Microscopic Shear Deformation Characteristics of the Lüders Front in a Metastable Austenitic Transformation-induced-plasticity Steel

Abstract

To elucidate the mechanisms of deformation and a state of plastic stability at the front of Lüders bands during a tensile test, metastable austenitic transformation-induced plasticity (TRIP) steels with different dislocation densities and ferritic steels were characterized via macroscopic-DIC-based stress–strain investigations and scanning electron microscopy (SEM). A direct correlation between stress–strain curves and measured strain distributions in the tensile specimen indicated that the Lüders front represents a transition region from a state of plastic instability to one of stability, whereby a general rule relating the Lüders strain () and increments in the true stress in the Lüders band () to a lower yield stress () can be described as irrespective of the amount of deformation-induced martensite in the band or crystal structure of the steel. The inclination angle of the Lüders front with respect to the tensile direction changed from 55° to 90° with a reduction in the measured strain ratio (−ε_yy/ε_xx) in the Lüders band, and the change agreed with the tendency calculated by the plasticity model, assuming the pure shear occurs under the minimum shear strain criterion. SEM observations of the sheet surface and the front cross-section in the TRIP steel showed the formation of multiple inclined ~20 µm-wide shear deformation zones that accompanied a reduction in thickness. All the observed geometrical characteristics of the Lüders front were qualitatively described by a mechanism involving minimizing the misalignment from the fixed tensile axis caused by ‘shear’ deformation.

1. Introduction

The use of high-strength steel (HSS) sheets in automotive bodies has reduced energy consumption and improved vehicle crashworthiness.¹⁾ The adoption of HSSs is steadly increasing, and gigapascal-grade steel sheets with excellent press formability is expected to further expand the use of HSSs.²⁾ High-strength transformation-induced plasticity (TRIP) steels with medium-to-high alloy compositions with main or secondary phases comprising metastable austenite (γ)^2,3,4,5,6,7) are the candidates to satisfy these requirements, and those with yield strengths (YSs) of ~1.5 GPa have recently been reported.^8,9) However, TRIP steels with high YSs tend to exhibit discontinuous yielding during tensile testing, which may cause undesirable surface strain marks and fracture in unfavorable scenarios during press forming. Understanding the mechanism of discontinuous yielding and the factors influencing it is therefore crucial for the development of steels.

When discontinuous yielding occurs during tensile tests, inhomogeneous deformation zones known as Lüders bands appear in the parallel section of a dog-bone specimen and propagate throughout the specimen. The local strain in these inhomogeneous deformation zones is called the Lüders strain, the magnitude of which along the tensile direction is in principle equivalent to the yield point elongation (YPE). The effects of microstructure on discontinuous yielding in ferritic steels have been extensively studied:^10,11,12,13) conditions to increase the yield stress, including grain refinement^11,12) and strain aging,¹³⁾ result in discontinuous yielding. The magnitude of YPE increases linearly with increasing yielding stress when the composition of the steel is constant,^11,12,14) while a proportional, composition-independent relationship is reported to exist between YPE and the inverse of the work hardening rate at lower yield stresses.¹⁴⁾ Microstructural control to increase yield stress via grain refinement^{4,15,16,17,18,19)} and increasing the dislocation density in the matrix phase by ausforming^3,20,21) enhances the discontinuous yielding that is observed in TRIP steels, along with Lüders band propagation. In TRIP steels, martensitic transformation often occurs within the Lüders bands^{3,15,17,19,20,22,23,24)} in addition to the plastic deformation of the parent austenite phase. The magnitude of Lüders strain is affected by the chemical and mechanical stability of austenite^17,21,24) which reflect the potential of metastable austenite to resist the martensitic phase transformation. The magnitude of Lüders strain in TRIP steels varies with the austenite stability even at a similar lower yield stress;¹⁷⁾ hence no simple correlation between the Lüders strain and yield stress has been found. However, assuming that the force-balancing between Lüders band and non-deformed region in tensile specimens is satisfied, there presumably exists a universal criterion to correlate the yielding stress with the Lüders strain in terms of mechanics. However, no consensus has yet been established on the most appropriate criterion to relate the yield stress with the magnitude of the Lüders strain in TRIP steels.

Concentrated local deformation zones appear during tensile testing under conditions of plastic instability. Both local necking after uniform deformation and the occurrence of Lüders bands are considered plastic instabilities.²⁵⁾ If the flow stress is a function of strain alone, the Considère criterion, which is expressed by Eq. (1), governs the plastic instability caused during tensile testing:^26,27,28)   

$${\sigma }^{*}>d{\sigma }^{*}/d{\epsilon }^{*},$$(1)

where ε^* and σ^* denote the true strain and true stress, respectively. To prevent necking and fracturing of the concentrated local deformation zone, the deformed region must be work-hardened until the condition of plastic stability is satisfied, i.e., the force in the deformed region should balance that in the non-deformed region. This force-balancing condition for Lüders bands is expressed by Eq. (2):   

