Дослідження впливу кристалічних решіток на фрактальні характеристики матеріалів
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Date
2021
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НМетАУ, Дніпро
Abstract
UKR: Значний інтерес представляють кристалічні тіла, які характеризуються впорядкованим розташуванням атомів. До кристалів можна віднести мінерали, всі метали, солі, більшість органічних сполук і велику кількість інших твердих тіл. Закономірна і досконала геометрія кристалів говорить про наявність закономірностей і в їх внутрішній будові. Ці впорядковані структурні частинки, що розташовані правильними рядами в строгій ієрархічній послідовності і мають ось симетрії другого, третього, четвертого і шостого порядків. Строго математично доведено, що відмічені порядки осей в тому або іншому поєднанні для кристалів єдино можливі. Інших порядків осей симетрії, поворот навколо яких переводив би грати кристала самі в себе, в класичній кристалографії не існує. Тому є сенс побудувати поверхні другого, третього, четвертого порядків симетрії та провести аналіз їх фрактальних характеристик. Цікавим дослідженням є побудова фрактальних розподілень таких поверхонь.
ENG: In the work, natural and computer crystalline and quasi-crystalline fractal surfaces were investigated. For each surface, fractal dimension was calculated and fractal distribution was constructed. During the study, PenrouseFS software is developed, which allows the fractal surfaces on crystalline grates with different rotary symmetry and quasi-crystalline lattices. The fractal surfaces of crystalline and quasi-crystalline structures are constructed using a random shear method. In the course of the work, a numerical number of fractal surfaces of crystalline and quasi-crystalline structures was built using the given method: the crystalline structures of the second, third, fourth, sixth, sixth forms of symmetry and quasi-crystalline structures with a different number of iterations (from 4 to 9), which are responsible for the degree of surface detailation, was constructed. and analyzed their fractal characteristics. Analysis of crystalline surfaces with symmetry of different orders allowed to determine their fractal dimensions, to construct the distribution of local fractal dimensions. During the study, the growth of fractal dimension with an increase in the number of iterations was determined. In almost all types of surfaces, a slight decrease in the values of fractal dimension in the amount of iterations equal to 5 and 6. is the smallest value of fractal dimension with the smallest detail of the surface. The greatest values of fractal dimension have crystalline surfaces of the 4th order of symmetry. The crystalline surfaces of the 3rd and 6th order of symmetry have very close fractal dimensions. In this case, their value of fractal dimension is smaller than the crystalline surfaces of the 2nd and 4th order of symmetry. In the study of fractal properties, about 50 images of a variety of snowflakes belonging to different groups were analyzed. In the study, the fractal dimension of microstructures was determined by the Box Couning method. To construct fractal distributions, a method of sliding window was used. When calculated fractal properties of snowflakes, a number of images of "plates" and "Star Dendrites" type were analyzed. As a result of the study, it was found that for a group of snowflakes "Plates" range of fractal dimension values from 1.54 to 1.81, and the number of modification modes 6, and for the group of snowflakes "Star Dendrid" range of fractal dimension values from 1.55 to 1.72, and the number of modification modes 6 . The image of the Quasi crazi-crazinite has 5 modes of distribution. The same amount of distribution mods have a quasi-crystalline surfaces of pentroose generated by a computer route. The fractal dimension of the quasi-crystal of shekhtmanite has the same meaning as a computer model of a quasi-crystalline surface at the highest degree of detail. During the research, it has been found that fractal distributions almost for all types of crystalline and quasi-crystalline surfaces have a multimodal distribution and a wide spread of fractal dimension values, which indicates complex images. Fractal distribution of each of the built surfaces has a number of mods that coincides with the order of symmetry.
ENG: In the work, natural and computer crystalline and quasi-crystalline fractal surfaces were investigated. For each surface, fractal dimension was calculated and fractal distribution was constructed. During the study, PenrouseFS software is developed, which allows the fractal surfaces on crystalline grates with different rotary symmetry and quasi-crystalline lattices. The fractal surfaces of crystalline and quasi-crystalline structures are constructed using a random shear method. In the course of the work, a numerical number of fractal surfaces of crystalline and quasi-crystalline structures was built using the given method: the crystalline structures of the second, third, fourth, sixth, sixth forms of symmetry and quasi-crystalline structures with a different number of iterations (from 4 to 9), which are responsible for the degree of surface detailation, was constructed. and analyzed their fractal characteristics. Analysis of crystalline surfaces with symmetry of different orders allowed to determine their fractal dimensions, to construct the distribution of local fractal dimensions. During the study, the growth of fractal dimension with an increase in the number of iterations was determined. In almost all types of surfaces, a slight decrease in the values of fractal dimension in the amount of iterations equal to 5 and 6. is the smallest value of fractal dimension with the smallest detail of the surface. The greatest values of fractal dimension have crystalline surfaces of the 4th order of symmetry. The crystalline surfaces of the 3rd and 6th order of symmetry have very close fractal dimensions. In this case, their value of fractal dimension is smaller than the crystalline surfaces of the 2nd and 4th order of symmetry. In the study of fractal properties, about 50 images of a variety of snowflakes belonging to different groups were analyzed. In the study, the fractal dimension of microstructures was determined by the Box Couning method. To construct fractal distributions, a method of sliding window was used. When calculated fractal properties of snowflakes, a number of images of "plates" and "Star Dendrites" type were analyzed. As a result of the study, it was found that for a group of snowflakes "Plates" range of fractal dimension values from 1.54 to 1.81, and the number of modification modes 6, and for the group of snowflakes "Star Dendrid" range of fractal dimension values from 1.55 to 1.72, and the number of modification modes 6 . The image of the Quasi crazi-crazinite has 5 modes of distribution. The same amount of distribution mods have a quasi-crystalline surfaces of pentroose generated by a computer route. The fractal dimension of the quasi-crystal of shekhtmanite has the same meaning as a computer model of a quasi-crystalline surface at the highest degree of detail. During the research, it has been found that fractal distributions almost for all types of crystalline and quasi-crystalline surfaces have a multimodal distribution and a wide spread of fractal dimension values, which indicates complex images. Fractal distribution of each of the built surfaces has a number of mods that coincides with the order of symmetry.
Description
А. Журба: ORCID 0000-0002-9983-3971
Keywords
фрактальна розмірність, фрактальні розподілення, кристалічна решітка, квазікристали, шехтманіт, обертальна симетрія, порядок симетрії, мозаїка Пенроуза, кристалічна та квазікристалічна фрактальна поверхня, fractal dimension, fractal distributions, crystal lattice, quasicrystals, Schechtmanite, rotational symmetry, symmetry order, Penrose mosaic, crystalline and quasicrystalline fractal surface, КІТС
Citation
Журба А. О. Дослідження впливу кристалічних решіток на фрактальні характеристики матеріалів. Сучасні проблеми металургії. Дніпро, 2021. № 24. С. 21–34. DOI: 10.34185/1991-7848.2021.01.03.