Bridging between Heterogeneous Local Strain Distribution and Macroscopic Stress-strain Curves

Abstract

A modified continuum composite model was utilized to analyze the stress–strain behavior of as-quenched steels with a fully martensitic microstructure. The model was employed to express the stress–strain curves obtained by a simple tensile test and those obtained by forward and backward loading using a simple shear test machine. The study confirmed that the model can represent the stress–strain behavior under both forward and backward deformation. In addition, the evolution of local strain distributions during plastic deformation was investigated using a digital image correlation method. The strain distributions and their evolution during deformation were qualitatively represented using this model. The discrepancies between the model calculations and experiments are due to the limitations of the iso-work assumption and the impact of slip deformation on the macroscopic work-hardening behavior of martensitic steels. Highly strain-concentrated regions aligned along the longitudinal direction of the lath and block, known as in-lath-plane slips, may not play an important role in the work-hardening behavior of as-quenched martensitic steels. However, the other slips, namely the out-of-lath slip, may play a significant role in the work hardening of as-quenched martensitic steels.

1. Introduction

Martensitic steels are commonly used in machine structural parts to prevent abrasion, plastic deformation, and fatigue failure, owing to their high strength resulting from their carbon content. Martensite is also one of the major microstructural components in high-strength press-formable steel sheets for automotive applications, such as dual-phase steels. The hardness of martensitic steels is a characteristic mechanical property that increases with the carbon content.¹⁾ The stress-strain curves of as-quenched martensitic steels exhibit small elastic limits, substantial strain hardening dependent on the carbon concentration of the steel, and relatively low ductility. Several mechanisms have been proposed to explain the very low elastic limits of the as-quenched martensitic steels. The basic ideas of these studies are very similar in that they introduce the sequential yielding of different localized elements or regions as a generalized Masing method^2,3) based on a continuum composite feature of martensite. This method is referred to as the continuum composite approach (CCA).⁴⁾ The first approach involves the introduction of yield strength spectra for local elements,^4,5,6) facilitating sequential yielding contingent on the local yield strength of each element. The second approach involves the incorporation of preexisting internal shear residual stresses randomly distributed in different regions^7,8) because of the sequential formation of martensite laths via the shear transformation mechanism. Although the macroscopic deformation behavior of the as-quenched martensitic steels can be represented by the CCA model, the effect of the microstructural features of martensite on its work-hardening behavior is unclear. Martensite in Fe–C alloys shows a complex microstructural feature comprising lath, block, packet, and prior austenite grain boundaries. When the carbon concentration of the steel is increased, refinements in the size of the microstructural units⁹⁾ and an increase in strength have been reported. In a previous work,¹⁰⁾ the CCA model proposed by Allain et al.⁴⁾ was modified by introducing the iso-work assumption¹¹⁾ for strain redistribution among the elements and work hardening of the yielded elements. The modified CCA can represent the experimentally observed stress–strain curves and provide an idea of the size of the microstructural unit that contributes to the macroscopic stress–strain curves. However, the characteristic sizes or spacings for heterogeneous strain distribution were referred to as the prior austenite grain size,¹²⁾ block size,^13,14) or lath size⁵⁾ without any conclusive experimental evidence. Model analysis¹⁰⁾ revealed that the possible sizes that represent the heterogeneity of strain due to deformation are of the same order of magnitude as the lath width of martensite, and decrease with increasing carbon content.¹⁰⁾ The result was confirmed qualitatively using high-spatial-resolution digital image correlation (DIC), in which the strain concentrations were observed at some of the lath boundaries, as well as at the packet and prior austenite grain boundaries.¹⁰⁾ However, it is necessary to assess the strain distribution quantitatively, in addition to the positions of the strain concentration in the microstructures, to clarify the relationship between the microstructure and macroscopic deformation behavior.

