Clear band formation simulated by dislocation dynamics
Dislocation Dynamics simulations of dislocations gliding across a random populations of Frank loops are presented. Specific local rules are developed to reproduce elementary interaction mechanisms obtained in Molecular Dynamics simulations. It is shown that absorption of Frank loops as helical tu...
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| Veröffentlicht in: | Вопросы атомной науки и техники |
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| Datum: | 2009 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2009
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| Zitieren: | Clear band formation simulated by dislocation dynamics / T. Nogaret, D. Rodney, M. Fivel, C. Robertson // Вопросы атомной науки и техники. — 2009. — № 4. — С. 97-108. — Бібліогр.: 22 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859708807240744960 |
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| author | Nogaret, T. Rodney, D. Fivel, M. Robertson, C. |
| author_facet | Nogaret, T. Rodney, D. Fivel, M. Robertson, C. |
| citation_txt | Clear band formation simulated by dislocation dynamics / T. Nogaret, D. Rodney, M. Fivel, C. Robertson // Вопросы атомной науки и техники. — 2009. — № 4. — С. 97-108. — Бібліогр.: 22 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Dislocation Dynamics simulations of dislocations gliding across a random populations of
Frank loops are presented. Specific local rules are developed to reproduce elementary interaction
mechanisms obtained in Molecular Dynamics simulations. It is shown that absorption of Frank
loops as helical turns on screw dislocations governs the process of clear band formation,
because: (1) it transforms the loops into jogs on dislocations, (2) when the dislocations unpin, the
jogs are transported along the dislocation lines, leading to a progressive clearing of the band and
(3) the dislocations are re-emitted in a glide plane different from the initial one, allowing for a
broadening of the band. It is also shown that a pile-up of dislocations is needed to form a clear
band of finite thickness.
У термінах дислокаційної динаміки представлено моделювання дислокацій, що
перетинають розташовану випадковим чином сукупність петель Франка. Розроблені
локальні правила для відтворення елементарних механізмів взаємодії, що отримані при
моделюванні методом молекулярної динаміки. Показано, що поглинання петель Франка у
вигляді гелікоїдальних витків на гвинтових дислокаціях визначає процес утворення
вільних зон, оскільки: 1) воно перетворює петлі у східці на дислокаціях, 2) у випадку
відкріплення дислокації східці переносяться вздовж ліній дислокацій і 3) дислокації знову
надходять у площину ковзання, яка відрізняється від вихідної, забезпечуючи тим самим
розширення вільної зони. Крім того, показано, що скупчення дислокацій необхідне для
утворення вільної зони з кінцевою товщиною.
В терминах дислокационной динамики представлено моделирование дислокаций,
пересекающих расположенную случайным образом совокупность петель Франка.
Разработаны локальные правила для воспроизведения элементарных механизмов
взаимодействия, полученных при моделировании методом молекулярной динамики.
Показано, что поглощение петель Франка в виде геликоидальных витков на винтовых
дислокациях определяет процесс образования свободных зон, поскольку: 1) оно
преобразует петли в ступеньки на дислокациях, 2) в случае открепления дислокации
ступеньки переносятся вдоль линий дислокаций и 3) дислокации вновь поступают в
плоскость скольжения, отличающуюся от исходной, обеспечивая тем самым расширение
свободной зоны. Кроме того, показано, что скопление дислокаций необходимо для
образования свободной зоны с конечной толщиной.
|
| first_indexed | 2025-12-01T04:33:44Z |
| format | Article |
| fulltext |
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2009. №4-1.
Серия: Физика радиационных повреждений и радиационное материаловедение (94), с. 97-108. 97
CLEAR BAND FORMATION SIMULATED BY DISLOCATION
DYNAMICS
Thomas Nogaret1,2, David Rodney1, Marc Fivel1, Christian Robertson2
1SIMAP-GPM2, INP Grenoble, CNRS/UJF, BP46, 38402 Saint Martin d’Heres,
France;
2SRMA, CEA DEN/DMN Saclay, 91191 Gif-Sur-Yvette, France
Dislocation Dynamics simulations of dislocations gliding across a random populations of
Frank loops are presented. Specific local rules are developed to reproduce elementary interaction
mechanisms obtained in Molecular Dynamics simulations. It is shown that absorption of Frank
loops as helical turns on screw dislocations governs the process of clear band formation,
because: (1) it transforms the loops into jogs on dislocations, (2) when the dislocations unpin, the
jogs are transported along the dislocation lines, leading to a progressive clearing of the band and
(3) the dislocations are re-emitted in a glide plane different from the initial one, allowing for a
broadening of the band. It is also shown that a pile-up of dislocations is needed to form a clear
band of finite thickness.
1. INTRODUCTION
Neutron irradiation causes a degradation of
the mechanical properties of metals: a
pronounced hardening, a reduction of
ductility and plastic instabilities are observed
[1]. In stainless steels, hardening is usually
ascribed to the creation of a high density of
nanometer-sized irradiation defect clusters, in
the form of interstitial Frank Loops [2-4].
