Suppression of Band-Type Inhomogeneous Plastic Deformation in Thin Steel Plate
Research Center for Structural Materials, National Institute for Materials Science
Principal Researcher, Hai Qiu
1．Introduction
Plastic instability occurs in the deformation process of some crystalline materials in
the form of single or multiple plastic bands. Piobert [1] and Lüders [2] first reported
that plastic instability took place when a mild steel transited from an elastic to a plastic
state. Portevin and Le Châtelier [3] found that plastic localization existed in a certain
range of a plastic deformation process in aluminum-based alloys and low-carbon steels.
Their results showed that plastic instability can take place not only in the elastic-toplastic transition region, but also during the process of plastic deformation. The plastic
instability occurring in the former period is referred to as the Lüders phenomenon, andthat in the latter period as the Portevin-Le Châtelier (PLC) phenomenon. Somematerials have only one of the two types of plastic instabilities [4,5], and some have both[6]. The type of plastic instability can be identified from the typical characteristic on thetensile curve: a yield plateau for the Lüders deformation [4,5] or a jerky flow (a series ofserrations) for the PLC effect [7].
The Lüders deformation process is schematically shown in Fig. 1. Micro-yielding
occurs at one site (Stage I), and followed by the formation of a plastic band at that site
(Stage II). The band propagates across the whole area (Stage III). The formation of aplastic band was regarded to be associated with the pinning and unpinning of
dislocations around the interstitial C and N.

Stage I
σ1

Stage II
σ1

σ1

σ1

Micro-yielding
site
Inhomogeneous onsetof yielding

Stage IIIσ1

σ1
Band propagation

Band formation

Band propagation

Fig. 1 Schema of inhomogeneous plastic deformation (Lüders deformation)
Based on the understanding on the micromechanism of the formation of a plastic bandand the exhibition of the Lüders deformation on the stress-strain curve, twoconventional measures (Method 1 and Method 2) were taken to suppress the Lüders

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phenomenon (cf. Fig. 2). In method 1, IF steel in which the amount of C and N wascontrolled as low as possible was developed. Although decreasing the amount of C andN can lower the possibility of the formation of plastic bands, the Lüders phenomenon isstill present. In method 2, by using skin-pass mill, plastic pre-strain was introduced
over the whole bulk material to make the material exceed the yield plateau region. The
skin-pass processed steel has a round stress-strain curve, and it completely suppressedthe Lüders phenomenon. However, the large plastic pre-strain (2.23%) (almost two
times the Lüders elongation) decreases the resistance to fracture.

Fig. 2 Conventional methods for suppressing Lüders phenomenon
Steel has several elementary microstructures, such as ferrite, austenite, bainite,martensite, and pearlite. The Lüders phenomenon is known to occur in the ferrite [8] oraustenite [9] phase, and the strain-induced phase transformation from metastable
austenite to martensite leads to the PLC effect [10]. Due to the presence of ferrite oraustenite, plastic instability can also take place in multi-phase steels containing either
phase,

e.g.,

ferrite/pearlite

steel[4,5],

ferrite/austenite

steel[2,6],

andferrite/austenite/martensite steel [10]. However, the possible occurrence of plastic
instability in phases other than ferrite and austenite has not been reported in theliterature. The tempered martensite is a common phase for medium carbon steel. The
investigation on the inhomogeneous plastic deformation in medium carbon martensitic
steel is one object of the present study.
In metal forming process, occurrence of inhomogeneous plastic deformation,
especially band-like plastic deformation (i.e., Lüders phenomenon), lowers the surface
quality. Therefore, this kind of inhomogeneous plastic deformation is expected to be
suppressed. Because the conventional measures shown in Fig. 2 have someshortcomings, a new measure is expected. In commercial steels, less local sites are
suitable to form microyielding. Due to this reason, microyielding often takes place at
specific local sites, and correspondingly macroscopic yielding occurs and propagates at－ 26 －

these local sites, resulting in inhomogeneous plastic deformation. Based on this fact, aconcept is proposed in Fig. 3 that if sufficient local sites are provided for microyielding,microyielding will take place simultaneously at everywhere in the elastic region insteadof a local site, and as a result, the formation of a plastic band will be suppressed, andplastic deformation proceeds homogeneously. In this study, verifying the validity of thisconcept in tempered martensite is another object.

