The relationship between the fractal dimension and shape properties of particles

  • Seracettin Arasan
  • Suat Akbulut
  • A. Samet Hasiloglu
Geotechnical Engineering

Abstract

Due to their irregularity, the shape of particles is not accurately described by Euclidian geometry. However, fractal geometry uses the concept of fractal dimension, DR, as a way to describe the shape of particles. In this study, the fractal dimensions and shape properties of particles were determined using image analysis. Exponential relationships between the fractal dimension and roundness, sphericity, angularity, convexity were described. A set of empirical correlations were also presented which clearly demonstrated the link between fractal dimension and shape properties of particles. Additionally, a new classification chart proposed for use in describing and comparing particle shape and fractal dimension.

Keywords

fractal dimension particle shape roundness image analysis 

References

  1. Akbulut, S. (2002). “Fractal Dimensioning of sand grains using image analysis system.” Pamukkale University Journal of Engineering Science, Vol. 8, No. 3, pp. 329–334.Google Scholar
  2. Al-Rousan, T., Masad, E., Tutumluer E., and Pan, T. (2007). “Evaluation of image analysis techniques for quantifying aggregate shape characteristics.” Construction and Building Materials, Vol. 21, No. 5, pp. 978–990.CrossRefGoogle Scholar
  3. Alshibli, K. A. and Alsaleh, M. I. (2004). “Characterizing surface roughness and shape of sands using digital microscopy.” Journal of Computing in Civil Engineering, Vol. 18, No. 1, pp. 36–45.CrossRefGoogle Scholar
  4. Arasan, S. and Akbulut, S. (2008). “Determination of grain-size distribution of soils wýth image analysis.” 12. National Soil Mechanic and Foundation Engineering Congress, pp. 323–332 (in Turkish with an English summary).Google Scholar
  5. Arasan, S., Yener, E., Hattatoglu, F., Hinislioglu, S., and Akbulut, S. (2010a). “The correlation between shape of aggregate and mechanical properties of asphalt concrete: Digital image processing approach.” Road Materials and Pavement Design, Vol. 12, No. 2, pp. 239–262.CrossRefGoogle Scholar
  6. Arasan, S., Akbulut, S., and Hasiloglu, A.S. (2010b). “Shape properties of natural and crushed aggregate using image analysis.” International Journal of Civil and Structural Engineering, Vol. 1, No. 2, pp. 221–233.Google Scholar
  7. Arasan, S., Akbulut, S., and Hasiloglu, A. S. (2010c). “Effect of particle roundness on the maximum and minimum void ratios of granular soils.” 13. National Soil Mechanic and Foundation Engineering Congress, (in Turkish with an English summary).Google Scholar
  8. Arasan, S., Yener, E., Hattatoglu, F., Akbulut, S., and Hinislioglu, S. (2010d). “The relationship between the fractal dimension and mechanical properties of asphalt Concrete.” International Journal of Civil and Structural Engineering, Vol. 1, No. 2, pp. 165–170.Google Scholar
  9. Arasan, S., Akbulut, S., and Hasiloglu, A. S. (2010e). “Fractal dimension and void ratios of granular materials.” (under review).Google Scholar
  10. Barrett, P. J. (1980). “The shape of rock particles, a critical review.” Sedimentology, Vol. 27, pp. 291–303.CrossRefGoogle Scholar
  11. Chan, C. Y. and Page, W. P. (1997). “Particle fractal and load effects on internal friction in powders.” Powder Technology, Vol. 90, No. 3, pp. 259–266.CrossRefGoogle Scholar
  12. Cho, G. C., Dodds, J., and Santamarina, J. C. (2006). “Particle shape effects on packing density, stiffness and strength-natural and creshed sands.” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 132, No. 5, pp. 591–602.CrossRefGoogle Scholar
  13. Cox, E. A. (1927). “A method for assigning numerical and percentage values to the degree of roundness of sand grains.” Journal of Paleontology, Vol. 1, No. 3, pp. 179–183.Google Scholar
  14. Cubrinovski, M. and Ishihara, K. (1999). “Empirical correlation between SPT N-values and relative density for sandy soils.” Soils and Foundations, Vol. 40, No. 4, pp. 103–119.Google Scholar
  15. Cubrinovski, M. and Ishihara, K. (2002). “Maximum and minimum void ratio characteristics of sands.” Soils and Foundations, Vol. 42, No. 6, pp. 65–78.Google Scholar
  16. Erdogan, S. T. (2005). Determination of aggregate shape properties using X-ray tomographic methods and the effect of shape on concrete rheology. PhD Dissertation, University of Texas at Austin.Google Scholar