Another important feature of martensitic steels is the significant Bauschinger effect. Understanding the stress–strain relationship under non-monotonous loading paths is important not only for cold-press forming, but also for comprehending the deformation behavior of vehicles during crash events. In the CCA model, the soft phases control the microscopic yielding, and the hard phases, which remain elastic, provide a high work-hardening behavior. Hence, the model can naturally express the internal stress and is useful for representing the large Bauschinger effect of martensitic steels as discussed by Allain et al.⁴⁾

The aim of this study is to confirm that the modified CCA model proposed by Sakaguchi and his co-workers¹⁰⁾ can represent the macroscopic monotonous and non-monotonous stress–strain relations and to discuss the local strain distribution and evolution of strain distribution during the plastic deformation of martensitic steels using the modified CCA model. Possible improvements to the model are also discussed.

2. Experimental Procedure

Three laboratory-melt Fe–0.1C steels with 3.0 mass% Mn (3Mn alloy), 5.0 mass% Mn (5Mn alloy), and 10.5 mass% Ni (10Ni alloy), as listed in Table 1, were hot rolled, cold rolled, and heat treated at temperatures in the austenite single-phase region. The alloys were then quenched in water to obtain fully martensitic microstructures. The 3Mn and 5Mn alloys were heat treated for 10 min at 1323 K, and the 10Ni alloy was heat treated for 20 min at 1273 K. The tensile specimens shown in Fig. 1 were machined from the heat-treated coupon samples. The samples were subjected to uniaxial tensile deformation to obtain the stress–strain curves. The heterogeneous strain distribution in the martensitic microstructures during straining was analyzed using the DIC. The surfaces of the specimens used for DIC were ground and finished with colloidal silica. Speckle patterns were introduced using water-solvent Ag nanoparticles. To visualize the heterogeneous strain distribution on a scale smaller than the block size, the subset and step sizes were selected to be 101 and 21 pixels, respectively. This step corresponds to a spatial resolution of approximately 78 nm, which is sufficiently small for discussing the heterogeneous strain distribution inside the blocks. Secondary electron images from scanning electron microscopy (SEM) were used for DIC, and the images were analyzed using Vic-2D software. The sample with the speckle pattern was subjected to a uniaxial tensile deformation with an engineering strain of 0.026, and SEM images were captured before and after straining. Electron backscattering diffraction (EBSD) observations were conducted on the tensile specimens prior to straining, and areas for DIC were selected inside the EBSD observation area.

Table 1. Alloy chemistries of materials used.

	C	Si	Mn	Ni	P	S
Ni alloy	0.10	0.01	0.01	10.5	0.006	0.001
3Mn alloy	0.10	<0.01	3.03	–	<0.002	0.001
5Mn alloy	0.09	<0.01	5.00	–	<0.002	0.002

Fig. 1. Specimen for uniaxial tensile test. (Online version in color.)

A planar simple shear test machine¹⁵⁾ was used to obtain the stress–strain relationships during forward and backward loading to analyze the Bauschinger effect of the materials. The planar simple shear test did not involve strain inhomogeneities or buckling during the deformation along different paths. A schematic of the specimen used for the planar simple shear test is shown in Fig. 2.¹⁵⁾ A specimen with a width and length of 23 and 38 mm, respectively, was first shear-deformed in the forward direction and then in the backward direction. The stress–strain relationship during the cyclic deformation was analyzed.

Fig. 2. Specimen for planar simple shear test. (Online version in color.)

3. Modified CCA Model

Sakaguchi and his coworkers¹⁰⁾ reported a modified CCA model. They used the same probability function of the accumulated yield strength spectrum F{σ} employed in the original model proposed by Allain et al.⁴⁾ They assumed a yield strength spectrum f{σ} that can be expressed as follows:

  

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

  

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

where σ_min is the threshold stress below which all the elements remain elastic, and therefore, f{σ}=0, and F{σ}=0. The parameters σ₀ and n are determined by adjusting the calculation based on the experimentally observed stress–strain curves. They also implemented the iso-work assumption in the model. To avoid confusion, we define the expression of strain as follows. A strain owned by an element i at a straining step m is expressed as ${}^i{\epsilon }_X^m$. The subscript X is “t” for total strain, “e” for elastic strain, and “p” for plastic strain. The total strain increments owned by elements i and j at a straining step m, i.e., ${\mathrm{\Delta }}^i{\epsilon }_t^m$ and ${\mathrm{\Delta }}^j{\epsilon }_t^m$ satisfy the following equation for all i,j pairs:

