of the cross section of this plate confirmed the absence of Ni segregation. Subsequently, austenitization treatment was performed for 30 min in the temperature range of 750 to 1000°C to adjust the austenite grain size, and then it was cooled with water. The Ms point of this alloy was 300 ± 10°C. Saturation magnetization measurement confirmed the absence of austenitization residue in the water-cooled specimen after the austenitization. Tensile tests were conducted using specimens with a thickness of 1 mm, a parallel section width of 3 mm, and a parallel length of 20 mm at an initial strain rate of 1.7 × 10−4/s. The microstructure was observed by crystal orientation mapping with a scanning electron microscope.
Specimens with dimensions of 15 × 15 mm, which were cut out from heat-treated plates, were used for X-ray diffraction. The surfaces of the cut and rolled specimens were made flat using sandpaper and then electropolished by 50 µm in thickness to eliminate the effects [17] of the surface polishing. Cu-KαI (wavelength ω = 0.15405 nm) was used for the X-ray diffraction, with a scan speed of 0.8°/min, divergence and scattering slit degrees of 1°, a receiving slit of 0.3 mm, and no collimator. The crystal orientation distribution was not completely isotropic; however, the diffraction intensities of the peaks other than {220} were sufficient to measure the full width at half maximum. Commercially available LaB6 was used to correct the instrument functions, and the pseudo-Voigt function was applied [18]. Young’s modulus and shear modulus of the Fe-18%Ni alloy obtained by the dynamic vibration method were 173 GPa and 64.0 GPa, respectively. Poisson’s ratio obtained from these values is 0.352. The elastic stiffness ratio c12/c44 is 1.808 and the elastic anisotropy parameter Ai is 2.326. The contrast factor Ch00 for the {h00} plane corresponding to these values based on the data of Ungár et al. [19] yielded a value of 0.290 for both screw dislocation (Ch00S) and edge dislocation (Ch00E). The coefficient q, which represents the extent of the dependence on crystal orientation for the {hkl} plane contrast factor Chkl, was 2.638 for screw dislocation (qS) and 1.502 for edge dislocation (qE).
Figure 1 shows the crystal orientation maps of the specimen with solution treatment temperatures varying between 750°C and 1000°C. The maps show that masses in the same color form a block, which is a group of laths. They also show that as the austenite grain size decreases, the substructure in the grain interior becomes finer. The prior austenite grain size was measured by a comparison method using a saturated picric acid solution for preferentially corroding the boundaries of austenite grains. The grain sizes in specimens at austenitization temperatures of 750°C, 800°C, 900°C, and 1000°C were 5 µm, 10 µm, 55 µm, and 70 µm, respectively. In order to clarify the relationship with the microstructure, austenite grain size is hereinafter used as an index representing the microstructure. Figure 2 shows the tensile deformation behavior of specimens with different austenite grain sizes. As shown in the figure, almost no effect of microstructure on the yield process is observed for elongation below 2%. Local necking deformation occurs in areas where elongation is greater than 2%. The finer the substructure in the austenite grains, the higher the necking deformation stress and the total elongation tend to be. In other words, it is assumed that the microstructure of the matrix has almost no effect on the yielding behavior of lath martensite, but affects necking deformation and ductile fracture behavior. The 2% proof stress strictly corresponds to the tensile strength; however, it is defined as the macroscopic yield stress in this study for convenience because the amount of deformation is small.
Effect of austenitization conditions on the crystal orientation map of Fe-18%Ni alloy.
Effect of austenite grain size on the tensile deformation behavior of Fe-18%Ni alloy.
The elastic limit, 0.6% proof stress, and 2% proof stress were determined to investigate the effect of microstructure on yielding behavior. Figure 3 shows their relationship with austenite grain sizes. Dislocations introduced by martensitic transformation are very unstable and many dislocations are expected to start moving when the applied stress exceeds the frictional force [10, 15]; therefore, the elastic limit can be regarded as the frictional force for lath martensitic steels. The frictional force is an intrinsic value of materials not depending on microstructure, although it varies depending on the temperature and strain rate. However, it is difficult to accurately determine the elastic limit from the tensile deformation curve of steels having a lath martensitic structure, and various values, such as 125 MPa [14], 255 MPa [15], and 300 MPa [10], have been reported by researchers. In this study, the elastic limit was almost the same as that reported by Nakajima [10]. For martensitic steels, the stress increases with a slight plastic strain when it exceeds the elastic limit; therefore, the 0.6% proof stress is often taken as the yield stress as a matter of convention. This study reconfirmed that the 0.6% proof stress is not affected at all by the austenite grain size or the microstructure.
