2024 Volume 64 Issue 5 Pages 859-867
Medium-manganese steel that undergoes Lüders deformation exhibits good uniform elongation owing to large elongation with a yield plateau. To accurately predict the deformation behavior in engineering applications, the yield stresses of medium-manganese steel (5% Mn), exhibiting the transformation-induced plasticity (TRIP) effect were investigated during elongation under a multi-axial stress state (MSS). Compact tensile tests with real-time diameter measurements were conducted on smooth and notched, tiny round-bar specimens to evaluate the local yield stress and strain without the Lüders band propagation effect. Consequently, the true stress plateau was measured without the upper yield point for the smooth round-bar specimen, and the cross-sectional average true stress of the blunt notched round-bar specimens had the same plateau as the smooth round-bar specimen. The sharp-notched round-bar specimen exhibited a two-stage linear increase in true stress. The true stresses of the three specimens at the initial yield point were almost identical. Under the MSS, the hydrostatic stress typically increases the true stress at the initial yield point. The independence of the MSS indicates that the yield stress during elongation was independent of the shear-dominant crystal slip resistance. Finite element (FE) analysis using the Mises yield locus did not express the true stress plateau and its independence of the MSS. Additionally, the transformation rate of retained austenite was measured for mechanistic analysis; however, the TRIP effect did not contribute to this unusual independence because it started at the intermediate yield elongation stage. Thus, the stress criterion for the generation of mobile dislocations can determine yield stress.
Medium-Mn steels have attracted significant attention as the next-generation high-tensile steels for automotive applications. Medium-Mn steels are low-carbon steels containing approximately 3–10% Mn, exhibit microstructure consisting of ferrite matrix and retained austenite, and also exhibit superior ductility owing to the transformation induced plasticity (TRIP) effect of the retained austenite.1,2) Especially, some medium-Mn steels that have been annealed after cold rolling exhibit significant Lüders deformation (inhomogeneous deformation accompanied by Lüders bands) of approximately 10% or even more. In such materials, the work-hardening curves appear offset by the degree of Lüders deformation, and despite their high strength, they demonstrate extremely high uniform elongation.3,4,5) However, Lüders deformation may cause defects forming such as stretcher strains during press forming,6) and an accurate constitutive model is necessary when considering countermeasures.
Lüders deformation and the accompanying yield elongation are long-standing fundamental characteristics of yield phenomena in steel. During Lüders deformation, yielding begins when the applied stress reaches the upper yield point, and the nominal stress immediately drops to the lower yield point. Plastic deformation proceeds while maintaining a constant stress without work hardening until a certain degree of elongation (yield elongation) is reached. Although much about its mechanism remains unclear, it is understood as a phenomenon caused by the dislodging of the pinning between carbon atoms and dislocations,7) or by the rapid proliferation of mobile dislocations.8,9)
Lüders bands are shear zones caused by inhomogeneous deformation due to a decrease in yield stress. They originate near the gripping area of the tensile test specimen during yield elongation and propagate along the longitudinal direction multiple times. Therefore, the propagation behavior of Lüders bands and the internal state of the metal structure during plastic deformation are correlated with the stress plateau and following decrease.10,11,12,13) Indeed, as the elongation increases with the propagation of the Lüders bands, the behavior of the shear bands significantly affects the tensile test results (relationship between stress and strain). The measured yield elongation has been reported to strongly depend on the gauge length and strain rate, and the yield elongation decreases as the gauge length shortens and the strain rate decreases.14,15)
From a solid-mechanics perspective, the hidden higher-order deformation mode during uniform deformation is regarded as manifestation of Lüders (shear) bands.16) The manifestation of such higher-order modes is known as bifurcation. As a bifurcation phenomenon, an occurrence of a shear band is unavoidable when the work-hardening rate falls below a certain value. Hence, the occurrence of shear bands is inevitable in situations where the work-hardening rate becomes zero or negative, such as in Lüders deformation. Constitutive laws introducing a term for the reduction in flow stress based on these theories have already been proposed,17,18) and the occurrence and propagation of Lüders bands and the resulting yield elongation have been reproduced by incorporating them into finite element (FE) simulations.19) According to these proposed constitutive laws, the yield elongation is considered to be zero, with only the yield stress experiencing a decrease. In these numerical analyses, no local state exists with a constant yield stress analogous to the yield elongation. Instead, yield elongation is considered to emerge during the Lüders band propagation across the test specimen.
