Fractal Dimensions of Granular Materials Based on Grading Curves
Publication: Journal of Materials in Civil Engineering
Volume 30, Issue 6
Abstract
A grading curve is an important measure for illustrating the grain size distribution of granular materials. In general, there exists a statistical fractal relationship between the cumulative number of particles and the grain size for granular materials in nature. The objective of this paper is to calculate fractal dimensions for different types of grading curves of granular materials. The common grading curves are classified into four categories: concave, convex, combinational, and gapped. The formulas of the concave and convex grading curves are derived from the same statistical fractal relation. The fractal dimensions can then be obtained from the known grading curves by a linear fitting approach. For combinational and gapped grading curves, two different fractal dimensions are used to describe the fine and coarse grain parts. In addition, a criterion to determine the optimal demarcation point of a combinational grading curve is suggested. All types of grading curves of common granular materials in civil engineering are considered in this study. Their statistical fractal dimension can be used to study the physical and mechanical properties of materials containing granular ingredients to facilitate the consideration of the complexity of grain distribution.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (Grant No. 51479112).
References
Aberg, B. (1993). “Washout of grains from filtered sand and gravel materials.” J. Geotech. Eng., 36–53.
Adler, P. M. (1996). “Transports in fractal porous media.” J. Hydrol., 187(1–2), 195–213.
Adler, P. M., Jacquin, C. G., and Quiblier, J. A. (1990). “Flow in simulated porous media.” Int. J. Multiphase Flow, 16(4), 691–712.
Arasan, S., Akbulut, S., and Hasiloglu, A. S. (2011). “The relationship between the fractal dimension and shape properties of particles.” KSCE J. Civil Eng., 15(7), 1219–1225.
Bartoli, F., Bird, N. R. A., Gomendy, V., Vivier, H., and Niquet, S. (1999). “The relation between silty soil structures and their mercury porosimetry curve counterparts: Fractals and percolation.” Eur. J. Soil Sci., 50(1), 9–22.
Bartoli, F., Philippy, R., Doirisse, M., Niquet, S., and Dubuit, M. (1991). “Structure and self-similarity in silty and sandy soils: The fractal approach.” J. Soil Sci., 42(2), 167–185.
Burenkova, V. V. (1993). “Assessment of suffusion in non-cohesive and graded soils.” Filters in Geotechnical and Hydraulic Engineering: Proc., 1st Int. Conf. 'Geo-filter’, J. Brauns, M. Heibaum, and U. Schuler, eds., A.A. Balkema, Rotterdam, Netherlands, 357–360.
Cividini, A., Bonomi, S., Vignati, G. C., and Gioda, G. (2009). “Seepage-induced erosion in granular soil and consequent settlements.” Int. J. Geomech., 187–194.
Dathe, A., Eins, S., Niemeyer, J., and Gerold, G. (2001). “The surface fractal dimension of the soil-pore interface as measured by image analysis.” Geoderma, 103(1–2), 203–229.
Dathe, A., and Thullner, M. (2005). “The relationship between fractal properties of solid matrix and pore space in porous media.” Geoderma, 129(3–4), 279–290.
Ghilardi, P., Kai, A. K., and Menduni, G. (1993). “Self-similar heterogeneity in granular porous media at the representative elementary volume scale.” Water Resour. Res., 29(4), 1205–1214.
Giménez, D., Allmaras, R. R., Huggins, D. R., and Nater, E. A. (1998). “Mass, surface, and fragmentation fractal dimensions of soil fragments produced by tillage.” Geoderma, 86(3–4), 261–278.
Hasmy, A., and Olivi-Tran, N. (1999). “Diffusivity and pore distribution in fractal and random media.” Phys. Rev. E, 59(3), 3012–3015.
Huang, G., and Zhang, R. (2005). “Evaluation of soil water retention curve with the pore-solid fractal model.” Geoderma, 127(1–2), 52–61.
Istomina, V. S. (1957). Filtration stability of soils, Gestroizdat, Moscow (in Russian).
Kenney, T. C., and Lau, D. (1985). “Internal stability of granular filters.” Can. Geotech. J., 22(2), 215–225.
Kezdi, A. (1979). Soil physics: Selected topics, Elsevier, Amsterdam, Netherlands.
Lehmann, P., Stähli, M., Papritz, A., Gygi, A., and Flühler, H. (2003). “A fractal approach to model soil structure and to calculate thermal conductivity of soils.” Transp. Porous Media, 52(3), 313–332.
Mandelbrot, B. B. (1982). The fractal geometry of nature, W. H. Freeman and Company, New York.
Nadim Hassoun, M., and Al-Manaseer, A. (2005). Structural concrete: Theory and design, Wiley, NJ.
Neimark, A. (1992). “A new approach to determination of surface fractal dimension of porous solids.” Phys. A, 191(1–4), 258–262.
Perfect, E., and Kay, B. D. (1991). “Fractal theory applied to soil aggregation.” Soil Sci. Soc. Am. J., 55(6), 1552–1558.
Perfect, E., Kay, B. D., and Rasiah, V. (1993). “Multifractal model for soil aggregate fragmentation.” Soil Sci. Soc. Am. J., 57(4), 896–900.
Perrier, E., Bird, N., and Rieu, M. (1999). “Generalizing the fractal model of soil structure: The pore-solid fractal approach.” Geoderma, 88(3–4), 137–164.
Richard, P., Nicodemi, M., Delannay, R., Ribière, P., and Bideau, D. (2005). “Slow relaxation and compaction of granular systems.” Nat. Mater., 4(2), 121–128.
Shepard, J. S. (1993). “Using a fractal model to compute the hydraulic conductivity function.” Soil Sci. Soc. Am. J., 57(2), 300–306.
Terzaghi, K., Peck, R. B., and Mesri, G. (1996). Soil mechanics in engineering practice, Wiley, NJ.
Turcotte, D. L. (1997). Fractals and chaos in geology and geophysics, Cambridge University Press, New York.
Wan, C. F., and Fell, R. (2008). “Assessing the potential of internal instability and suffusion in embankment dams and their foundations.” J. Geotech. Geoenviron. Eng., 401–407.
Xu, Y. F., and Dong, P. (2004). “Fractal approach to hydraulic properties in unsaturated porous media.” Chaos Solitons Fractals, 19(2), 327–337.
Yeggoni, M., Button, J. W., and Zollinger, D. G. (1996). “Fractals of aggregates correlated with creep in asphalt concrete.” J. Transp. Eng., 22–28.
Young, I. M., and Crawford, J. W. (1991). “The fractal structure of soil aggregates: Its measurement and interpretation.” J. Soil Sci., 42(2), 187–192.
Yu, B. (2008). “Analysis of flow in fractal porous media.” Appl. Mech. Rev., 61(5), 50801.
Information & Authors
Information
Published In
Copyright
©2018 American Society of Civil Engineers.
History
Received: May 12, 2017
Accepted: Oct 27, 2017
Published online: Mar 20, 2018
Published in print: Jun 1, 2018
Discussion open until: Aug 20, 2018
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.