Microstructural Ductile Fracture Analysis of 1180-MPa Class Martensite-matrix Dual-phase Steel via in situ Tensile Test

Abstract

Martensite-matrix dual-phase (DP) steel is increasingly used for high-strength automobile parts owing to its excellent compatibility, ductility, and tensile strength. However, its higher fracture strain, reflected by the hole expansion ratio, hinders further adoption of this material. Therefore, in this study, we conducted a microscale investigation of the ductile fracture behavior of 1180-MPa class martensite-matrix DP steel to obtain a guideline for microstructural design and improve fracture strain. In situ tensile test was conducted simultaneously with scanning electron microscopy (SEM) and crystal plasticity finite element analysis (CP-FEA). The in situ tensile test results indicated that microcracks initiated at certain martensite packets, not propagating into other packets. The CP-FEA results revealed that the martensite crystal orientation caused this behavior to induce remarkable stress and strain localization at interfaces within the vicinity of ferrite islands, relaxing the stress and strain localization at distant martensite packets. Although the cracks observed around the ferrite–martensite interfaces were similar to those observed in conventional ferrite-matrix DP steels, such matrix-phase cracks have rarely been reported, except for those immediately before final fracture. Thus, the optimization of the ferrite island distribution to suppress the formation of stress and strain localization sites was identified as the key aspect of martensite-matrix DP steel microstructural design. This design can be achieved using a combination of data science and CP-FEA.

1. Introduction

Weight reduction is an essential requirement for reducing carbon dioxide emissions from automotive structural components. Weight reduction requires thinner components and at the same time, high crash safety performance must be ensured. High tensile steel sheets have been developed to achieve thinner components with high crash safety performance and are now widely used commercially.¹⁾ However, the press formability of high tensile steel sheets is worse than that of mild steel sheets because of several issues such as shape fixability,^2,3) damage on the die and workpiece surface,^4,5) and high press forces.^6,7) In particular, material fracture during press forming is a serious problem. The decrease in ductility associated with higher strength causes this material to fracture during press forming.^8,9,10)

Ferrite–martensite dual-phase steels (hereinafter referred to as “DP steels”) were developed to address the aforementioned issues. In DP steels, soft ferrite ensures ductility while hard martensite provides strength, resulting in high strength and ductility.^10,11) Owing to their excellent properties, DP steels are widely used as high tensile strength steels for automobiles. Although DP steels are highly ductile materials that elongate well, their hole expansion limits¹²⁾ are lower than those of single-phase steels.^8,13) This causes frequent cracking at the steel sheet edges during flange-up, which is known as stretch-flange forming. This is due to the low local plastic strain leading to fracture (hereinafter referred to as “fracture strain”).^14,15,16) The fracture strain was evaluated as the equivalent plastic strain calculated from the rate of cross-sectional reduction immediately before fracture in a tensile test.^15,16,17,18)

The mechanism behind the lower fracture strain of DP steels compared with that of single-phase steels has been analyzed by observing the behavior of microvoids formed during tensile deformation.^{19,20,21,22,23)} To summarize the results, microscopic cracks were generated in the DP steels and opened near the ferrite–martensite boundary or at the neck of the martensite owing to the deformation concentration, resulting in microvoid formation. Deformation concentration zones are formed on the ferrite surface that passes through the “island” martensite, and microvoids that initially grow in the tensile direction grow and connect along these deformation concentration zones. Eventually, a large crack forms, which separates the material. Thus, in DP steels, the deformation concentration in ferrite caused by martensite promotes fractures. Based on these findings, as a material design guideline to improve the hole expansion limit (and hence, fracture strain), regulation of the martensite shape^24,25) and tuning of the hardness difference between ferrite and martensite have been performed to suppress the deformation concentration within ferrite.^23,26,27) The most significant result of reducing the hardness difference is the improvement of the hole expansion limit of single-phase steels.²⁸⁾

However, these results are limited to DP steels with island martensite in the ferrite matrix (hereinafter referred to as F-DP steels). F-DP steels have a tensile strength of 590–780 MPa, and those with a tensile strength of 980 MPa or higher often exhibit a structure consisting of a martensite matrix with scattered islands of ferrite. It is unlikely that microscopic ductile fracture would occur in ultra-high strength DP steels with such a reversed ferrite–martensite relationship by a mechanism similar to that observed in F-DP steels. Therefore, it is currently unclear whether the hard martensite of martensite-matrix DP steels (hereinafter referred to as M-DP steels) causes a deformation concentration similar to that of the ferrite matrix of F-DP steels, or whether the ductile fracture of the necks in the ferrite forms microvoids. To obtain a guideline for material development with the aim of increasing the fracture strain, it is necessary to clarify the microscopic ductile fracture mechanism of M-DP steels.

With this in mind, the purpose of this study is to clarify the microscopic ductile fracture mechanism of M-DP steels. To achieve this, in situ tensile tests and scanning electron microscopy (SEM) were performed on an 1180-MPa M-DP steel, which was coarse-grained via heat treatment. The origin and development of microscopic ductile fractures in M-DP steels were analyzed by observing the crack formation behavior in the microstructure under tension. In this study, crystal plasticity finite element analysis (CP-FEA) was performed using a model based on the actual microstructural image of the tensile specimen surface to evaluate the relationship between the measured crystal orientation and ductile fracture. We also conducted an analysis assuming that all crystal grains were martensite and compared the results with the analysis of DP steel, thereby elucidating the influence of island ferrite on the deformation of the martensite matrix phase.

