Microstructural Size Effect on Strain-Hardening of As-Quenched Low-Alloyed Martensitic Steels

Abstract

Quenched martensitic steels are known to show the characteristic feature of stress–strain relations, with extremely low elastic limits and very large work-hardening. The continuum composite approach is one way to express this characteristic feature of stress–strain curves. Although the overall stress–strain curves, as a function of alloy chemistries of steels, were well represented by this approach, the relationship between the macroscopic deformation behaviors and microstructural information is yet to be clarified. A high-spatial-resolution digital image correlation analysis was conducted to demonstrate the possible unit size corresponding to the microstructure. The continuum composite approach model was also modified to consider the size effect of the microstructure on the stress–strain curves of the as-quenched martensitic steels. Strain concentrations were observed at various boundaries, including lath boundaries, and the characteristic microstructural size was also predicted by the present model, which is smaller than the reported spacing between adjacent strain-concentrated regions.

1. Introduction

Martensite is a well-known hardening microstructural component in carbon steels and has been widely applied in the production of high and very high strength steels. Martensite is usually too hard to be plastically deformed. Hence, the microstructure is acceptable for tools, gears, and bearings to resist abrasion and deformities. Martensite is also one of the major microstructural components in the high-strength press formable steel sheets for automotive applications, such as Dual-Phase steels. Even in this case, the plastic deformability of steels is considered to be mainly owed to the deformation of the soft matrix phase, ferrite. When steels with mixed microstructures, including martensite, are strained, non-uniform strain and stress distributions are expected. The strain and stress caused by the microstructural components may depend on their comparative strength, size, crystallographic, and geometrical features. The most common feature of the mechanical properties of martensitic steels is their hardness as a function of the carbon content.¹⁾ The higher the carbon content, the higher the hardness of the martensitic steel. Martensite in low carbon steels shows complex microstructural feature composed of lath, block, and packet in a prior austenite grain. As the carbon content of the steel is increased, a refinement in the size of martensite microstructural units occurs.²⁾ The Hall–Petch relation between the yield strength of the martensitic steels and their block size has also been proposed.³⁾ However, the effect of microstructural features of martensite on its work-hardening behavior has not yet been clarified. The stress–strain curves of as-quenched martensitic steels are well recognized to show the typical features. They have small elastic limits, very large strain-hardening depending on the carbon concentration of steels, and relatively small ductility. Several mechanisms have been proposed to explain the very low elastic limits of as-quenched martensitic steels. Although 0.2% yield strength is often discussed as yielding, it is worth noting that 0.2% strain is not small compared with their uniform elongation in tensile tests of martensitic steels. It is particularly inappropriate to use 0.2% yield strength as a measure of strength when a certain amount of martensite tempering is expected. This is because tempering can change both, the elastic limit and the strain-hardening behavior in different ways. Metallurgical routes that lower the elastic limits of as-quenched martensitic steels could be a decrease in dislocation density⁴⁾ of martensite due to straining, the effect of transformation-induced plasticity⁵⁾ introduced by retained austenite untransformed during quenching, randomly distributed residual stresses⁶⁾ introduced during the martensitic transformation, and a continuum composite feature with different local yield strengths of martensite⁷⁾ due to heterogeneous distributions of dislocations and carbon atoms. The mechanism of the very high strain-hardening of martensitic steels, together with their very low elastic limits, is still being investigated. Different approaches have been proposed for the expression of the low elastic limits and very large strain-hardening behaviors of as-quenched martensitic steels. The basic ideas of these studies are very similar, in that they introduce sequential yielding of different localized elements or regions as a generalized Masing method^8,9) based on a continuum composite feature of martensite. The first approach is the introduction of a yield strength spectra for local elements,^7,10,11) which allows for sequential yielding—depending on the local yield strength of the elements. They assumed a continuous yield strength spectrum and elastic-perfect-plastic stress–strain relationship for each element. Although the strain and stress partitioning among the elements are not constant in general, they assumed an arbitrary constant partitioning parameter to avoid the iso-strain assumption. The second approach is the adoption of pre-existing internal shear residual stresses distributed randomly in different regions^6,12) as a result of the sequential formation of martensite laths by the shear transformation mechanism. The residual stresses due to the martensitic shear transformation were observed as relaxation strains obtained by the micro-scale focused-ion beam ring-core milling technique.¹³⁾ The residual shear stresses with flat-top distributions were assumed to be set randomly for all the elements within the microstructure. They also adopted the elastic-perfect-plastic stress–strain relation, iso-strain assumption, and Tresca criterion as the yield condition for each element.

