Dislocation kinetics during plastic deformation of two-dimensional polycrystals
The dislocation-kinetic approach is applied to the study of plastic flow of plate specimens of two-dimensional polycrystals of high purity metals under uniaxial tension with a constant strain rate at moderate temperatures. A dislocation-kinetic equation is formulated. It takes into account the role...
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nasplib_isofts_kiev_ua-123456789-1949322025-02-09T10:14:14Z Dislocation kinetics during plastic deformation of two-dimensional polycrystals Дислокаційна кінетика при пластичній деформації двовимірних полікристалів Дислокационная кинетика при пластической деформации двумерных поликристаллов Badiyan, E.E. Tonkopryad, A.G. Ftomov, Ye.V. Shekhovtsov, O.V. Physics of radiation damages and effects in solids The dislocation-kinetic approach is applied to the study of plastic flow of plate specimens of two-dimensional polycrystals of high purity metals under uniaxial tension with a constant strain rate at moderate temperatures. A dislocation-kinetic equation is formulated. It takes into account the role of the free surface of a plate specimen, which is the source and sink of dislocations, and the strengthening effect of through grain boundaries in a twodimensional polycrystal. To calculate tensile stress-strain curves, the kinetic equation was transformed using the Taylor strain hardening law and an analytical solution was obtained for this equation. Using the example of plate specimens of two-dimensional polycrystals of high purity aluminium (99.999 at.%) it was shown that the calculation results are in good agreement with experimental data. Дислокаційно-кінетичний підхід застосовано до дослідження пластичної течії плоских зразків двовимірних полікристалів чистих металів в умовах одноосного розтягу з постійною швидкістю деформації при помірних температурах. Сформульовано дислокаційно-кінетичне рівняння, в якому враховані роль вільної поверхні плоского зразка, яка є джерелом і стоком дислокацій, і зміцнююча дія наскрізних меж зерен у двовимірному полікристалі. Для розрахунку кривої деформації кінетичне рівняння перетворено з використанням закону деформаційного зміцнення Тейлора і отримано аналітичне рішення цього рівняння. На прикладі плоских зразків двовимірних полікристалів чистого алюмінію (99,999 ат.%) показано, що результати розрахунків досить добре узгоджуються з експериментальними даними. Дислокационно-кинетический подход применен к исследованию пластического течения плоских образцов двумерных поликристаллов чистых металлов в условиях одноосного растяжения с постоянной скоростью деформации при умеренных температурах. Сформулировано дислокационно-кинетическое уравнение, в котором учтены роль свободной поверхности плоского образца, являющейся источником и стоком дислокаций, и упрочняющее действие сквозных границ зерен в двумерном поликристалле. Для расчета кривой деформации кинетическое уравнение преобразовано с использованием закона деформационного упрочнения Тейлора и получено аналитическое решение этого уравнения. На примере плоских образцов двумерных поликристаллов чистого алюминия (99,999 ат.%) показано, что результаты расчетов достаточно хорошо согласуются с экспериментальными данными. 2019 Article Dislocation kinetics during plastic deformation of two-dimensional polycrystals / E.E. Badiyan, A.G. Tonkopryad, Ye.V. Ftomov, O.V. Shekhovtsov // Problems of atomic science and technology. — 2019. — № 2. — С. 25-29. — Бібліогр.: 18 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/194932 539.4.01 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Physics of radiation damages and effects in solids Physics of radiation damages and effects in solids |
| spellingShingle |
Physics of radiation damages and effects in solids Physics of radiation damages and effects in solids Badiyan, E.E. Tonkopryad, A.G. Ftomov, Ye.V. Shekhovtsov, O.V. Dislocation kinetics during plastic deformation of two-dimensional polycrystals Вопросы атомной науки и техники |
| description |
