To the question of channeling dislocation
The model of plastic deformation evolution and localization processes in irradiated materials is proposed. This models takes into account the dislocation distribution function dependence on dislocation velocity in an ensemble. It is shown that the fraction of dislocation overcoming radiation defects...
Gespeichert in:
| Veröffentlicht in: | Вопросы атомной науки и техники |
|---|---|
| Datum: | 2001 |
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
|
| Schlagworte: | |
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/80041 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | To the question of channeling dislocation / V.V. Krasil’nikov, A.A. Parkhomenko, V.N. Robuk, V.V. Sirota // Вопросы атомной науки и техники. — 2001. — № 6. — С. 318-320. — Бібліогр.: 12 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-80041 |
|---|---|
| record_format |
dspace |
| spelling |
Krasil’nikov, V.V. Parkhomenko, A.A. Robuk, V.N. Sirota, V.V. 2015-04-09T16:23:27Z 2015-04-09T16:23:27Z 2001 To the question of channeling dislocation / V.V. Krasil’nikov, A.A. Parkhomenko, V.N. Robuk, V.V. Sirota // Вопросы атомной науки и техники. — 2001. — № 6. — С. 318-320. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 61.72.Ss https://nasplib.isofts.kiev.ua/handle/123456789/80041 The model of plastic deformation evolution and localization processes in irradiated materials is proposed. This models takes into account the dislocation distribution function dependence on dislocation velocity in an ensemble. It is shown that the fraction of dislocation overcoming radiation defects with high velocities in the dynamical regime grows with increasing radiation hardening. Authors thank the Corresponding Member of Ukrainian NAS, prof. I.M. Neklyudov and Academician of Petri Primi Academia Scientiarum et Artium, prof. N.V. Kamyshanchenko for useful discussion of this work. One of the authors (V.V. Krasil’nikov) thanks the RFBR for the financial support by grant №00-02-16337. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory To the question of channeling dislocation К вопросу о каналировании дислокаций Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
To the question of channeling dislocation |
| spellingShingle |
To the question of channeling dislocation Krasil’nikov, V.V. Parkhomenko, A.A. Robuk, V.N. Sirota, V.V. Kinetic theory |
| title_short |
To the question of channeling dislocation |
| title_full |
To the question of channeling dislocation |
| title_fullStr |
To the question of channeling dislocation |
| title_full_unstemmed |
To the question of channeling dislocation |
| title_sort |
to the question of channeling dislocation |
| author |
Krasil’nikov, V.V. Parkhomenko, A.A. Robuk, V.N. Sirota, V.V. |
| author_facet |
Krasil’nikov, V.V. Parkhomenko, A.A. Robuk, V.N. Sirota, V.V. |
| topic |
Kinetic theory |
| topic_facet |
Kinetic theory |
| publishDate |
2001 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
К вопросу о каналировании дислокаций |
| description |
The model of plastic deformation evolution and localization processes in irradiated materials is proposed. This models takes into account the dislocation distribution function dependence on dislocation velocity in an ensemble. It is shown that the fraction of dislocation overcoming radiation defects with high velocities in the dynamical regime grows with increasing radiation hardening.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/80041 |
| citation_txt |
To the question of channeling dislocation / V.V. Krasil’nikov, A.A. Parkhomenko, V.N. Robuk, V.V. Sirota // Вопросы атомной науки и техники. — 2001. — № 6. — С. 318-320. — Бібліогр.: 12 назв. — англ. |
| work_keys_str_mv |
AT krasilnikovvv tothequestionofchannelingdislocation AT parkhomenkoaa tothequestionofchannelingdislocation AT robukvn tothequestionofchannelingdislocation AT sirotavv tothequestionofchannelingdislocation AT krasilnikovvv kvoprosuokanalirovaniidislokacii AT parkhomenkoaa kvoprosuokanalirovaniidislokacii AT robukvn kvoprosuokanalirovaniidislokacii AT sirotavv kvoprosuokanalirovaniidislokacii |
| first_indexed |
2025-11-25T22:42:31Z |
| last_indexed |
2025-11-25T22:42:31Z |
| _version_ |
1850569431116152832 |
| fulltext |
TO THE QUESTION OF CHANNELING DISLOCATION
V.V. Krasil’nikova), A.A. Parkhomenkob), V.N. Robuka), V.V. Sirotaa)
a)Belgorod State University, Belgorod, Russia
e-mail: kras@bsu.edu.ru
b) National Science Center “Kharkov Institute of Physics and technology”, Kharkov, Ukraine
The model of plastic deformation evolution and localization processes in irradiated materials is proposed. This
models takes into account the dislocation distribution function dependence on dislocation velocity in an ensemble. It
is shown that the fraction of dislocation overcoming radiation defects with high velocities in the dynamical regime
grows with increasing radiation hardening.