$${\sigma }_0^{*}\cdot A_0=({\sigma }_0^{*}+\mathrm{\Delta }{\sigma }_L^{*})\cdot (A_0+\mathrm{\Delta }A)$$(2)

where ${\sigma }_0^{*}$ and A₀ are the stress and cross-sectional area in the non-deformed region, respectively, and $\mathrm{\Delta }{\sigma }_L^{*}$ and ΔA are the stress increment and decrement in cross-sectional area of the Lüders bands, respectively. Assuming constant-volume conditions before and after the local deformation, Eq. (2) can be rewritten as:   

$${\sigma }_0^{*}=\mathrm{\Delta }{\sigma }_L^{*}/\mathrm{\Delta }{\epsilon }_L^{*},$$(3)

where $\mathrm{\Delta }{\epsilon }_L^{*}$ is the true strain generated in the Lüders bands. Because Eq. (3) is derived solely from the mechanical force balance, the relationship is expected to be valid regardless of the material or grade of steel.

Shear deformation accompanied by geometric softening is one mechanism that may occur under plastic instability.^28,29,30) With the formation and propagation of Lüders bands under plastic instability, shear deformation is expected to occur at the Lüders front;^10,11,31) however, the microscopic features of the Lüders front are not clear. Varanasi et al.²⁴⁾ observed parallel plastic zones approximately 30 μm wide perpendicular to the tensile direction at the Lüders front in ultrafine-grained medium-Mn TRIP steels by SEM-electron channeling contrast imaging, leading them to suggest the possibility of shear deformation. However, direct evidence of microscopic shear deformation has not yet been obtained, and the reasons for the formation of spaced parallel plastic zones remain unclear.

This study therefore verifies the criteria for the formation of Lüders bands and their propagation in metastable γ-TRIP steels by analyzing stress–strain curves and macroscopic strain distributions obtained via digital image correlation (DIC). To evaluate the effects of yield stress, deformation-induced martensitic transformation, and crystal structure on the formation and propagation of Lüders bands, we prepared four γ-TRIP steels with different dislocation densities, and two ferritic steels. The microscopic deformation behavior at the Lüders front in TRIP steels was characterized from two directions (sheet surface and cross-section) by scanning electron microscopy (SEM)/electron backscatter diffraction (EBSD) and SEM-based microscopic DIC (μDIC) strain investigations during the tensile tests to clarify the deformation mode. Additionally, because the deformation angle typically identifies the deformation mode, the inclination angles at the Lüders front measured by DIC were compared with those calculated by the shear deformation model to verify the occurrence of shear deformation at the Lüders front.

2. Experimental

Three steels with compositions of 0.13C-12.1Ni-6.1Mn, 0.07C-1.0Mn, and 0.59C-1.0Mn (mass%)—designated as 12Ni6Mn, CMn1 and CMn2, respectively—were prepared and analyzed. The steels were melted in a vacuum furnace, and the ingots were hot-forged and homogenized at 1280°C to minimize the microscopic segregation of Mn and Ni, which could affect the local variation in austenite stability, and subsequently hot rolled. To assess the effects of yield stress and mechanical stability of austenite on the initiation and propagation of Lüders bands, four austenitic TRIP steels with various dislocation densities were prepared using 12Ni6Mn. The 12Ni6Mn steel was heated at 950°C, cooled to ~25°C, and ausformed by warm rolling at 450°C to introduce dislocations in austenite at reduction ratios of 35%, 60%, and 74% (AF35, AF60, and AF74 steels, respectively). A 12Ni6Mn steel with the lowest dislocation density (AF0 steel) was prepared by heating at 950°C, cold rolling at 20–50°C with a 75% reduction ratio, and annealing at 610°C for 10 min. The four 12Ni6Mn steels were confirmed to consist of a single face-centered cubic (fcc) austenite phase by EBSD. CMn steels were prepared to assess the effect of the crystal structure on the Lüders bands initiation and propagation. The CMn1 and CMn2 steels were cold-rolled at a reduction ratio of 80%, annealed at 800°C and 650°C for 3 min, respectively, and furnace-cooled to ~25°C to obtain a body-centered cubic (bcc) ferrite phase.