Ductility reduction and plastic instabilities are
associated with the localization of the
deformation into defect-free shear bands
called clear bands [5, 6]. Clear bands are
characterized by a constant thickness that
depends on the resistance of the defects.
Weaker defects, such as SFTs, lead to wider
clear bands than stronger obstacles, such as
Frank loops: �100 nm for the former
compared to �20 nm for the latter [7]. The
formation mechanism of clear bands remains
however not well understood; namely, the
clearing and broadening mechanisms of the
bands are still unknown.
Recently, important modelling and
experimental efforts have been devoted to
investigate the elementary interaction
mechanisms between dislocations and
irradiations defects. Systematic studies were
performed by Molecular Dynamics (MD)
simulations [8, 9]. The interaction
mechanisms with Frank loops and SFTs were
found in strong analogy. In both cases, edge
and screw dislocations behave differently.
Screw dislocations mainly absorb defects as
helical turns whereas edge dislocations shear
the defects at low applied stresses. Screw
dislocations are strongly pinned by helical
turns because the latter can glide only in the
screw direction. When screw dislocations
unpin, they are reemitted in a glide plane
parallel to the initial glide plane, because of
the three-dimensional structure of the helical
turn. Similar behaviors were observed in in-
situ Transmission Electron Microscopy
(TEM) [10]: screw dislocations are mainly
responsible for defect removal, absorb defects
as helical turns and are re-emitted in new
glide planes upon unpinning. TEM
observations showed also that clear bands are
formed by screw dislocation pile-ups emitted
from heterogeneities, such as grain
boundaries [6]. Conventional Frank Read
sources are indeed strongly pinned by the
formation of dense clouds of defects during
the irradiation [11] and remain inactive.
Nevertheless, the role of the above nano-
scale interaction mechanisms in forming
micron-scale clear bands has yet to be
demonstrated. Dislocation Dynamics (DD)
simulations are suitable for addressing this
issue. In the early 2000’s, DD simulations
were performed in order to study clear band
formation by dislocations generated from a
Frank Read source and interacting with a
population of irradiation defects [12-15].
However the local rules used to model the
short-range interactions between dislocations
and irradiation defects were very simple: no
distinction between screw and edge
dislocation was made and the defects were
systematically removed from the simulation
cell, after interaction.
In this paper, an existing DD code is
modified in order to accurately reproduce the
MD interaction mechanisms. The different
behaviors between screw and edge
dislocations are reproduced with realism by
using specific local rules of interaction and
using an undecorated initial dislocation
source, positioned at the simulation cell
boundary. In Section 2, TEM observation of
ion irradiated stainless steel specimens are
first described. Information from these
experiments are used in for the simulation
settings to be described Section 3, where the
simulation techniques are presented, i.e. the
parameters and the configurations, the local
rules of interaction and the elementary
interaction mechanisms. In Section 4, the
cases of isolated dislocation and collective
dislocation interaction with a random
population of defects are presented. In
Section 5, the results are discussed with
respect to experiments and previous
simulations.
2. TEM OBSERVATION OF
CLEAR BANDS IN ION
IRRADIATED STAINLESS STEEL
Flat shaped tensile specimens were
machined from solution annealed AISI 316L
steel plates. The stacking fault energy (SFE)
of this alloy is about 30 mJ/m2, i.e. it is
consistent with the interatomic potential used
in the MD simulations to be described in the
next section. Some specimens were irradiated
at 350 °C using 2.1 MeV Kr ions, yielding a
0.5 µm thick irradiated surface layer, with
peak damage around 3 dpa. Other specimens
were irradiated at 350 °C using 95 MeV Xe
ions, yielding a 10 µm thick irradiated surface
layer with a mean irradiation dose around
1 dpa. All the specimens were then strained in
uni-axial tension at 350 °C with a 10-4
s-1
strain rate up to 8% plastic strain then, finally
thinned down to 100 µm by mechanically
polishing the un-irradiated surface.
Then, 3 mm discs were punched out and
TEM thin foils were obtained by a back-side
electro-polishing technique.
The irradiation defects visible in the TEM
samples are interstitial Frank loops with a
diameter around 10 nm, i.e. the same defects
and the same size as considered in the MD
simulations. The loop density is about
1023 min-3 the 3 dpa irradiated samples, and
about 1022
m-3
in the 1 dpa irradiated samples,
also consistent with the MD simulation
setting to be described in Section 3.
After plastic straining, clear bands of
thickness 20-80 nm were observed in the
1 dpa irradiated samples. Two examples of
clear bands are shown in Fig. 1: the two
bands are located in the same grain and are
parallel to 2 distinct {111} glide planes. The
observed channel width in stainless steel is
about one order of magnitude smaller than in
irradiated Cu deformed at the same
temperature [5]. It is believed that thinner
channels are obtained in stainless steel
because cross-slip is more difficult than in
Cu, owing to the lower SFE of the steel
(SFE316L < SFECu), at the same temperature1
.