  

$${\mathrm{\Delta }}^i{\epsilon }_t^m{\cdot }^i{\sigma }^{m-1}={\mathrm{\Delta }}^j{\epsilon }_t^m{\cdot }^j{\sigma }^{m-1}$$(3)

where ⁱσ^m−1 and ^jσ^m−1 are the stress magnitudes of elements i and j after the previous strain step. The macroscopic stress ${\sigma }_t^m$ and total strain ${\epsilon }_t^m$ after the straining step m are expressed as follows:

  

$${\sigma }_t^m=\sum _{\left(i\right)}f{\left\{{}^i{\sigma }_L\right\}}^i{\sigma }^m$$(4)

  

$${\epsilon }_t^m={\epsilon }_t^{m-1}+\mathrm{\Delta }{\epsilon }_t^m$$(5)

where ⁱσ_L is the yield strength of element i. The macroscopic total strain increment $\mathrm{\Delta }{\epsilon }_t^m$ can be expressed as follows:

  

$$\mathrm{\Delta }{\epsilon }_t^m=\sum _{\left(i\right)}f\left\{{}^i{\sigma }_L\right\}\cdot {\mathrm{\Delta }}^i{\epsilon }_t^m=\sum _{\left(i\right)}f\left\{{}^i{\sigma }_L\right\}\cdot {\mathrm{\Delta }}^j{\epsilon }_t^m\cdot \frac{{}^j{\sigma }^{m-1}}{{}^i{\sigma }^{m-1}}$$(6)

The strain increment of element j at a straining step m can then be expressed for a given macroscopic strain increment $\mathrm{\Delta }{\epsilon }_t^m$, as

  

$${\mathrm{\Delta }}^j{\epsilon }_t^m=\frac{\mathrm{\Delta }{\epsilon }_t^m}{{\sum }_{\left(i\right)}f\left\{{}^i{\sigma }_L\right\}\cdot \frac{{}^j{\sigma }^{m-1}}{{}^i{\sigma }^{m-1}}}$$(7)

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

  

$${}^j{\epsilon }_t^m={}^j{\epsilon }_t^{m-1}+{\mathrm{\Delta }}^j{\epsilon }_t^m$$(8)

Sakaguchi and his coworkers¹⁰⁾ also introduced an expression to account for the strain hardening of an element i at a straining step m after yielding, which adopts the forest-hardening model of ferrite to express the evolution of the flow stress, as follows:^16,17,18)

  

$$\frac{{\mathrm{\Delta }}^i{\sigma }^m}{{\mathrm{\Delta }}^i{\epsilon }_p^m}=\frac{1}{2}\beta MK\cdot \frac{exp\left(-\beta M^i{\epsilon }_{p-sh}^m\right)}{\sqrt{1-exp\left(-\beta M^i{\epsilon }_{p-sh}^m\right)}}$$(9)

Here, $K=\alpha M\mu \sqrt{\frac{b}{\beta \Lambda }}$; M is the Taylor factor; b is the magnitude of the Burgers vector; μ is the shear modulus of ferrite; α is a constant equal to 0.5; Λ is the mean free distance of dislocation movement; and β is a parameter for the annihilation rate of dislocations. An illustration of the strain hardening is shown in Fig. 3. As illustrated in Fig. 3, the strain-hardening curve is shifted left to express the strain-hardening behavior of the element i where the elastic curve meats the strain-hardening curve of the element i at its yield stress, ⁱσ_L. Therefore, the strain used to calculate the magnitude of strain-hardening in Eq. (9) is ${}^i{\epsilon }_{p-sh}^m$ in Fig. 3 and can be obtained by the following equation:

  

$${}^i{\epsilon }_{p-sh}^m={}^i{\epsilon }_t^m+{}^i{\epsilon }_L-\frac{{}^i{\sigma }_L}{Y}={}^i{\epsilon }_t^m-\frac{1}{\beta M}ln\left\{1-{\left(\frac{{}^i{\sigma }_L}{K}\right)}^2\right\}-\frac{{}^i{\sigma }_L}{Y}$$(10)

where Y is the Young’s Modulus, and ⁱε_L is the strain at which the magnitude of strain-hardening reaches ⁱσ_L. They obtained the parameters in Eq. (2) by adjusting the model based on the experimentally observed stress–strain curves. In their analysis, the friction stress expressed by Eq. (11)¹⁹⁾ was subtracted from the experimentally observed stress–strain curves to directly compare the stress–strain curves of alloys with different chemistries.

  

$$\begin{array}{l}{\sigma }_{friction}=60+33\mathrm{\hspace{0.17em} }mass\%\mathrm{\hspace{0.17em} }Mn+81\mathrm{\hspace{0.17em} }mass\%\mathrm{\hspace{0.17em} }Si\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}+48\mathrm{\hspace{0.17em} }mass\%\mathrm{\hspace{0.17em} }Cr+48\mathrm{\hspace{0.17em} }mass\%\mathrm{\hspace{0.17em} }Mo+0\mathrm{\hspace{0.17em} }mass\%Ni\end{array}$$(11)

It is now possible to calculate the total strain and stress values of all the elements without any fitting parameter. Using Eqs. (4), (5), (7), (8), (9) and (10), macroscopic stress–strain relations can be calculated. This enables to evaluate the distribution of strain and stress among constituent elements with varying yield strengths, in addition to accounting for the work-hardening effects that occur in each element after yielding. As demonstrated by Sakaguchi et al.,¹⁰⁾ the stress–strain curves during forward tensile deformation were reproduced accurately by the model.

Fig. 3. Illustrative diagram for a yield strength spectrum and strain hardening model for yielded elements.

In addition, the elements in a specimen were expected to undergo two types of deformation stages at the end of plastic deformation. One was still elastic, while the other was plastic and strain-hardened. Therefore, different amounts of internal stress were expected for each element at the end of the deformation. Non-monotonous stress–strain curves can then be calculated as discussed by Asaro²⁰⁾ and Allain et al.⁴⁾

4. Experimental Results

4.1. Microstructure and Mechanical Properties

The typical martensitic microstructures obtained for the three studied alloys are shown in Fig. 4. XRD measurements confirmed that the amount of austenite retained in the three alloys was negligible. The prior austenite grain sizes were 85, 70, and 35 μm for 3Mn, 5Mn, and 10Ni alloys, respectively.

Fig. 4. IPF-maps of a) 10Ni alloy quenched at 1273 K, b) 3Mn and c) 5Mn alloys quenched at 1323 K.

The true stress-true strain curves of the alloys are shown in Fig. 5. Although the amount of solution hardening differed, the differences between the three stress–strain curves were small. The carbon content of 5Mn alloy is lower than the other two alloys. This also affected the stress–strain curve of the alloy.

Fig. 5. Experimentally obtained true stress–true strain curves.

The 10Ni alloy was subjected to a simple shear test using a planar simple-shear test machine.¹⁵⁾ The experimentally obtained shear-stress–shear-strain relations, σ_S−ε_S, with maximum shear strains of 0.02, 0.04, 0.07 and 0.10 in forward deformation are plotted in Fig. 6.

Fig. 6. Experimentally observed shear-stress–shear-strain curves with the maximum shear strain of 0.02, 0.04, 0.07 and 0.10 in forward deformation.

