Fractal model of estimating quality of cold worked fuel cladding tubes
A possibility was considered concerning estimation of grain anisomery in the structure of fuel cladding tubes of corrosion-resistant 026Cr16Ni15Mo3Nb steel of austenitic class rolled according to two flow charts: regular and intensive technologies using fractal formalism. Role of grain boundary hard...
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Vakhrusheva, V.S. Volchuk, V.M. Hruzin, N.V. Tiutieriev, I.A. 2023-12-05T10:34:07Z 2023-12-05T10:34:07Z 2021 Fractal model of estimating quality of cold worked fuel cladding tubes / V.S. Vakhrusheva, V.M. Volchuk, N.V. Hruzin, I.A. Tiutieriev // Problems of Atomic Science and Technology. — 2021. — № 5. — С. 57-63. — Бібліогр.: 21 назв. — англ. 1562-6016 PACS: 61.82.Bg, 96.30.nd, 81.40.Gh, - Ef, 81.70.-q DOI: https://doi.org/10.46813/2021-135-057 https://nasplib.isofts.kiev.ua/handle/123456789/195444 A possibility was considered concerning estimation of grain anisomery in the structure of fuel cladding tubes of corrosion-resistant 026Cr16Ni15Mo3Nb steel of austenitic class rolled according to two flow charts: regular and intensive technologies using fractal formalism. Role of grain boundary hardening during cold plastic deformation was analyzed by studying the effect of the fractal dimension of grains D and their boundaries Dg on σ₀.₂, σw, and δ₅. The best correlation among those that were considered was observed between relative elongation and fractal dimensions of the grain structure (R² = 0.90). The smallest correlation was observed with the yield stress (R² = 0.64). It is because of variation of plastic flow processes towards a decrease in the degree of hardening in the material rolled according to the intensive technology. Cold deformation results in refining of the average grain size from 15.50 to 15.42 µm. In this case, extent of the grain boundary length L increased by 17.62% at an iteration step δ commensurate with the average grain size which is indicated by a change in the fractal dimension according to L ∼ δ¹⁻ᴰ. Degree of the grain structure inhomogeneity was estimated using ratios of self-similarity of regions of fractal dimensions of the structure. The obtained results on the level of mechanical properties of fuel cladding tubes made of austenitic steel indicate advantage of the intensive technology over regular one that was confirmed by results of fractal modeling. Розглянуто можливість оцінки різнозернистості структури труб-оболонок твел з корозійно-стійкої сталі аустенітного класу 026Х16Н15М3Б, прокатаних за двома технологічними схемами, яка діє зараз та за інтенсивною технологією, із застосуванням фрактального формалізму. Проаналізовано роль зернограничного зміцнення при холодній пластичній деформації на прикладі дослідження впливу фрактальної розмірності зерен D і їх границь Dg на σ₀.₂, σw і δ₅. Встановлено, що найкраща кореляція серед розглянутих спостерігається між відносним подовженням і фрактальними розмірностями зеренної структури (R² = 0.90), а найменша – з межею плинності R² = 0.64, що обумовлено зміною процесів пластичної течії матеріалу, що прокатаний за інтенсивною технологією, в сторону зниження ступеня зміцнення. Холодна деформація призводить до подрібнення середнього розміру зерен з 15,50 до 15,42 мкм. При цьому збільшується протяжність довжини границь зерен L на 17,62 % при кроці ітерації δ порівнянним із середнім розміром зерна, що фіксується зміною фрактальної розмірності згідно з L ∼ δ¹⁻ᴰ. За допомогою співвідношень областей самоподібності фрактальних розмірностей зерен структури оцінено ступінь її неоднорідності. Отримані результати за рівнем механічних властивостей оболонок твел з аустенітної сталі вказують на перевагу інтенсивної технології перед штатною, що підтверджується результатами фрактального моделювання. Рассмотрена возможность оценки разнозернистости структуры труб-оболочек твэл из коррозионностойкой стали аустенитного класса 026Х16Н15М3Б, прокатанных по двум технологическим схемам – действующей и интенсивной, с применением фрактального формализма. Проанализирована роль зернограничного упрочнения при холодной пластической деформации на примере исследования влияния фрактальной размерности зерен D и их границ Dg на σ₀.