  17. Erdogan, S. T, Quiroga, P. N.,. Fowler, D. W, Saleh, H. A., Livingston, R. A., Garboczi, E. J., Ketcham, P. M., Hagedorn, J. G., and Satterfield, S. G. (2006). “Three-dimensional shape analysis of coarse aggregates: New techniques for and preliminary results on several different coarse aggregates and reference rocks.” Cement and Concrete Research, Vol. 36, pp. 1619–1627.CrossRefGoogle Scholar
  18. Gori, U. and Mari, M. (2001). “The correlation between the fractal dimension and internal friction angle of different granular materials.” Soils and Foundations, Vol. 41, No. 3, pp. 17–23.Google Scholar
  19. Hasiloglu, A. S., Arasan, S., and Akbulut, S. (2010). “Determination of fractal dimensions of coarse grained soils with image analysis.” 9 th International Congress on Advances in Civil Engineering, Trabzon, Turkey.Google Scholar
  20. Holubec, I. and D’Appolonia, E. (1973). Effect of particle shape on the engineering properties of granular soils, ASTM Special Technical Publication, pp. 304–318.Google Scholar
  21. Hoyez, B. (1994). “The roughness of sand grains: an application of Fourier analysis and of fractal dimension.” Ann. Soc. Géol. du Nord, Vol. 3, No. 2ème série, pp. 73–83. (In French).Google Scholar
  22. Huber, G. A. and Heiman, G. H. (1987). “Effect of asphalt concrete parameters on rutting performance: A field investigation.” Proceedings of the Association of Asphalt Paving Technologist, Vol. 56, pp. 33–61.Google Scholar
  23. Hudson, B. (1999). “Modification to the fine aggregate angularity test.” Proceedings, Seventh Annual International Center for Aggregates Research Symposium, Austin, TX.Google Scholar
  24. Hyslip, J. P. and Vallejo, L. E. (1997). “Fractal analysis of roughness and size distribution of granular materials.” Engineering Geology, Vol. 48, Nos. 3–4, pp. 231–244.CrossRefGoogle Scholar
  25. Jamkar, S. S. and Rao, C. B. K. (2004). “Index of aggregate particle shape and texture of coarse aggregate as a parameter for concrete mix proportioning.” Cement and Concrete Research, Vol. 34, No. 11, pp. 2021–2027.CrossRefGoogle Scholar
  26. Kalcheff, I. V. and Tunnicliff, D. G. (1982). “Effects of crushed stone aggregate size and shape on properties of asphalt concrete.” Proceedings of Association of Asphalt Paving Technologists, Vol. 51, pp. 453–483.Google Scholar
  27. Kaye, B. H. (1978). “Specification of the ruggedness and/or texture of a fine particle profile by its fractal dimension.” Powder Technology, Vol. 21, No. 1, pp. 1–16.CrossRefGoogle Scholar
  28. Kennedy, S. K. and Lin, W.-H. (1992). “A comparison of Fourier and fractal techniques in the analysis of closed forms.” Journal of Sedimentary Petrology, Vol. 62, No. 5, pp. 842–848.Google Scholar
  29. Kolay, E. and Kayaball, K. (2006). “Investigation of the effect of aggregate shape and surface roughness on the slake durability index using the fractal dimension approach.” Engineering Geology, Vol. 86, No. 4, pp. 271–284.CrossRefGoogle Scholar
  30. Krumbein, W. C. (1941). “Measurement and geological significance of shape and roundness of sedimentary particles.” Journal of Sedimentary Petrology, Vol. 11, No. 2, pp. 64–72.Google Scholar
  31. Krutz, N. C. and Sebaaly, P. E. (1993). “Effect of aggregate gradation on permanent deformation of asphaltic concrete.” Proceedings of the Association of Asphalt Paving Technologists; Vol. 62, pp. 450–473.Google Scholar
  32. Kuo, C. Y. and Freeman, R. B. (2000). “Imaging indices for quantification of shape, angularity, and surface texture of aggregates.” Transportation Research Record, Vol. 1721, pp. 57–65.CrossRefGoogle Scholar
  33. Kuo, C. Y, Frost, J. D., Lai, J. S., and Wang, L.B. (1996). “Threedimensional image analysis of aggregate particles from orthogonal projections.” Transportation Research Record, Vol. 1526, pp. 98–103.CrossRefGoogle Scholar
  34. Kwan, A. K. H., Mora, C. F., and Chan, H. C. (1999). “Particle shape analysis of coarse aggregate using digital image processing.” Cement and Concrete Research, Vol. 29, No. 9, pp. 1403–1410.CrossRefGoogle Scholar
  35. Lade, P. V., Liggio, C. D., Jr., and Yamamuro, J. A. (1998). “Effects of non-plastic fines on minimum and maximum void ratios of sand.” Geotechnical Testing Journal, Vol. 21, No. 4, pp. 336–347.CrossRefGoogle Scholar
  36. Li, M. C. and Kett, I. (1967). “Influence of coarse aggregate shape on the strength of asphalt concrete mixtures.” Highway Research Record; Vol. 178, pp. 93–106.Google Scholar
