2. gated. The influence of monotonic tension preload on cyclic plas-
tic deformation and the persistence of inhomogeneous plastic
deformation during multiple-step cyclic loading were studied. At-
tempts were made to study the dislocation structures and the evo-
lution of dislocations during inhomogeneous deformation by us-
ing transmission electron microscopy ͑TEM͒. The mechanisms
involved in the initiation and propagation of inhomogeneous cy-
clic plastic deformation were discussed.
2 Experiments
The as-received SAE 1045 steel was in a hot rolled state with
the following chemical compositions: 0.46 wt.% C, 0.27 wt.% Si,
0.70 wt.% Mn, 0.019 wt.% P, 0.027 wt.% S, with balance of Fe.
Cylindrical specimens were machined after normalizing the mate-
rial at 815°C for 8 hours. The specimens were identical in material
and heat treatment to those used in a previous study ͓3͔. A typical
microstructure after normalization is shown in Fig. 1. The volume
fraction of pearlite was approximately 55% and the mean ferrite
grain size was 10 m. The uniform gage section of the specimen
had a diameter of 12 mm and a gage length of 25.4 mm. The
gauge length surface was machined followed by polishing with
fine sand papers with grit size 600.
To measure local strains, a Micro-Measurements Group EA-06-
062MD-120 10-element strip gage was bonded on the surface
within the gage section of the specimen. The strip strain gage
length was 20 mm. Each gage element had a resistance of 120 ⍀
and the gage factor was 2.01. Each strain gage had an area of
1.57ϫ1.57 mm, with a spacing between strain gages in the strip of
2.1 mm. The row of strain gages was aligned with the specimen
axis identical to that used in a previous study ͓3͔. A 25.4 mm gage
length extensometer was attached to the specimen in the gage
section to measure the average deformation.
Experiments were conducted at room temperature in air. Cyclic
testing was conducted at a frequency of 0.25 Hz. During an ex-
periment, the load, extensometer displacement, and the outputs of
all the 10 strain gages were recorded simultaneously. Two hundred
data points were recorded for each variable during a loading
cycle. To distinguish the deformation measured by the extensom-
eter and the strain gages, the strain obtained from the extensom-
eter is referred to as ‘‘extensometer strain’’ while the strain gage
measurements are denoted as ‘‘local strains’’.
Thin foil TEM samples were prepared with the material cut
from the designated areas in the testing specimen using a low
speed diamond saw. The normal direction of the thin foils was
parallel to the specimen axis. The thin foils were mechanically
ground on both sides with fine sand papers down to a thickness of
100 m. Discs of 3 mm diameter were punched from these foils.
The discs were then electropolished using a twin jet technique in
a solution of 10 vol.% perchloric acid mixed with 90 vol.% eth-
anol at Ϫ30 °C until perforation. The samples were examined
with a JEOL 200CX TEM operated at 200 KV.
3 Experimental Results
3.1 Lu¨ders Bands Under Monotonic Loading. Monotonic
tension was conducted under stroke ͑displacement͒-controlled
loading mode. The extension rate was 0.002 mm/s that corre-
sponded to an approximately 5ϫ10Ϫ5
/s strain rate. The results
are shown in Fig. 2. The upper yield stress is 420 MPa and the
lower yield stress is 400 MPa. The distinct plateau in the stress-
extensometer strain curve corresponds to the formation and propa-
gation of Lu¨ders bands. The full Lu¨ders strain ͑the length of the
plateau͒ is 0.98%. The dots and numbers in the stress-
extensometer strain curve ͑Fig. 2͑a͒͒ denote the moment that the
corresponding strain gage on the specimen showed the onset of
plastic deformation. In other words, a dot represents the specific
value of the extensometer strain at which the Lu¨ders front reached
the area covered by the numbered strain gage. The onset of plastic
deformation of each strain gage is indicated by an abrupt increase
in local strain, as shown in Fig. 2͑b͒. The area under Gage 10 was
the last area that experienced plastic deformation. The particular
moment did not correspond to the end of the plateau in the stress-
strain curve ͑Fig. 2͑a͒͒. Two factors are responsible for this dis-
crepancy. The extensometer gage length ͑25.4 mm͒ was longer
than the strip strain gage length ͑20 mm͒. More importantly, the
strain at the Lu¨ders front was significantly smaller than the full
Lu¨ders strain.