₂, σw и δ₅. Установлено, что наилучшая корреляция среди рассматриваемых характеристик наблюдается между относительным удлинением и фрактальными размерностями зеренной структуры (R² = 0.90), а наименьшая – с пределом текучести R² = 0.64, что обусловлено изменением процессов пластического течения прокатываемого материала по интенсивной технологии в сторону снижения степени упрочнения. Холодная деформация приводит к измельчению среднего размера зерен с 15,50 до 15,42 мкм. При этом увеличивается протяженность длины границ зерен L на 17,62% при шаге итерации δ, соизмеримым со средним размером зерна, что фиксируется изменением фрактальной размерности согласно L ∼ δ¹⁻ᴰ. С помощью соотношений областей самоподобия фрактальных размерностей зеренной структуры оценена степень ее неоднородности. Полученные результаты по уровню механических свойств оболочек твэл из аустенитной стали указывают на преимущество интенсивной технологии перед штатной, что подтверждается результатами фрактального моделирования. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Thermal and fast reactor materials Fractal model of estimating quality of cold worked fuel cladding tubes Фрактальна модель оцінки якості холоднодеформованих труб-оболонок твел Фрактальная модель оценки качества холоднодеформированных труб-оболочек твэл Article published earlier |
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| title |
Fractal model of estimating quality of cold worked fuel cladding tubes |
| spellingShingle |
Fractal model of estimating quality of cold worked fuel cladding tubes Vakhrusheva, V.S. Volchuk, V.M. Hruzin, N.V. Tiutieriev, I.A. Thermal and fast reactor materials |
| title_short |
Fractal model of estimating quality of cold worked fuel cladding tubes |
| title_full |
Fractal model of estimating quality of cold worked fuel cladding tubes |
| title_fullStr |
Fractal model of estimating quality of cold worked fuel cladding tubes |
| title_full_unstemmed |
Fractal model of estimating quality of cold worked fuel cladding tubes |
| title_sort |
fractal model of estimating quality of cold worked fuel cladding tubes |
| author |
Vakhrusheva, V.S. Volchuk, V.M. Hruzin, N.V. Tiutieriev, I.A. |
| author_facet |
Vakhrusheva, V.S. Volchuk, V.M. Hruzin, N.V. Tiutieriev, I.A. |
| topic |
Thermal and fast reactor materials |
| topic_facet |
Thermal and fast reactor materials |
| publishDate |
2021 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Фрактальна модель оцінки якості холоднодеформованих труб-оболонок твел Фрактальная модель оценки качества холоднодеформированных труб-оболочек твэл |
| description |
A possibility was considered concerning estimation of grain anisomery in the structure of fuel cladding tubes of corrosion-resistant 026Cr16Ni15Mo3Nb steel of austenitic class rolled according to two flow charts: regular and intensive technologies using fractal formalism. Role of grain boundary hardening during cold plastic deformation was analyzed by studying the effect of the fractal dimension of grains D and their boundaries Dg on σ₀.₂, σw, and δ₅. The best correlation among those that were considered was observed between relative elongation and fractal dimensions of the grain structure (R² = 0.90). The smallest correlation was observed with the yield stress (R² = 0.64). It is because of variation of plastic flow processes towards a decrease in the degree of hardening in the material rolled according to the intensive technology. Cold deformation results in refining of the average grain size from 15.50 to 15.42 µm. In this case, extent of the grain boundary length L increased by 17.62% at an iteration step δ commensurate with the average grain size which is indicated by a change in the fractal dimension according to L ∼ δ¹⁻ᴰ. Degree of the grain structure inhomogeneity was estimated using ratios of self-similarity of regions of fractal dimensions of the structure. The obtained results on the level of mechanical properties of fuel cladding tubes made of austenitic steel indicate advantage of the intensive technology over regular one that was confirmed by results of fractal modeling.