The dislocation-kinetic approach is applied to the study of plastic flow of plate specimens of two-dimensional polycrystals of high purity metals under uniaxial tension with a constant strain rate at moderate temperatures. A dislocation-kinetic equation is formulated. It takes into account the role of the free surface of a plate specimen, which is the source and sink of dislocations, and the strengthening effect of through grain boundaries in a twodimensional polycrystal. To calculate tensile stress-strain curves, the kinetic equation was transformed using the Taylor strain hardening law and an analytical solution was obtained for this equation. Using the example of plate specimens of two-dimensional polycrystals of high purity aluminium (99.999 at.%) it was shown that the calculation results are in good agreement with experimental data. |
| format |
Article |
| author |
Badiyan, E.E. Tonkopryad, A.G. Ftomov, Ye.V. Shekhovtsov, O.V. |
| author_facet |
Badiyan, E.E. Tonkopryad, A.G. Ftomov, Ye.V. Shekhovtsov, O.V. |
| author_sort |
Badiyan, E.E. |
| title |
Dislocation kinetics during plastic deformation of two-dimensional polycrystals |
| title_short |
Dislocation kinetics during plastic deformation of two-dimensional polycrystals |
| title_full |
Dislocation kinetics during plastic deformation of two-dimensional polycrystals |
| title_fullStr |
Dislocation kinetics during plastic deformation of two-dimensional polycrystals |
| title_full_unstemmed |
Dislocation kinetics during plastic deformation of two-dimensional polycrystals |
| title_sort |
dislocation kinetics during plastic deformation of two-dimensional polycrystals |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2019 |
| topic_facet |
Physics of radiation damages and effects in solids |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/194932 |
| citation_txt |
Dislocation kinetics during plastic deformation of two-dimensional polycrystals / E.E. Badiyan, A.G. Tonkopryad, Ye.V. Ftomov, O.V. Shekhovtsov // Problems of atomic science and technology. — 2019. — № 2. — С. 25-29. — Бібліогр.: 18 назв. — англ. |
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Вопросы атомной науки и техники |
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ISSN 1562-6016. PASТ. 2019. №2(120), p. 25-29.
DISLOCATION KINETICS DURING PLASTIC DEFORMATION
OF TWO-DIMENSIONAL POLYCRYSTALS
E.E. Badiyan, A.G. Tonkopryad, Ye.V. Ftomov, O.V. Shekhovtsov
V.N. Karazin Kharkiv National University, Kharkov, Ukraine
E-mail: Evgeny.E.Badiyan@univer.kharkov.ua,
Phone: +38(057)707-53-47
The dislocation-kinetic approach is applied to the study of plastic flow of plate specimens of two-dimensional
polycrystals of high purity metals under uniaxial tension with a constant strain rate at moderate temperatures. A
dislocation-kinetic equation is formulated. It takes into account the role of the free surface of a plate specimen,
which is the source and sink of dislocations, and the strengthening effect of through grain boundaries in a two-
dimensional polycrystal. To calculate tensile stress-strain curves, the kinetic equation was transformed using the
Taylor strain hardening law and an analytical solution was obtained for this equation. Using the example of plate
specimens of two-dimensional polycrystals of high purity aluminium (99.999 at.%) it was shown that the calculation
results are in good agreement with experimental data.
PACS: 61.72.Cc, 61.82.Bg, 62.20.Fe
INTRODUCTION
The widespread use of various polycrystalline solids
as structural materials necessitates an understanding of
the physical nature of their strength and ductility. It is
known that grain boundaries are not only barriers to the
movement of dislocations that control the mechanical
properties of polycrystals, but also effective sources and
sinks of dislocations and other radiation defects
providing enhanced radiation resistance 1–3.Two-
dimensional polycrystals are a successful model object
for research. They contain only one layer of grains and
have through “vertical” grain boundaries. All grains are
“surface” in the sense that they have access to the free
surface of the sample. In two-dimensional polycrystals,
it is possible to determine the crystallographic
orientation of all grains and the crystal-geometric
parameters of their boundaries. Rotational 4, 5 and
other effects 6, 7 associated with plastic deformation
are specifically manifested due to the lack of tightness
of the “surface” grains in the “vertical” direction, which
is perpendicular to the tensile axis. At the same time,
two-dimensional polycrystals are finding independent
practical application as polycrystalline films, foils and
plates, which are operated under the action of
mechanical stresses.