Tensile testing of the specimens was performed using specimens measuring 2 mm-wide an 0.4–0.5 mm-thick with a parallel section of 10 mm. The surfaces of the tensile specimens were electropolished in an electrolyte containing 5% perchloric acid and 95% acetic acid to remove surface martensitic layers introduced by mechanical gliding. Uniaxial tensile tests were performed at a constant tensile speed of 7.2 μm/s in both air and an SEM apparatus using a compact tensile testing stage. In the air-based tensile tests, random speckle patterns were spray-painted on the surface of the sheet specimens, and the changes in macroscopic strain distribution were monitored during discontinuous yielding by DIC using photographs taken with an industrial CMOS camera every 0.5 s. Observations of the tensile specimens via SEM-based tensile tests were performed at the center of the sheet surface, and analysis of the microscopic-strain-distribution of the Lüders front was conducted using densely dispersed colloidal silica particles with a diameter of 20–100 nm as a marker for DIC. The tensile tests to be observed by SEM were interrupted when the propagating Lüders front was positioned in the middle of the tensile specimen, and the SEM and EBSD observations were made to identify the phase transformation behavior at the Lüders front. After the observation, the specimens from the interrupted tensile test were removed from the SEM apparatus and split in half along their tensile direction by wire electrical discharge machining to investigate the deformation and transformation behavior at the cross section of the Lüders front. After mechanical and electrical polishing, the cross section of the Lüders front was characterized by EBSD to determine the quasi-three-dimensional deformation characteristics and transformation behavior at the Lüders front. SEM-EBSD was performed at an accelerating voltage of 15 kV using a Hitachi SU-5000 system equipped with an AMETEC Hikari EBSD detector. The crystal structure was analyzed by EBSD using EDAX OIM analysis software, and the DIC strain analysis was performed using Correlated Solutions Vic2D software, in which the tensile axis was considered the x-direction. The subset size of DIC analysis, in which more than 10 markers for analysis are included, was selected to 300 × 300 μm² (pixel size: 11 × 11 μm²) for photographs taken by industrial CMOS camera and 1.1 × 1.1 μm² (pixel size: 0.03 × 0.03 μm²) for SEM images.

3. Estimation of Shear Deformation Angle under Tensile Loading

To verify the relevance of Lüders banding to the shear deformation, we estimated the shear deformation angle during tensile testing of the sheet specimen. Although the localized necking angles under uniaxial tension have been estimated under various conditions,^32,33,34) the following formulation (Eqs. (4), (5), (6), (7)) aims to determine the effect of the ratio of strain along the tensile and width directions on the inclination angle of the shear band.

The model considers the shear deformation in a sheet specimen under tensile loading. The angles α and β describe the inclination of the shear band with respect to the tensile direction in the sheet plane and the inclination with respect to the tensile direction in the thickness plane, respectively (Fig. 1). Two coordinate systems were established: a sample coordinate system {x y z} and shear band coordinate system {x′ y′ z′}, with coordinate transformation as:   

$$\left\{\begin{array}{c}x\\ y\\ z\end{array}\right\}=\left[\begin{array}{ccc}{l}_1& m_1& n_1\\ l_2& m_2& n_2\\ l_3& m_3& n_3\end{array}\right]\left\{\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right\}$$(4)

where l_i m_i n_i (i = 1,2,3) denote the directional cosine between the main axis of each coordinate system. In the shear band coordinate system, the x′ and y′ directions were taken as the shear direction in the shear plane and the normal direction to the shear plane, respectively (Fig. 1(b)). With a shear strain given by γ (Fig. 1(b)), six components in the strain tensor of the specimen coordinate system are represented by:   

$$\begin{array}{l}{\epsilon }_{xx}=\gamma \mathrm{\hspace{0.17em} }{l}_1\mathrm{\hspace{0.17em} }{m}_1,\\ {\epsilon }_{yy}=\gamma \mathrm{\hspace{0.17em} }{l}_2\mathrm{\hspace{0.17em} }{m}_2,\\ {\epsilon }_{zz}=\gamma \mathrm{\hspace{0.17em} }{l}_3\mathrm{\hspace{0.17em} }{m}_3,\\ {\epsilon }_{xy}=\gamma (l_1{m}_2+l_2{m}_1)/2,\\ {\epsilon }_{yz}=\gamma (l_2{m}_3+l_3{m}_2)/2,\\ {\epsilon }_{xz}=\gamma (l_3{m}_1+l_1{m}_3)/2\end{array}$$(5)

Here, we introduce the ratio of the strain along the width and tensile directions (q) which takes a value between 0 and 0.5:   

$$q=-{\epsilon }_{yy}/{\epsilon }_{xx}=1-{\epsilon }_{zz}/{\epsilon }_{xx}$$(6)

In isotropic materials, conditions of plane strain and uniaxial tension are represented by q=0 and q=0.5, respectively. Combining q and the angles α and β, the relationship between the shear strain (γ) and strain along the tensile direction (ε_xx) can be written as:   

$$\gamma =\frac{cd(1-q)}{sin\alpha cos\beta }{\epsilon }_{xx},$$(7)

where c and d are the absolute values of vectors parallel to the y′-axis and x′-axis, respectively.

Fig. 1.

Shear bands formation in sheet specimens under tensile loading. (Online version in color.)