4.2. Local Strain Distribution

As discussed by Sakaguchi et al.,¹⁰⁾ the heterogeneous ε_XX strain distribution in the specimen after tensile straining was identified using DIC, with an average engineering strain of 0.026, as shown in Fig. 7(b). The strain ε_XX in Fig. 7(b) is the total strain along the tensile direction at each point analyzed using Vic-2D software before unloading. They observed strain concentrations at the prior austenite grain boundaries, block boundaries, and some lath boundaries. They also observed strain-concentrated regions that were not well-aligned along the boundaries. The strain data obtained by the DIC were analyzed in detail. Three points were selected in the study area, as shown in Fig. 7(b). The strain development at these three points against the average strains of the area are plotted in Fig. 8. The strain at these points increases with the average strain at different gradients. This indicates an enhancement in the strain heterogeneity during deformation. The strain at point A reached a value ten times larger than that at point C, when the average strain was 0.026. Although the reason for this is unclear, the increase in strain at each point is almost linearly dependent on the average strain. The strain distributions observed by the DICs in two areas¹⁰⁾ at average strains of 0.0079, 0.0100, 0.0137, 0.0186, and 0.0255 are shown in Fig. 9. As the average strain increased, the strain distribution broadened. This clearly shows that the strain heterogeneity during tensile straining was enhanced. It should be noted that the strain distribution obtained in this study corresponds only to a limited area. Although further data collection is required, qualitative information can be obtained from these analyses.

Fig. 7. (a) EBSD boundary maps before straining and (b) strain distribution obtained by the DIC analysis at 0.026 of macroscopic engineering strain.¹⁰⁾

Fig. 8. Strains at the points A, B, and C from Fig. 7(b) as a function of average strain.

Fig. 9. Strain distributions at average strains of 0.0079, 0.0100, 0.0137, 0.0186, and 0.0255.

5. Discussion

5.1. Confirmation of the Applicability of the Modified CCA Model to the Alloys Studied

According to Eq. (4), the addition of Ni has no effect on the friction stress. This was based on the data reported by Pickering.²¹⁾ However, Takeuchi²²⁾ claimed that the addition of Ni affected the yield stress. Therefore, it is meaningful to reassess the effect of Ni addition on the friction stress. Yield stress, σ_L, as a function of the grain diameter, d, is expressed as follows:

  

$${\sigma }_L={\sigma }_{friction}+k{d}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},$$(12)

where σ_friction is the friction stress of the alloy. This relationship is known as the Hall-Petch relationship. Therefore, σ_friction can be determined as the y-intercept of the plot of σ_L as a function of $d^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$. The effect of Ni on the Hall-Petch relationship was reported by Akama et al.²³⁾ and Morrison and Leslie.²⁴⁾ They conducted the experiments on carefully prepared Fe–Ni alloys. The y-intercepts obtained by them are plotted in Fig. 10 as a function of Ni concentration of steels. The friction stress is expressed as a function of the Ni concentration in the steel.

  

$${\sigma }_{friction}\left\{mass\%\mathrm{\hspace{0.17em} }Ni\right\}=52.32+14.59\cdot mass\%\mathrm{\hspace{0.17em} }Ni\mathrm{\hspace{0.17em} } \left(MPa\right)$$(13)

Although the maximum limit of Ni addition in Eq. (13) is 3.08%, we assume that Eq. (13) can be extrapolated to 10.5 mass% without proof. Together with Eq. (11), the stress–strain curves of the studied alloys were calculated and compared with the experimentally observed stress–strain curves in Fig. 11. Because the friction stress is subtracted from the stress–strain curves, the stress–strain curves depend only on the carbon concentration of the steels. A modified CCA calculation was conducted for martensitic steels with 0.1 and 0.09 mass% of carbon. The effect of 0.01 mass% difference in carbon concentration can be seen in the experimental stress–strain curves of the 3Mn and 5Mn alloys and is well represented by the modified CCA calculations.

Fig. 10. Effect of Ni concentration on the friction stress of steel.

Fig. 11. Comparison between experimental and calculated stress–strain curves of the three alloys studied. Calculated stress–strain curves of 3Mn and 10Ni alloys are identical.

