Characterization of Evolution of Microscopic Stress and Strain in High-Manganese Twinning-Induced Plasticity Steel

Abstract

Electron backscatter diffraction was used to observe the microstructure of an austenitic high-manganese twinning-induced plasticity steel and investigate the crystal orientation of grains in this steel. The results showed that mechanical twins are formed in a grain with a high Schmid factor during the tensile test. The orientation data obtained were used to estimate the anisotropic elasticity of the grains in the steel. The microscopic stress and strain evolved in the microstructure of the steel unloaded after plastic deformation were estimated using finite element method simulation in which the elastic anisotropy of the steel was taken into account. The simulation indicated that the evolution of microscopic stress and strain in the microstructure is considerably influenced by the crystal orientation of the grains. Furthermore, white X-ray diffraction with microbeam synchrotron radiation was used to characterize the evolution of microscopic stress and strain in the grains of the steel. The stress analysis during white X-ray diffraction indicated the formation of residual microscopic stress after tensile deformation, which was found to be distributed heterogeneously in the steel. It was also shown that the direction of the maximum principal stresses at different points in the microstructure under loading were mostly oriented along the tensile direction. These results are fairly consistent with the results obtained by the simulation, although absolute values of the real principal stresses may be influenced by the heterogeneously evolved strain and the several assumptions used in the simulation.

1. Introduction

It is well known that austenitic steels containing more than about 20 mass% of manganese exhibit characteristic mechanical properties, such as high elongation and ductility at room temperature.^1,2,3,4) As the mechanical properties of these steels are often related with the formation of mechanical twins by deformation, these steels are called twinning-induced plasticity (TWIP) steels. The mechanical properties,^{5,6,7,8,9,10,11,12,13,14,15)} deformation mechanisms associated with stacking fault energies (SFE),^{16,17,18,19,20,21,22,23)} the thermodynamic properties,^24,25) and the microstructure^{26,27,28,29,30,31)} of the TWIP steels have been extensively investigated. Since a stacking fault may act as an embryo of a twin, the SFE or the chemical composition of the alloying elements, such as manganese, carbon, silicon, and aluminum, is an important factor affecting the twinning processes caused by plastic deformation. The mechanical properties and possible mechanisms of twinning have been previously discussed in a review paper.³²⁾

Among previous studies, special focus has been laid on the microstructure, which induces mechanical twinning in TWIP steels; however, a number of these studies have been performed by using electron microscopy.^{26,27,28,29,30)} According to these studies, the mechanical properties of the TWIP steels are associated with grain refinement and texture formation by twinning, and the microstructure or twinning is correlated with SFE. However, since the steels are composed of polycrystalline grains, in discussing the mechanical properties of the TWIP steels, it is important to consider the conditions under which twinning in the microstructure occurs. Therefore, the effect of SFE on the microstructure, specifically the crystallite size, during the deformation of TWIP steels has been also studied by analyzing line broadening in X-ray diffraction patterns.³¹⁾ These results showed that the stacking fault probability is correlated with the crystallite sizes of different TWIP steels.

Nevertheless, most studies indicate that in addition to mechanical twin formation, the plastic deformation of the TWIP steels is also influenced by slip deformation due to dislocations. Figures 1(a) and 1(b) illustrate the ideal twin formation in a face centered cubic (fcc) structure as in the case of austenitic steels. The stress required for the twin formation is dependent on the orientation of the crystal with respect to the stress direction. Typically, mechanical twins are relatively easily formed in a grain in which the [144] is almost aligned toward the stress direction, in which the maximum Schmid factor of 0.5 is achieved. The Schmid factors for the formation of mechanical twins and the occurrence of slip in the fcc structure are shown in Fig. 1(c), which shows a contour plot of the Schmid factors for twin and slip deformations in the fcc structure. The orientation of the maximum Schmid factor for twinning differs from that for slip deformation because the shear directions of twinning and slip deformation are [121] and [110], respectively. The differences in the Schmid factors for different deformation modes arise from the differences in the shear directions for the slip and twinning deformations. Despite the simple model for mechanical twin formation, it has been shown that the orientation of the crystal to the deformation direction plays a key role in twinning.^27,28) There have been several advances in techniques used for morphology observation and chemical composition analysis; however, techniques for simulating stress and strain distribution in the microstructure of materials have also become popular. For instance, to understand the mechanical properties of polycrystalline materials, modeling of the microstructure of the materials has been carried out using FEM simulation.³³⁾

Fig. 1.

