The methods for the calculation of the elastic interaction of point defects with a dislocation loops in hexagonal crystals
Green's function method is used for hexagonal crystals according to Lifshitz-Rosenzweig and Kröner. Analytical expressions have been derived for the energy of elastic interaction of radiation point defects with dislocation edge loops of two types: c-loop (in the basal plane; the Burger’s vector...
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| Zitieren: | The methods for the calculation of the elastic interaction of point defects with a dislocation loops in hexagonal crystals / O.G. Trotsenko, P.N. Ostapchuk // Вопросы атомной науки и техники. — 2017. — № 2. — С. 83-90. — Бібліогр.: 17 назв. — англ. |
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Trotsenko, O.G. Ostapchuk, P.N. 2018-06-15T18:00:29Z 2018-06-15T18:00:29Z 2017 The methods for the calculation of the elastic interaction of point defects with a dislocation loops in hexagonal crystals / O.G. Trotsenko, P.N. Ostapchuk // Вопросы атомной науки и техники. — 2017. — № 2. — С. 83-90. — Бібліогр.: 17 назв. — англ. 1562-6016 PACS: 62.20.Dc; 62.20.Fe https://nasplib.isofts.kiev.ua/handle/123456789/136012 Green's function method is used for hexagonal crystals according to Lifshitz-Rosenzweig and Kröner. Analytical expressions have been derived for the energy of elastic interaction of radiation point defects with dislocation edge loops of two types: c-loop (in the basal plane; the Burger’s vector bᴰ=½[0001]) and α-loop (in the plane {112¯0}, the Burger’s vector bᴰ=⅓[112¯0]). In the case of the basal loop an analogous expression is obtained by the independent solution of the equilibrium equations by the Eliot’s method. Numerical comparison of the expressions for zirconium showed complete identity of these approaches. Методом функцій Гріна для гексагональних кристалів у підходах Ліфшиця-Розенцвейга і Кренера отримані аналітичні вирази енергії пружної взаємодії радіаційних точкових дефектів з дислокаційними крайовими петлями двох видів: c-петлею (базисна площина залягання, вектор Бюргерса bᴰ=½[0001]) і α-петлею (площина залягання {112¯0}, вектор Бюргерса bᴰ=⅓[112¯0]). У разі базисної петлі аналогічний вираз отримано незалежно рішенням рівнянь рівноваги методом Еліота. Чисельне порівняння отриманих результатів для цирконію показало повний збіг розглянутих підходів. Методом функций Грина для гексагональных кристаллов в подходах Лифшица-Розенцвейга и Кренера получены аналитические выражения энергии упругого взаимодействия радиационных точечных дефектов с дислокационными краевыми петлями двух видов: c-петлей (базисная плоскость залегания, вектор Бюргерса bᴰ=½[0001]) и α-петлей (плоскость залегания {112¯0}, вектор Бюргерса bᴰ=⅓[112¯0]). В случае базисной петли аналогичное выражение получено независимо решением уравнений равновесия методом Элиота. Численное сравнение полученных результатов для циркония показало полное совпадение рассмотренных подходов. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Проблемы современной ядерной энергетики The methods for the calculation of the elastic interaction of point defects with a dislocation loops in hexagonal crystals Методи розрахунку пружної взаємодії точкових дефектів з дислокаційними петлями в гексагональних кристалах Методы расчета упругого взаимодействия точечных дефектов с дислокационными петлями в гексагональных кристаллах Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
The methods for the calculation of the elastic interaction of point defects with a dislocation loops in hexagonal crystals |
| spellingShingle |
The methods for the calculation of the elastic interaction of point defects with a dislocation loops in hexagonal crystals Trotsenko, O.G. Ostapchuk, P.N. Проблемы современной ядерной энергетики |
| title_short |
The methods for the calculation of the elastic interaction of point defects with a dislocation loops in hexagonal crystals |
| title_full |
The methods for the calculation of the elastic interaction of point defects with a dislocation loops in hexagonal crystals |
| title_fullStr |
The methods for the calculation of the elastic interaction of point defects with a dislocation loops in hexagonal crystals |
