Multi-scale morphological descriptors from the fractal analysis of particle contour

  • Giulia GuidaEmail author
  • Giulia M. B. Viggiani
  • Francesca Casini
Research Paper


The increasing understanding of the connection between particle morphology and mechanical behaviour of granular materials has generated significant research on the quantitative characterisation of particle shape. This work proposes a simple and effective method, based on the fractal analysis of their contour, to characterise the morphology of soil particles over the range of experimentally accessible scales. In this paper, three new non-dimensional quantitative morphological descriptors are introduced to describe (1) overall particle shape at the macro-scale, (2) particle regularity at the meso-scale, and (3) particle texture at the micro-scale. The characteristic size separating structural features and textural features emerges directly from the results of the fractal analysis of the contour of the particle, and is a decreasing fraction of particle dimension. To explore the meaning of the descriptors, the method is applied first to a variety of Euclidean smooth and artificially roughened regular shapes and then to four natural and artificial sands with different levels of irregularity. Relationships are established between the new morphological descriptors and other quantities commonly adopted in the technical literature.


Fractals Granular material Particle-scale Shape 



  1. 1.
    Al-Raoush R (2007) Microstructure characterization of granular materials. Physica A Stat Mech Appl 377(2):545–558Google Scholar
  2. 2.
    Alshibli K (2013) The University of Tennessy. Retrieved July 24, 2017, from
  3. 3.
    Altuhafi F, O’sullivan C, Cavarretta I (2013) Analysis of an image-based method to quantify the size and shape of sand particles. J Geotech Geoenviron Eng 139(8):1290–1307Google Scholar
  4. 4.
    Altuhafi FN, Coop MR, Georgiannou VN (2016) Effect of particle shape on the mechanical behavior of natural sands. J Geotech Geoenviron Eng 142(12):04016071Google Scholar
  5. 5.
    Arasan S, Akbulut S, Hasiloglu AS (2011) The relationship between the fractal dimension and shape properties of particles. KSCE J Civ Eng 15(7):1219Google Scholar
  6. 6.
    Bagheri GH, Bonadonna C, Manzella I, Vonlanthen P (2015) On the characterization of size and shape of irregular particles. Powder Technol 270:141–153Google Scholar
  7. 7.
    Bareither CA, Edil TB, Benson CH, Mickelson DM (2008) Geological and physical factors affecting the friction angle of compacted sands. J Geotech Geoenviron Eng ASCE 134(10):1476–1489Google Scholar
  8. 8.
    Barrett PJ (1980) The shape of rock particles, a critical review. Sedimentology 27(3):291–303Google Scholar
  9. 9.
    Beucher S, Meyer F (1992) The morphological approach to segmentation: the watershed transformation. Opt Eng N Y Marcel Dekker Inc 34:433Google Scholar
  10. 10.
    Bhushan B (2001) Nano-to microscale wear and mechanical characterization using scanning probe microscopy. Wear 251(1):1105–1123Google Scholar
  11. 11.
    Boulanger J (1992) The “Motifs” method: AN interesting complement to ISO parameters for some functional problems. Int J Mach Tools Manuf 32(1–2):203–209Google Scholar
  12. 12.
    Bowman ET, Soga K, Drummond TW (2001) Particle shape characterization using Fourier descriptors analysis. Géotechnique 51(6):545–554Google Scholar
  13. 13.
    Cabalar AF, Dulundu K, Tuncay K (2013) Strength of various sands in triaxial and cyclic direct shear tests. Eng Geolol 156(1):92–102Google Scholar
  14. 14.
    Casini F, Viggiani GMB, Springman SM (2013) Breakage of an artificial crushable material under loading. Granul Matter 15(5):661–673Google Scholar
  15. 15.
    Cavarretta I (2009) The influence of particle characteristics on the engineering behaviour of granular materials. PhD thesis, Imperial College, LondonGoogle Scholar