Schematic illustration of mechanical twin formation in an fcc structure induced by shear deformation as observed from views (a) and (b). (c) The contour plots of Schmid factors for twin (green) and slip (red) deformations.

Contrary to studies on the mechanical properties and microstructure of steels, analytical methods for investigating microscopic stresses have been also developed owing to advances in diffraction techniques. Typically, the microscopic stress and strain have been investigated using X-ray diffraction with microbeam synchrotron radiation.^{34,35,36,37,38,39)} White X-ray diffraction method with microbeam synchrotron radiation has been also applied to observe the microstructure and to analyze the microscopic stress in steels.^40,41) This method has been employed to analyze the microscopic stresses evolved in the microstructure of iron-based shape memory alloys. The analysis showed that characteristic microscopic stresses are evolved by tensile deformation at room temperature and by subsequent annealing.⁴²⁾

In the light of the previous studies mentioned above, the objective of the present study is to investigate the stress distribution in the initiation of mechanical twins in polycrystalline TWIP steels. In particular, to understand the conditions for the formation of mechanical twins in TWIP steels, we have focused on the crystal orientation of grains and the stress distribution around grain boundaries in the microstructure. This is because complicated stress distribution occurs in polycrystalline samples even by simple tensile deformation. In this study, electron backscatter diffraction (EBSD) was used to determine the crystal orientation of the grains in the TWIP steel. The stress and strain distributions were estimated using the finite element method (FEM) simulation, in which the elastic anisotropy of the fcc structure was taken into account. Since atomistic deformation processes in twining and slip are very complicated as seen in the dislocation theories, it is actually impossible to consider or simulate such atomistic deformation processes by using the FEM. Nevertheless, the FEM simulation is useful to observe the heterogeneous stress and strain distribution formed in polycrystalline materials consisting of many grains. Along with the FEM simulation, the stress distribution in a differently deformed sample was investigated by using white X-ray diffraction with microbeam synchrotron radiation in this study. The microscopic stress analyzed by the white X-ray diffraction is correlated with the results estimated by the FEM simulation.

2. Material and Methods

2.1. Sample Preparation

A 100 g button of an Fe-24%Mn-0.48%C (mass%) alloy was prepared by arc melting in argon atmosphere according to a previous method.³⁰⁾ The button of coarse grains was cut to prepare small samples with a gauge size of 3 × 0.3 × 1 mm³ for tensile tests. After annealing these tensile test samples at 1223 K, they were electrochemically polished. The samples showed a large average grain size of approximately 200 μm, which enabled us to perform X-ray diffraction analysis using synchrotron microbeam for only one grain in the direction of the sample thickness.

2.2. Measurements

EBSD was used to observe the microstructure and to analyze the crystal orientation of the grains in the microstructure of the sample before and after deformation. The EBSD patterns were measured using a Nordlys II EBSD detector attached to a Hitachi SU-6600 scanning electron microscope. From the crystal orientation obtained for each grain, anisotropic elastic constants, such as the Young’s modulus and Poisson’s ratio for the grains of the fcc structure were estimated using the elastic parameters according to the previously reported method.^42,43)

These anisotropic elastic constants were used for FEM simulation, in which the plane stress conditions were assumed to estimate the stress and strain distributions in the polycrystalline sample. The FEM simulation was carried out using the software released by ANSYS Inc. Although a simplified model under the plane stress condition was used in the FEM simulation, the influence of the anisotropic elastic constants on the deformation of the TWIP steel with fcc structure was considered. The deformation conditions of FEM simulation are shown as a stress-strain curve in Fig. 2.

Fig. 2.

