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Random heterogeneous microstructure construction of composites via fractal geometry

  • Siwen Wang
  • Zhansheng GuoEmail author
Article
  • 13 Downloads

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 

Chinese Library Classification

O29 O341 

2010 Mathematics Subject Classification

68W99 82D20 

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References

  1. [1]
    MANDELBROT, B. B. The Fractal Geometry of Nature, W. H. Freeman and Company, New York (1982)zbMATHGoogle Scholar
  2. [2]
    BALANKIN, A. S. A continuum framework for mechanics of fractal materials II: elastic stress fields ahead of cracks in a fractal medium. The European Physical Journal B, 88(4), 1–6 (2015)MathSciNetGoogle Scholar
  3. [3]
    HAN, C. and CHENG, P. Micropore variation and particle fractal representation of lime-stabilised subgrade soil under freeze-thaw cycles. Road Materials and Pavement Design, 16(1), 19–30 (2014)CrossRefGoogle Scholar
  4. [3]
    YANG, X. and WANG, F. Random-fractal-method-based generation of meso-model for concrete aggregates. Powder Technology, 284, 63–77 (2015)CrossRefGoogle Scholar
  5. [5]
    CHEN, X. and YAO, G. An improved model for permeability estimation in low permeable porous media based on fractal geometry and modified Hagen-Poiseuille flow. Fuel, 210, 748–757 (2017)CrossRefGoogle Scholar
  6. [6]
    PIA, G. and SANNA, U. An intermingled fractal units model and method to predict permeability in porous rock. International Journal of Engineering Science, 75, 31–39(2014)CrossRefzbMATHGoogle Scholar
  7. [7]
    GAO, Y., FENG, P., and JIANG, J. Analytical and numerical modeling of elastic moduli forcement based composites with solid mass fractal model. Construction and Building Materials, 172, 330–339(2018)CrossRefGoogle Scholar
  8. [8]
    SHEN, X., LI, L., CUI, W., and FENG, Y. Improvement of fractal model for porosity and permeability in porous materials. International Journal of Heat and Mass Transfer, 121, 1307–1315 (2018)CrossRefGoogle Scholar
  9. [9]
    SALEMI, M. and WANG, H. Image aided random aggregate packing for computational modeling of asphalt concrete microstructure. Construction and Building Materials, 177, 467–476 (2018)CrossRefGoogle Scholar
  10. [10]
    CHEN, J., WANG, H., and LI, L. Determination of effective thermal conductivity of asphalt concrete with random aggregate microstructure. Journal of Materials in Civil Engineering, 27(12), 04015045 (2015)CrossRefGoogle Scholar
  11. [11]
    DE STEFANO, M. Simulating geophysical models through fractal algorithms. Geophysical Prospecting, 66(1), 26–33 (2017)CrossRefGoogle Scholar
  12. [12]
    ROBBINS, S. J. The fractal nature of planetary landforms and implications to geologic mapping. Earth and Space Science, 5(5), 211–220 (2018)CrossRefGoogle Scholar
  13. [13]
    LU, S., TANG, H., ZHANG, Y., GONG, W., and WANG, L. Effects of the particle-size distribution on the micro and macro behavior of soils: fractal dimension as an indicator of the spatial variability of a slip zone in a landslide. Bulletin of Engineering Geology and the Environment, 77(2), 665–677 (2017)CrossRefGoogle Scholar
  14. [14]
    MUTO, J., NAKATANI, T., NISHIKAWA, O., and NAGAHAMA, H. Fractal particle size distribution of pulverized fault rocks as a function of distance from the fault core. Geophysical Research Letters, 42(10), 3811–3819 (2015)CrossRefGoogle Scholar
  15. [15]
    JU, Y., ZHENG, J., EPSTEIN, M., SUDAK, L., WANG, J., and ZHAO, X. 3D numerical reconstruction of well-connected porous structure of rock using fractal algorithms. Computer Methods in Applied Mechanics and Engineering, 279, 212–226 (2014)CrossRefzbMATHGoogle Scholar
  16. [16]
    ZHOU, X. and XIAO, N. A hierarchical-fractal approach for the rock reconstruction and numerical analysis. International Journal of Rock Mechanics and Mining Sciences, 109, 68–83 (2018)CrossRefGoogle Scholar
  17. [17]
    ZHOU, X. and XIAO, N. Analyzing fracture properties of the 3D reconstructed model of porous rocks. Engineering Fracture Mechanics, 189, 175–193 (2018)CrossRefGoogle Scholar
  18. [18]
    KHLYUPIN, A. N. and DINARIEV, O. Y. Fractal analysis of the 3D microstructure of porous materials. Technical Physics, 60(6), 805–810 (2015)CrossRefGoogle Scholar
  19. [19]