Assuming that the orientation of the shear deformation is predominantly determined by the minimum shear strain (γ_min), a set of solutions (q, α, β) was obtained with the representative values of (0, 90°, 45°) and (0.5, 54.7°, 54.7°). The estimated angle α in the sheet plane under plain strain tension (q=0) and uniaxial tension (q=0.5) are in good agreement with previous reports.^32,33,34)

4. Results

4.1. Correspondence between Stress–strain Curves and the Formation and Propagation of Lüders Bands

The engineering stress–strain curves of austenitic TRIP steels (12Ni6Mn) and ferritic steels (CMn1 and CMn2) shows discontinuous yielding, while the YS of the austenitic TRIP steels increased as the reduction ratio by warm rolling increased (Fig. 2). The correlation between discontinuous yielding and plastic instability was investigated by monitoring the change in the instantaneous work-hardening rate (dσ^*/dε^*) with true strain (ε^*) and comparing this to the σ^* curve. A representative low strain region of the ε^*−dσ^*/dε^* and ε^*−σ^* curves of AF74 steel are shown in Figs. 3(a) and 3(b). With increasing strain, dσ^*/dε^* rapidly decreased and satisfied the plastic instability condition (σ^* > dσ^*/dε^* and 0 ≥ dσ^*/dε^*) at point (ii) in Fig. 3(b). In this instance, an inhomogeneous deformation zone with a high strain rate spanning the sample width appeared at one end of the tensile specimen (Fig. 3(c-ii)), followed by a second deformation zone appearing at the left of the first zone with an interval of ~0.3 mm (Fig. 3(c-iii)), accompanied by a small load drop (Fig. 3(b)). The dσ^*/dε^* value subsequently increased and satisfied the macroscopic plastic stability condition (σ^* ≤ dσ^*/dε^*) at point (iv), thereby arresting the load drop. With a further increase in strain, a deformation zone comprising two separated zones began propagating into the non-deformed region (Fig. 3(c-v)), while dσ^*/dε^* oscillated as the deformation zone propagated along the longitudinal direction of the tensile specimen. This oscillation was attributed to the non-steady propagation of deformation zones in the Lüders front. The inhomogeneous Lüders deformation was triggered all the steels investigated, including ferritic steels, when the macroscopic plastic instability condition (Eq. (1)) was satisfied, while the zones started to propagate when the macroscopic plastic stability condition (σ^* ≤ dσ^*/dε^*) was satisfied.

Fig. 2.

Engineering stress–strain curves of austenitic TRIP steels (AF0, AF35, AF60, AF74) and ferritic steels (CMn1 and CMn2). (Online version in color.)

Fig. 3.

(a) True strain (ε^*)–instantaneous work-hardening rate (dσ^*/dε^*) and ε^*–true stress (σ^*) curves in the low-strain region of AF74 steel, and enlarged ε^*–σ^* curve, (b) enlarged graphs of the low-strain region of (a), and (c) the corresponding strain rate (dε_xx/dt) maps observed during tensile tests. (Online version in color.)

The force balance between the propagative deformation zone and the non-deformed region was assumed to comply with the conditions expressed in Eq. (3), which contains three parameters: the applied stress in the cross-sectional area of the non-deformed region; the stress increment at the deformation zone; and the strain in the deformation zone. The stress applied to the non-deformed region during the propagation of the Lüders band is approximated as the macroscopic true stress immediately after the plastic stability condition is satisfied, that is, ${\sigma }_{y0}^{*}$ (point (iv) in Fig. 3(b)). The stress increment in the propagating Lüders deformation zone, $\mathrm{\Delta }{\sigma }_L^{*}$, is obtained from the difference in the true stress at the end and beginning of Lüders banding (Fig. 3(a)) because the stress applied to the deformation zone corresponds to the macroscopic true stress immediately after the completion of the Lüders band propagation in the parallel section of the tensile specimens (point (vi) in Fig. 3(a)). The Lüders strain $\mathrm{\Delta }{\epsilon }_L^{*}$ is obtained from the difference in true strain at the beginning and end of the Lüders band propagation. Table 1 summarizes the $\mathrm{\Delta }{\sigma }_L^{*}$ and $\mathrm{\Delta }{\epsilon }_L^{*}$ values obtained from the stress–strain curves of the 12Ni6Mn TRIP steels and CMn ferritic steels. A plot of $\mathrm{\Delta }{\sigma }_L^{*}$/$\mathrm{\Delta }{\epsilon }_L^{*}$ against ${\sigma }_{y0}^{*}$ (Fig. 4) confirmed that all of the steels investigated satisfied the condition ${\sigma }_{y0}^{*}$ ≈ $\mathrm{\Delta }{\sigma }_L^{*}$/$\mathrm{\Delta }{\epsilon }_L^{*}$. The measured fraction of α’-martensite formed in the Lüders band ($f_L^{{\alpha }^{\prime }}$) of TRIP steels is also summarized in Table 1. The fraction of α’-martensite varies with the reduction ratios of warm-rolling, but no simple $\mathrm{\Delta }{\epsilon }_L^{*}$ – $f_L^{{\alpha }^{\prime }}$ relationship was observed. These results implied that the magnitude of the Lüders strain ($\mathrm{\Delta }{\epsilon }_L^{*}$) is dominated by Eq. (3) regardless of the amount of deformation-induced martensite and crystal structure of steel. The high $\mathrm{\Delta }{\sigma }_L^{*}$/$\mathrm{\Delta }{\epsilon }_L^{*}$ values of the ausformed 12Ni6Mn steels may arise from the hard martensite formed during inhomogeneous deformation, which can sustain high stresses, allowing the inhomogeneous deformation zone to propagate without necking even at high YSs (${\sigma }_{y0}^{*}$). Tsuchida et al. identified a proportional relationship between the Lüders strain dε_L and the inverse of the work-hardening rate dσ^*/dε^* at lower yield strengths, which was established using K and n values from fits of the stress–strain curve of ferrite, that is, σ^* = K·ε^*n.¹⁴⁾ However, this relationship could not be confirmed in the TRIP steels considered herein. In summary, the discontinuous yielding that accompanies the inhomogeneous deformation band begins under conditions of plastic instability, and the band propagates into the non-deformed region in tensile specimens that satisfy the force-balancing condition expressed in Eq. (3). These results indicate that the Lüders front represents a transition region from plastically unstable to stable state. Finally, Eq. (3) can be applied as a general rule that relates Lüders strains to lower yield stresses irrespective of composition or type of steels.