However, as reported, the yield elongation cannot simply be captured as a decrease in the yield stress and the accompanying occurrence and propagation of shear bands. In round-bar tensile tests, measuring the deformation resistance at a gauge length of zero is possible by measuring the diameter changes instead of the elongation.20,21,22,23) Even in measurements with a gauge length of zero, deformation under a constant stress state equivalent to yield elongation (not exactly elongation, but for simplicity, this deformation is also called yield elongation) has been observed.20,21) This result indicates that the yield elongation occurs regardless of the behavior of the shear bands.
Against this background, in this study, the local deformation resistance during Lüders deformation of medium-Mn steel was measured. Similar to previous reports,20,21) we obtained the local flow stresses in inhomogeneous deformation fields by measuring the diameter changes during tiny round-bar tensile tests. A distinctive feature of this study was the implementation of tensile tests using tiny notched round-bar specimens on medium-Mn steel containing a retained austenite phase. Even in materials that exhibit Lüders deformation, the minimum cross-section at the notch bottom preferentially deforms in tensile test specimens with notches.24) Hence, if the diameter of the notched part is measured, the Lüders bands do not affect the stress or strain measurement results. Using this method, we once again confirmed whether yield elongation manifests in the flow stresses and further elucidated the impact of the multi-axial stress state (MSS) induced by the notches on Lüders deformation. Owing to the induction of hydrostatic stress in a MSS, the true stresses assessed from notched specimens were higher for the same amount of strain than those from smooth specimens.21) Furthermore, hydrostatic stress enhances the TRIP effect,25) leading to a higher work-hardening rate in MSSs such as in notched (or necked) parts.23) If the TRIP effect manifests from the onset of yielding, Lüders deformation can be eliminated. This potential behavior may offer insights for developing forming methods to mitigate the defects caused by Lüders deformation. Nevertheless, in steels demonstrating the TRIP effect, such as medium-Mn steels, no instances of measuring the flow stresses during yield elongation in MSSs have been reported. If the plateau flow stress during yield elongation is determined by either dislocation pinning or proliferation, the validity of the yield criterion will be questioned. This criterion is defined by the equivalent stress based on the critical slip shear stress and excludes hydrostatic stress.
Moreover, in this study, FE simulations and measurements of the austenite volume fraction history during tensile deformation were performed using synchrotron X-ray diffraction (XRD) analyses. These methodologies were employed to consider the observed stress-strain behavior under Lüders deformation in a MSS.
Medium-Mn steel containing approximately 5% Mn was used in this study, and its chemical composition is summarized in Table 1. The ingot, once vacuum-melted, was hot rolled to a thickness of 8 mm at 1200°C (finishing temperature was maintained above 850°C), followed by air cooling. Subsequently, it was cold rolled to a thickness of 2 mm. To create a material that exhibits Lüders deformation, the cold-rolled material was annealed at 650°C for 30 min.
| C | Si | Mn | P | S | Al | O | N |
|---|---|---|---|---|---|---|---|
| 0.092 | 0.04 | 4.91 | <0.002 | 0.002 | 0.024 | 0.005 | 0.001 |
The microstructure of this steel is shown in Fig. 1. The scanning electron microscope (SEM) image (Fig. 1(a)) and the inverse pole figure (IPF) map obtained from electron back scattering diffraction (EBSD) analysis (Fig. 1(b)) reveal the formation of a fine-grained microstructure with grain sizes of less than 1 μm. The phase map obtained from EBSD (Fig. 1(c)) indicated that the retained austenite fraction in the observed field of view was measured to be 26%. This steel is the same as the one used in previous studies,11,12) and for more detailed microstructural analysis, the reader is referred to previous reports. As the retained austenite fraction was approximately 20% in previous studies,11,12) the measured values of retained austenite fraction were considered to fluctuate depending on the selected field of view.