Stage I

Stage II

Homogeneous onset ofyielding by increasing
yielding sites

σ1

Homogeneous plastic
deformation

σ2

σ1

yielding site

Small

σ2

StrainLarge

Fig. 3 Concept of suppression of inhomogeneous plastic deformation
2．Experimental
2.1 Preparation of thin steel plate
An ingot with the chemical composition of 0.3 C, 1.5 Mn, and the balance Fe (in wt%)
was hot rolled to produce a thin steel plate with the cross-section of 60 mm wide and 1.6
mm thick. The hot-rolled thin steel plates 120 mm long × 60 mm wide × 1.6 mm thickwere heat treated to prepare the following samples.
(1) The samples for evaluating the inhomogeneous plastic deformation in martensite
Tempered martensite steel plates (QT steel) were prepared by the heat treatment
shown in Fig. 4. Since Lüders phenomenon in ferritic steel has been well investigated,
ferritic steel was selected as a reference steel. Ferritic steel plates (F steel) were also
produced (cf. Fig. 4).

RT, room temperature
WQ, water quenchingFC, furnace cooling

850℃, 5 min

600℃, 20 min

WQ

Sample size:
120 mm × 40 mm × 1.6 mm
Obtained samples: F steel, QT steel

800℃, 5 min

RT

FC

FC

F steelQT steelFig. 4 The heat treatment processes and the obtained samples－ 27 －

(2) The samples for verifying the proposed measure of suppressing the Lüders
phenomenon
Martensite phases with different dislocation densities were produced by the heat
treatment processes shown in Fig. 5. Four types of samples (S1 to S4) were prepared.
850℃, 5 min

RT, room temperature
WQ, water quenchingWC, water cooling

500℃, 0.5 min
WQ

Sample size:
120 mm × 40 mm × 1.6 mm
Obtained samples: S1, S2, S3, S4

500℃, 10 min
500℃, 2 min

RT

S1

S2 S3

S4

Fig. 5 The heat treatment processes and the obtained samples (S1 to S4)
The microstructures of the steel plates were examined by EBSD.
2.2 Tensile test
Dog-bone-type specimens with a parallel part 30 mm long, 8 mm wide, and 0.8~0.9mm thick (cf. Fig. 6) were machined from QT steel (two specimens, and their numbers:
QT-1 and QT-2) and F steel (two specimens, and their numbers: F-1 and F-2). Tension
tests were performed at room temperature and at a crosshead speed of 0.01 mm/s.

Fig. 6 The size of the dog-bone-type specimen and the spackle patterns on
the front surface of the specimen
2.3 Plastic deformation measurement by digital image correlation
The front surfaces of the tensile specimens were sprayed with white and black paintto make speckles for DIC analysis. An extensometer with a gauge length (GL) of 30 mm(equal to the length of the parallel part of the specimen) was attached to the back surface.

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The deformation process on the front surface was recorded successively at a timeinterval of 0.5 s using a camera. The digital images (area: 30 mm × 8 mm) obtained were
processed using VIC-2D software with a subset size of 9 pixels × 9 pixels (246 μm × 246μm) and a step of 5 pixels (137 μm) to produce the displacement field, strain field, andstrain-rate field. In the DIC operation, the displacement uncertainty is 0.02 pixels. The
DIC measurement area covers the whole GL. In the present study, the two results wereobtained from tension tests: (1) macroscopic stress-strain curves showing the global
image of tensile property; and (2) the evolution of plastic deformation in terms of thestrain field and strain-rate field.
3．Results
3.1 Lüders deformation in tempered martensite
Two tension tests were carried out for each steel. The obtained macroscopic stressstrain curves that show the average deformation behavior over a gauge length of 30 mm
are given in Fig. 7. To clearly identify the individual curves of one steel, the two curves(F-1 and F-2, QT-1 and QT-2) were intentionally placed 0.05 away from each other along
the macroscopic strain axis. The microstructures of F steel and QT steel are shown in
Fig. 7.