Розглянуто можливість оцінки різнозернистості структури труб-оболонок твел з корозійно-стійкої сталі аустенітного класу 026Х16Н15М3Б, прокатаних за двома технологічними схемами, яка діє зараз та за інтенсивною технологією, із застосуванням фрактального формалізму. Проаналізовано роль зернограничного зміцнення при холодній пластичній деформації на прикладі дослідження впливу фрактальної розмірності зерен D і їх границь Dg на σ₀.₂, σw і δ₅. Встановлено, що найкраща кореляція серед розглянутих спостерігається між відносним подовженням і фрактальними розмірностями зеренної структури (R² = 0.90), а найменша – з межею плинності R² = 0.64, що обумовлено зміною процесів пластичної течії матеріалу, що прокатаний за інтенсивною технологією, в сторону зниження ступеня зміцнення. Холодна деформація призводить до подрібнення середнього розміру зерен з 15,50 до 15,42 мкм. При цьому збільшується протяжність довжини границь зерен L на 17,62 % при кроці ітерації δ порівнянним із середнім розміром зерна, що фіксується зміною фрактальної розмірності згідно з L ∼ δ¹⁻ᴰ. За допомогою співвідношень областей самоподібності фрактальних розмірностей зерен структури оцінено ступінь її неоднорідності. Отримані результати за рівнем механічних властивостей оболонок твел з аустенітної сталі вказують на перевагу інтенсивної технології перед штатною, що підтверджується результатами фрактального моделювання.
Рассмотрена возможность оценки разнозернистости структуры труб-оболочек твэл из коррозионностойкой стали аустенитного класса 026Х16Н15М3Б, прокатанных по двум технологическим схемам – действующей и интенсивной, с применением фрактального формализма. Проанализирована роль зернограничного упрочнения при холодной пластической деформации на примере исследования влияния фрактальной размерности зерен D и их границ Dg на σ₀.₂, σw и δ₅. Установлено, что наилучшая корреляция среди рассматриваемых характеристик наблюдается между относительным удлинением и фрактальными размерностями зеренной структуры (R² = 0.90), а наименьшая – с пределом текучести R² = 0.64, что обусловлено изменением процессов пластического течения прокатываемого материала по интенсивной технологии в сторону снижения степени упрочнения. Холодная деформация приводит к измельчению среднего размера зерен с 15,50 до 15,42 мкм. При этом увеличивается протяженность длины границ зерен L на 17,62% при шаге итерации δ, соизмеримым со средним размером зерна, что фиксируется изменением фрактальной размерности согласно L ∼ δ¹⁻ᴰ. С помощью соотношений областей самоподобия фрактальных размерностей зеренной структуры оценена степень ее неоднородности. Полученные результаты по уровню механических свойств оболочек твэл из аустенитной стали указывают на преимущество интенсивной технологии перед штатной, что подтверждается результатами фрактального моделирования.
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| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/195444 |
| citation_txt |
Fractal model of estimating quality of cold worked fuel cladding tubes / V.S. Vakhrusheva, V.M. Volchuk, N.V. Hruzin, I.A. Tiutieriev // Problems of Atomic Science and Technology. — 2021. — № 5. — С. 57-63. — Бібліогр.: 21 назв. — англ. |
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ISSN 1562-6016. PASТ. 2021. №5(135), p. 57-63.
SECTION 2
THERMAL AND FAST REACTOR MATERIALS
https://doi.org/10.46813/2021-135-057
FRACTAL MODEL OF ESTIMATING QUALITY OF COLD WORKED
FUEL CLADDING TUBES
V.S. Vakhrusheva, V.M. Volchuk, N.V. Hruzin, I.A. Tiutieriev
Department of Materials Science and Treatment of Materials, State Higher Educational;
Institution “Prydnіprovska State Academy of Civil Engineering and Architecture”,
Dnipro, Ukraine
E-mail: vs062@ukr.net
A possibility was considered concerning estimation of grain anisomery in the structure of fuel cladding tubes of
corrosion-resistant 026Cr16Ni15Mo3Nb steel of austenitic class rolled according to two flow charts: regular and
intensive technologies using fractal formalism. Role of grain boundary hardening during cold plastic deformation
was analyzed by studying the effect of the fractal dimension of grains D and their boundaries Dg on 0.2, w, and 5.
The best correlation among those that were considered was observed between relative elongation and fractal
dimensions of the grain structure (R
2
= 0.90). The smallest correlation was observed with the yield stress (R
2
= 0.64).