    CHEN, Z. L., WANG, N. T., SUN, L., TAN, X. H., and DENG, S. Prediction method for permeability of porous media with tortuosity effect based on an intermingled fractal units model. International Journal of Engineering Science, 121, 83–90 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    LIU, J., LI, Y., YAN, S., ZHANG, Z., HUO, W., ZHANG, X., and YANG, J. Optimal design on the mechanical and thermal properties of porous alumina ceramics based on fractal dimension analysis. International Journal of Applied Ceramic Technology, 15(3), 643–652 (2017)CrossRefGoogle Scholar
  21. [21]
    JU, Y., HUANG, Y., ZHENG, J., QIAN, X., XIE, H., and ZHAO, X. Multi-thread parallel algorithm for reconstructing 3D large-scale porous structures. Computers & Geosciences, 101, 10–20 (2017)CrossRefGoogle Scholar
  22. [22]
    VERMA, A. and PITCHUMANI, R. Fractal description of microstructures and properties of dynamically evolving porous media. International Communications in Heat and Mass Transfer, 81, 51–55 (2017)CrossRefGoogle Scholar
  23. [23]
    MITIĆ, V. V., KOCIĆ, L., PAUNOVIĆ, V., BASTIĆ, F., and SIRMIĆ, D. The fractal nature materials microstructure influence on electrochemical energy sources. Science of Sintering, 47, 195–204 (2015)CrossRefGoogle Scholar
  24. [24]
    CHOWDHURY, N. R. and KANT, R. Theory of generalized gerischer impedance for quasi-reversible charge transfer at rough and finite fractal electrodes. Electrochimica Acta, 281, 445–458 (2018)CrossRefGoogle Scholar
  25. [25]
    ZHENG, J., LIU, H., WANG, K., and YOU, Z. A new capillary pressure model for fractal porous media using percolation theory. Journal of Natural Gas Science and Engineering, 41, 7–16 (2017)CrossRefGoogle Scholar
  26. [26]
    YANG, X., WANG, F., YANG, X., and ZHOU, Q. Fractal dimension in concrete and implementation for meso-simulation. Construction and Building Materials, 143, 464–472 (2017)CrossRefGoogle Scholar
  27. [27]
    YANG, X., WANG, F., YANG, X., ZHU, F., and CHI, B. Quantity and shape modification for random-fractal-based 3D concrete meso-simulation. Powder Technology, 320, 161–178 (2017)CrossRefGoogle Scholar
  28. [28]
    ZHANG, L., WANG, H., and REN, Z. Computational analysis of thermal conductivity of asphalt mixture using virtually generated three-dimensional microstructure. Journal of Materials in Civil Engineering, 29(12), 04017234 (2017)CrossRefGoogle Scholar
  29. [29]
    CHEN, J., WANG, H., DAN, H., and XIE, Y. Random modeling of three-dimensional heterogeneous microstructure of asphalt concrete for mechanical analysis. Journal of Engineering Mechanics, 144(9), 04018083 (2018)CrossRefGoogle Scholar
  30. [30]
    CHEN, J., ZHANG, M., WANG, H., and LI, L. Evaluation of thermal conductivity of asphalt mixture with heterogeneous microstructure. Applied Thermal Engineering, 84, 368–374 (2015)CrossRefGoogle Scholar
  31. [31]
    CHEN, J., WANG, H., and LI, L. Virtual testing of asphalt mixture with two-dimensional and three-dimensional random aggregate structures. International Journal of Pavement Engineering, 18(9), 824–836 (2015)CrossRefGoogle Scholar
  32. [32]
    MIAO, T., YU, B., DUAN, Y., and FANG, Q. A fractal analysis of permeability for fractured rocks. International Journal of Heat and Mass Transfer, 81, 75–80 (2015)CrossRefGoogle Scholar
  33. [33]
    MITIC, V. V., KOCIC, L., PAUNOVIC, V., LAZOVIĆ, G., and MILJKOVIC, M. Fractal nature structure reconstruction method in designing microstructure properties. Materials Research Bulletin, 101, 175–183 (2018)CrossRefGoogle Scholar
  34. [34]
    SCHMIDT, D., KAMLAH, M., and KNOBLAUCH, V. Highly densified NCM-cathodes for high energy Li-ion batteries: microstructural evolution during densification and its influence on the performance of the electrodes. Journal of Energy Storage, 17, 213–223 (2018)CrossRefGoogle Scholar
  35. [35]
    JAVAHERIAN, H. Virtual Microstructure Generation of Asphaltic Mixtures, M. Sc. dissertatioin, University of Nebraska Lincoln (2011)Google Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesShanghai UniversityShanghaiChina
  2. 2.Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering ScienceShanghai UniversityShanghaiChina
  3. 3.Shanghai Key Laboratory of Mechanics in Energy EngineeringShanghai UniversityShanghaiChina

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