| title_full_unstemmed |
The methods for the calculation of the elastic interaction of point defects with a dislocation loops in hexagonal crystals |
| title_sort |
methods for the calculation of the elastic interaction of point defects with a dislocation loops in hexagonal crystals |
| author |
Trotsenko, O.G. Ostapchuk, P.N. |
| author_facet |
Trotsenko, O.G. Ostapchuk, P.N. |
| topic |
Проблемы современной ядерной энергетики |
| topic_facet |
Проблемы современной ядерной энергетики |
| publishDate |
2017 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Методи розрахунку пружної взаємодії точкових дефектів з дислокаційними петлями в гексагональних кристалах Методы расчета упругого взаимодействия точечных дефектов с дислокационными петлями в гексагональных кристаллах |
| description |
Green's function method is used for hexagonal crystals according to Lifshitz-Rosenzweig and Kröner. Analytical expressions have been derived for the energy of elastic interaction of radiation point defects with dislocation edge loops of two types: c-loop (in the basal plane; the Burger’s vector bᴰ=½[0001]) and α-loop (in the plane {112¯0}, the Burger’s vector bᴰ=⅓[112¯0]). In the case of the basal loop an analogous expression is obtained by the independent solution of the equilibrium equations by the Eliot’s method. Numerical comparison of the expressions for zirconium showed complete identity of these approaches.
Методом функцій Гріна для гексагональних кристалів у підходах Ліфшиця-Розенцвейга і Кренера отримані аналітичні вирази енергії пружної взаємодії радіаційних точкових дефектів з дислокаційними крайовими петлями двох видів: c-петлею (базисна площина залягання, вектор Бюргерса bᴰ=½[0001]) і α-петлею (площина залягання {112¯0}, вектор Бюргерса bᴰ=⅓[112¯0]). У разі базисної петлі аналогічний вираз отримано незалежно рішенням рівнянь рівноваги методом Еліота. Чисельне порівняння отриманих результатів для цирконію показало повний збіг розглянутих підходів.
Методом функций Грина для гексагональных кристаллов в подходах Лифшица-Розенцвейга и Кренера получены аналитические выражения энергии упругого взаимодействия радиационных точечных дефектов с дислокационными краевыми петлями двух видов: c-петлей (базисная плоскость залегания, вектор Бюргерса bᴰ=½[0001]) и α-петлей (плоскость залегания {112¯0}, вектор Бюргерса bᴰ=⅓[112¯0]). В случае базисной петли аналогичное выражение получено независимо решением уравнений равновесия методом Элиота. Численное сравнение полученных результатов для циркония показало полное совпадение рассмотренных подходов.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/136012 |
| citation_txt |
The methods for the calculation of the elastic interaction of point defects with a dislocation loops in hexagonal crystals / O.G. Trotsenko, P.N. Ostapchuk // Вопросы атомной науки и техники. — 2017. — № 2. — С. 83-90. — Бібліогр.: 17 назв. — англ. |
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| first_indexed |
2025-11-25T22:22:40Z |
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2025-11-25T22:22:40Z |
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1850563294671142912 |
| fulltext |
ISSN 1562-6016. PASТ. 2017. №2(108), p. 83-90.
THE METHODS FOR THE CALCULATION OF THE ELASTIC
INTERACTION OF POINT DEFECTS WITH A DISLOCATION LOOPS
IN HEXAGONAL CRYSTALS
O.G. Trotsenko, P.N. Ostapchuk
Institute of Electrophysics and Radiation Technologies,
National Academy of Sciences of Ukraine, Kharkov, Ukraine
E-mail: ostapchuk@kipt.kharkov.ua
Green's function method is used for hexagonal crystals according to Lifshitz-Rosenzweig and Kröner. Analytical
expressions have been derived for the energy of elastic interaction of radiation point defects with dislocation edge
loops of two types: c -loop (in the basal plane; the Burger’s vector 1/ 2 0001=D
b ) and a -loop (in the plane
1120 , the Burger’s vector 1/ 3 1120=D
b ). In the case of the basal loop an analogous expression is obtained
by the independent solution of the equilibrium equations by the Eliot’s method. Numerical comparison of the
expressions for zirconium showed complete identity of these approaches.