  16. 16.
    Cavarretta I, O’Sullivan C, Coop MR (2016) The relevance of roundness to the crushing strength of granular materials. Géotechnique 67(4):301–312Google Scholar
  17. 17.
    Chapuis RP (2012) Estimating the in situ porosity of sandy soils sampled in boreholes. Eng Geol 141–142(19):57–64Google Scholar
  18. 18.
    Cho GC, Dodds J, Santamarina JC (2006) Particle shape effects on packing density, stiffness, and strength: natural and crushed sands. J Geotech Geoenviron Eng ASCE 132(5):591–602Google Scholar
  19. 19.
    Clark MW (1981) Quantitative shape analysis: a review. Math Geol 13(4):303–320Google Scholar
  20. 20.
    Clayton CRI, Abbireddy COR, Schiebel R (2009) A method of estimating the form of coarse particulates. Géotechnique 59(6):493–501Google Scholar
  21. 21.
    Demirmen F (1972) Mathematical search procedures in facies modeling in sedimentary rocks. In: Mathematical models of sedimentary processes. Springer, Boston, MA, pp 81–114Google Scholar
  22. 22.
    Devarrewaere W, Foqué D, Heimbach U, Cantre D, Nicolai B, Nuyttens D, Verboven P (2015) Quantitative 3D shape description of dust particles from treated seeds by means of X-ray micro-CT. Environ Sci Technol 49(12):7310–7318Google Scholar
  23. 23.
    Dobkins JE, Folk RL (1970) Shape development on Tahiti-nui. J Sediment Res 40(4):1167–1203Google Scholar
  24. 24.
    Ehrlich R, Weinberg B (1970) An exact method for characterization of grain shape. J Sediment Res 40(1):205–212Google Scholar
  25. 25.
    Fonseca J, O’Sullivan C, Coop M, Lee P (2012) Non-invasive characterization of particle morphology of natural sands. Soils Found 52(4):712–722Google Scholar
  26. 26.
    Hanaor DA, Gan Y, Einav I (2013) Effects of surface structure deformation on static friction at fractal interfaces. Géotechn Lett 3(2):52–58Google Scholar
  27. 27.
    Herle I, Gudehus G (1999) Determination of parameters of a hypoplastic constitutive model from properties of grain assemblies. Mech Cohes Frict Mater 4:461–486Google Scholar
  28. 28.
    ISO (2008) ISO 9276-6:2008: representation of results of particle size analysis—part 6: descriptive and qualitative representation of particle shape and morphology. ISO, GenevaGoogle Scholar
  29. 29.
    Kandasami RK, Murthy TG (2014) Effect of particle shape on the mechanical response of a granular ensemble. In: Soga K, Kumar K, Biscontin G, Kuo M (eds) Geomechanics from micro to macro. CRC Press, London, pp 1093–1098Google Scholar
  30. 30.
    Krumbein WC (1941) Measurement and geological significance of shape and roundness of sedimentary particles. J Sediment Res 11(2):64–72Google Scholar
  31. 31.
    Krumbein WC, Sloss LL (1963) Stratigraphy and sedimentation, 2nd edn. WH Freeman & Co., San FranciscoGoogle Scholar
  32. 32.
    Kuenen PH (1956) Experimental abrasion of pebbles: 2. Rolling by current. J Geol 64(4):336–368Google Scholar
  33. 33.
    Lees G (1964) A new method for determining the angularity of particles. Sedimentology 3(1):2–21Google Scholar
  34. 34.
    Lin CL, Miller JD (2005) 3D characterization and analysis of particle shape using X-ray microtomography (XMT). Powder Technol 154(1):61–69Google Scholar
  35. 35.
    Mandelbrot BB (1967) How long is the coast of Britain. Science 156(3775):636–638Google Scholar
  36. 36.
    Mandelbrot BB (1975) Stochastic models for the Earth’s relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands. Proc Natl Acad Sci 72(10):3825–3828MathSciNetGoogle Scholar
  37. 37.
    Matlab 2015b (2015) [Computer software] MatworksGoogle Scholar
  38. 38.
    Meloy TP (1977) Fast Fourier transforms applied to shape analysis of particle silhouettes to obtain morphological data. Powder Technol 17(1):27–35Google Scholar
  39. 39.