Non-monotonous stress–strain curves were also calculated following the forward deformations as discussed by Allain et al.⁴⁾ To compare the experimental stress–strain curves with those of the modified CCA model, the results of the simple shear test were converted to uniaxial tensile equivalent stress–strain curves. Shirakami et al.¹⁵⁾ reported how shear-stress–shear-strain relations can be translated into equivalent stress–strain relations during uniaxial tensile deformation. They assumed the following two relationships:

  

$$\overline{\sigma }={\kappa }_{\sigma }{\sigma }_S$$(14)

  

$$d{\overline{\epsilon }}_p={\kappa }_{\epsilon }d{\epsilon }_{p-s}$$(15)

Here $\overline{\sigma }$ and $d{\overline{\epsilon }}_p$ are the maximum principal stress in uniaxial tensile deformation and the incremental plastic strain along the direction of the principal stress, respectively. σ_S and dε_p–s are experimentally obtained shear stress and plastic shear strain increment, respectively, along the shear direction. Two parameters, κ_σ and κ_ε were introduced to relate shear stress and uniaxial tensile stress, and plastic shear strain and plastic uniaxial tensile strain, respectively. Assuming that the work performed in the two different deformation modes are identical, the following relationship is established:

  

$$\overline{\sigma }\cdot d{\overline{\epsilon }}_p={\sigma }_S\cdot d{\epsilon }_{p-s}$$(16)

Using Eqs. (14), (15), and (16), we obtain:

  

$$d{\overline{\epsilon }}_p=\frac{{\sigma }_S}{\overline{\sigma }}d{\epsilon }_{p-s}=\frac{1}{{\kappa }_{\sigma }}d{\epsilon }_{p-s}$$

Therefore:

  

$${\kappa }_{\epsilon }=\frac{1}{{\kappa }_{\sigma }}$$(17)

As a result, one parameter relates the simple shear deformation to the uniaxial tensile deformation. Comparing experimentally obtained shear-stress–shear-strain curve and uniaxial tensile stress–strain curve for 10Ni alloy, κ_σ=1.82 was obtained.

The maximum strain in the forward tensile deformation at which the deformation direction is reversed was selected to be equal to that in the experiment. The maximum strains are 0.0568, 0.0401, 0.0230, and 0.0116. The calculated stress–strain curves were compared with the experimentally observed uniaxial tensile equivalent stress–strain curves in Fig. 12. The experimentally observed stress–strain curves were well represented by the modified CCA model calculation without any fitting, apart from the selection of the appropriate maximum strains in the forward deformation as discussed by Allain et al.⁴⁾ However, the discrepancy between the experiment and calculation increases with the maximum strain in the forward deformation at the beginning of strain hardening. This may be due to the effect of the forward deformation on the dislocation substructure, which could potentially change the average distance of the dislocation movement and strain-hardening behavior during backward deformation. Although it is not clear how to cope with this effect, the development of the substructure and its effect on the average distance of the dislocation movement should be considered.

Fig. 12. Comparison between experimental and calculated forward and backward stress–strain curves of 10Ni alloy.

5.2. Effect of Deformation on the Strain Distribution

The effect of strain during tensile deformation on the strain distribution, as shown in Fig. 9, is analyzed using the modified CCA model. At a step m during the tensile deformation with the macroscopic total strain of ${\epsilon }_t^m$, there are two categories of elements. One is an element that is still elastic and the other is an element that has already yielded. The elements that are still in elastic, have the same elastic strain, ${\epsilon }_e^m={\sigma }_e^m/Y$, and ${\sigma }_e^m$ is the stress shared by the elastic elements after the m-th straining step. By applying Eq. (2), the fraction of the elastic region $V_e^m$, can be calculated using Eq. (18).

  

$$V_e^m\left\{{\sigma }_e^m\right\}=exp\left[-{\left(\frac{{\sigma }_e^m-{\sigma }_{min}}{{\sigma }_0}\right)}^n\right]$$(18)

The other elements are in the plastic region and have different strains and stresses depending on their yield strength. Considering a group of elements, denoted as i, where each element has the same yield strength of ⁱσ_L, shared stress of ⁱσ^m, and total strain of ${}^j{\epsilon }_t^m$, the fraction of group i can be expressed as the derivative of Eq. (2).