It is because of variation of plastic flow processes towards a decrease in the degree of hardening in the material
rolled according to the intensive technology. Cold deformation results in refining of the average grain size from
15.50 to 15.42 µm. In this case, extent of the grain boundary length L increased by 17.62% at an iteration step
commensurate with the average grain size which is indicated by a change in the fractal dimension according to
L
1-D
. Degree of the grain structure inhomogeneity was estimated using ratios of self-similarity of regions of
fractal dimensions of the structure. The obtained results on the level of mechanical properties of fuel cladding tubes
made of austenitic steel indicate advantage of the intensive technology over regular one that was confirmed by
results of fractal modeling.
PACS: 61.82.Bg, 96.30.nd, 81.40.Gh, - Ef, 81.70.-q
INTRODUCTION
Quality requirements to nuclear fuel element
materials used in nuclear power engineering including
structure and mechanical properties are ever toughening
in the connection with intricate conditions of their
operation [1]. Tubes made of austenitic chromium-
nickel steel and chromium-nickel-molybdenum steel,
for example, 03Сr18Ni11, 03Сr17Ni14Мo3Nb, occupy
a special place among the critical purpose materials.
Treatment of various types is used to improve material
service characteristics [2] in connection with current
requirements.
Cold-worked thin-walled tubes of
026Сr16Ni15Mo3Nb steel of austenitic class are used
for the most critical units of power plants [3]. Due to the
multi-parameter and multi-criteria nature of the
technology used for production of tubes from
026Сr16Ni15Mo3Nb steel, their structure is
characterized by chemical and structural heterogeneity.
Anisotropy, anisomery, grain number, grain boundaries
are structural characteristics that have a significant
effect on mechanical properties of austenitic alloy steel.
In this case, Geometric configuration of the grain shape
is not usually taken into account because of
transformation of grains after plastic working which is
insufficiently taken in consideration by conventional
techniques of quantitative metallography.
According to the present-day concepts, surface of
many materials is of fractal nature which [4] is the most
common case in nature [5]. It may happen that the
neglect of fractal dimension [6] of the metal structure
elements can introduce an error in establishing a
relationship between structure and properties [7].
In order to take into account influence of
“deformity” of grain structure in thin-walled tubes of
structurally sensitive 026Сr16Ni15Mo3Nb steel after
cold rolling on their mechanical characteristics, it was
proposed to apply the Mandelbrot’s fractal formalism
[8]. Practical implementation of this formalism allows
one to regularize conditionally incorrect problems of
metal science [9]. The use of fractal (fractional)
dimensions in materials science has made it possible to
determine average grain size in metals [10], predict
mechanical properties of low-carbon steel after heat
treatment [11], estimate residual life of aircraft
structures [12], develop a fractal accelerator based on a
corrugated plasma waveguide [13], etc. Application of
the fractal theory confirms a more accurate estimation
of quality characteristics of various materials (cast iron
rolls of cold rolling mills [14], metals [15], concretes
[16]) due to their ranking which contributes to
stabilization of the multi-parameter technology of their
production.
The abovementioned and many other publications
indicate prospects for using fractal modeling to describe
structural transformations of materials.
1. MATERIAL AND TECHNIQUE
Chemical composition of the tube steel under study
meets requirements of TU 14-1-1641 (Table 1).
Experimental samples of 5.8×0.3 mm tubes of
026Сr16Ni15Mo3Nb steel manufactured both by the
regular (current) technology and intensive technology
(with intensive plastic deformation in the working
passes) must meet requirements of Technical Condition.
In this connection, all tubes were tested for compliance
with this technical specification in industrial production
conditions.
Table 1
Chemical composition of 026Cr16Ni15Мo3Nb steel
Element content, wt.%
C Si Mn Cr Ni Mo Nb
0.017 0.06 0.82 16.48 15.59 2.57 0.45
S P N V Co Cu –
0.01 0.01 0.012 0.094 0.005 0.06 –
A standard flowchart for the production of cladding
tubes requires application of a significant number of
cyclic operations at main units and a large number of
auxiliary operations. In the manufacture of tubes, the
existing schedule includes 8 cycles and 220–230
operations. To reduce the production cycle, improve the
set of mechanical properties and reduce structural
heterogeneity, an intensive technology of tube
processing has been implemented [3]. In this case, the
diagram of stressed state in the deformation zone was
altered. The intensive technology consists in the use of
two-row deformation schemes at KhPT cold tube rolling
mills using two pairs of rolls arranged in series (four-
high stands) and KhPTR cold tube rolling (roller) mills
with a two-row separator. Simultaneous reduction of
billets in two pairs of rolls (rollers) determines a
specific cyclic force interaction of two deformation
zones, which significantly changes plastic flow of the
rolled material towards a decrease in the degree of
hardening, makes it possible to more fully use the metal
plasticity resource and practically halve cycle city of the
tube production process [3].
