Multi Scale Random Sets: From Morphology to Effective Behaviour

  • Dominique Jeulin
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 17)


Complex microstructures in materials and in biology often involve multi-scale heterogeneous textures, that we model by random sets derived from Mathematical Morphology. Our approach starts from 2D or 3D images; a complete morphological characterization is performed, and used for the identification of a model of random structure. Simulations of realistic microstructures are introduced in a numerical solver to compute appropriate fields (electric, elastic, velocity, …) and to estimate the effective properties by numerical homogenization, accounting for scale dependent statistical fluctuations of the fields.


Representative Volume Element Percolation Threshold Effective Property Random Medium Poisson Point Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balberg, I., Anderson, C.H., Alexander, S., Wagner, N.: Excluded volume and its relation to the onset of percolation. Phys. Rev. B, 30, N ∘  7, 3933 (1984)Google Scholar
  2. 2.
    Beran, M.J., Molyneux, J.: Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous materials. Q. Appl. Math. 24, 107 (1966)Google Scholar
  3. 3.
    Bretheau, T., Jeulin, D.: Caractéristiques morphologiques des constituants et comportement élastique d’un matériau biphasé Fe/Ag. Revue Phys. Appl. 24, 861–869 (1989)Google Scholar
  4. 4.
    Cailletaud, G., Jeulin, D., Rolland, Ph.: Size effect on elastic properties of random composites. Eng. Comput. 11, N 2, 99–110 (1994)Google Scholar
  5. 5.
    Eyre, D.J., Milton, G.W.: A fast numerical scheme for computing the response of composites using grid refinement. Eur. Phys. J. Appl. Phys. 6, 41 (1999)Google Scholar
  6. 6.
    Hashin, Z.: The elastic moduli of heterogeneous materials. J. Appl. Mech., Trans. ASME 29, 143 (1962)Google Scholar
  7. 7.
    Jean, A., Jeulin, D., Forest, S., Cantournet, S., N’Guyen, F.: A multiscale microstructure model of carbon black distribution in rubber. J. Microscopy. 241(3), 243 (2011)Google Scholar
  8. 8.
    Jean, A., Willot, F., Cantournet, S., Forest, S., Jeulin, D.: Large scale computations of effective elastic properties of rubber with carbon black fillers, International Journal of Multiscale Computational Engineering 9(3), 271 (2011)Google Scholar
  9. 9.
    Jean, A., Jeulin, D., Forest, S., Cantournet, S., N’Guyen, F.: A multiscale microstructure model of carbon black distribution in rubber. J. Microsc. 241(3), 243 (2011)Google Scholar
  10. 10.
    Jeulin, D.: Modeling heterogeneous materials by random structures, Invited lecture, European Workshop on Application of Statistics and Probabilities in Wood Mechanics, Bordeaux (22–23 March 1996), N-06/96/MM, Paris School of Mines Publication (1996)Google Scholar
  11. 11.
    Jeulin, D.: Random texture models for materials structures. Stat. Comput. 10, 121 (2000)Google Scholar
  12. 12.
    Jeulin, D.: Random structure models for homogenization and fracture statistics. In: Jeulin, D., Ostoja-Starzewski, M. (eds.) Mechanics of Random and Multiscale Microstructures, p. 33. Springer, Berlin (2001)Google Scholar
  13. 13.
    Jeulin, D.: Random structures in physics. In: Bilodeau, M., Meyer, F., Schmitt, M. (eds.) Space, Structure and Randomness, Series: Lecture Notes in Statistics, vol. 183, p. 183. Springer, Berlin (2005)Google Scholar
  14. 14.
    Jeulin, D.: Multi scale random models of complex microstructures. In: Chandra, T., Wanderka, N., Reimers, W., Ionescu, M. (eds.) Thermec 2009, Materials Science Forum, vol. 638–642, pp. 81–86 (2009)Google Scholar
  15. 15.
    Jeulin, D.: Analysis and modeling of 3D microstructures. In: Talbot, H., Najman, L. (eds.) Mathematical Morphology: From Theory to Applications, Chapter 19. ISTE/Wiley, New York (2010)Google Scholar
  16. 16.
    Jeulin, D., Moreaud, M.: Multi-scale simulation of random spheres aggregates-application to nanocomposites. In: Proceedings of the 9th European Congress on Stereology and Image Analysis, vol. I, p. 341. Zakopane, Poland (2005)Google Scholar
  17. 17.