Detailed local strains shown in Fig. 2͑b͒ can provide a better
insight of the Lu¨ders band propagation. Both local strains and
extensometer strain are shown with respect to time. A number
shown on the side of each curve denotes a specific strain gage.
The strain gages in the strip were numbered starting from the top
of the gage section. After the extensometer strain reached 0.002,
which corresponded to the upper yield stress, the area covered by
Gage 2 and Gage 3 started to deform plastically. The process
continued with the plastic deformation occurring in the areas cov-
ered by Gages 4, 5, 6, 7, 8, 9, and 10. Except Gage 1, all the other
gages displayed a certain pattern. Among the ten strain gages,
eight gages ͑Gage 3ϳ10͒ experienced the abrupt deformation in
Fig. 1 Microstructure of SAE 1045 steel after normalization
Fig. 2 Monotonic tension behavior of 1045 steel: „a… mono-
tonic stress-extensometer strain curve; and „b… variations of
local strains
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3. such a way that the end of the strain burst was followed by the
start of the strain burst of the neighboring strain gage ͑Fig. 2͑b͒͒.
Considering the finite length of the strain gage, this observation
indicates that the value of the burst strain is approximately 0.66%.
Since the strain burst was followed by strain hardening in the
subsequent loading for a given location, a simple mathematic
speculation can lead to the conclusion that the ‘‘real’’ strain at the
Lu¨ders front should be a little smaller in value than that measured
using a small but finite length strain gage.
A vertical line in Fig. 2͑b͒ denotes the end of the plateau in the
stress-extensometer strain curve. A common conclusion from
early research is that the end of the plateau is the starting point of
homogeneous deformation in the gage section ͓1͔. However, the
results shown in Fig. 2͑b͒ clearly indicate that at the end of the
plateau, the local deformation was still inhomogeneous. The inho-
mogeneous deformation persisted in the work-hardening zone.
3.2 Localized Cyclic Plastic Deformation. A fully re-
versed stress-controlled tension-compression experiment was con-
ducted for 1550 cycles and the results are summarized in Fig. 3.
The stress amplitude was 320 MPa which was less than the lower
yield stress ͑400 MPa͒ of the material. The variations of local
strain amplitudes and mean strains with the number of loading
cycles were used to characterize the cyclic strain history. The
cyclic deformation process consisted of three stages: an incuba-
tion stage between 1ϳ230 cycles, a propagation stage between
230ϳ750 cycles, and a saturation stage after 750 cycles. During
the incubation stage, the material displayed almost purely elastic
deformation. After about 230 cycles, cyclic plastic deformation
was initiated in the area covered by Gage 2 ͑upper part of the gage
section͒ and propagated gradually into the two opposite directions
with increasing number of loading cycles. The shaded bars in-
serted in Fig. 3͑a͒ illustrate the propagation of the local plastic
deformation. The length of a bar represents the gage length cov-
ered by the strain gage strip ͑20 mm͒ and the shaded area denotes
the plastically deformed region. A threshold of 0.02% plastic
strain amplitude was used to define the onset of cyclic plastic
deformation. The local deformation was significantly inhomoge-
neous during the propagation stage. Inhomogeneous and nonzero
local mean strains developed simultaneously although symmetric
cyclic stress was applied.
After approximately 750 loading cycles, the strain amplitude
and mean strains became stabilized and the whole gage section
experienced cyclic plastic deformation. However, the stabilized
strain amplitudes and mean strains within the gage section were
not uniform ͑Fig. 3͑b͒͒. At Gage 2, where cyclic plastic deforma-
tion occurred first, the strain amplitude was the largest while the
mean strain was negative. At Gage 10, where cyclic plastic defor-
mation occurred last, the strain amplitude was the smallest while
the mean strain was positive. The non-uniform strains persisted
during subsequent cycling.
The influence of stress amplitude and mean stress on incuba-
tion, propagation, and saturation strain amplitude is listed in Table
1. A larger stress amplitude leads to a shorter incubation stage, a
shorter propagation stage, and a larger saturation strain amplitude.
A tensile mean stress slightly shortens the incubation period but
has no significant influence on the saturation strain amplitude.