2. In situ Tensile Test

2.1. Material and Specimen

In this study, M-DP steel with coarsened crystal grains was used, as shown in the micrograph in Fig. 1. Grain coarsening of the specimen reduced the number of crystals, allowing ease of observation.²⁹⁾ This is because the number of crystals that can be modeled in the CP-FEA is constrained by computational time considerations. This aspect is further elaborated in the following section.

Fig. 1. Micrograph of the Nital-etched DP steel, in which the white ferrite islands are embedded in the martensite matrix. (Online version in color.)

The composition of the steel specimens is listed in Table 1. The commercially available JSC 1180YL steel (initial yield stress (0.2% proof stress): 852 MPa, tensile strength: 1200 MPa) was prepared through heat treatment in a nitrogen atmosphere at 1050°C for 2 h, followed by air cooling (grain-coarsening annealing) and then water cooling (two-phase annealing) after holding at 750°C for 30 min.

Table 1. Chemical compositions [mass%].

C	Si	Mn	P	S	Sol-Al	N
0.17	2.0	2.6	0.008	0.001	0.035	0.0028

Table 2 lists the mechanical properties of the steel specimen. The mechanical properties in Table 2 were measured using tensile specimens with 30-mm long, 10-mm wide, and 1.4-mm thick parallel sections, with a gauge length of 20 mm. The initial yield stress (0.2% proof stress) of the coarse-grained steel specimens was 682 MPa and the tensile strength was 1267 MPa. The initial yield stress of the coarse-grained specimens decreased and the tensile strength increased relative to that of the original material before coarsening.

Table 2. Mechanical properties.

Yield stress [MPa]	Tensile strength [MPa]	Uniform elongation [%]	Total elongation [%]
682	1267	9.4	12.6

A microtensile specimen,³⁰⁾ as shown in Fig. 2, were used as the tensile test specimen. This test specimen was designed with a centrally located notch (width: 0.2 mm) to ensure that the entire deformation region of the specimen was within the field of view during SEM observations. The stress triaxiality in tension was high at the center of the sheet thickness, which was the starting point for crack formation. For specimens with a cross-sectional aspect ratio close to one, such as square bars, the triaxiality in the center was even higher, and the material might separate with a crack in the center of the thickness without a crack on the surface. Therefore, the thickness was reduced to 0.11 mm to suppress the stress triaxiality at the center of the thickness and to allow cracks to form on the specimen surface.

Fig. 2. Micro specimen used for in-situ tensile testing. Dimensions are in mm.

In addition, as shown in Fig. 2, diamond grinding and colloidal silica polishing were performed on the test specimen surface to facilitate differentiation between ferrite and lath martensite during SEM observations.

2.2. Experimental Conditions

Tensile tests were performed with SEM using a compact tensile tester MT300 manufactured by Deben, as shown in Fig. 3, where a 2.2-mm wide section of the specimen (Fig. 2) was held in a holder, and tension was applied to the specimen by moving the holder to one side (the lower side of the image). The SEM image (backscattered electron image) of the microspecimen before the tensile test is shown in Fig. 4. As expected, the full width of the notch (deformation region) of the microtensile specimen was within the field of view, and the shapes of the martensite matrix and island ferrite grains were discernible. Two ferrite grains are shown in Fig. 4 and are given the IDs F1 and F2 for use in later discussion. In addition to the SEM images, electron backscatter diffraction (EBSD) measurements were performed before tensile tests. The inverse pole figure (IPF), augmented with the image quality (IQ) map, was used as a reference for modeling in the CP-FEA, as described in the following section.

Fig. 3. Micro-specimen set in the mini-tensile testing machine embedded on the SEM platform. (Online version in color.)

Fig. 4. Micro-specimen observation prior to tensile testing. (Online version in color.)

The tensile test was terminated when a crack was observed on the surface. This is because the EBSD measurements around the crack initiation zone were performed later. The holder movement was stopped thrice for imaging before the end of the test. As shown in Fig. 4, the elongation at the point of test termination was evaluated by considering the distance between foreign particles adhered to the test specimen as the gauge length (the gauge section contained almost no ferrite), resulting in a strain of 4.0%. Although the strain rate could not be controlled in this test, the tensile test time, excluding observations, only lasted a few minutes, and based on the 4.0% elongation, it can be assumed that the strain rate was under 10⁻³/s.

2.3. Results

Figure 5 shows the microstructure of the entire notch of the specimen after the tensile test. Compared to the initial microstructure in Fig. 4, the overall structure was elongated, and cracks can be observed in some regions as dark areas. Cracks occurred near the ferrite–martensite boundary and in the martensite matrix.

Fig. 5. Micro-specimen observation after 4.0% elongation measured using the gauge shown in Fig. 3. (Online version in color.)

Here, we will focus on explaining the two areas where crack initiation was observed. First, Fig. 6(a) shows the magnified image of a crack near the ferrite–martensite boundary in Area 1 of Fig. 5. The local elongation calculated from the width of the ferrite grains was 9.7%, as shown in Fig. 6(a-1). This value was more than twice the elongation obtained from the arbitrarily selected foreign particle distance, as shown in Fig. 4, indicating a high degree of deformation inhomogeneity due to the mixture of ferrite and martensite. In contrast to the predeformation state shown in Fig. 6(a-1), cracks appeared at the boundary of martensite at the convexities of the ferrite grains protruding from the top and bottom after deformation, as shown in Fig. 6(a-2). Although generally located at the interface along the boundary, a crack can be assumed to be on the ferrite because the crack edge lies inside the ferrite, thus deviating from the interface. All cracks occurred at the boundaries nearly perpendicular to the tensile axis. A small crack extending from the ferrite–martensite boundary to the martensite was also observed in the lower part.