PACS: 61.72.Ss
Radiation hardening and embrittlement caused by it
are one of the most actual directions in the reactor
material science. Material radiation hardening manifests
itself in increasing an yield point and lowing hardening
velocity of materials and also in forming “fluidity tooth”
and the fluididty area of Chernov – Luders’ kind [1,2].
Plastic instability of materials is due to these effects.
Usual curves of deformation are shown in Fig. 1 for
reactor steels at test temperatures below 0.3 Tm (Tm is
melting temperature). Curve 1 is an initial material,
curve 2 corresponds to a lower dose than curve 3 does.
Our analysis [3] displayed that (curve 2) a lot of
materials has such a type of strain already at radiation
dose ≤ 10-2÷10-1 dpa (displacement per atom). The
minimum or “area” of curve 2 is a result of manifesting
the effects of plastic instability, namely, dislocation
channeling. The stage corresponding to “area” of curve
2 indirectly transits to the material destruction stage at
higher doses of radiation (≥1…10 dpa, curve 3).
Up – to – date approach to plastic deformation as a
collective dislocation process is supposed to describe
the effects of dislocation localization and self-
organization on the basis of studying the evolution of
dislocation ensembles in deformed materials. In the
works [4], the kinetic processes of dislocation ensemble
are considered theoretically in details within the
synergetical approach, and the models are supposed to
explain forming channels without defects in
nonirradiated crystals and localization of deformation in
irradiated materials.
Earlier, the models [5] were proposed to consider
arising the effects of plastic instability and plastic
deformation localization on the basis of individual
dislocation behavior. But many of plastic deformation
processes are a result of stochastic motions of
dislocations. There are some models (see, for instance,
[6]) that start from dislocation ensemble defined by a
dislocation distribution function depending on radius –
vector r and time t .
However, we’ll consider the dislocation distribution
function depends on not only radius – vector r , time t
but and on velocity v and its orientation in a space, as
material plastic deformation is caused by the mobile
dislocations. The dislocation distribution functions
averaged over orientation of dislocation lines in the
space are considered in this work. Upon that the
dislocations of ensemble can be considered as a set of
dislocation line segments [11].
Let the mobile dislocations interact with fixed
obstacles of different nature and pass through it, moving
in channeling regime [2]. Upon that it is supposed on
the basis of experimental facts that the ensemble
dislocations have the velocities near 0,1 of sound
velocity in a irradiated deformed material. This situation
corresponds to initial stages of irradiated material
deformation when the dislocation ensembles overcome
the obstacles represented by small clusters, loops and
micro voids.
Fig. 1. The typical strain curves (σ is strain, ε is
deformation) of reactor steels at the test temperatures
below 0.3 Tm (Tm is melting temperature). 1 – initial
(nonirradiated) material, 2 – material irradiated by
small doses (10-2÷10-1 dpa), 3 – material irradiated by
the doses upwards of 1 dpa
We’ll investigate the evolution processes of plastic
deformation on the basis of the general kinetic equation
for the dislocation distribution function n t( , )r v, :
,)),v,r(),v,r((V4
1
jva
tntndNv
div
nn
t
n
′−∫ ′Ω
π
=+
∂
∂
+
∂
∂
+
∂
∂
v
v r
(1)
where a is a dislocation acceleration caused by an
external force F , v′Ωd is an element of solid angle in
velocity space, N is a density of the immovable
obstacles interacting with dislocations. It is supposed
that the collision frequency of dislocations moved by
62 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 62-320.
velocity v with obstacles is equal Nv where v=v
by analogy with a gas charged particle scattered by an
immobile molecule in plasma.
The presence of a divergent term in Eq. (1) is
caused by supposing the Fokker – Planck form of the
collision term for dislocations [10].