However, a tensile mean stress reduces the propagation period
greatly. These results are consistent with those reported previously
͓3–8͔.
A two-step experiment was designed to examine the influence
of the monotonic tension preload on cyclic plastic deformation.
Monotonic tension was terminated when the extensometer strain
reached 0.5%. After the monotonic loading, the Lu¨ders bands had
propagated over a half of the gage section covering Gage 1 to
Gage 6 ͑Fig. 4͑a͒͒. The rest of the area covering Gage 7 to Gage
10 remained plastically undeformed.
Following the monotonic preload, a fully reversed stress-
controlled cyclic loading with a stress amplitude of 320 MPa was
applied. The magnitudes of local strains in the cyclic loading are
shown in Figs. 4͑b͒ and ͑c͒. One noticeable observation is the
elimination of the incubation period due to the monotonic preload.
In the area that had been plastically deformed in the monotonic
preload, the strain amplitude decreased slightly to the saturation
values. In the area that had not been plastically deformed in the
monotonic loading, incubation stage disappeared. The propagation
of cyclic plastic deformation occurred immediately upon the ap-
plication of the cyclic loading and the strain amplitudes of Gages
7, 8, and 9 increased gradually to the saturation values. The propa-
gation stage was noticeably shortened. The saturation strain am-
plitudes were nearly identical to those shown in Fig. 3, knowing
that the cyclic load magnitudes for the two cases were identical.
Results shown in Fig. 4͑c͒ reveal that the local mean strains were
scattered and distributed irregularly.
A three-step stress-controlled cyclic tension-compression ex-
periment was used to examine the influence of prior cyclic plas-
ticity history on further cyclic deformation at a different mean
stress. The stress amplitude was 300 MPa for all the three loading
steps and the mean stresses were 100 MPa, 0, and Ϫ100 MPa in
Step 1, Step 2, and Step 3, respectively. The numbers of loading
cycles performed in Step 1, Step 2, and Step 3 were 920, 1250,
and 1250, respectively. The results are summarized in Fig. 5 and
Table 2.
Cyclic plasticity in Step 1 eliminated the incubation and propa-
gation stages in Step 2 and Step 3. In each loading step, the strain
amplitudes do not change significantly, but the mean strain in-
Fig. 3 Cyclic deformation of 1045 steel under fully reversed
stress-controlled uniaxial loading: „a… variations of local strain
amplitudes and propagation of plastic zone; and „b… variations
of local mean strains
Table 1 Influence of stress amplitude and mean stress on cy-
clic plastic deformation
⌬/2 ͑MPa͒ m ͑MPa) Ni ͑Cycles) Np ͑Cycles) ⌬/2 ͑%͒
350 0 12 48 0.35
320 0 230 470 0.25
270 0 6000 8000 0.15
320 80 160 ϳ5 0.25
300 100 240 5 0.23
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4. creased gradually in the direction of mean stress ͑ratcheting͒. The
local mean strains were non-uniform. The heterogeneity persisted
and increased with the loading steps. It was reported ͓7,20͔ that
under multiple-step loading, materials exhibited a strong memory
of the previous loading history, and such memory had a great
influence on the subsequent ratcheting.
Also shown in Figs. 3 and 4 are the average local strains and
the extensometer strains. The average local strain was the average
of the strain values taken from the 10 strain gages. One purpose to
compare the two values was to monitor the possible breakage or
debonding of strain gages during an experiment. The small differ-
ence between the average local strain value and the extensometer
measurement was mainly due to the difference between the exten-
someter gage length ͑25.4 mm͒ and the length of the strain strip
͑20 mm͒, knowing that the deformation within the gage section
was non-uniform.
3.3 Dislocation Substructures. Under fully reversed stress-
controlled tension-compression loading with a stress amplitude of
320 MPa, four companion specimens were tested. The experi-
ments were terminated at different cycles and dislocation sub-
structures at different strain gage sites were examined. The loca-
tions where the TEM samples were taken and the number of
loading cycles tested are indicated in Fig. 3͑a͒. The loading cycles
corresponded to the incubation, propagation, and saturation
stages.