Fig. 6. Magnified views before and after tensile testing. Areas 1 and 2 are identified in Fig. 5. (Online version in color.)

The cracks in the martensite matrix in Area 2 are discussed below. Figure 6(b) shows an enlarged image of the martensite matrix crack in Area 2 before and after deformation. The elongation was estimated from the distance between two points in the martensite grain (Fig. 6(b-1)), which was about 9.5%. This magnitude of elongation was also larger than that estimated from the distance between the foreign particles in Fig. 4 and was equivalent to the deformation of the ferrite grains in Fig. 5. This indicates that the deformation within specific martensite grains was larger than average, and its magnitude was equivalent to the deformation observed in the soft ferrite grains. Here, owing to coarse annealing, it was difficult to distinguish the packet or block boundaries of the martensite in the tested steel. Thus, the term “martensite grain” is used for convenience. Figure 6(b-2) shows that multiple cracks were generated perpendicular to the tensile axis. The cracks were concentrated within a single grain, with a few propagating to other adjacent martensite grains.

3. Image-based Crystal Plasticity Finite Element Method

3.1. Settings of the CP-FEA

CP-FEA was performed to interpret the results of Section 2 from a stress–strain perspective. In the analysis, a portion of the notch area was extracted to reduce the computational time. Figure 7 shows the location of the analysis area as well as the IPF and IQ images measured by EBSD. The analysis area was slightly less than half of the notch width from the edge. The total number of martensite and ferrite grains in the IPF/IQ map in Fig. 7 was 45, two of which were ferrite grains. Based on this IPF/IQ map, a finite element model (total number of elements: 24888) was developed, as shown in Fig. 8.

Fig. 7. IPF/IQ region map for crystal plasticity analysis. (Online version in color.)

Fig. 8. Finite element model with three-layer hexagonal meshes in the direction of material thickness. (Online version in color.)

The purpose of this analysis was to determine the degree to which the presence of ferrite grains enhanced or mitigated the deformation concentration caused by different crystal orientations between the martensite grains. Therefore, the model was simplified as follows to achieve a balance between the purpose of the analysis and computational time. First, the martensite grains were treated as crystals with a single orientation. The martensite grains in Fig. 7 exhibited a banded structure composed of lath or block regions, and some cementite appeared to be present. However, for the purpose of CP-FEA, these features were disregarded for simplification, as shown in Fig. 8. The presence of these microstructures was represented in the model as the high deformation resistance of martensite. As shown in Fig. 4, F1 and F2 in Fig. 8 were the ferrite grains identified in the SEM images. M represents the cracked martensite grains in Area 2. An 8-node bilinear solid element with selective reduced integration was used for discretization. Based on a previous study,³¹⁾ three layers were created along the thickness direction. It was difficult to measure the differences in crystal orientation in the thickness direction. Therefore, in this study, each grain was assumed to be a columnar grain penetrating in the thickness direction, and the same crystal orientations as those of the surface layer were assigned to all three layers in the thickness direction. A previous study reported that even if a columnar grain material with grains penetrating in the thickness direction is used as a specimen, it is difficult to quantitatively predict its strain distribution by CP-FEA.^e.g.32) Therefore, in this analysis, we will not quantitatively evaluate the results of the crystal plasticity analysis; rather, we will focus on qualitative aspects of the analysis.

Within each grain, the initial crystal orientation was assumed to be constant, and each element was assigned the average crystal orientation of the martensite grains obtained from the IPF/IQ map. We conducted two types of analysis to distinguish the deformation partitioning behavior based on the orientation of each martensite grain and the hard/soft regions: one using a single-phase assumption model, where the virtual deformation resistance was assigned to F1 and F2 representing martensite, and the other using a DP assumption model where the deformation resistance of ferrite was assigned to F1 and F2, similar to the actual case. The symmetric surface boundary conditions are given on the left and top surfaces of Fig. 8, and a tensile displacement of 2.1% of the entire model was applied to the right surface at a strain rate (quasi-static) of 10⁻⁵/s. Here, the deformation of 2.1% was small compared to the measured elongation of 4.0% explained in Section 2.2. This is because the computational time for the 2.1% deformation exceeded two weeks, given the current computing environment (Kyoto University supercomputer system/Xeon Broadwell 2.1 GHz, 24 cores). To resolve this issue, it is necessary to improve the computational method using data science, as described in the following section. However, this issue needs to be addressed in future studies.

The CP-FEA program developed by Hama et al.^{e.g.,33,34,35)} was used for the analysis. As the details of the formulation have been described in detail in previous papers, only a summary is provided here.