PACS: 62.20.Dc; 62.20.Fe
1. INTRODUTION AND THE BASIC
Irradiation by high-energy particles of the
constructions of nuclear facilities significantly change
their physical properties and entails a number of
phenomena that limit the duration of their operation [1]
(radiation embrittlement, radiation growth and creep,
radiation swelling). They adversely affect on the
performance of the elements of the reactors designs,
disabling them in unscheduled time. The physical
mechanisms underlying these phenomena are connected
with the evolution of the defect structure of the original
material, which is caused by diffusion fluxes of
radiation point defects (PD) (SIAs, vacancies) for
specific components of this structure. Density of flow of
PD to internal sinks S , which are the elements of the
defect structure of the material, contains two terms at a
constant temperature T and an isotropic coefficient of
diffusion D :
int ( )( )
( )
S
S ED DC
C
kT
rr
r rj . (1)
The first (stochastic) connected with the gradient of
the concentration ( )C r of PD in the material volume
and at the sink boundary, the second (drift current)
connected with the interaction of PD with the sink
elastic field int ( )SE r . Therefore, the theoretical
description of the sinks system evolution implies the
ability to calculate the value int ( )SE r . The most
important element of the defect structure of any metal is
a system of dislocations in the form of single loops or
clusters (network in cubic and rows or layers in
hexagonal close-packed crystals). With regard to
clusters, it is the task of the future, but the version of a
single loops is quite observable today. We are
interested, first of all zirconium, so the proposed
material is focused on hcp crystals. Thus, the purpose of
the article is to show how we can analytically calculate
the energy of the elastic interaction of PD with the
specific dislocation loops in real hexagonal crystal. As
an example, we considered the edge dislocation loops of
two types: c -loop (in the basal plane, the Burger’s
vector 1/ 2 0001=D
b ) and – a -loop (in the plane
1120 , the Burger’s vector 1/ 3 1120=D
b ). In
addition, for c -loop required energy directly calculated
by solution of the equilibrium equations.
In the theory of elasticity PD is described by the
volume distribution of dipole forces without moments,
i. e. by the expression of the form:
( ) ( )i i j jf P r r , i j j iP P . (2)
According Eshelby [2], the energy of interaction of such
an object with a source of stress S (
S
u ,
S
iju ,
S
ij ) in an
elastic medium is given by
int ( ) ( )S
ij ijE P u r r , (3)
where r is coordinate of PD. If ij ijP P , i. e. PD is
modeled by the dilation center, and source of stress S
is dislocation loop, for the energy of their interaction
finally we have:
int ( ) / ( )D
ijE P Spu rr , (4)
where P is the power of dilatation center; and the
coordinate system is associated with dislocation. Our
task is the calculation of the value
D
ijSpu for a
particular dislocation loop.
There are two methods for calculating the elastic
field of the loops
D
iju . The first method is the solution
of the quilibrium equations in the terms of
displacements with appropriate boundary conditions.
The second method is the Green's function method
allowing to calculate the displacement caused by the
dislocation of any form in an arbitrary anisotropic
elastic medium, according to the classical formula [3, 4]
D
i jD D D
i jklm m l
kS
G
u C b n dS
x
r r
r , (5)
where jklmC is the tensor of the elastic moduli of the
medium simulating the crystal;
D
mb is mth component of
the Burger’s vector of the dislocation;
D
ln is l th
component of the normal vector to any surface
DS ,
lying on the dislocation line; r is the coordinate of the
point on the surface
DS ; r is the coordinate of the point
of observation and i jG is the tensor Green's function
(TFG ) of the elastic medium. According Elliott [5], the
equilibrium equations are solved in terms of two stress
functions. As for the TFG, two approaches are cited in
the literature: Lifshitz-Rosenzweig (1947) [6] and
Kröner (1953) [7] with a modified coefficient [8]. The
first approach is valid for any unlimited elastic
anisotropic medium, and the second is valid seemingly
only for the hexagonal system.