The results of the standard tensile test measuring elongation using JIS13B test specimens are shown in Fig. 2. In terms of nominal stress, after reaching an upper yield point of 772 MPa, the yield stress decreased to 707 MPa, maintaining this level of stress until about 0.09 of nominal strain. The maximum tensile strength was approximately 880 MPa, and it reached a uniform elongation of 0.25. This is considered an exceptionally high level of uniform elongation for steel of this strength class.
We employed a custom-made compact tensile testing machine manufactured by Miyakojima-Seisakusho Corporation, Ltd.23,26) As shown in Fig. 3, a compact tensile testing machine measured the minimum cross-sectional diameter of the tiny round-bar specimen from two directions using two sets of LED projection digital shape-measuring devices (TM3000 manufactured by Keyence) (Fig. 3(a)). An example of the measurement screen is shown in Fig. 3(b). In these tensile tests, the diameters of the minimum cross section measured from two directions, Da and Db, were recorded in real time along with the force, F. The specimens were placed such that the thickness direction of the original steel sheet corresponded to Db. If the material deforms anisotropically, the cross section reduces to an elliptical shape, with Db serving as the short axis. Therefore, by assuming the minimum cross section as an ellipse (a circle when Da=Db), the average true stress σt at the minimum cross section can be calculated as follows:
| (1) |
Owing to the miniaturization of the tensile testing machine, a mechanism for measuring elongation was not incorporated. Instead, cross-sectional area reduction (CSAR) ρ was evaluated. Using the initial specimen diameter D0 and the measured values of Da and Db, ρ was calculated as follows:
| (2) |
Since the cross-section of the specimen decreases with each increment of strain, assuming that the volume of the plastic deformation region remains constant, the ρ would correspond to the true strain. As mentioned in Section 1, the true stress increases with the hydrostatic stress component after the specimen necks. Therefore, a correction must be applied to σt when evaluating flow stresses. However, this correction was not implemented in this study since reproduction of the σt−ρ curve via FE simulations is impossible, as will be discussed later. Additionally, in the experiments conducted, the ratio Da/Db was at most ~1.03 until ρ=0.09, which corresponds to yield elongation; thus, we can consider Da≈Db. Although this was beyond the strain range considered in this study, this ratio increased to approximately 1.3 by the time of specimen fracture.
Tensile tests were conducted under quasi-static conditions at room temperature at a strain rate of 2 μm/s.
For the tiny round-bar tensile specimens, we used a smooth round-bar with a straight section diameter of 1.0 mm, and two types of tiny notched round bars with notch radii of 10 mm (Notch A) and 3.4 mm (Notch B). Their shapes are shown in Fig. 4. All were cut from the 2 mm thick test steel sheet so that the tensile direction coincided with the direction perpendicular to the rolling. Since cutting out of a specimen gripping portion with a diameter of 2.5 mm from the thickness of the test steel sheet is not possible, the shape included a part of the gripping portion where the original sheet surface remained.
Figure 5 shows the σt−ρ curve obtained from the tiny smooth round-bar tensile test. For reference, the σt vs. true strain εt curves obtained from the JIS13B sheet type tensile test are also included. Despite some noise from the digital micrometers, a yield elongation similar to that obtained from the elongation measurement during the JIS13B specimen tensile test was observed in the σt−ρ curve. Notably, no upper yield point was observed in the σt−ρ curve. In the σt−εt curve, the σt dropped from the upper yield point to the lower yield point, and then slowly rose back to the upper yield point during yield elongation. However, in the σt−ρ curve, the deformation continued while maintaining a nearly constant σt (approximately equal to the upper yield point in the σt−εt curve), and work hardening began after the yield elongation. Here, the yield elongation in the σt−ρ curve was ~0.01 less than in the σt−εt curve. The work hardening rate after the yield elongation was also higher in the σt−ρ curve than in the σt−εt curve. A distinct Lüders band was observed in the JIS13B tensile test, the details of which have been introduced in previous reports.11,12)
The σt−ρ curve obtained from the tiny notched round-bar tensile test is shown in Fig. 6. Focusing on the yield elongation region, Notch A specimen (with a notch radius of 10 mm) exhibited almost identical σt−ρ as the tiny smooth round-bar tensile test. That is, no increase in σt owing to the notch effect was observed. The yield elongations of the tiny smooth round bar and Notch A specimens were also identical. After yield elongation, the σt of Notch A was higher than that of the tiny smooth round bar.