Although both models have succeeded in representing experimentally observed stress–strain curves of as-quenched martensitic steels, assuming the heterogeneities of the local mechanical properties, there are some challenges to meet the experimentally observed microstructural heterogeneities due to deformations.

The first is the size of the elements discussed in the continuum composite approach (CCA) models. As the models do not consider the size of the elements, they cannot directly connect the macroscopic stress–strain relationships to the microstructural factors of the as-quenched martensite. Therefore, it is not possible to analyze that which particular microstructural elements contribute to the macroscopic stress–strain relationship. The heterogeneous deformation behaviors of martensitic microstructures have been discussed, for example, using high-resolution time-of-flight neutron diffraction,^14,15) and it was found that there are two different slip systems: the in-lath-plane slip system and the out-of-lath-plane system, referred to as soft- and hard-packet orientation components. The heterogeneous deformation behavior was also visualized using the digital image correlation (DIC) method. DIC is a non-contact optical method used for the visualization of local strain distributions due to deformations.¹⁶⁾ Using different DIC methods, the strain concentrations in the martensitic microstructures introduced by deformations were observed. The spacings of the high-strain-concentrated regions were reported to be close to the prior austenite grain size¹⁷⁾ or block size.^18,19) Badinier¹⁰⁾ reported the importance of individual laths as elements of the CCA model. Therefore, it is not yet clear as to what key element of the microstructure contributes to the heterogeneity of the local strain distribution and, as a result, to the macroscopic stress–strain curves of as-quenched martensitic steels.

The second point is the strain partitioning among the elements. A simple strain and stress partitioning, such as the iso-strain assumption adopted by Hutchinson and his coworkers,⁶⁾ which is commonly accepted in Masing approaches, cannot express the heterogeneous strain distributions observed either by DIC or by neutron diffraction. Allain et al.,⁷⁾ however, assumed a fixed partitioning rate of stress and strain. Although this assumption can avoid extreme assumptions such as iso-strain, there are no theoretical or experimental reasons for the assumption of a constant partitioning rate.

The last point is the assumption of the elastic-perfect plastic deformation behavior assumed in both models. This assumption may be supported by compression tests of micropillars prepared from a single block of lath martensite,²⁰⁾ in which the plastic deformation is conducted by a pronounced single slip band crossing the whole pillar. However, it was also observed that the stresses after yielding increased in the case of the deformation of pillars that contained block or packet boundaries. An increase in the average dislocation density during straining was also observed,¹⁵⁾ although the observed increase was relatively small compared to that generally observed in ferritic steels. Moreover, a large strengthening of the hard-packet orientation component was detected.¹⁵⁾ Therefore, it is natural to introduce the work-hardening nature of elements after yielding.

The aim of this work is to consider the effect of microstructural unit sizes on the strain-hardening of as-quenched martensitic steels, along with an appropriate strain and stress partitioning among different elements in the martensitic microstructures. In the present work, an extension of the model proposed by Allain et al.⁷⁾ is discussed, considering the stress and strain partitioning, as well as the work-hardening of each yielded element. The direct measurements of the heterogeneous strain distributions in a martensitic microstructure during plastic deformation are also conducted to discuss the possibilities of further model improvements.