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Fig. 7 Macroscopic stress-strain curves of (a) ferrite/pearlite steel(F steel) and (b) tempered martensite steel (QT steel)Two tests were performed for each steel(F-1 and F-2 for F steel; QT-1 and QT-2 for QT steel)
The stress-strain curve of F steel shows a typical characteristic of Lüders deformation,i.e., a yield plateau. This means that the Lüders phenomenon inevitably occurs in F
steel. In contrast to F steel, the stress-strain curve of QT steel does not show clear
evidence of Lüders deformation and the PLC effect. This indicates that the occurrenceof plastic instability in QT steel cannot be identified only from the stress-strain curve.
Digital image correlation was used to analyze the plastic deformation behavior in thefollowing section. It is noted that a region of the stress-strain curve of QT steel isenclosed with a dotted rectangle. It was found that plastic instability takes place in therectangle, which will be described in detail later.
Lüders deformation or the PLC effect is characterized by the plastic band. Previous
studies [4,5] showed that the strain-rate field can effectively identify the moving plastic
bands. In the present study, the deformation process on the front surface of tension
specimens was digitalized by a camera, and the obtained digital images over the whole－ 30 －

tension process were used to analyze the strain and strain-rate field using twodimensional DIC. The analysis via the strain-rate field for the F steel and QT steel
shows that the plastic band occurs in two regions: (1) the elastic-to-plastic transition
region (i.e., the Lüders phenomenon), and (2) the region after the onset of the neckingof the tension specimen. This means that before the onset of the necking of the specimen,
only the Lüders phenomenon exists in both steels. It is well known that the necking of
the specimen induces plastic instability. This kind of plastic instability is not ourconcern, and it will not be discussed in the present study.
For the F steel, the Lüders phenomenon occurs mainly on the yield plateau. Elevenimages (image ① to image ⑪) on the yield plateau of the F-1 specimen (cf. Fig. 8)

Fig. 8 Evolution of plastic bands in ferrite/pearlite steel (F-1 specimen)
Eleven images (① to ⑪) on the macroscopic
stress-strain curve were taken－ 31 －

were selected, and their strain-rate fields over a gauge length of 30 mm are shown in
Fig. 8. It is generally believed that the pinning and the unpinning of dislocations cause
the formation of a Lüders band. To form a Lüders band, a certain level of stress isrequired. Apparently, at a given applied stress level, a site with a high stressconcentration more easily meets this critical stress condition than do other sites withlow stress concentrations. The shoulder of a specimen produces a high stressconcentration, as a result, providing an appropriate site for Lüders band nucleation. As
shown in image ①, two plastic bands (B-1 and B-2) have been formed near the rightand left shoulders of the specimen. It can be seen that the position of image ① on the
stress-strain curve is ahead of the yield point. This indicates that plastic bands havebeen formed ahead of the yield point, which agrees with the observations of the previous
study [4,5].
After band nucleation, B-1 propagates from right to left, while B-2 propagates toward
the right. Lüders band propagation is characterized by the moving of a leading band
front into the adjacent elastic region. It is also a repeated process of pinning andunpinning of dislocations. Naturally, the applied stress is required to exceed a criticalstress level. Band propagation velocity has been reported to be related to the magnitudeof applied stress [4]. As shown in Fig. 7, the applied stress on the yield plateau
essentially remains constant, but it fluctuates significantly at some points on the yieldplateau, and the applied stresses greatly decrease at those points. The applied stresscorresponding to image ③ is too low to exceed the required critical stress for unlocking
the dislocations, resulting in band disappearance. The strain-rate field of image ③
verifies the disappearance of the two plastic bands. When the applied stress recovers to
its original value, the two bands appear again at the original sites. In Fig. 8, four arrows
show four low stress levels. The two bands also disappear around the four stress levels.
B-1 and B-2 propagate oppositely, until coalescing with each other (cf. image ⑨). The
position of image ⑪ is nearly the end of the yield plateau, and the coalesced band
almost disappears at this point.
The deformation process of QT steel over the whole stress-strain curve was examined
via DIC. It was found that plastic instability occurred only within a certain range of thestress-strain curve. The range is enclosed in Fig. 7 by a dotted rectangle. The QT-1
specimen was taken to show in detail the evolution of the plastic instability in QT steel.
Twenty typical points on the macroscopic stress-strain curve of the QT-1 specimen wereselected (cf. Fig. 9). The strain-rate fields corresponding to the 20 points (image ① toimage ⑳) are given in Fig. 9. It can be seen that two bands (B-1 and B-2) have formednear the left and right shoulders of the specimen in image ①, respectively. The position－ 32 －