2. EDGE LOOP IN THE BASAL PLANE
OF THE HEXAGONAL CRYSTAL
We consider a circular vacancy loop of radius R
lying in the plane 0z (basal plane) of the cylindrical
coordinate system ( r , , z ), the Burger’s vector which
is perpendicular to the plane of the loop and has the
only z -component
D(0,0,b )=D
b . The normal
vector to the plane of the loop (0,0,1)=n coincides
with the positive direction of the axis “ z ”, which is
also the axis of symmetry of the crystal. An example of
such an object can serve, according to [9], the so-called
c -loop b 1/ 2 0001= in zirconium under electron
irradiation at the temperature 715T K .
2.1. SOLUTION OF THE EQUILIBRIUM
EQUATIONS (ELLIOTT’S APPROACH)
Since the problem is axially symmetric, the angular
dependence is absent, and the stress state is uniquely
determined by four components of the stress tensor:
rr , , zz , rz , which satisfy the equations of
equilibrium
0
rrrr rz
r z r
; (6)
0rz zz rz
r z r
. (7)
In terms of displacement, taking into account the
substitution ru
r
; zu k
z
they take the form
2
2
2
44 13 44
11
( )
0;
k
r z
C C C
C
2
2 33
2
44 13 44
0
C k
z C k C C z
, (8)
where
2
2
2
1
r rr
and 11C , 12C , 13C , 33C ,
44 55C C is the minimum number of non-zero elastic
moduli of hexagonal crystal. Equations (8) have a
solution under the condition
33
44 13 44
44 13 44
11
( )k C k
C k C C
C C C
C
.
This condition reduces (8) in the form
2
2
2
( , ) 0r z
z
, 1, 2 , (9)
where coefficients are the roots of a quadratic
equation
2 2
44 33
33
11 13 44 13 11
44
2
0.
C C
C
C C C C C
C
(10)
Meanwhile,
2
1
ru
r
;
2
1
zu k
z
;
33 44
13 4411 44
13 44
k
C C
C CC C
C C
(11)
and the desired quantity
D
ijSpu in terms of the
functions is given by the expression
22
2
1
( )D
ijSpu k
z
. (12)
Hankel’s transform [10] reduces (9) to ordinary
differential equations
2
2
2
( , ) 0
d
G z
dz
, (13)
whose solution is trivial
( , ) ( )exp( / )
( )exp( / ).
G z A z
B z
In our case we consider the solutions when all
components of the displacement and stress converges to
zero, when z . Then we have 0B , and the
coefficients A are determined from the boundary
conditions of the problem in the plane 0z . They are
as follows [11]:
1
( ,0)
2
D
zu r b , 0 r R ;
( ,0) 0zu r , r R , ( ,0) 0rz r . (14)
Using Hankel’s inversion theorem [10], we obtain
for shear stress:
2
3
44 1
0
1
( , )
1
( )exp( / ) ( ) ,
rz r z
k
C A z J r d
(15)
where
0J ,
1J are Bessel functions of the first kind.
And then the second boundary condition (14) gives the
relationship 21
2 1
2 1
1
( ) ( )
1
k
A A
k
. Acting in a
similar way for z -component of the displacement we
receive the following expression
2
2
0
0
1
( , )
( )exp( / ) ( ) .
zu r z
k
A z J r d
(16)
It is convenient to use dimensionless variables
t R ; /r R . Then the first boundary condition
(14) gives two integral equations
2
1 0
0
3
2
1
1 2
( / ) ( )
1
, 0 1;
2
D
t A t R J t dt
k b R
k k
2
1 0
0
( / ) ( ) 0t A t R J t dt
, 1 ,
which are satisfied under the condition
3
2 2
1 1 1
1 2
1
( / ) ( )
2
Dk b R
t A t R J t
k k
. As a result, the
function takes the form:
1
2
1 1 0 1
0
1 2
( , )
1 1
( ) ( )exp( / ) ;
2
D
r z
k b R
J R J r z d
k k
(17)
2
1
2 1 0 2
0
1 2
( , )
1 1
( ) ( )exp( / ) .