2. Extension of Allain’s CCA Model

A CCA model is adopted in this work, as with previous works by Allain et al.⁷⁾ and Hutchinson et al.;⁶⁾ to ensure a sequential yielding, the distribution of elements with different elastic limits are required. In the case of Allain’s model, a yield strength spectrum plays the role, as a flat distribution of elastic shear residual stress between −FS/2 and +FS/2 does in the case of Hutchinson’s model. Both models assumed the elastic-perfect plastic behavior of the stress response during the deformation. In addition, Hutchinson’s model allowed for the strength of the elements to increase by a linear hardening rate during the successive deformation steps after yielding, until it reached the macroscopic tensile strength, FS. This is owing to the elastic accommodation of the element to the surrounding elements after yielding. As the Hutchinson’s model uses the iso-strain model and elastic-perfect plastic behavior for all elements, the stress component of all elements along the tensile axis becomes FS when the last elements yield. Therefore, the FS value is another fitting parameter in Hutchinson’s model to represent the experimentally observed stress–strain curves. This difficulty is overcome by the introduction of a yield strength spectrum that spreads up to very large values of the yield strength of the elements in Allain’s model. Therefore, they do not need to introduce any further increase in the stresses of the elements after yielding in Allain’s model.

Once the work-hardening of the elements after yielding is adopted, the critical yield strength for the Tresca criterion assumed by Hutchinson’s model cannot be determined. Together with the fact that Hutchinson’s model cannot deal with the heterogeneity of strain distribution during the deformation of martensitic steels, we decided to extend Allain’s CCA model by introducing the iso-work assumption²¹⁾ and strain-hardening in the element after yielding.

Although the origin of the heterogeneity of local yielding has not been experimentally clarified, the CCA model with a yield strength spectrum was adopted here. As discussed by Allain et al.,⁷⁾ function F{σ} is defined as follows:   

$$F\left\{\sigma \right\}={\int }_{-\infty }^{\sigma }f\left\{x\right\}dx$$(1)

where, F{σ} = 1 when σ→∞.

It is also assumed that there is a minimum value, σ_min, below which f{σ} = 0 and F{σ} = 0. Although there is no physical reason, the function F{σ} is assumed to be expressed as follows, as used in Allain’s model:⁷⁾   

$$F\left\{\sigma \right\}=1-exp\left[-{\left(\frac{\sigma -{\sigma }_{min}}{{\sigma }_0}\right)}^n\right]$$(2)

for the value σ>σ_min. The parameter σ_min was set to be the experimentally observed elastic limit, as discussed by Allain et al.⁷⁾ The parameter σ₀ determines the strength level of the material and is expressed as a function of the UTS or carbon content of the material. This plays a similar role to FS in Hutchinson’s model.⁶⁾ The exponent n determines the shape of the calculated stress–strain curves.

When the martensitic microstructure is considered as a whole, the matrix contains a very high density of dislocations, which may be similar to the features of a heavily cold-worked ferritic microstructure. In addition, the carbon atoms in the solid solution are expected to contribute to the strength of martensite. Given this feature, it is natural to expect a low strain-hardening rate during straining. However, the CCA model assumes a yield strength spectrum that spreads from very low-soft regions to extremely strong-dislocated regions. The heterogeneity in the microstructure may have been caused by dislocation movements and carbon diffusion during quenching as an auto-tempering of martensite. Therefore, it may be natural to introduce a strain-hardening law of ferrite into the elements after yielding.