Microstructure and mechanical properties of the
cladding tubes produced by both technologies have been
studied (Fig. 1).
Mechanical properties of tubes rolled according to
two flow charts are given in Table 2. Microstructure of
finished tubes is shown in Fig. 1.
Table 2
Mechanical properties of fuel rod cladding tubes
Deformation variant
Tube dimensions
(diameter wall), mm
Теmperature, +20 С
0.2, N/mm
2
в, N/mm
2
5, %
intensive 5.80.3 320…365 610…620 46…48
regular 5.80.3 335…345 600…610 43…45
ТU 14-3-550
not less
– – 520 37
Fig. 1. Microstructure of finished (5.80.3 mm) tubes: regular technology (а); intensive technology (b) 100
Austenitic grain size was measured using an RPA
installation and metallographic examination of thin
sections was made using a Neophot-21 microscope.
Average grain size corresponded to size numbers 9–10
in tubes rolled using regular technology and size
numbers 10–11 in tubes rolled using intensive
technology according to GOST 5639 [17].
2. THE TECHNIQUE OF FRACTAL
ANALYSIS
To determine fractal dimension of austenitic
structure in the tubes obtained by the two rolling
variants described above, a patented technique was used
[18] according to the following scheme (Fig. 2).
Fig. 2. General scheme of determining fractal
dimensions
The developed technique was computer
implemented. It is based on determining the
convergence of values of the Hausdorff cellular
dimension (1) [19] and point (2) [20] dimension:
ln
)(Nln
limD
0
, (1)
where N () is the number of cells of size by which
the investigated fractal element is covered.
The point dimension was determined as follows:
K
1m
)L,m(P)m/1()(N
~ , (2)
where )(
~
N is the average value of the number of cells
of size that contain m points of the fractal (pixels for a
computer).
The results of calculating the fractal dimension of
the grain structure (see Fig. 1, a) are shown in Fig. 3.
Dependence 1 in Fig. 3 describes the fractal
dimension of boundaries of austenite grains calculated
by the cellular method and dependence 2 describes their
point dimension. Dependences 3 and 4 are cellular and
point dimensions of austenite grains, respectively.
b a
Fig. 3. Dependence of the fractal dimension on the cell
size (in pixels)
It follows from Fig. 3 that the best convergence of
values of fractal dimensions is observed at the 3
rd
step
of iteration for grain boundaries:
Dg = (Dk + Dt) = (1.528+1.434) = 1.481
and at the 13
th
step of calculations for grains:
D = (1.951+1.907) = 1.929.
3. RESULTS AND DISCUSSION
When controlling structure in tubes, a great
importance is attached to the grain size which is
determined by a strictly regulated complex of
properties, especially mechanical properties, which are
largely ensured by the grain structure of metal.
This is because of the fact that many of the
properties are structurally sensitive, for example, в,
0,2, 5, long-term strength, electrical resistance,
tendency to intergranular corrosion, etc. The processes
of precipitation of carbides occurring at operating
temperatures of the reactor are also associated with
grain size and the extent of the grain boundary. In their
turn, these processes significantly affect nature of creep
and high-temperature strength.
The study of efficiency of the fractal model of grain
structure in estimating mechanical properties (0,2, в,
and 5) of 5.80.3 mm cladding tubes was carried out
by comparison of its fractal dimensions with the results
of conventional Euclidean (diameter, area) and
statistical (number of grains, dispersion, coefficient of
variation) characteristics of the structure.
Quantitative estimation of microstructure of the
cladding tubes rolled according to the regular and
intensive technologies was carried out by conventional
methods using an Epiquant structure analyzer with
subsequent computer processing. The study results are
shown in Table 3.