The {110} <111> and {112} <111> slip systems were considered and the activity of each slip system was assumed to follow Schmid’s law. Body-centered cubic metals are known to have large elastic anisotropy at the crystal level. However, since it was difficult to identify the anisotropic parameters of this material, the material was assumed to be isotropic for simplicity in this study, with a Young’s modulus of 210 GPa and Poisson’s ratio of 0.3. As a constitutive law, the following strain rate-dependent exponential law³⁶⁾ was applied to the slip rate ${\dot{\gamma }}^{\alpha }$ of the slip system α:

  

$$\frac{{\dot{\gamma }}^{\alpha }}{{\dot{\gamma }}_0}={\left|\frac{{\tau }^{\alpha }}{{\tau }_Y^{\alpha }}\right|}^{\frac{1}{m}}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\tau }^{\alpha })$$(1)

where ${\dot{\gamma }}_0$ is the reference slip rate, m is the strain rate sensitivity exponent, and τ^α is the resolved shear stress. ${\dot{\gamma }}_0$ and m are the material parameters, set at 0.001 and 0.02/s, respectively. τ^α is given by the following equation:

  

$${\tau }^{\alpha }=\sigma :({\mathit{s}}^{\alpha }\otimes {\mathit{m}}^{\alpha })$$(2)

where σ is the Cauchy stress tensor, and s^α and m^α are unit vectors representing the slip direction and the normal to the slip plane of the slip system α, respectively. The evolution of ${\tau }_Y^{\alpha }$ is given by the following extended Voce hardening law:³⁷⁾

  

$${\dot{\tau }}_Y^{\alpha }=\sum _{\beta }{h}_{\alpha \beta }\left|{\dot{\gamma }}^{\beta }\right|,\mathrm{\hspace{0.17em} }{h}_{\alpha \beta }=q_{\alpha \beta }h(\overline{\gamma })$$(3)

  

$$h(\overline{\gamma })={\theta }_1+\left({\theta }_0-{\theta }_1+\frac{{\theta }_0{\theta }_1}{{\tau }_1}\overline{\gamma }\right)exp\left(-\frac{{\theta }_0\overline{\gamma }}{{\tau }_1}\right)$$(4)

where q_αβ is the latent hardening matrix and τ₁, θ₁, and θ₀ are material parameters related to hardening. $\overline{\gamma }$ is the cumulative slip and is expressed as follows:

  

$$\overline{\gamma }=\sum _{\alpha }{\int }_0^t\left|{\dot{\gamma }}^{\alpha }\right|$$(5)

where t denotes time. In DP steels, it is challenging to classify and evaluate the properties of each phase for implementation in CP-FEA because the evaluation method is not well established.^e.g.,38,39) Therefore, in this study, we used fitted parameters to replicate the macroscopic stress–strain relationship in a virtual dual-phase microstructure voxel model (representative volume element), including a ferrite volume fraction identical to that of actual DP steel, based on the characteristics observed in ferrite and martensite single-phase materials.⁴⁰⁾ The material parameters for ferrite and martensite are presented in Table 3. The improvement in the accuracy of characterization for each phase is an issue to be addressed in the future. For simplicity, the components of the latent-hardening matrix were set to unity. The same parameters were assigned for the {110} <111> and {112} <111> slip systems.

Table 3. Material parameters for extended Voce law.

	τ0	τ1	θ0	θ1
Martensite	310.0	400.0	30000.0	600.0
Ferrite	160.0	200.0	190.0	0.1

3.2. Results of CP-FEA

Figure 9 shows the results of the CP-FEA. Figure 9(a) shows a contour plot of the equivalent plastic strain calculated from the plastic strain increment tensor at each step. In the results of the single-phase model shown in Fig. 9(a-1), plastic strain localization appeared, presumably because of the difference in crystal orientations. The plastic deformation was pronounced in some martensite grains, including martensite grain M, where cracks were generated. Plastic deformation remained the highest at the notch located at the lower end of the specimen. In the DP model shown in Fig. 9(a-2), the degree of concentration of plastic deformation was more pronounced than that in the single-phase model. The plastic strains were large, mainly in the ferrite grains of F1 and F2, and the deformation propagated to the surrounding martensite grains. There were fewer martensite grains with a higher plastic strain than those in the single-phase model. Among them, the martensite grain M resulted in a more concentrated plastic deformation in the DP model.

Fig. 9. Results of crystal plasticity FE analysis. (Online version in color.)

Figure 9(b) shows a contour plot of the hydrostatic stress. The higher the hydrostatic stress on the compressive side (negative value), the higher the fracture strain. Therefore, by evaluating it together with the plastic strain, we can estimate the possibility of ductile fractures in a certain region.⁴¹⁾ The hydrostatic stress in Fig. 9(b-1) under the single phase is clearly nonuniform in distribution and is influenced by the difference in crystal orientation of the martensite grains or the nonuniform distribution of the grain shapes. Regions of high hydrostatic stress occurred in certain grains, but the martensite grain M, which caused the crack formation, was not under high hydrostatic stress compared with the surrounding grains. Some regions of high hydrostatic stress were maintained in the DP model, as shown in Fig. 9(b-2). However, under the DP assumption, the ferrite grains F1 and F2 remained under more compressive hydrostatic stress than other grains, and the surrounding martensite grains were also under compressive hydrostatic stress. The grains to the left of F1 induced greater compressive hydrostatic stress than the ferrite grains. In addition, the surrounding martensite grains experienced tensile hydrostatic stress to maintain balance. This is shown in the upper-right corner of Fig. 9(b-2).