Schematic nominal stress-strain curve showing conditions of the sample measured by EBSD and white X-ray diffraction. EBSD measurements were for the sample under unloading (A, B), and white X-ray diffraction was carried out for the sample under unloading (B) and loading (C). This nominal stress-strain curve was also used for the FEM simulations. The yield stress is 250 MPa and the increment of a nominal stress by a given strain after yielding is 4.5 MPa/%. The microscopic stress distribution was simulated in the sample under unloading after 10% tensile deformation (B) and the sample under loading after subsequent deformation up to 13% (C).

The X-ray diffraction experiments were performed at the beam line 28B2 of SPring-8 (Hyogo, Japan), where a high-energy white X-ray microbeam is available. The X-ray energy was more than 50 keV and the beam size was collimated to 15 × 15 μm². Laue diffraction patterns from a tensile sample with a low diffraction angle (2θ) of around 0.1° to 20° in the transmission Laue geometry were recorded during tensile testing using a flat panel detector.^41,42) For these measurements, small samples were prepared for the tensile test. The experimental set-up mainly consisted of an X-ray microbeam source, a tensile sample stage, a flat panel detector, and a Ge solid state detector (SSD), as reported previously.^41,42)

In order to analyze the stress distribution in the sample under unloading after plastic deformation and under subsequent tensile loading, the Laue diffraction patterns were recorded for a sample unloaded after about 10% tensile deformation and for a sample loaded after subsequently being subjected to about 13% tensile deformation, as illustrated in Fig. 2. EBSD of the unloaded sample before and after deformation was performed in conditions (A, B). The sample was deformed using a laboratory tensile machine. On the other hand, X-ray diffraction measurements with synchrotron radiation were carried out for the sample under unloading (B) and loading (C). The sample was deformed using a compact tensile machine for in-situ deformation.^40,41)

The nominal tensile strain was measured from the displacement of marks indented on the tensile sample using an optical microscope. In white X-ray diffraction by in-situ tensile deformation, the applied load was measured using a load cell attached to the sample grip. X-ray energy spectra for several Laue reflection spots were measured using a Ge solid-state detector.^36,37) The lattice spacing of the (hkl) planes, d^hkl, was calculated from the measured energy spectra. The lattice strain in the (hkl) planes, ε^hkl, was calculated from the change in d^hkl with respect to the initial lattice spacing d₀^hkl before deformation. In order to obtain images of the polycrystalline grains in the sample, the diffraction experiments were conducted by scanning the sample stage. Since the microbeam white X-ray diffraction patterns are sensitive to the orientation of the grains, the diffraction pattern changed significantly when the X-ray microbeam crossed grain boundaries in the sample during the stage scanning. By comparing these diffraction patterns numerically, we could obtain the grain images.⁴⁰⁾

The value of d^hkl was calculated from the obtained energy E^hkl according to Bragg’s law.⁴¹⁾ The value of ε^hkl was calculated from the difference between the lattice spacing d^hkl measured after deformation and the initial lattice spacing d₀^hkl. The resolution of the lattice strain was estimated to be ±0.0004 (±0.04%) from the strain measurement of a Si single crystal. The lattice spacing d₀^hkl was reasonably determined to be 0.36143 nm from the measured data of about 330 Laue spots for an undeformed sample by statistical treatment.

The stress was calculated assuming a two-dimensional stress state, in accordance to a previous study.⁴¹⁾ The following equation was used for the calculation.   

$${\epsilon }^{hkl}=A^{hkl}{\sigma }_x+B^{hkl}{\tau }_{xy}+C^{hkl}{\sigma }_y$$(1)

In the above equation, σ_x and σ_y are the normal stresses along the tensile direction (x) and the transverse direction (y), respectively. τ_xy is the shear stress. A, B, and C are constants determined by the elastic stiffness, crystal coordinate system, diffraction plane coordinate system or laboratory coordinate system, and specimen coordinate system. Three sets of lattice strain data are required to calculate the three stress tensors (σ_x, σ_y, τ_xy). For the elastic compliance, the elastic constants (C₁₁ = 174 GPa, C₁₂ = 85 GPa, and C₄₄ = 99 GPa) of a high-manganese TWIP steel obtained in a previous work were used.⁴³⁾

The stress calculation was carried out using a stress calculation program developed by SPring-8, which has been reported in the previous study.⁴¹⁾ The program was used for indexing Laue spots, strain/stress calculations, and visualization of the principal stress data (magnitude and direction). This method has been successfully applied for analyzing shape memory alloys which are high-manganese steels.⁴²⁾