According to the procedure of quantitative analysis
of structure (GOST 5639), in the case of determining
the grain size in an anisomerous structure, average
dimensions (diameter, area of the grain) are not
characteristics of its estimation. For example, it was
found that the maximum grain size decreased with
intensive processing technology by only 10% (from 50
to 45 µm) compared to the regular technology. In this
case, average austenite grain size during transition from
the regular technology to the intensive one decreased
only by (15.50–15.42) μm = 0.08 μm or 0.52% which is
practically leveled by the values of root-mean-square
deviation of its sizes 8.63 and 8.10 μm within the
experiment. The decrease in scatter of the grain size
values is evidenced by the coefficients of their
variation: 0.56 for the regular and 0.53 for intensive
technologies, respectively (see Table 3).
According to the ASTME system, average grain size
in 112 5.80.3 mm tubes produced by intensive
technology changed by ½ number (from 9 to 8.5).
According to GOST 5639, grain size in the metal of
tubes rolled using intensive technology also decreased
to 8–9 numbers in comparison with numbers 8–10 in
the metal of tubes rolled using regular technology. This
is probably due to a more uniform development of
recrystallization processes in metal with an increase in
the degree of plastic deformation in passes.
In the region of large deformations, density of
dislocations and excess of dislocations of the same sign
are so high that a large number of primary
recrystallization centers are rapidly formed which
involves the entire volume of metal during annealing,
Large degrees of deformation is one of conditions
for obtaining a more uniform grain structure upon
subsequent annealing. A graph of grains size
distribution in cladding tubes rolled according to two
technological schemes is given in Fig. 4.
The data presented indicate less blur of the curve (1)
in the case of rolling tubes by the intensive technology
which also indicates a decrease in the austenite
anisomery.
Within the experiment, number of grains changed by
only 0.50% in 1 mm
2
and it also changed insignificantly
(by 0.75%) in a unit of volume (1 mm
3
).
The obtained geometric and statistical estimates of
anisomerous austenitic structure within the experiment
indicate their insufficient sensitivity to changes in
mechanical properties of the cladding tubes in the
studied parameter range.
In this regard, in order to search for more effective
estimates of anisomerous structure of cladding tubes, it
was proposed to apply the fractal theory. Effect of
fractal dimension of austenite grains and their
boundaries on mechanical properties, i.e. their role in
hardening of grain boundary, was estimated in this
work. This choice of characteristics was determined by
the fact that the effect of grains and their boundaries on
plasticity, toughness, and crack resistance of metal is
very significant [1, 2]. In addition, grain boundaries not
only prevent crack initiation but also inhibit their
development giving rise to an additional energy
dissipation [21].
Figs. 5 and 6 show graphs describing relationship
between fractal dimensions of austenite grains and their
boundaries and mechanical properties of the cladding
tubes made of 026Cr16Ni15Мo3Nb steel. An increase
in the fractal dimension of grains is associated with a
decrease in grain size, i.e. with an increase in
compactness of filling the considered two-dimensional
space of the thin section. Tubes with higher strength
characteristics rolled using the regular technology
correspond to a grain size number 9, and those with
lower strength characteristics correspond to a grain size
number 10.