Further, we consider the spatially homogeneous
case
0
),,(
=
r
vr
∂
∂ tn
. (2)
Eq. (2) means that dnn α< <−=∆ 21 (d is a
distance between the obstacles, α is coefficient of the
order of unit) that is the distribution function of
dislocation ensemble does not change on the length
equal to the order of a distance between the obstacles. It
is easy to obtain in spherical coordinates v, θ, ϕ in the
velocity space with a polar axis along an external force
F drawing the acceleration а
,)sin
sin
1cos( 22
2 n
v
nv
vv
a
v
n ⋅
∂
∂−
∂
∂−=
∂
∂ θ
θθ
θa
θθ
θθ
j
v
jv
vv
div v ⋅
∂
∂+
∂
∂= sin
sin
11 2
2jv ,
where θjjv , are corresponding components of density
dislocation flow in spherical coordinates. We average
Eq. (1) with account Eq. (2) over angles. Upon that we
suppose 1cos ≈θ . As vv ′= , then the right side of
Eq. (1) is equal zero. In the result, we obtain the kinetic
equation for the averaged distribution function
∫ Ω= dvnn ),(
4
1 θ
π
as
01 2
2
2
2 =
∂
∂+
∂
∂+
∂
∂
vjv
vv
nv
vv
a
t
n
, (3)
where vj is the flow density radial component caused
by the collisions between the dislocations. Analysis of
the collision Fokker – Planck term gives the next
expression of the flow vj :
,)(
v
nTnMvfj ddv ∂
∂+−= (4)
where ddf is the collision frequency of the fast
dislocations with the slow ones, M is mass of a
dislocation quasi particle and Т is temperature of a
material sample.
As seen from Eq. (3), the total flow density vJ in
the velocity space consists of the collision part vj and
the part caused by the external force
najJ vv +=
As the flow of the fast sliding dislocation doesn’t
change practically during time, we consider the
distribution of these dislocations as stationary
0=∂∂ tn and come to the relationship
qconstJv v ≡=2 . (5)
According to Eq. (4), Eq. (5) is the differential equation
for the distribution function n . We consider that
3~ −vf dd for collisions of mobile dislocations with
immobile ones. Similarly to classic mechanics, the
dislocation considered as a quasi particle scatters
elastically by the potential field 1−r . As known, in
this case, the effective differential section of elastic
scattering (and, consequently, the collision frequency
too) is proportionally 4−v (see [7]). On the other
hand, it is known that moving dislocations can interact
with the immobile obstacles (for instance, immobile
dislocations) accordingly the law r/1~ where r is a
distance from the obstacle up-to the dislocation axis as
it places, for instance, for an edge dislocation in the case
of impurity Cottrell atmosphere [8], or in interacting
two screw dislocations (see, for instance, [12]).
We suppose
,32vM
Ta
f c
dd =
M
f
ac
int= ,
where intf is the internal strain force. Denoting
aM
Ta
v c
c =2 ,
cv
vx =
we go to the dimensionless form of Eq. (5):
,)1( 2 Cnx
dx
nd
xa
a
c
=−−− (6)
where С=const.
The solution of Eq. (6) takes the form
))
4
)2(
exp(
)(
4
)2(
exp(
24
0
1
24
xdx
a
xxa
C
a
a
C
a
xxa
n
c
x
c
c
−
−
−
−
=
∫
(7)
The integration constant C1 can be defined by the set
distribution at х=0, for instance, Maxwell distribution:
)
2
exp(
)2(
2
2
30 a
ax
MT
Nn cd −=
π
,
where dN is a dislocation density.
On the other hand, as the function n must be finite at
∞→x , then the expression in brackets of Eq. (7) goes
to zero. It is the condition for the constant С to be
found:
1
0
24
2
3
])
4
)2(
exp([
)2(
−
∞
∫
−
−×
×=
xdx
a
xxa
MTa
aNC
c
c
d
π
(8)
Evaluating the integral of Eq. (8) by the method of
saddle points we expand the exponent index near its
maximum point x=1. As result, it is easy to see that a
number of the dislocations passing through the obstacles
can be defined by the expression
63
)exp(~
a
aq c−ρ , (9)
where ρ is the average density of the dislocations.
The dependence )(sqq = takes the form shown on
Fig. 2, where the quantity aas c= is put as the
abscissa axis. According to the data obtained by us and
other authors [9] the relative increasing the flow stress
of a material in 4-20 times is observed in a lot of the
model and reactor materials already at the doses of
10-2...10-1 dpa. Besides, it is seen that the fraction of
dislocations overcoming obstacles in the dynamic
regime becomes already essential other things being
equal in the irradiated materials (accordingly [5], getting
dislocation velocities ∼ 0.1 of the sound velocity c is a
criterion of the dynamical or “pseudo – relativistic”
regime). Fig. 2 shows also that the dynamical (pseudo –
relativistic) regime of deformation can be getting at the
lower dislocation velocities.