The principal stresses (σ₁ and σ₂) and the principal stress direction (θ_p) can be calculated from the σ_x, σ_y, and τ_xy obtained. The direction of the principal stress is calculated from an angle where the shear stress becomes zero or when the stress operates along the normal direction. Then, the principal stresses are calculated using the following equation.   

$${\sigma }_{1,2}=\frac{{\sigma }_x+{\sigma }_y}{2}\pm \sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}{}^2}$$(2)

The principal stress direction is calculated from the following relationship.   

$$tan(2{\theta }_p)=\frac{\text{2}{\tau }_{xy}}{{\sigma }_x-{\sigma }_y}$$(3)

3. Results and Discussion

3.1. EBSD Analysis

Figures 3(a) and 3(b) show the inverse pole figure (IPF) maps of the microstructures of samples deformed to 0% and 10%, respectively. The color of the crystal grains in these maps denotes their crystal orientation with respect to the tensile direction and the color corresponds to the orientation in the stereographic triangle given in Fig. 3(a). The IPF map of the sample subjected to tensile deformation shows the heterogeneous evolution of plastic strain in the microstructure obtained by 10% deformation. As the misorientation angle between the measured neighboring points can be calculated from the crystal orientation at each point measured by EBSD, the local misorientation map for the sample deformed to 10% is shown in Fig. 3(c). The misorientation angle between the yellow and blue colors is about one degree. This misorientation map indicated that deformation results in microscopic plastic strain, which significantly occurs close to grain boundaries, particularly around grain boundary triple points. This implies that the grain boundaries play an important role in occurrence of the stress concentration in the microstructure of the polycrystalline sample, although it is difficult to quantitatively evaluate the local plastic strain in different grains.

Fig. 3.

IPF maps toward the tensile direction of samples subjected to (a) 0% and (b) 10% tensile deformation. A stereo triangle showing the crystal orientation is given in the figure. (c) Local misorientation map of the sample subjected to 10% tensile deformation.

In order to find the mechanical twins in the microstructure, a local area with relatively large misorientation was observed under high magnification, and the mechanical twins were observed only at point T in Fig. 3. Figure 4(a) shows a high magnification IPF map acquired from the area T denoted in Fig. 3(b). This IPF map indicated the creation of plate-like mechanical twins close to a grain boundary triple point in grain E with a large Schmid factor for twinning, as shown in Fig. 1(c). On the other hand, although the mechanical twins were not observed in grains with a relatively small Schmid factor such as grain G, it is known that the plastic deformation by dislocation slip was dominant in such grains.³¹⁾ It is noted that there are some local misorientations in the matrix, as shown in Fig. 4(b), implying that slips due to dislocations as well as twinning occur in the matrix. Thus, the stress concentration and crystal orientation are considered to be important factors in the formation of mechanical twins, since mechanical twins were hardly observed in other grains with other orientations under this strain condition.

Fig. 4.

(a) The IPF map along the tensile direction of the sample subjected to 10% tensile deformation. (b) The local misorientation map of the sample subjected to 10% tensile deformation.

In order to investigate the changes in the crystal orientation by tensile deformation, the crystal orientations of seven grains (A, B, C, D, E, F, and G) with respect to the tensile direction in samples deformed to 0% and 10%, (shown in Fig. 2) were plotted in the stereotriangles of Figs. 5(a) and 5(b), respectively. The orientations to the tensile direction can be compared with the contour of the Schmid factors for twin and slip deformations in the fcc structure, as shown in Fig. 1(c). Since grain E has a large Schmid factor for twinning, mechanical twins are expected to be formed by tensile deformation. Actually, mechanical twins were observed in grain E, as shown in Fig. 3(a). This result is consistent with previous results on mechanical twins observed in highly strained TWIP steels.^17,26) On the other hand, mechanical twins were not observed in grain B, which shows a large Schmid factor. This may arise from the dominant slip deformation in grain B rather than from the formation of mechanical twins because the initial orientation of grain B with respect to the tensile direction is rotated to the slip orientation, as shown in Fig. 5(b). Therefore, the formation of mechanical twins is considered to be influenced by not only the Schmid factor of each grain but also by the slipping of dislocations and the stress concentration enhanced by various grain boundaries.