Table 3
Computer processing of the structure in 5.80.3 mm tubes
Regular rolling technology Intensive roling technology
Structure of initial plane section
Structure of initial plane section
Structure after computer processing
Structure after computer processing
Number of grains, pcs 110 Number of grains, pcs 111
Maximum grain size, µm 50.00 Maximum grain size, µm 45.00
Average diameter of flat section, µm 15.50 Average diameter of flat section, µm 15.42
Dispersion, µm 8.63 Dispersion, µm 8.10
Coefficient of variation 0.56 Coefficient of variation 0.53
Total area of grains, mm
2
0.02895693 Total area of grains, mm
2
0.02907737
Number of grains in 1 mm
2
, pcs 3798 Number of grains in 1 mm
2
, pcs 3817
Number of grains in 1 mm
3
, pcs 234062 Number of grains in 1 mm
3
, pcs 235821
Average grain size number
(GOST 5639)
10
Average grain size number
(GOST 5639)
9
Average grain size number
(ASTME 112)
9
Average grain size number
(ASTME 112)
8.5
Fig. 4. Grain size distribution in 5.80.3 mm tubes
rolled by intensive (1) and regular (2) technologies
For tubes rolled using the intensive technology,
higher properties were recorded in those sections that
corresponded to a lower grain size number 8 compared
to the size numbers 9 and 10. A decrease in the grain
size leads to an increase in their length L per unit area,
which is fixed using their fractal dimension and
confirmed by the following relation [8]:
L = L0/
D-
, (3)
where L is the extent of the grain length calculated by
conventional method using a palette with a step . At
0, fractal dimension D of the curve L0 becomes
equal to one. As the grain size decreased from number
10 to number 8, fractal dimension of their length
increased from 1.458 to 1.633. In this case, length of the
grains estimated by formula (3) increased. For metal of
tubes rolled according to the regular technology, length
of the grain boundaries was 74.034 mm/mm
2
, and for
the tubes rolled according to the intensive technology,
length of the grain boundaries was 89.868 mm/mm
2
and
the difference in their length was 17.62%. An increase
in length of the grain boundaries per unit area was due
to the specific cyclic force interaction of two
deformation zones which leads to a smaller decrease in
the degree of hardening by changing the processes of
plastic flow of the material being rolled. From this point
of view, fractal dimension of the grain boundaries can
be an indicator of a change in the length L of grain
boundaries due to the influence of plastic deformation.
As a result of analysis of the relations given in Figs.
5 and 6, fractal models for predicting mechanical
properties of tubes (4–6) at +20
0
C were obtained.
0.2 = 2420.91+859.36D+739.61Dg, R
2
=0.64; (4)
B = 150.00+105.31D+169.06Dg, R
2
=0.84; (5)
= 314.4995.73D57.33Dg, R
2
=0.90. (6)
а b c
Fig. 5. Relationships between fractal dimension of grain boundaries Dg (1), grains D (2) and yield point (a);
ultimate strength (b); relative elongation (c) of cladding tubes rolled according to the regular technology
а b c
Fig. 6. Relationships between fractal dimension of the grain boundaries Dg (1), grains D (2) and yield point (a);
ultimate strength (b); relative elongation (c) of cladding tubes rolled using intensive technology
Based on analysis of the coefficients of fractal
models (4–6), it was found that contribution to the grain
boundary hardening of austenite grains for strength
characteristics of the cladding tubes made of
026Cr16Ni15Mo3Nb steel after cold rolling was
38...54% and it was 46...62% for their boundaries
(Fig. 7). Contribution of the grain size was 63% for the
plasticity characteristics (), and, respectively,
contribution of their boundaries was equal to 0.37%.
This was largely due to alteration of the plastic flow
processes in the rolled material towards a decrease in
the degree of hardening when using the intensive
technology [3].
Fig. 7. Estimation of contribution of austenite grains
and their boundaries to grain boundary hardening
using the fractal theory
Heterogeneity of anisomerous microstructure was
estimated in this study using fractal formalism (Fig. 8).
The coefficient of self-similarity of the microstructure
was determined from the ratio of fractal dimensions of
the regions of austenite grains and their boundaries.
It follows from the graph shown in Fig. 8 that the
region of self-similarity of grain boundaries Dg (2) is
the most sensitive to a change in anisomery of grains in
the tube metal (2). It ranged from 0.972 to 1.031
indicating the prospects for its application to estimation
of structural heterogeneity in comparison with the self-
similarity of grains.
Fig. 8. Relationship between the grain size according to
ASTME 112 and ratios of their fractal dimensions
Comparative analysis of the effect of cold plastic
working of fuel cladding tubes on their mechanical
properties using conventional and fractal structure
estimates indicates the prospects for using the fractal
approach. It follows therefrom that the fractal dimension
can be used as an indicator of structural transformations
of anisomerous structure the tube metal and mechanical
properties of the tubes.
CONCLUSIONS
Application of the fractal theory in modeling the
anisomerous structure and properties of the fuel
cladding tubes made of austenitic 026Cr16Ni15Mo3Nb
steel is useful in the following.
1. Assessment of the degree of influence of cold
deformation on microstructure by assessing its
contribution to grain-boundary hardening based on
dynamics of changes in the fractal dimension of the
grain structure.
2. Assessment of inhomogeneity of the grain
structure by comparing the regions of its self-similarity.