The results of the above analysis were compared with the EBSD measurements after 4.0% deformation, as shown in Fig. 10. The IPF/IQ image in Fig. 10(a) shows that the upper part of martensite grain M exhibited crystal rotation, and the grain reference orientation deviation (GROD) value, as shown in Fig. 10(b), was also high in this area. Here, GROD is a value representing the deviation from the average crystal orientation in the grains and is used as an index of plastic strain.⁴²⁾ Martensite grains adjacent to the right of the martensite grain M had a higher GROD, and this trend was similar to the plastic strain distribution in the DP model in Fig. 9(b-2). Figure 10(a) shows that ferrite grain F2 had a large elongation in the tensile direction due to deformation and was flattened compared with the initial microstructure shown in Fig. 7. The degree of elongation of the microstructural shape showed that the notch edges, including F2, were generally highly deformed. This can be confirmed by the GROD value in Fig. 10(b). The plastic deformation region near the notch edge was larger as it extended to immediately below the martensite grain M, and this trend was closer to the plastic strain distribution under the DP assumption in Fig. 9(a-2) than under the single-phase assumption in Fig. 9(a-1).

Fig. 10. EBSD maps after tensile deformation. (Online version in color.)

4. Discussion

4.1. Observations of Crack Formation under Tension

In Section 2, the M-DP steel specimen exhibited distinctly different crack formation behavior compared with conventional F-DP steels. In the early and middle stages of deformation, F-DP steels did not exhibit cracks in the matrix.^19,21,23) Crack formation in the martensite matrix, as shown in Area 2 (Fig. 5), is unique to the M-DP steel. Cracks near the ferrite–martensite interface, as shown in Area 1, were similarly observed in F-DP steels. However, the cracks near the interface of the F-DP steels did not propagate to the island martensite.^19,20) The M-DP steel exhibited crack propagation behavior both in the martensite matrix and island ferrite, which was also different from that of F-DP steels. Furthermore, in F-DP steels, cracks propagated in a stitch-like manner between the island-like martensite regions immediately before fracture.²²⁾ Until then, the cracks opened only along the tensile axis, and no behavior such as crack propagation outside the boundary was observed.^20,23) Similar results were observed for other specimens made of the same steel, although these are omitted owing to space constraints.

From the above results, the ductile fracture of the matrix was considered to play a greater role in the final fracture in the M-DP steel than in F-DP steels. Naturally, the final fracture occurred when the matrix failed in both DP steels. However, in the M-DP steel, the matrix cracked at a relatively early stage. In F-DP steels, crack formation is caused by deformation concentration owing to the difference in hardness within vicinity of martensite. However, in the M-DP steel, the martensite matrix not adjacent to the island ferrite breaks down, as shown in Area 2. Compared with F-DP steels, the crystal orientation of martensite is considered to have a stronger effect on microscopic crack formation. As shown in Fig. 6(b), the cracks in Area 2 were concentrated in the specific martensite grain M with negligible crack propagation into the adjacent martensite grains. In addition, the cracks observed in Area 1 in Fig. 6(a) propagated into the martensitic matrix, but most remained in the ferrite grains. Based on this observation and the fact that large cracks occur in regions not adjacent to ferrite grains, such as the martensite grain M, the effect of microscopic cracks originating from ferrite grains on the macroscopic fracture strain is small. Therefore, the effect of enhancing the macroscopic fracture strain obtained by controlling the hardness ratio between the two phases and the topology of the island-like ferrite is considered small in the M-DP steel. Instead, for the M-DP steel, controlling the crystal orientation of the martensite matrix phase and improving the microscopic fracture strain is important.

4.2. CP-FEA

Figure 11 shows a cross-sectional image of the CP-FEA region after deformation. Figure 11 shows that the metallographic structure observed on the surface layer occupied only a limited area of the surface layer and was not sufficiently large to penetrate the thickness direction. For example, the ferrite grain F1 had a thickness of 20 μm or less in the sheet thickness direction. Therefore, it is not appropriate to discuss the consistency between the experimental values and analysis that simulates only the surface crystalline structure in Section 3. Meanwhile, because the analytical results of the DP assumption are closer to the plastic strain distribution obtained from the EBSD than the single-phase model, as described in Section 3.2, the qualitative effect of ferrite on the plastic strain distribution is considered captured. In addition, the plastic deformation near the notch was smaller than the actual deformation because the notch in the actual specimen was only partially included in the analysis domain. Therefore, in the following discussion, the role of ferrite is analyzed qualitatively using CP-FEA, and the observed crack-formation behavior is discussed.

Fig. 11. Cross section of micro-specimen (after deformation). (Online version in color.)

As shown in Section 3.1, the fraction of ferrite in the FE model is small (two in 45 grains). However, the results in Fig. 7 clearly show a marked difference in the trends of plastic strain and hydrostatic stress between the single-phase and DP models. Interestingly, the martensite adjacent to ferrite induces compressive hydrostatic stress. It is well known that fracture is suppressed under compressive hydrostatic stress. Although its position relative to the tensile axis must be considered, ferrite plays a role in mitigating the deformation concentration of martensite grains by assuming plastic deformation.

Furthermore, the crystal orientation of the martensite matrix affected the deformation concentration in the analysis region to a small extent, as seen in the results of the analysis based on single-phase assumption (Fig. 9(a-1)). This is due to the deformation localization at the notch tip and the resulting overall heterogeneous deformation, as well as the deformation partitioning based on the Schmid factors associated with the crystal orientations of each martensite grain. To obtain evidence for this, the orientation difference between the martensite grains surrounding martensite grain M is shown. Figure 12 shows the equivalent plastic strain distribution around the martensite grain M extracted from Fig. 9(a-1) and the IDs of the surrounding grains. The orientation difference $\mathrm{\Delta }\tilde{\theta }$ between the crystal grains labeled with these IDs and the martensite grain M is shown in Table 4. The grains in contact with martensite grain M were grain IDs 1, 2, and 3 on the left side, and grain IDs 4, 5, and 6 on the right side. $\mathrm{\Delta }\tilde{\theta }$ of these grains on the left and right sides are generally large tilt angles exceeding 15°, except grain ID 2. It can be inferred that the deformation was concentrated in the martensite grain M because it was sandwiched between grains with different orientations. The orientation difference with the left-neighbor grain ID 2 is small (5.2°), but it is not considered to have a significant effect on the concentration of deformation in the martensite grain M owing to the small contact area.