Effect of austenite grain size d on the tensile properties of Fe-18%Ni alloy.
Annealed ferritic steels have almost no dislocation sources in the grain interior; therefore, macroscopic yielding is caused by the release of dislocations from grain boundaries (grain boundary dislocation yielding) [20]. When grain boundary dislocation yielding occurs, the Hall–Petch relationship is established, causing grain size dependence on yield stress. However, it is known that processing annealed material introduces dislocations to the grain interior, and the movement of the dislocations in the grain interior causes yielding (grain-interior dislocation yielding) [20]. That is, the mechanism of yielding is changed by machining. When dislocation yielding occurs in the grain interior, the yield stress increases in proportion to the square root of the dislocation density according to the Bailey–Hirsch relationship [20]. It is also known that the smaller the grain size, the higher the dislocation density and the higher the yield stress, even if the materials are machined in the same way [21]. In the past, on the assumption that the amount of dislocation strengthening is constant without evaluating dislocation density, the increase in dislocation strengthening, which is caused by grain size, was misinterpreted as the effect of crystal grain refinement strengthening. This study employed the mWH method to investigate the effects of austenite grain size and microstructure on dislocation density.
3.2 Dislocation analysis by X-ray diffraction methodTable 1 shows the values of diffraction angle 2θhkl and full width at half maximum Whkl, which were obtained after the correction of device functions, for the specimens used in this study. The mWH method uses the following parameters expressed as functions of ω, Whkl, and θhkl:
| \begin{equation} K_{\textit{hkl}} = 2\sin \theta_{\textit{hkl}}/\omega \end{equation} | (1) |
and
| \begin{equation} \varDelta K_{\textit{hkl}} = W_{\textit{hkl}}\cos \theta_{\textit{hkl}}/\omega . \end{equation} | (2) |
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Figure 4 summarizes the relationship between Khkl and ΔKhkl obtained for each specimen. For ⟨200⟩ and ⟨310⟩ with small Young’s modulus, the value of ΔKhkl is relatively large; on the contrary, for ⟨222⟩ with large Young’s modulus, the value of ΔKhkl is relatively small; however, the four specimens showed almost the same data. On the other hand, the orientation parameter Γhkl of the crystal plane {hkl} is given by the following equation:
| \begin{align} \varGamma_{\textit{hkl}} &= (h^{2}k^{2} + k^{2}l^{2} + l^{2}\underline{\text{h}}^{2})/(h^{2} + k^{2} + l^{2})^{2}\\ &\quad (0 \leq \varGamma_{\textit{hkl}} \leq 1/3) \end{align} | (3) |
Relation between the parameters Khkl and ΔKhkl in Fe-18%Ni alloy quenched from the selected temperature.
Ungár et al. proposed a method to correct for elastic anisotropy corresponding to crystal planes using the contrast factor Chkl defined by the following equation [16]:
| \begin{equation} C_{\textit{hkl}} = C_{h00}(1 - q\varGamma_{\textit{hkl}}) \end{equation} | (4) |
For Fe-18%Ni alloys, the values of Ch00 and q depend on the ratio S (0 ≤ S ≤ 1) of the screw component of the dislocation, as given by the following equations:
| \begin{equation} C_{h00} = C_{h00}{}^{E} + S(C_{h00}{}^{S} - C_{h00}{}^{E}) = 0.29 \end{equation} | (5) |
and
| \begin{equation} q = q^{E} + S(q^{S} - q^{E}) = 1.502 + 1.136 \times S \end{equation} | (6) |
Correction for elastic anisotropy uses the following mWH equation [16]:
| \begin{equation} \varDelta K_{\textit{hkl}} = \alpha + \varphi K_{\textit{hkl}}\sqrt{C_{\textit{hkl}}} + OK_{\textit{hkl}}{}^{2}C_{\textit{hkl}} \end{equation} | (7) |
where α is a parameter that depends on the crystallite size, and φ and O are constants that depend on the dislocation density and the distribution state of dislocations. φ is given as a function of the dislocation density ρ, the Burgers vector of dislocation b, and a parameter A that depends on the dislocation distribution [16] as given by the following equation:
| \begin{equation} \varphi = (\pi /2)^{1/2}Ab\rho^{1/2} \end{equation} | (8) |
The more uniform the dislocation distribution, the larger parameter A. Incidentally, it has been reported that A = 0.5 [22] for 10 to 80% rolled iron, A = 0.6 for cold-rolled martensitic steel, and A = 0.8 for quenched martensite [23]. This study used 0.8 as the value of parameter A because the subject of this analysis is as-quenched martensitic steel.