It was indeed explained above that cross-slip
is of prime importance for defect absorption,
during MD simulations (see Section 3).
Dislocation lines present in the vicinity of
the clear bands are shown in Fig. 2, at higher
magnification. The dislocations are heavily
jogged and appear wiggly in the micrographs.
These line shapes are most probably due to
the absorption of Frank loops and their
transformation into jogs along the dislocation
lines, as is observed in the MD simulations
[9]. Information from the various
micrographs provides insight regarding the
clear band formation process.
In Fig. 3 two distinct and parallel clear
bands are visible in the same grain. The width
of the two clear bands is nevertheless quite
different: one is rather wide and well
developed while the other is much narrower.
1 Conversely, the channel width in irradiated Cu at
room temperature (and lower) is comparable (to within
a factor 2) to the channel width observed in irradiated
316L steel deformed at 350 °C.
98
Fig. 1. Clear bands observed after deformation of samples irradiated to 1 dpa
b a
Fig. 2. Dislocations observed after deformation of samples irradiated to 1 dpa
Fig. 3. Two clear bands observed in a same grain after deformation of samples
irradiated to 1 dpa
In addition, no dislocation is visible in the
wide clear band whereas a dislocation pile-up
can be clearly seen in the narrow clear band. If
the formation of a pile-up structure is
interpreted as a prior step for clear band
formation, the observations prove that the
overall process is rather progressive and
involves the passage of many dislocations.
Analysis of the dislocations observed in clear
bands show that they mostly adopt the screw
character (see Fig. 2,a). These observations are
consistent with in-situ TEM analyses [6] and
with the scenario proposed from the MD
simulations described in Section 3. Indeed,
during the formation of a clear band, the first
dislocations to glide in the band absorb many
defects and become heavily jogged and
wiggly, as illustrated in Fig. 4,b. If the applied
99
stress is removed during this first stage of clear
band formation, the dislocations remain pinned
in the band, because they are heavily jogged.
Then, these dislocations are visible in post-
mortem samples. By way of contrast, the
dislocations that travel in a well formed clear
band see hardly any defects. These
dislocations therefore remain free of jogs and
either glide back upon specimen unloading or
are eliminated afterwards, during the TEM foil
preparation. Well-developed clear bands
therefore contain no dislocations in post-
mortem observations.
a b
Fig. 4. Pile-up of screw dislocations observed in a thin shear band (a), Simplified illustration of
the progressive process of clear band formation (b)
In conclusion, clear bands and wiggled
dislocations were observed in ion irradiated
specimens, after tensile straining at 350 °C.
Clear band formation is believed to be a
progressive, three-dimensional process,
involving numerous dislocation passages.
3. THE COMPUTATIONAL MODEL
3.1. The simulation cell
The Dislocation Dynamics code used in this
work was first developed by Verdier et al. and
is described in details in [16]. Only the points
specific to the present study are addressed
here. Dislocation lines are discretized in edge
and screw segments that glide on a discrete
lattice homothetic to the underlying
crystallographic structure. The segments are
treated as elastic inclusions, generating long-
range stress fields included in the calculation
of the local resolved shear stress, acting in the
simulation volume. Usually, the parameter of
the discrete lattice is 10b, where b is the
magnitude of the Burgers vector. In the present
work, in order to model nanometric defects
and sub-nanometric jogs on dislocations, a
smaller parameter 0.08b is used.
Consequently, the time step has been reduced;
down to 5·10−14
s. Elasticity is isotropic and
corresponds to a copper crystal in agreement
with the MD simulations [9]. The adopted
simulation cell is shown in Fig. 5. Its
dimensions are 0.6x0.6x0.24 m. The borders
act as impenetrable grain boundaries and
cannot be crossed by the dislocations.
Horizontal planes are Z = (111) glide planes,
while the Y axis is along the [10-1] Burgers
vector direction. In order to account for
dislocation emission from heterogeneities as
observed experimentally in irradiated materials
[6,19], a dislocation source is placed along a
border of the cell and model a grain boundary
source. Different types of sources that emit
either edge or screw dislocations were tested,
as well as sources of different lengths (see
Section 4.1). The applied stress tensor is
composed of only the σYZ shear component
which is feedback controlled in order to
impose a constant strain-rate of 1.2·103 s-1. It is
worth recalling that no thermally-activated
mechanisms (including cross-slip) are allowed
in the presented simulations.
100
Fig. 5. Simulation cell. The loops are in
grey, a screw dislocation appears in the
upper left border
The adopted simulation method is therefore
best adapted to analyse details of post-
irradiation plastic deformation at rather low
temperature (below creep threshold). This
method cannot reproduce high temperature in-
flux situations like irradiation creep; unless a
specific dislocation climb treatment is
implemented.