Table 1. Yield stress (YS), tensile stress (TS), Lüders strain ($\mathrm{\Delta }{\epsilon }_L^{*}$), and the increase in true stress during Lüders band propagation ($\mathrm{\Delta }{\sigma }_L^{*}$) evaluated using tensile tests, and the fraction of α’-martensite formed in the Lüders band ($f_L^{{\alpha }^{\prime }}$) measured by SEM/EBSD analysis.

Steel		Phase	YS (MPa)	TS (MPa)	$\mathrm{\Delta }{\epsilon }_L^{*}$	$\mathrm{\Delta }{\sigma }_L^{*}$ (MPa)	$f_L^{{\alpha }^{\prime }}$ (%)
12Ni6Mn	AF74	γ	839	1057	0.106	91	~54
	AF60	γ	779	1073	0.114	94	–
	AF35	γ	706	1047	0.102	74	~32
	AF0	γ	463	853	0.020	9	~0.2
ferrite	CMn1	α	270	333	0.029	7	–
	CMn2	α	736	796	0.022	16	–

Fig. 4.

Relationship between lower yield strength (${\sigma }_{y0}^{*}$) and the $\mathrm{\Delta }{\sigma }_L^{*}$/$\mathrm{\Delta }{\epsilon }_L^{*}$ values of 12Ni6Mn metastable-austenitic steels and CMn ferritic steels. (Online version in color.)

Snapshots of the macroscopic strain rate maps (dε_xx/dt) during the propagation of Lüders bands in all of the investigated steels reveal that a deformation band appeared at one side of the tensile specimens and continuously propagated to the other side of the specimen (Fig. 5). The angle was clearly observed in the strain rate maps, even in those of CMn steels with small Lüders strain, whose deformation front was diffuse^10,31) and difficult to observe directly by the naked eye or optical camera. The angle of the Lüders front with respect to the tensile axis in the sheet plane, α, varied during propagation in accordance with the composition and type of steels. The front angle α was approximately 90° in AF74 steel but inclined to the tensile axis in the AF35, AF0 and CMn2 steels. In AF0 and CMn2 steels, the propagation of the front was accompanied by two symmetrical bands inclined by α~58° to the tensile axis (Figs. 5(c) and 5(d)). Strain rate splitting in the form of thin stripes with an interval of 0.3–0.4 mm (Figs. 5(a) and 5(b)) was also observed in the three warm-rolled 12Ni6Mn steels. Table 2 summarizes the measured average front angle α, Lüders strain along tensile (${\epsilon }_{L,xx}^{dic}$) and sheet width directions (${\epsilon }_{L,yy}^{dic}$), and q (Eq. (6)) evaluated by the macroscopic DIC measurements. In uniaxial tensile tests of isotropic materials, q is often taken as 0.5;^32,33,34) however, the observed values in the Lüders band were considerably smaller and decreased with increasing ${\epsilon }_{L,xx}^{dic}$.

Fig. 5.

Strain rate (dε_xx/dt) maps acquired by macroscopic DIC during the propagation of the Lüders band in (a) AF74, (b) AF35, (c) AF0, and (d) CMn2 steels. (Online version in color.)

Table 2. The average angle of the Lüders front with respect to the tensile in sheet plane (α), and Lüders strain along the tensile direction (${\epsilon }_{L,xx}^{dic}$) and sheet width direction (${\epsilon }_{L,yy}^{dic}$) evaluated by macroscopic DIC measurements. q is the ratio calculated by −${\epsilon }_{L,yy}^{dic}$/${\epsilon }_{L,xx}^{dic}$.