2
D
r z
k b R
J R J r z d
k k
Accordingly, for the desired quantity
D
ijSpu from (12)
we obtain:
1 12 1 1 1 2 2
0 0
1 2 1 21 1 2 2
1 1
, , ,
2 4
D D
D
ij
k k k kb r z r z b r z
Spu I I I
R k k R k k R R R RR R
, (18)
where
1
0
, ( )expn n
m m
r z r z
I t J t J t t dt
R R R R
and thus, the problem is solved.
2.2. THE GREEN’S FUNCTIONS METHOD: LIFSHITZ-ROSENZWEIG’S APPROACH
From formula (5) we have:
2
2
13 , 33 3,3
1
( ) ( ) ( )
D
D D
i i i i
S
ij b d r C G C GSpu
r r r r r ,
,ij k ij
k
G G
x
, (19)
where the index “i” is summation from 1 to 3. Analytical universal expressions for the component of TFG of
hexagonal crystal are given in [12]
2 2
3 3 33 3
1
( ))
4
( ( ) kk k FG
r r
r r
, k k
k
x x
r r
;
2 2 2
3 3 3 3 3
1
( ) ( ) ( )( )
4
k k k kG
r r
r r
. (20)
As a result,
2
2 2 2 2
3 3 3 33 2
3
(1 3 ) ( ) 2 (1 )
4
( )
D
D
D
ij
S
b d r dQ
QSp
d
u
r
r r
;
2 2 2 2
3 13 3 33 3 13 33 3( ) ( ) ( ) ( ) ( )Q C C W C C V ;
2 2 2 2 2
3 3 3 3 3 2
3
( ) (1 3 ) ( ) 2 (1 )
d
V
d
;
2 2 2
3 3 3 2
3
( ) ( ) 2(1 )
dF
W F
d
;
2 2 2 2 2 2 2
3 3 3 3 3 3 32 2
3 3
( ) ( ) 2 3 ( ) 2 (1 )
d d
d d
. (21)
Functions
2
3( ) ,
2
3( )F ,
2
3( ) , and
2
3( ) are quite cumbersom and carried to appendix.
2.3. THE GREEN’S FUNCTIONS METHOD: KRÖNER’S APPROACH
Original formula is (19), but the components of TFG are other here. According to Kröner we have:
2 2 2 2 23
11 1/2 42 2
1
( )1 k
k k
k
k
x z y r z
G A B
rr z
,
2 2 2r x y ;
2 2 2 2 23
22 1/2 42 2
1
( )1 k
k k
k
k
y z x r z
G A B
rr z
;
3
33 1/2
2 2
1
k
k
k
D
G
r z
;
2 2
3
12 1/24 2 2
1
2k
k
k
k
r zxy
G A
r r z
;
3
31 1/22 2 2
1
k
k
k
Cxz
G
r r z
;
3
32 1/22 2 2
1
k
k
k
Cyz
G
r r z
. (22)
Performing the appropriate summation in (19), we obtain
2
2 2 2
13 1 3 33 2 3 13 33 3 33
(( ) ( ) ( ) ( ))
4
D
D
D
i
S
j
b d r
C C C CSpu
r
r r
;
2
1
2
2
3
1
( )A F
;
2
2
2 3
1
( )D F
;
33
2
2
1
( )C F
;
2 2
2 3 3
3 5/2
2 2
3 3
2 (1 )
( )
(1 )
F
, (23)
where are the roots of the same quadratic equation (10), and the coefficients in (22) and (23) have the form
2
66 11 33 44 13 44 /A C C C C C C E
, 66 11 12 / 2C C C ;
44 11 44 66 /D C C C C E ; 13 44 44 66 /C C C C C E ;
1 11 44 66 1 3 1 2E C C C ; 2 11 44 66 2 3 2 1E C C C . (24)
Thus, the problem reduces to the integration of (19)
and (23) in a plane of the circular loop of radius R
(recall that 0z since the loop lies in the basal plane
of the crystal).
So, we have three independent variants for solving
the same problem. Their numerical comparison was
carried out with respect to zirconium. The experimental
values of its modulus of elasticity are following [12]
(Mbar):
11 1.554C ;
12 0.672C ;
13 0.646C ;
33 1.725C ;
55 44 0.363C C . The comparison
results are presented in Figs. 1, 2 in dimensionless
cylindrical coordinates; /r R ; /z R .