The dislocation density evolution during straining, $\raisebox{1ex}{$d\rho $}\!\left/ \!\raisebox{-1ex}{$d\gamma $}\right.$, is expressed as the competition between the accumulation and annihilation of dislocations, as follows:   

$$\frac{d\rho }{d\gamma }=\frac{1}{bd}-f\rho$$(3)

where γ is the applied shear strain expressed as the macroscopic tensile strain, ε, multiplied by the Taylor factor, M; b is the magnitude of the Burgers vector; d is the mean free distance of the dislocation movement; and f is a parameter for the annihilation rate of dislocations. Integrating Eq. (3) and combining the classical relations between the flow stress, σ, and dislocation density,^22,23) Petitgand and Bouazis,²⁴⁾ reported the expression for the evolution of the flow stress as the forest-hardening model of ferrite, as follows:   

$$\begin{array}{l}\sigma \left\{\epsilon \right\}={\sigma }_f+\alpha M\mu \sqrt{\frac{b}{fd}}\cdot \sqrt{1-exp\left(-fM\epsilon \right)}\\ \hspace{1em}\hspace{1em}={\sigma }_f+K\cdot \sqrt{1-exp\left(-fM\epsilon \right)}\end{array}$$(4)

where σ_f is the friction stress of the material and μ is the shear modulus of ferrite. When the friction stress is subtracted from the stress evolution, the second term of Eq. (4) can be used to express the strain-hardening of each element after yielding. The strain-hardening rate is expressed by Eq. (5).   

$$\frac{d\sigma \left\{\epsilon \right\}}{d\epsilon }=\frac{1}{2}fMK\cdot \frac{exp\left(-fM\epsilon \right)}{\sqrt{1-exp\left(-fM\epsilon \right)}}$$(5)

It is evident from Eq. (5) that the strain-hardening rate, $\raisebox{1ex}{$d\sigma $}\!\left/ \!\raisebox{-1ex}{$d\epsilon $}\right.$, decreases rapidly with increasing strain, ε. The strain-hardening behavior of Eq. (4) for σ_f = 0 and the change in the strain-hardening rate of Eq. (5) for three different mean free distances of dislocation movement are compared in Fig. 1. The shorter the mean free distance of the dislocation movement, the larger the strain-hardening.

Fig. 1.

Effect of the mean free distance d(μm) of dislocation movement on strain-hardening behavior of ferrite. (Online version in color.)

When the yield strength of element i is ${\sigma }_i^L$, the element behaves elastically until the stress of the element exceeds the value ${\sigma }_i^L$. After yielding, the element is assumed to be strain-hardened, as expressed in Eq. (5). When the element yielded, the strain of the element was equal to $\raisebox{1ex}{${\sigma }_i^L$}\!\left/ \!\raisebox{-1ex}{$Y$}\right.$, where Y is the Young’s modulus of the material. The work-hardening rate of the element at a strain of ε_i larger than $\raisebox{1ex}{${\sigma }_i^L$}\!\left/ \!\raisebox{-1ex}{$Y$}\right.$ can be expressed by the following equation:   

$$\frac{d\sigma \left\{{\epsilon }_i\right\}}{d\epsilon }=\frac{1}{2}fMK\cdot \frac{exp\left(-fM\left\{{\epsilon }_i-\raisebox{1ex}{${\sigma }_i^L$}\!\left/ \!\raisebox{-1ex}{$Y$}\right.+{\epsilon }_i^L\right\}\right)}{\sqrt{1-exp\left(-fM\left\{{\epsilon }_i-\raisebox{1ex}{${\sigma }_i^L$}\!\left/ \!\raisebox{-1ex}{$Y$}\right.+{\epsilon }_i^L\right\}\right)}}$$(6)

with the strain ${\epsilon }_i^L$, which satisfies the equation below.   

$${\sigma }_i^L=\alpha M\mu \sqrt{\frac{b}{fd}}\cdot \sqrt{1-exp\left(-fM{\epsilon }_i^L\right)}$$(7)

Therefore, the stress evolution of the element i is   

$$\begin{array}{l}\sigma \left\{{\epsilon }_i\right\}=E\cdot {\epsilon }_i,\mathrm{\hspace{0.17em} } \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{\hspace{0.17em} } {\epsilon }_i\le {\epsilon }_i^L\\ \sigma \left\{{\epsilon }_i\right\}={\sigma }_i^L+{\int }_{{\epsilon }_i^L}^{{\epsilon }_i-\raisebox{1ex}{${\sigma }_i^L$}\!\left/ \!\raisebox{-1ex}{$Y$}\right.+{\epsilon }_i^L}\frac{d\sigma \left\{\xi \right\}}{d\epsilon }d\xi ,\mathrm{\hspace{0.17em} } \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{\hspace{0.17em} } {\epsilon }_i>{\epsilon }_i^L\end{array}$$(8)

as illustrated in Fig. 2.