    Mitchell JK, Soga K (2005) Fundamentals of soil behaviour, 3rd edn. Wiley, New YorkGoogle Scholar
  40. 40.
    Miura K, Maeda K, Furukawa M, Toki S (1997) Physical characteristics of sands with different primary properties. Soils Found 37(3):53–64Google Scholar
  41. 41.
    Mollon G, Zhao J (2012) Fourier–Voronoi-based generation of realistic samples for discrete modelling of granular materials. Granul Matter 14(5):621–638Google Scholar
  42. 42.
    Mollon G, Zhao J (2013) Generating realistic 3D sand particles using Fourier descriptors. Granul Matter 15(1):95–108Google Scholar
  43. 43.
    Orford JD, Whalley WB (1983) The use of the fractal dimension to quantify the morphology of irregular-shaped particles. Sedimentology 30(5):655–668Google Scholar
  44. 44.
    Otsu N (1979) A threshold selection method from gray-level histograms. IEEE Trans Syst Man Cybern 9(1):62–66Google Scholar
  45. 45.
    Otsubo M, O’Sullivan C, Hanley KJ, Sim WW (2016) The influence of particle surface roughness on elastic stiffness and dynamic response. Géotechnique 67(5):452–459Google Scholar
  46. 46.
    Park J, Santamarina JC (2017) Revised soil classification system for coarse-fine mixtures. J Geotech Geoenviron Eng 143(8):04017039Google Scholar
  47. 47.
    Powers MC (1953) A new roundness scale for sedimentary particles. J Sediment Res 23(2):117–119Google Scholar
  48. 48.
    Richardson LF (1961) The problem of contiguity. Gen Syst Yearb 6:139–187Google Scholar
  49. 49.
    Santamarina JC, Cascante G (1998) Effect of surface roughness on wave propagation parameters. Géotechnique 48(1):129–136Google Scholar
  50. 50.
    Stachowiak GW (1998) Numerical characterization of wear particles morphology and angularity of particles and surfaces. Tribol Int 31(1):139–157MathSciNetGoogle Scholar
  51. 51.
    Swan B (1974) Measures of particle roundness: a note. J Sediment Res 44(2):572–577Google Scholar
  52. 52.
    Sympatec (2008) QICPIC. Windox-operating instructions release Sympatec, Clausthal-Zellerfeld, GermanyGoogle Scholar
  53. 53.
    Vallejo LE (1995) Fractal analysis of granular materials. Géotechnique 45(1):159–163Google Scholar
  54. 54.
    Von Koch H (1906) Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire pour l’étude de certaines questions de la théorie des courbes planes. Acta Math 30:145–174MathSciNetGoogle Scholar
  55. 55.
    Wadell H (1932) Volume, shape, and roundness of rock particles. J Geol 40(5):443–451Google Scholar
  56. 56.
    Wentworth CK (1922) A scale of grade and class terms for clastic sediments. J Geol 30(5):377–392Google Scholar
  57. 57.
    Yang H, Baudet BA (2016) Characterisation of the roughness of sand particles. Procedia Eng 158:98–103Google Scholar
  58. 58.
    Yang H, Baudet BA, Yao T (2017) Characterization of the surface roughness of sand particles using an advanced fractal approach. Proc R Soc Lond A 472:20160524Google Scholar
  59. 59.
    Youd TL (1973) Factors controlling maximum and minimum densities of sands. In: Evaluation of relative density and its role in geotechnical projects involving cohesionless soils, ASTM STP, vol 523, pp 98–112Google Scholar
  60. 60.
    Zheng J, Hryciw R (2015) Traditional soil particle sphericity, roundness and surface roughness by computational geometry. Géotechnique 65(6):494–506Google Scholar
  61. 61.
    Zhou B, Wang J (2015) Random generation of natural sand assembly using micro X-ray tomography and spherical harmonics. Géotech Lett 5(1):6–11Google Scholar
  62. 62.
    Zhou B, Wang J, Wang H (2018) Three-dimensional sphericity, roundness and fractal dimension of sand particles. Géotechnique 68(1):18–30Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Università degli Studi Niccolò CusanoRomeItaly
  2. 2.University of CambridgeCambridgeUK
  3. 3.Università degli Studi di Roma Tor VergataRomeItaly

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