Fig. 1. Dependence of the functions ( , )EI (solid
line), ( , )LRI (dotted line) ( , )KI (points)
according (18), (21), and (23) on the relative
distance /r R in the plane 0.1 of zirconium
Fig. 2. Dependence of the functions ( , )EI (solid
line), ( , )LRI (dotted line) ( , )KI (points)
according (18), (21), and (23) on the relative
distance /r R in the plane 0.5 of zirconium
Formulas (18), (21), (23) are reduced to a united
form ( , )
4
D
ij
Db
I
R
Spu
, so in Figs. 1, 2 the
function ( , )EI (solid line) corresponds to the
formula (18), ( , )LRI (dotted line) corresponds to the
formula (21) (Lifshitz-Rosenzweig), ( , )KI (points)
corresponds to the formula (23) (Kröner). They show
the dependence of the functions
EI ,
LRI ,
KI on the
relative distance for the two planes 0.1 and
0.5 above the plane of the loop 0 . We see the
perfect agreement of all three approaches. The
qualitative nature of the interaction does not change
(vacancy loop, the formula (4)): vacancy ( 0P ) is
pushed out from the inside of the loop 1 , and
attracted in the external region 1 (SIA on the
contrary). However, if on a sufficiently large distances
from the plane of the loop ( 0.5 ) interaction
smoothly changes its sign at the boundary of the loop
1 , then with the approaching to the plane of loop
( 0.1 ) the interaction varies substantially step-wise
on its boundary.
3. EDGE LOOP IN PRISMATIC PLANE
OF THE HEXAGONAL CRYSTAL
Consider a circular vacancy loop of radius R lying
in the plane 0x of the Cartesian coordinate system
(prismatic plane of hexagonal crystal). The Burger's
vector is perpendicular to the plane of loop and has only
x -component
D(b ,0,0)=D
b . The vector of normal
to the plane of the loop (1,0,0)=n coincides with
the positive direction of the axis “x”. An example of
such an object may be, according to [9], the so-called
a -loops 1/ 3 1120=D
b in zirconium. It is the
dominant form of loops under neutron irradiation as the
vacancy as and the interstitial in nature. The axial
symmetry in this case is absent so well-developed
methods for solving equilibrium equations are not
applicable. It remains TFG’s method. From (5) we
again have:
2
2
12 , 13 3,3
1
2
11 12 1,1
( )
( ) ( ) ,
D
D
D D
ij i i i i
S
D
i i
S
Spu b C G C G d r
b C C G d r
r r
r
(25)
where the repeated suffix “i”, as previously, is summed
over the values 13. Note that the first term in (25)
coincides up to the coefficient with the expression (19),
for which the result is already known both Lifshitz-
Rosenzweig (21), and by Kröner (23). Therefore, only
the sum
1,1i iG to be calculated. The result of
calculations by Lifshitz-Rosenzweig is following:
2 2
2 2 2 2
3 11 12 1 3 33 3 2
3
( ) (( 3 ( ) 2
4 4
) )
D D
D
D
i
D
S S
j
b d r b d r dY
Q C C Ypu
d
S
r
r r r r
; (26)
2 2 2 2 2 2 2 2 2
3 3 12 3 13 3 3 3 12 3 13 3 11 12 32
3
( ) (1 3 ) ( ) ( ) 2 (1 ) ( ) ( ) ( ) ( );
d
Q C Y C C Y C C C Y
d
2 2 2
3 3 3( ) ( ) ( )V W ,
2 2 2
3 3 3( ) ( ) ( )Y V .
The functions
2
3( ) ,
2
3( )W ,
2
3( )V are the same as in (21). A similar result for Kröner is:
2
2 2 2
12 1 3 13 2 3 12 13 3 33
2
2 2
11 12 1 33
( ) ( ) ( ) ( )
4
( )
( )
( , );
4
D
D
D
S
D
i
S
D
j
b d r
C C C C
b r
C
u
d
p
C
S
r
r r
r r
(27)
2 22
2 2 1
1 3 3/2 2 22 2
1 3 33 3
( ) 3
( , ) 1
(1 )(1 )
A C B
.