During the incubation stage, the dislocation substructure had no
obvious change as compared to that of the annealed condition. A
typical dislocation arrangement is shown in Fig. 6 for a sample
taken from the specimen that has been loaded for 200 loading
cycles. The dislocation density in ferrite was low. Individual dis-
locations could be identified clearly. There was no detectable in-
dication of plastic deformation in the ferrite grains.
Fig. 4 Influence of monotonic tension preload on cyclic defor-
mation: „a… local tensile strains under monotonic tension pre-
load; „b… variations of local strain amplitudes with loading
cycles in subsequent cyclic loading; and „c… variations of local
mean strains with loading cycles in subsequent cyclic loading
Fig. 5 Cyclic deformation of 1045 steel under three-step
stress-controlled cyclic loading: „a… variations of strain ampli-
tudes; and „b… variations of mean strains
Table 2 Influence of mean stress and strain history on ratch-
eting deformation
Step m ͑M Pa) Ni ͑Cycles) Np ͑Cycles) ⌬/2 ͑%͒ m (%)
1 100 240 5 0.23 1.4ϳ1.6
2 0 0 0 0.23 0.5ϳ0.85
3 Ϫ100 0 0 0.2 Ϫ0.58ϳϪ1.2
Fig. 6 Typical dislocation arrangement during the incubation
stage
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5. Figure 7 shows the typical dislocation arrangements during the
propagation stage. The experiment was terminated after 420
cycles, which was within the stage of cyclic plastic propagation
͑Fig. 3͑a͒͒. The local strains were significantly inhomogeneous
and different dislocation substructures were observed at different
strain gage sites ͑see Fig. 3͑a͒ and Fig. 7͒. Since ferrite grains in
the material were randomly oriented, the resolved shear stress
amplitudes acting on the primary slip systems of individual ferrite
grains were different. In addition, dislocation arrangements in in-
dividual ferrite grains differed even at the same strain gage site.
Here, the representative dislocation arrangements in the most se-
riously deformed ferrite grains are presented. On the area covered
by Gage 10 where the plastic deformation was the smallest as
compared to the rest of the area of the gage section, only a few
ferrite grains were plastically deformed. Shown in Fig. 7͑a͒ is a
typical dislocation configuration in plastically deformed ferrite
grains. Loose dislocation tangles indicated the initiation of plastic
deformation. These dislocation tangles were distributed randomly
within the ferrite grains, indicating that the locations for the ini-
tiation of plastic deformation were randomly distributed within
the ferrite grains. Figure 7͑b͒ shows the dislocation configuration
on the area covered by Gage 6 where the strain amplitude was
medium. Dislocation density increased sharply by dislocation
multiplication and thick dislocation multipole veins distributed
randomly within the ferrite grains.
At Gage 2, plastic deformation was the largest over the gage
section. Most ferrite grains were plastically deformed. A typical
dislocation arrangement in the ferrite grains is shown in Fig. 7͑c͒.
Further plastic deformation resulted in an increase in the diameter
and dislocation density of the veins. The process reached a point
where local instabilities in the veins occurred due to dislocation
interaction and annihilation. This was followed by the evolution
of dislocation substructures into thinner yet denser dislocation
walls. The dislocation density between the walls was very low.
Although the build-up of macroscopic strain was complete at
the end of the propagation stage, continued cyclic loading led to a
further development of dislocation structures. The dislocation
configuration at the saturation state at a cycle number of 1550 is
shown in Fig. 8. In general, long and dense walls, elongated cells,
and equiaxial cells were observed. There were no obvious quali-
tative differences among dislocation arrangements for the material
within the gage length.
At the stage of cyclic plastic deformation propagation, the de-
formation process can be accomplished by single slips and the
influence of the grain/phase boundaries is not significant. How-
ever, in the subsequent transition to the cell structures, multiple
slip process is a necessity. At this point, the grain/phase bound-
aries play an important role. The plastic deformation in grain/
phase boundaries requires more activated slip systems due to the
constraints introduced. TEM observations confirmed that the mul-
tiple slip systems were always activated at first at the grain/phase
boundaries that resulted in the formation of the cell structures. A
typical TEM micrograph at the saturation stage is shown in Fig. 9.
Arrow A points to a grain boundary and Arrows B and C indicate
phase boundaries. With continued cyclic loading, the dislocation
cells were developed further into the central zone of the ferrite
grains.