Fig. 2.

Illustration showing the yield strength (YS) spectrum and the strain-hardening model for yielded elements. (Online version in color.)

The iso-work assumption²¹⁾ is now applied to the model for calculating the magnitude of strain owned by each element at all steps during straining. Although the iso-work assumption has not yet been justified theoretically, a good agreement with experimental results is reported.²¹⁾ The strains owned by elements i and j at a straining step k, i.e., $\mathrm{\Delta }{\epsilon }_i^k$ and $\mathrm{\Delta }{\epsilon }_j^k$ satisfy the following equation for all i,j pairs:   

$$\mathrm{\Delta }{\epsilon }_i^k\cdot {\sigma }_i^{k-1}=\mathrm{\Delta }{\epsilon }_j^k\cdot {\sigma }_j^{k-1}$$(9)

where ${\sigma }_i^{k-1}$ and ${\sigma }_j^{k-1}$ are the stress magnitudes of elements i and j after the previous strain step. Then, the macroscopic stress ${\sigma }_{tot}^k$ and strain ${\epsilon }_{tot}^k$ at straining step k are expressed as follows:   

$${\sigma }_{tot}^k=\sum _{\left(i\right)}f\left\{{\sigma }_i^L\right\}{\sigma }_i^k$$(10)

  

$${\epsilon }_{tot}^k={\epsilon }_{tot}^{k-1}+\mathrm{\Delta }{\epsilon }_{tot}^k$$(11)

For a given macroscopic strain increment $\mathrm{\Delta }{\epsilon }_{tot}^k$, the strain of element j can be expressed as   

$$\mathrm{\Delta }{\epsilon }_j^k=\frac{\mathrm{\Delta }{\epsilon }_{tot}^k}{{\sum }_{\left(i\right)}f\left\{{\sigma }_i^L\right\}\frac{{\sigma }_j^{k-1}}{{\sigma }_i^{k-1}}}$$(12)

Therefore, the strain of the j-th element after the k-th straining step, ${\epsilon }_j^k$ is expressed as follows, using the strain value at the (k−1)-th straining step, ${\epsilon }_j^{k-1}$, stresses of all the elements at the (k−1)-th straining step together with the yield strength spectrum function.   

$${\epsilon }_j^k={\epsilon }_j^{k-1}+\mathrm{\Delta }{\epsilon }_j^k$$(13)

It is now possible to calculate the strain and stress values of all the elements without any fitting parameter. It should be emphasized that the parameter β introduced by Allain et al. for the model calculation is not required to be set as a constant. The value of β for each element at any straining step can be evaluated numerically using the present model.

3. Application of the Model

The experimentally observed stress-strain curves reported by Allain et al. were adopted to adjust the model to the experimental data.

3.1. Effect of Iso-work Assumption on the Elastic-perfect Plastic Model

The first attempt was made to assess the value of β assumed by Allain et al.,⁷⁾ under the assumption of elastic and perfect plastic deformation of the elements. To analyze the effect of the introduction of the iso-work assumption, adjusting parameters in the present model were selected to match the calculated stress–strain curves with that of the experiments. The only difference between the present calculation and Allain’s is the assumption of strain and stress partitioning during straining. Parameter σ_min was set to be 300 MPa as in Allain’s model. The change in σ₀ as a function of carbon content is almost identical to Allain’s proposal, as can be seen in Fig. 3, although the value of n was found to be 1.35 for all the steels studied instead of the 1.82 in Allain’s proposal. Only the value of n, which controls the shape of the stress–strain curves, is different between the models with constant β and the iso-work assumption. The calculated stress–strain curves of the two models were compared with the experimentally observed stress–strain curve for Fe-0.215C (mass%) steel in Fig. 4. and a similar level of reproducibility was observed. The summations of the squares of the deviations of the calculated stresses from the experiments for 65 data points were 1128 in the present model and 6423 in Allain’s model. The value β was defined as follows using the macroscopic stress ${\sigma }_{tot}^k$ and strain ${\epsilon }_{tot}^k$ at the k-th step of straining, as illustrated in Fig. 5.   