In absence of detailed knowledge on
exactly how actual grain boundary sources
operate, a simple emission criterion is used:
the source emits a new dislocation when the
applied stress σYZ reaches a critical value,
called the nucleation stress τnucl. During a
simulation, this stress is the maximum value
that the applied stress may reach because in
such a case, a dislocation is emitted, leading to
an increment in plastic deformation that
decreases the applied stress. The emitted
dislocations belong to the .<10-1>{111}
system, to be called the primary slip system.
The cross-slip system is .<10-1>{1-11}. Both
systems share the primary Burgers vector
.<10-1>. The MD simulations show that when
a Frank loop is unfaulted by interaction with a
dislocation, it systematically obtains the
Burgers vector of the incoming dislocation [9].
Thus, in order to keep a simple computational
model, the Frank loops are modelled as
interstitial prismatic loops with the primary
.<10-1> Burgers vector. The initial loop shape
is parallelepipedic, composed of 2 segments in
the primary system and two segments in the
cross-slip system. The length of all the fixed
segments (and thus, the size of all the loops)
is set to D = 10 nm, representative2 of the
irradiation conditions as described in
Section 2. The loops are placed at random
positions in the simulation cell, with a density
N=3.7·1022
m-3, in agreement with typical
TEM observations in irradiated stainless steels
(4). The associated mean inter-loop interval
(projected in the glide planes) is then
L=1/√(N·D) = 52 nm, which also corresponds
to the distance adopted in MD simulations.
3.2. Elementary interaction mechanisms
Frank loops are sessile because they contain
a stacking fault. Such loops are unfaulted and
become glissile through the interaction with
screw dislocations, while they remain faulted
and are simply sheared when interacting with
edge dislocations. In order to reproduce these
elementary interactions, the loops in the
simulation cell are initially frozen, i.e. their
segments are immobile. When a dislocation
comes in contact with a loop, its character is
identified by computing the angle between the
local tangent to the dislocation line and the
Burgers vector. If this angle is ±20
o
, the
dislocation is declared screw and the loop
segments are "freed", i.e. they are allowed to
move according to the forces acting on them.
As will be seen in next paragraph, a helical
2 The adopted simulation volume dimensions fix the
maximum loop size that can be modeled. The
simulation volume must be large enough to contain the
loops and to accommodate the dislocation/loop reaction
mechanisms as shown in fig. 6. With the simulation
volume used here, Dmax should be close to 50 nm.
101
turn then forms spontaneously. If the
dislocation is not screw, the loop remains
frozen and the contacting dislocation segments
do not react with the loop. The dislocation is
allowed to cross the loop when its arms on
both sides of the loop reach a critical angle
that was set to 100o in order to match the
resistance obtained in MD simulations, i.e. a
critical shear stress of 130 MPa for an inter-
loop distance of 50 nm [9].
The interaction between a screw dislocation
and Frank loops is sketched in Fig. 6. The
incoming screw dislocation contacts a Frank
loop. At this moment, the segments forming
the loop are freed. They react with the
dislocation and spontaneously form a helical
turn (Fig. 6,b), made of 20 nm long segments:
3 super-jogs in (1-11) cross-slip planes and 2
segments in (111) primary planes located
above and below the initial glide plane. The
initial dislocation thus ends up with a 3D
structure. It does not belong to the initial glide
plane anymore because the helical turns
expanded along the dislocation line in order to
minimize the dislocation length and the
associated line tension energy.
The dislocation is pinned by the helical
turns because the super-jogs in cross-slip
planes can glide only in the [10-1] direction of
the Burgers vector, i.e. along the dislocation
line, and not in the initial [-12-1] glide
direction. Dislocation unpinning requires the
activation of a 20 nm long super-jog in a (111)
glide plane (Fig. 6,d). The activated segment
belongs to a (111) plane located above (along
the [111] direction) the initial glide plane.
Indeed, it can be shown from a line tension
approximation of a helical turn that upon
increasing shear stress, the segment located
furthest in the glide direction becomes
unstable first. Consequently, a dislocation that
glides in the [-12-1] (resp. [1-21]) direction is
re-emitted in an upper (resp. lower) (111)
plane. In the following, it is shown that this
unpinning mechanism plays a central role in
clear band broadening, as will be demonstrated
next.
The interaction between an edge dislocation
and three Frank loops is depicted in Fig. 7. As
a stress is applied, the mobile edge dislocation
bows out and comes into contact with one of
the loops (Fig. 7,a). The dislocation is then
blocked and bows out. When the applied stress
reaches 130 MPa, the angle between the
dislocation arms pinned on the central loop
reaches the critical angle of 100
o
(Fig. 7,b) and
the dislocation is allowed to go through the
loop. The latter remains frozen and is left
unchanged, since it was observed in MD
simulations that the step created on the loop is
mobile and annihilates on the loop border, thus
reforming the initial loop configuration.