4 Discussions
Strain gages provide a practical and accurate method to mea-
sure the local deformation. The results shown in Fig. 2͑b͒ are
different from those of previous research on Lu¨ders bands in sev-
Fig. 7 Typical dislocation arrangements during the propaga-
tion stage „at cycle 420…: „a… at Gage 10 „low local strain ampli-
tude: 0.165%…; „b… at Gage 6 „medium local strain amplitude:
0.2%…; and „c… at Gage 2 „high local strain amplitude: 0.23%…
Fig. 8 Typical dislocation arrangement at the saturation stage
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6. eral aspects. It is generally asserted that during propagation of
Lu¨ders bands, plastic strain in the plastically deformed zone is
equal to the full Lu¨ders strain. Upon the start of work-hardening,
the deformation is uniform. The current work reveals that inho-
mogeneous deformation persists into the work-hardening stage.
The Lu¨ders bands results reported in the current investigation
are inconsistent with those of the previous studies. The testing
material, the specimen geometry, and the techniques used to mea-
sure the local strains may have contributed to the discrepancy. A
thin and flat specimen was often used in the investigation of Lu¨d-
ers bands ͓21–32͔. In the current investigation, specimens of a
normalized 1045 steel with a circular cross section were tested.
An obvious advantage in using a specimen with a flat surface is
the convenience and suitability for surface observations. The pos-
sible influence of the testing specimen geometry on the Lu¨ders
band observations may require further investigation. Usually, sur-
face scratches ͓21–23,33͔, surface grid ͓24,25͔, contrast interfer-
ometer ͓22,23,26͔, stresscoat ͓34͔, electroplating of the specimen
prior to testing ͓27͔, optical ‘‘To¨pler Schlieren’’ method ͓21,33͔,
critical macro-illumination ͓28͔, gridreflection ͓26͔, clip-
extensometer ͓29,35͔, high speed camera ͓30͔, and infrared cam-
era ͓31͔ were used to observe the Lu¨ders bands and to identify the
local strains. These methods do not provide an accurate quantita-
tive measurement of the local strain. It is noted that by using four
small strain gages bonded on the specimen, a similar phenomenon
to that shown in Fig. 2͑b͒ was observed in an annealed aluminum
alloy under monotonic tension ͓32͔.
The formation and propagation of Lu¨ders bands in irons and
carbon steels under monotonic tension are generally attributed to
the strong interaction of interstitial atoms with dislocations,
known as Cottrell atmospheres ͓1͔. In the Cottrell atmospheres,
interstitial atoms segregate to dislocations and pin them in posi-
tions to lower the lattice distortion energy. In a normalized 1045
steel, the yield stress of pearlite is much higher than that of ferrite
͓36–38͔. In the current investigation, the strain amplitude was low
and plastic deformation took place mainly in the ferrite grains.
The ferrite grains contain small amount of carbon and nitrogen
atoms in solid solution and the Cottrell atmospheres play an im-
portant role in plastic deformation. In monotonic tension, the
abrupt increase of strain at the Lu¨ders front can be ascribed to the
dislocations unpinned from the Cottrell atmospheres. Their mo-
tion leads to an abrupt multiplication of new dislocations. Subse-
quently, dislocations multiply further but slowly.
Previous studies have confirmed that cyclic plasticity of low
carbon steels can eliminate Lu¨ders bands phenomenon in a sub-
sequent monotonic tension ͓10–14,39–41͔. This elimination oc-
curs in the first half cycle for stress amplitude higher than the
yield point. For stress amplitude below the yield point, the Lu¨ders
strain decreases continuously with increasing number of cycles.
The current investigation reveals that a monotonic tension preload
eliminates the incubation stage and either shortens or completely
eliminates the propagation stage of the cyclic plastic deformation
in the subsequent cyclic loading. It is reasonable to infer from the
observations that the formation and propagation of Lu¨ders bands
under monotonic tension and the inhomogeneous cyclic plastic
deformation phenomenon should be closely related in terms of
mechanism.
Klesnil and Luka´sˇ ͓10,11͔ used the Cottrell atmospheres to in-
terpret successfully the mechanism of the cyclic softening of low
carbon steels under stress-controlled cyclic loading with the stress
amplitude being below the yield stress. A low carbon steel ͑0.07%
C, 0.48% Mn, 0.3% Si, 0.018% P, 0.019% S͒ with a mean grain
size of ferrite of 40 m was used. The yield stress of the material
was 278 MPa. Stress was cycled between 228 MPa and 250 MPa.