Fig. 9 Evolution of plastic bands in tempered martensite steel(QT-1 specimen) E. D., extension direction; P. D., propagation directionof image ① in the stress-strain curve indicates that a band nucleated ahead of the yield
point. The variation in the value of the strain rate within B-1 from image ① to image⑥ shows the evolution of B-1, in that the band first grows (①→②), and then decays,
and finally completely disappears (②→⑥). The position of B-1 in images ①→⑤ isalmost unchanged. This indicates that the band formation, growth, and disappearanceof B-1 took place at almost the same site, and apparently band propagation did not occur.
After band formation, B-2 grows along one direction from image ① to image ②, and－ 33 －

then extends along another direction, shown by an arrow in image ②, instead of in its
original direction. Image ③ shows the appearance of B-2 after the change in the
direction of extension. B-2 propagates from right to left from image ③ to image ④. In
image ④, a new band (B-6) on the left side of B-2 has been formed. B-6 and B-2
coalesced and extended across the width of the specimen to produce a new band withseveral branches (B-7) in image ⑤. The position of B-7 is almost unchanged from image⑤ to image ⑮. This indicates that B-7 did not propagate after band formation.
A band (B-3) was formed in image ④, which is after the upper yield point. This band
extends across the width of the specimen from image ④ to image ⑥ and thenpropagates from left to right. In image ⑧, a new band (B-5) was split out of B-3, and B3 continues to propagate until coalescing with B-7 in image ⑯. The coalesced band
gradually decays and finally disappears. The split band (B-5) gradually decays withoutband propagation and finally disappears in image ⑭. B-4 experiences a process of bandformation, growth, and disappearance from image ⑦ to image ⑬ that is similar to B1 and B-5. It can be seen from Figs. 8 and 9 that the Lüders deformation process in QT
steel is more complicated than that in F steel.
Because Lüders deformation occurs in the elastic-to-plastic transition region, theplastic region and the elastic region are simultaneously present in the Lüders
deformation process. This indicates that the strain distribution over the specimen isheterogeneous. In Fig. 10, one point on the stress-strain curve of (a) F steel (F-1
specimen) and (b) QT steel (QT-1 specimen) that is nearly in the middle of Lüders
deformation process is selected. The strain heterogeneity in the two steels is revealedby analyzing the strain distribution at this point. The experimental data of the F-1
specimen and the QT-1 specimen at this point are summarized in columns (a) and (b),respectively.
The strain-rate (ε� � ) field of the F-1 specimen (cf. Fig. 10(a2)) shows that two moving

plastic bands exist. The corresponding strain (ε� ) field was described in terms of two-

dimensional contour (Fig. 10(a3)). The strain field is roughly divided into three regions:the middle is the elastic region, and the others are plastic regions. To quantitatively
describe the strain distribution, a center line (line AB) is drawn in the two-dimensionalstrain contour, and the strain (ε� ) along the line is extracted and shown in Fig. 10(a4).

Three reference points, which were directly derived from the macroscopic stress-strain

curve, are also given: the Lüders strain (ε� ), the average strain over the gauge length(GL), and the elastic limit. The Lüders deformation process is characterized by theformation of Lüders bands (single or multiple bands), followed by band propagation
over the whole specimen. The strain corresponding to the ending point of the Lüders