  37. Mandelbort, B. B. (1977). Fractals form, change and dimension, Freeman, San Francisco, p. 273.Google Scholar
  38. Mandelbrot, B. B. (1983). The fractal geometry of nature, W.H. Freeman, San Francisco, CA.Google Scholar
  39. Masad, E., Olcott, D., White, T., and Tashman, L. (2001). “Correlation of fine aggregate imaging shape indices with asphalt mixture performance.” Transportation Research Record, Vol. 1757, pp. 148–156.CrossRefGoogle Scholar
  40. Masad, E. (2004). Aggregate Imaging System (AIMS) basics and applications, Report no. FHWA/TX-05/5-1707-01-1, Texas Department of Transportation and Federal Highway Administration, Washington, D.C.Google Scholar
  41. Masad, E. and Button, J. (2000). “Unified imaging approach for measuring aggregate angularity and texture.” Journal of Computer-Aided Civil and Infrastructure Engineering, Vol. 15, No. 4, pp. 273–280.CrossRefGoogle Scholar
  42. Masad, E., Saadeh, S., Rousan, T. A., Garboczi, E., and Little, D. (2005). “Computations of particle surface characteristics using optical and X-ray CT images.” Computational Materials Science, Vol. 34, No. 4, pp. 406–424.CrossRefGoogle Scholar
  43. Miura, K., Maeda, K., Furukawa, M., and Toki, S. (1997). “Physical characteristics of sands with different primary properties.” Soils and Foundations, Vol. 37, No. 3, pp. 53–64.Google Scholar
  44. Oduroh, P. K., Mahboub, K. C., and Anderson, R. M. (2000). “Flat and elongated aggregates in superpave regime.” Journal of Materials in Civil Engineering, Vol. 12, No. 2, pp. 124–130.CrossRefGoogle Scholar
  45. Ozol, M. A. (1978). Test and properties of concrete aggregates: Chapter 35-Shape, surface texture, surface area, and coatings. STP169BEB, 584–628Google Scholar
  46. Pettijohn, F. J. (1949). Sedimentary rocks, Harper and Brothers, New York, p. 526.Google Scholar
  47. Powers, M. C. (1953). “A new roundness scale for sedimentary particles.” Journal of Sedimentary Petrology, Vol. 23, pp. 117–119.Google Scholar
  48. Rao, C. and Tutumluer, E. (2000). “Determination of volume of aggregates: New image-analysis approach.” Transportation Research Record, Vol. 1721, pp. 73–80.CrossRefGoogle Scholar
  49. Rittenhouse, G. (1943). “A visual method of estimating twodimensional sphericity.” Journal of Sedimentary Petrology, Vol. 13, pp. 79–81.Google Scholar
  50. Russell, R. D. and Taylor, R. E. (1937). “Roundness and shape of Mississippi River sands.” Journal of Geology, Vol. 45, No. 3, pp. 225–267.CrossRefGoogle Scholar
  51. Santamarina, J. C. and Cho, G. C. (2004). “Soil behaviour: The role of the particle shape.” Proceedings Skempton Conference, March, London.Google Scholar
  52. Shklarsky, E. and Livneh, M. (1964). “The use of gravels for bituminous mixtures.” In: Proceedings of The Association of Asphalt Paving Technologists; Vol. 33, pp. 23–65.Google Scholar
  53. Stephens, J. E. and Sinha, K. C. (1978). “Influence of aggregate shape on bituminous mix character.” Journal of the Association of Asphalt Paving Technologists, Vol. 47, pp. 434–456.Google Scholar
  54. Topal, T. and Sengoz, B. (2005). “Determination of fine aggregate angularity in relation with the resistance to rutting of hot-mix asphalt.” Construction and Building Materials, Vol. 19, pp. 155–163.CrossRefGoogle Scholar
  55. Vallejo, L. E. (1995). “Fractal analysis of granular materials.” Geotechnique, Vol. 45, pp. 159–163.CrossRefGoogle Scholar
  56. Vallejo, L. E. and Zhou, Y. (1995). “The relationship between the fractal dimension and Krumbein’s roundness number.” Soils and Foundations, Vol. 35, No. 1, pp. 163–167.Google Scholar
  57. Xu, Y. F. and Sun, D. A. (2005). “Correlation of surface fractal dimension with frictional angle at critical state of sands.” Geotechnique, Vol. 55, No. 9, pp. 691–695.CrossRefGoogle Scholar
  58. Yilmaz, Y. (2009). “A study on the limit void ratio characteristics of medium to fine mixed graded sands.” Engineering Geology, Vol. 104, Nos. 3–4, pp. 290–294CrossRefGoogle Scholar
  59. Youd, T. L. (1973). Factors controlling maximum and minimum densities of sands, ASTM Special Technical Publication, pp. 98–112.Google Scholar

Copyright information

© Korean Society of Civil Engineers and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Seracettin Arasan
    • 1
  • Suat Akbulut
    • 1
  • A. Samet Hasiloglu
    • 2
  1. 1.Dept. of Civil EngineeringAtaturk UniversityErzurumTurkey
  2. 2.Dept. of Computer EngineeringAtaturk UniversityErzurumTurkey

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