$$\beta =-\frac{{\sigma }_{tot}^k-{\sigma }_i^k}{{\epsilon }_{tot}^k-{\epsilon }_i^k}=-\frac{{\sigma }_{tot}^k-{\sigma }_k^L}{{\epsilon }_{tot}^k-\raisebox{1ex}{${\sigma }_k^L$}\!\left/ \!\raisebox{-1ex}{$E$}\right.}$$(14)

where ${\sigma }_k^L$ is the yield strength of an element that yields at the k-th step of straining. Stress–strain pairs for elements that are elastic and plastic at a certain straining state can be calculated using the present model with the iso-work assumption. The calculated results for the Fe-0.215C (mass%) alloy at macroscopic strains of 0.011, 0.021, and 0.031 are shown in Fig. 6. The calculated macroscopic stresses and strains for these three conditions are plotted in Fig. 6. It is clear that there is no linear relationship between the stress and strain pairs, as shown in Fig. 6. If we use the definition of parameter β, the value β cannot be constant and decreases with increasing macroscopic strain.

Fig. 3.

Parameter σ₀ obtained for the model with iso-work assumption, as a function of carbon content of steels. The solid line is the relation proposed by Allain et al.⁷⁾

Fig. 4.

Comparison among the experimentally observed stress–strain relation for Fe-0.215C alloy and the two model calculations with and without iso-work assumption. (Online version in color.)

Fig. 5.

Illustration of the definition of the parameter β in Allain’s model.⁷⁾ (Online version in color.)

Fig. 6.

Relations among the stress and strain pairs of elements at different macroscopic strains calculated by the model with iso-work assumption for Fe-0.215C (mass%) alloy. (Online version in color.)

3.2. Effect of Work-hardening after Yielding

There are four parameters in the model: n, σ₀ and σ_min for the cumulative probability function F{σ}, and the mean free distance of the dislocation movement, d. The same values as those proposed by Allain et al. for Young’s modulus E = 200 GPa and the minimum yield strength σ_min = 300 MPa were used for the adjustment. The exponent n and parameters σ₀ and d were adjusted to the experimental stress–strain curves. A comparison between the present model calculations and experimentally observed tensile behaviors of as-quenched martensitic steels with different carbon contents is shown in Fig. 7(a). The exponent n was found to be 1.35 for all the alloys considered. It is not surprising that the fitting is extremely good, as shown by Allain et al. for a wide range of stresses and the derivative of stress, as shown in Fig. 7(b).

Fig. 7.

Comparisons between the calculated (lines) by the present model, with iso-work assumption and work-hardening of elements after yielding, and experimentally observed (plots) (a) stress–strain curves and (b) strain-hardening rates for steels studied. (Online version in color.)

The values of σ₀ and d varies with the carbon content of the steel. The magnitude of σ₀ represents the ultimate strength of steel. Therefore, σ₀ increases with the increase in carbon content of the steel, as shown in Fig. 8. As long as the carbon content up to 0.4 mass% is concerned, a linear dependence of σ₀ on the carbon content of the steels can be observed. The last adjusting parameter is the mean free distance of the dislocation movement, d. Adjusted values are plotted against the carbon content of the steels in Fig. 9. The mean free distance of the dislocation movement, d, in the present model decreased with the increase in steel carbon content. This tendency may contribute to the increase in the strain-hardening rate of martensite with increasing carbon content.