3. Forecast of strength and plastic properties (0.2,
в, and 5) according to fractal dimensions of the grain
structure (R
2
= 0.64...0.90).
4. Application of the fractal approach as an
alternative in quality control of the cladding tubes when
sensitivity of conventional estimates of anisomerous
structure is insufficient to establish a correlation with
mechanical characteristics.
The obtained results in terms of level of mechanical
properties show advantage of the intensive technology
and the prospects of its application in connection with
the problems of operability of fuel element cladding in
reactors iwhere the cladding loses its plastic and
strength properties by the campaign end.
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Article received 05.08.2021
ФРАКТАЛЬНАЯ МОДЕЛЬ ОЦЕНКИ КАЧЕСТВА ХОЛОДНОДЕФОРМИРОВАННЫХ
ТРУБ-ОБОЛОЧЕК ТВЭЛ
В.С. Вахрушева, В.Н. Волчук, Н.В. Грузин, И.А. Тютерев
Рассмотрена возможность оценки разнозернистости структуры труб-оболочек твэл из коррозионно-
стойкой стали аустенитного класса 026Х16Н15М3Б, прокатанных по двум технологическим схемам –
действующей и интенсивной, с применением фрактального формализма. Проанализирована роль
зернограничного упрочнения при холодной пластической деформации на примере исследования влияния
фрактальной размерности зерен D и их границ Dg на 0,2, в и 5. Установлено, что наилучшая корреляция
среди рассматриваемых характеристик наблюдается между относительным удлинением и фрактальными
размерностями зеренной структуры (R
2
= 0,90), а наименьшая – с пределом текучести R
2
= 0,64, что
обусловлено изменением процессов пластического течения прокатываемого материала по интенсивной
технологии в сторону снижения степени упрочнения. Холодная деформация приводит к измельчению
среднего размера зерен с 15,50 до 15,42 мкм. При этом увеличивается протяженность длины границ зерен L
на 17,62% при шаге итерации , соизмеримым со средним размером зерна, что фиксируется изменением
фрактальной размерности согласно L
1-D
. С помощью соотношений областей самоподобия фрактальных
размерностей зеренной структуры оценена степень ее неоднородности. Полученные результаты по уровню
механических свойств оболочек твэл из аустенитной стали указывают на преимущество интенсивной
технологии перед штатной, что подтверждается результатами фрактального моделирования.
ФРАКТАЛЬНА МОДЕЛЬ ОЦІНКИ ЯКОСТІ ХОЛОДНОДЕФОРМОВАНИХ
ТРУБ-ОБОЛОНОК ТВЕЛ
В.С. Вахрушева, В.М. Волчук, Н.В. Грузін, І.А. Тютєрєв
Розглянуто можливість оцінки різнозернистості структури труб-оболонок твел з корозійно-стійкої сталі
аустенітного класу 026Х16Н15М3Б, прокатаних за двома технологічними схемами, яка діє зараз та за
інтенсивною технологією, із застосуванням фрактального формалізму. Проаналізовано роль
зернограничного зміцнення при холодній пластичній деформації на прикладі дослідження впливу
фрактальної розмірності зерен D і їх границь Dg на σ0,2, σв і δ5. Встановлено, що найкраща кореляція серед
розглянутих спостерігається між відносним подовженням і фрактальними розмірностями зеренної
структури (R
2
= 0,90), а найменша – з межею плинності R
2
= 0,64, що обумовлено зміною процесів
пластичної течії матеріалу, що прокатаний за інтенсивною технологією, в сторону зниження ступеня
зміцнення. Холодна деформація призводить до подрібнення середнього розміру зерен з 15,50 до 15,42 мкм.
При цьому збільшується протяжність довжини границь зерен L на 17,62 % при кроці ітерації δ порівнянним
із середнім розміром зерна, що фіксується зміною фрактальної розмірності згідно з L ~ δ
1-D
. За допомогою
співвідношень областей самоподібності фрактальних розмірностей зерен структури оцінено ступінь її
неоднорідності. Отримані результати за рівнем механічних властивостей оболонок твел з аустенітної сталі
вказують на перевагу інтенсивної технології перед штатною, що підтверджується результатами
фрактального моделювання.
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