Fig. 6. Interaction between a screw dislocation and Frank loop.
The dislocation glides in a (111) plane and comes into contact with the loop in its middle (a);
absorbs the loop as a helical turn (b). The same reaction as in (a) is shown in (c),
from a different viewing angle. In (d), it is clearly seen that a dislocation segment located in an
upper (111) plane is activated at the time of unpinning from the helical turn
102
Fig. 7. Interaction between an edge dislocation and a Frank loop. An edge character dislocation
glides towards an immobile Frank loop (a). The incoming dislocation is blocked and bows out
until the critical bowing angle of 100° is reached on the loop (b). The sheared loop remains
“frozen” after the mobile dislocation unpins, i.e. it stays at the same initial location as before the
interaction
4. GLIDE IN RANDOM LOOP
ENVIRONMENTS
In this section, the glide of dislocation(s)
through a random population of Frank loops is
simulated. Two glide regimes are studied by
changing the magnitude of the nucleation
stress τnuc. The case where the nucleation
stress is much larger than the loop resistance is
first evaluated, in Section 4.1. A single
dislocation then glides through the simulation
cell, driven by the applied stress only. In the
second case, treated in Section 4.2, the
nucleation stress is lower than the loop
resistance and so, no isolated dislocation can
glide on its own. More dislocations are then
nucleated until a pile-up is formed, leading to
collective interaction effects, that enable the
dislocations to glide through the cell at an
applied stress lower than when isolated.
4.1. Glide of single dislocations
A nucleation stress of 1000 MPa is used,
i.e. much larger than the loop resistance
evaluated in Section 3.2 (in MD simulations,
for instance). The applied stress needed to
accommodate the imposed plastic strain rate is
always lower than the nucleation stress and
only one dislocation glides through the cell.
Dislocation sources emitting 200 nm long
dislocations of either edge or screw characters
are tested. Let’s consider first the case of an
edge source, as shown in Fig. 8. In Fig. 8, a
which shows a [111] top view of the
simulation cell, the edge part emitted from the
source glides mainly by shearing loops. It
produces on its sides two long dislocations of
screw character. The latter are wavy and
composed of segments in the primary glide
plane as well as in cross-slip planes.
Fig. 8. Gride of a single edge dislocation:
a – [111] top view (untouched loops are in ligt
grey, segments in (111) planes in blue,
segments in cross-slip planes in orange;
b – stress/strain curve; c – [1-21] side view
(the green arrow shows the direction of glide)
These segments form helical turns created
on the dislocation line by the unfaulting and
absorption of Frank loops, following the same
interaction mechanism as described in Section
103
3.2. While the edge segment is mobile, the two
screw dislocations are strongly pinned. The [1-
21] side view of Fig. 8, b shows that
dislocation glide is planar on average,
although segments in cross slip planes
(belonging to helical turns), are also visible.
The accompanying stress/strain curve (not
shown) reveals that the stress required for the
glide of a single edge dislocation is between
130 and 160 MPa, depending on the local loop
density met by the dislocation along its path.
The screw case is shown in Fig. 9. As the
screw dislocation emitted from the source
advances through the simulation cell, it creates
edge parts that glide easily until they reach the
cell borders, while the screw segment unfaults
and absorbs loops as helical turns (see Fig.
9, a). The dislocation soon adopts the shape of
a long screw segment that traverses the whole
cell in the [10-1] direction, with edge segments
stacked on the cell boundaries. Thus, edge and
screw sources lead to similar microstructures
made of screw dislocations that extend over
the entire length of the simulation cell.
Fig. 9. Glide of a single screw dislocation :
a – [111] top view;
b – stress/strain curve
Using a screw type source, the stress required
for dislocation glide is between 200 and
260 MPa. The screw dislocation advances
through the simulation cell by a mechanism
close to the elementary mechanism described
in Section 3.2: a succession of formation of
helical turns that pin the dislocation followed
by the activation of segments in the weakest
zones along the dislocation, i.e. the zones
where the jog density is the lowest. The
activated segments glide on about 100 nm
before being pinned again by helical turns.
During this process, edge segments are created
and glide easily towards the simulation cell
borders. The edge segments, while travelling
towards the cell borders, sweep the dislocation
line and push the jogs on a finite distance
towards the cell borders. This mechanism
allows for a partial and progressive clearing of
the swept zone.
As in the elementary reactions, when the
screw dislocation unpins, it is systematically
re-emitted in an upper (111) plane.
Consequently, as seen in the [10-1] side view
in Fig. 9, b, the dislocation glides in an
average non-crystallographic plane, inclined
with respect to the initial (111) plane, in
contradiction with experimental observations.
Moreover, no clearing is observed in Fig. 9, b.
Plasticity is thus limited by screw dislocations
and both edge and screw dislocation sources
lead to the same anisotropic microstructure
with strongly pinned screw segments that
extend over the entire length of the simulation
cell. Thus in the following, only the case of a
screw dislocation source with a length equal to
that of the simulation cell will be examined.