All functions and constants are as defined above
(23), (24). Here we must remember that the loop lies in
the plane «yz» of Cartesian coordinate system, i.e.
1 /x r r , ( 0x ), and 3 ( ) /z z r r .
Therefore the dimensionless cylindrical coordinates
used for the numerical comparison of formulas (26) and
(27) are following: /x R , /r R
(
2 2 2r y z , cosy r , sinz r , and is
an azimuth angle, but in the plane of the loop “yz”). As
earlier, formulas (26), (27) are reduced to the standard
form ( , , )
4
D
ij
Db
I
R
Spu
, however, in order not
to overload the article by the pictures, we give them a
qualitative description. Firstly, we emphasize again full
coincidence of Lifshitz-Rosenzweig’s and Kröner’s
approaches. Secondly, we note that, unlike the base loop
the functions
LRI (26) and
KI (27) include a weak
dependence on the azimuthal angle in the plane of the
loop, but the character of interaction (their sign on both
sides of the conditional boundary loop) from is
independent. Moreover, the dependence of the functions
LRI ,
KI on the relative distance for different
planes const almost coincides with a similar for
the base loop (see Figs. 1, 2). It is quite natural in view
of the weak sensitivity of these functions to the angle
.
4. DISCUSSION OF RESULTS
So, the different methods are used to derive
analytical expressions for the energy of elastic
interaction of PD with the specific dislocation loops in
real hexagonal crystal. As an example, we were taken
vacancy edge loops of two types: c - and a -loop in
zirconium. Simplifies circumstance in cases considered
above is that the normal vector to the plane of the loop
and its Burgers vector has only one component in a
Cartesian coordinate system. In general, it is not. And
the initial formula (5) then become much more
complicated. However, the method of calculation
remains the same and does not contain principal
mathematical difficulties.
Knowing the energy of elastic interaction PD with a
loop allows one to calculate the diffusion fluxes of
radiation PD on it [13, 14], as well as its bias factor to a
specific type of PD. The bias is the most important
characteristic of dislocations which made possible to
explane such phenomena as swelling, radiation creep,
radiation hardening structural materials of nuclear
reactors [1517], etc. Here, however, we should note
the following. In all theories bias factor of dislocation
was calculated assuming an elastically isotropic crystal
when the energy of interaction of PD with a straight
dislocation or a loop is harmonic function. For a
hexagonal crystal it is not. This follows, for example,
from the expression (18) for the base prismatic loop.
Therefore, the corresponding diffusion problem [13, 14]
is complicated by the presence of the additional term,
and the authors' conclusions regarding of the depending
of the bias factor from the radius of the loop, the type
and ratio of dilatation volume of TD, can change.
REFERENCES
1. В.Н. Воеводин, И.М. Неклюдов. Эволюция
структурно-фазового состояния и радиационная
стойкость конструкционных материалов. Киев:
«Наукова думка», 2006, 376 с.
2. Дж. Эшелби. Континуальная теория
дислокаций. М.: «Наука», 1963, 215 с.
3. Л.Д. Ландау, Е.М. Лифшиц. Теория
упругости. М.: «Наука», 1987, 246 с.
4. Дж. Хирт, И. Лоте. Теория дислокаций. М.:
«Атомиздат», 1972, 600 с.
5. H.A. Elliott // Proc. Cambridge Phil. Soc. 1948,
v. 44, p. 522; 1949, v. 45, p. 621.
6. И.М. Лифшиц, Л.Н. Розенцвейг // ЖЭТФ.
1947, т. 17, с. 783.
7. E. Kröner // Zeitschrift fur Phyzik. 1953, v. 136,
p. 402.
8. J.R. Willis // Quart. J. Mech. Appl. Math. 1965,
v. 18, p. 419.
9. M. Griffiths // J. Nucl. Mater. 1988, v. 159,
p. 190.
10. Ian. N. Sneddon. Fourier Transforms. Mc
Graw Hill, New York, 1951, 542 p.
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p. 284.
12. П.Н. Остапчук, О.Г. Троценко // ФТТ. 2016,
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Nucl. Mater. 2005, v. 336, p. 11.
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p. 286.