It was noted that cyclic plasticity was initiated at the first loading
cycle for the cases under consideration ͓10,11͔. There exist a few
ferrite grains in the material where certain dislocations are less
effectively pinned by interstitials and can move at a stress level
below the yield stress. In the first half cycle, a small number of
such dislocations in some isolated grains are unpinned. This pro-
cess is repeated in each loading cycle followed. As a result, the
number of grains with free dislocations and bulk plastic strain
amplitude increase with increasing loading cycles. On the other
hand, fatigue hardening process, namely the increase of disloca-
tion density, and formation of dislocation structure occur in these
grains that lead to a decrease in plastic strain amplitude. The su-
perposition and interaction of these two processes result in the
observed cyclic softening and hardening phenomenon.
It is noticed that the microstructure and cyclic deformation be-
havior of the 1045 steel used in the current investigation are dif-
ferent from those of the low-carbon steel. In the current study, the
size of the ferrite grains is much smaller and they are separated
and constrained by the pearlite grains. The yield stress of the
ferrite grains is much higher than that in the low carbon steel. For
the 1045 steel under stress-controlled cyclic loading with stress
amplitude much lower than the lower yield stress, the incubation
stage is long and cannot be ignored. During the cyclic softening
stage, the distribution of local strains is significantly inhomoge-
neous. The local deformation depends not only on the number of
the ferrite grains with free dislocations but also on the extent of
plastic deformation of these ferrite grains. Therefore, the mecha-
nism of cyclic plasticity of the 1045 steel should be different from
that of the low-carbon steel.
For the 1045 steel under stress-controlled cyclic loading with a
stress amplitude much lower than the lower yield stress, deforma-
tion is almost purely elastic during the incubation stage. The dis-
location configuration is identical to that in the original normal-
ized condition with no observable change in the dislocation
density. Traditionally, phase/grain boundaries are viewed as the
most possible sites of stress concentration where pre-yield micros-
train may occur ͓42͔. However, during the incubation stage, the
concentration and pile-up of dislocations at the phase/grain
boundaries were rarely observed. As shown in Fig. 7, the loose
dislocation tangles, indicators of the initiation of plastic deforma-
tion, distribute randomly within the ferrite grains. Significant
changes in dislocation density and configuration can be observed
only during the propagation phase. During the saturation stage,
dislocation cells are found to form at first at phase/grain bound-
aries. The formation of the cells gradually propagates into the
interior of the grains.
The microscopic mechanism involved in the inhomogeneous
cyclic plastic deformation can be discussed based on the experi-
mental observations. In the incubation stage, the dislocations
pinned by the Cottrell atmospheres relax gradually under cyclic
stressing and become free and mobile dislocations. At the propa-
gation stage, the movement of free dislocations leads to the mul-
tiplication, interaction, and annihilation of dislocations. The dis-
Fig. 9 Dislocation cells are formed at first at the grainÕphase
boundary „at 1550th cycle in a saturation stage…
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7. location configurations evolve into dislocation tangles, veins, and
dense walls. In the later phase of propagation and the early phase
of saturation, multiple slip systems are activated at first near the
phase/grain boundaries and then propagate into the interior of the
ferrite grains. The dislocation arrangements eventually evolve into
cell structures.
The viewpoint is supported by the experimental results from the
two-step loading experiment shown in Fig. 4. In the plastically
deformed zone, the dislocations in the ferrite grains had been
unpinned and multiplied sufficiently. As a result, the incubation
and propagation stages were eliminated in the subsequent cyclic
loading. In the area where deformation remained macroscopically
elastic after the tension preload, the incubation stage disappeared
in the second loading step. This suggests that during the propaga-
tion process of Lu¨ders bands under the monotonic tension, the
dislocations in the plastically undeformed zone have been un-
pinned from the Cottrell atmospheres. However, these dislocations
do not move or only make local short-ranged movement, and
immediate generation of new dislocations has not yet taken place.