4.2. Collective dislocation motion
Now, in contrast with previous simulations,
a low nucleation stress of 90 MPa is
considered, i.e. below the critical stress for
edge or screw dislocation glide. Consequently,
a single dislocation cannot glide alone, and
collective effects are needed to keep on
deforming the simulation cell at the prescribed
strain rate. Fig. 10 illustrates the obtained glide
mechanism. The first dislocation nucleated
acquires helical turns and becomes pinned.
The stress in the simulation cell increases and
reaches the nucleation stress, as shown by an
arrow in Fig. 10, b. A second dislocation is
then nucleated. It produces some plastic strain,
а
b
104
allowing the applied stress to drop. This
second dislocation gets pinned as well, and the
applied stress rises again, triggering the
nucleation of a third dislocation, materialized
by an additional arrow in Fig. 10,b.
b
a c
Fig. 10. Glide of a dislocation pile-up: a – [111] top view; b – stress/strain curve; c – [10-1]
side view on a 200 nm thick thin foil
Dislocation pile-ups, as the one clearly
visible in Fig. 10, a, thus form progressively.
In this case, the 4 leading dislocations are
wavy and heavily jogged. They are responsible
for clearing the band by forming helical turns
and pushing the jogs towards the cell borders
upon unpinning. Accumulation of jogs is
visible on the cell periphery in Fig. 10, a.
Trailing dislocations contain very few jogs
because they glide in the region cleared by the
leading dislocations. The role of the trailing
dislocations is to produce the pile-up effect
and to concentrate the stress on the leading
dislocations. Some heavily jogged dislocations
are left behind, as seen in Fig. 10, a. They will
presumably remain in the clear band. As in
previous Section, the leading dislocations
unpin in upper (111) planes and remain pushed
by the pileup effect as long as they are not too
far away from the initial central glide plane.
As a consequence, as seen in Fig. 10, c, a
cleared region of finite thickness develops
parallel to the central (111) glide plane, in
agreement with experiments. The pile-ups
keep on advancing in the cell thanks to
collective effects that include a stress
concentration due to the pile-up effect, short
range interaction mechanisms (arm exchange)
and avalanches of dislocation glide. These
interactions are described in detail in [20].
5. DISCUSSION
For the present study, Dislocation
Dynamics simulations were adapted to the
nanometer scale, in order to reproduce with
realism the elementary interaction mechanisms
observed in MD simulations. As a result, the
computation load becomes very large and only
the first stage of the clear band formation was
simulated in a small grain (see Fig. 10, c).
However, this study allows us to draw some
conclusions about the mechanisms controlling
the whole process.
The simulations confirm earlier MD results
on the central role played by helical turns in
clear band formation. Screw dislocations
transform Frank loops into helical turns. The
helical turns are then transported along the
dislocation lines when they unpin leading to a
progressive clearing of the band and to the
accumulation of jogs and prismatic loops
aligned in the edge direction. These loops have
the same Burgers vector as the emitted
dislocations, i.e. the primary Burgers vector.
Helical turns are also central to clear band
widening because upon unpinning, screw
dislocations are re-emitted in new glide planes.
This is equivalent to a double cross-slip over a
height set by the loop size. This mechanism is
consistent with the work of Neuhauser and
Rodloff [19] who observed on the surface of
105
irradiated and deformed copper a distance
between slip lines on the order to the defect
size. Note that all the effects observed in the
simulations were obtained while thermally-
activated cross-slip and climb were switched-
off. Hence, thermal activation is not a
necessary condition for clear band formation.
The microstructure obtained in the
simulations, composed of long screw
dislocations with accumulations of jogs and
prismatic loops on the sides, is consistent with
the TEM observations made by Sharp [5] who
reported inside clear bands the presence of
dense clusters of heavily jogged prismatic
loops with a low density of screw dislocations,
all sharing the same Burgers vector. To our
knowledge, Sharp’s work is the only case
where a thin foil was prepared parallel to a
clear band, making possible a detailed analysis
of the microstructure inside a clear band. All
other TEM studies of clear bands in irradiated
materials used thin foils perpendicular to the
clear band, which is the best orientation to
locate a clear band, but the worst to study the
microstructure inside the band.
Fig. 11. Dislocation sources emitted at the tip of a nano-indenter in austenitic stainless steel.
Irradiated 316L stainless steel, using 700 keV Kr3+ ions at 300 °C, 3 dpa (a).
Non irradiated 316L steel (b). The indenter penetration depth is 30 nm in both cases
The present work shows the central role
played by dislocation pile-ups. Hence, isolated
dislocations can not form clear bands because
band clearing is very progressive. In addition,
isolated screw dislocations glide on non-
crystallographic planes, owing to their
systematic re-emission in upper (111) planes
(see Fig. 10, c). In contrast, when dislocations
glide in pileups, they remain along the central
(111) plane (see Fig. 10, c) and generate stress
concentration that contribute to re-activate
(temporarily locked) dislocation arms located
into different, parallel glide planes3. That is
the reason why clear of finite thickness only
form in presence of dislocations pile-ups.