A dislocation-kinetic approach can be used to
describe the plastic flow of a crystalline material. It is
based on equations describing the evolution of the
dislocation density in a material with increasing degree
of plastic strain. This approach makes it possible to
obtain the dependence of the flow stress on an average
grain size, transverse sample size, temperature, degree
and strain rate 8–14.
1. DISLOCATION-KINETIC EQUATION
FOR PLATE SPECIMENS
OF TWO-DIMENSIONAL POLYCRYSTALS
This paper is devoted to the study of plastic
deformation under conditions of uniaxial tension at
moderate temperatures of plate specimens of two-
dimensional polycrystals of high purity metals with a
thickness from or more and with an average
grain size from to macroscopic values (nano-
and micro-sized specimens with nano- and
microgranular structures were investigated in 12–14).
Here, plate specimens are understood as specimens with
a rectangular cross section whose sizes are related by
the ratio , where D is the specimen thickness
(size in the “vertical” direction), w and l are width and
length of the working area, respectively (Fig. 1). In such
specimens, the surface-to-volume ratio is
.
The kinetic equation describes the evolution of the
average dislocation density in a material with
increasing shear strain . It should contain the product
and the terms describing the processes of
accumulation of dislocations in the material and
reduction of their density. We will write down the
dislocation-kinetic equation for plate specimens of two-
dimensional polycrystals using data from 8–14. In this
case, the specificity of the studied specimens should be
taken into account. It consists in the fact that in such
objects a large role is played by the free surface of the
specimen, which is the source and sink of dislocations.
In addition, all grain boundaries are cross-cutting, which
determines the characteristics of the strain hardening
associated with them.
Fig. 1. Scheme of a plate specimen of a two-dimensional polycrystal
mailto:Evgeny.E.Badiyan@univer.kharkov.ua
The accumulation of dislocations in a three-
dimensional polycrystal, the thickness D of which
greatly exceeds the average grain size d, due to the
presence of grain boundaries is described by ,
where b is the Burgers vector, is the coefficient
determining the intensity of dislocation accumulation in
grains because the grain size d limits the mean free path
length of dislocations. The coefficient expresses the
relative proportion of grains enclosed in the bulk of the
specimen and not having access to the free surface of
the specimen. However, in a two-dimensional
polycrystal all the grains are “surface”, therefore .
Here, the through “vertical” grain boundaries have a
strengthening effect and we need to take it into account.
According to 7, the “vertical” boundaries of the
“surface” grains create obstacles for the movement of
dislocations only in those areas of the grains that adjoin
these boundaries. Thus, in the “surface” grain the
volume of which is equal to , there is a part of it with
volume , in which dislocations have difficulty for
ove ent, eeting with “vertical” grain boundaries,
whereas moving dislocations have no such obstacles in
the rest of the grain volume . The relative part of
the grain which is adjacent to the "vertical" boundaries
determines the factor . Then, by analogy with
the expression , we can write the expression
which characterizes the strengthening effect of
the “vertical” grain boundaries in a two-dimensional
polycrystal. The mean free path length of
dislocations in the “surface” grain before the eeting
with the vertical boundary (Fig. 1) needs to be
determined.
If we assume that the “surface” grains have the
shape of a square with a side on a free “horizontal”
surface and the size in the transverse
(“vertical”) direction then in the case of we are
dealing with a grain structure that is “needle” if
. For it, and where is the
angle between the slip plane and the tensile axis (the
length l of the working area of the specimen is measured
along this axis). Therefore, in this case
. For the grain structure with the parameters
, which is “pancake” for , we get
tg and sin , but the relation
remains unchanged. Thus, in a
dislocation-kinetic equation, grain-boundary hardening
in a two-dimensional polycrystal is described by the
term .