Specific analysis methods are shown below. We have confirmed that removing the third term on the right side in eq. (7) has almost no effect on the analytical results because it is significantly smaller than the other terms [24]; therefore, eq. (7) can be simplified as follows:
| \begin{equation} \varDelta K_{\textit{hkl}} = \alpha + \varphi K_{\textit{hkl}}\{C_{h00}(1 - q\varGamma_{\textit{hkl}})\}^{1/2} \end{equation} | (9) |
After transposing the parameter α to the left side, both sides can be rewritten by dividing by Khkl and squaring as follows:
| \begin{equation} (\varDelta K_{\textit{hkl}} - \alpha)^{2}/K_{\textit{hkl}}{}^{2} = \varphi^{2}C_{h00}(1 - q\varGamma_{\textit{hkl}}) \end{equation} | (10) |
The most distinctive characteristic of the mWH method is that it assigns a given value to α to determine the α-value so that the distribution of data on the left side with respect to Γhkl is optimal. The data showing the relationship between the left side and Γhkl is called an mWH plot. Figure 5 shows mWH plots optimized using the data in Fig. 4. Slight differences are observed among the four specimens; however, all specimens resulted in similar plots. The value of the intercept and the slope correspond to φ2Ch00 and φ2Ch00q, respectively. For Fe-18%Ni alloys, Ch00 is constant at 0.29, independent of the properties of the dislocations; therefore, the values of φ and q can be obtained from the value of the intercept and the slope, respectively. Equation (8) for dislocation density ρ can be rewritten as:
| \begin{equation} \rho = 2\varphi^{2}/(\pi A^{2}b^{2}) \end{equation} | (11) |
Modified Williamson-Hall plots in Fe-18%Ni alloy quenched from the selected temperature.
The b-value for the Fe-18%Ni alloy was found to be 0.24856 nm based on the X-ray diffraction angle. Assuming A = 0.8, the ρ-value is obtained by substituting the φ- and b-values into the equation above.
3.3 Effect of austenite grain size on dislocation densityThe frictional force of the microstructure is the same and the 2% proof stress is constant at 720 MPa for all the specimens used in this study. If the crystal grain refinement strengthening caused by the substructure acts as an additive for the 2% proof stress, the amount of dislocation strengthening must be lower as the microstructure becomes finer. In other words, the finer the austenite grains, the lower the dislocation density must be. As discussed below, specimens were investigated for the effect of austenite grain size on dislocation density and properties of dislocations using dislocation analysis by the mWH method. Figure 6 shows the relationship between the α-value obtained by the mWH method and the austenite grain size. The α-values are almost constant, with an average value of approximately 0.0028 nm−1. The Scherrer constant is 0.9, and the crystallite size in this alloy is estimated to be approximately 320 nm (= 0.9/0.0028). The lath width in lath martensitic steels is reported to be 0.1 to 1 µm [25], with an average value of approximately 0.3 µm [26]. The lath is the basic unit of martensitic transformation, and its aggregate is called a block. Basically, the laths in a block are the same variants, but they slightly differ in crystal orientation. Crystallites are defined as “single crystal grains in a polycrystalline body” in JIS K 0131 “General rules for X-ray diffraction analysis”, and the result that the size of crystallites in a lath martensitic steel corresponds to lath size is reasonable. This result suggests, although indirectly, that the lath size is approximately the same, being independent of the austenite grain size. The size of the lath, the smallest unit of martensitic transformation, is sufficiently small with respect to the austenite grain size, and it is natural that the austenite grain size does not affect the size of the lath generated during transformation.
Effect of austenite grain size d on the parameter α in Fe-18%Ni alloy.