Steel		α (°)	${\epsilon }_{L,xx}^{dic}$	${\epsilon }_{L,yy}^{dic}$	q
12Ni6Mn	AF74	86.3	0.123	−0.018	0.15
	AF60	79.8	0.129	−0.019	0.15
	AF35	78.4	0.114	−0.023	0.20
	AF0	57.8	0.025	−0.008	0.41
ferrite	CMn1	69.9	0.032	−0.008	0.26
	CMn2	57.5	0.025	−0.009	0.36

4.2. Microscopic Deformation and Transformation of Lüders Front

Secondary electron (SE) images and EBSD phase maps at the Lüders front in AF74 steel show bright bands with a width and interval of ~20 and ~50 μm, respectively, at the Lüders front, with the band contrast being less evident in the region behind the Lüders front (Fig. 6(a)). Because the bright contrast arises mainly from the surface roughness caused by plastic deformation, the image indicated that microscopically localized deformation zones were formed at the Lüders front. The angle of the microscopic deformation zones relative to the tensile axis, α, was approximately 90°, which is consistent with the angle observed by macroscopic DIC strain measurements (Fig. 5(a)). Figure 6(b) shows the phase map of the region indicated by a blue dotted line in Fig. 6(a), where α’-martensite and austenite are represented by red and green, respectively. α’-martensite was not observed in non-deformed regions (left side of Fig. 6(b)) but was formed at and behind the Lüders front. The bright regions in SE image correspond to localized martensitic phases, including the localized bands and the region behind them (Fig. 6(a)). Taken together, the SE image and EBSD demonstrate that the microscopically localized deformation zones at the Lüders front are accompanied by the formation of α’-martensitic phases. These features are remarkably similar to those observed at the Lüders front of ultrafine duplex TRIP steels.²⁴⁾ The cross-sectional SE image at the location defined by the black dashed dotted line in Fig. 6(a) clearly demonstrates that the thickness of the sheet specimens decreased at the Lüders front, especially in the region showing a contrast in the bright localized band in the sheet surface (Fig. 6(c)). The distribution of deformation-induced martensite in the cross-section at the Lüders front was clear and inclined to the tensile axis by β=44–54°. This observation supports the hypothesis that the localized deformation bands at the Lüders front in AF74 steel are formed via shear deformation mainly in the mode that reduced the thickness of the specimen. The length of the region with reduced thickness along the propagating direction was ~200 μm (length A or B in Fig. 6(c)), and the region contained at least four microscopically localized shear bands. Symmetrical shear deformation was expected because the reduction in thickness occurred symmetrically with respect to the central axis of the tensile specimen. The macroscopic X-shaped distribution of martensite in the thickness plane (yellow dashed lines in Fig. 6(d)) may reflect the simultaneous activation of the symmetric shear band. Moreover, when the shear-induced reduction in thickness occurred in an X-shaped manner (Fig. 6(d)), two distinct macroscopic deformation regions (denoted ‘A’ and ‘B’ in Fig. 6(d)) are expected to appear on the surface of sheet-type tensile specimens with an interval of ~300 μm. The strain rate splitting observed in the Lüders fronts of AF74 and AF35 steels takes the form of thin stripes (Figs. 5(a) and 5(b)) likely reflecting the simultaneous activity of the symmetrical macroscopic shear.

Fig. 6.

(a,c) Secondary electron (SE) images and (b,d) EBSD phase maps of the Lüders front in AF74 steel on the sheet surface (a,b) and on the cross-section (c,d) cut from the location denoted by the black dashed dotted line in (a). The green and red colors in the phase maps represent austenite and α’-martensite, respectively. At least four ~20 μm-wide inclined deformation bands can be observed at the Lüders front where a reduction in thickness occurs. The deformation bands accompanying martensitic transformation are inclined at β=44–54° to the tensile direction. A and B in (c,d) denote two distinct macroscopic shear-deformation regions that cause a symmetrical reduction in thickness. (Online version in color.)

A magnified cross-sectional enlarged phase map of the upper portion of the Lüders front shown in Fig. 6(d) reveals that α’-martensite was mainly dispersed in the inclined shear-deformation zones (Fig. 7). Each martensite phase (shown by arrows in Fig. 7(b)) had an elongated shape in two different directions—right and left inclines with the inclination angle, θ, of 32–40° relative to the tensile direction. These elongated directions are consistent with those of the microbands observed as black contrast in the image quality (IQ) map (Fig. 7(c)). The inclined fine black contrast was observed even in the non-deformed region (left side of Fig. 7(c)); therefore, most of these substructures likely represent microbands that formed in grains during warm rolling prior to the tensile tests. Because the deformation mode that reduced the thickness during the tensile tests was similar to that during rolling, the microbands in austenite grains introduced by warm rolling were presumably activated during the tensile tests, with the deformation-induced martensite being preferentially formed along the microbands.

Fig. 7.

(a) Enlarged cross-sectional phase map of the upper side of the Lüders front in AF74 steel shown in Figs. 6(d), 6(b) enlarged section of the phase map in (a), and (c) the corresponding image quality (IQ) map. Each α’-martensite segment is elongated along two different directions: left (i, ii) and right (iii, iv) inclines. The elongated directions are consistent with the directions of the microbands that appear as black contrast in the IQ map, as indicated by the arrows in (b) and (c). (Online version in color.)