Fig. 5.

The orientation of grains A, B, C, D, E, F and G with respect to the tensile direction in the samples subjected to (a) 0% and (b) 10% tensile deformation.

3.2. Stress Distribution in the Sample Unloaded after Small Plastic Deformation

To understand why heterogeneous strain was evolved in the present sample, as shown in Fig. 3(c), the stress and strain distributions in the microstructure were estimated using FEM simulation. Figures 6(a) and 6(b) present a simplified model of the two-dimensional microstructure and meshes used for the FEM simulation. The tensile direction of the model sample was horizontal in Figs. 6(a) and 6(b). The elastic constants of individual grains, which are calculated from the single crystal elastic constants (c₁₁= 206, c₁₂= 133, c₄₄ = 119 GPa) of an austenite stainless steel⁴⁴⁾ through the Kröner model,⁴⁶⁾ are summarized in Table 1.

Fig. 6.

(a) The microstructure model and (b) mesh used for the FEM simulations. The Young’s modulus and Poisson’s ratios calculated from the previous data were used in the simulation.

Table 1. Young’s modulus and Poisson’s ratio for grains A–G used for the present FEM calculation.

Grain	A	B	C	D	E	F	G
Young’s modulus (GPa)	234.17	214.31	209.76	227.88	198.50	185.59	150.63
Poisson’s ratio	0.2937	0.2729	0.2777	0.2586	0.2897	0.3033	0.3404

It is practically considered that the slope of the stress/strain curve after yielding can change depending on the constraints among grains and the crystallographic orientation in polycrystalline materials. The grain constraints in the present study were two-dimensional, and the type and the number of surrounding grains are rather small. Thus, the grain-constraints under the present boundary conditions would be weaker than three-dimensional constraint of general polycrystalline materials. Since it is rather difficult to evaluate the slope of the stress/strain curve after yielding for the specific grain model as shown in Fig. 2(a), we approximately employed those values obtained from the tensile test of the polycrystalline sample for FEM simulation. It was assumed that the yield stress is 250 MPa and the ratio of the increment of a nominal stress by a given strain after yielding or the apparent work hardening rate is 4.5 MPa/%, irrespective of the orientation. These values were used to estimate the heterogeneous stress and strain evolved in the microstructure under the present boundary conditions, although microscopic deformation processes, such as twining and slip in an fcc structure, should be taken into account in a more realistic model.

First, the stress and strain distributions were analyzed for the sample unloaded after undergoing tensile deformation to 10% using the FEM simulation. Figures 7(a) and 7(b) show contour maps of the residual stress and strain in the microstructure with respect to the tensile direction after 10% deformation, respectively. Grains A, B, C, D, E, F, and G shown in Fig. 7 are identical to those provided in Fig. 3. The stress and strain levels in the contour maps are by colored scales in the lower parts of the maps. The direction of the principal stress to the tensile direction in the sample unloaded after tensile deformation was nearly parallel to the tensile loading direction, although the variation of residual stress and strain with respect to the tensile direction are shown here. The stress estimated is heterogeneously distributed in the microstructure of the polycrystalline sample and several stress concentrations are observed close to the grain boundary triple points, which are denoted by circles of broken lines. Mechanical twins were created close to such a grain boundary triple point with the stress concentration shown in Fig. 3. In addition, it is found that the strain evolved after 10% tensile deformation is also influenced by the crystal orientation of each grain. Typically, the strain is large in grain D, while is the strain is small in grain G, as shown in Fig. 7(b). This phenomenon can be attributed to the crystallographic anisotropy of the Young’s modulus in the fcc structure, i.e., the modulus is low for [001], while is the value is high for [111]. The qualitative stress and strain distribution was calculated by the FEM simulation under simple assumptions in which the effect of the heterogeneous crystalline plasticity on deformation was not taken into account. Nevertheless, the simulation results indicated importance of the geometrical relationship between the arrangements of the grain boundaries and the external stress direction. For instance, these FEM simulation results are consistent with the facts that the stress concentration easily occurred near the crossing points of different grain boundaries and the local plastic deformation induced by the stress concentration was observed in the local misorientation maps obtained by EBSD.