Figure 7 shows the q- and φ-values obtained by substituting the optimal α-value. The q-values represent the average value for all dislocations present in the specimen and are close to the screw dislocation in all specimens. This result indicates that most of the dislocations introduced during martensitic transformation are screw dislocations. The elastic strain energy of a dislocation of unit length for edge dislocations is approximately 1.4 times greater than that for screw dislocations; therefore, it is reasonable that screw dislocations are preferentially introduced. On the other hand, φ-values, which reflect dislocation density and dislocation distribution, are almost constant for all specimens. The average value of φ is 0.01261. Assuming A = 0.8, the dislocation density is found to be 2.56 × 1015/m2. This result suggests that the amount of dislocation strengthening is the same for all specimens. In lath martensitic transformation, dislocations are spontaneously introduced to offset the lattice deformation caused by diffusionless transformation from fcc to bcc. The smallest unit of lath martensitic transformation is the lath, which is much smaller than austenite grains; therefore, the austenite grain size is considered to have no effect on the formation of the lath. In contrast, machining ferritic steel forcibly introduces dislocations by applying external shear stress; therefore, the ferrite grain size has an effect on the movement of dislocations [21]. In other words, when discussing the relationship between dislocation density and grain size, it is important to well understand the mechanism by which dislocations are introduced.
Effect of austenite grain size d on the parameter q (a) and φ (b) in Fe-18%Ni alloy.
As discussed above, the amount of dislocation strengthening is considered to be the same for all specimens. Assuming that dislocation strengthening and crystal grain refinement strengthening act to increase the yield stress of lath martensitic steels, the yield stress must increase as the austenite grain size becomes smaller. However, as mentioned above, the 0.6% and 2% proof stresses are constant independent of the austenite grain size and microstructure. Therefore, it is concluded that the yield stress σy of a lath martensitic steel containing no carbon is given by
| \begin{equation} \sigma_{y} = \sigma_{0} + \varDelta \sigma_{d} \end{equation} | (12) |
where σ0 is the frictional force including solid solution strengthening by Ni and Δσd is the amount of dislocation strengthening. Past studies discussing strengthening mechanisms of metals have often evaluated crystal grain refinement and dislocation strengthening as additive factors, but they should rather be considered to be in a competitive relationship because they fundamentally differ in the principle of strengthening [20]. As discussed below, we attempted to quantitatively estimate the frictional force σ0 and dislocation strengthening Δσd for the Fe-18%Ni alloy used in this study.
As is well known in discussions of solid solution strengthening, the amount of strengthening is proportional to the one-half power of the amount of alloying elements (1/2 power law) and the 1/2 power law holds for the high concentration range of 10 to 22% [27]. Ni is one of the elements used to adjust the quenchability and transformation temperature of steel, and several researchers reported solid solution strengthening of ferritic steels [28–30]. Among them, Speich prepared specimens with a uniform ferrite grain size of 20 µm and varied only the Ni content to investigate in detail the relationship between 0.2% proof stress and Ni content for alloys containing up to 9% Ni [30]. For the 0.2% proof stress σ0.2 of annealed ferritic steel, the following Hall–Petch relationship holds for the grain size d:
| \begin{equation} \sigma_{0.2} = \sigma_{0} + k_{y}/\sqrt{d} \end{equation} | (13) |
where σ0 is the frictional force and ky is a constant called the Hall–Petch coefficient. Speich seems to have considered ky to be constant; however, the ky-value becomes larger as the Ni content increases [31]; therefore, this effect needs to be taken into account to estimate the amount of crystal grain refinement strengthening. For example, the Hall–Petch coefficient is 150 MPa·µm1/2 for pure iron and 300 MPa·µm1/2 for Fe-3%Ni alloy [31]. The amount of crystal grain refinement strengthening corresponding to a ferrite grain size of 20 µm can be estimated as 33.5 MPa and 67.0 MPa, respectively. Because the maximum Hall–Petch coefficient in ferritic steel is approximately 600 MPa·µm1/2 [32], the maximum amount of crystal grain refinement strengthening corresponding to a ferrite grain size of 20 µm is estimated to be 134 MPa·µm1/2. It is difficult to form a ferrite structure in Fe-18%Ni alloys; therefore, measuring the Hall–Petch coefficient is also difficult. However, assuming that a ferrite structure was formed in this alloy, the Hall–Petch coefficient should have reached its saturation value. Figure 8 shows the 0.2% proof stress reported by Speich (dashed line) and the frictional force estimated by subtracting the value of the above crystal grain refinement (solid line). Assuming that the 1/2 power law holds for solid solution strengthening up to around 20% concentration, the frictional force of the Fe-18%Ni alloy can be estimated to be approximately 300 MPa. With the 2% proof stress of the Fe-18%Ni alloy being 720 MPa, the amount of dislocation strengthening is estimated to be approximately 420 MPa.