The deformation state of the parent austenite phase at the Lüders front was subsequently examined. Figure 8(a) shows an enlarged top-view SE image of the foremost region of the front, which corresponds to the area outlined with a white dotted line in Fig. 6(a). Fine stripe-like rough surface features perpendicular to the tensile axis were observed with intervals of 1 μm or less. The microscopic strain concentration (ε_xx) of 3%–12% along the striated rough features (Fig. 8(b)) indicated that the roughness was caused by localized slipping of austenite, which likely occurred along the microbands in warm-rolled austenite grains. Moreover, the phase map (Fig. 8(c)) indicated that α’-martensite was present in some of the localized slip zones of austenite, suggesting that local deformation, including the shear deformation of austenite, preceded the austenite-to-martensite transformation at the Lüders front. The microscopic strain along the tensile direction (ε_xx) in the region transformed from austenite to martensite was more than 9% (Fig. 8(b)), clearly larger than the isotropic volume strain caused by the transformation from austenite to martensite (~3 vol%). Because the deformation-induced formation of martensite occurred at high stress (${\sigma }_{y0}^{*}$ = 939 MPa in AF74 steel), the larger local elongation of the martensite phase may be due to the local TRIP.^35,36)

Fig. 8.

(a) A top-view SE image, (b) strain distribution map (ε_xx) acquired via in situ SEM-based μDIC, and (c) phase map of the frontal region in AF74 steel, which corresponds to the area shown by the dotted-line in Fig. 6(a). Localized slip lines perpendicular to the tensile axis are observed, along which deformation-induced martensite is present. The step size of EBSD analysis shown in (c) was 0.03 μm. (Online version in color.)

5. Discussion

Analysis of the stress–strain curves confirmed that the discontinuous yielding accompanying the inhomogeneous deformation band began under plastic instability, while the band started to propagate when the force-balancing condition expressed in Eq. (3) satisfied the conditions of plastic stability. This indicates that the Lüders front represents a transition region from a state of plastic instability to one of stability. Moreover, the microstructural investigation revealed the formation of multiple ~20 μm-wide inclined deformation bands accompanying a large reduction in thickness at the Lüders front. Shear deformation with geometric softening is likely to occur under conditions of plastic instability;^28,29,30) therefore, the shear deformation may be the general deformation mode that occurs at the Lüders front. To verify this, the inclination angles at the Lüders front measured by DIC analysis were compared with those estimated by the shear deformation model (section 3). Figure 9 shows the relationship between q and the shear angles α (dashed line) and β (dotted line) in the sheet specimen under tensile load calculated under minimum shear strain. A shear angle α of 90° is expected when q=0. This angle decreases with increasing q, reaching 54.7° when q=0.5. The measured front angles in four TRIP steels and two ferritic steels are plotted in Fig. 9. The measured α decreases from an initial 90°, approaching 54.7° as q increases. The inclination angle of microscopic deformation zones, β, observed in the thickness plane of AF74 steel ranged from 44–54° (Fig. 6(d)). This angle was similar to that estimated by the shear deformation model at q=0.15 (β=47.3°). The above comparison strongly supports the hypothesis that the predominant feature at the Lüders front is shear deformation. Striated-contrast-type structures on the surface or multiple deformation zones nearly perpendicular to the tensile direction have been observed at the Lüders front of sheet-type tensile specimens in fine-grained low-alloy TRIP steel,²²⁾ fine-grained duplex TRIP steels,^24,37) and Mg alloys.³⁸⁾ The aforementioned features differ from previously reported microstructural models of Lüders fronts, which involve successive increases in the slip deformation region³⁹⁾ or successive additions of deformation ledges that simultaneously move along the interface.⁴⁰⁾ Further microscopic observations of various metals are required to confirm whether the formation of multiple localized shear-deformation bands is a general feature of Lüders fronts. Herein, SEM/EBSD observations of AF74 steel revealed microscopic shear-like deformations accompanied by an elongated α’-martensite phase with an inclination angle θ of 32–40° were observed in multiple localized shear-deformation bands. These microscopic shear-like deformations may be derived from the warm-rolled microstructure of the investigated steels (Section 4.2). The shear-like substructure in multiple shear bands may therefore be a feature specific to the investigated warm-rolled austenitic steel. Figure 10 illustrates the preceding discussion.

Fig. 9.

Relationship between q and the shear angles α (dashed line) and β (dotted line) in the sheet specimen under tensile load under the assumption that the orientation of the shear deformation is predominantly determined by the minimum shear strain. The error bar represents the range of the front angle during the propagation of the Lüders band. The angle β (AF74) is the inclination angle of the deformation band observed in the thickness plane of AF74 steel (Fig. 6(d)).

Fig. 10.

Illustrations of the microscopic characteristics of the Lüders front in AF74 steel, along with descriptions concerning the change in plastic stability. (a) Overall Lüders front and (b) enlarged features at the forward region of the front. (Online version in color.)