Sources of dislocations at the origin of clear
bands must therefore emit a large number of
3 Stress concentration can be computed using analytical
solutions or numerically, by adding the stress fields
coming from selected dislocations (see Section 3).
dislocations. Since the sources prior to the
irradiation are locked by decoration, the most
probable sources are grain boundaries or other
stress concentrators, such as inclusions (21) or
singularities at grain boundaries. Note that
hardening is not described in the present
simulations because the resistance of the initial
source, which controls the flow stress, is given
as preset parameter. Clearly, more atomistic
information is needed to better understand how
heterogeneities may act as dislocation sources.
Insight about the formation sources can also be
obtained the other way around, by using TEM
observation of dislocation emission from
heterogeneous stress fields, in ion-irradiated
materials (see Fig. 11 and [22]). In absence of
this knowledge, a very simplified criterion was
used in the present DD simulations.
The mechanisms controlling the width of
clear bands have not been examined in details,
owing to the computational load of the
106
simulations. However, one possible origin is
the decay of the stress concentration away
from the plane of the pile-up. Indeed, the
leading dislocations of the pile-ups are heavily
jogged and need a stress concentration to
advance in the cell. As illustrated in Fig. 12,
the stress on the dislocations (τdislocation) is the
sum of the applied stress (τapplied) and the stress
concentrated by the pile-up (τpile−up). The
former is constant inside the simulation cell
while the latter decreases away from the plane
of the pile-up. When the leading dislocations
unpin and are re-emitted in upper glide planes,
they are subjected to a decreasing stress. There
is thus a critical distance from the pile-up
plane, which sets the band thickness, where
the stress on the dislocations just balances the
resistance due to the helical turns (τunpinning)
and the dislocations stop.
In their displacement, the leading
dislocations have started to clear the band and
the trailing dislocations can move forward and
they eventually glide away from the plane of
the pile-up until they are stopped and so on.
This scenario predicts that heavily jogged
screw dislocations should be left in one side of
the clear band (in this case, the upper glide
plane). Although such arrays of screw
dislocations have been observed [6], more
TEM studies are needed. The present scenario
also explains the experimental observation that
the band width decreases when the resistance
of defects or the resolved shear stress [7] or
the defect density [5] increase.
Fig. 12. Schematic representation of the equilibrium between the stress on the dislocations
τdislocation (equal to the sum of the applied stress τapplied and the pile-up stress concentration τpile-up
and the defect resistance τunpinning, which likely controls the band thickness
Indeed, for a given size of pile-up, the band
width decreases if τunpinning increases, i.e. if the
defects are intrinsically stronger or if they are
in higher density; it also decreases if the
resolved shear stress (τapplied) decreases.
6. CONCLUSION
The present work concludes a multiscale
simulation study of the formation of clear
bands. An existing DD code was modified to
reproduce accurately MD results on
elementary interaction mechanisms at the
nanometer scale. The DD simulations in
random loop environments confirm the central
role played by helical turns in the formation of
clear bands. It also shows that clear bands can
not form without dislocation pile-ups. From
the simulations, it is predicted that well-
developed clear bands should contain heavily
jogged screw dislocations as well as dense
concentrations of prismatic loops. In order to
confirm these predictions, detailed TEM
analysis of the dislocation microstructure
inside clear bands is needed.
This work was funded by the European
PERFECT project (No. FI60-CT-2003-
508840).
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22. Communication SMORE meeting.
МОДЕЛИРОВАНИЕ МЕТОДАМИ ДИСЛОКАЦИОННОЙ ДИНАМИКИ
ОБРАЗОВАНИЯ СВОБОДНЫХ ЗОН
Томас Ногарет, Дэвид Родни, Марк Файвел, Кристиан Робертсон
В терминах дислокационной динамики представлено моделирование дислокаций,
пересекающих расположенную случайным образом совокупность петель Франка.
Разработаны локальные правила для воспроизведения элементарных механизмов
взаимодействия, полученных при моделировании методом молекулярной динамики.
Показано, что поглощение петель Франка в виде геликоидальных витков на винтовых
дислокациях определяет процесс образования свободных зон, поскольку: 1) оно
преобразует петли в ступеньки на дислокациях, 2) в случае открепления дислокации
ступеньки переносятся вдоль линий дислокаций и 3) дислокации вновь поступают в
плоскость скольжения, отличающуюся от исходной, обеспечивая тем самым расширение
свободной зоны. Кроме того, показано, что скопление дислокаций необходимо для
образования свободной зоны с конечной толщиной.