  

$$V_i^m\left\{{}^i{\sigma }^m\right\}=\frac{n}{{\sigma }_0}{\left(\frac{{}^i{\sigma }^m-{\sigma }_{min}}{{\sigma }_0}\right)}^{n-1}exp\left[-{\left(\frac{{}^i{\sigma }^m-{\sigma }_{min}}{{\sigma }_0}\right)}^n\right]$$(19)

Therefore, it is possible to obtain the strain distribution at the macroscopic total strain of ${\epsilon }_t^m$.

An illustration of the amounts of strain for each element at different levels of average total strain are shown in Fig. 13 together with the yield strength spectrum. When the average total strain is ε_A, below the value $\raisebox{1ex}{${\sigma }_{min}$}\!\left/ \!\raisebox{-1ex}{$Y$}\right.$, all the elements are elastic, the strains of them are identical as illustrated in the Fig. 13. Therefore, a uniform elastic strain distribution is obtained. When the average total strain becomes ε_B, above the value $\raisebox{1ex}{${\sigma }_{min}$}\!\left/ \!\raisebox{-1ex}{$Y$}\right.$, a strain distribution appears as illustrated in Fig. 13. The elastic elements have an identical strain that is smaller than the average total strain ε_B because the strength of the elastic elements is higher than those of the yielded elements. The magnitudes of strain vary between the strain of the elastic elements and that of the element with the minimum yield strength. The magnitudes of strain of yielded elements depend on their strengths, which are the summation of the yield strength and the amount of work hardening. When the total strain is further increased, the fraction of the elastic region decreases and the strain distribution becomes broader as shown in Fig. 13.

Fig. 13. Illustrative diagram for a yield strength spectrum and strains of each element at different average total strains.

The calculated strain distributions were compared with the experimentally observed strain distributions at the average total strains of 0.0079, 0.0138, and 0.0255, as shown in Fig. 14. The evolution of strain distribution with the average strain is well represented by the calculations. However, it is noteworthy to examine the strain evolution in elements with different yield strengths. As a group of elements is defined by their yield stress, three yield stresses of 300, 668, and 1608 MPa, were selected, and the calculated strain evolutions are plotted in Fig. 15. The local strain evolutions were almost linearly dependent on the average strain, as observed in the experiment. There were two clear differences between experiments and calculations when the strain distribution and the strain evolution are compared. To clarify these discrepancies, the strain distributions obtained at an average total strain of 0.0255 are plotted in Fig. 16. The smallest strain obtained from the calculation was for the elastic elements. The calculated smallest strain is larger than the smallest strain observed in the experiments, as indicated by arrow A in Fig. 16. This discrepancy may indicate restrictions on the plastic deformation, caused by the discrete distributions of shear planes on which the sliding of dislocations occurs. Excluding the regions with a limited number of active slip planes, the average strain may decrease. The Iso-work assumption treats all elements as a continuum and contributes to deformation depending on the strength. Therefore, this assumption may not be suitable for expressing the features. Another discrepancy arises in the higher-strain region. The maximum strain obtained in the calculation was the strain of the first-yield elements. The maximum strain obtained by the DIC was larger than the calculated maximum strain, as indicated by arrow B in Fig. 16. These points correspond to in-lath slip deformations. In-lath slip is reported to be a soft-oriented-packet components that do not mainly contribute to the work hardening of martensitic steels, as discussed by Harjo and his co-workers.^25,26) According to their analysis, the work hardening of martensitic steels is mainly controlled by the deformation of hard-oriented packet components that correspond to out-of-lath slip deformation. Because the model can express the work hardening of steels with as-quenched martensitic microstructures, the calculated strain distributions may correspond to elements that exert significant influence on the work-hardening behavior of martensitic steels, thereby expressing the strain evolution associated with the out-of-lath slip deformation. Detailed analyses of the slip deformation behavior of martensitic steels are required to completely understand the plastic deformation behavior of steels possessing as-quenched martensitic microstructures.