However, as shown in Fig. 6, in Notch B (notch radius of 3.4 mm), σt did not significantly increase until ρ reached ~0.03, showing a rapid rise thereafter, leading to the formation of a yield plateau. Following the yield plateau, σt demonstrated a linear increase up to ρ≈0.09, and gradually asymptoted to a Swift-type power-law curve when ρ>0.09. The σt after the yield plateau was higher in Notch B than in Notch A. However, the σt at the yield plateau, namely the σt at the initial yield point, was virtually identical for the other two specimens. That is, the results showed that even in Notch B with a sharper notch, no increase in σt was observed due to the notch effect. This behavior contradicts the well-known results obtained for another type of steel, SS400.21)
Incidentally, we used two specimens per type in the experiments to validate the repeatability of this unusual behavior.
The σt−ρ curve, measured in the tiny smooth round-bar tensile test, clearly indicated the existence of a plastic region where the test steel deformed locally at a constant yield stress. The increase in σt beyond the upper and lower yield points, as measured in the JIS13B tensile test, did not manifest in the σt−ρ curve. Therefore, the behaviors observed in the JIS13B tensile test can be interpreted as gauge-length-dependent misinterpretations prompted by non-uniform deformations from the propagation of the Lüders band. Note that the term “misinterpretation” is strictly applied in the context of local yielding behavior.
Under conditions in which the work-hardening rate was nearly zero, such as during yield elongation, the specimens underwent necking. Thus, the use of the σt−ρ curve directly as flow stress curve is inadequate. The onset of necking introduces inhomogeneity in the strain distribution, which triggers a notch effect. Generally, this notch effect results in σt values higher than Y.
Therefore, we attempted to carry out an elasto-plastic finite element (FE) simulation that synchronized the tiny smooth round-bar tensile test for identification of Y−
Although work softening laws have been developed to reproduce yield elongation by Hahn17) and Onodera,29) static implicit simulations with work softening models often encounter convergence problems. Therefore, given that the σt−ρ curve of the test steel did not exhibit an upper yield point (Fig. 5), we employed the Swift work hardening law,30,31) which was simply offset by
| (3) |
where Yp denotes the initial yield stress, and F, ε0, n are material parameters in Swift law. Yp was determined to be 777.2 MPa from the σt−ρ curve shown in Fig. 5, and was increased linearly by 0.001 MPa up to
Figure 7 presents the σt−ρ curves in measurement and FE simulation for the smooth round-bar case, and also the Y−
As expected, the FE simulations were attributed to necking of the tensile specimen. Figure 8(a) illustrates the distribution of the equivalent stress analyzed by the FE simulation at point (a) in Fig. 7. The contour maps at this point show an almost uniform distribution of equivalent stress with some necking. During the FE simulation, temporary necking (strain localization) and subsequent flattening were observed, resembling a process equivalent to the propagation of Lüders bands in sheet-type tensile tests. The notch effect increased owing to this temporary necking, resulting in a gradual increase in σt. Additionally, as the deformation progressed and reached point (b), it exhibited an evident nonuniform stress distribution, as depicted in Fig. 8(b). The
Based on a comparison of the measurements with FE simulations, two hypotheses were suggested: 1) the decrease in Y counteracted the increase in σt caused by the notch effect; 2) the notch effect itself did not manifest in the deformation mechanism during the yield elongation of this material. Considering that the initial yield stress in the σt−ρ curve obtained from the tensile test of the tiny notched round bar was almost identical to that of the smooth type, the latter possibility is more likely. Figure 9 illustrates the results of FE simulations (σt−ρ curve) conducted on the tiny notched round-bars, employing the work-hardening law described in Eq. (3). The presence of the notch significantly increased the initial yield stress, unequivocally manifesting the notch effect. However, these experimental findings deviated from the FE simulation, indicating that the notch effect did not manifest in reality. The von Mises yield criterion is based on the shear strain energy, implying that the hydrostatic stress component, independent of the shear stress, does not play a significant role in the yielding process. Consequently, the hydrostatic stress was added to the flow stress, leading to an increase in σt under MSSs. This phenomenon is known as the notch effect. Conversely, the absence of the notch effect suggests that the yield stress during the yield elongation of this material was not governed by shear-related mechanisms, thereby refuting factors such as the critical slip shear stress.
