Random heterogeneous microstructure construction of composites via fractal geometry

doi:10.1007/s10483-019-2522-5

Applied Mathematics and Mechanics (English Edition) ›› 2019, Vol. 40 ›› Issue (10): 1413-1428.doi: https://doi.org/10.1007/s10483-019-2522-5

Random heterogeneous microstructure construction of composites via fractal geometry

Siwen WANG¹, Zhansheng GUO^2,3

1. Department of Mathematics, College of Sciences, Shanghai University, Shanghai 200444, China;
2. Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, China;
3. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China

收稿日期:2019-01-15 修回日期:2019-05-10 发布日期:2019-09-17
通讯作者: Zhansheng GUO E-mail:davidzsguo@shu.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 11972218 and 11472165)

Random heterogeneous microstructure construction of composites via fractal geometry

Siwen WANG¹, Zhansheng GUO^2,3

1. Department of Mathematics, College of Sciences, Shanghai University, Shanghai 200444, China;
2. Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, China;
3. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China

Received:2019-01-15 Revised:2019-05-10 Published:2019-09-17
Contact: Zhansheng GUO E-mail:davidzsguo@shu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 11972218 and 11472165)

摘要/Abstract

摘要： The microstructures of a composite determine its macroscopic properties. In this study, microstructures with particles of arbitrary shapes and sizes are constructed by using several developed fractal geometry algorithms implemented in MATLAB. A two-dimensional (2D) quadrilateral fractal geometry algorithm is developed based on the modified Sierpinski carpet algorithm. Square-, rectangle-, circle-, and ellipse-based microstructure constructions are special cases of the 2D quadrilateral fractal geometry algorithm. Moreover, a three-dimensional (3D) random hexahedron geometry algorithm is developed according to the Menger sponge algorithm. Cube-and sphere-based microstructure constructions are special cases of the 3D hexahedron fractal geometry algorithm. The polydispersities of fractal shapes and random fractal sub-units demonstrate significant enhancements compared to those obtained by the original algorithms. In addition, the 2D and 3D algorithms mentioned in this article can be combined according to the actual microstructures. The verification results also demonstrate the practicability of these algorithms. The developed algorithms open up new avenues for the constructions of microstructures, which can be embedded into commercial finite element method softwares.

关键词: microstructure, fractal geometry algorithm, arbitrary shape and size, arbitrary quadrilateral, arbitrary hexahedron

Abstract: The microstructures of a composite determine its macroscopic properties. In this study, microstructures with particles of arbitrary shapes and sizes are constructed by using several developed fractal geometry algorithms implemented in MATLAB. A two-dimensional (2D) quadrilateral fractal geometry algorithm is developed based on the modified Sierpinski carpet algorithm. Square-, rectangle-, circle-, and ellipse-based microstructure constructions are special cases of the 2D quadrilateral fractal geometry algorithm. Moreover, a three-dimensional (3D) random hexahedron geometry algorithm is developed according to the Menger sponge algorithm. Cube-and sphere-based microstructure constructions are special cases of the 3D hexahedron fractal geometry algorithm. The polydispersities of fractal shapes and random fractal sub-units demonstrate significant enhancements compared to those obtained by the original algorithms. In addition, the 2D and 3D algorithms mentioned in this article can be combined according to the actual microstructures. The verification results also demonstrate the practicability of these algorithms. The developed algorithms open up new avenues for the constructions of microstructures, which can be embedded into commercial finite element method softwares.

Key words: microstructure, fractal geometry algorithm, arbitrary shape and size, arbitrary quadrilateral, arbitrary hexahedron

中图分类号:

68W99
82D20

Siwen WANG, Zhansheng GUO. Random heterogeneous microstructure construction of composites via fractal geometry[J]. Applied Mathematics and Mechanics (English Edition), 2019, 40(10): 1413-1428.

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Random heterogeneous microstructure construction of composites via fractal geometry

Random heterogeneous microstructure construction of composites via fractal geometry

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