In addition to the grain-boundary hardening
considered above, contributions to the accumulation of
dislocations are also provided by the operation of
surface dislocation sources with density and the
dislocation generation by double cross-slip of screw
dislocations on forest dislocations. In the kinetic
equation, they are described by the terms
and
, respectively. Here, the coefficient
describes the intensity of dislocation multiplication on
forest dislocations ( ). It should be noted
that the mechanism of dislocation generation on forest
dislocations is characteristic of so-called ordinary
polycrystalline metals with grain sizes from
and more, but it does not work in nanocrystalline
materials 12, 14. For specimens with a rectangular
cross section, the surface-to-volume ratio is determined
by the expression , which for
plate specimens ( ) takes the form
13.
The loss of dislocations from the process of
generation due to their exit from the specimen onto its
surface leads to a decrease in the average density . In
the kinetic equation, such a process takes into account
the term . In addition, the dislocation
density in the material decreases due to the annihilation
of the screw sections of the dislocation loops, which
takes into account the term
, where is the
coefficient of annihilation of screw dislocation.
As a result, for plate specimens of two-dimensional
polycrystals of high purity metals with a thickness from
or more and with an average grain size from
to macroscopic values under uniaxial tension
with a constant strain rate ̇ at moderate temperatures
(in the absence of diffusion mechanisms of dislocation
annihilation) the dislocation-kinetic equation can be
written as
(
) (
) (
) (
)
(1)
2. PLASTIC FLOW OF PLATE SPECIMENS
OF TWO-DIMENSIONAL POLYCRYSTALS
The dependence of the stress on the degree of
plastic strain characterizes the plastic flow of a
material. To obtain stress-strain curves in the case
of uniaxial tensile of plate specimens of two-
dimensional polycrystals, we transform the kinetic
equation (1) in the same way as was done in 8, 10, 14
using the expressions , , where m is the
orientation factor, is the flow stress which is
determined by the interaction of dislocations with each
other in accordance with the Taylor equation 15
, (2)
in which is the constant of interaction of dislocations
with each other, is the shear modulus. Having
executed transformations, we receive
, (3)
where
,
,
.
Integrating (3), we obtain the dependence of the
deforming stress on the degree of plastic strain in
the implicit form:
ln| | ln| | )ln|
|
√
arctg √
. (4)
The integration constant C is determined from the condition . The parameters in (4) are:
(√
√( √
)
( √
)),
where
( √ )
( √ )
,
,
.
The quantities , , , are defined as the solution of the system of equations
,
,
,
,
wherein
√
, √
.
Further, as an example, the stress-strain curves
are presented in the case of uniaxial tensile of plate
specimens of two-dimensional polycrystals of high
purity aluminium (99.999 at.%). The tensile stress-strain
curves are shown by the lines in Fig. 2. They were
calculated on the basis of (4) and “stitched” to the linear
section corresponding to the elastic strain at stress and
strain values of 0.07 MPa and , respectively,
according to 16. The values of the parameters used in
the calculation of these tensile stress-strain curves by
the formula (4) were selected in accordance with the
data of 7, 12, 13, 17 and are presented in Table.
Experimental data are taken from 7 and are presented
in Fig. 2 by dots.
a b
Fig. 2. The tensile stress-strain curves for plate specimens of two-dimensional polycrystals of high purity aluminium
(99.999 at.%) a) with the thickness and a various average grain size : 1 , 2 ,,
3 , 4 ; b – with the average grain size and a various thickness :
5 , 6 , 7 , 8 .