Fig. 12. Grain IDs around the martensite grain M. (Online version in color.)

Table 4. Misorientation $\mathrm{\Delta }\tilde{\theta }$, based on the crystal orientation of the martensite grain M. The grain IDs are shown in Fig. 12.

Grain ID	1	2	3	4	5	6
$\mathrm{\Delta }\tilde{\theta }$ [deg]	44	5.2	35	23	25	42

Note that $\mathrm{\Delta }\tilde{\theta }$ is obtained from the trace of rotation tensor R determined from symmetry, considering the Euler angles ϕ, θ, and ψ. Assuming that the rotation tensors corresponding to the Euler angles of the martensite grain M and the surrounding grains are R_M and R_s, respectively, then the rotation tensor corresponding to the orientation difference between the two grains is ${\mathit{R}}_s{\mathit{R}}_M^T$. From this rotation tensor and the Rodriguez rotation formula, the azimuthal difference $\mathrm{\Delta }\tilde{\theta }$ is obtained as follows:

  

$$cos(\mathrm{\Delta }\tilde{\theta })=\frac{1}{2}\left(trace({\mathit{R}}_s{\mathit{R}}_M^T)-1\right)$$(6)

  

$$\mathrm{\Delta }\tilde{\theta }=arccos\left(\frac{1}{2}\left(trace({\mathit{R}}_s{\mathit{R}}_M^T)-1\right)\right)$$(7)

where the rotation tensor R is expressed using ϕ, θ, and ψ as follows.

  

$$\begin{array}{l}R=\\ \left[\begin{array}{ccc}cos(\varphi )cos(\psi )-sin(\varphi )cos(\theta )sin(\psi )& -cos(\varphi )sin(\psi )-sin(\varphi )cos(\theta )cos(\psi )& sin(\varphi )sin(\theta )\\ sin(\varphi )cos(\psi )+cos(\varphi )cos(\theta )sin(\psi )& -sin(\varphi )sin(\psi )+cos(\varphi )cos(\theta )cos(\psi )& -cos(\varphi )sin(\theta )\\ sin(\theta )sin(\psi )& sin(\theta )cos(\psi )& cos(\theta )\end{array}\right]\end{array}$$(8)

The difference in grain shape likely promoted the deformation concentration, but the CP-FEA was unable to isolate this effect. However, because grains with various morphologies were mixed, as shown in Figs. 7 and 8, it is presumed that the grain shape has some influence. For example, an intricate shape, such as the boundary between martensite grains M and grain IDs 4 and 5 (circled area in Fig. 12) can be considered to induce a stronger deformation concentration than a flat shape, such as the boundary with grain ID 3. In fact, the equivalent plastic strain at the boundary between grain ID 3 and the martensite grain M was smaller than that at the boundary between grain IDs 4 and 5.

Under the DP assumption, the addition of ferrite significantly changed the degree of deformation concentration. Figure 9(a-2) shows that the plastic deformation of the martensite matrix near the ferrite became larger compared with that for the single-phase case. However, the deformation concentration of the far martensite matrix was mitigated. In fact, focusing on the cracked martensite grain M, the plastic strain in the lower half of the grain was smaller in the DP model than in the single-phase model. In F-DP steels, inhomogeneous deformation due to the difference in deformation resistance between ferrite and martensite induced a deformation concentration zone in the matrix^22,25) and microcracks along the deformation concentration zone.²²⁾ Although the same phenomenon was expected to occur in the M-DP steel, interestingly, the amount of plastic strain was reduced in martensite grain M with microcracks, suggesting that the deformation concentration zone may be adversely suppressed by the presence of ferrite. The presence of a different phase at the crack initiation site may counteract the deformation concentration caused by other factors such as the crystal orientation (or grain shape) of the matrix, thus delaying fracture.

The results of the above analysis reaffirm the observation that the effect of the ductile fracture of the martensitic matrix during deformation on the macroscopic fracture strain was greater in the M-DP steel than in F-DP steels. The distribution of plastic strain and stress due to the crystal orientation and grain shape may determine which martensite grains cause fracture. However, the results of the analysis suggest that the degree of deformation concentration can be mitigated by ferrite. These findings can be used as a material design guideline for M-DP steels.

The design (optimization) of the ferrite phase distribution to mitigate the martensite deformation concentration, as described above, cannot be achieved without CP-FEA. However, it is impractical to apply this analysis to the design of commercially available M-DP steels with fine microstructures, considering the time frame of two weeks for a single case in this study. We believe that the use of data science is effective in this case; for example, the use of a machine learning framework for the FEA results, such as surrogate analysis, and the analysis of large microstructural deformations of M-DP steels from the analysis results of small grains and small deformations are promising.

In particular, image-based analysis, such as that used in this study, can only analyze a minimal area of the surface layer. Although synchronization between image-based analysis and experiments can be very effective in identifying mechanical parameters, there are doubts regarding the accuracy of identification because of problems in the analysis domain. In performing image-based analysis for such purposes, the analytical logic of inferring the behavior of the entire system from limited regions becomes crucial, even more so than the mechanical model itself or computational speed improvement techniques. This issue should be addressed in future studies.