－ 34 －

Fig. 10 Local strain (εx) distributions in (a) F steel (F-1 specimen) and (b) QTsteel (QT-1 specimen) over the gauge length at a point on the macroscopic stressstrain curve located at almost the middle of the Lüders deformation process. The
local strain (εx) distributions along the AB and CD lines were extracted. The
average strain over the GL is the macroscopic strain of the stress-strain curve
corresponding to the point of interest. εL, Lüders strain.
deformation process is referred to as the Lüders strain. For the steel with a clear yieldplateau, the ending point of the Lüders deformation process is generally around theending point of the yield plateau. This point in F-1 and QT-1 specimens was directly
determined by the strain-rate field in the present study. The average strain over the GL
is the macroscopic strain of the stress-strain curve of the F-1 specimen corresponding to
the point of interest. Careful examination of the macroscopic stress-strain curve in themacroscopic elastic region (i.e., from point zero to the upper yield point) shows that－ 35 －

stress linearly increases with strain only within the initial region (from point zero to acertain stress level); beyond this region, stress gradually deviates from this straight line.
The maximum macroscopic strain of the linear range is denoted as the elastic limit.
Fig. 10(a4) shows that the middle region is in the elastic state, and the correspondingstrain is almost equal to the elastic limit. The width of the two moving plastic bands
with respect to Line AB is marked by an arrow (↔) in Fig. 10(a2). The range of Line AB
enclosed by the band width is shown by two vertical dotted lines in red in Figs. 10(a3)and (a4). Fig. 10(a4) shows that the strain varies significantly within the band widthfrom an elastic strain to a large plastic strain (close to the εL). The average strain overthe GL is the average value of the elastic and plastic regions.
The strain heterogeneity in the QT-1 specimen was analyzed in a similar way in Fig.
10(b). As shown in Fig. 10(b2), the shape of the moving bands is more complicated thanthat in the F-1 specimen. The local strain along Line CD (cf. Fig. 10(b3)) is shown in Fig.
10(b4). The strain in the elastic region (middle region) is close to the elastic limit. The
width of three moving bands with respect to Line CD is shown by an arrow (↔) in Fig.
10(b2). The strain variation within the band width is similar to that in the F-1 specimen.
The maximum plastic strain within the band width also approaches the εL. The strain
distribution in the QT-1 specimen is more complicated than that in the F-1 specimen.
3.2 Suppression of Lüders deformation
Four samples (S1 to S4) were prepared. They have the same microstructure(martensite), but their dislocation density is different (the dislocation density of S1 >that of S2 > that of S3 > that of S4). The plastic deformation processes in the four
samples were measured via tension test. The macroscopic stress-strain curves areshown in Fig. 11. It can be seen that for the four samples, as the dislocation density
decreases, the strength gradually decreases, and the shape of the stress-strain curvechanges.

－ 36 －

2000

S1
S2

Stress (MPa)

1500
S3

1000
S4

500
0
0.00

0.02

0.04
0.06
Strain0.08

0.10

Fig. 11 Macroscopic stress-strain curves of various martensite phases
The evolution of plastic deformation over the whole front surfaces of tensile specimens
was measured with the DIC technique. It was found that Lüders phenomenon was notpresent in S1 and S2, but was present in S3 and S4. The proof stress point (0.2) on eachstress-strain curve shown in Fig. 11 was selected. The strain rate field corresponding tothis point is shown in Fig. 12. It is shown that plastic band is only present in S3 and S4.
This result indicates that plastic instability can be suppressed by controlling dislocation
density.

Fig. 12 The plastic strain distribution in S1 ~ S4 samples
at the stress point of the proof stress (0.2)－ 37 －

4．Conclusions
The plastic deformation behavior of medium-carbon tempered martensite steel atroom temperature has been investigated. The results obtained regarding the plasticity
of tempered martensite were as follows.
(1) The Lüders deformation phenomenon is present, but the Portevin-Le Châtelier
phenomenon is not found in the medium-carbon tempered martensite steel.
(2) The elastic and plastic regions are simultaneously present in the Lüders
deformation process. The local strain in the elastic region is close to the elastic limit.
The variation in strain within a Lüders band is significant, monotonously increasingfrom an elastic strain to a large plastic strain that is close to the Lüders strain.
(3) The local strain distribution in tempered martensite steel during the Lüders
deformation process is more complicated than that in ferrite steel.
(4) Lüders deformation can be suppressed by controlling the dislocation density.
Acknowledgments
This study was supported by the Technical Research Aid of JFE 21st Century
Foundation. The grants are greatly appreciated.
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