Parameters of plate specimens of two-dimensional polycrystals of high purity aluminium (99.999 at.%) used in the
calculations of the tensile stress-strain curves in Fig. 2 in accordance with the data of 7, 12, 13, 17
Curve
number
, GPa b, nm ka
1 95 4 120 2.60 27 0.286 0.32 9.7 1.00
2 95 4 183 2.65 27 0.286 0.32 9.7 0.85
3 97 4 205 2.52 27 0.286 0.32 9.7 1.50
4 95 4 1000 2.60 27 0.286 0.32 9.7 1.00
5 50 4 379 2.69 27 0.286 0.32 9.7 1.00
6 100 4 379 2.69 27 0.286 0.32 9.7 1.00
7 266 4 379 2.69 27 0.286 0.32 9.7 1.00
8 700 4 379 2.69 27 0.286 0.32 9.7 1.00
The strain hardening of plate specimens of two-
dimensional polycrystals depends according to (1) and
(4) both on the average grain size and on the
thickness D. The tensile stress-strain curves in Fig. 2
clearly show it. It should also be noted that these curves
are limited by the degree of strain of %. For large
plastic strain, it is necessary to take into account the
features of the formation of cellular and fragmented
dislocation structures and their contribution to strain
hardening.
CONCLUSIONS
In the framework of the dislocation-kinetic
approach, the plastic flow of plate specimens of two-
dimensional polycrystals of high purity metals to the
stage of developed plastic strain has been investigated.
Based on the data available in the literature, a kinetic
equation has been formulated that describes the
evolution of the dislocation density with increasing
degree of strain of a plate specimen of two-dimensional
polycrystal with a thickness and an average grain size
from to macroscopic values under uniaxial
tension with a constant strain rate at moderate
temperatures. To calculate a tensile stress-strain curve,
the kinetic equation was transformed using the Taylor
strain hardening law and an analytical solution of this
equation was obtained. As an example, the tensile
stress-strain curves for plate specimens of two-
dimensional polycrystals of high purity aluminium
(99.999 at.%) are presented. They are in fairly good
agreement with experimental data. The proposed model
allows one to quantitatively describe the strain
hardening of plate specimens of two-dimensional
polycrystals depending on the average grain size and
thickness of the deformable specimens.
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Article received 11.02.2019
ДИСЛОКАЦИОННАЯ КИНЕТИКА ПРИ ПЛАСТИЧЕСКОЙ ДЕФОРМАЦИИ
ДВУМЕРНЫХ ПОЛИКРИСТАЛЛОВ
Е.Е. Бадиян, А.Г. Тонкопряд, Е.В. Фтемов, О.В. Шеховцов
Дислокационно-кинетический подход применен к исследованию пластического течения плоских
образцов двумерных поликристаллов чистых металлов в условиях одноосного растяжения с постоянной
скоростью деформации при умеренных температурах. Сформулировано дислокационно-кинетическое
уравнение, в котором учтены роль свободной поверхности плоского образца, являющейся источником и
стоком дислокаций, и упрочняющее действие сквозных границ зерен в двумерном поликристалле. Для
расчета кривой деформации кинетическое уравнение преобразовано с использованием закона
деформационного упрочнения Тейлора и получено аналитическое решение этого уравнения. На примере
плоских образцов двумерных поликристаллов чистого алюминия (99,999 ат.%) показано, что результаты
расчетов достаточно хорошо согласуются с экспериментальными данными.
ДИСЛОКАЦІЙНА КІНЕТИКА ПРИ ПЛАСТИЧНІЙ ДЕФОРМАЦІЇ
ДВОВИМІРНИХ ПОЛІКРИСТАЛІВ
Є.Ю. Бадіян, А.Г. Тонкопряд, Є.В. Фтьомов, О.В. Шеховцов
Дислокаційно-кінетичний підхід застосовано до дослідження пластичної течії плоских зразків
двовимірних полікристалів чистих металів в умовах одноосного розтягу з постійною швидкістю деформації
при помірних температурах. Сформульовано дислокаційно-кінетичне рівняння, в якому враховані роль
вільної поверхні плоского зразка, яка є джерелом і стоком дислокацій, і зміцнююча дія наскрізних меж зерен
у двовимірному полікристалі. Для розрахунку кривої деформації кінетичне рівняння перетворено з
використанням закону деформаційного зміцнення Тейлора і отримано аналітичне рішення цього рівняння.
На прикладі плоских зразків двовимірних полікристалів чистого алюмінію (99,999 ат.%) показано, що
результати розрахунків досить добре узгоджуються з експериментальними даними.
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