5. Conclusions

In this study, the microscopic mechanism of ductile fracture was clarified by in situ tensile tests on M-DP steel with a martensite matrix. To analyze the observed crack initiation behavior from the perspective of stress concentration, simplified CP-FEA was conducted based on the observed images. The key findings of this study are as follows:

(1) During tensile deformation in the M-DP steel, microcracks occurred in the martensite matrix and ferrite at the ferrite–martensite boundary. This behavior is distinct from that of F-DP steels with a ferrite matrix phase. Cracks in the matrix were rarely observed in the F–DP steels.

(2) In the M-DP steel, the microcracks in the martensite matrix were confined to single martensite grains. This observation suggests that the deformation concentration caused by the crystal orientation of the martensite grains has a significant effect on crack formation.

(3) The results of the CP-FEA confirmed that the deformation concentration occurred depending on the grain shape, geometry of the specimen, and crystal orientation of each martensite grain. Comparing the analytical results with those of the single-phase and DP models, the ferrite reduced the deformation concentration of the far martensite. This finding is useful as a material design guideline for M-DP steels.

(4) Based on the above guidelines, optimization of material microstructures using CP-FEA is effective; however, it requires a method to estimate the overall deformation behavior of a few crystal grains. Considering the limited computational time and observation area, it is necessary to develop a method that combines data science and CP-FEA, such as a surrogate analysis.

Acknowledgments

This work was supported by a Grant-in-Aid for Scientific Research (KAKENHI) (JP 20H02484) from the Japan Society for the Promotion of Science.