    Jeulin, D., Moreaud, M.: Percolation of multi-scale fiber aggregates. In: Lechnerova, R., Saxl, I., Benes, V. (eds.) S4G, 6th International Conference Stereology, Spatial Statistics and Stochastic Geometry. Prague, 26–29 June 2006, Union Czech Mathematicians and Physicists, p. 269 (2006)Google Scholar
  18. 18.
    Jeulin, D., Moreaud, M.: Percolation d’agrégats multi-échelles de sphères et de fibres: Application aux nanocomposites. In: Matériaux 2006. Dijon (2006)Google Scholar
  19. 19.
    Jeulin, D., Moreaud, M.: Volume élémentaire représentatif pour la permittivité diélectrique de milieux aléatoires. In: Matériaux 2006, Dijon (2006)Google Scholar
  20. 20.
    Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D.: Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Sol. Str. 40, 3647 (2003)Google Scholar
  21. 21.
    Kanit, T., N’Guyen, F., Forest, S., Jeulin, D., Reed, M., Singleton, S.: Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry. Comput. Methods Appl. Mech. Eng. 195, 3960–3982 (2006)Google Scholar
  22. 22.
    Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)Google Scholar
  23. 23.
    Matheron, G.: The theory of regionalized variables and its applications. Paris School of Mines publications, Paris (1971)Google Scholar
  24. 24.
    Matheron, G.: Random sets and Integral Geometry. Wiley, New York (1975)Google Scholar
  25. 25.
    Matheron, G.: Estimating and Choosing. Springer, Berlin (1989)Google Scholar
  26. 26.
    McCoy, J.J.: In: On the displacement field in an elastic medium with random variations of material properties, Recent Advances in Engineering Sciences, vol. 5, p. 235. Gordon and Breach, New York (1970)Google Scholar
  27. 27.
    Mecke, K., Stoyan, D.: The Boolean model: from Matheron till today, space, structures, and randomness. In: Bilodeau, M., Meyer, F., Schmitt, M. (eds.) Space, Structure and Randomness, Series: Lecture Notes in Statistics, vol. 183, p. 151. Springer, Berlin (2005)Google Scholar
  28. 28.
    Milton, G.W.: Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solid. 30, 177 (1982)Google Scholar
  29. 29.
    Moulinec, H., Suquet, P.: A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comp. Meth. Appl. Mech. Eng. 157, 69 (1998)Google Scholar
  30. 30.
    Paciornik, S., Gomes, O.F.M., Delarue, A., Schamm, S., Jeulin, D., Thorel, A.: Multi-scale analysis of the dielectric properties and structure of resin/carbon-black nanocomposites. Eur. Phys. J. Appl. Phys. 21, 17 (2003)Google Scholar
  31. 31.
    Parra-Denis, E., Barat, C., Jeulin, D., Ducottet, Ch.: 3D complex shape characterization by statistical analysis: application to aluminium alloys. Mater. Char. 59, 338 (2008)Google Scholar
  32. 32.
    Rintoul, M.D., Torquato, S.: Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model. J. Phys. A. Math. Gen. 30, L585–L92 (1997)Google Scholar
  33. 33.
    Savary, L., Jeulin, D., Thorel, A.: Morphological analysis of carbon-polymer composite materials from thick sections. Acta. Stereol. 18(3), 297 (1999)Google Scholar
  34. 34.
    Serra, J.: Image analysis and Mathematical Morphology, Academic Press, London (1982)Google Scholar
  35. 35.
    Smith, J.C.: The elastic constants of a particulate-filled glassy polymer: comparison of experimental values with theoretical predictions. J. Res. NBS, 80A, 45 (1976)Google Scholar
  36. 36.
    Torquato, S., Lado, F.: Effective properties of two-phase disordered composite media: II. Evaluation of bounds on the conductivity and bulk modulus of dispersions of impenetrable spheres. Phys. Rev. B 33, 6428 (1986)Google Scholar
  37. 37.
    Willot, F., Jeulin, D.: Elastic behavior of composites containing Boolean random sets of inhomogeneities. Int. J. Eng. Sci. 47 313 (2009)Google Scholar
  38. 38.
    Willot, F., Jeulin, D.: Elastic and Electrical Behavior of Some Random Multiscale Highly Contrasted Composites, International Journal of Multiscale Computational Engineering 9(3), 305 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Centre de Morphologie MathématiqueMathématiques et SystèmesFontainebleauFrance

Personalised recommendations