Fig. 14. Comparison between experimental and calculated strain distributions at average strains of 0.0079, 0.0138, and 0.0255.

Fig. 15. Calculated strain development of elements with 300, 668, and 1608 MPa of yield strength.

Fig. 16. Comparison between experimental and calculated strain distributions at an average strain of 0.0255.

6. Conclusions

The modified CCA model proposed by Sakaguchi and his co-workers¹⁰⁾ was applied to the stress–strain curves of the three martensitic steels. The model was confirmed to be applicable for expressing not only the stress–strain curves in simple uniaxial tensile deformation, but also those in successive forward and backward deformations in the simple shear test. The model was found to represent the strain distributions and their evolution during deformation.

However, it is necessary to consider the changes in the dislocation substructure that affect the average distance of dislocation movement. It is also important to analyze the details of slip deformations in martensitic steels to improve the modified CCA model.

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.

Nomenclature

f{σ}: Yield strength spectrum as a function of stress, σ (-)

F{σ}: Accumulated yield strength spectrum (-)

σ_min: Threshold stress below which all the elements remain elastic (MPa)

σ₀: Parameter in F{σ} (MPa)

n: Parameter in F{σ} (-)

${}^i{\epsilon }_X^m$: Strain owned by an element i at a straining step m. Subscript X is “t” for total strain, “e” for elastic strain, and “p” for plastic strain. (-)

${\mathrm{\Delta }}^i{\epsilon }_X^m$: Total (X = t), elastic (X = e), or plastic (X = p) strain increment owned by elements i at a straining step m (-)

ⁱσ^m: Stresses owned by an element i after a straining step m (MPa)

${\sigma }_t^m$: Macroscopic stress after a straining step m (MPa)

${\epsilon }_t^m$: Macroscopic total strain after a straining step m (-)

$\mathrm{\Delta }{\epsilon }_t^m$: Macroscopic total strain increment at a straining step m (-)

ⁱσ_L: Yield strength of element i (MPa)

Δⁱσ^m: Magnitude of strain-hardening of an element i at a straining step m after yielding (MPa)

M: Taylor factor (-)

b: Magnitude of the Burgers vector (m)

α: Constant equal to 0.5

μ: Shear modulus of ferrite (MPa)

Λ: Mean free distance of dislocation movement (m)

β: Parameter for the annihilation rate of dislocations (-)

Δσ: Magnitude of strain-hardening as a function of plastic strain ε (MPa)

Y: Young’s modulus of ferrite (MPa)

ⁱε_L: Strain at which the magnitude of strain-hardening reaches ⁱσ_L (-)

${}^i{\epsilon }_{p-sh}^m$: Magnitude of plastic strain at which the strain-hardening of an element i at a straining step m is calculated (-)

σ_friction: Friction stress (MPa)

ε_XX: Total strain along the tensile direction obtained by DIC analysis using Vic-2D software (-)

σ_L: Macroscopic yield strength (MPa)

d: Grain diameter of ferrite (m)

k: Parameter in the Hall-Petch relationship (MPa/m^−1/2)

$\overline{\sigma }$: Maximum principal stress in uniaxial tensile deformation (MPa)

σ_S: Experimentally obtained shear stress (MPa)

ε_S: Experimentally obtained shear strain (-)

$d{\overline{\epsilon }}_p$: Incremental strain along the direction of the principal stress (-)

dε_p−s: Shear strain increment (-)

κ_σ: Parameter to relate shear stress and uniaxial tensile stress (-)

κ_ε: Parameter to relate plastic shear strain and plastic uniaxial tensile strain (-)

${\sigma }_e^m$: Stress shared by the elastic elements after a strain step m (MPa)

$V_e^m\left\{{\sigma }_e^m\right\}$: Fraction of elastic region at a stress of ${\sigma }_e^m$ (-)

$V_i^m{\{}^i{\sigma }^m\}$: Fraction of element i at its sheared stress of ⁱσ^m (-)

ε_A, ε_B, ε_C: Arbitrary average macroscopic strains (-)

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