Fig. 7.

(a) The residual stress and (b) residual strain along the tensile direction and (c) the direction of the maximum principal stress in the sample unloaded after 10% tensile deformation. The residual strain is small grains with low Young’s modulus (e.g., grain G), while is the strain is large in grains with high Young’s modulus (e.g., grain D). The maximum principal stress direction is nearly orientated to the tensile direction in the unloaded sample.

The realistic microscopic stress distribution needs to be analyzed by the diffraction method experimentally to investigate the microscopic heterogeneous stress distribution in the sample deformed to 10%. Figure 8 shows a grain image obtained by the microbeam X-ray diffraction of the sample unloaded after tensile deformation to 10%. In this image, the maximum and minimum principal stresses analyzed are shown. Since the grain boundaries in this sample are not necessarily perpendicular to the sample surface, the grain boundaries are unclear in this image. Nevertheless, Laue-spot image for single crystal were recorded in this microstructure. The stress analysis showed that the maximum and minimum principal stresses vary from one point to another point, indicating that the microscopic stress was released by the local plastic deformation. It should be mentioned that the principal stress evaluated by the FEM simulation was smaller than the experimental values and its direction was almost parallel to the tensile direction. This indicates that the actual deformation would occur in a complex manner inside of grains. Therefore, the present FEM simulation does not always provide the exact situation of the stress distribution but to observe an influence of orientation dependence of elastic constants on deformation of grains. The stress at point T near the grain-boundary triple point is rather small, whereas the stress estimated by the FEM simulation in Fig. 7 is large at the point. This discrepancy is attributed to the evolution of mechanical twins as observed in Fig. 3. The large stress close to the grain boundary triple point was probably released by the local twining deformation. The occurrence of the mechanical twins observed at point T in Fig. 2 is primarily due to the large Schmid factor of grain E for the formation of mechanical twin. Indeed, the FEM simulation in Fig. 7 suggests that the large stress take places close to point T, which is likely to induce the local twining deformation. As listed in Table 1, grain D and grain E reveal high and low Young’s modulus, respectively. These grains are two of the three grains around the grain-boundary triple point near point T. The large stress appearing around point T is considered to be due to the difference in elastic properties between grain D and grain E. Therefore, the crystallographic feature for the formation of mechanical twins is important to observe the mechanical twinning near a grain boundary triple point.

Fig. 8.

The maximum principal stress measured at various points in the microstructure of the sample unloaded after 10% tensile deformation.

In case of grain G, large principal stresses were observed near a grain boundary. This large principal stress may correspond to the large stress concentration observed in Fig. 7(a). In spite of the large stress in grain G, mechanical twins were not formed. This is mainly due to the small Schmid factor for the mechanical twins, and the stress was not released by the plastic deformation. For instance, the small stress measured near point T, where mechanical twins were observed, may be influenced by a stress release caused by twining. In addition, large principal stresses were observed near a grain boundary in grain G. Thus, the stress heterogeneously evolved in the polycrystalline sample unloaded after 10% deformation in complicated manner.

3.3. Stress Distribution in the Sample under Loading

The results shown above are concerned with the stress distribution in the sample unloaded after plastic deformation. On the other hand, to investigate the stress distribution during loading and the stress relaxation by deformation, it is feasible to carry out the stress analysis in the sample under loading. Therefore, the sample was further strained by a small amount of deformation, that is additional 3% or total 13%, and the stress analysis was performed under loading by in-situ tensile deformation. Stress analysis by FEM simulation and X-ray diffraction were also carried out for the sample subsequently deformed to 13%. The stress analysis of the sample was performed under loading. The assumptions used in the FEM simulation identical to those used in the sample deformed to 10%. Figures 9(a) and 9(b) show the contour maps of the stress and strain along the tensile direction in the sample deformed to 13%, respectively. The simulation results of the stress and strain with respect to the tensile direction are displayed in these maps. The simulated stress is again heterogeneously distributed in the microstructure of the polycrystalline sample and stress concentration is observed close to the grain boundary triple points. The stress and strain levels in the sample under loading and unloading are reversed in comparison to those observed in the unloaded sample. Typically, the strain is low in grain D with a high Young’s modulus, while the strain is high in grain G with a low Young’s modulus, as shown in Fig. 9(b). This may be mainly attributed to the crystallographic anisotropy of the Young’s modulus in the fcc structure. Such a phenomenon of reversal of stress caused by loading and unloading significantly appears close to portions in the microstructures in which there is stress concentration. This implies that microscopic plastic deformation takes place preferentially around neighboring grains with a large difference in the elastic anisotropy during cyclic deformation.

