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Random heterogeneous microstructure construction of composites via fractal geometry

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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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Shanghai University, Shanghai, 200444, China

    Siwen Wang

  2. Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200072, China

    Zhansheng Guo

  3. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, 200072, China

    Zhansheng Guo

Authors
  1. Siwen Wang
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  2. Zhansheng Guo
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Correspondence to Zhansheng Guo.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11972218 and 11472165)

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Wang, S., Guo, Z. Random heterogeneous microstructure construction of composites via fractal geometry. Appl. Math. Mech.-Engl. Ed. 40, 1413–1428 (2019). https://doi.org/10.1007/s10483-019-2522-5

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  • DOI: https://doi.org/10.1007/s10483-019-2522-5

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Chinese Library Classification

2010 Mathematics Subject Classification

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