Fig. 8.

Effect of carbon content on the parameter σ₀ for the present model, with iso-work assumption and work-hardening of elements after yielding, adjusted to the experimental data for as-quenched martensitic steels.

Fig. 9.

Effect of carbon content on the parameter d for the present model adjusted to the experimental data for as-quenched martensitic steels.

4. Discussions

The continuum composite approach, proposed by Allain et al. was adopted in the present work together with original modifications of the stress and strain partitioning among elements, assuming iso-work behavior in the system and a strain-hardening model after yielding. The strength of each element increased even after yielding during straining. This feature is one of the major differences between the present model and previous models, with the assumption of elastic and perfect plastic deformation behavior of each element.

The parameter introduced to consider the effect of work-hardening after yielding is the mean free distance, d, of the dislocation movement. As shown in Fig. 9, d decreased with an increase in the steel carbon content. The mean free distance of the dislocation movement is compared with the published average lath widths^{25,26,27,28,29)} in Fig. 10. The mean free distance, d, is almost the same order of magnitude as the average lath width reported. The mean free distance may correspond to the average spacing of strain-concentrated regions. The spacings between strain-concentrated regions were reported to be approximately 12 μm by Koga et al.,¹⁷⁾ and approximately 20 to 30 μm estimated from the strain distribution obtained by DIC reported by Sugiyama et al.¹⁹⁾ The mean free distance of the dislocation movement obtained in the present work seems to be much smaller than the reported average spacing of the strain-concentrated regions. However, a denser distribution of strain-concentrated regions other than block boundaries can also be observed in Sugiyama et al.¹⁹⁾ Therefore, it may be meaningful to try to visualize the strain distribution during the plastic deformation of martensitic steels on a scale smaller than the block sizes.

Fig. 10.

Comparison between the parameter d for the present model and the average lath width as a function of carbon content.

To clarify the effect of the microstructure on the heterogeneous strain distribution in the martensitic microstructure during straining, a DIC analysis was conducted. The surfaces of the specimens for DIC analysis were ground and finished using colloidal silica. Speckle patterns were introduced by the Ag water solvent nanoparticles. To visualize heterogeneous strain distributions on a scale smaller than the block sizes, the subset and step sizes were selected as 101 pixels and 21 pixels, respectively. This step corresponds to a spatial resolution of approximately 78 nm, which is small enough to discuss the heterogeneous strain distribution inside the blocks. Secondary electron images of scanning electron microscope (SEM) were used for DIC analysis and analyzed using Vic-2D software. Tensile specimens were machined from heat-treated Fe-0.1C-10.5Ni (mass%) coupon samples, which were vacuum melted, hot-rolled, cold-rolled, and heated at 1273 K for 20 min before being quenched in water. The sample was subjected to a uniaxial tensile deformation of 0.026 in the engineering strain, and the SEM images were taken before and after straining. Electron back scattering diffraction (EBSD) observations were conducted on the tensile specimen prior to straining, and the areas for the DIC analyses were selected inside the EBSD observation area.

A typical martensitic microstructure is shown in Fig. 11, which shows an inverse pole figure (IPF) map of the specimen quenched at 1273 K. An EBSD IPF map was obtained after the speckle patterning. A typical speckle pattern is shown in Fig. 12, where randomly dispersed Ag particles are visible. Two areas were selected for the DIC analyses, indicated in the figure as area-1 and area-2. We can see regions where too many particles are clustered, and clear Kikuchi patterns are not observed. Areas for DIC analyses were selected to avoid clustered regions. As shown in Fig. 11, area-1 contains prior austenite grain boundaries, packet boundaries, block boundaries, and lath boundaries, whereas almost no boundaries other than lath boundaries exist in area-2.

Fig. 11.

Inverse pole figure (IPF) of the martensite studied. The EBSD observation was conducted after the speckle patterning.

Fig. 12.