References

• 1)   S.  Keeler and  P.  Ulintz: Met. Form., 45 (2010), 24.
• 2)   K.  Mori,  K.  Akita and  Y.  Abe: Int. J. Mach. Tools Manuf., 47 (2007), 321. https://doi.org/10.1016/j.ijmachtools.2006.03.013
• 3)   T.  Yoshida,  K.  Sato,  E.  Isogai and  K.  Hashimoto, K.: Nippon Steel Tech. Rep., 103 (2013), 4. https://www.nipponsteel.com/en/tech/report/nsc/pdf/103-02.pdf, (accessed 2022-09-30).
• 4)   M.  Aramaki,  N.  Yamada and  O.  Furukimi: ISIJ Int., 51 (2011), 1137. https://doi.org/10.2355/isijinternational.51.1137
• 5)   T.  Matsuno,  Y.  Ochiai,  Y.  Okitsu,  M.  Iga,  A.  Kohri and  T.  Mikami: Int. J. Adv. Manuf. Technol., 116 (2021), 2873. https://doi.org/10.1007/s00170-021-07675-2
• 6)   K.  Mori,  T.  Maeno and  Y.  Maruo: CIRP Ann. - Manuf. Technol., 61 (2012), 255. https://doi.org/10.1016/j.cirp.2012.03.124
• 7)   H.  So,  D.  Faßmann,  H.  Hoffmann,  R.  Golle and  M.  Schaper: J. Mater. Process. Technol., 212 (2012), 437. https://doi.org/10.1016/j.jmatprotec.2011.10.006
• 8)   M.  Takahashi: Nippon Steel Tech. Rep., 88 (2003), 2. https://www.nipponsteel.com/en/tech/report/nsc/pdf/n8802.pdf, (accessed 2022-04-18).
• 9)   J.-B.  Yan,  J. Y. R.  Liew, M.-H. Zhang H. and J. -Y. Wang: Mater. Des., 61 (2014), 150. https://doi.org/10.1016/j.matdes.2014.04.057
• 10)   A.  Sakai,  S.  Matsumoto,  M.  Yano,  S.  Tamaru,  N.  Arai,  S.  Yoshizaki,  Y.  Shoji and  A.  Suenaga: Mazda, Tech. Rev., 36 (2019), 192 (in Japanese). https://doi.org/10.34338/mazdagihou.36.0_192
• 11)   Y. L.  Su and  J.  Gurland: Mater. Sci. Eng., 95 (1987), 151. https://doi.org/10.1016/0025-5416(87)90507-6
• 12)  ISO 16630: 2017, Metallic materials −Sheet and strip− Hole expanding test.
• 13)   A.  Konieczny and  T.  Henderson: SAE Trans., 116 (2007), 20. https://doi.org/10.4271/2007-01-0340
• 14)   T.  Matsuno: Sokeizai, 53 (2012), 2 (in Japanese).
• 15)   Y.  Nakagawa,  T.  Matsuno,  T.  Hama,  T.  Naito,  Y.  Okitsu,  S.  Hayashi and  K.  Takada: Proc. 71st Jpn. Joint Conf. Technology of Plasticity, The Japan Society for Technology of Plasticity, Tokyo, (2020), 355 (in Japanese).
• 16)   B. M.  Hance and  T. M.  Link: IOP Conf. Ser. −Mater. Sci. Eng., 651 (2019), 012061. https://doi.org/10.1088/1757-899X/651/1/012061
• 17)   T.  Matsuno,  T.  Hojo,  I.  Watanabe,  A.  Shiro,  T.  Shobu and  K.  Kajiwara: Sci. Technol. Adv. Mater.−Methods, 1 (2021), 56. https://doi.org/10.1080/27660400.2021.1922207
• 18)   T.  Matsuno,  T.  Matsuda,  H.  Shoji and  M.  Ohata: Metals, 12 (2022), 776. https://doi.org/10.3390/met12050776
• 19)   T.  Matsuno,  D.  Maeda,  H.  Shutoh,  A.  Uenishi and  M.  Suehiro: ISIJ Int., 54 (2014), 938. https://doi.org/10.2355/isijinternational.54.938
• 20)   G.  Avramovic-Cingara,  Y.  Ososkov,  M. K.  Jain and  D. S.  Wilkinson: Mater. Sci. Eng. A, 516 (2009), 7. https://doi.org/10.1016/j.msea.2009.03.055
• 21)   M.  Kikuzuki,  K.  Hayashi,  Y.  Tsunemi,  S.  Yabu and  T.  Matsuno: Tetsu-to-Hagané, 106 (2020), 244 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2019-111
• 22)   J.  Kadkhodapour,  A.  Butz and  S. Z.  Rad: Acta Mater., 59 (2011), 2575. https://doi.org/10.1016/j.actamat.2010.12.039
• 23)   M.  Azuma,  S.  Goutianos,  N.  Hansen,  G.  Winther and  X.  Huang: Mater. Sci. Technol., 28 (2012), 1092. https://doi.org/10.1179/1743284712Y.0000000006
• 24)   S.  Sun and  M.  Pugh: Mater. Sci. Eng. A, 335 (2002), 298. https://doi.org/10.1016/S0921-5093(01)01942-6
• 25)   T.  Matsuno,  T.  Yoshioka,  I.  Watanabe and  L.  Alves: Int. J. Mech. Sci., 163 (2019), 105133. https://doi.org/10.1016/j.ijmecsci.2019.105133
• 26)   K.  Takashima,  K.  Hasegawa,  Y.  Toji and  Y.  Funakawa: ISIJ Int., 57 (2017), 1289. https://doi.org/10.2355/isijinternational.ISIJINT-2017-086
• 27)   X.  Fang,  Z.  Fan,  B.  Ralph,  P.  Evans and  R.  Underhill: J. Mater. Process. Technol., 132 (2003), 215. https://doi.org/10.1016/S0924-0136(02)00923-8
• 28)   S.  Park,  J.  Jung,  K. I.  Kim,  H.  Kim,  S.-I.  Kim,  K. H.  Oh,  M.-G.  Lee and  H. N.  Han: Int. J. Mech. Sci., 156 (2019), 297. https://doi.org/10.1016/j.ijmecsci.2019.04.010
• 29)   T.  Hama,  T.  Nishi,  M.  Oka,  T.  Matsuno,  Y.  Okitsu,  S.  Hayashi,  K.  Takada and  H.  Takuda: ISIJ Int., 61 (2021), 1971. https://doi.org/10.2355/isijinternational.ISIJINT-2020-712
• 30)   T.  Matsuno,  C.  Teodosiu,  D.  Maeda and  A.  Uenishi: Int. J. Plast., 74 (2015), 17. https://doi.org/10.1016/j.ijplas.2015.06.004
• 31)   P.  Baudoin,  T.  Hama and  H.  Takuda: Int. J. Plast., 115 (2019), 111. https://doi.org/10.1016/j.ijplas.2018.11.013
• 32)   E.  Demir and  I.  Gutierrez-Urrutia: Int. J. Plast., 136 (2021), 102898. https://doi.org/10.1016/j.ijplas.2020.102898
• 33)   T.  Hama and  H.  Takuda: Int. J. Plast., 27 (2011), 1072. https://doi.org/10.1016/j.ijplas.2010.11.004
• 34)   T.  Hama,  K.  Kojima,  M.  Kubo,  H.  Fujimoto and  H.  Takuda: ISIJ Int., 57 (2017), 866. https://doi.org/10.2355/isijinternational.ISIJINT-2017-011
• 35)   T.  Hama: ISIJ Int., 60 (2020), 1849. https://doi.org/10.2355/isijinternational.ISIJINT-2020-011
• 36)   R. J.  Asaro and  A.  Needleman: Acta Metall., 33 (1985), 923. https://doi.org/10.1016/0001-6160(85)90188-9
• 37)   C.  Tome,  G. R.  Canova,  U. F.  Kocks,  N.  Christodoulou and  J. J.  Jonas: Acta Metall., 32 (1984), 1637. https://doi.org/10.1016/0001-6160(84)90222-0
• 38)   W.  Liu,  J.  Lian,  N.  Aravas and  S.  Münstermann: Int. J. Plast., 126 (2020), 102614. https://doi.org/10.1016/j.ijplas.2019.10.002
• 39)   Y.  Tamura,  T.  Kanno,  T.  Matsuno,  T.  Naito,  Y.  Okitsu and  T.  Hama: CAMP-ISIJ, 35 (2022), 581, CD-ROM (in Japanese).
• 40)   Y.  Tamura,  K.  Hatakeyama,  T.  Matsuno,  Y.  Okitsu,  K.  Takada,  S.  Hayashi and  T.  Hama: Proc. 72st Jpn. Joint Conf. Technology of Plasticity, The Japan Society for Technology of Plasticity, Tokyo, (2021), 145 (in Japanese).
• 41)   J. J.  Lewandowski and  P.  Lowhaphandu: Int. Mater. Rev., 43 (1998), 145. https://doi.org/10.1179/imr.1998.43.4.145
• 42)   M. N.  Gussev and  K. J.  Leonard: J. Nucl. Mater., 517 (2019), 45. https://doi.org/10.1016/j.jnucmat.2019.01.034