МОДЕЛЮВАННЯ МЕТОДАМИ ДИСЛОКАЦІЙНОЇ ДИНАМІКИ
УТВОРЕННЯ ВІЛЬНИХ ЗОН
Томас Ногарет, Девід Родні, Марк Файвел, Крістіан Робертсон
У термінах дислокаційної динаміки представлено моделювання дислокацій, що
перетинають розташовану випадковим чином сукупність петель Франка. Розроблені
локальні правила для відтворення елементарних механізмів взаємодії, що отримані при
моделюванні методом молекулярної динаміки. Показано, що поглинання петель Франка у
вигляді гелікоїдальних витків на гвинтових дислокаціях визначає процес утворення
вільних зон, оскільки: 1) воно перетворює петлі у східці на дислокаціях, 2) у випадку
відкріплення дислокації східці переносяться вздовж ліній дислокацій і 3) дислокації знову
надходять у площину ковзання, яка відрізняється від вихідної, забезпечуючи тим самим
розширення вільної зони. Крім того, показано, що скупчення дислокацій необхідне для
утворення вільної зони з кінцевою товщиною.
108
МОДЕЛИРОВАНИЕ МЕТОДАМИ ДИСЛОКАЦИОННОЙ ДИНАМИКИ ОБРАЗОВАНИЯ СВОБОДНЫХ ЗОН
|
| id | nasplib_isofts_kiev_ua-123456789-96341 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-01T04:33:44Z |
| publishDate | 2009 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Nogaret, T. Rodney, D. Fivel, M. Robertson, C. 2016-03-15T10:16:10Z 2016-03-15T10:16:10Z 2009 Clear band formation simulated by dislocation dynamics / T. Nogaret, D. Rodney, M. Fivel, C. Robertson // Вопросы атомной науки и техники. — 2009. — № 4. — С. 97-108. — Бібліогр.: 22 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/96341 Dislocation Dynamics simulations of dislocations gliding across a random populations of Frank loops are presented. Specific local rules are developed to reproduce elementary interaction mechanisms obtained in Molecular Dynamics simulations. It is shown that absorption of Frank loops as helical turns on screw dislocations governs the process of clear band formation, because: (1) it transforms the loops into jogs on dislocations, (2) when the dislocations unpin, the jogs are transported along the dislocation lines, leading to a progressive clearing of the band and (3) the dislocations are re-emitted in a glide plane different from the initial one, allowing for a broadening of the band. It is also shown that a pile-up of dislocations is needed to form a clear band of finite thickness. У термінах дислокаційної динаміки представлено моделювання дислокацій, що перетинають розташовану випадковим чином сукупність петель Франка. Розроблені локальні правила для відтворення елементарних механізмів взаємодії, що отримані при моделюванні методом молекулярної динаміки. Показано, що поглинання петель Франка у вигляді гелікоїдальних витків на гвинтових дислокаціях визначає процес утворення вільних зон, оскільки: 1) воно перетворює петлі у східці на дислокаціях, 2) у випадку відкріплення дислокації східці переносяться вздовж ліній дислокацій і 3) дислокації знову надходять у площину ковзання, яка відрізняється від вихідної, забезпечуючи тим самим розширення вільної зони. Крім того, показано, що скупчення дислокацій необхідне для утворення вільної зони з кінцевою товщиною. В терминах дислокационной динамики представлено моделирование дислокаций, пересекающих расположенную случайным образом совокупность петель Франка. Разработаны локальные правила для воспроизведения элементарных механизмов взаимодействия, полученных при моделировании методом молекулярной динамики. Показано, что поглощение петель Франка в виде геликоидальных витков на винтовых дислокациях определяет процесс образования свободных зон, поскольку: 1) оно преобразует петли в ступеньки на дислокациях, 2) в случае открепления дислокации ступеньки переносятся вдоль линий дислокаций и 3) дислокации вновь поступают в плоскость скольжения, отличающуюся от исходной, обеспечивая тем самым расширение свободной зоны. Кроме того, показано, что скопление дислокаций необходимо для образования свободной зоны с конечной толщиной. This work was funded by the European PERFECT project (No. FI60-CT-2003-508840). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Clear band formation simulated by dislocation dynamics Моделювання методами дислокаційної динаміки утворення вільних зон Моделирование методами дислокационной динамики образования свободных зон Article published earlier |
| spellingShingle | Clear band formation simulated by dislocation dynamics Nogaret, T. Rodney, D. Fivel, M. Robertson, C. |
| title | Clear band formation simulated by dislocation dynamics |
| title_alt | Моделювання методами дислокаційної динаміки утворення вільних зон Моделирование методами дислокационной динамики образования свободных зон |
| title_full | Clear band formation simulated by dislocation dynamics |
| title_fullStr | Clear band formation simulated by dislocation dynamics |
| title_full_unstemmed | Clear band formation simulated by dislocation dynamics |
| title_short | Clear band formation simulated by dislocation dynamics |
| title_sort | clear band formation simulated by dislocation dynamics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/96341 |
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