Effect of Ni content on 0.2% proof stress and friction stress in Fe-Ni alloys. 0.2% proof stress was obtained in ferritic steels with the grain size of 20 µm.
Shear deformation tests using iron single crystals have confirmed that dislocation strengthening of iron is unlikely to occur in the direction of easy slip and that the strengthening is likely to occur when shear stress is applied under the condition that multiple slip occurs [33]. This result suggests that the dislocation pinning strengthening model, which states that dislocation strengthening is caused by the entanglement of dislocations in a short range, is reasonable [34]. In lath martensitic transformation, high-density dislocations are spontaneously introduced to the lath interior to offset the transformation strain caused by lattice deformation from fcc to bcc; therefore, the dislocation distribution within the lath is relatively uniform. Figure 9 shows images of a specimen (a) quenched at 900°C and a specimen (b) cut out from a parallel section of a tensile test specimen (plastic deformation: 2%) obtained from a transmission electron microscope. The black linear area with dense dislocations corresponds to the lath boundary, and the area between the lath boundaries corresponds to the lath interior. The quenched material (a) has a structure where linear dislocations are randomly distributed within the lath. In contrast, in the 2% tensile deformed material (b), dislocations in the lath are tangled with each other, forming dislocation cells with diameters ranging from 20 to 100 nm. Swarr et al. observed that dislocation cells with diameters ranging from 20 to 140 nm were formed within the lath of 0.2%C martensitic steel processed after quenching and reported an average value of approximately 60 nm [35]. The results in Fig. 9 indicate that dislocation rearrangement occurs during deformation up to at least 2%, and these rearrangement phenomena are also confirmed by the sharp decrease in M and A values, which reflect the distribution of dislocations [10, 11]. It has also been found that the properties of dislocations and dislocation density show no change even after 5% cold-rolling [10, 11].
Transmission electron microstructure of specimens as-quenched (a) and tensiled by 2% (b) in Fe-18%Ni alloy.
On the other hand, it has been found that a non-uniform dislocation cell structure is formed in the grains of processed ferritic steel and that the yield stress can be explained by a mixed measurement of the strength inside cells having low dislocation density and the strength of cell walls having high dislocation density [36]. It has also been found that the strength in cell walls having high dislocation density can be reasonably estimated by applying a three-dimensional lattice model with respect to dislocation distribution [36]. Therefore, this study employed the three-dimensional lattice model to estimate the amount of dislocation strengthening in lath martensitic steels. In the three-dimensional lattice model, with the dislocation density ρ, the spacing L between entangled dislocations is given by the following equation:
| \begin{equation} L = \sqrt{3} /\sqrt{\rho } \end{equation} | (14) |
As discussed above, the dislocation density of Fe-18Ni alloys is 2.56 × 1015/m2, and the L-value corresponding to it is approximately 34.2 nm. In contrast, Morito et al. measured the dislocation density of Fe-Ni alloys by direct observation with a transmission electron microscope [37], and based on their experimental results, they estimated the dislocation density to be approximately 8 × 1014/m2. The L-value corresponding to this dislocation density is 61.2 nm. The reason the dislocation density significantly varies depending on the measurement method remains a topic for future study, but the L-value can be estimated to be between 34.2 and 61.2 nm. Therefore, this study attempted to theoretically estimate the amount of dislocation strengthening by assuming the lower and upper limits of L value as 34.2 nm and 61.2 nm, respectively.