The measurements presented in this study evidently suggested that the tested steel exhibited yielding at a specific σt value, irrespective of the shear component at that moment. This indicated that the flow stresses during yield elongation were governed by a mechanism different from that of conventional intracrystalline slip. The occurrence of deformation through intracrystalline slip has been well established and observed in numerous previous studies. However, our focus in this discussion is specifically on the stress required for deformation during yield elongation. The fact that the notched specimens were independent from the shear component represents highly unusual behavior in the plastic deformation representation of steel materials based on the von Mises yield criterion. The underlying mechanism behind this unusual behavior remains elusive at the current stage of this study. Possible explanations include scenarios in which the stress fields necessary for the detachment of dislocations are primarily controlled by principal stresses, where a stress criterion governing the proliferation of mobile dislocations exists, or where the transformation of retained austenite driven by hydrostatic stress can induce such an unusual behavior. Further investigation is required to fully elucidate this mechanism.
Moreover, in the absence of the notch effect, an increase in σt was anticipated owing to the partial plastic deformation exceeding
As described in Section 4.1, mechanisms other than resistance to slip deformation could play a role in determining the flow stresses during yield elongation. One potential mechanism involves plastic deformation associated with the transformation of retained austenite. Therefore, we measured the austenite volume fraction during the yield elongation (ρ≤0.08) with a particular focus on Notch B specimen, which was expected to exhibit a prominent notch effect.
The measurements were performed at the synchrotron radiation facility SPring-8 using the beamline BL28B2. White X-ray diffraction (diffraction angle 2θ = 10°) was employed to calculate the austenite volume fraction from the integrated intensities of the γ(2 0 0), γ(2 2 0), and γ(3 1 1) peaks using an averaged evaluation. The gauge volume, with a thickness of 50 μm along the tensile direction and a rectangular shape of 0.856 mm × 0.3 mm in the orthogonal direction, was set at the minimum cross-sectional area at the center of the specimen. For detailed measurement settings, refer to our previous report.23) Apart from the gauge volume size, the conditions were consistent with those reported previously.
Notably, the retained austenite in the test steel was prone to transformation. Therefore, we performed unloading at each XRD measurement step after the tensile loading and reloaded the specimen for the next step. Additionally, X-ray radiographs were acquired to determine the CSAR ρ during tensile deformation. As mentioned in Section 2.2, the aspect ratio of the cross section was assumed to be 1.0, allowing the calculation of ρ from the unidirectional diameter change using Eq. (2).
Figure 10 illustrates the XRD measurements of the austenite volume fraction during the yield elongation of the Notch B specimen. For a reference, the σt−ρ curve from Fig. 6 is also depicted in this figure. As observed from Fig. 10, no substantial decrease was observed in the retained austenite fraction up to ρ=0.022, maintaining a consistent level of approximately 17%. This volume fraction was lower than the previously reported values in Fig. 1(c) (26%) and earlier studies11,12) (20%) using EBSD measurements. However, given the differences in measurement techniques and potential variations within the confined observation field, the 17% determined by the XRD measurement was considered a reasonable initial value. Beyond ρ=0.06, a distinct decrease in the austenite fraction was detected, decreasing to 12.9% at ρ=0.08. As detailed in Section 3, σt demonstrated a second linear increase beginning at ρ=0.03. Therefore, this trend correlated with the point at which the retained austenite fraction began to decrease.