Fig. 9.

(a) The stress and (b) strain along the tensile direction and (c) the direction of the maximum principal stress in the sample subjected to total 13% tensile deformation. The stress and strain values were calculated under loading conditions. The strain was large in grains with low Young’s modulus, while the strain was small in a grain with high Young’s modulus. The maximum principal stress direction is orientated to the tensile direction.

Figure 10 shows a grain image obtained by microbeam X-ray diffraction of the sample after 13% tensile deformation. In this image, the maximum and minimum principal stresses analyzed are provided. The image position of the loaded sample is slightly different from that of the unloaded sample because of the additional plastic strain. As the image was slightly changed, the points measured for stress analysis are not always same as the points shown in Fig. 8. Stress analysis by X-ray diffraction showed that the direction of the maximum principal stress measured under loading was mostly aligned along the tensile direction. This indicates that the stress is distributed in the microstructure according to external loading. This is fundamentally consistent with the result obtained from the FEM simulation. Although the FEM simulation in Fig. 7 suggest that the stress around the grain-boundary triple point close to point T is high in grain E, the observed stress at point T in Fig. 10 was not necessarily highest in grain E, indicating that the stress near point T would be partially released by the development of mechanical twins. Furthermore, it is to be noted that there are some points in which the maximum principal stress is not along the tensile direction. This suggests that microscopic stress at such points was released by local plastic deformation due to slip of dislocations.

Fig. 10.

The measured maximum principal stress along the tensile direction in the sample under loading after total 13% tensile deformation.

Finally, the present results showed that the conditions for the formation of mechanical twins are influenced by Schimd factors of each grain and so on. It was also shown that the stress distribution estimated by the FEM simulation is fundamentally consistent with the microscopic stress analyzed by the microbeam white X-ray diffraction, although there are some differences in the stress levels estimated by these methods. These differences arise primarily from the heterogeneous microscopic plastic deformation due to the dislocations around the grain boundaries.⁴⁶⁾ Also, it is known that a large number of heterogeneous dislocations, which are geometrically necessary dislocations,^47,48) are generated in the proximity of the interfaces, such as grain boundaries and twin boundaries. The heterogeneous dislocation generation and twining during plastic deformation cause fluctuations in the stress levels in the analyzed area bearing in mind that the overall average lattice spacing was analyzed in this study. The heterogeneous local deformation can also induce local stress concentration in grains in which mechanical twins are initiated in the TWIP steel.

4. Summary

EBSD observations showed that mechanical twins are formed in grains with a large Schmid factor for twin formation in polycrystalline TWIP steel. FEM simulations indicated that the stress and strain distributions in the microstructure are influenced by the crystal orientation of the grains and grain boundaries. White X-ray diffraction with microbeam synchrotron radiation was used to analyze the evolution of microscopic stress and strain in the coarse grains of the TWIP steel. Stress analysis by microbeam white X-ray diffraction indicated that the direction of the maximum principal stresses in the steel during tensile loading is mostly oriented along the tensile direction and the residual stress and strain were distributed heterogeneously in the steel during tensile deformation. These results are fairly consistent with the results obtained from the FEM simulation, although the real principal stresses may include the effect of heterogeneous local plastic deformation on the stress distribution and simple assumptions are made in the FEM simulations undertaken in the present study.

Acknowledgments

This study was supported by a Grant-in-Aid for Scientific research of the Japan Society for the Promotion of Science and Nippon Steel & Sumitomo Metal Corporation. The synchrotron radiation experiments were performed on the BL28B2 at SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute.

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