15. P.T. Heald, M.V. Speight // Philos. Mag. 1974,
v. 29, p. 1075.
16. L.K. Mansur // Philos. Mag. A. 1979, v. 39,
p. 497.
17. V.I. Dubinko, S.A. Kotrechko, V.F. Klepikov //
Radiat. Eff. Defects Solids. 2009, v. 164, p. 647.
APPENDIX
2
3 2
1 2 3
2
( )
( )
i a b
z z A
;
2 2
3 32
1 2 3 1 2
2 2
( )
( )
i a b
F b a b
z z A z z
;
2 2
2 3 3
3 2 2 2
1 2 3 3 3
( )2
( )
( ) ( ) ( ) ( )(1 )
R bi
z z b A b P b
;
2 2 2
2 3 3 3
3 2 2 2 2 2
1 2 3 3 3 3
( ) ( )2
( )
( ) 1 ) ( ) ( ) ( )(1( )
P bi S
z z A b P bb
;
2 2
2 3 3
3 1 22 2
1 2 3 3
( ) ( )
( )
2 (1 )
( )( )
( )) (
a b A P b
z z
b
bz
R
z P
;
2 2
2 2 2 2 23 3
3 3 1 2 1 2 1 2 32
1 2 3
( )( )
(
( ) ( )1 2
(
2 ))
)
(1
A P b b B
S z z z z z z
z z P P
b
A
a b
b
;
2 2 2 2
3 3 3( ) 2 (1 ) (1 )A k l m ;
2 2
3 3( ) 2 (1 )B k l
2 2
3 3( ) (1 )P b ;
2k a b b ;
2
2m a b ; 2 2 2l a b b ;
12a C ; 11 12 66
1
( )
2
b C C C ; 13 12C C ;
44 11 12
1
( )
2
C C C ; 11 33 44 134 2C C C C ;
1 2 2
3
2
( )
k
z z
A
;
2
3
1
1/
2 2 2
3 3
2
( )2
2
( ) ( )
Bk
z
A A
z i
.
Article received 29.12.2016
МЕТОДЫ РАСЧЕТА УПРУГОГО ВЗАИМОДЕЙСТВИЯ ТОЧЕЧНЫХ ДЕФЕКТОВ
С ДИСЛОКАЦИОННЫМИ ПЕТЛЯМИ В ГЕКСАГОНАЛЬНЫХ КРИСТАЛЛАХ
О.Г. Троценко, П.Н. Остапчук
Методом функций Грина для гексагональных кристаллов в подходах Лифшица-Розенцвейга и Кренера
получены аналитические выражения энергии упругого взаимодействия радиационных точечных дефектов с
дислокационными краевыми петлями двух видов: c -петлей (базисная плоскость залегания, вектор Бюргерса
1/ 2[0001]=D
b ) и a -петлей (плоскость залегания 1120 , вектор Бюргерса 1/ 3 1120=D
b ). В
случае базисной петли аналогичное выражение получено независимо решением уравнений равновесия
методом Элиота. Численное сравнение полученных результатов для циркония показало полное совпадение
рассмотренных подходов.
МЕТОДИ РОЗРАХУНКУ ПРУЖНОЇ ВЗАЄМОДІЇ ТОЧКОВИХ ДЕФЕКТІВ
З ДИСЛОКАЦІЙНИМИ ПЕТЛЯМИ В ГЕКСАГОНАЛЬНИХ КРИСТАЛАХ
О.Г. Троценко, П.М. Остапчук
Методом функцій Гріна для гексагональних кристалів у підходах Ліфшиця-Розенцвейга і Кренера
отримані аналітичні вирази енергії пружної взаємодії радіаційних точкових дефектів з дислокаційними
крайовими петлями двох видів: c -петлею (базисна площина залягання, вектор Бюргерса 1/ 2[0001]=D
b )
і a -петлею (площина залягання 1120 , вектор Бюргерса 1/ 3 1120=D
b ). У разі базисної петлі
аналогічний вираз отримано незалежно рішенням рівнянь рівноваги методом Еліота. Чисельне порівняння
отриманих результатів для цирконію показало повний збіг розглянутих підходів.
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