4.3 Estimation of dislocation strengtheningThe pinning strengthening Δσ is given by
| \begin{equation} \varDelta \sigma = 2M\beta Gb\sin \theta /L \end{equation} | (15) |
where M is the Taylor factor, β is the linear tension coefficient of dislocation, G is the shear modulus, b is the Burgers vector of dislocation, θ is the pinning angle of dislocation, and L is the pinning spacing of dislocation. Ishimoto et al. investigated in detail the yield and deformation behavior of martensite, finding that blocks with a Schmid factor close to 0.5 in the tensile direction are relevant to yield and deformation [38]. This result indicates that it is reasonable to assume M ≈ 2 for the yield of lath martensitic steels. The G- and b-values of the Fe-18%Ni alloys are 64.0 GPa and 0.24856 nm, respectively. L is 34.2 nm or 61.2 nm. The linear tension coefficient β of dislocation is expressed as a function of the radius R0 of the dislocation core, the radius R of the elastic stress field of dislocation, and the dislocation constant k as given by the following equation:
| \begin{equation} \beta = \ln (R/R_{0})/4\pi k \end{equation} | (16) |
This study employed a value of approximately 0.7 nm for R0, which has been reported to be reasonable [39]. R is reported to be approximately equal to the mean dispersion spacing of dislocations [40]. Because the exclusive area per dislocation is 1/ρ, assuming that it is equal to the area of a circle with the diameter R, R is given by the following equation:
| \begin{equation} R = (4/\pi)^{1/2}/\sqrt{\rho } \approx 1.128/\sqrt{\rho } \end{equation} | (17) |
When the dislocation density is assumed to be 8 × 1014/m2 and 2.56 × 1015/m2, the R-value is estimated to be 39.9 nm and 22.3 nm, respectively. The dislocation constant k is expressed as a function of the angle θ between the dislocation line and the Burgers vector by the following equation [40]:
| \begin{equation} k = 1/\{\cos^{2}\theta + \sin^{2}\theta /(1 - \nu) \} \end{equation} | (18) |
For the Fe-18%Ni alloy, ν = 0.352. For screw dislocations, k = 1 because θ = 0. For edge dislocations, k = 0.648 because θ = 90°. For mixed dislocations, the values range from 0.648 to 1 depending on the value of the screw component.
It is important to note that the property and R-value of dislocations change as the pinning dislocations bow out. As discussed above, because the screw component in Fe-18%Ni alloys is close to 1, the following discussions assume that the screw dislocations are pinned. The property of the pinned dislocations changes continuously according to eq. (18), and the dislocation turns to an edge dislocation when θ reaches 90°. In other words, as θ increases, the k-value changes continuously from 1 to 0.648, and the linear tension of the dislocation gradually increases. Furthermore, because the two dislocations bowing out from a single pinning point have opposite signs, the radius R of the elastic stress field decreases as the two dislocations approach each other. When θ reaches 90°, the two dislocations with different signs merge and disappear, when R reaches 0. Near the pinning points, the distance between two pinning dislocations becomes narrower in proportion to cos θ; therefore, we assumed the initial value of R to be R* and approximated the radius R of the elastic stress field of dislocations by the following equation:
| \begin{equation} R = R^{\ast}\cos \theta \end{equation} | (19) |
For Fe-18%Ni alloys, R* = 39.9 or 22.3 nm. Figure 10 shows the change in the β-value obtained by substituting the k- and R-values into eq. (16). It has been found that when the pinned dislocation is a screw dislocation, the β-value does not change much in the area where the bow-out angle is below 60°, but when it is 60° or more, the β-value decreases sharply due to the decrease in the R-value.
Changes of the line tension factor β as a function of the bowing angle θ.
Figure 11 shows the change in the pinning strength Δσ obtained from eq. (15), where dislocation densities are 8 × 1014/m2 (L = 61.2 nm) and 2.56 × 1015/m2 (L = 34.2 nm). The figure also shows the dislocation strengthening (420 MPa) estimated by tensile testing. The amount of dislocation strengthening reaches its maximum value (Δσmax) when the bow-out angle reaches 60 to 70°. In other words, Δσmax corresponds to the threshold stress during the movement of pinning dislocations, which can be regarded as the amount of dislocation strengthening. When the dislocation density is 8 × 1014/m2 and 2.56 × 1015/m2, Δσmax is 345 MPa and 504 MPa, respectively, and the experimental values are in between these two values. The single dashed-dotted curve in the figure shows the values calculated based on assumed dislocation densities of 1.5 × 1015/m2 (L = 44.7 nm), and Δσmax is almost the same as the experimental value. Incidentally, the A-value obtained based on this dislocation density is 1.045. Further study is needed on the evaluation of dislocation density in martensitic steels. A possible method would be an approach based on strengthening mechanisms. Although the L-value varies to some extent depending on how the dislocation density is estimated, it is always 0.1 µm or less. Assuming the yield stress is determined by the dislocation behavior in a small area within the lath, it is natural that neither austenite grain size nor structure affects the yield behavior.
Changes of the amount of dislocation strengthening Δσ as a function of the bowing angle θ at the given dislocation density.
This study investigated the effects of austenite grain size and microstructure on the yield and deformation behavior of an ultra-low carbon lath martensitic steel (Fe-18%Ni alloy), and the following conclusions were drawn.
We sincerely thank Professor Tsuchiyama and Associate Professor Masumura of Kyushu University for their cooperation in preparing the materials used in this study and conducting the experiments.