The aforementioned discussion was concentrated on the inho-
mogeneous cyclic plasticity, the associated dislocation substruc-
tures, and possible mechanisms. However, the underlying mecha-
nism for localized cyclic plasticity has not been fully explained.
The mechanism governing the propagation of cyclic plastic defor-
mation from one area to another in the specimen remains unclear.
The mechanism operative in the localized monotonic plasticity
associated with the formation of Lu¨ders bands in irons and carbon
steels probably can provide some enlightenment.
During monotonic tension of irons and carbon steels, Lu¨ders
bands are believed to nucleate at points of high local stresses, such
as fillets, precipitates, and grain boundaries ͓1͔. The propagation
of Lu¨ders bands depends on factors such as crystal structure, grain
size, shape of the sample, composition, and microstructure ͓43͔.
The propagation of Lu¨ders bands in single crystals probably is
engendered by the local constraints and stress concentrations set
up between the deformed and undeformed regions. In a fine-
grained mild steel, the edge of Lu¨ders bands appears sharp. It is a
reasonable speculation that deformation is spreading grain by
grain through the specimen. Dislocations in the yielding grain run
up to the grain boundary, and form pile-up arrays of dislocations,
whose presence increases the stress in the neighboring grain until
a dislocation source there becomes unpinned and the grain yields.
As the grain size increases, the edge of Lu¨ders band becomes
diffuse. As pointed by Hall ͓1͔, ‘‘This effect arises approximately
at the point where Hutchison’s curve ͓44͔ for the upper yield
stress with respect to grain size intersects the lower yield stress. In
other words, the stress differences are here so low that any small
inhomogeneity in grain size will cause part of the band front to
progress far in advance of the remainder.’’
The inhomogeneous cyclic plastic deformation has a similarity
to the propagation of the Lu¨ders band. However, whether or not
similar or identical mechanisms exist for the two phenomena re-
quires further investigation. In addition, the effect of the stress
state on the inhomogeneous cyclic plasticity is also a subject of
future research.
5 Conclusions
From the experimental observations on the inhomogeneous
plastic deformation of 1045 steel, the following conclusions can
be drawn:
1. Under monotonic tension, the burst strain of Lu¨ders front is
much lower than the full Lu¨ders strain. The local deformation
remains inhomogeneous even at the work-hardening stage after
the termination of the Lu¨ders band propagation.
2. For stress-controlled tension-compression cyclic loading
with the maximum stress lower than the lower yield stress of the
material, cyclic plastic deformation is characterized by three
stages: incubation, propagation, and saturation. The local strains
are significantly inhomogeneous during the propagation stage and
the inhomogeneity persists at the saturation stage.
3. The monotonic tension preload can eliminate or shorten the
incubation and propagation stages of the cyclic plasticity in the
subsequent cyclic loading.
4. Inhomogeneous cyclic plasticity persists under multiple-step
loading. The inhomogeneity is evidenced mainly by the persistent
difference in local mean strains.
5. During the incubation stage, the dislocation structures do not
change significantly. With increasing loading cycles, the disloca-
tion structures evolve in a sequence of loose tangles, thick veins,
long walls, elongated cells, and equiaxial cells.
6. Under cyclic stressing, the dislocations pinned by the Cot-
trell atmospheres in the ferrite grains are unpinned gradually and
become free and mobile dislocations. The movement of free dis-
locations leads to the multiplication, interaction, and annihilation
of dislocations. Dislocation cells are formed at first at the grain
boundaries with the activation of multiple slip systems followed
by the cells propagation into the interior of the ferrite grains.
Acknowledgment
Financial support was provided by the National Science Foun-
dation ͑CMS-9984857͒. TEM experiments were conducted at the
National Center for Electron Microscopy at Lawrence Berkeley
National Laboratory. The assistance of Drs. Eric Stach, Tamara
Radetic, and Chengyu Song is greatly appreciated.
Nomenclature
⌬/2ϭ stress amplitude
m ϭ mean stress
R ϭ stress ratio ͑minimum stress over maximum stress͒
Ni ϭ number of loading cycle of the incubation stage
Np ϭ number of loading cycle of the propagation stage
ϭ extensometer strain
⌬/2 ϭ stabilized strain amplitude
m ϭ mean strain
L1 ϭ strain at the Lu¨ders front
L ϭ full Lu¨ders strain
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