Fig. 2. The dependence of the fraction of the
dislocations overcoming through the obstacles on the
external force characterized by the quantity s. q1, q2, q3,
q4 corresponds to the values of the obstacle
concentration increasing under irradiation
As shown in work [5], the pseudo – relativistic
effects must be taken into account already for the
dislocation densities ≈1010сm-2 that is the velocity of
dislocation motion can approach to the near sound one
(≤ 0.1 c). For instance, in the case of irradiated nickel,
nuclear steel, this dislocation density corresponds to the
strain ≥100Mpa. So, the similar effects can manifest at
the initial deformation stages corresponding to the
interval of Chernov – Luders deformations. A lot of the
experiments showed that the high dislocation densities
are observed in deformation channels forming in
irradiated materials near the yield point. This is
connected with the plastic instability of Chernov –
Luders’ kind [1,2].
The model represented by this work can be related
indirectly with the problem of embrittlement of the
irradiated nuclear steels. Experimental investigations
reveal that the processes of deforming and destroying
near steels are accompanied by the dynamical processes
of dislocation channeling and destruction of the smallest
defects such as micro void, loops and isolation in
nuclear steels. The localized deformation channels near
an intersurface can cause the sharp strain concentration
proportional to the total value of a dislocation “charge”
and favour forming the micro cracks.
Thus, in the represented model, the evolution of
plastic instability is considered in an irradiated
deformed material for allowing the dependence of the
dislocation distribution in the ensemble on velocities. It
is shown the sharp increasing the fraction of the
dislocations overcoming the obstacles in the dynamical
regime can be observed. Upon that this effect can be get
for lower deformation velocities in increasing a power
of embrittlement (irradiation dose).
ACKNOWLEDGMENTS
Authors thank the Corresponding Member of
Ukrainian NAS, prof. I.M. Neklyudov and Academician
of Petri Primi Academia Scientiarum et Artium, prof.
N.V. Kamyshanchenko for useful discussion of this
work. One of the authors (V.V. Krasil’nikov) thanks the
RFBR for the financial support by grant №00-02-16337.
REFERENCES
1. I.M. Neklyudov, N.V. Kamyshanchenko. Radiation
hardening and embrittlement of metals. In the book:
Structure and radiation damageness of construction
materials. M.: “Metallurgy”, 1996, 168 p.
2. A.V. Volobuyev, L.S. Ozhigov, A.A. Parkhomenko
// Problems of atomic science and technology.
Series: physics of radiation damagenesses and
radiation material science (64). 1996, №1, p. 3.
3. I.M. Neklyudov, L.S. Ozhigov, A.A. Parkhomenko,
V.D. Zabolotny. Physical phenomena in solid state.
Proceedings of second scientific conference.
Kharkov, KSU, 1995, p. 132.
4. G.A. Malygin. Selforganization of dislocation and
sliding localization in plastically deformed crystals //
Fizika Tverdogo Tela. 1995, v.37, №1, p. 3-42.
5. L.E. Popov, L.Ya. Pudan, S.N. Kolupayeva, V.S.
Kobytev, V.A. Starchenko. Mathematical modeling
plastical deformation. Tomsk: “Iz-vo Tomsk.
Univ.”, 1990, 184 p.
6. Sh.Kh. Khannanov. Fluctuations of dislocation
density under plastical flow of crystals // Fizika
metallov i metallobedenie. 1994, v. 78, №1, p. 31-
39.
7. L.D. Landau, E.M. Liphshits. Theoretical physics. V.1.
Mechanics. Moscow: “Nauka”, 1965, 204 p.
8. V.I. Vladimirov. Physical nature of metal
destrucation.. M.: “Metallurgy”, 1984, 368 p.
9. V.F. Zelensky, I.M. Neklyudov, L.S. Ozhigiv,
E.A. Reznichenko, V.V. Rozhkov,
T.P. Chernyayeva. Certain problems of physics of
material radiation damageness. Kiev: “Naukova
Dumka”, 1979, 330 p.
10. V.A. Likhachev, A.E. Volkov, V.E. Shudegov.
Continual theory of defects. Leningrad: iz-vo
Leningr. Univer. 1986, 232 p.
11. G.A. Malygin. Selforganization process of
dislocation and crystal plasticity // Usp. Fiz. Nauk.
1999, v. 169, №9, p. 979-1010.
12. J. Fridel. Dislocations. Pergamon Press, Oxford–
London–Edinburgh–New York–Paris–Frankfurt,
1964.
64
|