The factor that determines the propagation angle of the Lüders front was subsequently investigated. The front angle α with respect to the tensile direction is reportedly in the range of 55°–87° in sheet-type tensile specimens;^11,38,41) the angle α observed in the present study is also within this range (Table 2, Fig. 9). The angles α of 90° and 55.74° on the sheet surface correspond to the shear under a plane strain tension (i.e. q=0), and under uniaxial tension (q=0.5), respectively. The former reduces only the sheet thickness, while the latter reduces both the sheet thickness and width; therefore, the observed intermediate front angle, which was between 90° and 55.74°, indicates that the strain in the actual specimen lies somewhere between these two extremes. According to Butler,¹¹⁾ the front angle varies with the magnitude of Lüders strain, and the change in the shear direction occurs to minimize the shear-induced misalignment at the Lüders front (Fig. 11). Because the misalignment increases with Lüders strain, the preferred shear direction must shift from across the sheet width to that across the sheet thickness to more easily correct the misalignment via sheet bending.^11,31) Figure 12 shows a plot of the front angle α against the Lüders strain Δε_L of the 12Ni6Mn, CMn1 and CMn2 steels, in addition to the ferrite-related data reported by Butler.¹¹⁾ The angle clearly approached 90° as the magnitude of the Lüders strain increased. The change in the front angle during the propagation of the Lüders band can now be addressed. In AF35 steel (Fig. 5(b)), the initial front angle of 75° gradually changed direction culminating in an opposite direction (−78°). This variation can be explained by the minimization of the misalignment from the fixed tensile axis caused by the shear mode across the sheet width. The appearance of a symmetrical X-shaped band in the sheet surface of AF0 and CMn2 steels (Figs. 5(c) and 5(d)) and in the thickness plane of AF74 steel (Fig. 6(d)) can be also explained in the same manner. All of the geometrical characteristics of the Lüders front, including the angle dependence of the magnitude of Lüders strain, change in front angle during propagation, and concurrent appearance of a symmetrical X-shaped band, can be explained by the premise that the materials behave in such a way as to minimize or eliminate the misalignment caused by ‘shear’ deformation.

Fig. 11.

Schematics of the shear-deformation-induced misalignment from the tensile axis: (a) top view and (b) side view of a sheet tensile specimen. (Online version in color.)

Fig. 12.

Correlation between the Lüders front angle α and Lüders strain Δε_L in 12Ni6Mn metastable-austenitic steels and CMn ferritic steels. The data on ferritic steels reported by Butler¹¹⁾ is provided for comparison. (Online version in color.)

6. Conclusion

The deformation and transformation occurring at Lüders fronts in both metastable austenitic TRIP steels with various dislocation densities and ferritic steels were microscopically investigated to elucidate the mechanism of inhomogeneous Lüders deformation, along with a state of plastic stability. The following conclusions were drawn:

(1) The direct correspondence between the stress–strain curves with the strain distribution confirmed that the discontinuous yielding accompanying the inhomogeneous deformation band began under plastic instability, and the band propagates with satisfying the force-balancing between the band and the non-deformed region. This indicates that the Lüders front represents a transition region from a state of plastic instability to one of stability. Accordingly, a general rule relating Lüders strains $\mathrm{\Delta }{\epsilon }_L^{*}$ to lower yield stress ${\sigma }_{y0}^{*}$ can be defined as ${\sigma }_{y0}^{*}$ = $\mathrm{\Delta }{\sigma }_L^{*}$/$\mathrm{\Delta }{\epsilon }_L^{*}$, irrespective of the amount of deformation-induced α’-martensite in Lüders band or crystal structure of the steel.

(2) The measured inclination angle α of the Lüders front with respect to the tensile direction along the sheet surface changed from 55° to 90°; (i) with a reduction in the measured strain ratio q (=−ε_yy/ε_xx) in the Lüders band, and also (ii) with an increase in the magnitude of the Lüders strain Δε_L (≈ε_L,xx). The former change in α versus q agreed with the tendency calculated by the plasticity model, assuming the pure shear occurs under the minimum shear strain. The latter change in α versus Δε_L was attributed to the minimization of the misalignment from the fixed tensile axis induced by shear deformation at the Lüders front, which is achieved by changing the major shear from that across the width of the tensile specimens to that across their thickness. These behaviors demonstrate the formation of shear zones may be a general characteristic of Lüders fronts irrespective of the composition or type of steel.

(3) Observations of the sheet surface and cross-section of the Lüders front in austenitic TRIP steels by SEM/EBSD/μDIC confirmed that the formation of multiple ~20 μm-wide shear deformation zones accompanied the reduction in thickness. In addition, the deformation of austenite preceded the austenite-to-martensite transformation. However, the hierarchical shear-like substructure accompanied by an elongation of the α’-martensite in the numerous shear bands was considered a unique feature of the warm-rolled austenitic steels investigated herein.

Acknowledgements

The authors are grateful to Prof. Masaaki Sugiyama of Osaka University for inspiring us to begin the research on the Lüders banding in TRIP steels, and to Mr. Tomohito Tanaka in Nippon Steel Corporation for the support in the cooperative research between Osaka University and Nippon Steel Corporation.

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