Typical secondary electron micrograph showing the speckle pattern before straining.

After tensile straining of the specimen with an average engineering strain of 0.026, DIC analyses were conducted, and the results are shown in Fig. 13 for areas -1 and -2. In both cases, the heterogeneous distributions of strain can be clearly recognized. In area-1, the strain concentrations at the prior austenite grain boundaries and block boundaries are indicated by the white arrows. However, it is worth emphasizing that strain concentrations inside the blocks, indicated by yellow arrows, can also be clearly observed. The strain concentrations inside the blocks appear to correspond to the lath boundaries. The strain concentrations inside a block are also clear in area-2. These strain concentrations also seemed to coincide with the lath boundaries. However, the strain concentration is not observed at all of the lath boundaries. The reason for this heterogeneous selection of lath boundaries for strain concentration is currently not clear. Therefore, further investigation is required.

Fig. 13.

EBSD boundary maps for area-1 (a) and -2 (c) before straining and strain distributions obtained by the DIC analysis for area-1 (b) and area-2 (d) at 0.026 of macroscopic engineering strain.

As a result, the spacing between the strain-concentrated regions was found to be approximately between 0.7 and 2 micron-meters. Although further analysis is needed to clarify the spacing of the strain concentration regions, the spacing is in the order of 1 μm, which is one order of magnitude smaller than the spacing reported.

It is also important to focus on the distribution of the plastic strain observed by DIC analyses. The strain-concentrated regions tend to lie along the lath boundaries, which are the habit planes of the martensite transformation. However, we also observed strain-concentrated regions that were not well aligned along the lath boundaries. This may indicate that the out-of-lath slip of dislocations can also be active, although the major slip system may correspond to in-lath planes. It is also worth noting that the mountain ranges of the strain concentration do not always show a uniform height, but are rather heterogeneous. This may also indicate that mixtures of different slip systems work together during the plastic deformation of the martensite.

The model proposed herein deals with work-hardening based on ferrite. Therefore, it is naturally assumed that the directions of the dislocation movement lie randomly in the microstructure, depending on the texture of the material. Tabata et al.³⁰⁾ studied the plastic deformation behavior of a ferritic microstructure using the DIC technique. Because the slip systems for bcc crystals are {110}<111>, the number of active slip systems in each ferrite grain differ from grain to grain owing to its crystallographic orientation. The slip systems preferred by the Schmid factor are selected during plastic deformation and, as a result, a typical alignment of strain-concentrated regions appears. This feature is similar to that observed in the present work on as-quenched martensite. A simple work-hardening model for ferrite introduced in the present work for fully martensitic microstructures may provide a better idea of the microstructural units that control the large work-hardening of as-quenched martensitic steels.

5. Conclusions

Continuum composite analysis was adopted to understand the microstructural unit size for large work-hardening behaviors in the stress–strain curves of the as-quenched martensitic steels. The proposed model implements the iso-work assumption, which enables the calculation of strain and stress partitioning among elements with different yield strengths, as well as the effect of the work-hardening of each element after yielding. The proposed model can predict the average distance of the dislocation movement, which contributes to the work-hardening of martensitic steels. The distance depends on the carbon concentration of the steels and decreases with increasing carbon content. The value obtained for the distance is much smaller than the spacing between adjacent strain-concentrated regions reported previously, but almost the same order of magnitude as the average lath width reported.

A high-spatial-resolution DIC analysis revealed that the strain concentrations seem to occur at some of the lath boundaries, as well as block, packet, and prior austenite grain boundaries, although it is not clear at present which boundaries are selected for the strain concentration.

Acknowledgements

This work was a part of the Joint Research Project on Innovative High-Performance Structural Steel supported by Nippon Steel Corporation. The authors would like to thank Emeritus Professor Minoru Nishida, Associated Professor Masaru Itakura, and Dr. Hiroshi Akamine for the provision of the in-situ-tensile-test apparatus for the DIC analyses.

References

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