As evidenced by Fig. 10, the retained austenite largely remained constant during the initial deformation stage (ρ≤0.02), indicating that the TRIP effect did not contribute substantially to the initial flow stresses during yield elongation. This suggests that the flow stresses during this stage were predominantly dictated by the behavior of ferrite and austenite before their transformation. In contrast to the tiny smooth round bar and Notch A specimens, the Notch B specimen exhibited an increase in σt of up to ρ=0.08 (Fig. 10), which is attributable to the partial increases in
In the case of the smooth round bar, the results indicated a decrease in the flow stresses in the later stage of yield elongation. At this stage, a considerable portion of the austenite was transformed, suggesting that the decrease in the flow stress was mainly attributed to ferrite behavior. One hypothesis is that the austenite shows typical work hardening, whereas the remaining 80% of ferrite exhibits a decrease in flow stress and notch insensitivity. However, this requires further investigation, and obtaining the true stress using biaxial stress measurements is challenging owing to the instability of austenite in the test steel. Developing a method for rapid biaxial stress measurement is crucial for obtaining equivalent stress data.
As previously discussed, the yield elongation in a MSS exhibits a complex behavior that is influenced by various factors. Traditional plastic deformation mechanics, which are based on crystal slips and include the Mises yield criterion, cannot be applied here. Until now, minor yield elongations such as the 1–2% observed in Lüders deformation could be ignored without significantly affecting the analysis. However, this approach is not applicable for medium-Mn steels, like the test steel used in this study, which exhibit a significant yield elongation of nearly 10%. Therefore, establishing an appropriate constitutive model for effective utilization of such materials is critical. In actual scenarios like press forming or automobile crushing processes, considering more complex yielding behaviors, such as abrupt strain path changes,32) is necessary. Although this study provides some insights, the identification of mechanical factors that govern these complex phenomena remains a challenge for future research.
In this study, we investigated the local deformation resistance during Lüders deformation in a MSS, focusing on a medium-Mn steel containing 5% Mn. The test steel demonstrated a TRIP effect and displayed a high uniform elongation, combined with approximately 10% yield elongation. We conducted a real-time diameter measurement tensile test on tiny smooth and notched round bars of this material and successfully obtained the local average true stress and reduction in area (approximate true strain) at the minimum cross-sectional area under conditions not affected by Lüders bands. The results were as follows:
(1) Even in local areas without the Lüders band propagation effect, the material exhibits yield elongation, in which the deformation proceeds with a constant true stress.
(2) No increase in the initial yield stress due to the notch was observed. Both the smooth and notched round bars transitioned from elastic to plastic behavior at a true stress of approximately 780 MPa. This implies that expressing flow stresses during yield elongation using a yield criterion based on shear stress, such as Mises, is impossible. Indeed, FE simulations based on the von Mises yield criterion cannot reproduce the true stress behavior during yield elongation.
(3) This notch-independent phenomenon is believed to be due to factors other than the crystal slip resistance. However, we could not identify the underlying mechanism. We measured the transformation rate of austenite using synchrotron X-ray diffraction analysis; however, no effects of austenite transformation other than the promotion of inhomogeneous deformation were observed. Identifying the factors that influence flow stresses during yield elongation is essential for formulating a constitutive model that accurately represents material deformation behavior, including Lüders deformation of the medium-Mn steel.
This study was supported by the Amada Foundation (AF-2021003-A3). Additionally, the measurements using synchrotron XRD were part of the research results obtained with the aid of JSPS KAKENHI(JP20H02484) and were carried out under proposal numbers 2020A1097 and 2022A1385 at the large synchrotron radiation facility SPring-8. We would like to express our gratitude to Dr. Kentaro Kajiwara of JASRI and the following alumni and current members of the Solid Mechanics Laboratory at Tottori University for their dedicated efforts in conducting the experiments: Taiki Fujita, Daiki Kondo, Tomoya Takahashi, Yuya Ueda, Taichi Shimizu, Takahiro Sato, and Hidenari Yokota. The test steel and stress-strain curves obtained using JIS13B tensile test specimens were provided by Mutsumi Yoshitake of Nippon Steel Corporation as samples and data for the Iron and Steel Institute of Japan research committee on “Inhomogeneous Deformation Microstructure and Mechanical Properties of Steel Materials.” We hereby extend our acknowledgment to all those mentioned for their valuable contributions.
